Computation of Nonlinear Structures: Extremely Large Elements for Frames, Plates and Shells 9781118996959, 111899695X, 9781118996829, 1118996828, 9781118996867, 1118996860

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Computation of Nonlinear Structures Extremely Large Elements for Frames, Plates and Shells Debabrata Ray

COMPUTATION OF NONLINEAR STRUCTURES

COMPUTATION OF NONLINEAR STRUCTURES EXTREMELY LARGE ELEMENTS FOR FRAMES, PLATES AND SHELLS Debabrata Ray PhD (Univ. of California, Berkeley) ME, BE (Bengal Engineering College, Shibpur) Principal, Institute for Dynamic Response

This edition first published 2016 © 2016 John Wiley & Sons, Ltd Registered office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com. The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. It is sold on the understanding that the publisher is not engaged in rendering professional services and neither the publisher nor the author shall be liable for damages arising herefrom. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Library of Congress Cataloging-in-Publication Data applied for. A catalogue record for this book is available from the British Library. ISBN: 9781118996959 Set in 10/12pt Times by Aptara Inc., New Delhi, India 1 2016

To the memory of my dad & mom, Provanshu and Sushila Roy, with whose altruistic love it was nurtured & the best thing ever happened to me, my wife, Anjana Ray, M.D., with whose infinite patience and eternal support it blossomed & the budding analysts and researchers like my sons, Dipanjan Ray, PhD and Shonket Ray, PhD, to whom it is offered

Contents Acknowledgements

xi

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

1 1 2 9 9 10 11 12 14 15

Introduction: Background and Motivation What This Book Is All About A Brief Historical Perspective Symbiotic Structural Analysis Linear Curved Beams and Arches Geometrically Nonlinear Curved Beams and Arches Geometrically Nonlinear Plates and Shells Symmetry of the Tangent Operator: Nonlinear Beams and Shells Road Map of the Book References

Part I ESSENTIAL MATHEMATICS

19

2 2.1 2.2 2.3 2.4

Mathematical Preliminaries Essential Preliminaries Affine Space, Vectors and Barycentric Combination Generalization: Euclidean to Riemannian Space Where We Would Like to Go

21 21 33 36 40

3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15

Tensors Introduction Tensors as Linear Transformation General Tensor Space Tensor by Component Transformation Property Special Tensors Second-order Tensors Calculus Tensor Partial Derivatives of Tensors Covariant or Absolute Derivative Riemann–Christoffel Tensor: Ordered Differentiation Partial (PD) and Covariant (C.D.) Derivatives of Tensors Partial Derivatives of Scalar Functions of Tensors Partial Derivatives of Tensor Functions of Tensors Partial Derivatives of Parametric Functions of Tensors Differential Operators

41 41 44 46 50 57 62 74 74 75 78 79 80 81 81 82

Contents

viii

3.16 3.17 3.18 3.19

Gradient Operator: GRAD(∙) or ∇(∙) Divergence Operator: DIV or ∇∙ Integral Transforms: Green–Gauss Theorems Where We Would Like to Go

82 84 87 90

4 4.1 4.2 4.3 4.4 4.5 4.6 4.7

Rotation Tensor Introduction Cayley’s Representation Rodrigues Parameters Euler – Rodrigues Parameters Hamilton’s Quaternions Hamilton–Rodrigues Quaternion Derivatives, Angular Velocity and Variations

91 91 100 107 112 115 119 125

ESSENTIAL MESH GENERATION

133

Part II 5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13

Curves: Theory and Computation Introduction Affine Transformation and Ratios Real Parametric Curves: Differential Geometry Frenet–Serret Derivatives Bernstein Polynomials Non-rational Curves Bezier–Bernstein–de Casteljau Composite Bezier–Bernstein Curves Splines: Schoenberg B-spline Curves Recursive Algorithm: de Boor–Cox Spline Rational Bezier Curves: Conics and Splines Composite Bezier Form: Quadratic and Cubic B-spline Curves Curve Fitting: Interpolations Where We Would Like to Go

135 135 136 139 145 148 154 181 185 195 198 215 229 245

6 6.1 6.2 6.3 6.4 6.5 6.6 6.7

Surfaces: Theory and Computation Introduction Real Parametric Surface: Differential Geometry Gauss–Weingarten Formulas: Optimal Coordinate System Cartesian Product Bernstein–Bezier Surfaces Control Net Generation: Cartesian Product Surfaces Composite Bezier Form: Quadratic and Cubic B-splines Triangular Bezier–Bernstein Surfaces

247 247 248 272 280 296 300 306

Part III ESSENTIAL MECHANICS

323

7 7.1 7.2 7.3 7.4

325 325 326 337 351

Nonlinear Mechanics: A Lagrangian Approach Introduction Deformation Geometry: Strain Tensors Balance Principles: Stress Tensors Constitutive Theory: Hyperelastic Stress–Strain Relation

Contents

ix

Part IV A NEW FINITE ELEMENT METHOD

365

8 8.1 8.2 8.3 8.4 8.5 8.6

367 367 369 386 402 438 446

C-type Finite Element Method Introduction Variational Formulations Energy Precursor to Finite Element Method c-type FEM: Linear Elasticity and Heat Conduction Newton Iteration and Arc Length Constraint Gauss–Legendre Quadrature Formulas

Part V APPLICATIONS: LINEAR AND NONLINEAR

457

9 9.1 9.2 9.3 9.4 9.5 9.6

Application to Linear Problems and Locking Solutions Introduction c-type Truss and Bar Element c-type Straight Beam Element c-type Curved Beam Element c-type Deep Beam: Plane Stress Element c-type Solutions: Locking Problems

459 459 460 465 484 498 509

10 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 10.11 10.12 10.13 10.14 10.15 10.16 10.17 10.18 10.19 10.20

Nonlinear Beams Introduction Beam Geometry: Definition and Assumptions Static and Dynamic Equations: Engineering Approach Static and Dynamic Equations: Continuum Approach – 3D to 1D Weak Form: Kinematic and Configuration Space Admissible Virtual Space: Curvature, Velocity and Variation Real Strain and Strain Rates from Weak Form Component or Operational Vector Form Covariant Derivatives of Component Vectors Computational Equations of Motion: Component Vector Form Computational Derivatives and Variations Computational Virtual Work Equations Computational Virtual Work Equations and Virtual Strains: Revisited Computational Real Strains Hyperelastic Material Property Covariant Linearization of Virtual Work Material Stiffness Matrix and Symmetry Geometric Stiffness Matrix and Symmetry c-type FE Formulation: Dynamic Loading c-type FE Implementation and Examples: Quasi-static Loading

523 523 530 534 539 555 560 570 580 587 590 596 607 614 627 630 639 655 658 673 685

11 11.1 11.2 11.3 11.4 11.5

Nonlinear Shell Introduction Shell Geometry: Definition and Assumptions Static and Dynamic Equations: Continuum Approach – 3D to 2D Static and Dynamic Equations: Continuum Approach – Revisited Static and Dynamic Equations: Engineering Approach

721 721 727 746 763 771

Contents

x

11.6 11.7 11.8 11.9 11.10 11.11 11.12 11.13 11.14 11.15 11.16 11.17 11.18 11.19 11.20 Index

Weak Form: Kinematic and Configuration Space Admissible Virtual Space: Curvature, Velocity and Variation Real Strain and Strain Rates from Weak Form Component or Operational Vector Form Covariant Derivatives of Component Vectors Computational Equations of Motion: Component Vector Form Computational Derivatives and Variations Computational Virtual Work Equations Computational Virtual Work Equations and Virtual Strains: Revisited Computational Real Strains Hyperelastic Material Property Covariant Linearization of Virtual Work c-type FE Formulation: Dynamic Loading c-type FE Formulation: Quasi-static Loading c-type FE Implementation and Examples: Quasi-static Loading

783 788 799 810 817 820 830 841 851 861 864 877 891 914 930 967

Acknowledgements I would like to sincerely thank my friend, Anil K. Chopra, Johnson Professor of Structural Engineering, Department of Civil and Environmental Engineering, University of California, Berkeley, who introduced me to structural dynamics, for his encouragement and generous support in presenting my work to Eric Willner, Executive Commissioning Editor of John Wiley & Sons Ltd, who graciously agreed to publish the book, for which I am deeply appreciative. I must also thank Anne Hunt, Associate Commissioning Editor, Mechanical Engineering, for her unstinting support, and, Clive Lawson, Project Editor, Content Capture, Natural Sciences, Engineering & Stats, Professional Practice and Learning, Wiley. My special thanks for meticulous scrutiny of the manuscript to copy editor, Paul Beverley, LCGI, and Baljinder Kaur, Project Manager, Professional Publishing, Aptara. During the writing of the manuscript for the book, I was paralysed as a result of a botched surgical procedure; however, from the initial period of rehabilitation to the date of this writing, I have been extremely fortunate to be surrounded by innumerable friends and well-wishers. I would like to thank them all; especially, my deep appreciation goes to my friend, Prof. Amitabha Basu of Dept. of South Asian Studies, University of California, Berkeley, and, to my care-givers, Levi Soler and John Viray. Finally, however, the faults and mistakes, if any, are entirely mine. DR August 31, 2014

1 Introduction: Background and Motivation 1.1

What This Book Is All About

The book introduces linear and nonlinear structural analysis through a combination of of mesh generation, solid mechanics and a new finite element methodology called c-type finite element method (Ray, 1999, 2003, 2004, 2005, 2007, 2008). The ultimate objective is to present the largest possible (curved) beam, plate and shell elements undergoing extremely large displacement and rotation, and to apply these to solve standard industrial problems. Any finite element method is only as strong as its weakest link. In other words, the book is not just about unification of mesh generation and finite element methodology but it strives to serve as a reference for budding researchers, engineers, analysts, upper division and graduate students and teachers by demonstrating what various interdisciplinary machinery has to be accurately harnessed to devise a solid and conducive theoretical framework upon which to build a robust, reliable and efficient numerical methodology for linear and nonlinear static and dynamic analysis of beams, plates and shells. As indicated, the principal goal of the book is to produce the largest possible arbitrary shaped elements (a) defined and restricted solely by the requirements of geometry, material, loading and support conditions, (b) avoiding computational problems such as locking in the conventional finite element methods and (c) presenting new, accurate and explicit expressions for resolution of the symmetry issue of the tangent operator for beams, plates and shells in areas of extreme nonlinearity. The ‘mega-sized’ elements may result in substantial cost saving and reduced bookkeeping for the subsequent finite element analysis, and a reduced engineering manpower requirement for the final quality assurance. For example, the explicit algebraic and symmetric expressions of the tangent operator, as presented in the book, are an absolute necessity for computational cost efficiency, especially in repetitive calculations that are commonly associated with nonlinear problems. It must be recognized that the requirements for numerical convergence should be purely adaptive and subservient to the main delineating factors already mentioned. However, this strategy of computer generation of mega-elements of arbitrary shape, as it turns out, takes its toll on the analyst. Firstly, only accurate theoretical formulation can be used for the underlying continuum or solid mechanics principles without unnecessary ‘short-circuiting’ by proliferation of ad hoc numerical manipulations. Secondly, it demands that the applicable finite element method be devised to successfully accept computer generated elements with arbitrarily distorted shapes, with edges (or faces) consisting of up to truly 3D curved boundaries (or surfaces)

Computation of Nonlinear Structures: Extremely Large Elements for Frames, Plates and Shells, First Edition. Debabrata Ray. © 2016 John Wiley & Sons, Ltd. Published 2016 by John Wiley & Sons, Ltd.

Computation of Nonlinear Structures

2

with natural twist and bend (e.g. for shell elements). Thirdly, for these ‘hyperelements’ with conformity, the finite element method must be able to accommodate effortlessly and naturally C0 , C1 or C2 inter-element continuity on demand.

1.2

A Brief Historical Perspective

Every meaningful structural analysis is an exercise in abstraction about a structural system in the real world, so just as with any other natural or man-made phenomenon, the viability and safety of the structural system is intricately associated with the methodology underpinning such abstraction. More specifically, for a structural system, abstractions lie in the geometric modeling of its material body, its relevant support conditions and its imposed loadings, and finally its material properties; we associate the sum total of these abstractions with a structural theory. Moreover, each of these abstractions defines the extent to which a particular structural theory can efficiently and logically predict and control the response of a structural system. To paraphrase Einstein’s incisive comment to Heisenberg that led to the latter’s discovery of the uncertainty principle, every theory, like a mirror or a horse’s blinder, filters and determines what we can see of the real world. Naturally, lest we miss out on important real-world phenomena, as structural analysts, we have to critically evaluate structural theories, propounded in both the distant and the recent past, so that we can be successful in the ultimate goal of our exercise in abstraction, namely, the prediction and control of structural response to external stimuli. Translated into actual methodology for solid bodies, these abstractions reduce to two fundamentally complementary disciplines: solid mechanics and numerical analysis – each determining and harnessing the strength of the other.

1.2.1

Operational Mechanics

For hundreds of years, even before the digital age, the basic theoretical premises of linear and nonlinear solid mechanics involving the study of the deformation of a body, transmission of force through it and the characterization of its material properties, have been well established, and they were perfected during the twentieth century (Eringen, 1962; Green and Zerna, 1968; Ogden, 1997; Malvern, 1969). However, without the computational power of modern computers, the forms of the various equations in solid mechanics, while accurately describing real world problems, were not of much use for finding numerical solutions for material bodies of very complicated geometry and intricate support conditions, subjected to complex loading systems. Thus, before the computer era, solid mechanics had no unifying numerical methodology, and so structural problems were solved on a case-by-case basis using a variety of different analytical methods.

1.2.2

Conventional Finite Element Methods

With the coming of the digital era, numerical methods became much more dominant. Over 60 years ago, an energy-based methodology called the finite element method (Turner, et al., 1956) made its triumphant entry into the realm of numerical structural analysis, its theory having been established earlier (Courant, 1943). Its chief advance was to choose triangular elements with a complete set of basis functions following Pascal’s triangle, but soon after, the application of similar basis functions to quadrilateral elements proved to be a poor choice because arbitrarily curtailed and incomplete polynomials resulted in interpolation problems requiring various ad hoc numerical artifacts such as under integration and reduced integration, to rectify numerous locking issues that resulted. Both the conventional finite element methods – h-type with Lagrangian and

Introduction: Background and Motivation

3

Hermite basis functions (Zienkiewicz and Taylor, 2000) and p-type with Legendre basis functions (Szabo and Babuska, 1991), – suffered from these ill-conceived ideas. While the general theoretical formulation of the displacement finite element method of analysis is based on rigorous variational method, the practice of finite element method is another story – a harrowing experience because, in an ad-hoc manner, it tried in vain to tear apart the fundamental and inalienable dictum of the Rayleigh–Ritz–Galerkin method that says: “ . . . method is a ‘package deal’, and neither requires nor permits the user to make independent decisions about different parts of the problem” (Strang and Fix, 1973, p. 33). Based on the most common construction mechanism of global basis or Ritz functions from appropriate mapping of the chosen elemental or local basis functions, the ultimate success or failure of a practiced or computer-implemented finite element method essentially depends on the choice of the local basis functions (also known as the elemental shape functions), the adequacy of the mapping functions and the evaluations of the integrals. It is no wonder then that for more than 30 years, the primary focus of all notable fundamental researches in the field of finite element analysis has been on devising the “best” shape functions and associated mapping functions. It is instructive to briefly review the interpolation failure and concomitant patchy, ad hoc remedies with the ill side effects and to present some extremely important conceptual problems that remained unsolved by the conventional finite element methods, insofar as interpolation is concerned. h-type Methodology: Hermitian shape functions, C1 and C2 , are used for situations where flexural strain exists such as in the case of beam. The nodes in these cases are what can be considered as multiple nodes; that is, apart from the values, the slopes and/or curvatures are also taken as degrees of freedom at a physical location. The simplest element in the family has two end nodes with two degrees of freedom each for a total of four degrees of freedom. A cubic Hermite is used for this. The inter-element continuity that can be imposed is that the slope or the first derivatives of the functions are continuous, that is, C1 continuity. To speed up convergence where the solution is smooth or has curved flexural elements, one can use a quintic Hermite with curvature or bending moment continuity, that is, C2 continuity. Note that because of the inclusion of the derivative components as degrees of freedom, the interpolation loses its barycentric nature or convexity. The isoparametric elements are obtained following the assumption and procedure that the functional representation of the deformational behavior is employed in representation of the element geometry. In other words, the displacement vector components and the geometry use the same shape functions. This, in turn, forces the conventional element to have as many internal nodes as necessary to make geometry description isoparametric to the displacement function. For example, for a quadratic interpolation function for a truss element, it becomes a three-node element with one internal node. For a conventional isoparametric element, the accuracy increases with the corresponding element size measured by h, the diameter of the smallest circle encompassing the element, and, hence, also identified as h-type elements. The conventional h-type elements in two dimensions are interpolated with polynomials of degree one, two and sometimes three. The main elements, that have been known and used for quite some time, can be categorized into two major groups: Lagrangian and serendipity. These are obtained by taking the tensor product of the one-dimensional Lagrangian elements. The mapping functions are linear. For quadratic elements, say, the Lagrangian tensor product element consists naturally of nine nodes – four corner nodes, four edge nodes and one interior node. For the serendipity element, in this case, the interior node is dropped in favor of incomplete interpolation. The higher-order h-type elements are obtained analogously and may be found in any of the standard texts already mentioned. p-type Methodology: From later developments come the so-called hierarchical elements of increasing polynomial degree p, and hence, identified as p-type elements. The main idea behind these elements is that the next improved element of higher degree is achieved by retaining the earlier expressions of all degrees less than the improved one. In other words, to obtain an element

4

Computation of Nonlinear Structures

of degree p, p ≥ 2, only higher-order additional terms are introduced to the shape functions of the element of p − 1 degree. Moreover, these additional shape functions must vanish at old nodes, then since the pth derivative of an old (p − 1) degree shape function always vanishes, this condition is chosen to insert a new node, and finally the new shape function is scaled so that it assumes the value of unity at this generalized displacement node. The combination is not barycentric or convex, and the degrees of freedom consist of both functions and their derivatives as in the mixed formulations. For p-type quadrilateral elements, there is more than one variant of the polynomial spaces. The shape functions are categorized in those spaces into three groups: nodal, side and interior modes. The basic idea behind the shape functions is the same as in one dimension, that is, choosing Legendre polynomials as the primary functions. However, in order to enforce the fundamental property of basis functions, namely that each assumes a value of unity at one node and vanishes at the rest of the nodes, blending functions similar to h-type elements are used. Triangular Elements: The triangular elements, by contrast, have long been described in the barycentric coordinates. The root geometry has been described in three coordinates, suggesting a three-dimensional figure. However, a closer look will reveal that the root element is still planar, that is, a subset of two-dimensional space, R2 , because only two of the coordinates are independent and all three are related; the barycentric coordinates are essentially the area coordinates in two dimensions. In other words, any point in two dimensions can be expressed as convex combinations of the vertex of a triangle containing the point. The coefficients of combinations are the barycentric coordinates of the point. As shown in this book, the three coordinates are proportional to the three smaller triangles generated by joining the point to each vertex of the triangle. All shape functions can be defined on the standard triangle in terms of these barycentric coordinates. For higherdegree h-type triangular elements, edge nodes (e.g. for degree 2) and inside nodes (e.g. for degree three) are included in the element. For the hierarchic p-type elements, for p = 1, 2, the shape functions for the triangular elements are exactly the same as those for the h-type elements. The mapping functions are chosen as linear for straight edge triangles. For large curved triangles, mapping functions are isoparametric for p = 2. For higher-order triangular elements, mapping functions are developed based on a blending method. For the purpose of critical evaluation to be introduced later, this concludes a brief presentation of the characteristics of the shape functions and the mapping functions for both families, namely the h-type and the p-type, of conventional finite elements.

1.2.2.1

Problems of the Conventional Finite Element Methods

We start by referring to the comment by MacNeal (1994) that the “days of pioneering” or the days of “heuristics, hunches and experimental data to guide . . . design choices” is over, and that we should take design of finite elements “ . . . less as an art and more as a science”. The noteworthy applied mathematicians such as Strang and Fix appeared to be too apologetic, maybe because the engineers beat the mathematicians and discovered the finite element method; they also appeared to have been temptingly permissive by announcing that the serious violations of the theory are mere “variational crimes”: “ . . . it [the rule] is broken every day, and for good reason” (MacNeal, 1994). No wonder then that MacNeal is forced to believe that “mathematical rigor falls well short of the goal of converting finite element design into science”. In any case, because of these conflicting signals, scores of books keep appearing that simply repeat the same old (“variational crime”-ridden) information relating to the shape functions. The main goal, therefore, will be first to identify the various problems that crept into the mainstream conventional finite element method. However, in order for us to recognize the violations of the conventional finite element method, we need to reiterate the main dictates of the variational theorems: geometry must be accurately

Introduction: Background and Motivation

5

interpolated with extreme care for curved boundaries; the trial functions must have adequate derivatives across the element boundaries required by the inter-element continuity requirement; essential boundary conditions must be honored; and the integrals must be computed accurately. Power Series as Basis Functions: One look at the history of the development of finite elements will reveal that primarily the power series made up of monomials of Cartesian position coordinates constituted the basis functions. Naturally, the Pascal triangle had been the guiding post for determining the completeness of the polynomials made up of such monomials. But, innocuous and familiar as these are, the polynomials constructed from these monomials, in their indiscriminate use as basis functions, produced some very distressing phenomena. Induced Anisotropy: Once such commitment to power series has been made – the notion of completeness of the polynomials making up the basis functions became important for both convergence and suitability of the shape functions. Depending on the displacement solution function, convergence of incomplete polynomial basis functions to the solution is not guaranteed. Moreover, the basis functions based on incomplete polynomials can easily induce anisotropy for triangular or tetrahedral elements in the description of, say, the stiffness matrix, that is, the definition has to depend on the orientation of the coordinate axes. Similar loss of symmetry and hence induced anisotropy may result for arbitrary quadrilateral or hexahedral elements. Incomplete Strain States: Incomplete polynomials as basis functions fail to invoke all the internal strain states that may result due to all the possible imposed conditions. For example, a four-node linear conventional quadrilateral element can only represent two internal linear strain states as opposed to required six. In fact, this condition afflicts all serendipity and Lagrange elements in two or three dimensions. Absence of complete strain states gives rise to what is known as locking which we shall deal with in the chapter on linear application. Based on the preceding discussion, polynomial basis functions made up of monomials do not seem to be a good choice. In fact, p-type hierarchical elements dropped monomials in favor of Legendre polynomials and thus avoided these problems. However, completeness is not the only dominating issue in the choice of the basis functions and mapping functions. We now embark on one such important issue. Inter-element Continuity: As indicated in this book about the theory of Rayleigh–Ritz– Galerkin or finite element method, the trial and test spaces are finite dimensional subspaces of the solution space. The accuracy of the solution increases with the corresponding increase in the dimensionality of the subspaces. Furthermore, the energy in the error is equal to the error in the energy, and the approximated strain energy corresponding to any finite dimensional approximation is always less than that of the actual strain energy associated with exact solution. This, of course, is easy to understand when one considers that the description of an infinite dimensional displacement function by a finite set of basis functions only imposes additional constraints on the system. The additional stiffness, in turn, reduces the displacement and hence the strain energy of the system. The strain energy increases with increase in the number of basis functions. For this to continue to happen, however, the basis functions must satisfy some continuity requirements. For example, for plane stress problems in two dimensions, the trial function, for the Rayleigh–Ritz–Galerkin method, must be such that its first derivative has finite energy, that is, that it must be at least continuous, so that the virtual work expression is meaningful. In general, the trial functions must have finite energy (in the energy norm) in their mth derivatives when the system-governing differential equation is of order 2m. In finite element analysis, the geometry is reconstructed as an assemblage of finite elements. The above condition, then, becomes that the basis functions must be of class Cm−1 across the element boundary. For plane stress analysis, m = 1, so the trial functions must be at least C0 across the boundary, whereas for plate analysis, m = 2 and the corresponding continuity requirement is C1 , that is, the first derivative of the displacement function must be continuous. Any element that violates this requirement is known

6

Computation of Nonlinear Structures

as the non-conforming element (Taylor et al., 1976); otherwise, it is conforming. In general, to establish, say, C0 continuity at any edge between any two elements, the shape functions for a node connected to the edge must be identical for the two elements, and the shape functions for the rest of the nodes must vanish at the edge. For example, it can easily be shown that the constant strain, three-node triangle is conforming, and so are triangular or tetrahedral elements with edge nodes, as long as the edges are straight and all edges have same number of nodes. However, for the general four-node quadrilateral element, a simple coordinate transformation by rotation will show that the element is non-conforming. The only quadrilateral element that is conforming is the four-node rectangular element. All other general quadrilateral elements with edge nodes on the straight edges are non-conforming. Finally, if the edges of the elements, triangular or quadrilateral, are curved, the elements are non-conforming. All these elements made up of basis functions by incomplete power series of position coordinates lack the barycentric or convexity property of interpolation, and thus are devoid of invariance under affine – for example, rotational – transformations. In fact, as will be shown later, the problem associated with the general curved conventional elements of today’s finite element method is one of interpolation that translates into lack of inter-element continuity in general. Historically speaking, however, these earlier problematic h-type elements found their partial savior in what is known as parametric mapping and rigid body rotation. Parametric Mapping and Rigid Body Rotation: In the theoretical formulation of the isoparametric finite element method, it is assumed that the elemental basis functions are expressed in a root element, and the displacement functions are then mapped isoparametrically with the geometry. For the case where the geometry is straight and can be expressed by a linear root element, the displacement functions could be of higher order, needing conventionally edge or interior nodes; here, the mapping of the root element is known as subparametric. In the reverse situation of curved geometry with simple strain states, the number of basis functions necessary to completely describe the geometry may be higher than that needed for displacement description; this mapping goes by the name of superparametric. For superparametric elements, the representation of rigid body rotation and constant strain states is affected adversely. The history of construction of conventional finite elements with variable nodes appeared to have failed to realize one fundamental mathematical relationship: the conversion in description of non-parametric functions to parametric description automatically and inescapably fixes all the internal nodes. The displacement functions or the geometry coordinates, to start with, are non-parametric functions. To generate isoparametry between them, both needed to be first transformed into parametric entities. This then, by the principle just stated, fixes the intermediate nodes, be it just the edge nodes for one dimension, or the edge, face or internal nodes in their higher dimensional counterpart. Any discussion on distortion of the intermediate nodes is irrelevant and futile. In fact, if complete isoparametry is established, guaranteeing a barycentric or convex combination of the basis functions of the root element, the ensuing finite element, irrespective of any elemental distortion, will be conforming and complete. Short of guaranteed completeness of the polynomials, we would like to have a device by which various conventional elements can be measured for their acceptance. We now engage in discussing one such tool. Patch Test: In earlier days, when terms were being added or dropped for designing new finite elements in ad hoc fashion, a standard numerical test was necessary for evaluating their worth. One such test is the patch test. The idea behind the patch test is simple. A chunk of elements representing a system is plucked out and subjected some known boundary conditions for which the internal solutions are known. It is expected, then, for acceptance of the elements, that the corresponding computed internal quantities should be the same as those known expectations. In other words, if a linear black box (the element) is correct, the output and input must correspond. To make calculations easy, the boundary is taken simple as well, such as square or cube for

Introduction: Background and Motivation

7

two and three dimensions, respectively. The constant strain patch test is considered standard for two- dimensional in-plane load conditions, because as the elements reduce sufficiently in size, the state of the strain (or derivative of the displacement) becomes constant in the limit. So, for convergence, the element should be able to represent such a state of strain. For plate bending, this strain becomes the bending curvature. In any case, linear displacement-equivalent forces are applied on the external nodes and solved for internal node displacements. The derivatives of this solution must match the known constant strains. There are several versions of patch tests. A patch test was initially conceived as a numerical experimentation for element validation. For smooth solution problems, the patch test was later shown mathematically to be a necessary and sufficient test for convergence of h-type elements. It appears quite clear to the author that the patch test is, in the main, an interpolation check. However, it serves to detect other failures such as integration or equilibrium failures. Because of the conformity (satisfying equilibrium requirements) and completeness (satisfying interpolation requirements) of the conventional linear isoparametric elements having only corner nodes, generation of constant strain or linear displacement state is extremely easy. Similarly, the conventional variable-node higher-order isoparametric elasticity elements with straight edges or faces with evenly spaced nodes can pass a higher-order patch test. As rectangular elements with evenly spaced edge nodes, the serendipity elements fail the patch test while its counterpart full-noded Lagrangian elements do not. Because of incomplete quadratic terms, isoparametric three- or four-noded bending elements cannot pass the constant curvature or quadratic displacement patch test. For curved elements, triangular (and tetrahedral) or quadrilateral (and hexahedral) with variable nodes arbitrarily placed cannot pass the quadratic displacement or constant patch test as necessary for plate and shell elements. As will be shown in the subsequent chapters, this is clearly an interpolation failure associated with elements with variable nodes under a curved situation. Finally, of course, the linear elements fail to pass the constant curvature patch test. In other words, these elements should not be used to model, say, plate or shell systems. Locking Problems: The discussion on the patch test has been included merely to show the innate interpolation problems that have been afflicting conventional finite elements. More specifically, these translated into other serious numerical maladies known as locking. We will therefore now look at the phenomenon of locking only in conventional rectangular elements. Other elements such as triangular elements experience the same locking phenomena, so we will dscribe the remedial methods that have been tried, but not altogether successfully. The futility crept in because none of these so-called tricks ever tried to correct the innate interpolation problems. In fact, these remedies, which include reduced and selective integration, incompatible modes and so on, do not appear to be numerically robust in the classical sense of consistency, convergence and stability. As a result, we are burdened with the additional problem of spurious modes introduced by reduced or selective integration. Simple solutions based on exact interpolation theory that allow the reader to appreciate the strength of the new c-type finite element method are presented in Chapter 9 about linear applications. Shear and Membrane Locking and Shape Dependence: The major debacle of interpolation failure is locking, that is, elements showing unacceptably high stiffness, almost bordering on rigidity, in certain displacement states. In other words, the low order or variable node elements are failing to respond flexibly enough for or failing to recognize as it should, certain imposed conditions. Moreover, it was recognized that the severity of this phenomenon of locking depends on the shape of the element. Finally, the shape may be boiled down to a specific parameter that allegedly perpetrates such locking. Now let us briefly examine the conventional state-of-the-art remedies. Reduced/Selective Integration, Bubble Mode and Drilling Freedom: The linear four-node element failed to interpolate the quadratic term, and thus experienced dilatation locking.

8

Computation of Nonlinear Structures

Likewise, inability to handle quadratic term gave rise to spurious shear strain. This, in turn, contributed to the shear locking. As we already know, the numerical integration to evaluate, say, the stiffness matrix, is performed by selecting Gauss or other quadrature points. Thus, an immediate remedy to shear locking was offered by selective under-integration (Zienkiewicz et al., 1971) by evaluating the integral involving shear at only one Gauss point, which effectively wiped out the shear contribution to the stiffness matrix. The problem with this remedy is that the reduced integration, is obviously less accurate. More importantly, it creates undesirable spurious modes, that is, artificial singularities or eigenmodes induced by possible reduction in the rank of the stiffness matrix. Another ad hoc remedy was to include quadratic interpolation terms: so-called bubble modes (Wilson, 1973), without introducing any additional node, to handle both shear and dilatation locking. This, of course, renders the element non-conforming as two nodes cannot possibly represent a quadratic term. In line with the addition of modes of flexibility just described, modes known as drilling freedoms have been devised to resolve locking problems. Drilling freedoms (Allman, 1984) are nodal rotational degrees of freedom about an axis perpendicular to the plane of the element. These in-plane rotational degrees of freedom, in improving membrane performance, may create problems for bending performance of curved shell elements, where these degrees of freedom are already coupled with the bending modes. But, in general, most of these patchy remedies that worked on regular rectangular shaped element, fail to eradicate locking when applied to general h-type isoparametric curved elements of arbitrary shape. Moreover, these varieties of so-called remedial elements for specific displacement states increase the number of elements available commercially to such an extent that an ordinary user can be easily confused as to their use. It is extremely important that the element library be simple and user-friendly. Plate and Shell Elements: For our purpose, we need only to note that for arbitrary shaped plate and curved shell elements, the interpolation problems and proposed solutions are similar to those already described. The remedies that have been offered for offsetting interpolation failure are not robust and are highly dependent on the element shapes. For example, trapezoidal plate or shell elements can still create problems of transverse shear locking.

1.2.3

Essential Mesh Generation: Curves and Surfaces

At around the same time as the introduction of the finite element method to structural analysis, a different numerical revolution, unnoticed by the finite element analysts, was underway primarily in the car industry (Coons and Herzog, 1967; Bezier, 1972; de Casteljau, 1986; Gordon and Riesenfeld 1974a,b), to improve upon the production of automotives with arbitrarily curved bodies: namely, computer-aided geometric design (CAGD). Of all the new methods of CAGD, solid modeling (Mantyla, 1988; Hoffmann, 1989) and boundary representation (B-Rep) methods (Barnhill, 1974; Faux and Pratt, 1979; Bohm, 1981, 1984; Foley and van Dam, 1982; Yamaguchi, 1988; Farin, 1992; Piegl and Tillman, 1995) took the leading role. Solid modeling is based on using geometrically simple elements as building blocks with set-theoretic operations of union, intersection, complementarity and so on, but without any information on inter-element continuity. On the other hand, B-Rep methods are all about building the geometry from simple shapes, from curves to surfaces to solids, with information and control over the inter-element boundaries. The foundations of these methods are based on complete Bernstein polynomials (Davis, 1975) as shape factors in Bezier control vector form and Schoenberg B-spline (Schoenberg, 1946) curves in de Boor–Cox (de Boor, 1972; Cox, 1972) control forms. The most important aspect of this revolution, in so far as it could relate to possible application to finite element analysis, is that Bsplines are constructed geometrically as opposed to algebraically (Prenter, 1975) and thus provide

Introduction: Background and Motivation

9

the necessary local support for each finite element; in the ultimate analysis, these polynomials are complete for interpolation and approximation with local support.

1.3

Symbiotic Structural Analysis

To solve nonlinear structural problems such as beams, plates and shells under large displacement and rotation, it is not enough to engage the first part, namely, the mesh generation technique, and, the second part, namely, the c-type finite element method, free of ‘variational crimes’ that lead to locking, ad hoc numerical artefacts, and so on, without reformatting the fundamental axioms of solid mechanics to corresponding computational forms. In other words, the transformed governing equations of solid mechanics applied to beams, plates and shells form the third essential part of the system for possible production of largest finite elements. The reason for using such a transformation is primarily that finite rotations, unlike displacements of Euclidean space, traverse a configuration space that is a Riemannian curve space necessitating covariant derivatives as opposed to Gateaux derivatives; this book tries to draw together these three components in developing the largest possible nonlinear beam, plate and shell static and dynamic finite elements.

1.4

Linear Curved Beams and Arches

The study of curved beams in finite element development is extremely important for anticipating problems for a given proposed method, when extended to the more general case of curved elements, namely, shells. In the linear regime, for many years, Timoshenko- or Mindlin-type curved beam and arch elements reportedly suffered from what is commonly known as shear and membrane locking in the thin regime (Ashwell & Sabir, 1971; Ashwell et al., 1971). This, has triggered, for thirty years, a range of interpolationally inadequate approaches with limited success in terms of accuracy, rate of convergence and general applicability. Unfortunately, there are textbooks (e.g. Cook, 1994; Dow, 1999), that constantly refer to these methods as incurable observations, and this is misleading for the aspiring analysts. Nonetheless, we list a few such attempts with their associated pitfalls: the application of reduced integration of shear and membrane energy producing spurious modes of deformations in thin limits (Stolarski and Belytschko, 1982; Panadian, 1989); the truncation of so-called field-inconsistent strains resulting in low convergence rate (Babu and Prathap, 1986); the application of mixed polynomial-trigonometric displacement field restricting its general applicability to non-circular curved beams (Cantin & Clough, 1968; Ashwell, 1971); the utilization of higher-order quintic-quintic independent displacement fields (Dawe, 1974); the problem-specific penalty relaxation method (Tessler & Spiridigliozzi, 1986); the application of so-called quasi-conforming technique (Shi and Voyiadjis, 1991); and the circular curve-specific application of quadratic curvature interpolation (Lee and Sin, 1994). Finally, there is a method in which so-called material finite elements (MFE) are generated by analysis applicable only to linear circularly curved beams resulting in polynomial distributions for the field variables that are consistent with the equilibrium equations (Raveendranath et al., 1999). The elements are lock-free with good predictability for field variables but unfortunately the shear strain distribution is restricted to being constant, resulting in only approximately constant shear force over each element. All these methods mentioned are either numerically ineffective or problem-specific. Also, there continues to be an unnecessary general consensus that to be locking free, a consistent formulation must have the order of interpolation for tangential displacement one order higher than that of normal displacement (Meck, 1980). For formulation in the linear regime of truly three-dimensional curved beams, we refer to the original works of Love (1944,

10

Computation of Nonlinear Structures

1972). For warping with seven degrees of freedom – three translations, three rotations and one twist, we refer to Washizu (1964).

1.5

Geometrically Nonlinear Curved Beams and Arches

One main focus of this book, as already indicated, is to demonstrate the versatility, simplicity and accuracy of the c-type method as applied to three-dimensional, beams and arches. The beams may be shear-deformable and extensible, straight or truly three-dimensional space curved with natural twist and bend, and finally geometrically nonlinear with extreme rotations and displacements without warping. The three-dimensional curved beam problem in the geometrically nonlinear regime – that is, for finite deformations – was first reported by Reissner (1973) and then in (1981) with an improved formulation. However, the analysis is restricted to second-order approximations on the rotation parameter and presents no numerically explicit expressions for stiffness, and so on. The work of Argyris (1982) is, in an unduly long and confusing sense (Spring, 1986), an introduction to theoretical kinematics for the mechanics community. For formulation in the extreme geometrically nonlinear regime, see the work of Simo (1985). In the analytical formulation, the possibility of a naturally twisted and bent situation is suggested. However, the numerical examples presented in the referenced work modeled curved beams as an assemblage of straight elements, resulting in a larger than necessary number of elements as will be demonstrated later. Simo and Vu-Quoc (1986) succeeded in enriching this method by considering finite rotation as an algebraic structure, namely, a Lie group. Unfortunately, this paper missed the well-known role of covariant derivatives in parameters that travel on a differentiable manifold, that is, generalized multi-dimensional surfaces. However, this otherwise outstanding paper was taken as a standard by several other authors to justify their conclusions about the symmetry property of the tangent stiffness matrix. For example, Crisfield (1990), following a co-rotational formulation, wrongly asserted that even for a conservative system, the tangent stiffness matrix is unsymmetrical everywhere except at an equilibrium state. In view of this, we refer later to a simple lemma from Kreyszig (1959) to bring out the essence of the matter: in summary, the Gateaux derivatives are only appropriate on linear vector spaces, not on manifolds, for linearization and symmetry of the virtual functional, because, unlike ‘flat-spaces’ (e.g. Euclidean) where vectors are ‘free’ vectors, the Riemannian and differentiable manifolds accommodate only ‘bound’ vectors. An explicit proof for the multi-dimensional surface was eventually reported by Simo (1992). The tangent spaces are linear vector spaces, thus restriction of the rotation parameterization to elements of these spaces can avoid issues relating to covariant derivatives and allow the use of the familiar Gateaux derivatives needed for linearization. Following this idea, a series of papers appeared, starting with Cardona and Geradin (1988), where the symmetry of the geometric stiffness was correctly assumed, but without any proof. Similarly, in a related paper (Ibrahimbegovic, 1995), because of a less than instructive explicit expression of the same, the symmetry could only be verified but not proved. Another paper (Gruttmann et al., 2000) presented a proof but because of the Cartesian nature of the analysis, the expression for the geometric stiffness remained rather convoluted. Of course, covariant derivatives are always needed for a curvilinear formulation, say, in the development of the equilibrium equations as we will show. We attend to this problem by choosing all the parameters as vector fields measured in curvilinear coordinates for various reasons. The first is rooted in our intention to unite the geometric mesh generation and the subsequent finite element analysis. Thus, the information – Bezier geometric controls, curvature and twist in Frenet frames – produced by a mesh generation provides the starting point for our analysis. Secondly, applications of the covariant derivatives for the stress resultant and stress

Introduction: Background and Motivation

11

couples in deriving the equilibrium equations and subsequent virtual work principle result in a direct proof and an elegant yet explicit expression for both the material and the geometric stiffness. Thirdly, the elements with only six degrees of freedom per node can be directly incorporated into the standard direct stiffness based finite element packages. Finally, in the numerical examples presented in the literature, almost invariably, curved beams are modeled as an assemblage of straight elements resulting in more than the necessary number of elements. We treat curved beams as mega-elements with truly three-dimensional attributes of natural bend and twist, and thus use a minimum number of elements for analysis, as will be demonstrated. The book presents the nonlinear beam theory for both quasi-static and dynamic loading conditions. For the quasistatic problems a Newton-type iteration scheme has been employed, similar to (Crisfield, 1981), to solve various benchmark problems; no attempt has been made in this book to apply the theory to dynamic problems.

1.6

Geometrically Nonlinear Plates and Shells

Another important focus of the book, lies in a demonstration of the versatility, simplicity and accuracy of the c-type method as applied to shells. The shells may be shear-deformable and extensible, and geometrically cylindrical, spherical or truly arbitrary surfaces, and finally geometrically nonlinear with extreme rotations and displacements and subjected to quasi-static or dynamic loading. Computationally speaking, the methodologies for modeling and solution of shell structures can be generally categorized into four groups: (1) the earliest finite element approach consisted of facet-type plate elements with loss of slope continuity for Love–Kirchhoff-type shells that need -continuity, and that are plagued by the shear locking problem when dealing with Mindlin– Reissner-type shells with shear strains; (2) three-dimensional solid (brick) elements are used as shell elements based on 3D elasticity equation with a large number of nodes and degrees of freedoms. For thinner shells, these elements broke down with erratic behavior and locking problem and ill-conditioning; (3) A finite element modeling known as degenerative shell elements from three-dimensional shell brick elements based on Mindlin–Reissner-type shell theory with reduced integration; and (4) shell elements modeled after various old classical shell theories (Wempner, 1989; Ahmad et al., 1970; Zienkiewicz et al., 1971; Pawsey and Clough, 1971). For theoretical developments, we can divide the various efforts into two main groups: (1) in the direct approach the shell is taken at the very start as a 2D object and all the laws are developed for this, including the constitutive laws; Cosserat shells belong to this category (Naghdi, 1972); (2) the derived approach, where various suitable constraints are applied to the three-dimensional shell-like body. If the main assumption is that the normals to the surface remain normal during deformation and inextensible, this results in what is known as the Love–Kirchhoff theory of shells with only three degrees of freedom at each point; otherwise, if normals just remain straight and inextensible, we have the Mindlin–Reissner model of shells with five degrees of freedom. Finally, inclusion of extensibility accommodates the full set of six degrees of freedom and goes by the name of Timoshenko-type shell theory (Pietraszkiewicz, 1979; Simo et al., 1990). All these approaches suffer from adhocism in either their theoretical underpinning or their numerical evaluation. In this book, a modified method is developed following Reissner (1960) and Libai & Simmonds (1998). In this concept, we start from the exact three-dimensional balance laws and derive the variational functional exactly; any approximation is contained where it should naturally belong, that is the constitutive theory, which is at best empirical. The variational functional is linearized for numerical evaluation by a Newton-type iterative method by recognizing that the rotation tensor travels on a curve space and hence the absolute necessity of the covariant derivative of

12

Computation of Nonlinear Structures

curve spaces over the Gateaux derivative of so-called ‘flat’ Euclidean space; for the corresponding constitutive theory, we follow Chroscielewski et al., (1992).

1.7

Symmetry of the Tangent Operator: Nonlinear Beams and Shells

In nonlinear computational mechanics the prevalent Newton-type methods require linearization of the virtual functional for subsequent finite element formulation. The ad hoc practice of symmetrizing the tangent operator away from equilibrium, based on an application of Gateaux derivatives (a simple generalization of partial derivatives) in the linearization of parameters on a Riemannian surface, resulted in the claim that the geometric stiffness is non-symmetric away from equilibrium even for conservative systems. From a computational point of view, even for feasibility of producing a convergent solution without shear locking, and so on with truly curved and large beam or shell element, it is imperative that the concomitant theoretical formulation be derived in the most accurate and cost-effective sense. For beam and shell formulations in the geometrically nonlinear regimes, this brings us to the discussion of the following topics that show the essential motivation of our book. Symmetry of the Geometric Stiffness – Conservative Systems: The three-dimensional curved beam formulation with a rotating or spinning coordinate system, extended in shell theory by Simmonds and Danielson (1970), in the geometric nonlinear regime (i.e. for finite deformations) was first reported by Reissner (1973) and subsequently again in (1981), with an improved formulation. However, the analysis is restricted to second-order approximations on the rotation parameter and presents no numerically explicit expressions for the residual and stiffness needed for a finite element development. Simo and Vu-Quoc (1986) enriched this by consideration of finite rotation as an algebraic structure, namely, a Lie group. Unfortunately, their paper overlooked the well-known role of the covariant (absolute) derivative in consistent linearization with respect to parameters that travel on a smooth manifold, that is, generalized multi-dimensional surfaces. To quote Simo and Vu-Quoc (1986 p. 80): “we show . . . the global geometric stiffness arising from the (consistently) linearized weak form is non-symmetric, even for conservative loading, at a non equilibrium configuration . . . Argyris and coworkers [5–8] point out that this lack of symmetry inevitably arises at the element (local) level” (emphasis added). The mistake is rooted in the application of the Gateaux derivative (a generalized partial derivative applicable only to Euclidean or “flat” surfaces) to define and compute the Hessian from the virtual work functional defined on Riemannian or “curved” surfaces. As a result, to overcome the inevitable dilemma, they introduced an ad hoc procedure that has since become known as the symmetrization of the geometric stiffness. Simo recovered from this adhocism in a subsequent paper (1992) published four years later. However, Simo’s original assertion was taken as a standard by several other authors to justify their mistaken conclusions about the symmetry property of the tangent stiffness operator. For example, Crisfield stated, ( p. 148, Crisfield, 1990): “The tangent matrix is . . . nonsymmetric. . . . for the conservative problems analyzed, symmetry is recovered as an equilibrium state is reached” (emphasis added). Based on this, the paper goes off at a tangent to recommend future research: “a study of the geometric stiffness matrix in order to discover which terms may be reasonably neglected without serious cost to the performance” (Crisfield, 1990 p. 148). Symmetry of the Geometric Stiffness – Non-conservative Systems: For general threedimensional situations, it will be recognized that, as indicated by Ziegler (1977), a moment loading (i.e. either distributed or end moments) makes the system non-conservative and so the resulting geometric stiffness is non-symmetric. But, as we will show, an identification of the terms responsible for the asymmetry suggests that the non-symmetrical structure may be computationally rather weak. Thus, in the absence of moment loading or where the end boundary

Introduction: Background and Motivation

13

conditions are purely kinematic with no distributed body moments, the system is conservative, and thus the geometric stiffness is definitely symmetric as discussed in the previous section. For two-dimensional or planar systems with planar deformations, the reduction of the 3D expression of the geometric stiffness to the 2D situation will reveal that all moment-related terms, including the reactive and externally applied distributed or end moments, vanish identically, leaving the geometric stiffness unconditionally symmetric. Theoretical motivation: One important result of our theoretical motivation, is that we show that the symmetry of the tangent operator is always guaranteed for a conservative system, in equilibrium or away from the equilibrium, if, instead of the Gateaux derivative, the concept of the covariant derivative is applied in the definition of consistent linearization. Note that in any repetitive numerical solution (as in nonlinear problems): (a) the computational saving in working with a symmetric matrix, as opposed to a non-symmetric matrix, could be considerable and thus, always optimal, (b) an exact explicit definition, as opposed to an implicit one such as a derivative form or an approximate one, is always advantageous computationally. Thus, for a finite element formulation in the nonlinear regime with very large elements, it is not enough to prove that the tangent stiffness is symmetric for a conservative system. It is also imperative that the tangent stiffness present itself in its most simple and computationally effective form. In this sense, all papers published in the literature to date are not optimally suitable. Simo and Vu-Quoc (1986) and other similar papers presented the tangent stiffness in differential form. Cardona and Geradin (1988), where the symmetry of the geometric stiffness was correctly assumed, had to settle for an approximate expression of the tangent stiffness (on p. 2428, the comment after equation (146)). Similarly, in a related paper (Ibrahimbegovic, 1995), because of the disordered nature of the derived expression, the symmetry could not be proved. Gruttmann et al. (2000) presented a proof, but because of the Cartesian nature of the analysis, the expression for the geometric stiffness remained somewhat convoluted. Computational motivation: As a principal result of our computational motivation, we present in this book a crucial central lemma (Section 10.1) involving the virtual rotation and incremental rotation of a beam that avoids awkward derivation [which forced Cardona and Geradin (1988) to settle for an approximation] of second derivatives of tensors on a Riemannian surface. With the help of this lemma, along with other results, we present an accurate expression for the tangent stiffness in an optimal, cost-effective operational form. In this book, for several reasons, we choose all the parameters to be vector or tensor fields measured in curvilinear coordinates. This is first rooted in our intention of uniting the geometric mesh generation and the subsequent finite element analysis. Thus, the information, namely, Bezier geometric controls, curvature and twist in Frenet frames, produced by a mesh generation provides the starting point of our analysis. Secondly, applications of the covariant derivatives for the stress resultant and stress couples in deriving the equilibrium equations and subsequent linearization of the virtual work functional result in a direct proof of symmetry and an elegant yet explicit expression for both the material and the geometric part of the stiffness. Thirdly, the elements with only six degrees of freedom per node can be directly incorporated into a standard displacement-based finite element system. In conclusion, with the goal of developing a truly three-dimensional, geometrically nonlinear curved finite element, a geometrically exact curvilinear, nonlinear formulation for the truly three-dimensional curved beams/arches and shells is presented in the book. The symmetry question of the geometric stiffness matrix is fully treated and shows that the tangent operator is symmetric for a conservative system. The moment loading is discussed thoroughly by identification of specific terms of non-symmetry. The two-dimensional specialization establishes the unconditional symmetry of the operator. Of utmost importance, considering the requirement of repetitive generation of the tangent operator, is the presentation of a closed form expression for the associated geometric stiffness in its most simple, direct and explicit form compared to those available in the published literature.

Computation of Nonlinear Structures

14

Finally, a few words about the chronological history is in order. The original work by the author of geometrically applying Bernstein–Bezier and B-spline bases to beams was initiated in 1995, and a paper (Ray, 1999) contained the c-type method, noting the importance of choosing appropriate basis functions for a finite element method. Note that about 13 years before any paper containing a similar finite element method (the iso-geometric method) was published, a single conference paper (Reus, 1992) was found to exist through an extensive literature search; the present author, having been completely unaware of the paper, developed the c-type method independently.

1.8

Road Map of the Book

Based on the discussions of this chapter on background and motivations, the book is designed to accomplish its goals through a logical sequence of chapters, the content of each of the chapters is as follows: Chapter 1: this introductory chapter. Chapter 2: presents some essential mathematical entities from real analysis such as sets, group, algebra and functions and continuity, normed spaces, the Sobolev space, the Vainberg principle, and Gram–Schmidt orthogonalization. Chapter 3: introduces tensors in general, and then particularizes to second-order tensors; it identifies tensors both as linear functionals and dyadics or polyadics; it presents various properties and the differential and integral calculus of tensors so that all structural equations can be written in compact absolute notation devoid of cumbersome indicial description except where absolutely necessary. Chapter 4: discusses in detail the representation and properties of rotational tensors necessary for analysis of structures with finite rotational responses; it covers both the theoretical and computational details pertaining to rotation tensor, and thus introduces the mathematical object called the quaternion. Chapter 5: deals with the theory and computation of real curves in anticipation of its application to beams; in particular, it presents curves in the form of Bernstein polynomials and Bezier controls. It then includes B-spline representation geometrically by composite Bezier curves for local support. Chapter 6: deals with the theory and computation of real surfaces in anticipation of its application to plates and shells; more specifically, it presents surfaces in the form of Bernstein polynomials and Bezier controls. It extends further to include B-spline representation geometrically by composite Bezier surfaces for local support. Chapter 7: presents briefly the essential elements of nonlinear solid mechanics: deformation (strain), transmission of force (stress), balance laws and the constitutive theory of hyperelastic materials. Chapter 8: introduces finite element theory as an energy method, and it gives a presentation of the new c-type finite element method. Chapter 9: applies c-type finite element method to linear structural problems involving rods, straight and curved beams, plane stress and plane strain elements; it also provides solutions for various locking problems.

Introduction: Background and Motivation

15

Chapter 10: develops the theory for nonlinear beams subjected to quasi-static and dynamic loading, resulting in extremely large displacement and rotation responses, and applies the c-type finite element method to solving numerical problems with extremely large beam elements. Chapter 11: develops the theory for nonlinear plates and shells subjected to quasistatic and dynamic loading resulting in extremely large displacement and rotation responses and applies c-type finite element method to solve numerical problems with extremely large plate and shell elements.

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Foley, J. and van Dam, A. (1982) Fundamentals of Interactive Computer Graphics, Addison-Wesley. Gordon, W.J. and Riesenfeld, R.F. (1974a) Bernstein-Bezier methods for the computer-aided design of free-form curves and surfaces. Journal of the Association for Computing Machinery, 21(2), 293–310. Gordon, W.J. and Riesenfeld, R.F. (1974b) B-spline curves and surfaces, in Computer Aided Geometric Design, Academic Press, NY. Green, A.E. and Zerna, W. (1968) Theoretical Elasticity, 2nd edn, Oxford University Press. Gruttmann, F. et al. (2000) Theory and numerics of three-dimensional beams with elastoplastic material behavior. International Journal for Numerical Methods in Engineering, 48, 1675–1702. Hoffmann, C. (1989) Geometric & Solid Modeling – An Introduction, Morgan Kaufmann Publishers, Inc., CA. Huddleston, J.V. (1968) Finite deflections and snap-through of high circular arches. Journal of Applied Mechanics, 35(4), 763–769. Ibrahimbegovic, A. et al. (1995) Computational aspects of vector-like parametrization of three-dimensional finite rotations. International Journal for Numerical Methods in Engineering, 38, 3653–3673. Kreyszig, E. (1959) Differential Geometry, University of Toronto Press. Lang, S. (1968) Linear Algebra, 3rd Printing, Addison-Wesley Publishing Co., Inc. Lee, P.G. and Sin, H.C. (1994) Locking-free curved beam element based on curvature. International Journal for Numerical Methods in Engineering, 37, 989–1007. Libai, A. and Simmonds, J.G. (1998) Nonlinear Theory of Elastic Shells, 2nd edn, Cambridge University Press. Love, A.E.H. (1944) The Mathematical Theory of Elasticity, Dover, New York. MacNeal, R.H. (1994) Finite Elements: Their Design and Performance, Mercel Dekker, Inc. Malvern, L.E. (1969) Introduction to the Mechanics of Continuous Medium, Prentice Hall, Inc., NJ. Mantyla, M. (1988) An Introduction to Solid Modeling, Computer Science Press, Helsinki. Meck, H.R. (1980) An accurate polynomial displacement function for finite ring elements. Computers & Structures, 11, 265–269. Naghdi, P.M. (1972) The theory of plates and shells, in Handbuch der Physik Band VIa/2, Springer-Verlag, pp. 425–640. Ogden, R.W. (1997) Non-Linear Elastic Deformations, Dover Publications, Inc., NY. Palazotto, A.N. and Dennis, S.T. (1992) Nonlinear analysis of shell structures. AIAA Education series, Washington, DC, pp. 205–206. Panadian, N. et al. (1989) Studies on performance of curved beam finite elements for analysis of thin arches. Computers & Structures, 31, 997–1002. Pawsey, S.E. and Clough, RW. (1971) Improved numerical integration of thick shell finite elements. International Journal for Numerical Methods in Engineering, 3, 545–586. Piegl, L. and Tiller, W. (1995) The NURBS Book, Springer-Verlag, Berlin. Pietraszkiewicz, W. (1979) Finite Rotations and Lagrangian Description in the Nonlinear Theory of Shells, PWN, Warszawa=Poznan. Prathap, G. (1985) The curved beam/deep arch finite ring element revisited. International Journal for Numerical Methods in Engineering, 21, 389–407. Prathap, G. and Shashirekha, B.R. (1993) Variationally correct assumed strain field for the simple Curved beam Element. International Journal for Numerical Methods in Engineering, 37, 989–1007. Prenter, P.M. (1975) Splines and Variational Methods, John Wiley & Sons. Raveendranath, P. et al. (1999) A two-noded locking-free shear flexible curved beam element. International Journal for Numerical Methods in Engineering, 44, 265–280. Ray, D. (1999) Beam flexure and plane stress (unpublished) Ray, D. (2003) C-type method of unified CAMG & FEA. Part I: Beam and arch mega-elements – 3D linear and 2D non-linear. International Journal for Numerical Methods in Engineering, 58, 1297–1320. Ray, D. (2004) c-Type Method of Unified CAMG & FEA: Part II: Beam & Arch Mega-Elements – True 3D Curved, and Geometrically Nonlinear: An Exact Total & Incremental Curvilinear Formulation. (unpublished) Ray, D. (2004) c-Type Method of Unified CAMG & FEA: Geometrically Nonlinear Curved Beams, Plates & Shells, Indian Association for Computational Mechanics. International Congress on Computational Mechanics and Simulation. (ICCMS-04), December, IIT-Kanpur. Ray, D. (2004) Part III: On the Symmetry of the Tangent Operator of Nonlinear Beams: An Exact Curvilinear Formulation, (unpublished) Ray, D. (2007) Optimal Nonlinear Beam, Plate & Shell Elements & c-type FEM. 9th U.S. Congress on Computational Mechanics, San Francisco. Ray, D. (2008) Largest Geometrically Exact Nonlinear Beam, Plate & Shell Elements & c-type FEM. Proceedings of 6th International Conference, IASS-IACM 2008 – 6th ICCSSS, Cornell Univ., NY. Reddy, J.N. and Liu, C.F. (1985) A higher order shear deformation theory of laminated elastic shells. International Journal of Engineering Science, 23(3), 319–330

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Reissner, E. (1960) On some problems in shell theory. Proceedings of First Symposium Naval Structural Mech., pp. 74–114. Reissner, E. (1972) On one-dimensional finite strain beam theory: The plane problem. Journal of Applied Mathematics and Physics, 23, 795–804. Reissner, E. (1973) On one-dimensional large displacement finite strain beam theory. Studies in Applied Mathematics, 52, 87–95. Reissner, E. (1981) On finite deformations of space-curved beams. Journal of Applied Mathematics and Physics, 32, 734–744. Reus, J. (1992) Mechanical Deformations of Hyperpatch Solids. Second. International Conference On Computer Graphics and Visual Technology, December, pp. 147–158. Schoenberg, I.J. (1946) Contributions to the problem of approximation of equidistant data by analytic functions. Quarterly Applied Mathematics, 4, 45–99. Shi, G. and Voyiadjis, G.Z. (1991) Simple and efficient shear flexible Two-node arch/beam and four-node cylindrical shell/plate elements. International Journal for Numerical Methods in Engineering, 31, 759–776. Simmonds, J.G. and Danielson, D.A. (1970) Nonlinear shell theory with finite rotation vector. Proceedings of Koninklijke Nederlandse Akademie, Wetenschappen, B, 73, 460–478. Simo, JC. (1985) A finite strain beam formulation. The three-dimensional dynamic problem. Part I. Computer Methods in Applied Mechanics and Engineering, 49, 55–70. Simo, J.C. and Vu-Quoc, L. (1986) A three-dimensional finite strain rod model. Part II: Computational Aspects. Computer Methods in Applied Mechanics and Engineering, 58(1), 79–116. Simo, J.C. et al. (1990) On a stress resultant geometrically exact shell model. Part III: Computational aspects of the nonlinear theory. Computer Methods in Applied Mechanics and Engineering, 79, 21–70. Simo, J.C. (1992) The (symmetric) Hessian for geometrically nonlinear models in solid mechanics: Intrinsic definition and geometric interpretation. Computer Methods in Applied Mechanics and Engineering, 58(1), 189–200. Spring, KW. (1986) Euler parameters and the use of quaternion algebra in the manipulation of finite rotations: a review. Mechanism and Machine Theory, 21(5), 365–373. Stolarski, H. and Belytschko, T. (1982) Membrane locking and reduced integration for curved elements. Journal of Applied Mechanics, 49, 172–178 Strang, G. and Fix, G. (1973) An Analysis of the Finite Element Method, Prentice Hall. Szabo, B. and Babuska, I. (1991) Finite Element Analysis, John Wiley & Sons, Inc. Taylor, RL, et al. (1976) A nonconforming element for stress analysis. International Journal for Numerical Methods in Engineering, 10, 1211–1219. Taylor, R.L. and Ray, D. (2005) Rods – Some Developments for Large Displacements. 5th International Conference on Computation of Shell and Spatial Structures, Salzburg, Austria. Tessler, A. and Spiridigliozzi, L. (1986) Curved beam elements with penalty relaxation. International Journal for Numerical Methods in Engineering, 23, 2245–2262. Timoshenko, S and Goodier, J, (1951) Theory of Elasticity, 2nd Edition, McGraw-Hill, Inc. Turner, M.J., Clough, R.W., Martin, H.C., and Topp, LJ. (1956) Stiffness & deflection analysis of complex structures. Journal of Aerospace Science, 23, 805–823. Vainberg, M.M. (1964) Variational Methods for the Study of Nonlinear Operators, Holden-Day. Inc. Washizu, K. (1964) Some considerations on a naturally curved and twisted slender beam. Journal of Mathematics and Physics, 43(2), 111–116. Wempner, G. (1989) Mechanics and finite elements of shells. Applied Mechanics Reviews, 42, 129–142. Wilson, E. et al. (1973) Incompatible displacement models, in Numerical and Computer Methods in Structural Mechanics (eds S.T. Fenves et al.), Academic Press, pp. 43–57. Yamaguchi, F. (1988), Curves and Surfaces in Computer Aided Geometric Design, Springer-Verlag, Berlin. Ziegler, H. (1977) Principles of Structural Stability, 2nd edn, Berkhauser, Basel. Zienkiewicz, O.C, Taylor, R.L., and Too, J.M. (1971) Reduced integration techniques in general analysis of plates and shells. International Journal for Numerical Methods in Engineering, 3, 275–290. Zienkiewicz, O.C. and Taylor, R.L. (2000) The Finite Element Method - Basic Formulattion and Fundamentals, Butterworth.

Part I Essential Mathematics

2 Mathematical Preliminaries 2.1

Essential Preliminaries

In this chapter, we will briefly touch upon some of the mathematical ideas that are important for our subsequent investigation of problems in structural engineering, structural mechanics and computer-aided geometric design. Various mathematics and numerical analysis books can be also consulted.

2.1.1

Groups

First, let us introduce some operations and structures needed for the definition of groups:

r Binary Operation on set S: f r r r r r r

: S × S → S, S ≠ {𝜙}; the + operation on Z is an example, while the − operation on Z + is not; z, z+ are integer and positive integer sets, respectively. Commutative binary operation on set S: the binary operation is commutative if ∀a, b ∈ S, a ∙ b = b ∙ a; clearly, + operation is commutative but the − operation is not since a − b ≠ b − a. Binary Structure: (S, ∙), set S with binary operation; (Z, +) is an example, while (Z + , −) is not. Semi-group: (S, ∙) with associative property: ∀a, b, c ∈ S, (a ∙ b) ∙ c = a ∙ (b ∙ c); (Z, +), (Z + , +) is an example, while (Z + , −) is not; (a − b) − c ≠ a − (b − c). Identity element: ∀a ∈ S, and (S, ∙), a binary structure, e is an identity if there exists e ∈ S such that a ∙ e = e ∙ a = a. For (Z, +), e = 0 is the (additive) identity element because: ∀a ∈ Z, a + 0 = 0 + a = a; for (Z, ×), e = 1 is the (multiplicative) identity element because: ∀a ∈ Z, a × 1 = 1 × a = a. Monoid: a semi-group with an identity. (Z, +) with e = 0, and (Z, ×) with e = 1 are monoids, while (Z + , +) with e = 0 is not, because 0 ∉ (Z + , +). Inverse element: ∀a ∈ S, and (S, ∙), a monoid with e as the identity element, ã is an inverse if there exists ã ∈ S such that a ∙ ã = ã ∙ a = e. For (Z, +) with e = 0, −2 is the inverse of 2 since 2 + (−2) = −2 + 2 = 0.

Now, we are in a position to define a group; a group is a monoid, (S, ∙) with identity element e such that ∀a ∈ S, there exists an inverse element, ã . Thus, a group is a set of elements with a binary operation satisfying four properties: closure, associativity, existence of identity element and inverse elements. Clearly, (Z, +) with e = 0 form a group; but, (Z, ×) with e = 1 does not form a

Computation of Nonlinear Structures: Extremely Large Elements for Frames, Plates and Shells, First Edition. Debabrata Ray. © 2016 John Wiley & Sons, Ltd. Published 2016 by John Wiley & Sons, Ltd.

Computation of Nonlinear Structures

22

group because ã ∉ Z for ∀a ∈ Z. Similarly, (Q∗ , +) is not a group since e = 0 ∉ Q∗ (Q∗ = Q − {0} with Q, rational number set). In any event, in order to check for a group, we must see that the identity element and the inverse elements for all elements of the binary structure with associative property exist. A commutative group is known as an Abelian group. Clearly, (Z, +) and (Q∗ , ×) are Abelian groups. We will be interested in matrix groups, and particularly in rotational matrix groups under multiplication, denoted as special orthogonal group, SO(3), (see Chapter 4 on rotation tensors); this belongs to what is known as the Lie group, which has the geometry of a smooth manifold (a topological space resembling points and their neighborhoods as Euclidean spaces) with the algebraic structure of smooth continuous mapping; thus, the global group property can be constructed from the local group on the linear space known as Lie algebra.

2.1.2

Linear Vector Space

A vector space is defined as follows: given any two (members of the defining set) vectors, a, b, of a vector space, and scalars, 𝛼, 𝛽, the linear (i.e. homogeneity: scalar multiplication and additivity) combination of vectors given by: 𝛼a + 𝛽b is also a vector (closure property) belonging to the vector space; a vector space contains an origin, the vector 0, that follows by choosing: a = b; 𝛼 = 1; 𝛽 = −1. A vector addition is both commutative (a + b = b + a) and associative (a + (b + c) = (a + b) + c). For example, the forces can be represented as vectors, but the confounding nature of finite rotations does not permit them to be vectors: although a single rotation can be described by a vector-like directed segment called an axial vector along the axis of rotation, it does not obey the vector addition property; in fact, it takes a tensor (see Chapter 4 on rotation) to represent rotation as will be seen in the discussion on rotations. A set of vectors are linearly dependent if they can be linearly combined to add up to the zero vector without all of the coefficients of combination being simultaneously zero: in an n + 1-dimensional vector { } space, if c0 v0 + c1 v1 + ⋯ + cn vn = 0 with not all ci equal to zero, then the set of vectors vi are linearly dependent; otherwise, the set of vectors is known as linearly independent: in an n + 1-dimensional vector space. A set of n + 1 vectors that are linearly independent forms what is known as a basis vector set with each vector known as a base vector. A basis vector set spans the space, that is, every vector of the space can be expressed as a linear combination of this basis vector set. Let us consider a vector, v, of a vector space of n + 1 dimensions such that v = c0 v0 + c1 v1 + ⋯ + cn vn = ci vi with not all coefficients,ci , i = 0, 1, 2, … , n, simultaneously zero, then the vector, v, and the set of n + 1 vectors form a linearly dependent set of vectors; the set of vectors, vi , i = 0, 1, … , n form a basis vector set.

2.1.3

Real Coordinate Space, Rn

The real coordinate space, Rn , consisting of n− tuples of real numbers is an example of a n-dimensional linear vector space over a real number field. Indicial notation: all scalar components of a vector, a, are denoted in indicial notations for the purpose of both analysis and computation. For example, instead of designating components as x, y, z, …, these will be subscripted such as a1 , a2 , a3 , …, or in compact form as: ai , i = 1, n; as will be seen later, for curvilinear coordinate systems, these could be either subscripted or superscripted indices such as: aj , j = 1, n. Thus, a column vector, x, is written component-wise as: x = (x1 , x2 , … , xn )T ; the superscript T stands for vector transposition. The origin is the zero vector, 0, with all its components zero: O = (0, 0, … , 0)T . The linear combination of two vectors, x and y is given component-wise as: 𝛼x + 𝛽y = (𝛼x1 + 𝛽y1 , 𝛼x2 + 𝛽y1 , … , 𝛼xn + 𝛽yn )T .

Mathematical Preliminaries

2.1.4

23

Convex Set and Convex Hull

A subset C of a linear space X is called convex if C contains all the elements on the line segment joining any two elements on the subset, that is, ∀a, b ∈ C ⇒ c ∈ C where c = ta + (1 − t)b, t ∈ [0, 1]. For example, all circles are a convex set in R2 . The convex hull of a set S is the intersection of all the convex sets containing S. Einstein summation convention: for the example in the previous paragraph, in compact form, we could introduce and use the convention of summing over repeated indices. In Cartesian coordinate systems, the repeated indices are all subscripted; thus: ci vi = c0 v0 + c1 v1 + ⋯ + cn vn , or aj aj = a21 + a22 + a23 in three dimensions. However, for curvilinear coordinate systems, the repeated indices must be pair-wise diagonally placed, that is, if one is superscripted, the other must ∑ be subscripted; for example, aj bj = a1 b1 + a2 b2 + a3 b3 , and a𝛼𝛽 b𝜂𝛽 = 3𝛽=1 a𝛼𝛽 b𝜂𝛽 = a𝛼1 b𝜂1 + a𝛼2 b𝜂2 + a𝛼3 b𝜂3 in three dimensions. Finally, in compact form, a single equation representing the length, l, of a vector, a = (a1 , a2 , … , an )T can be written as: l2 = ai ai whereas, in three dimensions, ai = mbi hides three equations in it. The real coordinate space, Rn , has n standard bases given as: i1 = (1, 0, 0, … , 0)T , i2 = (0, 1, 0, … , 0)T , … , in = (0, 0, 0, … , 1)T . Thus, a vector, x, can be written, using the summation ∑ convention, as: x = ni=1 xi ii = xi ii where xi ’s are the components of the vector x.

2.1.5

Classes of Functions: Continuity and Differentiability

For both interpolations and approximations, we will deal almost exclusively with functions of real variables over finite intervals. The degree of smoothness of these functions determines in general the strength with which the analytical properties of interpolation and approximation can validate the process. In other words, the smoother the function, the better is the approximation. In the mathematical literature, functions of varying smoothness abound. Thus, we may organize functions depending on how smooth they are: a continuous function, f (x), x ∈ [a, b], belongs to the class of C0 [a, b] functions. It is important to note that we have implied both the interior and the boundary of the domain in the definition; if the property of the function holds strictly in the interior of the domain, we will denote it by C0 (a, b). For any n ≥ 0 integer, if the same function and its derivatives up to and including order n are continuous, then it belongs to Cn [a, b]; of course, this classification holds for any vector-valued functions as long as all the component functions individually satisfy the property. An infinitely smooth function belongs to the class denoted by C∞ [a, b]. Clearly, every polynomial function is a member of the class, C∞ [a, b]. A step function belongs to C−1 [a, b]. In finite element analysis, as we shall see, the constructed global basis functions are infinitely smooth polynomials in the interior of a single element but may be zero on the boundary and over all other elements, that is, all derivatives on the boundary are zero; these functions are denoted as class of C0∞ . Another class of important functions, for b

our purpose, belong to the set, Lp [a, b], for which ∫a |f (x)|p dx < ∞, where the integration is a b

b

Lebesgue integral (defined by ∫a f (x) dx = lim ∫a fn (x) dx) as opposed to a Riemann integral. n→∞ Of particular interest is the situation where p = 2, that is the L2 [a, b] space; all continuous functions belong to this class, However, more importantly, continuous functions which have discontinuities at a finite number of points belong to this class. To sum up, there are several reasons to include this class: (a) classes of functions which are not Riemann integrable but are Lebesgue integrable, that is, functions that are continuous except at a finite number of points, can still be investigated, (b) while C[a, b] is not complete (i.e. does not contain all the limit points

Computation of Nonlinear Structures

24

of Cauchy sequences of functions on them), L2 [a, b] is; in other words, the class of continuous functions are completized by inclusion into L2 [a, b]. This is important because it ensures that in any adaptive approximation procedure such as the finite element methods, any sequence of continuous functions approximating the solution will be guaranteed to converge eventually to the actual solution. One of the most important and ‘decent’ class of functions is the space of polynomial functions, Pn [a, b], of degree n or less in the interval [a, b]; they are continuously infinitely differentiable and nicely structured so that term-by-term operations of calculus and algebra are easy. Now, for our purposes here, we will need linear spaces with more structure than just the linear vector spaces defined above; thus, we will briefly introduce several spaces that we will refer to in the finite element methodology described here.

2.1.6

Inner Product Space: Hilbert Space

A real linear space X will be called an inner product space if for any x, y ∈ X, there exists a real-valued function, designated by (x, y) with the following properties: (1) Positivity: (x, x) ≥ 0, (x, x) = 0, iff x = 0, (2) Symmetry: (x, y) = (y, x), (3) Linearity: (x + y, z) = (x, z) + (y, z), ∀z ∈ X, (4) Homogeneity: (𝛽x + y) = 𝛽(x, y), ∀𝛽 real; (x, y) is called the inner product of x, y ∈ X. A complete inner product space is called a Hilbert space. Examples: It can be easily seen that the n-dimensional Euclidean space, En , is an inner ∑ product space with inner product defined as: (x, y) = ni=1 xi yi , x = (x1 , … , xn ) ∈ En , and so b

on. So is the space, say, C2 [a, b] with inner product defined as: (f ,g) = ∫a f (t) g(t) dt, or ) b( (f ,g) = ∫a f g + f ,t g,t dt,f , g ∈ C2 [a, b], and so on, Now, for an inner product space, each √ inner product defines a norm: ‖x‖ = (x, x); that all norm properties are satisfied is easily shown by Schwarz (triangular) inequality. Finally, an inner product space encompasses all the geometric properties such as the parallelogram theorem, angular symmetry, the Pythagoras theorem, projection, and so on. For example, we can define orthogonality, a geometric concept, of vectors: iff (x, y) =0, we say x is orthogonal to y; a set S ⊂ X, X inner product space, is called an orthonormal set if ∀x, y ∈ S, (x, y) = 0 if x ≠ y, and equal to 1 if x = y. One important theorem is that any finite set of non-zero orthogonal elements are linearly independent and thus can form the basis set of the subspace known as an orthogonal subspace. Now, the converse is not true, but every finite set of linearly independent elements can be orthogonalized (Lang, 1968), which we apply in our study of mesh generation; we briefly describe in 3D, without any loss of generalization, the construction of the process algorithmically.

2.1.7

Gram–Schmidt Orthogonalization

Step 0: given linearly independent x1 , x2 , x3 ∈ X, an inner product space. ‖ Step 1: set y1 = x1 and z1 = y1 ∕ ‖ ‖y1 ‖; Step 2: set y2 = x2 −

(y1 ,x2 ) y , (y1 ,y1 ) 1

that is, y2 is x2 minus the projection of x2 on y1 , ‖y 1 ‖(z1 ,x2 )   ‖ ‖y and z2 = y2 ∕ ‖ 1 ‖z1 = x2 − (z1 , x2 ) z1 ; note that: ‖y2 ‖; thus, y2 = x2 −  2  ‖y‖ (x ,x2 ) ) = 0, that is, y is perpendicular to y . (y2 , y1 ) = (x2 , x1 ) − (x1 , x (x1  1 2 1 1 ,x 1)

Mathematical Preliminaries

Step 3: set y3 = x3 −

25

(y1 ,x3 ) y (y1 ,y1 ) 1

−

(y2 ,x3 ) y (y2 ,y2 ) 2

= x3 − (z1 − x3 ) z1 − (z2 − x3 ) z2 , that is, ‖ y3 is x3 minus x3 ’s projection on the plane containing y1 and y2 ; and z3 = y3 ∕ ‖ ‖y3 ‖. Then, z1 , z2 , z3 are completely orthonormalized; since orthogonalized vectors are linearly independent, they form a basis set for the space spanned by them; of course, orthonormal vectors depend on the definition of the inner product of the space. Remark

r A complete inner product space is called a Hilbert space. r Every space, clearly, is not an inner product space. There are spaces in which the distance measure, that is, norm, cannot be induced by an inner product; these are known as normed linear spaces, thus, every inner product space is a normed linear space but the converse is not true. For example, a normed linear space with, say, max – norm, cannot define an inner product. In our finite element analysis, we will encounter normed linear spaces, thus we first introduce a normed linear space and then briefly describe a special and important normed linear space: Sobolev space.

2.1.8

Normed Linear Space: Banach Space

A linear space, X, is called a normed linear space if for each element x ∈ X, there exists a real number, designated ‖x‖ and called the norm of x with properties: (1) positivity: ‖x‖ ≥ 0, (2) definiteness: ‖x‖ = 0, iff x = 0, (3) homogeneity: ‖𝛽x‖ = |𝛽| ‖x‖, ∀𝛽 scalar, (4) triangular inequality: ‖x + y‖ ≤ ‖x‖ + ‖y‖. It is easily seen that a normed linear space is a metric space with ∑ the metric, d(x, y) = ‖x − y‖. Examples of normed linear spaces abound: Rn with ‖x‖2 = ni=1 xi2 , the square or Euclidean norm; another example is C[a, b] with ‖f ‖ = max f (x). A complete ⏟⏟⏟ x∈[a,b]

linear space is called a Banach space.

2.1.9

Sobolev Space

In finite element analysis, we encounter varieties of functions, for example, displacement functions, strain functions as derivatives of the displacement functions, and so on. We need to understand these functions as to how they are different from and similar to each other. Mathematically, this tantamount to devising an appropriate norm, that is, a distance measure or a measuring scale. One natural way is to devise a norm, known as the L2 -norm and denoted by ‖∙‖0 , that measures the distance between a function and the identically zero function in an average sense over the domain of the function: ( ‖f ‖0 =

)1 ∫

2

f dx

2

(2.1)

However, we may have a function, for example, a small saw-tooth function, which may have values very close to the zero function and hence appear similar, that is, a short distance apart under L2 -norm, but their derivatives may be wide apart pointwise, making them in reality very different, that is, far apart from each other. Thus, the L2 -norm which has no information on the

Computation of Nonlinear Structures

26

derivatives may not be the most suitable norm (or sieve) for the situation. Looking from another angle, instead of solving a mechanics problem, for example, the Dirichlet problem, involving differential equations in the interior and derivatives conditions at the boundary of a material body, a finite element method, which is a variational method, as we shall see in later chapters, converts it into a problem involving an integral functional and then finding the solution as a minimization process; the main idea behind this is that instead of seeking the solution function satisfying pointwise the differential equation and the boundary conditions, from the stringent Cn class, we plan to make it easier by weakening the derivative (averaging or smoothing by an integral operation) requirements by choosing the function from a wider class. This motivates devising what is known as the Sobolev norm, ‖∙‖n , of degree n ≥ 0, in one dimension, and the space defined by this norm is known as the Sobolev space denoted by  n whereby we define: ( ‖f ‖n =

∫ (

=

( f2 +

(

𝜕f 𝜕x

(

)2 +∙∙∙+

𝜕nf 𝜕xn

)2 )

)1 2

dx

‖( 𝜕f )‖2 ‖ ( 𝜕 n f ) ‖2 ‖ ‖ ‖ ‖ 2 ‖f ‖0 + ‖ ‖ +∙∙∙+‖ ‖ ‖ 𝜕x ‖ ‖ 𝜕xn ‖ ‖ ‖ ‖0 ‖0

)1 2

(2.2)

Clearly then L2 =  0 . The Sobolev space,  n [a, b], can be thought of as the completion of the class Cn [a, b] with the Sobolev norm, ‖∙‖n . In other words, every function belonging to a Sobolev space either belongs to Cn [a, b] space or it can be generated as a sequence of functions in Cn [a, b] space. Finally, a function that is a good approximation to the zero function  0 -norm may not be a bad approximation in  1 -norm.

2.1.10

Linear Transformations: Functionals and Operators

The motivation for the introduction of the linear functionals and linear operators arises from our need to deal with real-valued functions whose arguments could be functions or field variables themselves. For example, in the calculus of variations dealing with solid mechanics, the strain energy of a body may be expressed as a function of an associated displacement field or a function that defines the deformed body. Similarly, in mathematics, the definite integral of a function will serve as an example. In linear structural mechanics, the differential equations of equilibrium may be viewed as a linear operator equation involving a differential operator. In other words, any function that operates on its arguments (domain) that are functions to produce (range) real numbers is called a functional. For computational purposes, we are primarily interested in linear functionals defined over normed linear spaces.

2.1.11

Linear Functionals

Let us, then, briefly summarize the following definitions and results about functionals:

r f (x) is a functional on a linear vector space, Z, if a scalar is assigned to every vector x ∈ Z. For example, the space may be a vector function space where every point in the space represents a vector function, say, x = x(t), t ∈ [a, b]. In other words, a functional is a function of functions. Moreover, it is a linear functional if f (𝛼x + 𝛽y) = 𝛼f (x) + 𝛽f (y) for any x, y ∈ Z and 𝛼, 𝛽 arbitrary scalars,

Mathematical Preliminaries

27

r Given any linear vector space, Z, and the definition of inner products on it, every inner product, r r r r

(z, x) ≡ z ∙ x, where z ∈ Z is a fixed vector and x ∈ Z any argument vector, is a continuous (hence bounded) linear functional, say, f (x), The converse is not trivial but true: every continuous linear functional, f (x), has an inner product representation, that is, there exists a vector z ∈ Z such that f (x) = (z, x). The set of linear functionals on a linear space, X, themselves form a linear space, X ∗ , called an algebraic conjugate space defined by: L1 , L2 ∈ X ∗ ⇒ (aL1 + bL2 )x = aL1 (x) + bL2 (x) for any a, b scalar, and ∀x ∈ X. Orthogonal transformation: a linear transformation, L : X → X, where X is an inner product space, is called an orthogonal transformation if (Lx, Ly) = (x, y), where x, y ∈ X; the rotational transformation serves as an example (see Chapter 4 on rotation tensors). Energy inner product: every positive definite (i.e. (Lx, x) > 0 unless x = 0), symmetric (i.e. (Lx, y) = (x, Ly)) linear operator, L : X → Y, X an inner product space and Y ⊆ X, provides a second inner product, designated by (∙, ∙)L , defined by (x, y)L = (Lx, y), ∀x, y ∈ X.

Example LF1: Vectorial linear functional Let T be a linear transformation: T ∈ (R2 , R3 ) such that b = T(a) = (a1 + 2a2 , a1 − a2 , 2a1 + 3a2 ) with a(= ai Ei ) ∈2 and b(= (a1 + 2a2 )E1 + ( a1 − a2 )E2 + (2a1 + 3a2 )E3 ) ∈ R3 under standard bases, Ei , for both the domain (i = 1, 2) and range (i = 1, 2, 3), with: E1 = (1, 0, 0) , E2 = (0, 1, 0) , E3 = (0, 0, 1).

Example LF2: Integral linear functional Let the linear vector space X = C[a, b], the class of continuous functions on [a, b] ∈ R1 , the b real line. Then, L(f ) = ∫a x2 f (x)dx is a linear functional, because, for all f , g ∈ C[a, b], and for all [a, b] ∈ R1 , we have, by definition of functions and the linearity property of integration: L(𝛼f + 𝛽g) =

b

∫a

x2 (𝛼f + 𝛽g)(x)dx =

b

∫a

x2 (𝛼f (x) + 𝛽g(x))dx =

b

∫a

b

x2 f (x)dx +

∫a

x2 g(x)dx

= 𝛼L(f ) + 𝛽L(g).

Example LF3: Differential linear functional Let the linear vector space X = p2 [a, b], the class of polynomials of degree two or less on [a, b] ∈ 𝜕p(t) 1 R , the real line. Then, L(p) = (tD + 2)p = t 𝜕t + 2, p ∈ X,t ∈ [a, b] is a linear functional,

because, for all f , g ∈ p2 [a, b], and for all 𝛼, 𝛽 ∈ R1 , we have, by definition of functions and linearity property of differentiation: ) {( ) } {( ) } 𝜕 𝜕 𝜕 + 2 (𝛼f (t) + 𝛽g(t)) = 𝛼 + 2 f (t) + 𝛽 + 2 g(t) 𝜕t 𝜕t 𝜕t = 𝛼L(f ) + 𝛽L(g)

L(𝛼f + 𝛽g) =

2.1.12

(

Linear Operators

We note that the functionals are merely scalar-valued functions of generally vector arguments which are themselves functions of some vector argument. In other words, functionals are scalar

Computation of Nonlinear Structures

28

functions of functions. The operators, on the other hand, are vector functions of functions. Let us, then, briefly summarize the following definitions and results about operators:

rA r r r

function L : X → Y, X, Y linear vector spaces, is called a linear operator, if, for all x, y ∈ X, and for all 𝛼, 𝛽 scalars, L(𝛼x + 𝛽y) = aL(x) + 𝛽L(y). In other words, the operator or mapping is linear if the image of a linear combination of any set of vectors is the linear combination of the image vectors. Positive definite Linear Operator: a function L : X → X, X an inner product space, is called positive definite if (Lx, x) > 0 unless x = 0, x ∈ X. Symmetric Linear Operator: a function L : X → X, X an inner product space, is called symmetric if (Lx, y) = (Ly, x) > 0, ∀x, y ∈ X. Energy inner product: every positive definite, symmetric linear operator, L : X → X, X linear space, and Y ⊃ X, having an inner product, provides a second inner product, called energy inner product and designated as: (∙, ∙)L , and defined by (x, y)L = (Lx, y), ∀x, y ∈ X.

Example LO1: Matrix linear operator Let the linear vector spaces: X = En , Y = Em , be the Euclidean spaces of dimension n and m, respectively. Then, a matrix, A = (aij ), is a linear operator from X → Y. Example LO2: Integral linear operator Let X = Y = L2 [a, b], K(s, t) ∈ C[a, b] × C[a, b]. If u(s) ∈ L2 [a, b], then I(u) as defined under b is a linear operator: I(u) = ∫a K(s, t) u(s) ds Example LO3: Energy inner product { } Let X = u ∈ C2 [0, 1] | u(0) = 0, u(1) = 0 and Y = C[0, 1] with inner product: (f , g) = ◦

∫0 f (t) g(t) dt, f , g ∈ Y. Now, define the differential operator, Du = − 𝜕𝜕t2u ≡ −u,tt (t). Then, X is linear subspace of Y because utt (t) is continuous and u(t) = 0 identically is contained in X. Now, 1 1 ∀u, v ∈ X, (Du, v) = − ∫0 u,tt (t) v(t) dt = ∫0 u,t (t) v,t (t) dt, by integration by parts and bound1

2

1

ary conditions. So, (Du, v) = (Dv, u), that is, is symmetric. Now, (Du, u) = ∫0 (u,t (t))2 dt > 0 unless u(t) = c, a constant. But, then, by boundary conditions, u(t) is identically 0. Thus, D is a symmetric, positive definite operator, and D defines an inner product on X, known as energy inner product.

2.1.13

Gateaux or Directional Derivative and Rate of Change of Fields and Functionals

We recall from our previous discussion that functionals are merely scalar-valued functions of generally vector or tensor arguments which are themselves functions of some vector or scalar argument. In other words, functionals are scalar functions of functions. The operators are, likewise, vector or tensor functions of functions, that is, multi-dimensional functionals. Like the scalar functions, the functionals may also assume extrema (maxima or minima) for certain argument functions. In order for us to determine such extrema, we have to be able to track the rate of change of the functional with respect to its argument function. It is just like finding the extrema of a scalar function – the vanishing of the first derivative determines an extremum location. Similarly, the corresponding second derivative determines the nature of the extremum – stationary, maximum or minimum. The situation is very similar for functionals except that we have to have proper definitions for differentiations of functionals.

Mathematical Preliminaries

2.1.14

29

Gateaux or Directional Derivative

Directional derivative is a generalization of the concept of derivatives of ordinary scalar-valued functions with scalar arguments and differentials. Consider an ordinary scalar-valued vector function, f (x), x ∈ R3 . Now, if we move away from vector x = (x1 , x2 , x3 )T along a vector x̄ = (̄x1 , x̄ 2 , x̄ 3 )T , the approximate change df in f is given by:

2.1.14.1

Method 1: “𝛁-method”

The partial differentiations as: df = f ,x1 x̄ 1 + f ,x2 x̄ 2 + f ,x3 x̄ 3 = ∇f (x) ∙ x̄ with f ,xj ≡

𝜕f 𝜕xj

(2.3)

where, ( ∇f (x) ≡

𝜕f 𝜕x1

𝜕f 𝜕x2

𝜕f 𝜕x3

)T ≡

𝜕f 𝜕x

(2.4)

is the gradient of the function f . We also note that if f (x) is linear in x, that is, f (x) = ax + b, 𝜕f a, b constants, then df = 𝜕x dx = a dx, that is, the differential df does not depend on x, but only on dx. Interpreted from another angle, if we move along a vector x̄ = (̄x1 , x̄ 2 , x̄ 3 )T such that we always stay on a surface given by: f (x) = 0, that is, x̄ is a tangent to the surface, then we have 0 = df = ∇f (x) ∙ x̄ , which implies that ∇f (x) is normal to the surface; if f (x) = c depicts a contour of a mountainous surface, then the gradient points toward the direction of”steepest ascent”.

2.1.14.2

Method 2: “𝜺- Method”

However, for the purpose of further generalization to the case of functionals and operators, we may consider another scheme to define differentials based on the Taylor series expansion of f (x + 𝜀̄x) in the neighborhood of x along the vector direction x̄ , then, we will have: f (x + 𝜀̄x) = f (x) + 𝜀(∇f (x) ∙ x̄ ) + O(𝜀2 )

(2.5)

Now, if we maintain x and x̄ fixed and thus consider f (x + 𝜀̄x) as an one-parameter function of 𝜀 alone, the first derivative, as defined as follows, gives the same expression for the differential, df : [

d f (x + 𝜀̄x) d𝜀

{

] 𝜀=0

} f (x + 𝜀̄x) − f (x) ≡ lim 𝜀→0 𝜀 } { 1 = lim f (x) + 𝜀(∇f (x) ∙ x̄ ) + O(𝜀2 ) − f (x) 𝜀→0 𝜀 = ∇f (x) ∙ x̄ = df

(2.6)

Computation of Nonlinear Structures

30

Evidently, the second definition of the derivative produces the same differential without requiring any predefinition such as gradient, for the derivatives of the function with respect to its vector argument, x, as in the previous definition of df . Now, let us replace f by its generalization: functional, G(u(x)), x ∈ Rm , u : Rm → Rn . That is, G is a scalar function of its vector argument, u, which itself is a n-dimensional vector function of its m-dimensional vector argument, x. But, dG = G,u du has no meaning because the derivative of a functional with respect to its argument vector functions, G,u ≡ 𝜕G , has no 𝜕u meaning as yet. But we can still invoke the second scheme, that is, “𝜀 -method”, to define the rate of change of functionals. Thus:

r The Gateaux or directional derivatives of a scalar, vector or tensor (see Chapter 3 on tensors) field in Cartesian coordinate system are defined as: ̄ x̄ ) DT(x) ∙ x̄ ≡ T(x, {T(x + 𝜀̄x)parallel transported − T(x)} d {T(x + 𝜀̄x)} = lim 𝜀→0 d𝜀 𝜀→0 𝜀

(2.7)

= lim

r

̄ x̄ ) is the directional or Gateaux derivative of a tensor field, T(x), at x where DT(x) ∙ x̄ ≡ T(x, along the vector direction x̄ . As to functionals, let G(u(x)) be a functional defined for an argument function u(x) ∈ Rn at ̄ ∈ Rn be another function at x. Then, the functional defined by the any x ∈ Rm . Now, let u(x) ̄ or DG(u) ∙ u, ̄ of G directional or Gateaux derivative of the functional, denoted as G,u (u, u), ̄ at u along the direction u is defined as: [

d ̄ = DG(u) ∙ ū ≡ G,u (u, u) G(u + 𝜀u) d𝜀 Δ

{

] 𝜀=0

= lim

𝜀−>0

̄ − G(u) G(u + 𝜀u) 𝜀

} (2.8)

where DG(u) is an operator. As with the Taylor series expansion of functions, if the operator has an expansion such as: ̄ 𝜀)n > ̄ = G(u) + 𝜀 dG + 𝜀n R(u, u, G(u + 𝜀u) ∼2 ̄ 𝜀) = 0, then, with the property that the residual: lim 𝜀n−1 R(u, u, 𝜀−>0

̄ ≡ DG(u) ∙ ū ≡ G,u (u, u) [

[

d ̄ G(u + 𝜀u) d𝜀

] 𝜀=0

} d { ̄ 𝜀) − G(u) = G(u) + 𝜀 dG + 𝜀n R(u, u, d𝜀

] 𝜀=0

(2.9) = dG

justifies the name differential. Remarks

r G,u (u, u) ̄ is also sometimes denoted in the literature as operator ΔG(u, u). ̄ r In the context of virtual functionals used in finite element analysis, notations found include 𝛿G(u, 𝛿u) ≡ DG(u) ∙ 𝛿u to indicate the first variation of G(u(x)) due to 𝛿u(x), the variation of u at point x of a domain.

Mathematical Preliminaries

31

r As with the linear functions discussed above, if a functional is linear, that is, G(u + 𝜀v) = G(u) + 𝜀 G(v), then G, = G(v) and thus does not depend on u.

u r As in the case of the functions, the product and chain rules of differentiation hold for functionals, ∙

that is, if, in turn, u = u(t) and v = v(t) with t ∈ R, then we have G(u, v) ≡ G,u (u, v) ∙

2.1.15

𝜕u 𝜕t

∙

𝜕 G(u(t), v(t)) 𝜕t

=

= G,u (u, v) ∙ u.

Vainberg Principle: Symmetric Operator and Potentialness

For problems in nonlinear structural mechanics and engineering, we encounter equations of equilibrium described by nonlinear differential operators. In variational methods such as finite element methods, as we shall see, a problem of such nature is solved in weak form by converting the equations into integral functional form – in an inner product space, say, Hilbert space, (see Chapter 8, on finite element method) – as a virtual work functional; then seeking the extremum or stationary point of this functional having, say, a Gateaux differential, and setting the differential to zero usually gives us the nonlinear equations (known as Euler equations) in operator form. For structural and mechanical problems, the virtual functional always exist for any problems. It is always beneficial to know whether there exists a functional whose gradient is the virtual work functional – but it does not always exist. If it does, it reveals what is known as the energy principle, and the functional is known as the energy functional. The Vainberg principle (Vainberg, 1964) provides the necessary and sufficient condition under which we can guarantee the existence of the energy functional; moreover, it provides the rule from which to construct the energy functional from a corresponding virtual work functional. According to this principle, the operator in the operator equation must be symmetric for existence of the energy functional known as the potential functional, and the virtual functional is the gradient of the potential functional. Before we introduce the construction algorithm, let us try to understand the Vainberg principle by using an example. Let us first consider a linear operator equation, in structural engineering and mechanics, given by: A d = f, where A may be a differential operator from a normed linear space, E, to its conjugate space,E∗ ; d ∈ E, the displacement vector, and f ∈ E is the force vector. ̄ the virtual functional, G(d, d) ̄ ∈ H, a Then, with the virtual displacement designated by d, ̄ = (A d − f, d). ̄ ∀d. ̄ Now, the Hilbert space, can be represented by the inner product: G(d, d) ̂ is given by: Gateaux derivative functional, 𝛿G, of the functional, G, along the direction, d, { } ̂ d) ̄ − (f, d) ̄ = (A d, ̂ d) ̄ = (A∗ d, ̄ d), ̂ where A∗ is the adjoint ̄ d) ̂ = lim d (A(d + 𝛽 d), 𝛿G(d, d; d𝛽 𝛽→0

operator of A. Now, if the operator, A, is symmetric, that is, self-adjoint: A = A∗ , then, from the ̄ d) ̂ = (AT d, ̄ d) ̂ = previous expression and the definition of Gateaux derivative, we get: 𝛿G(d, d; ̄ ̂ ̂ ̄ ̄ ̂ (A d, d) = 𝛿G(d,d; d), ∀d, d. In other words, the functional, 𝛿G, is symmetric with respect to ̂ In this situation, there exists a functional, I(d), called the potential (or, in structural d̄ and d. ̄ ∀d. ̄ For this, let us engineering, the energy functional), whose gradient turns out to be G(d, d), 1 1 1 1 define: I(d) = ∫0 G(td, d) dt = ∫0 (tAd, d) dt − ∫0 (f, d) dt = 2 (Ad, d) − (f, d). Using the definition for the Gateaux derivative and the symmetry property of { } the operator, A, we can prove d 1 ̄ ̄ ̄ ̄ The this as: 𝛿I(d) = lim d𝛽 2 (A(d + 𝛽 d), d + 𝛽 d) − (f, d + 𝛽 d) = (A d − f) d̄ = G(d, d). 𝛽→0

Vainberg principle works even for a nonlinear operator (for which it was originally proved) equation of structural engineering and mechanics problems. For our purposes, it basically says that if the derivative of the virtual functional is symmetric, then a functional, called potential or energy functional, exists whose gradient turns out to be the virtual functional itself. In other words, there

Computation of Nonlinear Structures

32

exists a potential functional, I, such that the virtual functional is given by 𝛿I = G, and its second derivative, 𝛿 2 I = 𝛿G is symmetric. For curved spaces, the Gateaux derivative must be replaced by the appropriate covariant derivative. Later, in our development of beam and shell theories, we will apply the Vainberg theorem to recover real strain tensors from its virtual counterparts. In this connection, and as a converse, a theorem from differential geometry is worth mentioning: the tensor obtained from the covariant derivative of a vector field that is the gradient of a scalar field is symmetric. In the language of structural mechanics, for conservative systems, an energy principle exists and the tangent operator (stiffness matrix), obtained by linearization (first-order derivatives) of the virtual functional which itself is a gradient of the energy functional, is symmetric. Algorithmically speaking, in order to apply the Vainberg principle to a virtual functional in virtual displacement method, the real field, d, is replaced by t d, t ∈ [0, 1], and the virtual field, ̄ is replaced by the real field, d, and then we integrate over t ∈ [0, 1]. d,

2.1.16

Successive Forward Difference Operator

There will be occasions where we need to apply a forward difference scheme and notations in the construction of curves and surfaces. Definition: Given a sequence of values f0 , f1 , …, the n-th forward difference is defined recursively as: Δfi = fi+1 − fi Δn fi = Δ(Δn−1 fi ) = Δn−1 fi+1 − Δn−1 fi with Δ0 fi = fi where Δ is called the forward difference operator. We can particularize this, to write down the following useful expressions: Δ1 fi = Δ0 fi+1 − Δ0 fi = fi+1 − fi Δ2 fi = Δ1 fi+1 − Δ1 fi = fi+2 − 2fi+1 + fi Δ3 fi = Δ2 fi+1 − Δ2 fi = fi+3 − 3fi+2 + 3fi+1 − fi In general, it can be written as: Δn fi =

n ∑ r=0

(−1)n−r

( ) n f r i+r

with

( ) n Δ n! = r r!(n − r)!

Remarks

r Linearity: the difference operator, Δn , is linear in the sense that {fi } and {gi } are two sequences, and a, b scalars, then, Δn (a f + b g ) = a Δn f + b Δn g , ∀n ∈ {0, 1, 2, …}.

i i i i r Annihilation: with evenly spaced knots, Δn annihilates all polynomials of degree (n − 1); that

is, Δn pn−1 (x) with evenly spaced knots of any polynomial of degree (n − 1) is identically equal to zero.

Mathematical Preliminaries

2.2

33

Affine Space, Vectors and Barycentric Combination

If we bring in the real coordinate space to Euclidean geometry, we give algebraic structure to the geometry that creates what is known as the Euclidean space. Euclidean geometry embeds such geometric concepts as the distance (length) between two points of a space and the angle between two sides (lines) obtained from the points in the space. There are two allowable transformations that affect the points in Euclidean space: translation and orthogonal transformation. Under translation, or shift of all points in the space, the distance between any two points remains invariant; under orthogonal transformation such as rotation about the origin, the angle between two lines remain invariant. Considering that a space is a set of points with a metric (distance function) defined on it, a Euclidean space is thus a real coordinate space equipped with operations such as the inner product (depicting the angle between two lines) that induces a metric and a norm, the vector product (signifying a special kind of rotation of lines) and tensor product delineating general transformation of lines or vectors. Thus, we proceed to summarize these operations as follows:

r Scalar r r r

product or dot product or inner product: a first kind of vector multiplication that yields a scalar: a ∙ b = ai bi ; it is commutative (a ∙ b = b ∙ a) and distributive (a ∙ (b + c) = a ∙ b + a ∙ c); this makes a Euclidean space an inner product space (i.e. a Hilbert space). Norm or length of a vector: the scalar allows us to define the Euclidean length or √ product √ norm, ‖∙‖, of a vector as: ‖a‖ = a ∙ a = ai ai ; this makes a Euclidean space a normed space. Euclidean √ metric, d(a, b): the norm, in turn, defines the metric (distance) as: d(a, b) ≡ ∑n 2 ‖a − b‖ = i=1 (ai − bi ) ; this makes a Euclidean space a metric space. Vector product or cross product: the second kind of vector multiplication yields a vector: c = a × b ≠ b × a where c is a vector normal to both a and b; thus, it is not commutative but distributive (a × (b + c) = a × b + a × c). A vector product is not associative: a × (b × c) ≠ (a × b) × c. The cross product can be seen as an operation that transforms one vector to another: in the above example, the a× operation transforms vector b into vector c; as it will turn out, seen from this viewpoint of transformation, the cross product fits the definition of a tensor, and as such has a tensor representation. Finally, since we will use these quite often, we introduce the following important identities: a × (b × c) = (a ∙ c)b − (a ∙ b)c (a × b) × c = (a ∙ c)b − (b ∙ c)a

r Tensor

(2.10)

product: there is a third kind of vector multiplication that yields tensor; later, the definition and the properties of a tensor product will be introduced in great detail. r Vector triple product: yields a scalar: a × b ∙ c = b × c ∙ a = c × a ∙ b, which gives the volume of a parallelepiped formed by three vectors: a, b and c as the three sides. r Permutation symbol: in order that the vector products in three dimensions can be represented in indicial notation, we need what is called the permutation symbol, eijk , with the following properties: r If any two indices repeat: eijk = 0; example: e113 = 0, r If indices are cyclic in 1, 2, 3: eijk = +1; example: e123 = e231 = e312 = +1, r If indices are not cyclic in 1, 2, 3: eijk = −1; example: e213 = e213 = e321 = −1, Now, the vector product: c = a × b can be written, in a right-handed coordinate system, component-wise as: ci = eijk aj bk .

Computation of Nonlinear Structures

34

Figure 2.1 Free vector: translation.

2.2.1

Vectors in Affine Space

Now, in order that we can mathematically describe physical entities in a coordinate-independent way, we resort to what is known as affine space, a Euclidean space that has forgotten, so to say, its origin by adding translation to linear maps. Unlike vector spaces, in an affine space:

r no special significance is attached to origin – it is like any other point that contains n real numbers as components

r only the subtraction operation on the elements of the space, that is the points, are allowed r a unique vector in an affine space is then defined by the difference of two points; a directed r

line segment beginning with one point called the initial point and ending in the other point known as the final point; for example, if a and b are two points, then we can define vector, v, such that: v ≡ b − a, with v pointing from a to b the addition of a point to a vector to obtain another point is a valid operation known as translation (Figure 2.1).

Conversely, however, given a vector, v, we can express it by infinite pairs of points; we can have: u )−( a + u ) where u is any arbitrary v = b − a =( b + ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟ vector

point

point

point

translation vector

point

translation vector

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

translated point

translated point

translation vector. Thus, the vectors are invariant under translation while the points are not; in other words, the vector components are position independent under translation, that is, the initial point of the vector can be arbitrarily chosen. Under translation alone (parallel transport in Euclidean space), vector components, and hence the vector length, stay invariant in a Euclidean (more accurately, affine) space, so the vectors in a Euclidean space are free vectors in the sense that they are position independent, that is, the initial point of a vector can be arbitrarily determined by translation. In other words, two vectors in a Euclidean space are identical if and only if they are parallel and have equal length and sense of direction. As we will see, the term “free” in free vectors of Euclidean space is introduced to distinguish these from bound vectors of curved spaces where vector initial point is position dependent and cannot be arbitrarily chosen or moved. Then, the question is: given that addition of points is not a valid operation, can we express, as with the vectors in a vector space, a point, a, of an affine space by a linear combination of other points, known collectively as affinely dependent points, such as: a = c0 a0 + c1 a1 + ⋯ + cn an = ci ai ? The answer is that although the addition operation is not defined for points, for a special kind of linear combination – barycentric or affine combinations where the coefficients of combinations

Mathematical Preliminaries

35

Figure 2.2 Barycentric combination as point transmission.

all add up to unity – the points can be made to look as if they admit addition. So let us consider the following expression, where a point is allowably added to vectors:

a=

n ∑ a0 + ci (ai − a0 ), ⏟⏟⏟ i=1 ⏟⏞⏟⏞⏟ point

n ∑

ci = 1

(2.11)

i=0

vector

As indicated below, we can express equation (2.11), equivalently, as a kind of summation of points: n ∑

a = a0 + ( = a0

ci (ai − a0 ) = a0 +

i=1

1−

n ∑ i=1

ci ai − a0

i=1

) ci

n ∑

+

n ∑

n ∑

ci

i=1

ci ai = a0 c0 +

i=1

n ∑ i=1

ci ai =

n ∑ i=0

ci ai ,

with

n ∑

(2.12) ci = 1

i=0

There are numerous examples of summation of points found in mathematics and science: an arbitrary point c on a line segment from point a to point b is given by the barycentric combination of the points as: c(t) = (1 − t)a + tb, with coefficients adding up to unity: (1 − t) + t = 1; clearly, the expression is independent of any choice of an origin. To obtain an equivalent allowable operation of summation of a point and a vector, we can rewrite it as (Figure 2.2): c(t) = a +t (b − a). ⏟⏟⏟ ⏟⏟⏟ point

vector

We may also note that if we choose any arbitrary point o as origin, then we can express: c(t) = (1 − t)a + tb = o + (1 − t)(a − o) + t(b − o), which is the scaled successive translations of ⏟⏟⏟ ⏟⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏟ ⏟⏟⏟ point

vector

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

vector

translated point

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ translated point

the point o; thus, a barycentric or affine combination is independent of the choice of the origin.

36

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Figure 2.3 Euclidean geometry.

2.2.2

Base or Coordinate Vectors for Cartesian Coordinate Systems

The vectors and tensors are coordinate-independent objects, and as such, the form of equations written in these objects are invariant, but for computational purposes we will have to resort to specific component descriptions of these objects. This, then, brings us to a basis vector set of a Euclidean space that determines the components of a vector for a specific coordinate system. Referring to the Figure 2.3, in the Cartesian coordinate system of a Euclidean space, a line segment can be considered as intersections of two planes, and points can be recognized as the intersection of three planes. For Cartesian coordinate systems, where the coordinate planes are mutually normal at any point, the normal to a coordinate plane is identical to the tangent along the coordinate line created by{ the } intersections of the other two of these planes – giving rise to only one kind of base vector, ii , i = 1, 2, 3, for a three-dimensional space: let r = r(X) = X1 i1 + X2 i2 + X3 i3 is a radius vector at any point, X with ii , i = 1, 2, 3 as the standard base vectors in three dimensions. Then, the derivatives along the coordinate directions, 𝜕r i = 1, 2, 3, are position independent. The base vectors of any other Cartesian r,i ≡ 𝜕X i = ii , coordinate system can be obtained by the rotation of the standard base vectors; referring to the Figure 2.4, then, we have: ei = A ii , i = 1, 2, 3. Note that we have conventionally chosen to describe these with subscripted indices. Now that we have the definition of vectors in place, we will move directly to the curved spaces to introduce these two kinds of sets of base vectors. Finally, we avoid a very similar introduction of tensors in Cartesian coordination system in favor of a more general discussion in the curvilinear coordinate system.

2.3

Generalization: Euclidean to Riemannian Space

Let us now examine how the concepts in Euclidean geometry are directly generalized to Riemannian spaces, that is, curved spaces with adequate differentiability. We start with Euclidean curvilinear coordinate system to establish connection between the two spaces. However, expressions for the equilibrium equations and solution processes are rather cumbersome for curved structures such as curved beams and shell surfaces when expressed in Euclidean coordinate

Mathematical Preliminaries

37

Figure 2.4 Cartesian basis transformation.

systems. So we quickly abandon such Euclidean connection in favor of a more general parametric representation for curved spaces. Thus, as shown in the Figure 2.5, for three-dimensional spaces, any point P of a curved space may be represented parametrically by a three-parameter set, {𝜉 i }, i = 1, 2, 3. As indicated above, it can be functionally related to the position vector, r, of a Euclidean Cartesian coordinate system as: ( )T r = r(𝜉 1 , 𝜉 2 , 𝜉 3 ) = X 1 (𝜉 1 , 𝜉 2 , 𝜉 3 ), X 2 (𝜉 1 , 𝜉 2 , 𝜉 3 ), X 3 (𝜉 1 , 𝜉 2 , 𝜉 3 )

(2.13)

For the parametric representation expressed in equation (2.13) to be allowable, it must be arbitrarily smooth and must satisfy that the scaling matrix, or Jacobian matrix, J, as defined below, must be of rank 3; that is, there must be at least one non-vanishing determinant of order three to be invertible at any point in the space: ⎡ 𝜕X 1 ⎢ 𝜕𝜉 1 ⎢ ⎢ 𝜕X 2 J≡⎢ 1 ⎢ 𝜕𝜉 ⎢ 𝜕X 3 ⎢ 1 ⎣ 𝜕𝜉

𝜕X 1 𝜕𝜉 2 𝜕X 2 𝜕𝜉 2 𝜕X 3 𝜕𝜉 2

𝜕X 1 𝜕𝜉 3 𝜕X 2 𝜕𝜉 3 𝜕X 3 𝜕𝜉 3

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

In other words, the function given by equation (2.13) is invertible and one-to-one.

Figure 2.5 Allowable parametric representation.

(2.14)

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38

Figure 2.6 Curved space and tangent vectors.

Example: We examine the two-dimensional polar coordinate system, (r, 𝜃), where 𝜉 1 ≡ r and ( )T 𝜉 2 ≡ 𝜃, and equation (2.13) becomes: r = X 1 , X 2 = (r cos 𝜃, r sin 𝜃)T , then, from equation [ ] cos 𝜃 −r sin 𝜃 (2.14), we have: J = . sin 𝜃 r cos 𝜃

2.3.1

Dual Base vectors and Vector Components

As shown in the Figure 2.6, we can think of affine curves at a point in the space, and the tangents to these curves at a point form a local tangent space. As in the case of Euclidean space, taking the tangents along the affine coordinate curves, we can locally construct the set of base or coordinate vectors at a point { }that are generally not mutually orthogonal and known as the set of covariant base vectors, Gi , where: Gi ≡

𝜕r , i = 1,2,3 𝜕𝜉 i

(2.15)

Note that the covariant base vectors are described subscripted because, as opposed to Euclidean geometry, we have a different situation in curved geometry: there exists a dual set of bound base vectors at a point, known as the set of contravariant base vectors, denoted superscripted as {Gi }, i = 1, 2, 3 and defined such that the matrix, formed by these base vectors as the columns is the inverse of the matrix formed by the covariant base vectors as columns, In other words: [G1 G2 G3 ] = [G1 G2 G3 ]−1 , or, ∙j (Gi , Gj ) ≡ Gi ∙ Gj = 𝛿i = 𝛿ij where 𝛿ij is the Kronecker delta tensor, that is, 𝛿ii = 1 and 𝛿ij = 0, i ≠ j.

(2.16)

Mathematical Preliminaries

39

Remarks

r Because of the definition of the covariant base vectors, as given by equation (2.15), the Jacobian

[ ] matrix, J, as given by equation (2.14), can be rewritten as the matrix, G1 G2 G3 , made up of the covariant base vectors as the columns; its determinant, known as the Jacobian determinant, J, is given as: J ≡ det[G1

G2

G3 ] = G1 ∙ (G2 × G3 ) = G2 ∙ (G3 × G1 ) = G3 ∙ (G1 × G2 )

(2.17)

r From equation (2.17) and the definition of the contravariant base vectors, as in equation (2.16), we get: G1 =

1 1 1 G × G3 , G2 = G3 × G1 , G3 = G1 × G2 J 2 J J

(2.18)

r The differential, dr, at a point with the Einstein summation convention, is given by: dr =

𝜕r d𝜉 i = Gi d𝜉 i 𝜕𝜉 i

(2.19)

r The invariant metric or the distance along an arc of a curve, ds, from equation (2.19), is then given by: ds2 = dr ∙ dr = Gi ∙ Gj d𝜉 i d𝜉 j ≡ Gij d𝜉 i d𝜉 j

(2.20)

where we have introduced the definition of a tensor, (Gij ), as the covariant metric tensor, given by: Gij ≡ Gi ∙ Gj

(2.21)

r We will present details of this very important tensor shortly. In the mean time, for the Euclidean

r r r

Cartesian coordinate system, the base vectors, Ei ’s, are given by: Ei = A ii , where A is a fixed rotation tensor and the ii ’s are fixed standard mutually perpendicular base vectors, Ei ∙ Ej = Ei = A ii ∙ A ij = ii ∙ AT A ij = Iij . In other words, for a Cartesian coordinate system, the metric tensor is the second-order identity tensor, I. Unlike the Euclidean metric, the metric in the equation (2.20) is of quadratic differential form and known as the Riemannian metric or distance; the geometry with a Riemannian metric is known as the Riemannian geometry – after E. Riemann, its originator – and the metric space is known as the Riemannian space. Geometrically speaking, as a generalization of the Euclidean spaces, the planes are replaced by curved surfaces and lines, in general by curved lines in Riemannian spaces as shown in Figure 2.7. From Figure 2.7, the covariant base vectors can be identified by the tangents to the coordinate curves, and the contravariant base vectors with the normals to the coordinate surfaces. Moreover, note that, unlike in a Euclidean space, the normals to the coordinate surfaces are generally not coincident with the tangents to the coordinate curves, and thus two types of base vectors.

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40

Figure 2.7 Riemannian geometry.

With all these materials available to us, we are ready to introduce the concepts and the properties related to tensors. In particular, we will first identify tensors as linear combinations followed by their coordinate-independent definition based on how the components respond to allowable coordinate transformations; next, we bring in various relevant properties along with important special tensors that are frequently used in all phases of our investigation. Finally, tensor calculus is introduced for our later use. Example: Continuing on with the two-dimensional polar coordinate system, we have: 𝜕r 𝜕r 𝜕r 𝜕r T T ‖ and G2 ≡ 𝜕𝜉 implying: ‖ G1 ≡ 𝜕𝜉 1 = 𝜕r = (cos 𝜃, sin 𝜃) 2 = 𝜕𝜃 = r (− sin 𝜃, cos 𝜃) ‖G1 ‖ =

G1 G2 1 1 ‖ ̂ = G1 and ‖ ‖G2 ‖ = r, G2 = ‖G2 ‖ = r G2 . Now, using the relations: G ∙ G1 = ‖G 1 ‖ 1, G1 ∙ G2 = 0, and G2 ∙ G1 = 0, G2 ∙ G2 = 1 solving for unknowns G1 and G2 , we get: G1 1‖ 1 ̂1 G1 = (cos 𝜃, sin 𝜃)T and G2 = 1r (− sin 𝜃, cos 𝜃)T , implying: ‖ ‖G ‖ = 1, G = ‖G1 ‖ = G and ̂2 = G2 = rG2 . Thus, G1 = G andG2 = 1 G . The determinant, J, of the Jaco‖G2 ‖ = 1 , G 1 ‖ ‖ r r2 2 ‖G 2 ‖ bian matrix is given as: J = r; the differential: dr = (cos 𝜃, sin 𝜃)T dr + r (− sin 𝜃, cos 𝜃)T d𝜃, implying that the metric is given by: ds2 = dr2 + r2 d𝜃 2 .

̂1 = 1, G

2.4

Where We Would Like to Go

In this chapter, we have briefly presented some of the mathematical preliminaries that are important for our purposes. However, there are two other topics that need to be immediately investigated with a more detailed exposition for which we should look up:

r Tensor analysis and calculus (Chapter 3) r Rotational tensors (Chapter 4)

3 Tensors 3.1 3.1.1

Introduction What We Should Already Know

Linear vector space, base vectors, real coordinate space, linear functional or operator.

3.1.2

Why We Are Here

Nature is oblivious to any chosen coordinate system when it comes to its behavior; thus, the laws of nature – or more specifically, the equations encoding these laws – should not depend on any preferred coordinate system to describe them. In other words, the objects representing the physical entities and the mathematical form of the equations describing the natural phenomena must be independent of all possible coordinate systems. For any scalars involved, we know that we do not need to introduce any coordinate system, that is, the value of a scalar is the same in any coordinate system. In other words, a scalar is what is known as an invariant under transformation of coordinate systems. In the case of vectors (the geometric objects, not to be confused with the namesake members of a vector space), the question is: is it possible to talk about vectors without worrying about the coordinate system? The answer is yes. In fact, we can obtain coordinatefree mathematical objects such as vectors by defining in appropriate space: affine space (a Euclidean space with no preferred point for origin, as will be seen later) of Euclidean geometry; equations revealing natural laws that are written using vectors have invariant forms that do not depend on the coordinate system in this space. For practical computational purposes, however, we have to choose a specific coordinate system: the vectors are defined on the basis of how their components transform as we change from one coordinate system to another by allowable coordinate transformation: affine transformation. For computational purposes where we need these component description, we can put an equation in any coordinate system of our choice. Thus, from the practical point of view, the use of vectors in invariant or absolute form allows us to describe a physical phenomenon in a coordinate-independent invariant form (i.e. using mathematical objects in absolute notation), from which we can easily derive the component relations for any chosen coordinate system. However, there are phenomena such as the description of surfaces, the constitutive relations of materials, and Einstein’s relativity theory, for which vectors alone are not sufficient to describe them in a coordinate-independent invariant form; we have to deal with multidimensional arrays of scalars (equivalently, columns of vectors) as opposed to single rows or columns of scalars Computation of Nonlinear Structures: Extremely Large Elements for Frames, Plates and Shells, First Edition. Debabrata Ray. © 2016 John Wiley & Sons, Ltd. Published 2016 by John Wiley & Sons, Ltd.

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(equivalently, a single row or column vector). From this point of view, we also need to generalize the idea of vectors to higher-order mathematical entities: tensors (which, as we shall see shortly, are of great effectiveness); defining vectors based on the component transformation properties not only renders these objects coordinate-independent but also provides the added advantage of generalization of vectors to tensors; we can talk about tensors in a coordinate-independent way, and just as with the vectors, a tensor can be defined, among other possibilities, on the basis of how its components transform under allowable coordinate transformations. Before we delve into tensors, we may recall that a vector function is an ordered one-dimensional array of n scalar functions of variables, with n determining the dimensionality of the vector function. Each scalar function constitutes what is known as a component of the vector function which, in turn, can be a scalar (resulting in a scalar-valued vector function) or a vector (resulting in a vector-valued vector function) or a combination thereof. As a generalization of the concept of vectors or vector functions, a tensor function will turn out to be an ordered multi-dimensional matrix array of scalar or scalar functions of variables that can be, in turn, a scalar (resulting in scalar-valued tensor function) or a vector (resulting in vector-valued tensor function) or a tensor (resulting in tensor-valued tensor function) or a combination thereof. The fact that we can talk about a tensor as a matrix array of functions is another way of saying that a tensor is a linear transformation (or operator). More specifically, a tensor function can be seen as an operator whose components are functionals, that is, functions of a function. As we will see, an important property of linear functionals allows us to have an operationally powerful representation of tensors: the dyadic notation. Most importantly, however, any arbitrary one-dimensional collection of scalar functions cannot be a vector unless its components satisfy the condition of coordinate transformation. From this, we can relate to a scalar as simply an one-dimensional, (i.e. n = 1) vector. Similarly, any arbitrary multi-dimensional array of scalar functionals cannot be a tensor unless its components satisfy the condition of allowable coordinate transformation. Thus it will follow that a vector is a tensor of order 1 (i.e. one row or column of components), and that a scalar is a tensor of order 0 (i.e. just one component); thus, tensor analysis is a unifying concept. Moreover, the components of vectors and tensors themselves undergo generalizations as we move from the Cartesian coordinate systems of the Euclidean ‘flat’ space to curvilinear coordinate systems and more general spaces such as Riemannian or differentiable ‘curved’ spaces. We often have to deal with such coordinate systems or spaces: for example, cylindrical or spherical geometry, more general curved surfaces, or, in structural mechanics, the displacements related to, say, the rigid body cross-sectional rotations of beams, and curved surface geometries of shell structures. By visualizing a curved space as the union of patches of deformed Euclidean spaces, the differential geometry of curved spaces (for example, surfaces) is locally related to tensors, the linear operators. However, unlike the Cartesian coordinate systems of an Euclidean space where we have only one set of components for the basis vectors and basis tensors, in curvilinear coordinate systems, there exist dual basis vectors and three types of basis tensors, so the components of the vectors and tensors as coefficients of linear combination of these basis vectors and tensors may be either covariant, contravariant or mixed (for tensors) as we will see shortly. It is important to recall here that the fifth or the parallel postulate of Euclid’s geometry ceases to hold in curved geometry: the sum of the two internal angles subtended by a transversal does not stay at 180◦ , and that the parallel lines may meet at some point in the space, as shown in Figure 3.1, for example in the case of a spherical surface. In a Euclidean Cartesian coordinate system, parallelism is central to translation of vectors in the space giving rise to what is known as parallel transport. More specifically, as we will see later, a vector remains as an invariant geometric object under translation because the components of a vector are preserved by parallel transport. Now, we can think of another geometric property that defines parallelism in Cartesian coordinate system; the invariance of the angle between any

Tensors

43

Figure 3.1 Spherical surface.

transversal and a vector can induce parallel translation. In a curved space, such as a spherical surface, translation is not well defined and components of a vector are not preserved because the coordinate system itself is changing from one point to a neighboring point; thus, the concept of parallel transport can only be achieved by maintaining the angle between an infinitesimal vector on the tangent space at a point of a curve on the surface and the coordinate at the point of the curve. Geometrically speaking, the curve acts as a transversal, and thus, parallel transport depends on the curve along which a vector is translated. Finally, as applications of tensors in mechanics, we will find that the general deformations of, and the force transmission through, a material body need to be locally described effectively by tensors called the strain tensor and the stress tensor, respectively. These tensors, that involve transformation of vectors to vectors, belong to one of the most important classes of tensors for our purposes: second-order tensors. On the other hand, for the constitutive theory of materials in mechanics that involves transformation of the second-order tensors such as stress and strain tensors, we will have to consider higher-order tensors – more specifically – tensors of order 4. Clearly, then, we need to do the following:

r define tensors as linear transformations and introduce the tensor product, r define tensors based on how the components must transform under coordinate transformation, r r r r

so that we can test the validity of an array of scalars as a tensor: the geometric interpretation of tensors, organize the domain of a tensor function in terms of one-, two- or multi-dimensional mathematical entities such as scalars, vectors or tensors, respectively, know about the structure of a tensor so that we can evaluate its (range) value for any point in its domain; this brings us to the definition of tensors in absolute, component-free notation through the introduction of an operation known as the tensor product of component vectors of a tensor, define various algebraic operations, introduce tensor calculus.

3.1.3

How We Get It

Recalling real coordinate space and Euclidean geometry, a geometric object such as a vector is defined in a coordinate-independent way by introducing an affine space; the Euclidean ‘flat’

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44

space is further generalized into Riemannian ‘curved’ space in anticipation of tackling problems relating to curved structures such as shells. Recalling the linear functionals and operators, the coordinate-independent definition of the tensors as dyadic and polyadic are introduced; the vectors are identified as tensors of order one and scalars as tensors of order zero. Alternatively, tensors are described by their component transformation law, and various properties of tensors are noted. Finally, tensor calculus (differential and integral) is introduced for both Euclidean and Riemannian spaces.

3.1.4

What We Need to Recall

We need to recall the linear vector space, the real coordinate space. For tensor development, likewise, we recall linear functionals and linear operators. Finally, for generalization to covariant derivatives of vectors and tensors in Riemannian curved spaces, we evoke the directional or Gateaux derivative of Euclidean space.

3.2

Tensors as Linear Transformation

A tensor may be defined as a continuous linear transformation (and hence bounded), transforming a set of ordered “vectors” of a vector space to another set of ordered “vectors”; we have put quotation mark around the word “vector” because it is intended to be interpreted in its most general sense, which includes tensors themselves, as will be seen later. Let us again note that the linearity of transformations guarantees the existence of a linear space formed by the domain vector spaces and the range vector spaces. In an abstract sense, by definition, if T is a tensor, i.e., a linear transformation, we may write its operational relation as: T : a → b, simply as: b = T a, where a and b are vectors belonging to the domain and range, respectively. Noting that linear transformations themselves form vector spaces, the domain and the range could be vector spaces of linear transformations. In other words, a and b vectors belonging to the domain and range are used in the general sense of members belonging to vector spaces. Obviously, we need more structure to our definition of tensors for computational purposes. The abundance of linear transformation is experienced in all phases of linear or nonlinear (locally linear) analysis in engineering – strain to stress transformation (constitutive relations), displacement to strain transformation (compatibility relations), displacement to force transformation, or temperature to heat flow transformation to name but a few.

3.2.1

Second-Order Tensor

The simplest kind of tensor (generally adequate for our purposes, with some exceptions in the need for higher-order tensors for, say, constitutive theory) is envisioned with both the domain and the range of transformation consisting of vector spaces of one-dimensional column vectors, that is, a and b mentioned above are one-dimensional vectors; the domain and range can belong to the same vector space. This gives rise to what is known as second-order tensors. We have already encountered a very special second-order tensor, the metric tensor, and the rotation mentioned above, which transforms a vector to another vector, is also a second-order tensor. For another example, the familiar vector cross product is a linear transformation of the form, T ≡ (a ×), and given a vector, a, it transforms any other vector, say b, into another vector, c = T b ≡ (a ×) b, or the zero vector as it tries to transform itself, that is, b = a ⇒ (a ×) a = 0. Note that the vector cross product is clearly a linear transformation. Thus, the vector cross product is an example of a tensor.

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45

But, it is clearly a very restrictive and special kind of transformation as it only produces vectors that are normal to the plane containing the given vector and any other vector being transformed. So, in what follows, we proceed to identify more general tensors.

3.2.2

Tensor Product (Dyadic) Notation

Now, let T be a linear operator over vector space, Z. Then, for any x ∈ Z, Tx ∈ Z, and we can express the image vector as linear combinations of, say, the covariant base vectors, {Gi } as: Tx = x1 (x) G1 + x2 (x) G2 + ⋯ + xn (x) Gn

(3.1)

where n is the dimensionality of the vector space. Note that we have indicated by xi (x) in equation (3.1) the fact that the scalar coefficients of combinations are all functions of x. Thus, the coefficients xi (x) are functionals, that is, mapping vector functions to scalar functions. We will now show that the coefficients are, in fact, linear functionals: Note that for x, y ∈ Z and arbitrary scalars, 𝛼, 𝛽, we have, from equation (3.1): T(𝛼x + 𝛽y) = x1 (𝛼x + 𝛽y) G1 + x2 (𝛼x + 𝛽y) G2 + ⋯ + xn (𝛼x + 𝛽y) Gn

(3.2)

and because of the linearity of T: T(𝛼x + 𝛽y) = 𝛼T(x) + 𝛽T(y) = 𝛼x1 (x) G1 + 𝛼x2 (x) G2 + ⋯ + 𝛼xn (x) Gn +𝛽x1 (y) G1 + 𝛽x2 (y) G2 + ⋯ + 𝛽xn (y) Gn

(3.3)

Thus, it follows from equation (3.2) and equation (3.3): xi (𝛼x + 𝛽y) = 𝛼xi (x) + 𝛽xi (y),

i = 1, 2, … , n

(3.4)

In other words, the coefficient functionals are linear functionals. Then, they have a scalar product representation, that is, there exist vectors, zi ∈ Z for i = 1, 2, … , n such that: xi (x) = (zi , x). Thus, we can rewrite equation (3.1) as: Tx = G1 (z1 , x) + G2 (z2 , x) + ⋯ + Gn (zn , x)

(3.5)

Now, if we represent x and base vectors Gi as column vectors and denote them as x and Gi , and row vectors as zi , we can rewrite equation (3.5) again as: Tx = G1 (z1 , x) + G2 (z2 , x) + ⋯ + Gn (zn , x) = G1 (z1 + G2 )(z2 + ⋯ + Gn )(zn , x)

(3.6)

Noting that x) is any arbitrary vector, we finally have Dirac’s dyad notation for tensors as: T = G1 (z1 + G2 )(z2 + ⋯ + Gn )zn

(3.7)

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46

3.2.3

Tensor Product

Now, instead, we can introduce a rule of operation as: (a ⊗ b)c = (b ∙ c)a

(3.8)

where a, b, c ∈ Z and operation ⊗ is known as the tensor product of any two vectors, a, b ∈ Z that transforms vector c ∈ Z to a vector a ∈ Z scaled by (b ∙ c). Then, clearly from equation (3.5), we have: Tx = (G1 ⊗ z1 ) x + (G2 ⊗ z2 ) x + ⋯ + (Gn ⊗ zn ) x = {(G1 ⊗ z1 ) + (G2 ⊗ z2 ) + ⋯ + (Gn ⊗ zn )} x

(3.9)

Finally, noting that x is arbitrary, from equation (3.9), we get the tensor product definition of the tensor as: T = (G1 ⊗ z1 ) + (G2 ⊗ z2 ) + ⋯ + (Gn ⊗ zn )

(3.10)

To sum up:

r a tensor product, denoted by ⊗, of two vectors, a and b, defines a simple second-order tensor T ≡ a ⊗ b with the rule that for any vector c belonging to the same vector space as b, we have: Tc = (a ⊗ b)c ≡ (b ∙ c)a.

Clearly, by definition, a ⊗ b ≠ b ⊗ a that is it does not satisfy the commutative property. However, because of the linearity property of the dot product, the tensor product demonstrates both the distributive (a ⊗ (b + c) = a ⊗ b + a ⊗ c) and the associative (𝛼a ⊗ b = a ⊗ (𝛼b) = 𝛼(a ⊗ b) properties for any scalar 𝛼 and vectors a, b, c. If we settle for column vectors only, as is customary in mechanics, then the above tensor product can also be expressed simply as T ≡ abT since by this definition, we also get: Tc ≡ abT c = (bT c)a = (b ∙ c)a. Note that only b and c need to belong to the same vector space in the above example. Finally, tensor product between two vectors (dyad) may be seen as the backbone of secondorder tensors (various second-order tensors are presented later) in deriving various properties. Similarly generalizing the idea, simple arbitrary-order (polyad) tensors can be formed out of more than two vectors with appropriate rules of operation. For example, a third-order tensor, S, may be generated as: S ≡ a ⊗ b ⊗ c.

3.3

General Tensor Space

As indicated in the case of general tensors, a formal definition comes from a generalization of the representation of vectors in a vector space. Just as a vector space is spanned by its base vectors, a tensor space (i.e. a linear vector space whose elements are tensors) can be spanned by its base tensors. Also, just as the vector components are dependent on the choice of the base vector set, the tensor components depend on the choice of base tensor set; given a base tensor set, the components yield the matrix representation of a tensor. Of course, as indicated before, we have assumed that the linear transformations form a linear vector space as they do. The base tensors are formed by the tensor product of underlying covariant and/or contravariant base vectors; then, the base tensors are of the form: gi ⊗ gj , gi ⊗ gj , gi ⊗ gj , gi ⊗ gj . This can easily be seen,

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47

for example, if we substitute for any arbitrary vectors a = ai gi and b = bj gj in the tensor definition: T ≡ a ⊗ b, then by the associative law and definition T ij ≡ ai bj , we have T ≡ T ij gi ⊗ gj , where we have used the Einstein summation convention. We may note that just like the vectors, T is an invariant object. Clearly, in three dimensions with nine base tensors, a second-order tensor has nine components. In general, then, depending on pure covariant or contravariant or mixed base tensors, j component-wise we can describe by: T ≡ T ij gi ⊗ gj = Tij gi ⊗ gj = Tji gi ⊗ gj = Ti gi ⊗ gj where j

T ij ≡ ai bj , Tij ≡ ai bj , Tji ≡ ai bj , Ti ≡ ai bj . Thus, just as with the vectors, a general second-order tensor is represented by the base tensors of the underlying tensor space. Definition: A tensor is an invariant object which may be described as a linear combination of the base tensors with the constants of combination recognized as the components of the tensor, yielding a matrix representation of the tensor corresponding to the chosen set of base tensors.

3.3.1

Component Description: Matrix Representation

The physical significance of any component, say Tij , can be seen by first using the definition of the tensor product to operate the tensor on a covariant base vector gj , and then performing the vector scalar product with the resulting transformed vector and gi , that is gi ∙ (Tgj ) = gi ∙ j Tmn (gn ∙ gj )gm = Tmn 𝛿jn (gi ∙ gm ) = Tmj 𝛿im = Tij , where 𝛿i = 𝛿ij = 𝛿 ij = 𝛿ji are the components of a special tensor called an identity tensor. In other words, we can see that Tij is the ith component of the vector Tgj and is known as the covariant component of a tensor. These equivalences are also written in more compact form without the scalar product symbol: gi Tgj ≡ gi ∙ (Tgj ). Similar operations can be performed with base tensors formed by pure covariant base vectors or a mix of covariant and contravariant base vectors to describe contravariant component (T ij ) or mixed j components (Tji , Ti ) of a tensor. Now, we are ready to see, component-wise, how a tensor operates on any arbitrary vector, say, a = aj gj . Following the pattern above, we can easily verify that Ta = Tmn aj (gn ∙ gj )gm = Tmn aj 𝛿jn gm = (Tmn an )gm , where we have used the summation convention. For example, for a transformation of an N-dimensional vector space to an M-dimensional vector space, m runs from 1 to M, and n runs from 1 to N. Clearly, the resulting entity is a M-dimensional vector. Because of our choice of the base tensors and hence the components of the tensor, the resulting vector turned out to be expressed as a linear combination of the contravariant base vectors. If we express for the resulting vector: b ≡ Ta = bm gm , then the components of the transformed vector are given by: bm = Tmn an with summation implied for n over the range.

r Thus, every tensor, T, as a continuous linear operator, has a matrix representation, that is, there exist vectors, Gi , Gj , Gi , Gj ∈ Z such that Tij = Gi ∙ T Gj or T ij = Gi ∙ T Gj or Tji = j

r r

Gi ∙ T Gj or Ti = Gi ∙ T Gj , where Tij , T ij are the covariant and the contravariant components, respectively, and T j , Tji are mixed components of the matrix. i The matrix elements are nothing other than the components of a tensor when both the domain and the range base vectors or base tensors are already chosen; thus, the matrices are coordinate dependent while the corresponding tensors are not. Because the matrix representation of a tensor is dependent on both the domain and the range basis sets, any change in either of these will change the components of the matrix. Let T be a second-order tensor such that b = T a where a and b are vectors belonging to the domain and range, respectively; with domain and range bases chosen, let br = Tm ad , where Tm is the

Computation of Nonlinear Structures

48

matrix representation of the tensor, ad is the domain coordinate vector and br is the range coordinate vector. Now suppose that the basis sets of both the domain and range are changed such that: ad = Qm ad and br = Pm br where ad , br are the new domain and range coordinate vectors, respectively, and Qm , Pm are the corresponding matrices reflecting the basis changes. Then, using the new relationships, we have: Pm br = br = Tm Qm ad . Thus, the change of bases result in a new matrix representation: br = Tm ad , given by:

Tm = (Pm )−1 Tm Qm

(3.11)

Example MR1: Matrix representation of a tensor Continuing from Example LF1 (Chapter 2), for T, the linear transformation or tensor, we , under standard bases, Ei , for both the domain would like to get (a) its matrix representation, TE E g (i = 1, 2) and range (i = 1, 2, 3), and (b) its matrix representation, TG , under domain bases: G1 = (1, 1), G2 = (0, −1), and range bases: g1 = (1, 1, 1), g2 = (1, 0, 1), g3 = (0, 0, 1). (a) For standard bases, it is easy to get the matrix representation by observation of the tensor definition: ⎡1 2 ⎤ ⎢ 1 −1 ⎥ b(= (a1 + 2a2 )E1 + ( a1 − a2 )E2 + (2a1 + 3a2 )E3 ) ∈ R3 TE = E ⎢ ⎥ ⎣2 3 ⎦ [ ] ⎡1 2 ⎤ ⎡1 1 0 m ⎢ ⎥ (b) Following equation (3.11), = = 1 −1 , Q = , and Pm = ⎢ 1 1 −1 ⎢ ⎢ ⎥ ⎣2 3 ⎦ ⎣1 1 0⎤ ⎡ 0 Now, noting that (Pm )−1 = ⎢ 1 −1 0 ⎥, it follows from equation (3.11): ⎢ ⎥ ⎣ −1 0 1 ⎦ Tm

TE E

⎡ 0 Tm = (Pm )−1 Tm Qm = ⎢ 1 ⎢ ⎣ −1

1 −1 0

0⎤⎡1 2 ⎤[ 1 0 ⎥ ⎢ 1 −1 ⎥ ⎥⎢ ⎥ 1 1⎦⎣2 3 ⎦

] ⎡0 0 = ⎢3 −1 ⎢ ⎣2

1 0 1

0⎤ 0 ⎥. ⎥ 1⎦

1 ⎤ −3 ⎥ . ⎥ −1 ⎦

Example MR2: Matrix representation of differential linear functional Continuing from Example LF3 (Chapter 2), for the linear vector space X = p2 [a, b], the class of polynomials of degree two or less on [a, b] ∈ R1 , the real line, let b = T a where a, b ∈ X, + 2 is the tensor function. are polynomials of degree two or less, and T(∙) = (tD + 2)(∙) = t 𝜕(∙)(t) 𝜕t Now the problem is (a) to find vector b if a = 1 + 2t2 , and the matrix representation of T and (b) when both the domain and range basis sets are given by the monomials of up to degree two: ( 1 t t2 ), (c) when both the basis sets are changed to Bernstein bases (used extensively in computer-aided geometric design): ( (1 − t)2 2t(1 − t) t2 ). ( ) (a) b = T a = t 𝜕t𝜕 + 2 (1 + 2t2 ) = 2 + 8t2

Tensors

49

(b) with monomial bases: a = ( 1

similarly, b = ( 1 t

t2 )bmon

t

t2 )amon

where the coordinate vector:

bmon

where the range coordinate vector,

determined. Now,

b = (1 t

( ) 𝜕 t2 )bmon = T a = t + 2 ( 1 t 𝜕t = (2

3t

amon

⎧1⎫ ⎪ ⎪ = ⎨ 0 ⎬; ⎪2⎪ ⎩ ⎭

⎧?⎫ ⎪ ⎪ = ⎨ ? ⎬ yet to be ⎪?⎪ ⎩ ⎭

t2 )amon

4t2 )amon = ( 1 t

⎡2 t2 ) ⎢ 0 ⎢ ⎣0

0 3 0

0⎤ 0 ⎥ amon ⎥ 4⎦

mon where Tmon Comparing both sides of the above expression, we get: bmon = Tmon mon a mon is the matrix representation, under monomial bases, of the tensor, T, given by: Tmon mon ≡ ⎡2 0 0⎤ ⎢ 0 3 0 ⎥. We can check the invariance of the range vector, b, as: b = ( 1 t t2 )bmon = ⎢ ⎥ ⎣0 0 4⎦ ⎡2 0 0⎤ ⎡2 0 0⎤⎧1⎫ ⎪ ⎪ ( 1 t t2 ) ⎢ 0 3 0 ⎥ amon = ( 1 t t2 ) ⎢ 0 3 0 ⎥ ⎨ 0 ⎬ = 2 + 8t2 , as before. ⎢ ⎢ ⎥ ⎥⎪ ⎪ ⎣0 0 4⎦ ⎣0 0 4⎦⎩2⎭ (c) Now we can relate Bernstein polynomial bases with the monomial bases as: Ber ( (1 − t)2 2t(1 − t) t2 ) = ( 1 t t2 )PBer mon where the basis transformation matrix, Pmon , 0 0⎤ ⎡ 1 ⎡1 0 0⎤ ) ( ⎢ −2 2 0 ⎥ with PBer −1 = ⎢ 1 1 0 ⎥. Using the change of basis = is given as: PBer mon mon 2 ⎥ ⎢ ⎢ ⎥ ⎣ 1 −2 1 ⎦ ⎣1 1 1⎦ formula as in equation (3.11), we then have the new matrix representation, bBer = TBer aBer where:

Ber

T

=

−1 mon Ber (PBer mon ) Tmon Pmon

⎡1 = ⎢1 ⎢ ⎣1

0 1 2

1

0⎤⎡2 0⎥⎢0 ⎥⎢ 1⎦⎣0

0 3 0

0⎤⎡ 1 0 0 ⎥ ⎢ −2 2 ⎥⎢ 4 ⎦ ⎣ 1 −2

0⎤ ⎡ 2 0 ⎥ = ⎢ −1 ⎥ ⎢ 1⎦ ⎣ 0

0 0⎤ 3 0⎥. ⎥ −2 4 ⎦

We can now express aBer , the coordinate vector of a, in Bernstein polynomial bases from the relation between the monomial bases and the Bernstein polynomial bases:

a = (1 t

t2

)amon

= (1

t

t2

⎧1⎫ ⎪ ⎪ ) ⎨ 0 ⎬ = ( (1 − t)2 ⎪2⎪ ⎩ ⎭

2t(1 − t)

t2 )aBer = ( 1 t

Ber t2 )PBer mon a

Computation of Nonlinear Structures

50

implying:

Ber

a

1 ( )−1 mon ⎡ ⎢1 = PBer a = mon ⎢ ⎣1

0 1 2

1

0⎤⎧1⎫ ⎧1⎫ ⎪ ⎪ ⎪ ⎪ 0⎥⎨0⎬ = ⎨1⎬ ⎥⎪ ⎪ ⎪ ⎪ 1⎦⎩2⎭ ⎩3⎭

So, we get:

bBer = TBer aBer

⎡ 2 = ⎢ −1 ⎢ ⎣ 0

0 3 −2

0⎤⎧1⎫ ⎧ 2 ⎫ ⎪ ⎪ ⎪ ⎪ 0⎥⎨1⎬ = ⎨ 2 ⎬ ⎥⎪ ⎪ ⎪ ⎪ 4⎦ 3 10 ⎩ ⎭ ⎩ ⎭

We can again check for the invariance geometric object, b, from the Bernstein component vector, bBer , as:

b = ( (1 − t)2

Ber

2t(1 − t) t2 )b

= ( (1 − t)2

⎧ 2⎫ ⎪ ⎪ 2t(1 − t) t2 ) ⎨ 2 ⎬ = 2 + 8t2 , ⎪ 10 ⎪ ⎩ ⎭

as expected.

3.3.2

Pseudo-vector Representation of Tensors

For second-order tensors, vector-type notation of components and base vectors – that is, as a linear combination of base vectors with components as vectors – greatly simplifies tensor analysis. For example, a second-order tensor, T ≡ T ij gi ⊗ gj , can also be represented as: T ≡ Tj ⊗ gj , where Tj ’s are the pseudo-vectors defined by Tj ≡ T ij gi .

3.4

Tensor by Component Transformation Property

As desired earlier, we can define tensors without reference to any chosen coordinate system. That is, we can define general tensors as invariant entities by identifying how their components respond to coordinate transformation. From this point of view, the scalars and the vectors can also be considered as tensors as follows:

r A scalar, with only one component that is invariant under coordinate transformation is identified as a tensor of order 0.

r A vector, with only one column of components that transforms under coordinate transformation under a certain law as described below, is identified as a tensor of order 1.

r A tensor, with multi-dimensional components that transforms under coordinate transformation under a certain law as described below, is identified as a second-order tensor if the dimension is 2, as a third-order tensor if the dimension is 3, and so on. Thus, in order to understand the component transformation properties that determine tensors, we first need to discuss the relevant notions about allowable coordination transformations.

Tensors

3.4.1

51

Allowable Coordinate Transformation

We can have parameterization by introduction of suitable, generally nonlinear, transformation such as: 𝜉 i = 𝜉 i (𝜉̄i , 𝜉̄2 , 𝜉̄3 ),

i = 1, 2, 3

(3.12)

with the assumption that the Jacobian (i.e. the determinant of the Jacobian matrix), as defined below, is non-vanishing at all points: ⎡ 𝜕𝜉 ⎢ 𝜕 𝜉̄1 ⎢ ⎢ 𝜕𝜉 2 𝜕(𝜉 1 , 𝜉 2 , 𝜉 3 ) D= ≡ det ⎢ ̄1 𝜕(𝜉̄1 , 𝜉̄2 , 𝜉̄3 ) ⎢ 𝜕𝜉 ⎢ 𝜕𝜉 3 ⎢ ⎣ 𝜕 𝜉̄1 1

𝜕𝜉 1 𝜕 𝜉̄2 𝜕𝜉 2 𝜕 𝜉̄2 𝜕𝜉 3 𝜕 𝜉̄2

𝜕𝜉 1 ⎤ 𝜕 𝜉̄3 ⎥ ⎥ 𝜕𝜉 2 ⎥ ⎥≠0 𝜕 𝜉̄3 ⎥ 𝜕𝜉 3 ⎥⎥ 𝜕 𝜉̄3 ⎦

(3.13)

Remarks

r We have restricted our discussion to three dimensions; the arguments and derivations for any other dimensions are exactly the same, and the results are obtained in a similar way.

r Because of the invertibility of the transformation given by equation (3.12), there exists the following transformation: 𝜉̄i = 𝜉̄i (𝜉 i , 𝜉 2 , 𝜉 3 ),

i = 1, 2, 3

(3.14)

r Because of the generalization of the allowable coordinate transformations to nonlinear mapping, r

the base vectors (defined either by tangents or normals which are coordinate-dependent) in a curved space turn out to be bound vectors in the sense that they are position dependent; for example, in the tangent, vectors to the curves are pointwise fixed or bounded. Now, considering 𝜉 i and 𝜉 j independent, we note that by chain rule: 𝜕𝜉 i 𝜕𝜉 i = 𝜕𝜉 j 𝜕 𝜉̄k 𝜕 𝜉̄i 𝜕 𝜉̄i = k 𝜕𝜉 𝜕 𝜉̄j

𝜕 𝜉̄k = 𝛿ji , 𝜕𝜉 j

(3.15)

𝜕𝜉 k = 𝛿ji 𝜕 𝜉̄j

For linear transformations, we have to resort to transformations locally of differentials d𝝃̄ to d𝜉̄ and vice versa; from equation (3.12), we get: d𝜉 i =

𝜕𝜉 i ̄j d𝜉 , 𝜕 𝜉̄j

or,

d𝜉̄i =

𝜕 𝜉̄i j d𝜉 𝜕𝜉 j

(3.16)

where we have used the Einstein summation convention for repeated diagonal indices. Thus the allowable transformation as given by equation (3.12) induces the linear transformation properties of the differentials as in equation (3.16). In other words, Riemann spaces are made of locally deformed patches of Euclidean spaces.

Computation of Nonlinear Structures

52

Remarks

r We note for the covariant base vectors, the invariant transformation relation as: j

Gi ≡

j

𝜕𝜉 𝜕 𝜕 𝜕𝜉 = j i = i Gj , 𝜕𝜉 i 𝜕𝜉 𝜕𝜉 𝜕𝜉

and,

Gj ≡

𝜕 𝜕𝜉

j

=

𝜕𝜉 k 𝜕 𝜕𝜉 k = G j k 𝜕𝜉 k 𝜕𝜉 j 𝜕𝜉

(3.17)

r Considering the infinitesimal arc length, ds, to be an invariant geometric object, we have from coordinate transformation given by equations (3.12) and (3.14), and inserting equation (3.16) into the expression for the infinitesimal arc length, ds: ds2 = Gij d𝜉 i d𝜉 j = Gij

𝜕𝜉 i 𝜕𝜉 j ̄m ̄n ̄ mn d𝜉̄m d𝜉̄n d𝜉 d𝜉 = G 𝜕 𝜉̄m 𝜕 𝜉̄n (3.18)

i j ̄ mn = Gij 𝜕𝜉 𝜕𝜉 ⇒ G 𝜕 𝜉̄m 𝜕 𝜉̄n

r Similarly, we have: ̄i ̄j ̄ ij d𝜉̄i d𝜉̄j = G ̄ ij 𝜕 𝜉 𝜕 𝜉 d𝜉 m d𝜉 n = Gmn d𝜉 m d𝜉 n ds2 = G 𝜕𝜉 m 𝜕𝜉 n ⇒ Gmn

3.4.2

̄ ij =G

𝜕 𝜉̄i 𝜕 𝜉̄j 𝜕𝜉 m 𝜕𝜉 n

(3.19)

Contravariant Components of a Vector (First-order Tensor) Transformation Law

Let a = ai Gi = ā j Gj be a vector with as yet undefined type components, ai , ā j , described in two different coordinate systems; then from equation (3.17), we have: ai Gi = ā j Gj = ā j

𝜕𝜉 i 𝜕𝜉

G, j i

i

and,

ā i Gi = aj Gj = aj

𝜕𝜉 G 𝜕𝜉 j i

(3.20)

Thus, from equation (3.20), we finally get the transformation relations for the contravariant components of a vector, as:

ai =

𝜕𝜉 i 𝜕𝜉

ā j , j

i

and,

ā i =

𝜕𝜉 j a 𝜕𝜉 j

(3.21)

Now, comparing the transformation relation given by equation (3.21) with the transformation relations for the differentials as in equation (3.16), we are ready to define these components as follows.

Tensors

53

r For an object to be a first-order tensor, that is, a vector, the contravariant components of the vector must transform similarly to the differentials of the coordinates, that is, according to equation (3.16).

3.4.3

Covariant Base Vector Transformation Law

For covariant base vectors, the transformation relationships follow from those of the contravariant components of a vector, as follows. Let a = ai Gi = ā j Gj be a vector with contravariant components, ai , ā j , described in two different coordinate systems; then from equation (3.21), we have: j

a = ā j Gj = ai

⇒ Gi =

3.4.4

𝜕𝜉 G = ai Gi , 𝜕𝜉 i j 𝜕𝜉 j 𝜕𝜉

and,

a = aj Gj = ā i

𝜕𝜉 j 𝜕𝜉

i Gj

= ā i Gi ,

j

i Gj ,

and,

(3.22)

𝜕𝜉 Gi = i Gj 𝜕𝜉

Length of a Vector

Considering an invariant geometric object such as a vector, u, written in terms of the covariant base vectors, {Gi }, and the contravariant base vectors, {Gi }, we have: u = ui Gi = ui Gi , where ui = u ∙ Gi ’s and ui = u ∙ Gi ’s are the covariant and the contravariant components, respectively, of the vector. Finally, for the length of the vector, ‖ ∙ ‖, we have: ‖u‖2 = u ∙ u = ui Gi ∙ uj Gj = ui uj 𝛿ji = ui ui

(3.23)

As a scalar object, the length of a vector is invariant under coordinate transformation, which helps us to define the covariant component transformation law for vectors as follows.

3.4.5

Covariant Components of a Vector (First-order Tensor) Transformation Law

Similar transformation properties for the covariant components of a vector may be obtained by considering scalar-valued vector functions; let us consider the invariant scalar function, 𝜙(𝜉 1 , 𝜉 2 , 𝜉 3 ), such that, under allowable coordinate transformation, we have: 𝜙(𝜉 1 , 𝜉 2 , 𝜉 3 ) = 𝜙(𝜉̄1 , 𝜉̄2 , 𝜉̄3 )

(3.24)

Then, the gradients of the function transform as: 𝜕𝜙 𝜕 𝜉̄j 𝜕𝜙 = i , i 𝜕𝜉 𝜕𝜉 𝜕 𝜉̄j

and,

𝜕𝜙 𝜕𝜉 j 𝜕𝜙 = 𝜕 𝜉̄i 𝜕 𝜉̄i 𝜕𝜉 j

(3.25)

Computation of Nonlinear Structures

54

Now, inserting the contravariant component transformation relation given by equation (3.21) in the expression for the invariant length of a vector, a, as in equation (3.23), we have: ai ai = ai

𝜕𝜉 i 𝜕𝜉

j

ā j = ā j ā j ,

ā i ā i = ā i

and,

𝜕 𝜉̄i ⇒ aj = j ā i , 𝜕𝜉

ā j =

and,

𝜕 𝜉̄i j a = aj aj 𝜕𝜉 j (3.26)

𝜕𝜉 i 𝜕𝜉

a j i

Now, comparing the transformation relation given by equation (3.26) with the transformation relations for the gradients as in equation (3.25), we are ready to define these components as follows.

r For an object to be a first-order tensor, that is, a vector, the covariant components of the vector must transform similarly to the gradient of a scalar function, that is, according to equation (3.25).

3.4.6

Contravariant Base Vector Transformation Law

We note that, for contravariant base vectors, the transformation relationships follow from those of the covariant components of a vector, as follows. j

Let a = ai Gi = ā j G be a vector with covariant components, ai , ā j , described in two different coordinate systems; then from equation (3.26), we have: j

a = ā j G = ai ⇒ Gi =

3.4.7

𝜕𝜉 i 𝜕𝜉

𝜕𝜉 i 𝜕𝜉

j

G = ai Gi , j j

G, j

a = aj Gj = ā i

and, i

and,

G =

i 𝜕 𝜉̄i j G = ā i G , j 𝜕𝜉

(3.27)

𝜕 𝜉̄i j G 𝜕𝜉 j

Second-orderOrder Tensor Component Transformation Laws

We can describe various transformation laws pertaining to Tij , the covariant, T ij , the contravariant, j and Tji , Ti , the mixed components of a second-order tensor, T = a ⊗ b, given as: T ≡ T ij gi ⊗ gj = j

Tij gi ⊗ gj = Tji gi ⊗ gj = Ti gi ⊗ gj where T ij ≡ ai bj ,

3.4.7.1

Tij ≡ ai bj ,

Tji ≡ ai bj ,

j

Ti ≡ ai bj .

Covariant Components of a Second-order Tensor Transformation Law

Noting that Tij ≡ ai bj , and using the transformation law for covariant components of a vector as in equation (3.26), we get: Tij ≡ ai bj =

𝜕𝜉 m 𝜕𝜉 n i

𝜕𝜉 𝜕𝜉

a b = j i j

𝜕𝜉 m 𝜕𝜉 n i

𝜕𝜉 𝜕𝜉

j

Tij

(3.28)

Tensors

3.4.7.2

55

Contravariant Components of a Second-order Tensor Transformation Law

Noting that T ij ≡ ai bj , and using transformation law for contravariant components of a vector as in equation (3.21), we get: i

T ij ≡ ai bj =

3.4.7.3

j

i

j

𝜕𝜉 𝜕𝜉 m n 𝜕𝜉 𝜕𝜉 a b = m n T mn 𝜕𝜉 m 𝜕𝜉 n 𝜕𝜉 𝜕𝜉

(3.29)

Mixed Components of a Second-order Tensor Transformation Law j

Noting that Tji ≡ ai bj , Ti ≡ ai bj , and using the transformation law for covariant and contravariant components of a vector as in equations (3.21) and (3.26), we get: i

i Tj

3.4.8

𝜕𝜉 𝜕𝜉 n m = m T , 𝜕𝜉 𝜕 𝜉̄j n

j

and,

j Ti

𝜕𝜉 m 𝜕𝜉 n ≡ T 𝜕 𝜉̄i 𝜕𝜉 n m

(3.30)

Higher-order Tensor Component Transformation Laws

The second-order tensor component transformation laws can be readily generalized to higherorder tensor component transformation law as follows: ( ij∙∙∙ T ab∙∙

=

) i j 𝜕𝜉 𝜕𝜉 𝜕𝜉 p 𝜕𝜉 q mn∙∙∙ ∙∙∙ ∙ ∙∙ Tpq∙∙∙ 𝜕𝜉 m 𝜕𝜉 n 𝜕 𝜉̄a 𝜕 𝜉̄b

(3.31)

Remark

r For

Cartesian coordinate systems, the difference between the covariant and contravariant components disappears, and the mixed components do not exist.

3.4.9

Tensor Operations

Given tensors of various orders, new tensors of different orders can be generated by tensor operations such as simple contraction and double contraction, as follows.

3.4.9.1

Simple Contraction

Given a tensor A ≡ a ⊗ b and another tensor B ≡ c ⊗ d, the composite second-order tensor C by simple contraction is given by: C = AB = (a ⊗ b)(c ⊗ d) = (b ∙ c)a ⊗ d. The simple contraction is also denoted by: C = A ∙ B. For column vectors, it can be easily seen that: C = AB = (abT )(cdT ) = a(bT c)dT = (bT c) adT = (b ∙ c)a ⊗ d.

r A simple contraction between any two tensors of arbitrary order reduces the order of the resulting tensor by two less than the order of the tensor product of the argument tensors.

Computation of Nonlinear Structures

56

For example, a simple contraction between a tensor of order 3 and a tensor of order 4 will result in a tensor of order 5 (= 3 + 4 − 2). In terms of component-wise descriptions, if A = Aij gi ⊗ gj and B = Bkl gk ⊗ gl , then, we have: C = AB = Aij gi ⊗ gj ∙ Bkl gk ⊗ gl = Aij Bkl 𝛿jk gi ⊗ gl = Aij Bjl gi ⊗ gl ; now, using the indices lowering or raising properties of the metric tensor, {gij }, we have: gi ⊗ gj . Thus, if we express C = Cij gi ⊗ gj , we get the C = AB = Aim Bml gi ⊗ gjl gj = Aim Bm j covariant components of C as: Cij = Aim Bm . j It is possible to apply simple contraction to a higher-order tensor from both sides. For example, if a and b are vectors (i.e. tensors of order 1), then a simple contraction, say, to a second-order tensor, C = Cij gi ⊗ gj , from both sides will result in a scalar: aCb = a ∙ C ∙ b = ai gi ∙ Cjk gj ⊗ gk ∙ bl gl = ai Cjk bl (gi ∙ gj )(gk ∙ gl ) = ai Cjk bl 𝛿ij 𝛿kl = aj Cjk bk . Conversely, we note that given a scalar, aj Cjk bk , an equivalent expression in absolute notation is: aj Cjk bk = aCb = a ∙ C ∙ b.

3.4.9.2

Double Contraction

Assuming column vectors, and noting that trace(abT ) = (ab) = aT b, the double contraction is given by: C = A : B = a ⊗ b : c ⊗ d = (a ∙ c)(b ∙ d) = trace(acT )(bT d) = trace(abT dcT ) = trace(ABT ). We may note that C is a scalar as expected of a double contraction which is a generalization of the vector scalar product. A double contraction of second-order tensors results in a scalar.

r A double contraction between any two tensors of arbitrary order reduces the order of the resulting tensor by four less than the order of the tensor product of the argument tensors. For example, a double contraction between a tensor of order 3 and a tensor of order 4 will result in a tensor of order 3 (= 3 + 4 − 4). Thus: P : Q = Pijkl gi ⊗ gj ⊗ gk ⊗ gl : Qmns gm ⊗ gn ⊗ gs = Pijkl Qmns (gk ∙ gm )(gl ∙ gn )gi ⊗ gj ⊗ gs = Pijkl Qmns 𝛿km 𝛿ln gi ⊗ gj ⊗ gs = Pijml Qmls gi ⊗ gj ⊗ gs A double contraction is usually not commutative but obeys both distributive and associative rules. Finally, just as in simple contractions, a scalar-valued function can be written in a symbolic notation by double contraction: u = Cijrs eij ers , i, j, r, s = 1, 2, 3, is equivalent to: u = Cijrs eij ers = e : C : e = eij gi ⊗ gj : Cklmn gk ⊗ gl ⊗ gm ⊗ gn : ers gr ⊗ gs . Fact 1: for an arbitrary second-order tensor, T, and another second-order tensor, c ⊗ d, we have: T : (c ⊗ d) = cTd; that is, a double contraction can be replaced by simple contractions from both sides of a second-order tensor.

Proof. Let T = a ⊗ b; then, T : (c ⊗ d) = (a ⊗ b) : (c ⊗ d) = (a ∙ c)(b ∙ d) = c ∙ (a ⊗ b)d = c ∙ (Td) = cTd

(3.32)

Tensors

3.4.10

57

Quotient Rule and Invariance

As indicated earlier, a simple collection or matrix array of functionals does not form a tensor unless the transformation property of the tensor components is manifested by such an array. However, we can invoke the invariance property to determine the tensor characteristic of an array and hence bypass any investigation into its transformation property; such an invariance property is contained in what is known as the quotient rule: if a simple contraction with a tensor and an as yet undetermined array results in a tensor, then the array must also be a tensor. In other words, if a simple contraction with an invariant object and an array produces an invariant object, then the array must also be an invariant object. For example, the length, ‖u‖2 = ui ui , an invariant property of a vector, u, as given by equation (3.23), is obtained by simple contraction of ui , the covariant components, and ui , the contravariant components of the vector. More, importantly, if one object is covariant, the other must be contravariant. As a practical application, for work, w = ui f i , to be invariant, if ui , the displacement is taken as covariant, then f i , the force effecting such displacement must be contravariant. As another example, let q be a tensor of order 1, that is, a vector with qi , its contravariant component, and P(i, j), be a two-dimensional array such that a multiplication results in a tensor, Pj , as given by Einstein’s summation convention: P(i, j) qi = Pj

(3.33)

then, we can show that P(i, j) is a tensor of type Pij , that is, a second-order tensor. Proof. Since P(i, j) qi is a tensor of order 1 (i.e. a vector) with covariant component type, it must satisfy the transformation law given by equation (3.26): P(i, j) q̄ i = P̄ j =

𝜕𝜉 n 𝜕𝜉

P = j n

𝜕𝜉 n 𝜕𝜉

j

P(m, n) qm

(3.34)

Now, noting that qm , the contravariant component transforms according to equation (3.21) such m that qm = 𝜕𝜉 i q̄ i , from equation (3.34), we get: 𝜕𝜉

( P(i, j) −

𝜕𝜉 m 𝜕𝜉 n i

𝜕𝜉 𝜕𝜉

j

) P(m, n)

q̄ i = 0

(3.35)

Now, with q̄ i ≠ 0 an arbitrary tensor, we have: P(i, j) =

𝜕𝜉 m 𝜕𝜉 n i

𝜕𝜉 𝜕𝜉

j

P(m, n)

(3.36)

which is a covariant component transformation law for a second-order tensor of Pij type as given by equation (3.28). Thus, P(m, n) = Pmn are components of a tensor.

3.5

Special Tensors

We list below several special tensors that will be used quite frequently in our structural analysis.

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3.5.1

Metric or Fundamental Tensor

Following up on the definition given by equation (3.15), Gij ≡ Gi ∙ Gj , i, j = 1, 2, 3 are the covariant components of what is known as the second-order metric or fundamental tensor; it is so called as it is capable of providing all the geometric information such as the distances, the curvatures and the angular separations of vectors in curved spaces. Now, considering a vector, u, written in terms of the covariant base vectors, {Gi }, and the contravariant base vectors, {Gi }, we have: u = ui Gi = ui Gi , where ui = u ∙ Gi ’s and ui = u ∙ Gi ’s are the covariant and the contravariant components, respectively, of the vector; then, ui = u ∙ Gi = Gi ∙ uj Gj = Gi ∙ Gj uj ≡ Gij uj where we have defined the contravariant components of the metric tensor as: Gij ≡ Gi ∙ Gj

(3.37)

Fact 1: The metric tensors are symmetric: Gij = Gji and Gij = Gji . Proof. Since the inner product of two vectors is symmetric, we have: Gij ≡ Gi ∙ Gj = Gj ∙ Gi ≡ Gji . Similarly, Gij ≡ Gi ∙ Gj = Gj ∙ Gi = Gji . Fact 2: The metric tensor has the property of raising and lowering of indices. For example, vi = Gij vj and Tij = Gik Tjk . Proof. For a vector, v: vi = v ∙ Gi = vj Gj ∙ Gi = Gij vj ; thus, the covariant component of a vector is obtained by lowering the index of the contravariant component by multiplying with the covariant components of the metric tensor. Similarly, for a second-order tensor, T: Tij = Gi ∙ T ∙ Gj = Gi ∙ T.lk Gk ⊗ Gl ∙ Gj = T.lk (Gi ∙ Gk )(Gl ∙ Gj ) = T.lk Gik 𝛿lj = Gik T.jk Fact 3: Gij Gik = 𝛿i.k shows the connection between the covariant and the contravariant components of the metric tensor.

Proof. Using definition of Gij as in equation (3.18) and raising of index property of the metric tensor as in

Fact 2: we have: Gij Gik = (Gi ∙ Gj )Gik =

Gi Gik ⏟⏟⏟

∙ Gj = Gk ∙ Gj = 𝛿j.k

(3.38)

raising of index

Fact 4: If G ≡ det(Gij ), then det(Gij ) = sponding tensors.

1 G

where det stands for the determinant of the corre-

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59

Proof. Using the relation as in Fact 3, we have: det(Gij Gik ) = det(Gij ) det(Gik ) = det(𝛿i.k ) = 1 ⇒ det(Gik ) =

(3.39)

1 1 = det(Gij ) G

Fact 5: G = det(Gij ) = J 2 , where J = det[ G1 G2 G3 ] = G1 ∙ (G2 × G3 ) is the Jacobian determinant of the Jacobian matrix made up of the covariant base vectors as the columns of the matrix.

Proof. We can express the metric tensor [Gij ] in matrix form by the columnar covariant base vectors as: ⎡ G1 ∙ G1 ⎢ ⎢ G2 ∙ G1 ⎢G ∙ G ⎣ 3 1

G1 ∙ G3 ⎤ ⎧ GT1 ⎫ ⎥ ⎪ ⎪ G2 ∙ G3 ⎥ = ⎨ GT2 ⎬ [ G1 T⎪ G3 ∙ G3 ⎥⎦ ⎪ ⎩ G3 ⎭

G1 ∙ G2 G2 ∙ G2 G3 ∙ G2

G2

G3 ]

(3.40)

Then, using the result from equation, the product rule for determinants of matrices and the fact that, for any matrix, A : det A = det AT , we get: ⎧ GT1 ⎫ ⎪ ⎪ G = det(Gij ) = det ⎨ GT2 ⎬ det[ G1 ⎪ GT ⎪ ⎩ 3⎭

3.5.2

G2

G3 ] = J 2 ,

or, J =

√ G

(3.41)

Metric Tensor Transformation Laws

From equation (3.17), we get the transformation relation for the covariant components of the metric tensor as: k

Gij ≡ Gi ∙ Gj =

l

k

l

𝜕𝜉 𝜕𝜉 𝜕𝜉 𝜕𝜉 G ∙ G ⇒ Gij = i j Gkl 𝜕𝜉 i k 𝜕𝜉 j l 𝜕𝜉 𝜕𝜉

(3.42)

In fact, equation (3.42) depicts the transformation law for the covariant components of any second-order tensor. In other words,

r for

an object to be a second-order tensor, the covariant components of the object must transform according to equation (3.42); this, then, serves to define a second-order tensor.

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Similarly, using equation (3.22), we have the transformation relation for the contravariant components of the metric tensor as: Gij ≡ Gi ∙ Gj =

3.5.3

𝜕𝜉 i ̄ k 𝜕𝜉 j l 𝜕𝜉 i 𝜕𝜉 j kl G ⇒ Gij = k G G ∙ k ̄l 𝜕 𝜉̄l 𝜕𝜉 𝜕𝜉 𝜕 𝜉

(3.43)

Second-order Identity Tensor, I

Consider a tensor I such that, for any vector x, we have: I x = x. Now, from vector properties, the vector x can be expressed by its contravariant components as: x = xi Gi where Gi ’s are the covariant base vectors and xi = Gi ∙ x where Gi ’s are the reciprocal contravariant base vectors. Then, we clearly have: I x = x = xi Gi = (Gi ∙ x)Gi = (Gi ⊗ Gi ) x. Thus, I = (Gi ⊗ Gi ). This is known as the second-order identity tensor. Note that we have used the Einstein diagonal summation convention. Alternatively, considering that the vector x can also be expressed by its covariant components as: x = xi Gi where Gi ’s are the reciprocal contravariant base vectors and xi = Gi ∙ x where Gi ’s are the covariant base vectors, we have: I x = x = xi Gi = (Gi ∙ x)Gi = (Gi ⊗ Gi ) x leading to I = (Gi ⊗ Gi ). Now, using the index lowering and raising property of the metric tensor, G = {Gij = Gi ∙ Gj }, we get: I = (Gi ⊗ Gi ) = (Gi ⊗ Gij Gj ) = Gij (Gi ⊗ Gj ) = (Gi ⊗ Gi )

(3.44)

Using the tensor product of the mixed base vectors, other equivalent expressions for the identity tensor can be generated. Fact 1: For any second-order tensor, A, we get: IA = AI = A

Proof. with A = Amn Gm ⊗ Gn , we have: i Gi ⊗ Gn = Ain Gi ⊗ Gn = A IA = (Gi ⊗ Gi )Amn Gm ⊗ Gn = Amn 𝛿m

Fact 2: For any two vectors, a and b, the scalar product can be expressed as double contraction through the second-order identity tensor: a ∙ b = I : (a ⊗ b) = a I b, that is, a scalar product can be expressed as a double contraction through the identity tensor, I.

Proof. Using equation (3.32), I : (a ⊗ b) = a I b = a ∙ (I b) = a ∙ b = ai bi . Alternatively, from the definition of the identity tensor: I : (a ⊗ b) = (gi ⊗ gi ) : (a ⊗ b) = (gi ∙ a) (gi ∙ b) = ai bi = (a ∙ b)

Fact 3: For any second-order tensor, A, the trace is defined as: tr A ≡ A : I. Then, we get: tr A = A.ii

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61

Proof. with A = Amn Gm ⊗ Gn , we have: tr A ≡ A : I = (Gi ⊗ Gi ) : Amn Gm ⊗ Gn = Amn Gim 𝛿ni = A.ii

(3.45)

4

3.5.4

Fourth-order Identity Tensor, I

As a generalization of the second-order identity tensor, we have the fourth-order identity tensor, 4

I, defined as: 4

I ≡ I ⊗ I = Gi ⊗ Gi ⊗ Gj ⊗ Gj

(3.46)

4

I is extensively utilized in material modeling as in the stress–strain transformation relation of continuum mechanics. Now, as indicated earlier, the tensor A can be expressed by its pseudocontravariant components, Ai , as: A = Aij Gi ⊗ Gj = Aj ⊗ Gj where Gi ’s are the covariant base vectors and Aj ≡ Aij Gi . Then, we clearly have: 4

I : A = Gi ⊗ Gi ⊗ Gj ⊗ Gj : Akl Gk ⊗ Gl = Akl (Gj ∙ Gk )(Gj ∙ Gl )Gi ⊗ Gi j j .j = Gjk Akl 𝛿l I = A.lj 𝛿l I = Aj I = (tr A) I

(3.47)

where tr A is the abbreviation for the trace of a second-order tensor, A, defined by the double contraction with I, the second-order identity tensor, that is, j

tr A ≡ A : I = Aj

(3.48)

4

Fact 1: For any second-order tensor, A, we get: A : I : A = (tr A)2 4

Proof. Using equations (3.47) and (3.48), A : I : A = A : (tr A)I = (tr A)2

3.5.5

Third Order Permutation Tensor, E

In vector calculus, the cross product of any two vectors, a and b, is given by: a × b = ai bj gi × gj = 𝜀ijk ai bj gk = 𝜀ijk ai bj gk where gi × gj = 𝜀ijk gk , and, gi × gj = 𝜀ijk gk

(3.49)

and 𝜀ijk and 𝜀ijk are the covariant and contravariant components, respectively, of the third-order permutation tensor, E, that is, E ≡ 𝜀ijk gi ⊗ gj ⊗ gk = 𝜀ijk gi ⊗ gj ⊗ gk ,

with 𝜀ijk = −𝜀jik = 𝜀jki = −𝜀kji

(3.50)

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Fact 1: gi × gj = 𝜀ijk gk = E : (gi ⊗ gj ) = −gi E gj , that is, the vector cross product has a tensorial representation.

Proof. Using equations (3.32), (3.49) and (3.50), we have: j

E : (gi ⊗ gj ) = 𝜀ijk gi ⊗ gj ⊗ gk : (gi ⊗ gj ) = 𝜀ijk 𝛿jk 𝛿i gi = 𝜀ijk gk and −gi E gj = −gi ∙ 𝜀ijk gi ⊗ gj ⊗ gk ∙ gj = −𝜀ijk 𝛿jk gj = −𝜀ikj gk = 𝜀ijk gk = gi × gj = E:(gi ⊗ gj )

3.6

Second-order Tensors

As indicated in our discussion on tensors as linear transformations, a simple second-order tensor space can be defined in an invariant component-independent form from what is known as the tensor product (or dyadic) of two vectors: Definition:

r A tensor product of any two vectors, a and b, may define a second-order tensor T ≡ a ⊗ b such

that, for any vector c, the transformed vector d is determined by: d = Tc = (a ⊗ b)c = (b ∙ c)a.

In other words, according to this rule of transformation, the vector c is transformed by the tensor into a scaled vector along the direction of a, as shown in Figure 3.2: If we settle for column vectors only, as is customarily done in mechanics, then the above tensor product can be expressed as T ≡ abT since by this representation we also get: Tc ≡ abT c = (bT c)a = (b ∙ c)a as above. Note that only b and c need to belong to the same vector space to have an operational definition of tensors by tensor product. However, as indicated in the case of general tensors, a formal definition comes from a generalization of the representation of vectors in a vector space. Let us recall that just as a vector space is spanned by its base vectors, tensor spaces (i.e. linear vector spaces whose elements are tensors) are spanned by their base tensors. Of course, as indicated earlier, we have assumed that the linear transformations form a linear vector space as they do. The base tensors are formed by the tensor product of underlying the covariant and/or contravariant base vectors.

Figure 3.2 Second-order tensor.

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63

That is the base tensors are of the form: gi ⊗ gj , gi ⊗ gj , gi ⊗ gj , gi ⊗ gj . This can be easily seen, for example, if we substitute for any arbitrary vectors a = ai gi and b = bj gj in the tensor definition, then by associative law, and defining T ij ≡ ai bj , we have T ≡ T ij gi ⊗ gj , where we have used the Einstein summation convention of summing over the repeated diagonal superscripts and subscripts. We may recall, from our discussion of tensors as linear transformations, that just like the vectors, T is an invariant object independent of the choice of coordinate systems. Clearly, in three dimensions with nine base tensors, a second-order tensor has nine components. In general, then, depending on pure covariant or contravariant or mixed base tenj sors, component-wise we can describe it by: T ≡ T ij gi ⊗ gj = Tij gi ⊗ gj = Tji gi ⊗ gj = Ti gi ⊗ gj j where T ij ≡ ai bj , Tij ≡ ai bj , Tji ≡ ai bj , Ti ≡ ai bj . Thus, just like the vectors, a general secondorder tensor is represented by the base tensors of the underlying tensor space. So any second-order tensor is an invariant object described as a linear combination of the base tensors with the constants of combination recognized as the matrix components of the tensor corresponding to the chosen set of base tensors. With the tensor components thus identified, we can easily see that for any d = Tc, we have for covariant components of the transformed vector: di gi = (Tij gi ⊗ gj )(ck gk ) = Tij ck (gj ∙ gk )gi = j j Tij ck 𝛿k gi = (Tij cj )gi . Thus, di = Tij cj . Similarly, for mixed components of tensors: di = Ti cj , i i j i ij d = Tj c , and for contravariant components of tensors: d = T cj .

3.6.1

Inverse Tensors

The second-order inverse tensor, denoted as A−1 , of a tensor, A, is defined such that: A−1 A = A A−1 = I

(3.51)

where I is the second-order identity tensor.

r clearly: I−1 I = I I−1 = I, that is, the identity tensor is the inverse of itself, r (A−1 )−1 A−1 = I ⇒ (A−1 )−1 = A, r (AB)−1 AB = I ⇒ (AB)−1 ABB−1 A−1 = B−1 A−1 ⇒ (AB)−1 = B−1 A−1 ⎵ ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⎵ =I

3.6.2

Transposed Tensors

The second-order tensor transposition is an operation that is routinely utilized to determine symmetry, or lack of it, for tensors analysis. The transposed tensor, denoted by TT , of a simple second-order tensor T ≡ a ⊗ b is defined by TT ≡ b ⊗ a. For column vectors: T ≡ abT ⇒ TT = (abT )T = baT . This definition can be easily extended to a general second-order tensor. Let a ≡ ti = Tij gi and b ≡ gj . Then, we have a general second-order tensor: T ≡ a ⊗ b = tj ⊗ gj = Tij gi ⊗ gj with covariant components. The transposed tensor is, then, defined by: TT ≡ (TT )ij gi ⊗ gj = b ⊗ a = Tij gj ⊗ gi = Tji gi ⊗ gj

(3.52)

Note that in the last equality, we have only switched the dummy indices. Thus, the transposed tensor components are given by: (TT )ij = Tji . Similarly, we can get the transposed contravariant j j and mixed components: for mixed: (TT )ij = Tji and (TT )i = Ti , and for contravariant: (TT )ij = T ji .

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Fact 1: Given any two vectors, c and d, and a tensor T = tj ⊗ gj , we have: cTd = c ∙ (Td) = dTT c = d ∙ (TT c)

(3.53)

Proof. cTd = c(tj ⊗ gj )d = (c ∙ tj )(gj ∙ d) = (d ∙ gj )(tj ∙ c) = d(gj ⊗ tj )c = dTT c. In the equalities, we used the properties of vector scalar product and definition of a transposed tensor.

Fact 2: Given any three tensors, A, B and C, we have: A : BC = BT A : C = ACT : B

(3.54)

Proof. Let A = ai ⊗ gi , B = bj ⊗ gj and C = ck ⊗ gk . Then, A : BC = A : (bj ⊗ gj )(ck ⊗ gk ) = (ai ⊗ gi ) : (bj ⊗ gk )(gj ∙ ck ) = (ai ∙ bj )(gi ∙ gk )(gj ∙ ck ). Now, we can rewrite: (ai ∙ bj )(gi ∙ gk )(gj ∙ ck ) = (bj ∙ ai )(gj ⊗ gi ) : (ck ⊗ gk ) = (gj ⊗ bj )(ai ⊗ gi ) : C = BT A : C. In the equalities, we used the definition of transposed tensor, single contraction and double contraction.

r By similar arguments, we get: A : BC = ACT : B r By utilizing the linear properties of vectors, we get: (A + B)T = AT + BT and (AB)T = BT AT Fact 3: For the identity tensor, I = gi ⊗ gi , we have I = IT = gi ⊗ gi . j

Proof. for any vector: h = hj gj , we get IT h = (gi ⊗ gi )hj gj = hj (gi ∙ gj )gi = hj 𝛿i gi = hj gj = h = Ih. Thus, I = IT .

Fact 4: The inverse and transpose operations on a tensor are order independent, that is, for any tensor A, we have: (AT )−1 = (A−1 )T = A−T . Proof. with the definition of inverse, we have: AA−1 = I. Now, taking transpose of both sides and noting the transpose property of a composite tensor, we have: (AA−1 )T = (A−1 )T AT = IT = I. Thus, we find that (A−1 )T is inverse of AT . In other words, (A−1 )T = (AT )−1 . So, the parenthesis is irrelevant.

3.6.3

Orthogonal and Rotation Tensors

A tensor Q is called an Orthogonal tensor if: QT = Q−1 ⇒ QQT = QT Q = Q−1 Q = QQ−1 = I

(3.55)

The orthogonality imposes 6 conditions (3 equations due to orthogonality: QQT = I, and 3 due to symmetry: QQT = QT Q, the tensor Q is left with only 3 independent variables. An orthogonal transformation preserves both the lengths and in-between angle of any two vectors. To wit, for any two vectors, a and b with â = Qa and b̂ = Qb, we get: â ∙ b̂ = Qa ∙ Qb = a ∙ QT Qb = a ∙ b. This property, clearly, demonstrates that a rotational transformation, that preserves both the lengths and in-between angle of two vectors, is an orthogonal tensor.

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65

For a detailed presentation on rotation tensor that is of utmost importance in our geometrically nonlinear analysis of beams, plates and shells, we would follow up in Chapter 4. Another important orthogonal tensor is the reflection tensor, say, P, defined as: P=I−

2 (a ⊗ a) (a ∙ a)

(3.56)

where I is the identity tensor, a is the normal to the plane of reflection of a vector. Let b = c + d be a vector such that it is represented by its components: c = 𝛼a parallel to the normal a and d perpendicular to it, that is, contained in the plane; then, ( Pb = I −

) 2 (a ⊗ a) (𝛼a + d) (a ∙ a)

= 𝛼a + d −

2  = 𝛼a + d − 2𝛼a = −𝛼a + d  {𝛼(a ∙ a)a +  (a  ∙ d)a} (a ∙ a)

Thus, the component perpendicular to the plane is reversed and the one along the plane is not disturbed, that is, the vector b is reflected by the plane with normal given by the vector, a.

3.6.4

Symmetric Tensors

A tensor is symmetric if its transposed is same as itself, that is, if A = AT . Symmetric tensors abound in mechanics. For example, for non-polar mechanics, various stress tensors and strain tensors are symmetric; the identity tensor is symmetric and so on. Symmetric tensors can be formed by various ways from a general non-symmetric tensor. For example, a simple contraction of a tensor by its transposed generates a symmetric tensor. For example, for any A, we have AAT symmetric since by definition of transpose of composite tensors: (AAT )T = (AT )T AT = AAT . Fact 1: As an example, in continuum mechanics, the symmetric Cauchy stress tensor, 𝝈 is related to the second Piola–Kirchhoff stress tensor, S, through the deformation tensor, F, by the formula: S ≡ F−1 𝝈F−T to a constant. If 𝝈 is symmetric, so is S. Proof. We have ST = (F−1 𝝈F−T )T = (F−T )T 𝝈 T (F−1 )T by transposition properties of composite tensors. Now, using symmetry of 𝝈 and order independence of inverse and transpose, we get: ST = (F−T )T 𝝈 T (F−1 )T = F−1 𝝈F−T = S. Fact 2: For any symmetric tensor A, the covariant components have the property: Aij = Aji . Proof. From the component relations with transposed tensor, we get for the covariant components of any symmetric tensor A = Aij gi ⊗ gj = AT = (AT )ij gi ⊗ gj = Aji gi ⊗ gj . With base tensors arbitrary, we get: Aij = Aji . j j j Similarly, it is easy to show that for mixed: Ai = Ai = Ai , and for contravariant: Aij = Aji for A symmetric. Note also that the order of superscripted and subscripted indices is immaterial for mixed components of a symmetric tensor. Finally, for any A, B ≡ (A + AT ) is symmetric.

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3.6.4.1

Eigen Properties of Symmetric Tensors

By definition, for a symmetric tensor, S, with covariant components, the off-diagonal components, Sij , i ≠ j, are identical for any interchange of the indices. Then, for eigenanalysis of S, we have to solve for the eigenvalues, 𝜆, and subsequently determine the corresponding eigenvectors, x, in the eigenequation: S x = 𝜆x ⇔ (S−𝜆I)x = 0

(3.57)

For a non-trivial solution of equation (3.65), we must have the determinant |S − 𝜆I| vanish identically. Following the Cayley–Hamilton theorem, the characteristic polynomial is given by: p(𝜆) ≡ det(S − 𝜆I) = −𝜆3 + I1 𝜆2 − I2 𝜆 + I3 = 0

(3.58)

where, the three invariants, I1 , I2 , I3 of a second-order symmetric tensor are given as: I1 = tr S = Sii 1 j j {(tr S)2 − tr S2 } = Sii Sj − Sji Si 2 I3 = det S = ||Sji || I2 =

(3.59)

From equation (3.57), we have, for a symmetric tensor: xi S xi = xi ∙ 𝜆i xi = 𝜆i ‖xi ‖2 ≥ 0

(3.60)

This implies that the eigenvalues, 𝜆i , of a symmetric tensor are real and positive. Again from equation (3.57), we have, for a symmetric tensor: (𝜆i − 𝜆j )xi xj = xj S xi − xi S xj = 0

(3.61)

This implies that because 𝜆i ≠ 𝜆j , the eigenvectors of a symmetric tensor are orthogonal to each other. Thus, the eigenvalues of a general symmetric tensor are real and positive, and the eigenvectors are mutually orthogonal.

3.6.5

Skew-symmetric Tensors

A tensor is defined as skew-symmetric if its negative transpose is the same as itself. A tensor A is skew-symmetric if A = −AT , that is, A + AT = 0 where 0 is the null tensor. Clearly, Aij + Aji = 0 or Aij + Aji = 0 implies that the diagonal elements of a skew-symmetric tensor under covariant or contravariant component representation, are identically zero. Thus, in these cases, the number of independent components is given by 12 n(n − 1) where n is the underlying dimension of the domain and range spaces. However, for the mixed components of a j j skew symmetric tensor, we have: Ai = −Ai . Noting that in general, Aii ≠ Aii , the diagonal elements of a skew-symmetric tensor under mixed component representation is generally non-zero. Now, given an arbitrary tensor, C = a ⊗ b, we can extract the skew-symmetric part of the tensor as: 𝛀 ≡ 12 (C − CT ) = 12 (a ⊗ b − b ⊗ a) since it can be easily shown that 𝛀 + 𝛀T = 0.

Tensors

3.6.5.1

67

Axial Vectors

An axial vector is intimately related to the second-order skew-symmetric tensors in three dimensions. If we restrict the space of skew-symmetric tensors to the non-relativistic physical world, that is, an n = 3 dimensional space, the number of independent covariant or contravariant components of a skew-symmetric tensor becomes also 3 (= 12 n(n − 1)). This makes it possible to treat equivalently operations by skew-symmetric tensors by means of vector algebra. That is, for any three-dimensional vector c and a given skew-symmetric tensor: 𝛀 = 12 (a ⊗ b − b ⊗ a), we have

𝛀c = 𝝎 × c, where 𝝎 ≡ 12 (b × a). Using the definition of a skew-symmetric tensor, we can easily show: 𝛀c = 12 (a ⊗ b − b ⊗ a)c = 12 {(c ∙ b)a − (c ∙ a)b} = 12 c × (a × b) = 12 (b × a) × c = 𝝎 × c

(3.62)

where we have used the well-known vector identity: a × (b × c) = (a ∙ c)b − (a ∙ b)c for any three vectors. Clearly, then, a simple contraction 𝛀c is equivalent to the vector operation 𝝎 × c. Thus, there is a one-to-one correspondence between the tensor operation by 𝛀 and vector cross product operation by 𝝎. This equivalence is denoted by 𝛀 = 𝛀× = [𝛀] , and tensor algebra anywhere for a skew-symmetric tensor can be replaced by a corresponding vector operation. Hence, for orthonormal bases with components of 𝝎 given as: 𝝎 = { 𝜔1 𝜔2 𝜔3 }T , the component matrix of the tensor 𝛀 has only three independent components and the following representation: ⎡ 0 𝛀 = ⎢ 𝜔3 ⎢ ⎣ −𝜔2

−𝜔3 0 𝜔1

𝜔2 ⎤ −𝜔1 ⎥ ⎥ 0 ⎦

(3.63)

As an example, for a shell coordinate system, if 𝛀 is applied to a fixed surface normal, g3 , then 𝝎 has only two components since 𝛀g3 = 𝝎 × g3 = 𝜔i gi × g3 = 𝜔𝜎 g𝜎 , where the Latin indices run over 1 to 3, and the Greek indices over 1 and 2. Recall the summation and notation convention that Greek indices run through only two in three-dimensional tensors. For n > 3, this correspondence between a skew-symmetric tensor and a vector does not exist and hence we do not have any other option than to carry on with tensor algebra. For n = 2, a skew-symmetric tensor is equivalent to a scalar. Note also that by the definition of the permutation tensor E, we have a symbolic tensor representation: 𝝎 ≡ 12 (b × a) = 12 E : (b ⊗ a)

3.6.5.2

(3.64)

Eigenproperties of Skew-symmetric Tensors

By definition, for a skew-symmetric tensor, S, with covariant components, all the diagonal components {Sii } are zero, and the off-diagonal elements have the property: Sij + Sji = 0, i ≠ j. ⎡ 0 −s3 s2 ⎤ 0 −s1 ⎥. Now, let s = ( s1 s2 s 3 )T be the corresponding axial vector such that S = ⎢ s3 ⎥ ⎢ 0 ⎦ ⎣ −s2 s1

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68

Then, for eigenvalues of S, we have to solve for 𝜆 in the eigenequation: S x = 𝜆x ⇔ (S−𝜆I)x = 0

(3.65)

For a non-trivial solution of equation (3.65), we must have the determinant |S − 𝜆I| vanish identically. In other words, the characteristic polynomial is given by: p(𝜆) ≡ det(S − 𝜆I) = 0. Upon simplification, we have: ( ) 𝜆3 + s21 + s22 + s23 𝜆 = 0, { ( )} ⇒ 𝜆 𝜆2 + s21 + s22 + s23 = 0

or, 𝜆3 + ‖s‖2 𝜆 = 0

(3.66)

Clearly, then, the eigenvalues of a general skew-symmetric tensor are either zero or purely imaginary numbers.

3.6.5.3

Important Identities of Skew-symmetric Tensors

For later use, in finite element analyses, we list here some important identities for skew-symmetric tensors. Since every tensor satisfies its own characteristic polynomial, we have, from equation (3.66), the important identity: S3 + s2 S = 0, where ⇒ S3 = −‖s‖2 S

s ≡ ‖s‖ =

√ s21 + s22 + s23

(3.67)

Now, for all the following identities, we assume A and B as skew-symmetric tensors with a, b as the corresponding axial vectors, that is, A = [a] and B = [b]. h ∈ R3 is any arbitrary vector. Note that Aa = a × a = 0 and Bb = b × b = 0. Additionally, taking all vectors as column vectors, we have: (b ⊗ a) = b aT , where (b ⊗ a) is recalled as the tensor product of two vectors, a and b. Fact 1: Ab + Ba = 0 Proof. Ab + Ba = a × b + b × a = a × b − a × b = 0 Fact 2: AB = (b ⊗ a) − (a ∙ b)I Proof. ABh = a × (b × h) = (a ∙ h)b − (a ∙ b)h = [(b ⊗ a) − (a ∙ b)I]h Fact 3: A2 = (a ⊗ a) − ||a||2 I Proof. substitution of B by A in Fact 2 completes the proof. Fact 4: A2 B + B A2 = −(a ∙ b)A − ||a||2 B

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69

Proof. A2 B + B A2 = A(AB) + B(a ⊗ a) − ||a||2 B = A(b ⊗ a) − (a ∙ b)A + B(a ⊗ a) − ||a||2 B = A(b ⊗ a) + B(a ⊗ a) − (a ∙ b)A − ||a||2 B ( ( (( =( ({Ab +( Ba} ⊗ a) − (a ∙ b)A − ||a||2 B = −(a ∙ b)A − ||a||2 B (( Fact 5: ABA = −(a ∙ b)A.  ⊗ b) − (a ∙ b)A = −(a ∙ b)A Proof. Using Fact 2, we have: ABA = A[(a ⊗ b) − (a ∙ b)I] = ( Aa Fact 6: AB A2 = A2 BA = −(a ∙ b)A2 . Proof. Using Fact 5, we have: AB A2 = (ABA)A = −(a ∙ b)A. Similarly, for the second equality. Fact 7: A2 B A2 = ||a||2 (a ∙ b)A. Proof. Using Fact 6, and noting that A satisfies its characteristic equations, that is, ||a||2 A + A3 = 0, we have: A2 B A2 = A(A B A2 ) = −(a ∙ b)A3 = ||a||2 (a ∙ b)A. Fact 8: AB − BA = (b ⊗ a) − (a ⊗ b) = [a × b]. Proof.  − (b ∙ h)a + (a     (AB − BA)h = a × (b × h) − b × (a × h) = (a ∙ h)b −  (a  ∙ b)h  ∙ b)h 3 = {(b ⊗ a) − (a ⊗ b)}h, ∀h ∈ R Also, it follows that: (AB − BA)h = (a ∙ h)b − (b ∙ h)a = (h ∙ a)b − (h ∙ b)a = h × (b × a) = (a × b) × h = [a × b]h,

Fact 9: hAh = 0,

∀h ∈ R3 .

Proof. hAh = AT h h = −Ah h = −h Ah,

3.6.6

∀h ∈ R3

∀h ∈ R3 completes the proof.

Contractions

Given a tensor A ≡ a ⊗ b and another tensor B ≡ c ⊗ d, the composite second-order tensor C by simple contraction is given by: C = AB = (a ⊗ b)(c ⊗ d) = (b ∙ c)a ⊗ d. The simple contraction is also denoted by: C = A ∙ B. A simple contraction of second-order tensors results in a secondorder tensor. Moreover, with the column vector restriction and noting that trace(a ∙ bT ) = (ab) = aT b, the double contraction is given by: C = A : B = a ⊗ b : c ⊗ d = (a ∙ c)(b ∙ d) = trace(acT )(bT d) = trace(abT dcT ) = trace(ABT ). We may note that C is a scalar, as expected of a double contraction,

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70

which is a generalization of vector scalar product. A double contraction of second-order tensors results in a scalar.

3.6.7

Trace, Norm and Determinant

As it turns out, the trace and determinant of a tensor appear as two of the three invariant scalars that define the associated characteristic polynomial resulting from the eigenanalysis of a tensor. The trace in turn helps to generalize the length concept of a vector through the definition of the norm, an invariant measure of a tensor.

3.6.7.1

Trace

Denoting a tensor, A, in the dyadic form: A = Aj ⊗ gj = Ak ⊗ gk where Aj ≡ Aij gi and Ak = Amk gm are the contravariant and covariant vector representation, respectively, the trace of A, denoted by trA, is defined as: trA ≡ I : A

(3.68)

where I is the three-dimensional identity tensor. Thus, trA ≡ I : A = gm ⊗ gm : An ⊗ gn = gm ∙ An 𝛿nm = gm ∙ Am n = gm ∙ Anm gn = gmn Anm = Am m = An

(3.69)

Thus, denoting a tensor in dyadic form, we have from the application point of view: trA = n gm ∙ Am = Am m = An . Moreover, with the column vector restriction, we have trA = tr(a ⊗ b) = T T tr(ab ) = a b = (a ∙ b).

r trAT = tr(b ⊗ a) = (b ∙ a) = (a ∙ b) = trA, r The trace is a linear function of tensors: tr(A + B) = trA + trB. Fact 1: tr(ABT ) = tr(BAT ) = A : B = BT : AT Proof. Let A = a ⊗ b and B = c ⊗ d, then, ABT = (a ⊗ b) ∙ (d ⊗ c) = (b ∙ d)(a ⊗ c). Similarly, BAT = (b ∙ d)(c ⊗ a). Thus, tr(ABT ) = (b ∙ d) tr(a ⊗ c) = (b ∙ d) (a ∙ c) = (b ∙ d) (c ∙ a) = (d ∙ b) tr(c ⊗ a) = tr(BAT ). Note that we have used the definition of simple contraction for composite tensors. The trace can be directly verified by application of the concept of double contraction since: A : B = (a ⊗ b) : (c ⊗ d) = (b ∙ d)(a : c). Note also that tr(AB) ≠ trA trB. j

j

Fact 2: for any symmetric A, (trA)2 − tr(A2 ) = Aii Aj − Ai Aij j

Proof. we have: (trA)2 = trA trA = Aii Aj . j

Now, tr(A2 ) = tr(AA) = tr(AAT ) = A : A = Aj ⊗ gj : Ak ⊗ gk = Aj ∙ Ak 𝛿k = Aj ∙ Aj = Aij gi ∙ j

j

Am gm = Aij Ai . Now, switching the dummy indices completes the proof.

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71

3.6.7.2

Norm

Just as in vectors (first-order tensors), the norm of a second-order tensor is a measure of its ‘generalized length’ and, as in Hilbert space, may be induced from the definition of the underlying scalar product. The scalar product for tensors is generalized to double contraction. Thus, the norm, denoted by ‖ ⋅ ‖, of a general second-order tensor A may be defined as: ‖A‖ = Fact √ 1: The √ A:A B:B

Schwarz

√

A:A=

√

inequality:

tr(AAT ) =

√ tr(AT A)

‖AB‖ ≤ ‖A‖ ‖B‖ =

(3.70) √

√ tr(AAT ) tr(BBT ) =

Fact 2: From invariants study, the trace of a composite tensors in simple contraction is given by: tr(ABT ) = A : B. Then, for a symmetric tensor A, we have: j

tr(A2 ) = Ai Aij

(3.71)

Proof. j

j

j

tr(A2 ) = tr(AA) = tr(AAT ) = A : AT = Ai gi ⊗ gj : Akl gk ⊗ gl = Ai 𝛿ki 𝛿jl Akl = Ai Aij . j

j

Fact 3: for any symmetric A, (trA)2 − tr(A2 ) = Aii Aj − Ai Aij j

Proof. (trA)2 = (trA)(trA) = Aii Aj , and using equation (3.71): tr(A2 ) = tr(AA) = tr(AAT ) = j

A : AT = Ai Aij completes the proof.

3.6.7.3

Determinant

The definition of the determinant, also known as the third invariant, of a second-order tensor depends on the choice of the tensor components. To facilitate connection with the vector cross product through the permutation symbol, we may choose to define i:t by the mixed components of a tensor. Thus, for a second-order tensor A = Aij gi ⊗ gj , the determinant is defined as: | | j det A ≡ |Aij | = eijk Ai1 A2 Ak3 | |

(3.72)

where eijk is the third-order permutation tensor associated with the Cartesian coordinate systems. j j Given |Ai | as the determinant of a coefficient matrix with Ai as the i th row and j th column element, j j this definition is motivated by the verifiable relations: eijk Ari Asj Atk = |Ai |erst and eijk Air As Akt = j

|Ai |erst .

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72

3.6.8

Decomposition of Tensors

As it turns out, the double contraction of a symmetric and a skew-symmetric tensor results in the scalar zero, that is, it annihilates it. To wit: by definition, if A is symmetric, then Aij = Aji , and for a skew-symmetric tensor B, we have: Bkk = 0, and Bkl = −Blk for k ≠ l. Thus, A : B = Aij gi ⊗ gj : Bkl gk ⊗ gl = Aij Bkl (gj ∙ gl )(gi ∙ gk ) = Aij Bkl 𝛿jl 𝛿ik = Aij Bij = 0

(3.73)

This result is especially useful in tensor algebraic manipulations that arise frequently in mechanics. Thus, there is a need for an additive decomposition of a general second-order tensor for simplification as well as insight into the nature of the tensor. Also, the polar decomposition serves to identify the translational and rotational aspect of deformations, say, in strain tensors. Finally, for both analysis as in constitutive theory and numerical formulations such as mixed finite element analysis, stress and strain tensors can be profitably viewed from the volumetric (spherical) and the shearing (deviatoric) stresses and deformations.

3.6.8.1

Additive Decomposition: Symmetric and Skew-symmetric Tensors

A general second-order tensor A can always be represented as an additive sum of a symmetric tensor and a skew-symmetric tensor: A = Aj ⊗ gj = 12 (Aj ⊗ gj + gj ⊗ Aj ) + 12 (Aj ⊗ gj − gj ⊗ Aj ) =

1 (A + AT ) + 12 (A − AT ). Recall that by our notation convention about contravariant vectors, 2 Aj ≡ Aij gi . Now, by definition of a transposed tensor, the first part of the decomposition is a symmetric tensor, and we use this to denote: symA ≡ 12 (A + AT ) . Likewise, by the same argu-

ment, the second part is a skew-symmetric tensor and we denote: skewA ≡ 12 (A − AT ) . Thus, we have a representation: A = symA + skewA .

3.6.8.2

Polar Decomposition

A general second-order tensor F can always be represented as a multiplicative sum of another second-order tensor U or V and an orthogonal tensor R as: F = RU = VR Recalling that for any orthogonal tensor R, we have: RRT = RT R = I, it can easily be verified that: F = IF = (RRT )F = R(RT F) = RU. Similarly, F = FI = F(RT R) = (FRT )R = VR. We have clearly used the definitions: U = RT F and V = FRT . However, for a total of 12 independent variables for the decomposed tensors, we only have 9 equations due to the original tensor F. Thus, the decomposition is under-determined and hence, non-unique. In order to make the decomposition unique, we may impose three more symmetry constraints on U or V in the form: U = UT or V = VT for the determination of R. Thus, a general second-order tensor F can always be represented as a multiplicative sum of a symmetric second-order tensor U or V and an orthogonal tensor R as: F = RU = VR .

3.6.8.3

Spherical and Deviatoric Decomposition

A general second-order tensor E can always be represented as an additive sum of two tensors known as the spherical tensor sph E and the deviatoric tensor dev E as: E = sphE + devE . The

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73

equality can be easily verified by recalling the meaning of trace and double contraction along with the following definitions: sph E ≡ 13 (tr E)I = 13 (E : I)I = 13 Ekk gij gi ⊗ gj

(3.74)

dev E ≡ E − sph E = (Eij − Ekk gij )gi ⊗ gj

(3.75)

and

Clearly, sph E is a symmetric tensor. Fact 1: For any A, sph (sph A) = sph A. Proof. Recalling that tr I = I : I = 𝛿ii = 3, and applying the definitions, we have: sph (sph A) = 1 (tr A)sph(I) = 13 (tr A) 13 (tr I)I = 13 (tr A) 13 (tr I) I = sph A 3 ⏟⏟⏟ =1

Fact 2: For any A, sph (dev A) = 0.

Proof. Using Fact 1: sph (dev A) = sph (A − sph A) = sph A − sph (sph A) = sph A − sph A = 0

Fact 3: For any A and B, sph B : dev A = 0. Proof. Recalling that tr I = I : I = 𝛿ii = 3, and applying the definitions, we have: sph B : dev A = 1 (tr B)I : (A − 13 (tr A)I) = 13 (tr B)(tr A) − 19 (tr B)(tr A)(tr I) = 0 3 Fact 4: For any A and B, sph B : A = sph B : sph A = B : sph A.

Proof. From Fact 1 and Fact 3, we have: sph B : A = sph B : (sph A + dev A) = sph B : sph A. Finally, sph B : A = sph B : sph A = (sph B + dev B) : sph A = B : sph A

Fact 5: Show that for any A and B, dev B : A = B : dev A = dev B : dev A V. Using Fact 3, we have: dev B : A = dev B : (sph A + dev A) = dev B : dev A.

Fact 6: Show that for any A and B, B : A = dev B : dev A + sph B : sph A.

Proof. Using Fact 3, Fact 4 and Fact 5, we have: B : A = (sph B + dev B) : A = sph B : sph A + dev B : dev A.

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74

3.7

Calculus Tensor

Now that we have some idea about the geometry and algebraic operations of tensors along with vectors (tensors of order 1) and scalars (tensors of order 0), we conclude our treatment with the calculus of tensors. As a start, we define a field as a function over some continuous region or domain. A scalar field is defined when a scalar is assigned at each point of the region; similarly, a vector field is defined when a vector is assigned at each point of the region, and finally, a tensor field is obtained when the range is a tensor space. On the other hand, the domain itself could be a scalar, vector or tensor space. A scalar field over a vector is exemplified by, say, the temperature distribution over a three-dimensional body; a force distribution likewise defines a vector field over a two- or three-dimensional body. A tensor field over a tensor domain appears in the constitutive theory of mechanics where the second-order stress tensor is a function of the second-order strain tensor. In any case, for any or all of the above fields, we will be interested in the rate at which these fields change, leading us to differential calculus, or in the determination of the average or summing operation resulting in integral calculus. Furthermore, we need to recognize that our discussion must anticipate curved surfaces such as shells where, as will be revealed shortly, we will encounter derivatives other than the familiar partial derivatives – namely, covariant or absolute derivatives.

3.8

Partial Derivatives of Tensors

Partial derivatives arise practically in all three aspects of mechanics: for example, the balance principles, the equations of motion, the compatibility conditions and constitutive theory. In fact, partial derivatives with respect to scalars, vectors and tensors are needed in developing these theories.

3.8.1

Partial Derivatives of Base Vectors

Partial derivatives of the covariant base vectors, denoted as Gi ,j ≡ 𝜕𝜉𝜕i 𝜕𝜉 j , with respect to the coordinate variables, are also vectors, and hence have a vector representation: 2

Gi ,j ≡

𝜕2 = Γkij Gk , 𝜕𝜉 i 𝜕𝜉 j

i, j, k = 1, 2, 3

(3.76)

where we have introduced: {Γkij }, i, j, k = 1, 2, 3, known as the Christoffel symbols of the second kind. Remarks

r For Euclidean Cartesian coordinate system, the base vectors, Ei ’s are given by: Ei = A ii where A is a fixed rotation tensor and the ii ’s are fixed standard base vector; thus, Ei ,j = 0. In other words, the Christoffel symbols vanish identically: Γkij = 0 for all i, j, k.

r As an example, G1 ,2 ≡ 𝜕1 2 2 = Γ1 G1 + Γ2 G2 + Γ3 G3 . 12 12 12 𝜕𝜉 𝜕𝜉 r By definition, Christoffel symbols are symmetric in lower indices: Gi ,j ≡ Gj ,i ⇒ Γk = Γk , ij ji r From equations (2.16) and (3.76), : Γk = Gi ,j ∙ Gk ij

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75

r Again using equations (2.16) and (3.76), we also get the partial derivatives of the contravariant base vectors: Gi ,k ∙ Gj + Gi ∙ Gj ,k = 0

j

⇒ Gi ∙ Gj ,k = −Gi ,k ∙ Gj = −Γm G ∙ Gj = −Γik ik m

(3.77)

implying: j

Gj ,k = −Γik Gi

3.8.2

(3.78)

Remark

Although the Christoffel symbols appear as components of a third-order tensor, the do not form a tensor as they do not satisfy the transformation properties of a tensor. Example: Using two-dimensional polar coordinate system, we have: G1 = (cos 𝜃, sin 𝜃)T ⇒ and G2 = 1r (− sin 𝜃, cos 𝜃)T ⇒ G2 ,1 = r12 (sin 𝜃, − cos 𝜃)T ; similarly, G1 ,1 = (0, 0)T ,

G1 ,2 = (− sin 𝜃, cos 𝜃)T and G2 ,2 = − 1r (cos 𝜃, sin 𝜃)T . Thus, the Christoffel symbols are given as: Γ111 ≡ −G1 ,1 ∙ G1 = (0, 0) ∙ (cos 𝜃, sin 𝜃) = 0, Γ112 = Γ121 ≡ −G1 ,1 ∙ G2 = (0, 0) ∙ r(− sin 𝜃, cos 𝜃) = 0, and Γ122 ≡ −G1 ,2 ∙ G2 = (− sin 𝜃, cos 𝜃) ∙ r(− sin 𝜃, cos 𝜃) = r; similarly, Γ212 ≡ −G2 ,2 ∙ G1 = 1r , Γ222 ≡ −G2 ,2 ∙ G2 = 0 and Γ211 ≡ −G2 ,1 ∙ G1 = 0.

3.9

Covariant or Absolute Derivative

Referring to the definition given by equation (2.18), to successfully complete the difference ̄ − G(u), we implied that u and u + 𝜀u, ̄ belong to the same vector space – operation: G(u + 𝜀u) an assumption that is only valid for an Euclidean ‘flat’ space: we can translate or make a parallel transport (i.e. dragging a vector by keeping each component length of the vector intact) of ū to u and then perform the 𝜀-scaled addition because (i) vectors are invariant under translation in an Euclidean space and (ii) both u and ū belong to the same vector space, as discussed before. Similar assumptions have also been made for the Gateaux derivative of tensor fields as given by equation (2.7). But, these assumptions are not generally true for Riemannian ‘curved’ spaces: the tangent space, which serves locally as a linear vector space at each point on a curved space, is ̄ − G(u) operations different from point to point, and as such, makes both u + 𝜀ū and G(u + 𝜀u) generally invalid unless some connection is devised among the points in a neighborhood, that is, a parallel transport needs to be redefined. One such redefinition of parallelism is offered by the Levi–Civita connection: instead of preserving the coordinate components of a vector intact for parallel transport as in Euclidean Cartesian coordinate systems, here we require that another invariant property of parallel lines be retained: that a transversal subtends equal angles with a line that is being moved in parallel fashion as in Figure 3.3. For the curved spaces, a curve generalizes the transversal with the angle defined between the tangents to the curve and the vector being moved. Thus, while the Gateaux derivative is intimately related to tensor calculus in Cartesian coordinate systems of a Euclidean space, for curvilinear coordinate systems or curved spaces, the concept of the Gateaux derivative fails to preserve the tensor characteristic of a geometric object; for example, while a differentiation

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76

Figure 3.3 Parallel transport.

(gradient) of a scalar (tensor of order 0) field is a covariant vector field, the partial differentiation of a vector (tensor of order 1) field is not a tensor field. So, in the light of Levi–Civita connection, it is required to be further generalized to what is known as the covariant derivative – covariant in the sense that it satisfies that objects in geometry such as vectors and tensors should remain so after derivative operations, independent of their description in a particular coordinate system. In other words, the covariant derivative is the generalization of the Gateaux directional derivative of the Euclidean Cartesian coordinate systems to the curvilinear coordinate systems, or the curved spaces where v(x + 𝜌̄x) and v(x) of the definition do not belong to the same vector (tangent) space, and hence the difference operation does not have a meaning. In fact, the vector, itself, does not have a meaning other than as a tangent operation to an affine curve. Thus, the generalization is effected through the choice of affinely parameterized unique curves on the curved surface; this makes the definition of the covariant derivative path-dependent. Finally, in the analysis, by describing the vectors and tensors in coordinate-independent absolute form, we can effectively avoid use of the derivatives of the base vectors and stay with the familiar partial derivatives of a functional or vector or tensor field, since the covariant derivatives of the base vectors are identically zero. Here is the coordinate independent definition of the covariant derivative:

r The covariant derivatives, denoted by ∇x T(x) ∙ x̄ , of a vector or tensor field on a curved space or manifold are given by: ( ) 𝜕 𝜕 ̄ ∇x T(x) ∙ x̄ ≡ T x ≡ , x̄ ≡ 𝜕𝜆 𝜕𝜉 d = lim {T(x + 𝜀̄x)} = lim 𝜀→0 d𝜀 𝜀→0

r

{T(𝜆0 + 𝜀𝜉) transported along

an affine curve to 𝜆0

− T(𝜆0 )}

(3.79)

𝜀

where ∇x T(x) ∙ x̄ is the covariant derivative of T(x) at x along the vector direction x̄ ; note that T(x) could be a vector or a tensor field. ̄ of a functional on a curved space or manifold are The covariant derivatives, ∇u G(u, u), given by:

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77

Let G(u(x)) be a functional or operator defined for an argument function u(x) = at any x ∈ ̄ ∈ Rn be another function at x. Then, the covariant derivative, denoted as Rm . Now, let u(x) ̄ or DG(u) ∙ u, ̄ of G at u along the direction ū is defined as: G,u (u, u), ∇u (u) ∙ ū ≡

[

d ̄ G(u + 𝜀u) d𝜀

] 𝜀=0

(3.80)

However, as a practical scheme for computing the covariant derivatives of the components of a vector field (or tensor field), all we need to do is to include a correction in the form of the partial derivatives of the base vectors (or base tensors) as they are the tangents (or the tensor product of these) to the coordinate curves on the manifold. Thus, in order to preserve the tensor characteristic of a tensor, a translation operation must respect the change of basis rule reflected in the derivative of the base vectors, as in equation (3.76) and (3.78), through introduction of the Christoffel symbols.

3.9.1

Covariant or Absolute Derivatives of the Components of Vectors

With the definition of the partial derivatives of the covariant base vectors, {Gi }, as given by equation (3.76), given a vector function a(𝜉 1 , 𝜉 2 , ⋯) = ai (𝜉 1 , 𝜉 2 , ⋯) Gi (𝜉 1 , 𝜉 2 , ⋯) on a curved space, the absolute or covariant derivative of the contravariant components of a vector, denoted by ai |j , is given as: ai |j ≡

ai ,j + am Γimj ⏟⏟⏟ ⏟⏟⏟ not a tensor

(3.81)

not a tensor

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ whole expression is a tensor

Thus: ) ( a|j = ai ,j Gi + ai Gi ,j = ai ,j Gi + ai Γkij Gk = ai ,j +am Γimj Gi ≡ ai ||j Gi

(3.82)

Similarly, expressing a(𝜉 1 , 𝜉 2 , ⋯) = ai (𝜉 1 , 𝜉 2 , ⋯) Gi (𝜉 1 , 𝜉 2 , ⋯), and using equation (3.77), the absolute or covariant derivative of the covariant components of a vector is denoted by ai |j and given as: ai ||j ≡ ai ,j − ak Γkij

(3.83)

There are other alternative notations for the covariant derivative: Dj ai ≡ ai ; j ≡ ai |j , and so on. { } k Similarly, the Christoffel symbols of the second kind is alternatively written as: ≡ Γkij . i j The covariant derivatives of higher-order tensors are similarly given as: jk… | jk… j ji… jk… jk… k i i Tqr… | ≡ Tqr… ,m + Γim Tik… qr… + Γim Tqr… + − Γqm Tir… − Γrm Tqi… − |m

(3.84)

Note that in equation (3.84), for each contravariant index a Christoffel symbol term is added, while for each covariant index, a Christoffel symbol term is subtracted. Finally, note that for the

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differential geometry of thin surface-like structures such as shells, we can develop expressions without explicitly resorting to the Christoffel symbols.

3.9.2

Covariant Derivative of the Base Vectors and the Metric Tensor

Generalizing on the covariant derivatives of vector components as in equation (3.81), for, Tj , the contravariant pseudo-vector components of a tensor, T ≡ Tj ⊗ gj , we can express the covariant derivatives as: Tj ||k ≡ Tj ,k + Tm Γkm j

(3.85)

Similarly, for, Tj , the covariant pseudo-vector components of a tensor, T ≡ Tj ⊗ gj , and generalizing equation (3.83), we have the covariant derivatives as: | Tj | ≡ Tj ,k − Tm Γm jk |k

(3.86)

Now, applying equation (3.86) to the base vectors themselves and noting the partial derivatives of the base vectors as in equation (3.76), we get the absolute or covariant derivative of the covariant base vectors, {Gi }, we get Gi ||j = 0. Thus: Gi ||j = Gi ,j − Gm Γm = Γkij Gk − Gm Γm =0 ij ij

(3.87)

Now, noting the definition of the metric tensor, the covariant derivatives, Gij ||k , of the components, Gij , of the metric tensor are identically zero; so, using equation (3.87), we have: Gij ||k = Gi ||k ∙ Gj + Gi ∙ Gj ||k = 0

(3.88)

In summary:

r The covariant derivatives of both the base vectors and the metric tensor are identically zero; that is, under covariant derivative, they both act like constants.

3.10

Riemann–Christoffel Tensor: Ordered Differentiation

Using the covariant derivatives of a vector, as in equations (3.83) and (3.84), and taking again the covariant derivative of the covariant derivative of the covariant component of a vector, we have: Di Dj ak = = =

( ) Di ak ,j − Γm am kj ( ) ( ) ( ) ak ,j − Γm a , − Γnki an ,j − Γm a − Γnji ak ,n − Γm a nj m kj m i kn m ak ,ji − Γm a , − Γm , a − Γnki an ,j + Γnki Γm a nj m kj m i kj i m n n m − Γji ak ,n + Γji Γkn am

(3.89)

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79

where we have used Di as the covariant derivative with respect to the i coordinate direction. Now, by permutation of i and j in equation (3.89), we get: { } n m m m m Di Dj ak − Dj Di ak = Γnki Γm nj − Γkj Γni + Γki ,j − Γkj ,i am ≡ Rijk am ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

(3.90)

Δ

=Rm ijk

Δ

m m where we have defined, Rm = Γnki Γm − Γnkj Γm ni + Γki ,j − Γkj ,i , the components of a fourth-order nj ijk

tensor known as the Riemann–Christoffel curvature tensor. Remarks

r It can be shown that the Rm ’s are the components of a tensor, that is, they obey the transformation ijk

r r r r

law for a tensor. Since the Christoffel symbols depend solely on the metric tensor, the Riemann–Christoffel tensor belongs to the class of fundamental tensors. Since the Christoffel symbols, Γkij , identically vanish in Cartesian coordinate systems, the Riemann–Christoffel tensor is identically zero for all Cartesian coordinate systems in Euclidean space. Obviously, in Riemannian space, the Riemann–Christoffel tensor does not vanish, so the order of differentiation cannot be interchanged. From equation (3.90), the difference between the covariant derivatives of the covariant components of a second-order tensor can be given as: a + Rm a Di Dj akl − Dj Di akl = Rm kji ml lji km

3.11

(3.91)

Partial (PD) and Covariant (C.D.) Derivatives of Tensors

Using the derivatives of the base vectors as in equation (3.76), the partial derivatives of a tensor, say, a second-order tensor, T ≡ T ij gi ⊗ gj , can be expressed as: T,k ≡

𝜕T = T ij ,k gi ⊗ gj + T ij gi ,k ⊗gj + T ij gi ⊗ gj ,k 𝜕𝜉 k

= T ij ,k gi ⊗ gj + T ij Γm g ⊗ gj + T ij gi ⊗ Γm g ik m jk m ) ( j = T ij ,k +T mj Γimk + T im Γmk gi ⊗ gj = T ij ||k gi ⊗ gj

(3.92)

where T ij ||k ≡ T ij ,k +T mj Γimk + T im Γmk j

Now, because the covariant derivative of the base vectors are zero as given by equation (3.87), the covariant derivatives, T|k , of the tensor, T, are identical to the partial derivatives of the tensor: ) ( j T,k = T|k = T ij ||k gi ⊗ gj = T ij ,k +T mj Γimk + T im Γmk gi ⊗ gj

(3.93)

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Thus, let us note that: The partial derivatives of a tensor are identical to the corresponding covariant derivatives of the tensor (equation 3.93) but the partial derivatives of the components of a tensor are different from the corresponding covariant derivatives of the components of the tensor (equation 3.92).

3.12

Partial Derivatives of Scalar Functions of Tensors

Now, we may consider a scalar (tensor of order zero) valued function f (A) of a tensor. Then, the partial derivative of f with respect to second-order tensor, A, is a second-order tensor defined as:

f ,A ≡

𝜕f 𝜕f i 𝜕f gi ⊗ gj = g ⊗ gj = 𝜕A 𝜕Aij 𝜕Aij

(3.94)

A similar expression can be obtained in terms of the mixed components of A. In any event, the definition ensures that the partial derivative tensor is invariant, irrespective of the choice of components. The partial derivatives of various scalar functions – for example, the trace, one of the invariants, of a second-order tensor – are routinely needed in analysis, as follows. Fact 1: For any A ≡ Aj ⊗ gj , (trA),A = I . j

Proof. Let us recall that we have: trA = A : I = Ai ⊗ gi : gj ⊗ gj = gij (Ai ∙ gj ) = gij Aij = Aj . j

Then, (trA),A =

𝜕Aj p 𝜕Aq

j q

gp ⊗ gq = 𝛿p 𝛿j gp ⊗ gq = gj ⊗ gj = I.

Fact 2: For any A ≡ Aj ⊗ gj , (trA)2 ,A = 2(trA)I . Proof. Clearly, by Fact 1 and the chain rule: (trA)2 ,A = 2(trA)I. Fact 3: For any A ≡ Aj ⊗ gj , tr(A2 ),A = 2AT . j

j

j

Proof. Recalling that tr(A2 ) = tr(Aij gi ⊗ gj ∙ Akl gk ⊗ gl ) = tr(Aij Akl 𝛿k gi ⊗ gl ) = Aij Al 𝛿il = Aij Ai , we have: tr(A2 ),A =

j 𝜕(Aij Ai ) p 𝜕Aq

q j

j q

q

gp ⊗ gq = 𝛿pi 𝛿j Ai gp ⊗ gq + Aij 𝛿p 𝛿i gp ⊗ gq = 2Ap gp ⊗ gq = 2Aq ⊗

gq . Now, noting the definition: A ≡ Aj ⊗ gj , we clearly have: tr(A2 ),A = 2AT .

Exercise: Show that for any A, tr(An ),A = nAT

Tensors

3.13

81

Partial Derivatives of Tensor Functions of Tensors

Now, the above definition can be extended to the second-order tensor with the coefficients being functions of another tensor. Let B = Bij (A) gi ⊗ gj be the tensor-valued tensor function. Then, the partial derivative of B with respect to A is given by a fourth-order tensor: B,A ≡

𝜕B 𝜕Bij g ⊗ gj ⊗ gk ⊗ g l = 𝜕A 𝜕Akl i

(3.95)

𝜕A = 𝜕A gi ⊗ gj ⊗ gk ⊗ gl = 𝛿 ik 𝛿 jl gi ⊗ gj ⊗ gk ⊗ gl = gi ⊗ gj ⊗ With B ≡ A, we have: A,A ≡ 𝜕A 𝜕A kl gi ⊗ gj . Now, recalling from equation (3.46) a fourth-order identity tensor from equation that ij

4

is defined by only the base vectors: Ĩ ≡ gi ⊗ gj ⊗ gi ⊗ gj , we may have a compact symbolical definition: A,A ≡

𝜕A 𝜕A

4

= Ĩ . A repeat partial differentiation of a scalar valued tensor function will

result in a fourth-order tensor: f ,A⊗A ≡ = where Bijpq ≡

𝜕2f 𝜕 = 𝜕A 𝜕A 𝜕A Bijpq gi

(

𝜕f 𝜕A

) =

𝜕2f g ⊗ gj ⊗ gp ⊗ gq 𝜕Aij 𝜕Apq i

(3.96)

⊗ gj ⊗ gp ⊗ gq

𝜕2 f . As before, similar expressions can be written in terms of the covariant and 𝜕Aij 𝜕Apq

mixed components or their combinations.

3.14

Partial Derivatives of Parametric Functions of Tensors

For a scalar function of a tensor with the tensor having parametric dependence, the derivative of the function with respect to the parameter is obtained by a double contraction, equivalently describing the usual chain rule of differentiation. Thus, with f = f (A(t)), we have: ∙

f ≡

∙ 𝜕f = f ,A : A 𝜕t

(3.97)

Fact 1: From the definition of norm of a tensor, we have ||A||2 ≡ A : A = tr(AAT ). Then, for a symmetric tensor A, the partial derivative of the norm with respect to A is given by: ‖A‖ ,A ≡

𝜕 ‖A‖ A = 𝜕A ‖A‖

(3.98)

Proof. ‖A‖2 = tr(AAT ) = tr(A2 ). Now, noting from invariants study that (trA2 ),A ≡ 2A, and taking derivatives of both sides with respect to A, we get: 2||A||||A||,A = 2A. Now, diving both sides by 2||A|| completes the proof.

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3.15

Differential Operators

The differential operators are a generalization of derivatives of scalar functions of scalars (zeroorder tensors) to higher-order arguments. Interestingly, just like the vector multiplications that come under three different flavors – dot product (resulting in scalar), cross product (resulting in vector) and tensor product (resulting in tensor) – the vector differentiations manifest themselves as either divergence (resulting in a scalar field), curl (resulting in a vector field) or gradients (resulting in a tensor field). These, in turn, are conveniently employed through the introduction of the so-called Nabla operator.

3.15.1

Del or Nabla Operator: ∇

From the differential geometry of surfaces, let 𝜙(𝜉 1 , 𝜉 2 , 𝜉 3 ) = const define a surface. For example, a spherical surface with (𝜉 1 ≡ r, 𝜉 2 ≡ 𝜃, 𝜉 3 ≡ 𝜑) such that x1 = 𝜉 1 sin 𝜉 3 cos 𝜉 2 , x2 = 𝜉 1 sin 𝜉 3 sin 𝜉 2 , x3 = 𝜉 1 cos 𝜉 3 may be defined by 𝜙(𝜉 1 , 𝜉 2 , 𝜉 3 ) = 𝜉 1 = r = const. Then, if we 𝜕𝜙 i move a bit d𝜉 on the surface, the change in d𝜙 = 𝜕𝜉 i d𝜉 = 0. We define a pseudo-vector operator: ∇(∙) ≡ Gi

𝜕(∙) 𝜕𝜉 i

(3.99)

With this definition, the above change in d𝜙 may be effectively represented by a scalar product of 𝜕𝜙 i a vector, d𝜉 = d𝜉 k Gk and a pseudo-vector, ∇𝜙 = 𝜕𝜉 i G resulting in d𝜙 = ∇𝜙 ∙ d𝜉 = 0 . Thus, if d𝜉 is a change on the surface, more precisely, on the tangent plane at the point on the surface, then ∇𝜙 must be a vector along the normal to the surface. Thus, ∇, the pseudo-vector operator changes a scalar function on to a vector function along the normal to the surface. ∇𝜑 is known as the gradient of the scalar function 𝜙. For the spherical surface example above, ∇𝜙 = G1 is the (radial) base vector normal to the so-called 𝜉 1 = const surface. Example: Continuing with the two-dimensional polar coordinate system with G1 = G1 and 𝜕 𝜕 𝜕 𝜕 + G2 𝜕𝜃 = G1 𝜕r + r12 G2 𝜕𝜃 , implying that in polar coorG2 = r12 G2 , we have: ∇ ≡ Gi 𝜕𝜉𝜕 i = G1 𝜕r dinates: ̂r ∇=G

1̂ 𝜕 𝜕 + G 𝜕r r 𝜃 𝜕𝜃

(3.100)

̂r , G ̂𝜃 are the unit covariant base vectors. where G

3.16

Gradient Operator: GRAD(∙) or 𝛁(∙)

For scalar-valued vector functions, the ∇ operator is a logical generalization of the definition of differentiation of a scalar-valued scalar function. However, when we try to extend this to a vector- and tensor-valued function, or to other forms of differentiation (such as divergence, to be introduced shortly), then the ∇-operator, with column vectors in mind, is a bit confusing because it involves the transposed tensors. For this reason, we will introduce definitions that are more uniform for all orders of tensors.

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83

Gradient of Scalar Functions: GRAD 𝜙

3.16.1

The gradient of a scalar function 𝜙(𝜉 i ), i = 1, 2, … is a vector function, denoted as GRAD 𝜙, defined by: grad 𝜙 ≡

𝜕𝜙 i G = ∇𝜙 𝜕𝜉 i

(3.101)

Thus, the gradient of a zero-order tensor (scalar) function results in a first-order tensor (vector) function. Also, by definition, the GRAD and ∇ operators turn out to be the same. Remark

r In

nonlinear finite element analysis, which involves numerical minimization of an energy expression, we often need to determine the second derivative (i.e. the Hessian) of this scalar entity. For numerical efficiency, we further need to know whether the Hessian tensor is symmetric. For this information, we include the following important lemma.

3.16.2

Hessian Symmetry Lemma

The Hessian tensor obtained by taking the covariant derivative of the covariant components of the gradient vector of a scalar function is always symmetric. 𝜕𝜙 i Proof. Let ai = 𝜕𝜉 i ’s be the covariant components of the gradient vector of a scalar function, 𝜙(𝜉 ). Then, by equation (3.83), aij , the components of a Hessian obtained by the covariant derivative of ai is given by: aij = ai ||j ≡ ai ,j − ak Γkij . Now, by symmetry of the Christoffel symbols in lowercase

indices, that is, Γkij = Γkji , and noting the interchangeability of the partial derivatives, that is,

𝜙 𝜕 𝜙 k k ai ,j = 𝜕𝜉𝜕 i 𝜕𝜉 j = 𝜕𝜉 j 𝜕𝜉 i = aj ,i we get: aij − aji = ai ,j − ak Γij − aj ,i + ak Γji = 0, that is, noting that any tensor can be decomposed as a sum of a symmetric and an anti-symmetric tensor, the anti-symmetric part of the Hessian is identically zero, which completes the proof. 2

3.16.3

2

Gradient of Vector Functions: GRAD ψ

The gradient of a vector function 𝝍(𝜉 i ), i = 1, 2, … is a second-order tensor function, denoted as GRAD ψ, defined by: GRAD ψ ≡

𝜕ψ ⊗ Gj = ψ,j ⊗ Gj 𝜕𝜉 j

(3.102)

Now, noting that the covariant derivatives of the base vectors are zero by equation (3.87), we have: contravariant

GRAD ψ = ψ|j ⏟⏞⏟⏞⏟ ⏟⏟⏟ invariant

covariant

⏞⏞⏞ ⊗ Gj

(3.103)

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84

Thus, by the quotient rule, GRAD ψ is invariant and hence a tensor. We can write the componentwise description as: GRAD ψ = 𝜓i |j Gi ⊗ Gj . Finally, with the definition of the del operator as above, we have: ∇ψ = Gi ⊗ 𝜕𝜕𝜉ψi = Gi ⊗ ψ,i . Thus, comparing with GRAD ψ, we find that: GRAD ψ = ∇ψT .

3.16.4

Gradient of Tensor Functions: GRAD A

The gradient of a second-order tensor function, A(𝜉 i ), i = 1, 2, …, is a third-order tensor function, denoted as GRAD A, defined by: GRAD A ≡

𝜕A ⊗ Gj = A,j ⊗ Gj 𝜕𝜉 j

(3.104)

Now, noting that the covariant derivatives of the base vectors are zero by equation (3.87), we have: contravariant

GRAD A = A|j ⊗ ⏟⏞⏟⏞⏟ ⏟⏟⏟ invariant

⏞⏞⏞ Gj

(3.105)

covariant

Thus, by quotient rule, GRAD A is invariant and hence a tensor. We can write the componentwise description as: GRAD A = Aij |k Gi ⊗ Gj ⊗ Gk . Finally, comparing with the definition of del operator, we find that: GRAD A = ∇AT . Fact 2: For a scalar function 𝜑 and a vector function v: GRAD(𝜑v) = 𝜑 GRAD v + v ⊗ GRAD𝜑 ∇(𝜑v) = 𝜑 ∇v + v ⊗ ∇𝜑

(3.106)

Proof. GRAD(𝜑v) = (𝜑v),k ⊗ Gk = (𝜑,k v + 𝜑 v,k ) ⊗ Gk = v ⊗ 𝜑,k Gk + 𝜑 v,k ⊗ Gk = v ⊗ GRAD𝜑 + 𝜑 GRAD v

3.17

Divergence Operator: DIV or 𝛁∙

Because of the argument presented above as to the confusion related to the ∇ operator, we follow the gradient operator in introducing the definitions below that are consistent for tensors of order one and above.

3.17.1

Divergence of Vector Functions: DIV 𝜓

The divergence of a vector function ψ(𝜉 i ), i = 1, 2, … is a scalar or zero-order tensor function, denoted by DIV ψ, defined as: DIV ψ ≡ GRAD ψ: I

(3.107)

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85

Fact 1: For any vector function ψ: DIV ψ = ψ,j Gj ,

or, ∇ ∙ ψ = ψ,j Gj

(3.108)

Proof. From the definition of gradient and divergence of vector functions above and double contraction, we have: DIV ψ = GRAD ψ : I = (ψ,j ⊗ Gj ) : (Gi ⊗ Gi ) = (Gj ∙ Gi )(ψ,j ∙Gi ) = Gij (ψ,j ∙Gi ) = ψ,j ∙Gj

Fact 2: For any vector function ψ, and a scalar function, u, we have: DIV (ψu) = u DIV ψ + ψ ∙ GRAD u ∇ ∙ (ψu) = u (∇ ∙ ψ) + ψ ∙ ∇u

(3.109)

Proof. From equations (3.101) and (3.108), we have: DIV (uψ) = (uψ),j ∙Gj = u (ψ,j ∙Gj ) + ψ u,j ∙Gj = u (∇ ∙ ψ) + ψ ∙ u,j Gj = u (∇ ∙ ψ) + ψ ∙ ∇u

Fact 3: For any vector function ψ: DIV ψ = 𝜓 j |j

(3.110)

Proof. From the definition of the gradient above anda double contraction, and raising of indices by the contravariant components of the metric tensor, it follows that: DIV ψ = ψ,j ⊗ Gj : Gk ⊗ Gk = (Gi ∙ Gk )(𝜓i |j Gi ∙ Gk ) = Gjk (𝜓i |j 𝛿ki ) = Gjk 𝜓k |j = (Gjk 𝜓k )|j = 𝜓 j |j

Note that in the final inequality, we have used the fact that the covariant derivative of the components of the metric tensor is identically zero, that is, Gij ||k = 0. Let us also note here that the divergence operator reduces the order of the argument tensor by one. Finally, noting the definition of the ∇ operator, we have: DIV 𝝍 = ∇ ∙ 𝝍

(3.111)

Example: Continuing with the two-dimensional polar coordinate system, we first note that ̂ r, G ̂ 𝜃 as in equation (3.100), the nabla operator, ∇, written in terms of the unit base vectors, G k is not suitable for application of the Christoffel symbols, Γij ’s. So, representing a vector, v,

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86

̂r = Gr and G ̂ 𝜃 = rG𝜃 , we have, from ̂ i = vr G ̂ r + v𝜃 G ̂ 𝜃 , and using the relations: G as: v = vi G 𝜕v 1 𝜕v 𝜕v 1 𝜕v ̂1 ∙ ̂r ∙ + G ̂ ∙ ̂2 ∙ =G + rG with equation (3.100): ∇ ∙ v = G 𝜕r r 𝜃 𝜕𝜃 𝜕𝜉 1 𝜕𝜉 2 ) ( ̂1 ̂ 1 ∙ 𝜕v = vi ,1 G ̂ i + vi G ̂ i ,1 ∙ G G 𝜕𝜉 1

2 ̂ ̂ 2∙  = v1 ,1 G1 ∙ G1 + v1 G1 ,1 ∙ G1 + v2,1 G G 1 + v2 (rG ),1 ∙ G1 ( ) 2 = v1,1 + v1 (−Γ111 ) + v2 G2 + rG2 ,1 ∙ G1 = v1 ,1 +v2 ( G G2∙  1 + r Γ11 ) = vr ,r

and

( ) ̂2 ̂ i + vi G ̂ i ,2 ∙ G ̂2 ∙ 1 𝜕v = 1 vi ,2 G G r 𝜕𝜉 2 r ) (  1 1 1 1 1 = v1,2 G1∙ G2 + v1 G1 ,2 ∙ G2 + v2 ,2 rG2 ∙ G2 + v2 (rG2 ),2 ∙ G2 r r r r  r ( ) ( ) 1 1 1 1 1 1 1 − Γ122 + v2 ,2 + v2 Γ222 = v2 ,2 + v1 − (−r) = v1 r r r r r r r ) 1( v , +v = r 𝜃𝜃 r

Thus, for the divergence of a vector in polar coordinates, we finally have: 1 1 1 1 𝜕 1 𝜕v𝜃 ∇ ∙ v = vr ,r + (v𝜃,𝜃 ) = vr ,r + vr + v𝜃 ,𝜃 = (rvr ) + r r r r 𝜕r r 𝜕𝜃

3.17.2

(3.112)

Divergence of Tensor Functions: DIV A

The divergence of a tensor function A(𝜉 i ), i = 1, 2, … is a vector or first-order tensor function, denoted as DIV A, defined by: DIV A ≡ GRAD A : I

(3.113)

Fact 3: For any tensor function A, DIV A = A,j gj . Proof. From the definitions of gradient and divergence of tensor functions above and using double contraction, we have: DIV A = GRAD A : I = (A,j ⊗ Gj ) : (Gi ⊗ Gi ) = (Gj ∙ Gi )(A,j ∙Gi ) = Gij (A,j ∙Gi ) = A,j ∙Gj

(3.114)

Fact 4: For any tensor function A, div A = Aj |j , where Aj are the contravariant pseudovector components of the tensor obtained by vector-like notation for the second-order tensor: A = Aj ⊗ Gj

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87

Proof. From the definition of gradient above and using double contraction, and raising of indices by the contravariant components of the metric tensor, it follows that: DIV A = A,j ⊗ Gj : Gk ⊗ Gk = (Gi ∙ Gk )(Ai |j Gi ∙ Gk ) = Gjk (Ai |j 𝛿ki ) = Gjk Ak |j = (Gjk Ak )|j = Aj |j

(3.115)

Note that in the final inequality, we have used the fact that the covariant derivative of the components of the metric tensor is identically zero, that is, Gij ||k = 0. Let us also note that the divergence operator reduces the order of the argument tensor by one. Finally, noting the definition of the ∇ operator, we have: DIV A = ∇ AT . Fact 5: For a vector function. a, and a second-order tensor function, B: DIV(aB) = B : GRAD a + a ∙ DIV B

(3.116)

Proof. Using Fact 1, we have: DIV(aB) = (a ∙ B),j ∙ Gj = (a,j ∙ B) ∙ Gj + (a ∙ B,j ) ∙ Gj . Now, from Fact 3, we get (a ∙ B,j ) ∙ Gj = a ∙ (B,j ∙ Gj ) = a DIV B. Similarly with the definition of double contraction, we have: (a,j ∙ B) ∙ Gj = a,j ∙ B ∙ Gj = a,j ∙ Bi ⊗ Gi ∙ Gj = (a,j ∙ Bi )(Gi ∙ Gj ) = (Bi ∙ a,j )(Gi ∙ Gj ) = (Bi ⊗ Gi ) : (a,j ⊗ Gj ) = B : GRAD a

3.18

Integral Transforms: Green–Gauss Theorems

For virtual integral transforms in the finite element analysis, we will occasionally encounter scalar, vector- and tensor-valued functions or fields, or their derivatives such as gradients, divergence or curls, defined over a body, Ω, or its boundary, Γ. Thus, there will be need for relating their integrals over the body with those over the corresponding boundary. These relationships go by the general name of Green–Gauss theorems. Integral transforms range from where the body is onedimensional with two end points as boundary, and the functions are scalar-valued to cases where they are, correspondingly volume, surface and tensor-valued; then the result for the simplest case is familiarly known as the integration by parts.

3.18.1

Integration by Parts: One-dimensional Scalar-valued Functions

Let u(x), ū (x) ∈ C1 [a, b] be one-dimensional scalar-valued fields; then, by the chain rule for the dx = u,x dx, and dū = 𝜕𝜕xū dx = differential: d(u ū ) = u dū + ū du; we also note that: du = 𝜕u 𝜕x b

b

b

b

ū ,x dx. Now, integrating: ∫a d(u ū ) = ∫a u dū + ∫a ū du implying that u ū |ba = ∫a u dū + b ∫a ū

du, where u ū |ba ≡ u(b) ū (b) − u(a) ū (a). Thus, finally: b

∫a b

∫a

u dū = u ū |ba − u ū ,x dx = u

b

∫a

ū |ba

ū du b

−

∫a

(3.117) ū u,x dx

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3.18.2

Two- or Three-dimensional Scalar-valued Functions

Let f = u ū , where u(xi ), ū (xi ) ∈ C1 [Ω], i = 1, 2, 3 be scalar-valued fields on a volume, Ω, and boundary surface, Γ, with Ω as the closure; xi ’s are the covariant components of the coordinate vectors, x ∈ Ω, and let ni be the covariant components of the normal field, n, on the surface such that: n = ni Gi . Then, it can be easily shown by generalization of equation (3.117) that:

∫

f ,i dΩ =

3.18.3

∫

f ni dΓ ⇒

∫

Γ

Ω

u ū ,i dΩ =

∫

u ū ni dΓ −

Γ

Ω

∫

ū u,i dΩ

(3.118)

Ω

Vector-valued Functions: Divergence or Gauss Theorem

Let f ∈ C1 (Ω) be a vector-valued field on a volume, Ω, and boundary surface, Γ, with Ω as the closure, as shown in Figure 3.4; xi ’s are the covariant components of the coordinate vectors, and let ni be the covariant components of the normal field, n, on the surface such that: n = ni Gi . Then, by generalizing the first expression of equation (3.118), we get: ∫ (fi ,i dΩ = ∫ fi ni dΓ, Ω

Γ

implying:

DIV f dΩ =

∫

∫

(f ∙ n) dΓ,

Γ

Ω

or,

∫

(∇ ∙ f) dΩ =

∫

(f ∙ n) dΓ

(3.119)

Γ

Ω

Now, if f = ū u, where u is a vector field and ū is a scalar field, from equations (3.109) and (3.119), we get: ∫ DIV (̄u u) dΩ = ∫ (̄u DIV u + u ∙ GRAD ū ) dΩ = ∫ (̄u u) ∙ n dΓ, implying: Ω

∫

(DIV u) ū dΩ =

∫ Ω

∫

ū (u ∙ n) dΓ −

Γ

Ω

3.18.4

Γ

Ω

(∇ ∙ u) ū dΩ =

∫ Γ

∫

(u ∙ GRAD ū ) dΩ

Ω

ū (u ∙ n) dΓ −

∫

(3.120) (u ∙ ∇̄u) dΩ

Ω

Tensor-valued Functions: Green–Gauss Theorem

Let T ∈ C1 (Ω) be a second-order tensor-valued field on a volume, Ω, and boundary surface, Γ, with Ω as the closure; xi ’s are the covariant components of the coordinate vectors, and let ni be

Figure 3.4 Green–Gauss theorem.

Tensors

89

the covariant components of the normal field, n, on the surface such that: n = ni Gi . Then, we have:

∫

DIV T dΩ =

∫

(T n) dΓ,

or,

∫

Γ

Ω

(∇ ∙ T) dΩ =

∫

(T n) dΓ

(3.121)

Γ

Ω

Proof. Let h be a constant vector; then, applying the divergence theorem for vectors as in equation (3.119), we have:

∫

h ∙ (Tn) dΓ =

Γ

∫

n ∙ (TT h) dΓ =

Γ

∫

(∇ ∙ (TT h)) dΩ =

Ω

∫

h ∙ (∇ ∙ T) dΩ;

Ω

noting that the statement is true, for any constant vector, completes the proof. Now, noting from equation (3.115) that for any tensor function T, DIV T = Tj |j , where Tj are the contravariant vector components of the tensor obtained by vector-like notation for the second-order tensor: T = Tj ⊗ Gj , we get from equation (3.121):

∫

Tj |j dΩ =

(Tj nj ) dΓ

(3.122)

Γ

Ω

3.18.5

∫

An Important Lemma

Let T ∈ C1 (Ω) be a tensor field and ū ∈ C1 (Ω) be a vector function, then we have:

∫

DIV T ∙ ū dΩ =

Tn ∙ ū dΓ −

Γ

Ω

∫

∫

(∇ ∙ T) ∙ ū dΩ =

̄ dΩ (T:GRAD u)

Ω

∫

Tn ∙ ū dΓ −

Γ

Ω

∫ ∫

(3.123) ̄ dΩ (T:∇u)

Ω

Proof. Integrating the expression in equation (3.116) and applying divergence or Gauss relation of equation (3.119), and noting that, as a scalar by simple contraction of a second-order tensor ̄ ∙ n = Tn ∙ ū we get: by vectors from both sides: uT ∫

DIV T ∙ ū dΩ =

Ω

∫

̄ dΩ − DIV(uT)

Ω

=

∫ Γ

∫

̄ dΩ = (T:GRAD u)

Ω

Tn ∙ ū dΓ −

∫ Ω

̄ dΩ (T:GRAD u)

∫ Γ

̄ ∙ n dΓ − uT

∫ Ω

̄ dΩ (T:GRAD u)

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3.19

Where We Would Like to Go

The treatment of the geometry and mechanics of structures such as plates and shells become much more manageable by application of tensor analyses. There remain various other tensor topics such as Ricci or curvature tensor, Bianchi identity, and so on, that we have not included here, but they are of considerable importance for the geometric analysis of curved structures. Thus, for these and general geometric properties of curved structures such as shells, we must later look into the differential geometry of surfaces (Chapter 6). Finally, the finite rotation tensor that inherently belongs to a curved space poses special problems in both the theoretical and computational treatment of geometrically (i.e. with large displacement and rotation) nonlinear structures; these, in turn, need more detailed inspection of the nature of rotational tensor for which we must turn to: the rotation tensor (Chapter 4).

4 Rotation Tensor 4.1 4.1.1

Introduction What We Should Already Know

Configuration space, vector, matrices, determinants, traces, tensors, covariant base vectors, contravariant base vectors, linear transformation, orthogonal transformation, metric tensor, Cayley– Hamilton theorem, characteristic polynomial, eigenvalues, eigenvectors, skew-symmetric tensor, axial vectors, matrix Lie group, manifold, vector triple product, Gateaux derivative, covariant derivative.

4.1.2

Why We Are Here

The main goal here is to present a definition for rigid body rotations for use in structural and continuum mechanics such as finite rotations in geometrically nonlinear beam or shell analysis or polar decomposition of the deformation gradient. As an example of rigid body rotation in structural mechanics, we may cite finite deformations of a structural element such as a thin beam, where the cross-sections perpendicular to the centroidal curve of the beam are often modeled as rigid. This simplifies the configuration space of a beam to be characterized by the displacements (a translational movement) of the reference centroidal curve coupled with rigid body rotations of the corresponding cross-sections. The displacement and the rotation together at a point define what is known as the generalized displacement in computational mechanics. We present various ways that a rotation can be mathematically realized and represented; these offer various comparative possibilities for finite rotation parameterizations suitable for subsequent numerical evaluations in structural mechanics. As we shall see, Rodrigues parametrization ingeniously represents a rotation matrix with only three parameters; however, in this process, we have a singularity as the trace of the rotation matrix reaches −1. If we need to represent the rotation matrix in an unconditionally non-singular fashion, we have Euler–Rodrigues parametrization, but a vectorial parametrization of the rotation matrix is most important for our numerical purposes in the sense that the number of parameters (namely, 3) is the same as the usual rotational degrees of freedom of the structural elements such as beams, plates and shells, for computational schemes such as the finite element method. Although it is singular at some rotation, for most structural problems involving moderate finite rotations, it suffices. In the event that it is not, a numerical strategy based on a user-transparent combination of four-parameter parameterization – such

Computation of Nonlinear Structures: Extremely Large Elements for Frames, Plates and Shells, First Edition. Debabrata Ray. © 2016 John Wiley & Sons, Ltd. Published 2016 by John Wiley & Sons, Ltd.

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92

as Euler–Rodrigues or Hamilton–Rodrigues quaternions, in the background with the threeparameter Rodrigues vector – will be able to make rotation representation completely singularity-free.

4.1.3

How We Get It

Rotation is tractably associated with and recognized as an orthogonal transformation; in fact, as indicated in our tensor analysis, it is a linear transformation, that is, a tensor. As such, we get to discuss the tensorial description of a rotation tensor. This allows us to identify a rotation matrix with the components of a corresponding rotation tensor. We also determine that a pure rotation has the spherical displacement property. In other words, while a translational displacement belongs to a Euclidean space, the rotation travels on a Riemannian curved space: on the 3D surface of a 4D sphere. Realization of this property will eventually help us to devise a numerical scheme for finite rotational motions of structural elements. In order to help us with the parameterization of rotation for subsequent computational efficiency, we will look into the eigenproperties of a rotation matrix. Finally, we recognize rotation as forming a non-commutative group – that is, a matrix Lie group – and derive both algebraic and geometric insights into rotational transformation. Remark

r Unless otherwise mentioned, we will restrict our attention to the more interesting physical three-dimensional (3D) space for our purposes.

4.1.4

Orthogonal Transformation and the Rotation Tensor

If we consider the most general motion of a rigid body, it can be broken down into a translation and a rotation component. The two together, known as the generalized displacement, as indicated above, are also referred to as the screw motion in theoretical kinematics. For the motion of a rigid body, generalized displacement is an invariant distance-preserving (i.e. distance measured between two points in the body) and hence an angle-(measured between two lines in the body)preserving transformation. Now, in the absence of any translational displacement, a point in the configuration space (i.e. the space that contains the location of any point on the rigid body at all times of the motion) is completely characterized by only rotational transformation. In other words, as shown in Figure 4.1, if a point P in the initial state with coordinates given by the vector

Figure 4.1 Rigid body rotation.

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93

X, measured by a system fixed in the initial state of a rigid body, is rotated to a point p in the final state with coordinates given by the vector x measured by the same system in the initial state, we may represent the rotation of the vector by: x = RX

(4.1)

where R is the rotational transformation. Fact 1: Rotational transformation, R, is an orthogonal transformation, that is, RT R = R RT = I or R−1 = RT .

Proof. To guarantee that equation (4.1) represents a rigid transformation, the Euclidean distance between any two points on the body, with coordinates X and Y that are rotated to x and y, respectively, by R, must remain the same. In other words, |x − y| = |R(X − Y)| = 1

1

{(X − Y)T (RT R)(X − Y)} 2 must be same as |X − Y| = {(X − Y)T (X − Y)} 2 . The only way this is satisfied is when RT R = I ⇔ RT = R−1

(4.2)

that is, the rotational motion of a rigid body is completely characterized mathematically by an orthogonal transformation.

Fact 2: The determinant of an orthogonal transformation, R, is ±1. Proof. 1 = det(I) = det(RT R) = det2 (R) ⇒ det(R) = ±1 We now introduce two definitions. Definitions

r Orthogonal tensors with determinant +1 are called the direct or proper rotation tensors, r Orthogonal tensors with determinant −1 are called the indirect rotation tensors, or reflection tensors. Note that although reflections satisfy the rigid transformation constraint, only direct rotations are utilized to fix the position of a rigid body. Thus, from now on, any reference to a rotation tensor will mean a direct rotation tensor, that is, an orthogonal tensor with the determinant given by +1.

Fact 3: Rotational transformation, R, is linear, that is, R is a tensor. Proof. A pure rotational operation, R, on a rigid body preserves its size, shape and orientation. Mathematically expressed, it preserves dot product and cross product of any two vectors on a body, as they represent length, angle and orientation, and hence the determinant of a matrix. To wit, let vectors a,b,̄a,b̄ be such that ā = R a and b̄ = R b. Now, Ra ∙ Rb = a ∙ RT Rb = a ∙ b; so we need the following two corollaries to complete the proof:

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94

Corollary 1: For any matrix, A, and any three vectors, b1 ,b2 ,b3 , we have: Ab1 × Ab2 ∙ Ab3 = (det A)(b1 × b2 ∙ b3 )

(4.3)

Proof. Let the three columns of a 3 × 3 matrix, B ≡ ( b1 b2 b3 ) whose determinant is clearly given by a triple product: det B = b1 × b2 ∙ b3 . Now, det(AB) = det A det B = (det A)(b1 × b2 ∙ b3 ). Again, det(AB) = det(A( b1 b2 b3 )) = det( Ab1 Ab2 Ab3 ) = Ab1 × Ab2 ∙ Ab3 . The equality of the two previous expressions completes the proof. Corollary 2: For any matrix, A, and any two vectors, b1 , b2 , we have: AT (Ab1 × Ab2 ) = (det A)(b1 × b2 )

(4.4)

Proof. From Corollary 1, we have: AT (Ab1 × Ab2 ) ∙ b3 = (det A)(b1 × b2 ) ∙ b3 for any arbitrary vector b3 . Hence, AT (Ab1 × Ab2 ) = (det A)(b1 × b2 ). Now, continuing on with our proof of Fact 3, we note that for a proper rotation tensor,R, det R = +1, we have, from Corollary 2, RT (Ra × Rb) = (a × b) ⇒ (Ra × Rb) = R(a × b). So, the cross product is preserved under rotation. Finally, from Corollary 1, det(R( a b c )) = det( Ra Rb Rc ) = Ra ∙ Rb × Rc = det(R)(a ∙ b × c) = (a ∙ b × c). Thus, the determinant is preserved under the rotation operation. Since these preserved operations are linear, the rotation transformation is also linear, and hence by definition, a second-order tensor.

4.1.4.1

Tensorial Description of Rotational Transformation

If we consider a set of covariant base vectors, {Gi }, that undergoes a rigid body rotation to become the set of base vectors, {gi }, in the final state, then we have, by definition: gi = R Gi , for all i ∈ {1, 2, 3} for a three-dimensional space. By the relation: R−1 = RT , we also have: Gi = R−1 gi = RT gi , We can immediately recognize that the tensorial description of a rotation tensor through the tensor product is given by: R = gi ⊗ Gi

and

RT = Gi ⊗ gi

(4.5)

where Gi and gi are the contravariant base vectors corresponding to the initial and final states, respectively. Similarly, for the rotation of the contravariant base vectors: gi = R Gi and Gi = RT gi . By the relation: R−1 = RT , we then have: Gi == R−1 gi = RT gi , and: R = gi ⊗ Gi

4.1.4.2

and

RT = Gi ⊗ gi

(4.6)

Rotation Matrix: Components of a Rotation Tensor

Recalling that the matrix corresponding to a linear transformation is identified by the choice of the base vectors, we may have the rotation matrix as {Rij } or {rij }, based on the component definition: R = Rij Gi ⊗ Gj = rij gi ⊗ gj

(4.7)

The components of a rotation tensor are invariant with respect to the base vectors, that is, we have the following.

Rotation Tensor

95

Fact 4: {Rij } and {rij } are identical, that is, independent of the original or the rotated set of base vectors: Rij = rij

(4.8)

Proof. Using expression (4.5), R = Rij Gi ⊗ Gj ⇒ Rij = Gi R Gj = Gi (gk ⊗ Gk )Gj = Gi gj = RT gi gj = gi R gj = rij completes the proof. Thus, the matrix representation is one and the same, whether it is defined in the unrotated or rotated base vectors.

4.1.5

The Eigenproperties of a Rotation Tensor

The set of points in a body that do not rotate under the action of rotation, R, may be described by the equation: RX = 𝜆X ⇔ (R − 𝜆I)X = 0

(4.9)

that is, a point P, with coordinates given by vector X, retains its direction under the action of the rotation. For a non-trivial solution of equation (4.9), we must have the determinant |R − 𝜆I| vanish identically. Now, for a 3 × 3 matrix or second-order tensor in 3D, we can use Cayley–Hamilton theorem to describe its characteristic polynomial as: p(𝜆) ≡ IR ≡ IIR ≡ IIIR ≡

𝜆3 − IR 𝜆2 + IIR 𝜆 − IIIR = 0, tr R = R : I = R11 + R22 + R33 = 𝜆1 + 𝜆2 + 𝜆3 1 {(tr R)2 − tr R2 } = 𝜆1 𝜆2 + 𝜆2 𝜆3 + 𝜆3 𝜆1 2 det R = +1 = 𝜆1 𝜆2 𝜆3

(4.10)

We can immediately observe that 𝜆1 = 1 is the only real root or eigenvalue. Accordingly we can factorize equation (4.10) and represent as: p(𝜆) ≡ (𝜆 − 1){ 𝜆2 − 𝜆(R11 + R22 + R33 − 1) + 1} = 0

(4.11)

The other two roots may, then, be given as complex conjugates: 𝜆2 = cos 𝜃 + i sin 𝜃 = ei𝜃 𝜆3 ≡ 𝜆∗2 = cos 𝜃 − i sin 𝜃 = e−i𝜃

(4.12)

cos 𝜃 = 12 (𝜆2 + 𝜆3 ) = 12 {(R11 + R22 + R33 ) − 1} = 12 (tr R − 1)

(4.13)

with scalar 𝜃 determined by:

Now, let us introduce a new definition.

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96

Definition

r s as the real eigenvector associated with the real root, 𝜆1 = 1

Clearly, then, by equation (4.9): R s = 𝜆s = s, that is, all points along s vector are fixed in position under rotation. In other words, s is the instantaneous axis of rotation of the body. Now, let the complex eigenvectors p and p∗ be associated with the complex eigenvalues 𝜆2 and 𝜆3 . Then, by equation (4.9): s ∙ 𝜆2 p = R s ∙ R p = s ∙ (RT R) p =(s ∙ p) ⇒ (𝜆2 − 1)(s ∙ p) = 0. Now, since 𝜆2 ≠ 1, we must have: s ∙ p = 0. In other words, p is perpendicular to the axis of rotation, s. Similarly, p∗ is also perpendicular to the axis, s. Thus, the p and p∗ vectors define a plane perpendicular to the axis of rotation, s. We can easily construct a pair of real orthogonal vectors, q1 and q2 , from the complex conjugate pairs, p and p∗ , by setting: q1 ≡ 12 (p + p∗ )

(4.14)

q2 ≡ 12 i(p − p∗ )

The final rotated coordinates, Q1 and Q2 , measured in initial unrotated frame, of q1 and q2 is given by: Q1 = R q1 = 12 (R p + R p∗ ) = 12 (𝜆2 p + 𝜆∗2 p∗ ) = cos 𝜃 q1 + sin 𝜃 q2 Q2 = R q2 = 12 i(R p − R p∗ ) = 12 i(𝜆2 p − 𝜆∗2 p∗ ) = − sin 𝜃 q1 + cos 𝜃 q2

(4.15)

or, in matrix form: {

Q1 Q2

}

[

cos 𝜃 = − sin 𝜃

sin 𝜃 cos 𝜃

]{

q1 q2

} (4.16)

The above equation clearly shows a pure planar rotation of 𝜃 about s, of the real plane (see Figure 4.2) defined by the real eigenvectors, q1 and q2 and perpendicular to axis s.

Figure 4.2 Eigenproperties of rotation.

Rotation Tensor

4.1.5.1

97

Summary: The Eigenproperties

Let us then conclude that for all eigenvalues, 𝜆k , k = 1, 2, 3 of an orthogonal matrix:

r |𝜆k | = 1, r If 𝜆k is real, then 𝜆k = +1 or − 1, r If 𝜆k is imaginary, 𝜆k = ei𝜃 and e−i𝜃 , r An invariant line, representing the axis of rotation and corresponding to a real eigen value, is orthogonal to all invariant planes containing other eigenvectors.

r Finally, we see that our eigenanalysis tells us that: instead of the nine coefficients of matrix R, we may equivalently keep track of four elements: the angle of rotation, 𝜃, and the vector, s, the axis of rotation. Remark

r This brings us to the various ways of parameterizing rotation tensor for optimum numerical efficiency in our subsequent finite element analysis of various structural elements such as beams, plates and shells. If we want to work with the original nine parameters of a rotation matrix, we need to carry six side constraints on the nine parameters: three for the lengths of the columns and three for the orthogonality of the columns of the matrix, so that the rotation matrix always stays a rotation matrix upon multiplication. This is rather cumbersome and awkward numerically, hence the need for various other parameterizations.

4.1.5.2

Spherical Displacement

Now, as shown Figure 4.3, note that under pure rigid body rotation, because of length preservation of a vector in rotation, a point P of a body always remains on the surface of a sphere. This is why a pure rotation is often termed a spherical displacement in kinematics. We will discuss it more later; in the meantime, let us sum up:

r Pure rotational rigid body transformation can be identified as spherical kinematics through an orthogonal transformation.

Figure 4.3 Spherical displacement.

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4.1.6

Special Orthogonal Group,SO(3), Lie Group and Rotation Matrix

Motivated by the need to identify entities such as the tangent operator of the rotation transformation, the angular velocities, and so on, and to analyze and numerically treat rotational motion, we introduce here minimally the concept of the special orthogonal group, SO(3), and of the Lie group, and continue our discussion.

4.1.6.1

Special Orthogonal Group, SO(3)

Recall the algebraic structure of a group: a set with four properties: existence of identity, existence of inverse, associative property of products and closure. The 3 × 3 rotation matrix as orthogonal transformations clearly satisfies all the above properties under multiplication and thus forms a group known as the orthogonal group, O(3). To wit, I such that X = I X, that is, no rotation of any point X of a body, serves as the identity transformation; every R has an inverse given by R−1 = RT . Given any three rotational transformations: R1 , R2 , R3 , we have: R1 (R2 + R3 ) = Δ

R1 R2 + R1 R3 ; finally, for closure: if R1 , R2 ∈ O(3), under multiplication, R = R2 R1 belongs to O(3) since RT R = (R2 R1 )T R2 R1 = RT1 RT2 R2 R1 = I ∈ O(3). Moreover, since the determinant is +1, the rotation matrices in three dimensions under multiplication belong to what is known as the special orthogonal group, SO(3). To wit, if the rotation matrices R1 , R2 ∈ SO(3) such that Δ

R = R2 R1 , then det(R) = det(R2 R1 ) = det(R2 ) det(R1 ) = 1 ⇒ R ∈ SO(3).

4.1.6.2

Matrix Lie Group, SO(3)

Recall that a continuous manifold which is also an algebraic group is called a Lie group, bearing the name of its discoverer, Sophus Lie. A Lie group is endowed with both the algebraic structure of a group and the geometry of the Riemannian manifold; it is a non-commutative group. The rotation matrices form a Lie group under non-commutative multiplication, that is, R2 R1 ≠ R1 R2 . If we can show that the matrix product is continuous, we will have SO(3) as a Lie group. For establishing continuity, we need to define a suitable norm, ‖R‖, for a rotation matrix, R = {r1 r2 r3 }, where r1 , r2 , r3 are the column vectors. We choose the norm by replacing each column by the largest column, that is: ‖R‖ ≡

√ 3 max{‖ri ‖,

i = 1, 2, 3} =

√

3 ‖rmax ‖

(4.17)

Then, the metric – that is, the distance between any two rotation matrices, A, B – is given by: ‖A − B‖ ≡

√

3 max{‖ai − bi ‖,

i = 1, 2, 3} =

√ 3 ‖ai − bi ‖max

(4.18)

It can be shown that the definition given by equation (4.17) satisfies all the properties of a norm. Using this norm, it can be shown that if any two matrices are close together, then so are their corresponding elements. In other words, ‖aij − bij ‖ ≤ ‖A − B‖, i, j = 1, 2, 3, meaning that the elements are continuous. Now, since the elements of the product matrices, C = AB, are rational functions of each of the individual matrices, and rational operations such as addition, subtraction, multiplication and division preserve continuity, the product, C, is a continuous function of the two matrices, A, B. Similarly, det C and C−1 varies continuously as C. Thus, SO(3) is a continuous

Rotation Tensor

99

Figure 4.4 Simplest Lie group: circle.

manifold and forms a Lie subgroup. Now, with both the algebraic structure and the geometry of SO(3), we can begin to visualize, say, curves made up of rotation matrices, themselves as always belonging to or traveling on the 3D surface (denoted as 3-sphere) of a 4D spherical ball. In other words, a rotation matrix belongs to a curved surface as opposed to a flat Euclidean surface. In this context, let us recall that a 3D point under pure rotation travels on a 2D surface (2-sphere) of a 3D spherical ball. As will be shown, this way of looking into the rotation matrices helps us in devising various numerically efficient parameterizations of a rotation matrix and more accurate numerical integrations of the rotational motions. Example: The simplest of the Lie groups is the unit circle, S1 as shown in Figure 4.4. Clearly, S1 depicts a one-dimensional manifold. There are several ways to depict points on the circle:

r Points can be parameterized in the complex plane as: s = cos 𝜃+i sin 𝜃, r r

𝜃 ∈ [0, 2𝜋]; under complex multiplication, S1 forms an algebraic group. Points can be ]expressed through a plane rotation matrix parametrized by 𝜃 as: R(𝜃) = [ cos 𝜃 − sin 𝜃 ; the set of R forms a matrix group, SO(2), under matrix multiplication. sin 𝜃 cos 𝜃 Points can also be expressed map: s = ei𝜃 ; in matrix form, it [ ] through a scalar [ exponential ] 0 −𝜃 0 −𝜃 is given as: s = exp = eΘ , Θ ≡ where Θ, the skew-symmetric matrix 𝜃 0 𝜃 0 belongs to what is known as Lie algebra, that is, the tangent space, so(2), with eΘ = I + Θ + Θ2 + .... 2

Note that 𝜃 = 0 ⇒ eΘ = I, meaning that Lie algebra, so(2), is the tangent space at identity, and S1 has both the algebraic group structure and the geometry of a manifold, and thus forms a Lie group. In what follows, we use the above introductory notions, the algebraic structure and the geometry of SO(3) to shed light on the meaning of various parameterizations of the rotational motions, for example, as in Cayley’s parameterization.

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4.1.6.3

Euler Theorem

In this connection, it is interesting to note that in the year 1752, Euler discovered the closure property of the group without, of course, the then non-existing notion of groups. By his ingenious geometric construction, he proved that the position of a body after any two consecutive rotations can be obtained by a single rotation from the reference position, meaning that the product of any two consecutive rotations is also a rotation, as needed by the closure property of a group. 4.1.6.4

Representation of the 3D Rotation Matrix

We have identified rotation as an orthogonal transformation that contains nine components with the orthogonality condition providing three equations for interdependency of the parameters. But, for computational efficiency, we would like to seek any possible parameterization of a rotation matrix that results in further reduction in the number of parameters. For this, we have a representation theorem by Cayley that will put us in a favorable position to evaluate various parametrization possibilities. Concurrently, we will invoke Lie group properties of rotational transformations to eventually help us identify the best strategies for numerical treatment of nonlinear static and dynamic structural problems involving large finite rotations that travels on a curved Riemann surface: a 3-sphere, as we shall see. Clearly, then, for a fundamental representation of a rotation matrix, we must explore Cayley’s representation.

4.2

Cayley’s Representation

We introduce here a very important theorem by Cayley that allows us immediately to parameterize a rotation matrix by a vector: the Rodrigues vector. This opportunity profoundly increases numerical efficiency by considering only a three-dimensional vector (i.e. three parameters) instead of the computationally cumbersome nine components of a rotation tensor. Finally, the geometry of the theorem by Cayley that provides valuable insights for an eventual numerical strategy is appreciated by invoking the SO (3) and Lie group properties of the rotational transformations. As indicated above, we present an algebraic representation and a geometric interpretation for finite rotation suitable for subsequent numerical evaluations in structural mechanics. We may recall that rotation is tractably associated with and interpreted to be an orthogonal transformation belonging to SO (3) group. This, then, leads us to representation of rotation by an associated skew-symmetric matrix given by an important theorem by Cayley. Finally, we introduce axial vectors, corresponding to the skew-symmetric matrices in 3D, giving us the desired possibility of parametrization of rotation matrices by 3D vector quantities for optimum computational efficiency. The associated skew symmetric matrices can be identified as belonging to the tangent space at identity, so (3), which is known as the Lie algebra.

4.2.1

Cayley’s Theorem for Representation of an Orthogonal Matrix

Every orthogonal matrix, R, can be represented by a skew-symmetric matrix, S such that: R = (I − S)−1 (I + S)

(4.19)

Proof. Let p = R P define a rigid body rotation. For rigid body rotation, length is preserved, that is, p ∙ p − P ∙ P = 0 ⇒ (p + P) ∙ (p − P) = 0, that is,as shown Figure 4.5, the diagonals p + P and p − P of a rhombus with edges P and p intersect at right angles. So, if we define: m ≡ p + P and n ≡ p − P we have: m ∙ n = (p + P) ∙ (p − P) = 0 for all p. Eliminating p by application of

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Figure 4.5 Cayley’s theorem.

the rotation relation, we have: m = (R + I)P and n = (R − I)P. Now substituting P from m into n, we have: n = (R − I)(R + I)−1 m. Let us define: S ≡ [R − I][R + I]−1

(4.20)

So, n = S m. Noting from the above that: m ∙ n = n ∙ m = 0 and applying the relation: n = S m, we have: m ∙ n = mT S m = 0 and n ∙ m = mT ST m = 0. Now, adding these two relations, we have: mT (S + ST ) m = 0 for all m ∈ R3 . Thus, finally, we get: S + ST = 0 which means that S is a skew-symmetric matrix by definition. Now, solving S for R completes the proof. Remarks

r The inverse of (I − S) in equation (4.19) always exist. Otherwise, it will imply that det(I − S) = r

0 which means: for a h ∈ R3 , (I − S) h = 0 ⇒ S h = h, implying that 𝜆 = 1 is an eigenvalue of the skew-symmetric matrix,S. But, the only real eigenvalue of a skew-symmetric matrix is 0; so, this is a contradiction, and hence, matrix (I − S) is always invertible. The matrices: (I − S) and (I + S) commute. To wit, (I + S)(I − S) = (I − S)(I + S) = (I − S2 ). Now, pre-mumultiplying and post-multiplying these relations by (I − S)−1 , we can see that (I − S)−1 (I + S) = (I + S)(I − S)−1 , that is, (I − S)−1 and (I + S) also commute. Thus, we have an equivalent expression for Cayley’s theorem: R = (I + S)(I − S)−1

(4.21)

r Now, conversely, we would like to show that given a skew-symmetrix matrix, S ≡ (R − I)(R + I)−1 , we must have R as an orthogonal matrix. We have: R = (I − S)−1 (I + S) = (I + S)(I − S)−1 (because they commute as shown above). Now, from the definition of S, we have R = (I − S)−1 (I + S) ⇒ RT = (I + ST )(I − ST )−1 = (I − S)(I + S)−1 (in the last equality, we have used the property of skew-symmetric matrix: S = −ST ). Now, multiplying, we get: R RT = (I + S)(I − S)−1 (I − S)(I + S)−1 = I. Hence, R is an orthogonal matrix.

4.2.1.1

Summary: Cayley’s Theorem

r Every rotation matrix, R, is in a one-to-one correspondence with a skew-symmetric matrix, S, and related by: R = (I − S)−1 (I + S). Remarks

r For

direct rotation, that is, det(R) = +1, the formula (4.19) excludes the situation where the real eigenvalue of R becomes −1. However, the representation can be obtained by a

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limiting procedure as developed by Cayley. S being skew-symmetric, we need only 12 n(n − 1) parameters for a general n-dimensional space.

4.2.2

Three-dimensional Space, n = 3

Of particular interest to us is the case of the 3D space, i.e. n = 3. In this situation, the skewsymmetric matrix is completely determined by three coefficients above the diagonal. Thus, the number of parameters needed to describe a rotation matrix also becomes 3 (= 12 (3)(3 − 1)). This gives us an opportunity to associate the skew matrices with the corresponding three-dimensional vectors. Recall that a vector, a, is called the axial vector of a skew-symmetric matrix, A, if for all three-dimensional vectors, h ∈ R3 , we have: A h = a × h,

∀h ∈ R3

(4.22)

Note

r From now on, if a is an axial vector, the corresponding skew-symmetric matrix will be shown by [a]. Operationally, it is understood that [a] ≡ (a ×).

4.2.3

Closed-form Explicit Description of Rotation Matrix

For future numerical efficiency, we can use equation (4.21) to describe the rotation matrix, R, in a closed and explicit form in terms of the components of the skew-symmetric matrix, S. To this end, we need the inverse of the matrix (I − S). From direct solution, we have:

(I − S)

−1

⎡ 1 + s21 1 1 = (Δ I + S + S2 ) = ⎢s1 s2 − s3 Δ Δ⎢ ⎣s3 s1 − s2

s1 s2 − s3 1 + s22 s2 s3 + s1

s3 s1 + s2 ⎤ s2 s3 − s1 ⎥ ⎥ 1 + s23 ⎦

(4.23)

where, Δ ≡ 1 + ‖s‖2 = 1 + s21 + s22 + s23 . Now, by direct multiplication of matrices in equation (4.21), we have:

R≡

(I − S)−1

⎡ 1 + s21 1⎢ s s + s3 (I + S) = Δ⎢ 1 2 ⎣ s3 s1 − s2

s1 s2 − s3 1 + s22 s2 s3 + s1

s3 s1 + s2 ⎤ ⎡ 1 s2 s3 − s1 ⎥ ⎢ s3 ⎥⎢ 1 + s23 ⎦ ⎣−s2

−s3 1 s1

s2 ⎤ −s1 ⎥ , ⎥ 1 ⎦ (4.24)

Δ ≡ 1 + ‖s‖2 = 1 + s21 + s22 + s23 leading to: ⎡1 + s21 − s22 − s23 1⎢ 2(s1 s2 + s3 ) R= Δ⎢ ⎣ 2(s3 s1 − s2 )

2(s1 s2 − s3 ) 1 − s21 + s22 − s23 2(s2 s3 + s1 )

Δ ≡ 1 + ‖s‖2 = 1 + s21 + s22 + s23

2(s3 s1 + s2 ) ⎤ 2(s2 s3 − s1 ) ⎥ , ⎥ 1 − s21 − s22 + s23 ⎦

(4.25)

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Figure 4.6 Example: Cayley’s method.

Thus, we do not need to worry about the original and computationally cumbersome ninecomponent parameterization with six constraints of a 3D rotation matrix. The equivalent 3D skew-symmetric matrix, and hence 3D axial vector, representation helps us to consider only three parameters for the rotation matrix suitable for numerical efficiency. Example: As shown in Figure 4.6, a vector, a = (1, 0, 0), along the x-axis is rotated 90◦ about z-axis to produce the vector, b = (0, 1, 0) along the y-axis such that b = R a. We are interested in the expression for the rotation matrix, R, according to Cayley’s representation. Now, the instantaneous axial vector is given by s = (0, 0, 1). So, the corresponding skew-symmetric matrix ⎡0 S = ⎢1 ⎢ ⎣0 ⎡1 ⎢1 ⎢ ⎣0

−1 1 0

⎡0 ⎢1 ⎢ ⎣0

−1 0 0

−1 0⎤ ⎡ 1 0 0⎥; (I − S) = ⎢−1 ⎥ ⎢ 0 0⎦ ⎣ 0

1 1 0

0⎤ 0⎥ with (I − S)−1 = ⎥ 1⎦

0⎤ 0⎥. Thus, finally, we have: R = (I − S)−1 (I + S) = ⎥ 1⎦

0⎤ 0⎥ as expected. Given that the 𝜃 ⎥ 1⎦ ⎡cos 𝜋 − sin 𝜋 2 2 ⎢ cos 𝜋2 the rotation matrix. R = ⎢ sin 𝜋2 ⎢ 0 0 ⎣

4.2.4

1 2

⎡1 −1 0⎤ ⎢1 1 0⎥ and (I + S) = ⎢ ⎥ 0 0 2⎦ ⎣ 1 2

⎡1 −1 ⎢1 1 ⎢ 0 0 ⎣

0⎤ ⎡1 −1 0⎤ 0⎥ ⎢1 1 0⎥ = ⎥⎢ ⎥ 2⎦ ⎣0 0 1⎦

= 𝜋2 , we can compare with the familiar expression for 0⎤ ⎡0 ⎥ 0⎥ = ⎢1 ⎢ 1⎥⎦ ⎣0

−1 0 0

0⎤ 0⎥. ⎥ 1⎦

Instantaneous Rotation

It can easily be seen by the definition of a skew-symmetric matrix that for two-dimensional (planar) space, that is, [ n = 2,] we only need one[parameter, ] a, to describe a rotation. Thus, for 0 −a 1 −a 1 planar rotation, S = , (I − S)−1 = 1+a2 where det(I − S) ≡ Δ = 1 + a2 . Then a 0 a 1 [ ] 2 −2a 1 1−a −1 . Now, if we describe the rotation matrix is given by: R = (I − S) (I + S) = Δ 2a 1 − a2 a planar rotation about the third axis by 𝜃 such that a ≡ tan

( ) 𝜃 2

(4.26)

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Figure 4.7 A half turn.

2a then: 1−a = 2 cos 𝜃2 − 1 = cos 𝜃 and 1+a 2 = sin 𝜃. With these relationships in the background, 1+a2 we get our familiar expression for planar rotation as: 2

[ R=

4.2.4.1

cos 𝜃 sin 𝜃

− sin 𝜃 cos 𝜃

] (4.27)

Limiting Case: a → ∞ 2

2a Following from this, as the parameter a → ∞, we have: 1−a → −1 and 1+a 2 → 0 resulting in 1+a2 [ ] −1 0 R= . This rotation, R, corresponds to a half turn with eigenvalue 𝜆 = −1, as shown 0 −1

in Figure 4.7.

4.2.5

Lie Group, Tangent Operators, Exponential Map and Adjoint Map

Recall that a Lie group is endowed with both the algebraic structure of a group and the geometry of a Riemannian surface, that is, a differentiable curved surface. Recall also that the rotational transformations belong to SO (3), the special orthogonal group in three dimension, and form a matrix Lie group. In other words, SO (3) is a differentiable manifold.

4.2.5.1

The Tangent Operators of SO (3)

Let us consider on SO (3), a continuously differentiable affinely parametrized curve of rotational transformations, R(𝜆), 𝜆 ∈ [0, 1] with R(0) = I where I is the 3 × 3 identity matrix. Then, the rotational matrix function, R(𝜆) :→ SO(3), generates a continuous set of points, X(𝜆) ∈ R3 , of any point, x ∈ R3 through the relation: X(𝜆) = R(𝜆) x. The tangent direction to this curve is given by the derivative, denoted by (∙)′ , with respect to 𝜆, that is, X′ (𝜆) = R′ (𝜆) x = (R′ (𝜆) R−1 (𝜆)) X(𝜆) = (R′ (𝜆) RT (𝜆)) X(𝜆) where we have used for R(𝜆) the orthogonality condition: R(𝜆)T R(𝜆) = I. Noting the identity: R′ (𝜆) = (R′ (𝜆) RT (𝜆)) R(𝜆), we see that (R′ (𝜆) RT (𝜆)) also transforms the rotation matrix, R(𝜆), to give its derivative, R′ (𝜆). Now, also note that: (R′ (𝜆) RT (𝜆)) + (R′ (𝜆) RT (𝜆))T = R′ (𝜆) RT (𝜆) + R(𝜆)(R′ (𝜆) )T = (R(𝜆) RT (𝜆))′ = 0, because of the orthogonality condition: R(𝜆) RT (𝜆) = I. Thus the matrix, (R′ (𝜆) RT (𝜆)), is a skewsymmetric matrix and known as the tangent operator on SO(3).

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Figure 4.8 Cayley’s theorem: SO(3) and so(3).

4.2.5.2

Exponential Map: the Case of Constant Tangent Operator

Let us consider the case where the tangent operator of R(𝜆) is constant and given by the skewsymmetric matrix, S. Then, we have: R′ (𝜆) = S R(𝜆)

(4.28)

Note that if the initial condition is given as R(0) = I, the identity matrix, then, from equation (4.28), we have: S = R′ (0). In other words, S is the tangent operator at identity. Thus, the set of tangent operators on SO (3) is identical to the set of tangent directions at identity. The tangent space at identity, denoted by TI SO(3), is a vector space of skew-symmetric matrices (see Figure 4.8). Recalling that for any two skew-symmetric matrices, A, B, with a, b as the corresponding axial vectors, we have the identity: AB − BA =(b ⊗ a) − (a ⊗ b) = [a × b], we define a product, known as Lie product or Lie bracket, as: [A, B] = AB − BA. Recalling that a vector space with a product is known as algebra, the tangent space at identity is also recognized as a Lie algebra and is denoted by so(3). Now, focusing on equation (4.28) with initial condition: R(0) = I, we have the solution: R(𝜆) = e𝜆S = I + 𝜆S + 12 (𝜆S)2 + ⋯

(4.29)

Thus, as shown Figure 4.9, a rotation matrix can be generated from the corresponding skewsymmetric matrix through an exponential map: R = eS = I + S + 12 (S)2 + ⋯

and

RT = R−1 = e−S

(4.30)

If we normalize S by dividing it by the length s = ‖s‖ , of its axial vector, s, then we have: S = sN such that ‖N‖ = 1, that is, the axial vector, n, corresponding to the skew-symmetric matrix, N, is

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Figure 4.9 Cayley’s theorem: exponential map.

an unit vector. Noting the identity that a matrix satisfies its own characteristic polynomial as in the case of the skew-symmetric matrix, S: S3 + s2 S = 0 where

s ≡ ‖s‖ =

√ s21 + s22 + s23

(4.31)

we have: N3 + N = 0

(4.32)

Now, using identity (4.32) in equation (4.29), we get: R(𝜆) = e𝜆S = I + 𝜆S + 12 (𝜆S)2 + ⋯ = I + 𝜆sN + 12 (𝜆sN)2 + ⋯ ( ) ( { }) (𝜆s)4 (𝜆s)3 = I + 𝜆s − + ⋯ N + 1 − 1 − 12 (𝜆s)2 + −⋯ N2 3! 4! ⇒ R(𝜆) = I + (sin 𝜆s) N + (1 − cos 𝜆s)N2 ⇒ R = I + (sin s) N + (1 − cos s)N2 (4.33) Remark

r From our later discussion on Rodrigues parametrization of a rotation matrix by three parameters, we will be able to see the geometry of equation (4.33): the unit axial vector, n, of the unit skew-symmetric matrix, N, represents the axis of rotation with 𝜆s as the amount of rotation about it.

4.2.5.3

Adjoint Map

If R ∈ SO(3) and r ∈ so(3), then R [r] R−1 = R [r] RT ∈ so(3) where R [r] RT = [Rr] with the notation: [a] = a× for any vector a ∈ so(3).

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107

Proof. Recall that the cross product is preserved under rotation, that is, Ra × Rb = R (a × b) for any a, b ∈ R3 . So, (R[r]RT )h = R(r × RT h) = Rr × RRT h = Rr × h = [Rr]h for any h ∈ R3 completes the proof.

4.2.5.4

Discussion

As indicated before, Rodrigues parametrization ingeniously represents a rotation matrix with only three parameters; thus, for our subsequent computational descriptions of the rotation matrix, we now explore the Rodrigues vector and rotation parameterization.

4.3

Rodrigues Parameters

We take advantage of the earlier construction of the important theorem by Cayley that allows us to parameterize the rotation matrix by a vector, the Rodrigues vector. We now introduce the unit axial vectors corresponding to the skew-symmetric matrices in 3D giving us the possibility of parametrization of rotation matrices by vector quantities for optimum computational efficiency. Alternatively, we introduce Rodrigues parametrization through geometrical construction. This assertion greatly increases numerical efficiency by considering only a three-dimensional vector (i.e. three parameters) instead of the computationally cumbersome nine parameters of a rotation matrix with six constraints. Remark

r The Rodrigues vector was devised by Olinde Rodrigues in 1840, and Cayley’s theorem appeared in 1845. Thus, there is no need to resort to Cayley’s theorem to describe Rodrigues parameterization; this is clearly demonstrated towards the end, that is, where we see the geometry of the Rodrigues vector without resorting to Cayley’s theorem. In what follows, however, we choose the same geometric construction as we did in developing Cayley’s theorem as the starting point for introducing the Rodrigues vector. In so doing, we present our primary method for finite rotation parameterizations suitable for subsequent numerical evaluations in structural mechanics.

4.3.1

Rodrigues Vector, n𝜃 , in Three-dimensional Space

From Figure 4.10 and our discussion on Cayley’s theorem, recall that for x = R X, we find that x − X is perpendicular to x + X, and it follows that: x − X = S(x + X) where

S ≡ [R − I][R + I]−1

(4.34)

Now, from s as the axial vector of S it follows, by definition: x − X = s × (x + X)

(4.35)

If we consider the projection of X and x on a plane perpendicular to s, equation (4.35) still remains true if we replace X and x by their respective projections, that is, proj(x − X) = s × proj(x + X)

(4.36)

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Figure 4.10 Rodrigues vector.

By definition of a cross product and noting that s is perpendicular to proj(x + X), we have the following magnitude relation: ‖proj(x − X)‖ = ‖s‖ ‖proj(x + X)‖

(4.37)

Then, from Figure 4.10, we have: 1 ‖proj(x − X)‖ ( ) ‖proj(x − X)‖ 𝜃 2 = tan = 1 2 ‖proj(x + X)‖ ‖proj(x + X)‖ 2

(4.38)

So, from equations (4.37) and (4.38), we get for the magnitude of s: s = ‖s‖ = tan

( ) 𝜃 2

(4.39)

Thus, Cayley’s axial vector, s, may be written as: ( ) s = tan

𝜃 2

n𝜃 ,

‖n𝜃 ‖ = 1

(4.40)

Definition

r The unit axial vector, n𝜃 , that parameterizes a rotation matrix, is known as the Rodrigues vector.

4.3.2

3D Rotation Vector, θ, and the Rodrigues Vector, n𝜃

For numerical computations in our subsequent finite element analysis and following the magnitude of s from equation (4.39), we introduce:

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Figure 4.11 Rotation tensor and Rodrigues vector.

Definition A rotation vector, θ, along the Rodrigues vector is defined such that: s=

1 𝜃

( ) tan

𝜃 2

θ

(4.41)

√ where 𝜃 ≡ ‖θ‖ = (𝜃 ∙ 𝜃) is the magnitude of the rotation vector. We can easily verify that equation (4.41) implies equation (4.39). As shown in Figure 4.11, the rotation vector is the instantaneous axis of rotation of a rigid body. From equations (4.40) and (4.41), we have:

S=

4.3.3

( ) ( ) 1 𝜃 𝜃 Θ = tan N𝜃 tan 𝜃 2 2

where

N𝜃 ≡

Θ = [n𝜃 ], 𝜃

Θ = [θ] ,

Rotation Matrix, R, and Rodrigues Vector, n𝜃 ≡

n𝜃 =

θ , 𝜃

(4.42)

θ 𝜃

Now, inserting expression (4.42) for S in terms of the components of the Rodrigues vector, n𝜃 ≡ θ𝜃 , of equation (4.42), in equation (4.25), we get our final representation of a rotation matrix in terms of the components of the Rodrigues vector: 2(s1 s2 − s3 ) 2(s3 s1 + s2 ) ⎤ ⎡1 + s21 − s22 − s23 1⎢ 2 2 2 1 − s1 + s2 − s3 2(s2 s3 − s1 ) ⎥ 2(s1 s2 + s3 ) R (s(θ)) = ⎥ 𝜌⎢ 2(s2 s3 + s1 ) 1 − s21 − s22 + s23 ⎦ ⎣ 2(s3 s1 − s2 ) with 𝜌 ≡ 1 + s21 + s22 + s23 = 1 + ‖s‖2 , and, ( ) ( ) 1 𝜃 𝜃 θ, 𝜃 ≡ ‖θ‖ , ‖s‖ = tan s = tan 𝜃 2 2

(4.43)

Example: As shown in Figure 4.12, a vector, a = (1, 0, 0), along the x-axis is rotated 90◦ about z-axis to produce the vector, b = (0, 1, 0) along the y-axis such that b = R a. We are interested in the expression for the rotation matrix, R, according to the Rodrigues formula. Now, with 𝜃 = 𝜋2 and n𝜃 = (0, 0, 1), the Rodrigues vector is given by: s = tan 𝜃2 n𝜃 = (0, 0, 1). So, from

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Figure 4.12 Example: Rodrigues’ method.

equation (4.43), we have: ‖s‖ = tan

( ) 𝜃 2

= 1 ⇒ 𝜌 = 1 + ‖s‖2 = 2 and R =

1 2

⎡0 −1 0⎤ ⎢1 0 0⎥ as expected. ⎢ ⎥ 0 1⎦ ⎣0

4.3.4

⎡0 −2 ⎢2 0 ⎢ 0 ⎣0

0⎤ 0⎥ = ⎥ 2⎦

Geometry of the Rotation Matrix and the Rodrigues Vector

Let (X, Y, Z) be a global frame of reference. Now, referring to Figure 4.13, we choose – for simplicity and without any loss of generality – a local frame, (x, y, z), such that z-axis is aligned with the axis of rotation: θ = ( 𝜃X 𝜃Y 𝜃Z ), and xz-plane contains the initial unrotated vector, P that is rotated by rotation matrix, R, to the final rotated vector, p. Clearly, Px , the projection vector of the initial unrotated vector, P, along x-axis is given by: Px = P − (P ∙ n) n = P − (n ⊗ n) P = (I − (n ⊗ n)) P

(4.44)

Noting that n is a unit vector, we have: ‖Px ‖ = ‖pxy ‖ = ‖n × P‖. So, using equation (4.44), the projection vector, pxy , of the final rotated vector, p, is given by the components as: pxy = px + py = Px cos 𝜃 + (n × P) sin 𝜃 = Px cos 𝜃 + (n × P) sin 𝜃 = {(I − (n ⊗ n)) cos 𝜃 + [n] sin 𝜃} P

Figure 4.13 Geometry of Rodrigues vector.

(4.45)

Rotation Tensor

111

Now, by using equation (4.45), the final rotated vector, p, can be expressed in terms of the unrotated initial vector, P, as: p = pxy + (P ∙ n) n = {(I − (n ⊗ n)) cos 𝜃 + [n] sin 𝜃} P + (n ⊗ n) P = {I cos 𝜃 + [n] sin 𝜃 + (1 − cos 𝜃) (n ⊗ n)} P

(4.46)

If we define the rotation matrix, R, as: p = R P, then by equation (4.46), we have: R = I cos 𝜃 + [n] sin 𝜃 + (1 − cos 𝜃) (n ⊗ n)

(4.47)

{ } Finally, recalling that for any h ∈ R3 , I + N2𝜃 h = h + n × n × h = ̸ h + (n ∙ h) n − (n ∙ n)h = (n ⊗ n) h, that is, I + N2𝜃 = (n ⊗ n), where N𝜃 , as before, is the skew-symmetric matrix corresponding to the axial vector, n𝜃 , from equation (4.47), we have: R = I + sin 𝜃 N𝜃 + (1 − cos 𝜃) N2𝜃

(4.48)

If all the vectors are column vectors, we have equivalently, from equation (4.48): ( ) R = I + sin 𝜃 (n𝜃 ×) + (1 − cos 𝜃) n𝜃 nT𝜃 − I

(4.49)

Thus, the rotation is parameterized by three parameters: the components of the rotation axis or the Rodrigues vector. As we will see below, the three-parameter parametrization is singular at some rotations.

4.3.5

2D (Planar): Rodrigues Vector and Rotation Vector

It is instructive to bring out the familiar expression for rotation matrix for planar, that is, 2D rotation about the third axis with 𝜃1 = 𝜃2 ≡ 0 ⇒ ‖θ‖ = 𝜃3 ≡ 𝜃. From expression (4.41), ) ( ) 𝜃 𝜃 𝜃 1( we immediately have: s1 = s2 ≡ 0 ⇒ s3 ≡ tan = tan . Now, let us note: 1 − s23 = 2 𝜃 2 𝜌 𝜃 𝜃 1 − tan2 2 tan 2 = 2 cos2 𝜃 − 1 = cos 𝜃, and 1 (2s ) = 2 = 2 sin 𝜃 cos 𝜃 = sin 𝜃. Thus, the 𝜃 𝜃 2 𝜌 3 2 2 2 2 1 + tan 1 + tan 2 2 expression (4.43) reduces to the familiar instantaneous rotation matrix for 2D: ⎡1 − s23 −2s3 1⎢ 1 − s23 2s R= 𝜌⎢ 3 0 ⎣ 0 with 𝜌 ≡ 1 + s23

0 ⎤ ⎡cos 𝜃 0 ⎥ = ⎢ sin 𝜃 ⎥ ⎢ 1 + s23 ⎦ ⎣ 0

− sin 𝜃 cos 𝜃 0

0⎤ 0⎥ ⎥ 1⎦

(4.50)

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112

4.3.6

Singularity of the Rodrigues Rotation Matrix at ‖𝜃‖ = (2n + 1)𝜋, n = 0, 1, ...

For numerical computation, it is important to note that a rotation matrix as expressed by equation (4.43) is singular whenever tr R → −1, where tr stands for the trace of a matrix, or equivalently, ‖s‖ → ∞, or ‖𝜃‖ = (2n + 1)𝜋, n = 0, 1, .... To wit, let us first note that: ) ( 𝜃 (θ ∙ θ) 𝜃 = tan2 , where, recall that: 𝜃 ≡ ‖θ‖. Clearly, then, ‖s‖ → ‖s‖ = s ∙ s = tan2 2 ‖θ‖2 2 𝜃 1 ∞ ⇔ tan → ∞ ⇔ 𝜃 → (2n + 1)𝜋 for n = 0, 1, .... Now, from equation (4.43): (tr R − 1) = 2 2 𝜃 2 } { 1 − tan 1 (1 − ‖s‖2 ) 1 1 2 = cos 𝜃. Finally, knowing that cos 𝜃 → −1 (3 − ‖s‖2 ) − 1 = = 𝜃 2 𝜌 2 (1 + ‖s‖2 ) 1 + tan2 2 as 𝜃 → (2n + 1)𝜋 with n = 0, 1, ..., completes our results.

4.3.6.1

Discussion

A vectorial parameterization of the rotation matrix is most important for our purposes because the number of parameters (namely, three) is the same as that of the usual rotational degrees of freedom of structural elements such as beams, plates and shells, for computational schemes such as the finite element method. Although it is singular at some rotation, as shown above, for most structural problems with moderate rotations, it suffices. In the event that it is not, various other four-parameter parameterizations, such as the Euler– Rodrigues parameterization, can be combined transparently to make it singularity-free. Thus, for singularity-free parameterization, we consider next the Euler–Rodrigues parameters and rotation parameterization.

4.4

Euler – Rodrigues Parameters

The main goal here is to present a computationally singularity-free parameterization of rigid body rotations. More specifically, we concentrate on a very important parameterization of a rotation matrix by the four-dimensional Euler–Rodrigues parameter vector. This parameterization eliminates the singularity problems of parameterization by the Rodrigues vectors.

4.4.1

4D Euler–Rodrigues Vector, X ≡ {X1

X2

X3

X4 }T

Recalling our discussion on the Rodrigues rotation parameterization, the Rodrigues rotation matrix becomes singular at those θ’s for which ‖𝜃‖ = (2n + 1)𝜋, n = 0, 1, ... Furthermore, recalling that, although there are only three independent parameters (nine minus six due to orthonormality constraints) in a rotation tensor, it is not possible for a three-parameter set such as ( ) θ 𝜃 n𝜃 , ‖n𝜃 ‖ = 1 , n𝜃 ≡ , 𝜃 = ‖θ‖, to uniquely the Rodrigues parameter vector, s = tan 2 𝜃 describe all possible rotations. A non-singular one-to-one mapping takes at least five parameters while a non-singular many-to-one mapping requires at least four parameters.

Rotation Tensor

4.4.1.1

113

Cayley’s Representation and Euler–Rodrigues Parameters

Using the Rodrigues vector from equation (4.43) and inserting in Cayley’s representation as in equation (4.19), we have: ( R = (I − S)

−1

(I + S) = (I −

sin 𝜃2 cos 𝜃2

)

( N𝜃 )

−1

(I +

)−1 ( ) ( = cos 𝜃2 I − sin 𝜃2 N𝜃 cos 𝜃2 I + sin 𝜃2 N𝜃

)

sin 𝜃2 cos 𝜃2

N𝜃 ) (4.51)

where N𝜃 = [n𝜃 ] = n𝜃 ×. Now, we define: Definition

r The

four-dimensional unit Euler–Rodrigues parameter vector, X = {X1 defined as:

X2

X3

𝜃 𝜃 𝜃 𝜃 n , X ≡ sin n2 , X3 ≡ sin n3 , X4 ≡ cos , 2 1 2 2 2 2 ‖X‖ ≡ 1, with n𝜃 ≡ ( n1 n2 n3 ) and [n𝜃 ] = N𝜃

X1 ≡ sin

)−1 ( = cos 𝜃2 I + sin 𝜃2 N𝜃 + Noting that cos 𝜃2 I − sin 𝜃2 N𝜃 (4.51):

𝜃 2 𝜃 cos 2

sin2

X4 }T ,

(4.52)

( ) I + N2𝜃 , we have, from equation

R = I + 2 sin 𝜃2 cos 𝜃2 N𝜃 + 2 sin2 𝜃2 N2𝜃 = I + sin 𝜃 N𝜃 + (1 − cos 𝜃) N2𝜃 Note that we have used the characteristic polynomial identity: N3𝜃 + N𝜃 = 0

(4.53)

in the above

derivation. The representations of a rotation matrix as given by equation (4.53) are identical to what was obtained through Rodrigues parameterization. )−1 ( Exercise: prove that cos 𝜃2 I − sin 𝜃2 N𝜃 = cos 𝜃2 I + sin 𝜃2 N𝜃 +

𝜃 2 𝜃 cos 2

sin2

( ) I + N2𝜃

An orthogonal rotation matrix, R, with Euler–Rodrigues parameters, is characterized by a point on a three-dimensional unit spherical surface embedded in four dimensions, that is, a 3-sphere of radius 1, as shown in Figure 4.14, and is given by: 2(X1 X2 − X3 X4 ) ⎡X12 − X22 − X32 + X42 ⎢ −X12 + X22 − X32 + X42 R = 2(X1 X2 + X3 X4 ) ⎢ 2(X2 X3 + X1 X4 ) ⎣ 2(X1 X3 − X2 X4 ) with ‖X‖2 ≡ X12 + X22 + X32 + X42 = 1

2(X1 X3 + X2 X4 ) ⎤ 2(X2 X3 − X1 X4 ) ⎥ , ⎥ −X12 − X22 + X32 + X42 ⎦

(4.54)

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114

Figure 4.14 Euler–Rodrigues parameters.

with ) ) ) ( ( ( 𝜃 𝜃1 𝜃 𝜃2 𝜃 𝜃3 𝜃 X1 ≡ sin , X2 ≡ sin , X3 ≡ sin , X ≡ cos , 2 𝜃 2 𝜃 √ 2 𝜃 4 2 2 2 2 T θ = {𝜃1 𝜃2 𝜃3 } , 𝜃 ≡ ‖θ‖ = 𝜃1 + 𝜃2 + 𝜃3

(4.55)

From equations (4.54) and (4.55), rotations are non-singular two-to-one maps where globally both +X and −X represent the same rotation. Thus, a four-parameter Euler–Rodrigues vector is adequate against singularity instead of a full five-parameter representation. Finally, geometrically speaking, the finite rotations, R, move around on a 3D unit sphere; the set of instantaneous rotations define a tangent hyperplane at a point on this unit sphere. Example: As shown in Figure 4.15, a vector, a = (1, 0, 0), along the x-axis is rotated 90◦ about z-axis to produce the vector, b = (0, 1, 0) along the y-axis such that b = R a. We are interested in the expression for the rotation matrix, R, according to the Euler–Rodrigues formula. Now, with 𝜃 = 𝜋2 , we have, from equation (4.55), the Euler–Rodrigues parameters: X1 = 0, X2 = 0, X3 = 1 √ , X4 2

=

1 √ 2

⎡0 −1 0⎤ 0 0⎥ as and ‖X‖ = 1. Thus, finally using equation (4.54), we get R = ⎢1 ⎢ ⎥ 0 1⎦ ⎣0

expected.

Figure 4.15 Example: Euler–Rodrigues method.

Rotation Tensor

4.4.1.2

115

Discussion

As an alternative to Euler–Rodrigues parameterization, four-parameter Hamilton–Rodrigues quaternion non-singular parameterizations can be combined transparently to make numerical treatment singularity-free. But, before we can introduce Hamilton–Rodrigues quaternions and rotation parameterization, we will first familiarize ourselves with Hamilton’s quaternion.

4.5

Hamilton’s Quaternions

The main goal here is to introduce the concept of the quaternion as a precursor to its application as a locally singularity-free parameterization of rigid body rotations in structural and continuum mechanics. We introduce and discuss the important properties of the quaternions as devised by Hamilton. Particularly, the definitions for a conjugate quaternion, the norm of a quaternion, inverse quaternion, pure or vector quaternions, and for a unit quaternion are introduced. Finally, recognizing that quaternions form a non-commuting group, that is, a Lie group, a particular group operation reveals that quaternions are similar to rotation matrices, giving us the possibility of parameterization of rotation matrices by quaternions for optimum computational efficiency.

4.5.1

Four-dimensional Quaternions: Q ≡ {q1

q2

q3

q4 }T

William Rowan Hamilton was searching for the next higher dimensional extension to the twodimensional complex number: x = a + ib with the rule i2 = −1; he was unsuccessful for 15 years as he was concentrating on three dimensions. On an October stroll in 1843, he realized that the next successor dimension is four and not three; he named his mathematical entity a quaternion and described a quaternion, Q, as: Q ≡ a + bi + cj + dk with i2 = j2 = k2 = ijk = −1

(4.56)

It follows then: ij = k, ji = −k and so on. To wit, ijk = −1 ⇒ ijk2 = −k ⇒ ij = k because k2 = −1, and ji = −ji(ijk) = −j(i2 )jk = j2 k = −k. The multiplication of two quaternions with above rules as a four-dimensional entity is similar to that of complex numbers of two dimensions. For our purpose of geometrical interpretation and parameterization of the rotation matrices, we may regroup the parameters in the above description of a quaternion to rewrite it as a “scalar plus a vector”:

Q=(

V , s ) ⎵ ⎵

with v = (bi + cj + dk)

and

s=a

(4.57)

3D vector scalar

In order to explain the “scalar plus a vector” oddity, we may think of it as an addition of two underlying quaternions: Q = (v, s) = (v, 0) + (0, s) = Qv + Qs . Note that although, in modern mathematics, vectors are used far more than almost forgotten quaternions, historically speaking, quaternions preceded vectors. Only after the recognition of Euler–Rodrigues parameterization for rotational transformation has the quaternion received renewed interest among analysts.

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116

4.5.1.1

Quaternion Properties

r It follows, from Hamilton’s rules and equation (4.57), that notationally a quaternion is expressed component-wise as: Q = (v

s )T = ( q1 ≡ v1

q2 ≡ v2

q3 ≡ v3

q4 ≡ s )T

(4.58)

r Two quaternions are added (or subtracted) by simply adding (or subtracting) vector to vector and scalar to scalar.

r The multiplication rule is given by: Q1 = (v1 , s1 ) and Q2 = (v2 , s2 ) ⇒ Q ≡ Q1 Q2 = (s1 v2 + s2 v1 + v1 × v2 , s1 s2 − v1 ∙ v2 ) (4.59)

r

From equation (4.59), we can see that, except for the presence of the cross product, a quaternion multiplication has similar characteristics to the familiar (a1 + ( complex number multiplication: ) ib1 )(a2 + ib2 ) = i(a1 b2 + a2 b1 ) + (a1 a2 − b1 b2 ) = a1 b2 + a2 b1 , a1 a2 − b1 b2 . A pure or vector quaternion is a quaternion with the scalar part identically zero: Q ≡ (v, 0)

(4.60)

As will be seen later, this allows smooth transition from vector to quaternion and back.

r The conjugate, Q, ̃ of any quaternion, Q ≡ (v, s), is given by: ̃ ≡ (−v, s) Q

(4.61)

r The norm, ‖Q‖ of any quaternion, Q ≡ (v, s), is given by: ‖Q‖ =

√ v ∙ v + s2

 (( ̃ = ‖(( (+ It follows that: ‖QQ‖ −sv sv −  v × v,

s2 + v ∙ v)‖ = ‖(0,

(4.62) v ∙ v + s2 )‖ = ‖Q‖.

r The inverse quaternion, Q−1 of any quaternion, Q ≡ (v, s), is defined as: Q−1 ≡

r

̃ Q ‖Q‖

=

1 ‖Q‖

(−v, s)

(4.63)

‖ Q ̃ ‖ ‖ ‖Q‖ ‖QQ−1 ‖ = ‖ ‖Q ‖Q‖ ‖ = ‖Q‖ = 1 justifies the multiplicative inverse definition. ‖ ‖ √ A unit quaternion, Q, such that ‖Q‖ = v ∙ v + s2 = 1, converts to a rotation matrix under ̃ where the vector quaternion Qa embeds the vector a quaternion group operation, Q Qa Q that is undergoing rotation such that Qa ≡ (a, 0).

Note that under this quaternion operation, the resulting quaternion is also a vector ̃ = (b, 0) for some vector b. To wit, let us recall the idenquaternion, that is, Q Qa Q 2 2 tity: ‖v‖ I + V = (v ⊗ v) where v is the axial vector corresponding to the skew-symmetric matrix, V, that is, V = [v] (because, for any h ∈ R3 {‖v‖2 I + V2 }h = ‖v‖2 h + v × (v × h)

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117

 = (v ⊗ v) h) with (v ⊗ v) as the tensor product of two  2 ‖v‖ = h + (v ∙ h) v −  (v  ∙ v)h −v3 v2 ⎤ ⎡ 0 −v3 ⎡ 0 0 −v1 ⎥ ⎢ v3 0 (for column vectors, (v ⊗ v) = v vT ), and V2 = V V = ⎢ v3 ⎥⎢ ⎢ 0 ⎦ ⎣−v2 v1 ⎣−v2 v1 v1 v2 v3 v1 ⎤ ⎡−v22 − v23 ⎢ v1 v2 −v23 − v21 v2 v3 ⎥. ⎥ ⎢ v2 v3 −v21 − v22 ⎦ ⎣ v3 v1 So, now, noting that ‖v‖2 + s2 = 1 for the unit quaternion, Q = (v, s), we get:

vectors v2 ⎤ −v1 ⎥ = ⎥ 0 ⎦

̃ = (v, s)(a, 0)(−v, s) = (sa + v × a, −(v ∙ a))(−v, s) Q Qa Q = ((v ∙ a)v + s2 a + 2s(v × a) + v × (v × a),   −s(v ∙ a) +  s(v∙  a) +  (v  × a)  ∙v  = (((v ⊗ v) + 2sV + s2 I + V2 )a, 0) = ((‖v‖2 I + 2V2 + 2sV + s2 I)a, 0) = (((‖v‖2 + s2 )I + 2sV + 2V2 )a, 0) = ((I + 2sV + 2V2 )a, 0) = (b, 0) = Qb

(4.64)

where we have defined: b ≡ (I + 2sV + V2 )a = R a

(4.65)

with R ≡ I + 2sV + 2V2 ⎡1 − 2v22 − 2v23 = ⎢ 2(v1 v2 + sv3 ) ⎢ ⎣ 2(v1 v3 − sv2 )

2(v1 v2 − sv3 ) 1 − 2v21 − 2v23 2(v2 v3 + sv1 )

2(v1 v3 + sv2 ) ⎤ 2(v2 v3 − sv1 ) ⎥ ⎥ 1 − 2v21 − 2v22 ⎦

(4.66)

Notationally, this is equivalent to: ⎡ 1 − 2q22 − 2q23 2(q1 q2 − q3 q4 ) R = ⎢2(q1 q2 + q3 q4 ) 1 − 2q21 − 2q23 ⎢ ⎣2(q1 q3 − q2 q4 ) 2(q2 q3 − q1 q4 )

2(q1 q3 + q2 q4 )⎤ 2(q2 q3 − q1 q4 )⎥ ⎥ 1 − 2q21 − 2q22 ⎦

(4.67)

Or, noting that the quaternion, Q, is a unit quaternion, that is, ‖Q‖2 = ‖v‖2 + s2 = q21 + q22 + q23 + q24 = 1, we get: ⎡q21 − q22 − q23 + q24 R = ⎢ 2(q1 q2 + q3 q4 ) ⎢ ⎣ 2(q1 q3 − q2 q4 )

2(q1 q2 − q3 q4 ) −q21 + q22 − q23 + q24 2(q2 q3 − q1 q4 )

2(q1 q3 + q2 q4 ) ⎤ 2(q2 q3 − q1 q4 ) ⎥ ⎥ −q21 − q22 + q23 + q24 ⎦

(4.68)

Thus, from equations (4.64)–(4.68), we see that: ̃ , where Q is a unit quaternion, transforms the vector The quaternion group operation, Q Qa Q a to a vector b such that b = R a.

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118

Lemma: Prove that the matrix, R, is, in fact, a rotation matrix. Proof. Let us first recall that the characteristic polynomial, p(𝜆), of the skew symmetric matrix, V ≡ [v] ≡ (v×) of an axial vector, v, with length, ‖v‖, is given by: p(𝜆) = 𝜆3 + ‖v‖2 𝜆 = 0

(4.69)

and that the matrix, V ≡ [v] ≡ (v×), satisfies its own characteristic polynomial, that is, V3 = − ‖v‖2 V

(4.70)

V4 = − ‖v‖2 V2

(4.71)

From equation (4.70), it follows that,

Now, noting that the transpose of a skew-symmetric matrix, V ≡ [v] ≡ (v×), is given by: VT = −V, we have, from equation (4.66): RT ≡ I − 2sV + 2V2 . Thus, finally, using the relation given by equation (4.71) and noting that for the unit quaternion: ‖Q‖2 = ‖v‖2 + s2 = 1, we get: R RT = (I + 2sV + 2V2 )(I − 2sV + 2V2 )  3 3    + 2V2 +   − 4s2 V2 +  = I − 2sV −4sV 2sV 4sV + 2V2 + 4V4 ( (∥(v( = I −( ∥2 )4V2 = I (1 ( −( s2 − Thus, R is an orthogonal transformation, which completes the proof. Later, when we introduce the Hamilton–Rodrigues quaternion, we will be able to recognize that R is indeed a rotation matrix with the quaternion vector component, v, and the quaternion scalar part, s, associated with the axis and the amount of rotation, respectively.

r Also, just like a rotation matrix, a unit quaternion preserves dot products and cross products embedded in the quaternion product. To wit, given any two vector quaternions, Qa ≡ (a, 0) ̃ . It is interesting to note that ̃ Qb Q) ̃ = Q (Qa Qb ) Q and Qb ≡ (b, 0), we have: (Q Qa Q)(Q

r

it is the cross product in the quaternion multiplication rule that allows the rotation to confound axes. The quaternions form a non-commuting group, that is, a Lie group, under quaternion multiplication. However, quaternions with same vector parts commute because the cross product vanishes. To wit, if Q1 ≡ (v, s1 ) and Q2 ≡ (v, s2 ), then Q1 Q2 = ((s1 + s2 )v, s1 s2 − ‖v‖2 ) = Q2 Q1 . As we will be see, this tells us that any two consecutive rotations about the same axis are commutative, as expected.

Example: Show that if v in a unit quaternion, and Q ≡ (v, s), is an axis of rotation, then any rotation of a vector 𝜆v, a scaled vector of v, about v will keep 𝜆v undisturbed. To wit, let Q𝜆 ≡ (𝜆v, 0), then, noting that ‖Q‖2 = ‖v‖2 + s2 = 1, we have: ̃ = (v, s)(𝜆v, 0)(−v, s) = (𝜆sv, −𝜆 ‖v‖2 )(−v, s) Q Q𝜆 Q ( )     2 2 = (‖v‖2 + s2 )𝜆v, − 𝜆s + 𝜆s = (𝜆v, 0) ‖v‖ ‖v‖ Thus, Q𝜆 ≡ (𝜆v, 0) stays invariant under the quaternion operation, as expected.

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119

Discussion: A vectorial parameterization of a rotation matrix is most important for our purposes in the sense that the number of parameters (three) is same as the usual rotational degrees of freedom of structural elements such as beams, plates and shells for computational schemes like the finite element method. Although it is singular at some rotation, for most structural problems, it suffices. In the event that it is not, Hamilton–Rodrigues quaternions and parameterizations, as already mentioned, can be combined transparently to make it singularity free.

4.6

Hamilton–Rodrigues Quaternion

The main goal here is to present a computationally singularity-free parameterization of rigid body rotations. More specifically, we concentrate on a very important parameterization of a rotation matrix by the four-dimensional Hamilton–Rodrigues quaternion; this parameterization eliminates the singularity problems of parameterization by the Rodrigues vectors. Also, this possibility profoundly increases numerical efficiency and robustness by incorporating a strategy of transparently combining only a three-dimensional vector, say, Rodrigues vector, with this four-parameter Hamilton–Rodrigues parameterization of a rotation matrix instead of the computationally cumbersome nine parameters of a rotation matrix with six constraints.

4.6.1

4D Hamilton–Rodrigues Quaternion, Q = {q1

q2

q3

q4 }T

Recalling the discussion on the Rodrigues rotation parameterization, the Rodrigues rotation matrix becomes singular at those θ’s for which ‖𝜃‖ = (2n + 1)𝜋, n = 0, 1, ... Furthermore, recalling the discussion on rotation matrix, although there are only three independent parameters (nine minus six due to orthonormality constraints) in a rotation tensor, it is not possible for a three-parameter set such as the Rodrigues vector, s = tan( 𝜃2 ) n𝜃 ,

‖n𝜃 ‖ = 1 ,

θ

n𝜃 ≡ 𝜃 , 𝜃 = ‖θ‖, to uniquely describe all possible rotations; a non-singular one-to-one mapping takes at least five parameters while a non-singular many-to-one mapping requires at least four parameters. Definition

r The q3

four-dimensional unit Hamilton–Rodrigues quaternion (Figure 4.16), Q = {q1 q4 }T , defined as:

q1 ≡ sin ‖Q‖ ≡ 1

𝜃 n , 2 1

q2 ≡ sin

𝜃 n , 2 2

q3 ≡ sin

𝜃 n , 2 3

𝜃 q4 ≡ cos , 2

q2

(4.72)

where the unit vector, n𝜃 = (n1 n2 n3 )T = 𝜃1 (𝜃1 𝜃2 𝜃3 )T , is the Rodrigues axial vector along the rotation axis, θ, is a non-singular two-to-one map where globally both +Q and −Q represent the same rotation. Thus, a four-parameter Hamilton–Rodrigues quaternion will be adequate instead of a full five-parameter representation. The components of a point on the unit hypersphere (3-sphere) can be viewed as the components of a unit Hamilton–Rodrigues quaternion,Q. A

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120

Figure 4.16 Hamilton–Rodrigues quaternions.

comparison between the definitions of Euler–Rodrigues vector, X, and of Hamilton–Rodrigues quaternion, Q, as given by equation (4.72), shows that Q ≡ (q1 ≡ X1 , q2 ≡ X2 , q3 ≡ X3 , q4 ≡ X4 ), that is, Q ≡ {(sin 𝜃2 )

θ

𝜃

+ cos 𝜃2 },

̃ =1 ‖Q‖2 ≡ Q Q

(4.73)

̃ ≡ (̃q1 ≡ −X1 , q̃ 2 ≡ −X2 , q̃ 3 ≡ −X3 , q̃ 4 ≡ X4 ) as the conjugate quaternion of Q. with Q Remark

r With the above assignment, we see, as indicated, that the vector part of a quaternion, Q ≡ (v, s), represents some function of the axis of rotation, that is, v ≡ (sin 𝜃2 )

θ

𝜃

, and the scalar part

represents some function of the amount of rotation, that is, s ≡ cos 𝜃2 ; and the matrix R given by the equation (4.66) turns out to be a true rotation matrix. Then, any three-dimensional Euclidean point x ∈ R3 with the corresponding four-dimensional ⎛ ⎞ ⎜ ⎟ ̃ , vector or pure quaternion, Qx ≡ ⎜ x , 0⎟ , under quaternion operation given by Q Qx Q ⏟ ⏟ ⏟ ⎜ ⎟ ⎝ ∈R3 ⎠ will have an image y ∈ R3 lifted similarly to the four-dimensional vector quaternion, Qy ≡ ⎞ ⎛ ⎟ ⎜ ⎜ y , 0⎟ that lie on a 3-sphere (recall that a 3-sphere means a three-dimensional spherical ⎜⏟⏟⏟ ⎟ ⎟ ⎜ ⎠ ⎝ ∈R3 surface) embedded in a four-dimensional space such that: ̃ Qy = Q Qx Q

(4.74)

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121

Figure 4.17 Example: Hamilton–Rodrigues quaternion method.

with ⎧ y ⎫ ⎡q2 − q2 − q2 + q2 2 3 4 ⎪ 1⎪ ⎢ 1 ⎨ y2 ⎬ = ⎢ 2(q1 q2 + q3 q4 ) ⎪ y3 ⎪ ⎣ 2(q1 q3 − q2 q4 ) ⎩ ⎭

2(q1 q2 − q3 q4 ) −q21 + q22 − q23 + q24 2(q2 q3 − q1 q4 )

2(q1 q3 + q2 q4 ) ⎤ ⎧ x1 ⎫ ⎪ ⎪ 2(q2 q3 − q1 q4 ) ⎥ ⎨ x2 ⎬ , ⎥ −q21 − q22 + q23 + q24 ⎦ ⎪ x3 ⎪ ⎩ ⎭

y =R x

(4.75) Note that the matrix given in equation (4.75) is exactly the same as that given by equation (4.68) derived in the Hamilton quaternion section and the rotation matrix defined by the Euler– Rodrigues parameters. Thus, a spherical map, that is, a pure rotation, can always be described by the quaternion map given in equation (4.74). Additionally, note that the finite rotations, R, move around on a 3D unit spherical surface, i.e., a 3-sphere; the set of instantaneous rotations define a tangent hyperplane at a point on this unit sphere. Now, noting that ‖Q‖2 ≡ q21 + q22 + q23 + q24 = 1, we can express the rotation matrix, R, of equation (4.75), alternatively by: ⎡ 1 − 2q22 − 2q23 2(q1 q2 − q3 q4 ) R = ⎢2(q1 q2 + q3 q4 ) 1 − 2q21 − 2q23 ⎢ ⎣2(q1 q3 − q2 q4 ) 2(q2 q3 − q1 q4 )

2(q1 q3 + q2 q4 )⎤ 2(q2 q3 − q1 q4 )⎥ ⎥ 1 − 2q21 − 2q22 ⎦

(4.76)

Example As shown in Figure 4.17, a vector, a = (1, 0, 0), along the x-axis is rotated 90◦ about the z-axis to produce the vector, b = (0, 1, 0) along the y-axis such that b = R a. We are interested in the expression for the rotation matrix, R, according to the Hamilton–Rodrigues formula. Now, with 𝜃 = 𝜋2 , we have, from equation (4.72), the Hamilton–Rodrigues quaternions ) ) ( ( 1 1 1 ̃ x = 0 0 − √1 √ and Qa = (1 0 0 0). as: Qx = 0 0 √2 √2 , ‖Q‖ = 1, Q 2 2 Thus, using the quaternion(multiplication rule, as ) in equation (4.59), and i, j, k as the the unit base 1 1 ̃ x = (0i + j + 0k, 0), vectors, we get: Qx Qa = √2 i + √2 j + 0k, 0 , and finally, Qb = Qx Qa Q a unit vector quaternion, implying that b = (0 1 0)T as expected. Using the expression for Qx ⎡0 −1 0⎤ in equation (4.76), we get the rotation matrix as: R = ⎢1 0 0⎥ as expected. ⎢ ⎥ ⎣0 0 1⎦

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4.6.2

Transition: Rotation Matrix and Hamilton–Rodrigues Quaternion

As indicated before, a three-parameter vectorial parameterization, such as Rodrigues, of a rotation matrix is most important for our purposes because the number of parameters (three) is the same as the usual rotational degrees of freedom of structural elements such as beams, plates and shells for computational scheme such as the finite element method. Although it is singular at some rotation, as indicated above, for most structural problems, it suffices. In the event that it does not, a four-parameter Hamilton–Rodrigues non-singular quaternion parameterization, as presented above, can be combined user-transparently to make numerical treatment singularityfree. These are precisely the observations that suggest a smooth transition from the Rodrigues three-parameter θ-space to the Hamilton–Rodrigues four-parameter Q-quaternion space and back to obtain a singularity-free updating scheme for the rotation matrix for a typical nonlinear numerical analysis in structural engineering and mechanics.

4.6.2.1

Hamilton–Rodrigues Quaternion to Rotation Matrix

Given a unit Hamilton–Rodrigues quaternion, Q, we can construct the corresponding rotation matrix, R, by equation (4.76).

4.6.2.2

Rotation Matrix to Hamilton–Rodrigues Quaternion

Given the rotation matrix, R, as in equation (4.76), recovering the corresponding quaternion, Q, is a bit more involved but still a straightforward process. Algorithmically, we can describe it as follows: Step 0: Compute the corresponding scalar part,Q4 ≡ s ≡ X4 , of the quaternion as: Q24 ≡ s2 ≡ X42 = 14 (1 + trace(R)). If Q24 > 0, then, go to Step 1, else go to Step 2. Step 1: (It is immaterial whether we choose the positive or negative square root as both will represent the same rotation matrix; we will choose the positive square root), and Q4 ≡ s ≡ X4 =

1 2

Q2 ≡ v2 ≡ X2 =

√ (1 + trace(R)) 1 (R 4Q4 13

and

− R31 ) and

1 (R − R23 ) 4Q4 32 1 (R − R12 ) 4Q4 21

Q1 ≡ v1 ≡ X1 = Q3 ≡ v3 ≡ X3 =

STOP Step 2: Compute Q21 ≡ v21 ≡ X12 = − 12 (R22 + R33 ). If Q21 > 0, then, go to Step 3. else go to Step 4. Step 3: Q4 ≡ s ≡ X4 = 0 and Q2 ≡ v2 ≡ X2 =

1 R 2Q1 12

1 √ √ (−R22 − R33 ) 2 v3 ≡ X3 = 2Q1 R13 1

Q1 ≡ v1 ≡ X1 = and

Q3 ≡

Rotation Tensor

123

STOP Step 4: Compute Q22 ≡ v22 ≡ X22 = 12 (1 − R33 ). If Q22 > 0, then, go to Step 5. else go to Step 6. Step 5. Q4 ≡ s ≡ X4 = 0 and Q1 ≡ v1 ≡ X1 = 0 √ Q2 ≡ v2 ≡ X2 = √1 1 − R33 and Q3 ≡ v3 ≡ X3 = 2

1 R 2Q2 23

STOP Step 6: Q4 ≡ s ≡ X4 = 0 and Q1 ≡ v1 ≡ X1 = 0 Q2 ≡ v2 ≡ X2 = 0 and Q3 ≡ v3 ≡ X3 = 1

STOP

4.6.3

Interpolation and Approximation on a Rotational 3-Sphere

In interpolating the continuous rotational history from a few key rotations, or approximating the rotational history in a time series analysis in structural dynamics, we need to deal with the interpolation or approximation of a rotation tensor on a curved surface: a 3-sphere. We will be interested only in the linear interpolation of rotations for a nonlinear static problem or, say, in the linear approximation of rotational accelerations for nonlinear dynamic time history analysis.

4.6.3.1

Linear Interpolation or Approximation on 3-Sphere

Let A, B represent the unit Hamilton–Rodrigues quaternions for rotations at parameters t = 0 and t = 1, respectively, as shown in Figure 4.18. In The figure , the rotations are shown as points on a four-dimensional space; the rotation from the reference state, A, to a final state, B, is taken as a single rotation, which is always

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124

Figure 4.18 Spherically linear interpolation.

possible by Euler’s theorem. Now, let P(t), with ‖P‖ = 1, represents a rotation on this curve at an affine parameter t = 𝜆 ∈ [0, 1]. We are interested in expressing P in terns of A and B. Clearly, with ‖A‖ = ‖B‖ = ‖P‖ = 1, we have: A ∙ P = cos 𝜃 and A ∙ B = cos 𝜙 with parametric relation: 𝜃 = 𝜆𝜙. Now, let us assume that P is given as a linear combination of A and B by: P = 𝛼A + 𝛽B. Then, A ∙ P = 𝛼(A ∙ A) + 𝛽(A ∙ B) ⇒ cos 𝜃 = 𝛼 + 𝛽 cos 𝜙

(4.77)

With ‖P‖ = 1, we have: (𝛼A + 𝛽B) ∙ (𝛼A + 𝛽B) = 1 ⇒ 𝛼 2 + 2𝛼𝛽 cos 𝜙 + 𝛽 2 = 1. Replacing expression for 𝛼 in the previous expression by equation (4.77), we get: (cos 𝜃 − 𝛽 cos 𝜙)2 + 2𝛽(cos 𝜃 − 𝛽 cos 𝜙) cos 𝜙 + 𝛽 2 = 1 ⇒ 𝛽 =

Applying 𝛽 of equation (4.78) in equation (4.77), we get: 𝛼 =

P=

sin (1 − 𝜆) 𝜙 sin 𝜆𝜙 A+ B, sin 𝜙 sin 𝜙

sin 𝜃 sin 𝜙

(4.78)

sin (𝜙 − 𝜃) . Thus, finally: sin 𝜙

𝜆 ∈ [0.1]

(4.79)

Remark

r For

nonlinear structural dynamic problems involving finite rotations, we may choose the rotational acceleration distribution according to equation (4.79) for a predictor–corrector type of numerical analysis; the rotational velocity and rotation distribution may then be obtained by corresponding integration of the acceleration.

Rotation Tensor

4.6.3.2

125

Discussion

For numerical computations in the finite element phase, we will need the derivatives and the variational properties of a rotation matrix and the corresponding vector representations. Thus, for our subsequent computational needs, we must become familiar with the spatial and temporal derivatives and variations of rotation tensor field.

4.7

Derivatives, Angular Velocity and Variations

Here, our main goal is to determine derivative and variational properties of a rotation tensor field. In the beam finite rotation nonlinear analysis, we will encounter a one-parameter-rotation tensor field: R(S) where R is the rotation tensor with S as the parameter for the beam centroidal axis curve. Similarly, In the plate or shell finite rotation nonlinear analysis, we will have to deal with a two-parameter-rotation tensor field: R(S1 , S2 ) where R is the rotation tensor with S𝛼 , 𝛼 = 1, 2 as the arc length parameters along the lines of curvature of a plate or shell reference surface.

r These,

𝜕R in turn, necessitate evaluation of spatial derivatives: 𝜕R ≡ R,S ≡ R′ or 𝜕S 𝛼 ≡ R,S𝛼 , 𝜕S 𝛼 = 1, 2, of the rotation tensor field on a curved surface: SO(3), generating curvature tensor field.

Additionally, in the case of dynamic analysis, with rotation field given as: R(S, t) or R(S1 , S2 , t), where t ∈ R+ as the time variable,

r we will need 𝜕R tensor field.

𝜕t

∙

≡ R,t ≡ R, of the rotation tensor field generating rotational or angular velocity

Finally, for a variational method of numerical analysis, such as the finite element method, we will be required to define the admissible virtual space, corresponding to the beam, plate or shell configuration space consisting of the displacement vector field and the rotational tensor field, that identifies the spatial virtual strain fields and the corresponding spatial real strain fields in the weak form of equations of a beam centroidal axis or a shell reference surface.

r This, in turn, will again require, in essence, the one-parameter derivative of the rotation tensor field defining the variation of the rotation tensor. In each of these above situations, we will need the covariant derivative of the rotation tensor such that we can compose or combine rotations, so that the resulting object is still a rotation. Noting that the rotation field travels on a curved surface, the derivatives and the variations of the rotation field are carefully constructed with the help of the covariant derivative concept and the tangent space at identity, so(3) (which is also known as Lie algebra), the necessary differential geometric and the group properties of the rotation tensor as an orthogonal transformation of the special orthogonal matrix group, SO(3) which, because of its continuous topology and the algebraic structure, is also a Lie group.

4.7.1

Tangent Operator: Revisited

Let us recall that on a curved space, geometrical objects such as vectors and tensors must be defined locally instead of using a bi-local definition in the neighborhood, as in the flat Euclidean

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126

space. In other words, in an Euclidean (‘flat’) space, we can think of, say, a vector as joined by two neighboring points, but, in a curved space, a similar definition will yield a vector that may not even completely belong to the space. This necessitated the generalization from the definition of a vector as an arrow link between two neighboring points to a tangent operation to a curve (to be more specific, an affine curve for invariance or coordinate independence) at a point on the manifold. Thus, let the differentiable rotation tensor (or matrix) function R(𝜌) :→ SO(3) generates a continuous set of points defining a curve on SO(3), such that it passes through R(0) = R at 𝜌 = 0 | where R ∈ SO(3). Then,R,𝜌 , defined by the covariant derivative: R,𝜌 ≡ dR(𝜌) | , is the tangent d𝜌 |𝜌=0 tensor to SO(3) at R. The set of all tangent tensors at R, corresponding to various smooth curves passing though it, forms a vector space, denoted by TR SO(3). Now, by taking the derivative of the orthogonality property, RT R = I, and rearranging, we have:

R,𝜌 = K R ⇒ K ≡ R,𝜌 RT = R,𝜌 R−1

(4.80)

where we have introduced the following definition. Definition

r The

curvature tensor, K, the skew-symmetric matrix (because, with RRT = I, we have: K + KT = R,𝜌 RT + R R,T𝜌 = (RRT ),𝜌 = 0); K is thus the instantaneous and linearized skewsymmetric tensor at a rotation, R ∈ SO(3).

Because the tangent space is populated with the skew-symmetric matrices, the dimension of the tangent space is three (with only n(n − 1)∕2 elements in the matrix with n = 3 for the threedimensional space). Note also that the tangent space is a linear vector space.

4.7.1.1

Tangent Space at Identity: Revisited

Recall that the tangent space at identity, that is, at R = I, is given the special name, Lie algebra (recall that a vector space with a vector multiplication, [∙ , ∙], defined on it is an algebra) and denoted by so(3). The reason for this special treatment of the neighborhood at identity, I, is that the local structure of SO(3), the continuous rotation group, can be constructed out of this infinitesimal tangent space, so(3), by the exponential map given by equation (4.30); in other words, all the information of the group is contained in the algebra.

4.7.2

Adjoint Map and Left and Right Translations: Revisited

We have already seen that the elements of the Lie group, SO(3), are related to the corresponding elements of its Lie algebra, so(3), by the exponential map. Thus, we may define rotational variation ̄ = lim d (R𝜀 ) = lim d (R e𝜀 Φ̄ ) = R Φ, ̄ or through the exponential map in two different ways: R d𝜀 d𝜀 ̄ = lim R

d (R𝜀 ) 𝜀→0 d𝜀

̄ d 𝜀Φ (e 𝜀→0 d𝜀

= lim

𝜀→0

𝜀→0

̄ R where Φ ̄ ∈ so(3) is the instantaneous or infinitesimal R) = Φ

̄ ∈ SO(3) is the final rotation due to this variation. Recall now that another map variation and R known as the adjoint map also relates the elements of SO(3) with the corresponding elements

Rotation Tensor

127

of so(3). If an arbitrary rotation matrix, R ∈ SO(3), and its corresponding skew-symmetric matrix, S ∈ so(3), are given, then we have the skew-symmetric matrix,T, defined by the adjoint map: T ≡ R S RT , also as an element of so(3). We can easily see that because (T + TT = R S RT + R ST RT = R S RT − R S RT = 0) the symmetric part vanishes identically; it is a member of so(3). Thus, we have: T R = R S. Now, similar to equation (4.80), we have two ̄ In the first equality, ̄ by: R ̄ =Φ ̄ R = R Ψ. ways to superpose to represent the virtual rotation, R, ̄ is superposed on the current rotation; in the known as the left translation, the virtual rotation, Φ, second equality, known as the right translation, the current rotation is superposed on the virtual ̄ rotation, Ψ.

̄ Rotational Variation: R̄ and Φ

4.7.3

We choose the following form of superposition in all our subsequent applications: ̄ =Φ ̄ RT ̄ R ⇒ Φ ̄ =R R ( ) ̄ (R, 𝛿R) ≡ ∇𝜌 R ∙ 𝛿R ̄ (R, 𝛿Φ) ≡ (∇𝜌 R ∙ 𝛿Φ) where R and Φ

4.7.4

(4.81)

Material Curvature Tensor: R,𝜌 and K

Similarly, we can define the curvature tensor in two different ways as: K ≡ R,𝜌 RT or L ≡ RT R,𝜌 , and thus the spatial derivatives of the variables can have two representations. We choose the following form of superposition in all our subsequent applications:

R,𝜌 = K R

⇒

K = R,𝜌 RT

(4.82)

∙

Angular Velocity Tensor: R and Ω

4.7.5

Likewise, we can define the rotational or angular velocity tensor in two different ways as: ∙

∙

ΩS ≡ R RT or ΩM ≡ RT R, and thus the temporal derivatives of the variables can have two representations. We choose the following form of superposition in some of our subsequent applications: ∙

∙

R = ΩS R ⇒ ΩS ≡ R RT

4.7.6

(4.83)

Important Computational Identities

For future application in nonlinear analysis of beams, plates and shell structures, we introduce a collection of some useful results related to rotational transformation, that is parameterized by the Rodrigues vector, s = 𝜃1 tan( 𝜃2 ) θ, where θ is the familiar rotation vector.

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128

Fact D1: Given k, 𝜛 and 𝝓̄ as the axial vectors of the curvature tensor,K, the angular velocity ̄ respectively, we have: tensor, Ω, and the variation tensor,Φ, 2 K(𝜌) = ∑ (S S,𝜌 − S,𝜌 S + S,𝜌 ) 2 Φ(s, s̄ ) = ∑ (S S − S S + S)

and

∙ ∙ ∙ 2 Ω(t) = ∑ (S S − S S + S),

and

k(𝜌) = L1 (s) s,𝜌

𝝓̄ = L1 (s) s̄

and

(4.84) ∙

𝜛(t) = L1 (s) s

where L1 (s) ≡

2 (I + S) Σ

(4.85)

Proof D1. By taking direct one parameter, 𝜌, derivative of the rotation matrix in equation (4.43), applying the definition of K from equation (4.82), and through some algebra, we have the first of the equations. Then, in axial vector form, we have: k=

2 2 (s × s,𝜌 + s,𝜌 ) = (I + S) s,𝜌 = L1 (s) s,𝜌 Σ Σ ∙

as required. Similarly, replacing (),𝜌 by () and (̄)̄ in the above expressions, respectively, we get: ∙ 𝜛(t) = L1 (s) s and 𝝓̄ = L1 (s) s̄ , which completes the proof. Fact D2: L1 defined in equation (4.85) is given in terms of the rotation vector, θ, with the Rodrigues vector, s = 𝜃1 tan( 𝜃2 ) θ, as: ( L1 (θ) ≡ (1 + cos 𝜃) I + where Proof D2.

N≡

1 Θ, 𝜃

Following from equation (4.43):

and noting that: tan(𝜃∕2) = completes the proof.

2 sin(𝜃∕2) cos(𝜃∕2) 2 cos2 (𝜃∕2)

Θ = [θ], 2 ∑

=

) sin 𝜃 N (1 + cos 𝜃)

(4.86)

𝜃 = ‖θ‖

2 = 1+tan22 (𝜃∕2) = 2 cos2 (𝜃∕2) = 1 + cos 𝜃, 1+s∙s sin 𝜃 sin 𝜃 , and getting: S = (tan(𝜃∕2))N = 1+cos N, 1+cos 𝜃 𝜃

=

Fact D3: ∙

where

∙

̄ s,𝜌 = L2 (θ) θ,𝜌 , s = L2 (θ) θ, s̄ = L2 (θ) θ, ( ) sin 𝜃 − 𝜃 2 1 I− L2 (θ) = N 1 + cos 𝜃 𝜃

(4.87)

Proof D3. Using the definition in equation (4.43) and taking direct one parameter, 𝜌, derivative of (θ∙θ, ) the Rodrigues vector, s = 𝜃1 tan( 𝜃2 ) θ, and noting that ‖θ‖ ,𝜌 ≡ ‖θ‖′ = ‖θ‖𝜌 and n = ‖θ = θ𝜃 , θ‖

Rotation Tensor

129

we have: { { { ( ) } ( ) (θ ∙ θ,𝜌 ) } ( )} (θ ∙ θ,𝜌 ) 1 𝜃 𝜃 θ,𝜌 2 𝜃 s,𝜌 = tan θ− sec θ + tan 2 2 2 2 ‖θ‖ ‖θ‖2 ‖θ‖3 )( )) ( } ( ( ( )) { ( ) 𝜃 1 1 𝜃 𝜃 θ,𝜌 − 1 − = tan n ∙ θ,𝜌 n = tan 𝜃 2 sin 𝜃 𝜃 2 { ( ) } 𝜃 (n ⊗ n) θ,𝜌 I− 1− sin 𝜃 where n is the unit axial vector of N. Note that for column vectors: (n ⊗ n) = n nT . Now, using sin 𝜃 completes the the identity for skew-symmetric matrices: N2 = [n ⊗ n] − I and tan 𝜃2 = 1+cos 𝜃 ∙

proof of the first part. Similarly, replacing (),𝜌 by () and (̄)̄ in the above expressions, respectively, ∙

∙

̄ which completes the proof. we get: s = L2 (θ) θ and s̄ = L2 (θ) θ, Lemma L1: Given Fact D2 and Fact D3 as above, we have: ∙

k(𝜌) = W(θ) θ,𝜌 ,

𝜛(t) = W(θ) θ,

̄ = W(θ) θ̄ ̄ 𝝓(θ, θ)

(4.88)

where W(Θ) ≡ L1 L2 = I + c1 Θ{ + c2 Θ2 } W(θ) ≡ I + c1 [θ] + c2 (θ ⊗ θ) − ‖θ‖2 I 1 − cos 𝜃 𝜃 − sin 𝜃 where c1 ≡ and c2 ≡ 𝜃2 𝜃3

(4.89)

Proof L1. We directly pre-multiply by L1 , of equation (4.86), L2 , of equation (4.87), and recall that N, a skew-symmetric matrix with ∥ n ∥ = 1, has the characteristic equation given by: 𝜆3 + 𝜆 = 0. Now, noting that every matrix satisfies its characteristic equations, substitution of the relation: N3 + N = 0, and definition: N ≡ 𝜃1 Θ from Fact D2 completes the first expression. Now, with the use of the identity for the skew-symmetric tensor: Θ2 = (θ ⊗ θ) − ‖θ‖2 I where I is the three-dimensional identity tensor completes the proof. ̄ respectively, we have: Lemma LD1: Defining k and 𝝓̄ as the axial vectors of K and Φ, ̄ ̄ ̄ ̄ Φ(s), s = K + [K Φ − Φ K] ̄𝝓(s),s = k̄ + k × 𝝓̄

Proof LD1: Using relations (4.81) and (4.82), we have: ̄ s RT + (Φ ̄ s = R, ̄ R) RT ,s Φ, ̄ R RT + K Φ ̄ R RT + Φ ̄ R RT KT =K

(4.90)

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130

Now, noting the orthogonality property: R RT = I, and the skew-symmetry property: KT = −K completes the proof of the first part. Recognition of the Lie bracket property: for A, B skewsymmetric tensor, and a, b, corresponding axial vectors, [AB − BA] = [a × b], for the second part, completes the proof.

Remark

r W(θ)

in equation (4.89), fundamentally describes the first derivative of a rotation matrix parameterized by Rodrigues vector; for our future application, we will also need the second derivative (or the Hessian) of a rotation tensor, which leads us to the following lemma.

Lemma LD2: Given W(Θ) as in equation (4.89), and defining one parameter spatial derivative: [ ] ∙ 𝜕 d (∙), variation: ā (x, x̄ ) ≡ d𝜌 a(x + 𝜌̄x) and time derivative: (∙) ≡ 𝜕t𝜕 (∙), we have: (∙)′ ≡ 𝜕𝜌 𝜌=0

W′ (θ) ≡ c1 Θ′ + c2 {(θ ⊗ θ′ ) + (θ′ ⊗ θ)} + {(c2 − c1 )I + c3 Θ + c4 (θ ⊗ θ)}(θ ∙ θ′ ) W(θ, θ) ≡ c1 Θ + c2 {(θ ⊗ θ) + (θ ⊗ θ)} + {(c2 − c1 )I + c3 Θ + c4 (θ ⊗ θ)}(θ ∙ θ) ∙

∙

∙

∙

∙

W(θ) ≡ c1 Θ +c2 {(θ ⊗ θ) + (θ ⊗θ)} + {(c2 − c1 )I + c3 Θ + c4 (θ ⊗ θ)}(θ ∙ θ) c1 ≡

1 (1 − cos 𝜃) 𝜃2

c2 ≡

1 (𝜃 − sin 𝜃) 𝜃3

c3 ≡

1 (1 − 2c1 − 𝜃 2 c2 ) 𝜃2

c4 ≡

1 (c1 − 3c2 ) 𝜃2 (4.91)

̄ and taking variation of c Proof LD2. First noting that 𝜃⃖⃖⃖⃗2 ≡ ‖θ‖2 = (θ ∙ θ)2 ⇒ 𝜃̄ = 𝜃1 (θ ∙ θ) 1 ̄ and c̄ = c (θ ∙ θ) ̄ and taking variation of W in equation (4.89), and c2 , we have: c̄ 1 = c3 (θ ∙ θ) 2 4 ̄ Now, recalling the identities ̄ = −c θ̄ + c (ΘΘ ̄ + ΘΘ) ̄ + (c3 Θ + c4 Θ2 )(θ ∙ θ). ̄ we get: W(Θ, θ) 1 2 ̄ + [θ̄ ⊗ θ] − 2(𝜽θ)I, ̄ and replacing all super̄ + ΘΘ ̄ = [θ ⊗ θ] for skew-symmetric matrices: ΘΘ ∙

scripted barred variables, (∙), by superscripted dot, (∙), and primed, (∙)′ , variables completes the proof. Finally, we look at the central lemma for the nonlinear finite element analysis by Newton-type method.

Central Duality Lemma: Given W, c1 and c2 as in equation (4.89), we have: ̄ θ, = X(θ, θ, ) θ̄ ̄ W(θ, θ) 𝜌 𝜌 ΔW(θ, Δθ) θ,𝜌 = X(θ, θ,𝜌 ) Δθ

(4.92)

Rotation Tensor

131

where X ≡ −c1 Θ,𝜌 +c2 {(θ ⊗ θ,𝜌 ) − (θ,𝜌 ⊗ θ) + (θ ∙ θ,𝜌 )I} + V (θ,𝜌 ⊗ θ) where

V ≡ −c2 I + c3 Θ + c4 Θ2 1 {1 − 2c1 − 𝜃 2 c2 } 𝜃2 1 c4 ≡ 2 (c1 − 3c2 ) 𝜃

c3 ≡

(4.93)

where θ̄ and Δθ are virtual (variational) rotation and incremental rotation vectors, respectively.

Proof of Central Lemma: Noting the expression for W(θ, θ) from equation (4.91) and recall̄ + [θ̄ ⊗ θ] − 2(θ ∙ θ)I, ̄ and ̄ + ΘΘ ̄ = [θ ⊗ θ] ing the identities for skew-symmetric matrices: ΘΘ ̄ ̄ (θ ⊗ θ)θ,𝜌 = (θ ⊗ θ,𝜌 )θ, we have: ̄ θ, = (c Θ + c {(θ ⊗ θ) + (θ ⊗ θ)} + {(c − c )I + c Θ + c (θ ⊗ θ)}(θ ∙ θ)) θ, ̄ W(θ, θ) 𝜌 1 2 2 1 3 4 𝜌 = (−c1 Θ,𝜌 +c2 {(θ ⊗ θ,𝜌 ) − (θ,𝜌 ⊗ θ) + (θ ∙ θ,𝜌 )I} + {−c2 I + c3 Θ + c4 Θ2 }(θ,𝜌 ⊗ θ)) θ̄ = X(θ, θ,𝜌 ) θ̄ proves the first part of the equation (4.92). Now, replacing the (̄)̄ terms by the Δ() completes the proof.

4.7.7

Where We Would Like to Go

In the foregoing, we have introduced various parameterizations of rotation tensor and derived all the necessary relationships to be used in our beam, plate and shell analyses. To get familiarized with the overall computational strategy for these structures, we need to explore:

r the road map: nonlinear beams (Chapter 10) r the road map: nonlinear plates and shells (Chapter 11)

Part II Essential Mesh Generation

5 Curves: Theory and Computation 5.1 Introduction 5.1.1 What We Should Already Know Vectors, derivatives, vector dot product, vector cross product, vector triple product, basis functions, interpolation, approximation, conics, continuous functions, polynomials, forward difference operators, linear vector space, power basis.

5.1.2 Why We Are Here The main goal here is to present a topical discussion of the basic theory and computation of curves, and define relevant curve properties necessary for both mesh generation and finite element analysis phases of curved beams. However, as a start, let us remind ourselves that all of our theoretical discussions are primarily aimed at achieving robust computational methods and to guide us with useful practical tips. More specifically, we define and discuss the properties of parametric curves as opposed to their implicit description – to investigate computational Bezier–Bernstein–de Casteljau and Schoenberg–de Boor–Cox B-spline parametric curves. In the interactive mesh generation phase, we are interested in developing the final model of structures. Starting from a few key points of information, we would like to create the subassemblies of the model, and with the appropriate continuity conditions among the subassemblies eventually we will blend them into one integral mesh. In the finite element analysis phase, the mesh is used for structural integrity under various load conditions by determining key entities or field quantities such as displacements, stresses, and so on. Finally, for cost effectiveness in numerical analysis in the form of interpolations in the mesh generation phase and approximations in the finite element phases, we strive for elements that are as large as possible without compromising accuracy. Thus, in order that the interpolation and approximation algorithms can be easily understood, here we continue with the fundamental geometric entity: curves. There are two basic ways to represent a curve – implicit form or parametric form. For various reasons, the implicit description appears less promising than the parametric form for our purpose. Within the parametric domain, for several reasons, we prefer to represent curves in the Bezier–Bernstein–de Casteljau and Schoenberg-de Boor B-spline forms to various other possible forms such as piecewise Lagrangian and Hermite polynomials. The common shapes in conics such as circle, parabola and ellipse, need special attention under our choice of the parametric form. It turns out that in order to describe these shapes exactly, we must resort to rational Bezier and rational B-spline curves. Implementationally, Computation of Nonlinear Structures: Extremely Large Elements for Frames, Plates and Shells, First Edition. Debabrata Ray. © 2016 John Wiley & Sons, Ltd. Published 2016 by John Wiley & Sons, Ltd.

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Bezier curves and B-splines by composite Bezier construction act as the cornerstone of our computer-aided geometric modeling algorithms. We make these also the building blocks of our proposed finite element analysis:

r We begin with the differential geometry of real curves. r For an introduction to the fundamental concepts and properties of Bezier–Bernstein–de Casteljau, we introduce non-rational Bezier–Bernstein–de Casteljau curves.

r For parametric descriptions of conics and similar curves, we investigate rational Bezier and rational B-spine curves and NURBs.

r For an introduction to the fundamental concepts and properties of de Boor–B-Spline forms of representation of curves, we conclude with deBoor-Cox B-spline curves.

r For a unified implementation on the computer of Bezier curves and B-spine curves in Bezier form, we conclude with computer implementation of Bezier and B-spline curves.

5.1.3 How We Get It We introduce some fundamental concepts in differential geometry of parametric curves, such as the Frenet frame and Frenet–Serret derivatives with the definitions for curvature and twist; then we specialize in Bernstein–Bezier curves and de Boor–Cox B-spline curves, rational and non-rational, for computational purposes.

5.1.4

What We Need to Recall

In what follows, we touch upon some fundamental ideas and concepts that will be eventually utilized to introduce the concepts of computational curves. As indicated above, we need to recall the linear vector space, the real coordinate space and Euclidean ‘flat’ space for the introduction of an affine space.

5.2 Affine Transformation and Ratios 5.2.1 Affine Transformation Let us recall that the two-point (more specifically, the difference of two points) definition of vectors is easy to grasp geometrically, but it does not help generalization to higher-order invariant entities such as tensors. Thus, for an alternative definition, we resort to appropriate transformations and identify the transformation properties of vectors for their definition under such transformations. For this, we consider the affine transformations in a Euclidean geometry of the type: X′ = A X + b

and

Xi′ = Aij Xj + bi

(5.1)

where A is interpreted as a constant orthogonal transformation such that A AT = I and det A = 1, and b is a translation; X and X′ are the points before and after the transformation, respectively, but an alternative interpretation could be that X and X′ are the coordinates of the same point in two different Cartesian coordinate systems, before and after the above transformation. Clearly, equation (5.1) is not a linear map because of the presence of the translation term, b. However, since a vector remains invariant under translation, the linear transformation defining vectors may be written as: X′ = A X

(5.2)

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Component-wise, equation (5.2) can be expressed, with the Einstein summation convention, as: Xi′ = Aij Xj

(5.3)

Incidentally, for the component forms in equations (5.1) and (5.3), we have somewhat resorted to concepts that appeared earlier in the book: Aij ’s are the matrix components of tensor A expressed in Cartesian coordinate base tensors in dyadic notation as: A = Aij ii ⊗ ij ; in other words, the matrices to a tensor are similar to what the vector components are to a vector. Examples of affine transformations abound in computer-aided geometric design (CAGD) (referring to equation (5.1)):

r Translation: A = 0 and b non-zero. r Identity: A = I and b = 0. r Rotation: A = R and b = 0 with R as a rotation matrix: RRT = I and det R = 1. r Scaling: b = 0 and A is a diagonal matrix with entries defining scaling of a vector. r Shear: b = 0 and A is a matrix with diagonal entries all unity and the off-diagonal terms defining shearing of a vector. Now, we are in a position to redefine a vector based on transformation property:

r an n-dimensional vector is an ordered system of n numbers called its components that transfer according to equation (5.3) when subjected to an affine transformation given by equation (5.1). For example, if we are given two points, p and q, and transform them to points, p′ and respectively, such that: p′i = Aij pj + bi and q′i = Aij qj + bi , and take the difference, we get: ′ pi − q′i = Aij (pj − qj ), that is, since the component transformations satisfy equation (5.3), p − q must be a vector by the given definition; in other words, the difference of two points that previously defined a vector is identical to the current definition of a vector based on the transformation property. As we have seen, the vector definition by transformation can be generalized to higherorder invariant geometric objects such as tensors. It is important to note that a barycentric combination of points is preserved, that is, remains invariant, under the affine transformation given by equation (5.1). This is very important since geometric shapes, usually described by the barycentric combination of points in CAGD, remain invariant under affine coordinate transformation. To wit, let us consider a point, a, given as ∑n a barycentric combination as a = c0 a0 + c1 a1 + ⋯ + cn an = ci ai , i=0 ci = 1. Now, applying the affine transformation, we will get a point, ā , such that: ā = c0 a0 + c1 a1 + ⋯ + cn an = ∑n ci ai , i=0 ci = 1, because: q′ ,

) (∑ ) ∑ (∑ ci ai + ci b ā = Aa + b = A ci ai + b = A ∑ ∑ ∑ ∑ = ci Aai + ci b = ci (Aai + b) = ci ai

r Position

since

∑

ci = 1

or radius vector: Finally, for analytical and computational purposes, if we fix a coordinate system by choosing a point with all components identically zero, then the vector formed by the difference of any point and this point is known as the position or radius vector; the components of the position vector are identical to those of the other point. Thus, given a coordinate system, any point of an affine space is given by a position vector. We note too that in the absence of translation, the position vectors transform according to transformation of the type given by equation (5.3) and not by equation (5.1).

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Figure 5.1 Linear interpolation and ratio.

5.2.2 Linear Interpolation and Ratio Invariance One of the simplest barycentric combinations is exemplified by the linear interpolation, as shown in Figure 5.1; given any two points a and b as end points of a straight line segment, any point c(𝜉) may be generated as a linear interpolation of the end points, and parametrically expressed as: c(𝜉) = (1 − 𝜉)a + 𝜉b with parameter 𝜉 ∈ [0, 1]. We can also see this as a map of a point 𝜉 on the real line segment belonging to one-dimensional real space with end points 0 and 1: 𝜉 = (1 − 𝜉)0 + 𝜉1, to the three-dimensional affine space; clearly, the barycentric combination is preserved under this map. Thus, linear interpolation is an affine map; in other words, linear interpolation is affinely invariant. Δ

Δ

Now, for any three collinear points, a, b, c, the barycentric coefficients, 𝛼1 = (1 − 𝜉) and 𝛼2 = 𝜉 such that b = 𝛼1 a + 𝛼2 c, 𝛼1 + 𝛼2 = 1, can also be given as: 𝛼1 ≡

area (b, c) ‖c − b‖ = area (a, c) ‖c − a‖

and

𝛼2 ≡

area (a, b) ‖b − a‖ = area (a, c) ‖c − a‖

(5.4)

where “area” is included to accommodate definitions for planar geometry and stands for absolute length in one-dimensional objects. Then, what is known as ratio is defined as: ratio(a, b, c) ≡

𝜉 area (a, b) 𝛼2 = = area (b, c) 𝛼1 1−𝜉

(5.5)

Because, as shown above, the barycentric coefficients are invariant under affine transformation, the equation (5.5) states that ratios are preserved under barycentric combination. As will become clear later, the concept of ratio forms the backbone of the de Casteljau algorithm of non-rational curve generation by linear interpolation.

5.2.3 Four-dimensional Homogeneous Coordinate System As already indicated, an affine map is not linear because of the translational vector; in other words, given a function or map, f (x), and scalars, a, b, c, it does not satisfy homogeneity, that is, f (cx) = c f (x), or additivity, that is, f (ax + by) = a f (x) + b f (y). To wit, let the affine map be given by equation (5.1) with X′ = A X + b and Y′ = A Y + b; clearly, additivity is violated: A (X + Y) + b ≠ X′ + Y′ (= A X + A Y + 2b), and so is homogeneity. In view of this drawback, computer graphics that depend on matrix manipulations (i.e. linear operations) cannot be efficiently performed. A clever and ingenious device to circumvent this problem is to use a dimensionally augmented coordinate system known as a 4D homogeneous coordinate system;

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thus, the affine map given by equation (5.1) can be rewritten in homogeneous coordinate system, (X, w) as: ⎧ X1′ ⎫ ⎡ A11 ⎪ ′⎪ ⎢ ⎪ X2 ⎪ ⎢ A21 ⎨ ′⎬=⎢ ⎪ X3 ⎪ ⎢ A31 ⎪w⎪ ⎣ 0 ⎩ ⎭

A12

A13

A22

A23

A32

A33

0

0

b1 ⎤ ⎧ X1 ⎫ ⎪ ⎪ b2 ⎥ ⎪ X2 ⎪ ⎥⎨ ⎬ b3 ⎥ ⎪ X3 ⎪ ⎥ ⎪ 1 ⎦⎪ ⎩w⎭

(5.6)

As it turns out, for both non-rational and rational curves that we are about to delve into, we will utilize this 4D homogeneous coordinate system in computer coding as a most natural artifice or apparatus.

5.3

Real Parametric Curves: Differential Geometry

Let us consider three-dimensional Euclidean space (more precisely, affine space), and real curves. If an origin is chosen, then every point in space can be identified with a position vector, C = )T ( X ≡ X1 Y ≡ X2 Z ≡ X3 .

5.3.1 Real Curve A real curve is defined by a real vector function, C(𝜉), where 𝜉 ∈ [a, b], that is, a ≤ 𝜉 ≤ b is a real parameter, as shown in Figure 5.2. In component curve form: ⎧ X (𝜉) ⎫ ⎪ 1 ⎪ C(𝜉) = ⎨ X2 (𝜉) ⎬ , ⎪ X3 (𝜉) ⎪ ⎭ ⎩

𝜉 ∈ [a, b]

Figure 5.2 Real parametric curve.

(5.7)

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5.3.2

Derivative of a Curve

We will define the derivative of the curve at any parameter, 𝜉, by: C,𝜉 (𝜉) ≡

5.3.3

dC(𝜉) = d𝜉

(

dX1 (𝜉) dX2 (𝜉) dX3 (𝜉) d𝜉 d𝜉 d𝜉

)T (5.8)

Regular Curve

A regular curve will be defined as the curve, C(𝜉), where the derivatives of the component curves do not all simultaneously vanish at any parameter, 𝜉 ∈ [a, b], that is, C,𝜉 (𝜉) ≠ 0 vector. A point where a curve is regular is called an ordinary point, otherwise a singular point. As a counter example to regular curves, we may consider a curve: C(𝜉) = (1 − f (𝜉))C(𝜉0 ) + f (𝜉) C(𝜉1 ), 𝜉 ∈ df [𝜉0 , 𝜉1 ] such that the function f (𝜉) has stationary points, that is, d𝜉 = 0 at some 𝜉 ∈ [𝜉0 , 𝜉1 ]; then, df = (C(𝜉1 ) − C(𝜉0 )) d𝜉
= 0. As a general rule, if a curve has a cusp, then at that point,
either discontinuous or zero. dC d𝜉

5.3.4

dC d𝜉

is

Allowable Parametric Representation

A representation where a curve is regular and continuously differentiable r(≥ 1) many times, depending on a problem, is called an allowable parametric representation. The curve is said to be of class Cr . Thus, from now on, we will always assume our curves to be regular and of class Cr , r(≥ 1). In other words, we will assume that our curves will always be as smooth as necessary.

5.3.5 Allowable Parameter Transformations: Regular Reparameterization We define an allowable parameter transformation as of type: 𝜏 = 𝜏(𝜉), where 𝜏 is a differentiable function of 𝜉, and d𝜏 ≠ 0 for the entire interval that will not change the shape of the curve. d𝜉 Clearly, then, 𝜏(𝜉) is an invertible function and we can find: 𝜉 = 𝜉(𝜏).

5.3.6

Invariant Arc Length Parameterization

A particular parameter transformation is of interest to us: the arc length parameterization where S, the arc length is related to 𝜉 by: 𝜉

S=

∫a

||C,𝜉 (𝜉)|| d𝜉

(5.9)

√ where, ||C,𝜉 (𝜉)|| ≡ C,𝜉 (𝜉) ∙ C,𝜉 (𝜉) with ∙ the usual vector dot product. Because, C,𝜉 (𝜉) d𝜉 ≡ dC d𝜏 d𝜉 = dC d𝜉 = C,𝜏 (𝜏) d𝜏, the arc length parameterization does not depend on the original d𝜉 d𝜏 d𝜉 parameter, and hence, is an invariant parameterization. Also, by considering the above repreS sentation as: S = ∫0 dS, we get for an arc element – also called a linear element – of a curve: dS = ||C,𝜉 (𝜉)|| d𝜉 ≡

√ C,𝜉 (𝜉) ∙ C,𝜉 (𝜉) d𝜉

(5.10)

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Remarks

r Equation (5.10) gives us, component-wise, the usual relation: dS =

√ dX 2 + dY 2 + dZ 2 d𝜉

(5.11)

r In order to maintain that C,𝜉 (𝜉) ≠ 0 at all 𝜉, the arc length parameter, S, must monotonously increase, that is, 𝜉 > 𝜉 ⇒ S(𝜉 ) > S(𝜉 ).

1 0 1 0 r As a counter example to regular curves, we may consider a curve: C(𝜉) = (1 − f (𝜉))C(𝜉0 ) +

f (𝜉) C(𝜉1 ),

𝜉 ∈ [𝜉0 , 𝜉1 ] such that the function f (𝜉) has stationary points, that is,

some 𝜉 ∈ [𝜉0 , 𝜉1 ]; then,

dC d𝜉

df d𝜉

= 0 at

= (C(𝜉1 ) − C(𝜉0 )) = 0.

5.3.7 Curve Tangent Vector As shown in Figure 5.3, let us consider two points, C(𝜉) and C(𝜉 + Δ𝜉), on a curve at parameters, 𝜉 and 𝜉 + Δ𝜉. Now, if the curve is regular at 𝜉 and as Δ𝜉 → 0 in the limit, we get, by definition: C,𝜉 (𝜉) = lim C(𝜉+Δ𝜉)−C(𝜉) = dC is the tangent vector to the curve at 𝜉. Δ𝜉 d𝜉 Δ𝜉→0

5.3.8 Unit Tangent Vector, T Now, as shown in Figure 5.4, considering the arc length parameter, S, and relation (5.10), we get:

dC C,𝜉 d𝜉 d𝜉 dC dC = C′ (S) ≡ = = = unit vector along C,𝜉 dS dS d𝜉 dS ||C,𝜉 || d𝜉

(5.12)

In other words, the unit tangent vector at a point on a curve is given by the derivative of the position vector at the point with respect to the arc length parameter. The unit tangent vector will be denoted by T.

Figure 5.3 Tangent vector.

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Figure 5.4 Unit tangent vector.

Remarks

r From now on, notationally, we will use: (∙)′ ≡ with respect to arc length parameter.

d(∙) , dS

that is, the derivative at a point on a curve

Example: Let us consider the circle, shown in Figure 5.5, with points on it that are given by: C(𝜃) = a cos 𝜃 e1 + a sin 𝜃 e2 where e1 and e2 are two-dimensional unit Cartesian base vectors; then, the unit tangent vector is given as: Δ

T (𝜃) =

C,𝜃 −a sin 𝜃 e1 + a cos 𝜃e2 = − sin 𝜃e1 + cos 𝜃e2 = √ ‖C, ‖ ‖ 𝜃‖ a2 sin2 𝜃 + a2 cos2 𝜃

5.3.9 Normal Plane As shown in Figure 5.6, a plane that passes through a curve at a point, say, 𝜉 0 , and whose normal is the tangent to the curve at the point, 𝜉 0 , is called a normal plane. Clearly, then, if P is a point on the plane, the normal plane is given by equation: (P − C(𝜉 0 )) ∙ C,𝜉 (𝜉 0 ) = 0.

Figure 5.5 Example of a tangent vector of circle.

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Figure 5.6 Normal plane.

5.3.10 The Frenet Frame: Principal Normal Vector, M, and Bi-normal Vector, B The tangent at each point to a curve (we assume Cr , r ≥ 2) continuously changes as we move from point to point along the curve. It will be interesting to see how the tangents change along the curve. In fact, as we will see shortly, this will enable us to define a very important and suitable local Cartesian frame at a point on the curve. Based on Figure 5.7, by applying Taylor series approximation in the neighborhood of a point, C(𝜉), on the curve, we have: 1 1 C(𝜉 + Δ𝜉) = C(𝜉) + (Δ𝜉) C,𝜉 (𝜉) + (Δ𝜉)2 C,𝜉𝜉 (𝜉) + (Δ𝜉)3 C,𝜉𝜉𝜉 (𝜉) + higher order terms 2 6 (5.13) If we assume that C,𝜉 (𝜉), C,𝜉𝜉 (𝜉) and C,𝜉𝜉𝜉 (𝜉) are linearly independent then they can form an affine coordinate system.

Figure 5.7 Frenet frame by Gram–Schmidt orthogonalization.

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Remarks

r We note that if C,𝜉 (𝜉) and C,𝜉𝜉 (𝜉) are linearly dependent for all 𝜉, then the curve is a straight line.

Now, going back to our main discussion, we can apply Gram–Schmidt orthogonalization to these basis vectors to get a triad of orthonormal basis vectors forming a local Cartesian frame with origin at C(𝜉). Let us denote these basis vectors by the T, M and B defined by:

T= B=

C,𝜉 (𝜉) ||C,𝜉 (𝜉)|| C,𝜉 (𝜉) × C,𝜉𝜉 (𝜉)

(5.14)

||C,𝜉 (𝜉) × C,𝜉𝜉 (𝜉)||

M = B×T Definition

r The local Cartesian frame defined by T, M and B is known as the Frenet frame. r T is easily recognized as the unit tangent vector defined earlier. r M is known as the principal normal vector. r B is known as the bi-normal vector. Remarks

r From the orthogonalization process, we can write the direct expression for normal vector in terms of the curve properties as follows:

M=

(C,𝜉 ∙ C,𝜉 )C,𝜉𝜉 −(C,𝜉𝜉 ∙ C,𝜉 )C,𝜉 ||(C,𝜉 ∙ C,𝜉 )C,𝜉𝜉 −(C,𝜉𝜉 ∙ C,𝜉 )C,𝜉 ||

(5.15)

This equation is equally applicable to planar curves where we note that B coincides with the normal to the plane. As we shall see later, for both theoretical and computational purposes, the Frenet frame helps to bring out various properties of a curve in the most useful manner.

5.3.11 Osculating Plane and Rectifying Plane An osculating plane of a curve at a point is defined as the plane passing through the point and containing the tangent vector, T, and the principal normal vector, M. The osculating plane is given by the vector triple product: (P − C(𝜉 o )) ∙ (C,𝜉 (𝜉 o ) × C,𝜉𝜉 (𝜉 o )) = 0 where P is a point on the osculating plane and 𝜉 o is the point on the curve.

(5.16)

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Figure 5.8 Local axes on a curve.

As shown in Figure 5.8, a plane passing through a point on the curve and spanned by the normal vector, M and the bi-normal vector, B, is known as the rectifying plane.

5.3.12 Discussion

r In the foregoing, we have covered the definition and some of the basic properties of parametric curves and associated Frenet frames. However, for both finite element analysis and mesh generation, we would also be interested in how the Frenet frames are related as we move along the curve points, in other words, the derivatives of the Frenet frame. This gives us some very important properties, such as curvature and twist, of a general three-dimensional curve. Also, for computational purposes, both in mesh generation and finite element phases, a constructive description of curves is necessary that is easy to manipulate and calculate on computers.

5.4

Frenet–Serret Derivatives

Here, our main goal is to get acquainted with additional important properties – curvature and twist – of a curve, This, in turn, requires us to investigate the derivatives, known as the Frenet– Serret derivatives, of the Frenet frames of a general three-dimensional curve. In what follows, Δ d(∙) . dS

we will use the notation: (∙)′ =

5.4.1 Frenet–Serret Derivatives Let us consider the Frenet frame, T(S), M(S), B(S), as given by equation (5.14), with S as the arc length parameter, and all three axes being unit in length, that is, ‖T‖ = ‖M‖ = ‖B‖ = 1; clearly, then: ‖T‖′ = (T ∙ T)′ = T′ ∙ T = 0 ⇒ M′ ∙ M = 0 and B′ ∙ B = 0

(5.17)

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From equation (5.17), each of the primed vectors are perpendicular to the corresponding Frenet axis; specifically, T′ belongs to the M − B plane, M′ belongs to the B − T plane and B′ belongs to T − M plane. So, let us represent the primed vectors in terms of the Frenet frame as: T′ = 𝜅M + 𝜆B M′ = 𝜏B + 𝜉T B′ = 𝛾T + 𝜂M

(5.18)

With the Frenet axes perpendicular to each other, that is, T ∙ M = T ∙ B = M ∙ B = 0, we get:

T′ ∙ M = −T ∙ M′ ,

T′ ∙ B = −T ∙ B′

and

M′ ∙ B = −M ∙ B′

(5.19)

Using equation (5.19) in equation (5.18), we get: 𝜉 = −𝜅,

𝜂 = −𝜏

and

𝛾 = −𝜆

(5.20)

Now, applying equation (5.18) in equation (5.18) with 𝜆 ≡ 0, that is, choosing T′ along the principal normal, M, we get: T′ = 𝜅M M′ = 𝜏B − 𝜅T B′ = −𝜏M

(5.21)

In matrix form, we can rewrite equation (5.21) as: ′

⎧ T′ ⎫ ⎡ 0 ⎧T⎫ ⎧T⎫ ⎪ ⎪ d ⎪ ⎪ ⎪ ⎪ M ⎬ = ⎨ M ⎬ ≡ ⎨ M′ ⎬ = ⎢ −𝜅 dS ⎨ ′ ⎪ B ⎪ ⎢⎣ 0 ⎪B⎪ ⎪B⎪ ⎭ ⎩ ⎩ ⎭ ⎩ ⎭

k 0 −𝜏

0⎤⎧ T ⎫ ⎪ ⎪ 𝜏 ⎥⎨M⎬ ⎥⎪ ⎪ 0⎦ B ⎩ ⎭

(5.22)

It is clear, from equation (5.22), that the matrix is skew-symmetric and hence has a corresponding axial vector given by: ⎧ 𝜏(S) ⎫ ⎪ ⎪ d(S) ≡ ⎨ 0 ⎬ ⎪ 𝜅(S) ⎪ ⎩ ⎭

(5.23)

where the vector d(S) is known as Darboux’s vector; with a = T, M, B, we can express the = d × a = [d×] a. derivatives of the Frenet axes as: da dS 5.4.1.1

Curvature and Radius of Curvature

In equation (5.21), 𝜅 is known as the curvature at a point of a curve depicting the rate at which the curve departs from its tangent, as shown in Figure 5.9.

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Figure 5.9 Curvature of a curve.

The derivative of the unit tangent vector, T, by definition, is given by: T′ (S) = C′′ = lim

dS→0

T(S+dS)−T(S) dS

d𝜃 dS→0 dS

= lim

= 𝜌1 . Now, from Figure 5.9 and the Frenet frame

relation, as in equation (5.21), we get the relation between the curvature, 𝜅, and the radius of curvature, 𝜌, at a point on the curve as: ′ ‖ ‖ ′′ ‖ 𝜅 = ‖𝜅M‖ = ‖ ‖T (S)‖ = ‖C ‖ =

d𝜃 1 = dS 𝜌

(5.24)

Given the relationship between the arc length parameter, S, and the coordinate parameter, 𝜉, as: S = S(𝜉), and performing simple differentiation, and applying the definitions of the Frenet axes as given by equation (5.21), we get: C′ = C,S = C,𝜉 𝜉,S = T

and

C′′ = (C′ )′ = (C′ ).𝜉 𝜉,S = C,𝜉𝜉 𝜉,2S = T′ = 𝜅M

‖ ‖ ′ ′′ ‖ (C, 𝜉, ) × (C,𝜉𝜉 𝜉,2S )‖ ⇒ 𝜅 = ‖𝜅B‖ = ‖T × 𝜅M‖ = ‖ ‖C × C ‖ = ‖ ‖ 𝜉 S ‖ ‖ ‖ ‖ 3‖ = ‖(C,𝜉 × C,𝜉𝜉 )‖ ‖𝜉,S ‖ ‖ ‖‖ ‖ Now, from equation (5.10), noting that: 𝜉,S ≡

d𝜉 dS

=

(5.25)

1 , we also get, from equation (5.25), ||C,𝜉 (𝜉)||

an expression for the curvature, 𝜅, in terms of the original parameter, 𝜉, as: 𝜅=

‖(C,𝜉 × C,𝜉𝜉 )‖ ‖C,𝜉 ‖3

(5.26)

Remarks

r For a straight line, T is constant, implying that the curvature is zero, that is, 𝜅 ≡ 0, and 𝜌 = ∞. 5.4.1.2

Torsion or Twist

In equation (5.21), 𝜏 is known as the torsion at a point of a curve depicting the rate at which a twisted curve departs from its osculating plane. Now, given the relationship between the arc length parameter, S, and the coordinate parameter, 𝜉, as: S = S(𝜉), and performing simple

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differentiation, and applying the definitions of the Frenet axes as given by equation (5.21), we get: C′ = C,S = C,𝜉 𝜉,S = T C′′ = (C′ )′ = (C′ ).𝜉 𝜉,S = C,𝜉𝜉 𝜉,2S = T′ = 𝜅M C′′′ = (C′′ )′ = (C′′ ).𝜉 𝜉,S = C,𝜉𝜉𝜉 𝜉,3S = 𝜅 ′ M + 𝜅(𝜏B − 𝜅T)  + 𝜅(𝜏B −   𝜅T)) (C′ × C′′ ) ∙ C′′′ = (C,𝜉 × C,𝜉𝜉 ) ∙ C,𝜉𝜉𝜉 𝜉,6 = 𝜅B ∙ ( 𝜅′M

(5.27)

S

= 𝜅2𝜏 From equations (5.26) and (5.27), we finally get the expression for torsion, 𝜏, in triple product form as: 𝜏=

(C,𝜉 ×C,𝜉𝜉 ) ∙ C,𝜉𝜉𝜉 (C′ × C′′ ) ∙ C′′′ = 𝜅2 ||C,𝜉 (𝜉)||6 𝜅 2

(5.28)

Remarks

r For a plane curve, twist is zero, that is, 𝜏 ≡ 0, implying that the osculating plane does not change from point to point on the curve, From equation (5.28), the condition for a plane curve is given by the requirement that the triple product vanishes, that is, (C,𝜉 ×C,𝜉𝜉 ) ∙ C,𝜉𝜉𝜉 = 0. In matrix form, equivalently, we get: ⎡ X1 ,𝜉 (𝜉) X1 ,𝜉𝜉 (𝜉) ⎢ (C,𝜉 ×C,𝜉𝜉 ) ∙ C,𝜉𝜉𝜉 = det ⎢ X2 ,𝜉 (𝜉) X1 ,𝜉𝜉 (𝜉) ⎢ X , (𝜉) X , (𝜉) ⎣ 3𝜉 3 𝜉𝜉

X1 ,𝜉𝜉𝜉 (𝜉) ⎤ ⎥ X2 ,𝜉𝜉𝜉 (𝜉) ⎥ = 0 X3 ,𝜉𝜉𝜉 (𝜉) ⎥⎦

(5.29)

where det stands for the determinant of the matrix. 5.4.1.3

Discussion

This investigation has established the theoretical basis for parametric curves. However, for computational accuracy, cost-effectiveness and smoothness, we are specifically interested in some special classes of approximating and interpolating curves: polynomial and piecewise polynomial curves. In what follows, we prefer and investigate Bernstein polynomials and spline functions with special attention to a class of piecewise polynomials known as Schoenberg B-splines, over many other polynomials and piecewise polynomials such as Lagrangian and Hermite.

5.5 Bernstein Polynomials Here, our main goal is to present a short introduction to the Bernstein polynomials insofar as it is useful for both geometric and displacement modeling in mesh generation and finite element analysis phases, respectively. Bernstein polynomials form the backbone of modern-day CAGD through the boundary representation method and c-type finite element method which we introduced. These form the polynomial basis functions for a Bezier curve. Bernstein polynomials show up in a constructive proof of the Weierstrass approximation theorem on continuous functions. The inverse problem gives rise to the Bernstein–Bezier curves:

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Figure 5.10 A Bernstein polynomial.

5.5.1 Weierstrass Approximation Theorem One of the most fundamental theorems in approximation theory is the Weierstrass approximation theorem of 1885. In plane language, it states that every continuous function can be uniformly approximated by a sequence of polynomials over a closed interval. Of all the different proofs, Bernstein’s constructive proof is most relevant for us. The net contention of the proof is established through the following definition of the Bernstein polynomials.

5.5.2 Bernstein Polynomials Let q(𝜉) be a continuous curve as shown in Figure 5.10. Then the approximating nth Bernstein polynomial, CBn (q; 𝜉) for q(𝜉) is defined by the linear combination of the Bernstein basis functions with n + 1 sample points, q( ni ), at ( ni ), i ∈ {0, 1, … , n}, that is, equidistant points in the interval [0, 1], taken as the coefficients of combination. In other words: CBn (q; 𝜉) =

n ( ) ∑ q ni Bni (𝜉)

(5.30)

i=0

where, we have used the following definition. Definition

r The ith Bernstein basis function, Bn (𝜉), of the n-dimensional polynomial subspace of the class i

of continuous functions as:

Bni (𝜉)

( ) n = 𝜉 i (1 − 𝜉)n−i , i ∈ {0, 1, … , n}, i n! ( ) ⎧ ⎪ n i!(n − i)! where, ≡⎨ i ⎪ 0, ⎩

0≤i≤n otherwise

(5.31)

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Then, the Bernstein proof shows that: lim CBn (q; 𝜉) = q(𝜉),

n→∞

𝜉 ∈ [0, 1]

where the limit holds uniformly over the interval. Clearly, Bni (𝜉), i ∈ {0, 1, … , n} are all nth degree polynomials. By increasing n, the number of sample points, we can get as close to the curve, q(𝜉), as we choose. Here, we have introduced the Bernstein polynomial in terms of parametric curves, CBn (q; 𝜉). However, Bernstein’s original proof of the Weierstrass theorem was constructed in terms of the non-parametric Bernstein polynomial, pBn (f ; x) where f (x) is a target function, to be approximated, whose sample values are given by:

pBn (f ; x) =

n ( ) ∑ f ni Bni (x)

(5.32)

i=0

In the example shown by Figure 5.11, we are interested in approximating a continuous function f (x) by a quadratic Bernstein polynomial, pB2 (f ; x). Thus, from equation (5.32) with n = 2, we must have the sample points of f at locations 2i , i = 0, 1, 2. From Figure 5.11, it is easily seen that we can always establish the connection between a non-parametric function and its corresponding ∑ parametric form as a curve because of the linear precision property – n0 ( ni )Bni (𝜉) = 𝜉 – of the Bernstein polynomials. This, as we shall see later, is the underlying principal motivation for introducing the Bernstein–Bezier or B-spline-de Boor parametric curve form for the nonparametric displacement functions in finite element analysis. The most relevant information to remember about the proof is that there is a theoretical guarantee of uniform convergence in the approximation. With these in the background, we can now elaborate on the various properties of Bernstein polynomial curves.

Figure 5.11 Parametric curve and non-parametric function.

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5.5.3 End Point Interpolation CBn (q; 0) = q(0),

CBn (q; 1) = q(1)

and

This follows from the end point property of the Bernstein basis functions. This property is fundamentally important and tremendously advantageous in both geometric modeling and finite element space definition.

5.5.4 Forward Difference Form By applying the general definition of the forward difference operators, Δi q, we can write the forward difference form for the approximating Bernstein polynomial, CBn (q; 𝜉), as: CBn (q; 𝜉)

=

n ∑ i=0

( ) (n) Δ q ni 𝜉i i i

Exercise: Prove the above equality.

5.5.5 Simultaneous Approximation The Bernstein polynomials simultaneously approximate both a function and its derivatives if it is differentiable. In other words, the Bernstein polynomials follow the behavior of a function with a remarkable degree of closeness.

5.5.6 Partition of Unity If q(𝜉) = 1, using the property of successive forward difference (Δ0 q(0) = 1; Δ1 q(0) = Δ2 q(0) = ⋯ = 0), and the property of Bernstein basis functions, we can show that CBn (1; 𝜉) =

n ∑

Bni (𝜉) ≡ 1

i=0

In other words, it is the binomial expansion for 1n = {𝜉 + (1 − 𝜉)}n . This property has profound implication in the finite element method of approximation. Exercise: Prove the above equality.

5.5.7 Linear Precision If q(𝜉) = 𝜉, the property of successive forward difference (Δ0 q(0) = 0; Δ1 q(0) = 1n , Δ2 q(0) = ⋯ = 0) and the property of Bernstein basis functions show us: CBn (𝜉; 𝜉) =

n ( ) ∑ i i=0

n

Bni (𝜉) ≡ 𝜉

In other words, the initial linear function is reproduced. Exercise: Prove the above equality.

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Figure 5.12 Bezier curves of various degrees.

5.5.8

Bernstein Basis Functions: Properties

Here, our main goal is to present the essential properties of the Bernstein basis functions. The modern-day CAGD relies heavily on these. Our c-type finite element analysis, likewise, utilizes these properties. Example: Based on the definition given in equation (5.31), here are a few explicit expressions for the Bernstein basis functions for n = 1, 2, 3, 4. The corresponding graphs are also shown in Figure 5.12: Linear (n = 1) basis functions: B10 (𝜉) = 1 − 𝜉,

B11 (𝜉) = 𝜉

Quadratic (n = 2) 𝜉) 𝜉, B22 (𝜉) = 𝜉 2

B20 (𝜉) = (1 − 𝜉)2 ,

basis

functions:

Cubic (n = 3) basis functions: B30 (𝜉) = (1 − 𝜉)3 , 3 (1 − 𝜉) 𝜉 2 , B33 (𝜉) = 𝜉 3

B21 (𝜉) = 2 (1 −

B31 (𝜉) = 3 (1 − 𝜉)2 𝜉,

B32 (𝜉) =

Quartic (n = 4) basis functions: B40 (𝜉) = (1 − 𝜉)4 , B41 (𝜉) = 4 (1 − 𝜉)3 𝜉, B42 (𝜉) = 6 (1 − 𝜉)2 𝜉 2 , B43 (𝜉) = 4 (1 − 𝜉) 𝜉 3 , and B44 (𝜉) = 𝜉 4 ,

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5.5.8.1

153

Non-negativity and Unique Maximum

Bernstein basis functions, Bni (𝜉), are all non-negative and have one maximum, at 𝜉 = interval, 𝜉 ∈ [0, 1]. 5.5.8.2

i n

on the

Partition of Unity

For Bernstein basis functions, for any n, n ∑

𝜉 ∈ [0, 1]

Bni (𝜉) = 1,

(5.33)

i=0

5.5.8.3

Recursive Property

The Bernstein basis function of differing degrees are recursively connected: { Bni (𝜉)

=

n−1 (1 − 𝜉) Bn−1 i (𝜉) + 𝜉 Bi−1 (𝜉), i ∈ {0, 1, … , n} , 0, i ∉ {0, 1, … , n}

(5.34)

with B00 (𝜉) ≡ 1 n−1 It can be seen that Bni (𝜉) is obtained as the convex combination Bn−1 i (𝜉) and Bi−1 (𝜉) for all n and for all i ∈ {0, 1, … , n}.

5.5.8.4

Power Basis Connection

Power basis functions, {𝜉 i }, and the Bernstein basis functions are connected to each other: ( ) j n ∑ i 𝜉i = ( ) Bnj (𝜉) n j=i i

5.5.8.5

and

Bni (𝜉) =

n ∑

(−1)j−i

j=i

( )( ) n j 𝜉j j i

(5.35)

Subdivision Property Bni (c𝜉) =

n ∑

j

Bi (c) Bnj (𝜉)

(5.36)

i=0

5.5.8.6

Product Rule

This property is useful in the discretization of a specified traction to its nodal counterparts in the finite element analysis phase: (

)(

)

m n i j n ( ) Bm+n (𝜉) Bm i (𝜉) Bj (𝜉) = i+j m+n i+j

(5.37)

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154

5.5.8.7

Degree Elevation Formulas n + 1 − i n+1 Bi (𝜉) n+1 i + 1 n+1 𝜉 Bni (𝜉) = B (𝜉) n + 1 i+1 n + 1 − i n+1 i + 1 n+1 Bni (𝜉) = Bi (𝜉) + B (𝜉) n+1 n + 1 i+1

(1 − 𝜉) Bni (𝜉) =

5.5.8.8

(5.38)

The First Derivative

The first derivative is easily obtained as:

5.5.8.9

Bni,1 (𝜉) = n {Bn−1 (𝜉) − Bn−1 i (𝜉)} i−1

(5.39)

Bni (𝜉) = Bnn−i (𝜉)

(5.40)

Symmetry Property

5.5.8.10

Integral Property 𝜉

∫0

Bni (𝜍) d𝜍 =

n+1 1 ∑ n+1 B (𝜉) n + 1 j=i+1 j

and, 1

∫0

Bni (𝜍) d𝜍 =

(5.41)

1 n+1

5.6 Non-rational Curves Bezier–Bernstein–de Casteljau Remarks

r As to Bernstein polynomial, suppose that instead of the original curve, only a finite set of

r r r

n + 1 sample points, {qi }, are given such that, qi ≡ q( ni ), i ∈ {0, 1, … , n}. Then, because of the properties of the Bernstein basis functions, the approximating curve defined as above will closely follow the polygon through the given sample points. Finally, by changing the location of this finite set of the sample points, we will be able to visually control the shape of the approximating curve on a viewing surface while it will still mimic the polygon all the time. This is precisely the motivation for the definition of a Bezier curve, and serves to establish the connection between the Bernstein polynomials and the Bezier curves. As already indicated, most curves are amenable to representation by the Bernstein polynomials with the coefficients of combination being non-rational except for the conics, for which we will need rational coefficients, Thus, for a detailed discussion on Bezier curves, the underpinning tool of modern-day CAGD, we investigate first the non-rational Bernstein–Bezier curves followed by the rational Bernstein–Bezier curves, For more complicated curves that require composition of Bezier curves with certain smoothness, we present Bezier form of B-spline curves.

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Figure 5.13 Definition of an inverse problem. (See colour plate section)

r It is easy to identify a curve represented by a given control polygon, but in most practical r

applications where we are given points on a curve and required to generate the corresponding control polygon, as shown in Figure 5.13, things can get really involved; these are known as the inverse problems, and we will deal with them later. Finally, noting that the affine transformation is not linear because of the translation part, in computer graphics all computations are performed where the three-dimensional space is embedded into a four-dimensional homogeneous coordinate space; all our code snippets will be given in these coordinates.

5.6.1.1

Bezier–Bernstein Curves

Here, our main goal is to introduce Bezier curves in terms of the Bernstein polynomials – motivated by the need to discover several very important properties of a Bezier form of description of parametric curves. These properties have a pivotal role in both the algebraic and geometric understanding of the representation. Furthermore, these are essential for an application of the Bezier representation to interpolation in the mesh generation phase and approximation in the finite element phase of analysis. Let us recall that the Bezier curves are parametric polynomial curves. Thus, definition utilizes an important property that says that any Euclidean points belonging to an affine space can only be added meaningfully in barycentric combinations. We identify the Bernstein polynomials as the barycentric coefficients, and control points as the Euclidean points. In a non-parametric polynomial representation of continuous functions, as in the finite element analysis phase, the Bernstein polynomials can be seen as the basis functions of the underlying linear vector space. So, we will use these notions rather less rigorously and interchangeably, as long as we are aware of the context. 5.6.1.2

Bezier Curves with Bernstein Polynomials

We introduce here a Bezier curve as in the following definitions. Definition

r An nth degree Bezier curve, Cn (𝜉), is defined as: Cn (𝜉) =

n ∑ i=0

Bni (𝜉) qi ,

0≤𝜉≤1

(5.42)

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156

{ } where the basis (blending) functions, Bni (𝜉) , are the nth degree Bernstein polynomials: Bni (𝜉) =

( ) n 𝜉 i (1 − 𝜉)n−i , i ∈ {0, 1, … , n} i

where

( ) n n! ≡ i i!(n − i)!

(5.43)

r The coefficients of linear combination or Euclidean points, {qi }, are known as the control points of the curve.

r The polygon containing the control points, {qi }, is known as the control polygon of the curve. Let us note that 𝜉 ∈ [0, 1]. Also, from the definition of a parametric curve, recall that in two { }T { }T dimensions: C(𝜉) = X(𝜉) Y(𝜉) , and in three dimensions: C(𝜉) = X(𝜉) Y(𝜉) Z(𝜉) . Let us elucidate the above definitions by some simple examples. In mesh generation phase, usually curves with degree up to and including three are considered for numerical efficiency. For finite element analysis with mega-elements, it appears that up to quintic should be considered for, say, shell analysis. Example: Bezier curves of degree one – linear Let us consider the case of n = 1, that is, Bezier curves of degree one. We have, from equation (5.31), B10 (𝜉) = 1 − 𝜉 and B11 (𝜉) = 𝜉. Thus, from equation (5.42), we get: C1 (𝜉) = (1 − 𝜉) q0 + 𝜉 q1 , 0 ≤ 𝜉 ≤ 1. In other words, C1 (𝜉) describes a straight line segment between points q0 and q1 . Let us observe that the curve passes through the end control points. Example: Bezier curves of degree two – quadratics Let us consider the case of n = 2, that is, Bezier curves of degree two. We have, from equation (5.31), B20 (𝜉) = (1 − 𝜉)2 , B21 (𝜉) = 2(1 − 𝜉) 𝜉 and B22 (𝜉) = 𝜉 2 . Thus, from equation (5.42), we get: C2 (𝜉) = (1 − 𝜉)2 q0 + 2 (1 − 𝜉) 𝜉 q1 + 𝜉 2 q2 , 0 ≤ 𝜉 ≤ 1. In other words, C2 (𝜉) describes a parabolic arc between points q0 and q2 . Let us observe again that the curve passes through the end control points. Example: Bezier curves of degree three – cubics Let us consider the case of n = 3, that is, Bezier curves of degree three. We have, from equation (5.31), B30 (𝜉) = (1 − 𝜉)3 , B32 (𝜉) = 3(1 − 𝜉)2 𝜉, B31 (𝜉) = 3(1 − 𝜉) 𝜉 2 and 2 B2 (𝜉) = 𝜉 2 . Thus, from equation (5.42), we get: C3 (𝜉) = (1 − 𝜉)2 q0 + 3𝜉 (1 − 𝜉) q1 + 3𝜉 (1 − 𝜉)2 q2 + 𝜉 3 q3 , 0 ≤ 𝜉 ≤ 1. In other words, C3 (𝜉) describes a cubic arc between points q0 and q3 , as shown in Figure 5.14. Let us observe again that the curve passes through the end control points. Remarks

r Curves of degree up to two are planar curves, that is, a plane containing the curves fully in it can be identified. of degree greater than two are generally true three-dimensional curves, that is, in general, a plane cannot be identified that fully contains a curve. The end point interpolation by the Bezier curves is a very suitable feature for its application to finite element analysis in any dimension.

r Curves r

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Figure 5.14 A cubic Bezier curve.

5.6.1.3

Horner’s Algorithm

Let us consider a cubic polynomial curve, C3 (𝜉), expressed in monomial basis functions, {1, 𝜉, 𝜉 2 , 𝜉 3 }, such that: C3 (𝜉) = A + B 𝜉 + C 𝜉 2 + D 𝜉 3

(5.44)

The exponential powers associated with the basis functions make evaluation of the curve points numerically inaccurate as the degree of the curve increases. Computationally, a robust solution is offered by what is known as the Horner’s algorithm of nested multiplications by rewriting the expression (5.44) as: C3 (𝜉) = A + 𝜉(B + 𝜉(C + D 𝜉) )

(5.45)

and then, calculating the expression by evaluating the multiplication and addition operations from the innermost bracket outward. The matrix form of the Bernstein–Bezier curves as introduced next can also be cast in monomial basis functions with appropriate Bezier coefficient matrices for computational stability, accuracy and efficiency; this approach will be algorithmically documented when we deal with computational aspects of curves.

5.6.1.4

Matrix Form of a Bezier Curve

By collecting the control vectors into a matrix and the Bernstein polynomials into a vector, we can express a curve in matrix form. Thus, in the matrix form, the same Bezier curve, Cn (𝜉), may be described as: Cn (𝜉) = Q B(𝜉), m×1

m×n n×1

0≤𝜉≤1

(5.46)

For three-dimensional space, m = ] represents a Bezier curve, Cn (𝜉), [ 3, equation (5.46) with Bezier control matrix: Q = q0 q1 ⋯ qn , and Bernstein basis matrix: B = }T { n B0 (𝜉) Bn1 (𝜉) ⋯ Bnn (𝜉) . As indicated before, we would rather like the expression (5.46) further reduced to monomial form for computational purposes with applications in manipulation of Bezier curves on a digital computer.

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5.6.1.4.1 Linear Bezier Curves For n = 1, that is, linear Bezier curve, C1 (𝜉), can be rewritten in matrix form as: ] 1 −1 C1 (𝜉) = (1 − 𝜉)q0 + 𝜉q1 = q0 q1 0 1 ⏟⏞⏞⏟⏞⏞⏟ ⏟⏞⏞⏟⏞⏞⏟ ]

[

Geometry Matrix

[

{ } 1 𝜉 ⏟⏟⏟

(5.47)

Linear Bezier Monomial basis functions coefficient matrix of degree 1

] q1 is appropriately known as the geometry matrix as it determines the geometry [ ] 1 −1 of a linear curve, and , as the linear Bezier Coefficient matrix. 0 1

[ where q0

5.6.1.4.2 Quadratic Bezier Curves For n = 2, that is, quadratic Bezier curve, C2 (𝜉), can be rewritten in matrix form as: C2 (𝜉) = (1 − 𝜉)2 q0 + 𝜉(1 − 𝜉)q1 + 𝜉 2 q2 1⎤ ] ⎡ 1 −2 2 −2 ⎥ = q0 q1 q2 ⎢ 0 ⎥ ⏟⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏟ ⎢⎣ 0 0 1⎦ ⏟⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏟ Geometry [

matrix

⎧1⎫ ⎪ ⎪ ⎨𝜉 ⎬ ⎪ 𝜉2 ⎪ ⎩ ⎭ ⏟⏟⏟

(5.48)

Quadratic Bezier coefficient matrix Monomial basis functions of degree 2

] q2 is appropriately known as the geometry matrix as it determines the geom1⎤ ⎡ 1 −2 2 −2 ⎥, as the quadratic Bezier coefficient matrix. etry of a quadratic curve, and ⎢ 0 ⎢ ⎥ 0 1⎦ ⎣0 [ where q0

q1

5.6.1.4.3 Cubic Bezier Curves For n = 3, that is, cubic Bezier curve, C3 (𝜉), can be rewritten in matrix form as: C3 (𝜉) = (1 − 𝜉)3 q0 + 3𝜉(1 − 𝜉)2 q1 + 3𝜉 2 (1 − 𝜉)q2 + 𝜉 3 q3 1 ⎤ ⎡ 1 −3 3 ] ⎢ 0 3 −6 3 ⎥ = q0 q1 q2 q3 ⎢ 3 −3 ⎥ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⎢ 0 0 ⎥ 0 1 ⎦ ⎣0 0 Geometry ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ matrix [

Cubic Bezier coefficient matrix

⎧1⎫ ⎪𝜉 ⎪ ⎨ 𝜉2 ⎬ ⎪ 3⎪ ⎩𝜉 ⎭ ⏟⏟⏟

(5.49)

Monomial basis functions of degree 3

] q3 is appropriately known as the geometry matrix as it determines the 3 1⎤ ⎡ 1 −3 ⎢0 3 −6 3⎥ geometry of a cubic curve, and ⎢ , as the cubic Bezier coefficient matrix. 0 0 3 −3 ⎥ ⎢ ⎥ 0 0 1⎦ ⎣0

[ where q0

q1

q2

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Remarks

r In computer programming, it is also possible to consider the monomial basis function vectors in reverse order. For example, in the cubic case above, we can express the vector as: {𝜉 3 , 𝜉 2 , 𝜉 , 1}T ; of course, in those cases, the columns of the Bezier coefficient matrices also need to be reversed. In our example code snippet on curves, we will adopt this convention. 5.6.1.5

Sneak Preview of the de Casteljau Algorithm

Let us dig a bit further into the example above with n = 2, that is, the parabolic curve. We can rewrite the expression for the curve as: C2 (𝜉) = (1 − 𝜉)2 q0 + 2 (1 − 𝜉) 𝜉 q1 + 𝜉 2 q2 ,

0≤𝜉≤1

Linear combination Linear combination ⎛⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⎞ ⎛⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⎞ ⎜ ⎟ ⎜ ⎟ = (1 − 𝜉) ⎜(1 − 𝜉) q0 + 𝜉 q1 ⎟ + 𝜉 ⎜(1 − 𝜉) q1 + 𝜉 q2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

(5.50)

Linear combination

Now, we can clearly see that a parabolic curve, C2 (𝜉), is obtained by the linear combinations of curves one degree less (namely of degree one) than the degree of the intended curve. In other words, the parabolic curves are constructed by repeated linear interpolations of linear curves on the control polygons. This is not an isolated observation! In fact, it forms the basis of a very elegant geometric algorithm for construction of any Bezier curve and its other related properties known as de Casteljau algorithm. Remarks

r Here we have introduced a definition of Bezier curves in terms of Bernstein polynomials. This

r r

definition is purely algebraic and suitable for both analytical and computational use. However, it is of little use for geometric visualization. As suggested, for a geometric understanding to aid visualization and, equally importantly, for computational efficiency in the mesh generation phase, we need to look at the Bezier–de Casteljau–Bernstein connection. Note that here we have restricted ourselves merely to the definition of a Bezier curve. For computational efficiency in mesh generation and finite element analysis, we will need to utilize the various important properties of the Bezier curves. For a detailed discussion on the properties of a Bezier curve, we need to investigate the properties of Bezier curves, Finally, for manipulations of curves in the power basis, we need to be familiar with the matrix form of a Bezier Curve in power basis

5.6.1.6

Bezier–Bernstein–de Casteljau Connection

Here, our main goal is to present the de Casteljau algorithm for constructing Bezier or Bernstein polynomial curves. It is numerically more efficient and less error prone than computation in power basis form of the Bernstein basis functions obtained even through the Horner-type algorithm. In the mesh generation phase, the de Casteljau algorithm forms the backbone of modern-day CAGD through a boundary representation method that utilizes Bezier curves and surfaces. Let us consider the following simple case of quadratic Bernstein polynomial curve with n = 2, that is,

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the parabolic curve. Based on the Bernstein basis functions, we can write the expression for the curve as: C2 (𝜉) = (1 − 𝜉)2 q0 + 2 (1 − 𝜉) 𝜉 q1 + 𝜉 2 q2 ,

0≤𝜉≤1

(5.51)

However, by viewing it from a geometric point of view, we can rewrite the above equation as: Convex combination Convex combination ⎞ ⎛⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⎞ ⎛⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⎟ ⎜ ⎟ ⎜ C2 (𝜉) = (1 − 𝜉) ⎜(1 − 𝜉) q0 + 𝜉 q1 ⎟ + 𝜉 ⎜(1 − 𝜉) q1 + 𝜉 q2 ⎟ ⎟ ⎜ ⎟ ⎜ ⎠ ⎝ ⎠ ⎝ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

(5.52)

Convex combination

Now, we can clearly see that the parabolic curve, C2 (𝜉), is obtained by convex combination of curves of degree one less (i.e. of degree one) than the degree of the original curve. In other words, the parabolic curves are constructed by repeated linear interpolations of the segments of the control polygons which are linear curves. This is not an isolated observation! In fact, it forms the basis of the very elegant geometric algorithm for the construction of any Bezier curve and its other related properties such as tangents and curvatures, known as the de Casteljau algorithm. The central scheme is based on the recognition of the recursive relation of repeated linear interpolations for the Bernstein polynomials; for finding a point on a curve at any parameter point, we start with the given Bezier control polygon describing the curve; we locate a new control point in each segment of the polygon by linear interpolation (in the same ratio as the parameter point to the parameter interval) of the end control points of the corresponding segments. By continuing recursively a number of times equal to the degree of the curve, we recover the intended point on the curve. Before we present the de Casteljau algorithm, let us reintroduce: Definition

r The set {qi }, i ∈ {0, 1, … , n} are called the Bezier points or control points. r The polygon formed by joining successively qi and qi+1 , i ∈ {0, 1, … , n} is called the Bezier polygon or control polygon of the curve. 5.6.1.7

de Casteljau Algorithm

Let {qi }, i ∈ {0, 1, … , n} be a finite set of control points in 3 and 𝜉 ∈. Then, de Casteljau algorithm is defined by the following recursive relation via repeated linear interpolations: (𝜉) + 𝜉 qd−1 (𝜉), qdi (𝜉) = (1 − 𝜉) qd−1 i i+1 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ Convex combination

∀𝜉 ∈,

d ∈ {1, 2, … , n} (5.53)

i ∈ {0, 1, … , n − d} with q0i (𝜉) = qi Then, qni (𝜉), ∀𝜉 ∈ R defines a parametric Bezier curve, Cn (𝜉) of degree n. What better way to understand the details than actually going through an example:

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Example: Cubic (n = 3) Bezier curve We have: d ∈ {1, 2, 3} in equation (5.53). Now, let us explicitly record the steps: Step 0:

Recursion, d = 1: Recursion, d = 2: Recursion, d = 3:

q00 = q0 ,

q01 = q1 ,

q02 = q2

and q03 = q3 , for a given 𝜉 ∈ [0, 1]

⎧ q10 = (1 − 𝜉) q0 + 𝜉 q1 , ⎪ 1 ⎨ q1 = (1 − 𝜉) q1 + 𝜉 q2 ⎪ q1 = (1 − 𝜉) q + 𝜉 q 2 3 ⎩ 2 q20 = (1 − 𝜉) q10 + 𝜉 q11 , q21 = (1 − 𝜉) q11 + 𝜉 q12 , C3 (𝜉) = q30 = (1 − 𝜉) q20 + 𝜉 q21 , as shown in Figure 5.15.

The whole process can be seen schematically as in Figure 5.15. Note that the control point sets {q0 , q10 , q20 , q30 } and {q30 , q21 , q12 , q3 } subdivide the original cubic curve into two cubic curves at the parameter point: one on the left and one on the right. This aspect of the de Casteljau algorithm gives rise to the powerful notion of partition or subdivision of a Bezier curve, essential for various efficient applications such as computation of tangents, curvatures and twist. Note too that the entire algorithm is a collection of series of linear interpolations, that is, scaling and additions of control points. Thus, numerical errors are minimal and the process is fast. We can quickly reverse the process to see the Bernstein connection of a Bezier curve: C3 (𝜉) = q30 = (1 − 𝜉) q20 + 𝜉 q21 = (1 − 𝜉) {(1 − 𝜉) q10 + 𝜉 q11 } + 𝜉{(1 − 𝜉) q11 + 𝜉 q12 } 3 ∑ B3i (𝜉) qi = …,= i=0

This being a cubic expression in −∞ ≤ 𝜉 ≤ ∞, C3 (𝜉) is a cubic curve. Exercise: Fill in the intermediate steps for the reverse process of a cubic.

Figure 5.15 de Casteljau method for a cubic curve.

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5.6.1.8

Intermediate de Casteljau Curves

The intermediate de Casteljau curves have a suitable alternative expression that implies that these are indeed Bezier curves. Now, we present the expressions for these curves in the form of a lemma with proof by induction. Lemma qdi (𝜉)

=

d ∑

Bdj (𝜉) qi+j ,

d ∈ {0, 1, … , n},

i ∈ {0, 1, … , n − d}

(5.54)

j=0

Proof. Let us recognize that by the de Casteljau algorithm for d = 1, we have: q1i (𝜉) = ∑1 1 j=0 Bj (𝜉) qi+j , i ∈ {0, 1, … , n − 1}, that is, a set of pair-wise linear combinations of the original control points. Now, we will show that if it is true for d − 1 then it is also true for d. Then, by induction, the proof will be complete: Thus, assuming it is true for d − 1, we have: qd−1 (𝜉) = i

d−1 ∑

Bd−1 (𝜉) qi+j j

(by hypothesis)

j=0

=

i+d−1 ∑

Bd−1 j−i (𝜉) qj

(by setting k = i + j and renaming k → j)

j=i

=

i+d ∑

Bd−1 j−i (𝜉) qj

(added j = i term since Bd−1 (𝜉) ≡ 0 for d

j=i

all d ∉ {0, 1, … , d − 1} by definition) Similarly, we can show: (𝜉) = qd−1 i+1

i+d ∑

Bd−1 (𝜉) qj j−i−1

j=i

Now, by de Casteljau: (𝜉) + 𝜉 qd−1 (𝜉) = qdi (𝜉) = (1 − 𝜉) qd−1 i i+1

i+d ∑

d−1 {(1 − 𝜉) Bd−1 j−i (𝜉) + 𝜉 Bj−i−1 (𝜉)} qj

j=i

=

i+d ∑

Bdj−i (𝜉) qj

(by Bernstein recurrence relations)

Bdj (𝜉) qi+j

(by re-indexing j → j − i)

j=i

=

d ∑ j=0

This completes the proof. (𝜉) = Exercise: In the above proof, show that: qd−1 i+1

i+d ∑ j=i

Bd−1 (𝜉) qj j−i−1

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Figure 5.16 de Casteljau construction of a spline from cubic curve.

Remarks

r Let us note that each of these intermediate de Casteljau points defines a Bezier curve, that is, qi ,qi+1 , … ,qi+d define qdi (𝜉), a Bezier curve of degree d. For example, q0 ,q1 and q2 define a parabola, q20 (𝜉), 𝜉 ∈ [0, 1]. This observation is central to several crucial theorems that eventually lead up to generalization of the Bezier curves to a wider class of curves – B-spline curves that are of utmost importance both in computer-aided geometry modeling and finite element analysis.

5.6.1.9

Sneak Preview of B-spline Curve Construction

While we will have some detailed discussion on B-splines later, let us for now go back to the cubic example. We note that if we set: 3q1 − q0 = 3q2 − q3 = s, we may, as shown in Figure 5.16, describe the same curve as a curve of degree two, defined by the control points, {q0 , s, q3 }, as: C3 (𝜉) = (1 − 𝜉)2 q0 + 2(1 − 𝜉) 𝜉 s + 𝜉 2 q3 . It turns out that s represents a control point of a more versatile class of curves – the B-spline curves in de Boor geometric form.

5.6.2 Properties: Non-rational Bezier Curves Here, we provide various useful and relevant properties of a Bezier curve essential for both mesh generation and the finite element method. Our two different yet connected introductions to Bezier curves through the de Casteljau algorithm and the Bernstein polynomial basis functions nicely tie in with the two complementary ways of looking at any computational entity: by geometry and

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by algebra. Bezier by the de Casteljau algorithm provides the geometric flavor of the algebraic structures of Bernstein polynomial expansion. We have already indicated some of the relevant algebraic properties of Bernstein polynomials; strengthened by our geometric perspective, here we present most of the useful properties of Bezier curves:

r affine invariance r affine reparameterization invariance r end point interpolation r convex hull r symmetry r derivatives r integrals r degree elevation r partitioning or subdivision 5.6.2.1

Affine Invariance

To introduce the affine invariance property of a Bezier curve, which is important in developing an efficient mesh generation and viewing methodology, let us recall that we generate our finite element models from a finite number of control points. Because of the affine invariance property as will be explained, we can affinely transform only this finite number of control points rather than the entire infinite-dimensional curves or digitally finite but vast number of points on them to obtain the same transformed geometric object for, say, our viewing needs. We already know that barycentric combinations are invariant under affine transformations such as translation, rotations, scaling and reflection. Once we identify that Bernstein expansion is barycentric, our job is practically done – invariance proof utilizes the matrix form of a Bezier curve. Let us recall that an nth Bezier curve, Cn (𝜉), is defined as: Cn (𝜉) =

n ∑

Bni (𝜉) qi ,

0≤𝜉≤1

(5.55)

i=0

where the basis (blending) functions, {Bni (𝜉)}, are the nth degree Bernstein polynomials: Bni (𝜉)

( ) n = 𝜉 i (1 − 𝜉)n−i , i ∈ {0, 1, … , n} i

where

( ) n n! ≡ i i!(n − i)!

(5.56)

{ } with the coefficients of linear combination, qi , known as the control points of the curve. Let us note that the expansion given in equation (5.42) is a barycentric combination, since from ∑ the properties of Bernstein polynomials we have ni=0 Bni (𝜉) = 1. In the matrix form, the same Bezier curve, Cn (𝜉), may be described as: Cn (𝜉) = Q B(𝜉), m×1

m×n n×1

0≤𝜉≤1

(5.57)

n For three-dimensional space, m = ] represent a Bezier curve, C (𝜉), [ 3, equation (5.46) with Bezier control matrix: Q = q0 q1 ⋯ qn , and Bernstein basis matrix: B = }T { n B0 (𝜉) Bn1 (𝜉) ⋯ Bnn (𝜉) . Now, suppose that we define by an m × m matrix of affine,

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say, rotation, transformation, T, such that, after application of the rotation to the curve directly, we obtain C∗n (𝜉) as the transformed curve. We claim that: the same curve can be obtained by simply transforming the Bezier control matrix, Q to Q∗ = T Q, and using this new control matrix, Q∗ , to describe the curve, C∗n (𝜉). Because, C∗n (𝜉) = Q∗ B(𝜉) = T Q B(𝜉) = T Cn (𝜉), m×1

m×n n×1

m×n n×1

0≤𝜉≤1

(5.58)

as proposed. This is what is known as the affine invariance property of a Bezier curve. Example: A two-dimensional Bezier curves of degree 1: linear. Let us consider the case of m = 2 and n = 1, that is, a two-dimensional line segment. Suppose, { } 0 the segment is directed along the X-axis at the origin and of length unity. Then, clearly, q0 = 0 { } [ ] 1 0 1 and q1 = , which makes the control matrix: Q = . The Bernstein basis matrix 0 0 0 } [ ] { } { } { 0 B0 (𝜉) = 1 − 𝜉 0 1 1−𝜉 𝜉 . Thus, Cn (𝜉) = = , 0 ≤ 𝜉 ≤ 1. is given as: B = 0 0 𝜉 0 B10 (𝜉) = 𝜉 m×1 Now, suppose that we want the transformed description of the line segment when it[is rotated ] 0 −1 ◦ by, say, 90 towards the positive Z-axis. The transformation matrix is given by: T = . 1 0 for all We could apply the transformation directly to Cn (𝜉), [ that is,]transform { } { } 0 ≤ 𝜉 ≤ 1, to draw 0 −1 𝜉 0 = , 0 ≤ 𝜉 ≤ 1. But on the screen the new transformed line: C∗n (𝜉) = 1 0 0 𝜉 why? Instead, for far greater efficiency, we only need to[ transform vectors that ] [ the two ] control [ ] 0 −1 0 1 0 0 define the line segment to get new control vectors: Q∗ = = , and we 1 0 0 0 0 1 [ ]{ } { } 0 0 1−𝜉 0 have: C∗n (𝜉) = = , 0 ≤ 𝜉 ≤ 1. Note that its new control points will 0 1 𝜉 𝜉 { } { } 0 0 be given by: q0 = and q1 = as expected. Obviously, the higher the degree of the 0 1 curve, the more is the computational efficiency and hence more real-time response.

5.6.2.2

Affine Reparameterization Invariance

Here, our main goal is to introduce the affine reparameterization invariance property of a Bezier curve – motivated by the need to highlight the so-called “standard curves” instead of dealing with an arbitrarily given parameter range. Quite often, for convenience, a Bezier curve is given in what is known as the standard form, where the parameter 𝜉 travels from 0 to 1. The need for the convenient standard form arises for various reasons. Referring to Figure 5.17, if we are to describe an extracted part of a curve between 𝜉 = 𝜉 1 to 𝜉 = 𝜉 2 , for, say, numerical processing, it is most convenient to reparameterize with 𝜎 ∈ [0, 1], that is, 0 ≤ 𝜎 ≤ 1, like the original curve, for the sake of uniformity of application of the standard formulas already developed. For the extracted curve, 𝜉 ∈ [𝜉 1 , 𝜉 2 ] is known as the global parameter and 𝜎 ∈ [0, 1] is called the local parameter. We can think of another numerical situation: the de Casteljau algorithm for construction of a

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Figure 5.17 Reparameterization of curves.

Bezier curve is usually given assuming that the “standard” domain parameter runs from 0 to 1, although the actual domain parameter of the curve may be from a to b. Clearly, we would need a domain parameter transformation from an arbitrary range to the standard domain. So, the question is how do we reparameterize so that the curve degree and the shape remain invariant? In other words, how is an arbitrary domain parameter range related to the standard domain parameter range of [0, 1], so that the degree and the shape of the curve are not disturbed, that is, stays invariant. This is exactly what we set ourselves to achieve in the following. Let us recall that the barycentric or affine combinations are invariant under the affine transformations of translation, rotation, shearing, scaling and reflection. Once we identify that the Bernstein expansion of a Bezier curve is barycentric, and that we impose that the required transformation from the global to the local parameter (or, an arbitrary range to the standard range) is an affine map, our job is practically done, that is, the invariance of a Bezier curve under domain reparameterization is proved. Another way to see this is to recognize that a Bezier curve can be constructed by the de Casteljau algorithm, which is essentially a sequence of linear interpolations. But, a linear interpolation is an affine map, because the barycentric combination is invariant under linear interpolation, and hence a Bezier curve is invariant if the reparameterization itself is an affine map. 5.6.2.3

Affine Reparameterization Invariance: Standard Bezier Curve

Consider the Bezier curve as shown in Figure 5.18. The global parameter, 𝜎, runs from a to b. We would be interested to find the local (or standard) parameter, 𝜉 ∈ [0, 1], and its relationship with

Figure 5.18 Global–local coordinate relationship.

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the global (or arbitrary) parameter, 𝜎. We define the standard parameter, 𝜉, such that 0 ≤ 𝜉 ≤ 1, to describe a curve in the standard form. For invariance under this reparameterization, we seek to describe the curve with the standard parameter by the same degree Bezier curve as is done by the arbitrary parameter, 𝜎 ∈ [a, b]. In order that the degree and the shape of the curve remain unchanged, we must have the linear relationship between any global parameter, 𝜎, and the corresponding local parameter, 𝜉, as follows: 𝜎 = (1 − 𝜉)a + 𝜉 b = a + (b − a)𝜉

(5.59)

Clearly then 𝜉 = 0 and 𝜉 = 1 correspond to a and b, respectively. Thus, from equation (5.59), it follows that we may define the local parameter, 𝜉, in terms of the global parameter, 𝜎, as: 𝜉=

(𝜎 − a) (1) (−a) = 𝜎+ (b − a) (b − a) (b − a)

(5.60)

and 𝜉 assumes values from 0 and 1, that is, 0 ≤ 𝜉 ≤ 1. From the second equality, we may now recognize that the parameter transformation given by expression (5.60) is an affine map, that is, of the form: x̄ = cx + d where x is transformed to x̄ with c, d constants. Note that expression (5.60) is not a linear map as it contains an element of translation indicated (−a) above. by the part d = (b−a) Now, recall that an affine map is defined by the requirement that the barycentric combination remains invariant. Bezier curves are clearly given as a barycentric combination, as in equation ∑ (5.42), because of the partition property of the Bernstein basis functions, that is, ni=0 Bni (𝜉) = 1, 𝜉 ∈ [0, 1]. Thus, under affine transformation given by expression (5.60), the barycentric combination remains invariant. Thus, the degree and the shape of the curve remain invariant.

5.6.2.4

Effect of Reparameterization on the de Casteljau Algorithm

Based on our discussion above, for construction of a Bezier curve by de Casteljau algorithm, we simply need to plug in the local parameter, 𝜉, computed from a global (or arbitrary) to local (or standard) parameter transformation given by equation (5.60), and continue with the de Casteljau algorithm. Now, we will elaborate on this with a couple of examples. 5.6.2.4.1 Linear Interpolation: de Casteljau Scheme Referring to Figure 5.19, suppose that the arbitrary parameter range is given by: 𝜎 ∈ [a, b]. The two Bezier control points, for a linear curve, are q0 ≡ q00 and q1 ≡ q01 . Now, following the de Casteljau algorithm in terms of domain parameter, 𝜎 ∈ [a, b], we have:

C(𝜎) ≡ q10 =

{

b−𝜎 𝜎−a q + q b−a 0 b−a 1

} (5.61)

Note that the curve is given by a barycentric combination, because: b−𝜎 + 𝜎−a = 1. Clearly, b−a b−a however, expression (5.61) is rather awkward and cumbersome. So, if we plug in the parameter

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Figure 5.19 Linear interpolation.

transformation given by equation (5.59) in expression (5.61), we get de Casteljau in standard parameter, 𝜉 ∈ [0, 1]: 1 {(b − 𝜎)q0 + (𝜎 − a)q1 } b−a 1 = [{(1 − 𝜉)b − (1 − 𝜉)}q0 + (b − a)𝜉q1 ] = (1 − 𝜉)q0 + 𝜉q1 b−a

C(𝜉) =

(5.62)

The description of the same curve by equation (5.62) in terms of the standard parameter is obviously a barycentric combination. But, clearly, this time the expression is much nicer and simpler. 5.6.2.4.2 Quadratic Interpolation: de Casteljau Scheme Referring to Figure 5.20, suppose that the arbitrary parameter range is given by: 𝜎 ∈ [a, b]. The three Bezier control points, for a quadratic curve, are q0 ≡ q00 , q1 ≡ q01 and q2 ≡ q02 . Now, following the de Casteljau algorithm in terms of domain parameter, 𝜎 ∈ [a, b], we have, for first iteration: } } { { b−𝜎 0 𝜎−a 0 b−𝜎 0 𝜎−a 0 (5.63) q0 + q1 , q11 (𝜎) = q1 + q2 q10 (𝜎) = b−a b−a b−a b−a By applying the transformation given by equation (5.59) in equations (5.63), as we did in the previous linear interpolation, we get: } } { { q10 (𝜉) = (1 − 𝜉)q00 + 𝜉q01 , q11 (𝜉) = (1 − 𝜉)q01 + 𝜉q02

Figure 5.20 Quadratic interpolation.

(5.64)

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Now, for the second iteration of the de Casteljau algorithm, we have: { } b−𝜎 1 𝜎−a 1 C(𝜎) ≡ q20 = q0 (𝜎) + q1 (𝜎) b−a b−a

(5.65)

Note that the curve is given by a barycentric combination, because: b−𝜎 + 𝜎−a = 1. Clearly, b−a b−a however, expression (5.65) is rather awkward and cumbersome. So, if we plug in the parameter transformation given by equation (5.59) in expression (5.65), we get de Casteljau in standard parameter, 𝜉 ∈ [0, 1]: 1 {(b − 𝜎)q10 (𝜎) + (𝜎 − a)q11 (𝜎)} b−a 1 = [{(1 − 𝜉)b − (1 − 𝜉)}q10 (𝜉) + (b − a)𝜉q11 (𝜉)] b−a = (1 − 𝜉)q10 (𝜉) + 𝜉q11 (𝜉) = (1 − 𝜉)2 q0 + 2(1 − 𝜉)𝜉q1 + 𝜉 2 q2

C(𝜉) =

(5.66)

The description of the same curve by equation (5.66) in terms of the standard parameter is obviously a barycentric combination. But, clearly, this time the expression is much nicer and simpler. Hence, if we are given an arbitrary domain parameter range, we would simply apply the affine transformation to the standard parameter and then continue with the de Casteljau algorithm of construction of Bezier curves.

5.6.2.5

End Point Interpolation

The end point interpolation property tells us that a Bezier curve, given an ordered sequence of control points, will always pass through the two end points, that is, the first and the last points. This property puts Bezier curves at a great advantage compared to ordinary splines which do not interpolate the end points without a multiplicity of knots or nodes. Accordingly, in finite element method, the end Bezier control points clearly provide a natural definition of nodes of the displacement functions. Let us recall that an nth standard Bezier curve, Cn (𝜉), is defined as: Cn (𝜉) =

n ∑

Bni (𝜉) qi ,

0≤𝜉≤1

(5.67)

i=0

} { where the basis (blending) functions, Bni (𝜉) , are the nth degree Bernstein polynomials. Since, Bni (0) = Bni (1) = 1 if and only if i = n, and because the Bernstein polynomials are non-negative ∑ and make a partition of unity: ni=0 Bni (𝜉) = 1, we have Cn (0) = q0 as one end point and Cn (1) = qn as another end point, for any degree, that is, n.

5.6.2.6

Convex Hull

The convex hull property has an enormously cost-effective application in an otherwise very costly interference checking between objects. One can easily form the so-called min–max boxes wrapping around the control points of each object and check if these boxes interfere. If they do not, the curves certainly do not. The saving could be tremendous if we have thousands of objects to check against. Of course, if they do, further checks need to be made. Because of repeated

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Figure 5.21 Convex hull. (See colour plate section)

convex combinations of control points, as shown in the construction of a Bezier curve by de Casteljau algorithm, the resulting curve is fully contained within the convex hull of its control points. Because of this property, if the control points are planar, so is the curve as shown in Figure 5.21.

5.6.2.7

Symmetry of the Bezier Curve Representation

The symmetry property enables the description of a Bezier curve in terms of its control points, independent of the way the control polygon is traversed. This has its application in joining two or more Bezier curves or in the subdivision of a Bezier curve. It follows from the symmetry property{of the Bernstein polynomials. } Suppose that we are given an (n + 1) ordered set of control points: q0 q1 ⋯ qn−1 qn describing a Bezier curve. We can describe the same curve } { by reversing the order of the control point sequence to: qn qn−1 ⋯ q1 q0 . However, noting the symmetry property of the Bernstein polynomials as: Bni (𝜉) = Bnn−i (1 − 𝜉), we must have the curve characterized as: Cn (𝜉) =

n ∑

Bni (𝜉) qi =

n ∑

i=0

Bni (1−𝜉) qn−i ,

0≤𝜉≤1

(5.68)

i=0

Recall that the Bernstein polynomials are symmetric about 𝜉 = 12 .

5.6.3

Derivatives of Bezier Curves

Our main goal here is to provide a gateway to various ways of calculating the derivatives (i.e. tangents, curvatures, etc.) of a Bezier curve essential for both mesh generation and finite element method. Our two different yet connected introductions to Bezier curves through de Casteljau algorithm and Bernstein polynomial basis functions nicely tie up the two complementary ways of looking at any computational entity – by geometry and by algebra. Bezier by the de Casteljau algorithm provides the geometric flavor over the algebraic structures of the Bernstein polynomial expansion. Corresponding to each of these two different ways of describing a Bezier curve, naturally there is a method to find the derivatives of a Bezier curve:

r Method 1: Algebraically, by the application of the forward difference operators, resulting from the recurrence relations of Bernstein polynomials, to the control points of a Bezier curve. 2: As a by-product of the geometric de Casteljau algorithm by application of the difference operator to the control points of the intermediate auxiliary Bezier curves and then taking their difference.

r Method

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Furthermore, as we shall see, it is clearly more economical to subdivide (or partition or split) a curve at the point of interest and evaluate the end point derivatives by either of the two above methods of any one of the resulting subdivided curves. Finally, for most practical applications, we need the first derivatives (related to the tangents) and the second derivatives (related to the curvatures). However, for twist computation, we need the third derivative of a Bezier curve. Thus, we will now present two different ways of evaluating the derivatives of, and subdivision algorithm for, a curve.

5.6.3.1

Derivatives: Bernstein Form

Here, we present the derivative formulas for a Bezier curve in terms of the Bernstein basis functions. If we are interested only in the derivatives as in the finite element phase of element stiffness matrix generation with as yet unknown displacement control vectors, these forms seem appropriate for use. We apply directly the derivative formulas of the Bernstein polynomials to the definition of a Bezier curve of a given order and obtain the derivatives of a Bezier curve in a recurrence form involving lower-order Bezier curves. 5.6.3.1.1 First Derivative Let us recall the first derivative property of a Bernstein polynomial: ( ) (𝜉) − Bn−1 0≤𝜉≤1 Bni,𝜉 (𝜉) = n Bn−1 i (𝜉) , i−1

(5.69)

Let us also recall, from the Bernstein–Bezier connection, that a Bezier curve, Cn (𝜉), of degree n is given by: Cn (𝜉) =

n ∑

Bni (𝜉) qi ,

0≤𝜉≤1

(5.70)

i=0

Now applying equation (5.69) directly in equation (5.42), we get the first-order derivative, Cn,1 (𝜉), of a Bezier curve as: n ∑ (

) Bn−1 (𝜉) − Bn−1 i (𝜉) qi i−1 i=0 ( ) n n−1 ∑ ∑ =n (𝜉) qi − n Bn−1 Bn−1 i (𝜉) qi i−1 i=1 ( i=0 ) n−1 n−1 ∑ ∑ Bn−1 =n Bn−1 i (𝜉) qi+1 − n i (𝜉) qi

Cn,1 (𝜉) = n

=n

i=0 n−1 ∑

(

∴Bn−1 (𝜉), i−1

) i ∉ {1, 2, … , n − 1} (5.71)

(Re-index: i − 1 → i)

i=0

(qi+1 − qi ) Bn−1 i (𝜉)

i=0

Let us note the following.

r The first derivative of a Bezier curve is also a Bezier curve. Its degree is one less than that of r

the original one. Its control points are obtained by simply taking the difference of the original control points in a forward difference manner. The derivative expression greatly improves computational efficiency: we never have to take the actual derivative of the curve; we simply use the forward difference of the control points, and lower the degree of the Bernstein polynomial by one.

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In order to write equation (5.71) in a compact form, we can recall our definition of the first forward difference operator: Δ1 qi ≡ qi+1 − qi . Thus, we can rewrite the first derivative of a Bezier curve as:

Cn,1 (𝜉) = n

n−1 ∑

(Δ1 qi ) Bn−1 i (𝜉)

(5.72)

i=0

5.6.3.1.2 End Point First Derivatives The end point derivatives are of great importance because derivatives of a Bezier curve at any 𝜉 ∈ [0, 1] can be reduced to finding the corresponding end point derivative by subdividing the curve at 𝜉 and considering either of the two resulting curves. Noting the value of Bn−1 i (0) and (1), we can easily write the expressions for the end point derivatives: Bn−1 i Cn,1 (0) = n (Δ1 q0 ) = n (q1 − q0 ) Cn,1 (1) = n (Δ1 qn−1 ) = n (qn − qn−1 )

(5.73)

Let us observe the following.

r The first derivatives at end points involve one more control point than the end control points themselves.

r The tangents of the curve at end points will point along the direction of the end sides of the Bezier polygon. Example: For n = 3, that is, a cubic Bezier curve C3,1 (0) = 3 (q1 − q0 )

C3,1 (1) = 3 (q3 − q2 )

5.6.3.1.3 General Derivatives The higher-order derivatives can be similarly expressed through the use of higher-order forward difference operators and repeated application of the first-order derivatives. Thus, the dth derivative, Cn,d (𝜉), of a Bezier curve is: n! ∑ d (Δ qi ) Bn−d (𝜉), i (n − d)! i=0 n−d

Cn,d (𝜉) =

d ∈ {0, 1, … , n}

(5.74)

Example: For a cubic Bezier curve, C3 (𝜉), we would like to find the point, C3 ( 13 ), and the first

derivative, C3,1 (𝜉), at 𝜉 = 13 . With n = 3, d = 1 and 𝜉 = 13 , for the point on the curve at 𝜉 = 13 , we get: ( ) C3

1 3

| ≡ C3 (𝜉)|| 1 = {(1 − 𝜉)3 q0 + 3(1 − 𝜉)2 𝜉 q1 + 3 (1 − 𝜉)𝜉 2 q2 + 𝜉 3 q3 } 1 𝜉= |𝜉= 3 3 ( ) ( ) ( ) ( ) 8 1 q0 + 49 q1 + 29 q2 + 27 q3 = 27

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Then, using equation (5.74) with n = 3, d = 1 and 𝜉 = 13 , we have: C3,1 (𝜉)||

1 𝜉= 3

=3

2 ∑

(Δ1 qi ) B2i (𝜉)

i=0

= 3{(1 − 𝜉)2 (q1 − q0 ) + 2𝜉(1 − 𝜉)(q2 − q1 ) + 𝜉 2 (q3 − q2 )}

𝜉=

1 3

= 3{ 49 (q1 − q0 ) + 49 (q2 − q1 ) + 19 (q3 − q2 ) = − 43 q0 + q2 + 13 q3 5.6.3.1.4 End Point Higher Derivatives As indicated before the end point derivatives are extremely important for smoothness definition between adjoining Bezier curves, or inversely for subdivision (i.e. splitting of Bezier curves) algorithms central to both interactive modeling and finite element mesh adaptation and refinement. The end point derivatives are given as: Cn,d (0) =

n! (Δd q0 ), (n − d)!

Cn,1 (1) =

n! (Δd qn−d ) (n − d)!

(5.75)

Let us note the following.

r The dth

end point derivatives of a Bezier curve depend only on (d + 1) control points from the corresponding ends. This follows directly from the definition of the dth degree forward ( ) ∑ d qi+j difference which we may recall as: Δd qi ≡ dj=0 (−1)d−j j

Example: For n = 3, that is, a cubic Bezier curve, the first (d = 1) and second derivatives (d = 2) at starting end point (𝜉 = 0) are given as: C3,1 (0) = 3 (q1 − q0 ),

C3,2 (0) = 6 Δ2 q0 = 6 (q2 − 2q1 + q0 )

(5.76)

Thus the tangent at 𝜉 = 0 is defined by q0 and q1 . The curvature, likewise, is proportionally defined by only q0 , q1 and q2 . 5.6.3.1.5 Discussion r Here we have presented the derivative expressions for a Bezier curve with special reference to the first- and second-order derivatives, which are of more practical importance. However, in order to determine them simultaneously along with the Bezier curve points, a better alternative may be a utilization of the de Casteljau algorithm. This method of the derivative evaluation is introduced next.

5.6.3.2

Derivatives: de Casteljau Form

Next we will look at the simultaneous evaluation of a point and the derivatives of a Bezier curve through the de Casteljau algorithm. In the mesh generation phase, we often need information on both the point on a curve and its derivatives, as in the smooth joining of multiple Bezier curves. In such situations, the de Casteljau algorithm seems more suitable than the Bernstein form of the

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derivatives. We first observe that the forward difference operator is linear in control points. Then, starting from the derivative definition in terms of the Bernstein polynomials, we can eventually identify the derivatives with the difference of the intermediate de Casteljau control points. 5.6.3.2.1 General Derivatives Let us recall that the Bernstein form of the dth derivative, Cn,d (𝜉), for an n-degree Bezier curve is given as: n! ∑ d (Δ qi ) Bn−d (𝜉), i (n − d)! i=0 n−d

Cn,d (𝜉) =

d ∈ {0, 1, … , n}

(5.77)

Now noting the linearity property of a forward difference operator, we can rewrite Cn,d (𝜉) as:

Cn,d (𝜉) = = = = =

n−d ∑ n! (Δd qi ) Bn−d (𝜉), i (n − d)! i=0 } {n−d ∑ n! d n−d qi Bi (𝜉) Δ (n − d)! i=0 n! Δd Cn−d (𝜉) (n − d)! n! (𝜉) Δd qn−d 0 (n − d)! n! (𝜉) − Δd−1 qn−d (𝜉)) (Δd−1 qn−d 1 0 (n − d)!

d ∈ {0, 1, … , n} (linear operators Σ and Δcommute) (by definition) (by de Casteljau algorithm) (definition of Δd ) (5.78)

Thus, the de Casteljau form of the derivatives for a Bezier curve is given by: Cn,d (𝜉) =

n! (𝜉) − Δd−1 qn−d (𝜉)) (Δd−1 qn−d 1 0 (n − d)!

(5.79)

where, Δ is the forward-difference operator, qi ’s are the control points and Bni is the ith Bernstein basis functions for Bernstein polynomials of degree n. Let us note that

r In the de Casteljau algorithm, the polygons, qn−d (𝜉), mediate auxiliary polygons.

i

i ∈ {0, 1, .., n} are generated as inter-

5.6.3.2.2 First Derivatives The above observation provides an excellent opportunity to use the de Casteljau algorithm to compute curve points and the first derivatives at those points, if required simultaneously. Thus, for the computation of the first derivative, or equivalently, the tangent vectors at any 𝜉 ∈ [0, 1], we get with d = 1: ) ( Cn,1 (𝜉) = n qn−1 − qn−1 1 0

(5.80)

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Figure 5.22 de Casteljau construction of a point on cubic curve.

r The tangent vectors are given by the difference of the intermediate de Casteljau points scaled by n. In other words, among the intermediate polygons that are created successively towards evaluation of a point on the curve, the final polygon, which is a line segment, coincides with the tangent at that point.

5.6.3.2.3 Geometric Feel The above observation may be grasped better by presenting the de Casteljau algorithm in a geometric form. Let us now consider the de Casteljau algorithm as applied to the case of a cubic (n = 3) Bezier curve. Example: Recalling the same example that was solved by the Bernstein form of derivative computation, for a cubic Bezier curve, C3 (𝜉), we would like to find the point, C3 ( 13 ), and the first derivative, C3,1 (𝜉), at 𝜉 =

1 3

by the de Casteljau method. As shown in Figure 5.22, and using the

de Casteljau algorithm with n = 3, d = 1 and 𝜉 = 13 : ( ) ( ) ( ) = q0 q01 13 = q1 q02 13 = q2 q03 13 = q3 ( ) ( ) ( ) Iteration 1: q10 13 = 23 q0 + 13 q1 , q11 13 = 23 q1 + 13 q2 , q12 13 = 23 q2 + 13 q3 ( ) ( ) ( ) ( ) Iteration 2: q20 13 = 23 q10 13 + 13 q11 13 = 49 q0 + 49 q1 + 19 q2 , q21 13 = ( ) ( ) 2 1 1 1 1 1 + = 49 q1 + 49 q2 + 19 q3 q q 3 1 3 3 2 3 ( ) ( ) ( ) ( ) 8 1 Iteration 3: C3 13 ≡ q30 13 = 23 q20 13 + 13 q21 13 = 27 q0 + 49 q1 + 29 q2 + 27 q2

Initialize: q00

( ) 1 3

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And, finally, the derivative from terms in iteration 2: ( ) C3,1

1 3

) ( ) ( ) ( ) ( ) 3! {Δ0 q21 13 + Δ0 q20 13 } = 3{q21 13 − q20 13 } 2! [{ } { }] 4 4 1 4 4 1 =3 − = − 43 q0 + q2 + q + q + q q + q + q 1 2 3 0 1 2 9 9 9 9 9 9 (

=

1 q 3 3

We can easily verify that both methods give the same results.

5.6.4 Degree Elevation Next we consider an algorithm to degree elevate a Bezier curve, and thus allow us to profile a curve as will be explained shortlyl. Degree elevation is the numerical process of expressing a Bezier curve of a certain degree, say, n, defined by (n + 1) control points as a Bezier curve of (n + 1) degree with (n + 2) control points without changing the shape of the original curve. Degree elevation is central to both interactive modeling and finite element analysis. In the modeling phase based on boundary representation, the surfaces are formed by boundary curves of the same degree. Thus, if any of these curves is previously generated with a lower degree than the highest degree boundary curve, the lower degree curve needs to be degree elevated, that is, redefined in terms of the highest degree without changing the shape of the curve in any way for the surface definition. In another situation, where, for example, a curve of degree 1 (i.e. a line segment) is joined with a curve of degree 2 (i.e. a parabola) to generate a boundary curve of a surface, the line segment needs to be expressed as a special degenerate quadratic curve so that the entire profile curve is quadratic; an example of curve blending by degree elevation is shown in Figure 5.23.

Figure 5.23 Curve blending by degree elevation. (See colour plate section)

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177

Degree Elevation Proposition

Let {qi }, i ∈ {0, 1, 2, ∙ ∙ ∙, n} be the control vertices of a Bezier polygon defining a Bezier curve, Cn (𝜉), of degree n on 𝜉 ∈ [0, 1]. Then, let us define: {qei }, i ∈ {0, 1, 2, ∙ ∙ ∙, n + 1}, the new set of (n + 2) control vertices such that:

qei =

(

) ) ( i i qi−1 + 1 − q, n+1 n+1 i

i ∈ {0, 1, 2, ∙ ∙ ∙, n + 1}

(5.81)

that is, {qei }, i ∈ {0, 1, 2, ∙ ∙ ∙, n + 1} are obtained by piecewise linear interpolation of qi−1 and i , ∀i ∈ {0, 1, ∙ ∙ ∙, n + 1}. Then, both the sets of control points define qi at parameter value: 𝜉 = n+1 the same Bezier curve, Cn (𝜉). Proof Let Cen+1 (𝜉) be the curve generated by {qei }, i ∈ {0, 1, 2, ∙ ∙ ∙, n + 1}; we will show that Cen+1 (𝜉) = Cn (𝜉). By definition and using index j = i − 1, we get:

Cen+1 (𝜉) =

n+1 ∑

qei Bn+1 (𝜉) = i

i=0

n+1 ( ∑ i=0

n+1 ( ) ) ∑ i i qi−1 Bn+1 q Bn+1 (𝜉) 1− (𝜉) + i n+1 n+1 i i i=0

n+1 ( ) (n + 1)! ∑ (n + 1)! n+1−i 𝜉 i (1 − 𝜉)n+1−i qi−1 + 𝜉 i (1 − 𝜉)n+1−i qi (n + 1 − i)! i! n+1 (n + 1 − i)! i! i=0 i=0 } { n+1 } { n+1 ∑ ∑(n + 1 − i) n! n! i n−j i n+1−i + (1 − 𝜉) 𝜉 (1 − 𝜉) qj 𝜉 (1 − 𝜉) =𝜉 qi (n − j)! j! n+1 (n − i)! i! j=−1 i=0 { n } { n+1 } ∑ ∑ n n =𝜉 qi Bi (𝜉) + (1 − 𝜉) qi Bi (𝜉)

∑( n+1

=

i n+1

i=−1

)

i=0

Now, noting that Bni (𝜉) ≡ 0, ∀i ∉ {0, 1, … , n}, following through above expression, we { } n+1 {∑n } ∑ ∑ e n n qi Bi (𝜉) = ni=0 qi Bni (𝜉) = Cn (𝜉) comhave: Cn+1 (𝜉) = 𝜉 i=0 qi Bi (𝜉) + (1 − 𝜉) i=0

pletes the proof. Example DE1: Degree elevate the line segment shown in Figure 5.24 to the nth degree.

Figure 5.24 Example: degree elevation applied to a linear curve.

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We have: n = 1, q00 = a, q01 = b, and the forward difference operator: Δq0 = (q1 − q0 ) =b − a. Thus, application of the degree elevation algorithm once results in a quadratic, that is, n = 2, curve as: q20 = q0 1 1 1 q + q = q0 + Δq0 2 0 2 1 2 q22 = q1 q21 =

Now, application of another degree elevation will result in a cubic (n = 3) curve as: q30 = q0 1 2 q + 3 0 2 q32 = q21 + 3 q33 = q1 q31 =

2 2 q = q0 + 3 1 1 2 q = q0 + 3 2

1 Δq 3 0 2 Δq 3 0

By repeated applications of the degree elevation procedure up to degree n, it can be shown to be given by: qni = q0 + ni Δq0 . The resulting control points are shown in Figure 5.24. Example DE2: The Bezier curve shown in Figure 5.25 has been expressed as a quadratic curve as well as a cubic curve; establish their equivalence. ∑ Let the quadratic, as shown in Figure 5.25, be represented as: C2 (𝜉) = 2i=0 qi B2i (𝜉) = (1 − 𝜉)2 q0 + 2𝜉(1 − 𝜉)q1 + 𝜉 2 q2 . We now let: qe0 = q0 , qe3 = q2 , and qe1 = 13 (q0 + 2q1 ), qe3 = 1 (2q1 3

+ q2 ), then: Ce3 (𝜉) =

2 ∑

qei B3i (𝜉)

i=0

1 1 = (1 − 𝜉)3 q0 + 3𝜉(1 − 𝜉)2 (q0 + 2q1 ) + 3𝜉 2 (1 − 𝜉) (2q1 + q2 ) + 𝜉 3 q2 3 3 = (1 − 𝜉)2 q0 + 2𝜉(1 − 𝜉)q1 + 𝜉 2 q2 = C2 (𝜉)

Figure 5.25 Example: degree elevation applied to a quadratic curve.

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The process of degree elevation can be applied repeatedly to obtain higher-degree representation e of the same Bezier curve. It can be proved by induction that if {qi m }, i ∈ {0, 1, … , n + m} are the control vertices obtained after m repeated degree elevations, then:

e

qi m

( ) ⎤ ⎡ m n ⎢( ) ⎥ ∑ i − j ⎢ n ( = ) ⎥ qj , ⎢ j n+m ⎥ j=0 ⎥ ⎢ i ⎦ ⎣

∀i ∈ {0, 1, … , n + m}

(5.82)

5.6.5 Subdivision Algorithm Here, we offer an algorithm to subdivide or partition a Bezier curve at any given parameter point, and thus produce new Bezier curves. As an example, a point P, corresponding to any 𝜉 ∈ [0, 1] on a cubic Bezier curve, C3 (𝜉), divides the curve into two curves as shown in Figure 5.26. Looking at Figure 5.26, the question is: given the Bezier representation of the entire curve, and given the control points, do we have a Bezier representation, that is, can we find the Bezier control points of each part of the curve, Ln (𝜉) on the left of P, and Rn (𝜉) on the right of P? The answer is yes and the process of finding such representations is given by what is known as the subdivision algorithm. It has various applications. For example, in the model generation phase, the formula for evaluating the tangent vector at any arbitrary parameter point on the curve is rather involved and cumbersome. However, the formula for the tangents at the two ends of a curve is very simple and is given by the scaled difference of the two consecutive control points at the end. Thus, if we subdivide the curve at the parameter point, then we can evaluate the tangent vector by applying the end point formula to any of the partitioned curves. The subdivision algorithm is equally important in the adaptive refinement phase, say in contact–impact problems of finite element method where we may produce smaller elements from the original element on demand. For convenience and without any loss of generalization, we restrict ourselves to the de Casteljau algorithm for a cubic curve in the following discussion. Then, we explain the subdivision algorithm as a by-product of the de Casteljau algorithm. We provide a simple example of a cubic curve to demonstrate the various properties of the algorithm.

Figure 5.26 Subdivision: left and right half of a cubic curve.

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Figure 5.27 Subdivision: left and right control polygons of a cubic curve.

5.6.5.1

de Casteljau Algorithm for a Cubic Bezier

As shown in Figure 5.27, for a cubic (n = 3) Bezier Curve, we can explicitly go through the steps of the de Casteljau algorithm as follows: Step 0: q00 = q0 ,

q01 = q1 , q02 = q2 and q03 = q3 ,

for a given 𝜉 ∈ [0, 1]

Recursion 1: q10 = (1 − 𝜉) q0 + 𝜉 q1 , q11 = (1 − 𝜉) q1 + 𝜉 q2 , 𝜉 q3 Recursion 2: q20 = (1 − 𝜉) q10 + 𝜉 q11 ,

q12 = (1 − 𝜉) q2 +

q21 = (1 − 𝜉) q11 + 𝜉 q12 ,

Recursion 3: C3 (𝜉) = q30 = (1 − 𝜉) q20 + 𝜉 q21 , as shown in Figure 5.27. Note that the control point sets {q0 , q10 , q20 , q30 } and {q30 , q21 , q12 , q3 } subdivide the original cubic curve into two cubic curves at the parameter point: Ln (𝜉) curve on the left and Rn (𝜉) curve on the right with n = 3. Thus, we see that, for a cubic, the de Casteljau algorithm provides us with the control points for the two subdivided curves. Thus, it can be seen as another important by-product of the de Casteljau algorithm that generalizes into what is known as a subdivision algorithm and is fundamental to both interactive mesh generation (e.g. tangent and curvature evaluation) and finite element analysis (e.g. adaptive refinement).

5.6.5.2

Subdivision Algorithm for Bezier Curves

We first concentrate on the left part: in order to find the unknown control points, {Li }, of the left curve from the given control points, {qi }, i ∈ {0, 1, … , n}, we note that since both curves, Cn (𝜉) and Ln (𝜉), are parts of the same polynomial of degree n, they must be identical in all their derivatives everywhere – particularly at the 𝜉 = 𝜆 = 0 end. But, we already know that the dth derivative of a Bezier curve at a given end depends only on the (d + 1) consecutive control points from the end. Thus the dth derivative of Cn (𝜉) at 𝜉 = 𝜆 = 0 depends on {qi }, i ∈ {0, 1, … , d}, and of Ln (𝜉) on {L0 , L1 , … , Ld }. However, in the de Casteljau algorithm, the first (d + 1) control points (see the list of steps above as an example) also correspondingly define the qd0 (𝜉) and

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Ld0 (𝜆) curves; these two curves are identical since they have the same derivatives of all degrees at 𝜉 = 𝜆 = 0. Now, evaluating at 𝜉 = 𝜉0 or 𝜆 = 1, we get: Ld0 (1) = qd0 (𝜉0 ). Finally, noting that Ld0 (1) = Ld , we have: Ld = qd0 (𝜉0 ), d ∈ {0, 1, … , n}

(5.83)

Thus, Ld , the dth control point of the left part, is obtained by the de Casteljau intermediate control points, qd0 (𝜉0 ), of the original curve evaluated at the partition point, 𝜉 = 𝜉0 . For the right curve, recalling the symmetry property of the Bezier curve, we can easily deduce that Rd , the dth control point of the right part, counted from the right end point towards the partition point, is given by the de Casteljau intermediate control points, qdn−d , evaluated at the partition point, 𝜉 = 𝜉0 . Thus, we have: Rd = qdn−d (𝜉0 ), d ∈ {0, 1, … , n}

(5.84)

In other words, Rd , the dth control point of the right part is obtained by the de Casteljau intermediate control points, qdn−d (𝜉0 ) of the original curve evaluated at the partition point, 𝜉 = 𝜉0 . The expressions given by equations (5.83) and (5.84) are known as the subdivision formulas for a Bezier curve. This subdivision process is widely useful. For example:

r For r r r

rendering of curved surfaces generated by Bezier patches, they eventually have to be appropriately subdivided for application of planar algorithms of color, light reflection, and so on. For derivative evaluations at any point on a curve, the object may be momentarily subdivided to convert subject point to end point of a curve, for which the derivative expressions are very simple and thus cost-effective. For various intersection algorithms, the object may need to be subdivided repeatedly. For mesh refinement during adaptive finite element scheme such as in contact-impact problems, subdivision is very important as it allows change in element size on demand.

5.7

Composite Bezier–Bernstein Curves

A general mesh representation of a physical continuum of a structural system is expected to contain complex curves to start with. While Bezier representation is very good at mimicking the behavior of a given curve, the degree requirement on the interpolating Bezier curves for high complexity can be prohibitively large. For such cases, it is numerically wise to use the following. Definition

r A composite Bezier curve is a set of piecewise Bezier parametric polynomial r

curves of a particular degree between the junction points with the desired continuity at junction points (known as knots). Each component Bezier curve defines a Bezier element, or simply element. Thus far in our discussion we have almost exclusively defined a single Bezier curve as a transformation of a

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Figure 5.28 Composite Bezier curve.

r

unit real line segment, 𝜉 ∈ [0, 1], because of the invariance property of an affine parameter transformation. However, we have piecewise Bezier curves that we can still define the element or local parameter, 𝜉 ∈ [0, 1], defines the domain of the individual Bezier curve component. L ⋃ However, the composite Bezier curve may be defined over some union of intervals, [gi , gi+1 ], i=0

in which we designate the global parameter such that g ∈

L ⋃ i=0

[gi , gi+1 ] defines the domain of

the entire Bezier curve with the following definitions.

r Knots are defined as those parameter values, gi , i ∈ {0, 1, … , L} that define the intervals. r A knot sequence or knot vector is defined as the ordered set, 𝜋 = {g0 , g1 , … , gL }. For interpolation problems (to be discussed later) in structural mesh generation, representations by composite quadratic C1 curves for 2D geometry, and composite cubic C1 or C2 curves for 3D geometry seem reasonably adequate. Therefore, from now on, we will restrict our discussion primarily to focusing on cubic Bezier curves unless we need to touch upon general results. Figure 5.28 shows a composite cubic Bezier curve with local and global parameters, and knots. Obviously, for any ith Bezier element, i ∈ {0, 1, … , L}, the local parameter, 𝜉i , and the global parameter, g, are related by: 𝜉i =

g − gi , hi

i ∈ {0, 1, … , L}

(5.85)

where hi = gi+1 − gi , i ∈ {0, 1, … , L}, is the ith knot interval. This clearly makes 𝜉i ∈ [0, 1], i ∈ {0, 1, … , L}, as desired. For a composite Bezier curve written as C(g(𝜉)) or C(𝜉(g)), the derivative at a point on the curve, with respect to g or 𝜉, is related by chain rule as: dC = dg

( ) 1 dC(𝜉i ) hi d𝜉i

⇒

dC(g) dC = hi d𝜉 dg

(5.86)

Example: To find the end point first derivatives of a cubic composite Bezier curve with respect to the global parameter, g, we recall from equation (5.76) that for the first segment:

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dC3 (0) d𝜉

= 3 (q1 − q0 ), and for the last segment: (5.86), we get: dC3 (0) 3 (q − q0 ), = dg h0 1

5.7.1.1

dC3 (1) d𝜉

= 3 (q3L−1 − q3L ), so using equation

dC3 (1) 3 (q − q3L ) = dg hL−1 3L−1

(5.87)

Inter-element Continuity Class, Cd

In our discussion on the subdivision algorithm for a Bezier curve, we reinterpret each piece of the two subdivided Bezier curves as an element joined at the junction point of subdivision, and the entire Bezier curve as the composite curve of these two Bezier elements, and so we can safely deduce that these elements, being parts of the same polynomial curve of degree n which is at least Cn , themselves, in particular, are Cn , that is, n-times continuously differentiable, at all 𝜉 ∈ [0, 1]. Now, let us consider two elements of a composite Bezier curve of degree n, g−g as shown in Figure 5.28. Let us define a local parameter, 𝜉, such that 𝜉 = h i ; then the knot, gi+2 , with respect to 𝜉, is given by extrapolation as: 𝜉 =

hi +hi+1 . hi

i

Let {qi },

i ∈ {0, 1, … , n},

ith

and the (i + 1)th elements, and {qi }, i ∈ {n + 1, n + 2, … , 2n} be the control nets of the d respectively. Then, for C continuity, that is, d times continuous differentiability, at g = gi+1 , we must have, by the chain rule of differentiation, the following conditions: dd Ci+1 dd Ci = ⇒ dgd dgd

( )j ( )j 1 1 (Δj qn−j ) = (Δj qn ), hi hi+1

j ∈ {0, 1, … , d}

(5.88)

where Δj is the jth finite difference operator. In actual applications, however, we will be interested only in C0 , C1 and C2 continuity: Case C0 continuity: For d = 0, j = 0 ⇒ Δj ql = ql , so we will have: qn = qn

(5.89)

as expected, that is, both the elements should have the same control points at g = gi . Case C1 continuity: For d = 1, with j = 1 ⇒ Δj ql = ql+1 − ql , so we will have, apart from the C0 continuity condition given by equation (5.89), the following additional condition: ( ( ) ) 1 1 (qn − qn−1 ) = (qn+1 − qn ) hi hi+1

(5.90)

We recognize, from equation (5.89), that both the elements should have the same control points at g = gi , and from equation (5.90), that not only the directions of the tangent vectors – (qn − qn−1 ) of element i and (qn+1 − qn ) of element (i + 1), – at both sides of the knot, gi , must be parallel, but also their lengths must be proportional to the interval length, that is, to hi :hi+1 ; for equal intervals, that is, for hi = hi+1 , the tangent vectors are equal. Now, for future application of representing cubic spline curves by piecewise Bezier curves, as will be done shortly, for C1

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continuity between two adjoining pieces of Bezier curves at junction control point, qn , we must satisfy, from equation (5.90), the relation:

qn =

hi+1 hi q + q hi + hi+1 n−1 hi + hi+1 n+1

(5.91)

Case C2 continuity: For d = 2, with j = 2 ⇒ Δ2 qn = qn+2 − 2qn+1 + qn , and Δ2 qn−2 = qn−2 − 2qn−1 + qn , in equation (5.88), so we will have, apart from the C0 and C1 continuity conditions given by equation (5.89) and equation (5.90), respectively, the following additional condition: ( )2 ( )2 1 1 (qn − 2qn−1 + qn−2 ) = (qn+2 − 2qn+1 + qn ) hi hi+1

(5.92)

We recognize, from equation (5.89), that both the elements should have the same control points at g = gi , and from equation (5.90), both the elements should have same control points at g = gi , and that the tangent vectors must be collinear and with the same ratio, as before, and additionally from equation (5.92), we find, as expected, that C2 continuity is determined by two consecutive inner control points on either side of the junction control point. Although the above formula, as given by equation (5.92), for C2 continuity condition derived by derivative method is important on its own right algebraically, for our purposes of an eventual generalized and geometrically constructive representation of the B-spline curves by composite Bezier curves it is rather less useful. For this, we turn instead to the concept behind the subdivision algorithm and the de Casteljau construction presented earlier. Let us recall that for a Bezier curve of degree n, {qn−d , qn−d+1 , … , qn } on the left-hand side, and {qn , qn+1 , … , qn+d } for the right-hand side represent two Bezier curves of degree d on [gi−1 , gi ] and [gi , gi+1 ], respectively. And, they must together define a global Bezier curve of degree d on [gi−1 , gi+1 ], and thus, guarantee the existence of (d + 1) control points that include end control points, qn−d and qn+d . Now, for C2 continuity, with d = 2, there must exist a quadratic Bezier curve with three control points with two of them as, qn−2 and qn+2 . Let us denote the third control point as s, as shown in Figure 5.29.

Figure 5.29 C2 continuous composite Bezier curves.

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Then, the two polygons, {qn−2 , qn−1 , qn } and {qn , qn+1 , qn+2 } define a quadratic Bezier polygon, {qn−2 , s, qn+2 } such that: qn−1 = (1 − 𝜉i ) qn−2 + 𝜉i

s, qn+1 = (1 − 𝜉i ) s + 𝜉i qn+2

(5.93)

or, in other words, qn−1 =

hi+1 hi q + s, hi + hi+1 n−2 hi + hi+1

qn+1 =

hi+1 hi s+ q hi + hi+1 hi + hi+1 n+2

(5.94)

5.7.2 Discussions

r For higher-degree inter-element continuity, such explicit expressions as in equations (5.93) r

and (5.94) are effectively impossible to generate. For this, we need to turn to a more general class of curves with inter-element continuity built into them – splines, which are a natural generalization of the composite Bezier curves So we study next the explicit expressions as in equations (5.93) and (5.94), but they provide the crucial connection for representing cubic B-splines by composite Bezier curves where s stands for what will be described as one of the de Boor control points defining the B-spline curves.

Thus, we now turn to curves or functions – splines and B-splines – that are a natural generalization of the composite Bezier curves or functions. In our implementation phase, however, we always come back to representing the cubic B-splines by the composite Beziers using the C0 , C1 and C2 continuities introduced earlier.

5.8

Splines: Schoenberg B-spline Curves

In the chapter introduction, we alluded to limitations of piecewise Lagrange and Hermite polynomials, and even of Bezier spaces. A generalization that includes and improves upon these spaces as special cases will be discussed here that tries to avoid the limitations by converting the mixed problem of interpolation and smoothness to one of just interpolations; as it turns out, this is achieved by defining piecewise polynomials with junction smoothness embedded in them. Thus, in what follows, we introduce and investigate these polynomials after we briefly summarize the historical background of their genesis.

5.8.1 Physical Splines Historically speaking, a spline is a long thin flexible elastic strip of metal or wood used to design smooth curves passing through some given points describing parts in the ship or automobile industry, as shown in Figure 5.30. In Figure 5.30, the arrows represent what are known as weights or ducks. For an extremely thin strip, with small deformation, the shape of the deformed strip is such that the bending moment, M(x), between any two consecutive supports varies linearly and obeys the Bernoulli–Euler law, d2 y that is, M(x) = dx2 = y′′ . Thus, the deformed shape is cubic at each interval of the supports with not only the slopes but the curvatures being continuous at the weights; but, of course, the shear, V(x), which is proportional to the third derivative, y′′′ (x), is discontinuous at the weights. Thus, what we have is a deformed shape that could possibly be described by a function of class C2 that

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Figure 5.30 Physical spline with supports.

reduces to cubic polynomials (i.e. of degree 3) in each of the subintervals created by the weights. Note also that because these are simple supports with no clamping, the curvatures as well as the bending moments at both ends are zero, and hence the slopes at both ends are linear. This physical description of a spline can thus be translated and generalized into their mathematical counterparts known as the spline functions. 5.8.1.1

Spline Space, S(𝝅, n, k)

Apart from building on the physical splines, the need for a special class of functions arises in a natural way in trying to improve upon piecewise Lagrange, L(𝜋, n), or piecewise Hermite, H(𝜋, n), polynomials of degree n, with 𝜋 known as the partition or knot sequence. In our attempt to identify this class of functions, and thus the space thereof, the primary goal is to convert the mixed constraint problems of approximation – that is, containing both interpolations and smoothness constraints – to those having pure smoothness constraints. Thus, as we will be see, it is achieved by constructing the basic functions for the space with smoothness conditions already built into them. With this in mind, let us first introduce the spline functions and the space thereof, and then finally the basis functions of the space, with the clear understanding that the problem of approximation still remains as one of mixed constraints for which a solution is being sought. Definition: A{polynomial spline function of degree n (or order n + 1) with a partition or knot } sequence, 𝜋 = g0 , g1 , … , gL ∈ [a, b], is any function ∈ C1 [a, b] that reduces to a polynomial } { } { of degree n in each interval, 𝜋 = gi , gi+1 , i ∈ 1, … , gL−1 . Thus, a spline function consists of a number of polynomials of degree n connected together at knots where they are k-times continuously differentiable. Clearly, these functions form a linear space which we will denote as S(𝜋, n, k). By definition, every polynomial of degree n defined on [a, b] belongs to this space; it may readily be recognized that S(𝜋, n, 0) = L(𝜋, n), the piecewise Lagrange polynomial space, and that S(𝜋, n, 1) = H(𝜋, n), the piecewise Hermite polynomial space of degree n. Finally, S(𝜋, n, −1) is the space of all step functions with discontinuity at knots. Remarks

r However, given our main objective of constructing spline functions by generalization of the

r

Bezier functions, to be introduced and detailed later, we will restrict our attention to S(𝜋, n, n − 1), that is, the space of functions ∈ Cn−1 [a, b] that reduces to polynomials of degree n in each interval of 𝜋. In other words, we will consider only piecewise polynomials of degree n with smoothness condition on each knot defined by (n − 1) times continuous differentiability. We know that, from now on, the degree of smoothness is one less than that of the component polynomials, so we may rename our space as, B(𝜋, n): Definition: S(𝜋, n, n − 1) ≡ B(𝜋, n) .

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We would now like to know various relevant properties of B(𝜋, n) – its dimensions, variants, constructive characterization, the basis functions that span it, and so on.

5.8.1.2

Dimension of B(𝝅, n)

Suppose the partition, 𝜋 = {g0 , g1 , … , gL }, consists of (L + 1) distinct knots defining L subintervals of [a, b]. Now, it follows from the definition of B(𝜋, n) that there are L many n-degree polynomials connected at (L − 1) internal knots where they are (n − 1) times continuously differentiable. Then, L many n-degree polynomials, each requiring (n + 1) parameters to define them, need a total of L(n + 1) parameters. But (n − 1) differentiability constraints at each (L − 1) internal knot reduce the number of free (or independent) parameters by n(L − 1). Thus, the net-free degrees of freedom defining the dimension of the space is given by: L(n + 1) − n(L − 1) = L + n. Clearly, then, the dimension is given as: dim B(𝜋, n) = L + n

(5.95)

that is, it takes (L + n) parameters to uniquely determine a member of B(𝜋, n). In the above definitions, we have assumed that the knots are distinct, that is, simple knots. However, the definition of B(𝜋, n) does not restrict knots of multiplicity. In fact, as we shall see, multiplicity of knots crucially determines the end interpolation and intermittent smoothness: the more the multiplicity, the less is the smoothness in the interior knots. We will get back to this shortly, but first let us characterize its effect on the dimension of B(𝜋, n). Thus, suppose the polynomials are joined at knots in such a way that at knot, gi , i ∈ {0, 1, … , L}, the function is only (n − mi ) times continuously differentiable, then, by a similar argument as above, we have:

dim B(𝜋, n) = (n + 1) +

L−1 ∑

mi

(5.96)

i=1

We can easily verify that if mi = 1, ∀i ∈ {1, … , L − 1} – that is, the functions are (n − 1) continuously differentiable at all internal knots – the dimension of the space reduces to (L + n) as before in equation (5.95). Now, several observations are in order. Remarks

r The smoothness condition being already built into the spline function, there is no derivative r

data required for internal knots, and this eliminates one of the limitations of the members of H(𝜋, n), that is, the space of piecewise Hermites of degree n, For n = 2j + 1, that is, odd-degree polynomials with (L + 1) distinct knots, dim B(𝜋, 2j + 1) = L + 2j + 1. Thus, of (L + 2j + 1) free degrees of freedom, only (L + 1) interpolatory constraints are supplied by the data at (L + 1) knots. However, to uniquely determine a member function, we will still need 2j constraints; these are generally supplied as j-interpolatory derivative constraints at each of the end knots. In fact, depending on the problem at hand, variants of spline functions can be generated based on these end knot interpolatory conditions. For example, for n = 3, that is, a cubic spline function, j = 1; thus, we will still need either slope or curvature constraints at, say, each end to uniquely determine an interpolation spline that belongs to B(𝜋, 3).

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We will very soon present some variants based on this remark. But, for now, let us try to characterize B(𝜋, n).

5.8.1.3

Spline Function Representation

Before we set out to accomplish our stated goal, let us first extend the partition, 𝜋, to 𝜋̄ such that: Δ

𝜋̄ = {−∞, g−1 } ∪ 𝜋 ∪ {gL+1 , ∞}

(5.97)

and introduce a special class of functions belonging to B(𝜋, ̄ n), as follows. Definition: A truncated power function, (g − gi )n+ is defined as: Δ (g − gi )n+ =

{( )n g − gi , g ≥ gi 0, otherwise

(5.98)

It can be easily verified that a truncated power function for a given gi is continuous at g = gi , and Δ

̂ n), where 𝜋̂ = {−∞, gi , ∞} ⊂ is (n − 1) times continuously differentiable; thus (g − gi )n+ ∈ B(𝜋, [−∞, ∞]. 5.8.1.3.1 Schoenberg–Whitney Theorem Every spline function, f (g) ∈ B(𝜋, ̄ n), can be presented as:

f (g) = p(g) +

L+1 ∑

𝛼i (g − gi )n+

(5.99)

i=0

where the 𝛼i ’s are constants, and p(g) is a polynomial of degree n. Proof. Let pi (g) be a polynomial of degree n for g ∈ {gi , gi+1 }, with pdi (g) as the dth derivative for d ∈ {0, 1, … , n − 1}, that is, pi (g) = p0i (g), and by definition, pdi−1 (gi ) = pdi (gi ) for all d ∈ {0, 1, … , n − 1} and i ∈ {0, 1, … , L − 1} with p−1 (g) = p(g), an nth degree polynomial. Then, the difference function, pi (g) − pi−1 (g), is clearly of degree n, with n roots at gi , i ∈ {0, 1, … , L − 1}. Thus, the difference function may be represented as: pi (g) − pi−1 (g) = 𝛼i (g − gi )n , or equivalently pi (g) = pi−1 (g) + 𝛼i (g − gi )n . Applying this recurrence relation for each i ∈ {0, 1, … , L − 1}, and noting that p−1 (g) = p(g), we have: pi (g) = p(g) + ∑L−1 n g ∈ [gi , gi+1 ]. Now, utilizing the definition for the power function and recj=0 𝛼i (g − gj ) , ognizing that pi (g)’s are f (g) completes the proof.

5.8.2 Variants of Spline Functions As remarked earlier, the specification of the end conditions determines varieties of spline functions. Of course, the specifications depend on the problem at hand, that is, on the boundary value problem in differential forms. We describe later just a few among others that are relevant to interactive modeling and finite element analysis. For this, we will restrict ourselves to odd degree

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Figure 5.31 Cardinal spline.

splines, that is, n = 2m − 1 implying dim B(𝜋, 2m − 1) = L + 2m − 1 = (L + 1) + (2m − 2). This means that, apart from (L + 1) interpolatory constraints at (L + 1) knots, there need to be (2m − 2) additional constraints. Natural Splines: This class of splines is the exact generalization of the physical splines introduced earlier. Thus, we define this class as: a spline which is of (2j − 1) degree over partition, 𝜋, and at most (j − 1) degree at two end intervals of 𝜋, ̄ is called a natural spline. For example, for j = 2, we have a cubic natural spline over partition, 𝜋, and linear at two infinite end intervals outside 𝜋. We have already noted that the curvature at two ends is identically zero, thus the end slopes are linear and the spline stretches out linearly at each end. Hence, for cubic natural splines, an equivalent definition will be to impose free end conditions at each end, that is, the curvature requiring to be zero at each end. For the cubic example, the natural spline clearly describes the equilibrating shape of the physical splines. Thus, from the minimum potential energy principle, the cubic natural spline reflects the minimum potential energy configuration, that is, the natural shape of a physical spline; this justifies the name. Periodic Splines: A spline of degree n, which has equal slope and curvature at both end knots of 𝜋, is called a periodic spline. The usefulness of periodic splines for our purposes is in building closed curves where the end knots are equated to each other; this will be detailed later in our discussion on computational curves. Cardinal Splines: This class of splines is designed to mimic the Lagrange polynomial basis functions. Thus, given a partition, 𝜋 = {g0 , g1 , … , gL }, a cardinal spline, cnk (𝜋), is defined as a Kronecker delta function, 𝛿ik , over knots, that is, { cnk (𝜋) =

1, 0,

if k = i, k, i ∈ 𝜋 , otherwise

with suitable end conditions. One such cardinal spline is shown in Figure 5.31. Although these oscillate less than the corresponding Lagrange polynomials of degree n, the cardinal splines are non-zero over every subinterval of the partition, and thus are unsuitable for any numerical application because the resulting matrices – in both interpolation problems of computer modeling and approximation problems of finite element analysis – will be full and costly. Thus, for numerical effectiveness, we need basis functions with minimum (compact) support. It turns out, fortunately, that there exist just such polynomial splines – known as B-splines (for basis splines) – that form suitable basis functions for the polynomial spline space.

5.8.2.1

B-splines (Basis Functions)

Before we move on to discuss the most important class of spline functions for interpolations in computer-aided mesh generation and approximations in finite element analysis, we will introduce two more definitions.

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Extended Knot Sequence or Partition, 𝜋 e : Given a partition, 𝜋 = {g0 , g1 , … , gL }, and a spline of degree n, the extended partition is defined as: Δ

𝜋 e = 𝜋L ∪ 𝜋 ∪ 𝜋R

Δ

where 𝜋L = {g−n , g−(n−1) , … , g−1 } Δ

𝜋R = {gL+1 , gL+2 , … , gL+n }

(5.100)

that is, extended partition, 𝜋 e , consists of n additional knots on either side of the original partition, 𝜋. Knots of Multiplicity: If a knot gi equals (m − 1) other consecutive knots in value, then the knot gi is said to have a multiplicity of m; thus, a simple knot is said to have a multiplicity of 1. We reiterate that we are interested in the class of splines that help us to define a spline space with basis functions that span the space with minimum local support; B-splines turned out to be these basis functions. The B-splines can be defined in various different ways, based on the method of construction. One method based on forward difference and the truncated power function turned out to be numerically unsatisfactory. The method based on recurrence relation (due to deBoor, Cox and Bohm) is introduced here because it is most useful for generating piecewise Bezier curves and functions. Definition: Given a knot sequence, {gi }, the ith B-Spline, Nin , of degree n, is defined by the following four properties:

r convex or barycentric combination (i.e. partition of unity): ∑i N n (g) = 1, i r positive: N n (g) ≥ 0. ∀i, i r local support: N n (g) ≡ 0. if g ∉ [gi , gi+n+1 ], r continuity class:i N n (g) ∈ Cn−1 , that is, N n (g) is (n − 1) times continuously differentiable. i

i

5.8.2.1.1 Recursion Relation The B-splines are related by the following recursion relations: d ) N d−1 (g), Nid (g) = 𝛽id Nid−1 (g) + (1 − 𝛽i+1 i+1 { g − gi 1, d where 𝛽i ≡ with Ni0 (g) = 0, gi+d − gi

if g ∈ [gi , gi+1 ) otherwise

(5.101)

We can visualize the above recursion relation for cubic B-splines as shown in Figure 5.32. Example BS1: In Figure 5.33, we represent graphically the B-splines of degrees 0, 1, 2, 3. As is evident from the definition and the examples given, a B-spline, Nin (g), is defined with (gi+n+1 − gi ) as its support, consisting of (n + 1) consecutive subintervals starting at knot i.

5.8.2.2

B-splines and Knot Multiplicity

If m knots, gi = … = gi+m−1 , coincide, that is, m denotes the knot multiplicity, the B-spline curve may become only Cn−m continuous at gi , as shown in Figure 5.34. Thus, in order that a B-spline curve should have non-vanishing support, we must have m ≤ n + 1 so that the recursion formula applies.

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Figure 5.32 Cubic B-spline recursions.

Figure 5.33 Example: B-splines.

Figure 5.34 B-spline: effect of multiple knots.

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Example BS2: Let a knot vector with multiplicity be given as: {g0 = g1 = g2 = g3 = 0, g4 = g5 = g6 = g7 = 1}, that is, {0, 0, 0, 0, 1, 1, 1, 1}; we compute below recursively, using equation (5.101), the B-splines of degrees: 0, 1, 2, 3, for the given knot sequence: For degree 0: N00 (g) = N10 (g) = N20 (g) = 0,

∀g

N30 (g) = 1, = 0, 0 N4 (g) = N50 (g) = N60 (g) = 0,

∀g ∈ [g3 , g4 ) otherwise ∀g

For degree 1: g−g 𝛽i1 ≡ g −gi ⇒ 𝛽01 = 𝛽11 = 𝛽21 = 𝛽41 = 𝛽51 = 𝛽61 = 0,

𝛽31 = g

N01 (g) = 𝛽01 N00 (g) + (1 − 𝛽11 ) N10 (g) = 0,

∀g

i+1

N11 (g) N21 (g)

=

𝛽11 𝛽21

i

N10 (g) + (1 − 𝛽21 ) N20 (g) + (1 − 𝛽31 )

N20 (g) N30 (g)

= 0,

∀g

= = (1 − g), = 0, N31 (g) = 𝛽31 N30 (g) + (1 − 𝛽41 ) N40 (g) = g, = 0, 1 N4 (g) = 𝛽41 N40 (g) + (1 − 𝛽51 ) N50 (g) = 0,

∀g ∈ [g3 , g4 ) otherwise ∀g ∈ [g3 , g4 ) otherwise ∀g

N51 (g) = 𝛽51 N50 (g) + (1 − 𝛽61 ) N60 (g) = 0,

∀g

For degree 2: g−g 𝛽i2 ≡ g −gi ⇒ 𝛽02 = 𝛽12 = 𝛽42 = 𝛽52 = 0,

𝛽22 = 𝛽32 = g

N02 (g) = 𝛽02 N01 (g) + (1 − 𝛽12 ) N11 (g) = 0,

∀g

i+2

N12 (g)

𝛽12

i

N11 (g) + (1 − 𝛽22 )

N21 (g)

(1 − g)2 ,

= = = 0, N22 (g) = 𝛽22 N21 (g) + (1 − 𝛽32 ) N31 (g) = 2g(1 − g), = 0, N32 (g) = 𝛽32 N31 (g) + (1 − 𝛽42 ) N41 (g) = g2 , = 0, N42 (g) = 𝛽42 N41 (g) + (1 − 𝛽52 ) N51 (g) = 0, For degree 3: g−g 𝛽i3 ≡ g −gi ⇒ 𝛽03 = 𝛽43 = 0, i+3

N03 (g) = 𝛽03 = 0, 3 N1 (g) = 𝛽13 = 0, N23 (g) = 𝛽23 = 0, N33 (g) = 𝛽33 = 0,

i

∀g ∈ [g3 , g4 ) otherwise ∀g ∈ [g3 , g4 ) otherwise ∀g ∈ [g3 , g4 ) otherwise ∀g 𝛽13 = 𝛽23 = 𝛽33 = g

N02 (g) + (1 − 𝛽13 ) N12 (g) = (1 − g)3 ,

∀g ∈ [g3 , g4 ) otherwise 3 2 2 2 N1 (g) + (1 − 𝛽2 ) N2 (g) = 3g(1 − g) , ∀g ∈ [g3 , g4 ) otherwise N22 (g) + (1 − 𝛽33 ) N32 (g) = 3g2 (1 − g), ∀g ∈ [g2 , g3 ) otherwise N32 (g) + (1 − 𝛽43 ) N42 (g) = g3 , ∀g ∈ [g3 , g4 ) otherwise

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Remark

r From

the results of degree 3 above, we can easily recognize that the B-splines, N03 (g), N13 (g), N23 (g), N33 (g), with multiplicity of 4 at both ends, are the same as the cubic Bezier basis functions, B3i , i ∈ {0, 1, 2, 3}. This has a profound implication in the sense that the spline curves with appropriate multiplicity can be represented by piecewise Bezier curves; the influence of knot multiplicity is at the heart of the representation of splines by composite Bezier curves for interpolation as will be seen later.

Now, two important properties of the B-splines lend themselves to the usefulness of these curves, that is, allow B-splines to pass as basis functions of the spline space: minimal support and linear independence property (without proof), as we will introduce next. 5.8.2.3

Minimal Support Property

Proposition: Any piecewise polynomial, with the same smoothness properties over the same knot sequence and less support than Nin (g), is identically zero, that is, no such non-trivial piecewise polynomial exists, making the B-spline with minimal support. Proof. For any piecewise polynomial defined over [gi , gi+n+1 ], the support for Nin (g) are elements of the function space of dimension (2n + 1) – because (n + 1) polynomials of degree n freedoms reduced by (n − 1) differentiability and one value-matching condition at n internal nodes gives (n + 1) (n + 1) − (n) (n). Now, with one shorter interval, that is, with n polynomials of degree n reduced by n − 1 differentiability and one value-matching condition at (n − 1) internal nodes, we have dimension of 2n (i.e. n(n + 1) − n(n − 1)). So, for any polynomial with (n − 1) differentiability at two ends puts 2n restraints, leaving no room for any non-trivial value inside; thus, the polynomial must be identically zero completing our proof. 5.8.2.4

Linear Independence Property

Proposition:

Given a partition, 𝜋 = {g0 , g1 , … , gL }, and the extended partition defined as: Δ

Δ 𝜋e =

Δ

𝜋L ∪ 𝜋 ∪ 𝜋R where 𝜋L = {g−n , g−(n−1) , … , g−1 }, and 𝜋R = {gL+1 , gL+2 , … , gL+n }, the Bsplines of degree n, Nin (g), g ∈ 𝜋, i ∈ {0, … , L + n − 1}, are linearly independent. Exercise: Prove the above proposition.

We are now ready to express any spline function of degree n, of linear space, B(𝜋, n), as a linear combination of its basis functions – the B-splines, as given by the following theorem due to Curry and Schoenberg. Theorem: Any spline function, pSn (𝜋) ∈ B(𝜋, n), with 𝜋 = {g0 , g1 , … , gL }, can be expressed Δ

as a linear combination of the B-splines defined over the extended knot set, 𝜋 e = 𝜋 ∪ 𝜋R ,with Δ

𝜋R = {gL+1 , gL+2 , … , gL+n }, as:

pSn (g)

=

l+n ∑ i=l

Nin (g) si , g ∈ [gl , gl+1 ] ⊂ 𝜋, l ∈ [0, … , L − 1]

(5.102)

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Figure 5.35 Cubic Bezier by repeated knots in B-splines.

Remarks

r To compute a point on a spline at parameter point, g ∈ [gl , gl+1 ], with given knot vector and r r

control points, we will have to first compute the relevant B-spline basis functions, over its support, Nin (g), i ∈ [l, … , l + n], as in Example BS2 (above), and then perform the linear combinations as in equation (5.102). Because of local support, a particular segment of a B-spline curve may be changed without disturbing the entire curve, as in Bezier curves. For 𝜋 = {g0 , g1 , … , gL }, we have dim B(𝜋, n) = L + n, with: Number of B-spline control points = L + n, Support for each Basis function = [gi , gi+n ] Support for N0n (g), the first Basis function = [g0 , gn ] n (g), the last Basis function = [g Support for NL+n L+n , gL+2n ]

Number of total knots = L + 2n + 1 Number of distinct curve segments = L Number of distinct knots = L + 1 Number of repeated knots = total knots – distinct knots = (L + 2n + 1) − (L + 1) = 2n Remark: Normally, n repeated knots are used at each end, so that the end control points interpolate the curve ends, that is, they reside on the curve.

Example: consider a cubic (i.e. n = 3) B-spline curve with L = 4 segments with L + n (= 7) control points given by: {s0 , s1 , s2 , s3 , s4 , s5 , s6 }. The number of distinct knots is L + 1 (= 5) and the total number of knots is L + 2n + 1 (= 11); the number of repeated knots is n (= 3) at each end, as shown in Figure 5.35, so that the control points s0 and s6 interpolate the curve ends.

r For

our purposes, however, we will always represent a spline function by an appropriate composite (piecewise) Bezier function as will be seen later. As suggested by Example BS2, and as a special case for a single span cubic (n = 3) spline curve, with 4(= n + 1) multiple knots at each ends, we will obtain a single span cubic Bezier curve.

5.8.2.5

Splines, Differential Equations and Error Analysis

All the discussions presented so far about spline curves and functions, have their counterparts as the solution of ordinary differential equations when restricted to odd-degree polynomials.

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Figure 5.36 Best approximations by spline space.

The correspondence is easily established through interpolatory and smoothness conditions of splines as boundary conditions on the partition over which the differential equation is defined. For example, a cubic spline, c3 (g), can be viewed as an interpolate to a given function, f (g), related by the following 4th -order differential equation and the boundary conditions: D4 (c3 (g)) = 0, on Ω =

L−1 ⋃ i=0

[gi , gi+1 ) ⊂ [a, b],

c3 ,g (g0 ) = f ,g (g0 ), c3 (gi ) = f (gi ), c3 ,g (gL ) = f ,g (gL )

i ∈ {0, 1, … , L}

where partition: 𝜋 = {g0 , g1 , … , gL }. The proof due independently to Ahlberg and Greville that c3 (g) is, in fact, a cubic spline is omitted here. In fact, all the different splines introduced before – natural splines, periodic splines, and so on – have their corresponding differential equation definitions. It is interesting to note that the spline interpolates to functions are the best approximation in the sense that they satisfy the Pythagorean relationship under a particular definition of an inner product space with pseudo-norm or semi-norm, as follows.Let H = K2m [a, b], be the space of real-valued functions, f , defined over [a, b] such that f is absolutely (m − 1) times continuously differentiable with Lebesgue square integrability; H has an inner product b defined on it: (f , g) = ∫a f (m) g(m) dx where f (m) , g(m) are the mth derivatives of f , g ∈ H. The √ norm, |f |, is defined as: |f | = (f , f ) such that while |f | satisfies all other properties of a norm, |f | = 0 does not imply f ≡ 0, that is, |∙| is a pseudo- or semi-norm. Then, a spline interpolate, c2m−1 (x) ∈ B(𝜋, 2m − 1) to f ∈ H, satisfies the Pythagorean relation: |f |2 = |f − c|2 + |c|2 as shown in Figure 5.36.

5.8.2.6

Discussion

r This investigation establishes the theoretical basis for parametric curves. However, for practical curve constructions that are numerically robust and cost-effective on a computer, we need to discuss various numerical aspects and computational representation of the above parametric curves as applied to CAGD and the finite element method (FEM).

5.9

Recursive Algorithm: de Boor–Cox Spline

e As introduced partition or knot ] { } { }earlier ine equation (5.100), let 𝜋 Δ𝜋L U𝜋 U𝜋R [be an extended sequence, si , i ∈ 𝜋 be a finite set of points in R3 , and g ∈ gl , gl+1 ⊂ 𝜋 = g0 , g1 , … , gL , l ∈ [0, L − 1].

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5.9.1

Proposition

Let us define the following recursive relation through repeated linear interpolation: ) ( sdi (g) = 1 − 𝛼id sd−1 (g) + 𝛼id sd−1 (g) i i−1 Δ

with 𝛼id =

g − gi , gi+n+1−d − gi

d ∈ {1, 2, … , n − m}, i ∈ {l − n + d, … , l}

(5.103)

and s0i (g) = si where m, the multiplicity of g, is given by: { m=

Then, snl (g),

0, mi ,

if g is not a knot if g = gi , mi is the multiplicity of knot gi

(5.104)

∀l ∈ [0, L − 1], defines a parametric spline curve, CSn (g), of degree n.

Definitions

r The polygon, D, formed by joining si to si+1 , i ∈ {−n, … , L − 1} is called the de Boor polygon, or spline control polygon, of the curve, CSn (g), of degree n and is defined over 𝜋. i ∈ {−n, … , L − 1}, are known as the de Boor points, or spline control points.

r {si },

Remarks

r CS (g) is, in fact, a spline curve, that is, it is a piecewise polynomial of degree n, belonging to n

r

Cn−1 [𝜋], which can be proved. The de Boor definition of the spline curve by recursion can easily be seen as repeated linear interpolations applied to successive segments resulting from the original de Boor polygon; linear interpolation is invariant under affine transformation and so is repeated linear interpolation. Accordingly, a spline curve is invariant under affine transformation of its domain, that is, the real line intervals.

Example: Let us look at the interval, g ∈ [g2 , g3 ], of a quadratic spline curve, CSn (g) (i.e. of degree 2), with no multiplicity of knots, that is, n = 2, l = 2, m = 0, as shown in Figure 5.37. Then, s00 = s0 , s01 = s1 and s02 = s2 with d ∈ {1, 2} and i ∈ {d, … , 2} resulting in: ) ( s11 (g) = 1 − 𝛼11 s0 + 𝛼11 s1 , ) ( s12 (g) = 1 − 𝛼21 s1 + 𝛼21 s2 ,

g − g1 g2 − g1 g − g2 𝛼21 = g3 − g2 𝛼11 =

) ( C2 (g) = s22 (g) = 1 − 𝛼22 s11 (g) + 𝛼22 s21 (g),

(5.105) 𝛼22 =

g − g2 g3 − g2

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(a)

(b)

(c)

Figure 5.37 (a) Example: (B-spline as composite Beziers), (b) Scheme: (B-spline as composite Beziers), (c) Ratio: (B-spline as composite Beziers).

Let us look closer at C2 (g), as given by equation (5.105), by tracing back the recursive relations in reverse order, from which we get: C2 (g) =

)( )] ) ( )] [( [ ( 1 − 𝛼22 1 − 𝛼11 s0 + 𝛼11 1 − 𝛼22 + 𝛼22 1 − 𝛼21 s1 + 𝛼22 𝛼21 s2

(5.106)

Now, substituting for 𝛼22 , 𝛼11 in equation (5.106) from their values in equation (5.105), we get: )( ) (g − g)2 ( 1 − 𝛼22 1 − 𝛼11 = 3 (g3 − g2 )

(5.107)

Looking at the recursion relations of B-splines, as given by equation (5.101), we get: N00 (g) = N10 (g) = 0 because g ∉ [g0 , g2 ), and N20 (g) = 1 because g ∈ [g2 , g3 ), and g − g0 0 N (g) + g1 − g0 0 g − g1 0 N11 (g) = N (g) + g2 − g1 1

N01 (g) =

g2 − g 0 N (g) = 0, g2 − g1 1 g3 − g 0 g −g N (g) = 3 g3 − g2 2 g3 − g2

And, finally, N02 (g) =

g − g0 1 g −g 1 (g3 − g)2 N0 (g) + 3 N1 (g) = g2 − g0 g3 − g1 (g3 − g1 )(g3 − g2 )

(5.108)

which, when compared to equation (5.107), shows that N02 (g) matches the coefficient of s0 in equation (5.106); similarly, we can show that N12 (g) and N22 (g) match the coefficients of s1 and s2 in equation (5.106), respectively. Thus, we can express C2 (g) of equation (5.106) as:

C2 (g) =

2 ∑

Ni2 (g) si

(5.109)

i=0

that is, as the linear combinations of the B-spline or basis curves with the de Boor control points, {si }, as the coefficients of combination.

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Remarks: Several specific observations are in order in that they are indeed true for any spline in general:

r We have just shown that C2 (g) can be viewed as an approximation by the B-spline functions of degree 2 with de Boor control points as coefficients.

r Various de Boor polygons represent the same spline. For example, the control nets, {s0 , s1 , s2 } r r r

and {s0 , s11 , s12 , s2 }, both represent the same spline, C2 (g); this leads to the important phenomenon of degree elevation. The partitioned control nets, {s0 , s11 , s22 } and {s22 , s12 , s2 }, can be seen as the de Boor polygons of two sides of the spline curve, C2 (g): left and right of s22 , respectively; this generalizes into a subdivision algorithm similar to the Bernstein–Bezier curves. A general spline possesses most of the properties of Bezier curves discussed before – linear precision, convex hull property, and so on. For our purposes of representing spline curves by composite Bezier curves, as will be discussed later, the end knots of a spline curve of degree n, will always be assumed to have multiplicity of n + 1.

5.10 Rational Bezier Curves: Conics and Splines 5.10.1 Projective Map and Cross Ratio Invariance As indicated earlier, for non-rational curves, algorithms for curve generation based on repeated linear interpolation utilize the concepts of ratio and affine map for their invariance. However, for conics and other rational curves, we will need a more general map, known as a projective map, such as perspective in geometry and a generalization of the concept of ratio known as the cross ratio. To start with, for a projection, we have a center of projection, O, as shown in Figure 5.38, which projects, for example, a line segment, ABCD, to another line segment, abcd, onto a plane known as the image plane. We can clearly see that under the projective map of one line segment to another, ratios are not preserved because individual lengths are distorted. However, the cross ratio, denoted as cr (∙, ∙ , ∙ ,∙), of four collinear points is defined as: cr (a,b,c,d) =

ratio (a,b,d) ratio (a,c,d)

(5.110)

This can be shown to be preserved under projective map, that is, cr (a, b, c, d) = cr (A, B, C, D). Now, referring to Figure 5.38, we are going to show that under the projective map, a pre-image

Figure 5.38 Projection map and cross ratio.

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point, Θ, on line segment, ABCD, and the corresponding image point, 𝜃, on the line segment, abcd, are connected by a rational linear transformation. For this, we first note that unlike the affine map of line segments that is determined by two points – the end points – a projective map is determined by three pre-image and three image points. So, by invariance of cross ratios, we have: cr(a, b, 𝜃, c) = cr(A, B, Θ, C). For simplicity and without loss of generality, we may choose, a = A = 0 and c = C = 1; then, by applying the definition as in equation (5.110), we get:

Δ Δ ratio(0,b,1) . Now, letting: ratio(0, B, 1) = Π and ratio(0, b, 1) = ratio(0,𝜃,1) } { 𝜋Θ Π𝜃 . Finally, solving for Θ, we get: = (𝜃−1) expression: (Θ−1)

ratio(0,B,1) ratio(0,Θ,1)

previous

=

𝜃=

ΠΘ ΠΘ + 𝜋(1 − Θ)

𝜋, we get from

(5.111)

The transformation of a pre-image point, Θ, to the image point, 𝜃, as given by equation (5.111), is clearly a rational linear map; this is where the name rational Bezier curves came from, and they are based on the projective map which we will investigate next.

5.10.1.1

Rational Bezier Curves

It may be recalled from our derivation of the Bernstein basis representation of non-rational Bezier curves that the dimensionality of the underlying vector space containing a curve, Cn (𝜉), was not crucial. In fact, generalization of the Bernstein–Bezier representation to rational Bezier curves follows from this observation; accordingly, we have the following definition. Definition:

r A rational Bezier curve of degree n in three-dimensional real space, R3 , is the central projection, as shown in Figure 5.39, of a non-rational Bezier curve of the same degree in a four-dimensional real space, R4 . Thus, we are now in a position to characterize a rational Bezier curve in 3D in terms of its 4D reimage as a non-rational Bezier curve, which we have already discussed. Incidentally, note that conveniently for uniformity in computer coding, both the rational and the non-rational are represented in 4D as in a homogenous coordinate system.

Figure 5.39 Rational Bezier by projection from 4D to 3D.

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5.10.1.2

Representation Theorem

Let Cn (𝜉) ∈ R3 be a point on a rational Bezier curve of degree n; then, there exists {wi ∈ R} and {qi ∈ R3 } for ∀i ∈ {0, 1, … , n}, known as the weights and the control points of the curve, respectively, such that: n ∑

Cn (𝜉) =

wi qi Bni (𝜉)

i=0 n

∑

⎞ ⎛ ⎟ ⎜ n ⎜ wi Bi (𝜉) ⎟ ⎟ ⎜ n ⎟ ⎜∑ n wi Bi (𝜉) ⎟ ⎜ ⎠ ⎝ i=0 ⏟⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏟

∑ n

=

i=0

wi Bni (𝜉)

i=0

qi =

n ∑

Bwn (𝜉) i ⏟⏟⏟

i=0

qi ,

∀𝜉

Bernstein basis functions for rational Bezier

Bernstein basis functions for rational Bezier

(5.112) { } Cn (𝜉) Proof. Cn (𝜉) ∈ R3 can be seen as a point ∈ R4 . Now, because of above definition 1 { } Cn (𝜉) of a rational Bezier curve, can also be seen as a point on the curve, C4D n (𝜉) ≡ 1 { } Cn (𝜉) w(𝜉) with w(𝜉) so chosen that it is a non-rational Bezier curve of degree n. Then, w(𝜉) ∑n w(𝜉) must have a representation as: w(𝜉) = w Bn (𝜉) where {wi }, i ∈ {0, 1, … , n} are i=0 i i the Bezier coefficients. So, n ⎫ ⎧ ∑ ⎪ Cn (𝜉) wi Bni (𝜉) ⎪ ⎪ ⎪ i=0 C4D n ⎬ n (𝜉) ≡ ⎨ ∑ n ⎪ ⎪ wi Bi (𝜉) ⎪ ⎪ i=0 ⎭ ⎩

(5.113)

But, C4D n (𝜉) itself is a non-rational Bezier curve of degree n, and hence, has a representation: n ∑

{

q̂ i wi

} Bni (𝜉) with q̂ i ∈ R3

(5.114)

Thus, comparing equations (5.113) and (5.114), we have: ) ( n ) ( n ∑ ∑ wi Bni (𝜉) q̂ i Bni (𝜉) = Cn (𝜉)

(5.115)

C4D n (𝜉)

≡

i=0

i=0

i=0

which completes the proof. Remarks

r It can be easily shown that if all the weights, {wi }, degenerates into the non-rational Bezier curve.

i ∈ {0, 1, … , n}, are same, then C4D n (𝜉)

r To avoid singularity, we will require that the weights be all non-negative.

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r All the properties of a non-rational Bezier curve introduced before apply to its rational counterparts.

r The rational Bezier curve, as described, can be thought of as a representation by Bernstein (

basis functions defined by: form a basis set.

) wi Bni (𝜉) ∑n , ∀i n i=0 wi Bi (𝜉)

∈ {0, 1, … , n}; it can be proved that they actually

r The basis functions sum to unity, that is, they form partitions of unity, indicating a barycentric combination, and thus, invariant under affine transformations.

r Because of the restrictions that all weights be non-negative reals, the convex hull property is preserved.

r Rational Bezier curves retain end point interpolations. r Invariance under affine parameter transformation is, likewise, is maintained. 5.10.1.3

Reparameterization, Weights and Standard form

A rational linear parameter transformation (equivalently, the projective map) does not change the shape and the degree of a curve, only its control points and weights. So, we give the following definition. Definition: A rational Bezier curve is in standard form if its end point weights are unity, that is, w0 = wn = 1 A rational Bezier curve can always be transformed to its standard form by using rational linear parameter transformation without changing the shape and the degree of the curve. Equivalently, if we reparameterize a rational Bezier curve by changing the weights, {wi }, to new weights, {ŵ i }, such that: ( ŵ i =

w0 wn

)i∕ n

wi ,

i ∈ {0, 1, … , n}

(5.116)

then, from equation (5.116), we obtain: ŵ n = ŵ 0 = w0 ; now, dividing all weights by w0 gives us the curve in standard form.

5.10.2 Properties of Rational Bezier Curves As expected, all properties of a rational Bezier curve are very much analogous to their nonrational counterpart when the rational Bezier curve is treated as its 4D non-rational pre-image. In any event, we list below the important properties to show the effect of the weights in their characterization, and thus how they can be implemented with slight variations to their non-rational counterparts. 5.10.2.1

Subdivision Property

The subdivision algorithm for a rational Bezier curve is analogous to its non-rational counterpart, that is, the de Casteljau algorithm is used in 4D pre-image space of a 3D rational Bezier curve, Cn (𝜉). In other words, the algorithm is applied on both {wi qi } and {wi } curves for all i ∈ {0, 1, … , n}. The intermediate control points are projected onto the hyperplane, w = 1, to provide us with the left and right subdivided Bezier polygons.

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5.10.2.2

Derivative Property

Given: Cn (𝜉) =

N(𝜉) D(𝜉)

∑n ∑n with N(𝜉) = wi qi Bni (𝜉) and D(𝜉) = w Bn (𝜉) i=0 i=0 i i ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ Numerator

Denominator

The first derivative, Cn ,1 (𝜉), is given by: Cn ,1 (𝜉) =

N,1 (𝜉) D(𝜉) − N(𝜉) D,1 (𝜉) D2 (𝜉)

(5.117)

Now, noting that: at 𝜉 = 0, N(0) = w0 q0 , D(0) = w0 , N,1 (0) = w0 q0 − w1 q1 , and D,1 (0) = w1 − w0 , we get the first derivative at 𝜉 = 0 end: Cn ,1 (0) =

w1 w (q1 − q0 ) = 1 Δq0 w0 w0

(5.118)

with Δ as the forward difference operator. The derivatives at all other parameter points can be obtained by first subdividing the curve at the parameter point and then evaluating the first derivative, using equation (5.118), at the left end point of the right subdivision, or the right end point of the left subdivision. For higher derivatives such as curvature or torsion, it seems cost-effective to apply the subdivision algorithm and evaluate these by end point formulas which can be proved to be: area[q0 , q1 , q2 ] n(n − 1) w0 = n w1 w2 ‖q1 − q0 ‖3 ‖ ‖ volume[q0 , q1 , q2 , q3 ] 3(n − 2) w0 w3 torsion: 𝜏 = = 2n w1 w2 (area[q0 , q1 , q2 ])2

curvature: 𝜅 =

(5.119)

5.10.2.2.1 C++ Code: Tangent at Any Point on a Rational Bezier We present below the C++ code excerpts for computing the tangent at any point t ∈ [0, 1] of a rational Bezier curve: /////////////////////////////////////////////////////////// typedef int *pINT; typedef double *pDOUBLE; typedef struct { double x; double y; double z; } WORLD, *pWORLD; ///////////////////////////////////////////////////////////////////////////// int CCurve::Rat_Tangent_t(pWORLD Cons, pDOUBLE Wts, double t, int nDegree, WORLD& tangent) { //////////////////////////////////////////////////////////////////// // Returns tangent vector of rational Bezier curve at parameter t // Always: // Larger polygon selected for subdivision

Curves: Theory and Computation

203

// and its beginning tangent computed //////////////////////////////////////////////////////////////////// pWORLD NewC = new WORLD[nDegree+1]; pDOUBLE NewW = new double[nDegree+1]; /////////////////////////////////// double sign = 1.0; if(t3) // BS[L+2][3]

// = w_DT[L]; //

/////////////

} else

// CLOSED

{ ////////////////////////////////////////// First: 1 of them ————–| BS[0][0]

= Sol[L-1][0];

// d[-1] = d[L-1] // |

BS[0][1]

= Sol[L-1][1];

// |

BS[0][2]

= Sol[L-1][2];

// |

if(nDim>3) // BS[0][3]

//

= Sol[L-1][3]; //

////////////////////////////////////////// In-between: L of them ———| for(k=0;k3) // BS[k+1][3] }

// = Sol[k][3];

//

// /

////////////////////////////////////////// Last: 1 of them ———-/ BS[L+1][0]

= Sol[0][0];

// d[L] = d[0]

Computation of Nonlinear Structures

224

BS[L+1][1]

= Sol[0][1];

BS[L+1][2]

= Sol[0][2];

//////////////////////// if(nDim>3) // BS[L+1][3]

// = Sol[0][3]; //

} ////////// return 0; } Convert_BS2BZ(PDMA3 BS,pDOUBLE Knot,PDMA3 BZ,int L,BOOL bClosed,int nDim) { // NON-RATIONAL COUNTERPART //////////////////////////////////////////////////////////////// //

Converts cubic B-spline to Bezier –> still C2 class

//

1. Spline can be rational(nDim = 4) or non-rational(nDim = 3)

//

2. Spline can be open or closed.

//

KNOTS ARE SIMPLE

//////////////////////////////////////////////////////////////// // OPEN: L = nData – 1 = intervals //

Spline Dimension : L+3: (0, ... ,L+2)

//

Bezier Dimension : 3L+1: (0, ... ,3L)

// // CLOSE: L = nData = intervals: Periodic( Period = L) //

Spline Dimension : L: (0, ... ,L-1)

//

Bezier Dimension : 3L: (0, ... ,3L – 1)

//////////////////////////////////////////////////////////////// // //

h.i = Knot[i+1] – Knot[i] = Forward Difference

// //

BZ[0] = BS[0]; BZ[1] = BS[1];

//

BZ[2] = (h.1*BS[0]+h.0*BS[1])/h.0+h.1

// //

BZ[3i-1] = (h.i*BS[i]+(h.i-2+h.i-1)*BS[i+1])/(h.i-2+h.i-1+h.i)

//

BZ[3i+1] = (h.i*BS[i]+(h.i-1+h.i )*BS[i+1])/(h.i-2+h.i-1+h.i)

//

BZ[3i ] = h.i*BZ[3i-1]/(h.i-1+h.i)+h.i-1*BZ[3i+1]/(h.i-1+h.i)

// //

OPEN:

//

BZ[3L-2] = (h.L-1*BS[L]+h.L-2*BS[L+1])/(h.L-2+h.L-1)

//

BZ[3L-1] = BS[L+1]

//

BZ[3L ] = BS[L+2]

// //

CLOSED: L MUST BE >= 3

// BZ[3L-2] = (h.L-1*BS[L%Period]+h.L-2*BS[(L+1)%Period])/(h.L-2+h.L-1) //

BZ[3L-1] = BS[(L+1)%Period]

//

BZ[3L ] = modulo operation done in generation time

// /////////////////////////////////////////////////////////////////////////// //////////////// int k,j,Lt3; /////////////////////////////////////////////////////////////////// First points if(!bClosed)

Curves: Theory and Computation

225

// OPEN { for(j=0;j1) { hip1 = Knot[2] – Knot[1]; hi = Knot[1] – Knot[0]; sum0 = Knot[2] – Knot[0]; /////////////////////////// for(j=0;j2) { sum1 = Knot[3] – Knot[1]; sum = Knot[3] – Knot[0]; /////////////////////////// for(j=0;j2)

{ //////////////// double hi,him1,him2,hip1,sum0,sum1,sum2,sum3,sum4; //////////// him2 = Knot[L-1]

- Knot[L-2];

// h[-2] = h[L-2]

him1 = Knot[L]

- Knot[L-1];

// h[-1] = h[L-1]

hip1 = Knot[2]

- Knot[1];

hi = Knot[1]

- Knot[0];

// i = 0

sum0 = him2 + him1; sum1 = sum0 + hi; sum2 = hi + hip1; sum3 = him1 + sum2; sum4 = him1 + hi; /////////////////////////// for(j=0;j 0 (= m − 1). Thus, this boundary condition acts as a natural boundary condition.

8.4.8.7

The Energy Space and Norm: E(𝛀) and ||u||energy

Since the highest derivative in the integral expression for virtual work principle is 1, that is, m = 1, we may define the energy space, a normed linear space for the problem from where all the displacement functions, real or virtual, may be sought, as: { } E(Ω) = u ∈  1 (Ω) | ||u||energy < ∞

(8.166)

with energy norm, ||u||energy , defined by the strain energy functional,  (u), as:

||u||energy

def

=

√

1 1  (u) = (u, u) = 2 2 ∫0

1(

du(x) dx

)2 dx

(8.167)

C-type Finite Element Method

421

For this problem, with symmetric , the energy space is easily seen as an inner product space with the norm induced by the following, energy inner product (⋅ , ⋅) : def

1 (u , v) = ∫0

8.4.8.8

(

du dx

)(

dv dx

) dx,∀u, v ∈ E(Ω)

(8.168)

The Solution Set and Trial and Virtual Subspace: ESol (𝛀) and ESol (𝛀) Ess

0

Thus, the solution set, may be defined as: { } Sol (Ω) = u ∈ E(Ω) | u(1) = u EEss 1

(8.169)

Sol (Ω) is not a subspace – in fact, not even a vector space. It is a linear manifold. We Clearly, EEss apply the translation theorem of linear manifolds to represent u as the direct sum of u0 and any Sol (Ω), that is, arbitrary u⋆ ∈ EEss

u(x) = u0 (x) ⊕ u⋆ (x) where

{ } Sol (Ω) | u0 (1) = 0 u0 ∈ E0Sol (Ω) with E0Sol (Ω) = u0 ∈ EEss

(8.170)

Remark

r ESol (Ω) contains the displacement functions that satisfy essential boundary condition in its 0

homogeneous state. However, this happens to be the precise definition of the virtual space according to the virtual work principle. Thus, the direct sum representation allows us to make the trial space and the virtual space one and the same. In the Rayleigh–Ritz–Galerkin finite element formulation, this in turn allows us to choose the trial and virtual finite-dimensional subspace to be the same. Of course, for the final solution, we still have to determine the arbitrary u⋆ as a function to satisfy the inhomogeneous boundary condition; this is normally done in a way that is trivially simple and numerically efficient, as we shall see shortly. For convenience, let us now summarize the necessary entities for the given infinite-dimensional problem: the energy inner product, the strain energy, the energy space, the solution set which is the linear variety of the trial space and the virtual space, as: Energy inner product: Strain energy: Energy space: Solution set or linear variety: Trial subspace: Virtual subspace:

(u, v) ≡  (u) ≡ ||u||e ≡ E(Ω) ≡ Sol (Ω) ≡ EEss U0 ≡ V0 ≡

∫0 ( 𝜕u )( 𝜕û ) dx, ∀u, û ∈  1 (Ω) 𝜕x 𝜕x 1 (u, u), 2 √  (u), {u ∈  1 (Ω)| ||u||e < ∞}, {u ∈ E(Ω)| u(1) = ū 1 , ||u||e < ∞}, Sol , {u|u(1) = 0} ⊂ EEss U0 1

(8.171)

Computation of Nonlinear Structures

422

8.4.8.9

Finite Element Mesh: 𝚫 =

∪

e=1,2,…,N

𝛀e , 𝛀R , 

For this problem, we choose N = 1, that is, one element only. Thus, the geometric element, Ωe , coincides with the entire geometry, Ω. Since the bar is straight, and the length is unity, the root element, ΩR , and the physical element, Ωe , coincide. We denote the root element natural coordinates as 𝜉. Clearly, 𝜉 = x, that is, 𝜉 ∈ [0, 1]. Also, the bar being a straight element requires only linear Bernstein polynomials for geometry representation. In other words, the transformation,  , relating the root element to the physical element is given by: x(𝜉) T(𝜉) qx x(𝜉)

= T(𝜉)qx : in matrix form [ 1 ] 1 = B0 (𝜉) B1 (𝜉) { }T = qx0 qx1 = (1 − 𝜉) qx0 + 𝜉 qx1 : in component form

(8.172)

For the problem at hand, clearly, qx0 = 0 and qx1 = 1. Notice, that for a trivial geometry such as this, we have, by inspection, x = 𝜉. However, we choose to express this relation in the above elaborate form for the sake of uniformity that aids our subsequent exposition. The transformation set trivially contains one element, that is,  = {T}.

8.4.8.10

( ) Finite Element Trial and Virtual Subspace: SP 𝛀,𝚫, , ⊂ ESol (𝛀) 0

f S (𝜉),

The forcing function (we can think of it as a distributed surface traction), may be, for our example problem, an identically zero function or at a best linear continuous function. We will seek our trial and virtual displacements from among the set S2 of piecewise quadratic functions (p = 2) which are continuous at all nodes and satisfy the essential boundary condition at its homogeneous state. Now, each quadratic function describes a parabola for each element, requiring three free parameters for its definition. However, the continuity constraint at each physical end node reduces the number of free control vertices of each parabola ending in the node, to two. Thus, for the problem, the dimension of the finite element trial and virtual subspace is 2N, where N is the number of elements. For the above choice of one finite element, the number of free parameters should be 2Ne = 2.

8.4.8.11

Finite Element Basis Functions and Displacement Control Vector: B2i (𝝃) and qui

For a c-type finite element method, the local or elemental basis functions are Bernstein polynomials with Bezier } For quadratic functions, these become the quadratic Bernstein { control polygon. polynomials, B2i (𝜉),i = 0, 1, 2 . Now, the parameters serving as the coefficients of linear com} { bination of these basis functions are the as-yet-undetermined Bezier controls, qui ,i = 0, 1, 2 . In other words, any displacement function, u(𝜉), is given by a linear combination of this finite set of basis functions as:

u(𝜉) =

2 ∑ i=0

B2i (𝜉) qui

(8.173)

C-type Finite Element Method

423

Remark

r The actual number of physical nodes is two, with qu

and qu2 serving as the two end nodes. Knowing that B20 (0) = 1, B21 (0) = B22 (0) = 0, we have u(0) = qu0 . Accordingly, for satisfaction of the homogeneous counterpart of the essential boundary conditions, we only need to set qu0 = 0. As already explained, we allow the displacement trial function to satisfy the essential boundary conditions only in their homogeneous state, so that both the displacement trial subspace and the virtual subspace (functions of which, by choice, must vanish wherever essential boundary conditions are prescribed) can be one and the same with same basis functions spanning the subspace. This, in turn, allows generation of a numerically efficient symmetric stiffness matrix, etc. Thus, the as-yet-unknown parameters are qu1 and qu2 . 0

8.4.8.12

u0 (𝝃), and 𝚚 x , 𝚚 u and q̂ u Coerced Isoparametry: u0 (𝝃), ̂

Recall that the bar required only linear Bernstein polynomials, that is, of degree one, for geometry representation, that is, mesh generation. Recall also that the displacement solution function is already indicated to be quadratic, that is, of degree two. Thus, we may, although not necessary for computation, conceptually coerce the geometry transformation up by one application of the degree elevation technique. This then makes the geometry representation – that is, the transformation from root to physical element – to be: x(𝜉) = T(𝜉)qx [ ] T(𝜉) = B20 (𝜉) B21 (𝜉) B22 (𝜉) qx =

{

} 1 T 0 1 2

(8.174)

Most importantly, note that coercing did not introduce any new physical node. The element trial displacement function (see Figure 8.8 for control vertices) can be given isoparametrically as: u0 (𝜉) = T(𝜉)qu { }T qu = qu0 qu1 qu2 ≡ 0

Figure 8.8 Displacement control vertices.

(8.175)

Computation of Nonlinear Structures

424

Figure 8.9 Trivial solution for control vertices.

The virtual displacement function, likewise, can be given by: ̂ u0 (𝜉) = T(𝜉)̂ qu { u u }T ̂ q1 ̂ qu2 ≡ 0 q0 ̂ qu = ̂

8.4.8.13

(8.176)

Inhomogeneous Essential Condition: u0 ⊕ u⋆

We may construct the trivial solution, u⋆ (𝜉), due to the inhomogeneous boundary condition (see Figure 8.9 for trivial control vertices) as: u⋆ (𝜉) = T(𝜉) q⋆ { }T q⋆ = 0 0 q⋆ ≡ u1 2

(8.177)

u0 (𝜉) in equation (8.177), we see, Considering description of u0 (𝜉) in equation (8.175) and ̂ Sol FEM (𝜉), indeed, belongs to EEss (Ω), as follows: from equation (8.170), that u uFEM (1) = u0 (1) + u⋆ (1) ⎧0⎫ ⎧ qu ⎫ ⎪ ⎪ ⎪ 0u ⎪ = {0 0 1} ⎨ q1 ⎬ + {0 0 1} ⎨ 0 ⎬ ⎪ u1 ⎪ ⎪0⎪ ⎩ ⎭ ⎩ ⎭ = u1

(8.178)

where, we have used the property of the Bernstein polynomial of degree two: B20 (1) = (1 − r)2 |r=1 = 0, B21 (1) = 2(1 − r)r|r=1 = 0, B22 (1) = r2 |r=1 = 1,

(8.179)

C-type Finite Element Method

8.4.8.14

425

Strain–displacement Control Matrix: (𝝃)

The strain–displacement control matrix is defined by the finite element strain and displacement as: duFEM (𝜉) = (𝜉) qu d𝜉 dT(𝜉) (𝜉) = d𝜉

∫ FEM (𝜉) =

8.4.8.15

(8.180)

c-type Virtual Work Principle and Equilibrium Equation

Now, using definitions of u0 , ̂ u0 and (u0 , ̂ u0 ) in the equation of the virtual work principle, we get, in compact form: ( u )T [ S ( u )T ̂ K qu = ̂ q F q

−F0

] −Fa ,

∀̂ qu

(8.181)

where, K is the (3 × 3) unconstrained or full singular stiffness control matrix given by: ( 1 ∫0

dT(𝜉) d𝜉

)T (

dT(𝜉) d𝜉

)

K=

1 ∫0 B(𝜉)T

B(𝜉) d𝜉 =

=

⎧ B20 ,𝜉 1⎪ ∫0 ⎨ B21 ,𝜉 ⎪ 2 ⎩ B2 ,𝜉

⎫ ⎡ 2 −1 −1⎤ } ⎪{ 2 2 2 2 2 −1⎥ ⎬ B0 ,𝜉 B1 ,𝜉 B2 ,𝜉 d𝜉 = 3 ⎢⎢−1 ⎥ ⎪ 2⎦ ⎣−1 −1 ⎭

d𝜉 (8.182)

and the load control vectors are: 1

FS = ∫0 T(𝜉)T f S (𝜉) d𝜉 F0 = T(0)T f

0

⎧ −u1 ⎫ [ ] ⎪ ⎪ 2 1 F⋆ = ∫0 (𝜉)T (𝜉) d𝜉 q⋆ = K q⋆ = ⎨ −u1 ⎬ 3⎪ ⎪ ⎩ 2u1 ⎭

(8.183)

Finally, noting that the third element for both the trial and virtual displacement vectors to be identically zero, and that the virtual work principle is true for any kinematically admissible virtual displacement, we can write the constrained finite element equation of equilibrium from equation (8.181) in expanded matrix form as: [ 2 3

2 −1

]{

−1 2 where,

{

qu0 qu1 F1 F2

}

{ = {

} =

F1

}

F2 0

1

∫0 f S (𝜉) (1 − 𝜉)2 d𝜉 + f + 23 u1 1

2 ∫0 f S (𝜉) 𝜉(1 − 𝜉) d𝜉 + 23 u1

}

(8.184)

Computation of Nonlinear Structures

426

The equation (8.184) can be solved for the desired Bezier control vertices only if the body force, f B (𝜉), is given explicitly. Let us, then, consider two cases as under: Case I: f S (𝜉) ≡ 0, that is, homogeneous differential equation: In this case, the load vector, the right-hand side of equation (8.184), becomes: {

F1 F2

{

} =

0

f + 23 u1 2 u 3 1

} (8.185)

Now, solving equation (8.184) with the load vector given by equation (8.185), we get: {

qu0

{

} 1 = 2

qu1

}

0

2f + 2u1 0

(8.186)

f + 2u1

Then, using equation (8.186) for the known Bezier controls for u0 (𝜉) as in equation (8.175) with u⋆ (𝜉) as in equation (8.177) and 𝜉 = x in the expression for the finite element displacement, we finally get the desired displacement solution as: uFEM (x) = (1 − x)2 qu0 + 2x(1 − x) qu1 + x2 ū 1

= 12 (2f̄0 + 2̄u1 )(1 − x)2 + (f̄0 + 2̄u1 )x(1 − x) + x2 ū 1 = (1 − x)f̄0 + ū 1

(8.187)

= exact solution! The equation (8.187) represents the exact homogeneous part of the solution that may be simply obtained by a mere sequence of integrations of the equilibrium equation. Case II: f S (𝜉) ≡ f̄ S = constant, that is, inhomogeneous differential equation: In this case, we show only the additional part of the solution, that is, the particular solution of the differential equation due to this inhomogeneous term. The homogeneous part of the solution remains the same as expressed by equation (8.187). The generalized load control vector due to this is obtained from the integral expressions in equation (8.184) as: {

part }

F1

part

F2

1 = f̄ S 3

{ } 1 1

(8.188)

Now, solving equation (8.184) for this particular load vector given by equation (8.188), we get: {

part qu 0 part qu 1

} 1 = f̄ S 2

{ } 1 1

(8.189)

C-type Finite Element Method

427

Then, we finally get the desired particular part of the displacement solution as: part

2 u uFEM part (x) = (1 − x) q0

part

+ 2x(1 − x) qu1

1 = 12 f̄ S {(1 − x)2 + 2x(1 − x)} = (1 − x2 )f̄ S 2 = exact solution!

(8.190)

The equation (8.190) represents the exact particular solution of the differential equation; this completes the presentation of the current example.

8.4.9

Linear Elasto-dynamic Problems

For linear elasto-dynamic problems, we can continue as in the old finite element method presented in the appendix. The only thing that must be kept in mind, however, is that, unlike the old method, the generalized displacement consists of Bezier displacement control vectors. Let, 𝜌e be the effective mass density, and 𝜍 e the effective damping coefficients of the element, e Ω . Then, the generalized nodal mass, M, and damping, C, matrices can be similarly obtained if the acceleration and velocity distributions are assumed to be of the same isoparametric form as the coerced geometry and the displacement, respectively. They are formally defined as follows: M≡

]

[ ∑ e

∫ Ωe

𝜌e (Te )T Te dΩe

qd

e

(8.191)

and C≡

[ ∑ e

] ∫ Ωe

𝜍 e (Te )T Te dΩe

qd

e

(8.192)

In this case, the equilibrium equations become: M q̈ d + C q̇ d + K qd = F

(8.193)

where, q̇ d is the generalized nodal velocity control vector and q̈ d is the generalized nodal acceleration control vector, and all other terms are as described earlier, except for FB , the generalized nodal body load vector as follows: Remark

r It may be seen that for our unknown generalized nodal vectors, we have qd , the Bezier displacement controls which involve only the corner nodes of the old finite element methodologies. Furthermore, we have no need to include the ad hoc rotational degrees of freedom usually associated with the plate and shell analysis, irrespective of the kinematic relationship that makes these degrees of freedom necessary in the old finite element methodologies. In the derivation of the stiffness control matrix and load control vectors of the entire body, Ω, the summation of elements, of course, implied an appropriate rule of combination for the

Computation of Nonlinear Structures

428

corresponding elemental quantities. In this section, we explain these rules and thus introduce the new methodology for inter-element smoothness or continuity control. This phase of finite element formulation is widely known as the global assembly process. For practical applications to structural or continuum mechanics problems, we generally need to consider only C0 , C1 and C2 continuity across the element boundaries, which we discuss here.

8.4.10

Heat Conduction and Potential Flow Problems

The Rayleigh–Ritz–Galerkin method, and hence finite element methods, apply equally successfully to heat conduction and other potential flow problems. Obviously, our theory can be applied equally, with the exception that the state variables are represented as linear combinations of Bernstein polynomials with generalized unknown nodal coefficients as Bezier control vertices. For example, let us consider the heat conduction analysis where the state variable is the onedimensional, steady-state temperature distribution, 𝜃. We can apply everything discussed so far for linear elasticity to the heat conduction analysis, when we redefine various terms as follows: Ω the finite continuum or body, Γ, = the boundary of Ω, with Γ = Γ𝜃 ∪ ΓH ∪ ΓC , where Γ𝜃 , the boundary where temperature is prescribed, ΓH , the boundary where heat flux is prescribed, ΓC , ̄ the closure of Ω that is, Ω ̄ = Ω ∪ Γ; the boundary where convective conditions are prescribed, Ω Δ

d = θ, the column vector of the nodal generalized temperatures defining the temperature state ̄ under imposed boundary conditions: f = (fx , fy , fz )T , the imposed heat flow inputs along of Ω, coordinate directions at any point, c ∈ Γ; f may consist of any combination of: f B = (fxB , fyB , fzB )T , the body heat transfer per unit volume generated (or absorbed) by the distributed heat source (or sink) at any, c ∈ Ω, the surface heat transfer per unit area at any c ∈ Γ, f C = (fxC , fyC , fzC )T , the H H T concentrated heat flow input at any c ∈ Γ; qH = {qH x , qy , qz } = the heat flux per unit volume column vector; , the heat–temperature differential matrix (operator), , the heat conductivity matrix and , the heat–temperature control matrix. Now, the temperature distribution is expressed as: 𝜃(r, s, t) =

M ∑ N L ∑ ∑ i=0 j=0 k=0

⌢

⌢

⌢

⌢𝜃

BLi (r)BM (s)BN (t) qijke j k

(8.194)

⌢𝜃e

where qijk is the Bezier generalized nodal temperature control vertex at location ijk, and all other terms are as defined in linear elasticity. Then, applying the appropriate virtual work principle described in the previous sections, and the methodology used as in the linear elasticity, we get the global heat balance equation in the matrix form as: K q𝜃0 = F − F̂

(8.195)

where, K, the conductivity control matrix for the entire body, Ω, is given by: K≡

∑ e

Ke =

∑ e

∫ Ωe

(e )T e e dΩe

(8.196)

and F, the total generalized nodal heat flow control vector is defined by: F ≡ FB + FS + FC

(8.197)

C-type Finite Element Method

429

where, FB , the generalized nodal body heat flow control vector is given by: [ FB ≡

]

∑ e

∫ Ωe

(Te )T Te dΩe

B

qfe

(8.198)

FS , the generalized nodal surface heat flow control vector is given by: [ F ≡ S

]

∑ e

∫Γe

e T

e

(T ) T dΓ

e

S

qfe

(8.199)

̂ the FC , the generalized nodal concentrated heat flow control vector at nodes, and finally, F, generalized nodal inhomogeneous heat flow control vector is given by: F̂ ≡

∑ e

∫Ωe

e (e )T (calSθ̂ ) dΩe

(8.200)

Finally, the finite element temperature solution, θFEM , is given by: θFEM = θFEM + θ̂ 0

(8.201)

where, θFEM at any point in the body is obtained from the solved temperature control vectors, 0 q𝜃0 , by application of the transformation or mapping functions. Remark

r The convective heat boundary condition, like the springs in the elasticity situation, will contribute to both the global conductivity matrix and the heat flux parts of the equation (8.195).

r We have only considered the steady state solution of the heat transfer problem; the transient solution follows the theory presented earlier.

r Similar finite element solutions using the Bernstein elemental basis function and the Bezier control vertices representation of the state variables can be obtained for other field problems such as fluid flow, seepage and general flow problems.

8.4.11

Inter-element Continuity (IEC): Bezier to B-spline

A general mesh representation of a real physical continuum or a structural system can be expected to involve complex shapes. While Bezier representation amazingly mimics the behavior of a given shape, from the standpoint of the finite element philosophy which demands assemblage of simple elements, and from the mesh generation viewpoint, interpolation with Bezier elements with a very high degree is prohibitive. In such cases, the most natural choice is to introduce the concept of what may be called a composite Bezier super-element which is a composition of piecewise Bezier parametric elements with desired continuity at the junction with the neighboring elements. In other words, the finite element mesh defined by equation (8.116) can be imagined as one composite Bezier super-element. For finite element analysis, we will generally be interested in at most C1 continuity for, say, plate bending and shell problems involving fourth-order differential equations. However, higher continuity such as C2 will accelerate convergence wherever applicable. For

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elaboration and details on ideas relating to inter-element continuity, we refer to Chapter 5 on curves. For example, with the one-dimensional situation, to achieve these goals constructively in finite element analysis, we will need to recast the inter-element continuity equations, as follows.

8.4.11.1

Linear Elements and C0 Continuity

For connection between linear elements, only C0 continuity is meaningful, which is trivially achieved by renaming the junction displacement control vectors of each element so that they are one and the same.

8.4.11.2

Quadratic Elements and C1 Continuity

For connection between quadratic elements, additionally C1 continuity is meaningful. For example, let there be two elements, element 1 on the left and element 2 on the right, that are defined by the two displacement control vector sets, {qd0 , qd1 , qd2 } and {qd2 , qd3 , qd4 }, respectively. Notice that the C0 continuity between the two is already satisfied. Let h1 and h2 be the corresponding lengths measured in global parameter of the root elements; notice that if the elements are straight, these quantities represent their respective lengths. Now the C1 continuity is given by:

where,

qd2 = 𝛽 qd1 + 𝛼 qd3 , 𝛼≡

h1 h2 , and, 𝛽 ≡ h1 + h2 h1 + h2

(8.202)

For C1 continuity, we introduce the de Boor polygon vertices as: { d1 }T d d d sd1 ≡ s−1 ≡ qd0 , s01 ≡ qd1 , s11 ≡ qd3 , s21 ≡ qd4

(8.203)

It can be shown that with de Boor vertices constructed as above, and the equation (8.202) for C1 continuity at qd2 particularized as: d

d

qd2 = 𝛽 s01 + 𝛼 s11

(8.204)

guarantees that two quadratic Bezier displacement elements are converted to one quadratic Bspline displacement element. In vector-matrix form, the constraint defined by equation (8.204) along with the de Boor vector definition (8.203) can be expressed as: ⎧ qd ⎫ ⎪ 0⎪ ⎪ qd1 ⎪ ⎪ d⎪ ⎨ q2 ⎬ ⎪ qd ⎪ ⎪ 3⎪ ⎪ qd ⎪ ⎩ 4⎭ ⏟⏟⏟ Bezier–Bernstein

control vectors

⎡1 0 ⎢ ⎢0 1 = ⎢0 𝛽 ⎢ ⎢0 0 ⎢ ⎣0 0

0 0 𝛼 1 0

0⎤ ⎥ 0⎥ 0⎥ ⎥ 0⎥ ⎥ 1⎦

1 ⎫ ⎧ sd−1 ⎪ d1 ⎪ ⎪ s0 ⎪ ⎨ d1 ⎬ ⎪ s1 ⎪ ⎪ d1 ⎪ ⎩ s2 ⎭ ⏟⏟⏟

B-spline–de Boor

control vectors

or, qdBezier = TBS1 sd1

(8.205)

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In other words, inclusion of the C1 constraint converts the two quadratic Bezier elements into one quadratic B-spline element. The transformation matrix relating the unknown displacements in terms of the B-spline generalized values to those in terms of the Bezier control vectors is given by TBS . We can apply this matrix to both the global stiffness matrix and the load vector to obtain the equilibrium equation in terms of the B-spline control vectors as: KBS1 bd1 = FBS1 KBS1 ≡ (TBS1 )T K TBS1 , FBS1 = (TBS1 )T F

where

(8.206)

where K and F are our standard stiffness control matrix and load control vector, respectively, and defined in equation (8.144). Remark

r The transformation matrix, TBS1 , is extremely sparse, and in actual implementation, the constraints could be incorporated by means of sparse matrix technology, avoiding multiplication by zeros. Every nodal continuity constraint reduces the total assembled degrees of freedom by one in the system. Recall that our Bezier end nodes only interpolate the physical nodes.

8.4.11.3

Cubic Elements and C2 Continuity

For connection between cubic elements, additionally C2 continuity is meaningful, and when applied, defines a B-spline element in terms of the de Boor polygon vertices. For example, let there be two elements, element 1 on the left and element 2 on the right that are defined by the two displacement control vector sets, {qd0 , qd1 , qd2 , qd3 } and {qd3 , qd4 , qd5 , qd6 }, respectively. Notice that the C0 continuity between the two is already satisfied. Let h1 and h2 , as before, be the corresponding lengths measured in global parameter of the root elements; notice that if the elements are straight, these quantities represent their respective lengths. Now the C1 continuity, following equation (8.202), is given by: qd3 = 𝛽 qd2 + 𝛼 qd4

where

𝛼≡

h1 h2 and 𝛽 ≡ h1 + h2 h1 + h2

(8.207)

For C2 continuity, we introduce the de Boor polygon vertices as: { d2 }T d d d d sd2 ≡ s−1 ≡ qd0 , s02 ≡ qd1 , s12 , s22 ≡ qd5 , s32 ≡ qd6

(8.208)

It can be shown that, with the de Boor vertices constructed as above, the C2 continuity at qd3 , particularized as: d

d

qd2 = 𝛽 s02 + 𝛼 s12

and

d

d

qd4 = 𝛽 s12 + 𝛼 s22

(8.209)

guarantees that two cubic Bezier displacement elements are converted to one cubic B-spline displacement element. Now, substituting expressions for qd2 and qd4 from equation (8.209) above

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in equation (8.207) for C1 continuity, we get for the junction node: d

d

d

qd3 = 𝛽 2 s02 + 2𝛼𝛽 s12 + 𝛼 2 s22

(8.210)

Finally, we can write in matrix form, expanded or compact, the constraint conditions imposed by the C2 condition as: ⎧ qd ⎫ ⎪ 0⎪ ⎪ qd ⎪ ⎪ 1d ⎪ ⎪ q2 ⎪ ⎪ d⎪ ⎨ q3 ⎬ ⎪ d⎪ ⎪ q4 ⎪ ⎪ qd ⎪ ⎪ 5⎪ ⎪ qd ⎪ ⎩ 6⎭ ⏟⏟⏟

⎡1 ⎢0 ⎢0 ⎢ = ⎢0 ⎢0 ⎢0 ⎢ ⎣0

0 1 𝛽 𝛽2 0 0 0

0 0 𝛼 2𝛼𝛽 𝛽 0 0

0 0 0 𝛼2 𝛼 1 0

0⎤ 0⎥ 0⎥ ⎥ 0⎥ 0⎥ 0⎥ ⎥ 1⎦

d

2 ⎫ ⎧ s−1 ⎪ d2 ⎪ ⎪ s0 ⎪ ⎪ d2 ⎪ ⎨ s1 ⎬ ⎪ d2 ⎪ ⎪ s2 ⎪ ⎪ d2 ⎪ ⎩ s3 ⎭ ⏟⏟⏟

(8.211)

B-spline–de Boor control vectors

Bezier–Bernstein

control vectors

or, qdBezier = TBS2 sd2 In other words, inclusion of the C2 constraint converts the two quadratic Bezier elements into one quadratic B-spline element. The transformation matrix relating the unknown displacements in terms of the B-spline generalized values to those in terms of the Bezier control vectors is given by TBS2 . We can apply this matrix to both the global stiffness matrix and the load vector to obtain the equilibrium equation in terms of the B-spline control vectors as: KBS2 bd2 = FBS2 where KBS2 ≡ (TBS2 )T K TBS2

and

FBS2 = (TBS2 )T F

(8.212)

where K and F are our standard stiffness control matrix and load control vector, respectively, and defined in equation (8.144). Remark

r As before, the transformation matrix, TBS2 , is extremely sparse, and in actual implementation, the constraints could be incorporated by means of sparse matrix technology avoiding multiplication by zeros. Every nodal continuity constraint reduces the total assembled degrees of freedom by two in the system. Recall, as before, that the Bezier end nodes only interpolate the physical nodes. Let us now summarize the mechanisms by which inter-element continuity conditions are implemented for general situations: let Ωi and Ωj , i, j ∈ N(Δ) be two adjacent finite elements, sharing a boundary with each other. As to boundary, for the one-dimensional case, it is the common node between them, for two- and three-dimensional cases; they are the shared edge or face, respectively. As discussed earlier, depending on the degree of the Bernstein polynomials across the shared boundary, the element displacements are described by rows of hitherto unknown

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Bezier displacement control vectors parallel to the shared boundary. Now, we are ready to state the inter-element continuity conditions in term of these displacement controls. In actual application, these are as follows:

r For C0 r

continuity of displacements between elements, we simply rename the displacement control nodes that determine the displacements at the adjoining boundary of Ωi and Ωj . For C1 continuity of displacements between elements, in addition to imposition of C0 continuity, we introduce the constraint conditions in the form of linear combinations of those displacement control vectors that are on the first row parallel and nearest to the adjoining boundary control vectors of both Ωi and Ωj , such that the tangent or tangent plane, depending on the dimensionality of the problem, across the boundary is continuous.

For C2 continuity of displacements between elements, in addition to the imposition of C0 and continuity as above, we introduce additional constraint conditions. These take the form of linear combinations of those displacement control vectors that are on the first two rows parallel and nearest to the adjoining boundary control vectors of both Ωi and Ωj . Thus, the curvature of the surface, depending on the dimensionality of the problem, across the boundary is secured continuous. C1

Remark

r All of these conditions are, in practice, tantamount to a pre- and post-multiplication of the assembled stiffness control matrix, K, and pre-multiplication of the assembled load control vector, F, by a very sparse matrix. These can be performed very effectively by sparse matrix technology at the assembled (global) level. However, element (local) level implementation is also possible, since we will have a priori knowledge of the continuity conditions and the topology of the entire structure or continuum. In other words, the method of incorporation of these conditions is exactly the same as that of any constraint conditions. Now, we elaborate on the concept of inter-element continuity on demand by an example. For the old finite element method, the rate of convergence to the exact solution of a problem may be improved, normally, by:

r increasing the number of elements (h-type), r increasing the degree of the element (p-type). r

For a smooth solution, or where applicable, our new method introduces another control for improvement of the rate of convergence: inter-element continuity condition (c-type). We can now, finally, summarize to define our basis functions and the new finite element space.

8.4.12

The Basis Functions: ΦP,Q,R (Ω, Δ, T, C)

The basis functions in c-type finite element space are conceptually generated by first defining Bernstein basis functions on the root element of some degree P, Q and R for three global directions, and then mapping them to the ith physical elements, Ωei , through transformation, Ti , for i = 1, 2, ... Finally, the basis functions for Ω can be thought to be as these elemental basis functions joined appropriately, so that certain continuity, Cx , (x = 0, 1, 2 normally) is maintained at the

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joins. Thus, the basis functions depend on the mesh, Δ, the transformations, Ti , the Bernstein degree p, and finally, the continuity, Cx . We denote a typical member of the basis functions set as ΦP,Q,R (Ω, Δ, T, C). Then, of course, the displacement vectors, as usual, are given as linear combinations of this set of basis functions, except that, in our case, the coefficients of combination are Bezier displacement control vectors, qd .

8.4.13

The c-type Finite Element Space: SP,Q,R (Ω, Δ, T, C)

Finally, we are in a position to formally define the c-type finite element space: A finite element space consists of:

ra

partitioning or mesh, Δ, of the body, Ω, into a finite set of N bodies, Ωe , such that ⋃ Ωe ≡ Ω,

e∈{1,2,...N}

r a set of N r

transformations, Te : ΩR → Ωe , e ∈ {1, 2, … , N(Δ)}, where the Te ’s represent the complete set of Bernstein polynomials of degree, P, Q and R in three global directions, minimally and exactly required continuity conditions, C, between the adjacent finite elements.

Remarks

r For the first time, we have, through our new method, brought together the geometry (mesh r

r r

r r r

generation) and algebra (analysis), and local properties (displacement control vectors) of the finite difference and global properties (variational theory) of the finite element to solve general problems. The only way in which this finite element space with selective continuity fails to improve over all other the old methods is in the approximation of solutions which do not contain continuous derivatives, or solutions which have singularities. In the case of internal singularities, of course, it is essential not to impose excess smoothness of the trial space. This aspect will be further elucidated through examples, such as a typical beam problem loaded with nodal moment. The cubic finite elements, with C2 continuity across the element boundary, is the equivalent to B-spline space as described above. Most importantly, the assembly level continuity control is the ultimate in combining the global nature of the finite element with the local nature of the finite difference scheme. The use of the Bezier form with piecewise Bernstein polynomials as the finite element space uniquely provides the ground on which the finite element and finite difference scheme implicitly combine to achieve a solution of high accuracy with derivatives of high-order concomitantly approximated. A finite element space, SP,Q,R (Ω, Δ, T, C) is exactly and minimally conforming; that is, it constitutes the largest admissible subspace of the associated energy space. Later, we will show that it is necessary for it to be minimally conforming to arrive at the best approximation, as defined by the underlying variational principle. To be a finite element space, it must consist of finite-dimensional subspaces formed by the piecewise Bezier polynomials of certain degree, say, p, formed in Bernstein basis functions pieced together with Ck , k = 0, 1, 2..., continuity based on the solution space. c-type finite element space is totally different from all spaces published to date, yet it subsumes the old methodologies both conceptually and computationally. As a result, conventional, general purpose computer programs currently in use can be readily adapted to produce a problem-free convergent, controlled and cost-efficient procedure that unifies modeling and analysis.

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r Because of its novelty and the profound difference from the old methods, we demonstrate, now, in great detail, the promises that it presents, by examining the procedure in one, two and three dimensions – and in both linear and nonlinear regimes. We suggest that the usefulness of the new methodology will be greatest in problems with curved boundaries. Accordingly, we will look at solving one-dimensional problems that involve bodies with curved boundaries, that is, curved bars and beams along with their usual straight counterparts. However, we are more interested in seeing its usefulness in two- and three-dimensional problems – e.g. plate and shell problems – which, of course, happen to be the primary targets of the bulk of the finite element applications.

8.4.14

The c-type Finite Element Library

Finally, we are in a position to formally reveal our new finite element library. Here, it will be quickly seen that, unlike the old type finite elements with confusing arrays of elements in commercially available codes, our list of elements is very simple and short. The following general observations may be made.

r The elements depend primarily on the dimensionality and the degree of the basis functions. r All elements, curved or otherwise, are described in Cartesian coordinates. r The only physical nodes present are the end nodes in one dimension and the corner nodes in higher dimensions. In other words, there are no physical edge nodes nor interior nodes. So, let us go ahead and formally describe them in increasing dimensionality.

8.4.14.1

One-dimensional Elements

An element is characterized by:

r the root element, which is a unit straight line in geometry, r the transformation, which is subparametric or coerced isoparametric and defined by:

c(r) = and, d(r) =

M ∑ i=0 N ∑ i=0

All terms are as defined earlier in the text.

8.4.14.2

Two-dimensional Elements

An element is characterized by:

r the root element which is either: – an unit square, or – an isosceles triangle

c BM i (r) qi ,

(8.213) d BN i (r) qi ,

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(a)

(b)

Figure 8.10 (a) Cubic Bezier rectangular element; (b) Cubic Bezier triangular element.

r the transformation, which is subparametric or coerced isoparametric and defined by: – rectangular elements: L K ∑ ∑

c(r, s) = and,

d(r, s) =

i=0 j=0 N M ∑ ∑ i=0 j=0

c L BK i (r)Bj (s) qij ,

(8.214) N d BM i (r)Bj (s) qij ,

all terms are as defined earlier in the text; an example is shown in Figure 8.10(a). – triangular elements: c(r, s, t) =

∑ i+j+k=M

and,

d(r, s, t) =

∑

i+j+k=N

(r, s, t) qcijk , BM ijk (r, s, t) qdijk BN ijk

(8.215)

where, BXijk (r, s, t), X = M, N are the bi-variate triangular Bernstein polynomials in barycentric coordinates, given by: ( BN (r, s, t) = ijk ( ) N and, = ijk

N ijk

) r i sj t k

where

N! i!j!k!

An example is shown in Figure 8.10(b).

i + j + k = N, r + s + t = 1, (8.216)

C-type Finite Element Method

8.4.14.3

437

Three-dimensional Elements

An element is characterized by:

r the root element which is either: r

– a right unit hexahedron, or – a right unit tetrahedron, or – a right unit pentahedron the transformation, which is subparametric or coerced isoparametric and defined by: – hexahedral elements:

c(r, s, t) = and,

d(r, s, t) =

Q ∑ R P ∑ ∑ i=0 j=0 k=0 M ∑ N L ∑ ∑ i=0 j=0 k=0

Q BPi (r)Bj (s)BRk (t) qcijk ,

(8.217) (s)BN (t) BLi (r)BM j k

qdijk ,

all terms are as defined earlier in the text. – tetrahedral elements: ∑

c(r, s, t, p) =

i+j+k+l=M

and,

∑

d(r, s, t, p) =

i+j+k+l=N

(r, s, t, p) qcijkl , BM ijkl (r, s, t, p) qdijkl BN ijkl

(8.218)

where, BXijkl (r, s, t, p), X = M, N are the trivariate Bernstein polynomials, given by: (

) N = ri sj tk pl , ijkl where, i + j(+ k +)l = N, r + s + t + p = 1, N N! and, = ijkl i!j!k!l! (r, s, t, p) BN ijkl

(8.219)

– pentahedral elements:

c(r, s, t, p) = and,

d(r, s, t, p) =

∑

L ∑

i+j+k=K l=0 M ∑ ∑ i+j+k=N l=0

(r, s, t)BLl (p) qcijkl , BK ijk (8.220) (r, s, t)BM (p) BN l ijk

qcijkl

where, all terms are defined as before. Discussion:

r In the above description of the transformations, it is assumed that the loading functions are also interpolated isoparametrically to the geometry. This, of course, allows standardization of

438

r

r

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the loading input for any general-purpose finite element computer program. Arbitrary crosssectional and material properties can likewise be interpolated. However, for general-purpose programming, numerical integration is performed by appropriate quadrature rules that treat these properties as discretized at the quadrature points. It follows that to each geometry control point necessary to describe an element, there corresponds one displacement and loading control points. The above solutions with c-type bar or truss elements, straight and curved beam elements, and plane stress and plate elements indicate that use of the c-type elements will exactly represent all the element properties: displacements, strains, stresses and strain energies. Moreover, because of the existence of the physical nodes of our elements as only corner nodes, the conformity can be achieved on demand. Although the examples above demonstrate many specificities, these should not be construed as limiting the scope of the c-type finite element method but as mere illustrations of the strength and superiority of the c-type method over the old h-type and p-type finite element methods. In this section, for simplicity in application of the c-type method, the example problems are restricted to the linear regime, that is, small displacement, rotations and strains with homogeneous, isotropic material behavior. In later chapters, we remove some of these assumptions. For kinematic behavior, we eventually assume that a plane section remains plane but not necessarily perpendicular to the centroidal axis or plane, to include shear deformations. More importantly, we apply our c-type finite element method to complicated problems of beams, plates and shells in extreme geometrical nonlinearity, that is, very large displacements and rotations. The applications of the c-type method to beam, plates and shell problems in both linear and nonlinear regime are equally robust, stable and accurate; later, we will demonstrate these applications in detail.

8.5

Newton Iteration and Arc Length Constraint

Our main aim here is to devise a numerical scheme to trace nonlinear force–deformation path of a structural system subjected to quasi-static loads. For nonlinear structural problems, complicated or not, closed form solutions, more often than not, are non-existent. To trace the equilibrium (i.e. the force–deformation) path of a structural system subjected to quasi-static loadings, we need to resort to some kind of numerical method that is robust, stable and computationally cost-effective, and mimics the closed form solution. Most algorithms build on Newton’s method of solving the linearized (tangent operator) equations of the original set of nonlinear equations, and then drive the solution to its true value by some sort of iteration scheme. Here, we discuss and present one such method that we use extensively in subsequent application problems to our new c-type finite element method. As to the peculiarity of nonlinear structural equilibrium path tracing, generally there are two fundamentally different scenarios for loss of stability of a structure, as shown in Figure 8.11. These are related to the limit point associated with the snap-through phenomena, or the bifurcation point associated with the buckling situation. In the case of limit points, a basic Newton-type method will either overshoot near a limit point that is about to signal an unloading or undershoot near a limit point heralding a switch to a loading situation with no chance of correction or recovery. Similarly, out of two possible controls of tracing a load–displacement curve, we prefer load control to displacement control. That is, during numerical iteration, the load is incremented and the corresponding displacement increment is determined from the solution of the equilibrium equation; our discussion will be restricted to load control methods. Finally, for a practical problem,

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Figure 8.11 Snap-through and bifurcation buckling.

all that may be necessary is the first limit point, that is, the collapse load; even, in this situation, the simple Newton-type iteration method must be equipped with a mechanism to converge or predict the collapse load. Later, we introduce what is known as the arc length constraint mechanism for all numerical example of quasi-static proportional loading presented for nonlinear beams and shells. With these in the background, let us start by briefly reviewing the Newton’s method and the arc length constraint algorithm.

8.5.1

Newton’s Method

Let us consider a scalar-valued scalar function, G(d); we are interested, say, in the zeros of this function as shown in Figure 8.12. Geometrically speaking, we make an initial guess, say, d0 , and determine the tangent to the curve at this point; the intersection of this tangent with the abscissa gives us the next better guess for the zero of the function; the equivalent algebraic operations will be to determine the first-order ̄ approximation, G(d), to the function at the initial guess as: ̄ G(d) = G(d0 ) + Δd G′ (d0 ), Δd = d − d0

Figure 8.12 Newton’s method for scalar function.

(8.221)

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Figure 8.13 Locking in Newton method.

Δ

𝜕 where (∙)′ = 𝜕d (∙). For intersection of this linear function with the abscissa at, say, d = d1 , we ̄ 1 ) = 0, giving us the next better guess, d1 = d0 + Δd, from equation (8.221): take : G(d

̄ 1 ) = G(d0 ) + Δd G′ (d0 ) = 0 G(d

Δ

⇒ d1 = d0 + Δd = d0 −

G(d0 ) G′ (d0 )

(8.222)

Of course, we can keep on doing it iteratively and hope that it will get closer and closer to the zero of the function with a tolerance, 𝜀 > 0, for error prescribed to us a priori, such that we can stop the iteration process whenever ||G(dn )|| < 𝜀 for some finite n, but note that for this to happen, we need to beware of the following.

r The function should not have a point, di , at which the derivative vanishes. r We must have the starting guess close to the appropriate zero of the function, otherwise, we may settle into a situation that locks the iteration into an infinite loop, as shown in Figure 8.13.

r The Newton iteration method is easily generalized for a system of nonlinear equations; let

us suppose we have N nonlinear equations in N variables, given below in vector form as a vector-valued vector function: G(d) = 0,

G ∈ RN ,

d ∈ RN

(8.223)

then, the Newton method at the ith iteration is given as: ̄ i+1 ) = G(di ) + Δdi ∇d G(di ) = 0, i = 0, 1, 2, ... G(d ( )−1 ⇒ di+1 Δ = di + Δdi = di − ∇d G(di ) G(di )

(8.224)

𝜕G (d )

m i where ∇d G(di ) = 𝜕d Em ⊗ En is the N × N gradient tensor of the function, G(di ), at point n di . Now, assuming a suitable norm, ‖G‖, we may consider that the iteration has converged whenever [ ] ‖G‖ < 𝜀 with 𝜀 > 0 as the prescribed tolerance. Note that the inverse of the matrix, 𝜕Gm (di ) suggested by equation (8.224) is never computed in practical systems with large 𝜕d n

r

N; instead, the linear equation: ∇d G(di ) Δdi = −G(di ), i = 0, 1, 2, ... is solved for Δdi to update the iteration variable, as shown in the second of equation (8.224). As will be seen by the problems solved in nonlinear beams and nonlinear shells, the Newton method of iteration, if successful, converges quadratically.

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Figure 8.14 Load control in Newton method.

8.5.1.1

Algorithm For Newton Method of Iteration

We can sum up the previous presentation in the form of an algorithm: Step 0: Given G(d), set i = 0, and choose: starting point, di , and tolerance, 𝜀 > 0 ‖ Step 1: If the residual: ‖ ‖G(di )‖ < 𝜀, go to Step 3 ; else go to Step 2. Step 2: Solve incremental tangent equation: ∇d G(di ) Δdi = −G(di ), i = 0, 1, 2, ... ; update the Δ

estimate: di+1 = di + Δdi and set the iteration counter: i = i + 1; go to Step 1. Step 3: di is the desired solution; stop.

8.5.2

Non-symmetric Arc Length Constraint Method

The Newton method as presented above works fine as long as we are away from limit points. Particularly, for load–deformation equilibrium path tracing with load control, using the ordinary Newton scheme as above, we may overshoot the load increment around the limit points with no possible intersection of the tangent with the load–deformation curve, and thus we may not recover thereafter as suggested by Figure 8.14. Let us recall the nonlinear equilibrium equation obtained for a structural system under quasistatic proportional loading: G(d,𝜆) = 0

(8.225)

where G represents a system of N equations defined by N-dimensional generalized displacement vector and scalar 𝜆, the load proportionality factor. Suppose now that at each iteration we choose Δ𝜆 as a known increment from the previously equilibrium condition; then, we are left with N equations in N variables, given by equation (8.225), to solve for d. Of course, using the Newton algorithm, we can try to make ad hoc decisions to arbitrarily adjust the incremental load control to get to the limit point of which, however, we have no prior knowledge. Moreover, as we get closer and closer to a limit point, the gradient tensor become more and more ill-conditioned and eventually becomes singular (because it ends up having zero eigenvalue) at the limit point signaling the failure of the Newton method. On the contrary, we may treat load control, 𝜆, as an independent parameter and take note of the fact that there are N equations to solve for (N + 1) independent variables. Thus, for a well-determined equation solution, we may introduce one more equation, c(d, 𝜆) = 0, known as the constraint equation, such that we have the following (N + 1) system of equations in (N + 1) variables: G(d,𝜆) = 0 Original equations c(d,𝜆) = 0 Constraint equation

(8.226)

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Figure 8.15 Arc length constraint.

8.5.2.1

Choosing the Constraint Equation

The constraint equation may be chosen in several ways: generally speaking, all constraint algorithms are designed to constrain iteration steps by what is known as the arc length constraint; given the ith converged equilibrium state, (di , 𝜆i ), the next guess (see Figure 8.15) in the current iteration step, (d, 𝜆), is always constrained by a fixed length, l, expressed by a parameter, 𝛽, as: c(d, 𝜆) = 𝛽 {Δ𝜆}2 + ‖Δd‖2 − l2 , l > 0, 𝛽 = 0, 1 Δ𝜆 = 𝜆 − 𝜆i and Δd = d − di

(8.227)

where assuming a Euclidean norm, we have: ‖d‖2 = d ∙ d = dT d. With 𝛽 = 1, we get what is known as spherical constraint, because it can be seen as a spherical ball of constraint determined by the load and displacement increment at each iteration step with l as the spherical radius given by: l>0

c(d, 𝜆) = {Δ𝜆}2 + ‖Δd‖2 − l2 ,

(8.228)

With 𝛽 = 0, we get what may be characterized as the cylindrical constraint which involves only the displacement increment and may be expressed accordingly as: c(d) = ‖Δd‖2 − l2 ,

l>0

(8.229)

Now, Newton’s method is applied, assuming that we have an (N + 1) system of equations in (N + 1) independent variables: N dimensional displacement vector, d, and a scalar load parameter, 𝜆, as given by equation (8.226). If the constraint radius, l, is small enough, it is expected to intersect the equilibrium path at two points, one going backward and one forward of the last converged state; clearly, we are seeking the forward point on the path. To start with, at each iteration step, as before, we need to linearize (i.e. find the tangent operator) of both equations in equation (8.226) as: ̄ i+1 , 𝜆i+1 ) = G(d𝛼 , 𝜆𝛼 ) + Δd𝛼 ∇d G(d𝛼 , 𝜆𝛼 ) + Δ𝜆𝛼 ∇𝜆 G(d𝛼 , 𝜆𝛼 ) = 0 G(d i i i i i i i i c̄ (di+1 , 𝜆i+1 ) = c(d𝛼i , 𝜆𝛼i ) + Δd𝛼i ∇d c(d𝛼i , 𝜆𝛼i ) + Δ𝜆𝛼i ∇𝜆 c(d𝛼i , 𝜆𝛼i ) = 0 where ∇d G(d𝛼i , 𝜆𝛼i ) =

𝜕Gm (d𝛼i ,𝜆𝛼i ) Em 𝜕dn

(8.230)

⊗ En is the N × N gradient tensor of the function, G(di , 𝜆i ). 𝜕G (d𝛼 ,𝜆𝛼 )

with respect to displacement vector, d, and ∇𝜆 G(d𝛼i , 𝜆𝛼i ) = m 𝜕𝜆i i Em is the N × 1 gradient vector of the function, G(di , 𝜆i ). with respect to load parameter, 𝜆, at an intermediate point d𝛼i , 𝜆𝛼i

C-type Finite Element Method

443

at an intermediate point d𝛼i , 𝜆𝛼i , 𝛼 = 0, 1, 2, ..., in the main converged step i and the next desired converged step (i + 1); similar definitions apply to ∇d c(d𝛼i , 𝜆𝛼i ) and ∇𝜆 c(d𝛼i , 𝜆𝛼i ). If we collect appropriately, the matrix form of equation (8.230) to be solved at each intermediate step may be given as: { ]{ 𝛼 } } Δdi G(d𝛼i , 𝜆𝛼i ) ∇d G(d𝛼i , 𝜆𝛼i ) ∇𝜆 G(d𝛼i , 𝜆𝛼i ) =− ∇𝜆 c(d𝛼i , 𝜆𝛼i ) Δ𝜆𝛼i ∇d c(d𝛼i , 𝜆𝛼i ) c(d𝛼i , 𝜆𝛼i ) ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏟ [

(N+1)×(N+1)

(8.231)

(N+1)×1

Remark

r If the original tangent operator is symmetric, its symmetry is clearly destroyed by forming the augmented matrix given by (8.231) making it inefficient for computation when we are dealing with numbers of displacement degrees of freedom as high as the hundreds or thousands. It is possible to devise a separate numerical scheme using a similar idea, that is, arc length constraint, such that the symmetry of the tangent stiffness is preserved; this is what we use in our later example problems relating to nonlinear beams, plates and shells, since the tangent stiffness (i.e. material stiffness and geometric stiffness) is generally symmetric, and this we introduce next.

8.5.3

Symmetry Preserving Arc Length Constraint Iteration Method

In this scheme, the solution involving the tangent stiffness and that for the arc length constraint equation are kept separate, that is, at an intermediate point d𝛼i , 𝜆𝛼i , 𝛼 = 0, 1, 2, ..., in the main converged step i and the next desired converged step (i + 1). Let us recall that the governing covariantly linearized equation for a structural system undergoing quasi-static loading may be given by: G(di + Δd𝛼i , 𝜆i + Δ𝜆𝛼i ) = G(di , 𝜆i ) + ∇d G(di , 𝜆i ) Δd𝛼i + Δ𝜆𝛼i ∇𝜆 G(di , 𝜆i ) = G(di , 𝜆i ) + KT Δd𝛼i − Δ𝜆𝛼i P = 0 𝛼 ⇒ KT Δdi = Δ𝜆𝛼i P − G(di , 𝜆i ), 𝛼 = 1, 2, 3, ...

(8.232)

where KT , P and G(di , 𝜆i ) are the tangent stiffness, the external load vector and the unbalanced force vector, respectively. In this scheme, the solution, Δd𝛼i , for the iteration step is divided into two parts: Δdi , involving the unbalanced force and ΔdP due to the unscaled external load vector such that: Δdi = −K−1 G(di , 𝜆i ) T P ΔdP = K−1 T Δd𝛼i = Δ𝜆𝛼i ΔdP + Δdi ,

𝛼 = 0, 1, 2, ...

Out-of-balance displacement Tangential displacement Total incremental displacement

(8.233)

Remarks

r It is important to note that in this scheme the original tangent stiffness matrix at the beginning of each load step is used for all intermediate iteration steps, and ΔdP is solved only once in the beginning of a step; this scheme results in more iteration steps and fewer tangent stiffness computations.

Computation of Nonlinear Structures

444

r Depending on the severity of nonlinearity, we may choose to update the tangent stiffness at each iteration step; this normally results in fewer iteration steps. Now, the displacement and load parameter increments are updated as: Δdi+1 = Δdi + Δd𝛼i , Δ𝜆i+1 = Δ𝜆i + Δ𝜆𝛼i

𝛼 = 1, 2, 3, ...

(8.234)

Remarks

r The arc length constraint given by equation (8.228) may be replaced by the simplified version r

for practical problems at each intermediate iteration step and thus equation (8.229) may be used instead. Because the constraint equation is quadratic, there is a distinct possibility that some iteration steps may back up, doubling the previous solution; a strategy must be adopted to avoid such situation. Using equation (8.233) in equation (8.229), we get: ΔdTi+1 Δdi+1 = l2 ⇒ a (Δ𝜆𝛼i )2 + 2b Δ𝜆𝛼i + c = 0 where

Δ

a = ΔdTP ΔdP

(8.235)

Δ

b = (Δd𝛼i + Δdi )T ΔdP Δ

c = (Δd𝛼i + Δdi )T (Δd𝛼i + Δdi ) − l2 Remarks

r The solution of the quadratic equation (8.235) will have two roots, say, Δ𝜆𝛼

and Δ𝜆𝛼i2 : – If any of the roots is imaginary, we would cut the load increment as it is too large to intersect with the equilibrium path, so cut the arc length, l, in half and start the iteration again. – In order to avoid doubling back on some previous point on the equilibrium path, we would expect that the angle between the incremental displacement after iteration, Δdi+1 , and the displacement at the start of the iteration, Δdi , will be acute. So, we compute two estimates, Δd1i+1 and Δd2i+1 , corresponding to the roots, Δ𝜆𝛼i1 and Δ𝜆𝛼i2 , respectively; we compute: cos 𝜃k = (Δdki+1 ∙ Δdi ), k = 1, 2, and we choose Δ𝜆𝛼i = Δ𝜆𝛼ik based on the positive 𝜃k that is closest to the solution of the linear equation corresponding to equation (8.235), that is, i1

2b Δ𝜆𝛼i + c = 0 ⇒ Δ𝜆𝛼i = −

c 2b

(8.236)

r Now, the iteration can be initiated from the beginning by assuming an initial load increment, Δ𝜆0 ; then, the arc length, l, can be chosen as: √ l0 = Δ𝜆0 ΔdP ∙ ΔdP

(8.237)

C-type Finite Element Method

445

r For any subsequent iteration, we would like to choose the arc length in such a way that the number of iterations remain more or less the same at each load step; we would normally expect that the less the severity of nonlinearity in equilibrium path, the less is the number of iterations in each load step, and vice versa. So, if ni−1 and NS are the number of iterations at load steps, i − 1 and the assigned maximum number of iterations, respectively, and li−1 is the arc length at the (i − 1)th load step, then the arc length, li , at load step i may be set in inverse proportion as: ( li =

Nmax ni−1

) li−1

(8.238)

r For any load step other than the first one, we will choose the increment of load step as: li Δ𝜆i = ± √ ΔdP ∙ ΔdP

(8.239)

r The sign of the load increment in equation (8.239) depends on whether we are loading up

r r

on the equilibrium path or unloading after, say, a limit point; the indication comes from the determinant of the tangent stiffness matrix: during the loading cycle, the determinant should be positive as the tangent stiffness is positive definite, and it is negative during the unloading cycle for negative definitive tangent stiffness, The positive sign should be chosen for the loading cycle and negative for unloading; exactly at the limit point, the stiffness is singular but it is a very low probability of occurrence numerically. A tolerance for error of convergence criterion, 𝜀 > 0, and an arc length constraint parameter are input by the user of a computer code. Finally, for any load step i, the total displacement, di+1 , and the total load increment, 𝜆i+1 , after convergence, are given as: di+1 = di + Δdi

and

𝜆i+1 = 𝜆i + Δ𝜆i

(8.240)

Based on these discussions, we construct and choose the following algorithm for numerical iteration in all succeeding nonlinear problems.

8.5.3.1

Algorithm: Numerical Iteration

Step 0: Choose, an initial load increment, Δ𝜆0 , maximum iteration count, Nmax , maximum load step count, Lmax , a maximum load increment, 𝜆max , and a tolerance for convergence, 𝜀; set load step counter, i = 0, iteration counter, 𝛼 = 0. Step 1: If i ≠ 0, that is, not the first load step, go to Step 2; if the iteration counter, 𝛼 ≠ 0, that is, not first iteration, go to Step 3 else: compute ΔdP from equation (8.233); compute li from equation (8.237); compute Δd𝛼i from equation (8.233); go to Step 3. Step 2: If the iteration counter, 𝛼 ≠ 0, that is, not first iteration, go Step 4 else: compute li from equation (8.238), Step 3: Compute ΔdP from equation (8.233); set Δ𝜆i using equation (8.239) with sign chosen based on the determinant of the current tangent stiffness, KT ; go to Step 4. Step 4: If tangent stiffness, KT , is always kept constant during iteration, go to Step 5 else recompute ΔdP from equation (8.233), and go to Step 5.

Computation of Nonlinear Structures

446

Step 5: compute the out-of-balance displacement, Δdi , from equation (8.233); solve the quadratic l equation (8.235) for two roots, say, Δ𝜆𝛼i1 and Δ𝜆𝛼i2 : if any of the roots is imaginary, set li = 2i and 𝛼 = 0, and go to Step 3; else, compute: cos 𝜃k = (Δdki+1 ∙ Δdi ), k = 1, 2; if 𝜃1 > 0, Δ𝜆𝛼i = Δ𝜆𝛼i1 , and Δdi+1 = Δd1i+1 , where Δd1i+1 corresponds to the root, Δ𝜆𝛼i1 ; if 𝜃2 > 0, Δ𝜆𝛼i = Δ𝜆𝛼i2 , and Δdi+1 = Δd2i+1 , where Δd1i+1 corresponds to the root, Δ𝜆𝛼i2 ; if both 𝜃k , k = 1, 2, are positive, then choose Δ𝜆𝛼i = Δ𝜆𝛼ik based on positive 𝜃k that is closest to the solution of the linear equation 𝛼 𝛼−1 ‖ ‖ 𝛼‖ (8.236); update total displacement, d𝛼i = di + (Δd𝛼i − Δdi𝛼−1 ). If ‖ ‖di − di ‖ ≤ 𝜀 ‖di ‖, set 𝛼 di+1 = di , ni = n𝛼 and update Δ𝜆i+1 using equation (8.234), and go to Step 6; else set: 𝛼 = 𝛼 + 1, and go to Step 4. Step 6: If load step count, i ≥ Lmax , or load increment, 𝜆i ≥ 𝜆max , stop, else set: 𝛼 = 0, and go to Step 2.

8.6

Gauss–Legendre Quadrature Formulas

Here, our main goal is to introduce numerical schemes to evaluate integrals of functions; as indicated in previous sections on the finite element method, in computing the stiffness matrix, mass matrix and load vectors, integrals of the following general form need to be integrated: b

I=

∫a

w(x) f (x) dx

(8.241)

where w(x) are known as the weights; with user-supplied variable geometric properties such as thickness and area, and the inverse of the Jacobian function in the form of even rational polynomial as part of the integrand, f (r), the integration is invariably evaluated numerically.

8.6.1

Interpolatory Quadratures for Numerical Integration

We have already seen that a numerical integration scheme relies on approximating integration by finite summation; the interval of the integration is partitioned into a finite number of points, and the function values at these point serve as summands for the approximation by summation. In the most general scheme, the partition locations are variable, but in relatively simpler schemes such as the Newton–Cotes formulas, the partition points are fixed in position. So, considering one-dimensional integration, suppose (n + 1) distinct partition points are ordered as: x0 < x1 , ∙ ∙ ∙, < xn−1 < xn

(8.242)

With these interpolatory points, we can construct the interpolatory polynomial, Pn (x), of degree n, such that: f (xj ) = Pn (xj ),

j = 0, 1, … , n

(8.243)

Then, as an approximation of the integration given by equation (8.241), we set: b

Iapprox =

∫a

w(x) Pn (x) dx + error(f )

(8.244)

C-type Finite Element Method

447

Now, if we choose the Lagrange form for the polynomial, that is,

Pn (x) =

n ∑

n ln (x) f (xj ) with ln (x) ≡ Π (x − xi ) i=0 j=0 (x − xj ) ln ,x (xj )

(8.245)

where Π is the multiplicative symbol. Upon integration, we get from equation (8.244):

Iapprox =

n ∑ j=0

b

Aj f (xj )

where

Aj =

∫a

w(x)

ln (x) dx (x − xj ) ln ,x (xj )

(8.246)

where Aj ’s are known as the weights of the quadrature. Remark

r From the definition of wn (x) given in equation (8.245) and the Lagrangian error estimate, it

can be shown that the error in integration vanishes if f (x) is a polynomial of degree ≤ n; thus, a (n + 1) point interpolatory quadrature has degree of precision of at least n.

8.6.2

Newton–Cotes Formulas

These include end points as interpolatory points, and as such known as “closed” type formulas; in what follows, we assume:w(x) = 1, as indicated before, if f (x) = Pm (x), m ≤ n, the rules are exact.

8.6.2.1

Trapezoidal Rule: Two End Points b

∫a

8.6.2.2

b

b

h h h3 f (a) + f (b) − f (2) (𝜉), 2 2 12

h≡b−a

(8.247)

Simpson’s Rule: Two End Points and the Mid-point ∫a f (x) dx =

8.6.2.3

f (x) dx =

( ) } ( ){ h5 a+b h f (a) + 4f + f (b) − f (4) (𝜉), 3 2 90

h≡

b−a 2

(8.248)

3 Simpson’s th Rule: Four Equidistant Points Including Two End Points 8

∫a f (x) dx =

(

) 3h5 (4) 3h {f (a) + 3f (a + h) + 3f (a + 2h) + f (b)} − f (𝜉), 8 80

h≡

b−a 3 (8.249)

Computation of Nonlinear Structures

448

Remarks

r It is interesting to note that there exist f

∈ C0 [a, b] for which as n → ∞, the Newton–Cotes formulas do not converge; in other word, the above rules not necessarily converge as n → ∞.

8.6.3

Linear Transformation of Interval

Suppose two sets of intervals: x ∈ [a, b] and u ∈ [c, d] are given, and we know the rules for the interval, [a, b] such that: b

∫a

w(x) f (x) dx =

n ∑

Ak f (xk )

Now, suppose x = 𝛾u + 𝛽 ⇒ u = 𝛾1 (x − 𝛽) where 𝛾 = know the rule such that: b

d

w(x) f (x) dx =

∫a

(8.250)

k=1 b−a d−c

and 𝛽 =

ad−bc ; d−c

we would like to

̂ w(u) g(u) du

(8.251)

̂ w(𝛾u + 𝛽)

(8.252)

∫c

where g(u) = f (𝛾u + 𝛽), The rule can be easily found to be: d

∫c where

8.6.4

̂ w(u) g(u) du =

n ∑ k=1

 k g(uk )

1 Â k = Ak 𝛾 1 uk = (xk − 𝛽) 𝛾

(8.253)

Composite Formulas

We have already noted that the previous formulas do not necessarily converge, but if the interval is subdivided and the simple formulas are applied repeatedly, the convergence is achieved as n → ∞.

8.6.4.1

Repeated Trapezoidal Rule

With (n − 1) subintervals, we get: b

∫a

f (x) dx =

n−1 ∑ h h f (a + (k − 1)h) + f (b) + error(f ), f (a) + h 2 2 k=2

h≡

b−a n−1

(8.254)

C-type Finite Element Method

8.6.4.2

449

Repeated Mid-point Rule: “Open”, i.e. End Points Are Not Nodes

Based on the single interval mid-point. rule: (

b

∫a

f (x) dx = (b − a) f

a+b 2

) (8.255)

With n subintervals, for repeated mid-point. rule, we get: b

∫a

8.6.4.3

f (x) dx = h

) ( 1 h) + error(f ), f (a + k − 2 k=1

n ∑

h≡

b−a n

(8.256)

Repeated Simpson’s Rule

With (n − 1) subintervals, we get: b

∫a

f (x) dx =

( ) ( ) ( ) 4h 2h h f (a) + f (a + h) + f (a + 2h) 3 3 3 ( ) ( ) ( ) 2h 2h 4h f (a + 3h) + f (a + 4h) + ∙ ∙ ∙ + f (b − 2h) (8.257) + 3 3 3 ( ) ( ) b−a h 4h f (b − h) + f (b) + error(f ), h ≡ + 3 3 n−1

Remark

r The above repeated rules constitute what is known as the Riemann sum; thus, ∑ n →≤ ∞

8.6.5

→ ∫ as

Gauss – Legendre Quadratures

Let us recall that the Newton–Cotes formulas presented earlier with, say, n nodes, are exact for polynomial functions of only degree ≤ n − 1. The question is whether there exist quadratures that are exact for polynomials of higher degrees, and, if so, what is the highest degree of polynomial for which it remains exact. The answer is yes: if we make both the node locations of the nodes and the weights, that is, a total of 2n quantities, variable, then the quadrature constructed optimally can be exact for polynomials of degree ≤ 2n − 1: in other words, for a special choice of nodes, xi , i = 1, 2, … , n, we can find Ak , k = 1, 2, … , n so that b

∫a

w(x) f (x) dx =

is exact for polynomials of degree ≤ 2n − 1.

n ∑ k=1

Ak f (xk ) + error(f )

(8.258)

Computation of Nonlinear Structures

450

Remarks

r It is important to note that (2n − 1) is the highest degree of a polynomial that can be exactly represented by any quadrature formula with n nodes.

r These families of quadratures are known as Gaussian quadratures. r The degree of precision is defined as the highest degree of polynomials for which a particular quadrature is exact; the degree of precision for an n-point Gaussian quadrature is (2n − 1), and thus the Gaussian quadrature formulas have the highest degree of precision. The basic idea behind a Gaussian quadrature is to locate the nodes of quadrature, xi , i = 1, 2, … , n, at the zeros of an appropriate polynomial, Pn (x), of degree n, that is orthogonal to the interval of the integral with respect to the weighting function, w(x), to all polynomials, Qn−1 (x), of degree ≤ n − 1, that is, b

∫a

w(x) Pn (x) Qn−1 (x) dx = 0

(8.259)

Remarks

r It can be shown that the zeros of Pn (x) are real and distinct and reside in the open interval (a, b). r For w(x) = 1, and [a, b] = [−1, 1], the orthogonal polynomials are known as Legendre polynomials.

r All weights, Ak , k = 1, 2, … , n, of the Gaussian quadrature are positive. r Gaussian quadratures converge to an exact integral as n → ∞. r Geometrically speaking, the Legendre polynomials, say, being orthogonal to Qn−1 (x), are at r r r

the shortest distance from the space of all polynomials of degree ≤ n − 1, and thus minimize error and provide the best approximation. Similarly, if the integrand is a polynomial of degree d, then (2n − 1) should be at least equal to d, that is, the quadrature must have: n ≥ 12 (d + 1); for example, if d = 5, say, a quintic polynomial, the integration must be evaluated with at least a three-point Gaussian quadrature formula. The tables listing the Gauss–Legendre quadrature nodes and weights are widely available in any standard literature on numerical analysis, but the ones that we use here for c-type finite element analysis of beams, shells, etc., are listed later. The Gaussian quadratures are interpolatory; thus, the weights, Ak , k = 1, 2, … , n, may be expressed, among other ways, as:

Ak =

8.6.5.1

b P (x) 1 n dx, Pn ,x (xk ) ∫a (x − xk )

k = 1, 2, … , n

(8.260)

Linear Transformation of Interval

For the c-type finite element method, x ∈ [0, 1], instead of x ∈ [−1, 1] as in the old finite element methods, and this linear transformation is easily handled by Gaussian quadratures as before.

C-type Finite Element Method

451

With x ∈ [a, b] = [−1, 1] and u ∈ [c, d] = [0, 1], we get: 𝛾 = follows from equation (8.253):

d=1

̂ ∫c=0 w(u) g(u) du =

n ∑ k=1

b−a d−c

= 2 and 𝛽 =

uk =

= −1; so it

 k g(uk )

1 1 Â k = Ak = Ak 𝛾 2

where

ad−bc d−c

(8.261)

1 1 (x − 𝛽) = (xk + 1) 𝛾 k 2

An example will clarify these concepts. Example: Consider a function, f (x), that is a quintic polynomial, that is, of degree five, defined as: f (x) = 𝛼0 + 𝛼1 x + 𝛼2 x2 + 𝛼3 x3 + 𝛼4 x4 + 𝛼5 x5

(8.262)

The exact integration, I, is given by: 1

I=

∫−1

2 2 f (x) dx = 2𝛼0 + 𝛼2 + 𝛼4 3 5

(8.263)

The three parameters, 𝛼0 , 𝛼2 , 𝛼4 completely define the integral; we can construct the underlying Legendre polynomial, P3 (x), of degree three whose roots define the quadrature as follows: we know that P3 (x) is orthogonal to all polynomials of degree ≤ 2; thus, letting: P3 (x) = x3 + 𝛽2 x2 + 𝛽1 x + 𝛽0

(8.264)

and noting that the linear combination of the monomials, 1 = x0 , x, x2 , spans the subspace of polynomials of degree ≤ 2, and we have for orthogonality with xk , k = 0, 1, 2 1

∫−1

P2 (x) dx = 0

1

x P2 (x) dx = 0

⇒

2 2 + 𝛽 =0 5 3 1

x2 P2 (x) dx = 0

⇒

2 2 𝛽0 + 𝛽2 = 0 3 5

∫−1 1

∫−1

2 ⇒ 2𝛽0 + 𝛽2 = 0 3 (8.265)

Computation of Nonlinear Structures

452

Now, solving equation (8.265), we get: 𝛽0 = 𝛽2 = 0 and 𝛽1 = − 35 ; thus, the interpolating Legendre polynomial is given by: 3 P3 (x) = x3 − x 5

(8.266)

and its roots are given by: √ x = 0, +

√ 3 3 , − 5 5

(8.267)

that form the interpolating points, xk , k = 0, 1, 2, respectively. Using equation (8.260), for the weights, Ak , k = 0, 1, 2, we get: ( ) 3 x3 − x 1 5 dx = 8 A0 = ( ) 3 ∫−1 x 9 − 5 ( ) 3 √ ) ( ) 1( x3 − x 1 1 5 3 5 5 x2 ± A1 = A2 = ( ) x dx = ( √ ) dx = 6 ∫ 6 ∫−1 5 9 −1 3 x± 5 5 1

(8.268)

Finally, we can express the Gauss–Legendre quadrature as: ( ) ( ) 5 8 f (0) + f (x) dx = f ∫−1 9 9 1

(√ ) ( ) ( √ ) 5 3 3 + f − 5 9 5

(8.269)

Remarks

r This example has been worked out using the general principle of evaluating the nodes and the weights of Gauss–Legendre quadratures.

r The weights can be seen to be real, positive and distinct, and they reside inside the interval of integration.

r A polynomial of degree five has been exactly integrated by evaluating the function values at three points.

8.6.6

Higher Dimensional Gauss–Legendre Nodes and Weights

The numerical integration for higher dimensions – for example, two or three – can be achieved by straightforward extensions to the one-dimensional procedure described above. To wit, let us write the two-dimensional quadrature rule as: b

∫a ∫c

d

f (x, y) dx dy =

n ∑ k=1

d

∫c

Ak f (xk , y) dy =

m ∑ n ∑ l=1 k=1

Ak Al f (xk , yl )

(8.270)

C-type Finite Element Method

8.6.7

453

Tabular Examples of Gauss–Legendre Nodes and Weights

For future reference, we present below some values of the nodes and the weights for quadratures discussed above for various degrees that we actually use later in our applications: Real(kind=8) Function GaussPtVal(index,Nint,iPtVal) ! ! ! Notes: Nint should be at least 2 ! index = Index of Columns of Pts: 1 -> Nint ! Nint = Number of Gauss Points ! iPtVal = 1: Gauss Pt/ 2:Gauss Value ! Outputs ! GaussPtVal = Result based on iPtVal ! Implicit Real(kind=8) (a-h,o-z) ! Input Variables & Arrays !----------------------------------------------------------------------! Rk(8,8) = Gauss–Legendre Nodes Upto 8 Points ! Ak(8,8) = Gauss–Legendre Weights Upto 8 Points ! Recall: ! Our integration, r=[0,1] ! Conventional x=[-1,1] ! i.e. x = (2r-1) ! So, following values modified as: ! Rk = (1/2)(xk+1) ! Ak = (1/2)Axk !----------------------------------------------------------------------Real(kind=8) GLO16,GLO5,GLO7 Dimension GLO16(16,2),GLO5(5,2),GLO7(7,2) Real(kind=8) Rk,Ak,D,B,Add,DB Dimension Rk(10,9),Ak(10,9),D(3,3),B(3,32),Add(32),DB(4) DATA Rk/ 0.5D0, 0.0D0, ! 1-point & 0.0D0, 0.0D0, & 0.0D0, 0.0D0, & 0.0D0, 0.0D0, & 0.0D0, 0.0D0, & 0.211324865405187D0, 0.788675134594813D0, ! 2-point & 0.0D0, 0.0D0, & 0.0D0, 0.0D0, & 0.0D0, 0.0D0, & 0.0D0, 0.0D0, & 0.112701665379258D0, 0.5D0, ! 3-point & 0.887298334620742D0, 0.0D0, & 0.0D0, 0.0D0, & 0.0D0, 0.0D0, & 0.0D0, 0.0D0, & 0.069431844202974D0, 0.330009478207572D0, ! 4-point & 0.669990521792428D0, 0.930568155797026D0, & 0.D0, 0.0D0, & 0.D0, 0.0D0, & 0.D0, 0.0D0, & 0.0D0, 0.230765344947158D0, ! 5-point & 0.769234655052841D0, 0.046910077030668D0,

454

& & & & & & & & & & & & & & & & & & & & & & & & ! & & & & & & & & & & & & & & & & & & & & & & & & & & &

Computation of Nonlinear Structures

0.953089922969332D0, 0.0D0, 0.D0, 0.0D0, 0.D0, 0.0D0, 0.966234757101576D0, 0.033765242898424D0, 0.830604693233132D0, 0.169395306766868D0, 0.619309593041598D0, 0.380690406958402D0, 0.D0, 0.0D0, 0.D0, 0.0D0, 0.5D0, 0.297077424311301D0, 0.702922575688699D0, 0.129234407200303D0, 0.870765592799697D0, 0.025446043828620D0, 0.974553956171379D0, 0.D0, 0.D0, 0.0D0, 0.019855071751232D0, 0.980144928248768D0, 0.101666761293187D0, 0.898333238706813D0, 0.237233795041836D0, 0.762766204958164D0, 0.408282678752175D0, 0.591717321247825D0, 0.D0, 0.0D0, 0.5D0, 0.337873288298095D0, ! 9-point 0.662126711701904D0, 0.193314283649705D0, 0.806685716350295D0, 0.081984446336682D0, 0.918015553663318D0, 0.015919880246187D0, 0.984080119753813D0, 0.0D0 / DATA Ak/ 2.0D0, 0.0D0, 0.0D0, 0.0D0, 0.0D0, 0.0D0, 0.0D0, 0.0D0, 0.0D0, 0.0D0, 0.5D0, 0.5D0, 0.0D0, 0.0D0, 0.0D0, 0.0D0, 0.0D0, 0.0D0, 0.0D0, 0.0D0, 0.277777777777778D0, 0.444444444444444D0, 0.277777777777778D0, 0.0D0, 0.0D0, 0.0D0, 0.0D0, 0.0D0, 0.0D0, 0.0D0, 0.173927422568727D0, 0.326072577431273D0, 0.326072577431273D0, 0.173927422568727D0, 0.0D0, 0.0D0, 0.0D0, 0.0D0, 0.0D0, 0.0D0, 0.284444444444444D0, 0.239314335249683D0, 0.239314335249683D0, 0.118463442528095D0, 0.118463442528095D0, 0.0D0, 0.0D0, 0.0D0, 0.0D0, 0.0D0, 0.085662246189585D0, 0.085662246189585D0, 0.180380786524069D0, 0.180380786524069D0, 0.233956967286346D0, 0.233956967286346D0,

! 6-point

! 7-point

! 8-point

! 1-point

! 2-point

! 3-point

! 4-point

! 5-point

! 6-point

C-type Finite Element Method

& 0.0D0, 0.0D0, & 0.0D0, 0.0D0, & 0.208979591836734D0, 0.190915025252560D0, ! 7-point & 0.190915025252560D0, 0.139852695744638D0, & 0.139852695744638D0, 0.064742483084435D0, & 0.064742483084435D0, 0.0D0, & 0.0D0, 0.0D0, & 0.050614268145188D0, 0.050614268145188D0, ! 8-point & 0.111190517226687D0, 0.111190517226687D0, & 0.156853322938943D0, 0.156853322938943D0, & 0.181341891689181D0, 0.181341891689181D0, & 0.0D0, 0.0D0, & 0.165119677500630D0, 0.156173538520001D0, ! 9-point & 0.156173538520001D0, 0.130305348201468D0, & 0.130305348201468D0, 0.090324080347429D0, & 0.090324080347429D0, 0.040637194180787D0, & 0.040637194180787D0, 0.0D0 & / !----------------------------------------------------------------------DATA ((GLO5(i,j),j=1,2),i=1,5)/ & 0.9530899229693319, 0.1184634425280945, & 0.0469100770306680, 0.1184634425280945, & 0.7692346550528415, 0.2393143352496832, & 0.2307653449471584, 0.2393143352496832, & 0.5000000000000000, 0.2844444444444444 & / DATA ((GLO7(i,j),j=1,2),i=1,7)/ & 0.9745539561713792, 0.0647424830844349, & 0.0254460438286208, 0.0647424830844349, & 0.8707655927996972, 0.1398526957446383, & 0.1292344072003028, 0.1398526957446383, & 0.7029225756886985, 0.1909150252525595, & 0.2970774243113014, 0.1909150252525595, & 0.5000000000000000, 0.2089795918367347 & / DATA ((GLO16(i,j),j=1,2),i=1,16)/ & 0.9947004674958250D0, 0.0135762297058770D0, & 0.0052995325041750D0, 0.0135762297058770D0, & 0.9722875115366163D0, 0.0311267619693239D0, & 0.0277124884633837D0, 0.0311267619693239D0, & 0.9328156011939159D0, 0.0475792558412464D0, & 0.0671843988060841D0, 0.0475792558412464D0, & 0.8777022041775016D0, 0.0623144856277669D0, & 0.1222977958224985D0, 0.0623144856277669D0, & 0.8089381222013219D0, 0.0747979944082884D0, & 0.1910618777986781D0, 0.0747979944082884D0, & 0.7290083888286136D0, 0.0845782596975013D0, & 0.2709916111713863D0, 0.0845782596975013D0, & 0.6408017753896295D0, 0.0913017075224618D0, & 0.3591982246103705D0, 0.0913017075224618D0, & 0.5475062549188188D0, 0.0947253052275343D0, & 0.4524937450811813D0, 0.0947253052275343D0 & / !-----------------------------------------------------------------------

455

Computation of Nonlinear Structures

456

if (iPtVal == 1) Then if (Nint == 5) Then GaussPtVal = GLO5(index,1) else if (Nint == 7) Then GaussPtVal = GLO7(index,1) else if (Nint == 16) Then GaussPtVal = GLO16(index,1) else GaussPtVal = Rk(index,Nint) end if else if (iPtVal == 2) Then if (Nint == 5) Then GaussPtVal = GLO5(index,2) else if (Nint == 7) Then GaussPtVal = GLO7(index,2) else if (Nint == 16) Then GaussPtVal = GLO16(index,2) else GaussPtVal = Ak(index,Nint) endif end if ! return end

8.6.8

Where We Would Like to Go

We have introduced Gauss–Legendre quadrature for integrations that are routinely needed in structural system analysis as one of the final pieces of the puzzle for applying the c-type finite element method. So, to check the efficacy and robustness of the method, we should check out:

r applications of c-type finite element method (Chapter 9, Chapter 10 and Chapter 11)

Part V Applications: Linear and Nonlinear

9 Application to Linear Problems and Locking Solutions 9.1 Introduction 9.1.1 Why We Are Here Thus far we have presented in great detail our new c-type finite element method – a displacementbased finite element method, but nothing can make it more amenable for comprehension or more reliable for industrial usage than testing it with example problems of interest in structural engineering and structural mechanics. In each of these applications, the state is determined by the displacement distribution, a non-parametric function, while c-type finite element method is motivated by the Bernstein–Bezier parametric curves. We may now recall from our previous presentation that any non-parametric function can be represented by parametric Bernstein–Bezier ∑ curves because of the linear precision property: n0 ( ni )Bni (𝜉) = 𝜉. Thus, we recall that a displacement function, d(𝜉), can be transformed into parametric form as: { c(𝜉) =

x(𝜉)

}

y(𝜉)

⎧( i )⎫ ⎪ ⎪ qi ≡ ⎨ n ⎬ ⎪ qi ⎪ ⎭ ⎩ where, d(𝜉) =

n ∑ 0

{ =

𝜉 d(𝜉)

} =

n ∑ 0

Bni (𝜉) qi

(9.1)

Bni (𝜉) qi

that is, the control vertices are located at equal intervals. In order to evaluate the proposed c-type element library, several test problems, most of which are generally available in the literature, are solved and the results are presented and compared to demonstrate its numerical accuracy, robustness and stability. In all the numerical examples presented below, normal exact full Gauss quadratures were used for stiffness matrix computations avoiding ad hoc reduced or selective integration, and so on, of old methodologies. It may also be noticed that all problems are solved with the minimum number (compared to the old finite element methods) of mega-sized arbitrary Computation of Nonlinear Structures: Extremely Large Elements for Frames, Plates and Shells, First Edition. Debabrata Ray. © 2016 John Wiley & Sons, Ltd. Published 2016 by John Wiley & Sons, Ltd.

Computation of Nonlinear Structures

460

shaped (wherever applicable) elements that are central to the main motivation of the c-type method. In this chapter, we restrict applications to only linear problems; in the subsequent chapters, extreme nonlinearity is considered. Thus, in this chapter we will present the following. Section 2: The proposed c-type method with special reference to inter-element continuity on demand is exemplified by its application to one-dimensional straight truss or straight rod problems. Section 3: This discusses applications of c-type application to straight Euler– Bernoulli beams. Section 4: Here we investigate application of c-type Bezier and B-spline elements to curved beam problems, including pinched rings. The unification of mesh generation and finite element analysis in c-type is exemplified by a linear, truly three-dimensional curved beam element with natural twist and bend (the treatment of beams, plates and shells in extreme geometric nonlinearity is elaborated in the following two chapters). Section 5: The versatility, straightforward nature and ease of use of the method are demonstrated by looking at the problem of arbitrarily distorted and nearincompressible plane strain and plane stress deep beam elements that have, otherwise, remained in the literature and recent textbooks as an annoying nuisance for all other isoparametric finite element methods. Section 6: The solutions to usual locking problems: shear locking, dilatational locking, transverse shear locking, and so on, are generated with the c-type methodology.

9.2

c-type Truss and Bar Element

One-dimensional finite element analysis, although less consequential for practical applications, brings out most of the salient features of our c-type finite element formulation presented in the previous chapter; thus, we specialize our method by applying it to one-dimensional bars or rods (truss element) that involve second-order differential equations as the momentum balance equation. It is assumed at this point that a one-dimensional geometric assemblage has been generated at the interactive modeling phase; we must emphasize right at the outset that the discretization of a geometry in the modeling phase should not be performed without due regard to the loading and support conditions that define a particular problem for which the geometry is being discretized. For a one-dimensional truss element, the geometric property, the crosssectional area, A(x), the material property, the elasticity modulus, E(x), the loading, the body forces, f B (x), per unit volume will generally depend only on the axial coordinate, x. For linear truss elements, a hierarchical set of elements may be conceived through the polynomial degree allowed for displacement up to cubic for numerical efficiency. Given this limit, however, we will build two main categories of elements: Bernstein–Bezier elements characterized by Bezier control polygons, B-spline–de Boor elements formed by cubic composite Bernstein–Bezier polynomials connected by C2 continuity. With these in the background, let us consider the following onedimensional bar example problem. Example CFEM 1: One-dimensional bar with variable elasticity modulus Consider the bar or truss element of length, L, with constant cross-sectional area, A, as shown in Figure 9.1, but whose elasticity constant varies linearly along its length as E(x) = E0 (1 + Lx ) with A = E0 = L = P = 1 ; the bar is subjected to an axial load, P, at x = L end. We will try to identify for the problem: (a) stiffness control matrix, (b) load control vector, (c) displacement

Application to Linear Problems and Locking Solutions

461

Figure 9.1 Example CFEM1: 1D bar with variable elasticity.

distribution, (d) strain distribution and (e) strain energy of the bar, obtained under a hierarchy of c-type bar elements, along with varying inter-element continuity. For this problem, let ̄ = Ω ∪ Γ = [0, L], and the us recognize that: Ω = (0, L); Γd = {0}; Γ𝜎 = {L}; Γ = Γd ∪ Γ𝜎 ; Ω associated boundary value problem can be posed as: Differential equation:

(AE(x)u,x (x)),x = 0 in x ∈ Ω

Essential boundary condition: Natural boundary condition:

u(0) = 0 on Γd

(9.2)

AE(L) u,x (L) = P on Γ𝜎

The exact solution can be easily determined to be: ( Displacements: uExact (x) = ( Strains: 𝜀Exact (x) = ( 𝜀Exact (0) = Strain energy: UExact

PL AE0 P AE0 P AE0

1 = ln 2 2

) )

)

( ) ( ) ) ( PL x PL with uExact (L) = ln 2 ln 1 + < L AE0 AE0 1 ) with ( x 1+ L and

(

P2 L AE0

𝜀Exact (L) =

) = 0.3465736

1 2 (

(

P AE0 ) P2 L AE0

)

(9.3) 9.2.1.1

Case 1: One Linear Element – Constant Strain (CSL) ) [ 1 −1 ] ( AE Full singular stiffness matrix: K = 1.5 L 0 , Bezier controls (generalized Displace−1 1 ments): q0 = 0 q1 = 23 ; strain energy: U =

1 1 ∫ (1 + x) 2 0

(𝜀(x))2 dx = 0.333333.

Case 2: Two Linear Elements – C0 Continuity ( ) [ 1 −1 ] 5 AE0 Singular stiffness matrices: Elem #1: K1 = 2 L , −1 1 [ ] ( ) 1 −1 7 AE0 . 2 L −1 1 9.2.1.2

Elem

#2:

K2 =

Computation of Nonlinear Structures

462

0⎤ ) ⎡ 5 −5 ⎢ −5 12 −7 ⎥; assembled non-singular ⎢ ⎥ ⎣ [0 −7 ] 7 ⎦ ) 12 −7 ( AE = 2L0 ; load control vector: P = −7 7 (

Assembled singular stiffness matrix: KSIN =

AE0 2L

stiffness control matrix: with q0 = 0: KNON { } 0 ; Bezier controls (generalized displacements): q0 = 0, q2 = 1 U=

1 1 ∫ (1 + 2x ) 2 0

9.2.1.3

(𝜀(x))2 12 dx +

1 1 3 ∫ ( 2 0 2

24 ,q 35 1

=

7 q ; 12 2

strain energy:

+ 2x ) (𝜀(x))2 12 dx = 0.33428571429.

Case 3: One Quadratic Element

) ⎡ 5 −2 −3 ⎤ ⎢ −2 6 −4 ⎥; non-singular stiffness con⎢ ⎥ 7⎦ ⎣ −3 ]−4 [ { } ) ( 6 −4 0 AE0 ; s control vector: P = ; Bezier trol matrix: with q0 = 0: KNON = 3L −4 7 1 9 q1 = 23 q2 ; strain energy: U = controls (generalized displacements): q0 = 0 q2 = 13 [ ] 2 1 ∫0 2(1 + x) (1 − 2x)q1 + xq2 dx = 0.3461538462. (

Full singular stiffness matrix: KSIN =

9.2.1.4

AE0 3L

Case 4: Two Quadratic Elements – C0 Continuity

Assembled ⎡ 10 ( ) AE0 ⎢ −6 ⎢ 0 3L ⎢ ⎣ 0

non-singular stiffness control matrix: with q0 = 0: KNON = −6 0 0⎤ { }T 24 −6 −7 ⎥ load control vector: P = 0 0 0 1 ; Bezier con⎥ −6 14 −8 ⎥ −7 −8 15 ⎦ ( ) ( ) ( ) 365 513 3 q q q2 ; , q = , q = , q = strain energy: trols: q0 = 0, q4 = 1872 2 4 3 4 1 2701 624 624 5 [ [ ] ]2 2 1 1 U = ∫0 (2 + x) (1 − 2x)q1 + xq2 dx + ∫0 (3 + x) −(1 − x)q2 + (1 − 2x)q3 + xq4 dx = 0.3465383191.

9.2.1.5

Case 5: Two Quadratic Elements – C1 Continuity

Assembled singular stiffness control matrix and C1 continuity transformation matrix: ⎡1 0 0 0⎤ ⎢0 1 0 0⎥ T1 = ⎢ 0 .5 .5 0 ⎥ ⎢ ⎥ ⎢0 0 1 0⎥ ⎣0 0 0 1⎦

KSIN

̂ SIN K

⎡ 9 ̂ SIN T1 = ⎢⎢ −6.5 = TT1 K −2.5 ⎢ ⎣ 0

⎡ 9 −4 ⎢ −4 10 = ⎢ −5 −6 ⎢ 0 ⎢ 0 ⎣ 0 0

−6.5 −2.5 10 0 0 14 −3.5 −11.5

−5 −6 24 −6 −7

0 0 −6 14 −8

0 ⎤ −3.5 ⎥ −11.5 ⎥ ⎥ 15 ⎦

0 ⎤ 0 ⎥ −7 ⎥ ⎥ −8 ⎥ 15 ⎦

Application to Linear Problems and Locking Solutions

463

}T 0 0 0 1 ; assembled non-singular stiffness control 0 −7 ⎤ ) ⎡ 20 ( AE 70 28 −23 ⎥; Bezier controls: q0 = 0q4 = 101 matrix: with q0 = 0: KNON = 6L0 ⎢ 0 q2 = ⎢ ⎥ −7 −23 30 ⎦ ⎣ ( ) ( ) ( ) [ ]2 1 41 23 7 q q q2 ; strain energy: U = ∫0 (2 + x) (1 − 2x)q1 + xq2 dx + q = q = 4 3 4 1 70 28 20 [ ]2 1 ∫0 (3 + x) −(1 − x)q2 + (1 − 2x)q3 + xq4 dx = 0.3465346535.

load control vector: P =

9.2.1.6

{

Case 6: One Cubic Element (

Full singular stiffness control matrix: KSIN =

9AE0 60L

⎡ 14 ) ⎢ −6 ⎢ −5 ⎢ ⎣ −3

singular stiffness control matrix: with q0 = 0: KNON

−6 10 3 −7

⎤ ⎥ ⎥. Non⎥ ⎦ −7 ⎤ −12 ⎥. ⎥ 22 ⎦ energy: U=

−5 −3 3 −7 14 −12 −12 22 3 ) ⎡ 10 ( 9AE 14 = 60L0 ⎢ 3 ⎢ ⎣ −7 −12

99 62 q = 131 q3 q1 = 131 q3 ; strain controls: q0 = 0q3 = 131 189 2 [ ]2 1 1 ∫ (1 + x) 3[−(1 − x)2 q0 + (1 − 4x + 3x2 )q1 + (2x − 3x2 )q2 + x2 q3 ] dx 2 0

Bezier

⇒

U = 0.3465608466.

9.2.1.7

Case 7: Two Cubic Elements_ C0 - Continuity

Assembled non-singular stiffness control matrix:

KNON

⎡ 18 ⎢ ⎢ 5 ) ⎢ −11 ( 3AE = 20L0 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎣ 0

5

−11

0

0

22

−18

0

0

−18

72

−18

−13

0

−18

26

7

0

−13

7

30

0

−7

−15

−24

⎧0⎫ ⎧0 ⎪ ⎪ ⎪ 0.1661 ⎪0⎪ ⎪ ⎪ ⎪ ⎪ 0.2940 ⎪0⎪ ⎪ P = ⎨ ⎬ ⇒ Bezier controls: q = ⎨ 0.4055 ⎪0⎪ ⎪ 0.5164 ⎪0⎪ ⎪ ⎪ ⎪ ⎪ 0.6097 ⎪1⎪ ⎪ ⎩ ⎭ ⎩ 0.6931

strain energy: UElem

1

= 0.2027322404

UElem

2

0 ⎤ ⎥ 0 ⎥ −7 ⎥ ⎥ −15 ⎥ ⎥ −24 ⎥ ⎥ 46 ⎦

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭

= 0.1438410075 U = 0.3465732479 .

Computation of Nonlinear Structures

464

9.2.1.8

Case 8: Two Cubic Elements – C1 Continuity

Assembled non-singular stiffness control matrix:

KNON

⎡ 18 ⎢ 5 ) ⎢ −11 ( 3AE = 20L0 ⎢ ⎢ 0 ⎢ 0 ⎢ ⎣ 0

⎡1 ⎢0 ⎢ 0 T=⎢ ⎢0 ⎢0 ⎢ ⎣0

5 22 −18 0 0 0

−11 −18 72 −18 −13 −7

0 0 −18 26 7 −15

0 0 −13 7 30 −24

P = TT PNON

9.2.1.9

1

PNON

0 0 0 0⎤ 1 0 0 0⎥ ⎥ .5 .5 0 0 ⎥ 0 1 0 0⎥ 0 0 1 0⎥ ⎥ 0 0 0 1⎦

⎡ 2.7 ⎢ −0.075 K = TT KNON T = ⎢ −0.825 ⎢ ⎢ 0 ⎣ 0

strain energy: UElem

0 ⎤ 0 ⎥ ⎥ −7 ⎥ −15 ⎥ −24 ⎥ ⎥ 46 ⎦

−0.075 −0.825 3.3 0 0 3.9 −0.975 0.075 −0.525 −2.775 ⎧0 ⎫ ⎧0⎫ ⎪ 0.1660 ⎪ ⎪0⎪ ⎪ 0.2942 ⎪ ⎪ ⎪ ⎪ ⎪ = ⎨ 0 ⎬ ; q = ⎨ 0.4054 ⎬ ⎪0⎪ ⎪ 0.5166 ⎪ ⎪1⎪ ⎪ 0.6096 ⎪ ⎩ ⎭ ⎪ ⎪ ⎩ 0.6931 ⎭

= 0.2026499611

UElem

2

0 −0.975 0.075 4.5 −3.6

⎧0⎫ ⎪ ⎪ ⎪0⎪ ⎪0⎪ =⎨ ⎬ 0 ⎪ ⎪ ⎪0⎪ ⎪1⎪ ⎩ ⎭

0 ⎤ −0.525 ⎥ −2.775 ⎥ ⎥ −3.6 ⎥ 6.9 ⎦

= 0.1439231038 U = 0.3465730648 .

Case 9: Two Cubic Elements – C2 Continuity: One Cubic B-spline Element

KNON

PNON

⎡ 18 5 −11 0 ⎢ 5 22 −18 0 )⎢ ( 72 −18 3AE0 ⎢ −11 −18 = 20L 0 −18 26 ⎢ 0 ⎢ 0 0 −13 7 ⎢ 0 0 −7 −15 ⎣ ⎧0⎫ ⎡ 1 0 0 0⎤ ⎪ ⎪ ⎢ .5 .5 0 0 ⎥ ⎪0⎪ ⎢ ⎥ ⎪0⎪ .25 .5 .25 0 ⎥ = ⎨ ⎬ TBS = ⎢ 0 ⎢ 0 .5 .5 0 ⎥ ⎪ ⎪ ⎢ 0 0 1 0⎥ ⎪0⎪ ⎢ ⎥ ⎪1⎪ ⎣ 0 0 0 1⎦ ⎩ ⎭

0 0 −13 7 30 −24

0 ⎤ 0 ⎥ ⎥ −7 ⎥ −15 ⎥ −24 ⎥ ⎥ 46 ⎦

Application to Linear Problems and Locking Solutions

KBS

⎡ 3.45 ⎢ 0.375 = T KNON T = ⎢ −0.9 ⎢ ⎣ −0.2625 T

0.375 1.8 0.525 −1.65

⇒ de Boor controls : s = K−1 BS PBS

465

−0.9 0.525 5.55 −4.987

−0.2625 ⎤ −1.65 ⎥ −4.9875 ⎥ ⎥ 6.9 ⎦

PBS =

TTBS PNON

⎧0⎫ ⎪0⎪ =≥ ⎨ ⎬ 0 ⎪ ⎪ ⎩1⎭

⎧0 ⎧ 0 ⎫ ⎪ 0.1658 ⎪ 0.1658 ⎪ ⎪ 0.2944 ⎪ ⎪ ⎪ = ⎨ 0.4230 ⎬ Bezier controls:q = TBS s = ⎨ 0.4054 ⎪ 0.6098 ⎪ ⎪ 0.5164 ⎪ 0.6931 ⎪ ⎪ 0.6098 ⎩ ⎭ ⎪ ⎩ 0.6931

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

U = 0.3465728552 . strain energy: UElem 1 = 0.2026622896 UElem 2 = 0.1439105656 We show in Figures 9.2(a) and (b) the displacement and the strain distribution for various cases; the strains are not as easily satisfied as the displacements.

9.3

c-type Straight Beam Element

Here we present the element level stiffness control matrix and load control vector for an Euler– Bernoulli beam: The Euler–Bernoulli beam equation for the beam, shown in Figure 9.3, is given by: Momentum balance equation: EI w,ssss (s) = 0,

s ∈ (0, L)

Boundary conditions: EI w,sss (0) = P1 , EI w,sss (L) = −P2 ,

(9.4)

EI w,ss (0) = −M1 , EI w,ss (L) = M2 ̄ Now, using the virtual work principle with w(s) ≡ 𝛿w(s) as the virtual displacement, and applying integration by parts twice, we get: L

∫0

L L ̄ ss EI w,ss ds = −w̄ EI w,sss ||0 + w̄ EI w,ss ||0 w,

(9.5)

̄ ̄ ̄ s (0) M1 + w, ̄ s (L) M2 = w(0) P1 + w(L) P2 + w, s L 1 L2

Let us reparameterize the domain as: 𝜉 = d𝜉 𝜕 ds 𝜕𝜉

=

1 𝜕 L 𝜕𝜉

implying w,s =

1 L

w,𝜉 and w,ss =

such that 𝜉 ∈ [0, 1] with ds = L d𝜉 and

𝜕 𝜕s

=

w,𝜉𝜉 . So, equation (9.5) becomes:

L M M EI ̄T P ̄ 𝜉𝜉 w,𝜉𝜉 d𝜉 = w(0) ̄ ̄ ̄ 𝜉 (0) 1 + w, ̄ 𝜉 (1) 2 = w w, P1 + w(1) P2 + w, 3 ∫ L L L 0

(9.6)

Computation of Nonlinear Structures

466

(a)

(b)

Figure 9.2 (a) Example CFEM1: computed and exact displacements for various cases. (b) Example CFEM1: computed and exact strains for various cases.

Figure 9.3 Bernstein–Bezier beam element.

Application to Linear Problems and Locking Solutions

467

where

⎧ w(0) ⎫ ̄ ⎪ ⎪ ̄ 𝜉 (0) ⎪ ⎪ w, ̄ ≡⎨ w ⎬, ̄ ⎪ w(1) ⎪ ⎪ w, ⎪ ̄ (1) ⎩ 𝜉 ⎭

and

⎧ P1 L ⎫ ⎪ ⎪ 1 ⎪ M1 ⎪ P≡ ⎨ L ⎪ P2 L ⎬ ⎪ ⎪M ⎪ ⎩ 2⎭

(9.7)

We now represent both the displacement and the virtual displacement by cubic Bernstein– Bezier functions as:

w(𝜉) =

3 ∑ 0

B3i (𝜉) qi

and

̄ w(𝜉) ≡ 𝛿w(𝜉) =

3 ∑ 0

B3i (𝜉) 𝛿qi

(9.8)

In matrix form, let q = (q0 q1 q2 q3 )T , 𝛿q = (𝛿q0 𝛿q1 𝛿q2 𝛿q3 )T , and from equation (9.8), we get:

Δ

B(𝜉) =

d { 3 B0 d𝜉

B31

B32

B33

}T

= 3{−(1 − 𝜉)2 (1 − 4𝜉 + 3𝜉 2 ) (2𝜉 − 3𝜉 2 ) 𝜉 2 }T ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ 1×4

Δ d2

C(𝜉) =

d𝜉 2

{ 3 B0

B31

B32

B33

}T

= 6{(1 − 𝜉) − (2 − 3𝜉) (1 − 3𝜉) 𝜉}T ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ 1×4

(9.9)

Applying definitions given in equations (9.8) and (9.9) in equations (9.6) and (9.7), and by performing the integration on the left-hand side of the equation (9.6), we get in Bezier controls:

𝛿qT (KB q) = 𝛿qT PB ⎡ 2 ⎢ 6EI −3 where KB = 3 ⎢ L ⎢ 0 ⎢ ⎣ 1

−3 6 −3 0

0

1 ⎤ −3 0 ⎥ ⎥ 6 −3 ⎥ ⎥ −3 2 ⎦

and

⎧ P1 L − 3M1 ⎫ ⎪ ⎪ 3M1 ⎪ 1⎪ PB = ⎨ L ⎪ −3M2 ⎬ ⎪ ⎪ P L + 3M ⎪ 2⎭ ⎩ 2

(9.10)

Computation of Nonlinear Structures

468

(a)

(b)

Figure 9.4 (a) Example CFEM2: Cantilevered Euler–Bernoulli beam. (b) Example CFEM2: Control polygon.

Definition: KB is the unconstrained or singular Bernstein–Bezier beam element stiffness control matrix for an Euler–Bernoulli beam, PB is the Bernstein–Bezier beam element load control vector. Now, with 𝛿q arbitrary, we must have the following c-type finite element equation: KB q = PB ⎡ 2 −3 ⎢ 6EI −3 6 where KB = 3 ⎢ L ⎢ 0 −3 ⎢ ⎣ 1 0

0 −3 6 −3

1 ⎤ 0 ⎥ ⎥, −3 ⎥ ⎥ 2 ⎦

and

⎧ P1 L − 3M1 ⎫ ⎪ ⎪ 3M1 ⎪ 1⎪ PB = ⎨ L ⎪ −3M2 ⎬ ⎪ ⎪ P L + 3M ⎪ 2⎭ ⎩ 2

(9.11)

Remarks: Later, we will use equation (9.11) that describes a Bernstein–Bezier beam element to develop a B-spline beam element. But next let us look at a problem that we can solve in two ways: by directly applying the above element derived by the virtual work principle and by using the minimum potential energy principle. Example CFEM 2: One cubic Bernstein–Bezier element Let us consider the cantilevered Euler–Bernoulli beam, that is, bending in flexure with no shear strain, subjected to a tip load, P, as shown in Figure 9.4(a). Let w(s) be the deflection at any point s ∈ [0, L]; however, for the Bernstein–Bezier approx𝜕 𝜕 𝜕 d𝜉 = 𝜕𝜉 = L1 𝜕𝜉 . Thus, imation, we reparameterize by 𝜉 = Ls so that 𝜉 ∈ [0, 1], d𝜉 = L1 ds, and 𝜕s ds d2 w ds2

≡ w,ss =

1 w, . L2 𝜉𝜉

Now, the potential energy, Π, in flexure exists and is given by: L

Π=

1

1 EI EI(w,ss )2 ds − P w(L) = 3 (w,𝜉𝜉 )2 d𝜉 − P w(1) 2 ∫0 2L ∫0

(9.12)

Applying the minimum potential energy principle, that is, the first variation of the potential energy vanishing to zero, to equation (9.12) with w̄ ≡ 𝛿w as the virtual displacement, and by d ̄ 𝜉𝜉 , we get: ̄ 2𝜉𝜉 ≡ 𝛿(w,2𝜉𝜉 ) = lim d𝜀 {(w + 𝜀𝛿w),𝜉𝜉 }2 = 2w,𝜉𝜉 𝛿w,𝜉𝜉 = 2w,𝜉𝜉 w, Gateaux derivative, w, 𝜀→0

𝛿Π =

L

L

EI EI 𝛿(w,𝜉𝜉 )2 d𝜉 − P 𝛿w(1) = 3 w,𝜉𝜉 𝛿w,𝜉𝜉 d𝜉 − P 𝛿w(1) = 0 2L3 ∫0 L ∫0

(9.13)

Application to Linear Problems and Locking Solutions

469

We choose to approximate both the displacement function, w(𝜉), and the virtual displacement function, 𝛿w(𝜉), by the elements of the same polynomial space: the space of cubic Bernstein polynomials in Bezier representation. Thus,

w(𝜉) =

3 ∑

B3i (𝜉) qi

𝛿w(𝜉) =

and

0

In

matrix

d2 {B30 (𝜉) d𝜉 2

form,

let

3 ∑

B3i (𝜉) 𝛿qi

(9.14)

0

𝛿q = (𝛿q0 𝛿q1 𝛿q2 𝛿q3 )T ,

q = (q0 q1 q2 q3 )T ,

B31 (𝜉) B32 (𝜉) B33 (𝜉)}T = 6{(1 − 𝜉) − (2 − 3𝜉) (1 − 3𝜉) 𝜉}T . ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

and

Then,

Δ

B(𝜉) = applying

1×4

equation (9.14) to equation (9.13) and upon integrating, we get:

𝛿qT (KB q) = 𝛿qT (PB ) ⎡ 2 6EI ⎢ −3 = 3 ⎢ L ⎢ 0 ⎣ 1

where

KB =

−3 0 6 −3 −3 6 0 −3

EI L 3 ∫0

1 ⎤ 0 ⎥ −3 ⎥ ⎥ 2 ⎦

L

(B(𝜉)T B(𝜉)) d𝜉

PB =

and

{

0 0 0

P

}T

(9.15)

Definition: KB is the unconstrained or singular Bernstein–Bezier beam element stiffness control matrix for an Euler–Bernoulli beam, PB is the Bernstein–Bezier beam element load control vector. Now, noting that 𝛿q in equation (9.15) is arbitrary, we must have the finite element equation to solve for q as: ⎡ 2 6EI ⎢ −3 ⎢ L3 ⎢ 0 ⎣ 1

−3 6 −3 0

0 1 ⎤ ⎧0⎫ ⎪0⎪ −3 0 ⎥ q = ⎨ ⎬P 6 −3 ⎥ 0 ⎥ ⎪ ⎪ −3 2 ⎦ ⎩1⎭

(9.16)

With the fixed end essential boundary conditions: w(0) ⇒ q0 = 0, and w,s (s) = 3(q1 − q0 ) = 0 ⇒ q1 = 0, we get, from equation (9.16): [ 6EI 6 L3 −3

−3 2

]{

q2 q3

} =

{ } 0 P

(9.17)

resulting in (see Figure 9.4(b) for the control polygon): {

q2 q3

}

⎧1⎫ PL3 ⎪ 6 ⎪ = ⎬ EI ⎨ ⎪1⎪ ⎩3⎭

(9.18)

Computation of Nonlinear Structures

470

Finally, applying equation (9.18), we get: PL3 (Exact!) 3EI 1 3 PL2 (Exact!) = w,s (1) = w,𝜉 (1) = (q3 − q2 ) = L L 2EI

Tip displacement = w(1) = q3 = Tip slope

(9.19)

Discussion: For this problem, we have arrived at the solution by application of the minimum potential energy principle because the potential energy functional exists for the problem which involves a conservative system. However, not all systems possess a potential energy functional, but all systems do possess a virtual work principle. Thus, we could have used directly the result developed earlier to arrive at the same solution. Thus, in equation (9.10) with: P2 = P and P1 = M1 = M2 = 0, we get:

𝛿qT (KB q) = 𝛿qT PB

where

⎡ 2 6EI ⎢ −3 KB = 3 ⎢ L ⎢ 0 ⎣ 1

−3 6 −3 0

0 1 ⎤ −3 0 ⎥ 6 −3 ⎥ ⎥ −3 2 ⎦

and

⎧0⎫ ⎪0⎪ PB = ⎨ ⎬ 0 ⎪ ⎪ ⎩P⎭ (9.20)

which is exactly same as given by equation (9.15).

9.3.1

Two-node C1 Bernstein–Bezier Element

Here we present the solution for an Euler–Bernoulli beam, as in Figure 9.5, by selecting two two-node Bernstein–Bezier elements, as presented before, with C1 continuity between them. Example CFEM 3: Two cubic Bernstein–Bezier elements with C1 continuity The inter-element boundary condition with imposed moment M makes the beam curvature discontinuous at x = x1 , which in turn allows us only to consider up to C1 inter-element continuity; in fact the non-trivial natural boundary condition is given by: ( ) ( ) EI w,xx x1+ − EI w,xx x1− = −M

Figure 9.5 Example CFEM3: Two beams with junction moment loading.

(9.21)

Application to Linear Problems and Locking Solutions

471

The essential boundary conditions are: w(x0 ) = w(x2 ) = w,x (x2 ) = 0

(9.22)

The virtual work equation for the entire beam, as before, is given by: 1 ∑ i=0

L

∫0

̄ xx EI w,xx w,

| 1  |L 1 ∑ |  || ∑ ̄ xxx |+ ̄ x EI w,xx || w, dx = − wEIw, | i=0 | i=0 | |0  + − ̄ x (x1 ) EI w,xx −w, ̄ x (x1 ) EI w,xx = w,

(9.23)

where we have already accommodated the trivial applied shear condition. Based on the Bernstein– Bezier element stiffness matrix given by equation (9.11), the (7 × 7) C0 -assembled unconstrained Bezier stiffness matrix, K0B , for the entire beam, following the left-hand side of equation (9.23), is given as: 1 0 0 0 ⎤ ⎡ 2 −3 0 ⎢ −3 6 −3 0 0 0 0 ⎥ ⎢ 0 −3 6 −3 0 0 0 ⎥ ⎥ 6EI ⎢ 0 0 −3 4 −3 0 1 ⎥ KB = 3 ⎢ 1 L ⎢ 0 0 0 −3 6 −3 0 ⎥ ⎢ 0 0 0 0 −3 6 −3 ⎥ ⎢ ⎥ 0 0 1 0 −3 2 ⎦ ⎣ 0 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

(9.24)

7×7

Before we can proceed similarly with the right-hand side of equation (9.23) for the assembled load vector, we need to establish the connection between the applied moments and the control vertices; for the first derivative of the Bernstein–Bezier function we have: w,x (x1− ) =

3 (q − q2 ) L 3

and

w,x (x1+ ) =

3 (q − q3 ) L 4

(9.25)

If we apply the slope continuity condition to equation (9.25), that is, w,x (x1 ) = w,x (x1− ) = w,x (x1+ ) ⇒ q3 =

1 (q − q2 ) 2 4

(9.26)

then, for the right-hand side of equation (9.23), we get the assembled Bezier load vector, PB , for the entire beam as: [ 1 P0B = 0 ⏟⏟⏟ L

3 0 − M 2

0

3 M 2

]T 0 0

(9.27)

7×1

and the assembled C0 continuous c-type finite element control matrix equation is given as: K0B q0 = P0B

(9.28)

Computation of Nonlinear Structures

472

Now, for C1 continuity at the junction for the control vertices, we have the transformation relating q0 , the (7 × 1) C0 control vertices with q1 , the (6 × 1) C1 control vertices matrix, T01 BB as:

q0 = T01 BB

⎧ 0⎫ ⎪ q0 ⎪ ⎡ 1 0 0 0 0 0 ⎤ ⎧ 1 ⎫ ⎪ 0⎪ ⎢ ⎥ ⎪ q0 ⎪ ⎪ q1 ⎪ ⎢ 0 1 0 0 0 0 ⎥ ⎪ 1 ⎪ ⎪ 0⎪ ⎢ ⎥ ⎪ q1 ⎪ ⎪ q2 ⎪ ⎢ 0 0 1 0 0 0 ⎥ ⎪ 1 ⎪ ⎪ ⎪ ⎢ ⎥ ⎪ q2 ⎪ q1 , in expanded form: ⎨ q03 ⎬ = ⎢ 0 0 .5 .5 0 0 ⎥ ⎨ ⎬ 1 ⎪ 0⎪ ⎢ ⎥ ⎪ q3 ⎪ 0 0 0 1 0 0 q ⎪ 4⎪ ⎢ ⎥ ⎪ 1⎪ ⎪ 0⎪ ⎢ ⎥ ⎪ q4 ⎪ ⎪ q5 ⎪ ⎢ 0 0 0 0 1 0 ⎥ ⎪ 1 ⎪ ⎪ ⎪ ⎢ ⎥ ⎪ q5 ⎪ ⎪ q06 ⎪ ⎣ 0 0 0 0 0 1 ⎦ ⎩ ⎭ ⎩ ⎭ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏟⏟ ⏟⏟⏟ 6×1 7×6

(9.29)

7×1

Applying equation (9.29) to equation (9.28), we get K1B , the (6 × 6) C1 continuous unconstrained Bernstein–Bezier stiffness control matrix as: ⎡ 2 ⎢ −3 ⎢ 6EI ⎢ .5 K1B = T01T K0B T01 BB BB = L3 ⎢ .5 ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟ ⎢ 0 6×6 6×7 7×7 7×6 ⎢ ⎣ 0

−3 .5 6 −3 −3 4 0 −2 0 0 0 .5

.5 0 −2 4 −3 .5

0 0 ⎤ 0 0 ⎥ ⎥ 0 .5 ⎥ −3 .5 ⎥ 6 −3 ⎥ ⎥ −3 2 ⎦

(9.30)

and P1B , the (6 × 1) Bernstein–Bezier control vector as: 1[ 0 0 −1.5M P1B = T01T P0B = BB ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟ L 6×1

6×7

1.5M

0 0

]T

(9.31)

7×1

with the C1 - continuous unconstrained Bernstein–Bezier equation as: K1B q1 = P1B

(9.32)

Now, we can impose the essential boundary conditions as given in equation (9.22) to control the vertices: q10 = q14 = q15 = 0, so the reduced equation, from equation (9.32), is given as: ⎡ 6 −3 6EI ⎢ ⎢ −3 4 L2 ⎢ ⎣ 0 −2

⎧ 0 ⎫ 0 ⎤ ⎧ q11 ⎫ ⎥ ⎪ 1 ⎪ 3M ⎪ ⎪ −1 ⎬ −2 ⎥ ⎨ q2 ⎬ = 2 ⎨ ⎪ ⎪ ⎪ ⎥ ⎪ 1 4 ⎦ ⎩ q3 ⎭ ⎩ +1 ⎭

(9.33)

Application to Linear Problems and Locking Solutions

473

which upon solution for the C1 control vertices, and application of the transformation relating the C1 to the C0 control vertices, as given by equation (9.29), gives: ⎧ q0 ⎫ ⎧ 0 ⎫ ⎪ 0⎪ ⎪ ⎪ ⎪ q01 ⎪ ⎪ −4 ⎪ ⎪ 0⎪ ⎪ ⎪ ⎪ q2 ⎪ ⎪ −8 ⎪ ⎪ 0 ⎪ ML2 ⎪ ⎪ ⎨ q3 ⎬ = 96EI ⎨ −3 ⎬ ⎪ 0⎪ ⎪ 2 ⎪ ⎪ q4 ⎪ ⎪ ⎪ ⎪ q0 ⎪ ⎪ 0 ⎪ ⎪ 5⎪ ⎪ ⎪ ⎪ q0 ⎪ ⎪ 0 ⎪ ⎩ 6⎭ ⎩ ⎭

(9.34)

With the help of the Bernstein–Bezier functions and the computed control vertices as shown in equation (9.34), we can compute the displacements and slopes along the beam: w(x0 ) = q0 = 0 3ML2 96EI 2 3ML w(x1+ ) = q3 = − 96EI

w(x1− ) = q3 = −

w(x2 ) = q6 = 0

1 3 12ML q w, (0) = (q1 −  0) = − L 𝜉1 L 96EI 1 3 15ML q w,x (x1− ) = w,𝜉1 (1) = (q3 −  2) = L L 96EI 1 3 15ML w,x (x1+ ) = w,𝜉2 (0) = (q4 − q3 ) = L L 96EI 1 3 w,x (x2 ) = w,𝜉2 (0) = (q6 − q5 ) = 0 L L w,x (x0 ) =

(9.35)

We present the exact deformation as indicated by equation (9.35) graphically as shown in Figure 9.6. Now, for the force and moment distribution, let us first note the second (curvature) and third derivatives of the Bernstein–Bezier polynomials: w,𝜉1 𝜉1 = 6{(1 − 𝜉1 )(q2 − 2q1 +  q 0 ) + 𝜉1 (q3 − 2q2 + q1 )},

𝜉1 L ∈ [x0 , x1 ]

w,𝜉1 𝜉1 𝜉1 = 6{3q1 − 3q2 + q3 } q q q w,𝜉2 𝜉2 = 6{(1 − 𝜉2 )( 5 − 2q4 + q3 ) + 𝜉2 ( 6 − 2 5 + q4 )},

𝜉2 L ∈ [x1 , x2 ]

w,𝜉2 𝜉2 𝜉2 = 6{−q3 + 3q4 }

Figure 9.6 Example CFEM3: Exact deformed shape.

(9.36)

Computation of Nonlinear Structures

474

Figure 9.7 Example CFEM3: Exact force and moment distribution.

Using the expressions in equation (9.36), we can determine the force and moment distributions as: EI w,𝜉𝜉 (0) = 0 L2 EI 9M M(x1− = L− ) = EIw,xx (x1− ) = 2 w,𝜉𝜉 (𝜉1 = 1) = 16 L EI 7M + + + M(x1 = L ) = EIw,xx (x1 ) = 2 w,𝜉𝜉 (𝜉2 = 0) = − 16 L EI 2M M(x2 = 2L) = EIw,xx (x2 ) = 2 w,𝜉𝜉 (𝜉2 = 1) = 16 L EI 9M , x ∈ [0, 2L] F(x) = EIw,xxx (x) = 3 w,𝜉𝜉𝜉 (𝜉) = 16L L

M(x0 = 0) = EIw,xx (x0 ) =

(9.37)

We present the exact force and moment distributions as indicated by equation (9.37) graphically as a free-body diagram shown in Figure 9.7. Exercise: Apply two cubic Bernstein–Bezier elements with C1 continuity as before to solve the problem shown with the exact solution in Figure 9.8. Remark

r Let us note that two cubic Bernstein–Bezier elements assembled with C1 continuity reproduced the exact deformations, forces and moments; imposition of any additional continuity such as C2 continuity will spell disaster because the curvature is obviously discontinuous at the junction where a moment has been applied. However, we can refine our elements by considering B-spline elements, which we consider next.

9.3.2

Two-node B-spline Beam: Composite Bernstein–Bezier Form

We may recall from our discussion in Chapter 5 on curves that a cubic B-spline can be constructed by two piecewise cubic Bernstein–Bezier curves joined together by C2 connectivity. Let us

Figure 9.8 Exercise problem.

Application to Linear Problems and Locking Solutions

475

Figure 9.9 B-spline element in Bernstein–Bezier composite form.

consider the free-body of an Euler–Bernoulli beam, that is, bending in flexure with no shear strain, as shown in Figure 9.9. Let the control points of the Bezier element # 1 be given by: {q0 , q1 , q2 , q3 }, and for the Bezier element # 2: {q4 , q5 , q6 , q7 }, and for the resulting B-spline controls: {s0 , s1 , s2 , s3 , s4 }. Recalling that k = 4, with k = n + 1 where n, k are the degree and the order of a cubic curve, respectively, the original two cubic Bezier curves have lk = 2 × 4 = 8 controls where l = 2 is the number of Bezier segments; the B-spline, correspondingly, has: l + k − 1 = l + n = 2 + 3 = 5 control points which may be explained as: C0 continuity at the junction amounting to one constraint, and thus reduces the Bezier control number to l(k − 1) + 1 = 2 × 3 + 1 = 7; C1 continuity adds one extra constraint, reducing the number to 7 − 1 = 6. Finally, C2 continuity gives the number for the B-spline controls as 6 − 1 = 5 (which could have been directly obtained for B-spline controls as indicated earlier and as shown in Figure 9.10: Let us now recapture the effect of the constraints due and up to C2 - continuity, described before in Chapter 5 on curves, as follows: q0 = s0

q1 = s1

q2 =

h0 h1 s + s h0 + h1 1 h0 + h1 2

(C2 continuity)

q3 =

h1 q2 + h0 q4 h0 + h1

(junction pt. C1 continuity)

q4 =

h0 h1 s2 + s h0 + h1 h0 + h1 3

(C2 continuity)

q5 = s3

q1 = s4

Figure 9.10 Schematic description of two cubics to form one B-spline element.

(9.38)

Computation of Nonlinear Structures

476

We can write the transformation, given by equation (9.38), relating the Bezier controls to the B-spline controls in matrix form as: q = TBS s ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟ 7×5

7×1

(9.39)

5×1

where in expanded form: ⎧q ⎫ ⎡1 0 0 0 0⎤ ⎪ 0⎪ ⎢ ⎥ ⎪ q1 ⎪ ⎢ 0 1 0 0 0 ⎥ ⎧ s0 ⎫ ⎪ ⎪ ⎢ ⎪ ⎪ 𝛽0 0 0 ⎥⎥ ⎪ s1 ⎪ ⎪ q2 ⎪ ⎢ 0 𝛽1 ⎪ ⎪ ⎢ ⎪ ⎪ 2 2 ⎨ q3 ⎬ = ⎢ 0 𝛽1 2𝛽1 𝛽0 𝛽0 0 ⎥⎥ ⎨ s2 ⎬ ⎪q ⎪ ⎢ ⎪ ⎪ 𝛽1 𝛽0 0 ⎥ ⎪ s 3 ⎪ ⎪ 4 ⎪ ⎢0 0 ⎥ ⎪q ⎪ ⎢0 0 ⎪ 0 1 0⎥ ⎪ ⎩ s4 ⎭ ⎪ 5⎪ ⎢ ⎥ ⎪ q6 ⎪ ⎣ 0 0 0 0 1 ⏟⏟⏟ ⎩ ⎭ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⎦ 5×1 ⏟⏟⏟

(9.40)

7×5

7×1

With Δ

𝛽0 =

h0 h0 + h1

and

Δ

𝛽1 =

h1 h0 + h1

(9.41)

where for the interval length of each of the Bezier beam segments, we have: h0 = h1 = L2 . Then we get, the transformation matrix, TBS , from B-spline controls, s, to Bezier controls, q, as: ⎡1 ⎢ ⎢0 ⎢ ⎢ TBS = ⎢ 0 ⏟⏟⏟ ⎢ ⎢ 7×5 ⎢0 ⎢ ⎢0 ⎣

0 1 0

0

0

1 2 1 2

1 4 1 2 1 4 0

0

0

0

0

0 0 1 2 1 2 0

T

0 0⎤ ⎥ 0 0 ⎥⎥ ⎥ 0 0⎥ ⎥ ⎥ 1 0⎥ ⎥ 0 1 ⎥⎦

(9.42)

L∕2

Now, from equation (9.10), the unconstrained Bezier stiffness control matrix, KB , modified for 12 L long flexural beam, is given by:

L∕2 KB

⎡ 2 48EI ⎢ −3 = 3 ⎢ L ⎢ 0 ⎣ 1

−3 6 −3 0

0 1 ⎤ −3 0 ⎥ 6 −3 ⎥ ⎥ −3 2 ⎦

(9.43)

Application to Linear Problems and Locking Solutions

477

Thus, the (7 × 7) C0 -assembled unconstrained Bezier stiffness matrix, KB , for the entire beam is given as:

1 0 0 0 ⎤ ⎡ 2 −3 0 ⎢ −3 6 −3 0 0 0 0 ⎥ ⎢ 0 −3 6 −3 0 0 0 ⎥ ⎥ 48EI ⎢ 0 −3 4 −3 0 1 ⎥ KB = 3 ⎢ 1 L ⎢ 0 0 0 −3 6 −3 0 ⎥ ⎢ 0 0 0 0 −3 6 −3 ⎥ ⎢ ⎥ 0 0 1 0 −3 2 ⎦ ⎣ 0 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

(9.44)

7×7

L∕2

Similarly, from equation (9.10), the Bezier load control vector, PB , modified for 12 L long flexural beam, is given by:

L∕2

PB

=

1[ P L − 6M1 L 1

6M1

−6M2

P2 L + 6M2

]T

(9.45)

Thus, the (7 × 7) C0 -assembled Bezier load vector, PB , for the entire beam, with junction load and moments identically zero, is given as: 1[ P1 L − 6M1 PB = ⏟⏟⏟ L

6M1

0

0

0 −6M2

P2 L + 6M

]T

(9.46)

7×1

Thus, for the beam, we have the C0 -assembled Bezier element equation as:

KB q = PB

⎧ P L − 6M ⎫ 0 ⎤ 1⎪ ⎪ 1 ⎥ ⎪ ⎪ 6M1 6 −3 0 0 0 0 ⎥ ⎪ ⎪ ⎥ ⎪ ⎪ 0 −3 6 −3 0 0 0 ⎥ ⎪ ⎪ 1 0 0 −3 4 −3 0 1 ⎥ and PB = ⎨ ⎬ ⎥ L ⏟⏟⏟ ⎪ ⎪ 0 0 0 −3 6 −3 0 ⎥ 7×1 ⎪ ⎪ ⎥ ⎪ ⎪ −6M 0 0 0 −3 6 −3 ⎥ 2 ⎪ ⎪ ⎥ ⎪ P2 L + 6M2 ⎪ 0 0 1 0 −3 2 ⎦ ⎭ ⎩ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

⎡ 2 ⎢ ⎢ −3 ⎢ 0 ⎢ 48EI where KB = 3 ⎢ 1 L ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎣ 0

−3

0

1

0

0

7×7

(9.47)

Computation of Nonlinear Structures

478

Now, by pre-multiplying equation (9.47) by TTBS and applying the transformation equation, we finally get the c-type equation for the two-node B-spline flexural element: KS s = PS ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟ 5×5

where

5×1

(9.48)

5×1

KS is the unconstrained or singular B-spline element stiffness control matrix given as: ⏟⏟⏟ 5×5

⎡ 8 ⎢ −11 ⎢ 12EI KS = TTBS KB TBS = 3 ⎢ 2 L ⎢ ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟ ⎢ 1 7×7 5×5 7×5 5×7 ⎢ ⎣ 0

and

−11

2

1

16

−4

−2

−4

4

−4

−2

−4

16

1

2

−11

0 ⎤ 1 ⎥⎥ 2 ⎥ ⎥ −11 ⎥ ⎥ 8 ⎦

(9.49)

PS is the B-spline element load control vector given as: ⏟⏟⏟ 5×1

1[ P1 L − 6M1 PS = TTBS PB = ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟ L 5×1

5×7

6M1

0 −6M2

P2 L + 6M2

]T

(9.50)

7×1

Example CFEM 4: Two cubic B-spline elements with C1 connectivity Let us consider the beam in flexure with no shear strain as shown in Figure 9.11. Let the de Boor control polygon: {s00 , s01 , s02 , s03 , s04 } describes the B-spline element # 1, and the B-spline element # 2 is described by: {s10 , s11 , s12 , s13 , s14 }. Using the B-spline stiffness matrix equation (9.48) and applying direct stiffness assembly to two B-spline elements with P1 = 0, we get, for the entire system, the C0 control polygon, {s0 , s1 , s2 , s3 , s4 , s5 , s6 , s7 , s8 }, stiffness

Figure 9.11 Example CFEM4: Two B-splines with C1-junction for shear and moment.

Application to Linear Problems and Locking Solutions

matrix,

479

K0S , and load vector, P0S , related by: ⏟⏟⏟ ⏟⏟⏟ 9×9

9×1

K0S s0 = P0S ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟ 9×1

9×9

⎡ 8 ⎢ −11 ⎢ ⎢ 2 ⎢ ⎢ 1 ⎢ ⎢ 0 ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎣ 0

9×1

−11

2

1

0

0

0

0

16

−4

−2

1

0

0

0

−4

4

−4

2

0

0

0

−2

−4

16

−11

0

0

0

1

2

−11

16

−11

2

1

0

0

0

−11

16

−4

−2

0

0

0

2

−4

4

−4

0

0

0

1

−2

−4

16

0

0

0

0

1

2

−11

0 ⎫ ⎧ 0 ⎤ ⎪ s0 = s0 ⎪ ⎧ −6M1 ⎫ s1 = s01 ⎪ ⎪ 6M1 ⎪ 0 ⎥⎥ ⎪ ⎪ ⎪ ⎪ ⎪ 0 0 ⎥ ⎪ s2 = s2 ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎥⎪ 0 ⎥ ⎪ s3 = s03 ⎪ ⎪ −3M2 ⎪ ⎪ ⎪ ⎪ ⎥⎪ 0 ⎥ ⎨ s4 = s04 = s10 ⎬ = ⎨ P2 L ⎬ ⎪ ⎪ ⎪ 1 1 ⎥⎥ ⎪ ⎪ s5 = s1 ⎪ ⎪ 3M2 ⎪ 2 ⎥ ⎪ s6 = s1 ⎪ ⎪ 0 ⎪ 2 ⎪ ⎪ ⎪ ⎥⎪ −11 ⎥ ⎪ s7 = s1 ⎪ ⎪ 0 ⎪ 3 ⎪ ⎪ ⎪ ⎥⎪ 16 ⎦ ⎪ s = s1 ⎪ ⎩ 0 ⎭ 8 4 ⎭ ⎩

(9.51) Now, imposing the C1 continuity matrix, T01 – with s4 = 12 (s3 + s5 ) and thus eliminating SS control vertex, s4 – to equation (9.51), similar to the previous example, we get the corresponding C1 control polygon, s1 = {s0 , s1 , s2 , s3 , s5 , s6 , s7 , s8 }T , stiffness matrix, K1S = ⏟⏟⏟ 8×8

T01T K0S T01 P1S = T01T P0S , related by: SS SS , and load vector, SS ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟ 8×9

9×9

8×1

9×8

8×9

9×1

K1S s1 = P1S ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟ 9×1

9×9

⎡ 8 ⎢ ⎢ −11 ⎢ ⎢ 2 ⎢ 12EI ⎢ 1 L2 ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ 0 ⎣

9×1

−11

2

1

0

0

0

16

−4

−1.5

.5

0

0

−4

4

−3

1

0

0

9

−7

1

.5

−1.5 −3 .5

1

−7

9

−3

−1.5

0

0

1

−3

4

−4

0

0

.5

−1.5

−4

16

0

0

0

1

2

−11

⎫ ⎧ ⎫ ⎧ 0 ⎤ ⎪ s0 ⎪ ⎪ −6M1 ⎪ ⎥ ⎪ 6M1 0 ⎥ ⎪ s1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎪ 0 0 ⎥ ⎪ s2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎥ ⎪ s3 ⎪ ⎪ .5P2 L − 3M2 ⎪ ⎥⎨ ⎬ = ⎨ ⎬ 1 ⎥ ⎪ s5 ⎪ ⎪ .5P2 L + 3M2 ⎪ ⎥ ⎪ 0 2 ⎥ ⎪ s6 ⎪ ⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎪ 0 −11 ⎥ ⎪ s7 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 8 ⎥⎦ ⎪ s8 ⎪ ⎪ ⎪ ⎭ ⎩ ⎭ ⎩ (9.52)

Computation of Nonlinear Structures

480

Applying the essential boundary conditions: s0 = 0 for the simply supported end, and s7 = s8 = 0 for the fixed end to equation (9.52), we get:

⎡ 16 ⎢ −4 ⎢ 12EI ⎢ −1.5 L2 ⎢ ⎢ .5 ⎢ ⎣ 0

−4

−1.5

4

−3

−3

9

1

−7

0

1

⎧1⎫ ⎧ 0 ⎫ ⎧0⎫ 0 ⎤ ⎧ s1 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪0⎪ ⎪ 0 ⎪ ⎪0⎪ 1 0 ⎥⎥ ⎪ s2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎥ −7 1 ⎨ s3 ⎬ = P2 L ⎨ 1 ⎬ − 6M1 ⎨ 0 ⎬ + 3M2 ⎨ −1 ⎬ ⎥ 2 ⎪0⎪ ⎪ 1 ⎪ ⎪1⎪ s5 ⎪ 9 −3 ⎥ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪0⎪ ⎪ 0 ⎪ ⎪0⎪ −3 4 ⎦ ⎩ s6 ⎭ ⎩ ⎭ ⎩ ⎭ ⎩ ⎭ .5

(9.53) We can solve for the control vertices in equation (9.53) to get: ⎧ s1 ⎫ ⎪s ⎪ ⎪ 2⎪ L2 ⎨ s3 ⎬ = 96EI ⎪ s5 ⎪ ⎪s ⎪ ⎩ 6⎭

⎧8⎫ ⎧ −2 ⎫⎤ ⎧ 2 ⎫ ⎡ ⎪ 16 ⎪ ⎪ −6 ⎪⎥ ⎪ 6 ⎪ ⎢ ⎪ ⎪ ⎪ ⎪ ⎢ ⎪⎥ ⎪ ⎢P2 L ⎨ 7.5 ⎬ − M1 ⎨ 14 ⎬ + 3M2 ⎨ −1.5 ⎬⎥ ⎪ 10 ⎪ ⎪ −.5 ⎪⎥ ⎪ 6.5 ⎪ ⎢ ⎪4⎪ ⎪ 1 ⎪⎥ ⎪ 3 ⎪ ⎢ ⎣ ⎩ ⎭ ⎩ ⎩ ⎭⎦ ⎭

(9.54)

and additionally, from the C1 continuity condition, we get:

s4 =

1 L2 (s3 + s5 ) = {7P2 L − 12M1 − 3M2 } 2 96EI

(9.55)

We can now use Bernstein–Bezier and B-spline connection as given in equations (9.39) and (9.42), into equations (9.54) and (9.55) to describe each element in local Bernstein–Bezier control vertices: Element #1: Bernstein–Bezier control polygon for displacements and rotations ⎧ q1 ⎫ ⎪ 0⎪ ⎪ q11 ⎪ ⎪ 1⎪ ⎪ q2 ⎪ ⎪ 1⎪ L2 ⎨ q3 ⎬ = 96EI ⎪ 1⎪ ⎪ q4 ⎪ ⎪ q1 ⎪ ⎪ 5⎪ ⎪ q1 ⎪ ⎩ 6⎭

⎧ 0 ⎫ ⎧ 0 ⎫ ⎧ 0 ⎫⎤ ⎡ ⎪ ⎪ ⎪ ⎪ ⎪⎥ ⎢ ⎪ ⎪ 2 ⎪ ⎪ 8 ⎪ ⎪ −2 ⎪⎥ ⎢ ⎪ ⎪ ⎪ ⎪ ⎪⎥ ⎢ ⎪ ⎪ 4 ⎪ ⎪ 12 ⎪ ⎪ −4 ⎪⎥ ⎢ ⎪ 3⎪ ⎪ 1⎪ ⎪ 7 ⎪⎥ ⎢ ⎢P2 L ⎨ 5 8 ⎬ − M1 ⎨ 13 2 ⎬ + M2 ⎨ −4 8 ⎬⎥ ⎪61 ⎪ ⎪ ⎪ −5 3 ⎪⎥ ⎢ ⎪ ⎪ 4⎪ ⎪ 15 ⎪ ⎪ 4 ⎪⎥ ⎢ ⎪ 1⎪ ⎪ 14 ⎪ ⎪ 1 ⎪⎥ ⎢ 7 ⎪ 2⎪ ⎪ ⎪ −5 2 ⎪⎥ ⎢ ⎪ ⎪ 7 ⎪ ⎪ 12 ⎪ ⎪ −3 ⎪⎥ ⎢ ⎣ ⎩ ⎭ ⎩ ⎩ ⎭⎦ ⎭

(9.56)

Application to Linear Problems and Locking Solutions

481

Element #2: Bernstein–Bezier control polygon for displacements and rotations

⎧ q1 ⎫ ⎪ 0⎪ ⎪ q11 ⎪ ⎪ 1⎪ ⎪ q2 ⎪ ⎪ 1⎪ L2 ⎨ q3 ⎬ = 96EI ⎪ 1⎪ ⎪ q4 ⎪ ⎪ q1 ⎪ ⎪ 5⎪ ⎪ q1 ⎪ ⎩ 6⎭

9.3.2.1

⎧ 0 ⎫ ⎧ 0 ⎫ ⎧ 0 ⎫⎤ ⎡ ⎪ ⎪ ⎪ ⎪ ⎪⎥ ⎢ ⎪ ⎪ 2 ⎪ ⎪ 8 ⎪ ⎪ −2 ⎪⎥ ⎢ ⎪ ⎪ ⎪ ⎪ ⎪⎥ ⎢ ⎪ ⎪ 4 ⎪ ⎪ 12 ⎪ ⎪ −4 ⎪⎥ ⎢ ⎪ 3⎪ ⎪ 1⎪ ⎪ 7 ⎪⎥ ⎢ ⎢P2 L ⎨ 5 8 ⎬ − M1 ⎨ 13 2 ⎬ + M2 ⎨ −4 8 ⎬⎥ ⎪61 ⎪ ⎪ ⎪ −5 3 ⎪⎥ ⎢ ⎪ ⎪ 4⎪ ⎪ 15 ⎪ ⎪ 4 ⎪⎥ ⎢ ⎪ 1⎪ ⎪ 14 ⎪ ⎪ 1 ⎪⎥ ⎢ ⎪72 ⎪ ⎪ ⎪ −5 2 ⎪⎥ ⎢ ⎪ ⎪ 7 ⎪ ⎪ 12 ⎪ ⎪ −3 ⎪⎥ ⎢ ⎣ ⎩ ⎭ ⎩ ⎩ ⎭⎦ ⎭

(9.57)

Moment Distributions

Let us first recall that each of the B-spline elements is composed of two Bernstein–Bezier elements; thus, for the second half, that is, x ∈ ( L2 , L], of the B-spline element #1, the curvature ( )2 1 w,𝜉1 𝜉1 , and we get: is given by: w,xx = L∕2 Case M1 : Moment distribution )( ) ) M1 L2 4 { ( 1 I M(0) = EIw,xx (x1 = 0) =  6 q2 − 2q11 + q10 = −M1 E I 96 E L2 ( )( ) ( ) ) ( ( ) ) M1 L 2 L 5 L 4 { ( 1 I = EIw,xx x1 = = M 6 q3 − 2q12 + q11 = − E M1 2 2 2 8 I 96 E L )( ) ( { ( ) ( ) M1 L 2 1 4 − − I M1 6 q14 − 2q15 + q16 = − M(L ) = EIw,xx (x1 = L ) =  E 2 4 I 96 E L (

(

)(

Curvature matched @x = L )

( ) ) M1 L 2 4 { ( 2 1 M1 6 q0 − 2q21 + q22 = − 2 4 I 96 E L ( )( ) { ( ) ( ) M1 L 2 1 4 I 6 q24 − 2q25 + q26 = M1 M(2L) = EIw,xx (x2 = L) =  E 8 I 96 E L2 I M(L+ ) = EIw,xx (x2 = 0+ ) =  E

(9.58)

Computation of Nonlinear Structures

482

Case M2 : Moment distribution Similarly, we get: )( ) ) M2 L2 4 { ( 1 I M(0) = EIw,xx (x1 = 0) =  6 q2 − 2q11 + q10 = 0 E I 96 E L2 ( )( ) ) ( ( ) ) (9) M2 L 2 L L 4 { ( 1 I = EIw,xx x1 = = M2 M 6 q3 − 2q12 + q11 = E 2 2 32 I 96 E L2 )( ) ( { ( ) ( ) 2 L M 9 4 2 I 6 q14 − 2q15 + q16 = M2 M(L− ) = EIw,xx (x1 = L− ) =  E 2 16 I 96 E L (

(

)(

Curvature discontinuous @x = L )

) (7) M2 L 2 4 { ( 2 M2 6 q0 − 2q21 + q22 = 16 I 96 E L2 )( ) ( { ( ) ( ) M2 L 2 1 4 I 6 q24 − 2q25 + q26 = M2 M(2L) = EIw,xx (x2 = L) =  E 2 8 I 96 E L I M(L+ ) = EIw,xx (x2 = 0+ ) =  E

(9.59)

Case P2 : Moment distribution Similarly, we get: )( ) ) P2 L3 4 { ( 1 I M(0) = EIw,xx (x1 = 0) =  6 q3 − 2q12 + q11 = 0 E I 96 E L2 ( )( ) ( ) ) ( ( ) ) M2 L 2 L 5 L 4 { ( 1 I = EIw,xx x1 = = 6 q3 − 2q12 + q11 = − E P2 L M 2 2 2 32 I 96 E L ( ) )( ){ ( ( ) P2 L3 5 4 1 1 1 − − I 6 q4 − 2q5 + q6 = − P2 L M(L ) = EIw,xx (x1 = L ) =  E 16 I 96 E L2 Curvature matched @x = L ) ( ) ) ( ( ) P2 L3 5 4 { ( 2 I M(L+ ) = EIw,xx (x2 = 0∗ ) =  6 q0 − 2q21 + q22 = − P2 L E 2 16 I 96 E L )( ){ ( ( ) ( ) P2 L3 3 4 I 6 q24 − 2q25 + q26 = P2 L M(2L) = EIw,xx (x2 = L) =  E 2 8 I 96 E L (

(9.60)

Application to Linear Problems and Locking Solutions

9.3.2.2

483

Shear Distributions

Let us recall again that each of the B-spline elements are composed of two Bernstein–Bezier ( )3 1 elements; thus, for each Bezier element, the shear is constant is given by: w,xxx = L∕2 w,𝜉1 𝜉1 𝜉1 = 8 w, ; L3 𝜉1 𝜉1 𝜉1

let the shear be denoted as V(x), then, for moment loading:

Applied moment: V(x) = EIw,xxx =

) ( )( 8 ML2 8 w, = 6{q3 − 3q2 + 3q1 − q0 } 𝜉 𝜉 𝜉 96EI L3 1 1 1 L3

M {q − 3q2 + 3q1 − q0 } 2L 3 Applied force: ( )( 3 ) 8 PL 8 6{q3 − 3q2 + 3q1 − q0 } V(x) = EIw,xxx = 3 w,𝜉1 𝜉1 𝜉1 = 96EI L L3 P = {q3 − 3q2 + 3q1 − q0 } 2 =

(9.61)

Case M1 : Shear distribution } ( )M M1 { 27 3 1 − 36 + 24 − 0 = 2L 2 4 L } ( )M M { 27 3 1 = V(L− ) = EIw,xxx (x1 = L− ) = 1 −12 + 42 − 45 + 2L 2 4 L } ( )M M {9 3 1 − 21 + 30 − 12 = V(L+ ) = EIw,xxx (x2 = 0+ ) = 1 2L 2 4 L } ( )M M { 9 3 1 V(2L) = EIw,xxx (x2 = L) = 1 − + 21 − 30 + 12 = − 2L 2 4 L

V(0) = EIw,xxx (x1 = 0) =

(9.62)

Case M2 : Shear distribution Similarly, we get: } ( )M M2 { 39 9 2 + 12 − 6 − 0 = 2L 8 16 L } ( )M M { 33 69 39 9 2 = V(L− ) = EIw,xxx (x1 = L− ) = 2 −3 + − + 2L 2 4 8 16 L } ( )M M {3 3 3 9 2 V(L+ ) = EIw,xxx (x2 = 0+ ) = 2 − − +3 = 2L 8 4 2 16 L } ( )M M { 3 3 9 2 V(2L) = EIw,xxx (x2 = L) = 2 − + − 0 + 0 = 2L 8 2 16 L

V(0) = EIw,xxx (x1 = 0) =

(9.63)

Computation of Nonlinear Structures

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Case P2 : Shear distribution Similarly, we get: ( ) } P2 { 43 5 V(0) = EIw,xxx (x1 = 0) = − 12 + 6 − 0 = − P2 2 8 16 { } ( ) P2 45 8 43 5 − − V(L ) = EIw,xxx (x1 = L ) = 7− + − =− P2 2 2 4 8 16 { } ( ) P 25 57 39 11 P2 V(L+ ) = EIw,xxx (x2 = 0+ ) = 2 − + − +7 =− 2 8 4 2 16 { } ( ) P2 9 25 11 P2 V(2L) = EIw,xxx (x2 = L) = 0−0+ − =− 2 2 8 16

(9.64)

Remarks

r It can be easily verified that all distributions – displacement, rotation, shear and moment – are exact.

r From the free-body diagram, it can be seen that the equilibrium is preserved everywhere. 9.4

c-type Curved Beam Element

In the following example, we demonstrate imposition of the inter-element continuity conditions. In so doing, we show improvement in accuracy by introduction of a hierarchy of our c-type elements. It will also be shown constructively that lack of adequate inter-element continuity, otherwise required by the degree of the associated differential equation, spells disaster.

9.4.1

3D Linear Curved Beams with Natural Twist and Bend

For linear problems, as shown in Figure 9.12, the undeformed basis set, t, m, b, and the deformed basis set, n, mn , bn , are identical. It is assumed here that the curved beams are embedded in the general three-dimensional space with principal axes in the cross-sections aligned along the m and b directions. Let the generalized displacement and rotation components at the centroidal axis

Figure 9.12 Curved beam: Frenet frame for geometry and shear deformed frame for c-type FEM.

Application to Linear Problems and Locking Solutions

485

along t, m, b axes be given by u(𝜉), v(𝜉), w(𝜉), 𝜃t (𝜉) 𝜃m (𝜉) 𝜃b (𝜉), as functions only of 𝜉 ∈ [0, 1]. Under these assumptions and neglecting the warping effect, the Green’s strain components at any point (m, b) of the cross-section at 𝜉 are given by: emm = ebb = emb = 0 √ gett = 𝜀t + b𝜛m − m𝜛b √ 2 getm = 𝜀m − 𝜃b − b𝜛s √ 2 getb = 𝜀b + 𝜃m + m𝜛s where,

(9.65)

√ g = 1 − m𝜅 with g as the determinant of the metric tensor of the geometry, and 𝜀m = u,s −𝜏v + 𝜅w

𝜒m = 𝜃m ,s −𝜏𝜃b + 𝜅𝜃t

𝜀b = v,s +𝜏u

𝜒b = 𝜃b ,s +𝜏𝜃m

𝜀t = w,s −𝜅u

𝜒t = 𝜃t ,s −𝜅𝜃m

(9.66)

where, (∙),s = (∙),𝜉 ∕J. For comparison with published results in the literature, some of the following examples are planar curves and circular in shape with the plane of the circle coinciding with the osculating plane. With the additional assumption of cross-section symmetry, planar applied load and m𝜅 ≪ 1, we can reduce the above expressions by setting: 𝜏 = 0,

𝜃m = 𝜃t = 0,

v = 0, 𝜓 = 0,

𝜃b ≡ 𝜃 √ etb = 0, g≃1

(9.67)

If we denote, for simplicity, the uniform axial strain as: 𝜀 ≡ ett , bending strain as: 𝜒 = 𝜃,s and the shear strain as: 𝛾 = 2etm , we get: u 𝜀 = w,s − , R

𝛾 = u,s +

𝜒 = 𝜃,s ,

w −𝜃 R

(9.68)

For inextensional circular beam with no shear strain deformation, we have further simplification for the bending strain which now depends only on the radial displacement and is given by: 𝜒 = w,ss +

w R

(9.69)

Likewise, let Nt , Qm , Qb , Mt , Mm , Mb be the corresponding generalized axial force along the t axis, transverse shear forces in the m and b directions, torsional moment about the t axis, and bending moments about the m and b axes, respectively. The constitutive relations for a beam element are then given as: ⎧ N ⎫ ⎡ EA 0 ⎪ t ⎪ ⎢ ⎪ Qb ⎪ ⎢ 0 kG ⎪ Qm ⎪ ⎢ 0 0 ⎨M ⎬=⎢ 0 0 t ⎪ ⎪ 0 ⎪ Mb ⎪ ⎢⎢ 0 ⎪ Mm ⎪ ⎣ 0 0 ⎩ ⎭

0 0 kG 0 0 0

0 0 0 GJ 0 0

0 0 0 0 EIb 0

0 ⎤⎧ ⎪ 0 ⎥ ⎪ 𝛾tb ⎥⎪ 0 ⎥ 𝛾tm 0 ⎥⎨ ⎪ 0 ⎥⎪ ⎥⎪ EIm ⎦ ⎩

⎫ ett ⎪ = 2etb ⎪ = 2etm ⎪ ⎬ 𝜒t ⎪ 𝜒b ⎪ ⎪ 𝜒m ⎭

(9.70)

Computation of Nonlinear Structures

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where, E, G, A, Ib , Im and J are the usual elasticity and shear moduli, cross-sectional area, moments of inertia about the b and m axes and polar moment of inertia, respectively; k is the shear factor. In the absence of body forces and surface traction, the equilibrium equations involving the above stress resultants and stress couples, obtained by the application of the virtual work principle, are given as: Nt ,s − 𝜅Qm = 0 Qm ,s + 𝜅Nt − 𝜏Qb = 0 Qb ,s + 𝜏Qm = 0

(9.71)

Mt ,s − 𝜅Mm = 0 Mm ,s + 𝜅Mt − 𝜏Mb − Qb = 0 Mb ,s − 𝜏Mm + Qm = 0 The corresponding simplified expression for a circular beam is given by: M,s −Q = 0 N Q,s − = 0 R M,s + N,s = 0 R 9.4.1.1

(9.72)

Finite Element Interpolations

In general, the cubic Bernstein–Bezier polynomials in root variable, 𝜉, define the finite element interpolation fields for generalized displacements and rotations for straight beams. In matrix form, the finite element displacement distributions may be denoted, in matrix form and component-wise, respectively, as: d(𝜉) = T(𝜉)

qd

and

dj (𝜉) =

N ∑

d BN i (𝜉) qi ,

j = 1, … , 6

(9.73)

i=0

where T(𝜉) is the interpolation transformation matrix containing appropriate Bernstein polynomials, qd = { qu qv qw q𝜃t q𝜃b q𝜃m }T is the displacement control vector, and N denotes the appropriate degree of the polynomial. It may be reemphasized that for computational efficiency in the mesh generation phase, the geometry may be modeled subparametrically depending on the curve. For example, for the general three-dimensional space-curved beam, geometry may be represented at most by cubic Bernstein–Bezier with as many elements as needed. Similarly, for circular or conic shape other than parabolic beam, only the quadratic rational Bezier or cubic Bernstein–Bezier is needed. Finally, a quadratic polynomial exactly represents a parabolic beam. For linear curved beams, c-type quartic Bernstein–Bezier elements suffice. However, for extreme nonlinear problems considered next, quintic Bernstein–Bezier elements are necessary for so-called mega-elements.

9.4.1.2

Element Post-processing and Stress Recovery

A few words relating to stress recovery are in order. The displacements and rotations converge in the norm sense. Accordingly, the uncoupled rotations can recover the bending moment

Application to Linear Problems and Locking Solutions

487

Figure 9.13 Example CFEM5: Curved beam with inter-element continuity: control nodes for two cubics are shown.

distributions by constitutive relationship exactly. Because of the constraints imposed by the kinematic assumptions, the shear force and axial force cannot be recovered from the first three constitutive relations given by equations (9.70). For problems considered here, a simpler method based on the integration of the bending moments satisfying the equilibrium conditions given above by equations (9.71) produces accurate results for both elemental axial force and shear force distributions. For example, for each three-dimensional space-curved beam element in the following, after recovering the moment components by the usual application of the constitutive relations and the nodal reactions, the axial force computation preceded shear force distribution through integration of the equilibrium equations as given by: 𝜉

Nt (𝜉) = Nt (0) −

𝜅 Mb ,𝜉 d𝜉 −

∫0

𝜉

Qb (𝜉) = Qb (0) +

∫0 ∫0

∫0

𝜏 Mb ,𝜉 d𝜉 + 𝜉

Qm (𝜉) = Qm (0) −

𝜉

𝜏 Qb J d𝜉 −

𝜅𝜏 Mm J d𝜉 𝜉

∫0 𝜉

∫0

𝜏 2 Mm J d𝜉

(9.74)

𝜅 Nt J d𝜉

With these in the background, let us consider the application of inter-element continuity (IEC) on the following example of interest in structural engineering and solid mechanics. Example CFEM 5: Quadrant of a circular beam element For demonstration of inter-element continuity condition imposition, we choose the curved (circular) cantilever beam shown in Figure 9.13. The beam is assumed fixed at one end, and loaded vertically down at the other end. The beam is assumed very thin and long so that the extensional strain and shear strains are zero. One cubic c-type Bernstein–Bezier element suffices, but for demonstration of inter-element continuity, we continue to improve by increasing the number of cubic elements. The results of the hierarchic solutions for: (a) one Bernstein–Bezier cubic element, (b) two Bernstein–Bezier cubic elements with C1 continuity, and finally, (c) two cubic c-type B-spline elements with C1 continuity. The original circular arc can be geometrically

Computation of Nonlinear Structures

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exactly generated by a rational Bezier, or approximately by a cubic. For the last case, taking two cubic Bezier elements, and applying C2 continuity conditions between them, a cubic Bspline element is first formed. Two such cubic B-spline elements, connected by C1 continuity between them, generate the stiffness matrix. Although not necessary in general applications, as indicated before, the curvilinear coordinate system is chosen to make the presentation simple for our purpose with s denoting the position along the centroidal axis; the radial displacement, w, for circular beam, is defined by: s ∈ [0, L], L ≡

w(s),

𝜋R 2

(9.75)

The beam is assumed thin and long so that the following assumptions hold:

r inextensional, that is, extensional strain is zero everywhere: 𝜖𝜃 ≡ u,s +

w =0 R

⇒

u,s = −

w R

(9.76)

where, u(s) represents the hoop displacement.

r no shear strain, that is, 𝛾 ≡ 0

Now, let the bending strain, or the curvature, 𝜅 = 𝜓,s , where 𝜓 denotes the rotation of the normal to the neutral axis at any point s. Then, we have, applying equation (9.69), for bending strain: 𝜅(s) = w,ss −

u,s w = w,ss + 2 R R

(9.77)

The exact solution, for radial load P applied at s = L, is given by: wexact (L) =

𝜋R3 P , 4EI

uexact (L) =

PR3 , 2EI

𝜓exact (L) =

PR2 EI

(9.78)

The root element is defined by: r = Ls ⇒ r ∈ [0, 1]. The Jacobian, J(r) = L, is constant, and its determinant is, obviously, the same. Then, the potential energy functional, Π(w), exists for the system for a displacement state, w, and is given by: EI Π(w) = 2 ∫0

L[

w w,ss + 2 R

]2

EI ds − Pw(L) = 3 2L ∫0

1[

𝜋2 w w,rr + 4 R2

]2 dr − Pw(1)

(9.79)

Now, let the displacement function be as described in matrix form: w(r) = T q, w,r (r) = N q,

N ≡ T,r

w,rr (r) = H q,

H ≡ T,rr

(9.80)

where, T and q are the transformation matrix and Bezier displacement control vector, respectively, dependent on the choice of our c-type element. Let us consider several cases based on the number and type of element as under:

Application to Linear Problems and Locking Solutions

9.4.1.3

489

Case 1: One c-type Cubic Bernstein–Bezier Element

For this case, first note that the geometry in the curvilinear coordinate system is linear, while the displacement function is assumed cubic. Thus, although not necessary,we may coerce up the geometry representation to cubic also by requiring the location of the geometry Bezier control vector, qc , as following: 𝜋R {0 2

qc =

2 3

1 3

1}T

(9.81)

So with the transformation, T, defined as follows, we have a coerced isoparametric representation of the geometry and the displacement function: qd = {qd0 , qd1 , qd2 , qd3 }T T ≡ [(1 − r)3

3(1 − r)2 r

N ≡ 3[−(1 − r)2 H ≡ 6[(1 − r) w(1) =

T(1)qd

3(1 − r)r2

r3 ]

(1 − 4r + 3r2 ) (2r − 3r2 ) r2 ] − (2 − 3r)

= {0

(1 − 3r) 1}qd

0 0

≡ T1

(9.82)

r], qd

Now, taking the variation of the potential energy functional given by equation (9.79), we have the virtual work principle:

K

qd

=

TT1 P,

EI K≡ 3 L ∫0

where

1[

𝜋2 H+ T 4

]T [

] 𝜋2 H + T dr 4

(9.83)

where, K, is the system-full, unconstrained stiffness matrix, and all other terms are defined as before; given the matrices, it can be easily computed to be: ⎡ 18.791 ⎢ −20.526 K=⎢ 3.135 ⎢ ⎣ 7.524

−20.526 30.6 −19.089 3.135

3.135 −19.089 30.6 −20.526

7.524 ⎤ 3.135 ⎥ −20.526 ⎥ ⎥ 18.791 ⎦

(9.84)

Now, for solving for the Bezier displacement vector, qd in equation (9.83), we impose the essential boundary conditions (i.e. the fixed end boundary conditions) as follows: w(0) = qd0 = 0, ) 3( d w,r (0) = q1 − qd0 = 0 ⇒ qd1 = 0 L

(9.85)

We can solve for the remaining components of the unknown displacement control vectors, by considering only the lower (2 × 2) submatrix of the stiffness matrix, and get: {

qd2 qd3

}

{ =

0.134 0.199

}

PL3 EI

(9.86)

Computation of Nonlinear Structures

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We can then extract the desired radial displacement, w(L), at load point as: [ w(L) =

0.199𝜋 2 2

](

𝜋R3 P 4EI

)

( = 0.982

𝜋R3 P 4EI

) (9.87)

Now, for axial displacement, u(L), under the load point, after noting the boundary condition that u(0) = 0, we have, from equation (9.76), upon integration: 1

L w(r) dr R ∫0 ]( 3) ( 3) [ 4 } PR PR 𝜋 { d q0 + qd1 + qd2 + qd3 = −1.014 =− 8 2EI 2EI

u(L) = −

(9.88)

In equation (9.88) above, we have used the integration property of Bernstein polynomials: n 1∑

∫0

1 ∑ d q , n + 1 i=0 i n

T(r) qd dr =

i=0

n=3

(9.89)

Finally, for rotation of the normal to the centroidal axis, 𝜓(L), under the load point, we have: 𝜓(L) = w,s (L) −

u(L) = 0.99 R

(

PR2 2EI

) (9.90)

Remark

r We see that with only one element, we have achieved a pretty good estimate of the solution by our c-type method. However, we can continue to improve it by increasing the number of cubic elements. Let us try the following case. 9.4.1.4

Case 2: Two Cubic Bernstein–Bezier Elements with C1 continuity

Following the method described above, we can show that the assembled finite element equilibrium equations with:

r C0 continuity imposed between the elements r dropping off the first two rows and columns to reflect the fixed end boundary conditions, is given by: K0 q̃ d = P0 ⎡ 34.552 ⎢ −18.713 )⎢ ( 8EI ⎢ 0 where K0 = L3 ⎢ ⎢ 0 ⎢ ⎣ 0 q̃ d =

{qd2

qd3

qd4

qd5

−18.713

0

0

2(13.535)

−18.713

.751

−18.713

34.552

−18.346

.751

−18.346

34.552

6.373

0.751

−18.713

qd6 }T , P0

= {0

0 0 0 1}T P

⎤ 6.373 ⎥⎥ .751 ⎥ ⎥ −18.713 ⎥ ⎥ 13.535 ⎦ 0

(9.91)

Application to Linear Problems and Locking Solutions

491

Remark

r Any attempt to solve equation (9.91) will spell disaster because, for a variational solution, a system governed by fourth-order differential equation, as in our case, needs at least C1 continuity between the elements. So, let’s go ahead and impose the C1 continuity condition, between the elements. We apply to the C0 -assembled stiffness control matrix, K 0 , and the load control vector, P0 , and get the corresponding C1 -assembled stiffness control matrix and the load ontrol vector, denoted by K 1 and P1 , respectively, as given by: ⎡ 22.606 −11.946 ⎢ −11.946 22.606 K1 = CT K0 C = ⎢ ⎢ 0.375 −17.970 ⎢ ⎣ 3.186 3.937 P1 = CT P0 = {0 0 0 1},

0.375 −17.970 34.552 −18.713

3.186 ⎤ 3.937 ⎥ ⎥, −18.713 ⎥ ⎥ 13.535 ⎦

(9.92)

where, the C1 continuity condition matrix, C, is given by: q̃ d = C̃qd ,

or, in expanded form:

⎧ qd2 ⎫ ⎪ d⎪ ⎪ q3 ⎪ ⎪ d⎪ ⎨ q4 ⎬ ⎪ d⎪ ⎪ q5 ⎪ ⎪ qd ⎪ ⎩ 6⎭

⎡1 0 0 ⎢ .5 .5 0 ⎢ =⎢0 1 0 ⎢ ⎢0 0 1 ⎢ ⎣0 0 0

0⎤ d ⎧ q2 ⎫ 0 ⎥⎥ ⎪ d ⎪ ⎪q ⎪ 0 ⎥ ⎨ 4d ⎬ ⎥ q ⎪ 5 0⎥⎪ ⎪ ⎥⎪ qd 1⎦⎩ 6 ⎭

(9.93)

Remark

r The total number of degrees of freedom is reduced by one, that is, the junction degree of freedom, qd3 , between the two elements is eliminated from the equations by the condition of C1 continuity. Now, solving equation (9.91), and continuing with the procedure as before, element by element, we get the desired solution as: 𝜋R3 P , 4EI PR3 u(L) = −(0.99995) , 2EI PR2 𝜓(L) = (0.99932) EI w(L) = (0.9984)

(9.94)

Remarks

r The

above results almost match the exact solutions! It can be shown that improving the continuity condition does not improve the results anymore.

Computation of Nonlinear Structures

492

r However, as noted earlier, we can further increase the accuracy by changing the elements from simple cubic Beziers to next hierarchical elements: cubic B-spline elements connected by the C1 continuity condition. That is exactly what we propose to do next.

9.4.1.5

Case 3: Two c-type Cubic B-spline Elements with C1 Continuity

For this case, we do not intend to present our analysis in detail. We only highlight the major ideas as follows:

r As suggested by the definition, a cubic B-spline element is formed by taking two cubic Bezier elements, and applying C2 continuity conditions between them.

r Each cubic B-spline element will, then, have five degrees of freedom, that is, three less than the eight degrees of freedom of two cubic elements.

r The two cubic B-spline element when connected by C1 continuity between them, will have a total of eight degrees of freedom, that is, two less than ten for the two elements.

r For the problem at hand with fixed end boundary condition, the system total significant degrees of freedom are six. Finally, by carrying out the solution as before, we get: 𝜋R3 P , 4EI PR3 u(L) = −(1.00029) , 2EI PR2 𝜓(L) = (1.000219) EI w(L) = (1.000176)

(9.95)

Remark

r The

results, of course, speak for themselves – we have achieved almost the exact solution. We summarize below, in tabular form (see Figure 9.14), the tip response comparison between the analytical results and the solutions obtained by various hierarchic c-type finite elements.

Figure 9.14 Curved beam: c-type responses for various cases.

Application to Linear Problems and Locking Solutions

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Figure 9.15 Example CFEM 6: The 90◦ circular cantilevered beam with tip radial load.

Example CFEM 6: A 90◦ cantilevered circular ring with tip radial load As shown in Figure 9.15, and as a prelude to application of the c-type method to general shell analysis, we consider here the curved ring problem with both axial and shear deformation possibilities. The geometric control polygons√are generated for circular curve by both cubic Bernstein–Bezier polynomials with ki = 43 Ri ( 2 − 1), i = 1, 2, and rational quadratic Bezier polynomials. The results obtained by c-type formulation are shown for various aspect ratio, Rh , in a tabular form as shown in Figure 9.16: Remark

r Considering the remarkable accuracy of the responses, we may conclude that the interpolation problems known as shear, membrane and dilatation locking that severely and continuously plague the old finite element methodologies are absent in the results presented above. The results are compared with the corresponding tip responses obtained by application of Castigliano’s energy theorem. In comparing, however, it must be recognized that for a very thick beam, analytical solutions obtained by Castigliano’s theorem are only approximate when the model is treated as a simple beam. Example CFEM 7: Thin circular pinched ring As shown in Figure 9.17, a thin ring pinched by two diametrically opposite and identical forces is analyzed to assess the c-type curved beam element performance for modeling a deep arch

Figure 9.16 Example CFEM 6: c-type responses for several slenderness ratios.

494

Computation of Nonlinear Structures

Figure 9.17 Example CFEM 7: Pinched ring problem (a) geometry (b) finite elemnt model.

configuration. Because of symmetry, only one quadrant of the ring is modeled with appropriate boundary conditions and then analyzed twice: once with only one element and then with twoelement discretization. The same problem analyzed by many researchers has been reported in the literature where ad hoc treatments were needed to deal with the locking problems in curved beams. In Figure 9.18, the tip radial normalized displacement generated by the c-type method is compared with that given by others reported in the literature. For most cases, significantly more elements are needed to model the problem to obtain the same level of accuracy by the c-type curved beam element. The distributions of the bending moment, axial force and shear force are shown in Figures 9.19(a)–(c):

Figure 9.18 Example CFEM 7: Convergence comparison with various published results.

Application to Linear Problems and Locking Solutions

495

(a)

(b)

(c)

Figure 9.19 (a) Example CFEM 7: Bending moment distribution. (b) Example CFEM 7: Axial force distribution. (c) Example CFEM 7: Shear force distribution.

Computation of Nonlinear Structures

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Figure 9.20 Example CFEM 8: Thin circular arch with central moment discontinuity.

Remark

r Note that extreme accuracy is achieved for all stress resultants and stress couples; thus, the constant shear force assumption (Raveendranath, 1999) is unnecessary. So it seems appropriate to point out that for interpolationally correct elements that ensure convergence of displacements in the norm, field consistency (Prathap, 1985; Prathap and Shashirekha, 1993; Timoshenko and Goodier, 1951) consideration is irrelevant in the displacement-based finite element formulations. Rather, for general applicability, stress recovery must be treated with care, as noted earlier, by calling upon the equilibrium equations for consistency. Example CFEM 8: Thin circular arch with central moment discontinuity To qualify c-type curved beam elements for curvature discontinuity, a thin circular arch with central moment, as reported in the literature and shown in Figure 9.20, is considered here. The c-type element distributions of the bending moment, axial force and shear force are compared with the exact solution, as shown in Figures 9.21(a)–9.21(c): Remark

r As

can be seen from Figures 9.21(a)–(c), the c-type element responses almost mimic the exact solutions. As in the previous situation, the shear force distribution is obtained almost exactly without any constant shear force assumptions of MFE elements. Similarly, axial force distribution does not suffer from any discontinuity predicted by elements based on curvature,

Example CFEM 9: Nearly straight cantilever modeled as curved beam To test the predictability of the c-type curved beam elements for very large radius and short span, that is, almost straight beam configuration, the problem of a cantilevered beam with tip loading condition, shown in Figure 9.22, is considered. The entire beam is modeled by a single c-type curved beam element. The radial (near vertical) displacement and the rotation, and the bending moment, axial force and shear force distributions are compared and shown in Figures 9.23(a)–(c): Remark

r As can be seen, just a single element can predict the true responses extremely accurately. Example CFEM 10: Truly three-dimensional cantilevered curved beam element with natural twist and bend

Application to Linear Problems and Locking Solutions

497

(a)

(b)

(c)

Figure 9.21 (a) Example CFEM 8: Bending moment distribution. (b) Example CFEM 8: Shear force distribution. (c) Example CFEM 8: Axial force distribution.

Computation of Nonlinear Structures

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Figure 9.22 Example CFEM 9: Axial force distribution.

Finally, for general computer-aided curved beams or shell mesh generation through input of a few key data points, the curved boundaries of the elements may be truly three-dimensional. Thus, to test the predictability of the c-type for general 3D large curved beam elements, a truly three-dimensional beam in the linear regime is modeled as simple two-element configurations as shown in Figure 9.24. The model is initially generated as a single cubic Bernstein–Bezier element with control nodes, q1i , i = 0, … , 3. Then, an application of a subdivision algorithm followed by degree elevation, produces two quartic Bernstein–Bezier elements. The beam is subjected to simple tip load. The computed displacement and rotation distributions in the Cartesian frame, and the corresponding force and moment responses are shown in Figures 9.25(a),(b). Remarks

r The near-exact accuracy of the forces and moments, and displacements and rotations with only two quartic elements is self-evident.

r Note that it takes many more than two currently available elements to match the presented accuracy. Likewise, the bookkeeping and quality assurance needed for mesh generation and finite element analysis will result in a proportional increase in analyst/engineer man-hours.

9.5

c-type Deep Beam: Plane Stress Element

We have so far restricted ourselves to one-dimensional structures, with the main goal of demonstrating the new theory as simply as possible. However, we also want to demonstrate its usefulness to two- and three-dimensional structures which are the primary target of most finite element applications. We have already presented several interpolations for two-dimensional cases based on our new theory, and here we present another example from plane stress analysis in support of our theory. Example CFEM 11: Beam bending as plane stress problem For the beam shown in Figure 9.26, we consider one of our cubic–cubic elements, but note that the edges of the beam geometry are straight, requiring only linear–linear element for the representation of its geometry. So, although not necessary, we can apply degree-elevation and coerce up the geometry representation by cubic–cubic Bernstein polynomials so that it becomes

Application to Linear Problems and Locking Solutions

499

(a)

(b)

(c)

Figure 9.23 (a) Example CFEM 9: Vertical displacement distribution. (b) Example CFEM 9: Rotation distribution. (c) Example CFEM 9: Bending moment, shear and axial force distribution.

Computation of Nonlinear Structures

500

Figure 9.24 Example CFEM 10: Truly 3D curved beam with bend and twist.

isoparametric to the displacement function. The geometry is as shown in Figure 9.26 and the root element consists of unit square, that is, Ω ≡ {(x, y)|x ∈ (0, Lx ), y ∈ (0, Ly )} ⊂ R2 ΩR ≡ {(r, s)|r ∈ (0, 1), s ∈ (0, 1)} ⊂ R2

(9.96)

The Jacobian, J(r, s), and its determinant are given by: [ J(r, s) =

Lx 0

] 0 = constant ⇒ det J = Lx Ly = constant Ly

(9.97)

Now, for a cubic–cubic Bernstein–Bezier element, we set:

Geometry: c(r, s) =

3 3 ∑ ∑

{ B3i (r)

B3j (s)

q̃ cij ,

B3i (r)

B3j (s)

q̃ dij ,

c(r, s) =

i=0 j=0

Displacement: d(r, s) =

3 3 ∑ ∑ i=0 j=0

x(r, s) y(r, s)

{ d(r, s) =

}

u(r, s) v(r, s)

}

(9.98)

The transformation matrix is determined from equation (9.98) by appropriately packing the terms corresponding to the components of the geometry and the displacement control vectors denoted by qc and qd , respectively. These vectors are defined from q̃ cij and q̃ dij , respectively, by packing their all x-components first and then all y-components. Then, the strain–displacement control expressions can be easily found. Use of internal virtual work gives, as usual, the stiffness

Application to Linear Problems and Locking Solutions

501

(a)

(b)

Figure 9.25 (a) Example CFEM 10: Displacement and rotation distribution. (b) Example CFEM 10: Force and moment distribution.

Figure 9.26 Example CFEM 11: Deep beam bending as plane stress problem.

Computation of Nonlinear Structures

502

control matrix. The handling of the surface loading needs some elaboration. For surface loading, we have parabolic shear, f S (y), downward along the x = Lx edge such that the total load amounts to P, that is, f S (y) =

6P y(y − Ly ) Ly3

(9.99)

The loading is quadratic so it has a Bernstein–Bezier representation as: Matrix form: f S (s) = T qF

with T = {B20 (s) B21 (s) B22 (s)}

where, B20 (s) = (1 − s)2 , B21 (s) = 2(1 − s)s, { }T 3P 0 and, q̄ F = 0 Ly Expanded form: f S (s) =

2 ∑

B22 (s) = s2 , (9.100)

B2i (s) q̄ Fi ,

i=0

However, the element is cubic in the s-direction, so the surface load has to be degree elevated once, to obtain its cubic representation as: { } with T = B30 (s) B31 (s) B32 (s) B33 (s)

Matrix form f S (s) = T qF , Expanded form: f S (s) =

3 ∑

B3i (s) qFi ,

i=0

where:

B30 (s)

B31 (s) = 3(1 − s)2 s, B32 (s) = 3(1 − s)s2 , B33 (s) = s3 }T }T { 2P 2P 2 F 2 F F q̄ q̄ q̄ 2 = 0 0 3 1 3 1 Ly Ly

= (1 − s)3 ,

and: qF =

{

q̄ F0

(9.101) Now, given the transformation matrix and the surface load, the generalized nodal surface control loads are given by line integrals, which can be evaluated right away by numerical integration. However, for the sake of standardization of input of loads, we have represented the surface load isoparametrically. Then the nodal control surface load can be evaluated in closed form as we explain next. The surface control nodal load is given by: (

1

FS =

∫ Γ𝜎

where, q̂ F = {0

TT f s dΓ𝜎 = Ly

∫0

TT p(s) ds = Ly

∫0

)

1

TT T ds

( qF = 2P

∫0

)

1

TT T ds

q̂ F ,

1 1 0}T

(9.102)

Application to Linear Problems and Locking Solutions

503

This integration can be evaluated numerically by Gaussian quadrature formulas. But, we apply the useful properties of Bernstein polynomials for a closed form expression, as follows: (

)( ) m n i j n (s) where ) Bm+n Bm i (s) Bj (s) = ( i+j m+n i+j 1 1 and, Bn (s) ds = ∫0 i n+1

(

m i

) =

m! i!(m − i)!

(9.103)

Finally, using formulas (9.103) in equation (9.102), we get the generalized surface control load in two-dimensional notation as: ( )( ) ( )( ) 3 3 3 3 ( ) i 1 1 1 j i j 1 FijS = TTi Tj ds = B3i (s) B3j (s) ds = ( B6i (s) ds = ) ( ) ∫0 ∫0 ∫0 7 6 6 i+j i+j (9.104) The boundary condition is taken to be completely fixed at the x = 0 edge, that is, no warping is allowed. Finally, the problem is treated as a plane stress problem with the Poisson’s ratio, 𝜈 = 0.20. Now, solving for the particular case of: Lx =3 Ly

(9.105)

we get the tip displacement downward in the y-direction at mid-point of the loaded surface as: ) ( ) ( P v Lx , 1 Ly = 115.54 (computed) 2 Et ( ) P = 116.00 (exact) Et

(9.106)

where, E and t are the material elasticity modulus and the thickness, respectively, of the beam. For one element and large aspect ratio, the solution seems quite remarkable. Of course, the accuracy can be significantly improved with two elements. Example CFEM 12: Beam bending as plane stress problem (distortion and incompressibility) The beam shown in Figure 9.27 is supported at one end with warp possibility and is shear loaded vertically up on the other end. Let us consider a c-type cubic–cubic element in u and v displacements. Use of the virtual work principle, gives the stiffness control matrix and load control vector, as usual. However, the handling of the imposed surface load (likewise, the balancing shear load on the supported end) needs elaboration. The parabolic shear, f S (y), along the x = Lx edge, such that the total load amounts to P, that is, f S (y) = 6P3 y(y − Ly ), is treated in the same way as Ly

in the previous example to find the nodal load controls corresponding to cubic distribution. To

504

Computation of Nonlinear Structures

Figure 9.27 Example CFEM 12: Deep beam bending as plane stress problem – distortion and incompressibility.

further demonstrate the effectiveness of the c-type method, we continue with the two-element configuration of the same problem as introduced before. The tip displacements and extensional stresses of the extreme fiber at the common boundary of the two elements for various aspect ratios, distortion ratios and Poisson’s ratio are recorded in tabular form, as shown in Figure 9.28. The straight Timoshenko beam with the rotation of the normal to the neutral axis and the transverse displacement taken for cubic–cubic representation has also been considered. The results, although are not reproduced here, are equally excellent.

Figure 9.28 Example CFEM 12: Tip displacement and extensional stresses for various aspect ratio.

Application to Linear Problems and Locking Solutions

505

Figure 9.29 Example CFEM 13: Curved deep beam as a plane stress problem.

Example CFEM 13: Curved deep beam bending as a plane stress problem. Problem: Let us consider the deep beam in the form of one quadrant of a circle, shown in Figure 9.29, in plane stress. The problem is solved in Timoshenko and Goodier (1951) by way of using stress functions. The stresses are found to be: [ ] P a2 b2 a2 + b2 𝜎rr = r+ 3 − sin 𝜃 N r r [ ] P a2 b2 a2 + b2 𝜎𝜃𝜃 = 3r − 3 − sin 𝜃 N r r [ ] P a2 b2 a2 + b2 𝜏r𝜃 = r− 3 − cos 𝜃 N r r where

N ≡ a2 − b2 + (a2 + b2 ) log

(9.107)

b a

For the boundary conditions shown in Figure 9.29, the displacement solutions are found to be: ⎧ ⎫ a2 b2 (1 + 𝜈) ⎤ ⎪ ⎡ 1 (1 − 3𝜈)r2 − ⎪ ⎥ sin 𝜃 ⎪ 2r2 P ⎪ ⎢⎢ 2 ⎥ ur = ⎬ − K sin 𝜃 ⎦ ⎣ −(a2 + b2 )(1 − 𝜈) log r NE ⎨ ⎪ ⎪ ⎪ +(a2 + b2 )(2𝜃 − 𝜋) cos 𝜃 ⎪ ⎩ ⎭ ⎫ ⎧ a2 b2 (1 + 𝜈) ⎤ ⎪ ⎪ ⎡ 1 (5 + 𝜈)r2 − ⎥ cos 𝜃 ⎪ 2r2 P ⎪ ⎢⎢ 2 ⎥ u𝜃 = − ⎬ − K cos 𝜃 ⎣ +(a2 + b2 )[(1 − 𝜈) log r + (1 + 𝜈)] ⎦ NE ⎨ ⎪ ⎪ ⎪ ⎪ −(a2 + b2 )(2𝜃 − 𝜋) sin 𝜃 ⎭ ⎩ ] [ 2 b (1 + 𝜈) P 1 where ur (a, 𝜋∕2) = 0 ⇒ K ≡ (1 − 3𝜈)a2 − − (a2 + b2 )(1 − 𝜈) log a NE 2 2 𝜋P 2 with ur (r, 0) = − (a + b2 ) = constant NE (9.108)

Computation of Nonlinear Structures

506

Remark

r Since the radial displacement, ur , is constant at the 𝜃 = 0 shearing load boundary, we may choose to solve this problem for imposed displacement only rather than one of imposed tractions. For our example problem, we choose following properties: a = 5 (inner radius), b = 10 (outer radius), t = 1 (thickness) E = 10000 (elasticity modulus), 𝜈 = 0.25 (Poissons ratio) Boundary conditions: v = −0.01 on all points at the shear load edge

(9.109)

u(5, 𝜋∕2) = 0, v(r, 𝜋∕2) = 0 Exact total energy: Eexact =

1 (log 2 − 0.6) = 0.02964966844238 𝜋

In the mesh generation phase, the geometry of the structure can be represented either approximately by a non-rational cubic Bernstein–Bezier curve, or exactly by a rational quadratic Bernstein–Bezier curve. The strain–displacement relationship, likewise, can be developed in both Cartesian and polar coordinates, but for this problem we choose Cartesian coordinates as our basis of development. Finally, the problem is solved subparametrically, as opposed to isoparametry, as will be seen shortly; we will elaborate on the solution in some details because the problem may be taken as a prelude to more complicated membrane problems associated with structural surfaces such as shells, and such like. Solution We determine the accuracy of the c-type finite element method by choosing to solve this problem with increasing complexity in terms of geometry representation and element displacement distribution selection. Mesh generation phase: For non-rational geometry representation, let us recall, from Chapter 5 on curves, that a quadrant of a circle of unit radius can be represented approximately by a cubic Bernstein–Bezier curve. )T ( Thus, the geometry of the surface, S(𝜉 1 , 𝜉 2 ) = X(𝜉 1 , 𝜉 2 ) Y(𝜉 1 , 𝜉 2 ) , is represented always by the linear–cubic Bernstein–Bezier root element, linear in the radial direction and cubic in the circumferential direction, shown in Figure 9.30, as:

r a 1 × 3 non-rational Bernstein–Bezier surface, S(𝜉 1 , 𝜉 2 ), defined by: { S(𝜉 1 , 𝜉 2 )

=

X(𝜉 1 , 𝜉 2 ) Y(𝜉 1 , 𝜉 2 )

} ==

3 1 ∑ ∑ i=0 j=0

{ B1i (𝜉 1 ) B3j (𝜉 2 ) (qcij

=

qXij qYij

} ),

0 ≤ 𝜉1, 𝜉2 ≤ 1 (9.110)

where the basis (blending) function sets, {B1i (𝜉 1 )}, {B3j (𝜉 2 )}, are the Bernstein polynomials of degree one and three, respectively; for example: Bnj (𝜉 2 )

( ) ( ) n n n! 2 j 2 n−j = (𝜉 ) (1 − 𝜉 ) , j ∈ {0, 1, … , n} where ≡ j j j!(n − j)!

(9.111)

Application to Linear Problems and Locking Solutions

507

Figure 9.30 Example CFEM 13: Geometry control polygon and root element.

√ with 𝛼 = 43 ( 2 − 1), the control vertices are: qc00 qc10

{ } 0 = 0 { } b−a = 0

{ qc01 qc11

= { =

0 𝛼b

}

b−a 𝛼a

{

}

qc02 qc12

} b(1 − 𝛼) = b { } b − 𝛼a) = a

qc03 qc13

{ } b = b { } b = a

(9.112)

𝜕(X,Y) The Jacobian, J = 𝜕(𝜉 1 ,𝜉 2 ) , and the determinant of the Jacobian, det J, from equation (9.110), are computed in the mesh generation phase as follows:

Step 0: ⎧ 𝜕 ⎪ 𝜕𝜉 1 ⎨ ⎪ 𝜕 ⎩ 𝜕𝜉 2

⎫ ⎡ dX ⎪ ⎢ d𝜉 1 ⎬=⎢ ⎪ ⎢ dX ⎭ ⎣ d𝜉 2

dY d𝜉 1 dY d𝜉 2

⎤⎧ 𝜕 ⎥ ⎪ 𝜕X ⎥⎨ ⎥⎪ 𝜕 ⎦ ⎩ 𝜕Y

⎫ ⎧ 𝜕 ⎪ ⎪ 𝜕𝜉 1 ⎬⇔⎨ ⎪ ⎪ 𝜕 ⎭ ⎩ 𝜕𝜉 2

⎫ ⎧ 𝜕 ⎪ ⎪ 𝜕X ⎬ = J⎨ ⎪ ⎪ 𝜕 ⎭ ⎩ 𝜕Y

⎫ ⎪ ⎬ ⎪ ⎭

(9.113)

Note that, in actual implementation, the derivatives of the Bernstein–Bezier functions are computed by computationally economic difference operator. Step 1: det J =

dX dY dX dY − d𝜉 1 d𝜉 2 d𝜉 2 d𝜉 1

(9.114)

Step 2: The inverse of the Jacobian, needed for the strain–displacement relationship, is also stored during the mesh generation phase as: ⎧ 𝜕 ⎪ 𝜕X ⎨ ⎪ 𝜕 ⎩ 𝜕Y

⎫ ⎡ dY ⎪ 1 ⎢ d𝜉 2 ⎬ = det J ⎢ ⎪ ⎢ − dX ⎣ d𝜉 2 ⎭

dY d𝜉 1 dX d𝜉 1

−

⎤⎧ 𝜕 ⎥ ⎪ 𝜕𝜉 1 ⎥⎨ ⎥⎪ 𝜕 ⎦ ⎩ 𝜕𝜉 2

⎫ ⎪ ⎬ ⎪ ⎭

(9.115)

Computation of Nonlinear Structures

508

Figure 9.31 Example CFEM 13: Displacement control polygon and root element.

c-type finite element phase: Likewise, the displacement distribution, as shown in Figure 9.31, is represented by cubic–cubic non-rational Bernstein–Bezier functions, with as yet unknown control net, as: { d(𝜉 1 , 𝜉 2 )

=

u(𝜉 1 , 𝜉 2 ) v(𝜉 1 , 𝜉 2 )

} ==

3 3 ∑ ∑

( B3i (𝜉 1 ) B3j (𝜉 2 )

{ qdij

=

i=0 j=0

quij qvij

}) ,

0 ≤ 𝜉1, 𝜉2 ≤ 1 (9.116)

The (3 × 32) Cartesian strain–displacement matrix is obtained by computing u,X , u,Y , v,X , v,Y from equation (9.116) with the help of equation (9.115). With the constitutive matrix for plane stress and the strain–displacement matrix, the element singular stiffness control matrix may be computed. However, as indicated in the theory of finite element formulation, the integral is seldom calculated in closed form, but is numerically integrated by Gaussian quadrature formula. For the cubic–cubic displacement distribution, the full (4 × 4) Gaussian points are selected, and integration is performed by summation over these quadrature points. Before solving it, we need to impose the displacement constraints at appropriate degrees of freedom to generate the non-singular equation as follows:

u13 = u20 = u24 = u28 = u32 = 0, v17 = v21 = v25 = v29 = 0

(9.117)

The stiffness equation is then solved for the unknown displacement controls; the displacements, u and v are computed for any (𝜉 1 , 𝜉 2 ) using equation (9.116). Finally, the radial displacement, ur , and the circumferential displacement, u𝜃 , at any (r, 𝜃), are obtained by simple rotation of the

Application to Linear Problems and Locking Solutions

509

coordinate relationship. The problem is solved for two choices of geometry representation and various choices of displacement distributions: Case 1: One element: linear–cubic non-rational Bernstein–Bezier geometry with cubic–cubic non-rational Bernstein–Bezier displacement distribution, as described above, Case 2: Two equal elements: linear–cubic non-rational Bernstein–Bezier geometry elements with cubic–cubic non-rational Bernstein–Bezier displacement distribution connected circumferentially by C2 - condition, that is, a single B-spline element; the two cubic elements are obtained by subdivision algorithm. Case 3: Two equal elements: linear–cubic non-rational Bernstein–Bezier geometry with cubic–cubic non-rational Bernstein–Bezier displacement distribution, Case 4: A linear-quadratic Rational Bernstein–Bezier geometry with quintic-quintic non-rational Bernstein–Bezier displacement distribution. Case 5: A linear–cubic non-rational Bernstein–Bezier geometry with quartic-quartic non-rational Bernstein–Bezier displacement distribution; solved by 5 × 5 Gauss quadrature points. Case 6: A linear–cubic non-rational Bernstein–Bezier geometry with quintic-quintic non-rational Bernstein–Bezier displacement distribution; solved by 6 × 6 Gauss quadrature points. The radial and circumferential displacement distributions for the quintic–quintic case are shown in Figure 9.32(a), and, the total energy for the various cases mentioned above are shown and compared with the exact value in tabular form as in Figure 9.32(b).

9.6

c-type Solutions: Locking Problems

Recall that interpolation problems have been the nagging nuisance to all old (namely, h-type and p-type) finite element methods. Thus, as a further demonstration of superiority of the c-type finite element method, we now offer some solutions to outstanding and nagging locking problems of the these methods as presented in Chapter 1’s introduction and motivation, which may render old safety calculations dubious and unacceptably risky; we will use some practical problems that routinely appear in industry, and that are posed and considered solved by ad hoc means such as under-integration, reduced integration and arbitrary drilling degree of freedom in the old finite element methods. So, before we present our solutions for these locking problems, let us briefly recall the conventional state-of-the-art remedies.

9.6.1.1

Reduced/Selective Integration, Bubble Mode and Drilling Freedom

As we already know, the numerical integration used to evaluate, say, the stiffness matrix, is performed by selecting Gauss or other quadrature points. Thus, an immediate remedy to shear locking was offered by selective under-integration by arbitrarily evaluating the integral involving shear at only x = 0, that is, one point Gauss, which effectively wipes out the shear contribution to the stiffness matrix. The problem with this remedy is that the reduced integration is obviously less

Computation of Nonlinear Structures

510

(a)

(b)

Figure 9.32 (a) Example CFEM 13: Radial and circumferential displacement for a quintic-quintic case. (b) Example CFEM 13: Total energy for various cases compared to exact value.

Application to Linear Problems and Locking Solutions

511

accurate. More importantly, it creates undesirable spurious modes, that is, artificial singularities or eigenmodes, induced by possible reduction in the rank of the stiffness matrix. Another ad hoc remedy was to include quadratic interpolation terms, the so-called bubble modes, without introducing any additional node, to handle both shear and dilatation locking. This, of course, renders the element non-conforming as two nodes cannot possibly represent a quadratic term. In line with the addition of modes of flexibility just described, modes known as the drilling freedoms have been devised to resolve locking problems. The drilling freedoms are nodal rotational degrees of freedom about an axis perpendicular the plane of the element. These in-plane rotational degrees of freedom, in improving membrane performance, may create a problem for bending performance of the curved shell elements where these degrees of freedoms are already coupled with the bending modes. In general, most of these patchy remedies that were used on regular rectangular elements, fail to eradicate locking when applied to general h-type isoparametric curved element of arbitrary shape. Moreover, these varieties of so-called remedial elements for specific displacement states increase the number of elements available commercially to such an extent that an ordinary user can be easily confused as to their usefulness – it is important that the element library be simple and user-friendly. Thus, in what follows, it will be recognized that these locking problems are interpolational ones that crop up due to an incomplete set of basis functions, and hence are easily remedied by a complete set of ingenious polynomial functions with geometric representation such as the Bernstein–Bezier functions, which contain all the above modes (bubble, etc.) of deformation naturally. Example RSL: Remedy for shear locking: c-type four-node quadratic element For shear locking (as discussed in Chapter 1), consider the following state of stress imposed on a four-node old (i.e. h-type or p-type) rectangular membrane element: in-plane bending load due to a couple of moment applied at opposite ends of the element in plane stress situation, as shown in Figure 9.33. The state of the actual displacement is given by: u = cxy

and

v = − 12 cx2

and shown in Figure 9.34.

Figure 9.33 Example RSL: Remedy for shear locking problem.

(9.118)

Computation of Nonlinear Structures

512

Figure 9.34 Example RSL: Imposed u and v displacements.

The corresponding states of actual strain and stress for the plane stress situation with E, 𝜈 as the elasticity modulus and Poisson’s ratio, respectively, are given by: 𝜖x ≡

𝜕u 𝜕x

= cy

𝜖y ≡

𝜕v 𝜕y

=0

𝛾xy ≡

𝜕u 𝜕y

+

𝜕v 𝜕x

cEy 1 − 𝜈2 𝜈cEy 𝜎y = 1 − 𝜈2

𝜎x =

(9.119)

= 0 𝜏xy = 0

An application of the displacement state, given by the equation (9.118), to the joint nodes of the element and then subsequent calculation (by inspection) yields the element internal state of stress as: 𝜖x = cy 𝜖y = 0 𝛾xy = cx

cEy 1 − 𝜈2 𝜈cEy 𝜎y = 1 − 𝜈2 cEx 𝜏xy = 2(1 + v) 𝜎x =

(9.120)

From equation (9.120), we can easily see that the element has generated undesirable, spurious shear strain. The ill effect of this spurious strain on stiffness can be evaluated by observing the computed strain energy of the element. It can be easily shown that the ratio of the computed strain energy to the actual strain energy, which is same as the ratio of the element bending stiffness to actual bending stiffness, is given by: Element bending stiffness = 1 + 12 (1 − 𝜈)a2 Actual bending stiffness

(9.121)

Obviously, for a slender element with a reasonable a = 10, the element stiffness will be excessive and result in severe under-prediction of the displacements. This spurious shear strain brought in the name shear locking. As shown in Figure 9.35(a), the c-type unknown generalized displacement vectors are the Bezier control vectors, qu and qv in x and y directions, respectively. The mapping function is

Application to Linear Problems and Locking Solutions

513

(a)

(b)

Figure 9.35 (a) Example RSL: c-type quadratic membrane physical and root element. (b) Example RSL: Imposed displacement u-controls and v-controls.

composed of Bernstein polynomials; the root element is a square with r and s as the coordinate directions. The root to physical element inverse mapping is given by:

r=

x+a 2a

and

s=

y+1 2

(9.122)

In component form, we express the displacement vectors as a linear combination of the basis functions as:

u(r, s) =

2 2 ∑ ∑ i=0 j=0

B2i (r)B2j (s) quij

and

v(r, s) =

2 2 ∑ ∑

B2i (r)B2j (s) qvij

(9.123)

i=0 j=0

where, the second degree Bernstein polynomials for, say, r, are given by: B20 (r) = (1 − r)2 , B21 (r) = 2(1 − r)r, B22 (r) = r2

(9.124)

Computation of Nonlinear Structures

514

Most importantly, note that only the four corner nodes are physical nodes; the rest can be considered as generalized displacements. Based on the given displacement field and as shown in Figure 9.35(b), we must have the imposed control displacements corresponding to u, as:

qu00 = qu22 = −qu20 = −qu02 = ca qu01 = qu10 = −qu21 = −qu12 = qu11 = 0

(9.125)

and the imposed controls for v displacements are given as:

qv00 = qv02 = qv20 = qv22 = − 12 ca2 −qv11 = qv01 = −qv10 = qv21 = −qv12 = − 12 ca2

(9.126)

Note that the value of the displacements at the generalized nodes are not the same as the values of the generalized nodal control displacement values, except for the corner nodes where the control vectors interpolate the nodal displacements. This is a crucial observation to understand the new c-type method. We have approximated the u displacement by quadratic polynomials in two directions. A bilinear interpolation would have been sufficient. However, for v displacement field, we need a quadratic for exact interpolation. In order to keep the degrees the same for the example, we change the degree of the polynomial in the u direction to also being quadratic. This has been possible because of what we already know as the method of repeated degree elevation guaranteed by the linear precision property of the Bernstein polynomials. This is one of the cornerstones of the theory behind the c-type method. Most importantly, note that the linear combination given by equation (9.123) is barycentric because of the barycentric property of the Bernstein polynomials, that is, n ∑

Bni = 1

(9.127)

i=0

Now, for u displacement, substituting the values of the imposed generalized control displacements, given by equation (9.125), and the definition of Bernstein polynomials, as given by equation (9.124), in equation (9.123), we get: [{ }] (1 − r)2 (1 − 2s + s2 − s2 ) qu00 u(r, s) = } { − r2 (1 − 2s + s2 − s2 )

(9.128)

= (1 − 2r)(1 − 2s)qu00 = cxy = Exact solution! Thus, u displacement is interpolated exactly, and so will be the strains, stresses and element strain energy.

Application to Linear Problems and Locking Solutions

515

Similarly, for v displacement, substituting the values of generalized displacements, given by equation (9.126), and definition of Bernstein polynomials, as given by equation (9.124), in equation (9.123), we get: { } v(r, s) = (1 − r)2 1 − 2s + s2 + 2s − 2s2 + s2 qv00 { } + 2r(1 − r) 1 − 2s + s2 + 2s − 2s2 + s2 qv10 { } + r2 1 − 2s + s2 + 2s − 2s2 + s2 qv00

(9.129)

= (1 − 2r + 2r2 − 2r + 2r2 )qv00 = (1 − 2r)2 qv00 =

x2 v q = − 12 cx2 = Exact solution! a2 00

Thus, v displacement is interpolated exactly, and so will be the strains, stresses and element strain energy, irrespective of the aspect ratio representing shape sensitivity. Example RDL: Remedy for Dilatation Locking: c-type four-node cubic element incompressible state (𝜈 → 12 ) Incompressibility is associated, for example, with a practical bridge support made up of elastomeric rubber materials, or fluid flow problems as in modeling, say, reservoir water in dam safety computations. The rubber material and the fluid are nearly incompressible. For the dilatation locking problem arising out of it, consider the following state of stress imposed on a old four-node rectangular membrane element: in-plane pure bending state due to a couple of moment applied at the opposite end of the element in plane strain situation. The state of actual displacement is given by: u = cxy

v = cx x2 + cy y2 𝜈 and cy ≡ − c 2(1 − 𝜈)

and

where, cx ≡ − 12 c

(9.130)

The corresponding states of actual strain and stress for the plane strain situation (i.e. 𝜎z ≠ 0) with E, 𝜈 as the elasticity modulus and Poisson’s ratio, respectively, are given by: 𝜖x = cy 𝜖y = −

𝜎x =

𝜈 cy 1−𝜈

𝛾xy = 0

Ecy 1 − 𝜈2

𝜎y = 0

(9.131)

𝜏xy = 0

For future reference, let us recall, from Chapter 1, the following definitions related to volumetric strains and stresses: Dilatation, e: e ≡ 𝜖 x + 𝜖y + 𝜖z = 𝜈→

1 2

(

) 1 − 2𝜈 (cy), 1−𝜈

⇒e→0

(9.132)

Computation of Nonlinear Structures

516

Bulk modulus, K: K=

E 3(1 − 2𝜈)

(9.133)

Mean pressure, p: p = −(𝜎x + 𝜎y + 𝜎z ) = −K e =

−Ecy , 3(1 − v)

𝜈→

1 2

⇒ p=

−2Ecy M is computationally not recommended.

10.20.3.2

Rational Elements

For curve generation, the root element of an n-degree typical rational c-type geometric element interpolating the beam centroidal axis is a unit straight line and the geometric interpolation of the element itself can be written as:

C0 (𝜉) =

L ∑

Fi (𝜉) c qi

j=0

Fi (𝜉) ≡

wi BLi (𝜉) L ∑ j=0

wi BLi (𝜉)

(10.532)

Nonlinear Beams

689

where C0 (𝜉) ≡ {X(𝜉) Y(𝜉) Z(𝜉)}T , as before, defines the position vector of a point on the beam element, 𝜉 ∈ [0, 1]; Fi (𝜉) for i ∈ {0, 1, … , L} is the ith rational basis functions; wi for i ∈ {0, 1, … , L} are the weights at the control points.

r For generalized displacement interpolation, we still choose the non-rational curves as given in r

the equation (10.531) because, for unknown displacement distribution, we would like to avoid choosing ad hoc estimates for the corresponding required weights. In what follows, we will restrict our discussion to non-rational elements.

10.20.4

Bezier–Bernstein Geometric Curve Element

For a typical non-rational beam finite element, then, a nc -degree (i.e. Nc = nc + 1 as the order of the polynomial) typical c-type beam geometry can be written as: C0 (𝜉) = ⏟⏟⏟ 3×1

nc ∑

n

Bi c (𝜉)

i=0

⏟⏞⏞⏟⏞⏞⏟

c

qi , ⏟⏟⏟

(10.533)

3Nc ×1

3×3Nc c

where C0 (𝜉) ≡ {X(𝜉) Y(𝜉) Z(𝜉)}T , Bni (𝜉), c qi and nc are the geometric position vector, the Bernstein–Bezier basis functions, the geometry control vectors and the degree of the geometry element, respectively, at any point 𝜉 ∈ [0, 1]. The equation (10.533) can be rewritten in transformation form as: c C0 (𝜉) = c T(𝜉) q , ⏟ ⏟ ⏟ ⏟ ⏟ ⏟ ⏟⏟⏟ 3×3Nc

3×1

(10.534)

3Nc ×1

where we have: q = {qX0 ⋯ qXn qY0 ⋯ qYn qZ0 ⋯ qZn }T c c c ⏟⏟⏟ ⏟⏞⏟⏞⏟ ⏟⏞⏟⏞⏟ ⏟⏞⏟⏞⏟ c

3Nc ×1

1×Nc

1×Nc

(10.535)

1×Nc

̂ 𝜉) ̂ as the (3 × 3Nc ) geometry interpolation transthe (3Nc × 1) geometry control vector, and c T( formation matrix containing the Bernstein–Bezier polynomials, that is, n

n

n

n

c c Bncc Bncc 0... ⋯ ...0 ⎤ ⎡B0 B1 ⎥ ⎢ nc nc nc c ...0 B0 B1 Bnc T(𝜉) = ⎢ 0... ⎥ ⎢ nc nc nc ⎥ ⎣ 0... ...0 B0 B1 Bnc ⎦ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

(10.536)

3×3Nc

Note that Nc ≡ nc + 1 is the order of the geometric element with degree nc . Remarks

r It is recommended whenever possible to have geometry representation at most cubic, that is, nc ≤ 3 for ease and efficiency in the computation of tangents, curvatures and twists that are

Computation of Nonlinear Structures

690

r

subsequently necessary. So, as an example, for a cubic Bernstein–Bezier geometry polygon, that is, nc = 3, this results in the order of the geometry as: N = 4. Thus, we will have 12 (i.e. 3 sets of 4 geometry control nets) as the dimension of the geometry control vector, c q, and the interpolation transformation matrix is of dimension 3 × 4. Additionally, whenever conic curves (i.e. circles, ellipses, etc.) are involved, the geometric representation must be of rational Bernstein–Bezier form for accuracy.

Example: Let us consider a linear element, that is, n = 1, N = 2. We have: C0 (𝜉) = ⏟⏟⏟ 3×1

1 ∑

B1i (𝜉)

i=0

⏟⏞⏞⏟⏞⏞⏟

c

qi , ⏟⏟⏟

(10.537)

2×1

3×2

with: B10 (𝜉) ≡ 1 − 𝜉,

B11 (𝜉) ≡ 𝜉

(10.538)

Implying c q = { qX0 qX1 qY0 qY1 qZ0 qZ1 }T ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟ 6×1

1×2

1×2

(10.539)

1×2

and 0 0 ⎤ ⎡ T ⎢⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟⎥ 1×2 1×2 ⎥ ⎢ 1×2 ⎢ 0 T 0 ⎥ c T(𝜉) = ⎢⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟⎥ where T ≡ {(1 − 𝜉) ⎢ 1×2 1×2 1×2 ⎥ ⎢ ⎥ 0 T ⎥ ⎢ 0 ⎢⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟⎥ ⎣ 1×2 1×2 ⎦ 1×2 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

𝜉}

(10.540)

3×6

10.20.5

The c-type Beam Elements and State Control Vectors

Let us, then, introduce a n-degree c-type beam element as: ⌣

⌣

⌣

d(𝜉) = Td (𝜉) qd ,

⌣

d j (𝜉) =

n ∑ i=0

where we have introduced the following:

⌣dj

Bni (𝜉) q i ,

(10.541) j = 1, … , 6

Nonlinear Beams

691

Definitions

r ⌣qd = {qu qu ⋯ qu ⋯ q𝜃b ⋯ q𝜃nb }T be the (6N × 1) displacement control vector. n 0 0 1 r ⌣q̄ d be the (6N × 1) virtual displacement control vector, similar to ⌣qd , r Δ⌣qd , be the (6N × 1) incremental control vector, similar to ⌣qd , r T⌣d (𝜉) be the (6 × 6N) interpolation transformation matrix containing the Bernstein–Bezier polynomials, that is, ⎡Bn0 ⋯ Bnn 0 0 ⋯ ⋯ ⋯ 0 ⎤ ⎥ ⎢ ⎢ 0 0 0 Bn0 ⋯ Bnn 0 ⋯ 0 ⎥ ⌣ ⎥ ⎢ Td (𝜉) = ⎢ ⋯ ⋯ ⎥ ⎥ ⎢ ⋯ ⋯ ⎥ ⎢ ⎢0 0 ⋯ ⋯ n n 0 B0 ⋯ Bn ⎥⎦ ⎣ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

(10.542)

6×6N

Now, let us further introduce the following. Definition ⌣ r B(𝜉) be the (12 × 6N) augmented interpolation transformation matrix as: ⌣ ⎡ T d (𝜉) ⎤ ⎢ ⏟⏟⏟ ⎥ ⎢ 6×6N ⎥ ⎥ ⎢ ⌣ B(𝜉) ≡ ⎢− − − − −⎥ ⎥ ⎢ ⌣ ⎢ 1 Td , (𝜉) ⎥ 𝜉 ⎥ ⎢ J ⎢ ⏟⏞⏟⏞⏟ ⎥ ⎣ 6×6N ⎦

(10.543)

where Td ,𝜉̂ is computed by the first derivative recursive formula of the Bernstein basis functions at Gauss points as: ) ( d (𝜉) − Bn−1 (𝜉) B(𝜉) = n Bn−1 i i−1 d𝜉

(10.544)

̂ d𝜉, ̂ of the area integrals where Finally, we must express the infinitesimal length, dS = J(𝜉) ̂ is the determinant of the Jacobian matrix expressed in terms of the geometry coordinate J(𝜉) ̂ Based on the above definitions, we have: parameters, 𝜉. ⌣

⌣

⌣

⌣

⌣

⌣

⌣

⌣

d = Td (𝜉) qd

d̄ = Td (𝜉) q̄ d ⌣

Δd = Td (𝜉) Δqd

(10.545)

Computation of Nonlinear Structures

692

and ⌣

⌣

⌣

̄ = B(𝜉) H(d) q̄ d ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟ 12×1 ⌣

12×6N ⌣

6N×1

(10.546)

⌣

H(Δd) = B(𝜉) Δqd ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟ 12×1

12×6N

6N×1

r Note that both the T̂ d (𝜉) ̂ 𝜉) ̂ and B( ̂ matrices for all Gauss points can be computed and saved r r

r

once, and stored away for repetitive use in the Newton-type iterative algorithm; of course, in adaptive refinements the locations of the Gauss points change, requiring recomputation. For convergence, we must have the elements at most isoparametric, that is, nc = n; we must always avoid superparametry, that is, nc is never allowed to exceed n. For computational efficiency such as tangent and curvature computations, and so on, a subparametric representation, that is, nc strictly less than n, with nc = 3, that is, up to cubic spline representation for the non-rational geometry, seems an optimal choice. Similarly, for rational geometry such as circles, ellipses, and so on, the quadratic rational spline is the desired geometric representation. It turns out that for general 3D beam finite element analysis, the optimum choice for the generalized displacement distribution must be with a quintic Bernstein–Bezier net, that is, n = 5, resulting in the order of the element as: N = 6. Thus, we will have 36 (i.e. 6 × 6 control points) as the dimension of the generalized displacement control vector, d q, and the interpolation transformation matrix is of dimension 6 × 36; however, if desired, all the control points other than the end control points can easily be statically condensed to produce compact matrices.

10.20.6

Computational Symmetric Material Stiffness Matrix, KCM

By substituting equation (10.502) into equation (10.483), we get: 12×6 6×12 6×6 ⎡ ⎧ ⎫ ⎤ ⏞⏞⏞ ⏞⏞⏞ ⏞⏞⏞ ⎢ ⎪ ⎪ ⎥ 1 ⌣ ⌣ ⌣ ⌣ ⌣ ⌣ ⌣ ⌣ ⌣ ⎢ ⎪ ⎪ ⎥ ⌣ d T T T n ̄ Δd) ≡ q̄ M (d, d; D B(𝜉) ⎬ J(𝜉) d𝜉 ⎥ Δqd E(d) E(d) ⎢∫ ⎨ B(𝜉) ⏟⏟⏟ ⎢ 0 ⎪⏟⏟⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏟⏟⎪ ⎥ ⏟⏟⏟ 12×12 12×6N ⎪ 1×6N ⎢ ⎪ 6N×12 ⎥ 6N×1 ⎣ ⎦ ⎩ ⎭ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ 6N×6N ⌣

⌣d

= q̄ d T KC Δq M ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟ 1×6N

6N×6N

6N×1

(10.547) where, we have introduced:

Nonlinear Beams

693

Definition: The computational material stiffness matrix, KC , as: M 12×6 6×12 6×6 ⎧ ⎫ ⏞⏞⏞ ⏞⏞⏞ ⏞⏞⏞ ⎪ ⎪ 1 ⌣ ⌣ ⌣ ⌣ ⎪⌣ T ⎪ C T n KM ≡ D B(𝜉) ⎬ J(𝜉) d𝜉 E(d) E(d) ⎨ B(𝜉) ∫ 0 ⎪⏟⏟⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏟⏟⎪ ⏟⏟⏟ 12×12 12×6N ⎪ ⎪ 6N×12 6N×6N ⎩ ⎭ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

(10.548)

6N×6N

Exercise: Derive expression given by equation (10.548)

10.20.6.1

Code Fragment for E Matrix

Subroutine FormEMatrix(AMat,C0,nC,E,nE,W,R,AA,CK,nG,Iout) ! ---------------------------------------------------------------------------! Inputs ! AMat(nC,nC) = Rotation Matrix ! nG = 12 ! C0(nC) = init.Curvature vec = {torsion,0,curvature} = k0 ! nC = 3 ! nE = 6 ! W(nC,nC) = W Matrix ! R(nC,nC) = R Matrix ! AA(nC) = e + [k0]d ! CK(nC) = k0 + k = k0 + W(thetaPrimed) ! ! Outputs ! E(nE,nG) = E Matrix ! -----------------------------------------------------------Implicit Real(kind=8) (a-h,o-z) ! Real(kind=8) C0 ,E ,AMat Dimension C0(nC),E(nE,nG),AMat(nC,nC) Real(kind=8) E11 ,E12 Dimension E11(nC,nC),E12(nC,nC) Real(kind=8) E13 ,E22 ,E23 ,E24 Dimension E13(nC,nC),E22(nC,nC),E23(nC,nC),E24(nC,nC) Real(kind=8) CK ,AA Dimension CK(nC),AA(nC) Real(kind=8) W ,R Dimension W(nC,nC),R(nC,nC) ! DATA zero/0.D0/,one/1.0D0/,two/2.0D0/,three/3.0D0/ !- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Get EBlocks call FormEBlocks(C0,nC,W,R,AA,CK,AMat,E11,E12,E13,E22,E24,Iout)

694

Computation of Nonlinear Structures

!- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Form EMatrix E = zero ! -------do 40 i = 1,3 do 40 j = 1,3 E(i,j) = E11(i,j) E(i,j+3) = E12(i,j) E(i,j+6) = E13(i,j) E(i+3,j+3) = E22(i,j) E(i+3,j+9) = E24(i,j) 40 continue !- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - return End Subroutine FormEBlocks(C0,nC,W,R,AA,CK,AMat,E11,E12,E13,E22,E24, & Iout) ! ---------------------------------------------------------------------------! Inputs ! C0(nC) = init. Curvature vector = {torsion,0,curvature} ! nC = 3 ! AMat = Rotation Matrix ! W = matrix ! R = matrix ! AA = a Vector ! CK = Kc Vector ! ! Outputs ! E11(nC,nC) = E11 block ! E12(nC,nC) = E12 block ! E13(nC,nC) = E13 block ! E22(nC,nC) = E22 block ! E24(nC,nC) = E24 block ! -----------------------------------------------------------Implicit Real(kind=8) (a-h,o-z) ! Real(kind=8) C0 Dimension C0(nC) Real(kind=8) E11 ,E12 ,E13 ,E22 Dimension E11(nC,nC),E12(nC,nC),E13(nC,nC),E22(nC,nC) Real(kind=8) CK ,AA ,AMat ,TAMat Dimension CK(nC),AA(nC),AMat(nC,nC),TAMat(nC,nC) Real(kind=8) W ,R ,E24 ,sAA ,sKc Dimension W(nC,nC),R(nC,nC),E24(nC,nC),sAA(nC,nC),sKc(nC,nC) ! DATA zero/0.D0/,one/1.0D0/,two/2.0D0/,three/3.0D0/ !- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -AAMatTran TAMat = TRANSPOSE(AMat) !- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -E11Block

Nonlinear Beams

695

call FormSkew(C0,nC,E11,Iout) !skew[k0] E11 = MATMUL(TAMat, E11) !AMatTransposeE11 !- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -E12Block call FormSkew(AA,nC,sAA,Iout) !skew[a] E12 = MATMUL(sAA, W) ![a]W E12 = MATMUL(TAMat, E12) !AMatTransposeE12 !- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -E13Block E13 = TAMat !AMatTransposeE13 !- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -E22Block call FormSkew(CK,nC,sKc,Iout) !skew[Kc] E22 = R + MATMUL(sKc,W) !R + [Kc]W E22 = MATMUL(TAMat, E22) !AMatTransposeE22 !- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -E24Block E24 = W E24 = MATMUL(TAMat, E24) !AMatTransposeE24 !- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - return end

10.20.7

Computational Symmetric Geometric Stiffness Matrix, KGC

By substituting equation (10.546) into equation (10.529), we get: ⎡ ⎧ ⎫ ⎤ 1 ⌣ ⌣ ⌣ ⌣ ⎢ ⎪ ⎪ ⎥ ⌣ T ̄ Δd) ≡ q̄ G (d, d; G(d, F) B(𝜉) ⎬ J(𝜉) d𝜉 ⎥ Δqd ⎢∫ ⎨ B(𝜉) ⏟⏟⏟ ⎢ 0 ⎪⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟⎪ ⎥ ⏟⏟⏟ ⎦ 6N×1 ⎩ 6N×12 12×12 12×6N ⎭ 1×6N ⎣ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⌣

⌣

⌣

⌣

dT

(10.549)

6N×6N ⌣

⌣

= q̄ d T KGC Δqd ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟ 1×6N

6N×6N

6N×1

where, we have introduced the following. Definition: The computational geometric stiffness matrix, KGC , as: ⎧ ⎫ ⌣ ⌣ ⌣ ⎪⌣ T ⎪ ≡ G(d, F) B(𝜉) ⎬ J(𝜉) d𝜉 ⎨ B(𝜉) ∫ ⏟⏟⏟ ⏟ ⏟ ⏟ ⏟ ⏟ ⏟ 0 ⎪ ⏟⏟⏟ ⎪ ⎩ 6N×12 12×12 12×6N ⎭ 6N×6N ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ 1

KGC

6N×6N

(10.550)

Computation of Nonlinear Structures

696

where the symmetric geometric stiffness matrix, G, is given by: ⎡ 0 ⎢ T G ̂ F) ̂ ≡ ⎢⎢ 12 G(d, ⎢ 0 ⎢ ⎣ 0

G12

0

G22

G23

GT 23 GT 24

0

0 ⎤ ⎥ G24 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎦

0

(10.551)

with G12 ≡ [k0 ][ t F] W G22 ≡ 12 WT {[ t F ⊗ a] + [a ⊗ t F] − 2( t F a)}W + c2 (θ bf )I + 12 ([hb ⊗ θ] + [θ ⊗ hb ]) + 12 WT {[ t M ⊗ kc ] + [kc ⊗ t M] − 2( t M kc )}W + c2 (θ bm )I

(10.552)

+ 12 ([hm ⊗ θ] + [θ ⊗ hm ]) + WT [ t M] X + X [ t M]T W G23 ≡ WT [ t F] G24 ≡ WT [ t M] W + Y( t M) and c1 ≡ (1 − cos 𝜃)∕𝜃 2 , c2 ≡ (𝜃 − sin 𝜃)∕𝜃 3 , c̄ 1 ≡ 𝛿c1 ≡ c3 ≡ {(1 − 𝜃 2 c2 ) − 2c1 }∕𝜃 2 ,

(10.553)

c̄ 2 ≡ 𝛿c2 ≡ c4 ≡ (c1 − 3c2 )∕𝜃 2 a ≡ e + [k0 ]d = {(1̂ + d,s ) + [k0 ]d} kt ≡ W θ,s bf

≡

Ft

kc ≡ k0 + kt bm

× a

≡

Mt

×

(10.554)

kc

W ≡ I + c1 𝚯 + c2 𝚯2 X𝛼 ≡ −c1 𝚯,𝛼 +c2 {[θ ⊗ θ,𝛼 ] − [θ,𝛼 ⊗ θ] + (θ ∙ θ,𝛼 )I} + V [θ,𝛼 ⊗ θ]) V ≡ −c2 I + c3 𝚯 + c4 𝚯2 Y(x) ≡ −c1 [x] + c2 ([x ⊗ θ] − [θ ⊗ x]) + c2 (θ ∙ x)I + [θ ⊗ VT x], f

f

h𝛼 ≡ VT b𝛼

∀x ∈ R3

T m hm 𝛼 ≡ V b𝛼

(10.555)

Nonlinear Beams

697

Exercise: Derive expression given by equation (10.550).

10.20.7.1

Code Fragment for G Matrix

Subroutine FormGMatrix(HD,nG,FM,nF,C0,nC,G,W,R,AA,CK,Iout) ! ---------------------------------------------------------------------------! Inputs ! HD(nG) = {dis,rot,disPrimed,rotPrimed} ! nG = 12 ! FM(nC) = {force,moment} ! nF = 6 ! C0(nC) = init. Curvature vector = {torsion,0,curvature} ! nC = 3 ! ! Outputs ! G(nG,nG) = GMatrix ! -----------------------------------------------------------Implicit Real(kind=8) (a-h,o-z) ! Real(kind=8) HD ,FM ,C0 ,G Dimension HD(nG),FM(nF),C0(nC),G(nG,nG) Real(kind=8) G12 ,G22 ,G23 ,G24 Dimension G12(nC,nC),G22(nC,nC),G23(nC,nC),G24(nC,nC) ! DATA zero/0.D0/,one/1.0D0/,two/2.0D0/,three/3.0D0/ !- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Get GBlocks call FormGBlocks(HD,nG,FM,nF,C0,nC,W,R,AA,CK,G12,G22,G23,G24,Iout) !- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Form GMatrix G = zero ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Upper Triangle with Diagonals do 10 i = 1,3 do 10 j = 1,3 G(i,j+3) = G12(i,j) G(i+3,j+3) = G22(i,j) G(i+3,j+6) = G23(i,j) G(i+3,j+9) = G24(i,j) 10 continue !- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Symmetry:Lower Triangle do 50 i = 1,11 do 50 j = i+1,12 G(j,i) = G(i,j) 50 continue !- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - return End

&

Subroutine FormGBlocks(HD,nG,FM,nF,C0,nC,W,R,AA,CK, G12,G22,G23,G24,Iout)

698

Computation of Nonlinear Structures

! ---------------------------------------------------------------------------! Inputs ! HD(nG) = {dis,rot,disPrimed,rotPrimed} ! nG = 12 ! FM(nC) = {force,moment} ! nF = 6 ! C0(nC) = initCurvature vector = {torsion,0,curvature} ! nC = 3 ! ! Outputs ! Gij(nC,nC) = GBlocks ! -----------------------------------------------------------Implicit Real(kind=8) (a-h,o-z) ! Real(kind=8) HD ,FM ,C0 Dimension HD(nG),FM(nF),C0(nC) Real(kind=8) G12 ,G22 ,G23 ,G24 Dimension G12(nC,nC),G22(nC,nC),G23(nC,nC),G24(nC,nC) Real(kind=8) Di ,Dp ,Ro ,Rp ,Fo ,XM Dimension Di(nC),Dp(nC),Ro(nC),Rp(nC),Fo(nC),XM(nC) Real(kind=8) CK ,AA ,BB ,CC ,Hat1 Dimension CK(nC),AA(nC),BB(nC),CC(nC),Hat1(3) Real(kind=8) W ,R ,SM ,SF ,SC Dimension W(nC,nC),R(nC,nC),SM(nC,nC),SF(nC,nC),SC(nC,nC) Real(kind=8) FA ,XMK ,TA ,ATA Dimension FA(nC,nC),XMK(nC,nC),TA(nC,nC),ATA(nC,nC) Real(kind=8) WMR ,TMT ,SMt Dimension WMR(nC,nC),TMT(nC,nC),SMt(nC,nC) ! common/ThetaConstants/c1,c2,c1b,c2b ! DATA zero/0.D0/,one/1.0D0/,two/2.0D0/,three/3.0D0/ DATA Hat1/1.0D0,0.D0,0.D0/ ! -------------------------------------------------------------- - - - - H(dHat) ! call Extract_Vector(HD,nG,Di,nC,1,Iout) !displacement call Extract_Vector(HD,nG,Ro,nC,4,Iout) !rotation ! call Extract_Vector(HD,nG,Dp,nC,7,Iout) !displacementPrimed call Extract_Vector(HD,nG,Rp,nC,10,Iout)!rotationPrimed ! call Extract_Vector(FM,nF,Fo,nC,1,Iout) !Force call Extract_Vector(FM,nF,XM,nC,4,Iout) !Moment !- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -G22Block:Translational ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - FA = WTran(Fxa + axF -2(F.a)I)W call FormF_a_PartMatrix(Fo,AA,FA,3,1,Iout) !Fxa + axF -2(F.a)I ! call Form_WMatrix(Ro,W,nC,c1,c2,c1b,c2b,Iout) FA = MATMUL(FA, W) !(Fxa + axF -2(F.a)I)W FA = MATMUL(TRANSPOSE(W), FA) !WTran(Fxa + axF -2(F.a)I)W ! call CrossProduct(Fo,AA,BB,Iout) ! b = F x a

Nonlinear Beams

699

call DotProduct(Ro,BB,dot,nV,Iout) ! theta dot b C2TdotB = c2*dot ! c2*(theta dot b) ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - .5*FA + c2(theta dot b)I FA = 0.5D0*FA FA(1,1) = FA(1,1) + C2TdotB FA(2,2) = FA(2,2) + C2TdotB FA(3,3) = FA(3,3) + C2TdotB ! -------------------------------------------------------------- - - - - - - - - - - - - - - Rotational ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -XMK = WTran(Mxkc + kcxM 2(M.kc)I)W call FormF_a_PartMatrix(XM,CK,XMK,3,2,Iout)!Mxkc + kcxM -2(M.kc)I ! XMK = MATMUL(XMK, W) !(Mxkc + kcxM -2(M.kc)I)W XMK = MATMUL(TRANSPOSE(W), XMK)!WTran(Mxkc + kcxM -2(M.kc)I)W ! call CrossProduct(XM,CK,CC,Iout) ! c = M x kc call DotProduct(Ro,CC,dot,nV,Iout) ! theta dot c C2TdotC = c2*dot ! c2*(theta dot c) ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - .5*XMK + c2(theta dot c)I XMK = 0.5D0*XMK XMK(1,1) = XMK(1,1) + C2TdotB XMK(2,2) = XMK(2,2) + C2TdotB XMK(3,3) = XMK(3,3) + C2TdotB ! --------------------------------------------------------------AlfaCxRot + RotxAlfaC call FormThetaAlfaTrans_Matrix(Ro,CC,TA,nC,c1,c2,c1b,c2b,Iout) !RotxAlfa ATA = 0.5D0*(TA + TRANSPOSE(TA)) ! --------------------------------------------------------------AlfaBxRot + RotxAlfaB call FormThetaAlfaTrans_Matrix(Ro,BB,TA,nC,c1,c2,c1b,c2b,Iout) !RotxAlfa ATA = ATA + 0.5D0*(TA + TRANSPOSE(TA)) ! ---------------------------------------------------------WTranMR+RTranMTranW call FormSkew(XM,nC,SM,Iout) !skew[M] call FormR_Matrix(Ro,Rp,R,nC,c1,c2,c1b,c2b,Iout) !R matrix ! WMR = MATMUL(TRANSPOSE(W),SM) !WTran[M] WMR = MATMUL(WMR,R) !WTran[M]R ! WMR = WMR + TRANSPOSE(WMR) ! -------------------------------------------------------------- - - - - - - - - - G22Block G22 = FA + XMK + ATA + WMR !- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -G12Block !- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - --------------------------------call FormSkew(Fo,nC,SF,Iout) !skew[F] call FormSkew(C0,nC,SC,Iout) !skew[k0] ! G12 = MATMUL(SC,SF) ![k0][Ft] G12 = MATMUL(G12,W) ![k0][Ft]W

Computation of Nonlinear Structures

700

!- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -G23Block G23 = MATMUL(TRANSPOSE(W),SF) !WTran[Ft] !- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -G24Block TMT = MATMUL(TRANSPOSE(W),SM) !WTran[Mt] TMT = MATMUL(TMT,W) !WTran[Mt]W ! call FormS_Matrix(Ro,XM,SMt,nC,c1,c2,c1b,c2b,Iout) !S(Mt) Matrix ! G24 = TMT + SMt !- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - return end

10.20.8

Unbalanced Force

By substituting equation (10.498) into equation (10.530), we get: Unbalanced force

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⌣

⌣

⌣

⌣

⌣

̄ = q̄ d T R(𝜆0 , d0 ) G(𝜆0 , d0 ; d)

(10.556)

where, we have defined the following. Definition ⌣

{

⌣

R(𝜆0 , d0 ) ≡

1 ⌣

∫0

⌣

⌣

⌣

⌣

⌣

⌣

⌣

⌣

}

[BT ET n F − TT FB (𝜆0 , 𝜉) ]J(𝜉) d𝜉 − 𝜆0 (TT (1) FS (1) − TT (0) FS (0)) (10.557)

10.20.9

Incremental Applied Force

By substituting equation (10.498) into the applied force part of equation (10.524), the applied external force for a step can be expressed as: Incremental applied force

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ {( ) [⌣ ]𝜉=1 } 1⌣ ⌣ ⌣ ⌣ ⌣ ̄dT FB ,𝜆 (𝜆) J(𝜉) d𝜉 + d̄ T FS Δ𝜆 = q̄ d T p(𝜆) Δ𝜆 ∫0 𝜉=0

(10.558)

where, we have defined the following. Definition 1⌣

⌣

p(𝜆) ≡

∫0

⌣

⌣

⌣

⌣

⌣

TT (𝜉) FB (𝜆, 𝜉),𝜆 J(𝜉) d𝜉 + TT (1) FS (1) − TT (0) FS (0)

(10.559)

Nonlinear Beams

10.20.10

701

Geometrically Exact Element Linearized Incremental Equation

Now substituting the expressions given in equations (10.548), (10.550), (10.557), and (10.559) in equation (10.524), we have: { } ⌣ ⌣ ⌣ ⌣ ⌣d ⌣ C ⌣d ( q ) + K ( q ) Δqd = q̄ d T {p(𝜆) Δ𝜆 − R(𝜆0 , d0 )} q̄ d T KC M G

⌣

(10.560)

Noting that equation (10.560) above is true for any arbitrary admissible generalized virtual ⌣ control displacement, q̄ d , we can finally write the geometrically exact linearized finite element ⌣ equation in unknown incremental generalized displacement control, Δqd , and the incremental load factor, Δ𝜆, as: ⌣

⌣

⌣

⌣

KcT Δqd + p(𝜆) Δ𝜆 = −R(𝜆0 , d0 )

(10.561)

where, we have introduced the following. Definition: The computational tangent stiffness matrix, KcT : C KcT ≡ KC M + KG

10.20.11

(10.562)

Implementation: Accuracy and Symmetry Preservation

Our main goal here is to discuss various important issues and solutions relating to economical, robust and stable numerical implementation, and to provide several illustrative and representative examples for beams under quasi-static loading resulting in nonlinear (finite displacement and rotation) responses. As to numerical implementation, the following topics are in order:

r Numerical integration of the material stiffness and the geometric stiffness: the solution requires application of Gaussian quadrature of appropriate degree to the integrals. symmetry of the tangent operator during iteration for proportional loading: this is achieved by treating separately the linearized equations related to incremental displacement and rotations, (Δd, Δθ), from that involving the incremental proportionality factor, Δ𝜆. Newton-type iterative solution of nonlinear matrix equations.

r Maintaining r

10.20.12

Gauss Quadrature Points

For a quintic Bernstein–Bezier polygon, for example, over an element, that is, with n = 5, we have for the order of the element as: N = 6. Thus, for full integration, we will estimate a set of six Gauss quadrature points. With the locations of the Gauss quadrature points fixed, we need to construct the local Frenet frame at these points with the triad of base vectors, T, M, B; by taking derivatives of equation (10.533) with respect to the curve parameter, we get:

Computation of Nonlinear Structures

702

T= B=

C,𝜉 (𝜉) ||C,𝜉 (𝜉)|| C,𝜉 (𝜉) × C,𝜉𝜉 (𝜉)

(10.563)

||C,𝜉 (𝜉) × C,𝜉𝜉 (𝜉)||

M=B × T where we use the following first-order forward difference operator, Δ1 as: Δ1 c qi ≡ c q(i+1) − c qi

(10.564)

Note that the great advantage of using difference operators is that we only need to know and use the differences of the original Bezier control points of the geometry. With this background, we can express all the components of the desired curvature vector: k0 ≡ {𝜏 0 𝜅}T in terms of the coordinate parameters, 𝜉, with the local radius of curvature, 𝜌 = 𝜅1 , is given by 𝜌=

||C,𝜉 ||3 ||C,𝜉 × C,𝜉𝜉 ||

given by 𝜏 =

, 𝜅, the curvature, is defined as the reciprocal of 𝜌. The twist, 𝜏, of the curve is

C,𝜉 ×C,𝜉𝜉 ∙ C,𝜉𝜉𝜉 ||C,𝜉 × C,𝜉𝜉 ||2

. Finally, J(𝜉), the determinant of the Jacobian matrix is evaluated at

all the Gauss quadrature points of integration. Remark

r Thus, note that the reference curve properties: the initial curvature vectors, k0 along with the ̂ of the Jacobian matrix, can be computed at all Gauss points and saved once, determinant J(𝜉) and stored away for repetitive use in the Newton-type iterative algorithm; of course, in adaptive refinement, the locations of the Gauss points change, requiring recomputation.

10.20.12.1

Code Fragment for Reference Geometry Info

Subroutine Get_ReferenceInfo(Qc,nDim,nQc,RoCOld,CuCOld,nCInt, & Ri,jInt,R0,C0,CK,nC,T,B, & nDof,nDof2,nStif,Det,Iout,Nel,Nint) ! ---------------------------------------------------------------------------! Get Reference Geometric Info. ! ---------------------------------------------------------------------------! Inputs ! Qc(nDim,nQc) = Geometry Controls ! nDim = 3 ! nQc = order of Bezier ! RoCOld(nCInt) = rot Controls:Old for all Gauss point ! CuCOld(nCInt) = Curvature:Old ! nCInt = no. of Gauss Integration points * nDim (=3) ! = nDim*nInt

Nonlinear Beams

! ! ! ! ! ! ! ! ! ! ! ! ! !

!

!

Ri jInt nDof nDof2 nStif

703

= = = = =

Current Gauss location Pt. Current Gauss Index 6 12 6*nQc

Outputs R0(nDim) = rot Controls:Old for the current Gauss Pt. C0(nDim) = Ref Curvature:for the current Gauss Pt. T(nDof,nStif) = Dof Transformation Control matrix B(nDof,nStif) = Dof2 Transformation Control matrix Det = Determinant (Jacobian) -----------------------------------------------------------Implicit Real(kind=8) (a-h,o-z) Input Variables & Arrays Real(kind=8) Qc Dimension Qc(nDim,nQc) Real(kind=8) RoCOld ,CuCOld Dimension RoCOld(nCInt),CuCOld(nCInt) Working Variables & Arrays Real(kind=8) Br ,H1D ,Axes Dimension Br(nQc),H1D(nQc),Axes(3,3) Output Variables & Arrays Real(kind=8) B ,T Dimension B(12,nStif),T(6,nStif) Real(kind=8) R0 ,C0 ,CK Dimension R0(nDim),C0(nDim),CK(nDim) character*3 readstat

!========================================================== LOGICAL bReset common/LoadStepIter/iStep,iNut,bReset common/Scratch/iRotCur1,iRotCur2 !========================================================== DATA zero/0.D0/,one/1.0D0/,two/2.0D0/,three/3.0D0/ ! -----------------------------------------------------------call TMB_CuLn(Qc,nQc,nOrderR,Ri,Br,H1D,Det,Cur,Tor, & Axes,dxbardr,Nel,Iout,iPrt) C0(1) = Tor C0(2) = zero C0(3) = Cur AJI = one/det kr = (jInt-1)*nDim+1 call Extract_Vector(RoCOld,nCInt,R0,nC,kr,Iout) !rotationControlOld call Extract_Vector(CuCOld,nCInt,CK,nDIm,kr,Iout) CurvatureControlOld ! - - - - - - - - - - - - - - - - - - - initialize with geometric curvature if(iStep == 1) Then CK = C0 endif !- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Get !T(nDof,nDof*nQc),B(nDof2,nDof*nQc) call Form_B_and_Tmatrices(Br,H1D,T,nDof,B,nDof2,nStif,AJI, nQc,Iout) !- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - return

704

Computation of Nonlinear Structures

end ! ! ! ! ! ! ! ! ! ! !

Subroutine TMB_CuLn(Qc,nQc,R,Det,Cur,Tor,Axes,dxbardr) Inputs: Qc(1) = Control X-Coordinates = (qc0x qc1x ... qc3x) Qc(2) = Control Y-Coordinates = (qc0y qc1y ... qc3y) Qc(3) = Control Z-Coordinates = (qc0z qc1z ... qc3z) R = root domain value Outputs: Axes = T,M,B Cur = curvature Tor = torsion Det = determinant of The Jacobian

Implicit Real(kind=8) (a-h,o-z) Real(kind=8) Qc(3,nQc),Axes(3,3),Det,R,Rad Real(kind=8) dum,delCx,delCy,delCz,xMult,pi Real(kind=8) de2Cx,de2Cy,de2Cz,de3Cx,de3Cy,de3Cz Real(kind=8) Br(nQc), B1Dr(nQc),dxbardr,H1D(nQc),,H2D(nQc),H3D(nQc) Real(kind=8) xDot(3) ,xDo2(3) ,xDo3(3),AT(3),AM(3),AB(3),AJ, AJI !- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Displacement/Bernstein call Bernstein(Br,nQc,r,iOut) !- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Ist Derivative call Bernstein_IstDerivative(B1Dr,nQc,r,iOut) !- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Derivative of H wrt r and s do 10 i = 1,nQc H1D(i) = B1Dr(i) 10 continue !- - - - - - - - - - - - Jacobian delCx = 0.0D0 do 20 k = 1,nQc delCx = delCx + H1D(k)*Qc(1,k) 20 continue delCy = 0.0D0 do 30 k = 1,nQc delCy = delCy + H1D(k)*Qc(2,k) 30 continue delCz = 0.0D0 do 40 k = 1,nQc delCz = delCz + H1D(k)*Qc(3,k) 40 continue xDot(1) = delCx xDot(2) = delCy xDot(3) = delCz ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Determinant of Jacobian det = dsqrt((delCx*delCx)+(delCy*delCy)+(delCz*delCz)) if(det.gt.0.0000001) go to 45 stop ’det 0’ 45 continue !- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 2nd Derivative xn = Factorial(nQc-1)/Factorial(nQc-3) call Bernstein(B1Dr,nQc-2,r,iOut)

Nonlinear Beams

!

705

-------------H2D = 0.D0 ! ALL do 50 i = 1,nQc-2 H2D(i) = B1Dr(i) 50 continue de2Cx = 0.0D0 do 60 k = 1,nQc-2 d2Qc = Qc(1,k+2) – 2.D0*Qc(1,k+1) + Qc(1,k) de2Cx = de2Cx + xn*H2D(k)*d2Qc 60 continue de2Cy = 0.0D0 do 70 k = 1,nQc-2 d2Qc = Qc(2,k+2) – 2.D0*Qc(2,k+1) + Qc(2,k) de2Cy = de2Cy + xn*H2D(k)*d2Qc 70 continue de2Cz = 0.0D0 do 80 k = 1,nQc-2 d2Qc = Qc(3,k+2) – 2.D0*Qc(3,k+1) + Qc(3,k) de2Cz = de2Cz + xn*H2D(k)*d2Qc 80 continue xDo2(1) = de2Cx xDo2(2) = de2Cy xDo2(3) = de2Cz !- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 3rd Derivative (for torsion) xn = Factorial(nQc-1)/Factorial(nQc-4) call Bernstein(B1Dr,nQc-3,r,iOut) ! -------------H3D = 0.D0 ! ALL do 51 i = 1,nQc-3 H3D(i) = B1Dr(i) 51 continue de3Cx = 0.0D0 do 61 k = 1,nQc-3 d2Qc = Qc(1,k+3) – 3.D0*Qc(1,k+2) + 3.D0*Qc(1,k+1) – Qc(1,k) de3Cx = de3Cx + xn*H3D(k)*d2Qc 61 continue de3Cy = 0.0D0 do 71 k = 1,nQc-3 d2Qc = Qc(2,k+3) – 3.D0*Qc(2,k+2) + 3.D0*Qc(2,k+1) – Qc(2,k) de3Cy = de3Cy + xn*H3D(k)*d2Qc 71 continue de3Cz = 0.0D0 do 81 k = 1,nQc-3 d2Qc = Qc(3,k+3) – 3.D0*Qc(3,k+2) + 3.D0*Qc(3,k+1) – Qc(3,k) de3Cz = de3Cz + xn*H3D(k)*d2Qc 81 continue xDo3(1) = de3Cx xDo3(2) = de3Cy xDo3(3) = de3Cz ! --------------------------------------------------------------------------------- T AJ = det

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AJI = 1.D0/AJ ! Axes(1,1) = xDot(1) * AJI Axes(1,2) = xDot(2) * AJI Axes(1,3) = xDot(3) * AJI ! --------------------------------------------------------------------------------- B call CrossProduct(xDot,xDo2,AB,Iout) AJ = dsqrt(AB(1)*AB(1)+AB(2)*AB(2)+AB(3)*AB(3)) !- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - xDot & xDo2 not lin.indep if(AJ ==0.D0) Then Axes (3,1) = 0.D0 Axes (3,2) = 0.D0 Axes (3,3) = 0.D0 return endif AJI = 1.D0/AJ ! AB(1) = AB(1) * AJI AB(2) = AB(2) * AJI AB(3) = AB(3) * AJI ! Axes (3,1) = AB(1) Axes (3,2) = AB(2) Axes (3,3) = AB(3) ! --------------------------------------------------------------------------------- M call CrossProduct(AB,AT,AM,Iout) AJ = dsqrt(AM(1)*AM(1)+AM(2)*AM(2)+AM(3)*AM(3)) AJI = 1.D0/AJ ! Axes (2,1) = AM(1) * AJI Axes (2,2) = AM(2) * AJI Axes (2,3) = AM(3) * AJI ! - - - - - - find dxbar/dr ! = [t|m|b]transpose.{dx/dr dy/dr dz/dr}transpose dxbardr = Axes(1,1)*delCx + Axes(1,2)*delCy + Axes(1,3)*delCz ! ------------------------------------------------------------ Curvature & Torsion call CrossProduct(xDot,xDo2,AB,Iout) xNum = dsqrt(AB(1)*AB(1)+AB(2)*AB(2)+AB(3)*AB(3)) xDen = dsqrt(xDot(1)*xDot(1)+xDot(2)*xDot(2)+xDot(3)*xDot(3)) xDen = xDen*xDen*xDen ! Cur = xNum/xDen ! - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Torsion xDen = xNum*xNum xNum = AB(1)*xDo3(1)+AB(2)*xDo3(2)+AB(3)*xDo3(3) ! Tor = xNum/xDen ! --------------return end

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10.20.13

707

Stiffness Matrices and Residual Forces

Both the instantaneous material stiffness matrix and the geometric stiffness matrix are formed with appropriate substitution of various fundamental vectors: k0 , a, X and so on. Moreover, generally for a c-type finite element, the internal Bezier control points can be easily eliminated by static condensation through partial Gauss elimination algorithm.

10.20.14

Incremental Equation Solution and Updates

Now let us recall the geometrically exact linearized finite element equation in unknown incremental generalized displacement, Δq̂ d , and the incremental load factor, Δ𝜆, as: ̂ 0 , d̂ 0 ) ̂ KcT Δq̂ d + p(𝜆) Δ𝜆 = −R(𝜆

(10.565)

It may be recalled that the linearized equation (10.565) has been formulated to be solved by a Newton-type iterative solution procedure for the configuration path evaluation. The configuration path is sought for a set of discrete load level parameters: 𝜆 = 𝜆0 , 𝜆1 , 𝜆2 , …. Generally, a nonlinear configuration path may consist of both snap-through and snap-back buckling phenomena. A most suitable procedure for capturing these unstable situations is provided by embedding the arc length constraint algorithm in a Newton-type incremental/iterative method. At each step of an iterative solution procedure, both d, the displacement vector, and R, the rotation tensor, need to be updated for use in the next step of iteration, and for the determination of the beam configuration path, (d, R), of interest. For the displacement vector, being a member of a vector space, the updating is additively straightforward as given by: d(i+1) = d(i) + Δd(i)

(10.566)

where Δd(i) , d(i) , d(i+1) are the incremental displacement vector obtained as a solution of the linearized incremental equation (10.565), the known starting displacement vector at iteration, i, and the updated displacement vector, respectively. An additive update for the rotation tensor, being a member of a manifold, that is, a curved space (namely, SO(3), the special orthogonal group), is not allowed. However, we can locally parameterize the rotation tensor by θ, the Rodrigues rotation vector that belong to the tangent spaces. A tangent space being a vector space, we can have an additive updating scheme for the rotation vector, without leaving the space, similar to that of the displacement update of the reference surface as: θ(i+1) = θ(i) + Δθ(i)

(10.567)

where Δθ(i) , θ(i) , θ(i+1) are the incremental rotation vector obtained as a solution of the linearized incremental equation (10.565), the known starting rotation vector at iteration, i, and the updated rotation vector, respectively. For most structural problems of interest, the additive updating procedure suffices, resulting in extremely accurate approximations, close to the exact solution for moderately large rotations.

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Remark

r Recalling our discussion on the rotation tensor, the updating scheme given by equation (10.567) will not work for ||𝜃|| = (2n + 1)𝜋, n = 0, 1, …, as the Rodrigues rotation tensor becomes singular at those θ’s.

10.20.15

Singularity-free Update for the Configuration

The update for the translational configuration, that is, the displacement of the beam reference surface, is done as shown earlier by equation (10.566). Here, we are mainly concerned with a singularity-free rotational configuration update (see Chapter 4 on rotation tensors); although there are only three independent parameters in a rotation tensor, it is not possible for a three-parameter set such as the Rodrigues parameter vector, θ, to uniquely describe all possible rotations; a non-singular one-to-one mapping takes at least five parameters, while a non-singular many-toone mapping requires at least four parameters. Recall further that the 4D unit Euler–Rodrigues parameter vector, X = {X1 X2 X3 X4 }T , or the unit Rodrigues quaternion, Q = {v, s} ≡ {v1 v2 v3 s}T with s as the scalar part, and v as the vector part, are non-singular two-to-one maps where globally both +Q and −Q represent the same rotation. Thus, a four-parameter Euler– Rodrigues vector or a quaternion will be adequate instead of a full five-parameter representation. Recall that an orthogonal rotation matrix, R, with Euler–Rodrigues parameters, is characterized by a point on a unit spherical surface in 4D, that is, a 3-sphere of radius 1, and is given by: 2 2 2 2 ⎡X1 − X2 − X3 + X4 ⎢ R = ⎢ 2(X1 X2 + X3 X4 ) ⎢ ⎣ 2(X1 X3 − X2 X4 )

2(X1 X2 − X3 X4 ) −X12 + X22 − X32 + X42 2(X2 X3 − X1 X4 )

2(X1 X3 + X2 X4 ) ⎤ ⎥ 2(X2 X3 − X1 X4 ) ⎥ , ⎥ −X12 − X22 + X32 + X42 ⎦

(10.568)

||X||2 ≡ X12 + X22 + X32 + X42 = 1 with ) ) ) ( ( ( 𝜃 𝜃1 𝜃 𝜃2 𝜃 𝜃3 𝜃 X1 ≡ sin , X2 ≡ sin , X3 ≡ sin , X ≡ cos , 2 𝜃 2 𝜃 2 𝜃 4 2 √ θ = {𝜃1 𝜃2 𝜃3 }T , 𝜃 ≡ ||θ|| = 𝜃12 + 𝜃22 + 𝜃32

(10.569)

Note that the finite rotations, R, move around on a unit sphere; the set of infinitesimal or virtual rotations define a tangent hyperplane at a point on this unit sphere. The components of a point on the unit hypersphere can also be viewed as the components of a unit Rodrigues quaternion, Q. If we define the components of a unit quaternion by the Euler–Rodrigues parameter, X, such that Q ≡ (Q1 ≡ X1 , Q2 ≡ X2 , Q3 ≡ X3 , Q4 ≡ X4 ), that is, {( ) 𝜃 θ sin Q≡ 2 𝜃

𝜃 cos 2

}T ,

̃ =1 ||Q||2 ≡ Q Q

(10.570)

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̃ ≡ (Q ̃ 1 ≡ −X1 , Q ̃ 2 ≡ −X2 , Q ̃ 3 ≡ −X3 , Q ̃ 4 ≡ X4 ) = (−v, s) as the conjugate quaternion of with Q Q, then any 3D Euclidean point x ∈ ℝ3 with the corresponding 4D vector quaternion, Qx ≡ ̃ , will have an image y ∈ ℝ3 lifted ( x , 0), under quaternion operation given by Q Qx Q ⏟⏟⏟ ∈ℝ3

similarly to the 4D vector quaternion, Qy ≡ (

y , 0) that lie on a 4D hypersphere such that: ⏟⏟⏟ ∈ℝ3

̃ Qy = Q Qx Q

(10.571)

with ⎧y1 ⎫ ⎪ ⎪ ⎨y2 ⎬ = ⎪ ⎪ ⎩y3 ⎭

2 2 2 2 ⎡Q1 − Q2 − Q3 + Q4 ⎢ ⎢ 2(Q1 Q2 + Q3 Q4 ) ⎢ ⎣ 2(Q1 Q3 − Q2 Q4 )

2(Q1 Q2 − Q3 Q4 ) −Q21 + Q22 − Q23 + Q24 2(Q2 Q3 − Q1 Q4 )

2(Q1 Q3 + Q2 Q4 ) ⎤ ⎧x1 ⎫ ⎥⎪ ⎪ 2(Q2 Q3 − Q1 Q4 ) ⎥ ⎨x2 ⎬ , ⎥⎪ ⎪ −Q21 − Q22 + Q23 + Q24 ⎦ ⎩x3 ⎭

y=Rx (10.572) Note that the matrix given in equation (10.572) is exactly the same as the rotation matrix defined by the Euler–Rodrigues parameters of equation (10.568). Thus, a spherical map, that is, a pure rotation, can always be described by the quaternion map given in equation (10.571). Now, noting that ||X||2 ≡ X12 + X22 + X32 + X42 = 1, we can express the rotation matrix, R, of equation (10.572), alternatively by:

2 2 ⎡ 1 − 2Q2 − 2Q3 ⎢ R = ⎢2(Q1 Q2 + Q3 Q4 ) ⎢ ⎣2(Q1 Q3 − Q2 Q4 )

2(Q1 Q2 − Q3 Q4 ) 2(Q1 Q3 + Q2 Q4 )⎤ ⎥ 1 − 2Q21 − 2Q23 2(Q2 Q3 − Q1 Q4 )⎥ ⎥ 2(Q2 Q3 + Q1 Q4 ) 1 − 2Q21 − 2Q22 ⎦

(10.573)

√ Note that trace(R) = −1 + 4(Q4 )2 ⇒ Q4 = 12 (1 + trace(R)). These are precisely the observations that suggest a smooth transition from the Rodrigues three-parameter θ-space to the Rodrigues four-parameter Q-quaternion space and back to obtain a singularity-free updating scheme for the rotation matrix.

10.20.15.1

Rodrigues Quaternion to Rotation Matrix

Given a unit Rodrigues quaternion, Q, we can construct the corresponding rotation matrix, R, by equation (10.573).

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10.20.15.2

Rotation Matrix to Rodrigues Quaternion

Given the rotation matrix, R, as in equation (10.573), recovering the corresponding quaternion, Q, is a bit more involved but is a straightforward process. Algorithmically, we can describe as follows. Step 0: Compute the corresponding scalar part, Q4 ≡ s ≡ X4 , of the quaternion as: Q24 ≡ s2 ≡ X42 = 14 (1 + trace(R)). If Q24 > 0, then, go to Step 1. else go to Step 2. Step 1: (It is immaterial whether we choose positive or negative square root as both will represent the same rotation matrix; we will choose positive square root), and 1√ (1 + trace(R)) 2 1 (R − R23 ) Q1 ≡ v1 ≡ X1 = 4Q4 32 1 Q2 ≡ v2 ≡ X2 = (R − R31 ) 4Q4 13 1 Q3 ≡ v3 ≡ X3 = (R − R12 ) 4Q4 21 Q4 ≡ s ≡ X4 =

STOP Step 2: Compute Q21 ≡ v21 ≡ X12 = − 12 (R22 + R33 ). If Q21 > 0, then, go to Step, else go to Step 4. Step 3: Q4 ≡ s ≡ X4 = 0

1 √ Q1 ≡ v1 ≡ X1 = √ (−R22 − R33 ) 2 1 Q2 ≡ v2 ≡ X2 = R 2Q1 12 1 Q3 ≡ v3 ≡ X3 = R 2Q1 13 STOP Step 4: Compute Q22 ≡ v22 ≡ X22 = 12 (1 − R33 ). If Q22 > 0, then, go to Step, else go to Step 6. Step 5: Q4 ≡ s ≡ X4 = 0 Q1 ≡ v1 ≡ X1 = 0

1 √ Q2 ≡ v2 ≡ X2 = √ 1 − R33 2 1 Q3 ≡ v3 ≡ X3 = R 2Q2 23 STOP

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Step 6: Q4 ≡ s ≡ X4 = 0 Q1 ≡ v1 ≡ X1 = 0 Q2 ≡ v2 ≡ X2 = 0 Q3 ≡ v3 ≡ X3 = 1 STOP

10.20.15.3

Update for the Rotation Configuration

With the above conversions from the rotation matrix to the quaternion and back in place, let Δθ(i) be the incremental Rodrigues rotation vector. We will have in storage the current quaternion, Q(i) , for the iteration step i; we can retrieve and convert it to the rotation matrix, R(i) , by using equations (10.572) or (10.573). Then, we can generate the quaternion, ΔQ(i) , corresponding to Δθ(i) by using equation (10.570) by replacing θ by Δθ(i) , and similarly for the norm; using equation (10.573), we can construct the incremental rotation matrix, ΔR(i) from the quaternion ΔQ(i) . Now, we get the updated rotation matrix, R(i+1) , as R(i+1) = ΔR(i) R(i) . Finally, by application of the algorithm for the conversion of a rotation matrix to the corresponding quaternion, we convert R(i+1) to Q(i+1) and store it in the database. 10.20.15.4

Code Fragment for Tangent Stiffness Matrix

Subroutine Form_STIF(RoCOld,CuCOld,RoSav,CKSav,nCInt,Qd,Qc,nQc, & Nel,NelType,Nint,DMat, & Stif,BFors,StifK,nStif,Iout,Trace) !- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - --------------------------! Form Element Stiffness Matrix !- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - --------------------------! Notes: Nint should be atleast 2 ! ! Inputs ! Qd(nStif) = Latest displacement controls: state vector ! Qc(1,16) = Bezier X-Coordinates = (qc00x qc10x ... qc33x) ! Qc(2,16) = Bezier Y-Coordinates = (qc00y qc10y ... qc33y) ! Qc(3,16) = Bezier Z-Coordinates = (qc00z qc10z ... qc33z) ! Nel = Element Number ! NelType = Element Type: Beam ! Nint = Number of Gauss Points ! E = Elasticity Modulus ! PR = Poisson’s ratio ! Thk = Thickness ! nStif = Row or Col size of Stiffness matrix = nDof*nQc ! Iout = Error Output file# ! ! Outputs ! Stif(nStif,nStif) = Stiffness Matrix

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! ! grals ! !

bFors(nStif) StifK(nStif)

= Body force Vector Integrals = kLambda Stiffness last column Vector Inte-

Trace

= trace of the matrix

Implicit Real(kind=8) (a-h,o-z) ! ! !

Input Variables & Arrays Real(kind=8) Dimension Real(kind=8) Dimension Real(kind=8) Dimension Real(kind=8) Dimension integer

! ! !

Qc ,Qd Qc(3,nQc),Qd(nStif) RoSav ,CKSav RoSav(nCInt),CKSav(nCInt) RoCOld ,CuCOld RoCOld(nCInt),CuCOld(nCInt) DMat DMat(6,6) Nel,NelType,Iout,nStif

Output Variables & Arrays Real(kind=8) BFors ,StifK ,Stif ,Trace Dimension BFors(nStif),StifK(nStif),Stif(nStif,nStif) Real(kind=8) EDG ,B ,DB ,T Dimension EDG(12,12),B(12,nStif),DB(12),T(6,nStif) Real(kind=8) Add ,HD ,RoOld ,Rp Dimension Add(nStif),HD(12),HDOld(12),Rp(3) Real(kind=8) BVEC ,SKLM ,R0 ,RK ,C0 ,CK Dimension BVEC(nStif),SKLM(nStif),R0(3),RK(3),C0(3),CK(3) Real(kind=8) AM,BM,CM,TH,Ri,Si,Wt,sum

! LOGICAL bReset !============================================ common/LoadStepIter/iStep,iNut,bReset common/Scratch/iRotCur1,iRotCur2 !============================================ DATA zero/0.D0/,one/1.0D0/ DATA nC/3/,nDof/6/,nDof2/12/ !- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Stiffness Matrix Calculation BFors = 0.D0 !all StifK = 0.D0 !all Stif = 0.D0 ! all elements = 0. !======================================================= Restore inquire(iRotCur1,NEXTREC = nRec) if(nRec.GT.1) then nRec = Nel call PutGetRotCur(iRotCur1,RoCOld,CuCold,nCInt,nRec,2) !Read endif !========================================= Loop over Gauss Locations do 60 jInt = 1,Nint Ri = GaussPtVal(jInt,Nint,1) !- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Get

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Geometric Curvature Vector etc. call Get_ReferenceInfo(Qc,nC,nQc,RoCOld,CuCOld,nCInt, & Ri,jInt,R0,C0,CK,nC,T,B, & nDof,nDof2,nStif,Det,Iout,Nel,Nint) !- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - DisRots & Derivatives HD = MATMUL(B,Qd) !- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Get EDG(nDof2,nDof2) call Form_EtranDEplusGMatrix(R0,HD,nStif,DMat,C0,RK,CK, & T,B,BVEC,SKLM,EDG,3,nDof,nDof2,Iout) !- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Save Curvature do 10 ii = 1,nC ij = ii + (jInt -1)*nC RoCOld(ij) = RK(ii) CuCold(ij) = CK(ii) 10 continue !- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Form: Btran_EDG_B & Add to stiffness matrix Wt = GaussPtVal(jInt,Nint,2)*Det ! do 50 j = 1,nStif do 20 k = 1,nDof2 ! DB(k) = 0.0 do 20 l = 1,nDof2 ! DB(k) = DB(k) + EDG(k,l)*B(l,j) 20 continue do 40 i = j,nStif sum = 0.0 do 30 l = 1,nDof2 ! sum = sum + B(l,i)*DB(l) 30 continue Stif(i,j) = Stif(i,j) + sum*Wt 40 continue 50 continue !- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Load Vector: Residual & Body force part do 53 i = 1,nStif BFors(i) = BFors(i) + Wt*BVec(i) 53 continue !- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Form: Last Column of stiffness: KLambda Body Force part do 55 i = 1,nStif StifK(i) = StifK(i) + Wt*SKLM(i) 55 continue 60 continue ! ! symmetry: Upper Triangle ! do 70 j = 1,nStif do 70 i = j,nStif Stif(j,i) = Stif(i,j)

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70 continue !======================================================== Save ! nRec = (Nel-1)*Nint + jInt nRec = Nel call PutGetRotCur(iRotCur2,RoCOld,CuCold,nCInt,nRec,1) !Write ! Get Trace Trace = 0.D0 do 80 i = 1,nStif Trace = Trace + Stif(i,i) 80 continue ! Check each col. vector for Null Add = 0.D0 do 100 j = 1,nStif sum = 0.0 do 90 i = 1,nStif sum = sum + Stif(i,j) 90 continue Add(j) = sum 100 continue ! return end

10.20.16

Newton-type Iteration and Arc Length Constraint

Finally, a Newton-type method can be devised to solve problems and perform stability or buckling analysis with quadratic convergence. For proportional loading, the coefficient of proportionality for the loading, 𝜆, is gradually incremented through a Newton-type iterative method to trace the configuration path of the system. However, a straightforward application of the method fails near, at or beyond the stability limit point. Recall that there are generally two fundamentally different scenarios for loss of stability of a beam structure; these are related to the limit point associated with the snap-through phenomena, or the bifurcation point associated with the buckling situation. In the case of limit points, a basic Newton-type method will either overshoot near a limit point that is about to signal an unloading or undershoot near a limit point, heralding a switch to a loading situation with no chance of correction or recovery. Similarly, recalling two possible controls of tracing a load–displacement curve, we prefer load control to displacement control. Finally, for a practical problem, all that may be necessary is the first limit point, that is, the collapse load; even, in this situation, the simple Newton-type iteration method must be equipped with a mechanism to converge or predict the collapse load. Recalling our discussion about nonlinear iterative methods, we choose the arc length constraint mechanism for all the following numerical examples; usually, a tolerance for error of the convergence criterion and an arc length constraint parameter are input by the user of the computer code.

10.20.17

Numerical Examples

Now that we have the implementation details, we would like to apply these to various problems of interest, and examine and interpret the accuracy, robustness and numerical efficiency of the obtained results and compare with the published solutions by other methods, whenever available. It is important to note that in all of the following example full Gauss quadrature points were

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Figure 10.17 Example 1: Straight beam curling problem.

selected with absolute minimum number of mega c-type curved beam elements to show that locking is a non-issue and there is no need for any ad hoc so-called “under integration” or “reduced integration”; this was only possible by c-type finite elements that are backed by a balanced exact theoretical formulation and a concomitant numerical implementation with Bernstein–Bezier polynomials as the basis functions. Example 1: A 2D straight beam coiling to circle A straight cantilevered beam, shown in Figure 10.17, of unit cross-sectional width and depth, subjected to a moment loading at the free end has been reported in the published literature. The elasticity modulus, E, and the shear modulus, G, are 106 and 0.5 × 106 , respectively. The beam (displacement and rotation) is modeled as a single degenerate 3D quintic in X-Y plane with Z directional constraints. The deformed shapes for various tip moment loading are shown in Figure 10.17. Note that compared to the reported analyses that have used eight elements, a single mega quintic element suffices, and thus points towards great reduction in subsequent quality assurance efforts in practical endeavors. Example 2: Geometrically exact nonlinear curved arches To exemplify unification of geometry modeling and finite element analysis in both formulation and subsequent numerical solution, a geometrically exact nonlinear curved arch problem with shear deformation and extensibility is considered here. The load–deformation paths are shown I in Figure 10.18 for two Huddleston compressibility parameters (Huddleston, 1968), c = A(2bd) where, I is the moment of inertia, A represents cross-sectional area, and for b, d. One model analyzed with c = 2.55 × 10−6 , that is, flexibility is so low that it renders the arch relatively incompressible. The other has c = 0.01, that is, has very high I, and thus, making the arch substantially compressible. The exact deformed shapes for various load levels for c = 2.55 × 10−6 are shown in Figure 10.18(c). It may be noted that it is solved with only one c-type quintic element to accurately model the problem, while it takes 40 Q-type shell elements of Palazotto and Dennis (1992), to seriously underestimate the limit point and track the wrong equilibrium path. The Q-type degenerate shell element, with Poisson’s ratio vanishing, of Palazotto and Dennis is characterized by unnecessary Reddy-type quadratic transverse shear (Reddy and Liu, 1985) and quadratic in-plane displacement approximations resulting in 36 degree of freedoms. For flexible bodies, that is, beams, plates and shells, undergoing extreme geometric nonlinearity through large

716

Computation of Nonlinear Structures

Figure 10.18 (a) Example 2: nonlinear arch with Huddleston parameter. (b) Example 2: nonlinear arch with Huddleston parameter. (c) Example 2: nonlinear arch deformations at various load levels for parameter.

rotations and displacement, it seems unwise to compromise analytical formulation prematurely for numerical solution early in the procedure. Example 3: A 2D-curved beam element with 3D loading A cantilevered 45◦ curved beam in X-Y plane loaded with a force in the Z direction at the free end invoking bending and torsion, is considered here. It was originally proposed by Bathe and Bolourchi (1979), and since analyzed by various other researchers. The model and the deformations for various load levels are shown in Figure 10.19(a) and (b), respectively. It took only five load steps to arrive at a load level of 600, compared to 60 equal load steps reported in Bathe and Bolourchi (1979). Figure 10.19(b) shows the load–displacement curves for various components of load point displacement. A comparison of deformed tip positions for reported load levels are shown, in tabular form, in Figure 10.19(c). Note that while it took eight elements for most other researchers, a single mega c-type element suffices. Finally, there is no need for arbitrary under/reduced Gauss quadrature points for mythical shear locking.

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717

(a)

(b)

(c)

Figure 10.19 (a) Example 3: 2D rod problem with 3D loading. (b) Example 3: load–deformation curves. (c) Example 3: table of comparisons for number of elements needed.

Example 4: A 3D cantilevered curved beam element with natural twist and bend Finally, we note that for general computer-aided curved beam mesh generation through the input of few key data points, the curved boundaries of the elements may be truly 3D. Thus, to test the robustness of the c-type finite element method for general large 3D curved beams, a 3D beam (see Chapter 9 for linear solution) is modeled by two truly three-dimensional elements, as shown in Figure 10.20(a). The geometric shape is initially generated as a single cubic with control nodes, q1i , i = 0, … , 3; then, an application of the subdivision algorithm produces two cubic Bernstein– Bezier geometric elements. The displacement finite elements, then, are made up of two quintic Bernstein–Bezier elements. The beam is subjected to simple tip load. The corresponding exact geometrically nonlinear deformations following the present formulation for various load levels are shown in Figure 10.20(b); the load-deformation curves are shown Figure 10.20(c). Note that it takes 100 currently available elements (FEAP computer program) compared to two c-type elements used here to match the presented accuracy as shown in Figure 10.20(d), which compares the present result with that by Simo and Vu-Quoc (1986); the quadratic convergence is shown in Figure 10.20(e).

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Figure 10.20 (a) Example 4: 3D curved beam problem with natural twist and bend. (b) Example 4: nonlinear deformations of two quintic elements at various load levels. (c) Example 4: load–deformation curves. (d) Example 4: comparison of load–deformation histories with FEAP elements. (e) Example 4: quadratic convergence of displacement components.

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719

In summary, the concomitant bookkeeping and quality assurance in mesh generation and finite element analysis will result in a proportional decrease in analyst/engineer man-hours if the presented formulation is used.

10.20.18

Where We Would Like to Go

This section concludes our nonlinear beam exposition; the next logical step will be to study nonlinear plates and shells for which we must check out:

r nonlinear plates and shells (Chapter 11)

11 Nonlinear Shell 11.1

Introduction

11.1.1

Why We Are Here

In this chapter we develop and examine the nonlinear theory of general shells for both nonlinear quasi-static and dynamic loadings; we, then, specialize the theory for thin hyperelastic shells subjected to extremely large displacements and rotations by deriving the most efficient and robust computational form to accommodate and devise extremely large c-type finite elements. We briefly detail below the contents of the various sections to achieve our goal: Section 11.2: We define various geometric properties of an undeformed structural shell-like body for our subsequent analysis, and indicate the assumptions that are implicit in the definition. Section 11.3: We reduce the dynamic and static balance of equations of 3D continuum mechanics to the shell equations of motion of two dimensions. In so doing, we derive the shell dynamic and quasi-static vector momentum balance and moment of momentum balance equations. The main reason behind this starting point as a reduction from the 3D equations is to lend support to the accuracy, consistency and validity of the shell equations of motion that may also be derived directly and simply by following an engineering approach of assuming a shell-like body as a mere 2D surface (see the direct engineering approach referred to in the Section 11.5). Furthermore, it connects the shell kinematic and force boundary conditions to the actually prescribed 3D counterparts that is pre-eminently essential to establish the equivalence between the real problems of three-dimensional nature and the supposedly corresponding problems to be solved by a shell theory – a theory that models real body effectively as a linear or nonlinear, thick or thin, two-dimensional surface. Section 11.4: We specialize the dynamic and static balance equations of general shells to those of very thin and flexible shells. The case of thin and very flexible shells constitutes a class that is of great practical importance, and deserves special attention. The thinness attribute simplifies the shell equations for subsequent computational formulation. Section 11.5: We develop a simple and direct engineering way (as opposed to the reduction of the equations of motion of continuum mechanics from 3D to 2D) of Computation of Nonlinear Structures: Extremely Large Elements for Frames, Plates and Shells, First Edition. Debabrata Ray. © 2016 John Wiley & Sons, Ltd. Published 2016 by John Wiley & Sons, Ltd.

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applying the momentum balance and moment of momentum balance (i.e. angular momentum) principles to establish the shell static and dynamic equations of motion. Let us remind ourselves that because we are interested to include the case of extremely large (i.e. finite) displacements and rotations resulting in the nonlinear responses, all equilibrium equations must be considered in the deformed geometry of a shell. Hence, we will develop the balance equations on the current geometry for the nonlinear responses, and later, particularize by linearization for the case of infinitesimal deformations for the linear regime. Furthermore, in order to arrive at our stated goal, at the very outset we need to view a shell as a representative but fictitious thin two-dimensional curvilinear physical sheet, as opposed to the 3D continuum it really is, identified geometrically as the reference surface with equivalent cross-sectional and physical properties assigned to simulate real deformations that are expected of a shell-like 3D continuum. As it turns out that this way of viewing a shell as a two-dimensional surface produces the exact equations of motion obtained otherwise through an elaborate and rigorous reduction, from the 3D continuum viewpoint to an equivalent 2D surface except that the 3D connection provides insight into the real characters of the forces and moments, and the linear and angular momenta, introduced here. Thus, while the direct approach provides an easy and simple way to the balance laws, we must remember that consistent interpretation of the entities developed here must await the derivations that involve reduction from the 3D continuum balance principles. Section 11.6: We identify the most suitable shell kinematic variables and the configuration space, that is, the set of kinematic parameters containing all the information about the motion history (i.e. the configurations) of the body, from the shell weak form of the balance equations. Our goal is motivated by the requirement that for the determination of a shell configuration path, or the history of motion for a shell numerically by an energy-type procedure such as the c-type finite element method, we must first identify the equivalent 2D configuration parameters of a 3D shell-like body that is energetically consistent with the exact balance laws. Towards this, we start from the derived 2D exact equations of the momentum and the angular momentum balance laws of motion of a shell and by the standard method of selecting test functions, develop the weak form of the equations which in turn allows us to identify the kinematic parameters appropriate for the conjugate virtual strains. However, alternatively, it is possible to start directly with the integral equations of the stress or deformation power and an assumed kinematics and the deformation map for the shell-like body, and define equivalently both the forces and moments and identify the conjugate virtual strains. For the shell-like body, we will have occasion to perform these operations as an example. Philosophically, however, it seems to make more sense to adhere to the first method as the various shell theories with different kinematics crop up naturally in an exact fashion. Section 11.7: We define the curvature vectors and tensors, the angular velocity vectors and tensors and the admissible virtual space, corresponding to the shell configuration space consisting of the displacement vector field and the rotational tensor field, necessary to identify the spatial strain rate fields and the corresponding spatial real strain fields in the weak form of equations (and later for the computational virtual work equation) of a shell reference surface. In so doing, we describe in terms of the rotation field, the curvature vector and tensor, and the angular velocity vector and tensor field, and the variations of the kinematic variables, that is, the displacement

Nonlinear Shell

vector field and the rotation tensor field, constituting the configuration space with respect to the arc length parameters along the lines of curvature of a shell reference surface. Section 11.8: We identify the force and moment virtual strain rate fields from the internal virtual work functional, and the corresponding real strain rate fields whose variations validate the presumptions that the strain fields, derived from the weak form, in the internal virtual work expressions are actually virtual in nature, that is, there exist functions (real fields) whose variations are these fields. As will be apparent later in our discussion on the shell constitutive theory, our goal is also motivated by the need to develop the appropriate constitutive laws for the 2D shell surface by establishing the existence of an equivalent 2D surface strain energy functional. Section 11.9: We define the component or operational forms of various vectors associated with the shell vector balance equations. These definitions allow us to convert the invariant vector form of the shell equations of motion to the component form suitable for subsequent computational development. Recall that our interest in any theory is solely governed by its relevance to our computational endeavors. In other words, here we will develop a computationally efficient component or operational vector forms by first collecting the curvilinear components of displacements and rotations, of forces and moments and of linear and angular velocities and momenta, and then describing these as components of corresponding vectors that help to define both the undeformed and the deformed states of an arbitrary shell. To be sure, we are interested ultimately in the computation of the components of displacements and rotations, and of the reactive forces and moments at any point on a shell that has been subjected to some loading conditions. This, in turn, motivates us to directly describe the relevant parameters and equations of a shell in component vectors. Accordingly, in what follows here, we develop the notion of component vectors. More specifically, we define the displacement and rotation, and the force and moment component vectors. Finally, we establish the relations that exist between the force and moment component vectors as defined in the undeformed frame and a rotating or spinning deformed frame. Section 11.10: We define the covariant derivatives of the component or operational vectors necessary to convert the shell equations of motion in invariant vector form to a suitable component vector form for ease in subsequent computational development. Section 11.11: We convert the invariant vector form of the balance equations of motion of a thin shell to a modified and computationally efficient component vector or operational form suitable for subsequent computational tasks such as the virtual work equation development which is the cornerstone of a finite element procedure such as the c-type finite element method. For finite rotation – a situation for which we will develop our theory, the shell real configuration space being a Riemannian or curved surface, we will need to work with the covariant derivatives of the component vectors. Section 11.12: We define the computational curvature tensor and vector, the computational angular velocity tensor and vector and their linearization (as a precursor to covariant linearization of the dynamic virtual functional necessary for a Newton-type method of iterative solution technique) and the computational virtual space, corresponding to the shell configuration space consisting of the displacement vector field

723

724

Computation of Nonlinear Structures

and the rotational tensor field necessary to derive the computational virtual work functional for our subsequent numerical treatment by the c-type finite element method. This is also aimed at the identification of the virtual strain fields and the development of the linearized virtual work functionals necessary for a Newton-type iterative procedure for the numerical solution of shell problems by a variational scheme such as the c-type finite element method. Section 11.13: We develop the thin shell computational virtual work equations from the balance equations which we have specialized for thin shells and described as the computational shell balance equation. This is developed because, for the determination of a shell configuration path or the history of motion for a shell numerically by an energy-type procedure such as the c-type finite element method, we will have to derive, from the differential strong form of the shell equations, the integral virtual weak form of equations. Finally, we intend then to follow a semi-discrete finite element formulation in which the shell geometry and the spatial distribution of the deformations are discretized by the c-type finite element method and thus reduce the virtual work equation eventually into a matrix second-order ordinary differential equations. Under this type of formulation, the virtual space is taken as time-independent because the time is viewed only as an indexing parameter of the configuration space variables: the displacement vector and the rotation tensor, and thus, the time marching essentially traces a curve on the configuration space. Section 11.14: We express the virtual strains in a computationally efficient matrix form of description leading to a desirable articulation of the virtual work functional, and also to a definition of various generalized entities of relevance for notational compactness. Furthermore, in so doing, we rewrite the virtual work functional of a shell in the generalized form to prepare for our subsequent computational work for shells subjected to proportional loadings in the case of the quasi-static or static analyses, and the time history of loadings for dynamic analyses. Section 11.15: We develop the compatible real strain fields from the virtual strain fields that we have already derived as energetically conjugate to the real stress resultant and stress couple fields of a shell. The real strain field is necessary as one of the essential ingredients in the computational procedure; the real strains with the appropriate constitutive relationship allow us to compute the real internal reactive force (the stress resultants and the stress couples) fields of a shell. The contradiction, if any, between the internal and the applied force system drives an iterative method such as the Newton-type algorithm, in computational structural engineering and solid mechanics. Section 11.16: We introduce the elastic material properties, that is, the stress–strain relationship, or the constitutive laws, appropriate for an elastic shell. We do this to reduce the 3D stress–strain laws to a 2D form that could be suitably incorporated in the internal virtual work functional of a shell. Clearly, this will be an approximation, but then the entire subject of the constitutive theory by its very nature of fitting realworld experimental data is an empirical science. Although the virtual work principle is exact, irrespective of the material properties, here we restrict ourselves to the case of the elastic shells. Section 11.17: We accomplish the covariant linearization of the nonlinear virtual work functional that is necessary for the computation of the configuration history of a

Nonlinear Shell

shell digitally by a Newton-type method of iterative solution for the nonlinear quasistatic (proportional loading), and by a numerical integration method of solution for the nonlinear dynamic (time history of loading) cases under semi-discrete formulation. In other words, we would like to obtain the linearized equations for both the quasi-static case under the proportional loading system (for incremental load step) identified by 𝜆, the coefficient of load proportionality, and also the dynamic case under any time history of applied loading system (for incremental time step). Section 11.18: We present the semi-discrete c-type finite element formulation for shells with dynamic time history loading systems. We extend this section with the description of the residual forces with the introduction of the c-type finite elements and the notion of control vectors, and complete it with a computationally efficient exposition of the incremental virtual work equation for a time step of the motion of a shell finite element. Starting from the already-derived fully linearized virtual work functional, first we aim to transform a part – specifically the internal linearized virtual work functional – to its aforesaid matrix representation. The transformed matrix representations will subsequently give rise to what is known as the material stiffness matrix and the geometric stiffness matrix of a shell. Next we focus on the part known as the inertial linearized virtual work functional, and transform it to the most suitable matrix representation, and the transformed matrix representations will subsequently give rise to what is known as the mass matrix, gyroscopic (damping) matrix and the centrifugal (stiffness) matrix of a shell undergoing dynamic motion. Using these developments, we present the c-type finite element formulation for the dynamic loading systems and generate the final form of the incremental dynamic virtual work equation; the previous form recalled later is not suitable for a subsequent finite element implementation since the sought-after independent incremental variables appear only in an implicit way, hidden behind the incremental strains and the virtual strain–displacement functions for the incremental internal virtual work functional. Also, the relevant matrices were not fully identified in that form for the incremental inertial virtual work functional. For the semi-discrete finite element formulation, we would like to have a form of the linearized virtual equation that is explicit in the incremental generalized displacement vector, the incremental generalized velocity vector, and the incremental generalized acceleration vector, so that the virtual work equation can be used digitally by a numerical time integration method in conjunction with a Newton-type iterative algorithm for tracing the dynamic motion history of a shell structure. Section 11.19: We present the c-type finite element formulation for shells with quasi-static proportional loading systems; we then describe the residual and applied proportional forces with the introduction of the c-type finite elements and the notion of control vectors, and complete it with a computationally efficient exposition of the incremental equation for each finite element. Starting from the linearized virtual work functional, here we first intend to transform a part – specifically the internal linearized virtual work functional – to its suitable matrix representation. The transformed matrix representations will subsequently give rise to what is known as the material stiffness matrix and the geometric stiffness matrix of a shell. Using these developments, we present the c-type finite element formulation for the quasi-static loading systems and generate the final form of the incremental virtual work equation. The previous form recalled later is not yet suitable for a subsequent finite element implementation since the sought-after independent incremental variables appear only in an implicit way

725

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726

hidden behind the incremental strains and the virtual strain–displacement functions. We would like to have a form of the linearized virtual equation that is explicit in the incremental generalized displacement vector, so that the virtual work equation can be solved digitally by a Newton-type iterative method. Section 11.20: We present the c-type shell finite element computer programming details on the finite element formulation of the material control stiffness matrix, the geometric control stiffness matrix, the applied loads and the residual loads, and so on, of a shell for subsequent direct stiffness method of assembly and solution by Newton-type algorithm to ensure solutions without any locking phenomena – shear, membrane, and so on. Finally, we present solutions for several nonlinear shell problems of practical importance by c-type finite element method.

Nonlinear Shell: A Road Map Mesh Geometry Bezier-Bernstein & B-spline & NURBs

Defined Stress Resultant & Couple

3D Balance Equations

Direct 2D Balance Equations

Define Real Parameters

2D Balance Equations Integrate by parts after multiplying with admissible virtual parameters

Identify Boundary Conditions

Newton-type Algorithm

Virtual work equation and Virtual Functional

Identify Incremental Parameters

Covariantly Linearize Virtual work

Mass matrix

Identify Stress Resultant & Couple

Integrate away thickness parameters

Gyroscopic Damping

Centrifugal Stiffness

Material Stiffness

Identify Virtual Parameters

Conjugate Virtual strains

Constitutive or Material Property

Geometric Stiffness

Apply Vainberg Principle Real Strains

Finite Element Discretization: Bezier-Bernstein and B-spline & NURBs Computational Mass, Gyroscopic & Centrifugal Matrices for Dynamic Loading Only

Code Fragments

Linear Incremental Finite Element Equations Numerical Set up

Loop Solve

material & Geometric Stiffness Matrix for Quasi-static and Dynamic Loading

Examples

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11.2 Shell Geometry: Definition and Assumptions 11.2.1 What We Need to Recall As will be seen later, because thin shell geometry is essentially captured by a general 2D surface, we must recall, from Chapter 6 on surfaces, the various geometric properties of a surface.

11.2.2

3D Shell-like Body

For analysis and computational purposes, a 3D body may be called a shell-like body if it has any two of its dimensions conspicuously dominant compared to the remaining one. The two dominant dimensions will be identified with the shell surfaces and the remaining direction is associated with the thickness of a shell. Thus, as opposed to viewing the body just as a general 3D continuum identified in a Euclidean space, the body will then be equivalently perceived as the union or family of shell surfaces along the thickness direction that occupies the same 3D space, and henceforth, will be called a shell space. Our Euclidean space will be endowed with an inertial frame. Then, any point on the Euclidean space can be identified with a unique position vector with respect to this frame.

11.2.3

Shell Undeformed Geometry

In what follows, we are interested in the response of a shell to static and dynamic loads. So, let us suppose that, as shown in Figure 11.1, a three-dimensional shell material body in an undeformed reference state occupies a regular region, B, in a Euclidean space with boundary denoted by 𝜕B at any time t0 . In other words, the shell body is considered embedded in a 3D Euclidean (flat) point space, although the shell surface itself constitutes a curved non-Euclidean space. We will assume that this shell space is differentiable to an order ≥ 2. We may introduce a curvilinear coordinate system, {𝜉̂i }, i = 1, 2, 3 to describe the shell geometry and with which we may identify ̂ ≡ C( ̂ 𝝃) ̂ 𝜉̂1 , 𝜉̂2 , 𝜉̂3 ), at any point, 𝝃̂ ≡ {𝜉̂1 , 𝜉̂2 , 𝜉̂3 }T , of the body. Now, with a position vector, C( the knowledge that geometrically the shell thickness direction is of different scale from those of the other two dimensions, and with an assignment of the first two curvilinear coordinates

Figure 11.1 3D body and Shell geometric definition.

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Figure 11.2 Shell space and map.

to the surface dimensions, we will redefine our curvilinear coordinates as: {𝜉̂𝛼 }, 𝛼 = 1, 2 to define surfaces, and 𝜂 ≡ 𝜉̂3 to define the thickness direction. Thus, as shown in Figure 11.2, any material point on the shell body is given by {𝜉̂1 , 𝜉̂2 , 𝜂}. Then, a coordinate system known as, a normal coordinate system, can be introduced by defining a shell reference surface and taking the thickness direction, 𝜂, to be coincident along the normal at a point on this surface. 11.2.3.1

Shell Reference Surface

For reasons indicated above and for efficient and computationally optimal theoretical formulation, the shell body may be conceived as the union of surfaces given by {𝜉̂1 , 𝜉̂2 , 𝜂 = const} along the thickness direction, 𝜂. While any one of these surfaces may serve as a shell reference surface, we choose our shell reference surface, denoted by M ⊂ B, as the one with 𝜂 = 0. Thus, any point on the shell reference surface is given by {𝜉̂1 , 𝜉̂2 , 0}. ̂ 𝜉̂1 , 𝜉̂2 , 𝜂), at Now, with the given arbitrary but fixed inertial frame and the position vectors, C( 1 2 ̂ ̂ a point {𝜉 , 𝜉 , 𝜂}, we will denote the position vector for any point on the shell reference surface ̂ 𝜉̂1 , 𝜉̂2 , 0), as shown in Figure 11.3. ̂ 0 (𝜉̂1 , 𝜉̂2 ) ≡ C( as: C 11.2.3.2

Shell Optimal Normal Coordinate System

From our study of surfaces in Chapter 6, and as shown in Figure 11.3, we have what is known as two sets of iso-curves: as 𝜉̂𝛼 = constant curves for 𝛼 = 1, 2, through any point on these parallel ̂ 𝜉̂𝛼 ) ̂ 𝛽 ≡ 𝜕C( at surfaces. The triad of tangent vectors forming a set of the local base vectors: G 𝜕 𝜉̂𝛽 a point defined by the tangents to these curves (and the unit normal to these base vectors, ̂ ≡G ̂ 3 = Ĝ 1 × Ĝ 2 ) are generally non-orthogonal to each other, and thus, their expressions are N ̂ 1× G ̂ 2‖ ‖G

coupled and their use can be cumbersome at best. Thus, these coordinates are not suitable for computationally optimal theoretical formulations; a better scenario is presented by the lines of curvature which are orthogonal at any point on a surface leading to a set of locally orthogonal base vectors. So we may opt for the curvilinear coordinates, denoted by {𝜉 𝛼 }, 𝛼 = 1, 2, along the lines of curvature on the shell reference surface. Then, a normal coordinate system can be introduced 1 2 ̄ 𝛽 (𝜉 1 , 𝜉 2 ) ≡ 𝜕C(𝜉 ,𝜉 ) defined by the tangents to the locally by a triad of base vectors at a point: G 𝜕𝜉 𝛽

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Figure 11.3 Shell reference surface.

̄ ̄ ×G G

̄ 1, 𝜉2) ≡ G ̄ 3 (𝜉 1 , 𝜉 2 ) = 1 2 . lines of curvature, and the unit normal to these base vectors, N(𝜉 ̄ 1× G ̄ 2‖ ‖G ̄ 1 , 𝜉 2 , 𝜂), defined by it, at any Then, given the fixed inertial frame and the position vectors, C(𝜉 point {𝜉 1 , 𝜉 2 , 𝜂}, we may denote the position vector for any point on the shell reference surface ̄ 1 , 𝜉 2 , 0). ̄ 0 (𝜉 1 , 𝜉 2 ) ≡ C(𝜉 as: C However, parameterization of surfaces by {𝜉 a }, a = 1, 2 along the lines of curvature is still not the best choice for either theoretical formulation or numerical evaluation. For example, the base vectors are not normalized, that is, not of unit lengths. Thus, the best choice is to replace {𝜉 𝛼 }, 𝛼 = 1, 2 of the shell reference surface by the corresponding invariant arc length parameterization, denoted by {S𝛼 }, 𝛼 = 1, 2, respectively, along the lines of curvature. Let us recall that the arc length parameter, S𝛼 , is related to the parameter 𝜉 𝛼 by the relations: S1 (𝜉 1 , 𝜉 2 ) = 𝜉1 ̄ 𝜉2 ̄ 1 ̄ 1 , 𝜉 2 , 0) is the position ∫0 ‖C(𝜌, 𝜉 2 , 0), 𝜌 ‖d𝜌 and S2 (𝜉 1 , 𝜉 2 ) = ∫0 ‖C(𝜉 , 𝜌, 0), 𝜌 ‖d𝜌 where C(𝜉 vector of a point on the shell reference surface, Finally, any point on the shell reference surface may then be given by {S1 , S2 , 0}. We will denote the position vector for any point on the shell reference surface as: C0 (S1 , S2 ) ≡ C(S1 , S2 , 0). Thus, any material point on the shell body is given by {S1 , S2 , 𝜂}. Now, we can define locally what we will call the optimal normal coordinate system by a triad of orthonormal base vectors at a point on the shell reference surface, M, as:

r the two coordinate unit tangent vectors, G𝛼 (S1 , S2 ) = C0 , 𝛼 (S1 , S2 ) ≡

𝜕 C (S1 , S2 ), 𝜕S𝛼 0

𝛼 = 1, 2, along the lines of curvature on the shell reference surface, and the unit vector N(S1 , S2 ) ≡ G ×G G3 (S1 , S2 ) = ‖G1 ×G2 ‖ = √1 G1 × G2 , normal to the shell reference surface, where G is the G 1 2 ] [ G11 G12 determinant of the 2 × 2 shell reference surface metric tensor given by: G21 = G12 G22 with G𝛼𝛽 = G𝛼 ∙ G𝛽 , 𝛼, 𝛽 = 1, 2. We have used the Lagrange identity: (a × b) ∙ (c × d) = √ (a ∙ c) ∙ (b ∙ d) − (a ∙ b)(c ∙ d) to imply ‖G1 × G2 ‖ = G in the above definition. Note that because of orthonormality: G𝛼 (S1 , S2 ) = G𝛼 (S1 , S2 ), 𝛼 = 1, 2, and G11 = G22 = 1, G𝛼𝛽 = 0, √ 𝛼 ≠ 𝛽 reducing the determinant to unity, that is, G = 1.

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Remarks: It is important to recognize the notational scheme that we have adopted for the derivatives of various entities.

r For derivatives with respect to the arc length parameters, {S1 , S2 }, we will sometimes use r r

shorthand such as: (∙), 𝛼 (S1 , S2 ) ≡ 𝜕S𝜕𝛼 (∙)(S1 , S2 ), 𝛼 = 1, 2. For derivatives with respect to {𝜉 1 , 𝜉 2 }, we may use shorthand such as: (∙),𝛼̄ (𝜉 1 , 𝜉 2 ) ≡ 𝜕 (∙)(𝜉 1 , 𝜉 2 ), 𝛼 = 1, 2. 𝜕𝜉 𝛼 For derivatives with respect to {𝜉̂1 , 𝜉̂2 }, we may use shorthand such as: (∙),𝛼̂ (𝜉̂1 , 𝜉̂2 ) ≡ 𝜕 (∙)(𝜉̂1 , 𝜉̂2 ), 𝛼 = 1, 2. 𝜕 𝜉̂𝛼

Exercise: show that: ‖G1 × G2 ‖ = ‖ ‖

√ G

(11.1)

Solution: Applying the Lagrange identity: (a × b) ∙ (c × d) = (a ∙ c)(b ∙ d) − (a ∙ b)(c ∙ d), ‖G1 × G2 ‖ 2 = (G1 × G2 ) ∙ (G1 × G2 ) = (G1 ∙ G1 )(G2 ∙ G2 ) − (G1 G2 )2 = G11 G22 − (G12 )2 = det (surface metric tensor) ≡ G completes the proof. 11.2.3.3

The Lines of Curvature from the Iso-curves on the Shell Reference Surface

Let us recall that given the iso-curves, the lines of curvature at a point for the shell reference surface can always be obtained through an application of the extremum principle leading to the following equations in 𝜆𝛼 , 𝛼 = 1, 2, for determining the directions of the lines of curvature: ⎡ 𝜆2 ⎢̂ det ⎢ G 11 ⎢ B̂ ⎣ 11

−𝜆 ̂ 12 G B̂ 12

1 ⎤ ̂ 22 ⎥⎥ = 0 G B̂ 22 ⎥⎦

(11.2)

̂ 𝛼𝛽 and B̂ 𝛼𝛽 , 𝛼, 𝛽 = 1, 2 are the components of the surface metric tensor and the curvawhere, G ture tensor, respectively, at any point on the shell reference surface described by the {𝜉̂𝛼 }, 𝛼 = 1, 2 coordinate system. Recall that the principal (i.e. along the lines of curvature) radii of curvatures, R𝛼 , at a point on the shell reference surface, are given by: (

1 R𝛼

)2

̂ 𝛼𝛽 − B̂ 𝛼𝛽 G

(

1 R𝛼

) +

B̂ =0 ̂ G

(11.3)

̂ 𝛼𝛽 are the components of the tensor reciprocal to the surface metric tensor defined by where G ̂ 𝛽 = 𝛿 𝛼 with 𝛿 𝛼 as the components of the identity tensor, and G ̂𝛼 ∙G ̂ and B̂ as the determinants G 𝛽 𝛽

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731

of the surface metric tensor and the curvature tensor on the shell reference surface, respectively. The mean curvature, H, and the Gaussian curvature, K, are then defined as: ( ) 1 1 1 1 + = B̂ .𝛼 2 R1 R2 2 𝛼 1 1 B̂ B̂ .2 B̂ .1 K≡ = B̂ .1 − B̂ .2 = 1 2 1 2 ̂ R1 R2 G

H≡

(11.4)

When the coordinate curves are the lines of curvature, we have B12 = B21 = G12 = G21 = 0, and G11 = G1 and G22 = G1 ; thus, the expressions for the mean and the Gaussian curvatures reduce 22 11 to: ) ) ( ( B 1 B11 1 1 1 + 22 = + 2 G11 G22 2 R1 R2 B11 B22 1 1 K= = G11 G22 R1 R2

H=

(11.5)

Finally, with the arc length parameters along the lines of curvature with G11 = G22 = 1, we have: H = 12 (B11 + B22 ) and K = B11 B22 .

11.2.3.4

Transformations: the Iso-curves to the Lines of Curvature

In our subsequent numerical evaluations of various quantities of interest by c-type finite element method, we would like to resort to the descriptions of these entities in terms of the optimal normal coordinate system: {S1 , S2 , 𝜂}, because of its most desired form of simplicity of expressions, and hence, computational efficiency. For example, the element stiffness matrices are numerically evaluated through integrations by quadrature rules at specific quadrature points; at these points on the shell reference surface, we can determine the lines of curvature as above for the simplest expressions for the involved derivatives with concomitant numerical efficiency, as will be detailed later. But, our geometry of the shell body is normally given to us in terms of the non-orthonormal coordinate system: {𝜉̂1 , 𝜉̂2 , 𝜂}. Thus, we need to know the necessary transformations for the ̂𝛼 ≡ derivatives in the above two coordinate system. We have on the shell reference surface: G ̂0 𝜕C 𝜕 𝜉̂𝛼

=

𝜕C0 𝜕S𝛽 𝜕S𝛽 𝜕 𝜉̂𝛼

𝛽

= ( 𝜕Ŝ𝛼 ) G𝛼 . Thus, because of orthonormality of G𝛼 , that is, with G𝛼 = G𝛼 : 𝜕𝜉

𝜕S𝛽 ̂ 𝛼 ∙ G𝛽 = G ̂ 𝛼 ∙ G𝛽 =G 𝜕 𝜉̂𝛼

(11.6)

Now, from equation (11.6), we have the desired relations: 𝜕(∙) 𝜕S𝛽 𝜕(∙) ̂ 𝛼 ∙ G𝛽 ) 𝜕(∙) ⇒ 𝜕(∙) = (G ̂ 𝛼 ∙ G𝛽 )−1 𝜕(∙) = = (G 𝜕S𝛽 𝜕S𝛽 𝜕 𝜉̂𝛼 𝜕 𝜉̂𝛼 𝜕S𝛽 𝜕 𝜉̂𝛼

(11.7)

Computation of Nonlinear Structures

732

In matrix form, we can express the desired transformation relations as: ⎧ 𝜕(∙) ⎫ [ ̂ 1 ∙ G1 G ⎪ 𝜕S1 ⎪ ⎨ 𝜕(∙) ⎬ = ̂ G2 ∙ G1 ⎪ ⎪ ⎩ 𝜕S2 ⎭

11.2.3.5

̂ 1 ∙ G2 G ̂ 2 ∙ G2 G

]−1 ⎧ 𝜕(∙) ⎫ ⎪ 𝜕 𝜉̂1 ⎪ ⎨ 𝜕(∙) ⎬ ⎪ ⎪ ⎩ 𝜕 𝜉̂2 ⎭

(11.8)

Shell Shifter Tensor

For a consistent reduction to, and development of, a 2D shell formulation in curved {𝜉 1 , 𝜉 2 , 𝜂 = 0} shell reference surface parameters from the 3D continuum mechanics principles, it is necessary that all the variables of interest in the shell space, that is, {𝜉 1 , 𝜉 2 , 𝜂 ≠ 0} in particular, be described in terms of the variables on the shell reference surface. This is achieved 𝜕C , known as the shifter tenby the introduction of a linear transformation, 𝜂 μ ≡ μ(𝜂) ≡ 𝜕C 0

sor. First, with 𝝃 ≡ { 𝜉 1 𝜉 2 𝜂 ≡ 𝜉 3 }T , let the undeformed shell space Jacobian, J(𝜂), at any 𝜂 ≠ 0 ∈ [𝜂 t , 𝜂 b ] be given by: J𝜂 ≡

̄2 ̄ 1× G G ̄ 1× G ̄ 2‖ , ‖G

𝜕C(𝜉 1 ,𝜉 2 ,𝜂) 𝜕𝝃

= A1 ⊗ Ei where A𝛼 =

𝜕C(𝜉 1 ,𝜉 2 ,𝜂) 𝜕𝜉 𝛼

𝛼 = 1, 2, and

the normal, N(𝜉 1 , 𝜉 2 ) = are the triad of base vectors at a point on the 𝜂-surface. In other words, we have Ai = J𝜂 Ei by definition, and J𝜂 may be represented in column form, that is, J𝜂 = [ A1 A2 A3 ≡ N ]. Note that N is generally not normal to A𝛼 , 𝛼 = 1, 2. 𝜕C ̄ i ⊗ Ei where Now, the reference surface Jacobian, J0 = J(𝜂 = 0), is given by: J0 ≡ 0 = G ̄𝛼 = G

𝜕𝝃

𝜕C0 (𝜉 1 ,𝜉 2 ) 𝜕𝜉 𝛼

𝛼 = 1, 2, and the normal, N, forms a triad of orthogonal base vectors at a point ̄ i = J0 Ei by definition, and J0 may be repreon the shell reference surface. As before, we have G ̄ 3 ]; recall that {Ei } is the set of orthonormal ̄1 G ̄2 N≡G sented in column form, that is, J0 = [ G bases associated with the fixed inertial reference frame. Now, we have a representation for the shifter, 𝜂 μ ≡ μ(𝜂), by the chain rules of derivatives as: 𝜕C 𝜕C ≡ = 𝜕C0 𝜕𝝃 J𝜂 = 𝜂 μ J0

𝜂μ

(

𝜕C0 𝜕𝝃

)−1 = J𝜂 J−1 0 .

(11.9)

Alternatively, we may also obtain it as: ̄ i = 𝜂μ G ̄ i ⇒ 𝜂 μ = Ai ⊗ G ̄i Ai = J𝜂 Ei = (J𝜂 J−1 )G 0

(11.10)

Referring to the column definitions above of J0 and J𝜂 , from equation (11.9), we have a column representation of μ as: μ = [ μ1 μ2 μ3 ≡ N ] in the shell reference surface optimal normal bases with μ1 and μ2 yet to be explicitly determined. In other words, the covariant base vectors, A𝛼 , 𝛼 = 1, 2 at any 𝜂-surface are related to those of the reference surface by the shifter, μ, and ̄ i through the contravariant base vectors, G ̄ i , of this has a tensor representation as: μ = Ai ⊗ G −1 −1 ̄ ̄ the reference surface. Also, from equation (11.10), we have: Gi = μ Ai ⇒ μ = Gi ⊗ Ai ⇒

Nonlinear Shell

733

̄ i . Thus, the contravariant base vectors at any 𝜂-surface are related to those of (μ−1 )T = Ai ⊗ G the reference surface by the shifter through: ̄i Ai = (μ−1 )T G

𝜕𝜂 ≡ 𝜂,C0 , and the position vector 𝜕C0 1 2 ̄ 3 (𝜉 1 , 𝜉 2 ), we get from the C0 (𝜉 , 𝜉 ) + 𝜂 G

̄ 3 = 𝜉 3 ,C ≡ Now, with the contravariant base vector G 0 of any point on a 𝜂-surface given by: C(𝜉 1 , 𝜉 2 , 𝜂) = definition of the shifter operator, μ: μ(𝜂) ≡

𝜕𝜉 3 𝜕C0

(11.11) =

𝜕C ̄ 3 ),C = G ̄𝛼⊗ G ̄𝛼 +G ̄ 3 ⊗ 𝜂,C +𝜂 G ̄ 3 ,C = (C0 + 𝜂 G 0 0 0 𝜕C0 i ̄ ̄ ̄ = Gi ⊗ G + 𝜂 G3 ,C0 = I − 𝜂 B

(11.12)

̄ i , is the identity tensor, and B is the symmetric surface curvature tensor, at ̄i⊗ G where, I ≡ G a point on the reference surface, derived as follows: noting that the position vectors C0 (𝜉 1 , 𝜉 2 ) ≡ C(𝜉 1 , 𝜉 2 , 0) on the reference surface, and applying the surface curvature relationship – that is, the Weingarten formula – we have: ̄ 3 , 𝛼 ≡ N, 𝛼 = −b.𝜎 G ̄ 𝜎 = −b𝛼𝛽 G ̄ 𝜎 = −b𝛼𝛽 G ̄𝛽 ̄ 𝜎𝛽 G G 𝛼

(11.13)

By using equation (11.13) and switching indices, we define the desired reference surface curvature tensor, B, as: ̄ 3 ,C = b𝛼𝛽 G ̄𝛽 ̄𝛼 ⊗G B ≡ −G 0

(11.14)

Now, by choosing our optimal coordinate system, that is, with arc length parameters, {S𝛼 }, along the lines of curvature, we have for the shifter, μ, the following matrix representation in curvilinear bases: 𝜂 ⎡1 − R 1 ⎢ μ(𝜂) = ⎢ 0 ⎢ ⎢ ⎣ 0

0 1−

𝜂 R2

0

0⎤ ⎥ ⎥ 0⎥ ⎥ 1⎦

(11.15)

In other words, we have: (

) 𝜂 G1 , R1 ( ) 𝜂 A2 = 1 − G2 , R2 A3 = N

A1 =

1−

(11.16)

Computation of Nonlinear Structures

734

Remark

r We

have taken time to get to equation (11.16) to establish it from the general curvilinear coordinate system. For coordinate systems along the lines of curvature with the arc length as the coordinate parameters, we could have directly used our specialized Weingarten formula to obtain the expression as:

C(S1 , S2 , 𝜂) = C0 (S1 , S2 ) + 𝜂 N(S1 , S2 ) A𝛼 ≡ C, 𝛼 = G𝛼 + 𝜂 N, 𝛼 = G𝛼 − 𝜂 B𝛼𝛽 G𝛽𝜎 G𝜎 ( ) B(𝛼𝛼) 𝜂 = G𝛼 − 𝜂 G𝜎 = 1 − G𝛼 (no sum) G(𝛼𝛼) R𝛼

⇒

(11.17)

and A3 ≡ C, 𝜂 = N where we have used: G11 =

1 ; G22 G22

=

1 ; G𝛼𝛽 G11

(11.18)

= 0, 𝛼 ≠ 𝛽, and

1 R𝛼

≡

B(𝛼𝛼) . Clearly the expresG(𝛼𝛼)

sions in equations (11.17) and (11.18) are same as those in equation (11.16).

11.2.3.6

Shell Shifter (Translator) Determinant

In our subsequent effort to describe various infinitesimal area and volume elements in the shell space, as well as on the shell reference surface, we will need the shifter determinant, denoted by μ, of the shifter tensor, μ. From the definition of the shell space Jacobian, J, and the shell reference surface Jacobian, J0 , we have their determinants as: det J(𝜂) = 𝜂 A1 × 𝜂 A2 ∙ (𝜂 A3 ≡ N) = 𝜀12 N ∙ N = √ det J0 = G1 × G2 ∙ (G3 ≡ N) = 𝜀12 N ∙ N = G

√

𝜂A

(11.19)

where, we have used the facts: N is orthonormal so√that A3 = A3 = G3 = G3 = N, and that √ √ 𝜀12 (𝜂) = A(𝜂) e12 , and 𝜀12 (𝜂 = 0) = A(𝜂 = 0) = G e12 are the components of the 2 × 2 surface covariant permutation tensors on the 𝜂-surface and the shell reference [ ] surface, respec0 1 tively; e12 is the component of the 2 × 2 permutation matrix given by . Now, from the 1 0 first of the equation (11.9), and equation (11.19), we get: 𝜂

𝜇 ≡ det 𝜂 μ = det (J𝜂 J−1 0 )=

det J𝜂 det J0

√ =

𝜂A

G

(11.20)

The determinant, 𝜇, of the shifter can also be given in terms of the mean curvature, H, and ̂ 1 , 𝜉 2 , 𝜂) = C0 (𝜉 1 , 𝜉 2 ) + 𝜂 N(𝜉 1 , 𝜉 2 ), we have: Gaussian curvature, K. To wit, noting that C(𝜉

Nonlinear Shell

735

A𝛼 = G𝛼 + 𝜂 N, 𝛼 = G𝛼 − 𝜂 B𝛼𝛽 G𝛽 . Thus, the components, A𝛼𝛽 , of the metric tensor at the 𝜂surface is given by: A𝛼𝛽 = A𝛼 ∙ A𝛽 = (G𝛼 − 𝜂 B𝛼𝜎 G𝜎 ) ∙ (G𝛽 − 𝜂 B𝛽𝜌 G𝜌 ) = G𝛼𝛽 − 2𝜂 B𝛼𝛽 + 𝜂 2 B𝜌𝛼 B𝛽𝜌 = G𝜎𝜌 (G𝛼𝜎 − 𝜂 B𝛼𝜎 )(G𝛽𝜌 − 𝜂 B𝛽𝜌 )

(11.21)

Now, let us note that for normal coordinates along the lines of curvature, the determinant of the contravariant metric tensor, G𝛼𝛽 is G1 where G = G11 G22 as G12 = 0. Similarly, the reference surface curvature tensor determinant is given by B = B11 B22 with B12 = 0. Now, with: (2 (G( −( 𝜂 B( det (G𝛼𝛽 − 𝜂 B𝛼𝛽 ) = (G11 − 𝜂 B11 )(G22 − 𝜂 B22 ) − ( 12 ( 12 ) { ( ) } B11 B22 B 2 = G 1 − 2𝜂 + + 𝜂 G = G(1 − 2𝜂H + 𝜂 2 K) G11 G22

(11.22)

where, we have used equation (11.5) for the definition of H and K. Thus, the determinant, A, from equation (11.21), is given by: 𝜂A

= det (G𝛼𝛽 ) {det(G𝛼𝛽 − 𝜂 B𝛼𝛽 )}2 =

1 2 G (1 − 2𝜂H + 𝜂 2 K)2 = G (1 − 2𝜂H + 𝜂 2 K)2 G

(11.23)

Finally, the determinant, 𝜇, using the above relation and equation (11.20), is given by: 𝜂𝜇

≡

det 𝜂 μ

=

det (J𝜂 J−1 ) 0

=

det J𝜂 det J0

√ =

𝜂A

G

= 1 − 2𝜂H + 𝜂 2 K

(11.24)

Moreover, for arc length parameterization along the lines of curvature, the determinant, μ, in equation (11.24) can be used by direct application of H and K using the radii of curvatures as indicated in equation (11.15).

11.2.3.7

Shell Boundary

For a theoretical formulation of the momentum balance and the moment of momentum balance laws useful for subsequent numerical evaluation, it will be necessary to produce a closed freebody; thus, we need to identify and particularize various components of boundaries that make up the total boundary, 𝜕B, of the 3D body in relation to the optimal normal coordinate systems on the shell reference surface; thus, we can think of it as being the union of three surfaces: shell top face, Mt , shell bottom face, Mb , and the shell side surface, MS .

11.2.3.8

Shell Top and Bottom Faces (Surfaces): Mt and Mb

The shell top and bottom faces are defined to be the extreme surfaces along the thickness direction defined by 𝜂 = 𝜂 t (S𝛼 ) and 𝜂 = 𝜂 b (S𝛼 ), respectively. The position vectors to a point on the top and bottom faces, as shown in Figure 11.4, are given by C(S1 , S2 , 𝜂 t ) and C(S1 , S2 , 𝜂 b ), respectively,

Computation of Nonlinear Structures

736

Figure 11.4 Shell top and bottom surface.

defined by: C(S1 , S2 , 𝜂 t ) = C0 (S1 , S2 ) + 𝜂 t (S1 , S2 ) N,

(S1 , S2 ) ∈ M

(11.25)

C(S1 , S2 , 𝜂 b ) = C0 (S1 , S2 ) + 𝜂 b (S1 , S2 ) N,

(S1 , S2 ) ∈ M

(11.26)

and

11.2.3.9

Shell Side Surface: MS

In order to define shell side surface, we first define a curve, 𝜕M, that completely encompasses and lies on the shell reference surface, M. The shell side surface is, then, defined by the union of normals, N(S1 , S2 ), (S1 , S2 ) ∈ 𝜕M, at and along the shell boundary, 𝜕M, and bounded by the shell top and bottom faces, as shown in Figure 11.5. For subsequent analysis, we may parameterize the boundary curve, 𝜕M, on the shell reference surface by an arc length parameter that will be denoted by S.

11.2.3.10

Shell Coordinates

With the above definitions, we can now sum up representation of the coordinates of all points in the 3D body as:

r a point {S1 , S2 , 𝜂} ∈ B, 𝜂 ≠ 0: C(S1 , S2 , 𝜂) = C0 (S1 , S2 ) + 𝜂 N,

𝜂 ∈ [−𝜂 b (S1 , S2 ), 𝜂 t (S1 , S2 )], (S1 , S2 ) ∈ M (11.27)

r a point on the shell reference surface: C0 (S1 , S2 ) ≡ C(S1 , S2 , 0) ,

(S1 , S2 ) ∈ M

(11.28)

Nonlinear Shell

737

Figure 11.5 Shell side surface.

r a point on the top surface: C(S1 , S2 , 𝜂 t ) = C0 (S1 , S2 ) + 𝜂 t (S1 , S2 ) N,

(S1 , S2 ) ∈ M

(11.29)

(S1 , S2 ) ∈ M

(11.30)

r a point on the bottom surface: C(S1 , S2 , 𝜂 b ) = C0 (S1 , S2 ) + 𝜂 b (S1 , S2 ) N,

r a point on the shell side surface: C(S, 𝜂) = C0 (S) + 𝜂(S) N,

𝜂 ∈ [−𝜂 b (S), 𝜂 t (S)],

S ∈ 𝜕M

(11.31)

where, S is the arc length parameter along the shell boundary, 𝜕M.

11.2.3.11

Shell Triads of Covariant Base Vectors

With the definitions: A𝛼 (S1 , S2 , 𝜂) ≡ 𝜕C0 (S1 ,S2 ) , 𝜕S𝛼

𝛼 = 1, 2, (S1 , S2 ) ∈ M, and

𝜕C(S1 ,S2 ,𝜂) = G𝛼 (S1 , S2 ) + 𝜂 N, 𝛼 𝜕S𝛼 1 ,S2 ) N, 𝛼 (S1 , S2 ) ≡ 𝜕N(S , we have 𝜕S𝛼

with

G𝛼 (S1 , S2 ) ≡

for:

r a point {S1 , S2 , 𝜂} on the parallel surfaces with 𝜂 ≠ 0: A𝜂𝛼 (S1 , S2 , 𝜂), N(S1 , S2 ),

𝜂 ∈ [−𝜂 b (S1 , S2 ), 𝜂 t (S1 , S2 )], (S1 , S2 ) ∈ M

(11.32)

r a point on the shell reference surface: G𝛼 (S1 , S2 ), N(S1 , S2 ),

(S1 , S2 ) ∈ M

(11.33)

(S1 , S2 ) ∈ M

(11.34)

r a point on the shell top surface: At𝛼 (S1 , S2 ), 𝝂 t (S1 , S2 ),

where, 𝝂 t is the outward unit normal at a point on the top surface,

Computation of Nonlinear Structures

738

r a point on the shell bottom surface: Ab𝛼 (S1 , S2 ), 𝝂 b (S1 , S2 ),

(S1 , S2 ) ∈ M

(11.35)

where, 𝝂 b is the outward unit normal at a point on the bottom surface,

r a point on the shell side surface:

t𝜂 (S, 𝜂), 𝝂 𝜂 (S, 𝜂), N(S), t0 (S) ≡ t𝜂 (S, 0),

𝜂 ∈ [−𝜂 b (S), 𝜂 t (S)],

S ∈ 𝜕M (11.36)

𝝂 0 (S) ≡ 𝝂 𝜂 (S, 0), where, S is the arc length parameter along the shell boundary, 𝜕M, and we have: 𝝂 𝜂 (S, 𝜂) ≡ t𝜂 (S, 𝜂) × N(S), 𝜂 ∈ [−𝜂 b (S), 𝜂 t (S)], S ∈ 𝜕M

11.2.3.12

Shell Reference Surface Differential Area and Interior Volume Elements

For our theoretical formulation in the Lagrangian form, we will subsequently need the description of a typical undeformed scalar differential area, dA, in the shell reference surface. Additionally, the area elements of the top, dAt , bottom, dAb , and side, dAs , surfaces will have to be described in terms of the corresponding reference surface and boundary curve elements. Finally, we will also need the description of a typical differential volume, dV, element belonging the shell space interior. 11.2.3.12.1 Reference Surface Oriented Differential Area: dA = N dA ̄ ̄ ̄𝛼 ̄𝛼 ̄ not The line element, dS𝜶̄ , along a coordinate curve is given by dS𝜶̄ = 𝜕𝜕𝜉C ̄𝛼 d 𝜉 = G𝛼̄ d 𝜉 with 𝛼 √ 𝜶 𝛼 ̄ 𝛼 summed. The magnitude, clearly, is given by: dS = G𝛼̄ 𝛼̄ d𝜉̄ . The line element, dS , in terms 𝜕C 𝛼 𝛼 of the arc length parameter, S𝛼 , is given by: dS𝜶 = 𝜕S 𝛼 dS = G𝛼 dS with 𝛼 not summed, and 𝛼 the magnitude by: dS with G𝛼𝛼 being unity. A typical differential area axial vector,√ using definition of the normal, N, to the shell reference surface, and the fact that: ‖G1 × G2 ‖ = G, is given by:

dA = N dA ≡ dS1 × dS2 = G1 dS1 × G2 dS2 = √ ⇒ dA = G dS1 dS2

√

√ G N dS1 dS2 = ( G dS1 dS2 ) N (11.37)

√ √ With G𝜶 , 𝛼 = 1, 2 being unit vectors along the lines of curvature, we have: G = G11 G22 = 1 with G12 = G21 = 0. We note, using equations (11.24) and (11.37), that the area element, dA𝜂 , for any 𝜂 in the shell interior, is given by: 𝜂

𝜂

dA = N dA = (

√

𝜂A

√ 1

2

dS dS ) N =

where 𝜂 𝜇 is defined as 𝜇 at any 𝜂 ∈ (−𝜂 b , 𝜂 t ).

𝜂A

G

√ ( G dS1 dS2 ) N = (𝜂 𝜇 dA) N

(11.38)

Nonlinear Shell

739

11.2.3.12.2 Shell Interior Differential Volume: dV 𝜼 Using equation (11.38) for dA𝜂 , we have: dV 𝜂 = dA𝜂 ∙ d𝜂 = N dA𝜂 ∙ N d𝜂 = 𝜂 μ dA d𝜂

(11.39)

11.2.3.12.3 Top Surface Oriented Differential Area: dAt = 𝝂 t dAt From equation (11.38), a typical differential area axial vector is given by: dAt = 𝝂 t dAt = (𝜇t dA) 𝝂 t where 𝜇t is defined as 𝜇 at 𝜂 = 𝜂 t . 11.2.3.12.4 Bottom Surface Oriented Differential Area: dAb = 𝝂 b dAb From equation (11.38), a typical differential area axial vector is given by: dAb = 𝝂 b dAt = (μb dA) 𝝂 b where 𝜇b is defined as 𝜇 at 𝜂 = 𝜂 b . 𝜼

𝜼

11.2.3.12.5 Side Surface Differential Area: dAS = 𝝂 o dAS and dAS = 𝝂 μS dAS There are several ways to relate the side surface area, dAS = 𝝂 o dAS , at 𝜂 = 0 with the surface 𝜂 𝜂 area, dAS = 𝝂 𝜇S dAS , at any 𝜂 ∈ [−𝜂 b , 𝜂 t ]. Before we present two such methods, note that, by definition, a typical differential area axial vector dAS = 𝝂 dAS at 𝜂 = 0 on the side surface, 𝜂 MS , is given by: dAS = dCS × dC𝜂 = C,S × C, 𝜂 dS d𝜂 = C,S × C, 𝜂 dAS where dAS ≡ dS d𝜂. Moreover, C(S, 𝜂) = C(C0 (S), 𝜂), C0 ,S = t0 , and as a triad of orthonormal bases, we have 𝝂 0 = t0 × N. Finally, with the component representation of t0 by t0 = t0𝛼 G𝛼 , we have with t0 = dS 𝛼 𝛼 C0 (S(S𝛼 )),S = ( dS 𝛼 ) C0 .𝛼 = t0 G𝛼 , the components as t0 ≡

dS . dS𝛼

dS 𝛼 Thus, N,S = N, 𝛼 dS 𝛼 = t0 N, 𝛼 ;

0

𝝂 0 = 𝜈0𝛽 G𝛽 . Now, 𝝂 0 = t0 × N = S, 𝛼 G𝛼 × N = S, 𝛼 𝜀𝛼𝛽 G𝛽 = 𝜈0𝛽 G𝛽 implying 0

0

0

0

S, 𝛼 𝜀𝛼𝛽 = 𝜈0𝛽 ⇒ S, 𝛼 𝜀𝛼𝛽 𝜀𝛽𝛼 = 𝜀𝛽𝛼 𝜈0𝛽 ⇒

0

𝜕S 𝛽 𝛿 = 𝜀𝛽𝛼 𝜈0𝛽 𝜕S𝛼 𝛼

(11.40)

11.2.3.12.6 Method 1 for Side Surface Area Transformation 𝜂 𝜂 Now, to have dAS in the form of 𝝂 𝜂 𝜇S dAS , we must have: 𝜂𝜇 𝝂

= 𝜂 T 𝝂0

(11.41)

where we have defined a transformation tensor, T, as: 𝜂T

≡ I + 𝜂 [(N,S ∙ t0 )I − (t0 ⊗ N,S )] = I + 𝜂 t0𝛼 [(N, 𝛼 ∙ t0 )I − (t0 ⊗ N, 𝛼 )]

(11.42)

Exercise: Show that: 𝜂 𝜇 𝝂 = 𝜂 T 𝝂 0

Solution 𝜂𝜇 𝝂

= C,S × C, 𝜂 = (t0 + 𝜂 N,S ) × N = 𝝂 0 + 𝜂 (N,S × 𝝂 0 × t0 ) = 𝝂 0 + 𝜂{(N,S ∙ t0 )𝝂 0 − (N,S ∙ 𝝂 0 )t0 } = {I + 𝜂 [(N,S ∙ t0 )I − (t0 ⊗ N,S )]} 𝝂 0 = 𝜂 T 𝝂 0

As a special case, if we have the side surfaces parallel to the lines of curvature, and use arc lengths, {S𝛼 }, as the reference surface parameter along the lines of curvature, then we have, say, for the side surface along the direction 1: t0 ≡ G1 ; 𝝂 0 ≡ G2 ; implying t01 = 1; N,1 = − R1 G1 = − R1 t0 ; 1

1

Computation of Nonlinear Structures

740

Figure 11.6 Shell surface along a line of curvature.

these imply that t01 (N,1 ∙ t0 ) = − R1 , and t01 (t0 ⊗ N,1 ) 𝝂 0 = − R1 (t0 ∙ 𝝂 0 ) t0 = 0. In this special 1

case, from equation (11.42), we have: 𝜂 T = (1 −

𝜂 )I R1

1

as shown in Figure 11.6. Similarly, for the

side surface parallel to the other line of curvature, direction 2, we get: 𝜂 T = (1 −

𝜂 )I. R2

11.2.3.12.7 Method 2: For Side Surface Area Transformation 𝜂 Now, a typical differential area axial vector dAS at any 𝜂 ≠ 0 on the side surface, MS , is obtained S ∈ 𝜕M by focusing on the expression for C(S, 𝜂 t ) = C0 (S) + 𝜂(S) N, 𝜂 ∈ [−𝜂 b (S), 𝜂 t (S)], as in equation (11.31). Note that 𝜂 A𝛼 × N = 𝜂 𝜀𝛼𝛾 𝜂 A𝛾 at any 𝜂, and 0 A𝛼 ≡ G𝛼 at 𝜂 = 0 with √𝜂 √ √ A 𝜂𝜀 𝜂A e = = G e𝛼𝛾 = 𝜂 μ 0 𝜀𝛼𝛾 . Furthermore, with the relation: 0 𝜀𝛽𝛼 0 𝜀𝛼𝛾 = −𝛿𝛾𝛽 , 𝛼𝛾 𝛼𝛾 G we can show that, on the side surface at any 𝜂: 𝝂 𝜂 dAS𝜂 = 𝜂 μ 𝜈0𝛽 𝜂 A𝛽 dAS0

(11.43)

Exercise: Prove the relation given by equation (11.43).

Solution: Using the relation given by equation (11.40), we get: 𝝂 𝜂 dAS𝜂 = dC(𝜂, S) × d𝜼 = C,S dS × N d𝜂 = S𝛼 ,S C, 𝛼 × N dAS0 = ( 0 𝜀𝛽𝛼 𝜈0𝛽 )( A𝛼 × N) dAS0 = ( 0 𝜀𝛽𝛼 𝜈0𝛽 )(−𝜂 𝜀𝛼𝛾 𝜂 A𝛾 ) dAS0 = −𝜈0𝛽 ( 0 𝜀𝛽𝛼 𝜂 𝜇 0 𝜀𝛼𝛾 ) 𝜂 A𝛾 dAS0 = 𝜂 𝜇 𝜈0𝛽 𝛿𝛾𝛽 𝜂 A𝛾 dAS0 = 𝜂 𝜇 𝜈0𝛽 𝜂 A𝛽 dAS0 In what follows, we use both method 1 and method 2 relations to show alternative definitions for a shell side-surface tractions.

11.2.4

Gauss–Weingarten Relations: Along the Lines of Curvature with Arc Length Parameterization

Among the most important formulas in the description of surfaces are the Gauss–Weingarten formulas that relate the base vector derivatives to the base vectors themselves – these are the

Nonlinear Shell

741

surface counterparts of the Frenet–Serret formulas for curves. However, for computational ease and efficiency, we must express these in the optimal coordinate system, that is, in terms of the base vectors along the lines of curvature, and parameterized in terms of the arc length parameters.

11.2.4.1

Normal Coordinate System

First, let us recall (see Chapter 6 on surfaces) the following Gauss–Weingarten formula in general ̂2 G ̂ 3 ≡ N }, and parameters (𝜉̂1 , 𝜉̂2 ), of a ̂1 G shell coordinates system, with base vectors { G 2-surface: ̂ 𝛽 ,𝛼̂ = Γ̂ 𝛾 G ̂ 𝛾 + B𝛽𝛼 N (Gauss) G 𝛽𝛼 ̂𝛽 N,𝛼̂ = −B𝛼. 𝛽 G

(11.44)

(Weingarten)

In matrix form, we can express this as: ̂1⎫ ⎧G ⎡ Γ̂ 1 ⎪ ⎪ ⎢ 1𝛼 ̂ 2 ⎬ , 𝛼 = ⎢ Γ̂ 1 ⎨G 2𝛼 ⎪N⎪ ⎢ −B̂ . 1 ⎣ 𝛼 ⎩ ⎭

Γ̂ 21𝛼 Γ̂ 2 2𝛼

−B̂ 𝛼. 2

̂ ⎫ B̂ 1𝛼 ⎤ ⎧ G ⎥⎪ 1⎪ ̂B2𝛼 ⎥ ⎨ G ̂2⎬ ⎥ ⎪ 0 ⎦⎩ N ⎪ ⎭

(11.45)

̂𝛾 , are given as the functions of where recall that the Christoffel symbols of the second kind, Γ 𝛽𝛼 the components of the metric tensor as: ̂ 𝜎𝛾 (G ̂ 𝛼𝜎 , ̂ + G ̂ 𝛾𝛽 ,𝛼̂ − G ̂ 𝛼𝛽 ,𝛾̂ ) Γ̂ 𝜎𝛼𝛽 = 12 G 𝛽 Γ̂ 3𝛼𝛽 ≡ B̂ 𝛼𝛽

(11.46)

̂ 𝜎𝛽 B̂ 𝛼𝜎 B̂ 𝛼. 𝛽 = G

11.2.4.2

Normal Coordinate System Along the Lines of Curvature

̄1 ≡ If we now consider the normal coordinate system, with base vectors set: {G ̄2 ≡ G

𝜕C0 𝜕𝜉 2

𝜕C0 𝜕𝜉 1

̄ 3 ≡ N}, and parameters (𝜉̄1 , 𝜉̄2 ) along the lines of curvature (which are, of G ̂ 𝛼𝛽 → G ̄ 𝛼𝛽 = 0, B̂ 𝛼𝛽 → B̄ 𝛼𝛽 = 0, 𝛼 ≠ 𝛽; also, course, perpendicular to each other), we have: G  ̄ 22 −  ̄ ̄ 12 ̄ ̄ =G ̄ 11 G G G with the determinant, G, of the 2 × 2 surface metric tensor, we have: G 21 ⇒ ̄ (22) G 1 1 (11) (22) ̄ ̄ G = = and G = with (𝛼𝛼) means no sum intended. Now, recalling that ̄ G

̄ (22) G B̄ ̄𝛼 ∙G ̄ 𝛼 = 𝛾𝛼 G𝛼 ∙ 𝛾𝛼 G𝛼 = 𝛾 2 where ̄ 𝛼𝛼 = G = = = = Ḡ (11) , and letting G 𝛼 (11) 𝜕C ̄ we have introduced unit base vector set {G𝛼 } ≡ { 𝜕S𝛼 } with 𝛾𝛼 as the length of G𝛼 , from the third 1 R1

B.11

̄ (11) G

̄ 𝛾1 B̄ 1𝛾 G

̄ (11) B̄ (11) G

of equation (11.45), we get:

𝛾 𝜕N ̄ 𝛽 = −B. 𝛼 G ̄ 𝛼 − B. 𝛽 G ̄𝛽 =− 1 G ̄ 𝛼 = − 𝛼 G𝛼 = −B.𝛼𝛽 G 𝛼 𝛼 𝛼 𝜕𝜉 R𝛼 R𝛼 ⏟⏟⏟ 𝛼≠𝛽

(11.47)

Computation of Nonlinear Structures

742

̄ 𝛼 = 𝛾𝛼 G𝛼 , we get: and from G 𝜕G𝛼 1 ̄ 1 ̄ = G , − 𝛾 , G 𝜕𝜉 𝛼 𝛾𝛼 𝛼 𝛼 𝛾𝛼2 𝛼 𝛼 𝛼

(11.48)

̄ 𝛼𝛽 = B̄ 𝛼𝛽 = 0, 𝛼 ≠ 𝛽, and G ̄ (𝛼𝛼) = Now, from the first of the relations in equation (11.45) with G 1 and so on, and noting that 𝜉 Γ̄ 𝜎𝛼𝛽 described in the {𝜉 𝛼 }-coordinate system, we get: ̄ G (𝛼𝛼)

( 𝜉Γ ̄𝜎 𝛼𝛽

1 ̄ 𝜎𝛾 = G 2

̄ 𝛾𝛽 ̄ 𝛼𝛽 ̄ 𝛼𝜎 𝜕 G 𝜕G 𝜕G + − 𝜕𝜉 𝛼 𝜕𝜉 𝛾 𝜕𝜉 𝛽

) (11.49)

1 ̄ 11 ,1̄ } = 1 1 G ̄ ̄ 11 G Particularizing equation (11.49): Γ̄ 111 = 12 {G ̄ 11 11 ,1̄ = 𝛾1 𝛾1 ,1̄ ; similarly, we get: 2G 𝛾 Γ̄ 211 = − 12 𝛾1 ,2̄ . Now, using equations (11.47) and (11.48), we get from the first of equation

(11.45):

𝛾2

) ( 1 ̄ 1 ̄ = 1 Γ̄ 𝛼 G ̄ + B(𝛼𝛼) N − 1 𝛾𝛼 ,𝛼̄ G ̄𝛼 G , − 𝛾 , G 𝛾𝛼 𝛼 𝛼̄ 𝛾𝛼2 𝛼 𝛼̄ 𝛼 𝛾𝛼 𝛼𝛼 𝛼 𝛾𝛼2 ( ) 𝛾𝛼2 1 ̄𝛼 ̄ 1 𝛽 ̄𝛽 + ̄𝛼 = N − 2 𝛾𝛼 ,𝛼̄ G Γ𝛼𝛼 G𝛼 + Γ̄ 𝛼𝛼 G 𝛾𝛼 R𝛼 𝛾𝛼

G𝛼 =

  𝛾 1 ̄ 𝛼 − 1 𝛾𝛼 ,𝛽̄ G ̄ 𝛽 + 𝛼 N − 1 𝛾𝛼 ̄𝛼 = 2 𝛾𝛼 ,𝛼̄ ,𝛼̄ G G 2 2 R𝛼 𝛾𝛼 𝛾𝛼 𝛾𝛽   =−

𝛾 1 𝛾𝛼 ,𝛽̄ G𝛼 + 𝛼 N, 𝛾𝛽 R𝛼

(11.50)

𝛼≠𝛽

Using equations (11.47) and (11.50), we can rewrite the Gauss–Weingarten formula as: ⎡ 0 ⎧ G1 ⎫ ⎢ ⎪ ⎢⎢ 1 𝜕 ⎪ G ⎨ 2 ⎬ = ⎢ 𝛾 𝛾1 ,2̄ 𝜕𝜉 1 ⎪ 2 ⎪ ⎩ N ⎭ ⎢ 𝛾1 ⎢ − ⎣ R1

−

1 𝛾 ,̄ 𝛾2 1 2 0 0

𝛾1 ⎤ R1 ⎥ ⎧ G ⎫ ⎥⎪ 1⎪ 0 ⎥ ⎨ G2 ⎬ ⎥⎪ ⎥⎩ N ⎪ ⎭ 0 ⎥ ⎦

(11.51)

and ⎡ 0 ⎧ G1 ⎫ ⎢ ⎪ ⎢⎢ 1 𝜕 ⎪ G ⎬ = ⎢ 𝛾 𝛾2 ,1̄ ⎨ 2 𝜕𝜉 2 ⎪ 1 ⎪ ⎩N⎭ ⎢ ⎢ 0 ⎣

1 𝛾 ,̄ 𝛾1 2 1 0 −

𝛾2 R2

0 ⎤ ⎥⎧ G ⎫ 𝛾2 ⎥ ⎪ 1 ⎪ ⎥ G2 ⎬ R2 ⎥ ⎨ ⎪ ⎥⎪ ⎩N⎭ 0 ⎥ ⎦

(11.52)

Nonlinear Shell

11.2.4.3

743

Optimal Normal Coordinate System

Now, let us consider the optimal normal coordinate system, that is, coordinates along the lines of curvature with arc length parameterization; let (S1 , S2 ) define a point on the shell undeformed ref𝛼 𝜕𝛾 𝜕𝛾 𝜕𝛾 𝜕𝜉 𝜕𝛾 𝜕𝛾 𝜕𝛾 = 𝛾2 𝜕S12 ≡ 𝛾2 𝛾1 ,2 erence surface. Noting that 𝜕S11 = 𝜕𝜉11 𝜕S11 = 𝛾1 𝜕𝜉11 , we get: 𝛾1 ,2̄ ≡ 𝜕𝜉12 = 𝜕S1𝛼 𝜕S 𝜕𝜉 2 1

G1 ,1̄ ≡

𝜕G1 𝜕G1 𝜕S𝛼 = = 𝛾1 G1 ,1 1 𝜕S𝛼 𝜕𝜉 1 𝜕𝜉

(11.53)

Using the first row of the relation in equation (11.51), we get: G1 ,1 ≡

𝜕G1 1 1 𝜕G1 = G , ̄≡ 1 1 𝛾1 𝛾1 𝜕𝜉 1 𝜕S1 ( ) 𝜕𝛾 1 1 𝛾1 1 = N − 𝛾2 12 G2 + 𝛾1 𝛾2 𝛾1 R1 𝜕S 1 𝜕𝛾1 1 = G + N 𝛾1 𝜕S2 2 R1

(11.54)

Modifications similar to equation (11.54) can be developed for each term of the equations (11.51) and (11.52). Thus, from equations (11.47) and (11.50), we get the desired Gauss–Weingarten formulas for the derivatives of the triad of base vectors in matrix form with respect to the arc length parameters, S𝛼 , 𝛼 = 1, 2, given by: ⎡ 0 ⎧ G1 ,1 ⎫ ⎧ G1 ⎫ ⎢ ⎪ ⎢ 1 ⎪ ⎪ 𝜕 ⎪ ⎨ G2 ,1 ⎬ ≡ 1 ⎨ G2 ⎬ = ⎢⎢ 𝛾 𝛾1 ,2 ⎪ N, ⎪ 𝜕S ⎪ N ⎪ ⎢ 1 ⎭ ⎩ 1 ⎭ ⎩ 1 ⎢ − ⎣ R1

−

1 𝛾 , 𝛾1 1 2 0 0

1 ⎤ R1 ⎥ ⎧ G ⎫ ⎫ ⎧ ⎥ ⎪ 1 ⎪ ⌢ ⎪ G1 ⎪ 0 ⎥ 0 ⎨ G2 ⎬ ≡ K1 ⎨ G2 ⎬ ⎥⎪ ⎪N⎪ ⎥⎩ N ⎪ ⎭ ⎭ ⎩ 0 ⎥ ⎦

(11.55)

0 ⎤ ⎥⎧ ⎧ G1 ⎫ G1 ⎫ ⎪ ⌢0 ⎪ ⎪ 1 ⎥⎥ ⎪ G ≡ K2 ⎨ G2 ⎬ R2 ⎥ ⎨ 2 ⎬ ⎪N⎪ ⎪ ⎥⎪ ⎩ ⎭ ⎩N⎭ 0 ⎥ ⎦

(11.56)

and ⎡ 0 ⎧ G1 ,2 ⎫ ⎧ G1 ⎫ ⎢ ⎪ ⎪ ⎪ ⎢ 1 𝜕 ⎪ ⎨ G2 ,2 ⎬ ≡ 2 ⎨ G2 ⎬ = ⎢⎢ − 𝛾 𝛾2 ,1 ⎪ N, ⎪ 𝜕S ⎪ N ⎪ ⎢ 2 ⎩ 2 ⎭ ⎩ ⎭ 0 ⎢ ⎣ ⌢

1 𝛾 , 𝛾2 2 1 0 −

1 R2

where K0𝛼 , 𝛼 = 1, 2 are the shell undeformed curvature matrices with R𝛼 (S1 , S2 ), 𝛼 = 1, 2, are the radii of curvatures at any point, {S𝛼 }, of the shell √ reference surface, and 𝛾𝛼 , 𝛼1 =2 1, 2, are dS𝛼 ̄ ̄ 𝛼 (𝜉 1 , 𝜉 2 ) ≡ 𝜕C(𝜉 ,𝜉 ,0) , with ̄ 𝛼𝛼 where G the lengths of the base vectors: 𝛾𝛼 ≡ d𝜉 𝛼 = ‖G𝛼 ‖ = G 𝜕𝜉 𝛼 no sum on 𝛼.

Computation of Nonlinear Structures

744

Remarks

r The curvature matrices are skew-symmetric. r The individual coefficients of the matrices can be easily identified with the Christoffel symbols of second kind, Γi .

r In our entire shelljkanalysis, we will completely avoid explicit use of these obscuring symbols by adhering to familiar geometric entities such as the radii of curvature, and so on, as we have done before, for ease of interpretation and clarity.

11.2.5

Initial Curvature Tensors and Vectors

On the other hand, the Gauss–Weingarten relation for the derivatives of the triad of base vectors on a surface in invariant tensor form with respect to the arc length parameters, S𝛼 , 𝛼 = 1, 2, may be obtained by first introducing the following. Definition: Shell initial curvature tensors, K0𝛼 , 𝛼 = 1, 2: ⌢

K0𝛼 ≡ (K0𝛼 )Tij Gi ⊗ Gj = −Gi ⊗ Gi , 𝛼 ,

𝛼 = 1, 2;

i, j = 1, 2, 3

(11.57)

To wit, for the second equality in equation (11.57), we note that because of anti-symmetry: ⌢

⌢

(K0𝛼 )Tij = −(K0𝛼 )ij . Then, we have: ⌢

⌢

⌢

K0𝛼 ≡ (K0𝛼 )Tij Gi ⊗ Gj = −(K0𝛼 )ij Gi ⊗ Gj = − Gi ⊗ (K0𝛼 )ij Gj = −Gi ⊗ Gi , 𝛼 , 𝛼 = 1, 2; i, j = 1, 2, 3 where we have used the result from equation (11.57). Note that the curvature tensor is skewsymmetric and thus: (K0𝛼 )T = −K0𝛼 . This can be easily verified: ⌢

⌢

(K0𝛼 )T ≡ (K0𝛼 )Tij Gj ⊗ Gi

= (K0𝛼 )Tji Gi ⊗ Gj ⏟⏟⏟ Switch indices

=

⌢

− (K0𝛼 )Tij Gi ⊗ Gj

= ⏟⏟⏟ Skew−symmetric

= −K0𝛼 , ⏟⏟⏟

𝛼 = 1, 2;

Definition

Definition: Shell initial curvature vector, k0𝛼 , 𝛼 = 1, 2: The axial vectors corresponding to the initial curvature tensors, K0𝛼 , 𝛼 = 1, 2, are given by: { k01 ≡ k02

≡

{

0 −

1 R1

1 R2

0

1 𝛾 , 𝛾1 1 2 }T 1 𝛾2 ,1 𝛾2

}T

−

(11.58)

Nonlinear Shell

745

Now, we can easily get from equation (11.57), the invariant tensor form as: Gi , 𝛼 = K0𝛼 Gi ,

𝛼 = 1, 2; i = 1, 2, 3

= k0𝛼 × Gi

(11.59)

Note that, in tensor form as in expression (11.57), unlike the matrix representations (11.55) and (11.56), each unit base vector derivative is given as a transformation of itself. As we shall see, this allows us to write expressions in absolute compact form free of matrix indices. Exercise: Verify the tensor form of the derivatives of base vectors given by equation (11.59).

Solution: = −(K0𝛼 )T Gi = −(−Gk , 𝛼 ⊗ Gk )Gi ⏟⏟⏟ ⏟⏟⏟ Definition Skew-symmetry = Gi , 𝛼 , 𝛼 = 1, 2; i = 1, 2, 3

K0𝛼 Gi

Exercise: Identify the components of the matrices in equations (11.55) and (11.56) as compoj nents of Christoffel symbol matrix, {Γi 𝛼 }, i = 1, 2, 3; j = 1, 2, 3; 𝛼 = 1, 2. Discussions

r Shell geometry representation goes hand in hand with the assumptions that we build into our r

r

method of analysis. This, in turn, entails consideration of the range of both geometric (such as thin or thick shell) and material properties (such as small or finite strain level) that the shell may predictably undergo. In classical shell theory, the shell kinematic parameters are restricted to only five components – three for displacements and two for rotations about the surface coordinate vectors at any point on the shell. We will be interested in parameterization by all six degrees of freedom, that is, three degrees of freedom each for displacement and rotation. In the ensuing nonlinear theory, we intend to hold onto these same six components as well for engineering applications. As we shall see, we have to resort to reasonable kinematic and material assumptions in our definition of a shell as a thin structural element undergoing small material property changes (such as hyperelastic strain), or alternately, allow it to be thick and subjected to finite strain, and all appropriate approximations are lumped in the declaration of the constitutive theory, which is at best empirical anyway. As an example, for kinematic behavior, it has been assumed that a normal section remains plane (i.e. no warping) but not necessarily perpendicular to the shell reference surface. Moreover, the plane cross-sections remain rigid in their own plane. This is commonly known as the Reissner– Mindlin hypothesis in which linear through-the-thickness deformation distribution is permitted. However, we must guarantee that this hypothesis is natural. In other words, adoption of this assumption is contingent upon the fact that the true 3D effects may be acceptably captured on an average sense over the cross-sections. The thin shells with surface dimensions conspicuously

Computation of Nonlinear Structures

746

dominant tend to behave in a way that makes this assumption seem reasonable. That is, with this assumption, we can still predict quite comfortably all the general behaviors of a shell: the axial, shear and bending characteristics.

11.2.6

Where We Would Like to Go

In the foregoing, we have defined shells suitable for computation based on the assumptions delineated above. In so doing, we have made extensive use of the general theory of surfaces. Clearly, then, we should review the chapter on:

r the theory and computation of surfaces. r Now, for proceeding onward with the shell analysis, we must move on to: r shell equation derivation: 3D to 2D. r Finally, to get familiarized with and be conscious about the overall computational strategy for nonlinear shells, we should explore:

r the road map: nonlinear shells.

11.3 Static and Dynamic Equations: Continuum Approach – 3D to 2D 11.3.1 What We Need to Recall

r A structural shell, by definition, is a 3D continuum with one dimension, namely the thickness,

r r

considerably smaller than the other two, that is, a continuum in the form of a 2D surface called the shell reference surface. Thus, as a surface, we will eventually be able to characterize the configurations (undeformed and deformed) of a shell as functions of two parameters of our choice: the arc lengths along the lines of curvature, on the reference surface of the shell. We characterize the shell undeformed reference surface by the arc length parameter S1 , S2 so that the position vectors on the shell reference surface may be given as C0 (S1 , S2 ) ≡ C(S1 , S2 , 0). S1 , S2 , as smooth reparameterization, are associated with 𝜉 1 , 𝜉 2 , the natural coordinate parameters along the lines of curvature, respectively, by the relations: S1 (𝜉 1 , 𝜉 2 ) = 𝜉1

r

r

𝜉2

∫0 ‖C(𝜌, 𝜉 2 , 0), 𝜌 ‖d𝜌 and S2 (𝜉 1 , 𝜉 2 ) = ∫0 ‖C(𝜉 1 , 𝜌, 0), 𝜌 ‖d𝜌 where C(𝜉 1 , 𝜉 2 , 0) is the position vector of a point on the shell reference surface. The undeformed 3D shell interior, B, is completely enclosed by the boundary 𝜕B which, in turn is made up of three disjoint surfaces: the top and bottom faces, denoted by M t and M b , respectively, and a side surface denoted by M S . Any undeformed thickness direction point away from the undeformed reference surface is fully parameterized by a parameter, 𝜂, along the normal, N(S1 , S2 ), to the reference surface at a point, (S1 , S2 ), on it; thus, any point on the shell body is identified by the coordinates (S1 , S2 , 𝜂) at any time t ∈ R+ . The 3D equations of motion in Lagrangian (material) formulation for a 3D body, B, with the surface undeformed surface area, 𝜕B, for dynamic loading conditions, can be written (see Chapter 7 on nonlinear mechanics) as:

∫

t dA +

𝜕B

∫ B

b dV =

d v 𝜌0 dV dt ∫ B

d c × t dA + c × b dV = c × v 𝜌0 dV ∫ ∫ dt ∫

𝜕B

B

(Momentum balance)

B

(11.60) (Moment of momentum balance)

Nonlinear Shell

r r r

747

where t and b are the surface traction vector measured per unit surface area and the body force vector measured per unit volume of the undeformed body, respectively, but acting on the current deformed body. In other words, we track the areas and the volumes of the neighborhoods around, t and b, respectively, in the current deformed body by identifying their corresponding areas and volumes in the undeformed body. v(S1 , S2 , 𝜂, t) is the current velocity at any point (S1 , S2 , 𝜂) and at the current time t ∈ R+ ; 𝜌0 is the mass density per unit volume of the undeformed body; c is the position vector of the point (S1 , S2 , 𝜂) on the body in the deformed configuration. According to the theory of surfaces, on the undeformed reference surface, the optimal normal curvilinear frame along the lines of curvature and the normal to the surface may be denoted by a triad of orthonormal base vectors G1 , G2 , G3 ≡ N. Additionally, for later use, we can equivalently write them in compact form as: Gk , k = 1, 2, 3. On the shell reference surface, the differential area is given by: dA = dS1 dS2 In the undeformed shell interior around any point (S1 , S2 , 𝜂), the differential volume, dV 𝜂 , is given by: dV 𝜂 = dA𝜂 ∙ d𝜂 = N dA𝜂 ∙ N d𝜂 = 𝜂 𝜇 dA d𝜂 √

r The shell top face differential area is given by: dAt = 𝜇t dA where 𝜇t =

𝜂t A

= 1 − 2𝜂 t H + with defined as 𝜇, the determinant of the shift tensor, at 𝜂 = and H and K as t the mean and Gaussian curvatures; A𝜂 and G are the surface metric determinant at 𝜂 = 𝜂 t and 𝜂 = 0, respectively. √ (𝜂 t )2 K

r The

𝜇t

𝜂t ,

G

shell bottom surface differential area is given by: dAb = 𝜇b dA where 𝜇b =

1 − 2𝜂 b H + (𝜂 b )2 K with 𝜇t defined as 𝜇 at 𝜂 = 𝜂 b .

r The shell side surface differential area at 𝜂 = 0 is given by: dAs

0

have: 𝝂 𝜂 dAs𝜂 = 𝜂 𝜇S 𝝂 𝜂 dAS0 where by method 1:

𝜂b A

G

=

= dS d𝜂, and at any 𝜂 ≠ 0, we

𝜂 𝜇S 𝝂

= 𝜂 T 𝝂0 𝜂 T = I + 𝜂 t𝛼 [(N, ∙ t )I − (t ⊗ N, )] 𝛼 0 0 𝛼 0

(11.61)

t0𝛼 = t0 ∙ G𝛼

and by method 2: 𝝂 dAs𝜂 = 𝜂 𝜇S 𝜈0𝛽 𝜂 A𝛽 dAS0 𝜈0𝛽 = 𝜈0 ∙ G𝛽 𝜂 A𝛽

≡

(11.62)

A𝛽 (S1 , S2 , 𝜂)

r With the representation of second-order tensor A as: A = A𝛼 ⊗

G𝛼 , we have:

DIV A = A𝛼 |𝛼

(11.63)

where DIV is the surface divergence operator.

r For any vector a and second-order tensor B, with the contravariant base vector set {G𝛼 }, we have: DIV(a × B) = a × DIV B + a, 𝛼 × B ∙ G𝛼

(11.64)

748

Computation of Nonlinear Structures

r Finally, as a notational reminder, in what follows, all lowercase Greek indices assume values 1 and 2, and the lowercase Latin indices runs through 1, 2, 3; also, for repeated diagonal indices, Einstein’s summation convention of summing over their ranges holds.

11.3.2

Dynamic Balance Equations: General Theory

As indicated above, we would like to establish the connection between the 2D shell balance equations and the balance equations of a shell viewed as a 3D continuum or body. For this, if we extract from the undeformed reference body a completely arbitrary and closed free-body with the interior given by Ω ⊂ B, and the enclosing boundary denoted by 𝜕Ω, we must have the Lagrange momentum balance and the Lagrange angular momentum balance laws satisfied for each equilibrium configuration of the free-body. We are free to choose the shape of the free-body with arbitrary boundary description; however, the choice is usually made such that the resulting derivation can be as simple as possible, and thus, naturally follows the undeformed geometry definition in the Lagrangian or material form of analysis. Accordingly, in the undeformed reference body, as shown in Figure 11.7, the boundary surface, 𝜕Ω, is made up of three disjointed surfaces: a top, 𝜕ΩT ⊂ M t , and a bottom face, 𝜕ΩB ⊂ M b , and a side surface, 𝜕ΩS . The side surface, denoted by 𝜕ΩS , is generated by the continuous set of normals, N(S, 0), at the reference time t = 0 drawn at any arc length parameter, S, along an arbitrary and closed curve, 𝜕ΩM ⊂ M, on the reference surface, perpendicular to the shell reference surface, ΩM ⊂ M, and intersecting the top and bottom faces. Now, let us recognize the surface traction, t, at any point, (S1 , S2 , 𝜂) at the current time t ∈ R+ , on the deformed boundary, as shown in Figure 11.8, of the free-body as: t = P ∙ Aj with P(S1 , S2 , 𝜂, t) as the two-point first Piola–Kirchhoff stress tensor (PK1) defined by: P = Pj ⊗ Aj with {Ai (S1 , S2 , 𝜂, t = 0)} as the set of covariant base vectors at a point, in the undeformed body, which when deformed to the current state becomes the traction point in consideration on the deformed surface. Note that: t ≡ Pj = P ∙ Aj is the traction at a point on the currently deformed surface that had Aj , the contravariant base vector at the point (S1 , S2 , 𝜂), as the normal in the undeformed state. Note that PK1 is expressed in a mixed basis form with the domain basis set as {Aj }, and the range basis

Figure 11.7 Shell undeformed free-body.

Nonlinear Shell

749

Figure 11.8 3D body: surface traction and body force.

set in the deformed body yet to be explicitly specified. The intended and computationally useful set is identified only after we recognize the nature of the underlying configuration space later. Considering that the shell bounding surface is made up of three surfaces, the tractions on each of these deformed surfaces can be identified as shown in Figure 11.7. With these definitions, we can specialize into the shell space the balance laws of equation (11.60) as:

∫

(tS ≡ P 𝝂 s ) dAs +

𝜕ΩS

(tt ≡ P 𝝂 t ) dAt +

∫

𝜕ΩT

∫

𝜕ΩS

c × P 𝜈 s dAs +

∫

∫

(tb ≡ P 𝝂 b ) dAb +

𝜕ΩB

c × P 𝝂 t dAt +

𝜕ΩT

∫

𝜕ΩB

∫

b dV 𝜂 =

Ω

c × P 𝝂 b dAb +

∫ Ω

c × b dV 𝜂 =

d v 𝜌0 dV 𝜂 dt ∫ Ω

d c × v 𝜌0 dV 𝜂 dt ∫ Ω

(11.65) where 𝝂 t , 𝝂 b and 𝝂 s are the normals at a point on the top face, 𝜕ΩT , the bottom face, 𝜕ΩB , and the side surface, 𝜕ΩS , respectively, of the undeformed body; dAt , dAb and dAs are the differential areas in the neighborhood of the point on the top face, the bottom face and the side surface, respectively, of the undeformed body; 𝜌0 (S1 , S2 , 𝜂, 0) is the mass density per unit volume of the undeformed body, that is, at the time t = 0; c(S1 , S2 , 𝜂, t) is the position vector of the point (with undeformed coordinates (S1 , S2 , 𝜂)) on the body in the deformed configuration at the current ∙ time t ∈ R+ ; v(S1 , S2 , 𝜂, t) ≡ c(S1 , S2 , 𝜂, t) is the current velocity at any material point at the current time t ∈ R+ that used to have the coordinates (S1 , S2 , 𝜂) in undeformed reference state; and dV 𝜂 = 𝜂 𝜇 dA d𝜂 is the differential volume of the interior, Ω, of the undeformed body as the volumetric neighborhood of the body force vector and the momenta at the point with undeformed coordinates (S1 , S2 , 𝜂) on the body.

11.3.2.1

Deformation Map

The theory we are trying to detail is expected to handle a general deformation map. So, let us assume that χ : B → E3 defines a sufficiently smooth deformation map for the shell body as

Computation of Nonlinear Structures

750

Figure 11.9 Shell deformation map.

shown in Figure 11.9. In other words, the deformed position vector, c(S1 , S2 , 𝜂, t), of a point that had a position vector, C(S1 , S2 , 𝜂, 0), in the undeformed reference state is related by: c = χ(C). On the shell deformed reference surface, χ(M), the position vectors are denoted by: c0 (S1 , S2 , t) ≡ c(S1 , S2 , 0, t). Now, we will assume that the deformation of the 3D shell-like body may be given by: c(S1 , S2 , 𝜂, t) = c0 (S1 , S2 , t) + h(S1 , S2 , 𝜂, t), h(S1 , S2 , 0, t) ≡ 0 c0

(S1 , S2 , t)

= C0

(11.66) (S1 , S2 ) + d(S1 , S2 , t)

where, h(S1 , S2 , 𝜂, t) is an as yet unknown and arbitrary vector-valued through-the-thickness deformation function, with the usual requirement of invertibility, that locates any arbitrary point on the deformed body, given the shell reference surface map χ(M), and d(S1 , S2 , t) is the displacement of a point (S1 , S2 ) on the reference surface at any time t ∈ R+ . For example, we may take h(S1 , S2 , 𝜂, t) = 𝜂 R(S1 , S2 , t) N(S1 , S2 ) where R(S1 , S2 , t) is a rotation tensor at the current time, t ∈ R+ and N(S1 , S2 ) is the normal at a point, (S1 , S2 ), on the undeformed shell reference surface. Clearly, this assumption implies that the unit normal to the undeformed reference surface remains inextensible, but not necessarily perpendicular to the deformed reference surface during deformation (because it rotates due to shear deformations), which is a commonly accepted underpinning of theories for very thin and flexible shells with shear deformations (reflected in the rotation of the reference surface) and goes by the name: Reissner–Mindlin-type kinematic assumptions. Thus, the choice of h(S1 , S2 , 𝜂, t) essentially helps categorizing the various shell theories with differing kinematic assumptions; more on this later.

11.3.2.2

Dynamic Balance Equations: 3D Continuum to Shell Space

In what follows, we will reduce the 3D continuum mechanics balance laws: the momentum balance and the angular momentum balance laws, given by equation (11.65), to shell space balance laws. We will be able to recognize the reduced equations as identical to those obtained by the direct engineering approach and thus establish the latter’s legitimacy as consistent with the laws of continuum mechanics.

Nonlinear Shell

11.3.3

751

Dynamic Momentum Balance Equations

Using the shell geometry and the deformation map as discussed above and rearranging the terms, we can rewrite the first of equation (11.65) as: ( ∫

𝜕ΩM

𝜂t

∫𝜂 b

(

) 𝜂 S

P 𝝂 𝜇 d𝜂 dS +

∫

ΩM

𝜂t

∫𝜂b

t) ( ) 𝜂t |𝜂 d | 𝜂 b 𝜇 d𝜂 + P 𝝂 𝜇| dA = v 𝜌0 𝜇 d𝜂 dA |b ∫𝜂 b dt ∫ |𝜂 ΩM

𝜂

(11.67) In order to get the final shell integral or differential equation, we introduce the following definitions for the surface stress resultant, surface linear momentum and the surface external applied forces:

11.3.3.1

Stress Resultants, F𝜶 , 𝜶 = 1, 2: by Method 1 Given by Equation (11.61)

From the first integral of equation (11.67), we have

F𝜈 (S, t) ≡ =

𝜂t

∫𝜂 b (

P(S, 𝜂, t) 𝝂(S, t) 𝜂 𝜇S d𝜂 𝜂t

∫𝜂b

) P(S, 𝜂, t) T(S, 𝜂, t) d𝜂 f P(S, t) ⏟⏟⏟

=

∙

𝝂 0 (S)

(11.68)

𝝂 0 (S) ⏟⏟⏟

Shell resultant stress tensor Shell boundary normal

where, in the last equality above, we have used the following definition: Stress resultant tensor:

f P(S, t)

≡

𝜂t

∫𝜂 b

P(S, 𝜂, t) T(S, 𝜂, t) d𝜂

(11.69)

Remark

r Note

r

the resemblance in the definition of shell edge traction between the last equality of equation (11.68) and the usual definition of a traction on a 3D body through the PK1-like relationship: a traction at a point on the deformed surface edge is given by a two-point surface stress tensor, f P, that has domain in the undeforgmed surface and the range in the current deformed surface, that is, a linear transformation that maps a normal, 𝝂 0 , at a point on the undeformed surface edge that became the traction point in the current and deformed state to the traction, F𝜈 (S, t). The unit normal, 𝝂 0 , is always defined in the tangent plane to any closed curve in the shell reference surface, that is, 𝝂 0 = 𝜈0𝛼 G𝛼 , and has no component along the normal, N.

Computation of Nonlinear Structures

752

So, the shell resultant stress tensor, f P(S, t), may have a representation: f P (S, t)

= F𝜈 (S, t) ⊗ 𝝂 0 = 𝝂 𝛼0 F𝜈 (S, t) ⊗ G𝛼 = F𝛼 (S, t) ⊗ G𝛼

(11.70)

where, we have introduced the following. Definition: Internal resistive stress resultant, F𝛼 (S, t), 𝛼 = 1, 2, per unit of undeformed length: F𝛼 (S, t) ≡ 𝝂 𝛼0 (S) F𝜈 (S, t) = f P (S, t) ∙ G𝛼 (S)

(11.71)

acting on the deformed shell surface as the projection of F𝜈 onto a tangent plane in the shell undeformed boundary containing the unit normal, 𝝂 0 . Noting that 1 = 𝝂 0 ∙ 𝝂 0 = 𝜈0𝛼 𝜈0𝛼 where 𝜈0𝛼 , 𝜈0𝛼 are the covariant and the contravariant components of the unit normal, 𝝂 0 , from the first equality of equation (11.71), we have: F𝛼 𝝂 0𝛼 ≡ 𝝂 0𝛼 𝝂 𝛼0 F𝜈 = F𝜈

11.3.3.2

(11.72)

Stress Resultant, F𝜶 (S, t), 𝜶 = 1, 2: by Method 2 Given by Equation (11.62) ( ) t t 𝜂

F𝜈 (S, t) = =

∫b (𝜂

𝜂

P(S, 𝜂, t) (𝝂(S, t) 𝜂 𝜇S ) d𝜂 = ) t

𝜂

∫𝜂 b

𝜂 𝛼

𝜂 S

P (S, 𝜂, t) 𝜇 d𝜂

∫𝜂b

𝜈0𝛼 =

{P(S, 𝜂, t) ∙ 𝜂 A𝛼 } 𝜂 𝜇S d𝜂

𝜈0𝛼 (11.73)

F𝛼 (S, t)

𝜈0𝛼

where, in the last equality above, we have used the following. Definition: Internal resistive stress resultant, F𝛼 (S, t), 𝛼 = 1, 2, per unit of undeformed boundary length: F𝛼

≡

𝜂t

∫𝜂b

𝜂 𝛼 𝜂 S

P

𝜇 d𝜂

(11.74)

where, of course, 𝜂 P𝛼 = P ∙ 𝜂 A𝛼 . Now, continuing further on equation (11.73), we have: F𝜈 (S, t) = F𝛼 (S, t) 𝜈0𝛼 = F𝛼 ⊗ G𝛼 ∙ 𝜈0𝛼 G𝛼 =

f P (S, t) ⏟⏟⏟

∙

𝝂 0 (S) ⏟⏟⏟

Shell resultant stress tensor Shell boundary normal

and where, in the last equality above, we have used again the following definition:

(11.75)

Nonlinear Shell

753

Stress resultant tensor: f P(S, t)

= F𝛼 (S, t) ⊗ G𝛼 (S)

(11.76)

whereby, we have: F𝛼 = f P ∙ G𝛼

(11.77)

Remarks

r Again, note the resemblance in the definition of shell edge traction between the last equality r

of equation (11.75) snd the usual definition of a traction on a 3D body through the Lagrange postulate-like relationship with a surface stress tensor, f P, and the normal, 𝝂 0 , to the boundary. Additionally, note that by equation (11.72) and the last equality of equation (11.73), the shell edge traction, F𝜈 (S, t), is equivalently described in both the methods.

11.3.3.3

Surface Linear Momentum, L(S1 , S2 , t) and Total Linear Momentum, LTOT (t)

From the integral on the right-hand side of equation (11.67), we have: ( d LTOT (t) ≡ dt ∫

𝜂t

∙

∫𝜂 b

ΩM

=

∙

∫

1

) v(S1 , S2 , 𝜂, t) 𝜌0 𝜂 𝜇 d𝜂

dA =

d L(S1 , S2 , t) dA dt ∫ ΩM

(11.78)

2

L(S , S , t) dA

ΩM ∙

where we have used the notation: (∙) ≡

d (∙), dt

and introduced:

Definitions

r Shell total linear momentum, LTOT (t): LTOT (t) ≡

∫

L(S1 , S2 , t) dA

(11.79)

ΩM

with

r Surface linear momentum, L(S1 , S2 , t), per unit of the undeformed reference surface area: L(S1 , S2 , t) ≡

𝜂t

∫𝜂 b

𝜂t

=

∫𝜂 b

v(S1 , S2 , 𝜂, t) 𝜌0 (S1 , S2 , 𝜂) 𝜂 𝜇(S1 , S2 , 𝜂) d𝜂 (11.80) ∙

𝜂

c(S , S , 𝜂, t) 𝜌0 (S , S , 𝜂) 𝜇(S , S , 𝜂) d𝜂 1

2

1

2

1

2

Computation of Nonlinear Structures

754

Figure 11.10 Shell surface forces and linear momentum.

∙

∙

∙

where v(S1 , S2 , 𝜂, t) ≡ c(S1 , S2 , 𝜂, t) = d(S1 , S2 , t) + h(S1 , S2 , 𝜂, t) is, as indicated earlier, the velocity at current time t ∈ R+ at a point with undeformed coordinates (S1 , S2 , 𝜂) in the interior of the shell-like body. Surface external applied force: We have from the second integral of equation (11.67) the following. Definition

r Surface external applied force, F(S ̂ 1 , S2 , t), per unit of undeformed reference surface area: ̂ 1 , S2 , t) ≡ F(S

𝜂t

∫𝜂 b

b(S1 , S2 , 𝜂, t) 𝜇𝜂 d𝜂 + t(S1 , S2 , 𝜂 x , t) 𝜇|𝜂x =𝜂b x

t

𝜂 =𝜂

(11.81)

̂ 1 , S2 , t) are the statically equivalent surface forces to the prescribed 3D surface Note that F(S tractions, t(S1 , S2 , 𝜂 x , t), x = b, t, on the top and bottom faces, and body forces, b(S1 , S2 , 𝜂, t). Thus, we can always apply the prescribed forces of a 3D shell-like body as equivalent forces on its 2D shell model, as shown in Figure 11.10.

11.3.3.4

Dynamic Momentum Balance Equations

Now, incorporating definitions (11.68) or (11.75), and (11.81) in equation (11.67), we get the integral form of the momentum balance equation: ∙

∫

𝜕ΩM

or,

∫

𝜕ΩM

f

F𝜈 (S, t) dS +

∫

̂ 1 , S2 , t) dA = LTOT (t) ≡ F(S

ΩM

P(S, t) 𝝂 0 (S) dS +

∙

∫ ΩM

∙

∫

ΩM

L(S1 , S2 , t) dA

̂ 1 , S2 , t) dA = LTOT (t) ≡ F(S

(11.82) ∙

∫ ΩM

1

2

L(S , S , t) dA

Nonlinear Shell

755

Now, by applying the surface Green–Gauss or divergence theorem to the first part of the second of equation (11.82), we can rewrite it to get the invariant form of the integral equation as: ̂ 1 , S2 , t)} dA = {DIV f P(S1 , S2 , t) + F(S

∫ ΩM

∙

∫

L(S1 , S2 , t) dA

(11.83)

ΩM

where DIV(∙) ≡ (∙), 𝛼 ⊗ G𝛼 is the material surface divergence operator. Noting that the shell surface area dA can be arbitrarily chosen, we have the invariant form of the differential equation as: ̂ 1 , S2 , t) = L(S1 , S2 , t) DIV f P(S1 , S2 , t) + F(S ∙

(11.84)

Remark

r Noting that with the representation of the stress resultant tensor, f P(S, t) = F𝛼 ⊗

G𝛼 , as given by the last equality of equation (11.70), or equation (11.76), we have by equation (11.63): DIV f P = F𝛼 |𝛼

(11.85)

Finally, using the relationship given by equation (11.85) in equation (11.84), we get the vector differential form of the momentum balance equations as: F𝛼 |𝛼 + F̂ = L ∙

(11.86)

We can now recognize equation (11.86) as identical to what is obtained by the direct engineering approach except that, as intended, we now have consistent interpretations for the stress resultant, F𝛼 (S, t), 𝛼 = 1, 2, the surface linear momentum, L(S1 , S2 , t), and the surface ̂ 1 , S2 , t), by equations (11.74), (11.80) and (11.81), respectively; this external applied load, F(S was not possible in the direct engineering approach whereby we could, at best, resort to ad hoc assumptions for the interpretations of the above shell surface entities.

11.3.4

Dynamic Angular Momentum Balance Equations: Integral and Differential Forms

Using the shell geometry and the deformation map as discussed above and rearranging the terms, we can rewrite the second of equation (11.65) as: ( ∫

𝜕ΩM

𝜂t

∫𝜂 b

) c×P 𝝂

𝜇𝜂S d𝜂

( dS +

d = (c × v 𝜌0 𝜂 𝜇 d𝜂)dA dt ∫ ΩM

∫

ΩM

𝜂t

∫𝜂b

t) |𝜂 | c × b 𝜇 d𝜂 + c × P 𝝂 𝜇| dA |𝜂 b

𝜂

(11.87)

Computation of Nonlinear Structures

756

In order to get the final shell integral or differential equation, we introduce the following definitions for the shell surface stress couple, the surface angular momentum and the surface external applied moments.

11.3.4.1

Stress Couples: M𝜶 (S1 , S2 , t), 𝜶 = 1, 2 by Method 1 Given by Equation (11.61)

From the first integral of equation (11.87), we have, with c = c0 + h: 𝜂t

c0 (S, t) ×

∫𝜂 b

P(S, 𝜂, t) 𝝂(S, t) 𝜇𝜂 d𝜂 + (

= c0 (S, t) × F𝜈 (S, t) +

𝜂t

𝜂t

∫𝜂 b

h(S, 𝜂, t) × P(S, 𝜂, t) 𝝂(S, t) 𝜇𝜂 d𝜂 )

h(S, 𝜂, t) × P(S, 𝜂, t) T(S, 𝜂) d𝜂

∫𝜂b

m P (S, t) ⏟⏞⏟⏞⏟

= c0 (S, t) × F𝜈 (S, t) +

𝝂 0 (S)

(11.88)

𝝂 0 (S) ⏟⏟⏟

∙

Shell boundary stress couple tensor Shell boundary normal

= c0 (S, t) × F𝜈 (S, t) + M𝜈 (S, t) where, we have used equation (11.68) for the definition of F𝜈 (S, t), and in the last equality above, we have used the following definition. ( M𝜈 (S, t) ≡

𝜂t

∫𝜂 b

) h(S, 𝜂, t) × P(S, 𝜂, t) T(S, 𝜂, t) d𝜂 m

=

P (S, t) ⏟⏞⏟⏞⏟

∙

𝝂 0 (S)

𝝂 0 (S) ⏟⏟⏟

(11.89)

Shell boundary stress couple tensor Shell boundary normal

with the introduction of the following. Definition: Stress couple tensor:

m P (S, t)

≡

𝜂t

∫𝜂b

h(S, 𝜂, t) × P(S, 𝜂, t) T(S, 𝜂, t) d𝜂

(11.90)

Remarks

r Note the resemblance in the definition of shell edge couple between the last equality of equation r

(11.89) and the usual definition of a couple on a 3D body through the Lagrange postulate-like relationship with a surface stress couple tensor, m P, and the normal, 𝝂 0 , to the boundary. Note also that the unit normal, 𝝂 0 , is always defined in the tangent plane to any closed curve in the shell reference surface, that is, 𝝂 0 = 𝜈0𝛼 G𝛼 and has no component along the normal, N.

Nonlinear Shell

757

Now, with M(S, t) = m P (S, t) ∙ 𝝂 0 from equation (11.89), the shell boundary stress couple tensor, m P (S, t), can have the representation: m P (S, t)

= M𝜈 (S, t) ⊗ 𝝂 0 (S) = 𝝂 𝛼0 M𝜈 (S, t) ⊗ G𝛼 (S) =

M𝛼

(11.91)

(S, t) ⊗ G𝛼 (S)

where, we have the following. Definition: Internal resistive stress couple, M𝛼 (S, t), 𝛼 = 1, 2, per unit of undeformed length: M𝛼 (S, t) ≡ 𝝂 𝛼0 (S) M𝜈 (S, t) = m P (S, t) ∙ G𝛼 (S)

(11.92)

as the projection of M𝜈 on a tangent plane of the shell boundary containing the unit normal, 𝝂 0 . Noting that 1 = 𝝂 0 ∙ 𝝂 0 = 𝜈0𝛼 𝝂 𝛼0 where 𝜈0𝛼 , 𝝂 𝛼0 are the covariant and the contravariant components of the unit normal, 𝝂 0 , from the first equality of equation (11.92), we have: M𝛼 𝝂 0𝛼 ≡ 𝝂 0𝛼 𝝂 𝛼0 M𝜈 = M𝜈

11.3.4.2

(11.93)

Stress Couples, M𝜶 (S1 , S2 , t), 𝜶 = 1, 2: by Method 2 Given by Equation (11.62) 𝜂t

M𝜈 (S, t) = =

∫𝜂b (

h(S, 𝜂, t) × P(S, 𝜂, t) (𝝂(S, t) 𝜂 𝜇S ) d𝜂 𝜂t

∫𝜂 b ( t 𝜂

=

∫𝜂 b

) 𝜂

𝛼

𝜂 S

h(S, 𝜂, t) × {P(S, 𝜂, t) A } 𝜇 d𝜂

𝜈0𝛼

) h(S, 𝜂, t) × 𝜂 P𝛼 (S, 𝜂, t) 𝜂 𝜇S d𝜂

(11.94)

𝜈0𝛼

= M𝛼 (S, t) 𝜈0𝛼 (S) where, in the last equality above, we have used the following. Definition: Internal resistive stress couple, M𝛼 (S, t), 𝛼 = 1, 2, per unit of undeformed boundary length: M𝛼 (S, t)

≡

𝜂t

∫𝜂 b

where, of course, 𝜂 P𝛼 = P ∙ 𝜂 A𝛼 .

h(S, 𝜂, t) × 𝜂 P𝛼 (S, 𝜂, t)𝜂 𝜇S (S, 𝜂, t) d𝜂

(11.95)

Computation of Nonlinear Structures

758

Remarks

r Note that a specific choice as to the nature of the function h(S, 𝜂) identifies a specific shell

theory. For example, in its simplest form: h(S, 𝜂) = 𝜂 n in equation (11.66), keeping the normal to the surface inextensible through the deformation history, Reissner–Mindlin shell theory may be developed. In this case, however, while we will have six degrees of freedoms, we end up with no so-called “drilling” couples in the deformed coordinate system: M 𝛼3 = n ∙ M𝛼 = 𝜂t n ∙ ∫𝜂b 𝜂 n × 𝜂 P𝛼 𝜂 𝜇S d𝜂 = 0 for 𝛼 = 1, 2. With the addition of one more through-thethickness degree of freedom, 𝜆(𝜂), we may relax into a higher-order shell theory by allowing the deformed normal extensible by choosing h(S, 𝜂) = 𝜆(𝜂) n. Clearly, other possibilities exist to model various materials such as rubber, and so on, in which cases appropriate constitutive theories demand representation of the more relaxed deformations.

Now, continuing further on equation (11.94), we have: M𝜈 (S, t) = M𝛼 (S, t) 𝜈0𝛼 (S) = M𝛼 ⊗ G𝛼 ∙ 𝜈0𝛼 G𝛼 =

m

P (S, t) ⏟⏞⏟⏞⏟

∙

𝝂 0 (S) ⏟⏟⏟

(11.96)

Shell stress couple tensor Shell boundary normal

and where, in the last equality above, we have again used the following. Definition

r Stress couple tensor: m P (S, t)

= M𝛼 (S, t) ⊗ G𝛼 (S)

(11.97)

whereby, we have: M𝛼 (S, t) =

m P(S, t) ∙ G𝛼 (S)

(11.98)

Remark

r Again,

we may note the resemblance in the definition of shell edge traction, between the last equality of equation (11.96), and the usual definition of a traction on a 3D body through the PK1-like relationship: a traction at a point on the deformed surface edge is given by a two-point surface stress couple tensor, m P, that has domain in the undeformed surface and the range in the current deformed surface, that is, a linear transformation that maps a normal, 𝝂 0 , at a point on the undeformed surface edge that became the traction point in the current and deformed state, to the traction, M𝜈 (S, t). Also note that by equation (11.93) and the last equality of equation (11.94), the shell edge couple, M𝜈 (S, t), is equivalently described in both the methods.

Nonlinear Shell

11.3.4.3

759

Surface Angular Momentum, A(S1 , S2 , t) and Total Angular Momentum, ATOT (t)

From the integral on the right-hand side of equation (11.87), we have, with c = c0 + h: ( d ATOT (t) ≡ dt ∫ ∙

ΩM

𝜂t

∫𝜂b

) 𝜂

c × v 𝜌0 𝜇 d𝜂 (

d c(S1 , S2 , t) × = dt ∫

𝜂t

∫𝜂b

ΩM

(

d c(S1 , S2 , 𝜂, t) × = dt ∫ ΩM

dA )

∙

𝜂

c(S , S , 𝜂, t) 𝜌0 (S , S , 𝜂) 𝜇(S , S , 𝜂) d𝜂 𝜂t

∫𝜂 b

1

2

1

2

1

2

dA )

c(S1 , S2 , 𝜂, t) 𝜌0 (S1 , S2 , 𝜂) 𝜂 𝜇(S1 , S2 , 𝜂) d𝜂 ∙

dA (11.99)

(

)

⎧ 𝜂t ∙ ⎪ c0 (S1 , S2 , t) × c(S1 , S2 , 𝜂, t) 𝜌0 (S1 , S2 , 𝜂) 𝜂 𝜇(S1 , S2 , 𝜂) d𝜂 ∫ ⎪ d 𝜂b = 𝜂t dt ∫ ⎨ ∙ ⎪+ 1 2 1 2 1 2 𝜂 1 2 ΩM ⎪ ∫ b h(S , S , 𝜂, t) × c(S , S , 𝜂, t) 𝜌0 (S , S , 𝜂) 𝜇(S , S , 𝜂) d𝜂 𝜂 ⎩ ) ( d = c0 (S1 , S2 , t) × L(S1 , S2 , t) + A(S1 , S2 , t) dA dt ∫

⎫ ⎪ ⎪ ⎬ dA ⎪ ⎪ ⎭

ΩM

where we have used the definition for the linear momentum, L(S1 , S2 , t), as given by equation (11.80), and introduced the total angular momentum, ATOT (t), and the surface angular momentum, A(S1 , S2 , t), as follows: Definitions

r Shell total angular momentum, ATOT (t): ATOT (t) ≡

) ( c0 (S1 , S2 , t) × L(S1 , S2 , t) + A(S1 , S2 , t) dA

∫

(11.100)

ΩM

with

r Surface angular momentum, A(S1 , S2 , t), per unit of the undeformed reference surface area: A(S1 , S2 , t) ≡

𝜂t

∫𝜂 b 𝜂t

=

∫𝜂 b

h(S, 𝜂, t) × v(S1 , S2 , 𝜂, t) 𝜌0 (S1 , S2 , 𝜂) 𝜂 𝜇(S1 , S2 , 𝜂) d𝜂 (11.101) ∙

𝜂

h(S, 𝜂, t) × c(S , S , 𝜂, t) 𝜌0 (S , S , 𝜂) 𝜇(S , S , 𝜂) d𝜂 1

2

1

2

1

2

Computation of Nonlinear Structures

760

Figure 11.11 Shell surface moment and angular momentum.

∙

∙

∙

where v(S1 , S2 , 𝜂, t) ≡ c(S1 , S2 , 𝜂, t) = d(S1 , S2 , t) + h(S1 , S2 , 𝜂, t) is the velocity at current time t ∈ R+ at a point with undeformed coordinates (S1 , S2 , 𝜂) in the interior of the shell-like body, as shown in Figure 11.11.

11.3.4.4

Surface External Applied Moment

From the second integral of equation (11.87), we have, with c = c0 + h:

1

𝜂t

2

c0 (S , S , t) ×

∫𝜂b

b(S , S , 𝜂, t) 𝜇(S , S , 𝜂, t) d𝜂 + 1

2

1

2

𝜂t

∫𝜂 b

h(S1 , S2 , 𝜂, t)

× b(S1 , S2 , 𝜂, t) 𝜇(S1 , S2 , 𝜂, t) d𝜂 + c0 (S1 , S2 , t) × (t(S1 , S2 , 𝜂, t) 𝜂 𝜇)|𝜂b + (h(S1 , S2 , 𝜂, t) × t(S1 , S2 , 𝜂, t) 𝜂 𝜇)|𝜂b 𝜂 𝜂 ( t t

= c0 (S1 , S2 , t) ×

𝜂

∫𝜂b

t

b(S1 , S2 , 𝜂, t)

𝜂

(11.102)

)

𝜇 d𝜂 + (t(S1 , S2 , 𝜂, t) 𝜂 𝜇)|𝜂b t

𝜂

̂ 1 , S2 , t) = c0 (S1 , S2 , t) × F(S ̂ 1 , S2 , t) + M(S ̂ 1 , S2 , t) + M(S ̂ 1 , S2 ), and in the last equality above, where we have used equation (11.81) for the definition of F(S we have introduced the following. Definition

r External applied moment, M(S ̂ 1 , S2 , t), per unit of undeformed reference surface area: ̂ 1 , S2 , t) ≡ M(S

𝜂t

∫𝜂b

h(S1 , S2 , 𝜂, t) × b(S1 , S2 , 𝜂, t)𝜇𝜂 d𝜂 + h(S1 , S2 , 𝜂, t) × t𝜇|𝜂b t

𝜂

(11.103)

Nonlinear Shell

761

Remark

r Note that M(S ̂ 1 , S2 ) is the statically equivalent shell surface moments to the prescribed 3D surface tractions, t, on the top and bottom faces, and body forces, b. Thus, we can always apply the prescribed moments of a 3D shell-like body as equivalent moments on its 2D shell model. Now, incorporating definitions (11.88) and (11.103) in equation (11.87), we get the integral form of the angular momentum balance equation:

∫

(M𝜈 + c0 × F𝜈 ) dS +

𝜕ΩM

or,

∫

∫

∙ ( ) ̂ dA = ATOT (t) ≡ d (c0 × F̂ + M) c0 × L + A dA ∫ dt

ΩM

( m P + c0 × f P) ∙ 𝝂 0 dS +

𝜕ΩM

ΩM

∙ ) ( d ̂ (c0 × F̂ + M)dA = ATOT (t) ≡ c0 × L + A dA ∫ dt ∫

ΩM

ΩM

(11.104) Noting that with the representation of (11.63):

m P (S, t)

as in equation (11.91), we have by equation

DIV m P = M𝛼 |𝛼

(11.105)

DIV (c0 × f P) = c0 × DIV f P + c0 , 𝛼 × ( f P ∙ G𝛼 )

(11.106)

and by equation (11.64), we have:

Now, by applying surface Green–Gauss or divergence theorem to the second of equation (11.104) and applying relationships given by equation (11.105) and equation (11.106), and taking derivative of the right-hand side of second of the equation (11.104) and rearranging terms, we can rewrite it to get: { ∫

} (( ∙ f (( ̂ dA (( DIV m P + c0 , 𝛼 × (f P ∙ G𝛼 ) + c0 × (( P + F̄ − L) + M DIV

ΩM

=

∫

(11.107)

) (∙ ∙ c 0 × L + A dA

ΩM

If the momentum balance equation (11.84) is satisfied identically, the third term of equation (11.107) drops out as shown. Now, applying the relationship given by equation (11.77) in the second term, we get what we will call the invariant form of the integral equation:

∫ ΩM

̂ dA = {DIV m P + c0 , 𝛼 × F𝛼 + M}

∫ ΩM

) (∙ ∙ c 0 × L + A dA

(11.108)

Computation of Nonlinear Structures

762

Noting that the shell free-body surface area ΩM can be arbitrarily chosen, we have the invariant form of the differential equation as: ̂ = c∙ 0 × L + A DIV m P + c0 , 𝛼 × ( f P ∙ G𝛼 ) + M ∙

(11.109)

Finally, using relationship given by equation (11.105), we get the vector form of angular momentum balance equations as: ̂ = c∙ 0 × L + A M𝛼 |𝛼 + c0 , 𝛼 × F𝛼 + M ∙

(11.110)

We can now recognize equation (11.110) as identical to what is obtained by the direct engineering approach except that, as intended, we now have consistent interpretations for the stress couple, M𝛼 (S, t), 𝛼 = 1, 2, the surface angular momentum, A(S1 , S2 , t), and the surface exter̂ 1 , S2 , t), by equationS (11.95), (11.101) and (11.103), respectively. nal applied moment, M(S This was not possible in the direct engineering approach, where we could, at best, resort to ad hoc assumptions for interpretations of the shell surface entities. Exercise:

Show that if P = P𝛽 ⊗ G𝛽 , then DIV P = P𝛽 |𝛽

Solution: Note that G𝛼 |𝛼 = 0, that is, the covariant derivative of a base vector is 0 (why?). So, by definition: DIV P ≡ GRAD P:I = P𝛼𝛽|𝛾 G𝛼 ⊗ G𝛽 ⊗ G𝛾 : G𝜎 ⊗ G𝜎 = P𝛼𝛽|𝛾 G𝛾𝛽 G𝛼 = P𝛼𝛾 |𝛾 G𝛼 = P𝛼𝛾 |𝛾 G𝜌𝛼 G𝜌 = P𝛼𝛽 |𝛽 G𝛼 = (P𝛼𝛽 G𝛼 )|𝛽 = P𝛽 |𝛽 .

11.3.5

Static Balance Equations: General Theory

The quasi-static balance laws can be easily obtained from their dynamic counterparts by assuming that the linear and the angular momentum vectors identically vanish, that is, L(S1 , S2 , t) ≡ 0

(11.111)

A(S1 , S2 , t) ≡ 0 Thus, from equation (11.86), we get: 11.3.5.1

Static Momentum Balance Equations (Force Equilibrium) F𝛼 |𝛼 + F̂ = 0,

𝛼 = 1, 2

(11.112)

and from equation (11.110): 11.3.5.2

Static Angular or Moment of Momentum Balance Equations (Moment Equilibrium) ̂ = 0, M𝛼 |𝛼 + c0 , 𝛼 × F𝛼 + M

𝛼 = 1, 2

(11.113)

Nonlinear Shell

11.3.6

763

General Shell Vector Dynamic Balance Equations of Motion

For easy reference, here we copy and collect the derived dynamic balance equations as follows: F𝛼 , 𝛼 + F̂ = L, ∙

𝛼 = 1, 2

Momentum balance

̂ = c∙ 0 × L + A, M𝛼 , 𝛼 + c0 , 𝛼 × F𝛼 + M ∙

𝛼 = 1, 2

Moment of momentum balance (11.114)

11.3.7

General Shell Vector Static Balance Equations of Motion

Here we copy and collect the derived static balance equations: F𝛼 |𝛼 + F̂ = 0,

𝛼 = 1, 2

Momentum balance

̂ = 0, M𝛼 |𝛼 + c0 , 𝛼 × F𝛼 + M

𝛼 = 1, 2

Moment of momentum balance

(11.115)

Remark

r We have deduced the shell balance equations consistently by starting from the 3D continuum mechanics balance laws. In this respect, the equations are exact. They are the true equations for any shell-like body, linear or nonlinear, thick or thin.

11.3.8

Where We Would Like to Go

So far, we have derived the balance equations by reducing the 3D equations of motion of continuum mechanics for general shells. However, for our purposes in the rest of the book, we will restrict ourselves to an important subclass of shells: thin and very flexible shells. Thus, we must specialize the above balance equations to thin shells; for this, we should check out:

r thin shell dynamic and static balance equations 11.4 Static and Dynamic Equations: Continuum Approach – Revisited 11.4.1 What We Need to Recall Let us recall the following for general shells.

r We characterize the shell undeformed reference surface by the arc length parameters S1 ,

S2 so ≡

(S1 , S2 )

r

that the position vectors on the undeformed shell reference surface may be given as C0 C(S1 , S2 , 0, 0). S1 , S2 , as smooth reparameterizations, are associated with 𝜉 1 , 𝜉 2 , the natural coordinate parameters along the lines of curvature, respectively, by the relations: S1 (𝜉 1 , 𝜉 2 ) =

r

∫0 ‖C(𝜌, 𝜉 2 , 0), 𝜌 ‖d𝜌 and S2 (𝜉 1 , 𝜉 2 ) = ∫0 ‖C(𝜉 1 , 𝜌, 0), 𝜌 ‖d𝜌 where C(𝜉 1 , 𝜉 2 , 0) is the position vector of a point on the shell reference surface. c(S1 , S2 , 𝜂, t) is the position vector of the point (with undeformed coordinates (S1 , S2 , 𝜂)) on the ∙ body in the deformed configuration at the current time t ∈ R+ ; v(S1 , S2 , 𝜂, t) ≡ c(S1 , S2 , 𝜂, t)

𝜉1

𝜉2

Computation of Nonlinear Structures

764

r

is the current velocity at any material point at the current time t ∈ R+ that used to have the coordinates (S1 , S2 , 𝜂) in the undeformed reference state; on the shell deformed reference surface, χ(M), the position vectors are denoted by: c0 (S1 , S2 , t) ≡ c(S1 , S2 , 0, t). The deformed position vector has a representation as: c0 (S1 , S2 , t) = C0 (S1 , S2 ) + d(S1 , S2 , t)

r r

(11.116)

where d(S1 , S2 , t) is the displacement vector of a point (S1 , S2 ) on the mid-surface at current time t ∈ R+ and C0 (S1 , S2 ) is the position vector of the same point on the undeformed mid-surface. P(S1 , S2 , 𝜂, t) as the two-point first Piola–Kirchhoff stress tensor (PK1) defined by: P = Pj ⊗ Aj with {Ai (S1 , S2 , 𝜂, t = 0)} as the set of covariant base vectors at a point in the undeformed body which, when deformed to the current state, become the traction point in consideration on the deformed surface. the deformation map of the 3D shell-like body may be given by: c(S1 , S2 , 𝜂, t) = c0 (S1 , S2 , t) + h(S1 , S2 , 𝜂, t),

(11.117)

h(S1 , S2 , 0, t) ≡ 0

r

where, h(S1 , S2 , 𝜂, t) is an as yet unknown and arbitrary vector-valued through-the-thickness deformation function, with the usual requirement of invertibility, that locates any arbitrary point on the deformed body, given the shell reference surface map χ(M). The stress resultant, F𝛼 (S, t), 𝛼 = 1, 2, per unit of undeformed boundary length, is defined as:

F𝛼 ≡

𝜂t

∫𝜂b

𝜂 𝛼 𝜂 S

P

𝜇 d𝜂

(11.118)

where, of course, 𝜂 P𝛼 = P 𝜂 A𝛼 .

r The effective surface linear momentum, L(S1 , S2 , t), per unit of undeformed surface area, is defined as: L(S1 , S2 , t) ≡

𝜂t

∫𝜂b 𝜂t

=

∫𝜂b

v(S1 , S2 , 𝜂, t) 𝜌0 (S1 , S2 , 𝜂) 𝜂 𝜇(S1 , S2 , 𝜂) d𝜂 (11.119) c(S1 , S2 , 𝜂, t) 𝜌0 (S1 , S2 , 𝜂) 𝜂 𝜇(S1 , S2 , 𝜂) d𝜂 ∙

∙

r

where v(S1 , S2 , 𝜂, t) ≡ c(S1 , S2 , 𝜂, t) is, as indicated earlier, the velocity at current time t ∈ R+ at a point with undeformed coordinates (S1 , S2 , 𝜂) in the interior of the shell-like body. ̂ 1 , S2 , t), per unit of undeformed reference surface area, The surface external applied force, F(S is defined as: ̂ 1 , S2 , t) ≡ F(S

𝜂t

∫𝜂 b

b(S1 , S2 , 𝜂, t) 𝜇𝜂 d𝜂 + t(S1 , S2 , 𝜂 x , t) 𝜇|𝜂x =𝜂b x

𝜂 =𝜂

t

(11.120)

Nonlinear Shell

765

r The stress couple, M𝛼 (S1 , S2 , t),

𝛼 = 1, 2, per unit of undeformed boundary length, is defined

as: M𝛼 (S, t) ≡

𝜂t

h(S, 𝜂, t) ×

∫𝜂b

𝜂 𝛼

P (S, 𝜂, t) 𝜂 𝜇S (S, 𝜂, t) d𝜂

(11.121)

r The effective surface angular momentum, A(S1 , S2 , t), per unit of undeformed surface area, is defined as: A(S1 , S2 , t) ≡

𝜂t

∫𝜂 b 𝜂t

=

∫𝜂 b

h(S, 𝜂, t) × v(S1 , S2 , 𝜂, t) 𝜌0 (S1 , S2 , 𝜂) 𝜂 𝜇(S1 , S2 , 𝜂) d𝜂 (11.122) ∙

𝜂

h(S, 𝜂, t) × c(S , S , 𝜂, t) 𝜌0 (S , S , 𝜂) 𝜇(S , S , 𝜂) d𝜂 1

2

1

2

1

2

∙

r

where v(S1 , S2 , 𝜂, t) ≡ c(S1 , S2 , 𝜂, t) is, as indicated earlier, the velocity at current time t ∈ R+ at a point with undeformed coordinates (S1 , S2 , 𝜂) in the interior of the shell-like body. ̂ 1 , S2 , t), per unit of undeformed reference surface The surface external applied moment, M(S area, is defined as: ̂ 1 , S2 , t) ≡ M(S

𝜂t

∫𝜂 b

h(S1 , S2 , 𝜂, t) × b(S1 , S2 , 𝜂, t) 𝜇𝜂 d𝜂 + h(S1 , S2 , 𝜂, t) × t 𝜇|𝜂b t

𝜂

(11.123)

r The vector dynamic momentum balance equations: F𝛼 , 𝛼 + F̂ = L, ∙

M𝛼 ,

𝛼

+ c0 , 𝛼 ×

F𝛼

𝛼 = 1, 2

Momentum balance

̂ = c0 × L + A, + M ∙

∙

𝛼 = 1, 2

Moment of momentum balance (11.124)

r The vector static momentum balance equations: M𝛼 |𝛼

+ c0 , 𝛼

F𝛼 |𝛼 + F̂ = 0, ̂ = 0, × F𝛼 + M

𝛼 = 1, 2

Momentum balance

𝛼 = 1, 2

Moment of momentum balance

(11.125)

r The spatial angular velocity, ω ≡ ωS , is the axial vector corresponding to the skew-symmetric tensor, 𝛀 ≡ 𝛀S , defined by:

∙

∙

R = 𝛀S R ⇔ 𝛀S = R RT 𝛀S ωS = 0

(11.126)

where R(S1 , S2 , t) is the rotation field at a point (S1 , S2 ) on the shell reference surface at time t ∈ R+ .

Computation of Nonlinear Structures

766

r Finally, as a notational reminder, in what follows, all lowercase Greek indices assume values 1 and 2, and the lowercase Latin indices runs through 1, 2, 3; also, for repeated diagonal indices, Einstein’s summation convention of summing over their ranges holds.

11.4.2

Thin and Very Flexible Shell: Specialization

First of all, for very thin and flexible shells:

r The mid-surface is chosen as the shell undeformed reference surface for computational purposes; then, the locations through the thickness along the normal to the mid-surface are given by 𝜂 ∈ [− H2 ≡ H − , + H2 ≡ H + ] where H is the constant thickness of the shall, that is,

r

𝜂 t = + H2 ≡ H + and 𝜂 b = − H2 ≡ H − . It is assumed that the undeformed mid-surface is also the center of mass surface at time t = 0. So, from 𝜂 = 0 on the mid-surface, and the definition of center of mass density, 𝜂, ̄ we have: H+

𝜂̄ =

∫H − H+

∫H − ⇒

11.4.2.1

H+

∫H −

𝜂 𝜌0 (S1 , S2 , 𝜂) 𝜂 𝜇(S1 , S2 , 𝜂) d𝜂 =0 𝜂

𝜌0 (S , S , 𝜂) 𝜇(S , S , 𝜂) d𝜂 1

2

1

2

(11.127)

𝜂 𝜌0 (S1 , S2 , 𝜂) 𝜂 𝜇(S1 , S2 , 𝜂) d𝜂 = 0

Deformation Map: Thin Shell

For the most general case, the deformation map of a 3D shell-like body characterized by h(S1 , S2 , 𝜂, t) was defined in equation (11.117). The choice of h(S1 , S2 , 𝜂, t) essentially helps in categorizing the various shell theories with differing kinematic assumptions. For thin and very flexible shells, as shown in Figure 11.12, h(S1 , S2 , 𝜂, t) is taken as: h(S1 , S2 , 𝜂, t) ≡ p(S1 , S2 , 𝜂, t) = 𝜂 n(S1 , S2 , t) = 𝜂 R(S1 , S2 , t) N(S1 , S2 )

Figure 11.12 Thin shell deformation map.

(11.128)

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where n(S1 , S2 , t) is the current unit normal obtained by rotation of N(S1 , S2 ), the unit undeformed normal to the undeformed mid-surface, such that n = R N with R(S1 , S2 , t) the rotation at a point on the mid-surface at the current time, t ∈ R+ . Noting that any point in the shell-like body at current time is obtained by pure rotation, the assumption implies that the unit normal to the undeformed mid-surface remains inextensible, but not necessarily perpendicular to the deformed mid-surface during deformation (because it may rotate due to shear deformations), which is a commonly accepted underpinning of theories for very thin and flexible shells with shear deformations (reflected in the rotation of the mid-surface) and goes by the name of Reissner– Mindlin-type kinematic assumptions. For such cases, instead of equation (11.117), we will have deformation map as: c(S1 , S2 , 𝜂, t) = c0 (S1 , S2 , t) + 𝜂 R(S1 , S2 , t) N(S1 , S2 ),

(11.129)

If we exclude considerations for all shear deformations and restrict ourselves only to flexural behaviors by requiring that the unit normal to the mid-surface always remains normal and inextensible to the deformed mid-surface, the involved theory is associated with the name: Love–Kirchhoff-type kinematic assumption. In what follows, we will be interested in specializing the balance equations to the Reissner–Mindlin-type shell configurations for computational purposes.

11.4.2.2

Effective Surface Linear Momentum

The effective surface linear momentum, L(S1 , S2 , t), defined in equation (11.119), can be rewritten as: H+

c(S1 , S2 , 𝜂, t) 𝜌0 (S1 , S2 , 𝜂) 𝜂 𝜇(S1 , S2 , 𝜂) d𝜂 ) ( H+ (((( ∙ 1 2 1 ( 2 (( 𝜂 (( 1 2 𝜂( 𝜌0( (S(, S , 𝜂) 𝜇(S , S , 𝜂) d𝜂 R(S , S , t) N(S1 , S2 ) = ∫( ( − ( ( H ( ) +

L(S1 , S2 , t) =

∙

∫H − (

H

+

∫H −

𝜌0 (S1 , S2 , 𝜂) 𝜂 𝜇(S1 , S2 , 𝜂) d𝜂

(11.130)

∙

c 0 (S1 , S2 , t)

= M0 (S1 , S2 ) v0 (S1 , S2 , t) where we have dropped a term based on equation (11.127) and introduced: Definition:

r Surface linear velocity, v0 (S1 , S2 , t): ∙

∙

v0 (S1 , S2 , t) = c0 (S1 , S2 , t) ≡ d(S1 , S2 , t)

(11.131)

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768

r Surface mass, M0 (S1 , S2 ), per unit surface area of the undeformed mid-surface: H+

M0 (S1 , S2 ) =

∫H −

𝜌0 (S1 , S2 , 𝜂) 𝜂 𝜇(S1 , S2 , 𝜂) d𝜂

(11.132)

A typical assumption for thin shell is that the mass density, 𝜌0 (S1 , S2 , 𝜂), is constant through the thickness, that is, 𝜌0 ≡ 𝜌0 (S1 , S2 ), and 𝜂 𝜇(S1 , S2 , 𝜂) ≈ 1. Thus, with the application of these assumptions to equation (11.132), we get: M0 (S1 , S2 ) = 𝜌0 H

(11.133)

r Effective surface linear momentum, L(S1 , S2 , t):

∙

L(S1 , S2 , t) = M0 (S1 , S2 ) v0 (S1 , S2 , t) ≈ 𝜌0 H d

11.4.2.3

(11.134)

Effective Surface Angular Momentum

Note that from the definition of the angular velocity, ω, from equation (11.126), we have: n = R N ⇒ R = n ⊗ N, ∙

∙

∙

∙

⇒ n = R N = R RT R N = 𝛀 n

⇒ n=𝛀n=ω × n,

∙

∙

𝛀 ω = 0 ⇒ (n ⊗ n) ω = 0 ⇒ (ω ∙ n) n = 0

⇒ (ω ∙ n) = 0 ,

(11.135)

∙  n ⇒ n × n∙ = ω n × n = n × (ω × n) = (n ∙ n) ω −  (n ∙ ω)

Then, with the last relation in equation (11.135), the effective (spatial) surface angular momentum, A(S1 , S2 , t), from equation (11.122), may be redefined as:

A(S1 , S2 , t) =

𝜂t

∫𝜂 b (

=

h(S, 𝜂, t) × c(S1 , S2 , 𝜂, t) 𝜌0 (S1 , S2 , 𝜂) 𝜂 𝜇(S1 , S2 , 𝜂) d𝜂 ∙

∙

H+

∫H −

⏞⏞⏞⏞⏞⏞⏞⏞⏞ 𝜂 n × (c0 + 𝜂 n) 𝜌0 𝜂 𝜇 d𝜂

)

() (((( ∙ 2 (𝜂((1 2 ( = 𝜂𝜌( , S , 𝜂) 𝜇(S , S , 𝜂) d𝜂 n × c0 (( 0 (S ( ∫ ( ( (H − + ) (

(11.136)

H+

1

H

+

𝜂 2 𝜌0 (S1 , S2 , 𝜂) 𝜂 𝜇(S1 , S2 , 𝜂) d𝜂

∫H −

∙

n×n

∙

= I0 (n × n) = I0 ω where we have dropped a term based on equation (11.127) and introduced:

Nonlinear Shell

769

Definition

r Surface angular velocity, ω(S1 , S2 , t): ∙

ω(S1 , S2 , t) ≡ n(S1 , S2 , t) × n(S1 , S2 , t), with n(S1 , S2 , t) = R(S1 , S2 , t) N(S1 , S2 ), ∙

∙

∙

n(S1 , S2 , t) = R(S1 , S2 , t) N(S1 , S2 ) = R RT R N = 𝛀(S1 , S2 , t) n(S1 , S2 , t) = ω × n, ω∙ n = 0

(11.137)

r Surface rotary inertia, I0 (S1 , S2 ), per unit surface area of the undeformed mid-surface: H+

I0 (S1 , S2 ) =

∫H −

𝜂 2 𝜌0 (S1 , S2 , 𝜂) 𝜂 𝜇(S1 , S2 , 𝜂) d𝜂

(11.138)

A typical assumption for thin shell is that the mass density, 𝜌0 (S1 , S2 , 𝜂), is constant through the thickness, that is, 𝜌0 ≡ 𝜌0 (S1 , S2 ), and 𝜂 𝜇(S1 , S2 , 𝜂) ≈ 1. Thus, with the application of these assumptions to equation (11.138), we get:

I0 (S1 , S2 ) = 𝜌0

H3 12

(11.139)

Effective surface angular momentum, A(S1 , S2 , t):

A(S1 , S2 , t) = I0 (S1 , S2 ) ω(S1 , S2 , t) ≈ 𝜌0

11.4.2.4

H3 ω 12

(11.140)

Thin Shell Dynamic Momentum Balance Equations

Now, from the first of equation (11.124), with definitions in equation (11.134), we can rewrite the momentum balance equation as: F𝛼 , 𝛼 + F̂ = L, ∙

𝛼 = 1, 2,

∙∙ ) ∙∙ ( ∙ ∙ L = M0 v0 = M0 d ≈ 𝜌0 H d

(11.141)

Computation of Nonlinear Structures

770

11.4.2.5

Thin Shell Dynamic Angular Momentum Balance Equations ∙

Noting that: c0 × L = v0 × M0 v0 = 0 , from the second of equation (11.124), and with definitions in equation (11.140), we can rewrite the angular momentum balance equation as: ̂ = A, M𝛼 |𝛼 + c0 , 𝛼 × F𝛼 + M ∙

𝛼 = 1, 2,

∙

∙

A(S1 , S2 , t) = I0 (S1 , S2 ) ω(S1 , S2 , t) ≈

( ) (11.142) ∙ H3 𝜌0 ω(S1 , S2 , t) 12

Finally, note that in equation (11.142), we have redefined the stress couple, M𝛼 (S, t), 𝛼 = 1, 2, of ̂ 1 , S2 , t), of equation (11.123), as follows. equation (11.121) and the external applied moment, M(S Definitions

r Internal resistive stress couple, M𝛼 (S, t), M𝛼 (S, t) ≡ n(S, t) ×

𝛼 = 1, 2, per unit of undeformed boundary length: 𝜂t

∫𝜂b

𝜂

𝜂 𝛼

P (S, 𝜂, t) 𝜂 𝜇S (S, 𝜂, t) d𝜂

(11.143)

r External applied moment, M(S ̂ 1 , S2 , t), per unit of undeformed reference surface area: ̂ 1 , S2 , t) ≡ n(S, t) × M(S

11.4.3

𝜂t

∫𝜂 b

𝜂 b(S1 , S2 , 𝜂, t) 𝜇𝜂 d𝜂 + 𝜂 n(S, 𝜂, t) × t 𝜇|𝜂b (11.144) t

𝜂

Thin Shell Dynamic Balance Equations of Motion

For easy reference, here we copy and collect the derived dynamic balance equations: F𝛼 , 𝛼 + F̂ = L, ∙

∙

M𝛼 ,

𝛼

F𝛼

+ c0 , 𝛼 ×

𝛼 = 1, 2

∙ M0 v0

L= ∙ ̂ = A, + M

Momentum balance ∙∙

∙∙

= M0 d ≈ 𝜌0 H d

𝛼 = 1, 2

∙

∙

Moment of momentum balance

A(S1 , S2 , t) = I0 (S1 , S2 ) ω(S1 , S2 , t) ≈ 𝜌0

11.4.4

(11.145)

H3 ∙ 1 2 ω(S , S , t) 12

Where We Would Like to Go

We have now derived the shell balance equations by reducing the 3D equations of motion of continuum mechanics. However, the shell equations of motion can be directly developed by considering a shell as a reference surface to start with. For this simple engineering method, we would like to check out the following. The direct engineering approach to shell balance equations: while the shell equations presented in equation (11.141) and equation (11.143) are very compact and thus useful for analysis, we

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771

will modify the current vector form to a component or operational vector form suitable for computational purposes where the appropriate components of the forces and moments are of primary importance. As prerequisites to such a task, we should check out:

r component or operational vectors r derivatives of component vectors. r

With these background materials, we will be able to modify the equations in terms of what may be called the component vector or operator form as described in: computational balance equations.

11.5 Static and Dynamic Equations: Engineering Approach 11.5.1 What We Need to Recall

r The undeformed coordinate system, hereinafter known as the tangent coordinate system, corresponding to the lines of curvature with arc length parameterization, {S1 , S2 } and to the unit normal to the shell reference surface, with base vectors: {G1 (S1 , S2 ), G2 (S1 , S2 ), G3 (S1 , S2 ) ≡ N(S1 , S2 )}, is an orthonormal system of coordinates where both the sets of the contravariant base vectors, {Gi }, and the covariant base vectors, {Gi }, are one and the same; arc length parameters, {S1 , S2 }, are essentially a smooth reparameterization of the root parame𝜉1

ters, {𝜉 1 , 𝜉 2 }, through the relationship: S1 (𝜉 1 , 𝜉 2 ) = ∫0 ‖C0 , 𝜂 (𝜂, 𝜉 2 )‖ d𝜂 and S2 (𝜉 1 , 𝜉 2 ) =

r r

𝜉2

∫0 ‖C0 , 𝜂 (𝜉 1 , 𝜂)‖ d𝜂. For the ensuing vector form of the balance equations, no kinematic assumptions are necessary; however, for the component form, we must bring in the displacement parameters for computationally meaningful expressions. The Gauss–Weingarten relation for the derivatives of the triad of base vectors in matrix form with respect to the arc length parameters, S𝛼 , 𝛼 = 1, 2, is given by: ⎡ 0 ⎧ G1 ,1 ⎫ ⎧ G1 ⎫ ⎢ ⎪ ⎢⎢ 1 ⎪ ⎪ 𝜕 ⎪ 𝛾 , G , G = ≡ ⎨ 2 1⎬ 1 ⎨ 2⎬ 𝛾 12 ⎪ N, ⎪ 𝜕S ⎪ N ⎪ ⎢⎢ 1 ⎭ ⎩ 1 ⎭ ⎩ ⎢ −1 ⎣ R1

−

1 𝛾 , 𝛾1 1 2 0 0

1 ⎤ R1 ⎥ ⎫ ⎫ ⎧ ⎧ ⎥ ⎪ G1 ⎪ ⌢ ⎪ G1 ⎪ 0 0 ⎥ ⎨ G2 ⎬ ≡ K1 ⎨ G2 ⎬ ⎥⎪ ⎪N⎪ ⎥⎩ N ⎪ ⎭ ⎭ ⎩ ⎥ 0 ⎦

(11.146)

0 ⎤ ⎥⎧ ⎧ G1 ⎫ G1 ⎫ ⎪ ⌢0 ⎪ ⎪ 1 ⎥⎥ ⎪ G ≡ K2 ⎨ G2 ⎬ R2 ⎥ ⎨ 2 ⎬ ⎪ ⎪N⎪ ⎥⎪ ⎭ ⎩ ⎩N⎭ 0 ⎥⎦

(11.147)

and ⎡ 0 ⎧ G1 ,2 ⎫ ⎧ G1 ⎫ ⎢ ⎪ ⎢ 1 ⎪ ⎪ 𝜕 ⎪ ⎨ G2 ,2 ⎬ ≡ 2 ⎨ G2 ⎬ = ⎢⎢ − 𝛾 𝛾2 ,1 ⎪ N, ⎪ 𝜕S ⎪ N ⎪ ⎢ 2 ⎭ ⎩ 2 ⎭ ⎩ ⎢ 0 ⎣ ⌢

1 𝛾 , 𝛾2 2 1 0 −

1 R2

where K0𝛼 , 𝛼 = 1, 2 are the shell undeformed curvature matrices with R𝛼 (S1 , S2 ), 𝛼 = 1, 2, are the radii of curvatures at any point, {S𝛼 }, of the shell reference surface, and 𝛾𝛼 , 𝛼 = 1, 2,

Computation of Nonlinear Structures

772

̄ 𝛼 : 𝛾𝛼 ≡ are the lengths of the root base vectors, G 𝜕C(𝜉 1 ,𝜉 2 ,0) , 𝜕𝜉 𝛼

dS𝛼 d𝜉 𝛼

̄ 𝛼‖ = = ‖G

√ ̄ 𝛼 (𝜉 1 , 𝜉 2 ) ≡ ̄ 𝛼𝛼 where G G

𝛼 = 1, 2 with {𝜉 1 , 𝜉 2 } as the root parameters along the lines of curvature of the shell

surface.

r On the other hand, the Gauss–Weingarten relation for the derivatives of the triad of base vectors on a surface in invariant tensor form with respect to the arc length parameters, S𝛼 , may be obtained by first introducing the following.

𝛼 = 1, 2,

Definition: Shell initial curvature tensors, K0𝛼 , 𝛼 = 1, 2: ⌢

K0𝛼 ≡ (K0𝛼 )Tij Gi ⊗ Gj = −Gi ⊗ Gi , 𝛼 ,

𝛼 = 1, 2;

i, j = 1, 2, 3

(11.148)

Definition: shell initial curvature vector, k0𝛼 , 𝛼 = 1, 2: The axial vectors corresponding to the initial curvature tensors, K0𝛼 , 𝛼 = 1, 2, are given by: k01 ≡ k02

≡

{ {

0

− R1

1

1 R2

0

− 𝛾1 𝛾1 ,2 1

1 𝛾 , 𝛾2 2 1

}T

}T

(11.149)

Now, we can easily get from equation (11.148), the invariant tensor form as: Gi , 𝛼 = K0𝛼 Gi = k0𝛼 × Gi ,

𝛼 = 1, 2; i = 1, 2, 3

(11.150)

r Finally, the covariant derivative of a base vector is the zero vector, that is, Gi |𝛼 = 0,

i= 1, 2, 3. Thus, the covariant derivative of a tensor function of any order is identical to its partial derivative. If we take a vector (a tensor of order 1) function, a(S𝛼 ), we then have: a|S𝛼 ≡ a|𝛼 = a, 𝛼 = t ai |𝛼 Gi = G t a|𝛼 where G ≡ [ G1 G2 G3 ] is the orthonormal matrix with base vectors as its columns, and t a(S𝛼 ) is the component vector corresponding to the vector function, a(S𝛼 ).

Exercise: Show that the partial derivative of a tensor function of any order is identical with its covariant derivative with respect to its arguments. Hint: For vector function G |𝛼 = t di |𝛼 Gi = d, 𝛼 . Why do you think Gi |𝛼 = 0? Because, by defid|𝛼 = t di |𝛼 Gi + t di  k nition, Gi |j = Gi ,j − Γij Gk = Γkij Gk − Γkij Gk = 0.

11.5.2

Deformed Geometry, Stress Resultants and Stress Couples

As already indicated, because we are interested to include in our analysis the case of finite displacements and rotations, all equilibrium equations must be considered for the current deformed geometry of a shell. Thus, as a starting point, we will attend to deformed geometry.

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773

Figure 11.13 Direct engineering equilibrium derivation.

11.5.2.1

Deformed Geometry

Let us consider a typical differential element of the shell reference surface as shown in Figure 11.13, and also suppose that we have determined the lines of curvature directions from the given coordinate iso-curves of the undeformed shell surface, and that we will parameterize the shell reference surface by the arc length parameterization (S1 , S2 ) along these lines of curvature (which are, of course, orthogonal to each other). Note first that we have intentionally chosen the shell element along the lines of curvature; this is not at all necessary for analysis and so we can choose any arbitrarily oriented element. However, it turns out that the equations are extremely efficient and simple for the most accurate computational development as we apply these expressions to our c-type finite element method. To jump ahead, as it will turn out computationally, all relevant information is needed only at the numerical integration points (i.e. the Gauss quadrature points) along the lines of curvature of the undeformed shell surface. Thus, without loss of any generality, we choose our elements along the lines of curvature to develop expressions for future computational use. Referring to Figure 11.13, the position vector, c0 (S1 , S2 , t) locates a point a on a deformed reference surface (representing the shell in a deformed state) at any time t ∈ R+ whose corresponding location, on the reference surface C0 (S1 , S2 ) ≡ c0 (S1 , S2 , 0) (representing the undeformed shell) at initial reference time t = 0, is denoted by A. Similarly, considering a differential element dS1 × dS2 along the lines of curvature at A on the surface, c0 (S1 + dS1 , S2 , t) locates a point b on the same deformed reference surface whose corresponding location on the undeformed reference surface C0 (S1 + dS1 , S2 ) at (S1 + dS1 , S2 ) is denoted by B. In other words, a differential element ABCD on the undeformed shell reference surface becomes abcd on a deformed state of the shell. For small enough dS1 and dS2 , we may approximate to the first-order partial derivatives of the vector functions C0 (S1 , S2 ) with respect to the arc length parameters, S1 and S2 , respectively, as: C0 (S1 + dS1 , S2 ) ≃ C0 (S1 , S2 ) + C0 (S1 , S2 ),S1 dS1 and C0 (S1 , S2 + dS2 ) ≃ C0 (S1 , S2 ) + C0 (S1 , S2 ),S2 dS2 . Finally, we indicate the displacement of the undeformed point A to its current point a by d(S1 , S2 , t) such that c0 (S1 , S2 , t) = C0 (S1 , S2 ) + d(S1 , S2 , t)

(11.151)

Computation of Nonlinear Structures

774

11.5.2.2

Force (Stress Resultant) and Moment (Stress Couple) Vectors: F𝜶 M𝜶 , 𝜶 = 1, 2

Referring to Figure 11.13, we introduce the force and moment vectors acting on the deformed shell surface as follows. Definition:

r F𝛼 (S1 , S2 , t) = the force vector acting outward on that edge of the deformed reference surface r

(which was the 𝛼 = const surface at the (S1 , S2 ) point of the undeformed shell reference surface for 𝛼 = 1, 2 at any time t ∈ R+ ; F𝛼 is defined per unit length along 𝛼 = constcross-section of the undeformed reference surface. M𝛼 (S1 , S2 , t) = the moment axial vector acting outward on that edge of the deformed reference surface which was the 𝛼 = const surface at the (S1 , S2 ) point of the undeformed shell reference surface for 𝛼 = 1, 2 at any time t ∈ R+ ; M𝛼 is defined per unit length along 𝛼 = constcrosssection of the undeformed reference surface.

Remark

r Let us note that at this time, in the absence of any kinematic assumptions, we do not have any computationally useful stress-level interpretations, and hence expressions, for either the invariant force (i.e. stress resultant) vector, F𝛼 (S1 , S2 , t), or the moment (i.e. stress couple) vector, M𝛼 (S1 , S2 , t). Clearly, the names suggested in parentheses allude to an assumed shell through-the-thickness deformation distribution which we refrain from discussing any further, and leave it for our derivation of the balance laws from the fundamental principles of continuum mechanics, as we have done in the section on 3D to 2D reduction. 11.5.2.3

Component Force and Moment Vectors in terms of Undeformed Frame: t F𝜶 (S1 , S2 , t) and t M𝜶 (S1 , S2 , t)

The invariant force vectors, F𝛼 , and the moment vectors, M𝛼 , can have component representation in any triad of base vectors, Cartesian or curvilinear. At this point of the analysis, we choose to describe the components in the undeformed curvilinear coordinate system (henceforth, known as the tangent coordinate system) with the triad of base vectors, {G1 , G2 , G3 ≡ N} on the shell reference surface. Thus, referring to Figure 11.14, we can write: F𝛼 (S1 , S2 , t) = t F 𝛼1 (≡ N𝛼1 ) G1 + t F 𝛼2 (≡ N𝛼2 ) G2 + t F 𝛼3 (≡ Q𝛼 ) N = t F 𝛼i (S1 , S2 , t) Gi (S1 , S2 ),

𝛼 = 1, 2;

i = 1, 2, 3

(11.152)

where, N11 , N22 are known as the in-plane extensional forces, N12 , N21 as the in-plane shear forces, and Q1 , Q2 as the transverse shear forces, as shown in Figure 11.15; and M1 (S1 , S2 , t) = t M 11 (≡ −M12 ) G1 + t M 12 (≡ M1 ) G2 + t M 13 (≡ M13 ) N = t M 1i Gi , M2 (S1 , S2 , t)

= =

M𝛼 (S1 , S2 , t) =

t M 21 (≡

i = 1, 2, 3

−M2 ) G1 + t M 22 (≡ M21 ) G2 t M 2i G , i = 1, 2, 3 i t M 𝛼i (S1 , S2 , t) G (S1 , S2 ), 𝛼 = 1, 2, i

+ t M 23 (≡ M23 ) N i = 1, 2, 3

(11.153)

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775

Figure 11.14 Shell force and moment components.

where, M1 , M2 are known as the bending moments, M12 , M21 are known as the in-plane twisting moments, and M𝛼3 as the transverse twisting moments on a shell cross-section with normal to it pointing to 𝛼 direction, 𝛼 = 1, 2 (for the classical shell theory, these components are identically zero. Conventionally and notationwise, these are described as such in engineering terminology as shown in Figure 11.14. In the final equalities of equations (11.152) and (11.153), we have introduced the following.

Figure 11.15 Engineering force and moment components.

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776

Remarks

r Notationally, the first indices of the shell consistent components indicate the direction of the r

normals to the cross-section, and the second indices point to the direction of the forces and moments. In the engineering community, the stress couples are expressed in the same way as the associated stresses, as shown in Figure 11.15, but note that the shell consistent moment components, t M 𝛼i , 𝛼 = 1, 2, i = 1, 2, 3, in the undeformed coordinate system {G1 , G2 , G3 ≡ N}, are different from their counterparts in the engineering community. Of course, these can be easily related by the following map that uses 2D permutation matrix, e𝛼𝛽 : t

M 𝛼𝛽 = e𝜂𝛽 M𝛼𝜂 ,

[ (e𝜂𝛽 ) =

0

1

] (11.154)

−1 0

We collect the components of F𝛼 and M𝛼 , and define the component vectors, respectively, as: t F1 (S1 , S2 , t)

≡

t F2 (S1 , S2 , t)

≡

t M1 (S1 , S2 , t)

≡

t M2 (S1 , S2 , t)

≡

{t {t

{t {t

F 11 ≡ N1 F 21 ≡ N21

M 11 ≡ −M12 M 21 ≡ −M2

t F 12

≡ N12

t F 13

≡ Q1

≡ N2

t F 23

≡ Q2

t F 22

}T

≡ M1

t M 13

≡ M13

≡ M21

t M 23

≡ M23

t M 12 t M 22

}T (11.155) }T }T

(11.156)

With the above definitions, we can express these in the column vector form as: F𝛼 (S1 , S2 , t) = G(S1 , S2 ) t F𝛼 (S1 , S2 , t) M𝛼 (S1 , S2 , t) = G(S1 , S2 ) t M𝛼 (S1 , S2 , t),

𝛼 = 1, 2

(11.157)

where G ≡ [ G1 G2 G3 ≡ N ] is the orthonormal matrix with base vectors as its columns; clearly, the determinant of the matrix G is unity: det G = G1 × G2 ∙ G3 = 1. 11.5.2.4

(Effective) Linear Momentum and (Effective) Angular Momentum Vectors: L(S1 , S2 , t) and A(S1 , S2 , t)

We introduce the linear momentum and the angular moment vectors acting on the deformed shell reference surface as follows. Definitions

r L(S1 , S2 , t) = the linear momentum vector per unit area of the undeformed shell reference r

surface acting at a point on the deformed shell reference surface (which was the (S1 , S2 ) point of the undeformed shell reference surface) at any time t ∈ R+ . A(S1 , S2 , t) = the angular momentum or the moment of momentum per unit area of the undeformed shell reference surface acting at a point on the deformed shell reference surface (which was the (S1 , S2 ) point of the undeformed shell reference surface) at any time t ∈ R+ .

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777

Remark

r Let us note that, at this time, in the absence of any kinematic assumptions, we do not have any computationally useful deformation-level interpretations, and hence, expressions for either the invariant linear momentum (i.e. effective) vector, L(S1 , S2 , t), or the angular momentum (i.e. effective) vector, A(S1 , S2 , t). However, the qualifiers suggested in the parenthesis allude to an assumed shell through-the-thickness rate of deformation.

11.5.2.5

Material Components of Linear Momentum and Angular Momentum Vectors in Terms of the Undeformed Frame: t L(S1 , S2 , t) and t A(S1 , S2 , t)

The invariant linear momentum vector, L(S1 , S2 , t), and the angular momentum vector, A(S1 , S2 , t), for any t ∈ R+ , can have component representation in any triad of base vectors, Cartesian or curvilinear. At this point of the analysis, we choose to describe the components in the undeformed curvilinear coordinate system (the tangent coordinate system) with the triad of base vectors, {G1 , G2 , N ≡ G3 } on the shell reference surface. Thus, we can write: L(S1 , S2 , t) = t L1 (≡ P1 ) G1 + t L2 (≡ P2 ) G2 + t L3 (≡ PN ) G3 = t Li (S1 , S2 , t) Gi (S1 , S2 ),

i = 1, 2, 3

(11.158)

and A(S1 , S2 , t) = = t A1 (≡ A1 ) G1 + t A2 (≡ A2 ) G2 + t A3 (≡ AN ) G3 = t Ai (S1 , S2 , t) Gi (S1 , S2 ),

i = 1, 2, 3

(11.159)

We collect the components of L and A, and define the component vectors, respectively, as follows. Definitions t L(S1 , S2 , t)

≡ { t L1

t L2

t L3

≡ LN }T

t A(S1 , S2 , t)

≡

t A2

t A3

≡ AN }T

{ t A1

(11.160)

With the above definitions, we can express these in the column vector form as: L(S1 , S2 , t) = G(S1 , S2 ) t L(S1 , S2 , t) A(S1 , S2 , t) = G(S1 , S2 ) t A(S1 , S2 , t)

(11.161)

∙ Δ

Likewise, we have the time derivatives, (∙) = 𝜕t𝜕 , of the of the linear momentum and the angular momentum in terms of their material component vectors: ∙

∙

∙

∙

L(S1 , S2 , t) = G(S1 , S2 ) t L(S1 , S2 , t) A(S1 , S2 , t) = G(S1 , S2 ) t A(S1 , S2 , t)

(11.162)

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Figure 11.16 Linear and angular momentum in undeformed frame.

11.5.3

Dynamic Balance Equations

Let us again emphasize that the forces and moments, and the linear and the angular momenta, are acting on the current deformed shell reference surface but defined per unit length in the undeformed shell reference surface along the lines of curvature at the point where they are acting, with the components given in the local basis directions of the undeformed curvilinear coordinate system. With the above preliminaries as the background and referring to Figures 11.14 and 11.16, we can directly apply the momentum balance and the moment of momentum balance (moment being taken about the point b of the deformed differential element ds1 × ds2 ), that is, the angular momentum balance, principles and we get the following.

11.5.3.1 Momentum Balance Equations (Force Equilibrium) ∑ F = F1 (S1 + dS1 , S2 , t)dS2 − F1 (S1 , S2 , t)dS2 + F2 (S1 , S2 + dS2 , t)dS1 −F2 (S1 , S2 )dS1 + F̂ dS1 dS2     = dS2 + F1 ,S1 dS1 dS2 −  dS2 +  dS1 + F2 ,S2 dS1 dS2 −  dS1 + F̂ dS1 dS2 F1 F1 F2 F2 ( 1 ) ̂ dS1 dS2 = d L(S1 , S2 , t) dS1 dS2 = F ,S1 + F2 ,S2 + F̂ dS1 dS2 = (F𝛼 , 𝛼 + F) dt Or, dividing the above equations all over by dS1 dS2 , we get F𝛼 , 𝛼 + F̂ = L, ∙

𝛼 = 1, 2

(11.163)

where we have the following notation and relation for any vector function, (∙)(S1 , S2 , t): (∙), 𝛼 ≡ (∙),S𝛼 = t (∙)i , 𝛼 Gi + t (∙)i Gi , 𝛼 = t (∙)i , 𝛼 Gi + 𝜿 0𝛼 × ≡

t (∙)| 𝛼

t (∙)i

Gi

(11.164)

for the covariant derivative of the component vector, t (∙), corresponding to the invariant vector function, (∙)(S1 , S2 , t), and Gi , 𝛼 depend on the shell surface metric tensor and the curvature tensor, as given by equations (11.146) and (11.147); we have also used Einstein’s summation convention

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779

for the contravariant and the covariant repeated indices with 𝛼 = 1, 2. In fact, as indicated before, all Greek indices will be assumed to run over 1 and 2, and all Latin indices on 1,2 and 3. Note that the momentum balance equation of motion does not involve any kinematic parameter.

11.5.3.2

Angular or Moment of Momentum Balance Equations (Moment Equilibrium)

We intend to take moments about the origin of the fixed inertial frame, so we have: ∑

M = M1 (S1 + dS1 , S2 , t)dS2 − M1 (S1 , S2 , t)dS2 + M2 (S1 , S2 + dS2 )dS1 − M2 (S1 , S2 , t)dS2 + c0 (S1 + dS1 , S2 , t) × F1 (S1 + dS1 , S2 , t)dS2 − c0 (S1 , S2 , t) × F1 (S1 , S2 , t)dS2 + c0 (S1 , S2 + dS2 , t) × F2 (S1 , S2 + dS2 , t)dS1 ̂ 1 , S2 , t) dS1 dS2 − c0 (S1 , S2 , t) × F2 (S1 , S2 , t)dS1 + M(S ̂ 1 + 1 dS1 , S2 + 1 dS2 , t)dS1 dS2 + c0 (S1 + 12 dS1 , S2 + 12 dS2 , t) × F(S 2 2 ( 𝛼 ) 1 2 ̂ dS dS = M ,𝛼 + M } { c0 (S1 + 12 dS1 , S2 + 12 dS2 , t) × L(S1 + 12 dS1 , S2 d dS1 dS2 = 1 2 1 2 dt + dS , t) + A(S , S , t)

(11.165)

2

Now, from equation (11.165), we intend to retain terms up to and including second differential order, such as dS1 dS2 , and neglect products of higher differential order terms. In order that we are consistent in our approximations, we must look into the following terms more closely: Thus, M1 (S1 + dS1 , S2 , t)dS2 − M1 (S1 , S2 , t)dS2 + M2 (S1 , S2 + dS2 )dS1 − M2 (S1 , S2 , t)dS2 (( (( 1 (S( 1 ,( 1 (S( 1 ,( =( S2( , t)dS2 + M1 (S1 , S2 , t),1 dS1 dS2 − ( S2( , t)dS2 + higher order M( M( (11.166) (( (( 2 (S( 1 ,( 2 (S( 1 ,( +( S2( , t)dS2 + M2 (S1 , S2 ),2 dS2 dS1 − ( S2( , t)dS2 M( M( = M1 (S1 , S2 , t),1 dS1 dS2 + M2 (S1 , S2 ),2 dS2 dS1 = M𝛼 (S1 , S2 , t), 𝛼 dS1 dS2 , 𝛼 = 1, 2 and c0 (S1 + dS1 , S2 , t) × F1 (S1 + dS1 , S2 , t)dS2 − c0 (S1 , S2 , t) × F1 (S1 , S2 , t)dS2 { } = c0 (S1 , S2 , t) + c0 ,1 (S1 , S2 , t)dS1 + higher order { } × F1 (S1 , S2 , t)dS2 + F1 ,1 (S1 , S2 , t)dS2 dS1 − c0 (S1 , S2 , t) × F1 (S1 , S2 , t)dS2 + higher order (((( ( 2 , t) ( 1( 1 , S2 , t)dS2 + c (S1 , S2 , t) × F1 , (S1 , S2 , t)dS2 dS1 ( , S( × F (S = c0 (S1( (11.167) ( 0 1 (( 1 2 1 1 1 2 2 + c0 ,1 (S , S , t)dS × F (S , S , t)dS ( (((( 2 , t) ( + c0 ,1 (S1 , S2 , t)dS1 × F1 ,1 (S1 , S2 , t)dS2 dS1 −c0 (S1( , S( F1( (S1 , S2 , t)dS2 (× ( ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ (( higher order

= c0 (S1 , S2 , t) × F1 ,1 (S1 , S2 , t)dS2 dS1 + c0 ,1 (S1 , S2 , t)dS1 × F1 (S1 , S2 , t)dS2

Computation of Nonlinear Structures

780

Similarly, following the same arguments as in equation (11.167), we get: c0 (S1 , S2 + dS2 , t) × F2 (S1 , S2 + dS2 , t)dS1 − c0 (S1 , S2 , t) × F2 (S1 , S2 , t)dS1 = c0 (S1 , S2 , t) × F2 ,2 (S1 , S2 , t)dS1 dS2 + c0 ,2 (S1 , S2 , t)dS1 × F2 (S1 , S2 , t)dS2

(11.168)

By combining equations (11.167) and (11.168), we get for 𝛼 = 1, 2: c0 (S1 + dS1 , S2 , t) × F1 (S1 + dS1 , S2 , t)dS2 − c0 (S1 , S2 , t) × F1 (S1 , S2 , t)dS2 + c0 (S1 , S2 + dS2 , t) × F2 (S1 , S2 + dS2 , t)dS1 −c (S1 , S2 , t) × F2 (S1 , S2 , t)dS1 { 0 1 2 } = c0 (S , S , t) × F𝛼 , 𝛼 (S1 , S2 , t) + c0 , 𝛼 (S1 , S2 , t) × F𝛼 (S1 , S2 , t) dS1 dS2

(11.169)

Next, we have: ̂ 1 + 1 dS1 , S2 + 1 dS2 , t)dS1 dS2 c0 (S1 + 12 dS1 , S2 + 12 dS2 , t) × F(S 2 2 ̂ 1 , S2 , t) + higher order}dS1 dS2 = {c0 (S1 , S2 , t) + higher order} × {F(S = c0

(S1 , S2 ,

(11.170)

̂ 1 , S2 , t) t) × F(S

Then, following the same arguments as in equation (11.170), we get: c0 (S1 + 12 dS1 , S2 + 12 dS2 , t) × l(S1 + 12 dS1 , S2 + 12 dS2 , t) dS1 dS2 ̂ 1 , S2 , t) + higher order} dS1 dS2 = {c0 (S1 , S2 , t) + higher order} × {F(S = c0

(S1 , S2 ,

t) ×

̂ 1 , S2 , F(S

t)

(11.171)

dS1 dS2

Now, using the relations in equations (11.166)–(11.171), we get from equation (11.165): ∑

̂ + c0 × M = M𝛼 , 𝛼 + c0 , 𝛼 × F𝛼 + M

} ∙ { ∙ ∙  , 𝛼+ F̂ − L = c 0 × L + A F𝛼 

(11.172)

where we have used the momentum balance equation (11.163) to strike out the terms in the ∙ second bracket and reused the notation for the time derivative: 𝜕t𝜕 (∙) ≡ (∙). Thus, using equation (11.151), we finally get the angular momentum balance equations: ̂ = d × L + A, M𝛼 , 𝛼 + c0 , 𝛼 × F𝛼 + M ∙

∙

𝛼 = 1, 2

(11.173)

Remark

r If, for a thin shell, we define intuitively (will be derived in dynamic equation by reduction from 3D) the linear momentum, L, as: ∙

L(S1 , S2 , t) = M0 (S1 , S2 ) d(S1 , S2 , t)

(11.174)

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781

where M0 (S1 , S2 ) is the effective mass at the point, (S1 , S2 ), measured per unit of the undeformed ∙ surface, then d × L = 0; thus the angular momentum balance equation (11.173) becomes: ̂ = A, M𝛼 , 𝛼 + c0 , 𝛼 × F𝛼 + M ∙

11.5.4

𝛼 = 1, 2

(11.175)

Static Balance Equations

The quasi-static balance laws can be easily obtained from their dynamic counterparts by assuming that the linear and the angular momentum vectors identically vanish, that is, L(S1 , S2 , t) ≡ 0

(11.176)

A(S1 , S2 , t) ≡ 0 Thus, we get the following from equation (11.163).

11.5.4.1

Static Momentum Balance Equations (Force Equilibrium) F𝛼 , 𝛼 + F̂ = 0,

𝛼 = 1, 2

(11.177)

and from equation (11.173):

11.5.4.2

Static Angular or Moment of Momentum Balance Equations (Moment Equilibrium) ̂ = 0, M𝛼 , 𝛼 + c0 , 𝛼 × F𝛼 + M

11.5.5

𝛼 = 1, 2

(11.178)

Vector Dynamic Balance Equations of Motion

For easy reference, here we copy and collect the derived dynamic balance equations: F𝛼 , 𝛼 + F̂ = L,

𝛼 = 1, 2

Momentum balance

̂ = A, M𝛼 , 𝛼 + c0 , 𝛼 × F𝛼 + M

𝛼 = 1, 2

Moment of momentum balance

∙

∙

(11.179)

Now, recalling that the partial derivatives and the covariant derivatives of a tensor of any order are the same, the above vector (tensor of order 1) equations can also be written as: F𝛼 |𝛼 + F̂ = L,

𝛼 = 1, 2

Momentum balance

̂ = A, + M

𝛼 = 1, 2

Moment of momentum balance

∙

M𝛼 |𝛼

+ c0 , 𝛼 ×

F𝛼

∙

(11.180)

Computation of Nonlinear Structures

782

11.5.6

Vector Static Balance Equations of Motion

Likewise, here we copy and collect the derived static balance equations:

M𝛼 ,

𝛼

+ c0 , 𝛼

F𝛼 , 𝛼 + F̂ = 0, ̂ = 0, × F𝛼 + M

𝛼 = 1, 2

Momentum balance

𝛼 = 1, 2

Moment of momentum balance

(11.181)

Now, recalling that the partial derivatives and the covariant derivatives of a tensor of any order are the same, the above vector (tensor of order 1) equations can also be written as:

M𝛼 |𝛼

+ c0 , 𝛼

F𝛼 |𝛼 + F̂ = 0, ̂ = 0, × F𝛼 + M

𝛼 = 1, 2

momentum balance

𝛼 = 1, 2

moment of momentum balance

(11.182)

Remarks

r Note that, as shown in equations (11.145) and (11.181), the vector shell balance equations of r

motions do not depend on the rotation parameters (which need to be introduced later) whether shear deformation is considered or not, although the displacement function is implicit in the second term of the moment of momentum balance equation. Finally, although we have already introduced in our analysis the force and moment components, and the momentum and the angular momentum, on a shell reference surface, we have deliberately avoided revealing their true character by connecting these to the kinematics and the stresses of a 3D shell-like body it really is supposed to represent. For a consistent treatment of this aspect of the analysis, we must deduce the balance equations from the basic balance laws of continuum mechanics which is, as we will show, detailed in the derivation of the balance equations by reduction from the 3D continuum mechanics counterparts to 2D shell balance equations.

11.5.7

Where We Would Like to Go

r We have directly developed the shell equations of motions by considering a shell as a 2D

r

geometric object (i.e. a surface) with the forces and moments, and the linear and angular momentum functions. However, to validate the consistency of these equations of motion, and to gain insight into the true natures of these quantities, by establishing the connection between the shell equations and the more fundamental balance equations of 3D continuum mechanics, these equations will be derived by starting from and reducing the 3D equations of motion of solid mechanics. For this development, we must investigate: r shell balance equations: 3D to 2D. While the shell equations presented in equations (11.145) and (11.181) are very compact and thus useful for analysis, we will use the modified component or operational vector form that are most suitable for computational purposes where the components of displacements, rotations, forces and moments, and so on, need to be determined as the primary variables of interest. As prerequisites to such a task in the case of reduction of shell balance equations from 3D to 2D we should check out: r configuration space and kinematics

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783

r component or operational vectors r derivatives of component vectors. r

Then, with these background materials, we will be able to have the fully interpreted, final and computationally most effective balance equations: computational balance equations.

11.6

Weak Form: Kinematic and Configuration Space

11.6.1

What We Need to Recall

r The shell vector balance equations:

F𝛼 , 𝛼 + F̂ = L,

𝛼 = 1, 2

momentum balance

̂ = A, + M

𝛼 = 1, 2

moment of momentum balance

∙

M𝛼 ,

𝛼

+ c0 , 𝛼 ×

F𝛼

∙

(11.183)

The internal resistive stress resultant, F𝛼 , is defined by:

F𝛼 (S, t) ≡

𝜂t

∫𝜂 b

𝜂 𝛼

P (S, 𝜂, t) 𝜂 𝜇S (S, 𝜂, t) d𝜂

(11.184)

The internal resistive stress couple, M𝛼 , is defined by:

M𝛼 (S, t) ≡

𝜂t

∫𝜂 b

𝜂 𝛼

P (S, 𝜂, t) 𝜂 𝜇S (S, 𝜂, t) d𝜂

(S, 𝜂, t) ×

(11.185)

The linear momentum, L(S1 , S2 , t), is defined by:

L(S1 , S2 , t) ≡

𝜂t

∫𝜂 b 𝜂t

=

∫𝜂 b

v(S1 , S2 , 𝜂, t) 𝜌0 (S1 , S2 , 𝜂) 𝜂 𝜇(S1 , S2 , 𝜂) d𝜂 (11.186) c(S1 , S2 , 𝜂, t) 𝜌0 (S1 , S2 , 𝜂) 𝜂 𝜇(S1 , S2 , 𝜂) d𝜂 ∙

∙

where v(S1 , S2 , 𝜂, t) ≡ c(S1 , S2 , 𝜂, t) is the velocity at current time t ∈ R+ at a point with undeformed coordinates (S1 , S2 , 𝜂) in the interior of the shell-like body. The angular momentum, A(S1 , S2 , t), is defined by:

A(S1 , S2 , t) ≡

𝜂t

∫𝜂b 𝜂t

=

∫𝜂b

c(S1 , S2 , 𝜂, t) × v(S1 , S2 , 𝜂, t) 𝜌0 (S1 , S2 , 𝜂) 𝜂 𝜇(S1 , S2 , 𝜂) d𝜂 (11.187) ∙

𝜂

c(S , S , 𝜂, t) × c(S , S , 𝜂, t) 𝜌0 (S , S , 𝜂) 𝜇(S , S , 𝜂) d𝜂 1

2

1

2

1

2

1

2

Computation of Nonlinear Structures

784

Definition: F𝜈 as: F𝜈 ≡ f P ∙ 𝝂 0 ⇐ F𝛼 ⊗ G𝛼 ∙ 𝜈0𝛽 G𝛽 ⇐ F𝛼 𝜈0𝛼

(11.188)

Definition: M𝜈 as: M𝜈 ≡

m

P ∙ 𝝂 0 ⇐ M𝛼 ⊗ G𝛼 ∙ 𝜈0𝛽 G𝛽 ⇐ M𝛼 𝜈0𝛼

(11.189)

using the vector triple product property: a × b ∙ c = b × c ∙ a = c × a ∙ b, ∀a, b, c vectors.

r In what follows, a barred quantity signifies that it is the variation of a corresponding real entity. r The definition of a rotation tensor, its matrix representation, and the algebra, geometry and topology of rotation.

11.6.2

Weak Form of the Vector Balance Equations

Let us consider the shell reference surface, ΩM ≡ M, bounded by a smooth closed curve boundary, 𝜕ΩM ≡ 𝜕M. Let (v, ω) be any sufficiently smooth vector fields defined on M × R+ . Now, following the standard model of the weak form generation, we multiply equation (11.183) appropriately by two vector fields: v and ω, and integrate over the arbitrary shell reference surface area, Ω, to ̂ Dyn , given as: get the shell weak form functional, G

̂ Dyn G

) ( ⎫ ⎧ F𝛼 | + F̂ − ∙ ∙ v + L 𝛼 ⎪ ⎪ ≡ − ∫ ⎨( }) {∙ ⎬ dA ∙ ̂ − c0 × L + A Ω M ⎪ M𝛼 | + c , ∙ ω⎪ × F𝛼 + M 𝛼 0 𝛼 ⎭ ⎩ = 0,

(11.190)

By an application of the Green–Gauss (or divergence) theorem for a surface: ∫ F𝛼 |𝛼 dA = ΩM

∫ DIV f P dA = ∮ F𝛼 ⊗ G𝛼 𝜈0𝛽 G𝛽 dS = ∮ F𝛼 𝜈0𝛼 dS ≡ ∮ F𝜈 dS where 𝝂 0 = 𝜈0𝛼 G𝛼 is 𝜕ΩM

ΩM

𝜕ΩM

𝜕ΩM

the outward unit normal on the tangent plane on the boundary curve, Γ, to the undeformed shell reference surface area, ΩM , and S is the arc length parameter along the boundary curve, Γ; F𝜈 and M𝜈 are given by equations (11.188) and (11.189), respectively. Then, ∫ (F𝛼 ∙ v)|𝛼 dA = ∮ (F𝛼 ∙ v) 𝜈0𝛼 dS = ∮ (F𝛼 𝜈0𝛼 ∙ v) dS = ∮ (F𝜈 ∙ v) dS. Now, 𝜕ΩM

ΩM

𝜕ΩM

𝜕ΩM

by applying the chain rule of covariant differentiation on the left-most term above and rearranging terms, we get: −

∫

(F𝛼 |𝛼 ∙ v) dA =

ΩM

∫ ΩM

(F𝛼 ∙ v|𝛼 ) dA −

∮

(F𝜈 ∙ v) dS

(11.191)

𝜕ΩM

Similarly, for the moments, we have: −

∫

ΩM

(M𝛼 |𝛼 ∙ ω) dA =

∫ ΩM

(M𝛼 ∙ ω|𝛼 ) dA −

∮

𝜕ΩM

(M𝜈 ∙ ω) dS

(11.192)

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785

Now, recalling the commutative property of a vector triple product, and applying equations ̂ Dyn , as: (11.191) and (11.192) in equation (11.190), we get the weak form, G Internal stored power

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ̂ Dyn ≡ G

∫

{( F𝛼 ∙ (v, 𝛼 + c0 , 𝛼 × ω) + M𝛼 ∙ ω, 𝛼 } dA

ΩM Internal power

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ +

∙

∫

∙

{L ∙v + (A) ∙ ω}dA (11.193)

ΩM External applied power

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⎫ ⎧ ⎪ ⎪ ̂ ̂ ̂ ̂ (F𝜈 ∙ v + M𝜈 ∙ ω) dS+ (F ∙ v + M ∙ ω) dA⎬ −⎨ ∫ ∮ ⎪ ⎪ M ΩM ⎭ ⎩𝜕Ω = PInternal + PInertia − PExternal =0 In the event that v and ω are the actual linear velocity and the angular velocity of the shell reference surface, we can identify, in the integral given by equation (11.193), in the context of a mechanical energy conjugate relationship, the internal stored power (deformation or stress power), the inertial power and the external applied power as follows. Definition: Internal deformation power, PInternal , as:

PInternal ≡

∫

{( F𝛼 ∙ (v, 𝛼 + c0 , 𝛼 × ω) + M𝛼 ∙ ω, 𝛼 } dA

(11.194)

ΩM

Inertial power, PInertia , as: { PInertia ≡

∫

∙

}

∙

L ∙ v + (A) ∙ ω

dA

(11.195)

ΩM

External applied power, PExternal , as:

PExternal ≡

∮

𝜕ΩM

̂ 𝜈 ∙ ω) dS+ (F̂ 𝜈 ∙ v + M

∫

ΩM

̂ ∙ ω) dA (F̂ ∙ v + M

(11.196)

Computation of Nonlinear Structures

786

11.6.3

Kinematic Parameters

The weak form of equation for the equivalent 2D shell – that is, the shell undeformed reference surface, given by equation (11.193) – holds whenever the strong form, given by equation (11.183), is true; in fact, the weak form is just as exact as the strong form.

11.6.3.1

New Moving or Spinning Coordinate System

Moreover, to be energy conjugate in the internal stored power, we may call v, 𝛼 + c0 , 𝛼 × ω, and ω, 𝛼 for 𝛼 = 1, 2, in equation (11.193), (as the force strain rate vector and the moment strain rate vector, respectively) as the 2D strain rate fields over the shell reference surface as functions of the parameters, (v(S1 , S2 , t), ω(S1 , S2 , t)), presuming that these functions exist necessarily as rates of some as yet undetermined 2D real strain fields over the reference surface as functions of some as yet undetermined real parameters. Naturally, then, this way of going about the problem will make sense only if we are able to determine those real parameters and 2D real strain fields. In other words, we have set our problem as one of determining the kinematics of a 2D shell reference surface. It will turn out, in the material coordinate system (i.e. in the event we describe the invariant force, F𝛼 , and moment, M𝛼 , vectors in terms of the undeformed basis vectors), {Gi }, that the kinematic translational expression, v, 𝛼 + c0 , 𝛼 × ω, and rotational expression, ω, 𝛼 , for 𝛼 = 1, 2 (that comprise the internal stored power) cannot be the expressions for the real strain rates since there exist no functions whose rates could produce these expressions. However, It will be found later that the expressions v, 𝛼 + c0 , 𝛼 × ω, and ω, 𝛼 for 𝛼 = 1, 2, in equation (11.193), are true rates only when these are interpreted in some, as yet unknown, spinning orthogonal basis set that will be called, in the case of static loading, the co-rotated bases, or in the case of dynamic loading, the co-spinning bases, defining, as shown in Figure 11.17, a new moving coordinate system; to be sure, we must understand that this new coordinate system is not what is commonly referred to as the convected coordinate system; more on this later. Now, the main point is that this, in turn, enables us to identify an additional kinematic variable in the form of a rotation tensor, R(S1 , S2 , t). Clearly, for the velocity vector, v, we can choose the kinematic variables, d(S1 , S2 , t), as the displacement vector of the translational movement that belongs to the flat Euclidean space, ∙

that is, v ≡ d. However, for the angular velocity vector, ω, it is not so obvious because of the very nature of the rotation tensor, R(S1 , S2 , t), that travels on a curved space, and thus we will have to wait until we consider the topic of the calculus of curved spaces.

Figure 11.17 Shell Kinematics.

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787

Thus, our configuration space consists of the history of an ordered pair (d(S1 , S2 , t), R(S1 , S2 , t)) with d(S1 , S2 , t), as the displacement vector, and R(S1 , S2 , t), as the rotation tensor describing the co-spinning bases, for all points of the 2D shell reference surface over all times of interest. Of course, it does not mean that we cannot choose to express the components of the force and the moment vectors, and the linear and angular momentum vectors in material coordinate bases; in fact, we would do just that in our component or operational form of the virtual work equations. However, in that case, we have to rotate the internal work equations to express these again in the co-rotated (or co-spun) coordinate bases. Remark

r Clearly, one more decision has been made that has overwhelming importance for the computational ease and accuracy. It is about the choice of independent kinematic variables. We can opt to settle, as has been done in the past, for the as yet unknown real translational and bending strain tensors as the independent kinematic variables for the numerical formulation of a subsequent finite element procedure. The major pitfall of this choice is that these real strain tensors must guarantee that a real deformed surface can be constructed with these tensors; in other words, these will then need to additionally satisfy the strain compatibility equations for the deformed reference surface, which is much like the integrability or the inverse problem for the metric and curvature tensors of constructing a 2D surface leading to the Gauss-Mainardi-Codazzi conditions. This can get too complicated, if not be impossible, for numerical comfort. Thus, from now on, our sought-after kinematic variables will be the real counterparts, (d(S1 , S2 , t), R(S1 , S2 , t)), ∙

of the real rate entities, (v ≡ d, ω). The real compatible strains and their rates, as we will see, can then be derived and computed from these parameters satisfying the strain compatibility a priori.

11.6.4

Configuration Space, C

Thus, given a shell problem with the shell having been defined by the equivalent and oriented shell reference surface, the shell kinematics is completely characterized by a displacement vector field, d(S1 , S2 , t), tracking each point on the reference surface, and an orthonormal rotational tensor field, R(S1 , S2 , t), that monitors rotations of the triad of base vectors (also referred to as the orientation defining directors) on the shell reference surface where S𝛼 , 𝛼 = 1, 2 are the arc length parameters along the lines of curvature of the undeformed shell reference surface for any time t ∈ R+ . Keeping these assumptions in mind, we can ascertain the location of any point on the shell at any time during its deformation history with these two fields. Thus, we may define:

r the shell configuration space as the space defined by (d(S1 , S2 , t),

R(S1 , S2 , t)), (S1 , S2 ) ∈ ΩM ,

t ∈ R+ . In other words, a configuration space tells us all about the possible deformations that a shell can undergo, and thus consists of all ordered pairs of (d(S1 , S2 , t), R(S1 , S2 , t)) for all points on the shell reference surface for the entire deformation history; a particular history (induced by a specific set of loading system) of motion of a shell reference surface traces out a unique curve on the configuration space. Also, assuming adequate smoothness properties on the fields defining the configuration space, we will also be able to establish later the corresponding virtual space

Computation of Nonlinear Structures

788

̄ R), ̄ of the configuration space that we are allowed that defines the admissible variations, (d, to consider in our subsequent computational virtual work principle application for our c-type finite element method. We just note that because of the inclusion of the rotational tensor field, R(S1 , S2 , t) (algebraically, a member of SO(3), the 3D special orthogonal group), the configuration space is not a linear space, but rather a geometrically curved space known to mathematicians as the Riemannian (we assume R(S1 , S2 , t) as smooth as necessary) differentiable manifold. As we shall see later, this has major implications in our treatment of the calculus (the derivatives and the variations) of the rotation tensor, and for the linearization of the virtual work equations for our intended computational scheme. The dimension of the configuration space comprises three displacement components, three independent rotational components (because there are six constraints of orthonormality of the rotational tensor) for every point on the shell for all times of interest – a huge dimension.

11.6.5

Kinematic Parameterization: Displacements and Rotations

Note that because of the choice of orthonormality (i.e. RT R = I where I is the identity tensor) for R(S1 , S2 , t), only three degrees of freedom for rotations can be chosen for parameterization. As discussed in the kinematics of large rotation in Chapter 10, for representation of the rotational motions, many possibilities exists for parameterization of finite rotation R(S1 , S2 , t) such as Rodrigues parameters, Euler Rodrigues parameters, Hamilton–Rodrigues quaternions and Euler angles; more on this later. Now, by accepting the shell configuration space made up of the kinematic variables (d, R), we have the immediate task of defining the real strains in term of these ∙

real variables, and then relating these to the real strain rates through the rate entities (v = d, ω) defining the weak form as given by the equation (11.193).

11.6.6

Where We Would Like to Go

As discussed, in order that we can describe the presumed virtual strains as the variations of some real strain fields, we must take up the next step of seeking the real strain fields, the variations of which will guarantee the virtual nature of the energy conjugate strain expressions in the internal virtual work similar to that given by equation (11.194). However, before we can get on with it, we need to equip ourselves with adequate mathematical preliminaries relating to the definitions for the spatial derivatives (i.e. curvatures), temporal derivatives (i.e. rates) and variations. Thus, we must next visit:

r curvatures, angular velocity and variations r real strains and strain rates 11.7 11.7.1

Admissible Virtual Space: Curvature, Velocity and Variation What We Need to Recall

r The vector triple product property: a × b ∙ c = b × c ∙ a = c × a ∙ b, ∀a, b, c vectors. r The vector triple cross product property: a × b × c = (a ∙ c)b − (a ∙ b)c, ∀a, b, c vectors. r The definition of a rotation tensor, its matrix representation, and algebra, geometry and topology of rotation.

r A rotation matrix, R, corresponding to the three-dimensional special orthogonal matrix group, SO(3), may be parameterized by the Rodrigues vector, s, the axis of the instantaneous rotation.

Nonlinear Shell

789

r The Gateaux derivatives on a vector space are: v̄ (x, x̄ ) = lim

𝜌→0

r

(11.197)

where v̄ (x, x̄ ) is the one-parameter variation of a vector or tensor field, say v(x), where x and x̄ belong to a linear vector space; in other words, v̄ (x, x̄ ) is the directional derivative of v(x) at x along the vector direction x̄ . The covariant derivatives on a curved space are: (

d d , x̄ ≡ v̄ x ≡ d𝜉 d𝜆

r

{v(x + 𝜌̄x) − v(x)} d {v(x + 𝜌̄x)} = lim 𝜌→0 d𝜌 𝜌

) =

{v(𝜆0 + 𝜌)parallel transported to 𝜆0 − v(𝜆0 )} dx = lim d𝜆 𝜌→0 𝜌

(11.198)

where v̄ (x, x̄ ) is the covariant derivative of v(x) at x along the vector direction x̄ . The covariant derivative is the generalization of the Gateaux directional derivative of the vector spaces to the curved spaces where v(x + 𝜌̄x) and v(x) of the definition (11.197) do not belong to the same vector (tangent) space, and hence the difference operation do not have a meaning. In fact, the vector itself does not have a meaning other than as a tangent operation to a curve. Thus, and for invariance, the generalization is effected through the choice of affinely parameterized unique curves, but this makes the definition of the covariant derivative path-dependent. However, as a practical scheme for computing the covariant derivatives of vector fields (or tensor fields), all we need to do is to include the derivatives of the base vectors (or base tensors) as they are the tangents (or tensor product of these) to the coordinate curves on the manifold. For notational uniformity, we denote all variations of entities by using a superscripted bar. Furthermore all entities measured in the G1 , G2 , N frame are superscripted as t (∙), and those in the rotating g1 , g2 , g3 coordinate system as n (∙). However, for the sake of uncluttered readability, sometimes the superscripts may be dropped, but the context should help determine the appropriate underlying coordinate system.

11.7.2

Curvature and Angular Velocity Fields: the Admissible Virtual Space

In the development of the shell virtual work principle necessary for numerical analysis by a variational method such as the c-type finite element method, we will need the derivatives and the variations of the parameters of the configuration space. The configuration space parameters are the continuously differentiable displacement vector field, d(S1 , S2 , t), and the rotation tensor field, R(S1 , S2 , t), of the shell reference surface, where S1 , S2 are the arc length parameters along the lines of curvature of the shell reference surface, and t ∈ R+ is the time parameter. For dynamic motion, we will need both the temporal and the spatial derivatives, and the variations of these fields; for quasi-static motion, the fields are independent of time, that is, d = d(S1 , S2 ) and R = R(S1 , S2 ), in which case we will need the variations and spatial derivatives of the fields for the subsequent real and the virtual strain measures, and so on.

11.7.2.1

Real Displacement Vector Field and Admissible Virtual Space

The real displacement vector field, d(S1 , S2 , t), describing the translational movement of the points on a arbitrary configuration of the surface, for all times of interest, belongs to the 3D Euclidean space, E3 . Thus, we can construct the displacement portion of the virtual space as the vector space containing the perturbed or the infinitesimal displacement vector field that is “superposed” on the

Computation of Nonlinear Structures

790

real displacement vector field to obtain the perturbed or virtual configuration; it will be identified ̄ 1 , S2 , t) ≡ 𝛿d(S1 , S2 , t); the method of superposition is clearly defined by the vector addition as d(S operation. 11.7.2.2

Real Rotation Tensor Field, Manifold Lie Group, SO(3) and Admissible Virtual Space

The real rotation field, describing the independent rotation of the triad of base vectors of a surface belongs algebraically to the special orthogonal group, SO(3), and geometrically to a curved space. Thus, the admissible virtual space for rotation is considerably complicated compared to its displacement counterpart described in the previous section. We will need to define the infinitesimal (later, it will also be the instantaneous) rotations that can serve as the required virtual rotations which can then be superposed on the real rotations to describe the perturbed or virtual configurations. This, clearly, needs identification of the space to which these infinitesimal rotations belong to, and additionally, the method of superposition. In other words, in a Euclidean (‘flat’) space, we have identified, before for the displacement vector field, superposition simply as the vector addition operation, the question for us to resolve is if we can do the same for the rotation tensor field. It turns out that with appropriate parameterization for the rotation field, the answer is yes, although certain singularity issues must first be tackled. Moreover, we will need, for future use, the curvature and angular velocity fields, that is, the spatial and temporal derivatives of the rotation field. Next, we try to precisely meet these goals. Now, recall that the SO(3) group is the set of all 3 × 3 real proper orthogonal matrices, that is, R ∈ SO(3) ⇒ RT R = I, with I as the identity matrix and rotation matrix has positive unit determinant, that is, det R = +1. The four properties of a group: the existence of the identity, the existence of the inverse element, the associative property of the product and closure, are all satisfied by the rotation matrices under the product rule given by the usual matrix multiplication. The identity element: I with det I = 1, the inverse element: R−1 = RT for each rotation R belonging to the group, and the associative property of the matrices, satisfy the first three conditions. Now for closure: with R, S ∈ SO(3), two proper orthogonal matrices, we have R S also orthonormal because (R S)T (R S) = ST (RT R) S = I, and because: det (R S) = det R det S = 1; in other words, R ∈ SO(3), S ∈ SO(3) ⇒ R S ∈ SO(3). Also, a set of invertible tensor field is a continuous manifold. Since a group which is also a continuous manifold is called a Lie group, SO(3) is a Lie group.

11.7.3

Tangent Operator and Exponential Map

Let us recall that on a curved space, geometrical objects such as vectors, tensors, and so on, must be defined locally instead of bi-local definition in the neighborhood as in the flat Euclidean space; in other words, in a Euclidean (‘flat’) space, we can think of, say, a vector as joined by two neighboring points, but in a curved space a similar definition will yield a vector that may not even completely belong to the space. This necessitated the generalization from the definition of a vector as an arrow link between two neighboring points to a tangent operation to a curve (to be more specific, an affine curve for invariance or coordinate independence) at a point on the manifold as shown next. 11.7.3.1

Spatial Tangent Operator

Let the one spatial parameter family of the rotation tensor (or matrix) function R(𝜌) :→ SO(3) generate a continuous set of points defining a curve on SO(3), such that it passes through R(0) = R

Nonlinear Shell

791

d at 𝜌 = 0 where R ∈ SO(3). Then, R, 𝜌 , defined by the covariant derivative: R, 𝜌 ≡ d𝜌 R(𝜌)|𝜌=0 , is a tangent vector to SO(3) at R. The set of all tangent vectors at R, corresponding to various smooth curves passing though it forms a vector space, denoted by TR SO(3). Now, by taking the derivative of the orthogonality property, RT R = I, and rearranging, we have:

R, 𝜌 = K R, K ≡ R, 𝜌 RT = R, 𝜌 R−1

(11.199)

where we have introduced the following. Definition

r The

tangent (linearized) operator, K, the skew-symmetric matrix (because, with RRT = I, we have: K + KT = R, 𝜌 RT + R R, T𝜌 = (RRT ), 𝜌 = 0); K is the instantaneous and linearized skew-symmetric tensor at a rotation, R ∈ SO(3).

Now, if we consider a constant tangent operator, K, that is, independent of the parameter 𝜌, we have from equation (11.199), R(𝜌), 𝜌 = K R(𝜌), which has a solution given by: R(𝜌) = e𝜌K ≡ (I + 𝜌K +

(𝜌K)2 + …) 2!

(11.200)

Thus each element of the tangent space is related to a corresponding element of the curve space by an exponential map given by equation (11.200). In other words, there always exists a skewsymmetric matrix that can represent a rotation matrix by this exponential map. More specifically, for a rotation matrix, R ∈ SO(3), there exists a linearized skew-symmetric matrix, K ∈ so(3) related by: R = eK 11.7.3.2

(11.201)

Temporal Tangent Operator

Similarly, let the one temporal (time) parameter family of the rotation tensor (or matrix) function R(t) :→ SO(3) generate a continuous set of points defining a curve on SO(3), such that it passes ∙

through R(0) = R at t = 0 where R ∈ SO(3). Then, R ≡ ∙

d R, defined by the covariant derivative: dt

dR(t) R ≡ dt |t=0 , is the tangent vector to SO(3) at R. The set of all tangent vectors at R, corresponding to various smooth curves passing though it forms a vector space, denoted by TR SO(3). Now, by taking the temporal derivative of the orthogonality property, RT R = I, and rearranging, we have:

∙

R = ω R, ∙

∙

ω ≡ R RT = R R−1 where we have introduced the following.

(11.202)

Computation of Nonlinear Structures

792

Definition

r The

tangent (linearized) operator, 𝛀, the skew-symmetric matrix (because, with RRT = I, ∙

∙

we have: 𝛀 + 𝛀T = R RT + R RT = dtd (RRT ) = 0); 𝛀 is the instantaneous and linearized skew-symmetric tensor at a rotation, R ∈ SO(3). Because the tangent space is populated with the skew-symmetric matrices, the dimension of the tangent space is 3 (with only n(n − 1)∕2 elements in the matrix with n = 3 for the 3D space). Note also that the tangent space is a linear vector space. Now, if we consider a constant tangent operator, 𝛀, that is, independent of the parameter t, we ∙

have from equation (11.202), R(t) = 𝛀 R(t), which has a solution given by: t𝛀

R(t) = e

( ) (t𝛀)2 ≡ I + t𝛀 + + ... 2!

(11.203)

Thus, each element of the tangent space is related to a corresponding element of the curved space by an exponential map given by equation (11.203). In other words, there always exists a skew-symmetric matrix that can represent a rotation matrix by this exponential map. More specifically, for a rotation matrix, R ∈ SO(3), there exists a linearized skew-symmetric matrix, 𝛀 ∈ so(3) related by: R = e𝛀

(11.204)

Remark

r Although equations (11.201) and (11.204) relate the rotation matrix with the corresponding skew-symmetric matrix of the tangent space through the exponential map, there is, as derived before, a better way of relating the two by expressing the exponential map by Rodrigues vector or Rodrigues quaternion parameters of the rotation matrix through Cayley’s theorem.

11.7.3.3

Tangent Space at Identity

The tangent space at identity, that is, at R = I, is given the special name, Lie algebra (recall that a vector space with a vector multiplication, [∙, ∙], defined on it is an algebra) and denoted by so(3). The reason for this special treatment of the neighborhood at identity, I, is that the local structure of SO(3), the continuous rotation group, can be constructed out of this infinitesimal tangent space, so(3), by the exponential map given by equation (11.203). In other words, all the information of the group is contained in the algebra. Thus, so(3) consists of 3 × 3 skewsymmetric matrices; as any 3 × 3 skew-symmetric matrix has a corresponding 3 × 1 axial vector, so(3) is isomorphic to the 3D Euclidean vector space. More generally, a Lie algebra is a vector space, , with a bilinear map, [∙, ∙] :  ×  → , known as the Lie bracket, so that for every a, b, c ∈  ⇒, 1) [a , a] = 0, 2) [a, [b, c]] + [b, [c, a]] + [c, [a, b]] = 0 (the Jacobi identity). From condition (1) and the bilinearity of the map, [a + b, a + b] = [a, b] + [b, a] = 0 implying [a, b] = −[b, a]. For square matrix Lie group and Lie algebra, the Lie bracket reduces to the matrix commutator: [A, B] = AB − BA, ∀A, B ∈ so(3). With the notation, [a] ∈ so(3) as the skew-symmetric matrix corresponding to the axial vector, a, we can easily show that AB − BA = [a × b].

Nonlinear Shell

11.7.4

793

Admissible Virtual Space

Based on our discussion about the tangent space at identity, so(3), the admissible virtual space is defined by this linear space of the configuration space made up of the union set of all the possible configurations of the surface defined by the ordered pair (d, R), that is, the displacement vector ̄ ≡ field and the rotation tensor field, respectively. In other words, every ordered pair, (d̄ ≡ 𝛿d, R ̄ 𝛿R) of the virtual of displacement vector, d, and the infinitesimal, skew-symmetric rotation tensor, ̄ belongs to this linear space. This, then, takes care of the part of our original questions about R, how to define admissible virtual space. But, we still have not determined how to superpose the virtual rotations to the current real rotation to obtain the perturbed or virtual configuration space necessary for the subsequent virtual work principle. As it turns out, there are two possible ways to accomplish this and we have to conveniently choose one; this aspect of the study we will discuss next.

11.7.5

Adjoint Map and Material and Spatial Variations

We have already seen through equation (11.203) that the elements of the Lie group, SO(3), is related to the corresponding elements of its Lie algebra, so(3), by the exponential map. Another map known as the adjoint map also relates the elements of SO(3) with the corresponding elements of so(3). If an arbitrary rotation matrix, R ∈ SO(3), and its corresponding skew-symmetric matrix, ̄ ∈ so(3), are given, then, we have 𝚽 ̄ defined by the adjoint map: 𝚽 ̄ ≡R𝚿 ̄ RT , also an element of 𝚿 T T T T ̄ +𝚽 ̄ =R𝚿 ̄ R +R𝚿 ̄ R == R 𝚿 ̄ RT − R 𝚿 ̄ RT = so(3). We can easily see that because (𝚽 ̄ R = R 𝚿, ̄ 0) the symmetric part vanishes identically, it is a member of so(3). Thus, we have: 𝚽 and using similar to equation (11.199), we have two ways to superpose to represent the virtual ̄ by: rotation, R, ̄ R=R𝚿 ̄ ̄ =𝚽 R

(11.205)

̄ is In the first equality of equation (11.205), known as the left translation, the virtual rotation, 𝚽, being superposed on the current rotation to obtain the updated rotation; we use this relationship in our subsequent analysis, but the components are measured in the undeformed and fixed material frame, G1 , G2 , G3 ≡ N. In the second equality of equation (11.205), known as the right translation, ̄ to obtain the updated rotation. the current rotation is being superposed on the virtual rotation, 𝚿, We do not use this relationship further in our subsequent analysis and hence will not discuss it any further. Finally, then, the finite rotation, R, belongs to the spherical curved space, SO(3), and the ̄ represented by a skew-symmetric matrix, instantaneous and infinitesimal virtual rotation, 𝚽, belongs to the tangent space at identity, the Lie algebra.

11.7.6

̄ and ϕ ̄ Rotational Variation: Φ

As indicated above, we choose the following form for rotational variation in all subsequent applications: ̄ R⇔𝚽 ̄ ≡R ̄ =𝚽 ̄ RT R

(11.206)

Computation of Nonlinear Structures

794

For the 3D rotation tensors, the corresponding skew-symmetric tensors only have three significant variables (i.e. the number of variables equal the dimension of the space), and thus can have axial }T { ̄ ≡ 𝜑̄ 1 𝜑̄ 2 𝜑̄ 3 be such axial vector corresponding to the skewvector representation. Let ϕ −𝜑̄ 3 𝜑̄ 2 ⎤ ⎡ 0 ̄ with 𝚽 ̄ ϕ ̄ = ⎢ 𝜑̄ 3 ̄ = 0. Knowing that 𝚽 0 −𝜑̄ 1 ⎥, the equality can symmetric tensor, 𝚽, ⎥ ⎢ − 𝜑 ̄ 𝜑 ̄ 0 ⎦ ⎣ 2 1 be readily verified.

11.7.7

Rotational Spatial Derivatives – Curvatures: K and k

Next, we can define the curvature tensor in two different ways as: K𝛼 ≡ R, 𝛼 RT or L𝛼 ≡ RT R, 𝛼 , and thus, the spatial derivatives of variables of the reference surface can have two representations; we choose the following spatial form in all subsequent applications: R, 𝛼 = K𝛼 R ⇔ K𝛼 ≡ R, 𝛼 RT

(11.207)

As before, let k be the axial vector corresponding to the skew-symmetric tensor, K, with K k = 0.

11.7.8

Rotational Time Derivative – Angular Velocity: Ω and ω ∙

Similarly, we can define the angular velocity tensor in two different ways as: 𝛀 ≡ R RT or ∙

∙

Ξ ≡ RT R, where (∙) ≡ dtd (∙), and thus, the time derivatives of variables of a Cosserat surface can have two representations. We choose the following form in all subsequent applications: ∙

∙

R = 𝛀 R ⇔ 𝛀 = R RT

(11.208)

As before, let 𝛀 be the axial vector corresponding to the skew-symmetric tensor, 𝛀, with 𝛀 ω = 0.

11.7.9

Coordinate Frame Rotation, Objective Variations and Derivatives

For future application, let us suppose an initial orthonormal coordinate frame, {Gi (S𝛼 )}, i = 1, 2, 3, attached at a material point S𝛼 , 𝛼 = 1, 2 to a material surface in space, and let us also suppose that in time t, a current coordinate system, {gi (S𝛼 , t)}, i = 1, 2, 3, is obtained from the initial coordinate frame by pure rotation such that gi = R Gi , i = 1, 2, 3 where R(S𝛼 , t) is the rotation tensor. Now, let x(S𝛼 , t) be any vector field attached to the body at point S𝛼 , 𝛼 = 1, 2. If an observer is sitting on the body such that it also co-spins as above, we would be interested in how it sees the variations and time rate of change in x(S𝛼 , t); we will call this the objective variations, and objective time rate, respectively, of vector fields. However, before we delve into the derivations of the objective rate and note the following.

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795

Variation of Current Base Vectors: {gi (S𝜶 , t)}, i = 1, 2, 3

11.7.9.1

From equation (11.206), the variation of gi is given as: ̄ R Gi = 𝚽 ̄ gi = ϕ ̄ Gi = 𝚽 ̄ × gi gi = R

(11.209)

{ } Spatial Derivative of Current Base Vectors: gi (S𝜶 , t) , i = 1, 2, 3

11.7.9.2

From equation (11.207), the spatial derivative of gi is given as: gi , 𝛼 = R, 𝛼 Gi = K𝛼 R Gi = K𝛼 gi = k𝛼 × gi

(11.210)

Time Derivative of Current Base Vectors: {gi (S𝜶 , t)}, i = 1, 2, 3

11.7.9.3

From equation (11.126), the time derivatives of gi is given as: ∙

∙

gi = R Gi = 𝛀 R Gi = 𝛀 gi = ω × gi

(11.211)

v

11.7.9.4

Objective Variation, (∙)

Now, let us also observe that for any vector field x described in the current rotated coordinate system, that is, x = xi gi , we have, using equation (11.209), for variation in spatial coordinate system: x̄ ⏟⏟⏟

̄ × xi gi = x̄ i gi + ϕ ̄ × x = x̄ i gi + xi ḡ i = x̄ i gi + ϕ

Total variation v

⇒x≡

x̄ i gi ⏟⏟⏟ Variation in rotated coordinate system, i.e. objective variation

x̄ − ⏟⏟⏟

=

Total variation

̄ × x ϕ ⏟⏟⏟

̄ = R{RT x̄ + (−RT Φ)x}

Spinning variation

⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ RT x⎟ ̄ T x} = R = R{RT x̄ + R ⏟⏟⏟ ⎜⏟⏟⏟ ⎟ Push forward ⎜Pull back ⎟ ⎜⏟⏞⏞⏟⏞⏞⏟⎟ ⎝ Variation ⎠ (11.212) From equation (11.212), we now introduce what is known as the objective variation or the co-rotated variation. v

Definition: Objective variation, (∙): v

̄ × (∙) (∙) ≡ (∙) − ϕ

(11.213)

Computation of Nonlinear Structures

796

t

Objective Time Rate, (∙)

11.7.9.5

Similarly, using equation (11.209), for the time derivative in the current rotated coordinate system, we have: ∙

x ⏟⏟⏟

∙

∙

∙

∙

= xi gi + xi gi = xi gi + ω × xi gi = xi gi + ω × x

Total derivative t

⇒x≡

∙

xi gi ⏟⏟⏟ Derivative in rotated coordinate system,i.e. objective derivative

∙

x ⏟⏟⏟

=

Total derivative

−

ω × x ⏟⏟⏟

∙

= R{RT x + (−RT ω)x}

Spinning observer correction

⎛ ⎞ ∙ ⎜ ⎟ ⎜⏞⏞⏞⏞⏞⏞⏞⏞⏞⎟ ∙ ∙ ⎜( RT x)⎟ = R{RT x + RT x} = R ⏟⏟⏟ ⎜ ⏟⏟⏟ ⎟ ⎟ Push forward ⎜ Pull back ⎜⏟⏞⏞⏞⏟⏞⏞⏞⏟⎟ ⎝ Derivative ⎠ (11.214) From equation (11.214), we now introduce what is known as the objective time rate, or the co-rotated time rate. t

Definition: Objective time rate, (∙): t

(∙) ≡

𝜕 (∙) = (∙) − ω × (∙) 𝜕t

(11.215)

Remarks

r The idea behind the objective variation (or objective time derivative) in the current rotated

r r

body attached coordinate system is that it is the net variation (or time derivative) in the co-rotated coordinate system, that is, the total (material) variation minus the variation of the spinning coordinate system itself; in other words, it is the variation (or time derivative) observed by an observer attached to the spinning coordinate system given by the base vectors, {gi }. The objective variation (or time derivative) may also be seen as the ordered sequence of operations as the “pulling back” to the material coordinate system, taking the variation, and finally “pushing forward” to the current coordinate system, as in the definition of the Lie derivative. Note, however, that, as will be seen in our subsequent computational efforts, the components of all involved vector and tensor quantities are measured in terms of the undeformed coordinate system.

Nonlinear Shell

11.7.9.6

797

̄ and ϕ ̄ and K and k Relations between 𝚽

From equations (11.206) and (11.207), and noting that R RT = I, I being the 3D identity tensor, we have for the variation of the curvature: ̄ 𝛼 = (R ̄ (RT KT ) ̄ RT ), 𝛼 = R, ̄ 𝛼 RT + R ̄ RT , 𝛼 = K𝛼 R RT + 𝚽R 𝚽, 𝛼 ̄ R RT K𝛼 = K𝛼 + K𝛼 𝚽 ̄ R RT − 𝚽 ̄ −𝚽 ̄ K𝛼 (11.216) = K𝛼 R RT + K𝛼 𝚽 ⏟⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏟ Lie bracket

implying, with the application of the definition of objective variation from equation (11.213) to equation (11.216), and the definition of the axial vectors and the Lie brackets, that: { } ̄ 𝛼 ̄ 𝛼 = k𝛼 − ϕ ̄ × k𝛼 = R RT k𝛼 + (−RT Φ)k ϕ, ⎛ ⎞ ⎜ ⎟ v ⎜ ⎟ { T } ⎜ RT k𝛼 ⎟ ≡ k𝛼 ̄ T k𝛼 = = R R k𝛼 + R R ⏟⏟⏟ ⎜⏟⏟⏟ ⎟ ⎟ Push forward ⎜Pull back ⎜⏟⏞⏞⏞⏟⏞⏞⏞⏟⎟ ⎝ Variation ⎠

11.7.9.7

(11.217)

Relations between 𝛀 and ω and K and k

From equations (11.207) and (11.126), and noting that R RT = I, I being the 3D identity tensor, we have for the spatial derivative of the angular velocity: ∙

⏞⏞⏞ 𝛀, 𝛼 = (R RT ), 𝛼 = R, 𝛼 RT + R RT , 𝛼 = K𝛼 R RT + 𝛀 R (RT KT𝛼 ) ∙

∙

∙

∙

= K𝛼 R RT + K𝛼 𝛀 R RT − 𝛀 R RT K𝛼

(11.218)

∙

= K𝛼 + K𝛼 𝛀 − 𝛀 K𝛼 ⏟⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏟ Lie Bracket

implying, with the application of the definition of objective time derivative from equation (11.215) to equation (11.218), and the definition of the axial vectors and the Lie brackets, that: ∙ ∙ { ( ) } ω, 𝛼 = k𝛼 − ω × k𝛼 = R RT k𝛼 + −RT ω k𝛼

⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ∙ t ∙ ⎜ RT k ⎟ ≡ k = R{RT k𝛼 + RT k𝛼 } = R 𝛼 ⏟⏟⏟ ⎜⏟⏟⏟ 𝛼 ⎟ ⎟ Push forward ⎜Pull back ⎜⏟⏞⏞⏞⏟⏞⏞⏞⏟⎟ ⎝ Variation ⎠

(11.219)

Computation of Nonlinear Structures

798

Taking variation of ω, 𝛼 in equation (11.219), and observing that both variation and derivative are linear operations (and hence the order of operation can be switched), we have: t

̄ 𝛼 = k𝛼 ω, 11.7.9.8

(11.220)

̄ and ϕ ̄ Relations Between 𝛀 and ω and 𝚽

From equations (11.206) and (11.126), and noting that R RT = I, I being the 3D identity tensor, we have for the variation of the angular velocity: ∙ ) ∙ (∙ ∙ ̄ R RT + 𝛀 R (RT 𝚽 ̄ T) ̄ = R RT = R ̄T = 𝚽 ̄ RT + R R 𝛀 ∙

∙

∙

̄ R RT + 𝚽 ̄ =𝚽 ̄ +𝚽 ̄ 𝛀− 𝛀𝚽 ̄ ̄ R RT − 𝛀 R RT 𝚽 =𝚽 ⏟⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏟

(11.221)

Lie bracket

implying, with the application of the definition of objective time derivative from equation (11.215) to equation (11.221), and the definition of the axial vectors and the Lie brackets, that: ∙ ∙ { ( ) } ̄ ̄ = 𝝋 ̄−ω × 𝝋 ̄ = R RT 𝝋 ̄ + −RT 𝛀 𝚽 ω

⎛ ⎞ ∙ ⎜ ⎟ ⎜⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⎟ t ∙ { T ∙ } ⎜( RT 𝝋) ̄ + RT 𝝋 ̄ = ̄ ⎟≡𝝋 ̄ =R R 𝝋 R ⏟⏟⏟ ⎜ ⏟⏟⏟ ⎟ ⎟ Push forward ⎜ Pull back ⎜⏟⏞⏞⏞⏞⏟⏞⏞⏞⏞⏟⎟ ⎝ Derivative ⎠

(11.222)

̄ in equation (11.222), and observing that both variation and Taking spatial derivative of ω derivative are linear operations (and hence the order of operation can be switched), we have: t

̄ 𝛼 = 𝝋, ̄ 𝛼 ω,

(11.223)

Thus, finally, from equations (11.220) and (11.223), we have: t

t

̄ 𝛼 = 𝝋, ̄ 𝛼 = k𝛼 ω,

11.7.10

(11.224)

Where We Would Like to Go

Here, we have introduced various parameters and derived all the necessary relationships anticipating the next step of identifying the spatial real strain and the real strain rate fields for the shell reference surface. Clearly, then, we should proceed by embarking on:

r spatial real strains and rates. r

Finally, to get familiarized with and be conscious about the overall computational strategy for nonlinear shells, we should explore: the road map: nonlinear shells

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11.8 Real Strain and Strain Rates from Weak Form 11.8.1 What We Need to Recall

r The shell internal stored power, PInternal , as: PInternal ≡

∫

) { 𝛼 ( } ( F ∙ v, 𝛼 + c0 , 𝛼 × ω + M𝛼 ∙ ω, 𝛼 dA

(11.225)

Ω

The resistive stress resultant, F𝛼 , is defined by:

F𝛼 (S, t) ≡

𝜂t

∫𝜂b

𝜂 𝛼 𝜂 S

P

𝜇 d𝜂

(11.226)

The resistive stress couple, M𝛼 , is defined by:

M𝛼 (S, t) ≡

𝜂t

∫𝜂b

(c(S, 𝜂, t) − c0 (S, t)) ×

𝜂 𝛼 𝜂 S

𝜇 d𝜂

P

(11.227)

where (S, 𝜂) defines the through-the-thickness deformation of the shell-like body. ∙

r r

(v ≡ d, ω) are the weak parameters, The shell configuration space is completely identified by the history of the kinematic parameters, (d, R) The derivatives of the initial base vector set, {Gi (S1 , S2 )}, i = 1, 2, 3, are given by: d G (S1 , S2 ) ≡ Gi (S1 , S2 ), 𝛼 = K0𝛼 Gi , dS𝛼 i

𝛼 = 1, 2

The temporal derivatives of the current base vector set, {gi (S1 , S2 , t)}, by: ∙

gi = ω × gi

(11.228) i = 1, 2, 3, are given

(11.229)

r For any vector field x described in the rotated coordinate system, that is, x = xi gi , we have for the variation in the coordinate system: v

x≡

x̄ i gi ⏟⏟⏟ Variation in rotated coordinate system,i.e. objective variation

v

x̄ − ⏟⏟⏟

=

Total variation

v

̄ × x ϕ ⏟⏟⏟ Spinning variation

̄ × (∙) . x is the objective variation of x, defined by: (∙) ≡ (∙) − ϕ

(11.230)

Computation of Nonlinear Structures

800

r For any vector field x described in the rotated (coordinate system, i.e. x = xi gi , we have for the rate in this coordinate system: ∙

t

x≡

xi gi ⏟⏟⏟

=

Derivative in rotated coordinate system, i.e. objective derivative

∙

x ⏟⏟⏟

ω × x ⏟⏟⏟

−

Total derivative

(11.231)

Spinning observer correction

r xt is the objective time rate of x, defined by: (∙)t ≡ 𝜕 (∙) = (∙) − ω × (∙) . 𝜕t r The relationship between the derivative of the angular velocity tensor, ω, and the curvature tensor, K𝛼 : ∙

∙

𝛀, 𝛼 = K𝛼 + K𝛼 𝛀 − 𝛀 K𝛼 ⇔ ω, 𝛼 = k𝛼 − ω × k𝛼 ⏟⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏟

(11.232)

Lie bracket

r The vector triple product property: a × b ∙ c = b × c ∙ a = c × a ∙ b, 11.8.2

∀a, b, c vectors.

Real Strains and Strain Rates in the Spinning Coordinate System

{ } Let us note that gi (S1 , S2 , t) = R(S1 , S2 , t) Gi (S1 , S2 ), where {gi }, and Gi are the base vector sets for the spinning coordinate system of the current deformed shell surface, and the undeformed coordinate shell reference surface, respectively. Now, with the relation given in equation (11.228), we can obtain for the derivatives of gi as: gi , 𝛼 = R, 𝛼 Gi + R Gi , 𝛼 = K𝛼 R Gi + R K0𝛼 Gi

= K𝛼 gi + R K0𝛼 RT R Gi ≡ (K𝛼 + R K0𝛼 RT ) gi ≡

Ktot 𝛼

gi =

ktot 𝛼

(11.233)

× gi

where we introduce the following. tot Definitions: The total curvature tensor, Ktot 𝛼 , and the corresponding axial vector, k𝛼 , as: 0 T tot 0 Ktot 𝛼 ≡ K𝛼 + R K𝛼 R ⇒ k𝛼 ≡ k𝛼 + R k𝛼

(11.234)

The result is that, for shells with initial curvature, the curvature must always be understood as the sum of one due to the current rotation, that is, K𝛼 ≡ R, 𝛼 RT , and another due to the initial tot curvature, that is, k0𝛼 , of the undeformed shell reference surface with Ktot 𝛼 and k𝛼 being the total curvature tensor and the total curvature vector, respectively.

11.8.2.1

Force Real Strain Field ∙

⏞⏞⏞ ∙ Note that with c0 (S1 , S2 , t), 𝛼 = G𝛼 (S1 , S2 ) + d(S1 , S2 , t), 𝛼 , we have as time rate: c0 , 𝛼 = d, 𝛼 = ∙

∙

⏞⏞⏞ ∙ ⏞⏞⏞⏞⏞⏞⏞ v, 𝛼 , and applying equation (11.229): d, 𝛼 − ω × g𝛼 = c0 , 𝛼 − g𝛼 = c0 , 𝛼 − g𝛼 . Now, with the ∙

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801

results in equation (11.231) applied to the presumed force strain rate expression in the internal stored power as given in equation (11.225), we get: v, 𝛼 + c0 , 𝛼 × ω ∙

t

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ = d, 𝛼 − ω × g𝛼 − ω × c0 , 𝛼 + ω × g𝛼 = (c0 , 𝛼 − g𝛼 ) − ω × (c0 , 𝛼 − g𝛼 ) ≡ (c0 , 𝛼 − g𝛼 ) ∙

∙ ∙ ⎫ ⎧ ⏞⏞⏞⏞⏞⏞ ⎫ ⎧ ⏞⏞⏞⏞⏞⏞ ⏞⏞⏞⏞⏞ ⏞⏞⏞⏞⏞ ∙ ( ) ⎪ ⎪ T ⎪ ⎪ T T T = R ⎨R (c0 , 𝛼 − g𝛼 ) + −R 𝛀 (c0 , 𝛼 − g𝛼 )⎬ = R ⎨R (c0 , 𝛼 − g𝛼 ) + R (c0 , 𝛼 − g𝛼 )⎬ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ ⎭ ⎩ ⎛ ⎞ ∙ t ⎜ ⎟ ⏞⏞⏞ ⎜⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⎟ ⎜ RT (c , − g𝛼 )⎟ = R β𝛼 = R ⎟ ⏟⏟⏟ ⎜⏟⏟⏟ 0 𝛼 ⎟ Push forward ⎜ Pull back ⎜⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⎟ ⎝ ⎠ Time rate

(11.235) where we have introduced the following. Definition

r The rotated force real strain, R β𝛼 : R

β𝛼 ≡ c0 , 𝛼 − g𝛼

(11.236)

Thus, we conclude that the expression v, 𝛼 + c0 , 𝛼 × ω, in equation (11.225), truly represents the rotated force strain rate as it can be obtained from the time derivative of a function: the rotated force real strain, R β𝛼 defined by equation (11.236); it embodies translational strains: the real extensional strain and the real shear strain, along the rotated coordinate system with the base vector set given by {gi }. Note that in the undeformed reference surface: as d → 0 and R → I, the force real strain, R β𝛼 → 0 as it should be.

11.8.2.2

Moment Real Strain Field ∙

By definition, the rotational velocity tensor, 𝛀 ≡ R RT , the curvature, K𝛼 ≡ R, 𝛼 RT , the initial curvature of the undeformed shell reference surface, K0𝛼 with Gi , 𝛼 = K0𝛼 Gi , and finally, recall the ̂ 0 with the corresponding axial vector, k̂ 0 , defined by: property: for any skew-symmetric tensor K 𝛼 𝛼 0 0 T 0 0 0 ̂ ̂ K𝛼 = R K𝛼 R ⇒ k𝛼 = R k𝛼 , where k𝛼 is the axial vector corresponding to the skew-symmetric

Computation of Nonlinear Structures

802

0 T tot 0 tensor K0𝛼 . Now, with Ktot 𝛼 ≡ K𝛼 + R K𝛼 R as the total curvature tensor with k𝛼 = k𝛼 + R k𝛼 as the corresponding total axial curvature vector, we have:

∙

∙

∙

∙

∙

0 T 0 T tot 0 T 0 T K𝛼 = Ktot 𝛼 − R K𝛼 R − R K𝛼 R = K𝛼 −𝛀 R K𝛼 R + R K𝛼 R 𝛀 ∙ ( ) ( ) 0 T = Ktot 𝛀 − 𝛀 R K0𝛼 RT 𝛼 + R K𝛼 R ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

(11.237)

Lie bracket

implying in vector form: ∙

∙

∙

0 tot 0 k𝛼 = ktot 𝛼 +(R k𝛼 ) × ω = k𝛼 −ω × (R k𝛼 )

(11.238)

Now, from the second of equation (11.232) and relation in equation (11.238), we have: t

⏞⏞⏞ ω, 𝛼 = k𝛼 − ω × k𝛼 ≡ k𝛼 { } { } ∙ ∙ ∙ ( T ) T T T = R R k𝛼 + −R ω k𝛼 = R R k𝛼 + R k𝛼 ∙

⎛ ⎞ t ⎜ ⎟ ⎜ ⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⎟ ⏞⏞⏞ ⎜ RT k ⎟ = R χ = R 𝛼 ⏟⏟⏟ ⎜ ⏟⏟⏟ 𝛼 ⎟ ⎜ ⎟ Push forward Pull back ⎜ ⏟⏞⏞⏞⏟⏞⏞⏞⏟ ⎟ ⎝Time derivative⎠

(11.239)

where we have introduced: Definition

r The rotated moment real strain: R

0 χ𝛼 ≡ k𝛼 = ktot 𝛼 − R k𝛼 ,

K𝛼 ≡ R, 𝛼 RT

(11.240)

Thus, we conclude that the expression ω, 𝛼 truly represents the rotated moment strain rate as obtained from the time derivative of a function: the rotated moment real strain, R χ𝛼 ; it embodies rotational strain: the real bending strain, along the rotated coordinate system with the base vector set given by {gi }.

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803

Note that in the undeformed reference surface: R → I, the moment real strain, R χ𝛼 → 0 as it should. Thus, by inserting results from equations (11.235) and (11.239) into the internal stored power expression of equation (11.225), it can be rewritten as: t

PInternal

t

⎧ ⏞⏞⏞ ⏞⏞⏞ ⎫ ⎪ 𝛼 R ⎪ 𝛼 ( F ∙ β𝛼 + M ∙ R χ𝛼 ⎬ dA ≡ ∫ ⎨ ⎪ ⎪ Ω ⎩ ⎭

(11.241)

Remark

r For our subsequent computational virtual work principle necessary for a c-type finite element formulation, we will develop the rotating coordinate system by first expressing the invariant force and moment vectors in the material coordinate system and then applying the rotation tensor to the components. This allows us to develop and track the most accurate numerical scheme with a cost-effective explicit identification of the relevant entities. Thus, for computational ease and accuracy, all the expressions pertaining to the weak form, the virtual strains, and so on, must be modified to the most suitable operational or component vector form, which we intend to do next. However, before we get into the computational description of the shell behavior, we will make a short detour to present an alternate way to develop the weak form, the stress resultants and the stress couple and the associated conjugate strains for historical connection. We have developed the exact weak form by starting from the exact balance laws of motion. In this alternative way of thinking, we instead starts by making some kinematic assumptions; then, the stress or deformation power (in the case of static analysis, the internal strain energy variations) provides the conjugate real strains and real strain rates upon suitable definition of the stress resultants and the stress couples. So, let us look into it with a kinematic assumption that is appropriate and commonly used for thin shells, as follows.

11.8.3

Alternative Derivation: Kinematics and Stress or Deformation Power

A simple 2D surface can never represent completely the behavior of a 3D shell-like body; an additional mechanism must be embedded in the 2D surface to capture, in some representative fashion, the through-the-thickness response of the 3D shell-like body. It turns out that the 2D kinematics that are energetically consistent with the 3D shell-like body is best captured by what is known as the Cosserat surface, which is an oriented surface, that is, a 2D surface with a triad of orthonormal vector fields, {gi (S1 , S2 )}, i = 1, 2, 3, known as the directors, acting as the base vectors of a coordinate system and identifying the deformed shell reference surface. This coordinate system will be termed as the co-rotating or spinning system, the meaning of which will be clear shortly. We choose these directors at the reference configuration to be coincident with the base vectors, {Gi (S1 , S2 )}, i = 1, 2, 3, of the shell reference surface where G1 , G2 are the unit tangent base vectors with S𝛼 , 𝛼 = 1, 2 as the arc length parameters along the lines of curvature, and G3 ≡ N is the unit normal to the shell reference surface as introduced earlier.

Computation of Nonlinear Structures

804

Figure 11.18 Kinematics and deformation map.

Then, as shown in Figure 11.18, the most general deformation map of a Cosserat shell surface is given by: c0 (S1 , S2 ) = C0 (S1 , S2 ) + d(S1 , S2 ) gi (S1 , S2 ) = R(S1 , S2 ) Gi (S1 , S2 ), i = 1, 2, 3

(11.242)

where d(S1 , S2 ) is the displacement vector field identifying the translational movement of the reference surface, R(S1 , S2 ) is the rotational tensor field designating the rotation of the directors independent of the deformation of the reference surface; as introduced earlier, C0 (S1 , S2 ) and c0 (S1 , S2 ) are the corresponding position vectors in the reference surface and the current deformed surface, respectively. Remark

r It

is important to note that the base vector set, {gi (S1 , S2 )}, i = 1, 2, 3 is not the the set of convected coordinate bases; recall that the convected coordinate bases are given by {c0 ,1 , c0 ,2 , c0 ,1 × c0 ,2 }, if shear deformation is neglected.

11.8.3.1

Weak Form: Kinematic Assumption and Reduction from the 3D Shell-like Body

Contrary to what we have done before, we use the deformation map as our starting point. As an example, in what follows, we restrict our exposition to the case of shells with a linear deformation distribution through the thickness of the shell body, that is, we take h(S1 , S2 , 𝜂) = 𝜂 g3 (S1 , S2 ) in equation (11.227), except that we do not allow any thickness change. Thus, the basic kinematic assumption is the inextensible one-director Cosserat surface. Now, let us suppose that a particle with the position vector, C(S1 , S2 , 𝜂), in the 3D shell-like undeformed body assumes under deformation a location with the position vector given by c(S1 , S2 , 𝜂); and suppose the corresponding position vectors on the shell reference surface to be C0 (S1 , S2 ) ≡ C(S1 , S2 , 0) and c0 (S1 , S2 ) ≡ c(S1 , S2 , 0), respectively. Thus, any configuration of the shell-like body under the kinematic assumption is given by: c(S1 , S2 , 𝜂) = c0 (S1 , S2 ) + 𝜂 g3

(11.243)

Nonlinear Shell

805

The rotated base vector set, {gi }, on the deformed reference surface is related to the base vector set, {Gi }, on the undeformed reference surface, as before, by: gi = R Gi . The reference configuration is exactly given by: C(S1 , S2 , 𝜂) = C0 (S1 , S2 ) + 𝜂 G3

(11.244)

Then, using equations (11.233) and (11.243), the deformation gradient, FD , is given by: 𝜕c = c,i ⊗ Ai = c, 𝛼 ⊗ A𝛼 + c, 𝜂 ⊗ G3 𝜕C { } 𝛼 = c0 , 𝛼 + ktot 𝛼 × (c − c0 ) ⊗ A + g3 ⊗ G3

FD ≡ GRAD c =

(11.245)

where A𝛼 , 𝛼 = 1, 2 are the contravariant base vectors at the particle of interest in the undeformed state. Now noting the time rate, from equation (11.229), we have, from equation (11.245), the time ∙

rate, F ≡

d F, dt

of the deformation map as:

∙ ⎧ ⏞⏞⏞⎫ ∙ ∙ ∙ ⎪ tot × (c − c )⎪ ⊗ A𝛼 + g ⊗ G FD = ⎨c0 , 𝛼 + ktot 0 ⎬ 3 3 𝛼 × (c − c0 ) + k𝛼 ⎪ ⎪ ⎩ ⎭ { } ∙ ∙ tot tot = c0 , 𝛼 + k𝛼 × (c − c0 ) + k𝛼 × (ω × (c − c0 ) ) ⊗ A𝛼 + (ω × g3 ) ⊗ G3 ∙

(11.246)

Fact 1: As a supporting formula, let us note that for any B = a ⊗ b, a symmetric tensor, that is, B − BT = 0, we have: a × b = 0 because: ∀h ∈ R3 , 0 = (B − BT )h = (a ⊗ b)h − (b ⊗ a)h = (h ∙ b)a − (h ∙ a)b = (a × b) × h where we have used the vector identity: a × (b × c) = (a ∙ c)b − (a ∙ b)c. Now, if P ≡ Pj ⊗ Aj is the first Piola–Kirchhoff stress tensor, we get from equation (11.246): ∙

∙

∙

𝛼 P : FD = P𝛼 ∙ c0 , 𝛼 + (c − c0 ) × P𝛼 ∙ ktot 𝛼 +P

+ ω ∙ (g3 × P3 )

{

} ktot 𝛼 × (ω × (c − c0 )) (11.247)

where we have used the definition that for any two tensors, A = a ⊗ b, and B = c ⊗ d, we have: A:B = (a ⊗ b):(c ⊗ d) = (a ∙ c)(b ∙ d). Now, let us recall that the angular momentum balance law gives the following symmetry relation: FD PT = P FTD

(11.248)

whereby we get FD PT = P FTD = c,i ⊗Ai ∙ Aj ⊗ Pj = c,i ⊗Pi as symmetric, and by the above supporting fact, we conclude: c,i × Pi = 0 ⇒ c, 𝛼 × P𝛼 = −c, 𝜂 × P3 = −g3 × P3

(11.249)

Computation of Nonlinear Structures

806

So, from equation (11.249), we get for the last term of equation (11.247): ω ∙ (g3 × P3 ) = − ω ∙ (c, 𝛼 × P𝛼 ) { ( )} = −P𝛼 ∙ ω × c0 , 𝛼 + ω × ktot × (c − c0 ) 𝛼

(11.250)

Inserting equation (11.250) in equation (11.247), we get: ∙

P : FD = P𝛼 ∙ + P𝛼

{ {

∙

}

∙

+ (c − c0 ) × P𝛼 ∙ ktot 𝛼 ( )} × (ω × (c − c0 ) ) − ω × ktot × (c − c0 ) 𝛼

c0 , 𝛼 − ω × c0 , 𝛼 ktot 𝛼

(11.251)

Now, let us note the following supporting vector identity. Fact 2: For any three vectors, a × (b × c) − b × (a × c) = (a × b) × c. Proof. Let Z ≡ (b ⊗ a) be a tensor, then from the previous supporting fact, we have for the skewsymmetric part: Z − ZT ≡ (b ⊗ a) − (a ⊗ b) = [a × b] = (a × b)×. Now, by the vector identity: a × (b × c) = (a ∙ c)b − (a ∙ b)c, we complete the proof with: a × (b × c) − b × (a × c) = (a ∙ c)b − (b ∙ c)a = {(b ⊗ a) − (a ⊗ b)}c = {Z − ZT }c = (a × b) × c. So, applying this supporting fact, we get for the last term of equation (11.251) as: { ( tot )} P𝛼 ∙ ktot k𝛼 × (c − c0 ) 𝛼 × (ω × (c − c0 ) ) − ω × {( tot ) } ( ) = P𝛼 ∙ k𝛼 × ω × (c − c0 ) = −(c − c0 ) × P𝛼 ∙ ω × ktot 𝛼

(11.252)

Now, inserting equation (11.252) in equation (11.251), we finally get, with help of equations (11.235) and (11.239), the expression for the stress or deformation power density as: ∙

P : FD =

P𝛼

{ ∙ } } {∙ ( ) 𝛼 tot tot ∙ c0 , 𝛼 − ω × c0 , 𝛼 + (c − c0 ) × P k𝛼 − ω × k𝛼 t

t

t

t ⏞⏞⏞ ⏞⏞⏞ ⏞⏞⏞ ⏞⏞⏞ R 𝛼 𝛼 ∙ Rβ 𝛼 ∙ = P +(c − c ) × P χ𝛼 = P ∙ c0 , 𝛼 +(c − c0 ) × P𝛼 ∙ ktot 𝛼 0 𝛼

(11.253) Now, incorporating the definition of F𝛼 and M𝛼 as in equations (11.226) and (11.227), respectively, we get, using equation (11.253), the expression for the internal stored power as: t

PInternal

t

⎧ ⏞⏞⏞ ⏞⏞⏞ ⎫ ⎪ 𝛼 ⎪ R 𝛼 (F ∙ ≡ β𝛼 + M ∙ R χ𝛼 ⎬ dA ∫ ⎨ ⎪ ⎪ Ω ⎩ ⎭

(11.254)

Which is exactly the same as that given by equation (11.241). Another compact way of finding the relation given by equation (11.253) is to consider the shell motion in terms of the configuration space; this is what we will do next.

Nonlinear Shell

807

Figure 11.19 Shell deformation map.

11.8.3.2

Deformation Gradient as the Tangent Map

We may choose to derive equation (11.253) by using the deformation gradient as the tangent map on the configuration manifold. A motion may be viewed as a curve on the configuration space passing through the undeformed state as well as other and current deformation states. We can then view the deformation gradient which carries local information as the tangent map that relates the tangent to the curve at the undeformed state to that at the current deformed state. In other words, If we denote C(S1 , S2 , 𝜂) = 𝜗0 (S1 , S2 , 𝜂) and c(S1 , S2 , 𝜂) = 𝜗(S1 , S2 , 𝜂) as the values of the functions 𝜗0 and 𝜗 as shown in Figure 11.19, we get 𝜒 the deformation map from the undeformed shell-like body to the current deformed body as: 𝜒 ≡ 𝜗◦𝜗−1 0

(11.255)

( )−1 Noting that, ∇𝜗 = gi ⊗ Ei , we have: gi = (∇𝜗) Ei = (∇𝜗) ∇𝜗0 Gi = F Gi where we have used the definition of the deformation gradient as: F ≡ gi ⊗ Gi . With the deformation map given by equation (11.243), we have: ∇𝜗 = 𝜗, 𝛼 ⊗ E𝛼 + 𝜗, 𝜂 ⊗ E3 = {c0 , 𝛼 + 𝜂 g3 , 𝛼 } ⊗ E𝛼 + g3 ⊗ E3 . The tangent map of the deformation map given by equation (11.255) is: F ≡ (∇𝜗)(∇𝜗0 )−1

(11.256)

Then, from equation (11.256), by taking the time derivative, we have: ∙ ∙ ⎫ ⎧⎛ ∙ ⏞⏞⏞ ⏞⏞⏞⎞ ∙ ⎪⎜⏞⏞⏞ 𝛼 3⎪ ⎟ F = ∇𝜗 (∇𝜗0 ) = ⎨ c0 , 𝛼 +𝜂 g3 , 𝛼 ⊗ E + g3 ⊗ E ⎬ (∇𝜗0 )−1 ⎟ ⎪ ⎪⎜⎝ ⎠ ⎭ ⎩ ∙

(11.257)

Now, noting that Gi = (∇𝜗0 )Ei ⇒ (∇𝜗0 )−1 Gi = Ei ⇒ (∇𝜗0 )−1 = Ei ⊗ Gi ⇒ (∇𝜗0 )−T = Gi ⊗ Ei , and that A:B = tr(A BT )where tr stands for the trace of a tensor, by definition, we have with

Computation of Nonlinear Structures

808

equation (11.257):

∙

⎧ ⏞⏞⏞⎫ ) }} { ( { ⎪ ⎪ −T (∇𝜗)T ⎬ = tr P (Gi ⊗ Ei ) ∙ E𝛼 ⊗ c0 , 𝛼 + 𝜂 g3 , 𝛼 + E3 ⊗ g3 P: F = tr ⎨P (∇𝜗0 ) ⎪ ⎪ ⎩ ⎭ ∙ ∙ ∙ ∙ ⎧⏞⏞⏞⏞⏞⏞⏞ ⎫ ⎛⏞⏞⏞ ⏞⏞⏞⎞ ⏞⏞⏞ ∙ ⎪ ⎪ 𝛼 ⎜ 3 tot 𝛼 𝛼 ⎟ = PG ∙ c0 , 𝛼 +𝜂 g3 , 𝛼 + PG ∙ g3 = P ∙ c0 , 𝛼 +P ∙ 𝜂 ⎨k𝛼 × g3 ⎬ + P3 ∙ (ω × g3 ) ⎜ ⎟ ⎪ ⎪ ⎝ ⎠ ⎩ ⎭ ∙

(11.258)

where we have used the tensor trace property: tr(a ⊗ b) = a ∙ b, and equations (11.229). Now, ∙ ∙ ∙ ∙ ⎧⏞⏞⏞⏞⏞⏞⏞⏞⏞⎫ ⏞⏞⏞ ⏞⏞⏞ ⏞ ⏞ ⏞ ⎪ ⎪ tot noting that: 𝜂 ⎨ktot g3 = ktot × 𝜂 g3 + ktot × (c − c0 ) + ktot 𝛼 × g3 ⎬ = k𝛼 𝛼 𝛼 × 𝜂 𝛼 × ⎪ ⎪ ⎩ ⎭ (ω × (c − c0 )), and using equation (11.250) for the third term, we get from equation (11.258):

∙

∙ ⎧⏞⏞⏞⏞⏞⏞⏞⏞⏞⎫ ⏞⏞⏞ ∙ ⎪ ⎪ 𝛼 𝛼 3 P : F = P ∙ c0 , 𝛼 + P ∙ 𝜂 ⎨ktot 𝛼 × g3 ⎬ + P ∙ (ω × g3 ) ⎪ ⎪ ⎭ ⎩ ∙ ∙ ⏞⏞⏞ ⏞⏞⏞ +P𝛼 ∙ ktot = P𝛼 ∙ c0 , 𝛼 + (c − c0 ) × P𝛼 ∙ ktot 𝛼 𝛼 × (ω × (c − c0 )) ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

(11.259)

2nd term

{ } + P𝛼 ∙ −ω × c0 , 𝛼 − ω × (ktot 𝛼 × (c − c0 )) ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ 3rd term

Finally, using Fact 1 for the third term in equation (11.259), we get:

∙

P : F = P𝛼

{

∙

c0 , 𝛼 − ω × c0 , 𝛼

}

{ + (c − c0 ) × P𝛼 ∙ t

} ∙ ( ) tot ktot − ω × k 𝛼 𝛼 t

t

t ⏞⏞⏞ ⏞⏞⏞ ⏞⏞⏞ ⏞⏞⏞ tot R 𝛼 𝛼 𝛼 𝛼 β𝛼 +(c − c0 ) × P ∙ R χ𝛼 = P ∙ c0 , 𝛼 +(c − c0 ) × P ∙ k𝛼 = P ∙

(11.260)

Clearly, the expressions in equation (11.253) and equation (11.260) are identical.

Nonlinear Shell

11.8.3.3

809

Reduced Internal Energy (Deformation or Stress Power): Conjugate Stresses and Strains

Using equation (11.253) or equation (11.260), we can identify the deformation or stress power, ∙

U , of a 2D shell reference surface as the reduced 3D stress power, Pinternal , of the corresponding shell-like body as: ( Pinternal =

𝜂t

(

𝜂t

M

( ∫ M

) 𝜂 S

𝜇 d𝜂

dA

) ⎧ ∙ ⎫ ⎪ ⎪⏞⏞⏞ 𝜇 d𝜂 ∙ ⎨ c0 , 𝛼 − ω × c0 , 𝛼 ⎬ dA ⎪ ⎪ ⎭ ⎩

𝛼 𝜂 S

P

∫𝜂 b

∫

+

P: F

∫𝜂 b

∫ M

=

∙

𝜂t

∫𝜂 b

∙ ) ⎧⏞⏞ ⎫ ⏞ ( ) ⎪ 𝛼 𝜂 S tot tot ⎪ (c − c0 ) × P 𝜇 d𝜂 ∙ ⎨ k𝛼 − ω × k𝛼 ⎬ dA ⎪ ⎪ ⎩ ⎭ t

(11.261)

t

⎧ ⏞⏞⏞ ⏞⏞⏞⎫ ∙ ( ) ⎪ 𝛼 ⎪ R 𝛼 F ∙ β𝛼 +M ∙ R χ𝛼 ⎬ dA = U R β𝛼 , R χ𝛼 = ∫ ⎨ ⎪ ⎪ M ⎩ ⎭ where we have inserted the definitions as given in equations (11.226) and (11.227) for the stress resultant and the stress couple, respectively, and in equations (11.236) and (11.240) for the rotated translational and bending strains, respectively. Remark

r Equation

(11.261) establishes the existence of an equivalent 2D strain energy functional, ( ) U R β𝛼 , R χ𝛼 , under conservative loading, or at least the existence of the strain energy functional as functions of the 2D shell reference surface strains only. With this knowledge we never need to look back to the originating 3D shell-like body again, when we develop the appropriate constitutive theory for the shell reference surface as will become apparent soon.

11.8.4

Where We Would Like to Go

In conformity with our principal interest in computational structural engineering and structural mechanics, we need to bring every shell equation down to a form where we can easily solve the shell problems numerically by the use of mega-shell elements of our finite element scheme, the c-type finite element method. This in turn, requires exact derivation of the compatible real strain fields in component or operational form along the configuration path from our knowledge of the above virtual strain fields; this endeavor will again validate our belief that the presumed virtual strains are truly virtual, that is, these can be shown to exist as variations of some real strain fields. Thus, before anything else, the expressions for the virtual strain fields in the internal virtual work

Computation of Nonlinear Structures

810

expression of equation (11.225) must all be modified to the component vector form. For this, we must next look at:

r component or operational vector form. 11.9 Component or Operational Vector Form 11.9.1 What We Need to Recall Our stated goal is achieved by use of the curvilinear description of a geometric surface embedded in a general 3D space, but first there are various facts to recall.

r For the undeformed coordinate system corresponding to the lines of curvature along and the r

r

normal to the shell reference surface with base vectors, {G1 , G2 , G3 ≡ N}, being an orthonormal system of coordinates, both the sets of the contravariant base vectors, {Gi (S1 , S2 )}, and the covariant base vectors, {Gi (S1 , S2 )}, are one and the same. It is assumed that the current deformation of a shell is completely characterized by the displacement vector field: d(S1 , S2 , t), and a rotation tensor field: R(S1 , S2 , t), both measured in the undeformed coordinate system, {G1 , G2 , G3 ≡ N} on the shell reference surface at time t ∈ R+ . However, we have another curvilinear co-spinning coordinate system (rotating synchronously with the rotation tensor, R(S1 , S2 , t)) and denoted as the normal coordinate system, with base vectors, {n G1 ,n G2 , N} in the undeformed configuration that becomes {n g1 ,n g2 , n} in the current deformed state at any point {S1 , S2 } ∈ , the reference surface of a shell at time t ∈ R+ . The undeformed and deformed base vectors have the relation: n = RN, n g𝛼 = R n G𝛼 , 𝛼 = 1, 2. In compact tensor notation, we can rewrite these relations as: n gi = R Gi , i = 1, 2, 3. Moreover, in the undeformed configuration of the shell, {n G1 ,n G2 , N} coincide identically with {G1 , G2 , G3 ≡ N}, respectively, The rotation matrix, t R, measured in undeformed coordinate system, and the Rodrigues vector, s ≡ { s1 s2 s3 }T , are related by: ⎡ 1 + s2 − s2 − s2 1 2 3 t R = 1 ⎢ 2(s s + s ) 1 2 3 𝜌 ⎢⎢ 2(s s − s 3 1 2) ⎣

2(s1 s2 − s3 ) 1 − s21

+ s22

− s23

2(s2 s3 + s1 )

2(s3 s1 + s2 ) ⎤ ⎥ 2(s2 s3 − s1 ) ⎥ 1 − s21 − s22 + s23 ⎥⎦

(11.262)

with 𝜌 ≡ 1 + s21 + s22 + s23

r The rotation vector, θ, measured in the reference coordinate system, and the Rodrigues vector, s, are related such that: s=

r

( ) 1 𝜃 θ tan 𝜃 2

(11.263)

√ where 𝜃 ≡ ‖θ‖ = (θ ∙ θ) is the magnitude of the rotation vector. The angular velocity vector, ω, measured in the reference coordinate system is given by: ∙

ω ≡ W(θ) θ

(11.264)

Nonlinear Shell

811

where W(θ) matrix is given by W(θ) ≡ I + c1 𝚯 + c2 𝚯2 , 1 − cos 𝜃 , 𝜃2 𝜃 − sin 𝜃 c2 (θ) = 𝜃3 c1 (θ) =

(11.265)

r In the tensor notation, for the rotating normal coordinate system, {n G1 ,n G2 , N}, in the unde-

formed configuration that becomes {n g1 ,n g2 , n}in the current deformed state at any point {S1 , S2 } ∈ , the reference surface of a shell, being orthonormal systems of coordinates, both the set of the contravariant base vectors, say, {Gi } for the undeformed and {gi } for the deformed states, and the covariant base vectors, say, {Gi } for the undeformed and {gi } for the deformed states, correspondingly, are one and the same.

11.9.2

Relevant Operational Matrix Representations

For our immediate use, we will represent the base vector sets in operational matrix form as: G ≡ [ G1

G2

G3 ≡ N ],

g ≡ [ n g1

ng 2

n]

(11.266)

where GT G = gT g = I with I as the 3 × 3 identity matrix. Now, the component representation of the rotation tensor, R = t Rij Gi ⊗ Gj , can be given as the matrix, t R, as: t

R = GT R G, ⇔ R = G t R GT

(11.267)

where ijth -th component of t R is given by t Rij . Finally, given the relations: n gi = R Gi , i = 1, 2, 3, we can use equations (11.266) and (11.267) to express these relation in operational matrix form as: G ≡ [ G1

G2

G3 ≡ N ],

g ≡ [ n g1 n g2 n ] = [ R G1 R G2 = R G = G t R GT G = G t R

11.9.3

R G3 ≡ R N ]

(11.268)

Component or Operational Vectors

As indicated in our stated goal, and as shown in Figure 11.20, we define here the component vectors by first identifying the coefficients (i.e. the components) of linear combinations and the shell base vectors that make up the corresponding invariant vectors, and then collect the components in an operational vector form.

11.9.3.1

Component Displacement and Rotation Vectors: t d and t θ

In terms of the linear combinations of the parameters measured in the undeformed tangent coordinate system, {G1 , G2 , G3 ≡ N}, the displacement vector, d, an invariant vector measuring

Computation of Nonlinear Structures

812

Figure 11.20 Displacement and rotation components.

the movement of a point on the undeformed reference surface to its current position on the deformed reference surface, can be written as: d = t d1 (≡ u)G1 + t d2 (≡ v)G2 + t d3 (≡ w)N = t di Gi

(11.269)

and similarly, the rotation vector, θ, as: θ = ( t 𝜃1 ≡ 𝜃1 )G1 + ( t 𝜃2 ≡ 𝜃2 ) G2 + ( t 𝜃3 ≡ 𝜃3 ) N = t 𝜃i Gi

(11.270)

where G ≡ [ G1 G2 G3 ] is the orthonormal matrix with the base vectors as its columns. Note that, although it is immaterial for the orthogonal frames, for the sake of consistency with the description in the curvilinear coordinates, the displacement and rotation components are shown as covariant components. We collect the components of d and θ, and define the component vectors t d and t θ, respectively, as follows, Definition: The component or operational vectors: t d and t θ t

t

d≡

θ≡

{

{

td 1

t𝜃

1

≡u

≡ 𝜃1

td 2

≡v

td 3

t𝜃

≡ 𝜃2

t𝜃

2

≡w

3

}T

≡ 𝜃3

}T

(11.271)

(11.272)

Note that the superscript t reminds us that the components are measured in the undeformed tangent coordinate system. With the above definitions, we have: d = G td θ = G tθ

(11.273)

Nonlinear Shell

11.9.3.2

813 ∙

Component Linear Velocity Vectors: t v ≡ t d ∙

Following the same ideas as above, the linear velocity, t v ≡ t d, can be written as: ∙

∗

∙

∙

∙

∙

∙

∙

v ≡ d = t d1 (≡ u)G1 + t d2 (≡ v)G2 + t d3 (≡ w)N = t di Gi ∗

(11.274)

∗

We collect the components of v ≡ d, and define the component vector t v ≡ t d, as: ∙

Definition: The component or operational vectors: t d t

∗

v≡ td≡

{ t

∙

∙

d1 ≡ u

t

∙

∙

d2 ≡ v

t

∙

∙

}T

d3 ≡ w

(11.275)

With the above definitions, we have: ∙

∗

v≡d=Gtd

11.9.3.3

(11.276)

Component Angular Velocity Vectors: t ω

The angular velocity, t ω, can be written as: ω = t 𝜔1 G1 + t 𝜔2 G2 + t 𝜔3 N = t 𝜔i Gi = G t ω

(11.277)

We collect the components of t ω, and define the component vector t ω, as: Definition: The component or operational vectors: t ω t

ω≡

{

t𝜔

1

t𝜔

2

t𝜔

}T

(11.278)

3

With the above definitions and equation (11.264), we have: t

11.9.3.4

∙

∙

ω ≡ G t W GT G t θ = G t W t θ

(11.279)

Component Force and Moment Vectors in Undeformed Frame: t F𝜶 and t M𝜶

In terms of the linear combinations of parameters measured in the undeformed tangent coordinate system, {G1 , G2 , G3 ≡ N}, the force vector, F𝛼 , an invariant vector measuring the current stress resultant on an 𝛼 = const. cross-section of the deformed shell, and the moment vector, M𝛼 , an invariant vector measuring the current stress couple resultant on an 𝛼 = const. cross-section of a deformed shell, can be written, respectively, as: F𝛼 = t F 𝛼1 (≡ N𝛼1 ) G1 + t F 𝛼2 (≡ N𝛼2 ) G2 + t F 𝛼3 (≡ Q𝛼 ) N = t F 𝛼i Gi ,

𝛼 = 1, 2;

i = 1, 2, 3

(11.280)

Computation of Nonlinear Structures

814

where, N11 , N22 are known as the in-plane extensional forces, N12 , N21 as the in-plane shear forces, and Q13 , Q23 are the transverse shear forces; conventionally and notation-wise, these are described as such in engineering terminology, and M1 = t M 11 (≡ −M12 ) G1 + t M 12 (≡ M1 ) G2 + t M 13 (≡ M13 ) N = t M 1i Gi , M2 M𝛼

i = 1, 2, 3

=

t M 21 (≡

=

t M 2i G , i t M 𝛼i G , i

=

−M2 ) G1 + t M 22 (≡ M21 ) G2 + t M 23 (≡ M23 ) N

(11.281)

i = 1, 2, 3 𝛼 = 1, 2,

i = 1, 2, 3

where, M1 , M2 are known as the bending moments, M12 , M21 as the in-plane twisting moments, and M𝛼3 are the transverse twisting moments on a shell cross-section with the normal to it pointing in the 𝛼 direction, 𝛼 = 1, 2 (for the classical shell theory, these components are identically zero); conventionally and notation-wise, these are described as such in engineering terminology. Remarks

r Notationally, the first indices of the shell consistent components indicate the directions of the r

normals to the cross-section, and the second indices point to the directions of the forces and moments. In engineering, the stress couples are expressed in the same way as the associated stresses. However, we must reiterate that the shell consistent moment components, t M 𝛼i , 𝛼 = 1, 2, i = 1, 2, 3, in the undeformed tangent coordinate system, {G1 , G2 , G3 ≡ N}, are different from their counterparts in engineering. Of course, these can be easily related by the following map that uses the 2D permutation matrix, e𝛼𝛽 : t

[

M 𝛼𝛽 = e𝜂𝛽 M 𝛼𝜂 ,

(e𝜂𝛽 ) =

0

1

−1

0

] (11.282)

We collect the components of F𝛼 and M𝛼 , and define component vectors, respectively, as follows. Definition: t F1

≡

t F2

≡

t M1

≡

t M2

≡

{t {t

{t {t

F 11 ≡ N1 F 21 ≡ N21

M 11 ≡ −M12 M 21 ≡ −M2

t F 12

≡ N12

t F 13

≡ Q1

≡ N2

t F 23

≡ Q2

t F 22

}T

≡ M1

t M 13

≡ M13

≡ M21

t M 23

≡ M23

t M 12 t M 22

}T (11.283) }T }T

(11.284)

With the above definitions, we have: F𝛼 = G t F𝛼 M𝛼 = G t M𝛼 ,

𝛼 = 1, 2

(11.285)

Nonlinear Shell

815

where G ≡ [ G1 G2 G3 ] is the orthonormal matrix with base vectors as its columns. Note that although it is immaterial for orthogonal frames, for the sake of consistency with the description in the curvilinear coordinate system, the force and moment components are shown as the contravariant components.

11.9.3.5

Component Force and Moment Vectors in Terms of Deformed Frame: n F𝜶 and n M𝜶

In terms of the parameters measured in the current normal coordinate system, {n g1 ,n g2 , n}, let the current stress resultants (or forces): n F𝛼 = n F 𝛼1 n g1 + n F 𝛼2 n g2 + n F 𝛼3 n g3 = n F 𝛼i n gi and the current resultant stress couple (or moments): n M𝛼 = n M 𝛼1 n g1 + n M 𝛼2 n g2 + n M 𝛼3 n g3 = n M 𝛼i n g . i We collect the components of F and M, and define component vectors, respectively, as follows. Definition: n 𝛼

F (S1 , S2 ) ≡

n

11.9.4

M𝛼 (S1 , S2 ) ≡

{

{

n F 𝛼1 (S1 , S2 )

n F 𝛼2 (S1 , S2 )

n M 𝛼1 (S1 , S2 )

n M 𝛼2 (S1 , S2 )

} n F 𝛼3 (S1 , S2 ) T

(11.286)

} n M 𝛼3 (S1 , S2 ) T

(11.287)

Relations Between: t F𝛼 and n F𝛼 , and t M𝛼 and n M𝛼 ,

𝛼 = 1, 2

Considering that the {n g1 ,n g2 , n} frame in the deformed shell reference surface, is defined by the orthogonal rotation tensor, R(S1 , S2 ), measured in the {G1 , G2 , G3 ≡ N} frame of the undeformed shell, the two sets of the stress resultants and stress couples component vectors are related by the following relations, for 𝛼 = 1, 2: t F𝛼 t M𝛼

= G t F𝛼 = g n F𝛼 = G t R n F𝛼 = G t M𝛼 = g n M𝛼 = G t R n M𝛼

(11.288)

where we have used equation (11.268). From equation (11.288), we can finally write the following relationships relating the components of forces and moments in undeformed and current spinning frames as: t F𝛼

= t R n F𝛼 t M𝛼 = t R n M𝛼 ,

11.9.4.1

𝛼 = 1, 2

(11.289)

Components of Linear Momentum and Angular Momentum Vectors in Terms of the Undeformed Frame: t L(S1 , S2 , t) and t A(S1 , S2 , t)

The invariant linear momentum vector, L(S1 , S2 , t), and the angular momentum vector, A(S1 , S2 , t), for any t ∈ R+ , can have component representation in any triad of base vectors, Cartesian or curvilinear.

Computation of Nonlinear Structures

816

Thus, we can write: L(S1 , S2 , t) = t L1 (≡ P1 ) G1 + t L2 (≡ P2 ) G2 + t L3 (≡ PN ) G3 = t Li (S1 , S2 , t) Gi (S1 , S2 ),

i = 1, 2, 3

(11.290)

and A(S1 , S2 , t) = = t A1 (≡ A1 ) G1 + t A2 (≡ A2 ) G2 + t A3 (≡ AN ) G3 = t Ai (S1 , S2 , t) Gi (S1 , S2 ),

i = 1, 2, 3

(11.291)

We collect the components of L and A, and define the component vectors, respectively, as follows. Definitions t L(S1 , S2 , t)

≡

t A(S1 , S2 , t)

≡

{t {t

L1

t L2

t L3

≡ LN

A1

t A2

t A3

≡ AN

}T }T

(11.292)

With the above definitions, we can express these in the column vector form as: L(S1 , S2 , t) = G(S1 , S2 ) t L(S1 , S2 , t) A(S1 , S2 , t) = G(S1 , S2 ) t A(S1 , S2 , t)

(11.293)

Likewise, we have the time derivatives of the linear momentum and the angular momentum in terms of their material component vectors: ∙

∙

∙

∙

L(S1 , S2 , t) = G(S1 , S2 ) t L(S1 , S2 , t) A(S1 , S2 , t) = G(S1 , S2 ) t A(S1 , S2 , t)

11.9.5

(11.294)

Where We Would Like to Go

r We have defined the component or operational vectors corresponding to the displacement, rotation and velocity vectors, and the force and moment vectors, and so on. These definitions play a vital role in converting the invariant vector form of the shell balance equations: 3D to 2D, or the shell balance equations: direct 2D to computationally more suitable component vector or operational form of balance equations. However, to understand this conversion fully, we would need and investigate next: r derivatives of component vectors r computational curvatures, angular velocity and variations

Nonlinear Shell

817

11.10 Covariant Derivatives of Component Vectors 11.10.1 What We Need to Recall

r The Gauss–Weingarten relation for the derivatives of the triad of base vectors in matrix form with respect to the arc length parameters, S𝛼 ,

𝛼 = 1, 2, is given by:

1 𝛾 , 𝛾1 1 2

⎡ 0 ⎧ G1 ,1 ⎫ ⎧ G1 ⎫ ⎢ ⎪ ⎢ 1 ⎪ ⎪ 𝜕 ⎪ ⎨ G2 ,1 ⎬ ≡ 1 ⎨ G2 ⎬ = ⎢⎢ − 𝛾 𝛾1 ,2 ⎪ N, ⎪ 𝜕S ⎪ N ⎪ ⎢ 1 ⎭ ⎩ 1 ⎭ ⎩ 1 ⎢ ⎣ R1

−

0 0

1 ⎤ R1 ⎥ ⎧ ⎥ ⎪ G1 0 ⎥ ⎨ G2 ⎥⎪ ⎥⎩ N 0 ⎥⎦

⎫ ⎧ G1 ⎫ ⎪ ⌢0 ⎪ ⎪ ⎬ ≡ K1 ⎨ G2 ⎬ ⎪ ⎪N⎪ ⎭ ⎭ ⎩

(11.295)

and ⎡ 0 ⎧ G1 ,2 ⎫ ⎧ G1 ⎫ ⎢ ⎪ ⎢⎢ 1 ⎪ ⎪ 𝜕 ⎪ 𝛾 , G , G = ≡ ⎨ 2 2⎬ 2 ⎨ 2⎬ 𝛾 21 ⎪ N, ⎪ 𝜕S ⎪ N ⎪ ⎢⎢ 2 ⎭ ⎩ 2 ⎭ ⎩ ⎢ 0 ⎣

−

1 𝛾 , 𝛾2 2 1 0 1 R2

0 ⎤ ⎥⎧ ⎧ G1 ⎫ G1 ⎫ ⎪ ⌢0 ⎪ ⎪ 1 ⎥⎥ ⎪ − G2 ⎬ ≡ K2 ⎨ G2 ⎬ ⎨ R2 ⎥ ⎪ ⎪N⎪ ⎥⎪ ⎭ ⎩ ⎩N⎭ ⎥ 0 ⎦

(11.296)

⌢

where K0𝛼 , 𝛼 = 1, 2 are the shell undeformed curvature matrices with R𝛼 (S1 , S2 ), 𝛼 = 1, 2, are the radii of curvatures at any point, {S𝛼 }, of the shell reference √ surface, and 𝛾𝛼 , 𝛼 = dS𝛼 ̄ 𝛼‖ = G ̄ 𝛼 (𝜉 1 , 𝜉 2 ) ≡ ̄ 𝛼𝛼 where G 1, 2, are the lengths of the base vectors: 𝛾𝛼 ≡ 𝛼 = ‖G d𝜉 𝜕C(𝜉 1 , 𝜉 2 , 0) , with no sum on 𝛼. Note that the curvature matrices are skew-symmetric. With 𝜕𝜉 𝛼 the notation for the unit base vectors: G1 , G2 , G3 ≡ N , we can rewrite the matrix relations in compact notation as: ⌢

Gi , 𝛼 = (K 0𝛼 )i j Gj ,

r

𝛼 = 1, 2; i = 1, 2, 3;

j = 1, 2, 3

(11.297)

Clearly, these matrix forms give the derivatives of a base vector as a linear combination of all the base vectors. The individual coefficients of the matrices can be easily identified with the Christoffel symbols of the second kind, Γijk . However, in our entire shell analysis, we will completely avoid explicit use of these obscuring symbols by adhering to familiar geometric entities such as the radii of curvature, and so on, as we have done above, for ease of interpretation and clarity. On the other hand, the Gauss–Weingarten relation for the derivatives of the triad of base vectors on a surface in invariant tensor form with respect to the arc length parameters, S𝛼 , 𝛼 = 1, 2, may be obtained by first introducing the following.

Definition: Shell initial curvature tensors, K0𝛼 , 𝛼 = 1, 2: ⌢

K0𝛼 ≡ (K0𝛼 )Tij Gi ⊗ Gj = −Gi ⊗ Gi , 𝛼 ,

𝛼 = 1, 2;

i, j = 1, 2, 3

(11.298)

Computation of Nonlinear Structures

818

To wit, for the second equality in equation (11.148), we note that because of anti-symmetry: ⌢

⌢

(K0𝛼 )Tij = −(K0𝛼 )ij . Then, we have: ⌢

⌢

K0𝛼 ≡ (K0𝛼 )Tij Gi ⊗ Gj = −(K0𝛼 )ij Gi ⊗ Gj ⌢

= − Gi ⊗ (K0𝛼 )ij Gj = −Gi ⊗ Gi , 𝛼 ,

𝛼 = 1, 2;

i, j = 1, 2, 3

where we have used the result from equation (11.297). Note that the curvature tensor is skewsymmetric and thus: (K0𝛼 )T = −K0𝛼 . This can be easily verified: ⌢

⌢

⌢

(K0𝛼 )T ≡ (K0𝛼 )Tij Gj ⊗ Gi = (K0𝛼 )Tji Gi ⊗ Gj = − (K0𝛼 )Tij Gi ⊗ Gj = −K0𝛼 ,

𝛼 = 1, 2;

Definition: Shell initial curvature vector, k0𝛼 , 𝛼 = 1, 2: The axial vectors corresponding to the initial curvature tensors, K0𝛼 , 𝛼 = 1, 2, are given by: {

1 1 − 𝛾1 ,2 ≡ 0 − R1 𝛾1 { }T 1 1 0 𝛾2 ,1 k02 ≡ R2 𝛾2

}T

k01

(11.299)

Now, we can easily get from equation (11.148), the invariant tensor form as: Gi , 𝛼 = K0𝛼 Gi = k0𝛼 × Gi ,

𝛼 = 1, 2; i = 1, 2, 3

(11.300)

Note that, in tensor form, unlike the matrix expression (11.297), each unit base vector derivative is given as a transformation of itself. As we shall see, this allows us to write expressions in absolute compact form free of matrix indices. Exercise: Verify the tensor form of the derivatives of base vectors given by equation (11.150). Solution:

K0𝛼 Gi = −(K0𝛼 )T Gi = −(−Gi , 𝛼 ⊗ Gi )Gi = Gi , 𝛼 ,

𝛼 = 1, 2; i = 1, 2, 3

r Finally, the covariant derivative of a base vector is the zero vector, that is, Gi |𝛼 = 0,

i= 1, 2, 3. Thus, the covariant derivative of a tensor function of any order is identical with its partial derivative. If we take a vector (a tensor of order 1) function, a(S𝛼 ), we then have: a|S𝛼 ≡ a|𝛼 = a, 𝛼 = t ai |𝛼 Gi = G t a|𝛼 where G ≡ [ G1 G2 G3 ] is the orthonormal matrix with base vectors as its columns, and t a(S𝛼 ) is the component vector corresponding to the vector function, a(S𝛼 ).

Exercise: Show that the partial derivative of a tensor function of any order is identical with its covariant derivative with respect to its arguments. Hint: For vector function Gi |𝛼 = t di |𝛼 Gi = d, 𝛼 . Why do you think Gi |𝛼 = 0? Because, by defid|𝛼 = t di |𝛼 Gi + t di  k nition, Gi |j = Gi ,j − Γij Gk = Γkij Gk − Γkij Gk = 0.

Nonlinear Shell

11.10.2

819

Covariant Derivatives of the Component or Operational Vectors

Now, with the above results in the background, we are in a position to express the derivatives of various component vectors with the help of the following lemma for a shell. Lemma: Given any vector function, a(S𝛼 ), on a shell reference surface, and its corresponding component vector, t a(S𝛼 ), the covariant derivative of the component vector is given as: t a| 𝛼 t a| 𝛼

= t a, 𝛼 + k0𝛼 × t a, =

t a,

𝛼

+

𝛼 = 1, 2

K0𝛼 t a

(11.301)

where × denotes vector cross product operation. Proof. Using results from above, we have: ( ) a|𝛼 = a, 𝛼 = t ai , 𝛼 Gi + t ai Gi , 𝛼 = G t a, 𝛼 + K0𝛼 t a ) (t = G a, 𝛼 + k0𝛼 × t a

(11.302)

Now, noting that a|𝛼 = a, 𝛼 = G t a|𝛼 completes the proof. 11.10.2.1

Covariant Derivatives of Component Displacement and Rotation Vectors: t d and t θ

Based on equation (11.301) of the lemma presented, we have: t d| 𝛼 t θ| 𝛼

11.10.2.2

= t d, 𝛼 + k0𝛼 × t d, =

t θ,

𝛼

+

k0𝛼

×

𝛼 = 1, 2

tθ

(11.303)

Covariant Derivatives of the Component Force and Moment Vectors: t F𝜶 and t M𝜶

Based on equation (11.301) of the lemma presented, we have: t F𝛼 | = t F𝛼 , + k0 × t F𝛼 , 𝛽 𝛽 𝛽 t M𝛼 | = t M𝛼 , + k0 × t M𝛼 𝛽 𝛽 𝛽

11.10.3

𝛼, 𝛽 = 1, 2

(11.304)

Where We Would Like to Go

We have defined the covariant derivatives of the component or operational vectors corresponding to the displacement and rotation vectors, and the force and moment vectors. These definitions play a vital role in converting the vector form of:

r the shell balance equations: 3D to 2D r the shell 2D direct balance equations

Computation of Nonlinear Structures

820

to computationally more suitable component vector form of balance equations. Thus, with these background materials, we will be able to modify the equations in terms of what may be called the component vector or operator form as described in:

r the computational balance equations. 11.11 Computational Equations of Motion: Component Vector Form 11.11.1 What We Need to Recall Let us recall the following for a thin shell.

r The invariant vector shell dynamic balance equations are given by: F𝛼 , 𝛼 + F̂ = L, ∙

∙

M𝛼 , 𝛼 + c0 , 𝛼

𝛼 = 1, 2

∙ M0 v0

L= ∙ ̂ = A, × F𝛼 + M

Momentum balance ( ) ∙∙ = M0 d ≈ 𝜌0 H d ∙∙

𝛼 = 1, 2

Moment of momentum balance ( ) ∙ ∙ ∙ H3 1 2 1 2 1 2 ω(S1 , S2 , t) A(S , S , t) = I0 (S , S ) ω(S , S , t) ≈ 𝜌0 12 (11.305)

r We have for the component or operational vectors: t d and t θ t

d ≡ { d1 ≡ u d2 ≡ v t

θ ≡ { 𝜃1

𝜃2

d3 ≡ w }T 𝜃3 }T

(11.306) (11.307)

with: d = G td θ = G tθ

(11.308)

r Similarly, for the forces and moments, we have: F𝛼 = t F 𝛼1 (≡ N𝛼1 ) G1 + t F 𝛼2 (≡ N𝛼2 ) G2 + t F 𝛼3 (≡ Q𝛼 ) N = t F 𝛼i Gi , 𝛼 = 1, 2; i = 1, 2, 3

(11.309)

M𝛼 = t M 𝛼1 (≡ −M𝛼2 ) G1 + t M 𝛼2 (≡ M𝛼1 ) G2 + t M 𝛼3 (≡ M𝛼3 ) N = t M 𝛼i Gi , 𝛼 = 1, 2, i = 1, 2, 3

(11.310)

F𝛼 = G t F𝛼 , 𝛼 = 1, 2 M𝛼 = G t M 𝛼

(11.311)

with:

Nonlinear Shell

821

r k0 ,

𝛼 = 1, 2, are the initial curvature vectors, that is, the axial vectors corresponding to the 𝛼 initial curvature tensors, K0𝛼 , 𝛼 = 1, 2, and are given by: { k01 ≡

0 − {

k02

≡

1 R2

}T { }T 1 𝛾1 ,2 ≡ 𝜏 1 𝜅21 𝜅31 𝛾1 }T { }T 1 𝛾2 ,1 ≡ 𝜅32 𝜏 2 𝜅22 𝛾2

1 R1

−

0

(11.312)

r For the covariant derivative of displacement component vectors, we have: d|𝛼 = d, 𝛼 = G t d|𝛼 ,

𝛼 = 1, 2 with: t d| 𝛼 t d| 𝛼

= t d, 𝛼 + k0 × t d = t d, 𝛼 + K0 t d

(11.313)

Similar results follow for the rotation component vector.

r Likewise, we have the covariant derivatives of forces and moments: F𝛼 , 𝛼 = G t F𝛼 |𝛼

(11.314)

= t F𝛼 , 𝛼 + k0𝛼 × t F𝛼 ,

(11.315)

with: t F𝛼 | 𝛼 t F𝛼 | 𝛼

= t F𝛼 , 𝛼 + K0𝛼 t F𝛼

and M𝛼 , 𝛼 = G t M𝛼 |𝛼

(11.316)

with: t M𝛼 | 𝛼 t M𝛼 | 𝛼

= t M𝛼 , 𝛼 + k0𝛼 × t M𝛼 = t M𝛼 , 𝛼 + K0𝛼 t M𝛼

(11.317)

r Likewise, we have the time derivatives of the linear momentum and the angular momentum in terms of the components measured in the undeformed coordinate system: L(S1 , S2 , t) = G(S1 , S2 ) t L(S1 , S2 , t), ∙

∙∙

∙∙

L = M0 t d ≈ (𝜌0 H) t d A(S1 , S2 , t) = G(S1 , S2 ) t A(S1 , S2 , t), ( ) ∙ ∙ H3 t ∙ t t ω A = I 0 ω ≈ 𝜌0 12 t

(11.318)

Finally, G = [ G1 G2 G3 ] is the 3 × 3 tensor that has a representation as an orthonormal matrix with the base vectors stored as its columns.

Computation of Nonlinear Structures

822

11.11.1.1

Covariant Derivative of the Deformed Position Vector

The derivative of the deformed position vector, c0 , in the second of the balance equation (11.305), can be rewritten as: c0 ||𝛼 = c0 , 𝛼 = C0 , 𝛼 + d, 𝛼 = G𝛼 + G t d|𝛼 = G (1̂ 𝛼 + t d, 𝛼 + k0 × t d) = G (e𝛼 +

k0𝛼

×

𝛼 t d) =

(11.319) 𝛼 = 1, 2

G a𝛼 ,

where we have introduced the following. Definitions 1̂ 1 ≡ { 1 1̂ 2 ≡ { 0

0

0 }T ,

1

0 }T

(11.320)

are the unit pseudo-vectors along the base vectors, {Gi }, i = 1, 2, 3, and e𝛼 ≡ 1̂ 𝛼 + t d, 𝛼 , 𝛼 = 1, 2 e1 ≡ 1̂ 1 + t d,1 = { (1 + u,1 ) v,1 e2 ≡ 1̂ 2 + t d,2 = { u,2 (1 + v,2 )

w,1 }T , w,2

(11.321)

}T

and a𝛼 ≡ (1̂ 𝛼 + t d|𝛼 ), 𝛼 = 1, 2 = (1̂ 𝛼 + t d, 𝛼 ) + k0𝛼 × t d a1 ≡ (1̂ 1 + t d,1 ) + k01 × t d = e1 + k01 × { = (1 + u,1 −𝜅31 v + 𝜅21 w) (v,1 + 𝜅31 u) a2 ≡ (1̂ 2 + t d,2 ) + k02 × t d = e2 + k02 × { = (u,1 −𝜅22 v) (1 + v,1 + 𝜅22 u − 𝜅32 w)

td

(w,1 − 𝜅21 u)

}T

td

(w,1 + 𝜅32 v)

,

(11.322)

}T

Then, from the second term in the angular momentum balance equation (11.305), by using the first relation in equation (11.311), we have: c0 , 𝛼 × F𝛼 = G a𝛼 × G t F𝛼 = (det G) G(a𝛼 × t F𝛼 ) = G(a𝛼 × t F𝛼 ),

𝛼 = 1, 2 (11.323)

Exercise: Show that for any two non-collinear vectors b1 , b2 , and an orthonormal tensor A: Ab1 × Ab2 = (det A) A(b1 × b2 ). Solution: Let b3 be any arbitrary vector non-collinear with b1 , b2 . Let us define a tensor B ≡ bi ⊗ Gi where Gi are the contravariant base vectors. Note that bi ≡ B Gi that is, the matrix representation of B has three columns as b1 , b2 and b3 . Thus, det B = b1 × b2 ∙ b3 . Note that AB = (Abj ⊗ Gj ) because for ∀h ∈ R3 , ABh = A(bj ⊗ Gj )h = A(Gj ∙ h)bj = (Abj )(Gj ∙ h) = (Abj ⊗ Gj )h. So, det(AB) = Ab1 × Ab2 ∙ Ab3 . But, by the product rule of determinants,

Nonlinear Shell

823

we have: det(AB) = det(A) det(B) = det(A)(b1 × b2 ∙ b3 ). So, we get: AT (Ab1 × Ab2 ) ∙ b3 = det(A)(b1 × b2 ) ∙ b3 . Now, noting that b3 is arbitrary and that AAT = I because A is orthonormal, completes the proof. Now, evaluating the expressions in equation (11.322), we have: ( ( ) ) 1 1 1 ⎫ ⎫ ⎧ ⎧ ) + 𝛾 , w − 𝛾 , v − v (1 + u, u, 1 2 ⎪ ⎪ ⎪ 𝛾1 1 2 R1 ⎪ 𝛾2 2 1 ⎪ ⎪ ⎪ ⎪ ( ( ) ) ⎪ ⎪ ⎪ ⎪ 1 1 1 a1 = ⎨ v,1 − 𝛾1 ,2 u 𝛾2 ,1 u − w ⎬ (11.324) ⎬ ; a2 = ⎨ (1 + v,2 ) + 𝛾1 𝛾2 R2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 1 ⎪ ⎪ ⎪ ⎪ w,2 + v w,1 + u ⎭ ⎭ ⎩ ⎩ R1 R2 Then, noting in equation (11.323), c0 , 𝛼 × F𝛼 in column vector form is given by a𝛼 × t F𝛼 as: ( ( ) ) ) ( 1 1 ⎫ ⎧ 𝛾 , u Q1 − w,1 + u N12 v,1 − ⎪ ⎪ 𝛾1 1 2 R1 ⎪ ⎪ ) ( ( ) ( ) ⎪ ⎪ 1 1 1 t 1 a1 × F = ⎨ 𝛾1 ,2 v − w Q1 w,1 + u N1 − (1 + u,1 ) + ⎬ R 𝛾 R 1 1 1 ⎪ ⎪ ( ) ( ( ( ) ) ) ⎪ ⎪ ⎪ (1 + u,1 ) + 1 𝛾1 ,2 v − 1 w N12 − v,1 − 1 𝛾1 ,2 u N1 ⎪ ⎭ ⎩ 𝛾1 R1 𝛾1

(11.325)

and similarly, ( ) ( ) ) ( 1 1 1 ⎫ ⎧ 𝛾2 ,1 u − w Q2 − w,2 + v N2 (1 + v,2 ) + ⎪ ⎪ 𝛾2 R2 R2 ⎪ ⎪ ( ) ) ) ( ( ⎪ ⎪ 1 1 t 2 a2 × F = ⎨ w,2 + v N21 − u,2 − 𝛾2 ,1 v Q2 ⎬ R2 𝛾2 ⎪ ⎪ ( ( ( ) ) ) ) ⎪ ⎪( 1 1 1 ⎪ u,2 − 𝛾2 ,1 v N2 − (1 + v,2 ) + 𝛾2 ,1 u − w N21 ⎪ ⎭ ⎩ 𝛾2 𝛾2 R2

11.11.1.2

(11.326)

Component Vector Form of Dynamic Balance Equations

Now, by applying equations (11.314), (11.316) and (11.162) in the equation (11.305), we get with equation (11.323): G( t F𝛼 |𝛼 + t F̂ 𝛼 − t L) = 0 ∙

∙

̂ 𝛼 − t A) = 0 G( t M𝛼 |𝛼 + a𝛼 × t F𝛼 + t M

(11.327)

Computation of Nonlinear Structures

824

11.11.1.3

Expanded Component-wise Force and Moment Covariant Derivatives

As a preliminary result, we may expand expressions in equation (11.315) component-wise: t F𝛼 | 𝛼

= t F𝛼 , 𝛼 + 𝜿 0𝛼 × t F𝛼 { } 1 1 1 = N1 ,1 + 𝛾1 ,2 N12 − 𝛾2 ,1 N2 + N21 ,2 − Q1 + F̂ 1 G1 𝛾1 𝛾2 R1 { } 1 1 1 2 ̂ + N2 ,2 + 𝛾2 ,1 N21 − 𝛾1 ,2 N1 + N12 ,1 − Q2 + F G2 𝛾2 𝛾1 R2 { } 1 1 + N1 + N2 + Q1 ,1 +Q2 ,2 +F̂ 3 N R1 R2

(11.328)

= t F𝛼 , 𝛼 + K0𝛼 t F𝛼 Similarly, from equation (11.317) with the definition of a𝛼 as above, we get: t M𝛼 | 𝛼

+ a𝛼 × t F𝛼 = t M𝛼 , 𝛼 + 𝜿 0𝛼 × t M𝛼 + {𝜿 0𝛼 × d + e𝛼 } × t F𝛼 1 1 1 ⎫ ⎧ ⎪ −M2 , 2 − M12 , 1 − R M13 + 𝛾 𝛾1 , 2 M1 + 𝛾 𝛾2 , 1 M21 + X1 ⎪ 1 1 2 ⎪ ⎪ ⎪ ⎪ 1 1 1 = ⎨ M1 , 1 + M21 , 2 − M13 + 𝛾1 , 2 M12 − 𝛾2 , 1 M2 + X2 ⎬ R2 𝛾1 𝛾2 ⎪ ⎪ ⎪ ⎪ 1 1 M13 , 1 + M23 , 2 − M12 + M21 + X3 ⎪ ⎪ R1 R2 ⎭ ⎩

(11.329)

= t M𝛼 , 𝛼 + K0𝛼 t M𝛼 + {(K0𝛼 D − D K0𝛼 ) + E𝛼 } t F𝛼 where we have used the following new definitions: ( ) ( ) ( ) ( ) 1 1 1 1 v, 1 − 𝛾1 , 2 u Q1 − w, 1 + u N12 + 1 + v, 1 + 𝛾2 , 1 u Q2 − w, 1 + v N2 𝛾1 R1 𝛾2 R2 ( ) ( ) ( ) 1 1 1 1 X2 ≡ w, 1 + u N1 − 1 + u, 1 + 𝛾1 , 2 v − w Q1 + w, 1 + v N21 R1 𝛾1 R1 R2 ( ) 1 − u, 1 − 𝛾2 , 1 v Q2 𝛾2 ( ) ( ) ( ) 1 1 1 1 X3 ≡ 1 + u, 1 + 𝛾1 , 2 v − w N12 − v, 1 − 𝛾1 , 2 u N1 + u, 1 − 𝛾2 , 1 v N2 𝛾1 R1 𝛾1 𝛾2 ( ) 1 − 1 + v, 1 + 𝛾2 , 1 u N21 𝛾2 X1 ≡

(11.330) In the above expression, we have used the following.

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825

Definitions

r D ≡ [d]

is the skew-symmetric tensor with the displacement vector, d = { u ≡ d1 v ≡ d2 w ≡ d3 }T taken as its axial vector, r E1 ≡ [e1 ] is the skew-symmetric tensor corresponding to the special displacement derivative vector e ≡ { (1 + u, 1 ) v, 1 w, 1 }T = 1̂ 1 + d, 1 taken as its axial vector. r E2 ≡ [e21] is the skew-symmetric tensor corresponding to the special displacement derivative vector e2 ≡ { u, 2 (1 + v, 2 ) w, 2 }T = 1̂ 2 + d, 2 taken as its axial vector. Finally, we have used the Lie bracket definition, a (commutator) property relating a skewsymmetric tensor to its axial vector. Exercise: Given any two skew-symmetric tensors, A and B with the corresponding axial vectors, a and b, respectively, we have the commutator (or Lie bracket) relationship: AB − BA = [a × b] = (a × b)× .  − (b ∙  Solution: For any h ∈ R3 , (AB − BA)h = a × (b × h) − b × (a × h) = (a ∙ h)b −  (a  ∙ h)b    h)a+  (a ∙ h)b = (a ∙ h)b − (b ∙ h)a = {(b ⊗ a) − (a ⊗ b)} h completes the first equality; also, (a ∙ h)b − (b ∙ h)a = h × (b × a) = (a × b) × h = [a × b]h establishes the rest of the equalities. In all of the derivations, we have used the vector identity: a × (b × c) = (a ∙ c)b − (a ∙ b)c. Exercise: Verify expressions given by equations (11.328) and (11.329).

11.11.1.4

Component-wise Dynamic Balance Equations

Now, using equations (11.328) and (11.329) in equation (11.327), we can express the combined balance equations component-wise as follows: G( t F𝛼 , 𝛼 + 𝜿 0𝛼 × t F𝛼 + t F̂ 𝛼 − t L) = 0 ∙

∙

̂ 𝛼 − t A) = 0 G( t M𝛼 , 𝛼 + 𝜿 0𝛼 × t M𝛼 + {𝜿 0𝛼 × d + e𝛼 } × t F𝛼 + t M

(11.331)

or, using the definition of the initial curvature vectors in equation (11.312) and of the force components in equation (11.309), we have for the first (momentum balance) of equation (11.331): {

} 1 1 1 1 ̂ N1 , 1 + 𝛾1 , 2 N12 − 𝛾2 , 1 N2 + N21 , 2 − Q1 + F G1 𝛾1 𝛾2 R1 { } 1 1 1 + N2 , 2 + 𝛾2 , 1 N21 − 𝛾1 , 2 N1 + N12 , 1 − Q2 + F̂ 2 G2 𝛾2 𝛾1 R2 { } ∙ 1 1 + N1 + N2 + Q1 , 1 + Q2 , 2 + F̂ 3 N = t L i Gi R1 R2

(11.332)

Now, in equation (11.332), noting that the base vectors are linearly independent, the coefficient of each has to independently vanish resulting in the following three momentum or force

Computation of Nonlinear Structures

826

balance equations: N1 , 1 +

∙ 1 1 1 𝛾1 , 2 N12 − 𝛾2 , 1 N2 + N21 , 2 − Q1 + F̂ 1 = L1 𝛾1 𝛾2 R1

N2 , 2 +

∙ 1 1 1 𝛾2 , 1 N21 − 𝛾1 , 2 N1 + N12 , 1 − Q2 + F̂ 2 = L2 𝛾2 𝛾1 R2

(11.333)

∙ 1 1 N1 + N2 + Q1 , 1 + Q2 , 2 + F̂ 3 = LN R1 R2

Next, using the similar arguments, we can find the expanded component-wise angular momentum balance equations. Thus, by combining the moment balance and the angular momentum balance equations, we have: N1 , 1 + N21 , 2 +

∙ 1 1 1 𝛾1 , 2 N12 − 𝛾2 , 1 N2 − Q1 + F̂ 1 = L1 𝛾1 𝛾2 R1

N2 , 2 + N12 , 1 +

∙ 1 1 1 𝛾2 , 1 N21 − 𝛾1 , 2 N1 − Q2 + F̂ 2 = L2 𝛾2 𝛾1 R2

Q1 , 1 + Q2 , 2 + −M2 , 2 − M12 , 1 −

∙ 1 1 1 ̂ 1 = A1 M13 + 𝛾1 , 2 M1 + 𝛾2 , 1 M21 + X1 + M R1 𝛾1 𝛾2

M1 , 1 + M21 , 2 −

∙ 1 1 1 ̂ 2 = A2 M13 + 𝛾1 , 2 M12 − 𝛾2 , 1 M2 + X2 + M R2 𝛾1 𝛾2

M13 , 1 + M23 , 2 − 11.11.1.5

∙ 1 1 N1 + N2 + F̂ 3 = LN R1 R2

(11.334)

∙ 1 1 ̂ 3 = AN M12 + M21 + X3 + M R1 R2

Matrix Form of Dynamic Balance Equations

Equation (11.334) may be expressed in matrix form as: 11.11.1.5.1

Momentum Balance Equations 1 𝛾 , 𝛾2 2 1

⎡ 0 ⎧ N1 , 1 ⎫ ⎢ ⎪ ⎪ ⎢ 1 ⎨ N12 , 1 ⎬ + ⎢⎢ − 𝛾 𝛾2 , 1 ⎪Q , ⎪ ⎢ 2 ⎩ 1 1⎭ ⎢ 0 ⎣ ⎡ 0 ⎧ N21 , 2 ⎫ ⎢ ⎪ ⎪ ⎢ 1 ⎨ N2 , 2 ⎬ + ⎢⎢ 𝛾 𝛾1 , 2 ⎪Q , ⎪ ⎢ 1 ⎩ 2 2⎭ ⎢ −1 ⎣ R1 and

0 − −

1 R2

1 𝛾 , 𝛾1 1 2 0 0

0 ⎤ ⎥⎧ N1 ⎫ ⎪ 1 ⎥⎥ ⎪ N + R2 ⎥ ⎨ 12 ⎬ ⎪ Q ⎥⎪ ⎩ 1⎭ 0 ⎥⎦ 1 ⎤ ⎧∙⎫ R1 ⎥ ⎧ N21 ⎫ ⎪ L1 ⎪ ⎥⎪ ⎪ ⎪∙⎪ 0 ⎥ ⎨ N2 ⎬ = ⎨ L ⎬ 2 ⎥⎪ ⎪∙⎪ ⎥ ⎩ Q2 ⎪ ⎭ ⎪L⎪ 0 ⎥⎦ ⎩N ⎭

(11.335)

Nonlinear Shell

11.11.1.5.2

827

Angular (or Moment of) Momentum Balance Equations

⎡ 0 ⎧ −M12 , 1 ⎫ ⎢ ⎪ ⎪ ⎢ 1 ⎨ M1 , 1 ⎬ + ⎢⎢ − 𝛾 𝛾2 , 1 ⎪ M , ⎪ ⎢ 2 ⎩ 13 1 ⎭ ⎢ 0 ⎣ ⎡ 0 ⎧ −M2 , 2 ⎫ ⎢ ⎪ ⎪ ⎢ 1 + ⎨ M21 , 2 ⎬ + ⎢ 𝛾1 , 2 𝛾 ⎪ M , ⎪ ⎢⎢ 1 ⎩ 23 2 ⎭ ⎢ −1 ⎣ R1

1 𝛾 , 𝛾2 2 1

0 ⎤ ⎥⎧ ⎧ N1 ⎫ −M12 , 1 ⎫ ⎪ ⎪ ⎪ 1 ⎥⎥ ⎪ M1 , 1 ⎬ + K01D ⎨ N12 ⎬ R2 ⎥ ⎨ ⎪ ⎪Q ⎪ ⎥⎪ ⎩ 1⎭ ⎩ M13 , 1 ⎭ 0 ⎥⎦

0 −

1 R2

−

1 𝛾 , 𝛾1 1 2 0 0

1 ⎤ ⎧ ∙ ⎫ R1 ⎥ ̂ 1 ⎫ ⎪ A1 ⎪ ⎫ ⎧ N21 ⎫ ⎧ M ⎧ ⎥ ⎪ −M2 , 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∙ ⎪ ̂2⎬=⎨A ⎬ 0 ⎥ ⎨ M21 , 2 ⎬ + K0 ⎨ N2 ⎬ + ⎨ M 2 2D ⎥⎪ ⎪ Q ⎪ ⎪M ⎪ ⎪ ∙ ⎪ ̂ ⎥ ⎩ M23 , 2 ⎪ ⎭ ⎩ 2 ⎭ ⎩ N ⎭ ⎪ AN ⎪ 0 ⎥⎦ ⎩ ⎭ (11.336)

where K0D , the matrix commutator, is given by: K01D ≡ K01 D − D K01 where ⎡ 0 ⎢ ⎢ 1 K01 = ⎢ − 𝛾2 , 1 ⎢ 𝛾2 ⎢ ⎢ 0 ⎣

1 𝛾 , 𝛾2 2 1 0 −

1 R2

0 ⎤ ⎥ 1 ⎥⎥ R2 ⎥ ⎥ 0 ⎥⎦

and

⎡ 0 ⎢ D=⎢ w ⎢ −v ⎣

−w v ⎤ ⎥ 0 −u ⎥ u 0 ⎥⎦

(11.337)

and K02D ≡ K02 D − D K02 where ⎡ 0 ⎢ ⎢ 1 K02 = ⎢ 𝛾1 , 2 ⎢ 𝛾1 ⎢ ⎢ −1 ⎣ R1

−

1 𝛾 , 𝛾1 1 2 0 0

1 ⎤ R1 ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥⎦

and

⎡ 0 −w ⎢ 0 D=⎢ w ⎢ −v u ⎣

v ⎤ ⎥ −u ⎥ 0 ⎥⎦

(11.338)

Exercise: Starting from equation (11.329), verify expressions given by equation (11.336). Remark

r It is interesting to note that the rotation tensor due to shell shear deformation does not appear in any of the balance equations. It is only when we develop, later, the virtual work functional in a co-rotated frame under the Reissner–Mindlin assumption that we begin to see the effect of rotation due to shear deformation. Under the Kirchhoff assumption of the normal at a point on

Computation of Nonlinear Structures

828

the shell reference surface remaining straight and perpendicular to the shell deformed reference surface (i.e. without any shear deformation), the functional will be completely characterized by the displacements alone.

11.11.2

Static Balance Equations

The quasi-static balance laws can be easily obtained from their dynamic counterparts by assuming that the linear and the angular momentum vectors identically vanish, that is, L(S1 , S2 , t) ≡ 0

(11.339)

A(S1 , S2 , t ≡ 0 Thus, we get: 11.11.2.1

Static Momentum Balance Equations (Force Equilibrium) F𝛼 , 𝛼 + F̂ = 0,

𝛼 = 1, 2

(11.340)

and 11.11.2.2

Static Angular or Moment of Momentum Balance Equations (Moment Equilibrium) ̂ = 0, M𝛼 , 𝛼 + c0 , 𝛼 × F𝛼 + M

11.11.2.3

𝛼 = 1, 2

(11.341)

Component Vector form of Static Balance Equations

We get from equation (11.327): G( t F𝛼 |𝛼 + t F̂ 𝛼 ) = 0 ̂ 𝛼) = 0 G( t M𝛼 |𝛼 + a𝛼 × t F𝛼 + t M

(11.342)

as the static component vector form of the linear momentum and angular momentum balance equations.

11.11.2.4

Component-wise Static Balance Equations

Now, from equation (11.331), we can express the static combined balance equations componentwise as:

G( t M𝛼 , 𝛼 + 𝜿 0𝛼

G( t F𝛼 , 𝛼 + 𝜿 0𝛼 × t F𝛼 + t F̂ 𝛼 ) = 0 ̂ 𝛼) = 0 × t M𝛼 + {𝜿 0𝛼 × d + e𝛼 } × t F𝛼 + t M

(11.343)

Nonlinear Shell

829

or

N1 , 1 + N21 , 2 +

1 1 1 𝛾 , N − 𝛾 , N − Q + F̂ 1 = 0 𝛾1 1 2 12 𝛾2 2 1 2 R1 1

N2 , 2 + N12 , 1 +

1 1 1 𝛾 , N − 𝛾 , N − Q + F̂ 2 = 0 𝛾2 2 1 21 𝛾1 1 2 1 R2 2 Q1 , 1 + Q2 , 2 +

1 1 N + N + F̂ 3 = 0 R1 1 R2 2

(11.344)

1 1 1 ̂1=0 −M2 , 2 − M12 , 1 − M13 + 𝛾1 , 2 M1 + 𝛾2 , 1 M21 + X1 + M R1 𝛾1 𝛾2 M1 , 1 + M21 , 2 −

1 1 1 ̂2=0 M + 𝛾 , M − 𝛾 , M + X2 + M R2 13 𝛾1 1 2 12 𝛾2 2 1 2

M13 , 1 + M23 , 2 −

11.11.2.5

1 1 ̂3=0 M + M + X3 + M R1 12 R2 21

Matrix Form of Shell Static Balance Equations

From equations (11.335) and (11.336), we get in matrix form as:

11.11.2.6

Momentum Balance Equations

⎡ 0 ⎢ ⎧ N1 , 1 ⎫ ⎢ ⎪ ⎪ ⎢ 1 ⎨ N12 , 1 ⎬ + ⎢ − 𝛾 𝛾2 , 1 ⎪ ⎪ ⎢ 2 ⎩ Q1 , 1 ⎭ ⎢ ⎢ 0 ⎣

1 𝛾 , 𝛾2 2 1 0 −

1 R2

0 ⎤ ⎥ ⎥ ⎧ N1 ⎫ ⎪ 1 ⎥⎪ N R2 ⎥ ⎨ 12 ⎬ ⎪ ⎥⎪ ⎥ ⎩ Q1 ⎭ 0 ⎥ ⎦ (11.345)

⎡ 0 ⎢ ⎧ N21 , 2 ⎫ ⎢ ⎪ ⎪ ⎢ 1 + ⎨ N2 , 2 ⎬ + ⎢ 𝛾1 , 2 𝛾 ⎪ ⎪ ⎢ 1 ⎩ Q2 , 2 ⎭ ⎢ 1 ⎢ − ⎣ R1 and

1 − 𝛾1 , 2 𝛾1 0 0

1 ⎤ R1 ⎥ ⎥ ⎧ N21 ⎫ ⎧ 0 ⎫ ⎪ ⎪ ⎪ ⎥⎪ 0 ⎥ ⎨ N2 ⎬ = ⎨ 0 ⎬ ⎪ ⎪ ⎪ ⎥⎪ ⎥ ⎩ Q2 ⎭ ⎩ 0 ⎭ 0 ⎥ ⎦

Computation of Nonlinear Structures

830

11.11.2.7

Angular (or Moment of) Momentum Balance Equations:

⎡ 0 ⎧ −M12 , 1 ⎫ ⎢ ⎪ ⎪ ⎢⎢ 1 M , ⎨ 1 1 ⎬ + − 𝛾 𝛾2 , 1 ⎪ M , ⎪ ⎢⎢ 2 ⎩ 13 1 ⎭ ⎢ 0 ⎣ ⎡ 0 ⎧ −M2 , 2 ⎫ ⎢ ⎪ ⎪ ⎢ 1 + ⎨ M21 , 2 ⎬ + ⎢ 𝛾1 , 2 𝛾 ⎪ M , ⎪ ⎢⎢ 1 ⎩ 23 2 ⎭ ⎢ −1 ⎣ R1

1 𝛾 , 𝛾2 2 1 0 −

1 R2

−

0 ⎤ ⎥⎧ ⎧ N1 ⎫ −M12 , 1 ⎫ ⎪ ⎪ ⎪ 1 ⎥⎥ ⎪ 0 M1 , 1 ⎬ + K1D ⎨ N12 ⎬ ⎨ R2 ⎥ ⎪ ⎪Q ⎪ ⎥⎪ ⎩ 1⎭ ⎩ M13 , 1 ⎭ ⎥ 0 ⎦

1 𝛾 , 𝛾1 1 2 0 0

1 ⎤ R1 ⎥ ̂ 1 ⎫ ⎧0⎫ ⎫ ⎧ N21 ⎫ ⎧ M ⎧ ⎥ ⎪ −M2 , 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ̂ 2 ⎬ = ⎨0⎬ 0 ⎥ ⎨ M21 , 2 ⎬ + K ⎨ N2 ⎬ + ⎨ M 2D ⎥⎪ ⎪ Q ⎪ ⎪M ⎪ ⎪ ⎪ ⎥ ⎩ M23 , 2 ⎪ ⎭ ⎩ 2 ⎭ ⎩ ̂ N ⎭ ⎩0⎭ ⎥ 0 ⎦ (11.346)

11.11.3

Where We Would Like to Go

We have now derived the thin shell computational balance equations. These serve as the starting point in developing

r virtual work functional necessary for the numerical formulation based on a finite element method of solution.

11.12 11.12.1

Computational Derivatives and Variations What We Need to Recall

r The vector triple product property: a × b ∙ c = b × c ∙ a = c × a ∙ b, ∀a, b, c vectors. r The vector triple cross product property: a × b × c = (a ∙ c)b − (a ∙ b)c, ∀a, b, cvectors. r The definition of a rotation tensor, its matrix representation and algebra, geometry and topology of rotation.

r A rotation matrix, R, corresponding to the 3D special orthogonal matrix group, SO(3), may be parameterized by the Rodrigues vector, s, the axis of the instantaneous rotation. s2 s3 }T , is orientated along the instantaneous axis of rotation at a point of the shell reference surface. The skew-symmetric matrix, S, corresponding to the Rodrigues vector, s, that is, S s = 0, is given by:

r The Rodrigues vector, s = { s1

⎡ 0 ⎢ S = ⎢ s3 ⎢ −s ⎣ 2

r The

−s3 0 s1

s2 ⎤ ⎥ −s1 ⎥ 0 ⎥⎦

(11.347)

rotation matrix, R(S1 , S2 , t), is related, through Cayley’s theorem, to the Rodrigues vector, s(S1 , S2 , t) = { s1 (S1 , S2 , t) s2 (S1 , S2 , t) s3 (S1 , S2 , t) }T , with the components in the

Nonlinear Shell

831

G1 , G2 , N frame at any point, (S1 , S2 ), parameterized by the arc lengths along the lines of curvature on a shell reference surface, at time t ∈ R+ , by: tR

= (I − S)−1 (I + S) ⎡ 1 + s2 − s2 − s2 1 2 3 1 ⎢ = ∑ ⎢ 2(s1 s2 + s3 ) ⎢ 2(s s − s ) 1 3 2 ⎣

where by:

∑

2(s1 s2 − s3 ) 1 − s21

+ s22

− s23

2(s2 s3 + s1 )

2(s1 s3 + s2 ) ⎤ ⎥ 2(s2 s3 − s1 ) ⎥ 1 − s21 − s22 + s23 ⎥⎦

(11.348)

≡ 1 + ∥ s ∥2 with the (I − S) matrix always invertible for any rotation matrix, defined

(I − S)−1

⎡ (1 + s2 ) (s1 s2 − s3 ) (s3 s1 + s2 ) ⎤ 1 ⎥ 1⎢ (1 + s22 ) (s2 s3 − s1 ) ⎥ = ⎢ (s1 s2 + s3 ) Σ⎢ (1 + s23 ) ⎥⎦ ⎣ (s3 s1 − s2 ) (s2 s3 + s1 )

(11.349)

r Finally, the rotation vector, θ, is the scaled Rodrigues vector related by: s=

( ) 1 𝜃 θ tan 𝜃 2

(11.350)

1

where 𝜃 ≡ ‖θ‖ = (θ ∙ θ) 2 is the length measure of rotation vector. Alternatively, defining θ the unit vector, n ≡ , and the corresponding skew-symmetric matrix, N, such that N h = 𝜃 n × h, ∀h ∈ R3 , we have: R = (I − S)−1 (I + S) ( ) 𝜃 n, s = tan 2 ( ) 𝜃 S = tan N 2

(11.351)

r The Rodrigues quaternion and Euler–Rodrigues parameters. r For notational uniformity, we denote all variations of entities by a superscripted bar. Furthermore all entities measured in G1 , G2 , N frame are superscripted as t (∙), and those in the rotating g1 , g2 , g3 coordinate system as n (∙). However, for the sake of uncluttered readability, sometimes the superscripts may be dropped, but the context should help determine the appropriate underlying coordinate system.

11.12.2

Curvature Matrices, t K𝛼 , 𝛼 = 1, 2, Curvature Vectors, k𝛼 , 𝛼 = 1, 2

t

Let R(S1 , S2 , t) be the rotation matrix at a point on the shell reference surface described in the arc length parameters, (S1 , S2 ), along the lines of the curvature direction, at any fixed time t ∈ R+ . Then, considering it as a one-parameter family along each direction, and taking the derivative,

Computation of Nonlinear Structures

832

in the sense described above, of the orthonormality relationship: R RT = I, with respect to this parameter, we have: (R, 𝛼 RT ) + (R, 𝛼 RT )T = 0,

𝛼 = 1, 2

(11.352)

Now, we introduce the following. Definitions

r The curvature matrices,

tK , 𝛼

𝛼 = 1, 2, as: t

K𝛼 ≡ t R, 𝛼 t RT ,

𝛼 = 1, 2

(11.353)

By introducing above definitions into equation (11.352), we have: t K𝛼 + (t K𝛼 )T = 0, 𝛼 = 1, 2, signifying that t K𝛼 , 𝛼 = 1, 2 are 3D skew-symmetric matrices; we can then define the corresponding axial vectors, t k𝛼 , 𝛼 = 1, 2, such that t K𝛼 h = t k𝛼 × h, 𝛼 = 1, 2 for ∀h ∈ R3 . Moreover, from the matrix representation, as given by equation (11.348), of the rotation tensor, we have, by taking straightforward term-by-term derivatives, the corresponding matrix representation of the curvature tensors, t K𝛼 , 𝛼 = 1, 2, as:

tK 𝛼

⎫ ⎧ ) ⎪ 2⎪ ( = ⎨ S S, 𝛼 − S, 𝛼 S +S, 𝛼 ⎬ Σ ⎪ ⏟⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏟ ⎪ ⎭ ⎩Commutator or Lie bracket ⎡ 0 (−s1 s2 , 𝛼 + s1 , 𝛼 s2 − s3 , 𝛼 ) 2 ⎢ 0 = ∑ ⎢ (s1 s2 , 𝛼 − s1 , 𝛼 s2 + s3 , 𝛼 ) ⎢ (s s , − s , s − s , ) (s s , − s , s + s , ) 1 𝛼 3 2 𝛼 2 3 𝛼 2 𝛼 3 1 𝛼 ⎣ 1 3 𝛼

(−s1 s3 , 𝛼 + s1 , 𝛼 s3 + s2 , 𝛼 ) ⎤ ⎥ (−s2 s3 , 𝛼 + s2 , 𝛼 s3 − s1 , 𝛼 ) ⎥ ⎥ 0 ⎦ (11.354)

where the third equality follows from the first equality by the following fact. Fact: (A B − B A) h = [a × b] h, ∀h ∈ R3 where [(∙)] denotes a skew-symmetric matrix corresponding to an axial vector (∙), and A, B are skew-symmetric matrices, and a, b are the corresponding axial vectors, respectively. Exercise: Show that matrix Lie bracket, A B − B A = [a × b]  − (b ∙ h)a−  Solution: (A B − B A)h = a × b × h − b × a × h, ∀h ∈ R3 = (a ∙ h)b −  (a  ∙ b)h   (b ∙ a)h = h × (b × a) = a × b × h = [a × b] h completes the proof.  Exercise: Derive equation (11.354).

Nonlinear Shell

833

Solution: By taking the derivative of equation (11.348) with respect to 𝛼 = 1, 2, and using the definition of the curvature matrix as in equation (11.353), we have: (I − S) t R, 𝛼 − S, 𝛼 t R = S, 𝛼 ⇒ (I − S) t R, 𝛼 t RT − S, 𝛼 t R t RT = S, 𝛼 t RT ⇒ t K𝛼 = (I − S)−1 {S, 𝛼 + S, 𝛼 t RT } Now, using the inverse relation given by equation (11.349) and direct multiplication completes the proof. 11.12.2.1

Curvature Vector, t k𝜶 ,

𝜶 = 1, 2

Now, the expressions for the axial curvature vectors, t k𝛼 , 𝛼 = 1, 2 corresponding to the curvature matrices, t K𝛼 , 𝛼 = 1, 2, such that t K𝛼 t k𝛼 = 0, 𝛼 = 1, 2, follow from equation (11.354), as: tk 𝛼

2 = ∑ (s × s, 𝛼 + s, 𝛼 ), 2 = ∑ (S s, 𝛼 + s, 𝛼 ) =

𝛼 = 1, 2 {

} 2 ∑ (I + S) s, 𝛼 = W1 (s) s, 𝛼

(11.355)

where we have introduced, for future use, the definition of the matrix: 2 W1 (s) ≡ ∑ (I + S)

(11.356)

2 𝜃 = and 2 1 + cos 𝜃 , we have, using the second relation of equation (11.351) in equation (11.356):

with I as the 3 × 3 identity matrix. Now, noting that Σ ≡ 1 + ‖s‖2 = 1 + tan2 tan

𝜃 sin 𝜃 = 2 1 + cos 𝜃

{ W1 (s) = (1 + cos 𝜃) I +

} sin 𝜃 N 1 + cos 𝜃

(11.357)

Note that this equation gives the expression in terms of the rotation vector, θ. 11.12.2.2

Curvature Vectors, t k𝜶 ,

𝜶 = 1, 2 in Terms of the Rotation vector t θ

Note that from the fourth equality in equation (11.355) that we only get t k𝛼 as the function of the derivatives, s, 𝛼 , of the Rodrigues vector. But, we would like to get its dependence on the rotation vector, t θ . For this, we take the derivative of the relation given by equation (11.350): } { {( } { } ) (θ ∙ θ, 𝛼 ) θ, 𝛼 1 𝜃 𝜃 2 𝜃 (θ ∙ θ, 𝛼 ) sec s, 𝛼 = tan θ− θ+ tan 2 2 2 𝜃 2 𝜃2 𝜃3 ){ )( ( ( ) } 𝜃 𝜃 1 θ, 𝛼 − 1 − = tan n ∙ θ, 𝛼 n 𝜃 2 sin 𝜃 ){ ( ) } ( 𝜃 𝜃 1 (n ⊗ n) θ, 𝛼 = W2 θ, 𝛼 , 𝛼 = 1, 2 I− 1− tan = 𝜃 2 sin 𝜃

(11.358)

Computation of Nonlinear Structures

834

where we have introduced the definition for the matrix: W2 ≡

(

𝜃 1 tan 𝜃 2

){

( I− 1−

) } 𝜃 (n ⊗ n) sin 𝜃

(11.359)

Alternatively, as previously done for equation (11.356), we can express the relationship in equation (11.359) in terms of the rotation vector, θ, as:

W2 =

1 1 + cos 𝜃

{

( I−

) } sin 𝜃 − 𝜃 N2 𝜃

(11.360)

where we have additionally used the identity: N2 = {(n ⊗ n) − I}. Exercise: Show: N2 = {(n ⊗ n) − I} Solution: R3

N2 h ≡ N[n × h] = n × n × h = (n ∙ h)n − (n ∙ n)h = {(n ⊗ n) − I} h,

∀h ∈

Now, using equation (11.358) with definition (11.360) in equation (11.355) with definition (11.357), we have: t

k𝛼 = W1 (s) s, 𝛼 = W1 W2 θ, 𝛼 ≡ W(θ) t θ, 𝛼 ,

𝛼 = 1, 2

(11.361)

where we have introduced the following definitions: Definitions W(θ) ≡ I + c1 𝚯 + c2 𝚯2 , 1 − cos 𝜃 , 𝜃2 𝜃 − sin 𝜃 c2 (θ) = 𝜃3 c1 (θ) =

(11.362)

Note that equation (11.361) with the definitions (11.265) describes the curvature vector t k𝛼 in terms of the derivatives, θ, 𝛼 , of the rotation vector for 𝛼 = 1, 2 where 𝚯 is the skew-symmetric matrix corresponding to rotation vector, θ, as the axial vector such that 𝚯 θ = 0, and 𝚯2 ≡ 𝚯𝚯 = [θ × θ ×]. Exercise: Prove the relationships in equation (11.265). Solution: Let us note that N, being a skew-symmetric matrix with the form given similar to equation (11.347), and that every matrix satisfies its characteristic equation; from its eigen

Nonlinear Shell

835

or characteristic equation, p(𝜆) ≡ 𝜆3 + 𝜆 = 0, we have: N + N3 = 0 ⇒ N3 = −N. Now, using relations (11.357) and (11.360) in equation (11.361), we have: { W ≡ W1 W2 = I +

sin 𝜃 N 1 + cos 𝜃

}{

( I−

) } sin 𝜃 − 𝜃 N2 𝜃

) sin 𝜃 sin2 𝜃 − 𝜃 sin 𝜃 3 sin 𝜃 − 𝜃 N2 + N− N 𝜃 1 + cos 𝜃 𝜃(1 + cos 𝜃) { } ) ( sin 𝜃 sin 𝜃 sin2 𝜃 𝜃 − sin 𝜃 N2 + = I+ + − cos 𝜃 𝜃(1 + cos 𝜃)  cos 𝜃 N 𝜃 1+ 1+  (

= I−

) ) ( 𝜃 − sin 𝜃 1 − cos 𝜃 Θ + Θ2 𝜃2 𝜃3 ⏟⏞⏞⏞⏞⏟⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏟⏞⏞⏞⏞⏟ (

= I+

≡c1

≡c2

Θ , with Θ as the skew-symmetric matrix corre𝜃 sponding to θ taken as the axial vector, completes the proof.

Noting that we have used the unit vector: N ≡

11.12.3

Angular Velocity Matrix, t Ω, Angular Velocity Vector, t ω

This time we take the rotation tensor, R(S1 , S2 , t), to be fixed at a point on the shell reference surface described in the arc length parameters, (S1 , S2 ), along the lines of curvature directions for any time t ∈ R+ . Then, considering it as a one-parameter family along the time coordinate, and following the steps used in the curvature derivation, but replacing the spatial derivative operations, ∙ d (∙), 𝛼 , 𝛼 = 1, 2, everywhere by the time derivative operation, (∙) ≡ (∙), and recognizing that: dt

t

∙

𝛀 ≡ R RT

(11.363)

we have from equation (11.363): ⎫ ⎧ ⎪ ⎪ ( ) ∙ ∙ ∙⎪ t𝛀 = 2 ⎪ + S⎬ S S− SS Σ⎨ ⎪ ⎪ ⎪ ⎪ ⏟⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏟ ⎭ ⎩Commutator or Lie bracket ⎡ 0 ⎢ ∙ ∙ ∙ 2 ⎢ = ∑ (s1 s2 − s1 s2 + s3 ) ⎢ ∙ ∙ ∙ ⎢ ⎣ (s1 s3 − s1 s3 − s2 )

(11.364)

∙ ∙ (−s1 s2 + s1 s2

∙ − s3 )

0 ∙ ∙ (s2 s3 − s2 s3

∙

+ s1 )

∙ ∙ (−s1 s3 + s1 s3 ∙ ∙ (−s2 s3 + s2 s3

0

∙ + s2 ) ⎤ ⎥ ∙ − s1 ) ⎥ ⎥

⎥ ⎦

Computation of Nonlinear Structures

836

11.12.3.1

Angular Velocity Vector, t ω

Now, from equation (11.364), we have: t

∙

ω = W1 (s) s

(11.365)

where W1 (s) is given by equation (11.357).

11.12.3.2

Angular Velocity Vector, t ω, in Terms of the Rotation Vector, t θ

Using equation (11.360) and replacing spatial derivatives by time derivatives in equation (11.361), we have: t

∙

∙

∙

ω = W1 (s) s = W1 W2 θ ≡ W(θ) t θ

(11.366)

where we have used the definition of W(θ) as given by equation (11.265).

11.12.4

̄ ≡ 𝛿 t Φ, Variational Vector, t ϕ ̄ ≡ 𝛿 tϕ Variational Matrix, t Φ

This time we take the variation of the rotation tensor, R(S1 , S2 , t), at a point on the shell reference surface described in the arc length parameters, (S1 , S2 ), along the lines of curvature directions for a time t ∈ R+ . Then, following the steps used in the angular velocity derivation but replacing the ∙

time derivative operations, (∙), everywhere by the variation operation, (∙) ≡ 𝛿(∙), and recognizing ̄ ≡ 𝛿𝚽 ≡ R ̄ RT ≡ 𝛿R RT , we have from equation (11.364): that 𝚽

t𝚽 ̄

⎫ ⎧ ( ) ⎪ 2⎪ = ⎨ +S̄ ⎬ S S̄ − S̄ S Σ⎪ ⏟⏞⏞⏞⏞⏟⏞⏞⏞⏞⏟ ⎪ ⎩Commutator or Lie bracket ⎭ ⎡ 0 (−s1 s̄ 2 + s̄ 1 s2 − s̄ 3 ) 2 ⎢ 0 = ∑ ⎢ (s1 s̄ 2 − s̄ 1 s2 + s̄ 3 ) ⎢ (s s̄ − s̄ s − s̄ ) (s s̄ − s̄ s + s̄ ) 1 3 2 2 3 2 3 1 ⎣ 1 3

(−s1 s̄ 3 + s̄ 1 s3 + s̄ 2 ) ⎤ ⎥ (−s2 s̄ 3 + s̄ 2 s3 − s̄ 1 ) ⎥ ⎥ 0 ⎦

(11.367)

where a superscripted bar over the variable signifies variation of the variable by the definition: d the variation of a along ā is given by: ā = lim (a + 𝜀̄a). 𝜀→0 d𝜀 11.12.4.1

̄ ≡ 𝜹 t𝝋 Rotational Variational Vector, t 𝝋

̄ corresponding to the variation Now, the expressions for the axial rotational variational vector, ϕ, ≡ 𝛿 t 𝚽, follow from equation (11.367), as:

t𝚽 ̄

t

̄ = W1 (s) s̄ ϕ

where we have W1 (s) as defined before in equation (11.356).

(11.368)

Nonlinear Shell

11.12.4.2

837

̄ in Terms of the Rotation Vector Variation, θ̄ ϕ

̄ for k𝛼 , By substituting ϕ

𝛼 = 1, 2, and θ̄ for θ, 𝛼 in equations (11.361) and (11.265), we get: t

̄ = W1 (s) s̄ = W1 W2 θ̄ ≡ W(θ) t θ̄ ϕ

(11.369)

where W(θ), c1 (θ), c2 (θ) have been defined earlier in equation (11.265). ∙

11.12.5

∙∙

Derivatives: W, 𝛼 (θ), 𝛼 = 1, 2 and W (θ), W (θ) and Variation: ̄ W(θ) ≡ 𝛿W(θ) of W(θ) ∙

∙∙

For future use, we will need the derivatives: W, 𝛼 (θ), 𝛼 = 1, 2, the rates: W (θ), W (θ), and the ̄ variation: W(θ) ≡ 𝛿W(θ) of the matrix W(θ). Upon differentiating by S𝛼 , 𝛼 = 1, 2 the expression for W(θ) as given by equation (11.265), we have: W, 𝛼 = c1 𝚯, 𝛼 + c2 (𝚯, 𝛼 𝚯 + 𝚯 𝚯, 𝛼 ) + c1 , 𝛼 𝚯 + c2 , 𝛼 𝚯2 = (c2 − c1 )(θ ∙ θ, 𝛼 )I + c1 𝚯, 𝛼 + c2 {(θ ⊗ θ, 𝛼 ) + (θ, 𝛼 ⊗ θ)} + c1 , 𝛼 𝚯 + c2 , 𝛼 (θ ⊗ θ) 1 (1 − 2c1 − 𝜃 2 c2 )(θ ∙ θ, 𝛼 ) = c3 (θ ∙ θ, 𝛼 ), 𝜃2 1 c2 , 𝛼 ≡ 2 (c1 − 3c2 )(θ ∙ θ, 𝛼 ) = c4 (θ ∙ θ, 𝛼 ) 𝜃 c1 , 𝛼 ≡

(11.370) where we have introduced the following definitions. Definitions 1 (1 − 2c1 − 𝜃 2 c2 ), 𝜃2 1 c4 ≡ 2 (c1 − 3c2 )(θ ∙ θ, 𝛼 ) 𝜃 c3 ≡

(11.371)

and applied the following identities: 𝚯2 = {(θ ⊗ θ) − 𝜃 2 I} (𝚯, 𝛼 𝚯 + 𝚯 𝚯, 𝛼 ) = {(θ ⊗ θ, 𝛼 ) + (θ, 𝛼 ⊗ θ) − 2(θ ∙ θ, 𝛼 )I} 1 𝜃, 𝛼 ≡ ‖θ‖, 𝛼 = (θ ∙ θ, 𝛼 ) 𝜃 Exercise: Prove the identities (11.372). Solution ∀h ∈ R3 , 𝚯2 h = θ × θ × h = {(θ ⊗ θ) − 𝜃 2 I}h

(11.372)

Computation of Nonlinear Structures

838

and (𝚯, 𝛼 𝚯 + 𝚯 𝚯, 𝛼 )h = θ, 𝛼 × θ × h + θ × θ, 𝛼 × h = (θ, 𝛼 ∙ h)θ − (θ, 𝛼 ∙ θ)h + (θ ∙ h)θ, 𝛼 − (θ ∙ θ, 𝛼 )h { } = (θ ⊗ θ, 𝛼 ) + (θ, 𝛼 ⊗ θ) − 2(θ ∙ θ, 𝛼 )I h complete the proof. Exercise: Derive relations given in equation (11.370). Solution: By taking the derivative with respect to S𝛼 , 𝛼 = 1, 2 of H(θ) in equation (11.265) and using relations in equation (11.372): W, 𝛼 = c1 𝚯, 𝛼 + c2 (𝚯, 𝛼 𝚯 + 𝚯 𝚯, 𝛼 ) + c1 , 𝛼 𝚯 + c2 , 𝛼 𝚯2 = c1 𝚯, 𝛼 + c2 {(θ ⊗ θ, 𝛼 ) + (θ, 𝛼 ⊗ θ) − 2(θ ∙ θ, 𝛼 )I} + c1 , 𝛼 𝚯 + c2 , 𝛼 {(θ ⊗ θ) − 𝜃 2 I} = c1 𝚯, 𝛼 + c2 {(θ ⊗ θ, 𝛼 ) + (θ, 𝛼 ⊗ θ)} + c1 , 𝛼 𝚯 + c2 , 𝛼 (θ ⊗ θ) + {−c1 + 3c2 − 2c2 }(θ ∙ θ, 𝛼 )I = (c2 − c1 )(θ ∙ θ, 𝛼 )I + c1 𝚯, 𝛼 + c2 {(θ ⊗ θ, 𝛼 ) + (θ, 𝛼 ⊗ θ)} + c1 , 𝛼 𝚯 + c2 , 𝛼 (θ ⊗ θ) ∙

∙

completes the derivation. Now, replacing θ, 𝛼 by θ and 𝚯, 𝛼 by 𝚯 everywhere in equations ∙

(11.370)–(11.372), we have the expression for W, the time derivative of W matrix: ∙

∙

∙

∙

∙

∙

W = c1 𝚯 +c2 (𝚯 𝚯 + 𝚯 𝚯) + c1 𝚯 + c2 𝚯2 ∙

∙

∙

∙

∙

∙

= (c2 − c1 )(θ ∙ θ)I + c1 𝚯 + c2 {(θ ⊗ θ) + (θ ⊗ θ)} + c1 𝚯 + c2 (θ ⊗ θ) ∙

c1 ≡

∙ ∙ ∙ ∙ ∙ 1 1 (1 − 2c1 − 𝜃 2 c2 )(θ ∙ θ) = c3 (θ ∙ θ), c2 ≡ 2 (c1 − 3c2 )(θ ∙ θ) = c4 (θ ∙ θ) 2 𝜃 𝜃

(11.373) ∙

∙∙

Taking the time derivative of W in equation (11.373), we have the expression for W, the second time derivative of W matrix: ∙∙

∙∙

{

W = c1 𝚯 +2 ∙∙

∙ ∙

∙

c1 ≡ c5 (θ ∙ θ)2 + c3 c5 ≡ −

∙

∙

∙

}

c1 𝚯 + c2 (𝚯 𝚯 + 𝚯 𝚯) {

∙

𝜃 2 + (θ ∙ θ)

1 (2c2 + 4c3 + 𝜃 2 c4 ), 𝜃2

∙∙

∙

∙∙

∙∙

∙∙

+ c2 (𝚯 𝚯 + 2 𝚯2 +𝚯 𝚯) + c1 𝚯 + c2 𝚯2

} ,

∙∙

∙

c2 ≡ c6 (θ ∙ θ)2 + c4 c6 ≡ −

{

∙

}

𝜃 2 + (θ ∙ θ)

1 (c3 − 5c4 ), 𝜃2 (11.374)

where we have used the properties of the constants as given by equations (11.265) and (11.371).

Nonlinear Shell

839

̄ everywhere in equations (11.370)–(11.372), we have Now, replacing θ, 𝛼 by θ̄ and 𝚯, 𝛼 by 𝚯 ̄ the expression for W, the variation of W matrix: ̄ + c2 (𝚯 ̄ 𝚯 + 𝚯 𝚯) ̄ + c̄ 1 𝚯 + c̄ 2 𝚯2 ̄ = c1 𝚯 W )} ( ) {( ) ( ̄ + c2 θ ⊗ θ̄ + θ̄ ⊗ θ + c̄ 1 𝚯 + c̄ 2 (θ ⊗ θ) = (c2 − c1 ) θ ∙ θ̄ I + c1 𝚯 1 ̄ = c (θ ∙ θ), ̄ c̄ ≡ 1 (c − 3c )(θ ∙ θ) ̄ = c (θ ∙ θ) ̄ c̄ 1 ≡ 2 (1 − 2c1 − 𝜃 2 c2 )(θ ∙ θ) 3 2 1 2 4 𝜃 𝜃2 (11.375)

11.12.6

∙

Angular Acceleration Vector, t ω, in Terms of the Rotation Vector, t θ

Taking the time derivative of the angular acceleration given in equation (11.366), we have: t

∙

∙

∙

∙∙

ω = W(θ) t θ + W(θ) t θ

(11.376)

∙

where W(θ) is given by equation (11.265), and W (θ) by equation (11.373).

11.12.6.1

An Important Lemma

We will also need the following important result, relating the derivatives of the variational vector, t 𝛼 , and the curvature vectors, k𝛼 :

t ϕ, ̄

̄ ̄ as the axial vectors of t K𝛼 , 𝛼 = 1, 2 and t 𝚽, Lemma LD1: Defining t k𝛼 , 𝛼 = 1, 2 and t ϕ respectively, we have: t 𝚽, ̄

𝛼

̄ 𝛼 ϕ, ⏟⏟⏟ t

Effective material curvature vector variation

̄ − t𝚽 ̄ t K𝛼 ], ̄ 𝛼 + [ t K𝛼 t 𝚽 = tK =

t̄

k𝛼 ⏟⏟⏟ Material curvature vector variation

+

t

k𝛼 ⏟⏟⏟ Material curvature vector

𝛼 = 1, 2 ×

̄ ϕ ⏟⏟⏟ t

(11.377)

Material rotation vector variation

̄ ≡ 𝛿 t𝚽 ≡ R ̄ RT ≡ 𝛿R RT , we have: Proof LD1: Using relations (11.353) and the definition of t 𝚽 ̄ t , 𝛼 = R, ̄ R) RT , 𝛼 = t K ̄ R RT + t 𝚽 ̄ R RT ( t K)T ̄ 𝛼 RT + ( t 𝚽 ̄ 𝛼 R RT + t K𝛼 t 𝚽 𝚽 Now, noting the orthogonality property of a rotation tensor: R RT = I, and skew-symmetry property: ( t K)T = − t K completes the proof of the first part. Recognition of the Lie bracket property: for A, B skew, and a, b, the corresponding axial vectors, [AB − BA] = [a × b], for the second part, completes the proof. Exercise: Detail the steps in the LD1 proof.

Computation of Nonlinear Structures

840

Solution t 𝚽, ̄

𝛼

̄ R RT + t 𝚽 ̄ R RT ( t K𝛼 )T = t K ̄ − t𝚽 ̄ t K𝛼 ̄ 𝛼 R RT + t K𝛼 t 𝚽 ̄ 𝛼 + t Kt 𝚽 = tK 𝛼 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏟ Lie bracket

̄ 𝛼 = t k̄ 𝛼 − t ϕ ̄ × t k𝛼 ⇒ t ϕ,

11.12.6.2

The Central Lemma

We will need the following most important lemma, relating the derivatives, θ, 𝛼 , and the variation, ̄ or the increment, Δθ (appears in the linearized virtual functional later in the treatment of a θ, Newton-type iterative algorithm), of the rotation vector, θ, for deriving the geometric stiffness operator and proving its symmetry under the conservative system of loading, presented later. Lemma LD2: Given W, c1 and c2 as in equation (11.265), we have: ̄ ̄ θ, 𝛼 = X𝛼 θ, 𝛼 = 1, 2 W ΔW θ, 𝛼 = X𝛼 Δθ

(11.378)

where X𝛼 ≡ −c1 𝚯, 𝛼 + c2 {[θ ⊗ θ, 𝛼 ] − [θ, 𝛼 ⊗ θ] + (θ ∙ θ, 𝛼 )I} + V [θ, 𝛼 ⊗ θ] 1 1 V ≡ −c2 I + c3 𝚯 + c4 𝚯2 , c3 ≡ 2 {(1 − 𝜃 2 c2 ) − 2c1 , c4 ≡ 2 (c1 − 3c2 ) 𝜃 𝜃

(11.379)

where θ̄ and Δθ are virtual (variational) and incremental rotation vectors, respectively. 1 ̄ and taking one parameter variation of c and c , Proof LD3: First noting that 𝜃̄ = (θ ∙ θ) 1 2 𝜃 ̄ and we have, from equations (11.370), (11.373) and (11.379): c1 , 𝛼 = c3 (θ ∙ θ, 𝛼 ), c̄ 1 = c3 (θ ∙ θ) ̄ Now, rewriting the variation of W as defined in the first equality c2 , 𝛼 = c4 (θ ∙ θ, 𝛼 ), c̄ 2 = c4 (θθ). of equation (11.373), and multiplying both sides by θ, 𝛼 , we get: ̄ θ, ̄ + 𝚯Θ) ̄ θ, 𝛼 + (c3 𝚯 + c4 𝚯2 )(θ ∙ θ) ̄ θ, 𝛼 + c2 (𝚯Θ ̄ θ, 𝛼 = −c1 𝚯 W 𝛼

(11.380)

̄ + [θ̄ ⊗ θ] − 2(θ ∙ θ)I ̄ ̄ ̄ + 𝚯Θ ̄ = [θ ⊗ θ] Noting the identities: 𝚯Θ and [θ ⊗ θ]θ, s = [θ ⊗ ̄ we have: θ, ]θ, s

̄ ̄ θ, 𝛼 = θ̄ × θ, 𝛼 = −θ, 𝛼 × θ̄ = −𝚯, 𝛼 θ; 𝚯 ̄ ̄ ̄ ̄ (𝚯Θ + 𝚯Θ) θ, 𝛼 = θ × θ × θ, 𝛼 + θ × θ × θ, 𝛼 (11.381) ̄ = (θ̄ ∙ θ, 𝛼 )θ − (θ̄ ∙ θ)θ, 𝛼 + (θ ∙ θ, 𝛼 )θ̄ − (θ ∙ θ)θ, 𝛼 { } ̄ = (θ ⊗ θ, 𝛼 ) − (θ, 𝛼 ⊗ θ) + (θ ∙ θ, 𝛼 )I θ̄ − (θ, 𝛼 ⊗ θ)θ; Now, inserting relations (11.381) in equation (11.380), we get: { ̄ θ, 𝛼 = W

−c1 𝚯, 𝛼 + c2 {(θ ⊗ θ, 𝛼 ) − (θ, 𝛼 ⊗ θ) + (θ ∙ θ, 𝛼 )I} +(−c2 I + c3 𝚯 + c4 𝚯2 )(θ, 𝛼 ⊗ θ)

} θ̄

(11.382)

Nonlinear Shell

841

Next, insertion of the definitions for X𝛼 , 𝛼 = 1, 2 and V of equation (11.379) in equation (11.382) completes the proof for the first part of equation (11.378), and replacement of (∙) terms by the Δ(∙), completes the proof for the increment.

11.12.7

Where We Would Like to Go

We have now introduced various parameters and derived all the necessary relationships anticipating the next step of developing the virtual work equation for the shell reference surface. Clearly, then, we should move on to:

r the computational virtual work principle. 11.13 Computational Virtual Work Equations 11.13.1 What We Need to Recall

r The real configuration space of a shell is characterized by ( t d(S1 , S2 , t), t R(S1 , S2 , t)). r The virtual space for a shell may be described by the virtual displacement vector and the

r

̄ ̄ 1 , S2 ) = G t R ̄ GT ), with the com̄ 1 , S2 ) = G t d, R(S virtual rotational tensor fields, (d(S t t t t t ̄ ≡ 𝛿 𝚽) where 𝚽(S ̄ 1 , S2 ), the instantaponent virtual entities defined as ( d̄ ≡ 𝛿 d, 𝚽 neous (variational) rotational skew-symmetric matrix describing the one parameter variation t R(S ̄ ≡ tR ̄ 1 , S2 ) ≡ 𝛿 t R(S1 , S2 ), of the rotation matrix t R defined by: t 𝚽 ̄ t RT with t ϕ ̄ as its 3 t t t t ̄ = ϕ ̄ ϕ ̄ × h, ∀h ∈ R , and 𝚽 ̄ = 0. axial vector, that is, 𝚽h The computational dynamic balance equations in component vector form are: G( t F𝛼 |𝛼 + t F̂ 𝛼 − t L) = 0 ∙

(11.383)

∙

̂ 𝛼 − t A) = 0 G( t M𝛼 |𝛼 + a𝛼 × t F𝛼 + t M or G( t F𝛼 , 𝛼 + 𝜿 0𝛼 × t F𝛼 + t F̂ 𝛼 − t L) = 0 ∙

∙

̂ 𝛼 − t A) = 0 G( t M𝛼 , 𝛼 + 𝜿 0𝛼 × t M𝛼 + {𝜿 0𝛼 × d + e𝛼 } × t F𝛼 + t M

r

(11.384)

where G = [ G1 G2 N ] is the orthonormal matrix with the base vectors as the columns of the matrix such that GT G = I. The force and moment component vectors in undeformed and current deformed coordinate system are related by the force and moment rotation transformation: t F𝛼

= t R n F𝛼 ,

t M𝛼

t R n M𝛼

=

𝛼 = 1, 2

(11.385)

where the superscript t over an object that reminds us that the object is expressed in the undeformed G1 , G2 , G3 ≡ N orthonormal frame, may be dropped occasionally for clarity; a superscript n identifies variables in the rotated frame, g1 = t R G1 , g2 = t R G2 , g3 = t R G3 where t R is the current rotation tensor measured in the undeformed coordinate system; the context should help us recall the appropriate frame of reference.

Computation of Nonlinear Structures

842

r The covariant derivatives of the displacement and rotation component vectors: t d| 𝛼 t θ| 𝛼

= t d, 𝛼 + k0𝛼 × t d,

𝛼 = 1, 2

= t θ, 𝛼 + k0𝛼 × t θ

(11.386)

r Likewise, we have the time derivatives of the momentum, L(S1 , S2 , t), and the angular momentum, A(S1 , S2 , t), in terms of the components measured in the undeformed coordinate system:

L(S1 , S2 , t) = G(S1 , S2 ) t L(S1 , S2 , t), ∙

∙∙

∙∙

L = M0 t d ≈ (𝜌0 H) t d 1 2 A(S , S , t) = G(S1 , S2 ) t A(S1 , S2 , t), ( ) ∙ ∙ H3 t ∙ t t ω A = I 0 ω ≈ 𝜌0 12 t

(11.387)

r The vector triple product property: a × b ∙ c = b × c ∙ a = c × a ∙ b, ∀a, b, c vectors. r In what follows notationally, all virtual entities are identified with a superscripted bar over them.

11.13.2

Boundary Conditions

Just as in the reduction of the 3D balance laws of a shell-like body by integration through the thickness direction produced the exact counterparts for a 2D shell surface that are valid for its interior, M, we need to determine the exact statically equivalent boundary conditions for the shell edge from the prescribed boundary conditions on the side surface, M S , of a 3D shell-like body. The boundary conditions on a 3D shell-like body can basically be of the following types:

r Traction boundary condition: Tractions are specified on all or part of the enclosing edge (the side-surface) of a shell-like body. If the specification involves the entire edge, some sort of constraint must be imposed against the rigid body movement. For tractions specified partially on some patches of the side surfaces, we collect all the areas and designate the aggregate area as the set MfS ⊂ M S . Let the specified 3D traction, t̂(S, 𝜂), (S, 𝜂) ∈ MfS ⊂ M S , be given which of course satisfies the relationship: t̂(S, 𝜂) = P(S, 𝜂) 𝜈(S, 𝜂) where 𝜈(S, 𝜂) is the outward normal at any point (S, 𝜂) on the side surface with S as the arc length parameter along the boundary curve defining the side surface with shell reference normal on it. In this case, we may obtain the 2D counterparts by the through-the-thickness integration as: F ∙ 𝜈0 = F𝛼 ⊗ G𝛼 ∙ 𝜈0𝛼 ⊗ G𝛼 = F𝛼 𝜈0𝛼 = F𝜈 = F̂ 𝜈 𝜂t

t̂(S, 𝜂) 𝜂 𝜇S d𝜂, S ∈ 𝜕Mf ∫𝜂b ̂𝜈 M ∙ 𝜈0 = M𝛼 ⊗ G𝛼 ∙ 𝜈0𝛼 ⊗ G𝛼 = M𝛼 𝜈0𝛼 = M𝜈 = M ≡

≡

𝜂t

∫𝜂b

h(S, 𝜂) × t̂(S, 𝜂) 𝜂 𝜇S d𝜂

(11.388)

Nonlinear Shell

843

where 𝜕Mf = M ∩ MfS is that part of the shell edge where the shell statically equivalent tractions will be considered; for a thin shell with h(S, 𝜂) = 𝜂 N(S), from equation (11.388), we will have: F ∙ 𝜈0 = F𝛼 ⊗ G𝛼 ∙ 𝜈0𝛼 ⊗ G𝛼 = F𝛼 𝜈0𝛼 = F𝜈 = F̂ 𝜈 ≡

𝜂t

∫𝜂 b

t̂(S, 𝜂) 𝜂 𝜇S d𝜂,

S ∈ 𝜕Mf

̂𝜈 M ∙ 𝜈0 = M𝛼 ⊗ G𝛼 ∙ 𝜈0𝛼 ⊗ G𝛼 = M𝛼 𝜈0𝛼 = M𝜈 = M ≡ N(S) ×

𝜂t

∫𝜂b

(11.389)

𝜂 t̂(S, 𝜂) 𝜂 𝜇S d𝜂

r Kinematic boundary condition: Displacements are specified on all or part of the side surface.

r

For displacements specified partially on some patches of the side surfaces, we collect all the areas and designate the aggregate area as the set MdS ⊂ M S . We will admit only those displacements that have finite strain energy. Now, if the displacements are homogeneous, we simply set: t d = 0, t R = 0, ( t d, t R) ∈ 𝜕Md where 𝜕Md ⊂ 𝜕M is that portion of the boundary curve, 𝜕M, that is also common with MdS . For inhomogeneous displacement specifications, ̂ t R = R, ̂ we will set: t d = d, ( t d, t R) ∈ 𝜕Md ; the hat superscript means the quantities are ̄ ≡ prescribed. In this case, the kinematically admissible virtual displacements, ( t d̄ ≡ 𝛿 t d, R 𝛿 t R) will be identically zero on 𝜕Md . Thus, in the subsequent virtual work equation, we must have the boundary integral on 𝜕Md identically vanished. The actual finite element solution for a shell with an inhomogeneous displacement condition will be given as an additive sum of two displacement functions with one part as the solution of the relevant virtual work equation and the other part as an arbitrary fixed admissible displacement function. Mixed traction and displacement boundary conditions: In the event that the side surface of the 3D shell-like body is in contact with other elastic bodies such as, say, elements modeled as distributed springs, the boundary conditions will be expressed as a linear combinations of both the traction and displacement vectors with the spring stiffness as given. This condition is essentially a combination of the previous two boundary conditions.

11.13.3

Initial Conditions

For a shell semi-discrete finite element formulation (which we have alluded to in our motivation), the virtual work equation eventually reduces to a matrix second-order of ordinary differential equations. Clearly, the typical initial conditions will be the prescribed initial displacement and velocity at the finite element nodes for calculating the time-evolution of the shell deformations and internal reactive stresses. Now, the initial conditions may be written as:

d(0) = d0 . θ(0) = θ0 . ∙

v(0) ≡ d(0) = v0 , ∙

ω(0) ≡ θ(0) = ω0

(11.390)

Computation of Nonlinear Structures

844

11.13.4

Virtual Work from the Vector Balance Equations

Let us consider a shell reference surface of area M bounded by a smooth closed curve, 𝜕M. Let ̄ ϕ) ̄ t ϕ) ̄ is the ̄ be a virtual vector set belonging to the admissible test or virtual space. If ( t d, (d, t t ̄ ̄ ̄ t ϕ) ̄ note that ( t d, ̄ is ̄ = G ϕ; corresponding component vectors, then we have d = G d and ϕ time-independent. Now, with the above background in mind, we multiply equation (11.383) or ̄ and integrate over an arbitrary equation (11.384) by the admissible virtual state vectors d̄ and ϕ, ̄ ϕ), ̄ given as: shell reference surface area, Ω, to get the shell virtual work functional, G(d, ϕ; d, { ̄ ϕ) ̄ ≡− G(d, ϕ; d,

∫

}

̄ ( t F𝛼 |𝛼 + t F̂ 𝛼 − t L) ∙ t d+ ∙

∙

̂ 𝛼 − t A) ∙ t ϕ ̄ ( t M𝛼 |𝛼 + (1̂ 𝛼 + t d|𝛼 ) × t F𝛼 + t M

M

dA (11.391)

̄ tϕ ̄ ∀ admissible t d,

= 0,

where we have applied the fact that GT G = I. Now, by an application of the Green–Gauss (or divergence) theorem for a surface: ∫ t F𝛼 |𝛼 dA = ∫ DIV f S dA = ∮ t F𝛼 ⊗ G𝛼 ∙ 𝜈0𝛽 G𝛽 dS = M

𝜕M

M

∮ t F𝛼 𝜈0𝛼 dS ≡ ∮ t F𝜈 dS where 𝝂 0 = 𝜈0𝛼 G𝛼 is the outward unit normal on the tangent plane

𝜕M

𝜕M

on the boundary curve, 𝜕M, to the shell reference surface area, M; we have also introduced the following definition. Definition:

tF 𝜈

as: t

F𝜈 ≡ f S ∙ 𝝂 0 ⇐ t F𝛼 ⊗ G𝛼 ∙ 𝜈0𝛽 G𝛽 ⇐ t F𝛼 𝜈0𝛼

(11.392)

We can have a similar expression for moments as: ∫ t M𝛼 |𝛼 dA = ∮ t M𝛼 𝜈0𝛼 dS ≡ ∮ t M𝜈 dS 𝜕M

M

with the: Definition:

tM

𝜈

as: t

Then, t d)dS. ̄

we

𝜕M

have:

M𝜈 ≡

m

S ∙ 𝝂 0 ⇐ t M𝛼 ⊗ G𝛼 ∙ 𝜈0𝛽 G𝛽 ⇐ t M𝛼 𝜈0𝛼

(11.393)

̄ 𝛼 dA = ∮ ( t F𝛼 ∙ t d) ̄ 𝜈0𝛼 dS = ∮ ( t F𝛼 𝜈0𝛼 ∙ t d)dS ̄ ∫ ( t F𝛼 ∙ t d)| = ∮ ( t F𝜈 ∙ 𝜕M

M

𝜕M

𝜕M

Now, by applying the chain rule of differentiation and rearranging terms, we get: −

∫

̄ dA = − ( t F𝛼 |𝛼 ∙ t d)

M

∫ M

̄ 𝛼 ) dA − ( t F𝛼 ∙ t d|

∮

̄ dS ( t F𝜈 ∙ t d)

(11.394)

̄ dS ( t M𝜈 ∙ t ϕ)

(11.395)

𝜕M

Similarly, for the moments, we have: −

∫ M

̄ dA = − ( t M𝛼 |𝛼 ∙ t ϕ)

∫ M

̄ 𝛼 ) dA − ( t M𝛼 ∙ t ϕ|

∮

𝜕M

Next, recalling the commutation property of vector triple product, and applying equations (11.191), (11.192) and (11.387) in equation (11.190) with boundary conditions given by equation

Nonlinear Shell

845

(11.388), we get: ̄ t ϕ) ̄ ≡ GInertial + GInternal − GExternal GDyn ( t d, t R; t d, Inertial

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ } { ∙∙ ∙ t̄ t ̄ t t = ( M0 d ∙ d + I0 ω ∙ ϕ dA ∫ M Internal

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ { t 𝛼 t } ̄ 𝛼 − tϕ ̄ × (1̂ 𝛼 + t d|𝛼 )] + t M𝛼 ∙ t ϕ| ̄ 𝛼 dA + ( F ∙ [ d| ∫ M External

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⎫ ⎧ ) ) ⎪ (t ⎪ (t̂ t̄ t ̂ t ̄ t t t ̂ ∙ ϕ ̄ dA⎬ −⎨ F𝜈 ∙ d + M𝜈 ∙ ϕ dS+ F̂ ∙ d̄ + M ∫ ∮ ⎪ ⎪𝜕M M ⎭ ⎩ f ̄ tϕ ̄ = 0, ∀ admissible t d, (11.396) where we have the angular velocity component vector, t ω, related to the Rodrigues’ rotation component vector, t θ, and identified the inertial virtual work, the internal virtual work and the external virtual work of the virtual work principle as follows: Definition: Inertial virtual work, GInertial , as: { GInertial ≡

∫

∙∙

t̄

∙

}

̄ ( M0 d ∙ d + I0 ω ∙ ϕ t

t

t

dA

(11.397)

M

Internal virtual work, GInternal , as: GInternal ≡

∫

{t

F𝛼 ∙

(t

) } ̄ 𝛼 − tϕ ̄ × ∙ (1̂ 𝛼 + t d|𝛼 ) + t M𝛼 ∙ t ϕ| ̄ 𝛼 dA d|

(11.398)

M

External virtual work, GExternal , as: GExternal ≡

∮

𝜕Mf

{t

̂ 𝜈 ∙ tϕ ̄ F̂ 𝜈 ∙ t d̄ + t M

}

dS+

∫

{t

̂ ∙ tϕ ̄ F̂ ∙ t d̄ + t M

}

dA

(11.399)

M

Remark

r Although from an apparent energy conjugation in the internal virtual work equation (11.398),

̄ 𝛼 − tϕ ̄ × (1̂ 𝛼 + t d|𝛼 ) and t ϕ| ̄ 𝛼 present themselves as the force and the expressions, t d| moment virtual strains, it turns out that there exist no functions whose variations can possibly be identified with these functions; in other words, these are not truly the virtual strains

Computation of Nonlinear Structures

846

derivable from some real strains. Fortunately, however, if we rotate the force and moment component vectors by pre-multiplying with t RT to obtain forces and moments as the components in the rotating deformed frame, we achieve our goal of establishing the true virtual nature of the newly generated and energy conjugated strain functions; this is precisely what we intend to do next. However, it is important to note that all relevant quantities are measured in the Lagrangian or material coordinate system.

11.13.5

Virtual Translational and Rotational Strain Vectors

Now, we introduce the rotation matrix, t R, to express the internal virtual work in the rotated frame, and by applying equation (11.385) in equation (11.194) and noting that t R is orthonormal, that is, t RT t R = I , we get: GInternal = ∫

{t

M

⎧ ⎪ ⎪ ⎪ =∫ ⎨ M ⎪ ⎪ ⎪ ⎩

) } ̄ 𝛼 − tϕ ̄ × (1̂ 𝛼 + t d|𝛼 ) + t RT t M𝛼 ∙ t R t ϕ| ̄ 𝛼 dA d| { } n 𝛼 ̄ 𝛼 − ( t RT t Φ) ̄ (1̂ 𝛼 + t d|𝛼 ) + ⎫ F ∙ t RT t d| ⏟⏟⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⎪ Real component ⎪ Virtual translational strain vector force vector along ⎪ rotated frame ⎬ dA n 𝛼 ̄ 𝛼 M ∙ t RT t ϕ| ⎪ ⏟⏟⏟ ⏟⏞⏞⏟⏞⏞⏟ ⎪ Real component Virtual rotational ⎪ moment vector along strain vector ⎭ rotated frame

RT t F𝛼 ∙ t RT

(t

(11.400) Next, applying the definition of the covariant derivatives as given by equation (11.386) and ̄ (being skew-symmetric: t Φ ̄ T = − t Φ) ̄ as the variation of the rotation matrix, interpreting t Φ t R, related by: t 𝚽 ̄ ≡ tR ̄ t R)T = t RT t 𝚽 ̄ T = − t RT t 𝚽, ̄ we can rewrite and ̄ t RT ⇒ ( t R) ̄ T = ( t𝚽 define the virtual strain vectors as follows. Definition

r The virtual translational (axial & shear) component strain vector, n β̄ 𝛼 , nβ ̄

𝛼

𝛼 = 1, 2:

̄ 𝛼 − ( t RT t Φ) ̄ (1̂ 𝛼 + t d|𝛼 ) ≡ t RT t d| = t RT (1̂ 𝛼 + t d|𝛼 ) + t RT (1̂ 𝛼 + t d|𝛼 ) = t RT (1̂ 𝛼 + t d|𝛼 )

(11.401)

and

r The virtual rotational (bending) component strain vector, n χ̄ 𝛼 , nχ ̄

𝛼

𝛼 = 1, 2:

̄ 𝛼 = t RT { t ϕ, ̄ 𝛼 + k0𝛼 × t ϕ} ̄ ≡ t RT t ϕ| t T t 0 ̄ ̄ = R { k𝛼 − Φ ( k + k𝛼 )} 𝛼

̄ ( k0 + k𝛼 ) = t RT k̄ 𝛼 + (− t RT t Φ) 𝛼

= t RT (k0𝛼 + k𝛼 ) + t RT ( k0𝛼 + k𝛼 ) = t RT ( k0𝛼 + k𝛼 )

(11.402)

Nonlinear Shell

847

̄ k𝛼 with k0 ̄ 𝛼 = k̄ 𝛼 − t Φ In equation (11.402), we have used the virtual derivative relation: t ϕ, 𝛼 𝛼 as the initial curvature vector, and k𝛼 as the current curvature vector along the S , 𝛼 = 1, 2, direction, respectively. Also, recall that all entities with barred superscript are virtual in nature. Finally, introducing equations (11.397) through (11.402) in equation (11.396), we get the desired shell virtual functional as: ̄ t ϕ) ̄ ≡ GInertial + GInternal − GExternal GDyn ( t d, t R; t d, Inertial

Internal

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ } { ∙∙ { n 𝛼 n } ∙ t̄ t ̄ t t ̄ + n M𝛼 ∙ n χ̄ = ( F ∙ β ( M0 d ∙ d + I0 ω ∙ ϕ dA + 𝛼 𝛼 dA ∫ ∫ M

M External

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⎫ ⎧ ) ) ⎪ (t ⎪ (t̂ t̄ t ̂ t ̄ t t t ̂ ∙ ϕ ̄ dA⎬ −⎨ F𝜈 ∙ d + M𝜈 ∙ ϕ dS+ F̂ ∙ d̄ + M ∫ ∮ ⎪ ⎪𝜕M M ⎭ ⎩ f t t ̄ ϕ ̄ = 0, ∀ admissible d, (11.403)

Note that in the internal virtual work, the forces and the moments, and the corresponding conjugate translational and bending virtual strain vectors, although measured in undeformed coordinates, are all expressed as the components in the current deformed (rotated) local coordinate system, that is, the co-spinning coordinate system. One of the main consequences is that the constitutive relationship must also be considered in this deformed coordinate system which is the co-spinning coordinate system: n G1 , n G2 , N, in the undeformed state that becomes g1 , g2 , n in the current deformed configuration.

11.13.6

Static Virtual Work Functional and Virtual Strain Tensors

Just as the virtual strain vectors in the co-rotated coordinates have been represented by the undeformed coordinates, their tensor counterparts, as we shall see, are also most suitably represented in the mixed bases. Remark

r For this, we will consider only the static part of the invariant form of the balance equations. Thus, let the static balance equation be: DIV f P + F̂ = 0 ̂ =0 DIV m P + c0 , 𝛼 × ( f P ∙ G𝛼 ) + M

(11.404)

Computation of Nonlinear Structures

848

̄ and As before, we multiply equation (11.404) by any admissible virtual state vectors d̄ and ϕ, integrate over an arbitrary shell reference surface area, M, and focus on the shell static virtual ̄ ϕ), ̄ given as: work functional, GStatic (d, R; d, ̄ ϕ) ̄ ≡− GStatic (d, R; d,

{ ∫ M

= 0,

} ̂ + ( DIV m P + ϕ ̂ ̄ ∙ c0 , 𝛼 × ( f P ∙ G𝛼 ) + M) d̄ ∙ ( DIV f P + F) dA

̄ ϕ ̄ ∀ admissible d, (11.405)

Let us recall the product rule of divergence of a vector times a tensor: −d̄ ∙ DIV f P = −DIV(d̄ ∙ f P) + f P: GRAD d̄ ̄ ∙ DIV m P = −DIV(ϕ ̄ ∙ m P) + m P: GRAD ϕ ̄ −ϕ

(11.406)

̄ 𝛼 ⊗ G𝛼 , and GRAD ϕ ̄ ≡ ϕ, ̄ 𝛼 ⊗ G𝛼 with GRAD where we have, by definition: GRAD d̄ ≡ d, as the material surface gradient, and : is the usual double contraction for tensors. Let us also recall the surface Green–Gauss or divergence theorem: ∫ DIV (∙) dA = ∮ (∙) ∙ 𝝂 dS where 𝝂 is 𝜕M

M

the outward unit normal on the shell tangent plane at an enclosing closed boundary, 𝜕M, of the shell reference surface area, M. Now applying definitions given by equations (11.392) and (11.393), and relationships given by equations (11.406), we have from equation (11.405) with the boundary conditions (11.388): ̄ ϕ) ̄ GStatic (d, R; d, =

∫

{f

P : GRAD d̄ +

m

} ̄ + ϕ ̄ ∙ c0 , 𝛼 × ( f P ∙ G𝛼 ) dA P : GRAD ϕ

M

−

̄ ∙ ( m P ∙ 𝝂)} dS − {d̄ ∙ ( f P ∙ 𝝂) + ϕ

∮

𝜕M

=

∫

{ ∫

} ̂ dA ̄ ∙ M d̄ ∙ F̂ + ϕ

Ω

{f

} ̄ + F̂ − ( DIV m P + ϕ ̄ ∙ c0 , 𝛼 × ( f P ∙ G𝛼 ) dA P : GRAD d̄ + m P : GRAD ϕ

M

−

∮

𝜕Mf

̂ 𝜈 } dS − ̄ ∙ M {d̄ ∙ F̂ 𝜈 + ϕ

{ ∫

} ̂ dA = 0, ̄ ∙M d̄ ∙ F̂ + ϕ

̄ ϕ ̄ ∀ admissible d,

Ω

(11.407) ̄ ⊗ G𝛼 = −ϕ ̄ × c0 , 𝛼 ⊗ G𝛼 = −ϕ ̄ × GRADc0 = GRADc0 × ϕ, ̄ we Now, noting that c0 , 𝛼 × ϕ have: ̄ ∙ c0 , 𝛼 × ( f P ∙ G𝛼 ) ϕ ̄ × c0 , 𝛼 ) = ( f P ∙ G𝛼 ) ⊗ G𝛼 : (ϕ ̄ × c0 , 𝛼 ) ⊗ G𝛼 (11.408) = ( f P ∙ G𝛼 ) ∙ (ϕ ̄ × GRAD c0 = − f P : GRAD c0 × ϕ ̄ = (G𝛼 ⊗ G𝛼 ) ⊗ f P : ϕ

Nonlinear Shell

849

Next, using equation (11.408) in equation (11.407), we get: ̄ ϕ) ̄ = GStatic (d, R; d,

∫

̄ + {f P : (GRAD d̄ + GRAD c0 × ϕ)

m

̄ dA P : GRAD ϕ}

M

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ Internal virtual work

−

∮

̂ 𝜈 } dS + ̄ ∙ M {d̄ ∙ F̂ 𝜈 + ϕ

𝜕Mf

∫

̂ dA ̄ ∙ M} { d̄ ∙ F̂ + ϕ

M

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ External virtual work

̄ ϕ ̄ ∀ admissible d,

= GInternal − GExternal = 0,

(11.409) Let us note that recalling relation f P ≡ F𝛼 ⊗ G𝛼 , as before in the translational internal work part in equation (11.403), we have: ∫

̄ } dA { f P : (GRAD d̄ + GRAD c0 × ϕ)

M

=

∫

̄ 𝛽 + c0 , 𝛽 × ϕ) ̄ ⊗ G𝛽 } dA {F𝛼 ⊗ G𝛼 : ( d,

M

=

∫

̄ 𝛼 + c0 , 𝛼 × ϕ) ̄ } dA = {F𝛼 ∙ ( d,

M

=

∫

∫

̄ 𝛼 + (G𝛼 + d, ̄ 𝛼 ) × ϕ) ̄ } dA {F𝛼 ∙ ( d,

M t̄

̄ 𝛼 ) × t ϕ) ̄ } dA { F ∙ ( d|𝛼 + (1̂ 𝛼 + t d| t 𝛼

(11.410)

M

=

∫

̄ 𝛼 + (−RT Φ)( ̄ 𝛼 )} } dA ̄ 1̂ 𝛼 + t d| { RT t F𝛼 ∙ { RT t d|

M

=

∫

̄ 𝛼 ) } dA = { n F𝛼 ∙ RT (1̂ 𝛼 + t d|

M

∫

̄ } dA { n F𝛼 ∙ n β 𝛼

M

with all terms and relations defined earlier in the investigation. Similarly, we can recover the expression for the rotational internal work part of equation (11.403) as: ∫

̄ } dA = { m P : (GRAD ϕ)

M

∫

̄ 𝛽 ⊗ G𝛽 } dA {M𝛼 ⊗ G𝛼 : ϕ,

M

=

∫

̄ 𝛼 } dA = {M𝛼 ϕ,

M

=

∫ ∫ M

̄ 𝛼 } dA { t M𝛼 ∙ t ϕ|

M Tt

{R M

𝛼

̄ 𝛼 } dA = R ϕ| Tt

M

=

∫

(11.411) ∫ M

{ n M𝛼

n̄

χ𝛼 } dA

{ n M𝛼 ∙ RT (k0𝛼 + k𝛼 ) } dA

Computation of Nonlinear Structures

850

11.13.6.1

Virtual Translational (Axial and Shear) and Rotational (Bending) Strain Tensors

Now we are in a position to define the virtual strain tensors in material bases as follows: ̄ Define: Virtual translational (axial & shear) strain tensor, n E: n

̄ ⊗ G𝛼 Ē ≡ n β 𝛼

(11.412)

̄ Virtual rotational (bending) strain tensor, n K: n

̄ ≡ n χ̄ 𝛼 ⊗ G𝛼 K

(11.413)

Finally, with the definitions given by equations (11.412) and (11.413), we have from equation (11.409) after applying equations (11.410) and (11.411): ̄ ϕ) ̄ = GStatic (d, R; d,

∫

̄ } dA {RT f P : n Ē + RT m P : n K

M

⎛ ⎞ ⎜ ⎟ ̄ ̂ ̂ ̄ ̂ ̂ ̄ ∙ M𝜈 } dS + ̄ ∙ M} dA⎟ −⎜ {d ∙ F𝜈 + ϕ {d ∙ F+ ϕ ∮ ∫ ⎜𝜕M ⎟ M ⎝ f ⎠ ̄ ϕ ̄ = 0, ∀ admissible d, (11.414)

Remarks

r We can easily see the equivalence of equation (11.414) described by the virtual strain tensors with equation (11.403) modified for the static situation involving the virtual strain vectors.

r The virtual work functional presented above is true and exact (in the sense of our definitions) irrespective of the shell material properties (i.e. the constitutive law) and the loading system.

11.13.7

Where We Would Like to Go

However, let us note that the virtual functional of equations (11.403) is expressed in terms of ̄ the variational axial vector corresponding to the instantaneous rotational variation vector, t ϕ, ̄ of the rotation matrix, t R. Since we parameterize the rotation matrix by variational matrix, t 𝚽, the Rodrigues rotation vector, t θ, we must express for computational purposes the virtual strains, angular accelerations and the virtual functional as functions of this rotation vector and its virtual ̄ Thus, we must check out: counterpart, t θ.

r computational virtual work: revisited

Nonlinear Shell

851

11.14

Computational Virtual Work Equations and Virtual Strains: Revisited 11.14.1 What We Need to Recall

r The virtual work equation is given as:

̄ t ϕ) ̄ ≡ GInertial + GInternal − GExternal G( t d, t R; t d, Inertial

Internal

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ } { ∙∙ { n 𝛼 n } ∙ t̄ t ̄ t t ̄ + n M𝛼 ∙ n χ̄ = ( F ∙ β dA ( M0 d ∙ d + I0 ω ∙ ϕ dA + 𝛼 𝛼 ∫ ∫ M

M External

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⎫ ⎧ ) ) ⎪ (t ⎪ (t̂ t̄ t ̂ t ̄ t t t ̂ ϕ ̄ dA⎬ −⎨ F𝜈 ∙ d + M𝜈 ∙ ϕ dS + F̂ ∙ d̄ + M ∫ ∫ ⎪ ⎪𝜕M M ⎭ ⎩ f = 0,

̄ tϕ ̄ ∀ admissible t d, (11.415)

Definitions

r The translational virtual strain fields, nβ ̄

𝛼

= t RT

(t

nβ ̄

𝛼,

𝛼 = 1, 2, are given as:

) ( ) ̄ t e𝛼 + t k0 × t d ̄ 𝛼 + t k0 × t d̄ − t RT t 𝚽 d, 𝛼 𝛼 (11.416)

=

t RT (1 ̂𝛼

+

r The bending virtual strain fields, n̄

t d| ) 𝛼

nχ ̄

𝛼,

𝛼 = 1, 2, are given as:

̄ 𝛼 = t RT { t ϕ, ̄ 𝛼 + k0𝛼 × t ϕ} ̄ ≡ t RT ( k0𝛼 + k𝛼 ) χ𝛼 ≡ t RT t ϕ|

(11.417)

r The definition of a rotation tensor, its matrix representation, and algebra, geometry and topology of rotation.

r A rotation matrix, r

t R,

corresponding to the 3D special orthogonal matrix group, SO(3), may be parameterized by the Rodrigues vector, s, the axis of the instantaneous rotation. The rotation matrix, t R(S1 , S2 , t), is related, through Cayley’s theorem, to the Rodrigues vector, s(S1 , S2 , t) = { s1 (S1 , S2 , t) s2 (S1 , S2 , t) s3 (S1 , S2 , t) }T , with the components in the

Computation of Nonlinear Structures

852

G1 , G2 , N frame at any point, (S1 , S2 ), and at a time, t ∈ R+ , parameterized by the arc lengths along the lines of curvature on a shell reference surface by: ⎡ 1 + s2 − s2 − s2 1 2 3 1 ⎢ R = ∑ ⎢ 2(s1 s2 + s3 ) ⎢ 2(s s − s ) 1 3 2 ⎣ where

∑

2(s1 s2 − s3 ) 1 − s21

+ s22

− s23

2(s2 s3 + s1 )

2(s1 s3 + s2 ) ⎤ ⎥ 2(s2 s3 − s1 ) ⎥ 1 − s21 − s22 + s23 ⎥⎦

(11.418)

≡ 1 + ∥ s ∥2

r The Rodrigues vector, s, is orientated along the instantaneous axis of rotation at a point of the shell reference surface. The skew-symmetric matrix, S, corresponding to the Rodrigues vector, s, is given by: ⎡ 0 ⎢ S = ⎢ s3 ⎢ −s ⎣ 2

−s3 0 s1

s2 ⎤ ⎥ −s1 ⎥ 0 ⎥⎦

(11.419)

r Finally, the Rodrigues rotation vector, θ, is the scaled Rodrigues vector related by: s=

( ) 1 𝜃 θ tan 𝜃 2

(11.420)

1

where 𝜃 ≡ ‖θ‖ = (θ ∙ θ) 2 is the length measure of rotation vector. Alternatively, defining θ the unit vector, n ≡ , and the corresponding skew-symmetric matrix, N, such that N h = 𝜃 n × h, ∀h ∈ R3 , we have: R = (I − S)−1 (I + S) ( ) 𝜃 n, s = tan 2 ( ) 𝜃 N S = tan 2

(11.421)

r The Rodrigues quaternion and Euler–Rodrigues parameters. r The finite rotation, R, belongs to the spherical curved space, SO(3), and the instantaneous

̄ represented by a skew-symmetric matrix, belongs to and infinitesimal virtual rotation, 𝚽, the tangent space at identity, Lie algebra, and they are related by an exponential map but, computationally, it is more convenient to apply Cayley’s formula: ̄ ̄ −1 (I + 𝚽) R = (I − Φ)

(11.422)

r For notational uniformity, we denote all variations of entities by a bar superscripted on it. Furthermore all entities measured in G1 , G2 , N frame is superscripted as t (∙), and those in the rotating g1 , g2 , g3 coordinate system as n (∙). However, for the sake of uncluttered readability, sometimes the superscripts may be dropped, but the context should help determine the appropriate underlying coordinate system.

Nonlinear Shell

853

r The material “curvature” tensors,

tK , 𝛼 t

𝛼 = 1, 2, and “curvature” vector, t k𝛼 , 𝛼 = 1, 2, as:

K𝛼 ≡ R, 𝛼 RT ,

𝛼 = 1, 2

(11.423)

𝛼 = 1, 2

(11.424)

W( t θ) ≡ I + c1 t 𝚯 + c2 t 𝚯2 , 1 − cos 𝜃 , c1 ( t θ) = 𝜃2 𝜃 − sin 𝜃 c2 ( t θ) = 𝜃3

(11.425)

t

k𝛼 = W(θ) t θ, 𝛼 ,

where we have:

r Definitions:

r

with t 𝚯 is the skew-symmetric matrix corresponding to the axial vector, t θ, such that = 0. ̄ ≡ 𝛿 t 𝚽, and first variation vector, t ϕ ̄ ≡ 𝛿 t ϕ, as: The material first variation tensor, t 𝚽 t𝚯 tθ

t

̄ ≡ 𝛿 t𝚽 ≡ R ̄ RT ≡ 𝛿R RT 𝚽

(11.426)

̄ = W(θ) t θ̄ ϕ

(11.427)

t

r The angular velocity component vector, vector,

t θ,

t ω,

is related to the Rodrigues rotation component

by: t

∙

ω = W(θ) t θ

r The angular acceleration component vector,

t

(11.428)

∙

ω, is related to the Rodrigues rotation component

vector, t θ, by: t

r Lemma LD1: Defining respectively, we have:

∙

∙

∙∙

(11.429)

̄ ̄ as the axial vectors of t K𝛼 , 𝛼 = 1, 2 and t 𝚽, 𝛼 = 1, 2 and t ϕ

tk , 𝛼

t 𝚽, ̄

∙

ω = W(θ) t θ + W(θ) t θ

𝛼

t ̄ ϕ, 𝛼 ⏟⏟⏟ Effective material curvature vector variation

̄ − t𝚽 ̄ t K𝛼 ], 𝛼 = 1, 2 ̄ 𝛼 + [ t K𝛼 t 𝚽 = tK t̄ t t ̄ ϕ = + k𝛼 × k𝛼 ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟ Material curvature vector variation

Material curvature vector

Material rotation vector variation

(11.430)

Computation of Nonlinear Structures

854

r The central lemma We will need the following most important lemma, relating the derivatives, θ, 𝛼 , and the ̄ or the increment, Δθ, (appears in the linearized virtual functional later in the variation, θ, treatment of a Newton-type iterative algorithm), of the rotation vector, θ, for deriving the geometric stiffness operator and proving its symmetry under the conservative system of loading, presented later Lemma LD2: Given W, c1 and c2 as in equation (11.265), we have: ̄ ̄ θ, 𝛼 = X𝛼 θ, 𝛼 = 1, 2 W ΔW θ, 𝛼 = X𝛼 Δθ

(11.431)

where X𝛼 ≡ −c1 𝚯, 𝛼 + c2 {[θ ⊗ θ, 𝛼 ] − [θ, 𝛼 ⊗ θ] + (θ ∙ θ, 𝛼 )I} + V [θ, 𝛼 ⊗ θ] V ≡ −c2 I + c3 𝚯 + c4 𝚯2 1 c3 ≡ 2 (1 − 𝜃 2 c2 ) − 2c1 𝜃 1 c4 ≡ 2 (c1 − 3c2 ) 𝜃

(11.432)

where θ̄ and Δθ are virtual (variational) and incremental rotation vectors, respectively.

r Notationally, the bracket [∙] denotes the skew-symmetric matrix corresponding to the argument, that is, [a] ≡ a× for an axial vector, a, such that [a] a = 0.

11.14.2

Virtual Functional and Virtual Strains Revisited

̄ from all the virtual strains and the virtual functional, and replace Here we intend to eliminate t ϕ t ̄ them with θ, the Rodrigues rotation vector.

11.14.3

Inertial Virtual Functional Revisited

Using the relationships in equation (11.427) and equation (11.429), we get: { GInertial ≡

∫

( M0

t

} ̄ d ∙ d + I0 W (W θ + W θ) ∙ θ dA

∙∙

t̄

T

∙

t

∙

t

∙∙

t

(11.433)

M

11.14.4

Internal Virtual Functional and Virtual Strains Revisited

We would also like to express the translational and the bending virtual strains in computationally suitable matrix form.

Nonlinear Shell

11.14.4.1

855

Computational Translational Virtual Strain in Matrix Form

̄ = W t θ̄ from equation (11.427), and using definitions a𝛼 ≡ e𝛼 + Using the relationship, t ϕ 0 k𝛼 × d where e𝛼 ≡ d, 𝛼 + 1̂ 𝛼 , 𝛼 = 1, 2 with 1̂ 1 ≡ { 1 0 0 }T , 1̂ 2 ≡ { 0 1 0 }T , we have: ̄ then, we can rewrite the expression for ̄ t e𝛼 + t k0 × t d) = RT [a𝛼 ] t ϕ ̄ = RT [a𝛼 ]W t θ; −RT t 𝚽( 𝛼 ̄ , 𝛼 = 1, 2, of equation (11.416) as: the translational virtual strain field, n β 𝛼

{

nβ ̄ nβ ̄

11.14.4.2

} 1

[ =

t RT

0

2

0 t RT

][

[k01 ] [a1 ]W

I

0 0

[k02 ] [a2 ]W 0 0

I

⎧ ⎪ ⎪ ]⎪ 0 ⎪ ⎨ 0 ⎪ ⎪ ⎪ ⎪ ⎩

td ̄1 ⎫ ⎪ tθ ̄ ⎪ 1 ⎪ t d, ̄ 1⎪ t θ, ̄ ⎬ 1⎪ t d, ̄ 2⎪ ⎪ t θ, ̄ ⎪ 2

(11.434)

⎭

Computational Rotational Virtual Strain in Matrix Form

̄ 𝛼 = t k̄ 𝛼 + t k𝛼 × t ϕ ̄ from the second of equation (11.430), t k𝛼 = Using the relationship, t ϕ, ̄ ̄ W θ, 𝛼 , and the central lemma: W θ, 𝛼 = X𝛼 θ, 𝛼 = 1, 2 of equation (11.378), we have: nχ ̄𝛼

} { ̄ t k0 = t RT t k̄ 𝛼 + ( t k𝛼 + t k0 ) × t ϕ ̄ 𝛼 − t RT t 𝚽 ̄ = t RT t ϕ, 𝛼 𝛼 } { ̄ + [ t k + t k0 ]W θ̄ ̄ θ, 𝛼 + W θ, = t RT W 𝛼 𝛼 𝛼 } { ̄ + [ t k + t k0 ]W θ̄ = t RT X𝛼 θ̄ + W θ, 𝛼 𝛼 𝛼 { } ̄ = t RT (X𝛼 + [ t k𝛼 + t k0𝛼 ]W)θ̄ + W θ, 𝛼

(11.435)

Now, introducing the following definition. Definition

r The total current “curvature” vectors, kc , 𝛼

𝛼 = 1, 2,

t c k𝛼

≡ t k𝛼 + t k0𝛼

(11.436)

we have in matrix form:

{

nχ ̄ nχ ̄

} 1 2

[ =

t RT

0

0

t RT

][

0 X1 + [ t kc1 ] W 0 X2 + [ t kc2 ] W

0 W 0 0

0 0

⎧ ⎪ ⎪ ]⎪ 0 ⎪ ⎨ W ⎪ ⎪ ⎪ ⎪ ⎩

td ̄1 ⎫ ⎪ tθ ̄ ⎪ 1 ⎪ t d, ̄ 1⎪ t θ, ̄ ⎬ 1⎪ t d, ̄ 2⎪ ⎪ t θ, ̄ ⎪ 2

⎭

(11.437)

Computation of Nonlinear Structures

856

11.14.4.3

Virtual Strain–Displacement

By introducing the following definitions, Definitions

r the r r

̂̄ ≡ {d̄ θ̄ d, ̄ ̄ 1 θ, ̄ (18 × 1) virtual generalized deformation function H(d) 1 d, 2 ̄θ, }T , 2 the total current “curvature” skew-symmetric matrices [kc𝛼 ] ≡ [k0𝛼 ] + [ t k𝛼 ], 𝛼 = 1, 2, ⌣ nχ nχ ̄ ̄ ̄ 1 nβ ̄ 2 }T , and we may express the the virtual generalized strain vector, ε̄ ≡ { n β 1 2 virtual strain–displacement relations in compact matrix form as: ⌣

̄ = E(d, θ) H(d, ̄ ̄ θ) ̄ θ) ̄ θ; d, ε(d,

(11.438)

where, we have introduced the following definition, Definition: The virtual strain–displacement matrix, E(d, θ): T

⎡R ⎢ 0 E(d, θ) = ⎢ ⎢ 0 ⎢ ⎣ 0

0

0

RT

0

0

RT

0

0

[a1 ]W I 0 0 0 ⎤ ⎡ [k01 ] ⎥ ⎢ c 0 X1 + [k1 ]W 0 W 0 0 ⎥⎢ [a2 ]W 0 0 I 0 ⎥ ⎢ [k02 ] ⎥⎢ c T X2 + [k2 ]W 0 0 0 R ⎦⎣ 0

0⎤ 0 ⎥ ⎥ 0⎥ ⎥ W⎦

(11.439)

where, R(θ), W(θ), X𝛼 (θ), 𝛼 = 1, 2 matrices are as defined in equations (11.348) with (11.350), (11.265), and (11.379), respectively, and I, is the 3 × 3 identity matrix.

11.14.5

External Virtual Functional Revisited

A definitive form of the external virtual work depends on the basic nature of the externally applied loading system. In what follows, we will consider two types of loading system.

11.14.5.1

Quasi-Static Proportional Loading

This type of loading system essentially results in the static analysis of a shell structure in the nonlinear regime; in this case, the linear and the angular momentum of a shell are assumed identically zero, that is, the inertial virtual work vanishes identically: GInertia ≡ 0. Thus, In order that we can produce a concrete solution strategy for the quasi-static analysis of a shell, we need to characterize the external virtual work more precisely. As indicated earlier, the exact expression for the external virtual work depends on the forces applied to a shell. We may take the distributed forces and the moments along the shell body, and the forces and moments along the edge boundary curve to be purely proportional, with the former being only functions of the shell arc length parameters. In other words, we assume that the prescribed forces and moments are functions of a single real parameter, 𝜆 ∈ R. Clearly, within this assumption of one parameter family of forces and moments, various possibilities exist such as the

Nonlinear Shell

857

combination of a constant, b1 (S1 , S2 , 𝜂), and a proportional distributed load, b2 (S1 , S2 , 𝜂), that is, b(𝜆, S1 , S2 , 𝜂) ≡ b1 (S1 , S2 , 𝜂) + b2 (𝜆, S1 , S2 , 𝜂) where 𝜆 is the constant of proportionality for the distributed loading. Similarly, displacement state dependent loadings, that is, non-conservative loadings such as the follower forces, and so on, can also be considered through an additional term reflecting the appropriate differentiations. We will only consider conservative loading system for static analysis. The utility of the choice of a one-parameter family of forces and moments is that by specializing the external virtual work equations for this family, we can nonlinearly trace the configuration path of a shell. With these in mind, we may include the load proportionality parameter, 𝜆, as a ̄ as: ̄ t θ), formal parameter, and rewrite the external virtual work functional, GExternal (𝜆; t d, ̄ ̄ t θ) GExternal (𝜆; t d, ⎧ ⎫ ) ⎪ (t̄ t̂ ⎪ 1 2 t̄ t ̂ 1 2 =⎨ d ∙ F(𝜆, S , S ) + W θ ∙ M(𝜆, S , S ) dA⎬ ∫ ⎪ f ⎪ ⎩M ⎭ ⎧ ⎫ ) ⎪ (t ⎪ t t t ̂ 𝜈 (𝜆, S) dS⎬ = 0, +𝜆 ⎨ d̄ ∙ F̂ 𝜈 (𝜆, S) + W θ̄ ∙ M ⎪ ∮f ⎪ ⎩𝜕M ⎭

̄ t θ̄ ∀ admissible t d,

(11.440) where the superscript t and n indicate that all quantities are expressed in the undeformed frame and the deformed rotated frame, respectively.

11.14.5.2

Time-dependent Dynamic Loading

This type of loading system essentially results in the full dynamic analysis of a shell structure in the linear and nonlinear regimes. For this case also, we will only consider conservative loading system. For these situations and using the relationships in equation (11.427), we rewrite the external virtual work functional as: ̄ = ̄ t θ) GExternal (𝜆; t d,

∫

𝜕Mf

(t

) ̂ 𝜈 ∙ t θ̄ dS + F̂ 𝜈 ∙ t d̄ + WT t M

∫

{t

} ̂ ∙ t θ̄ dA F̂ ∙ t d̄ + WT t M

M

(11.441)

11.14.6

Dynamic Computational Virtual Work Functional

Now, we can express the virtual functional of equation (11.415) in terms of the real displacement, t d, and rotation, t θ, and their virtual counterparts for dynamic analysis as:

Computation of Nonlinear Structures

858

̄ t θ̄ ) ≡ GInertial + GInternal − GExternal G( t d, t θ; t d, Inertial

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ { } ∙ ∙∙ ∙∙ ∙ ( M0 t d ∙ t d̄ + I0 WT (W t θ + W t θ) ∙ t θ̄ dA = ∫ M Internal

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ { n 𝛼 n } ̄ + n M𝛼 ∙ n χ̄ + ( F ∙ β dA 𝛼 𝛼 ∫ M External

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⎧ ⎫ ) (t ) ⎪ ⎪ (t̂ t̄ Tt ̂ t̄ t̄ Tt ̂ t̄ ̂ F𝜈 ∙ d + W M𝜈 ∙ θ dS + F ∙ d + W M ∙ θ dA⎬ − ⎨ ∫ ∫ ⎪𝜕M ⎪ M ⎩ f ⎭ ̄ t θ̄ = 0, ∀ admissible t d,

(11.442) with the initial conditions given as: d(0) = d0 . θ(0) = θ0 . ∙

v(0) ≡ d(0) = v0 ,

(11.443)

∙

ω(0) ≡ θ(0) = ω0

11.14.7

Static and Quasi-Static Computational Virtual Work Functional

Similarly, we can express the virtual functional of equation (11.415) in terms of the real displacement, t d, and rotation, t θ, and their virtual counterparts for static and quasi-static analysis as: ̄ ̄ t θ) G(𝜆, t d, t θ; t d, Internal

≡ GInternal − GExternal

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ { n 𝛼 n } ̄ + n M𝛼 ∙ n χ̄ = ( F ∙ β dA 𝛼 𝛼 ∫ M External

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⎫ ⎧ ) ) ⎪ (t ⎪ (t̂ t̄ t ̂ t̄ t t t ̂ ∙ W θ̄ dA⎬ −⎨ F𝜈 ∙ d + M𝜈 ∙ W θ dS + F̂ ∙ d̄ + M ∫ ∫ ⎪ ⎪𝜕M M ⎭ ⎩ f ̄ t θ̄ = 0, ∀ admissible t d, (11.444) with appropriate prescribed boundary conditions.

Nonlinear Shell

11.14.8

859

Generalized Definitions

For notational compactness and as a follow-up on our discussion of the shell configuration space, we may combine both the real vector displacement and rotation fields, t d and t θ, and the virtual ̄ respectively, for the following. vector displacement and rotation fields, t d̄ and t θ, Definitions

r (6 × 1) the generalized real state field, t ⌣d as: t ⌣d ≡ { t d t θ }T ⌣ r (6 × 1) the generalized virtual state field, t ⌣d as: t d̄ ≡ { t d̄ t θ̄ }T ∙

∙

r (6 × 1) the generalized real velocity field, t ⌣d as: t ⌣d ≡ { v r (6 × 1) the generalized real inertial force field, ∙

∙

ω }T ⌣

tF

∙∙

Iner

as:

∙∙

⌣

tF

Iner

≡ {M0 t d

I0

(WT (W t θ + W t θ))}T Similarly, we have the following. Definitions

r (12 × 1) the generalized real strain, n ⌣ε, as n ⌣ε ≡ { n β1 n χ1 n β2 ⌣ { r (12 × 1) the generalized virtual strain, n ⌣ε, ̄ ̄ ϕ) ̄ as: n ε(d, ̄ ϕ; d, ̄ ≡ nβ 1 ⌣ ⌣ { r (12 × 1) the generalized reactive force field, n F, as: n F ≡ n F1 ⏟⏟⏟

r (12 × 1) the generalized external force field,

12x1

⌣

t F, ̂

{

⌣

as:

t̂

F ≡ ⏟⏟⏟

tF ̂1

nχ

}T 2 nβ ̄

nχ ̄1 n M1

tM ̂1

2 n F2

tF ̂2

}T nχ ̄2 } n M2 T tM ̂2

}T

12x1

Similar generalizations apply for other force and moment terms.

11.14.9

Dynamic Virtual Work Functional in Generalized Form

With the above notational definitions and the load proportionality parameter, 𝜆, we may rewrite the ↼ ̄ t 𝜃), of equation (11.442) in compact generalized notation virtual work functional, G( t d, t θ; t d, as: ⌣

⌣

̄ ≡ GInternal + GInertial − GExternal = 0, G( t d; t d) GInternal =

⌣

∫

n̄ n

⌣

∀ admissible t d̄

⌣

ε F dA

M ⌣T

GInertial =

∫

t̄

d

t

(11.445)

⌣

FIner dA

M ⌣T

GExternal =

∫ M

t̄

d

⌣

t̂

F(S1 , S2 ) dA + 𝜆

⌣T

∫

𝜕M f

t̄

d

⌣

t̂

F(S) dS

Computation of Nonlinear Structures

860

with the applicable initial conditions on the dynamic state variables as: ⌣

⌣

d(0) = d0 . (11.446)

∙

t

11.14.10

⌣

⌣

d(0) = v 0 ,

Static Virtual Work Functional in Generalized Form

With the above notational definitions and the load proportionality parameter, 𝜆, we may rewrite ↼ ̄ t 𝜃), of equation (11.444) in compact generalized the virtual work functional, G(𝜆, t d, t θ; t d, notation as: ⌣

⌣

⌣

⌣

⌣

̄ ≡ GInternal ( t d; t d) ̄ − Wexternal (𝜆; t d) ̄ = 0, G(𝜆, t d; t d) GInternal =

⌣

∫

n̄ n

⌣

∀ admissible t d̄

⌣

ε F dA (11.447)

M ⌣T

GExternal =

∫

t̄

d

⌣B

t̂

F (𝜆, S1 , S2 ) dA + 𝜆

M

⌣T

∫

t̄

d

⌣S

t̂

F (S) dS

𝜕M f

where the superscripts t and n indicate that all quantities are expressed in the undeformed frame and the deformed frame, respectively.

11.14.11

Where We Would Like to Go

In conformity with our principal interest in computational solid mechanics and structural engineering, we need to bring all the shell equations down to a form where we can easily solve shell problems numerically by our finite element scheme: the c-type finite element method. This, in turn, requires the derivation of the compatible real strain fields along the configuration path from our knowledge of the above virtual strain fields; this endeavor validates our assumption that the derived virtual strains are truly variational, that is, these can be shown to exist as variations of some real strain fields. Our expressions for the virtual strain fields given in equations (11.416) and (11.417) already suggest as to what the real strain fields will be. However, we would like to record formally the expressions for real strain fields through an application of what is known as the Vainberg principle. Accordingly, the most logical route will be:

r the compatible real strains in component form. Moreover, as we can see from equations (11.445) and (11.447), the virtual functional, being generally highly nonlinear in displacements and rotations, may not be solved in a closed form in an analytical way. The solution, that is, the tracing of the shell motion or configuration path, will require some sort of iterative numerical method. Of all the possible schemes, the one that we intend to pursue is a Newton-type method. This, in turn, requires linearization of the equations at every iteration step. Clearly, then, we will have to further modify the virtual work equation for the ensuing computational strategy. Let us recall, however, that while the translational displacement

Nonlinear Shell

861

of the shell reference surface travels on a Euclidean (flat) configuration space, the shell rotation tensor belongs to a Riemannian (curved) configuration space. More specifically, as we will see, we will need to recognize this by resorting to covariant derivatives as opposed to standard Gateaux derivatives of the Euclidean space in our linearization of the virtual work functional. Thus, the next logical place to go from here will be:

r covariant linearization of virtual work functionals 11.15 Computational Real Strains 11.15.1 What We Need to Recall Now, let us recall that we have defined the virtual strains in terms of the displacement and rotation fields in invariant (defined from the weak form of balance equations) and component vector forms (defined from the virtual work equation) by:

11.15.1.1

Internal Virtual Work Functional GInternal ≡

{n

∫

̄ + F𝛼 ∙ n β 𝛼

n

} M𝛼 ∙ n χ̄ 𝛼 dA

(11.448)

M

11.15.1.2

Component Vector Form

r The translational virtual strain fields, n

̄ = t RT β 𝛼

(t

nβ ̄

𝛼 = 1, 2, given as:

𝛼,

) ( ) ̄ e𝛼 + k0 × t d ̄ 𝛼 + k0 × t d̄ − t RT t 𝚽 d, 𝛼 𝛼

(11.449)

Component-wise in spatial representation, that is, in rotated bases: n

̄ = β 𝛼

{

n 𝛽̄ 𝛼1

n 𝛽̄ 𝛼2

n 𝛽̄ 𝛼3

}T

,

𝛼 = 1, 2

(11.450)

For equation (11.449), we have the following definitions: e𝛼 ≡ 1̂ 𝛼 + t d, 𝛼 , t d| 𝛼

=

r The bending virtual strain fields,

nχ ̄

t d, 𝛼,

𝛼

+

k0𝛼

𝛼 = 1, 2, × t d,

𝛼 = 1, 2,

(11.451)

𝛼 = 1, 2, given as:

̄ k0 ̄ 𝛼 − t RT t 𝚽 χ𝛼 = t RT t 𝝋, 𝛼

n̄

(11.452)

Component-wise in spatial representation, that is, in rotated bases: n̄

χ𝛼 =

{

n 𝜒̄

𝛼1

n 𝜒̄

𝛼2

n 𝜒̄

𝛼3

}T

,

𝛼 = 1, 2

(11.453)

Computation of Nonlinear Structures

862

Furthermore, we need the following results: t

k𝛼 = W(θ) θ, 𝛼 ,

𝛼 = 1, 2

(11.454)

where: W(θ) ≡ I + c1 𝚯 + c2 𝚯2 , c1 (θ) =

1 − cos 𝜃 , 𝜃2

c2 (θ) =

𝜃 − sin 𝜃 𝜃3

(11.455)

̄ ̄ as the axial vectors of t K𝛼 , 𝛼 = 1, 2 and t 𝚽, Lemma LD1: Defining t k𝛼 , 𝛼 = 1, 2 and t ϕ respectively, we have: t 𝚽, ̄

𝛼

̄ 𝛼 ϕ, ⏟⏟⏟ t

Effective material curvature vector variation

̄ − t𝚽 ̄ t K𝛼 ], 𝛼 = 1, 2 ̄ 𝛼 + [ t K𝛼 t 𝚽 = tK t̄ t t ̄ ϕ = + k𝛼 × k𝛼 ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟ Material curvature vector variation

Material curvature vector

(11.456)

Material rotation vector variation

11.15.2

Component Vector Form: Real Strains and the Vainberg Principle

11.15.2.1

Virtual Strain Symmetry

As indicated, in order for us to apply the Vainberg principle of determining the real strains from the virtual ones, we need to guarantee the symmetry of the virtual strains on a linear vector space which is provided by the tangent space at identity; the components of the virtual strains ̄ , n χ̄ ) belongs to a linear vector space are expressed in the rotated bases. Noting that 𝜺̂̄ ≡ ( n β 𝛼 𝛼 (namely, the tangent space at identity), the following symmetry relations hold: ̃ t d, ̄ = 𝜺̃̂ ( t d, t θ; t d, ̄ t d, ̃ ̃ t θ; ̄ t θ) ̄ t θ; ̃ t θ) ̂̄ t d, t θ; t d, ε(

(11.457)

̃ t θ̄ belong to the tangent space at identity with t d, t θ as the real where the parameters t θ, t θ, t t t t ̃ ̄ ̃ ̄ parameters, d, θ; d, θ are the two sets of the virtual parameters. In other words, the virtual strains are symmetric for the conservative loading, and thus, by the Vainberg principle, the potential functional exists. In other words, there exist functions (we will call them the real strains) the variation of which will yield the virtual strains.

11.15.2.2

Component Form: Real Translational (Extensional and Shear) Strain Field n β𝜶 , 𝜶 = 1, 2

The real translational strain field, n β𝛼 , can be derived by an application of the integral expression in the definition of the Vainberg principle as follows (mnemonically, multiply the real variables

Nonlinear Shell

863

by 𝜌 ∈ [0.1] such as d → 𝜌d, and so on, and replace the virtual variables by the corresponding real one such as d̄ → d, and so on, and integrate over 𝜌 ∈ [0.1]): 1 nβ

𝛼

=

n

∫0

̄ (𝜌 t d, 𝜌 t ϕ; t d, t ϕ) d𝜌 = β 𝛼

1

∫0

̄ 1̂ 𝛼 + 𝜌 t d|𝛼 )} d𝜌 { t RT𝜌 t d|𝛼 + (− t RT𝜌 t 𝚽)(

1

=

d t T ̂ { R𝜌 (1𝛼 + 𝜌 t d|𝛼 )} d𝜌 = { t RT𝜌 (1̂ 𝛼 + 𝜌 t d|𝛼 )}𝜌=1 𝜌=0 ∫0 d𝜌

= t RT (1̂ 𝛼 + t d|𝛼 ) − 1̂ 𝛼 ,

𝛼 = 1, 2

In the above derivation, we have used relations in equation (11.451), and the following definition. Definition:

tR 𝜌

≡ t R(𝜌ϕ)

Facts:

r

t R ≡ t R(0) = I 0 t RT ≡ 𝛿( t RT )

11.15.2.3

= t RT (0) = t RT0 ̄ (see discussions on the derivatives and variations of rotation) = − t RT t 𝚽

Component Form: Real Bending Strain Field: n χ𝜶 ,

𝜶 = 1, 2

Likewise, the real bending strains, n χ𝛼 , can be derived by an application of the integral expression in the definition of the Vainberg principle as in the exercise below, n χ𝛼 = t RT kc𝛼 − k0𝛼 , 𝛼 = 1, 2 where we have used the following definitions. Definitions

r The current total curvature vector: kc

≡ k0𝛼 + k𝛼 , 𝛼 = 1, 2, with k𝛼 ≡ W θ, 𝛼 where W(θ) ≡ 𝜃 c2 ≡ 𝜃−sin with W defined in equation (11.455) and introI + c1 𝚯 + c2 𝚯 , c1 ≡ 𝜃3 duced in the section on the derivatives and variations of rotation tensor, t R, in which 𝚯 is the skew-symmetric tensor corresponding to rotation vector, θ, and 𝜃 stands for the length of θ; k0𝛼 , 𝛼 = 1, 2 are the initial curvature vectors of the shell reference surface along the lines of curvature. 1−cos 𝜃 , 𝜃2

2

𝛼

Exercise: Following the similar procedure used for the real translational strains, derive the real bending strain as: n χ𝛼 = RT kc𝛼 − k0𝛼 , 𝛼 = 1, 2. Solution 1 nχ

𝛼

=

∫0

n̄

χ𝛼 (𝜌 t d, 𝜌 t ϕ; t d, t ϕ) d𝜌

1

=

d t T 0 { R𝜌 (k𝛼 + 𝜌 k𝛼 )} d𝜌 = { t RT𝜌 (k0𝛼 + 𝜌 k𝛼 )}𝜌=1 𝜌=0 ∫0 d𝜌

= t RT (k0𝛼 + k𝛼 ) − k0𝛼 = t RT kc𝛼 − k0𝛼 ,

(11.458)

𝛼 = 1, 2

where we have recalled the definition for the total curvature vector as: kc𝛼 ≡ k0𝛼 + k𝛼 , 𝛼 = 1, 2.

Computation of Nonlinear Structures

864

11.15.3

The Real Translational and Bending Strains

Thus, we have the real translational (axial and shear) and the bending strains given as: nβ nχ

11.15.4

𝛼 𝛼

) ( = RT e𝛼 + k0𝛼 × d − 1̂ 𝛼 , =

RT kc𝛼

𝛼 = 1, 2

− k0𝛼

(11.459)

Where We Would Like to Go

So far, we have derived the shell real strain fields by applying the Vainberg principle to the virtual strain fields. Now that both the real and the virtual strain fields are known, we must particularize the virtual functional further by determination of the reactive forces and moments through the application of a relevant constitutive law. Thus, we must check out the special material properties and the loading system for which the analysis will be specialized and pursued henceforth for numerical implementation. Let us also recall that the internal virtual work functional of equation (11.448), has been expressed in terms of the real and virtual displacements and rotations, and the reactive forces and moments. For notational compactness and in favor of uncluttered presentation, it is useful to generalize the expressions by lumping various translational entities with their corresponding rotational counterparts – such as the displacements with the rotations (real and virtual), and so on. Thus, a logical place to go from here should be:

r shell material property (constitutive laws) r covariant linearization of virtual work equations. 11.16 Hyperelastic Material Property 11.16.1 What We Need to Recall

r The invariant form of the real strain vectors in rotated (spatial) frame, and the components in rotating bases, are given as: β𝛼 = c0 , 𝛼 − g𝛼 = n 𝛽𝛼𝜂 g𝜂 + n 𝛽𝛼3 (≡ 𝛽𝛼 ) g3 , 𝛼 = 1, 2

(11.460)

χ𝛼 = k𝛼 = kc𝛼 − R k0𝛼 = n 𝜒𝛼𝜂 g𝜂 + n 𝜒𝛼3 (≡ 𝜒𝛼 ) g3 , 𝛼 = 1, 2

(11.461)

r The invariant form of the internal virtual work functional is given by: { ̂ Internal ≡ G

∫

Δ 𝛼

n

𝛼

Δ

( F ∙ β𝛼 + M ∙ χ𝛼

} dA

(11.462)

Ω Δ

Δ

where β𝛼 and χ𝛼 are the co-rotated or objective variations of the force and moment strain fields, respectively;

Nonlinear Shell

865

r The resistive stress resultant, F𝛼 , is defined by: 𝛼

F ≡

𝜂t

𝜂 𝛼 𝜂 S

𝜇 d𝜂

P

∫𝜂 b

(11.463)

r The resistive stress couple, M𝛼 , is defined by: M𝛼 (S) ≡

𝜂t

∫𝜂b

𝜂 𝛼 𝜂 S

(S, 𝜂) ×

P

𝜇 d𝜂

(11.464)

where (S, 𝜂) defines the through-the-thickness deformation of the shell-like body. and component-wise, denoted by: F𝛼 = n F 𝛼𝛽 g𝛽 + n F 𝛼3 g3 , 𝛼 = 1, 2

(11.465)

M𝛼 = n M 𝛼𝛽 g𝛽 + n M 𝛼3 g3

r The component form of the real strain vectors, and the components in rotating bases, are given as: nβ

𝛼

nχ 𝛼

) ( = RT e𝛼 + k0𝛼 × d − 1̂ 𝛼 , =

RT kc𝛼

𝛼 = 1, 2

(11.466)

− k0𝛼

with components denoted by: nβ nχ

r

𝛼 𝛼

= =

{ {

n𝛽

n𝛽

𝛼1

n𝜒

𝛼2

n𝜒

𝛼1

𝛼2

n𝛽

𝛼3

n𝜒

}T

𝛼3

𝛼 = 1, 2

, }T

(11.467)

where a superscript n signifies that the object has a spatial representation in the rotating bases, {gi }, i = 1, 2, 3. ̄ ϕ), ̄ is given as: The internal virtual work functional, GInternal (d, R; d, ̄ ϕ) ̄ ≡ GInternal (d, R; d,

∫

{ n 𝛼 n ̄ + ( F ∙ β 𝛼

n

M𝛼 ∙ n χ̄ 𝛼

}

dA

(11.468)

Ω

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ GInternal

The shell reactive component or operational force vector, n F𝛼 , and the shell reactive component or operational moment vector, n M𝛼 , are component-wise denoted by: n F𝛼 n M𝛼

= { n F 𝛼1 ={

n M 𝛼1

n F 𝛼2 n M 𝛼2

n F 𝛼3 }T

= t RT t F𝛼 ,

n M 𝛼3 }T

=

t RT t M𝛼

𝛼 = 1, 2

(11.469)

Computation of Nonlinear Structures

866

r

where the superscript t and n indicate that all quantities are measures and expressed in the undeformed G1 , G2 , G3 ≡ N frame, and the deformed and rotated g1 , g2 , g3 frame, respectively. The reduced internal energy variation is given as: ( ̄ ≡ 𝛿W = W

∫ M

=

∫

{

𝜂t

∫𝜂b

) P : F̄ 𝜂 𝜇S d𝜂

dA

} F𝛼 ∙ β𝛼 + M𝛼 ∙ χ𝛼 dA = Ū ≡ 𝛿U

(11.470)

M

where Ū ≡ 𝛿U is the strain energy variation of a two-dimensional shell reference surface ̄ ≡ 𝛿W, of the corresponding shell-like identified as the reduced 3D strain energy variation, W body.

11.16.2

Elastic Shell Material Property

As mentioned earlier, and in our discussion on the constitutive laws, every constitutive theory is approximate by its very task of modeling and fitting experimental data; the shell constitutive properties will not be an exception and, in fact, will induce additional approximations through the reduction of the 3D theory to an equivalent 2D shell material properties. In any case, as usual, the shell constitutive modeling must also satisfy the four general principles of a consistent 3D theory which we may recall are: the principle of stress determinism, the principle of local action, the principle of material frame indifference, and the principle of material symmetry. As a reminder, we must note that the derived real translational and bending strains as given by equation (11.460) and (11.461), respectively, and the virtual work principle given by equation (11.468) is exact and does not depend on the material properties of a shell. 11.16.2.1

Principle of Equivalence

All the same, we need an equivalent 2D constitutive theory from a corresponding 3D one, that, on the one hand, is suitable for a shell, and replicates the response of its corresponding 3D counterparts, that is, a shell-like body, as close as possible, on the other. One way of measuring this equivalency is to require that the internal virtual work of a 3D shell-like body integrated through the thickness (just like the way we have derived the 2D virtual work principle and the boundary conditions) be as close as possible to the 2D internal virtual work expressed in equation (11.468). This has been achieved elsewhere earlier in the text as shown in equation (11.470). Recall that for an elastic material, by definition, the first Piola–Kirchhoff stress tensor, P = Pi ⊗ Gi , can be expressed as a function of the deformation gradient tensor, F, by: ̄ P = (F, X)

(11.471)

where X denotes a particle on the 3D shell-like body in the undeformed reference state; recall that the imposition of the material frame indifference guarantees that equation (11.471) can be expressed in terms of energy conjugate stress and strain variables and that with C ≡ FT F as the right Cauchy–Green stretch tensor, we can redefine equation (11.471) as: P = (C, X)

(11.472)

Nonlinear Shell

867

An application of the equivalence principle will generally result in the reduction of the 3D strain ̄ ≡ 𝛿W, and as such establish the existence of the equivalent 2D strain energy energy variation, W variation, Ū ≡ 𝛿U, as: ( ̄ ≡ 𝛿W = W

∫ M

=

∫

𝜂t

∫𝜂 b {

) P : F̄ 𝜇 d𝜂 𝜂 S

dA

} F𝛼 ∙ β𝛼 + M𝛼 ∙ χ𝛼 dA + possible error

(11.473)

M

= Ū ≡ 𝛿U Recall also that under the kinematic assumption of 1D inextensible director, we have already established this reduction exactly as shown by equation (11.470). However, if we generate the stress resultants and the stress couple, and the conjugate strains exactly, as we do in our theory, without any kinematic assumptions, the reduction in the form of equation (11.473) will generally entail possible error which conforms to our design of containing and restricting all the approximations in the constitutive theory of shells.

11.16.2.2

Isotropic, Hyperelastic Material

Now, if we restrict, as we do, our ensuing analysis to only isotropic hyperelastic, or Green-elastic shells, we can recall that, by definition, there exists a strain energy functional, W = 𝜌 Φ, per unit of undeformed volume, that depends on the deformation gradient or the strain tensor; 𝜌 is the current density and Φ is the specific strain energy (i.e. strain energy per unit mass), also known as the Helmholtz free energy, such that we have: (C, X) =

𝜕W 𝜕W = 2F 𝜕C 𝜕F

(11.474)

Now, considering equation (11.473) with the shell real strain measures given by β𝛼 , the real translational (axial and shear) spatial strain, and χ𝛼 , the real rotational (bending) strain, the shell strain energy, U, for the 2D surface, must be given by: U ≡ U(β𝛼 , χ𝛼 , S1 , S2 , 𝜂) =

𝜂t

∫𝜂 b

W(C, X) 𝜂 𝜇S d𝜂 + possible error

(11.475)

where we have included the explicit dependence on the position of a material particle by the arc length parameters, S1 , S2 along the lines of curvature, and 𝜂, the shell thickness. Then, by the postulate of hyperelasticity, there exist two vector valued functions,  ( β𝛼 , χ𝛼 , S1 , S2 , 𝜂) and (β𝛼 , χ𝛼 , S1 , S2 , 𝜂) such that F𝛼 , the spatial stress resultant, and M𝛼 , the spatial stress couple, are given by: F𝛼 =  (β𝛼 , χ𝛼 , S1 , S2 , 𝜂) ≡

𝜕U 𝜕 β𝛼

M𝛼 = ( β𝛼 , χ𝛼 , S1 , S2 , 𝜂) ≡

𝜕U 𝜕 χ𝛼

(11.476)

Computation of Nonlinear Structures

868

11.16.2.3

Homogeneous, Isotropic, Hyperelastic Material

If the material properties are independent of the position of a particle, then we can rewrite equation (11.476) as: F𝛼 =  (β𝛼 , χ𝛼 ) ≡

𝜕U 𝜕 β𝛼

M𝛼 = ( β𝛼 , χ𝛼 ) ≡ 11.16.2.4

(11.477)

𝜕U 𝜕 χ𝛼

Linear, Homogeneous, Isotropic, Hyperelastic Material

In what follows, we restrict the exposition only to linear, homogeneous, isotropic, hyperelastic materials. Let us recall that for such material properties, the strain energy is a quadratic function of the strains and thus, the constitutive law for a 3D body can be represented in tensor form or component-wise by: 1 E:C:E 2 } 1{ = Eij Ai ⊗ Aj : Cmnpq Am ⊗ An ⊗ Ap ⊗ Aq : Ekl Ak ⊗ Al 2 } 1 1 { i j k l mnpq = Eij Ekl = Cijkl Eij Ekl 𝛿 𝛿 𝛿 𝛿 C 2 m n p q 2 } { ijkl 𝜕W ⇒S= = C : E = C Ekl Ai ⊗ Aj , 𝜕E W=

(11.478)

Sij = Cijkl Ekl where S = Sij (S1 , S2 , 𝜂) Ai (S1 , S2 , 𝜂) ⊗ Aj (S1 , S2 , 𝜂) is the second Piola–Kirchhoff stress tensor, E =}Ekl (S1 , S2 , 𝜂) Ak (S1 , S2 , 𝜂) ⊗ Al (S1 , S2 , 𝜂) is the Green–Lagrange strain tensor, { and Ai (S1 , S2 , 𝜂) , i = 1, 2, 3 is the base vector set at any particle point (S1 , S2 , 𝜂) in the undeformed reference system. Now, recall that for linear isotropic material with the identity tensors: 4

I ≡ Ai ⊗ Aj ⊗ Ai ⊗ Aj and I ≡ Ai ⊗ Ai , the constitutive tensor is given by: ( ) 4 𝜈 C ≡ 2𝜇 I + I⊗I 1 − 2𝜈

(11.479)

resulting in the coefficients, Cijkl , i, j, k, l = 1, 2, 3 of the fourth-order constitutive tensor, C = Cijkl Ai ⊗ Aj ⊗ Ak ⊗ Al , in the curvilinear coordinates as given by: ( Cijkl ≡ 𝜇 Aik Ajl + Ail Ajk + where 𝜇 ≡ G = Poisson’s ratio. 11.16.2.5

2𝜈 Aij Akl 1 − 2𝜈

) (11.480)

E is the shear modulus with E as the elasticity modulus and 𝜈 as the 2(1 + 𝜈)

First Piola–Kirchhoff Stress Tensor in Mixed Bases

The stress–strain relations expressed in equations (11.478)–(11.488) are all in terms of the second Piola–Kirchhoff stress tensor, S, and the Green–Lagrange strain tensor, E; however, we are interested in relations in terms of the first Piola–Kirchhoff stress tensor, P = Pij ai ⊗ Aj , and

Nonlinear Shell

869

the deformation gradient tensor, F, defined by F ≡ ai ⊗ Aj where {ai } and {Ai } are the deformed and the undeformed base vector sets, respectively. Recall that P = F S. The important thing to note here is that we have represented P in a mixed basis form, that is, the domain basis in the undeformed reference state and the range basis in the deformed state. Under the mixed basis representation, we have: P = Pij c, i ⊗ Aj = F S =

c, i ⊗ Ai ⏟⏞⏟⏞⏟

∙

F, deformation gradient tensor

Sij Ai ⊗ Aj ⏟⏞⏞⏟⏞⏞⏟ S, second Piola-Kirchhoff tensor

(11.481)

Thus, the components, Pij , of P represented in the mixed basis are identical with the corresponding components, Sij , of S, and hence, symmetric. Now, noting, that P = F S, we can rewrite the constitutive relation of equation (11.478) as: Pij ai ⊗ Aj = P = F S = F C : E = ai ⊗ Ai ∙ Ckjmn Ak ⊗ Aj ⊗ Am ⊗ An : Epq Ap ⊗ Aq = Cijmn (Emn ai ⊗ Aj ) = (Cijmn Emn ) ai ⊗ Aj ⇒

Pij

=C

ijmn

(11.482)

Emn

From equation (11.482), we see that if the first Piola–Kirchhoff stress tensor is represented in the mixed basis, then for the stress–strain relations, we can continue to use the same constitutive relations, as between the second Piola–Kirchhoff and the Green–Lagrange strain tensor, with the effective strain tensor represented in the mixed form with the coefficients of the Green–Lagrange strain tensor.

11.16.3

Thin, Linear, Homogeneous, Isotropic, Hyperelastic Shell

In what follows, we restrict the exposition only to linear, homogeneous, isotropic, hyperelastic and very thin shells to derive the stress–strain relationship that will be used in our subsequent numerical simulations. Also, for the thin shells, we will assume that h(S, 𝜂) = 𝜂 g3 in equation (11.464), keeping the normal to the surface inextensible through the deformation history. Now, let us first recall that for our optimal normal shell coordinate system along the lines of curvature with the arc lengths as the coordinate parameters, for any particle with coordinates (S1 , S2 , 𝜂) in the undeformed shell like body, we have: C(S1 , S2 , 𝜂) = C0 (S1 , S2 ) + 𝜂 N(S1 , S2 ) ⇒ A𝛼 = G𝛼 + 𝜂 N, 𝛼 = G𝛼 − 𝜂 K𝛼𝛽 G𝛽 = G𝛼 − 𝜂 =

(G𝜎𝛽 K𝛼𝜎 )

( ) 𝜂 1− G𝛼 R𝛼

G𝛽 = G𝛼 − 𝜂

(

K𝛼𝛼 G𝛼𝛼

) G𝛼

(11.483)

where R𝛼 is the radius of curvature, G𝛼𝛽 and K𝛼𝛽 are the covariant components of the metric tensor and the curvature tensor of the undeformed shell surface with off-diagonal terms identically zero

Computation of Nonlinear Structures

870

for the coordinate system along the lines of curvature. We have also used the relationship between the covariant and the contravariant coefficients of the metric tensor for the same coordinate system. 11.16.3.1

Thin Shell Material Properties

𝜂 ≪1 R𝛼 resulting in A𝛼 ≃ G𝛼 for all practical purposes. Thus, we may approximate the coefficients of the constitutive tensor from equation (11.480) to:

The assumption of a thin shell may be mathematically expressed by the condition:

( Cijkl ≡ 𝜇 Gik Gjl + Gil Gjk +

2𝜈 Gij Gkl 1 − 2𝜈

) (11.484)

and in the optimal normal shell coordinates, we can identify and particularize the coefficients of equation (11.484) to get: ( C𝛼𝛽𝛾𝛿 = 𝜇 G𝛼𝛾 G𝛽𝛿 + G𝛼𝛿 G𝛽𝛾 +

C

𝛼𝛽33

C𝛼3𝛽3

C3333

) 2𝜈 G𝛼𝛽 G𝛾𝛿 , 1 − 2𝜈

⎞ ⎛ ⎜ 𝛼3   E𝜈 2𝜈   𝛼𝛽 33 ⎟ 𝛽3 𝛼3 𝛽3   = 𝜇 ⎜ G ⎟= G G𝛼𝛽 , G G + G G + ⏟ ⏟ ⏟ 1 − 2𝜈 (1 + 𝜈)(1 − 2𝜈) ⎟ ⎜ =1 ⎠ ⎝ ⎞ ⎛ ⎟ ⎜ 𝛼𝛽   E 2𝜈   33 𝛼3 𝛽3 𝛼3 𝛽3    = 𝜇 ⎜G G + G𝛼𝛽 , G G + G G ⎟=𝜇G= ⏟ ⏟ ⏟ 1 − 2𝜈 2(1 + 𝜈) ⎟ ⎜ =1 ⎠ ⎝ ( ) E(1 − 𝜈) 2𝜈 = 𝜇 2+ = (1 − 2𝜈) (1 + 𝜈)(1 − 2𝜈)

(11.485)

Now, for the shell-like body, if we continue to stay with the optimal normal coordinate system, that is, coordinates along the lines of curvature with arc length parameterization, and then noting that in this coordinate system: G𝛼𝛽 = 0, G𝛼3

= 0,

G33

=1

𝛼≠𝛽 (11.486)

we get: C𝛼𝛽𝛾3 = C𝛼𝛽3𝛾 = C𝛾3𝛼𝛽 = C3𝛾𝛼𝛽 = 0, C𝛼333 = C3𝛼33 = C33𝛼3 = C333𝛼 = 0

(11.487)

Using the relations given in equation (11.487), we can list the non-trivial stress components from equation (11.478) as: S𝛼𝛽 = C𝛼𝛽𝛾𝛿 E𝛾𝛿 + C𝛼𝛽33 E33 , S𝛼3 = 2C𝛼3𝛾3 E𝛾3 , S33

=

C33𝛾𝛿

E𝛾𝛿

+ C3333

(11.488) E33 ,

Nonlinear Shell

871

Obviously, by solving for E33 in the expression for S33 , we can eliminate it from the expression for S𝛼𝛽 in equation (11.488). For the plane stress assumption on the top and bottom faces, we will place the restriction, S33 ≡ 0, in which case we will have: E33 = −

C33𝛾𝛿 E𝛾𝛿 C3333

(11.489)

The coefficients, Sij , of the stress tensor can be similarly particularized. However, before we do so, we may impose the plane stress condition: S33 ≡ 0, on the top and the bottom faces of C33𝛾𝛿 the shell. Then, as before we get from equations (11.489) and (11.485): E33 = − 3333 E𝛾𝛿 = C 𝜈 − G𝛼𝛽 E𝛼𝛽 . Now, applying this relation to the expressions (10.16.29) for the stress com(1 − v) ponents, we finally get: S𝛼𝛽

( = C𝛼𝛽𝛾𝛿 −

) 𝜈 E𝜈 𝛼𝛽 𝛾𝛿 G G E𝛾𝛿 1 − 𝜈 (1 + 𝜈)(1 − 2𝜈) { ( )} 2𝜈 2𝜈 2 𝛼𝛽 𝛾𝛿 𝛼𝛾 𝛽𝛿 𝛼𝛿 𝛽𝛾 = 𝜇 G G +G G + E𝛾𝛿 = Ĥ 𝛼𝛽𝛾𝛿 E𝛾𝛿 , − G G 1 − 2𝜈 (1 − 𝜈)(1 − 2𝜈)

S𝛼3 = 2C𝛼3𝛾3 E𝛾3 (11.490) where we have introduced the definition: Ĥ 𝛼𝛽𝛾𝛿 =

( ) E 2𝜈 𝛼𝛽 𝛾𝛿 G𝛼𝛾 G𝛽𝛿 + G𝛼𝛿 G𝛽𝛾 + G G 2(1 + 𝜈) 1−𝜈

(11.491)

and C𝛼3𝛾3 is as given in equation (11.485).

11.16.3.2

Two-dimensional Strain Energy, U(β𝜶 , χ𝜶 )

Before we can formally write down the expression for the 2D strain energy functional, U(β𝛼 , χ𝛼 ) per unit of the shell surface area, we need to express the coefficients, Eij , of the Green–Lagrange strain tensors in terms of the coefficients as given by equations (11.460) and (11.461) of β𝛼 and χ𝛼 , the spatial translational and bending strain vectors of the shell surface; similarly, the components of the stress resultants, F𝛼 , and those of the stress couple, M𝛼 , as given by equations (11.465) must be expressed in terms of the coefficients, Sij , of the second Piola–Kirchhoff stress tensor. Let us recall some preliminaries: that A = Ai ⊗ Ai = A𝛼 ⊗ A𝛼 + N ⊗ N is the identity tensor in the undeformed optimal normal coordinate system; that for an arbitrary particle, the current particle position vector is given by c = c0 + 𝜂 g3 ; that, at an arbitrary particle, the deformed base vector set is given as: a𝛼 ≡ c, 𝛼 = c0 , 𝛼 + 𝜂 g3 , 𝛼 a3 ≡ c, 3 = c, 𝜂 = g3

(11.492)

Computation of Nonlinear Structures

872

with: 0 0 𝜂 g3 , 𝛼 = 𝜂ktot 𝛼 × g3 = 𝜂 k𝛼 × g3 + 𝜂 R k𝛼 × g3 = 𝜂 k𝛼 × g3 + 𝜂 R k𝛼 × R N

= 𝜂 χ𝛼 × g3 + 𝜂 R (k0𝛼 × N) = 𝜂 χ𝛼 × g3 + 𝜂 R N, 𝛼

𝜂 = 𝜂 χ𝛼 × g3 + R  G𝛼 = 𝜂 χ𝛼 × g3 R  𝛼 ⏟⏟⏟

(11.493)

higher order

( ) where we have used the identity: R k0𝛼 × R N = R k0𝛼 × N (det R) with det R = 1; that the deformation gradient, F, is given by: F ≡ a, i ⊗ Ai = c, 𝛼 ⊗ A𝛼 + c, 𝜂 ⊗ A3 = (c0 , 𝛼 + 𝜂 g3 , 𝛼 ) ⊗ A𝛼 + g3 ⊗ N

⇒ FT = A𝛼 ⊗ (c0 , 𝛼 + 𝜂 g3 , 𝛼 ) + N ⊗ g3

(11.494)

where the deformed shell reference surface spin coordinate base vectors are given as: gi = R Gi with {Gi } as the undeformed shell reference surface base vector set; that noting the strain expressions as given by equations (11.460) and (11.461), we have: c0 , 𝛼 = β𝛼 + g𝛼 ; now, considering only the linear terms we get: ∙ β c0 , 𝛼 ∙ c0 , 𝛽 = (β𝛼 + g𝛼 ) ∙ (β𝛽 + g𝛽 ) =  β𝛼 𝛽 + β𝛼 ∙ g𝛽 + β𝛽 ∙ g𝛼 + g𝛼 ∙ g𝛽 ⏟⏟⏟ =

n𝛽

𝛼𝛽

+

n𝛽

Nonlinear 𝛽𝛼

(11.495)

+ G𝛼𝛽

where we have used: g𝛼 ∙ g𝛽 = R G𝛼 ∙ R G𝛽 = G𝛼𝛽 ; now, noting that: 𝜂 g𝛼 ∙ RN, 𝛽 = 𝜂 RG𝛼 ∙ √ 𝜂 1 RN, 𝛽 = −𝜂 R(G𝛼 ∙ G𝛽 ) = − R( G𝛼𝛽 ) ≪ 1, and G = 1 ⇒ 𝜀𝛾𝜎 = e𝛾𝜎 , we get: R𝛽 R𝛽 𝜂(c0 , 𝛼 ∙ g3 , 𝛽 ) = 𝜂{(β𝛼 + g𝛼 ) ∙ (χ𝛽 × g3 + R N, 𝛽 )} = 𝜂 {(β𝛼 + g𝛼 ) ∙ (χ𝛽 × g3 ) + RK0𝛽 N} ⎧ ⎫ 0 ∙( (χ𝛽(×(g( β𝛼( 3 ) + 𝜂 β𝛼 ∙ RK𝛽 N ⎪ ⎪𝜂 ( ⎪ ⏟⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏟ ⎪ Nonlinear ⎪ ⎪ ≪1 =⎨ ⎬ 0 ⎪ + g𝛼 ∙ (χ𝛽 × g3 ) + 𝜂 g𝛼 ∙ RK𝛽 N ⎪ ⎪ ⏟⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏟ ⎪ ⎪ ⎪ ≪1 ⎩ ⎭ = 𝜂 g𝛼 ∙ (χ𝛽𝛾 g𝛾 × g3 ) = 𝜂 g𝛼 ∙ (χ𝛽𝛾 𝜀𝛾𝜎 g𝜎 ) = 𝜂 χ𝛽𝛾 e𝛾𝛼

(11.496)

√ where recall that the surface permutation tensor, 𝜀𝛾𝜎 = G e𝛾𝜎 with the surface metric √ determinant: G = 1 in the optimal normal coordinate system and e𝛾𝜎 is the coefficient of the [ ] 0 1 surface permutation matrix: e = . −1 0

Nonlinear Shell

873

11.16.3.3

Explicit Strain Components ) 1( T 1 F F − A = Eij Ai ⊗ Aj , with A𝛼 ≃ G𝛼 , we have in the Now, noting that E ≡ (C − A) = 2 2 shell optimal normal coordinates: E𝛼𝛽 =

1 (a ∙ a − G𝛼𝛽 ), 2 𝛼 𝛽

E𝛼3 =

1 (a ∙ a − G𝛼3 ), 2 𝛼 3

E33 =

1 (a ∙ a − G33 ) 2 3 3

(11.497)

with G33 = N ∙ N = 1. Now, then, E𝛼𝛽 =

=

= = =

1 {(c , + 𝜂 g3 , 𝛼 ) ∙ (c0 , 𝛽 + 𝜂 g3 , 𝛽 ) − G𝛼𝛽 } 2 0 𝛼 ⎫ ⎧  ⎪ 1⎪ 2  g3 , 𝛼 ∙ g3 , 𝛼 − G𝛼𝛽 ⎬ c0 , 𝛼 ∙ c0 , 𝛽 + 𝜂 (c0 , 𝛼 ∙ g3 , 𝛽 + c0 , 𝛽 ∙ g3 , 𝛼 ) +  𝜂  ⎨ 2⎪ ⏟⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏟ ⎪ ⎭ ⎩ Nonlinear ( ) } { 1 n 𝛽𝛼𝛽 + n 𝛽𝛽𝛼 + G + 𝜂 n 𝜒𝛽𝛾 e𝛾𝛼 + n 𝜒𝛼𝛾 e𝛾𝛽 − G 𝛼𝛽 𝛼𝛽 2 ( )} { 1 n 𝛽𝛼𝛽 + n 𝛽𝛽𝛼 + 𝜂 n 𝜒𝛽𝛾 e𝛾𝛼 + n 𝜒𝛼𝛾 e𝛾𝛽 2 C L ) 1 ( ) 1 (n 𝛽𝛼𝛽 + n 𝛽𝛽𝛼 + 𝜂 n 𝜒𝛽𝛾 e𝛾𝛼 + n 𝜒𝛼𝛾 e𝛾𝛽 ≡ E𝛼𝛽 +𝜂 E𝛼𝛽 2 2

(11.498)

where we have used the relations given by equations (11.495) and (11.496), and introduced the constant variation (related to shear and extensional stresses) of the strain component through the C ) 1 (n 𝛽𝛼𝛽 + n 𝛽𝛽𝛼 , and the linear variation (related to bending stresses) strain thickness as: E𝛼𝛽 ≡ 2 L ) 1 (n 𝜒𝛽𝛾 e𝛾𝛼 + n 𝜒𝛼𝛾 e𝛾𝛽 . We get: components as: E𝛼𝛽 ≡ 2

E𝛼3

⎫ ⎧ ⎪ 1 1⎪ = {(c0 , 𝛼 + 𝜂 g3 , 𝛼 ) ∙ g3 } = ⎨c0 , 𝛼 ∙ g3 + 𝜂  g3 ,𝛼 ∙ g 3⎬ 2 2⎪ ⏟⏞⏟⏞⏟⎪ ⎭ ⎩ =0 =

(11.499)

C 1 1 {(β𝛼 + g𝛼 ) ∙ g3 } = n 𝛽𝛼3 ≡ E𝛼3 2 2

where we have used the fact that: g3 , 𝛼 = ktot 𝛼 × g3 ⇒ g3 , 𝛼 ∙ g3 = 0. Similarly, E3𝛼 = Finally, for inextensional normal, we have: E33 = 0

1 n 𝛽3𝛼 . 2

(11.500)

For a thin shell, we will take the mid-surface as the reference shell surface with the thickness 𝜂 𝜂 defined as: h ≡ 𝜂t + 𝜂b and the shifter: 𝜂 𝜇S ≡ 1 − 2H𝜂 + K𝜂 2 → 1 as 𝜂H = + → 0 and R1 R2

Computation of Nonlinear Structures

874

𝜂2K =

𝜂 𝜂 → 0. Now, then, the 2D strain energy may be expressed as: R1 R2

U=

=

=

h 2 ∫− h 2 h 2 ∫− h 2 h 2 ∫− h 2

W 𝜂 𝜇S d𝜂 =

h 2 ∫− h 2

( 𝛼𝛽 ) S E𝛼𝛽 + 2S𝛼3 E𝛼3 d𝜂

( 𝛼𝛽𝛾𝛿 ) Ĥ E𝛾𝛿 E𝛼𝛽 + 4C𝛼3𝛾3 E𝛾3 E𝛼3 d𝜂 Ĥ 𝛼𝛽𝛾𝛿

h 2 + h ∫− 2

{

( C L ) } L L C L 2 E𝛾𝛿 E𝛼𝛽 +𝜂 E𝛾𝛿 E𝛼𝛽 + E𝛾𝛿 E𝛼𝛽 + 𝜂 E𝛾𝛿 E𝛼𝛽 d𝜂 C

(11.501)

C

( ) C C 4kS C𝛼3𝛾3 E𝛾3 E𝛼3 d𝜂 C

L

C

where we have used definitions for E𝛼𝛽 , E𝛼𝛽 as in equation (11.498) and for E𝛼3 as equation (11.499); finally, noting that:

h ∫ 2h − 2

d𝜂 = h,

h ∫ 2h − 2

𝜂 d𝜂 = 0, and

into consideration the definition in equation (11.491), we have: 1 U= 2

{(

C

C

L

L

)

DE E𝛾𝛿 E𝛼𝛽 +DM E𝛾𝛿 E𝛼𝛽

H

𝛼𝛽𝛾𝛿

h ∫ 2h − 2

+ 2DS C

𝜂 2 d𝜂 =

𝛼𝛾

C

C

h3 , and taking 12

}

E𝛾3 E𝛼3

(11.502)

where we have introduced the following definitions: Eh 1 − 𝜈2 Eh3 DB ≡ 12(1 − 𝜈 2 )

DM ≡

DS ≡ 2Gh = 2 H 𝛼𝛽𝛾𝛿 ≡

(Membrane stiffness) (Bending stiffness)

Eh = DE (1 − 𝜈) 2(1 + 𝜈)

(11.503) (Shear stiffness)

( ) 1−𝜈 2𝜈 𝛼𝛽 𝛾𝛿 G𝛼𝛾 G𝛽𝛿 + G𝛼𝛿 G𝛽𝛾 + G G 2 1−𝜈

From equation (11.503), let us note that: ( ) 1−𝜈 2𝜈 11 𝛾𝛿 G1𝛾 G1𝛿 + G1𝛿 G1𝛾 + G G 2 1−𝜈 ( ) ⎧ 1 − 𝜈 1 + 1 + 2𝜈 = 1, 𝛾 = 𝛿 = 1 ⎪ 2 1−𝜈 ⎪ 1 − 𝜈 2𝜈 =⎨ ( ) = 𝜈, 𝛾 = 𝛿 = 2 2 1−𝜈 ⎪ ⎪ = 0, 𝛾 ≠ 𝛿 ⎩

H 11𝛾𝛿 ≡

(11.504)

Nonlinear Shell

875

Similar expressions can be obtained for H 11𝛾𝛿 ; for H 12𝛾𝛿 , we have: ( )  1−𝜈 2𝜈 𝛾𝛿 G  G1𝛾 G2𝛿 + G1𝛿 G2𝛾 + G12 2 1−𝜈 ⎧ 0, 𝛾 = 𝛿 = 1 ⎪ ⎪ = ⎨ 1 − 𝜈, 𝛾 ≠ 𝛿 ⎪ 2 ⎪ ⎩ 0, 𝛾 = 𝛿 = 2

H 12𝛾𝛿 ≡

(11.505)

Using the relations given by equations (11.498) and (11.499), the 2D strain energy, U, can be expressed as functions of the components of the spatial strains, β𝛼 and χ𝛼 , and the stress resultants and the stress couples are given by: n F 𝛼𝛽 n M 𝛼𝛽

𝜕U 𝜕U 𝜕U , n F 𝛼3 = n 𝛼3 , n F 33 = n 33 , n 𝛼𝛽 𝜕 𝛽 𝜕 𝛽 𝜕 𝛽 𝜕U 𝜕U 𝜕U = n 𝛼𝛽 , n M 𝛼3 = n 𝛼3 , n M 33 = n 33 𝜕 𝜒 𝜕 𝜒 𝜕 𝜒

=

(11.506)

Now, performing the derivative operations on equation (11.502), and using equations (11.498) and (11.499), we finally get: n F 11

= DM ( n 𝛽 11 + 𝜈 n 𝛽 22 ),

n F 22

n F 𝛼3

= DM ( n 𝛽 22 + 𝜈 n 𝛽 11 ), ) ( 1 − 𝜈 n 12 = DM 𝛽 = D𝜈M n 𝛽 12 , 2 ) ( 1 − 𝜈 n 21 = DM 𝛽 = D𝜈M n 𝛽 21 , 2 = DS n 𝛽 𝛼3 ,

n M 12

= DB ( n 𝜒 12 − 𝜈 n 𝜒 21 ),

n M 21

= DB ( n 𝜒 21 − 𝜈 n 𝜒 12 ),

n M 11

= DB (1 − 𝜈) n 𝜒 11 = D𝜈B n 𝜒 11 ,

n M 22

= DB (1 − 𝜈) n 𝜒 22 = D𝜈B n 𝜒 22 ,

n F 12 n F 21

n M 𝛼3

(11.507)

= 0,

where we have introduced the definitions: D𝜈M ≡ DM

(

1−𝜈 2

) and

D𝜈B ≡ DB (1 − 𝜈) ,

(11.508)

Remark

r It may be noted that because of the inextensible normal to the shell reference surface,

n M 𝛼3

=0 for 𝛼 = 1, 2; in other words, the constant normal length constraint reduces the rotation degrees of freedom to only two, and the rotation tensor belongs to 2-sphere, or the S2 manifold instead

Computation of Nonlinear Structures

876

of SO3 manifold. If we want to continue with the original three degrees of freedom, we may approximate the torsional stress–strain relations for n M 𝛼3 and n 𝜒 𝛼3 by including a very small term such as: n M 𝛼3 = 𝜀 (1 − 𝜈) DB n 𝜒 𝛼3 ≡ 𝜀D𝜈B n 𝜒 𝛼3 with a small enough torsional parameter (Chroscielewski et al. 1992), say, 𝜀 ≤ 0.01. Thus, finally, we can express the stress– strain matrix, D, of a thin, linear, hyperelastic, homogeneous shell by:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

⎫ ⎧ ⎪ n N 12 ⎪ ⎪ ⎪ ⎪ n N 13 ⎪ ⎪ ⎪ n M 11 ⎪ ⎪ ⎪ n M 12 ⎪ ⎪ ⎪ ⎪ n M 13 ⎪ ⎪ ̂ = D⎨ n N 21 ⎬ ⎪ ⎪ n N 22 ⎪ ⎪ ⎪ ⎪ n N 23 ⎪ ⎪ ⎪ n M 21 ⎪ ⎪ ⎪ n M 22 ⎪ ⎪ ⎪ ⎪ n M 23 ⎩ ⎪ ⎭ n N 11

n𝛽

⎫ ⎪ 12 ⎪ n𝛽 ⎪ 13 n𝜒 ⎪ 11 ⎪ n𝜒 ⎪ 12 ⎪ n𝜒 13 ⎪ n𝛽 ⎬ 21 ⎪ n𝛽 ⎪ 22 ⎪ n𝛽 23 ⎪ n𝜒 ⎪ 21 ⎪ n𝜒 ⎪ 22 ⎪ n𝜒 ⎭ 23 11

n𝛽

(11.509)

̂ is defined as: where the (12 × 12) constitutive matrix, D, 0 0 0 0 0 0 𝜈DM 0 0 0 0 ⎤ ⎡ DM ⎢ 0 2D𝜈M 0 0 0 0 0 0 0 0 0 0 ⎥ ⎥ ⎢ 0 DS 0 0 0 0 0 0 0 0 0 ⎥ ⎢ 0 ⎥ ⎢ 0 0 2D𝜈B 0 0 0 0 0 0 0 0 ⎥ ⎢ 0 ⎢ 0 0 0 0 DB 0 0 0 0 −𝜈DB 0 0 ⎥ ⎥ ⎢ 0 0 0 0 𝜀D𝜈B 0 0 0 0 0 0 ⎥ ⎢ 0 ̂ ≡⎢ D ⎥ 0 0 0 0 0 2D𝜈M 0 0 0 0 0 ⎥ ⎢ 0 ⎢ 𝜈DM 0 0 0 0 0 0 DM 0 0 0 0 ⎥ ⎥ ⎢ 0 0 0 0 0 0 0 DS 0 0 0 ⎥ ⎢ 0 ⎢ 0 0 0 0 −𝜈DB 0 0 0 0 DB 0 0 ⎥⎥ ⎢ ⎢ 0 0 0 0 0 0 2D𝜈B 0 ⎥ ⎥ ⎢ ⎣ 0 0 0 0 0 0 0 𝜀D𝜈B ⎦ Note that the components of the stress resultants, the stress couples and the strains are all indicated in the rotated spin bases; the corresponding relations for the engineering components for the stress couples are easily obtained with the help of the simple permutation matrix discussed earlier in the text.

Nonlinear Shell

11.16.4

877

Where We Would Like to Go

Now for the next step of the computational mechanics of shells, as we can see from equation (11.468), the virtual functional in the integral or weak form of equation, being generally highly nonlinear in displacements and rotations, may not be solved in a closed form analytical way. The solution that is – the tracing of the shell motion or the configuration path – will require some sort of iterative numerical method. Of all the possible schemes, the one that we intend to pursue is a Newton-type method. This, in turn, requires linearization of the equations at every iteration step. Clearly, then, we will have to further modify the virtual work equation for computational strategy. Thus, a logical place to go from here will be:

r the covariant linearization of virtual functional. 11.17 Covariant Linearization of Virtual Work 11.17.1 What We Need to Recall

r The generalized virtual work equation for quasi-static proportional loading, for all admissible ⌣

tθ ̄ }T ,

generalized virtual displacement, t d̄ ≡ { t d̄ ⌣

⌣

⌣

⌣

is given as:

⌣

⌣

⌣

⌣

̄ ≡ GInternal ( t d; t d) ̄ − Gexternal (𝜆; t d) ̄ = 0, GStatic (𝜆, t d; t d) ̄ = GInternal ( t d; t d)

⌣

∫

⌣

⌣

⌣

∀ admissible t d̄

⌣

̄ T n F( t d) dA ε( d; t d)

n̄ t

(11.510)

M ⌣

̄ = GExternal (𝜆; t d)

⌣

⌣

∫

t ̄T t ̂

F(𝜆, S1 , S2 ) dA + 𝜆

d

∫

⌣

⌣

d

F𝜈 (S) dS

t ̄T t ̂

𝜕M f

M

r The generalized virtual work equation for dynamic loading, for all admissible generalized ⌣

virtual displacement, t d̄ ≡ { t d̄

tθ ̄ }T ,

is given as:

̄ ≡G ̄ t θ) GDyn ( t d, t θ; t d, Internal + GInertial − GExternal ⌣

⌣

̄ = GInternal ( t d; t d)

⌣

∫

⌣

⌣

⌣

⌣

̄ T n F( t d) dA ε( d; t d)

n̄ t

M ⌣

⌣

̄ = GInertial ( t d; t d)

⌣

∫ M

=

t ̄T t

d

{

t ̄T

d

∫

⌣

⌣

FIner ( t d) dA ∙∙

∙

T M0 t d + t θ̄ I0 (WT t ω)

(11.511)

} dA

M ⌣

̄ = GExternal ( t d)

⌣

∫ M

= 0,

t ̄T

d

⌣

t̂

F(S1 , S2 ) dA +

̄ t θ̄ ∀ admissible t d,

⌣

∫

𝜕M f

t ̄T

d

⌣

t̂

F𝜈 (S) dS

Computation of Nonlinear Structures

878

r

⌣

n ε, ̄

the generalized virtual strain field is given as: ⌣

nε ̄

≡

{

nβ ̄

nχ ̄1

nβ ̄

nχ ̄2

}T

, ) ( ( ) nβ ̄ = t RT t d, ̄ t e𝛼 + t k0 × t d = t RT (1̂ 𝛼 + t d|𝛼 ), ̄ 𝛼 + t k0 × t d̄ − t RT t 𝚽 𝛼 𝛼 𝛼 } { ( ) nχ ̄ = t RT t φ, ̄ + k0 × t φ ̄ = t RT k0 + k , ̄ 𝛼 ≡ t RT t φ| 𝛼 𝛼 𝛼 𝛼 𝛼 tφ ̄ = W t θ̄ 1

2

(11.512) where W is given as:

W( t θ) ≡ I + c1 t 𝚯 + c2 t 𝚯2 , c1 ( t θ) =

r

1 − cos 𝜃 , 𝜃2

c2 ( t θ) =

(11.513)

𝜃 − sin 𝜃 𝜃3

with t 𝚯 is the skew-symmetric matrix corresponding to the axial vector, t θ, such that t 𝚯 t θ = 0. ∙ W is given as: ∙

∙

∙

∙

∙

∙

W = c1 𝚯 +c2 (𝚯 𝚯 + 𝚯 𝚯) + c1 𝚯 + c2 𝚯2 ∙

∙

∙

∙

∙

∙

= (c2 − c1 )(θ ∙ θ)I + c1 𝚯 + c2 {(θ ⊗ θ) + (θ ⊗ θ)} + c1 𝚯 + c2 (θ ⊗ θ) ∙ ∙ ∙ ∙ ∙ ∙ 1 1 c1 ≡ 2 (1 − 2c1 − 𝜃 2 c2 )(θ ∙ θ) = c3 (θ ∙ θ), c2 ≡ 2 (c1 − 3c2 )(θ ∙ θ) = c4 (θ ∙ θ), 𝜃 𝜃 (11.514)

r W is given as: ∙∙

} ∙ ∙ ∙∙ ∙∙ ∙ ∙ ∙ ∙ ∙∙ ∙∙ c1 𝚯 + c2 (𝚯 𝚯 + 𝚯 𝚯) + c2 (𝚯 𝚯 + 2 𝚯2 +𝚯 𝚯) + c1 𝚯 + c2 𝚯2 } } { { ∙ ∙ ∙ ∙ ∙∙ ∙∙ 2 2 2 2 c1 ≡ c5 (θ ∙ θ) + c3 𝜃 + (θ ∙ θ) , c2 ≡ c6 (θ ∙ θ) + c4 𝜃 + (θ ∙ θ) ∙∙

∙∙

{

W = c1 𝚯 +2

c5 ≡ −

1 (2c2 + 4c3 + 𝜃 2 c4 ), 𝜃2

c6 ≡ −

1 (c3 − 5c4 ), 𝜃2 (11.515)

r

⌣

n F,

the generalized reactive force field given as: n

r

⌣

tF

Iner ,

⌣

F≡

{

nF 1

nM

1

nF 2

nM

}T 2

(11.516)

the generalized inertial force field given as: ⌣

tF Iner tω

}T { ∙ ∙∙ ≡ M0 t d I (WT t ω) , 0 ∙

≡ W t θ,

t

∙

∙

∙

∙∙

ω = (W t θ + W t θ)

(11.517)

Nonlinear Shell

879

where the angular velocity vector, t ω, is the axial vector corresponding to the skew-symmetric angular velocity matrix, 𝛀, given by: ∙

Ω ≡ R RT

r

(11.518)

⌣

t F, ̂

the generalized external applied force field given as: ⌣

t̂

F≡

{

tF ̂1

tM ̂

1

tF ̂2

tM ̂

}T

(11.519)

2

r Equation

⌣

(11.510) and equation (11.511) are generally nonlinear in t d ≡ { t d t θ }T , the generalized displacement state of the shell; similarly, all entities in equation (11.512) to equation ⌣

r

(11.519) are functions of generalized state variables, t d ≡ { t d t θ }T . For the quasi-static case, we have restricted our presentation to a weak form involving only proportional loading system with the assumption that the forces are truly external and hence ⌣

r

r

independent of the displacement state, t d ≡ { t d t θ }T , and as such can be expressed with a multiplicative variable of proportionality, 𝜆, as we have indicated in the above equation. Briefly, the basic traits of a Newton-type method of tracing a configuration path entail the following: starting from a known state or configuration (which is usually the unloaded and undeformed structure at the first step), we find the tangent to the equilibrium or configuration path (usually, as a rule for every load step for a quasi-static system, or for every time step for a time history dynamic analysis) that helps us set up the instantaneous linear equations to be solved for the incremental and independent variables iteratively for each step; the state variables are updated by the incremental state variables and the process is repeated. Finally, the superscript n in the above expressions that helps to identify the relevant entities in the current, rotated or deformed configuration will occasionally be dropped for notational clarity although the context will lend support to consistency.

11.17.2

Incremental State Variables

Thus, for numerical evaluation of the motion of a shell on a digital computer by a typical iterative Newton-type method, we will need to derive for every step of the iteration, the instantaneous linearized form of the virtual equations as a function of the incremental generalized displacement ⌣

state; we denote it hereon as t Δd ≡ { t Δd 11.17.2.1

t Δθ }T .

Quasi-static Proportional Loading

For quasi-static proportional load cases, in actual numerical evaluation with the incremental load step, denoted by Δ𝜆, as a preassigned entity in a typical load-control incrementation method, a Newton-type method without any special treatment such as arc length constraint, is inherently disastrous at the limit points or at the bifurcation points of a configuration path. Thus, for computational purpose (i.e. to maintain symmetry of the stiffness matrices), we consider a Newton-type method with an arc length constraint algorithm in which Δ𝜆 also is treated as an ⌣

additional independent variable as the displacement state variable, t d ≡ { t d t θ }T . Thus, the virtual work equation (11.510) must be linearized with respect to both the independent variable ⌣

displacement sets: t d ≡ { t d

t θ }T ,

and the coefficient of proportionality, 𝜆.

Computation of Nonlinear Structures

880

11.17.2.2

Dynamic Time History Loading

For dynamic time history load cases, the virtual work equation (11.511) must be linearized with ⌣

respect to only the independent variable displacement sets: t d ≡ { t d t θ }T , but we need to additionally linearize the inertial virtual work functional with particular attention to the angular ∙

acceleration, t ω. Finally, in attempting to linearize the virtual equations, we must be alert to the fact that the rotation tensor belongs to a curved space; hence, the linearization must be performed by the covariant differentiation in mind; the practical way to accommodate covariant differentiation is simply to include the differentiation of the associated base vectors (for the vector fields), or the base tensors (for the tensor field).

11.17.3

Linearization of the Virtual Work Equation

As indicated earlier, the virtual work – equation (11.510) for the quasi-static case or equation (11.511) for the dynamic case – must be linearized, that is, expressed in incremental form for an incremental quasi-static load step or a dynamic time step, to apply a Newton-type iteration scheme for tracing the motion of a shell. Let the known – but not necessarily in equilibrium – ⌣ ⌣ ⌣ ̄ may not satisfy equation configuration be given by t d0 at a load level 𝜆0 , that is, G(𝜆0 , t d0 ; t d) ⌣

⌣

̄ for the dynamic analysis). We will try (11.510) for the quasi-static case (similarly, G( t d0 ; t d) ⌣

to seek equilibrium in the neighborhood of (𝜆0 , t d0 ) for the quasi-static case (similarly, in the ⌣

⌣

neighborhood of t d0 for the dynamic analysis). More specifically, at (𝜆, t d) for the quasi-static ⌣ ⌣ ⌣ ⌣ ̄ = 0, ∀ t d̄ admissible (similarly, at t d for the dynamic analysis, case, that is, we want G(𝜆, t d; t d) ⌣

⌣

⌣

⌣

̄ = 0, ∀ t d̄ admissible). Expanding by Taylor series around (𝜆0 , d0 ) that is, we want G( t d; t d) ⌣

for quasi-static case (similarly, around d0 for the dynamic case) and retaining only the first⌣

⌣

⌣

order (linear) terms, that is, terms involving Δt d (≡ t d − t d0 ) and Δ𝜆 (≡ 𝜆 − 𝜆0 ), we have the following.

11.17.3.1

For Quasi-static Proportional Loading ⌣

⌣

⌣

⌣

⌣

⌣

⌣

t d; t d; ̄ Δ t d, Δ𝜆) ≡ G(𝜆0 , t d0 ; t d) ̄ + Quasi [G(𝜆0 , t d0 ; t d)] ̄ =0 ̃ G(𝜆, ⌣ ⌣

⌣

⌣

⌣

Δ t d ,Δ𝜆

⌣

⌣

⌣

⇒

⌣

̄ [G(𝜆0 , t d0 ; t d)]



⌣

̄ ≡ ∇𝜆0 G(𝜆0 , t d0 ; t d) ̄ ∙ Δ𝜆 + ∇ ⌣ G(𝜆0 , t d0 ; t d) ̄ ∙ Δt d [G(𝜆0 , t d0 ; t d)] t 0

Quasi Δ t d ,Δ𝜆 Quasi Δ t d ,Δ𝜆

=

̄ Δ t d) −G(𝜆0 , t d0 ; t d; ⌣

⌣

⌣

⌣

⌣

d

(11.520) ⌣

where ∇ t ⌣0 G and ∇𝜆0 G are the covariant derivatives of G with respect to t d and 𝜆 evaluated at ⌣

d

and 𝜆0 , respectively, and the linear part – that is, the first-order terms – of the virtual functional ⌣ ⌣ ⌣ ⌣ ̄ In a non-equilibrium state, G(𝜆0 , t d0 ; t d), ̄ known as the is denoted by Quasi [G(𝜆0 , t d0 ; t d)].

t d0

⌣

Δ t d ,Δ𝜆

Nonlinear Shell

881

total residual force functional, is non-zero. Noting that for conservative loading, the external loading does not depend on the current configuration, we can express the residual as: ⌣

⌣

⌣

⌣

⌣

̄ ≡ GInternal ( t d0 ; t d) ̄ − GExternal (𝜆0 ; t d) ̄ GStatic (𝜆0 , t d0 ; t d) Similarly, ⌣

of

the

⌣

⌣

coefficient

of

𝜆,

proportionality,

the

linear

operator,

̄ also consists of two parts: the internal and the external linear virtual [G(𝜆0 , t d0 ; t d)],

Quasi



because

(11.521)

Δ t d ,Δ𝜆

functional: ⌣

⌣

⌣

⌣

⌣

⌣

̄ ≡ ΔGInternal ( t d0 ; t d; ̄ Δ t d) + ΔGExternal (𝜆0 ; t d; ̄ Δ𝜆) Quasi [G(𝜆0 , t d0 ; t d)] ⌣ Δ t d ,Δ𝜆

⌣

⌣

⌣

(11.522)

̄ ∙ Δt d ΔGInternal ≡ ∇ t ⌣0 GInternal ( t d0 ; t d) d

⌣

̄ ∙ Δ𝜆 ΔGExternal ≡ ∇𝜆0 GExternal (𝜆0 ; t d)

11.17.3.2

For Dynamic Time History Loading ⌣

⌣

⌣

⌣

̄ ≡ G( t d0 ; t d) ̄ + ̃ t d; t d) G( ⇒

⌣

Δtd



⌣

Δtd

⌣

⌣

⌣

⌣

Dyn [G( t d 0 ; t d)] ̄ Dyn [G( t d 0 ; t d)] ̄

⌣ Δtd

⌣

⌣

⌣

⌣

Dyn [G( t d 0 ; t d)] ̄

=0

̄ = −G( t d0 ; t d)

(11.523) ⌣

⌣

⌣

̄ ∙ Δt d ≡ ∇ t ⌣0 G( t d0 ; t d) d

⌣

⌣

where ∇ t ⌣0 G is the covariant derivative of G with respect to t d evaluated at t d0 , and the linear d

Dyn

part – that is, the first-order terms – of the virtual functional is denoted by  ⌣

⌣

⌣

⌣

̄ In [G( t d0 ; t d)].

Δtd

⌣

̄ known as the residual force functional, is non-zero. Noting a non-equilibrium state, G( t d0 ; t d), that for conservative loading, the external loading does not depend on the current configuration, we can express the residual as: ⌣

⌣

⌣

⌣

⌣

⌣

⌣

̄ ≡ GInternal ( t d0 ; t d) ̄ + GInertial ( t d0 ; t d) ̄ − GExternal ( t d) ̄ GDyn ( t d0 ; t d) ⌣

(11.524)

⌣

0 t 0 t̄ ⌣ [G(𝜆 , d ; d)], consists only of two parts: the internal and the

Dyn

The linear operator, 

Δtd

inertial linear virtual functionals: 

⌣ Δtd

⌣

⌣

Dyn [G( t d 0 ; t d)] ̄

⌣

⌣

⌣

⌣

⌣

⌣

̄ Δ t d) + ΔGInertial ( t d0 ; t d; ̄ Δ t d) ≡ ΔGInternal ( t d0 ; t d; ⌣

⌣

⌣

⌣

⌣

⌣

̄ ∙ Δt d ΔGInternal ≡ ∇ t ⌣0 GInternal ( t d0 ; t d) d

(11.525)

̄ ∙ Δt d ΔGInertial ≡ ∇ t ⌣0 GInertial ( t d0 ; t d) d

In what follows, we provide a complete description of the various parts just mentioned.

Computation of Nonlinear Structures

882

11.17.4

Internal Virtual Functional: Covariant Linearization

Referring to equation (11.510) or equation (11.511), the internal virtual work functional is given as: ⌣

⌣

̄ = GInternal ( t d; t d)

⌣

⌣

⌣

⌣

⌣

̄ ∙ n F( t d) dA ε( d; t d)

n̄ t

∫

(11.526)

M

Taking the covariant derivative of the expression in equation (11.526), we have: ⌣

⌣

⌣

⌣

⌣

̄ Δ t d) ≡ ∇ ⌣ G( t d0 ; t d) ̄ ∙ Δt d̂ ΔG( t d0 ; t d; t d

=

⌣

⌣

ε̄ T Δn F dA +

∫ M

⌣

∫

Δn ε̄ T

n

⌣

F dA

M

⌣

⌣

⌣

⌣

⌣

(11.527)

Δn F ≡ ∇ t ⌣ n FT Δ t d, d

⌣

Δn ε̄ ≡ ∇ t ⌣ n ε̄ T Δ t d d

⌣

where ∇ t ⌣ G is the covariant derivatives of G with respect to t d, and we have introduced the d following. Definitions ⌣

⌣

⌣

The incremental generalized force: Δn F ≡ (∇n⌣ F)T Δ t d, td

and ⌣

⌣

⌣

̄ T Δ t d], The incremental generalized virtual strains: Δ ε̄ ≡ [(∇n⌣ ε) td

⌣

⌣

⌣

Now, noting that Δn F = n D Δn ε, the generalized reactive forces in the current configuration ⌣

of a linear, isotropic, homogeneous and hyperelastic material with n D as the constitutive matrix, we can rewrite the equation as: ⌣

⌣

⌣

̄ ∙ Δt d = ∇⌣0 G(𝜆0 , t d0 ; t d) d

⌣

∫ M

n ̄T n

ε

⌣

⌣

⌣

D Δn ε dA +

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ Material stiffness-specific

∫

⌣

̄ n F dA Δn εT

(11.528)

M

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏟

Geometric stiffness-specific

Remarks

r It may be noted that the internal virtual functional linearization consists of two parts. The first ⌣

⌣

⌣

part, ∫ n ε̄ T n D Δn ε dA , on the right-hand side of equation (11.528), involves the virtual strains Ω

and the incremental real strains derived from the linearization of the generalized reactive forces.

Nonlinear Shell

883

More specifically, it is the general nature of the nonlinear formulation that the incremental and the first variations of the real strains eventually give rise to what is known as the material stiffness functional, M : Definition: Material stiffness functional, M : ⌣

⌣

⌣

̄ Δt d) ≡ M ( t d, t d;

⌣

∫

n ̄T n

ε

⌣

⌣

D Δn ε dA

(11.529)

M

r Likewise,

⌣

⌣

̄ n F dA , as a general trait of a geometrically nonlinear the second part, ∫ Δn εT Ω

system, involves the second variations of the real strains or the first variation or increment of the virtual strains, derived from the linearization of the generalized strains and resulting in what is known as the geometric stiffness functional, G : Definition: Geometric Stiffness functional, G : ⌣

⌣

⌣

̄ Δ t d) ≡ G ( t d, t d;

⌣

⌣

⌣

⌣

⌣

̄ Δ t d) n F dA Δn ε̄ T ( t d, t d;

∫

(11.530)

M

r Clearly, then, if the real strain is linear, the second variation (or the Hessian) and thus, the second part vanish identically, that is, G ≡ 0. In other words, for linear shell responses, the geometric stiffness part drops out, with the shell tangent stiffness being purely due to the material stiffness contribution. Definition: Tangent Stiffness functional, T : T ≡ M + G

(11.531)

As indicated before, if the geometric stiffness functional is absent, as in the case of a linear shell system, the tangent stiffness functional is populated solely by the material stiffness functional.

11.17.5

Inertial Virtual Functional: Covariant Linearization

Referring to equation (11.511), the inertial virtual work functional is given as: ⌣

⌣

̄ = GInertial ( t d; t d)

⌣

∫ M

=

∫ M

t ̄T t

d

⌣

⌣

FIner ( t d) dA ∙∙

∙

{ t d̄ T M0 t d + t θ̄ I0 (WT t ω) }dA T

(11.532)

Computation of Nonlinear Structures

884

11.17.5.1

∙∙

Linearized Translational Acceleration Vector, 𝚫 t d ∙∙

∙∙

The translational acceleration vector, t d , is trivially linearized as Δ t d , since it is a member of a Euclidean vector space. Now, linearizing the expression in equation (11.532), we have: ⌣

⌣

⌣

⌣

⌣

d

=

∫

t ̄T

t

∙∙

⌣

⌣

̄ Δ t d) ≡ {∇ ⌣ GInertial ( t d; t d)} ̄ T Δ td = ΔGInertial ( t d; t d; t

∫

t ̄T

⌣

⌣

⌣

d Δ t FIner ( t d; Δ t d) dA

M ∙

t ̄T

∙

{ d (M0 Δ d ) + θ (I0 (ΔWT t ω +WT Δ t ω)) }dA

M

(11.533) Before we can generate the computationally efficient matrix form for the covariant linearization of the inertial virtual functional from equation (11.533), we need the following results. 11.17.5.2

∙

Linearized Angular Acceleration Vector, 𝚫 t ω

However, the skew-symmetric angular velocity matrix, ω, is related to the rotation matrix, R, that belongs to a curved space; hence, as before, linearization involves additional consideration as follows. Let ΔΨ define the incremental skew-symmetric matrix related to the linearized or incremental rotation matrix, ΔR, such that: Δ𝚿 ≡ ΔR RT

(11.534)

whereby the axial vector, Δψ, to the matrix, Δ𝚿, is related to the incremental rotation vector, Δθ, by: Δψ = W( t θ) Δ t θ

(11.535)

Now, starting from the relationship given by equation (11.518), and interchanging the order of the linear operations, the incremental or linearized angular velocity matrix, Δ t 𝛀, is given by: ∙

∙

Δ t 𝛀 = Δ R RT + R ΔRT ∙

⏞⏞⏞ ∙ ∙ = Δ𝚿 R RT + t 𝛀 R RT Δ𝚿T = Δ𝚿 R RT + Δ𝚿 R RT − t 𝛀 Δ𝚿 ∙ ∙ ( ) = Δ𝚿 +Δ𝚿 t 𝛀 R RT − t 𝛀 Δ𝚿 = Δ𝚿 + Δ𝚿 t 𝛀 − t 𝛀 Δ𝚿

(11.536)

where we have used the fact: R RT = I, with I as the 3 × 3 identity matrix. Noting the commutator or Lie-bracket property (i.e. for any two skew-symmetric matrices, A, B with a, b as the corresponding axial vectors in three dimensions, we have: AB − BA = [a × b], where the notation [c] ≡ c× for any vector c is implied) in equation (11.536), the incremental or linearized angular velocity vector, Δ t ω, the axial vector of Δ t 𝛀, is given by: ∙

Δ t ω = Δψ − t ω × Δψ

(11.537)

Nonlinear Shell

885

Taking the time derivative of the expression in equation (11.537), for the incremental or ∙

linearized angular acceleration, Δ t ω, we get: ∙∙

∙

∙

∙

Δ t ω = Δψ − t ω × Δψ − t ω ×Δψ

(11.538)

Now, we first note from equation (11.517) and equation (11.535): ∙

∙

∙

∙

∙

∙

∙

∙

ω × Δ ψ = W θ × (W Δθ + W Δ θ) = (W 𝚯 W) Δθ + (W 𝚯 W) Δθ

(11.539)

Similarly, we have: ∙

∙

∙

∙∙

∙

∙

∙∙

ω ×Δψ = (W θ +W θ) × W Δθ = (W 𝚯 W + W 𝚯 W) Δθ ∙

∙∙

∙

∙

∙∙

∙

(11.540) ∙

∙∙

Finally, noting that: Δ ψ = W Δ t θ + W Δ t θ and Δ ψ = W Δ t θ + 2 W Δ t θ +W Δ t θ , and using equations (11.539) and (11.540) in equation (11.538), we have the incremental or lin∙

earized angular acceleration, Δ t ω, in terms of the incremental or linearized rotation vector, Δ t θ, ∙

the incremental or linearized rotational velocity vector, Δ t θ, and the incremental or linearized ∙∙

rotational acceleration vector, Δ t θ, as: ∙∙

∙

∙

∙

Δ t ω = Δψ − t ω × Δψ − t ω ×Δψ ∙∙

=

∙

∙

∙

W Δ t θ + (2 W −W t 𝚯 W) Δ t θ ⏟⏟⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ MassGyroscopicspecific ∙∙

∙

∙

specific ∙

∙

(11.541) ∙∙

+ (W −{W t 𝚯 W + W t 𝚯 W} − W t 𝚯 W) Δ t θ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ Centrifugal-specific Now, from equation (11.533) and (11.541), we have: ∙∙

∙

∙

∙

∙

WT Δ ω = (WT W)Δ θ + WT (2 W −W t 𝚯 W) Δ θ ∙∙

∙

∙

∙

∙

∙∙

+ WT (W −{W 𝚯 W + W 𝚯 W} − W t 𝚯 W) Δθ ∙ ∙

(11.542)

∙∙

Again, from equation (11.533), letting angular acceleration, a ≡ W θ + W θ, we have: ( ) ΔWT a = {−c1 Δ𝚯 + c2 (Δ𝚯 𝚯 + 𝚯 Δ𝚯) + c3 𝚯 + c4 𝚯2 (θ ∙ Δθ)} a ) ( = {c1 [a] + −c2 I − c3 𝚯 + c4 𝚯2 (a ∙ θ) + c2 ((θ ⊗ a) − (a ⊗ θ) + (θ ∙ a) I} Δθ (11.543)

Computation of Nonlinear Structures

886

Exercise: Derive expression in equation (11.543). Finally, using the results in equation (11.542) and equation (11.543) in equation (11.533), we have the full linearization of the inertial virtual function in matrix form as: ⌣

⌣

⌣

⌣

⌣

⌣

⌣

⌣

⌣

⌣

⌣

⌣

̄ Δ t d) = ΔGM ( t d; t d; ̄ Δ t d) + ΔGG ( t d; t d; ̄ Δ t d) + ΔGC ( t d; t d; ̄ Δ t d) ΔGInertial ( t d; t d; ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ Mass functional

Gyroscopic functional

Centrifugal functional

(11.544) where

t

⌣

⌣

t̄

t

⌣

⌣

ΔGM ( d; d; Δ d) = ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ∫

⌣

⌣

t̄

d∙

t

⌣

⌣

Gyroscopic functional

⌣

⌣

t̄

t

I0

⌣

ΔGC ( d; d; Δ d) = ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ∫

[

0 0

⌣

W)

]

∙ ⌣

∙

WT (2 W −W 𝚯 W)

Δ t d dA ]

0

d∙

M

∙∙

Δ t d dA

0 ∙

0

M

t̄

(WT

0

d∙

⌣

Centrifugal functional

[

]

0

0

t̄

ΔGG ( d; d; Δ d) = ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ∫

t

M0 I

M

Mass functional

t

[

t̄

U(a) +

⌣

∙ ∙ ∙∙ ∙ ∙ WT {W −(W 𝚯 W + W 𝚯 W)

∙∙

− W 𝚯 W}

Δ t d dA

(11.545) where, from equation (11.543), we have defined operator, U, as: U(a) ≡ {c1 [a] + (−c2 I − c3 𝚯 + c4 𝚯2 )(a ∙ θ) + c2 ((θ ⊗ a) − (a ⊗ θ) + (θ ∙ a)I}, ∙ ∙

∙∙

a ≡ Wθ + Wθ

(11.546) ∙

∙∙

with W, W and W are given by equations (11.513), (11.514) and (11.515), respectively.

11.17.6

Quasi-Static External Virtual Functional: Covariant Linearization

Referring to equation (11.510), the quasi-static external virtual work functional for proportional loading is given as: ⌣

̄ = GExternal (𝜆; t d)

⌣

∫ M

⌣

t ̄T t ̂

d

F(𝜆, S1 , S2 ) dA + 𝜆

∫

𝜕M f

⌣

⌣

d

F𝜈 (S) dS

t ̄T t ̂

(11.547)

Nonlinear Shell

887

Then, we have the following symbolic representation of the first-order terms: ⌣

⌣

⌣

⌣

̄ Δ𝜆) ≡ ∇𝜆 GExternal (𝜆, t d; t d) ̄ ∙ Δ𝜆 ΔGExternal (𝜆, t d; t d; ⎧ ⎪ = −⎨ ∫ ⎪M ⎩

⌣

⌣

d

F, 𝜆 (𝜆, S , S ) dA +

t ̄T t ̂

1

2

⌣

∫

⎫ ⎪ F𝜈 (S) dS⎬ ∙ Δ𝜆 ⎪ ⎭ ⌣

t ̄T t ̂

d

𝜕M f

(11.548)

11.17.7

Quasi-static Linearized Virtual Work Equation

Thus, finally, using equations (11.528) and (11.548) in equation (11.520), we can introduce the following. Definition The linearized quasi-static virtual work equations: Material stiffness functional

Geometric stiffness functional

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⌣

∫

n ̄T n

ε

⌣

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⌣

⌣

D Δn ε dA +

∫

M

⌣

Δn ε̄ T n F dA

M Applied force functional

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⎫ ⎧ ⌣ ⌣ ⌣ ⌣ ⌣ ⌣ ⎪ ⎪ t ̄T t ̂ ̄ ̂ 𝜆 (𝜆0 , S1 , S2 ) dA + =− G(𝜆0 , t d0 ; t d) − ⎨ t d̄ T t F, d F𝜈 (S) dS⎬ ∙ Δ𝜆 ∫ ∫ ⎪ ⎪M 𝜕M f ⎭ ⎩ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ Unbalanced force functional

Residual force functional

(11.549) Or, Tangent functional

⎴ ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⎴ Material stiffness functional

⏞⏞⏞ M

Geometric stiffness functional

+

⏞⏞⏞ G

Applied force functional

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⎫ ⎧ ⌣ ⌣ ⌣ ⌣ ⎪ ⎪ t ⌣̄ T t ⌣̂ 0 t 0 t̄ 0 1 2 t ̄T t ̂ G(𝜆 , d ; d) =− −⎨ d F, 𝜆 (𝜆 , S , S ) dA + d F𝜈 (S) dS⎬ ∙ Δ𝜆 ∫ ∫ ⎪ ⎪M 𝜕M f ⎭ ⎩ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ Unbalanced force functional

Residual force functional

(11.550) where, we have introduced and collected, for convenience, the following.

Computation of Nonlinear Structures

888

Definitions

r The unbalanced force functional, G(𝜆0 , t ⌣d0 ; t d): ̄ ⌣

⌣

⌣

̄ ≡ G(𝜆 , d ; d) 0

0

⌣

∫ M

n ̄T n

ε

⎧ ⎪ D ε dA − ⎨ ∫ ⎪M ⎩ ⌣ ⌣ n

⌣

⌣

t ̄T t ̂

F(𝜆 , S , S ) dA + 𝜆

d

0

1

2

0

∫

𝜕M f

t ̂̄

d

T

⎫ ⎪ F𝜈 (S) dS⎬ ⎪ ⎭

⌣

t̂

(11.551)

r The residual force functional: Unbalanced force functional

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⌣

⌣

̄ G(𝜆0 , t d0 ; t d)

Residual force functional ≡ −

Applied force functional

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⎫ ⎧ ⌣S ⌣ ⎪ ⎪ t ⌣̄ T t ⌣̂ 0 1 2 t ̄T t ̂ ⎨∫ d F, 𝜆 (𝜆 , S , S ) dA + ∫ d F (S) dS⎬ Δ𝜆 ⎪ ⎪M 𝜕M f ⎭ ⎩

+

(11.552) Material stiffness functional, M : ⌣

⌣

⌣

̄ Δt d) ≡ M ( t d, t d;

⌣

∫

n ̄T n

ε

⌣

⌣

D Δn ε dA

(11.553)

M

Geometric stiffness functional, G : ⌣

⌣

⌣

̄ Δ t d) ≡ G ( t d, t d;

⌣

∫

⌣

⌣

⌣

⌣

̄ Δ t d) n F dA Δn ε̄ T ( t d, t d;

(11.554)

M

11.17.8

Dynamic Linearized Virtual Work Equation

Thus, finally, using equation (11.528) and equations (11.544)–(11.546) in equation (11.523), we can introduce and collect, for convenience, the following.

Nonlinear Shell

889

Definition The linearized dynamic virtual work equations: Material stiffness functional

Geometric stiffness functional

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⌣

∫

n ̄T n

ε

⌣

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⌣

⌣

⌣

D Δn ε dA +

Δn ε̄ T n F dA

∫

M

M Inertial functional

Unbalanced force functional

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⌣

⌣

⌣

⌣

⌣

⌣

⌣

⌣

⏞⏞⏞⏞⏞⏞⏞

⌣

⌣

⌣

̄ Δ t d) + ΔGG ( t d; t d; ̄ Δ t d) + ΔGC ( t d; t d; ̄ Δ t d) = − ̄ + ΔGM ( t d; t d; G( t d0 ; t d) ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ Residual force functional Mass-specific Gyroscopic Centrifugal

(11.555) Or, Tangent functional

⎴ ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⎴ Material stiffness functional

⏞⏞⏞ M

Geometric stiffness functional

⏞⏞⏞ G

+

Inertial functional

Unbalanced force functional

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⌣

⌣

⌣

⌣

⌣

⌣

⌣

⌣

⏞⏞⏞⏞⏞⏞⏞

⌣

⌣

⌣

̄ Δ t d) + ΔGG ( t d; t d; ̄ Δ t d) + ΔGC ( t d; t d; ̄ Δ t d) = − ̄ + ΔGM ( t d; t d; G( t d; t d) ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ Residual force functional Mass-specific Gyroscopic Centrifugal

(11.556) where, for convenience, we collect the linearized internal and inertial terms for the following. Definitions

r The residual force functional, G( t ⌣d; t d): ̄ ⌣

⌣ ⌣

̄ ≡ G(d; d)

∫M

n⌣

𝜺̄

T

⎫ ⎧ ) (t ̂ ∙ Wt θ̄ dA F̂ ∙ t d̄ + t M ⎪ ⎪ ⌣ ⎪ ⎪ ∫M n n⌣ D 𝜀 dA − ⎨ ⎬ ) (t ̂ v ∙ Wt θ̄ dS ⎪ ⎪ + F̂ v ∙ t d̄ + t M ∫𝜕M f ⎪ ⎪ ⎭ ⎩

(11.557)

⌣ r The linearized mass functional, ΔGM ( t ⌣d; t d; ̄ Δ t d): ⌣

⌣

⌣

t̄

t

⌣

⌣

ΔGM ( d; d; Δ d) ≡ t

∫ M

[

M0 I d∙ 0

t̄

0 I0 (WT W)

]

∙∙ t

⌣

Δ d dA

(11.558)

Computation of Nonlinear Structures

890

⌣ r The linearized gyroscopic (damping) functional, ΔGG ( t ⌣d; t d; ̄ Δ t d): ⌣

t

⌣

⌣

t̄

t

[

⌣

⌣

ΔGG ( d; d; Δ d) =

t̄

d∙

∫

0

]

0 ∙

⌣

Δ d dA

∙

0 WT (2 W −W 𝚯 W)

M

∙

t

(11.559)

⌣ r The linearized centrifugal (stiffness) functional, ΔGC ( t ⌣d; t d; ̄ Δ t d): ⌣

t

⌣

⌣

t̄

t

[

⌣

⌣

ΔGC ( d; d; Δ d) =

t̄

0

0

U( ω) + W {W −(W 𝚯 W + W 𝚯 W) − W 𝚯 W}

d∙

∫ M

]

0 t

∙∙

∙

⌣

∙

∙

∙

∙

Δ t d dA

∙∙

T

∙ ∙ ⎧ ⎫ 2 t t ⎪ c1 [ ω] + (−c2 I − c3 𝚯 + c4 𝚯 )( ω ∙ θ) ⎪ U( ω) ≡ ⎨ ⎬, ∙ ∙ ∙ ⎪ +c2 ((θ ⊗ t ω) − ( t ω ⊗ θ) + (θ ∙ t ω )I) ⎪ ⎩ ⎭ ∙

t

t

∙

∙

∙

∙∙

ω = (W t θ + W t θ)

(11.560) with [h] ≡ h×, ∀h ∈ R3 , and W( t θ) ≡ I + c1 t 𝚯 + c2 t 𝚯2 , 1 − cos 𝜃 , 𝜃2 𝜃 − sin 𝜃 c2 ( t θ) = 𝜃3

c1 ( t θ) =

∙

∙

∙

∙

∙

(11.561)

∙

W = c1 𝚯 +c2 (𝚯 𝚯 + 𝚯 𝚯) + c1 𝚯 + c2 𝚯2 ∙

∙

∙

∙

∙

∙

= (c2 − c1 )(θ ∙ θ)I + c1 𝚯 + c2 {(θ ⊗ θ) + (θ ⊗ θ)} + c1 𝚯 + c2 (θ ⊗ θ) ∙ ∙ 1 2 (1 − 2c − 𝜃 c )(θ ∙ θ) = c (θ ∙ θ), 1 2 3 𝜃2 ∙ ∙ ∙ 1 c2 ≡ 2 (c1 − 3c2 )(θ ∙ θ) = c4 (θ ∙ θ), 𝜃 ∙

c1 ≡

(11.562) and ∙∙

∙∙

∙ ∙

∙

∙

∙

∙∙

∙

∙∙

∙∙

∙∙

W = c1 𝚯 +2{c1 𝚯 + c2 (𝚯 𝚯 + 𝚯 𝚯)} + c2 (𝚯 𝚯 + 2 𝚯2 +𝚯 𝚯) + c1 𝚯 + c2 𝚯2 ∙∙

∙

∙

c1 ≡ c5 (θ ∙ θ)2 + c3 {𝜃 2 + (θ ∙ θ)}, c5 ≡ −

1 (2c2 + 4c3 + 𝜃 2 c4 ), 𝜃2

c6 ≡ −

∙∙

∙

∙

c2 ≡ c6 (θ ∙ θ)2 + c4 {𝜃 2 + (θ ∙ θ)} 1 (c3 − 5c4 ), 𝜃2 (11.563)

Nonlinear Shell

11.17.9

891

Where We Would Like to Go

Clearly, the generalized form of the linearized equation (11.549) or equation (11.555) is presented in functional form and thus is still not suitable for computational purposes; that is, the expressions ⌣

for the strains hide the sought-after incremental variable, Δ t d. We would like to have a form of ⌣

the virtual linearized equation that is totally explicit in the incremental variables Δ t d and Δ𝜆 for ∙

⌣

∙∙

⌣

⌣

quasi-static cases, and Δ t d, Δ t d and Δ t d for dynamic analyses so that it can be easily solved digitally by a Newton-type iterative method. In other words, we would like the matrix forms of the equations that are easily solved on digital computers. For this, we need to make a few more logical stops as follows. For the matrix form of the incremental equation (11.555) for dynamic time history loading we want:

r finite element formulation: dynamic loading r For the matrix form of the incremental equation (11.549) for quasi-static loading: r finite element formulation: quasi-static loading

11.18 c-type FE Formulation: Dynamic Loading 11.18.1 What We Need to Recall ↼ r The virtual work functional, GDyn ( t d, t θ; t d, ̄ t θ), in compact generalized notation:

⌣

⌣

̄ ≡ GInternal + GInertial − GExternal = 0, GDyn ( t d; t d)

GInternal =

⌣

∫

n̄ n

⌣

∀ admissible t d̄

⌣

ε F dA

M ⌣

GInertial =

∫

t ̄T t

d

(11.564) ⌣

FIner dA

M ⌣

GExternal =

∫

⌣

t ̄T t ̂

d

F(S1 , S2 ) dA + 𝜆

∫

⌣

⌣

d

F(S) dS

t ̄T t ̂

𝜕M f

M

with the applicable initial conditions on the dynamic state variables as:

⌣

⌣

d(0) = d0 . (11.565)

∙

t

⌣

⌣

d(0) = v 0 ,

Computation of Nonlinear Structures

892

The linearized dynamic virtual work equations: at a time t ∈ R+ ⌣

⌣

̄ ΔGInertial + ΔGInternal = residual force functional = R( t d(t); t d) Linearized internal functional, ΔGInternal

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ Material stiffness functional

Geometric stiffness functional

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⌣

∫

n ̄T n

ε

⌣

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

n⌣

D Δ ε(t) dA +

M

⌣

⌣

∫

Δn ε̄ T n F(t) dA

M

(11.566)

Inertial functional, ΔGInertial

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⌣

⌣

⌣

⌣

⌣

⌣

⌣

⌣

⌣

̄ Δ t d(t)) + ΔGG ( t d(t); t d; ̄ Δ t d(t)) + ΔGC ( t d; t d; ̄ Δ t d(t)) + ΔGM ( t d(t); t d; ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ Inertialmass

Gyroscopic

Centrifugal

Unbalanced force functional

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⌣

⌣

̄ G( t d(t); t d) =− ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ Residual force functional

or Tangent functional

⎴ ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⎴ Material stiffness functional

⏞⏞⏞ M (t)

Geometric stiffness functional

⏞⏞⏞ G (t)

+

Inertial functional

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⌣

⌣

⌣

⌣

⌣

⌣

⌣

⌣

⌣

̄ Δ t d(t)) + ΔGG ( t d(t); t d; ̄ Δ t d(t)) + ΔGC ( t d(t); t d; ̄ Δ t d(t)) + ΔGM ( t d(t); t d; ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ Mass-specific Gyroscopic Centrifugal

(11.567)

Unbalanced force functional

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⌣

⌣

̄ G( t d(t); t d) =− ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ Residual force functional

where the initial conditions on the generalized displacement and velocity are specified as: ⌣

d(0) ≡

{

t d(0) t θ(0)

}

⌣

= d̂ 0 ≡

{

td 0 tθ 0

} (11.568)

where, for convenience, we collect the linearized internal and inertial terms for the following.

Nonlinear Shell

893

Definitions ⌣ r The linearized inertial functional, ΔGInertial ( t ⌣d; t d; ̄ Δ t d): ⌣

⌣

⌣

⌣

⌣

⌣

⌣

⌣

⌣

⌣

⌣

⌣

⌣

̄ Δ t d) = ΔGM ( t d; t d; ̄ Δ t d) + ΔGG ( t d; t d; ̄ Δ t d) + ΔGC ( t d; t d; ̄ Δ t d) ΔGInertial ( t d; t d; ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ Mass functional

Gyroscopic functional

Centrifugal functional

(11.569) ⌣ r The linearized (symmetric) mass functional, ΔGM ( t ⌣d; t d; ̄ Δ t d): ⌣

⌣

⌣

t̄

ΔGM ( d; d; Δ d) ≡ t

t

⌣

⌣

∫

t̄

d∙

M

[

M0 I 0

]

0 I0

(WT

∙∙ t

W)

⌣

Δ d dA

(11.570)

⌣ r The linearized gyroscopic (damping) functional, ΔGG ( t ⌣d; t d; ̄ Δ t d): ⌣

t

⌣

⌣

t̄

t

⎡0 d∙⎢ ⎢ ⎣0

⌣

⌣

ΔGG ( d; d; Δ d) =

∫ M

t̄

∙ ⎤ ⌣ t ⎥ Δ d dA ∙ ∙ ⎥ I0 {WT (2 W −W 𝚯 W)} ⎦

0

(11.571)

⌣ r The linearized centrifugal (stiffness) functional, ΔGC ( t ⌣d; t d; ̄ Δ t d): ⌣

⌣

⌣

⎤ ⎡0 0 ⌣ ⎥ Δ t d dA d∙⎢ ∙ ∙ ∙ ∙ ∙∙ ∙ ∙ ∙ ⎢ 0 I {U( t ω) + WT { −(W 𝚯 + 𝚯 W) − W 𝚯 W}} ⎥ W W W ⎦ ⎣ 0

⌣

⌣

̄ Δ t d) = ΔGC ( t d; t d;

t̄

∫ M

∙ ∙ ⎧ ⎫ 2 t t ⎪ c1 [ ωS ] + (−c2 I − c3 𝚯 + c4 𝚯 )( ωS ∙ θ) ⎪ U( ωS ) ≡ ⎨ ⎬, ∙ ∙ ∙ ⎪ +c2 ((θ ⊗ t ωS ) − ( t ωS ⊗ θ) + (θ ∙ t ωS )I) ⎪ ⎩ ⎭ ∙

t

t

∙

∙

∙

∙∙

ωS = (W t θ + W t θ)

(11.572)

with [h] ≡ h×, ∀h ∈ R3 ,

Computation of Nonlinear Structures

894

⌣ r The residual force functional, R( t ⌣d(t); t d) ̄ = −G( t d(t); t d) ̄ is given as: ⌣

⌣

⌣

⌣

⌣

⌣

⌣

⌣

̄ ̄ = −G( t d(t); t d) ̄ = −G( t d(t), t θ(t); t d, ̄ t θ) R( t d(t); t d) ≡ −{GInternal + GInertial − GExternal } ̄ = GInternal ( t d(t); t d)

⌣

∫

⌣

⌣

⌣

⌣

̄ T n F( t d(t)) dA ε( d(t); t d)

n̄ t

M

GInertial

⌣

⌣

̄ ( t d(t); t d)

⌣

=

t ̄T t

d

∫

⌣

⌣

FIner ( t d(t)) dA (11.573)

M ∙∙

=

∫

t ̄T

{ t d̄ T M0 t d (t) +

θ

∙

I0 (WT t ωS (t)) }dA

M ⌣

̄ = GExternal ( t d)

⌣

t ̄T

d

∫

⌣

t̂

F(S1 , S2 ) dA +

M

̄ t θ̄ ∀ admissible t d,

= 0,

⌣

∫

t ̄T

d

⌣

t̂

F𝜈 (S) dS

𝜕M f

where W is given as: W( t θ) ≡ I + c1 t 𝚯 + c2 t 𝚯2 , c1 ( t θ) =

r

1 − cos 𝜃 , 𝜃2

c2 ( t θ) =

(11.574)

𝜃 − sin 𝜃 𝜃3

with t 𝚯 as the skew-symmetric matrix corresponding to the axial vector, t θ, such that = 0. is given as: W t𝚯 tθ ∙

∙

∙

∙

∙

∙

∙

W = c1 𝚯 +c2 (𝚯 𝚯 + 𝚯 𝚯) + c1 𝚯 + c2 𝚯2 ∙

∙

∙

∙

∙

∙

= (c2 − c1 )(θ ∙ θ)I + c1 𝚯 + c2 {(θ ⊗ θ) + (θ ⊗ θ)} + c1 𝚯 + c2 (θ ⊗ θ) ∙

c1 ≡

∙ ∙ 1 2 (1 − 2c − 𝜃 c )(θ ∙ θ) = c (θ ∙ θ), 1 2 3 𝜃2

∙

c2 ≡

∙ ∙ 1 (c − 3c )(θ ∙ θ) = c (θ ∙ θ), 1 2 4 𝜃2

(11.575) ∙∙ rW is given as: ∙∙

∙∙

∙ ∙

∙

∙

∙

∙∙

∙

∙∙

∙∙

∙∙

W = c1 𝚯 +2{c1 𝚯 + c2 (𝚯 𝚯 + 𝚯 𝚯)} + c2 (𝚯 𝚯 + 2 𝚯2 +𝚯 𝚯) + c1 𝚯 + c2 𝚯2 ∙∙

∙

∙

c1 ≡ c5 (θ ∙ θ)2 + c3 {𝜃 2 + (θ ∙ θ)}, c5 ≡ −

1 (2c2 + 4c3 + 𝜃 2 c4 ), 𝜃2

c6 ≡ −

∙∙

∙

∙

c2 ≡ c6 (θ ∙ θ)2 + c4 {𝜃 2 + (θ ∙ θ)} 1 (c3 − 5c4 ), 𝜃2 (11.576)

Nonlinear Shell

r

⌣

n ε, ̄

895

the generalized virtual strain field is given as: ⌣

nε ̄

nβ ̄

𝛼

≡

{n

̄ β 1

= t RT

(t

nχ ̄1

nβ ̄

nχ ̄2

2

}T

,

) ( ) ̄ t e𝛼 + t k0 × t d ̄ 𝛼 + t k0 × t d̄ − t RT t 𝚽 d, 𝛼 𝛼 (11.577)

= t RT (1̂ 𝛼 + t d|𝛼 ), nχ ̄

𝛼

tφ ̄

r

⌣

n F,

̄ = t RT ≡ t RT t φ| 𝛼

⌣

} ( ) ̄ + k0 × t φ ̄ = t RT k0 + k , φ, 𝛼 𝛼 𝛼 𝛼

= W t θ̄

the generalized reactive force field is given as: n

r

{t

tF Iner ,

⌣

F≡

{n

nM

F1

1

nF 2

nM

}T

(11.578)

2

the generalized inertial force field is given as: ⌣

∙

∙∙

tF Iner

≡ {M0 t d

tω

≡ W t θ,

T I0 (WT t ωS ) } ,

∙

S

t

∙

∙

∙

∙∙

(11.579)

ωS = (W t θ + W t θ)

where the angular velocity vector, t ω, is the axial vector corresponding to the skew-symmetric angular velocity matrix, ωS , given by: ∙

𝛀S ≡ R RT

(11.580)

The linearized internal functional, ΔGInternal : Linearized internal functional

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ Material stiffness functional

Geometric stiffness functional

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

ΔGInternal ≡

⌣

∫

n ̄T n

ε

⌣

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

n⌣

D Δ ε dA +

M

⌣

∫

⌣

Δn ε̄ T n F dA (11.581)

M Tangent functional

⎴ ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⎴ Material stiffness functional

=

⏞⏞⏞ M

Geometric stiffness functional

+

⏞⏞⏞ G

Computation of Nonlinear Structures

896

Definition: Material stiffness functional, M : ⌣

⌣

⌣

̄ Δt d) ≡ M ( t d, t d;

⌣

n ̄T n

ε

∫

⌣

⌣

D Δn ε dA

(11.582)

M

⌣

⌣

⌣

̄ Δ t d) ≡ G ( t d, t d;

⌣

⌣

⌣

⌣

⌣

̄ Δ t d) n F dA Δn ε̄ T ( t d, t d;

∫

(11.583)

M

the virtual strain–displacement relations in compact matrix form is: ⌣ ⌣ ⌣

⌣

⌣

̄ = E(d) H(d) ̄ ̄ d; d) ε(

(11.584)

where T

⎡R ⎢ ⌣ ⎢ 0 E(d) = ⎢ ⎢ 0 ⎢ ⎣ 0

0

0

RT

0

0

RT

0

0

[a1 ]W I 0 ⎤ ⎡ [k01 ] ⎥⎢ X1 + [kc1 ]W 0 0 ⎥⎢ 0 ⎥⎢ 0 0 ⎥ ⎢ [k2 ] [a2 ]W 0 ⎥ ⎢ RT ⎦ ⎣ 0 X2 + [kc2 ]W 0 ⌣

̄ ̄ ≡ { d̄ θ̄ d, ̄ 1 θ, and I, is the 3 × 3 identity matrix, H(d) 1 knowledge of the relevant derivatives and variations, we have:

0⎤ ⎥ W 0 0⎥ ⎥ 0 I 0⎥ ⎥ 0 0 W⎦

̄ 2 d,

0

0

(11.585)

̄ }T , and from our θ, 2

W ≡ I + c1 𝚯 + c2 𝚯2 X𝛼 ≡ −c1 𝚯, 𝛼 + c2 {[θ ⊗ θ, 𝛼 ] − [θ, 𝛼 ⊗ θ] + (θθ, 𝛼 )I} +V [θ, 𝛼 ⊗ θ]),

𝛼 = 1, 2

(11.586)

V ≡ −c2 I + c3 𝚯 + c4 𝚯2 c1 ≡ (1 − cos 𝜃)∕𝜃 2 , c2 ≡ (𝜃 − sin 𝜃)∕𝜃 3 ,

(11.587)

c1 ≡ 𝛿c1 ≡ c3 ≡ {(1 − 𝜃 2 c2 ) − 2c1 }∕𝜃 2 , c2 ≡ 𝛿c2 ≡ c4 ≡ (c1 − 3c2 )∕𝜃 2 ⌣

the (6 × 1) incremental generalized displacement state vector functions, Δd ≡ (Δ d Δθ )T , and ⌣

finally, n D is the symmetric hyperelastic constitutive matrix. In the expanded form, equation

Nonlinear Shell

897

(11.583) may be expressed as: Translational part

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ G ≡

∫

n

̄ ∙ n F𝛼 dA + Δβ 𝛼

Ω

Rotational part

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ∫

n

Δχ̄ 𝛼 ∙ n M𝛼 dA

(11.588)

Ω

where Δ, the incremental operations, involve the covariant derivatives of the vir⌣ ⌣ t d, ̄ ̄ }T , and H(𝚫 d) ≡ ̄ ≡ { t d̄ t θ̄ t d, ̄ 1 t θ, ̄ 2 t θ, tual strain functions, H( t d) 1 2 ⏟⏟⏟ ⏟⏟⏟ 18 × 1

18 × 1

{ 𝚫 t d 𝚫 t θ 𝚫 t d, 1 𝚫 t θ, 1 𝚫 t d, 2 𝚫 t θ, 2 }T with (∙), 𝛼 , 𝛼 = 1, 2 are derivatives with respect to the arc length parameters, S𝛼 , 𝛼 = 1, 2, along the lines of curvature.

r Superscript t reminds us that the relevant items are measured in the undeformed coordinate system, G1 , G2 , N; superscript n reminds us that the relevant items are measured in the rotated coordinate system, g1 (= R G1 ), g2 (= R G2 ), g3 ≡ n(= R N).

Remark

r The expressions given in equation (11.583) or equation (11.588) involve the first variation (incremental) of the virtual rotational strains or, equivalently, the second variation (Hessian) of the real rotational strains. It is this second variation that renders the derivation of the geometric stiffness matrix quite involved, and thus tedious. However, the subsequent reward in terms of high computational accuracy, ease and efficiency in situations of repetitive computations of the geometric stiffness, as we have in a nonlinear numerical analysis, makes it extremely worthwhile.

11.18.2

Material Stiffness Functional Revisited: M

As in the previous virtual expression, we group together the incremental state variables and their derivatives with respect to the arc length parameters, (S1 , S2 ), into an augmented ⌣ ⌣ ⌣ ̄ ≡ (18 × 1) incremental vector function (we can replace d̄ by Δd in the definition of H(d) ⏟⏟⏟ { d̄

̄ 1 θ̄ d,

̄ θ, 1

̄ 2 d,

18×1

̄ }T ) to introduce: θ, 2

Definition ⌣

Incremental generalized displacement function: H(Δd) ≡ {Δd ⏟⏟⏟

Δθ Δd, 1

Δθ, 1

18×1

Δd, 2 Δθ, 2 }T . Now, replacing the variational variables in equation (11.584) with incremental variables, with the above definition in mind, we get a similar expression as follows.

Computation of Nonlinear Structures

898

Definition Incremental strain–displacement relation: ⌣ ⌣

⌣

⌣ ⌣

⌣

⌣

⌣

Δ ε(d; Δd) = E(d) H(Δd)

(11.589)

Note that the incremental strain, Δ ε(d; Δd), is clearly linear in the incremental generalized dis⌣

⌣

placement, Δd, by being related to the strain–displacement matrix that is independent of Δd. By applying the virtual strain–displacement relation (11.584) and incremental strain–displacement relation of equation (11.589), we are now ready to transform equation (11.582) as: ⌣ ⌣

⌣

⌣

̄ Δd) ≡ M (d, d;

∫

⌣

⌣

⌣

̄ T (E(d)T n D ̂ E(d)) H(Δd) dA H(d)

(11.590)

M

Based on equation (11.590), we can redefine and present the material stiffness functional as follows. Definition: Material stiffness functional, M , is: ⌣ ⌣

⌣

⌣

̄ Δd) ≡ M (d, d;

∫

⌣

̄ T KM H(Δd) dA H(d)

(11.591)

M

where the matrix KM is defined as: ⌣

⌣

⌣

KM ≡ E(d)T n D E(d)

(11.592)

Remark

r Clearly, from equation (11.592), the matrix, KM , is symmetric in state or configuration space variables for a conservative system.

11.18.3

Geometric Stiffness Functional Revisited: G

In what follows, in order to avoid undue distraction, first we present the main results, and then we refer the interested reader to the sections on detailed derivations.

r Translational virtual work: linearization and symmetry We get as the final result: n Δβ ̄

𝛼

̄ 𝛼 ∙ ([ t F𝛼 ]T W)Δθ + Δd, 𝛼 ∙ ([ t F𝛼 ]T W)θ̄ ∙ n F𝛼 = d, + d̄ ∙ ([k0𝛼 ] [ t F𝛼 ] W)Δθ + Δd ∙ ([k0𝛼 ] [ t F𝛼 ] W)θ̄ + θ̄ ∙ Zβ Δθ Sym

(11.593)

Nonlinear Shell

899

with Sym

Z̄

β

= 12 {WT ([ t F𝛼 ⊗ a𝛼 ] + [a𝛼 ⊗ t F𝛼 ]) W + ([hf ⊗ θ] + [θ ⊗ hf ])} −( t F𝛼 ∙ a𝛼 )WT W + c2 (θ bf )I

r Rotational Virtual Work: Linearization and Symmetry We get as the final result: n Δχ ̄

̄ ∙ n M = Δθ ∙ {WT [ t M𝛼 ] W + Y( t M𝛼 )}θ, 𝛼 +θ̄ ∙ {WT [ t M𝛼 ] W + Y( t M𝛼 )}Δθ, 𝛼 Sym +Δθ ∙ Zχ̄

(11.594)

θ̄

with Symm

Zχ̄

≡ 12 {WT ([ t M𝛼 ⊗ kc𝛼 ] + [kc𝛼 ⊗ t M𝛼 ]) W + ([hm ⊗ θ] + [θ ⊗ hm ])} t 𝛼 T −( t M𝛼 ∙ kc𝛼 )WT W + c2 (θ ∙ bm )I + WT [ t M𝛼 ]X𝛼 + XT 𝛼 [ M ] W

and the Y(∙) function is defined later. Based on equation (11.593) and equation (11.594), we redefine and present here the geometric stiffness functional in its most uncluttered and explicit form. Definition: The geometric stiffness functional, G , is:

⌣ ⌣

⌣

̄ Δd) ≡ G (d, d;

⌣

⎧ ⎫ ⌣ ⌣ ⌣ ⎪ ⎪ ⌣T t H(d) G(d, F) H(Δd) ⎬ dA ∫ ⎨ ⏟⏟⏟ ⏟⏞⏟⏞⏟ ⏟⏟⏟⎪ ⎪ Ω ⎩ 1×18 18×18 18×1 ⎭

(11.595)

⌣

where the matrix G(d, t F), is given as: ⎡ 0 ⎢ ⎢ (G1 + G2 )T 12 ⎢ 12 ⎢ 0 ⌣ ⌣ G(d, t F) ≡ ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎣

G112 + G212

0

0

0

G122 + G222

G123

G124

G223

(G123 )T

0

0

0

(G124 )T (G223 )T (G224 )T

0

0

0

0

0

0

0

0

0

0 ⎤ ⎥ G224 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥⎦

(11.596)

Computation of Nonlinear Structures

900

where G𝛼12 ≡ [k0𝛼 ][ t F𝛼 ] W,

𝛼 = 1, 2 and no sum

G𝛼22 ≡ 12 WT {[ t F𝛼 ⊗ a𝛼 ] + [a𝛼 ⊗ t F𝛼 ] − 2( t F𝛼 ∙ a𝛼 )}W + c2 (θ ∙ b𝛼 )I f

+ 12 ([hb𝛼 ⊗ θ] + [θ ⊗ hb𝛼 ]) + 12 WT {[ t M𝛼 ⊗ kc𝛼 ] + [kc𝛼 ⊗ t M𝛼 ] − 2( t M𝛼 ∙ kc𝛼 )}W + c2 (θ ∙ bm 𝛼 )I

(11.597)

m T t 𝛼 t 𝛼 T + 12 ([hm 𝛼 ⊗ θ] + [θ ⊗ h𝛼 ]) + W [ M ] X𝛼 + X𝛼 [ M ] W

G𝛼23 ≡ WT [ t F𝛼 ] G𝛼24 ≡ WT [ t M𝛼 ] W + Y( t M𝛼 ) with c1 ≡ (1 − cos 𝜃)∕𝜃 2 , c2 ≡ (𝜃 − sin 𝜃)∕𝜃 3 , c̄ 1 ≡ 𝛿c1 ≡ c3 ≡ {(1 − 𝜃 2 c2 ) − 2c1 }∕𝜃 2 ,

(11.598)

c̄ 2 ≡ 𝛿c2 ≡ c4 ≡ (c1 − 3c2 )∕𝜃 2

a ≡ e + [k0 ]d = {(1̂ + d, s ) + [k0 ]d} kt ≡ W θ, s kc ≡ k0 + kt

(11.599)

bf ≡ Ft × a bm ≡ Mt × kc

W ≡ I + c1 𝚯 + c2 𝚯2 X𝛼 ≡ −c1 𝚯, 𝛼 + c2 {[θ ⊗ θ, 𝛼 ] − [θ, 𝛼 ⊗ θ] + (θ ∙ θ, 𝛼 )I} + V [θ, 𝛼 ⊗ θ]) V ≡ −c2 I + c3 𝚯 + c4 𝚯2 Y(x) ≡ −c1 [x] + c2 ([x ⊗ θ] − [θ ⊗ x]) + c2 (θx)I + [θ ⊗ VT x], f

f

h𝛼 ≡ VT b𝛼 T m hm 𝛼 ≡ V b𝛼

∀x ∈ R3

(11.600)

Nonlinear Shell

901

Exercise: Drawing upon the analyses referred to earlier for the translational and rotational geometric stiffness functional, derive the expression for G above. (Hint: look at Section 10.18 on the symmetry of the beam geometric stiffness matrix.)

Remark

r Clearly, from equation (11.596), the matrix, G, is symmetric in state or configuration space variables for a conservative system.

11.18.3.1

Implementational Note

Note that, as will be explained later, the computational finite element implementation for the internal work of a shell problem can be achieved essentially by using the corresponding beam formulations along each of the lines of curvature at numerical Gauss quadrature points in a typical Newton-type iteration method of solution under a time-marching formulation.

The material and the geometric stiffness functional parts in equations (11.591) and (11.595), respectively, need to be further modified in the subsequent finite element formulation as the direct sums of the element stiffness matrices for computational purposes. In so doing, we will be able to introduce the computational definition for the shell material and geometric stiffness matrices. Thus, for the final and computationally optimum form of the linearized virtual work, we will focus next on the c-type finite element formulation.

11.18.4

c-type Finite Element Formulation

Under the semi-discrete formulation, and following the standard definition of a finite element discretization, a shell initial surface, Ω, is broken into a disjointed finite number of elemental surfaces such that:

Ω=

e=no. of elements ⋃ e=1

Finite elemente =

e=no. of elements ⋃

Ωe

(11.601)

e=1

Then, the shell material and geometric stiffness matrices can be expressed as an assembly of the corresponding elemental matrices; a similar idea applies to the formation of the global unbalanced and the applied forces. As will be seen in our c-type finite element implementation phase, we will use what is known as the direct global stiffness and load assembly of the corresponding elemental entities. Thus, for now, we can express each of the following properties as a sum over all the corresponding entities belonging to the elements:

Computation of Nonlinear Structures

902

r Shell symmetric mass functional: t

⌣

⌣

t̄

t

⌣

ΔGM ( d; d; Δ d) =

∑

⌣

t ̄T

d ∫ ⏟⏟⏟ elements Ωe

1×6

[

M0 I

]

0

0 I0 (WT W) ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

∙∙ ⌣

Δ t d dA ⏟⏟⏟ 6×1

6×6

(11.602)

r Shell gyroscopic (damping) functional: ∙ ⎤ ⎡0 0 ⌣ t ⎥ ⎢ ΔGG ( d; d; Δ d) = Δ d dA d ∙ ∙ ∫ ⏟⏟⏟ ⎢ ⎥ elements e 0 I0 {WT (2 W −W 𝚯 W)} ⎦ ⏟⏟⏟ ⎣ Ω 1×6 6×1 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ t

⌣

⌣

t̄

t

∑

⌣

⌣

t ̄T

6×6

(11.603)

r Shell centrifugal (stiffness) functional: ⎤ ⎡0 0 ⎥ ⎢ ∙∙ ∙ ⌣ ⌣ ⌣ ⎫⎥ ⎧ t ⎢ t ̄T t ̄ Δ t d) = U( ω) d ΔGC ( t d; t d; Δ d dA ⎪⎥ ∫ ⏟⏟⏟ ⎢⎢ 0 I ⎪ ⏟⏟⏟ ⎬ ⎥ 0⎨ elements e ∙ ∙ ∙ ∙ ∙∙ ∙ ∙ Ω 1×6 6×1 ⎪ +WT {W −(W 𝚯 W + W 𝚯 W) − W 𝚯 W} ⎪ ⎥ ⎢ ⎭⎦ ⎩ ⎣ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ∑

⌣

⌣

6×6

⎫ ⎧ ∙ ∙ 2 t t ∙ ⎪ c1 [ ωS ] + (−c2 I − c3 𝚯 + c4 𝚯 )( ωS ∙θ) ⎪ U( t ωS ) ≡ ⎨ ⎬, ∙ ∙ ∙ ⎪ +c2 ((θ ⊗ t ωS ) − ( t ωS ⊗θ) + (θ ∙ t ωS ))I ⎪ ⎭ ⎩ t

∙

∙

∙

∙∙

ωS = (W t θ + W t θ)

(11.604)

r Shell material stiffness functional: ⌣ ⌣

⌣

̄ Δd) = M (d, d;

∑

⌣

⌣

⌣

̄ T KM (d) H(Δd) dA H(d) ∫ elements e ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟ Ω

1×18

18×18

(11.605)

18×1

r Shell geometric stiffness functional: ⌣ ⌣

⌣

̄ Δd) = G (d, d;

⎧ ⎫ ⌣ ⎪ ⎪ ⌣̄ T H(d) G H(Δd) dA ∫ ⎨ ⏟⏟⏟ ⏟⏟⏟⎬ ⎪ elements e ⎪⏟⏟⏟ Ω ⎩ 1×18 18×18 18×1 ⎭ ∑

(11.606)

Nonlinear Shell

903

r Shell unbalanced residual force functional: ⌣

⌣

⌣

⌣

⌣

⌣

̄ = −G( t d(t); t d) ̄ R( t d(t); t d) ̄ = GInternal + GInertial − GExternal G( t d(t); t d) ⌣ ⌣ ⎛ ⎞ n ̄T n n⌣ ⎜ ∫ ε D ε dA ⎟ ⎜ Ωe ⎟ ⎜ ⎟ ⌣ ⌣ ⎜+ ⎟ t ̄T t F dA d Iner ∑ ⎜ ∫ ⎟ = ⎜ Ωe ⎟ ⎟ elements ⎜ ⎫⎟ ⎜ ⎧ T T ⌣ ⌣ ⌣ ⎜ ⎪ ⌣̄ ̂ 1 2 ⎪⎟ ⎜ − ⎨∫ d F(S , S ) dA + ∫ d̄ F̂ 𝜈 (S) dS⎬ ⎟ ⎜ ⎪Ωe ⎪⎟ Γe ⎝ ⎩ ⎭⎠

(11.607)

Based on the assumption that the assembly (i.e. the summation) of the finite elements will be performed by the direct global assembly procedure where the properties of each element enter into the summing operation independent of the all other element data, we can now restrict our discussion to a typical finite element for the shell. Finally, we can access the available library of c-type shell elements of both curved quadrilateral and triangular shape. However, in what follows, we will restrict our discussion to only tensor product quadrilateral elements; similar arguments and ideas apply formally to the triangular elements too. The root element corresponding to a typical finite element is defined over the geometric 2D root parameter plane defined by the parameters (𝜉̂1 , 𝜉̂2 ) ∈ [0, 1] ∪ [0, 1]. Next, the parameters, (𝜉 1 , 𝜉 2 ), along the lines of curvature 𝜉1

is related to the arc length parameter, (S1 , S2 ), of a shell by: S1 (𝜉 1 , 𝜉 2 ) = ∫0 ‖ e C0 (𝜂, 𝜉 2 ), 𝜂 ‖d𝜂 𝜉2

and S2 (𝜉 1 , 𝜉 2 ) = ∫0 ‖ e C0 (𝜉 1 , 𝜂), 𝜂 ‖d𝜂, where e C0 (𝜉 1 , 𝜉 2 ), 𝜉 1 ∈ [0, 1], 𝜉 2 ∈ [0, 1] defines the reference surface position vector at any point (𝜉 1 , 𝜉 2 ) of the eth finite element, that is, the Finite elemente . Let us briefly recall the basic steps of a c-type finite element generation for future use. The coordinate geometry of a shell reference surface is discretized in the mesh generation phase whereby the root element is identified and fixed for an element; the position vector, C0 (𝜉̂1 , 𝜉̂2 , t), of any point, (𝜉̂1 , 𝜉̂2 ), on the element at the initial time t = 0 ∈ R+ is given by: ⎧ 1 ̂1 ̂2 ⎫ ⎪ C0 = X(𝜉 , 𝜉 ) ⎪ ⎪ ⎪ C0 (𝜉̂1 , 𝜉̂2 , 0) = ⎨ C02 = Y(𝜉̂1 , 𝜉̂2 ) ⎬ = c T(𝜉̂1 , 𝜉̂2 ) c q, ⎪ 3 ⎪ ⎪ C0 = Z(𝜉̂1 , 𝜉̂2 ) ⎪ ⎩ ⎭ C0k (𝜉̂1 , 𝜉̂2 , 0)

=

nc mc ∑ ∑

n m Bi c (𝜉̂1 ) Bj c (𝜉̂2 ) c qkij ,

(11.608)

k = 1, ⋯ , 3

i=0 j=0

where c T(𝜉̂1 , 𝜉̂2 ) is the root transformation matrix, Bni (𝜉̂1 ), Bm (𝜉̂2 ) are Bernstein–Bezier polynoj mials of degree nc and mc respectively, and c q, c qkij are the generalized coordinate control vector

Computation of Nonlinear Structures

904

and the component control net for the element, respectively. The discretization of the generalized ⌣

displacement field, d(t) ≡ (d(t), θ(t)), t ∈ R+ , for the finite element, is then performed over the geometric root element except that the degree of the displacement distribution polynomials (i.e. the local basis or shape functions) will be at least equal to that of the geometric coordinate distribution polynomials; in other words, the element is either subparametric or isoparametric.

11.18.5

c-type Shell Elements and State Control Vectors

Let us first recognize that shell geometry and the constituting finite elements are naturally discretized in terms of the parameters (𝜉̂1 , 𝜉̂2 ) as tensor product objects for quadrilateral elements. However, for analytical convenience, the linearized virtual work expression as given by equation (11.566) has been derived in terms of the arc length parameters, (S1 , S2 ), along the corresponding lines of curvature. Thus, for computational purposes, we must express all our equations back in the (𝜉̂1 , 𝜉̂2 ) parameters at the Gauss quadrature points of the numerical integration scheme. This is what is done in the ensuing finite element formulation. Let us, then, introduce a typical c-type shell element displacement interpolation for the semi-discrete finite element formulation as: ⌣

⌣

⌣ d(𝜉̂1 , 𝜉̂2 , t) = d T(𝜉̂1 , 𝜉̂2 ) d q(t), ⌣

⌣

d = { d1

⌣

d k (𝜉̂1 , 𝜉̂2 , t) =

⌣

d2

m n ∑ ∑

⋯

⌣

T

d6 }

(11.609)

k ̂2 d ⌣ Bni (𝜉̂1 ) Bm j (𝜉 ) q ij (t),

k = 1, … , 6

i=0 j=0

⌣

where d T(𝜉̂1 , 𝜉̂2 ) is the root transformation matrix, Bni (𝜉̂1 ), Bm (𝜉̂2 ) are Bernstein–Bezier polyj ⌣

⌣k

nomials of degree n and m respectively, and d q(t), d q ij (t) are the generalized time-dependent displacement control vector and the component control net for the element, respectively. Similar ⌣

distributions can be produced for the incremental generalized displacement, 𝚫 d; for the variations ⌣ ̄ the distribution is time independent and given as: in generalized displacements, d, ⌣

⌣

⌣

̄ 𝜉̂1 , 𝜉̂2 , t) = d T(𝜉̂1 , 𝜉̂2 ) d q, ̄ d(

⌣

d̄ k (𝜉̂1 , 𝜉̂2 , t) =

m n ∑ ∑

⌣k

̂2 d ̄ , Bni (𝜉̂1 ) Bm j (𝜉 ) q ij

(11.610) k = 1, … , 6

i=0 j=0

where we have introduced the following. Definitions

r

⌣

dq ̄

be the (6N 2 × 1) generalized time independent virtual displacement control vector, other⌣ wise, similar in the ordering of the components to d q.

Let us define N, M as the orders of the polynomials, respectively, given by: N ≡ n + 1 and M ≡ m + 1. As will be seen later, in actual implementation, we will enforce: n = m for each element. We may see that the tensor product of these polynomials serves as the local basis

Nonlinear Shell

905

functions for our c-type shell elements. Thus, a typical n-degree c-type shell element is given by: ⌣

⌣

⌣ d(𝜉̂1 , 𝜉̂2 , t) = d T(𝜉̂1 , 𝜉̂2 ) d q(t) , ⏟⏞⏞⏟⏞⏞⏟ ⏟⏞⏞⏟⏞⏞⏟ ⏟⏟⏟

6×1 ∙

⌣

6× 6N 2 ⌣

6N 2 ×1 ∙ d⌣

d(𝜉̂1 , 𝜉̂2 , t) = d T(𝜉̂1 , 𝜉̂2 ) q(t) , ⏟⏞⏞⏟⏞⏞⏟ ⏟⏞⏞⏟⏞⏞⏟ ⏟⏟⏟ 6×1

∙∙ ⌣

6× 6N 2 ⌣

(11.611)

6N 2 ×1 ∙∙ d⌣

d(𝜉̂1 , 𝜉̂2 , t) = d T(𝜉̂1 , 𝜉̂2 ) q(t) , ⏟⏞⏞⏞⏟⏞⏞⏞⏟ ⏟⏞⏞⏟⏞⏞⏟ ⏟⏟⏟ 6×1

6× 6N 2

6N 2 ×1

where we have introduced the following. Definitions d⌣

{

q(t) = ⏟⏟⏟ 6N 2 ×1

qu00 ⋯ qun0 qu01 ⋯ qun1 qu0n ⋯ qunn qv00 ⋯ qvn0 qv01 ⋯ qvn1 qv0n ⋯ qvnn ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

qw ⋯ qw 00 n0

1×N 2 w q01 ⋯ qw n1

𝜃1 𝜃1 qw ⋯ qw nn q00 ⋯ qn0 0n

1×NM 𝜃1 𝜃 q01 ⋯ qn11

𝜃

𝜃 𝜃 q002 ⋯ qn02

1×N 2 𝜃2 𝜃 q01 ⋯ qn12

𝜃 𝜃 𝜃 𝜃 q0n2 ⋯ qnn2 q003 ⋯ qn03

1×N 2 𝜃3 𝜃 q01 ⋯ qn13

𝜃

𝜃

q0n1 ⋯ qnn1 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

r

𝜃

q0n3 ⋯ qnn3 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ 1×N 2

is the (6N 2 × 1) }T

1×N 2

generalized time-dependent displacement control vector where all the components listed are time-dependent; a similar definition applies for ∙∙

∙ d⌣ q(t),

the (6N 2 × 1) generalized time-

⌣

dependent velocity control vector, and for d q(t), the (6N 2 × 1) generalized time-dependent ∙ ∙∙ acceleration control vector with all the coefficients d qij replaced by d q and d q , respectively; of ij

r

ij

course, other possibilities exist for sequencing the nodal variables and the degrees of freedoms. ⌣ d T(𝜉̂1 , 𝜉̂2 ) is the (6 × 6N 2 ) interpolation transformation matrix containing the Bernstein–Bezier polynomials, that is, ⌣

d T(𝜉̂1 ,

𝜉̂2 ) =

0 0 ⋯ ⋯ ⋯ 0 ⎡ B0 B0 B1 B0 ⋯ Bn Bn−1 Bn Bn ⎤ ⎢ ⎥ n n n n n n 0 0 B0 B0 B1 B0 ⋯ Bn Bn 0 ⋯ 0 ⎢ 0 ⎥ ⎢ ⎥ ⋯ ⋯ ⎢ ⎥ ⎢ ⎥ ⋯ ⋯ ⎢ ⎥ ⎢ ⎥ n n n n n n ⎣ 0 0 ⋯ ⋯ 0 B0 B0 B1 B0 ⋯ Bn Bn ⎦ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ n n

n n

n n

n n

6x6N 2

(11.612)

Computation of Nonlinear Structures

906 ⌣

d ̂1 ̂2 In other words, ( ) each row of T(𝜉 , 𝜉 ) is identical except that the non-zero contents are shifted 2 by 1 × 6N zero entries with increasing row numbers. Based on these definitions, we also have: ⌣

⌣

⌣ Δd(t) = d T(𝜉̂1 , 𝜉̂2 ) Δ d q(t),

∙

⌣

Δ d(t) =

⌣

d T(𝜉̂1 ,

∙∙ ⌣

∙

⌣ 𝜉̂2 ) Δ d q(t).

(11.613)

∙∙

⌣

⌣ Δ d(t) = d T(𝜉̂1 , 𝜉̂2 ) Δ d q(t),

where we have similarly introduced the following. Definitions

r

∙ ⌣ d Δ q(t)

∙∙ ⌣ d Δ q(t),

( ) and are the 6N 2 × 1 time-dependent incremental displacement, velocity and acceleration control vectors, with the components ordered in the same sequence ⌣ Δd q(t),

⌣

as the components of d q(t),

∙ d⌣ q(t)

and

∙∙ d⌣ q(t),

respectively.

Note that equations (11.611) and (11.612) imply that we have chosen to interpolate both the displacement components and the rotation components with the same degree control net. ⌣

Now, let us rewrite H( t d) ≡ { t d ⌣

⌣

S H( t d) ̄

⌣

tθ

t d,

t θ,

1

1

t d,

t θ,

2

T 2}

in generalized form as

⌣

≡ { t d t d, 1 t d, 2 }T , with the superscript S reminding us that the derivatives in the expression are with respect to the arc length parameters, S𝛼 , 𝛼 = 1, 2. Next, let us further introduce the following.

Definitions

r r

⌣

𝜉̂ H( d) ̄

⌣

⌣

⌣

̂

⌣

⌣

⌣

⌣

̄ ̂1 d, ̄ ̂2 }T , and similarly, 𝜉 H(Δd) ≡ { Δd Δd, 𝜉̂1 Δd, 𝜉̂2 }T d, 𝜉 𝜉 ⌣ ( ) 𝜉̂ B(𝜉̂1 , 𝜉̂2 ) is the 18 × 6N 2 augmented interpolation transformation matrix as: ≡ { d̄

⌣ ⎡ dT ⎢ ⏟⏟⏟ ⎢ 6×6N 2 ⎢ ⎢ d⌣ ⎢ T, 𝜉̂1 ̂𝜉 ⌣ ̂1 ̂2 B(𝜉 , 𝜉 ) ≡ ⎢ ⏟⏟⏟ ⏟⏞⏞⏟⏞⏞⏟ ⎢ 6×6N 2 18×6N 2 ⎢ ⌣ ⎢ d T, ̂2 ⎢ ⏟⏟𝜉⏟ ⎢ ⎣ 6×6N 2

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(11.614)

Nonlinear Shell

907

such that ⌣

𝜉̂

⌣

⌣

̄ = 𝜉̂ B(𝜉̂1 , 𝜉̂2 ) d q̄ H(d) ⏟⏟⏟ ⏟⏞⏞⏟⏞⏞⏟ ⏟⏟⏟ 18x1 ⌣

𝜉̂

18x6N 2

6N 2 ×1

(11.615)

̂⌣

⌣ H(Δd) = 𝜉 B(𝜉̂1 , 𝜉̂2 ) Δd q ⏟⏞⏟⏞⏟ ⏟⏞⏞⏟⏞⏞⏟ ⏟⏟⏟

18x1

18x6N 2

6N 2 ×1

with superscript 𝜉̂ in each term reminding us that the derivatives are with respect to parameters, 𝜉̂𝛼 , 𝛼 = 1, 2. Moreover, suppose the lines of curvature make angles of 𝛼1 and 𝛼2 with the 𝜉̂1 coordinate direction at any point (𝜉̂1 , 𝜉̂2 ) on the shell surface. Then, as indicated before, with the arc length parameters, (S1 , S2 ), along the corresponding lines of curvature, we can write the following transformation relationship for the virtual generalized displacement from (𝜉̂1 , 𝜉̂2 ) parameters to the arc length parameter, (S1 , S2 ): ⌣

⌣

⌣

̄ = S𝜉̂ B(𝜉̂1 , 𝜉̂2 ) 𝜉̂ H(d) ̄ H(d) ⏟⏟⏟ ⏟⏞⏞⏞⏟⏞⏞⏞⏟ ⏟⏟⏟ S

18×1

18×18 ⌣

18×1

̂⌣

(11.616)

⌣

̂ H(Δd) = S𝜉 B(𝜉̂1 , 𝜉̂2 ) 𝜉 H(Δd) ⏟⏞⏟⏞⏟ ⏟⏞⏞⏞⏟⏞⏞⏞⏟ ⏟⏞⏟⏞⏟ S

18×1

18×18

18×1

⌣

̂ with S𝜉 B(𝜉̂1 , 𝜉̂2 ) defined by:

0 ⎡ I ⎢ ⏟⏟⏟ ⎢ 3×3 ⌣ ⎢ 1 S𝜉̂ B(𝜉̂1 , 𝜉̂2 ) ≡ ⎢ 0 cos 𝛼1 I 𝛾1 ⏟⏞⏞⏞⏟⏞⏞⏞⏟ ⎢ 18×18 ⎢ 1 ⎢ 0 − sin 𝛼1 I ⎣ 𝛾2

⎤ ⎥ ⎥ ⎥ 1 sin 𝛼1 I ⎥ 𝛾1 ⎥ ⎥ 1 cos 𝛼1 I ⎥ ⎦ 𝛾2 0

(11.617)

𝜕C0 (𝜉 1 , 𝜉 2 ) dS𝛼 ̄ 𝛽 ‖, 𝛼 = 1, 2 with G ̄ 𝛽 (𝜉 1 , 𝜉 2 ) ≡ = ‖G defined by the tangents 𝛼 d𝜉 𝜕𝜉 𝛽 to the lines of curvature. Now, from equations (11.615) and (11.616), we have the transformation relations as:

with 𝛾𝛼 (𝜉 1 , 𝜉 2 ) ≡

⌣

⌣

⌣

̄ = B(𝜉̂1 , 𝜉̂2 ) d q̄ H(d) ⏟⏟⏟ ⏟⏞⏟⏞⏟ ⏟⏟⏟ S

18×1 ⌣

18×6N 2

6N 2 ×1

⌣

⌣ H(Δd) = B(𝜉̂1 , 𝜉̂2 ) d Δq ⏟⏞⏟⏞⏟ ⏟⏞⏟⏞⏟ ⏟⏟⏟

S

18×1

18×6N 2

6N 2 ×1

(11.618)

Computation of Nonlinear Structures

908 ⌣

where we have defined B(𝜉̂1 , 𝜉̂2 ) as: ⌣

B(𝜉̂1 , 𝜉̂2 ) ≡

S𝜉̂

⌣

̂⌣

B(𝜉̂1 , 𝜉̂2 ) 𝜉 B(𝜉̂1 , 𝜉̂2 )

(11.619)

√ ̂ d𝜉̂1 d𝜉̂2 , of the area integrals where ̂ = G Finally, we have to express the infinitesimal area, d A √ ̂ is the positive square root of the determinant of the surface metric expressed in terms of the G geometry coordinate parameters, 𝜉̂𝛼 , 𝛼 = 1, 2.

Computational Element Symmetric Material Stiffness Matrix, e KM

11.18.6

By substituting equation (11.618) in equation (11.591), we get for each finite element:

e

⌣ ⌣

̄

⌣

M (d, d; Δd)

18×12 12×12 12×18 ⎡ ⎧ ⎫ ⎤ ⏞⏞⏞ ⏞⏞⏞ ⏞⏞⏞ ⎢ 1 1⎪ ⎪ √ ⎥ ⌣ ⌣ ⌣ ⌣ ⌣ ⎢ ⎪ ⌣ ̂1 ̂2 T T n 1 ̂2 ⎪ 1 ̂2 ⎥ d ⌣ ̂ ̂ ̂ G d 𝜉 E( E( B( 𝜉 , 𝜉 ) d) D d) B( 𝜉 , 𝜉 ) d 𝜉 ≡ d q̄ T ⎢ ⎨ ⎬ ⎥ Δq(t) ⏟⏟⏟ ⎢∫0 ∫0 ⎪⏟⏞⏞⏟⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏟⏞⏟⎪ ⎥ ⏟⏟⏟ 18×18 18×6N 2 ⎪ 1×6N 2 ⎢ ⎪ 6N 2 ×18 ⎥ 6N 2 ×1 ⎣ ⎦ ⎩ ⎭ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ 6N 2 ×6N 2

=

⌣

⌣

d T e KM d Δq(t) q̄ ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟ 1×6N 2

6N 2 ×6N 2

6N 2 ×1

(11.620) where, we have introduced the following.

Definition

r The computational element symmetric material stiffness matrix,

eK , M

as:

18×12 12×12 12×18 ⎧ ⎫ ⏞⏞⏞ ⏞⏞⏞ ⏞⏞⏞ ⎪ ⎪ √ 1 1 ⌣ ⌣ ⌣ ⌣ ⌣ ⎪ ̂1 ̂2 T e T n 1 ̂2 ⎪ ̂ d𝜉̂1 d𝜉̂2 ̂ G KM ≡ D E(d) B(𝜉 , 𝜉 )⎬ ⎨B(𝜉 , 𝜉 ) E(d) ⏟⏟⏟ ∫0 ∫0 ⎪⏟⏞⏞⏟⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏟⏞⏟⎪ 18×18 18×6N 2 ⎪ ⎪ 6N 2 ×18 6N 2 ×6N 2 ⎩ ⎭ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ 6N 2 ×6N 2

(11.621)

Nonlinear Shell

909

Computational Element Symmetric Geometric Stiffness Matrix, e KG

11.18.7

By substituting equation (11.618) into equation (11.595), we get for each finite element: e

⌣ ⌣

̄

⌣

G ( d, d; Δ d)

⎡ ⎧ ⎫ ⎤ √ ⌣ ⌣ ⌣ ⎢ 1 1 ⎪ ⌣ ̂1 ̂2 T t 1 ̂2 ⎪ 1 ̂2 ⎥ d ⌣ ̂ ̂ ̂ G d𝜉 d𝜉 ⎥ Δq(t) ≡ q̄ ⎢ ⎨B(𝜉 , 𝜉 ) G(d, F) B(𝜉 , 𝜉 )⎬ ⏟⏟⏟ ⎢∫0 ∫0 ⎪⏟⏞⏞⏟⏞⏞⏟ ⏟⏞⏟⏞⏟ ⏟⏞⏟⏞⏟⎪ ⎥ ⏟⏟⏟ ⎦ 6N 2 ×1 ⎩ 6N 2 ×18 18×18 18×6N 2 ⎭ 1×6N 2 ⎣ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⌣

d T

6N 2 ×6N 2

=

⌣

⌣

d T e KG d Δq(t) q̄ ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟ 1×6N 2 6N 2 ×6N 2 6N 2 ×1

(11.622) where, we have introduced the following. Definition

r The computational element symmetric geometric stiffness matrix,

1

e KG ≡ ⏟⏟⏟ ∫0 ∫0 6N 2 ×6N 2

1

eK , G

as:

⎧ ⎫ √ ⌣ ⌣ ⌣ ⎪ ⌣ ̂1 ̂2 T t 1 2 T⎪ ̂ 𝜉̂1 d𝜉̂2 ⎨B(𝜉 , 𝜉 ) G(d, F) B(𝜉̂ , 𝜉̂ ) ⎬ Gd ⎪⏟⏞⏞⏟⏞⏞⏟ ⏟⏞⏟⏞⏟ ⏟⏞⏞⏟⏞⏞⏟⎪ ⎩ 6N 2 ×18 18×18 18×6N 2 ⎭ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

(11.623)

6N 2 ×6N 2

with the geometric stiffness matrix, G, defined by equation (11.596).

11.18.8

Computational Element Symmetric Mass Matrix, e M ∙∙ ⌣

By substituting expression for the generalized incremental acceleration vector, Δ d(t), and the ⌣ ̄ from equation (11.613) into equation (11.570), we get generalized virtual displacement vector, d, for each finite element:

ΔGM

⌣

⌣

⌣

̄ Δt d) (t d; t d;

where we have introduced the following.

∙∙

≡

⌣

tq ̄ T e MΔ t

⌣

d

(11.624)

Computation of Nonlinear Structures

910

Definition

r The computational element symmetric mass matrix,

e M,

as:

⎧ ⎫ ] [ ⎪ ⎪ √ 1 1 0 ⌣ ⎪ d ⌣ ̂1 ̂2 T M0 I d e 1 ̂2 ⎪ ̂ d𝜉̂1 d𝜉̂2 ̂ M ≡ T(𝜉 , 𝜉 ) T(𝜉 , 𝜉 )⎬ G ⏟⏟⏟ ∫0 ∫0 ⎨ ⏟⏞⏞⏞⏟⏞⏞⏞⏟⎪ 0 I0 (WT W) ⎪ ⏟⏞⏞⏞⏞⏟⏞⏞⏞⏞⏟ 6N 2 ×6N 2 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ 6N 2 ×6 6×6N 2 ⎪ ⎪ ⎩ ⎭ 6×6 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ 6N 2 ×6N 2

(11.625)

11.18.9

Computational Element Gyroscopic (Damping) Matrix, e CG ∙∙ ⌣

By substituting the expression for the generalized incremental acceleration vector, Δ d(t), and ⌣ ̄ from equation (11.613) into equation (11.571), the generalized virtual displacement vector, d, we get for each finite element: ⌣

⌣

⌣

⌣

∙

̄ Δ t d) ≡ t q̄ T e CG Δ t ⌣ ΔGG ( t d; t d; q

(11.626)

where we have introduced the following. Definition

r The computational element gyroscopic matrix, e CG , as: ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ √ ⎤ ⎡ 0 0 1 1 ⌣ ⎪ d ⌣ ̂1 ̂2 T ⎢ d e 1 2 ⎥ T(𝜉̂ , 𝜉̂ )⎪ ̂ d𝜉̂1 d𝜉̂2 CG ≡ ∙ ⎨ T(𝜉 , 𝜉 ) ⎢ ⎬ G ∙ ∫ ∫ ⎥ T ⏟⏞⏞⏞⏞⏟⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏟⏞⏞⏞⏟ 0 0 ⎪ ⏟⏟⏟ ⎪ ⎣ 0 W (2 W −W 𝚯 W) ⎦ 6N 2 ×6 6×6N 2 ⎪ ⎪ 6N 2 ×6N 2 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⎪ ⎪ 6×6 ⎩ ⎭ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ 6N 2 ×6N 2

(11.627)

11.18.10

Computational Element Centrifugal (Stiffness) Matrix, e KC ∙∙ ⌣

̂ By substituting the expression for the generalized incremental acceleration vector, Δ d(t), and ⌣

̄̂ from equation (11.613) into equation (11.572), the generalized virtual displacement vector, d,

Nonlinear Shell

911

we get for each finite element: ⌣

⌣

⌣

⌣

̄ Δ t d) ≡ t q̄ T e KC Δ t ⌣ ΔGC ( t d; t d; q

(11.628)

where we have introduced the following. Definition

r The computational element centrifugal stiffness matrix,

eK , C

as:

e KC ≡ ⏟⏟⏟ 6N 2 ×6N 2

⎧ ⎫ ⎪ ⎪ 0 ⎤ ⎡0 ⎪ ⎪ ⎥ ⎢ ⎪ ⎪ 1 1 ∙ ∙ ⎫ ⎥ d⌣ 1 2 ⎪ √ ⎧ ∙∙ ⎪ d ⌣ ̂1 ̂2 T ⎢ ̂ d𝜉̂1 d𝜉̂2 ̂ ̂ T(𝜉 , 𝜉 ) ⎢ ∙ ⎪ ⎥ T(𝜉 , 𝜉 )⎬ G ⎪ W −(W 𝚯 W t T ∫0 ∫0 ⎨ ⎬ ⎥ ⏟⏞⏞⏞⏟⏞⏞⏞⏟⎪ ⎪ ⏟⏞⏞⏞⏞⏟⏞⏞⏞⏞⏟ ⎢ 0 U( ωS ) + W ⎨ ∙ ∙ ∙∙ 6N 2 ×6 6×6N 2 ⎪ + W 𝚯 W) − W 𝚯 W ⎪ ⎥ ⎢ ⎪ ⎪ ⎭⎦ ⎩ ⎣ ⎪ ⎪ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⎪ ⎪ ⎩ ⎭ 6×6 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ 6N 2 ×6N 2

(11.629) with

U( t

∙ ∙ ⎧ ⎫ 2 t t ⎪ c1 [ ωS ] + (−c2 I − c3 𝚯 + c4 𝚯 )( ωS ∙θ) ⎪ ωS ) ≡ ⎨ ⎬, ∙ ∙ ∙ ⎪ +c2 ((θ ⊗ t ωS ) − ( t ωS ⊗θ) + (θ ∙ t ωS )I ⎪ ⎩ ⎭ ∙

t

∙

∙

∙

(11.630)

∙∙

ωS = (W t θ + W t θ)

where [h] ≡ h×, ∀h ∈ R3 .

11.18.11

⌣

Element Unbalanced Residual Force, R(d)

By substituting equation (11.618) into equation (11.573), we get for each finite element: Residual force

Unbalanced force

⏞⏞⏞ ⌣ ⌣

̄ R(d; d)

⏞⏞⏞⏞⏞

=−

e ⌣

⌣ ⌣

̄ G(d; d) ⌣ ⌣

̄ e R(d) = − d qT where, we have defined the following.

(11.631)

Computation of Nonlinear Structures

912

Definition ⌣ ⌣

R(d) ≡ ⏟⏟⏟ e

6N 2 × 1

⎧ ⎧ ⎫⎞ ⎪ 1 1⎛ ⌣ ⌣ ⌣ ⌣ ⎜ ⌣T ⎪ ⎪⎟ √ ̂ ̂1 ̂2 ⎪ T T t t̂ n̂ F F) G d𝜉 d𝜉 B E + T F − ⎜ ⎨ Iner ⎪ ∫0 ∫0 ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟⎬⎟ ⎜ ⎪ ⎪ ⎟ ⎪ ⎝6N 2 × 18 18 × 12 12 × 1 6N 2 × 6 ⎩ 6 × 1 6 × 1 ⎭⎠ ⎨ ⌣ ⎪ ⌣ ⎪− TT t F̂ 𝜈 (S) dS ∫ ⏟⏟⏟ ⏟⏟⏟ ⎪ ⎪ Γ 6N 2 × 6 6 × 1 ⎩

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ (11.632)

11.18.12

Geometrically Exact Element Incremental Dynamic Equation

Now substituting expressions given in equations (11.620), (11.622), (11.625), (11.627), (11.629) and (11.631) in equation (11.566), we have:

d

⌣

∙∙ d⌣

∙ d⌣

̄ M Δ q + CG Δ q + qT{ e

e

(e

⌣ ⌣ ) ⌣ ⌣ ̄ e R(d) KM + e KG + e KC Δd q} = − d qT

(11.633)

Noting that this equation is true for any arbitrary admissible generalized virtual control dis⌣

̄ we can finally write the 6N 2 × 1 geometrically exact fully linearized finite element placement, d q, ⌣

̂ equation for each element in unknown incremental generalized displacement control vector, Δ d q, ∙

generalized velocity control vector, any time t ∈ R+ , as:

Δd

∙∙

⌣

̂ and generalized acceleration control vector, q,

∙∙

6N 2 × 1

⌣

̂ for q,

∙

⌣ ⌣ ( ) ⌣ ⌣ ⌣ M Δ d q + e CG Δ d q + e KC + e KT Δ d q = − e R(d) ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟ ⏟⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏟ ⏟⏟⏟ ⏟⏟⏟

e

6N 2 ×6N 2

Δd

6N 2 ×6N 2

6N 2 × 1

6N 2 ×6N 2

6N 2 × 1

(11.634)

6N 2 × 1

where, we have introduced the following. Definition

r The computational tangent stiffness matrix, e

e Kc : T

KT ≡ e KM + e KG

of a shell, a concept familiar under quasi-static loading.

(11.635)

Nonlinear Shell

11.18.13

913

Direct Global Assembly

Thus far, we have detailed the development of the incremental second-order ordinary differential equation for a shell element at any time t ∈ R+ as given by equation (11.634). The corresponding incremental equation for the entire domain of a shell can be obtained by the standard direct global assembly procedure and is given as: ∙∙ d⌣

∙ d⌣

⌣

⌣ ⌣

M Δ q(t) + CG Δ q(t) + K Δ d q(t) = −R(d(t))

(11.636)

where the global matrices and vectors are obtained as: M≡ CG ≡

∑

e M , ⏟ ⏟ ⏟ element

∑

6N 2 ×6N 2

e CG , ⏟ ⏟ ⏟ element 6N 2 ×6N 2

K≡ ⌣

R≡

∑ ( e

) KM + e KG + e KC , element ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ∑

(11.637)

6N 2 ×6N 2 e

⌣ ⌣

R(d) element ⏟⏟⏟ 6N 2 × 1

An actual global assembly scheme, referred to in the finite element jargon as the direct global assembly, will normally be applied, which we will elaborate upon in the numerical implementation phase next.

11.18.14

Time Integration of the Semi-discrete Equation

So far, we have detailed the development of the global incremental matrix second-order ordinary differential equation as given by equation (11.636) under the semi-discrete c-type finite element formulation. The linearization has been introduced and devised with the understanding that a timeintegration algorithm for the ensuing matrix second-order ordinary differential equation will be developed in conjunction with a Newton-type iterative scheme which will require linearization of the nonlinear equations at each iterative step. The time integration must realize that the rotation vector marching and updating must be given special attention as it is constrained to travel over a curved space: the 3-sphere. Remark

r In this book, we have not implemented the dynamic treatment presented above, so for the r

following sections, we restrict ourselves to implementation of shell analysis to static and quasi-static loading conditions. However, for nonlinear structural dynamic problems involving finite rotations, we now suggest that the way to go may be the rotational acceleration with a spherically linear distribution (see Chapter 4 on rotation tensors) for a predictor–corrector type of numerical analysis; the rotational velocity and rotation distribution may then be obtained by corresponding integration of the acceleration.

Computation of Nonlinear Structures

914

Figure 11.21 Spherical interpolation and approximation.

11.18.14.1

Linear Interpolation or Approximation on 3-Sphere

Let A, B represent the unit Hamilton–Rodrigues quaternions for rotations at parameters t = 0 and t = 1, respectively, as shown in Figure 11.21. The rotation from the reference state, A, to a final state, B, is a taken as a single rotation which is always possible by Euler’s theorem. Now, let P(t), with ‖P‖ = 1, represent a rotation on this curve at an affine parameter t = 𝜆 ∈ [0, 1]. We are interested in expressing P in terns of A and B (see Chapter 4 on rotation tensors):

P=

11.18.15

sin (1 − t) 𝜙 sin t𝜙 A+ B, sin 𝜙 sin 𝜙

(11.638)

t ∈ [0.1]

Where We Would Like to Go

The main goal here was to present the finite element formulation of the dynamic fully linearized variational or virtual work equations. As indicated earlier, shell analysis subjected to dynamic loadings has not been implemented. We will continue with the implementational issues related to static and quasi-static analyses:

r finite element implementation for quasi-static analyses.

11.19 11.19.1

c-type FE Formulation: Quasi-static Loading What We Need to Recall

The linearized quasi-static virtual work equations:

Material stiffness functional

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⌣

∫ M

n ̄T n

ε

⌣

n⌣

D Δ ε dA +

Geometric stiffness functional

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⌣

∫ M

n ̄T n

Δ ε

⌣

F dA

Unbalanced force functional

⏞⏞⏞⏞⏞⏞⏞ ⌣

⌣

̄ =− G( t d0 ; t d) ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ Residual force functional

(11.639)

Nonlinear Shell

915

Or, Tangent functional

⎴ ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⎴ Material stiffness functional

Geometric stiffness functional

⏞⏞⏞ M

⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⌣

⌣

̄ =− G( t d0 ; t d) ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

⏞⏞⏞ G

+

Unbalanced force functional

Residual force functional

(11.640) The linearized internal functional, ΔGInternal : Linearized internal functional

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ Material stiffness functional

Geometric stiffness functional

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

ΔGInternal ≡

⌣

n ̄T n

ε

∫

⌣

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⌣

⌣

n⌣

D Δ ε dA +

∫

M

Δn ε̄ T n F dA (11.641)

M Tangent functional

⎴ ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⎴ Material stiffness functional

⏞⏞⏞ M

=

Geometric stiffness functional

⏞⏞⏞ G

+

Definition: Material stiffness functional, M : ⌣

⌣

⌣

̄ Δt d) ≡ M ( t d, t d;

⌣

n ̄T n

ε

∫

⌣

⌣

D Δn ε dA

(11.642)

M

Definition: Geometric stiffness functional, G : ⌣

⌣

⌣

̄ Δ t d) ≡ G ( t d, t d;

∫

⌣

⌣

⌣

⌣

⌣

̄ Δ t d) n F dA Δn ε̄ T ( t d, t d;

(11.643)

M

the virtual strain–displacement relations in compact matrix form is as follows: ⌣ ⌣ ⌣

⌣

⌣

̄ = E(d) H(d) ̄ ̄ d; d) ε(

(11.644)

where T

⎡R ⎢ ⌣ ⎢ 0 E(d) = ⎢ ⎢ 0 ⎢ ⎣ 0

0

0

RT

0

0

RT

0

0

0 ⎤ ⎡ [k01 ] ⎥⎢ 0 ⎥⎢ 0 ⎥⎢ 0 ⎥ ⎢ [k02 ] ⎥⎢ RT ⎦ ⎣ 0

[a1 ]W

I

0

0

X1 + [kc1 ]W

0

W

0

[a2 ]W

0

0

I

X2 + [kc2 ]W

0

0

0

0 ⎤ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ W⎦ (11.645)

Computation of Nonlinear Structures

916 ⌣

T ̄ ̄ ̄ ≡ { d̄ θ̄ d, ̄ ̄ 1 θ, and I, is the 3 × 3 identity matrix, H(d) 1 d, 2 θ, 2 } , and from our knowledge of the relevant derivatives and variations, we have:

W ≡ I + c1 𝚯 + c2 𝚯2 X𝛼 ≡ −c1 𝚯, 𝛼 + c2 {[θ ⊗ θ, 𝛼 ] − [θ, 𝛼 ⊗ θ] + (θθ, 𝛼 )I} +V [θ, 𝛼 ⊗ θ]),

(11.646)

𝛼 = 1, 2

V ≡ −c2 I + c3 𝚯 + c4 𝚯2

c1 ≡ (1 − cos 𝜃)∕𝜃 2 , c2 ≡ (𝜃 − sin 𝜃)∕𝜃 3 ,

(11.647)

c1 ≡ 𝛿c1 ≡ c3 ≡ {(1 − 𝜃 2 c2 ) − 2c1 }∕𝜃 2 , c2 ≡ 𝛿c2 ≡ c4 ≡ (c1 − 3c2 )∕𝜃 2

⌣

the (6 × 1) incremental generalized displacement state vector functions, Δd ≡ ⌣ ( )T Δ d Δθ , and finally, n D is the 12 × 12 symmetric elastic constitutive matrix. In the expanded form, equation (11.643) may be expressed as follows:

Translational part

Rotational part

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ G ≡

∫

n

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞

̄ n F𝛼 dA + Δβ 𝛼

M

n

∫

Δχ̄ 𝛼 n M𝛼 dA

(11.648)

M

where Δ, the incremental operations, involve the covariant derivatives of the virtual strain functions.

r The unbalanced force functional, G(𝜆0 , t ⌣d0 ; t d): ̄ ⌣

⌣

⌣

̄ ≡ G(𝜆 , d ; d) 0

0

⌣

∫ M

n ̄T n

ε

⎧ ⎪ D ε dA − ⎨ ∫ ⎪M ⎩ ⌣

n⌣

⌣

⌣

t ̄T t ̂

d

F(𝜆 , S , S ) dA + 𝜆 0

1

2

0

∫

𝜕M f

t ̂̄

d

T

⎫ ⎪ F𝜈 (S) dS⎬ ⎪ ⎭

⌣

t̂

(11.649)

Nonlinear Shell

917

r The residual force functional: Unbalanced force functional

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⌣

⌣

̄ G(𝜆0 , t d0 ; t d)

Residual force functional ≡ −

Applied force functional

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⎫ ⎧ ⌣S ⌣ ⎪ ⎪ t ⌣̄ T t ⌣̂ t ̄T t ̂ +⎨ d F, 𝜆 (𝜆0 , S1 , S2 ) dA + d F (S) dS⎬ Δ𝜆 ∫ ∫ ⎪ ⎪M 𝜕M f ⎭ ⎩ (11.650) ̂̄ ≡ { t d̄ H( t d) ⏟⏟⏟

tθ ̄

t d, ̄

1

t θ, ̄

1

t d, ̄

2

t θ, ̄

T 2} ,

and

̂ ≡ {𝚫 t d 𝚫 t θ 𝚫 t d, 1 H(𝚫d) ⏟⏟⏟

18 × 1

r r

18 × 1

𝚫 t θ, 1 𝚫 t d, 2 𝚫 t θ, 2 }T with (∙), 𝛼 , 𝛼 = 1, 2 are derivatives with respect to the arc length parameters, S𝛼 , 𝛼 = 1, 2, along the lines of curvature. Superscript t reminds us that the relevant items are measured in the undeformed coordinate system, G1 , G2 , N; superscript n reminds us that the relevant items are measured in the rotated coordinate system, g1 (= R G1 ), g2 (= R G2 ), g3 ≡ n(= R N). The expressions given in equation (11.643) or equation (11.648) involve the first variation (incremental) of the virtual rotational strains or, equivalently, the second variation (Hessian) of the real rotational strains. It is this second variation that makes the derivation of the geometric stiffness matrix extremely involved, and thus tedious. However, the subsequent reward in terms of high computational accuracy, ease and efficiency in situations of repetitive computations of the geometric stiffness as we have in a nonlinear numerical analysis, makes it well worth while.

Material Stiffness Functional Revisited: M

11.19.2

As in the virtual expression above, we group together the incremental state variables and their derivatives with respect to the arc length parameters, (S1 , S2 ), into an augmented ⌣ ⌣ ⌣ ̄ ≡ (18 × 1) incremental vector function (we can replace d̄ by Δd in the definition of H(d) ⏟⏟⏟ {

d̄ θ̄

̄ 1 d,

̄ θ, 1

̄ θ, 2

̄ 2 d,

}T

18×1

) to introduce the following.

Definition Incremental generalized displacement function: ⌣

H(Δd) ≡ {Δd ⏟⏟⏟

Δ𝜃

Δd,1

Δ𝜃,1

Δd,2

Δ𝜃,2 }T

(11.651)

18×1

Now, replacing the variational variables in equation (11.644) with incremental variables and keeping the above definition in mind, we get a similar expression as follows:

Computation of Nonlinear Structures

918

Definition Incremental strain–displacement relation:

⌣ ⌣

⌣

⌣ ⌣

⌣

⌣

⌣

Δ ε(d; Δd) = E(d) H(Δd)

(11.652)

Note that the incremental strain, Δ ε(d; Δd), is clearly linear in the incremental generalized dis⌣

⌣

placement, Δd, by being related to the strain–displacement matrix that is independent of Δd. By applying the virtual strain–displacement relation (11.644) and incremental strain–displacement relation (11.652), we are now ready to transform equation (11.642) as under: ⌣ ⌣

11.19.3

⌣

⌣

̄ Δd) ≡ M (d, d;

∫M

⌣

⌣

⌣

̄ T (E(d)T n D ̂ E(d))H(Δd)dA H(d)

(11.653)

Material Stiffness Matrix, KM

Based on equation (11.653), we can now redefine and present the material stiffness functional as follows: Definition: Material stiffness functional, M is: ⌣ ⌣

⌣

⌣

̄ Δd) ≡ M (d, d;

∫

⌣

⌣

̄ T KM (d) H(Δd) dA H(d)

(11.654)

M

where, we have introduced:

Definition: Material stiffness matrix, KM , as: ⌣

⌣

⌣

KM ≡ E(d)T n D E(d)

(11.655)

Remark

r Clearly,

from equation (11.655), the material stiffness matrix, K, is symmetric in state or ⌣

configuration space vector, d.

11.19.4

Geometric Stiffness Functional Revisited: G

Here, in order to avoid undue distraction, first we present the main results, and then refer the interested reader to the sections on detailed derivations.

Nonlinear Shell

919

r Translational virtual work: linearization and symmetry We get as the final result: n Δβ ̄

𝛼

̄ 𝛼 ∙([t F𝛼 ]T )Δθ + Δd,𝛼 ∙([t F𝛼 ]T W)θ̄ ∙ n F𝛼 = d, + d̄ ∙ ([k0𝛼 ][t F𝛼 ]W)Δθ + 𝛿d ∙ ([k0𝛼 ][t F𝛼 ]W)θ̄

(11.656)

Sym + θ̄ ∙ Zβ Δθ

with Sym

Z̄

β

= 12 {WT ([ t F𝛼 ⊗ a𝛼 ] + [a𝛼 ⊗ t F𝛼 ]) W + ([hf ⊗ θ] + [θ ⊗ hf ])} − ( t F𝛼 ∙ a𝛼 )WT W + c2 (θ ∙ bf )I

(11.657)

r Rotational virtual work: linearization and symmetry We get as the final result: n Δχ ̄ ∙ nM

̄ = Δθ ∙ {WT [ t M𝛼 ] W + Y( t M𝛼 )}θ, 𝛼 +θ̄ ∙ {WT [ t M𝛼 ] W + Y( t M𝛼 )}Δθ, 𝛼 Sym

+Δθ ∙ Zχ̄

(11.658)

θ̄

with Symm

Zχ̄

≡ 12 {WT ([ t M𝛼 ⊗ kc𝛼 ] + [kc𝛼 ⊗ t M𝛼 ]) W + ([hm ⊗ θ] + [θ ⊗ hm ])} t 𝛼 T − ( t M𝛼 ∙ kc𝛼 )WT W + c2 (θ ∙ bm )I + WT [ t M𝛼 ]X𝛼 + XT 𝛼 [ M ] W

(11.659) and Y(∙) function is defined later.

11.19.5

̂ F) ̂ Geometric Stiffness Functional, G and Matrix, G(d,

Based on equations (11.656) and (11.658), we can now redefine and present below the geometric stiffness functional in its most uncluttered and explicit form. Definition: Geometric stiffness functional, G , is:

⌣ ⌣

⌣

̄ Δd) ≡ G (d, d;

where

⎧ ⎫ ⌣ ⌣ ⌣ ⎪ ⎪ ⌣T t H(d) G(d, F) H(Δd) ⎬ dA ∫ ⎨ ⏟⏟⏟ ⏟⏞⏟⏞⏟ ⏟⏟⏟⎪ ⎪ M ⎩ 1×18 18×18 18×1 ⎭

(11.660)

Computation of Nonlinear Structures

920 ⌣

⌣

Definition: Geometric stiffness matrix, G(d, t F), is given as: ⎡ 0 ⎢( ) ⎢ G1 + G2 T 12 12 ⎢ ⎢ 0 ⌣ ⌣ G(d, t F) ≡ ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎣

G112 + G212

0

0

0

G122 + G222

G123

G124

G223

(G123 )T

0

0

0

(G124 )T

0

0

0

(G223 )T

0

0

0

(G224 )T

0

0

0

0 ⎤ ⎥ G224 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥⎥ 0 ⎥⎦

(11.661)

where [ ] G𝛼12 ≡ k0𝛼 [ t F𝛼 ] W,

𝛼 = 1, 2 and no sum

G𝛼22 ≡ 12 WT {[ t F𝛼 ⊗ a𝛼 ] + [a𝛼 ⊗ t F𝛼 ] − 2( t F𝛼 a𝛼 )}W + c2 (θ b𝛼 )I f

+ 12 ([hb𝛼 ⊗ θ] + [θ ⊗ hb𝛼 ]) + 12 WT {[ t M𝛼 ⊗ kc𝛼 ] + [kc𝛼 ⊗ t M𝛼 ] − 2( t M𝛼 kc𝛼 )}W + c2 (θ bm 𝛼 )I

(11.662)

m T t 𝛼 t 𝛼 T + 12 ([hm 𝛼 ⊗ θ] + [θ ⊗ h𝛼 ]) + W [ M ] X𝛼 + X𝛼 [ M ] W

G𝛼23 ≡ WT [ t F𝛼 ] G𝛼24 ≡ WT [ t M𝛼 ] W + Y( t M𝛼 ) with c1 ≡ (1 − cos 𝜃)∕𝜃 2 , c2 ≡ (𝜃 − sin 𝜃)∕𝜃 3 , c̄ 1 ≡ 𝛿c1 ≡ c3 ≡ {(1 − 𝜃 2 c2 ) − 2c1 }∕𝜃 2 ,

(11.663)

c̄ 2 ≡ 𝛿c2 ≡ c4 ≡ (c1 − 3c2 )∕𝜃 2

a ≡ e + [k0 ]d = {(1̂ + d, s ) + [k0 ]d} kt ≡ W θ, s kc ≡ k0 + kt bf ≡ Ft × a bm ≡ Mt × kc

(11.664)

Nonlinear Shell

921

W ≡ I + c1 𝚯 + c2 𝚯2 X𝛼 ≡ −c1 𝚯, 𝛼 + c2 {[θ ⊗ θ, 𝛼 ] − [θ, 𝛼 ⊗ θ] + (θ ∙ θ, 𝛼 )I} + V [θ, 𝛼 ⊗ θ]) V ≡ −c2 I + c3 𝚯 + c4 𝚯2 Y(x) ≡ −c1 [x] + c2 ([x ⊗ θ] − [θ ⊗ x]) + c2 (θ ∙ x)I + [θ ⊗ VT x], f

∀x ∈ R3

f

h𝛼 ≡ VT b𝛼 T m hm 𝛼 ≡ V b𝛼

(11.665) Exercise: Drawing upon the analyses referred to above for the translational and rotational geometric stiffness functional, derive the expression for G above. (Hint: look at Section 10.18 on geometric stiffness and its symmetry property of a beam.) Remark

r Clearly, from equation (11.661), the geometric stiffness matrix, G, is symmetric in state or configuration space variables for a conservative system.

r Because of total similarity in each of the lines of curvature, for derivation of the geometric stiffness and its symmetry property, look up the discussion on the geometric stiffness and its symmetry property of a beam. 11.19.5.1

Implementational Note

Note that, as will be explained later, the computational finite element implementation of a shell problem can be achieved essentially by using the corresponding beam formulations along each of the lines of curvature at numerical Gauss quadrature points in a typical Newton-type iteration method of solution strategy.

The material and the geometric stiffness functional parts in equations (11.653) and (11.660) need to be further modified in the subsequent finite element formulation for computational purposes. In so doing, we will be able to introduce the computational definition for the shell material and the geometric stiffness matrices. Thus, for the final and computationally optimum form of the linearized virtual work, we will focus next on the c-type finite element formulation.

11.19.6

c-type Finite Element Formulation

Following the standard definition of a finite element discretization, the shell surface, Ω, is broken into disjointed finite number of elemental surfaces such that:

Ω=

e=no. of elements ⋃ e=1

Finite elemente =

e=no. of elements ⋃ e=1

Ωe

(11.666)

Computation of Nonlinear Structures

922

Then, the shell material and geometric stiffness matrices can be expressed as an assembly of the corresponding elemental matrices; a similar idea applies to the formation of the global unbalanced and the applied forces. As will be seen in our c-type finite element implementation phase, we will use what is known as the direct stiffness and load global assembly of the corresponding elemental entities. Thus, for now, we can express each of the following properties as a sum over all the corresponding entities belonging to the elements:

r Shell material stiffness matrix functional: ⌣ ⌣

⌣

̄ Δd) = M (d, d;

∑

⌣

⌣

⌣

̄ T KM (d) H(Δd) dA H(d) ∫ ⏟⏟⏟ ⏟⏟ ⏟ ⏟⏟⏟ elements e Ω

1×18

18×18

(11.667)

18×1

r Shell geometric stiffness matrix functional: ⌣ ⌣

⌣

̄ Δd) = G (d, d;

⎧ ⎫ ⌣ ⎪ ⎪ ⌣̄ T H(d) G H(Δd) dA ∫ ⎨ ⏟⏟⏟ ⏟⏟⏟⎬ ⎪ elements e ⎪⏟⏟⏟ Ω ⎩ 1×18 18×18 18×1 ⎭ ∑

(11.668)

r Shell applied force functional: ⎞ ⌣ ∑ ⎛ ⌣ ⌣ ⎜ d̄ T F, ̂ 𝜆 (𝜆, S1 , S2 ) dA + d̄̂ T F̂ 𝜈 (S) dS⎟ ∫ ⎜∫ ⎟ elements ⎝ e ⎠ Γe Ω

(11.669)

r Shell unbalanced force functional: 0

G(𝜆 ,

⌣

d0

⌣

̄ = ; d)

⎛ ∑ ⎜ ⎜∫ elements ⎜ e ⎝Ω

⌣

n ̄T n

ε

⎧ ⎫⎞ ⌣ ⌣ ⎪ ⌣̄ T ⌣̂ 0 1 2 ⎪⎟ 0 T ̂ ̄ d F𝜈 (S) dS⎬⎟ D ε dA − ⎨ d F(𝜆 , S , S ) dA + 𝜆 ∫ ∫ ⎪Ωe ⎪⎟ Γe ⎩ ⎭⎠ ⌣

n⌣

(11.670) Based on the assumption that the assembly of the finite elements will be performed by the direct global stiffness assembly procedure where the properties of each element enter into the summing operation independent of the all other element data, we can restrict our discussion here to a typical finite element for the shell. Later, we will discuss the assembly of the global stiffness, and so on in our implementation phase. Finally, although we can use the available library of c-type shell elements of both curved quadrilateral and triangular shape, in what follows, we will restrict our discussion to only the Cartesian product quadrilateral elements; similar arguments and ideas apply formally to the triangular elements too. The root element corresponding to a typical finite element is defined over the geometric 2D root parameter plane defined by parameters (𝜉̂1 , 𝜉̂2 ) ∈ [0, 1] ∪ [0, 1]. Next, the parameters, (𝜉 1 , 𝜉 2 ), along the lines of curvature are related to the arc length parameter, (S1 , S2 ), 𝜉1

𝜉2

of a shell by: S1 (𝜉 1 , 𝜉 2 ) = ∫0 ‖ e C0 (𝜂, 𝜉 2 ), 𝜂 ‖d𝜂, and S2 (𝜉 1 , 𝜉 2 ) = ∫0 ‖ e C0 (𝜉 1 , 𝜂), 𝜂 ‖d𝜂 where e C (𝜉 1 , 𝜉 2 ), 𝜉 1 ∈ [0, 1], 𝜉 2 ∈ [0, 1] defines the reference surface position vector at any point 0

Nonlinear Shell

923

(𝜉 1 , 𝜉 2 ) of the eth finite element, that is, the Finite elemente . Let us briefly recall the basic steps of the c-type finite element generation for future use. The coordinate geometry of a shell reference surface is discretized in the mesh generation phase whereby the root element is identified and fixed for an element; the position vector, C0 (𝜉̂1 , 𝜉̂2 ), of any point on the element is given by: ⎧ 1 ̂1 ̂2 ⎫ ⎪ C0 = X(𝜉 , 𝜉 ) ⎪ ⎪ ⎪ C0 (𝜉̂1 , 𝜉̂2 ) = ⎨ C02 = Y(𝜉̂1 , 𝜉̂2 ) ⎬ = c T(𝜉̂1 , 𝜉̂2 ) c q, ⎪ 3 ⎪ ⎪ C0 = Z(𝜉̂1 , 𝜉̂2 ) ⎪ ⎩ ⎭ C0k (𝜉̂1 , 𝜉̂2 ) =

nc mc ∑ ∑

Bi c (𝜉̂1 ) Bj c (𝜉̂2 ) c qkij , n

m

(11.671)

k = 1, … , 3

i=0 j=0

where c T(𝜉̂1 , 𝜉̂2 ) is the root transformation matrix, Bni (𝜉̂1 ), Bm (𝜉̂2 ) are Bernstein–Bezier polynoj mials of degree nc and mc respectively, and c q, c qkij are the generalized coordinate control vector and the component control net for the element, respectively. ⌣

The discretization of the generalized displacement field, d(𝜆) ≡ (d(𝜆), θ(𝜆)), for the finite element, is then performed over the geometric root element, except that the degree of the displacement distribution polynomials (i.e. the local basis or shape functions) will be at least equal to that of the geometric coordinate distribution polynomials; in other words, the element is either subparametric or isoparametric.

11.19.7

c-type Shell Elements and State Control Vectors

Let us first recognize that the shell geometry and the finite elements are naturally discretized in terms of the parameters (𝜉̂1 , 𝜉̂2 ) as tensor product objects for quadrilateral elements. However, for analytical convenience, the linearized virtual work expression as given by equation (11.639) has been derived in terms of the arc length parameters, (S1 , S2 ), along the corresponding lines of curvature. Thus, for computational purposes, we must express all our equations back in the (𝜉̂1 , 𝜉̂2 ) parameters. This is what is done in the ensuing finite element formulation. Let us, then, introduce a typical c-type shell element displacement interpolation as ⌣

d(𝜉̂1 , 𝜉̂2 , 𝜆) =

⌣ d T(𝜉̂1 , 𝜉̂2 ) d ⌣ q(𝜆),

⌣

m n ∑ ∑

dk (𝜉̂1 , 𝜉̂2 , 𝜆) =

k

̂2 d ⌣ Bni (𝜉̂1 ) Bm j (𝜉 ) q ij (𝜆),

k = 1, ⋯ , 6

(11.672)

i=0 j=0 ⌣

where d T(𝜉̂1 , 𝜉̂2 ) is the root transformation matrix, Bni (𝜉̂1 ), Bm (𝜉̂2 ) are the Bernstein–Bezier j ⌣

⌣k

polynomials of degree n and m respectively, and d q, d q ij are the generalized displacement control vector and the component control net for the element, respectively. Similar distributions can be ⌣ ̄ and the incremental generalized produced for the variations in generalized displacements, d, ⌣

displacement, 𝚫 d. Let us define N, M as the orders of the polynomials, respectively, given by: N ≡ n + 1 and M ≡ m + 1. As will be seen, in actual implementation, we will enforce: n = m for

Computation of Nonlinear Structures

924

each element. We may see that the Cartesian product of these polynomials serve as the local basis functions for our c-type shell elements. Thus, a n degree typical c-type shell element is given by:

⌣

⌣

⌣ d(𝜉̂1 , 𝜉̂2 , 𝜆) = d T(𝜉̂1 , 𝜉̂2 ) d q(𝜆) ⏟⏞⏞⏞⏟⏞⏞⏞⏟ ⏟⏞⏞⏟⏞⏞⏟ ⏟⏟⏟

6×1

6×6N 2

(11.673)

6N 2 ×1

where we have introduced the following. Definitions {

d⌣

qu00 ⋯ qun0 qu01 ⋯ qun1 qu0n ⋯ qunn qv00 ⋯ qvn0 qv01 ⋯ qvn1 qv0n ⋯ qvnn ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

q(𝜆) = ⏟⏟⏟

r

6N 2 ×1 qw ⋯ qw 00 n0

1×N 2

𝜃

𝜃

𝜃

𝜃

𝜃

1×NM 𝜃

1 1 1 1 1 1 qw ⋯ qw qw ⋯ qw nn q00 ⋯ qn0 q01 ⋯ qn1 q0n ⋯ qnn 01 n1 0n ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

1×N 2 𝜃

𝜃

𝜃

1×N 2 𝜃

𝜃

𝜃

𝜃

𝜃

𝜃

𝜃

𝜃

𝜃

q002 ⋯ qn02 q012 ⋯ qn12 q0n2 ⋯ qnn2 q003 ⋯ qn03 q013 ⋯ qn13 q0n3 ⋯ qnn3 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ 1×N 2

r

is the (6N 2 × 1) }T

1×N 2

generalized displacement control vector. ⌣ d T(𝜉̂1 , 𝜉̂2 ) is the (6 × 6N 2 ) interpolation transformation matrix containing the Bernstein–Bezier polynomials, that is,

⌣

d T(𝜉̂1 ,

𝜉̂2 ) =

0 0 ⋯ ⋯ ⋯ 0 ⎤ ⎡ B0 B0 B1 B0 ⋯ Bn Bn−1 Bn Bn ⎢ ⎥ 0 0 Bn0 Bn0 Bn1 Bn0 ⋯ Bnn Bnn 0 ⋯ 0 ⎥ ⎢ 0 ⎢ ⎥ ⋯ ⋯ ⎢ ⎥ ⎢ ⎥ ⋯ ⋯ ⎢ ⎥ ⎢ ⎥ n n n n n n ⎣ 0 0 ⋯ ⋯ 0 B0 B0 B1 B0 ⋯ Bn Bn ⎦ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ n n

n n

n n

n n

6x6N 2

(11.674) ⌣

In other words, each row of d T(𝜉̂1 , 𝜉̂2 ) is identical except shifted by (1 × 6N 2 ) zero entries with increasing row numbers. Based on the above definitions, we have: ⌣

⌣

⌣ Δd(𝜆) = d T(𝜉̂1 , 𝜉̂2 ) Δ d q(𝜆)

where we have similarly introduced the following.

(11.675)

Nonlinear Shell

925

Definitions

r

⌣

Δd q(𝜆) is the (6N 2 × 1) load-dependent incremental displacement control vector, similar in ⌣ the ordering of components to d q(𝜆).

Note that equations (11.673) and (11.674) imply that we have chosen to interpolate both the displacement components and the rotation components with the same degree control net. Now, ⌣ ⌣ ̄ ≡ let us rewrite H( t d) ≡ { t d t θ t d, 1 t θ, 1 t d, 2 t θ, 2 }T in generalized form as S H( t d) ⌣

⌣

⌣

{ t d t d, 1 t d, 2 }T , with the superscript S reminding us that the derivatives in the expression are with respect to the arc length parameters, S𝛼 , 𝛼 = 1, 2. Next, let us further introduce the following. Definitions

r

𝜉̂ H( d) ̄

r

⌣ 𝜉̂ B(𝜉̂1 , 𝜉̂2 )

⌣

⌣

≡ { d̄

⌣

̄ ̂1 d, 𝜉

⌣

̂

⌣

⌣

̄ ̂2 }T , and similarly, 𝜉 H(Δd) ≡ { Δd d, 𝜉

⌣

Δd, 𝜉̂1

⌣

Δd, 𝜉̂2 }T

is the (18 × 6N 2 ) augmented interpolation transformation matrix as: ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⌣ ⎡ dT ⎢ ⏟⏟⏟ ⎢ 6×6N 2 ⎢ ⎢ d⌣ ⎢ T, 𝜉̂1 ̂𝜉 ⌣ ̂1 ̂2 B(𝜉 , 𝜉 ) ≡ ⎢ ⏟⏟⏟ ⏟⏞⏞⏟⏞⏞⏟ ⎢ 6×6N 2 18×6N 2 ⎢ ⌣ ⎢ d T, ̂2 ⎢ ⏟⏟𝜉⏟ ⎢ ⎣ 6×6N 2

(11.676)

such that 𝜉̂

⌣

⌣

⌣

̄ = 𝜉̂ B(𝜉̂1 , 𝜉̂2 ) d q̄ H(d) ⏟⏟⏟ ⏟⏞⏞⏟⏞⏞⏟ ⏟⏟⏟ 18×1 𝜉̂

18×6N 2 ⌣

̂⌣

6N 2 ×1 ⌣

H(Δd) = 𝜉 B(𝜉̂1 , 𝜉̂2 ) Δd q ⏟⏞⏟⏞⏟ ⏟⏞⏞⏟⏞⏞⏟ ⏟⏟⏟ 18×1

18×6N 2

(11.677)

6N 2 ×1

with superscript 𝜉̂ in each term reminding us that the derivatives are with respect to parameters, 𝜉̂𝛼 , 𝛼 = 1, 2. Moreover, suppose the lines of curvature make an angle of 𝛼1 and 𝛼2 with 𝜉̂1 coordinate direction at any point (𝜉̂1 , 𝜉̂2 ) on the shell surface. Then, as indicated earlier, with the arc length parameters, (S1 , S2 ), along the corresponding lines of curvature, we can write

Computation of Nonlinear Structures

926

the following transformation relationship for the virtual generalized displacement from (𝜉̂1 , 𝜉̂2 ) parameters to the arc length parameter, (S1 , S2 ): ⌣

⌣

⌣

̄ = S𝜉̂ B(𝜉̂1 , 𝜉̂2 ) 𝜉̂ H(d) ̄ H(d) ⏟⏟⏟ ⏟⏞⏞⏞⏟⏞⏞⏞⏟ ⏟⏟⏟ S

18×1

18×1

18×18 ⌣

̂⌣

(11.678)

⌣

̂ H(Δd) = S𝜉 B(𝜉̂1 , 𝜉̂2 ) 𝜉 H(Δd) ⏟⏞⏟⏞⏟ ⏟⏞⏞⏞⏟⏞⏞⏞⏟ ⏟⏞⏟⏞⏟ S

18×1

18×1

18×18

̂⌣

with S𝜉 B(𝜉̂1 , 𝜉̂2 ) defined by: ⎡ I ⎢ ⏟⏟⏟ ⎢ 3×3 ⌣ ⎢ ̂ S𝜉 B(𝜉̂1 , 𝜉̂2 ) ≡ ⎢ 0 ⏟⏞⏞⏞⏟⏞⏞⏞⏟ ⎢ 18×18 ⎢ ⎢ 0 ⎣

0 1 cos 𝛼1 I 𝛾1 −

⎤ ⎥ ⎥ ⎥ 1 sin 𝛼1 I ⎥ 𝛾1 ⎥ ⎥ 1 cos 𝛼1 I ⎥ ⎦ 𝛾2 0

1 sin 𝛼1 I 𝛾2

(11.679)

𝜕C0 (𝜉 1 , 𝜉 2 ) dS𝛼 ̄ 𝛽 ‖, 𝛼 = 1, 2 with G ̄ 𝛽 (𝜉 1 , 𝜉 2 ) ≡ = ‖G defined by the tan𝛼 d𝜉 𝜕𝜉 𝛽 gents to the lines of curvature. Now, from equations (11.677) and (11.678), we have the final transformation relations as:

with 𝛾𝛼 (𝜉 1 , 𝜉 2 ) ≡

⌣

⌣

⌣

̄ = B(𝜉̂1 , 𝜉̂2 ) d q̄ H(d) ⏟⏟⏟ ⏟⏞⏟⏞⏟ ⏟⏟⏟ S

18×1

18×6N 2

⌣

6N 2 ×1

⌣

⌣ H(Δd) = B(𝜉̂1 , 𝜉̂2 ) d Δqz ⏟⏞⏟⏞⏟ ⏟⏞⏟⏞⏟ ⏟⏟⏟

S

18×1

18×6N 2

(11.680)

6N 2 ×1

⌣

where we have defined B(𝜉̂1 , 𝜉̂2 ) as: ⌣

B(𝜉̂1 , 𝜉̂2 ) ≡

S𝜉̂

⌣

̂⌣

B(𝜉̂1 , 𝜉̂2 ) 𝜉 B(𝜉̂1 , 𝜉̂2 )

(11.681)

√ ̂ d𝜉̂1 d𝜉̂2 , of the area integrals where ̂ = G Finally, we must express the infinitesimal area, d A √ ̂ is the positive square root of the determinant of the surface metric expressed in terms of the G geometry coordinate parameters, 𝜉̂𝛼 , 𝛼 = 1, 2.

11.19.8

Computational Symmetric Element Material Stiffness Matrix, e KcM

By substituting equation (11.680) in equation (11.667), we get for each finite element:

Nonlinear Shell

e

⌣ ⌣

927

⌣

̄

M ( d, d; Δ d)

18×12 12×12 12×18 ⎡ ⎧ ⎫ ⎤ ⏞⏞⏞ ⏞⏞⏞ ⏞⏞⏞ ⎢ ⎪ ⎪ √ ⎥ ⌣ ⌣ ⌣ ⌣ ⌣ ⎢ 1 1 ⎪ ⌣ ̂1 ̂2 T d T n 1 ̂2 ⎪ 1 ̂2 ⎥ d ⌣ ̂ ̂ ̂ ̄ ⎢ G d𝜉 d𝜉 ⎥ Δq(𝜆) D E(d) B(𝜉 , 𝜉 )⎬ ≡ qT ⎨B(𝜉 , 𝜉 ) E(d) ⏟⏟⏟ ⎢∫0 ∫0 ⎪⏟⏞⏞⏟⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏟⏞⏟⎪ ⎥ ⏟⏟⏟ 18×18 18×6N 2 ⎪ ⎪ 6N 2 ×18 ⎥ 6N 2 ×1 1×6N 2 ⎢ ⎣ ⎦ ⎩ ⎭ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ 6N 2 ×6N 2

=

⌣

⌣

d e c ̄ KM d Δq(𝜆) qT ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟ 1×6N 2 6N 2 ×6N 2 6N 2 ×1

(11.682) where, we have introduced the following. Definition

r The computational symmetric element material stiffness matrix,

e Kc , M

as:

18×12 12×12 12×18 ⎧ ⎫ ⏞⏞⏞ ⏞⏞⏞ ⏞⏞⏞ ⎪ ⎪√ 1 1 ⌣ ⌣ ⌣ ⎪ ̂1 ̂2 T ⌣ T n ⌣ ⎪ ̂ ̂1 ̂2 e c 1 𝜉 d𝜉 KM ≡ B(𝜉 , 𝜉 ) E(d) D E(d) B(𝜉̂ , 𝜉̂2 )⎬ Gd ∫0 ∫0 ⎨ ⎪⏟⏞⏞⏟⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏟⏞⏟⎪ 18×18 18×6N 2 ⎪ ⎪ 6N 2 ×18 ⎩ ⎭ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

(11.683)

6N 2 ×6N 2

Computational Symmetric Elem. Geometric Stiffness Matrix, e KcG

11.19.9

By substituting equation (11.680) into equation (11.668), we get for each finite element: e

⌣ ⌣

̄

⌣

G ( d, d; Δ d)

⎡ ⎧ ⎫ ⎤ √ ⌣ ⌣ ⌣ ⎢ 1 1 ⎪ ⌣ ̂1 ̂2 T ⎪ t 1 2 1 2 ̂ d𝜉̂ d𝜉̂ ⎥⎥ d Δ⌣ ̄ ⎢ G ≡ qT q(t) ⎨B(𝜉 , 𝜉 ) G(d, F) B(𝜉̂ , 𝜉̂ )⎬ ∫ ∫ ⏟⏟⏟ ⎢ 0 0 ⎪⏟⏞⏞⏟⏞⏞⏟ ⏟⏞⏟⏞⏟ ⏟⏞⏟⏞⏟⎪ ⎥ ⏟⏟⏟ ⎦ 6N 2 ×1 ⎩ 6N 2 ×18 18×18 18×6N 2 ⎭ 1×6N 2 ⎣ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ d

⌣

6N 2 ×6N 2

=

⌣

⌣

d T e c d K Δq(t) q̄ ⏟⏟⏟ ⏟⏟G⏟ ⏟⏟⏟ 1×6N 2 6N 2 ×6N 2 6N 2 ×1

(11.684) where, we have introduced the following.

Computation of Nonlinear Structures

928

Definition

r The computational symmetric element geometric stiffness matrix,

e

KcG

≡

1

1

∫0 ∫0

e Gc ,

as:

⎧ ⎫ ⌣ ⌣ ⌣ ⎪ ⌣ ̂1 ̂2 T ⎪ √ ̂ ̂1 ̂2 t 1 2 ̂ ̂ B( 𝜉 , 𝜉 ) G( d, F) B( 𝜉 , 𝜉 ) ⎨ ⎬ G d𝜉 d𝜉 ⎪⏟⏞⏞⏟⏞⏞⏟ ⏟⏞⏟⏞⏟ ⏟⏞⏟⏞⏟⎪ ⎩ 6N 2 ×18 18×18 18×6N 2 ⎭ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

(11.685)

6N 2 ×6N 2

with the geometric stiffness matrix, G, defined by equation (11.661).

11.19.10

⌣

⌣

̄ Element Unbalanced Force, e G(𝜆0 , d0 ; d)

By substituting equation (11.680) into equation (11.670), we get for each finite element: Unbalanced force

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ e

⌣

⌣

⌣

⌣

⌣

̄ = d q̄ T e R(𝜆0 , d0 ) G(𝜆0 , d0 ; d)

(11.686)

where, we have defined: ⌣

⌣

R(𝜆0 , d0 ) ≡ ⏟⏞⏞⏞⏟⏞⏞⏞⏟ e

6N 2 × 1

⎛ ⎞⎫ ⎛ ⎞ ⎧ √ ⌣ ⌣ ⌣ ⌣ ⌣ ⎜ 0 ⎟⎪ ⎪ 1 1 ⎜ ⌣T t ̂ 0 ̂1 ̂2 ⎟ t̂ T T 1 ̂2 T n̂ ̂ ̂ B E T F + T F(𝜆 , 𝜉 , 𝜉 )⎟ G d𝜉 d𝜉 − ⎜𝜆 F𝜈 (S) dS⎟ ⎬ ⎨∫ ∫ ⎜⏟⏟ ∫ ⏟ ⏟ ⏟ ⏟ ⏟ ⏟ ⏟ ⏟ ⏟ ⏟ ⏟ ⏟ ⏟ ⏟⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏟⎟ ⏟⏟⏟ ⎜ Γ ⎟⎪ ⎪ 0 0 ⎜ 2 6N 2 × 6 6N 2 × 6 ⎝ ⎠⎭ ⎝ 6N × 18 18 × 12 12 × 1 ⎠ ⎩ 6×1 6×1

(11.687)

11.19.11

⌣

Element Incremental Generalized Applied Force, e p(𝜆)

By substituting equation (11.673) into equation (11.669), the applied external force for a step can be expressed as: Incremental applied force

⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ⎫ ⎧ ⌣ ⌣ ⎪ ⎪ t ⌣̄ T t ⌣̂ 0 1 2 t ̄T t ̂ ⎨∫ d F, 𝜆 (𝜆 , S , S ) dA + ∫ d F𝜈 (S) dS⎬ Δ𝜆 ⎪ ⎪Ω Γ ⎭ ⎩

(11.688) ⌣

⌣

= d q̄ T e p(𝜆) Δ𝜆

Nonlinear Shell

929

where, we have defined: √ 1 1 ⌣ ⌣ ̂ 𝜆 (𝜆0 , 𝜉̂1 , 𝜉̂2 ) ] G ̂ d𝜉̂1 d𝜉̂2 p(𝜆0 ) ≡ [TT (𝜉̂1 , 𝜉̂2 ) t F, ⏟⏟⏟ ∫0 ∫0 ⏟⏞⏞⏟⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏟ e⌣

6N 2 × 6

6N 2 × 1

+

⌣

(11.689)

t̂

T

T (S) F𝜈 (S) dS ∫ ⏟⏟⏟ ⏟⏟ ⏟ Γ

11.19.12

6×1

⌣

6N 2 × 6

6×1

Geometrically Exact Element Incremental Equation

Now substituting expressions given in equations (11.682), (11.684), (11.686) and (11.688) in equation (11.639), we have: ⌣

q̄

d T

{e

KcM + e KcG

}

⌣

⌣

⌣

⌣

Δd q̂ = d q̄ T {e p(𝜆) Δ𝜆 − e R(𝜆0 , d0 )}

(11.690)

Noting that this equation is true for any arbitrary admissible generalized virtual control dis⌣ ̄ we can finally write the geometrically exact linearized finite element equation for placement, d q, ⌣ each element in unknown incremental generalized displacement, Δd q, and the incremental load factor, Δ𝜆, as: e Kc T

⇒

⌣

Δd q −

e⌣ p(𝜆0 ) e Kc T

⌣

⌣

⌣

⌣

Δ𝜆 = −e R(𝜆0 , d0 ) ⌣

⌣

(11.691)

Δd q = − e R(𝜆0 , d0 ) + e p(𝜆0 ) Δ𝜆

where, we have introduced the following. Definition

r The computational tangent control stiffness matrix, e

11.19.13

KcT ≡

e

KcM +

e Kc : T e

KcG

(11.692)

Where We Would Like to Go

Here we presented the element-level finite element formulation of the linearized variational or virtual work equations. Considering that the shell geometric stiffness can be viewed as the direct sum of those along the direction of the lines of curvature in the beam formulation, we deliberately skipped the lengthy proofs for establishing both the translational and rotational symmetry of the geometric stiffness; for this, we refer to:

r beam geometric stiffness symmetry. Also, in our derivation, we relied – without detailing the numerical issues – on various topics such as the determination of the lines of curvature, global formulation of actually gathering the

Computation of Nonlinear Structures

930

r

r

element information, and so on. Thus, the next few logical steps should be the introduction to the numerical transformation of the coordinate frame to the curvilinear frame, a global assembly procedure of the finite element method, the numerical integration of the various stiffnesses and residual force expressions, and an iterative solution including the updating options by a Newton-type algorithm to set out the configuration path for a specific loading condition. Thus, for examination of one of the primary algorithms for a Newton-type solution in the nonlinear regime, we may refresh our memory by reviewing: a Newton-type method and the arc-length constraint algorithm. Finally, for shell element library and practical guidelines on all the relevant shell computational details, and with the details for the general iterative solution methodology behind us, we would like to apply these algorithms to various problems of interest, and to evaluate our shell theoretical and computational formulations; for this, we look into: c-type numerical implementation for quasi-static loading and example problems and solutions.

11.20 11.20.1

c-type FE Implementation and Examples: Quasi-static Loading c-type Shell Element Library

The library consists of the tensor product quadrilateral, and triangular elements; each is determined simply by the degree of the underlying polynomial of interpolation or the so-called shape functions. Within each category, both the non-rational Bernstein–Bezier and the rational Bernstein–Bezier interpolation types may be constructed. For general geometrical shapes, non-rational types are applicable; for quadric surfaces such as spheres and ellipsoids, but not paraboloids, we must use the rational counterparts for accuracy of representation; in fact, for large c-type mega-elements, it is absolutely crucial. For example, it is well known that an acute-angled portion of a spherical surface can be quite “closely” approximated geometrically by a bi-cubic net, but then the subsequent finite element analysis becomes erroneous for such elements of mega size. We will have occasion later to discuss it further.

11.20.1.1

Non-rational Tensor-product Quadrilateral Elements

The root element of a typical n-degree non-rational quadrilateral c-type finite element interpolating the shell geometry is a unit square, and the geometric interpolation of the element itself can be written as: C0 (𝜉̂1 , 𝜉̂2 ) =

L L ∑ ∑

BLi (𝜉̂1 )BLj (𝜉̂2 ) c qij

i=0 j=0 t d( ̂ 𝜉̂1 , 𝜉̂2 )

=

M M ∑ ∑

(11.693) ̂1 M ̂2 d BM i (𝜉 )Bj (𝜉 ) qij

i=0 j=0

where C(𝜉̂1 , 𝜉̂2 ) ≡ { X(𝜉̂1 , 𝜉̂2 ) Y(𝜉̂1 , 𝜉̂2 ) Z(𝜉̂1 , 𝜉̂2 ) }T defines the position vector of a point on the shell element, 𝜉̂1 ∈ [0, 1], 𝜉̂2 ∈ [0, 1]; BLi (𝜉̂1 ) and BLj (𝜉̂2 ), the ith and the jth Bernstein– Bezier interpolation polynomials of degree L, respectively, with the control points c qij for i, j ∈ {0, 1, … , L}; the tensor product BLi (𝜉̂1 )BLj (𝜉̂2 ) is the ijth non-rational basis functions, and the control points c qij for i, j ∈ {0, 1, … , L} form the Bernstein–Bezier net of control polygons. For

Nonlinear Shell

931

t d( ̂ 𝜉̂1 , 𝜉̂2 ),

M ̂2 th ̂1 the generalized displacement and rotation interpolations, BM i (𝜉 ) and Bj (𝜉 ), the i and jth Bernstein–Bezier interpolation polynomials of degree M, respectively, with the generalized ̂1 M ̂2 displacement control points d qij for i, j ∈ {0, 1, … , M}; the tensor product BM i (𝜉 )Bj (𝜉 ) is the th d ij non-rational basis functions, and the control points qij for i, j ∈ {0, 1, … , M} form the Bernstein–Bezier net of generalized displacement control polygons.

Remark

r We must have L ≤ M with the equality for isoparametry, and the inequality for subparametry; superparametry, that is, L > M is computationally not recommended.

11.20.1.2

Rational Quadrilateral Elements

For mesh generation, the root element of an n-degree typical rational quadrilateral c-type geometric element interpolating the shell geometry is a unit square, and the geometric interpolation of the element itself can be written as:

C0 (𝜉̂1 , 𝜉̂2 ) =

L ∑ L ∑

Fij (𝜉̂1 , 𝜉̂2 ) c qij

i=0 j=0

Fij (𝜉̂1 , 𝜉̂2 ) ≡

wij BLi (𝜉̂1 )BLj (𝜉̂2 ) L ∑ L ∑

(11.694)

wij BLi (𝜉̂1 )BLj (𝜉̂2 )

i=0 j=0

where C(𝜉̂1 , 𝜉̂2 ) ≡ { X(𝜉̂1 , 𝜉̂2 ) Y(𝜉̂1 , 𝜉̂2 ) Z(𝜉̂1 , 𝜉̂2 ) }T , as before, defines the position vector of a point on the shell element, 𝜉̂1 ∈ [0, 1], 𝜉̂2 ∈ [0, 1]; Fij (𝜉̂1 , 𝜉̂2 ) for i, j ∈ {0, … , L} is the ijth rational basis functions; wij for i, j ∈ {0, … , L} are the weights at the control points. Remarks

r Note that while the rational surface elements are obtained as the projections of the tensor product

r

of root elements, they are themselves not tensor product elements as the basis functions, Fij (𝜉̂1 , 𝜉̂2 ), cannot, in general, be expressed as the tensor product of the coordinate basis functions because of the presence of the denominator in the second expression of equation (11.694). For generalized displacement interpolation, we still choose the non-rational tensor products as given in the equation (11.693) because, for unknown displacement distribution, we would like to avoid choosing ad hoc estimates for the corresponding required weights.

11.20.1.3

Non-rational Triangular Elements

The root element of a typical n-degree non-rational triangular c-type finite element interpolating the shell geometry can be an equilateral triangle, and the geometric interpolation of the element

Computation of Nonlinear Structures

932

itself can be written as: C0 (r, s, t) =

∑

c BM ijk (r, s, t) qijk

i+j+k=M t d(r, s, t)

=

∑

(11.695)

d BM ijk (r, s, t) qijk

i+j+k=N

where, BM (r, s, t) and BN (r, s, t) are the bi-variate Bernstein polynomial of degree M and N, ijk ijk respectively, with N ≥ M, given by: ( BXijk (r, s, t) = where,

X ijk

i + j + k = X,

) ri sj tk ,

X = M, N (

r + s + t = 1,

and,

X

)

ijk

(11.696) X! = i ! j ! k!

Note that because of the constraint: r + s + t = 1, the Bernstein–Bezier polynomials effectively depend only on two variables. Remark

r In what follows, we will restrict our discussion to non-rational tensor product elements, that is, the quadrilateral finite elements; similar considerations for the triangular finite elements can be developed without much difficulty on the basis of the detailed discussion of the triangular geometric elements in the mesh generation phase of patches and surfaces.

11.20.2

Bernstein–Bezier Geometric Element

For a typical non-rational quadrilateral finite element, then, a typical n-degree c-type shell geometry can be written in tensor-product form as: C0 (𝜉̂1 , 𝜉̂2 ) = ⏟⏞⏞⏟⏞⏞⏟ 3×1

nc nc ∑ ∑

n n Bi c (𝜉̂1 ) Bj c (𝜉̂2 )

i=0 j=0

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ 3× 3Nc2

c

qij , ⏟⏟⏟

(11.697)

3Nc2 ×1

c c where C0 (𝜉̂1 , 𝜉̂2 ) ≡ { X(𝜉̂1 , 𝜉̂2 ) Y(𝜉̂1 , 𝜉̂2 ) Z(𝜉̂1 , 𝜉̂2 ) }T , Bni (𝜉̂1 ) Bnj (𝜉̂2 ), c qij and nc are the geometric position vector, the Bernstein–Bezier basis functions, the geometry control vectors and the degree of the geometry element, respectively, at any point (𝜉̂1 , 𝜉̂2 ). The equation (11.697) can be rewritten in transformation form as:

̂ 𝜉̂1 , 𝜉̂2 ) c q̂ , C0 (𝜉̂1 , 𝜉̂2 ) = c T( ⏟⏞⏞⏟⏞⏞⏟ ⏟⏞⏞⏟⏞⏞⏟ ⏟⏟⏟ 3×1

3× 3Nc2

3Nc2 ×1

(11.698)

Nonlinear Shell

933

where we have: q̂ = ⏟⏟⏟ c

3Nc2 ×1

{

qX00 ⋯ qXn 0 qX01 ⋯ qXn 1 qX0n ⋯ qXn n qY00 ⋯ qYn0 qY01 ⋯ qYn 1 qY0n ⋯ qYn n c c c c c c c c c ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ 1×Nc2

qZ00 ⋯ qZn 0 qZ01 ⋯ qZn 1 qZ0n ⋯ qZn n c c c c c ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

1×Nc2

}T

1×Nc2

(11.699) ̂ 𝜉̂1 , 𝜉̂2 ) as the (3 × 3N 2 ) geometry interpolation the (3Nc2 × 1) geometry control vector, and c T( c transformation matrix containing the Bernstein–Bezier polynomials, that is, c T( ̂ 𝜉̂1 ,

𝜉̂2 ) =

⎤ ⎡ Bnc Bnc Bnc Bnc Bnnc Bnnc 0⋯ ⋯ ⋯0 1 0 c c ⎥ ⎢ 0 0 n n n n n n ⎥ ⎢ 0⋯ ⋯0 B0c B0c B1c B0c Bncc Bncc ⎥ ⎢ nc nc nc nc nc nc ⎥ ⎢ 0⋯ ⋯ 0 B B B B B B nc nc ⎦ ⎣ 0 0 1 0 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ 3x3N 2

(11.700) Note that Nc ≡ nc + 1 is the order of the geometric element with degree nc . Remarks

r It is recommended whenever possible to have the geometry representation at most cubic, that

r

is, nc ≤ 3 for ease and efficiency in the computation of tangents, curvatures and twists that are subsequently necessary. So, as an example, for a cubic Bernstein–Bezier geometry net, that is, nc = 3, results in the order of the geometry as: N = 4. Thus, we will have 48(i.e. three sets of 4 × 4 geometry control nets) will be the dimension of the geometry control vector, c q, and the interpolation transformation matrix is of dimension 3 × 16. Additionally, whenever conic geometric surfaces (i.e. spheres, ellipsoids, etc.) are involved, the geometric representation must be of rational Bernstein–Bezier form for accuracy.

Example: Let us consider a bilinear element, that is, nc = 1, N = Nc = 2. We have:

C0 (𝜉̂1 , 𝜉̂2 ) = ⏟⏞⏞⏟⏞⏞⏟ 3×1

1 1 ∑ ∑

B1i (𝜉̂1 ) B1j (𝜉̂2 )

i=0 j=0

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ 3× 12

c

qij , ⏟⏟⏟ 12×1

(11.701)

Computation of Nonlinear Structures

934

with B10 (𝜉̂k ) ≡ 1 − 𝜉̂k ,

k = 1, 2 (11.702)

B11 (𝜉̂k ) ≡ 𝜉̂k implying q̂ = ⏟⏟⏟ c

{

qX00 qX10 qX01 qX11 qY00 qY10 qY01 qY11 qZ00 qZ10 qZ01 qZ11 ⏟⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏟

12×1

1×4

1×4

}T (11.703)

1×4

and 0 0 ⎤ ⎡ T ⎢ ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟ ⎥ 1×4 1×4 ⎥ ⎢ 1×4 ⎢ 0 T 0 ⎥ c T( ̂ 𝜉̂1 , 𝜉̂2 ) = ⎢ ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟ ⎥, ⎢ 1×4 1×4 1×4 ⎥ ⎥ ⎢ ⎢ 0 0 T ⎥ ⎢ ⏟⏟⏟ ⏟⏟⏟ ⏟⏟⏟ ⎥ ⎣ 1×4 1×4 ⎦ 1×4 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ T≡

11.20.3

{

(11.704)

3x12

(1 − 𝜉̂1 )(1 − 𝜉̂2 )

𝜉̂1 (1 − 𝜉̂2 )

(1 − 𝜉̂1 )𝜉̂2

𝜉̂1 𝜉̂2

}

Bernstein–Bezier Generalized Displacement Element

For a typical quadrilateral finite element, a typical n-degree c-type shell geometry can be written in tensor-product form as: ̂ 𝜉̂1 , 𝜉̂2 ) = d( ⏟⏞⏟⏞⏟ 6×1

n n ∑ ∑

Bni (𝜉̂1 ) Bnj (𝜉̂2 )

i=0 j=0

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

d

qij ⏟⏟⏟

(11.705)

6N 2 ×1

6× 6N 2

̂ Bn (𝜉̂1 ) Bn (𝜉̂2 ), d qij and n are the generalized (i.e. displacement and rotation) displacewhere, d, i j ment vector, the Bernstein–Bezier basis functions, the generalized displacement control vectors and the degree of the finite element, respectively, at any point (𝜉̂1 , 𝜉̂2 ). The equation (11.705) can be rewritten in transformation form as: ̂ 𝜉̂1 , 𝜉̂2 ) = d T( ̂ 𝜉̂1 , 𝜉̂2 ) d q̂ d( ⏟⏞⏟⏞⏟ ⏟⏞⏞⏟⏞⏞⏟ ⏟⏟⏟ 6×1

6× 6N 2

6N 2 ×1

(11.706)

Nonlinear Shell

935

q̂ = ⏟⏟⏟ d

6N 2 ×1 qw ⋯ qw 00 n0

{

qu00 ⋯ qun0 qu01 ⋯ qun1 qu0n ⋯ qunn qv00 ⋯ qvn0 qv01 ⋯ qvn1 qv0n ⋯ qvnn ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

qw ⋯ qw 01 n1

1×N 2 𝜃1 𝜃1 w q0n ⋯ qw nn q00 ⋯ qn0

𝜃

𝜃

𝜃

1×NM 𝜃

q011 ⋯ qn11 q0n1 ⋯ qnn1 where we have: ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ 1×N 2 𝜃

𝜃

𝜃

1×N 2 𝜃

𝜃

𝜃

𝜃

𝜃

𝜃

𝜃

𝜃

𝜃

q002 ⋯ qn02 q012 ⋯ qn12 q0n2 ⋯ qnn2 q003 ⋯ qn03 q013 ⋯ qn13 q0n3 ⋯ qnn3 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ 1×N 2

as }T

1×N 2

the (6N 2 × 1) generalized displacement control vector. Let d q̂̄ be the (6N 2 × 1) generalized ̂ be the (6N 2 × 1) generalized incremental control virtual displacement control vector, and d Δq, d d 1 2 ̂ ̂ ̂ vector, similar to q. Then, let T(𝜉 , 𝜉 ) be the (6 × 6N 2 ) interpolation transformation matrix containing the Bernstein–Bezier polynomials, that is, d T( ̂ 𝜉̂1 ,

𝜉̂2 ) =

0 0 ⋯ ⋯ ⋯ 0 ⎤ ⎡ B0 B0 B1 B0 ⋯ Bn Bn−1 Bn Bn ⎢ ⎥ 0 0 Bn0 Bn0 Bn1 Bn0 ⋯ Bnn Bnn 0 ⋯ 0 ⎥ ⎢ 0 ⎢ ⎥ ⋯ ⋯ ⎢ ⎥ ⎢ ⎥ ⋯ ⋯ ⎢ ⎥ ⎢ ⎥ n n n n n n ⎣ 0 0 ⋯ ⋯ 0 B0 B0 B1 B0 ⋯ Bn Bn ⎦ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ n n

n n

n n

n n

6x6N 2

(11.707) ̂ ̂ ̂1 ̂2 The derivatives involved in 𝜉 B( 𝜉 , 𝜉 ), the (18 x 6N 2 ) augmented interpolation transformation matrix:

⎡ dT ⎤ ⎢ ⏟⏟⏟ ⎥ ⎢ 6× 6N 2 ⎥ ⎢ d ⎥ ⎢ T, 𝜉̂1 ⎥ 𝜉̂ ̂ ̂1 ̂2 B(𝜉 , 𝜉 ) ≡ ⎢ ⏟⏟⏟ ⎥ ⏟⏞⏞⏟⏞⏞⏟ ⎢ 6× 6N 2 ⎥ ⎢ ⎥ 18 × 6N 2 ⎢ d T, ̂2 ⎥ ⎢ ⏟⏟𝜉⏟ ⎥ ⎢ ⎥ ⎣ 6× 6N 2 ⎦

(11.708)

where d T, 𝜉̂1 and d T, 𝜉̂2 are computed by the first derivative recursive formula of the Bernstein basis functions at Gauss points as: d n−1 ̂𝛼 ̂𝛼 B(𝜉̂𝛼 ) = n (Bn−1 i−1 (𝜉 ) − Bi (𝜉 )), d𝜉̂𝛼

𝛼 = 1, 2

(11.709)

√ ̂ d𝜉̂1 d𝜉̂2 , of the area integrals where ̂ = G Finally, we must express the infinitesimal area, d A √ ̂ is the positive square root of the determinant of the surface metric expressed in terms of the G geometry coordinate parameters, 𝜉̂𝛼 , 𝛼 = 1, 2.

Computation of Nonlinear Structures

936

Remarks

r Note that both the d T( ̂ 𝜉̂1 , 𝜉̂2 ) and 𝜉̂ B( ̂ 𝜉̂1 , 𝜉̂2 ) matrices for all Gauss points can be computed and r r

r

saved once and stored away for repeated use in the Newton-type iterative algorithm; of course, in adaptive refinements the locations of the Gauss points change requiring recomputation. For convergence, we must have the elements at most isoparametric, that is, nc = n; we must always avoid superparametry, that is, nc must never allowed to exceed n. For computational efficiency such as tangent and curvature computations, a subparametric representation, that is, nc strictly less than n, with nc = 3, that is, up to cubic spline representation for the non-rational geometry, seems an optimal choice. Similarly, for rational geometry such as spheres, ellipsoids, and so on, the quadratic rational spline is the desired geometric representation. It turns out that for shell finite element analysis, the optimum choice for the generalized displacement distribution must be with a quintic Bernstein–Bezier net, that is, n = 5, resulting in the order of the element as: N = 6. Thus, we will have 216(i.e. six sets of 6 × 6 control nets) will be the dimension of the generalized displacement control vector, d q, and the interpolation transformation matrix is of dimension 6 × 36.

11.20.4

Gauss Quadrature Points and the Lines of Curvature

Now for a quintic Bernstein–Bezier net, for example, over an element that is, with n = 5, we have for the order of the element as: N = 6. Thus, for full integration, we will have a set of 6 × 6 Gauss quadrature points. With the locations of the Gauss quadrature points fixed, our goal is to determine the lines of curvature directions through these points and other desired geometric properties. But, in order to accomplish this, we need to construct a local frame at these points with the triad of base ̂0 𝜕C ̂𝛽 ≡ 𝛽 = 1, 2, and the unit vectors: two tangent vectors along the coordinate iso-curves: G 𝜕 𝜉̂𝛽 ̂2 ̂1× G G ̂ ≡G ̂3 = ; these are generally non-orthogonal to normal to these base vectors, N ≡ N ̂1× G ̂ 2‖ ‖G each other. So, by taking derivatives of equation (11.697) with respect to the surface parameters, we get: ̂ 1 (𝜉̂1 , 𝜉̂2 ) ≡ G ⏟⏞⏞⏟⏞⏞⏟

c ∑ 𝜕C0 (𝜉̂1 , 𝜉̂2 ) == nc 𝜕 𝜉̂1 j=0

n

3×1 nc ∑ 𝜕C0 (𝜉̂1 , 𝜉̂2 ) 1 ̂2 ̂ ̂ G2 (𝜉 , 𝜉 ) ≡ == nc 𝜕 𝜉̂2 ⏟⏞⏞⏟⏞⏞⏟ i=0

{ n −1 c ∑

} 1,0 c

(Δ

n −1 qij )Bi c (𝜉̂1 )

Bj c (𝜉̂2 ). n

i=0

{ n −1 c ∑

} n −1 (Δ0,1 c qij )Bj c (𝜉̂2 )

(11.710) n Bi c (𝜉̂1 )

j=0

3×1

where we have introduced the following first-order forward difference operators, Δ1.0 and Δ0.1 as: Δ1,0 c qij ≡ c q(i+1)j − c qij Δ0,1 c qij ≡ c qi(j+1) − c qij

(11.711)

Nonlinear Shell

937

Note that the great advantage of using difference operators is that we only need to know and use the differences of original Bezier control points of the geometry. The normal vector is, then, easily obtained as a cross product of the tangent vectors given by equations (11.710) and (11.711); however, note that the normals at four corners are easily obtained as the partials that may be expressed as the difference of boundary points, as for example:

̂ 0) = N(0,

Δ1,0 c q00 × Δ0,1 c q00

(11.712)

‖Δ1,0 c q00 × Δ0,1 c q00 ‖

Now, for the determination of the lines of curvature direction, we will additionally need the curvatures at Gauss points, that is, the second derivatives of the position vectors, or equivalently, the first derivatives of the base vectors. This, in turn, will need us to introduce the second-order forward differences. So, by taking second derivatives of equation (11.697) with respect to the surface parameters, we get:

C0 (𝜉̂1 , 𝜉̂2 ), 𝜉̂1 𝜉̂1 == C0 (𝜉̂1 , 𝜉̂2 ), 𝜉̂2 𝜉̂2 C0 (𝜉̂1 , 𝜉̂2 ), 𝜉̂1 𝜉̂2

n! ∑ (n − 2)! j=0 nc

{ n −2 c ∑

} 2,0 c

(Δ

n −2 qij )Bi c (𝜉̂1 )

n Bj c (𝜉̂2 ).

i=0

{ −2 c ∑ n∑

} n! n −2 n == (Δ0,2 c qij )Bj c (𝜉̂2 ) Bi c (𝜉̂1 ) (n − 2)! i=0 j=0 { } nc −1 nc −1 ∑ ∑ nc −1 ̂2 n −1 1,1 c 2 == n (Δ qij )Bj (𝜉 ) Bi c (𝜉̂1 ) nc

i=0

(11.713)

j=0

where we have introduced the second-order forward difference operators as: Δ2,0 c qij ≡ Δ1,0 c q(i+1)j − Δ1,0 c qij Δ0,2 c qij ≡ Δ0,1 c qi(j+1) − Δ0,1 c qij Δ1,1 c qij

≡

Δ0,1 c q(i+1)j

−

(11.714)

Δ0,1 c qij

Again, note that the great advantage of using difference operators is that we only need to know and use the differences of the original Bezier control points of the geometry. Let us recall that ̂ 𝛼 = 0 for 𝛼 = 1, 2, we because the normal vector has unit length, and that N ∙ C0 , 𝜉̂𝛼 = N ∙ G have, by differentiation: C0 , 𝜉̂𝛼 ∙ N, 𝜉̂𝛽 = −C0 , 𝜉̂𝛼 𝜉̂𝛽 ∙ N for 𝛼, 𝛽 = 1, 2. Now, from our discussions d𝜉̂2 on differential geometry of surfaces, with 𝜆 ≡ = tan 𝛼1 , the curvature is given as a function d𝜉̂1 of 𝜆 by:

k(𝜆) ≡

L + 2M𝜆 + N𝜆2 E + 2F𝜆 + G𝜆2

(11.715)

Computation of Nonlinear Structures

938

where we have introduced the following definitions: ̂ 11 = C, ̂1 ∙ C, ̂1 E≡G 𝜉 𝜉 ̂ 12 = C, ̂1 ∙ C, ̂2 F≡G 𝜉 𝜉 ̂ 22 = C, ̂2 ∙ C, ̂2 G≡G 𝜉 𝜉 L ≡ −C, 𝜉̂1 ∙ N, 𝜉̂1 = C, 𝜉̂1 𝜉̂1 ∙ N ≡ B̂ 11

(11.716)

1 M ≡ − (C, 𝜉̂1 ∙ N, 𝜉̂2 + C, 𝜉̂2 ∙ N, 𝜉̂1 ) = C, 𝜉̂1 𝜉̂2 ∙ N ≡ B̂ 12 2 N ≡ −C, 𝜉̂2 ∙ N, 𝜉̂2 = C, 𝜉̂2 𝜉̂2 ∙ N ≡ B̂ 22 Note that B̂ 𝛼𝛽 for 𝛼, 𝛽 = 1, 2 are the components of the surface curvature tensor. Moreover, also recall that if C0 is an umbilical point, that is, L : M : N = E : F : G, then the curvature is 1 1 independent of 𝜆; otherwise, we can find the extreme values as the desired curvatures, and R1 R2 at the roots 𝜆1 , 𝜆2 (where these roots define the directions of lines of curvature) from vanishing of the following determinant: ⎡ 𝜆2 ⎢ det ⎢ E ⎢ ⎣L

−𝜆 F M

1⎤ ⎥ G⎥ = 0 ⎥ N⎦

(11.717)

With some manipulation, we can write the expressions of the roots as: 𝜆1 or 2 =

The curvatures,

−(NE − GL) ±

√

(NE − GL)2 − 4(FN − GM)(ME − FL) 2(FN − GM)

(11.718)

1 1 and , themselves are obtained as roots of: R1 R2 ⎡ 1E −L ⎢ R det ⎢ ⎢ 1F −M ⎣R

1 F −M⎤ ⎥ R ⎥=0 1 G − N ⎥⎦ R

(11.719)

resulting in: √ −b ± b2 − 4ac 1 1 or = , R1 R2 2a a ≡ 2FN − GM b ≡ 2FM − NE − GL c ≡ ME − FL

(11.720)

Nonlinear Shell

939

Note that if F ≡ 0 and M ≡ 0 (spherical surfaces, for example), the coordinate iso-curves are B̂ B̂ 1 1 N L already along the lines of curvature, then, 𝜆 = 0, and we get: or = , = 22 , 11 . ̂ 22 G ̂ 11 R1 R2 G E G Also, we can√ compute the determinant of the surface metric tensor necessary for element surface √ ̂ = EG − F 2 . Now, the unit base vectors (i.e. with arc length parameterization) integrals as: G along the lines of curvature can be easily related by a rotational matrix to the coordinate base vectors as: ]{ } { } [ ̂1 cos 𝛼1 sin 𝛼1 G1 G = (11.721) ̂2 G2 − sin 𝛼1 cos 𝛼1 G where 𝛼1 is the angle that the 𝜉 1 - line of curvature makes with the 𝜉̂1 coordinate curve. Finally, using equation (11.721), we can easily establish the derivative relationship with respect to the arc length parameters and the coordinate parameters as: ⎧ 𝜕 ⎪ 𝜕S1 ⎨ ⎪ 𝜕 ⎩ 𝜕S2

⎫ ⎡ 1 cos 𝛼1 ⎪ ⎢ 𝛾1 ⎬=⎢ ⎪ ⎢ − 1 sin 𝛼1 ⎭ ⎣ 𝛾2

⎧ 𝜕 1 sin 𝛼1 ⎤ ⎪ 1 𝛾1 ⎥ ⎪ 𝜕 𝜉̂ ⎥⎨ 𝜕 1 cos 𝛼1 ⎥⎦ ⎪ 𝛾2 ⎪ 𝜕 𝜉̂2 ⎩

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

(11.722)

̄ 𝛼 (𝜉 𝛼 )‖, where 𝛾𝛼 ≡ ‖G Now, note that edge that: B̂ 2𝛽 cos 𝛼1 .

𝜕𝛾𝛼 𝜕 𝜉̂𝛽

𝛼 = 1, 2 are the lengths of the base vectors along the lines of curvature. 𝜕𝛾 equation (11.722) can be used to compute 𝛼𝛽 for 𝛼, 𝛽 = 1, 2 with the knowl𝜕S G𝛼 ∙ G𝛼 , 𝜉̂𝛽 = , G1 , 𝜉̂𝛽 = B̂ 1𝛽 cos 𝛼1 + B̂ 2𝛽 sin 𝛼1 , and G2 , 𝜉̂𝛽 = −B̂ 1𝛽 sin 𝛼1 + 𝛾𝛼

Exercise: Show that

G𝛼 ∙ G𝛼 , 𝜉̂𝛽 𝜕𝛾𝛼 = . 𝛾𝛼 𝜕 𝜉̂𝛽

Solution: With 𝛾𝛼2 ≡ ‖G𝛼 ‖2 = G𝛼 ∙ G𝛼 , taking derivative: ̸ 2𝛾𝛼 the proof.

𝜕G𝛼 𝜕𝛾𝛼 ≠ 2G𝛼 ∙ completes 𝜕 𝜉̂𝛽 𝜕 𝜉̂𝛽

With the above background, we can express all the components of both the desired curvature { { }T }T 1 1 1 1 − 𝛾1 , 2 0 𝛾2 , 1 and k02 ≡ in terms of the coordivectors: k01 ≡ 0 − R1 𝛾1 R2 𝛾2 √ √ ̂ ≡ G ̂ 11 G ̂ 22 − G ̂ 2 , the positive square root of the nate parameters, 𝜉̂𝛼 , 𝛼 = 1, 2. Finally, G 12

determinant of the surface metric is evaluated at all the Gauss quadrature points of integration. Remark

r Thus, note that the reference surface properties – the initial curvature vectors, k0 and k0 along with the square root of the determinant

1 2 √ ̂ of the surface metric – can be computed at all Gauss G

Computation of Nonlinear Structures

940

points and saved once and stored away for repeated use in the Newton-type iterative algorithm; of course, in adaptive refinement (i.e. ch −, cp − and ck − type refinements), the locations of the Gauss points change requiring recomputation.

11.20.5

Algorithmic Code Snippets for Geometric Properties Subroutine Calc_ReferenceProps(Qc,ri,rj,jCInt)

! ! ! ! ! !

=========================================== n m-1 A1hat = m * sum { sum (Dif10*Qij*Bri(ri)) } Brj(rj) j=0 i=0

=========================================== Implicit Real(kind=8) (a-h,o-z) ================ include include include include

!

'SizeVar.h' 'files.h' 'Scratch.h' 'CurParams.h'

================ Real(kind=8) Qc Dimension Real(kind=8) Dimension Real(kind=8) Dimension Real(kind=8) Dimension

X11hat A1 A1c1

Qc(nDim,nQc1,nQc2) ,X12hat ,X22hat X11hat(nDim),X12hat(nDim),X22hat(nDim) ,A2 ,Atem ,Btem A1(nDim),A2(nDim),Atem(nDim),Btem(nDim) ,A1c2 ,A2c1 ,A2c2 A1c1(nDim),A1c2(nDim),A2c1(nDim),A2c2(nDim)

! DATA zero/0.D0/,one/1.0D0/,two/2.0D0/,three/3.0D0/ !=================================================== ! ri = 0.D0 ! rj = 0.D0 ! ----------------------------------------------------------metric Tensor nQcm = MAX(nQc1-1,nQc2-1) ! call Calc_A1_Hat(Qc,nDim,nQc1,nQc2,nQcm,ri,rj,A1hat) ! call VectorNorm(A1hat,A,3,Iout) ! call ScaleVector(A1hat,uA1hat,1.D0/A,3,Iout) ! call Calc_A2_Hat(Qc,nDim,nQc1,nQc2,nQcm,ri,rj,A2hat) ! call VectorNorm(A2hat,A,3,Iout) ! call ScaleVector(A2hat,uA2hat,1.D0/A,3,Iout) ! ---------------------------------------------------- unit normal call CrossProduct(A1hat,A2hat,xNor,Iout) call VectorNorm(xNor,A,3,Iout) call ScaleVector(xNor,xNor,1.D0/A,3,Iout) ! ---------------------------------------------------- RootAhat call DotProduct(A1hat,A1hat,E,3,Iout) call DotProduct(A1hat,A2hat,F,3,Iout) call DotProduct(A2hat,A2hat,G,3,Iout) !

Nonlinear Shell

!

941

RutAhat = DSQRT(E*G - F*F) ---------------------------------------------------- curvature Tensor X11hat = 0.D0 X12hat = 0.D0 X22hat = 0.D0

! if(nQc1.gt.2) then ! atleast Quadratic call Calc_X11_Hat(Qc,nDim,nQc1,nQc2,nQcm,ri,rj,X11hat) endif if(nQc2.gt.2) then ! atleast Quadratic call Calc_X22_Hat(Qc,nDim,nQc1,nQc2,nQcm,ri,rj,X22hat) endif if(nQc1.gt.2.AND.nQc2.gt.2) then ! atleast both Quadratic call Calc_X12_Hat(Qc,nDim,nQc1,nQc2,nQcm,ri,rj,X12hat) endif !

!

call DotProduct(X11hat,xNor,xL,3,Iout) ! B11hat call DotProduct(X12hat,xNor,xM,3,Iout) ! B12hat call DotProduct(X22hat,xNor,xN,3,Iout) ! B22hat ---------------------------------------------------- Lines of curvatures FEG =F/E/G if (dabs(FEG).le.1.0D-3.or.xM==0.d0) then ! already lines of curvatures Rho1 = zero ! direction Rho2 = zero ! direction Cu1 = xL/E ! 1/R1 Cu2 = xN/G ! 1/R2

! ThetaC sT cT

= zero

ENmGL ENpGL FNmGM EMmFL SQ FNGM2

= = = =

= zero = one

else E*xN E*xN F*xN E*xM

+ -

G*xL G*xL G*xM F*xL = DSQRT(ENmGL*ENmGL - 4.D0*FNmGM*EMmFL) = 2.D0*FNmGM

! Rho1 Rho2

= (-ENmGL + SQ)/FNGM2 = (-ENmGL - SQ)/FNGM2

! EGmFsq = E*G - F*F TEGmFsq = 2.D0*EGmFsq xLNmMsq = xL*xN - xM*xM TFMmENpGL= 2.D0*F*xM - ENpGL SQ = DSQRT(TFMmENpGL*TFMmENpGL - 4.D0*EGmFsq*xLNmMsq) Cu1 = (-TFMmENpGL + SQ)/TEGmFsq Cu2 = (-TFMmENpGL - SQ)/TEGmFsq ! ThetaC sT cT endif

= DATAN(Rho1) = DSIN(ThetaC) = DCOS(ThetaC)

Computation of Nonlinear Structures

942

!

------------ Bases: Ao1 & Ao2 & Ao12N Coord Xformation call ScaleVector(A1hat,aTem,cT,3,Iout) call ScaleVector(A2hat,bTem,sT,3,Iout) A1 = aTem + bTem

!

!

call ScaleVector(A1hat,aTem,-sT,3,Iout) call ScaleVector(A2hat,bTem,cT,3,Iout) A2 = aTem + bTem ----------------------------------------------- Lamda1 & Lamda2 call VectorNorm(A1,xLam1,3,Iout) call ScaleVector(A1,Ao1,1.D0/xLam1,3,Iout) call VectorNorm(A2,xLam2,3,Iout) call ScaleVector(A2,Ao2,1.D0/xLam2,3,Iout)

!

!

Ao12N(1,1) = Ao1(1) Ao12N(2,1) = Ao1(2) Ao12N(3,1) = Ao1(3) ----------------------------------------------- Xformation: Global Cartesian -> Lines of Curvs. Ao12N(1,2) = Ao2(1) Ao12N(2,2) = Ao2(2) Ao12N(3,2) = Ao2(3)

! Ao12N(1,2) = Ao2(1) Ao12N(2,2) = Ao2(2) Ao12N(3,2) = Ao2(3) !

!

Ao12N(1,3) = xNor(1) Ao12N(2,3) = xNor(2) Ao12N(3,3) = xNor(3) ----------------------------------------------- (Lamda_alfa,beta) call ScaleVector(X11hat,aTem,cT,3,Iout) call ScaleVector(X12hat,bTem,sT,3,Iout) A1c1 = aTem + bTem ! A1,1 call DotProduct(A1,A1c1,xLam11hat,3,Iout) ! Lamda1,1hat xLam11hat = xLam11hat/xLam1

! call ScaleVector(X12hat,aTem,cT,3,Iout) call ScaleVector(X22hat,bTem,sT,3,Iout) A1c2 = aTem + bTem call DotProduct(A1,A1c2,xLam12hat,3,Iout) xLam12hat = xLam12hat/xLam1

! A1,2 ! Lamda1,2hat

! call ScaleVector(X11hat,aTem,-sT,3,Iout) call ScaleVector(X12hat,bTem, cT,3,Iout) A2c1 = aTem + bTem call DotProduct(A2,A2c1,xLam21hat,3,Iout) xLam21hat = xLam21hat/xLam2

! A2,1 ! Lamda2,1hat

! call call A2c2 call

ScaleVector(X12hat,aTem,-sT,3,Iout) ScaleVector(X22hat,bTem, cT,3,Iout) = aTem + bTem DotProduct(A2,A2c2,xLam22hat,3,Iout)

! A1,2 ! Lamda2,2hat

Nonlinear Shell

xLam22hat = xLam22hat/xLam2 ! !

! !

!

xLam11 = (cT*xLam11hat + sT*xLam12hat)/xLam1 ! lamda1,1 xLam21 = ( cT*xLam21hat + sT*xLam22hat)/xLam1 ! lamda2,1 xLam12 = (-sT*xLam11hat + cT*xLam12hat)/xLam2 ! lamda1,2 xLam22 = (-sT*xLam12hat + cT*xLam22hat)/xLam2 ! lamda2,2 -----------------------------------------------Curvatures: ko1 & ko2 C01(1) = zero C01(2) = +Cu1 C01(2) = Cu1 C01(3) = +xLam12/xLam1

!

! !

!

C02(1) = -Cu2 C02(2) = zero C02(3) = -xLam21/xLam2 ------------------------------------------------------- Curvatures: ko1 & ko2 CORRECT ONE C01(1) = zero C01(2) = -Cu1 C01(2) = Cu1 C01(3) = -xLam12/xLam1

! C02(1) = Cu2 C02(2) = zero C02(3) = xLam21/xLam2 !

========================================== save in tape call PutGetRefPropsatGaussPt(iRefPro1,jCInt,1)

!

========================================== save in tape end return end

Subroutine Calc_X11_Hat(Qc,nDim,nQc1,nQc2,nQcm,ri,rj,X11hat) ! actual degree m = nQc1 -1 ! actual degree n = nQc2 -1 ! Because indices run from 1, ... nQc1 instead of i = 0, ... .m ! Because indices run from 1, ... nQc2 instead of j = 0, ... .n ! so, range 0 to m-2 becomes 1 to nQc1-2, etc. ! ======================================================= ! n m-2 ! X11hat = m*(m-1) * sum { sum (Dif20*Qij*Bri(ri)) } Brj(rj) ! j=0 i=0 ! ======================================================= Implicit Real(kind=8) (a-h,o-z) Real(kind=8) X11hat ,Dif20 Dimension X11hat(nDim),Dif20(nDim) Real(kind=8) Qc ,Bri ,Brj Dimension Qc(nDim,nQc1,nQc2),Bri(nQc1),Brj(nQc2) ! nQc1m1 = nQc1-1 !actual degree = m nQc1m2 = nQc1-2 !actual degree - 1 = m-1 ! --------------------------------------------------Basis Functions call Bernstein(Bri,nQc1m2,ri,iOut) call Bernstein(Brj,nQc2,rj,iOut)

943

Computation of Nonlinear Structures

944

!

------------------------------X11hat = 0.D0 do 30 j= 1,nQc2 do 10 i= 1,nQc1m2

===================

!

Dif20 = (Type 2)

call DifferenceOp_2(Qc,nDim,nQc1,nQc2,i,j, Dif20,2,iOut)

=================== do 20 k= 1,nDim

!

20 10 30

X11hat(k) = X11hat(k) + Dif20(k)*Bri(i)*Brj(j) continue continue continue

! X11hat = nQc1m1 * nQc1m2 * X11hat !

========== return end Subroutine DifferenceOp_1(Qc,nDim,nQc1,nQc2,i1,j2,Diff,iTyp,iOut)

! !

====================================================== Implicit Real(kind=8) (a-h,o-z) =============== real(kind = 8) Qc Dimension real(kind = 8) Diff Dimension integer

Qc(nDim,nQc1,nQc2) Diff(nDim) nQc1,nQc2,iOut

! Diff = 0.D0 if(iTyp == 1) then ! --------------------------- Dif10 = Q(i+1,j) – Q(i,j) (Type 1) do 30 k= 1,nDim Diff(k) = Qc(k,i1+1,j2) - Qc(k,i1,j2) 30 continue ! --------------------------- Dif01 = Q(i,j+1) – Q(i,j) (Type 2) elseif (iTyp == 2) then do 40 k= 1,nDim Diff(k) = Qc(k,i1,j2+1) - Qc(k,i1,j2) 40 continue endif !--------------------------return end Subroutine DifferenceOp_2(Qc,nDim,nQc1,nQc2,i1,j2, Dif2,iTyp,iOut) ! !

====================================================== Implicit Real(kind=8) (a-h,o-z) ==================== real(kind = 8) Dimension real(kind = 8) Dimension integer

Qc Diff

Qc(nDim,nQc1,nQc2) ,Dif2 Diff(nDim),Dif2(nDim) nQc1,nQc2,iOut

Nonlinear Shell

945

!-----------------------------------Dif2 = 0.D0 ! if(iTyp == 1) then ! --------------------- Dif11 = Dif10(i,j+1) – Dif10(i,j) (Type 1) ! -----------or: Dif11 = Dif01(i+1,j) – Dif01(i,j) (Type 2) SELECTED do 20 k= 1,nDim ! =================== call DifferenceOp_1(Qc,nDim,nQc1,nQc2,i1+1,j2, Diff,2,iOut) Dif2(k) = Diff(k) call DifferenceOp_1(Qc,nDim,nQc1,nQc2,i1,j2, Diff,2,iOut) Dif2(k) = Dif2(k) - Diff(k) 20 continue elseif (iTyp == 2) then ! --------------------- Dif20 = Dif10(i+1,j) – Dif10(i,j) (Type 1) do 40 k= 1,nDim ! =================== call DifferenceOp_1(Qc,nDim,nQc1,nQc2,i1+1,j2, Diff,1,iOut) Dif2(k) = Diff(k) call DifferenceOp_1(Qc,nDim,nQc1,nQc2,i1,j2, Diff,1,iOut) Dif2(k) = Dif2(k) - Diff(k) 40 continue elseif (iTyp == 3) then ! --------------------- Dif02 = Dif01(i,j+1) – Dif01(i,j) (Type 2) do 60 k= 1,nDim call DifferenceOp_1(Qc,nDim,nQc1,nQc2,i1,j2+1, Diff,2,iOut) Dif2(k) = Diff(k) call DifferenceOp_1(Qc,nDim,nQc1,nQc2,i1,j2, Diff,2,iOut) Dif2(k) = Dif2(k) - Diff(k) 60 continue endif !--------------------------------------return end Subroutine Calc_ReferenceProps_Hem(Qc,tmin,tmax,pmin,pmax, ri,rj,jCInt) !============================================================== !HARDCODED for LINES OF CURVATURES ALONG COORD.DIRS. 1 & 2 ! For the Hemisphere with Hole Problem in the sequel !============================================================== ! n m-1 ! A1hat = m * sum { sum (Dif10*Qij*Bri(ri)) } Brj(rj) ! j=0 i=0 ! ====================================================== Implicit Real(kind=8) (a-h,o-z) ! ==================== include 'SizeVar.h' include 'files.h' include 'Scratch.h' include 'CurParams.h' ==================== ! Real(kind=8) RutAhatAlt Real(kind=8) Qc

Computation of Nonlinear Structures

946

Dimension Real(kind=8) Dimension Real(kind=8) Dimension Real(kind=8) Dimension

X11hat A1 A1c1

Qc(nDim,nQc1,nQc2) ,X12hat ,X22hat X11hat(nDim),X12hat(nDim),X22hat(nDim) ,A2 ,Atem ,Btem A1(nDim),A2(nDim),Atem(nDim),Btem(nDim) ,A1c2 ,A2c1 ,A2c2 A1c1(nDim),A1c2(nDim),A2c1(nDim),A2c2(nDim)

! DATA zero/0.D0/,one/1.0D0/,two/2.0D0/,three/3.0D0/ !================================================== rad = 10.D0 ! HARDCODED for radius = 10 the = tmax - tmin phe = pmax - pmin ! ---------------------------------------Coords along Lines of Curvatures cT = one ! needed in A1hat routine etc sT = zero ! ------------------------------------------ metric Tensor tri = tmin + (tmax - tmin)*ri prj = pmin + (pmax - pmin)*rj ! A1hat(1) = -rad*dcos(prj)*dsin(tri)*the A1hat(2) = rad*dcos(prj)*dcos(tri)*the A1hat(3) = zero ! A2hat(1) = -rad*dsin(prj)*dcos(tri)*phe A2hat(2) = -rad*dsin(prj)*dsin(tri)*phe A2hat(3) = rad*dcos(prj)*phe ! -------------------------------------------unit normal call CrossProduct(A1hat,A2hat,xNor,Iout) call VectorNorm(xNor,A,3,Iout) call ScaleVector(xNor,xNor,1.D0/A,3,Iout) ! -------------------------------------------RootAhat call DotProduct(A1hat,A1hat,E,3,Iout) ! A11hat call DotProduct(A1hat,A2hat,F,3,Iout) ! A12hat call DotProduct(A2hat,A2hat,G,3,Iout) ! A22hat ! RutAhat = DSQRT(E*G - F*F) RutAhatAlt = rad*rad*dcos(prj)*the*phe ! -------------------------------Bases: Ao1 & Ao2 & Ao12N Coord Xformation A1 = A1hat A2 = A2hat ! -------------------------------------------Lamda1 & Lamda2 call VectorNorm(A1,xLam1,3,Iout) call ScaleVector(A1,Ao1,1.D0/xLam1,3,Iout) call VectorNorm(A2,xLam2,3,Iout) call ScaleVector(A2,Ao2,1.D0/xLam2,3,Iout) ! Ao12N(1,1) = Ao1(1) Ao12N(2,1) = Ao1(2) Ao12N(3,1) = Ao1(3) ! --------------------------Xformation: Global Cartesian -> Lines of Curvs. Ao12N(1,2) = Ao2(1) Ao12N(2,2) = Ao2(2)

Nonlinear Shell

947

Ao12N(3,2) = Ao2(3) ! Ao12N(1,2) = Ao2(1) Ao12N(2,2) = Ao2(2) Ao12N(3,2) = Ao2(3) !

!

Ao12N(1,3) = xNor(1) Ao12N(2,3) = xNor(2) Ao12N(3,3) = xNor(3) ------------------------------------------- (Lamda, alfa, beta) X11hat(1) = -rad*dcos(prj)*dcos(tri)*the*the X11hat(2) = -rad*dcos(prj)*dsin(tri)*the*the X11hat(3) = zero

! X22hat(1) = -rad*dcos(prj)*dcos(tri)*phe*phe X22hat(2) = -rad*dcos(prj)*dsin(tri)*phe*phe X22hat(3) = -rad*dsin(prj)*phe*phe !

!

X12hat(1) = rad*dsin(prj)*dsin(tri)*the*phe X12hat(2) = -rad*dsin(prj)*dcos(tri)*the*phe X12hat(3) = zero ============================ Surface Curvatures Tensor call DotProduct(xNor,X11hat,B11,3,Iout) ! B11 call DotProduct(xNor,X12hat,B12,3,Iout) ! B12 ! SHOULD BE ZERO call DotProduct(xNor,X12hat,B21,3,Iout) ! B21 ! SHOULD BE ZERO call DotProduct(xNor,X22hat,B22,3,Iout) ! B22

! Cu1 = B11/(xLam1*xLam1) Cu2 = B22/(xLam2*xLam2) !

! B11/A11 ! B22/A22

====================================== Curvature End call DotProduct(A1,X11hat,xLam11hat,3,Iout) xLam11hat = xLam11hat/xLam1 call DotProduct(A1,X12hat,xLam12hat,3,Iout) xLam12hat = xLam12hat/xLam1 call DotProduct(A2,X12hat,xLam21hat,3,Iout) xLam21hat = xLam21hat/xLam2 call DotProduct(A2,X22hat,xLam22hat,3,Iout) xLam22hat = xLam22hat/xLam2

! Lamda1,1hat w.r.t zi ! ! Lamda1,2hat ! ! Lamda2,1hat ! Lamda2,2hat

!

!

xLam21 = xLam21hat/xLam1 ! lamda2,1 w.r.t s1 xLam12 = xLam12hat/xLam2 ! lamda1,2 w.r.t s2 -------------------------------------------Curvatures: ko1 & ko2 C01(1) = zero C01(2) = Cu1 C01(3) = xLam12/xLam1

!

!

C02(1) = -Cu2 C02(2) = zero C02(3) = -xLam21/xLam2 ------------------------------Curvatures: ko1 & ko2 CORRECT ONE C01(1) = zero C01(2) = -Cu1 C01(3) = -xLam12/xLam1

Computation of Nonlinear Structures

948

! C02(1) = Cu2 C02(2) = zero C02(3) = xLam21/xLam2 !

============================================== save in tape

!

=========================================== save in tape end

call PutGetRefPropsatGaussPt(iRefPro1,jCInt,1) return end

11.20.6

Stiffness Matrices and Residual Forces

As indicated earlier, the implementation for the shell analysis flows directly along the line of our beam analysis implementation (see Chapter 10 on nonlinear beams). In this sense, the shells can be viewed as beams along the lines of curvature coupled by the constitutive relations and other conditions. Thus, both the instantaneous material stiffness matrix and the geometric stiffness matrix are formed in an additive way with appropriate substitution of various fundamental vectors: k0𝛼 , a𝛼 , X𝛼 and so on, for directions given by indices 𝛼 = 1, 2. Moreover, generally for a c-type finite element, the internal Bezier control points can be easily eliminated by static condensation through partial Gauss elimination algorithm.

11.20.6.1

&

Algorithmic Code Snippets for Material and Geometric Stiffness Matrices

Subroutine Tensor_STIF(RoCOld,CuCOld,RoSav,CKSav,Qd, Nel,DMat,B,T,Stif,BFors,StifK,nStif,Iout,Trace)

! ! Notes: Nint should be atleast 2 ! ! Inputs ! Qd(nStif) = Latest displacement controls: state vector ! Nel = Element Number ! NelType = Element Type: ! Nint = Number of Gauss Points ! E = Elasticity Modulus ! PR = Poisson's ratio ! Thk = Thickness ! nStif = Row or Col size of Stiffness matrix ! = nDof*nQd1*nQd2 ! Iout = Error Output file# ! ! Outputs ! Stif(nStif,nStif) = Desired Stiffness Matrix ! bFors(nStif) = Desired Body force Vector Integrals ! StifK(nStif) = Desired kLamda Stiffness last column Vector Integrals ! Trace = trace of the matrix ! Implicit Real(kind=8) (a-h,o-z) ! ===========================

Nonlinear Shell

include include include include include include include include

949

'Material.h' 'CurParams.h' 'SizeVar.h' 'LoadStepIter.h' 'Scratch.h' 'ShellParams.h' 'LogParams.h' 'GaussIndex.h'

!============================================ ! ! Input Variables & Arrays ! Real(kind=8) Qd Dimension Qd(nStif) Real(kind=8) RoSav ,CKSav Dimension RoSav(nCInt),CKSav(nCInt) Real(kind=8) RoCOld ,CuCOld Dimension RoCOld(nCInt),CuCOld(nCInt) Real(kind=8) DMat Dimension DMat(nCMat,nCMat) ! ! Working Variables & Arrays ! ! Real(kind=8) DD ,RR ,DP1 ,Rp1 ,Dp2 ,Rp2 ! Dimension DD(3),RR(3),DP1(3),Rp1(3),Dp2(3),Rp2(3) ! ! Output Variables & Arrays ! Real(kind=8) B ,T Dimension B(nDof2,nStif),T(nDof,nStif) Real(kind=8) BFors ,StifK ,Stif ,Trace Dimension BFors(nStif),StifK(nStif),Stif(nStif,nStif) Real(kind=8) EDG ,DB Dimension EDG(nDof2,nDof2),DB(nDof2) Real(kind=8) Add ,HD ,Rp Dimension Add(nStif),HD(nDof2),Rp(3) Real(kind=8) BVEC ,SKLM ,R0 ,RK Dimension BVEC(nStif),SKLM(nStif),R0(3),RK(3) Real(kind=8) AM,BM,CM,TH,Ri,Rj,Si,Wt,sum LOGICAL bReset DATA zero/0.D0/,one/1.0D0/,onm/-1.0D0/ DATA NC/3/ !================================================= ! Stiffness Matrix (Tangent Operator) Calculation ! BFors = 0.D0 !all StifK = 0.D0 !all Stif = 0.D0 ! all elements = 0. !================================================= ! Rewind CurvatureInfo, B & T Scratch files if(Nel == 1) then nRec = 0 rewind iRefPro1

950

Computation of Nonlinear Structures

rewind iRefPro2 endif ! Loop over Gauss Locations do 60 iGau = 1,Nint1 do 60 jGau = 1,Nint2 ! ------------------------------------------ gauss positions ri = GaussPtVal(iGau,Nint1,1) rj = GaussPtVal(jGau,Nint2,1) !-----------------------------------——– Get Geometric Curvature Vector etc. nRec = nRec + 1 T = zero B = zero call Get ReferenceInfoataGaussPt & (T,B,nDof,nDof2,nStif,Nel,nRec) !---------------------------------------------------- DisRots & Derivatives call Split_Dis_Rot_Deriv(Qd,nStif,B,Iout) ! ----------------------------------------------------- Update Rotation ! do 10 i = 1,nC ! 10 RK(i) = R0(i) + RR(i) ============================== ED+G(nDof2,nDof2) call Form_EtranDEplusGMatrix(DMat,T,B,BVEC,SKLM,EDG,Iout) !=========================================== save ! if(iNut.LE.1) Then ! do 10 ii = 1,nC ! ij = ii + (jGau -1)*nC ! RoCOld(ij) = RK(ii) ! CuCold(ij) = CK(ii) ! 10 continue ! nRec = (Nel-1)*Nint + jGau ! if(iRotCur == iRotCur1) Then ! call PutGetRotCur(iRotCur2,RoCOld,CuCold,nCInt,nRec,1) ! else ! call PutGetRotCur(iRotCur1,RoCOld,CuCold,nCInt,nRec,1) ! endif ! endif !---------------------------------------------------------- Form: Btran_ED+G_B ! Add to stiffness matrix Wt = GaussPtVal(iGau,Nint1,2)*GaussPtVal(jGau,Nint2,2)*RutAhat do 50 j = 1,nStif do 20 k = 1,nDof2 ! DB(k) = 0.0 do 20 l = 1,nDof2 ! DB(k) = DB(k) + EDG(k,l)*B(l,j) 20 continue do 40 i = j,nStif sum = 0.0 do 30 l = 1,nDof2 ! sum = sum + B(l,i)*DB(l) 30 continue Stif(i,j) = Stif(i,j) + sum*Wt 40 continue 50 continue !----------------------------------------- Load Vector: Residual & Body force part

Nonlinear Shell

do 53 i = 1,nStif BFors(i) = BFors(i) + Wt*BVec(i) 53 continue !------------------------ Form: Last Column of stiffness: KLamda Body Force part do 55 i = 1,nStif StifK(i) = StifK(i) + Wt*SKLM(i) 55 continue 60 continue ! symmetry: Upper Triangle do 70 j = 1,nStif do 70 i = j,nStif Stif(j,i) = Stif(i,j) 70 continue ! Get Trace Trace = 0.D0 do 80 i = 1,nStif Trace = Trace + Stif(i,i) 80 continue !======================================================= return end

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

Subroutine FormEMatrix_1(EMat,nE,nG,nC,iDir,Iout) ---------------------------------------------------------– Inputs AMat(nC,nC) = Rotation Matrix nG = 18 C0i(nC) = initial Curvature vector = k0i for iCurDir i nC = 3 (= nDim) nE = 12 W(nC,nC) = W Matrix R1(nC,nC) = R1 Matrix R2(nC,nC) = R2 Matrix AAi(nC) = e + [k0i]d CKi(nC) = k0i + ki = k0i + W(thetaPrimed) Outputs EMat(nE,nG) = EMat Matrix (12,18) --------------------------------------------Implicit Real(kind=8) (a-h,o-z)

!==================================== include 'Material.h' include 'LoadStepIter.h' include 'CurParams.h' include 'ThetaConstants.h' include 'BeamParams.h' include 'ShellParams.h' !============================================== Real(kind=8) EMat Dimension EMat(nE,nG) E11 ,E12 Real(kind=8) Dimension E11(nC,nC),E12(nC,nC) Real(kind=8) E13 ,E22 ,E23 ,E24

951

Computation of Nonlinear Structures

952

Dimension Real(kind=8) Dimension

CoA

E13(nC,nC),E22(nC,nC),E23(nC,nC),E24(nC,nC) ,CoB CoA(3,3),CoB(3,3)

! DATA zero/0.D0/,one/1.0D0/,two/2.0D0/,three/3.0D0/,onem/-1.0D0/ !---------------------------------------------------- Get EBlocks for dir 1 ! ====================================== call FormEBlocks(nC,E11,E12,E13,E22,E24,1,Iout) !--------------------------------------------------------------- Form EMatrix EMat = zero ! -------do 10 i = 1,3 do 10 j = 1,3 EMat(i ,j ) = E11(i,j) ! [C0] EMat(i ,j+3) = E12(i,j) ! [a1]W EMat(i ,j+6) = E13(i,j) ! I EMat(i+3,j+3) = E22(i,j) ! R+[Cn]W EMat(i+3,j+9) = E24(i,j) ! W 10 continue !---------------------------------------------------------return End Subroutine FormEMatrix_2(EMat,nE,nG,nC,iDir,Iout) ! ---------------------------------------------------------------! Inputs ! AMat(nC,nC) = Rotation Matrix ! nG = 18 ! C0i(nC) = initial Curvature vector = k0i for iCurDir i ! nC = 3 (= nDim) ! nE = 12 ! W(nC,nC) = W Matrix ! R1(nC,nC) = R1 Matrix ! R2(nC,nC) = R2 Matrix ! AAi(nC) = e + [k0i]d ! CKi(nC) = k0i + ki = k0i + W(thetaPrimed) ! ! Outputs ! EMat(nE,nG) = EMat Matrix (12,18) ! -----------------------------------------------------------Implicit Real(kind=8) (a-h,o-z) !====================================== include 'Material.h' include 'LoadStepIter.h' include 'CurParams.h' include 'ThetaConstants.h' include 'BeamParams.h' include 'ShellParams.h' !========================================= Real(kind=8) EMat Dimension EMat(nE,nG) Real(kind=8) E31 ,E32 Dimension E31(nC,nC),E32(nC,nC)

Nonlinear Shell

Real(kind=8) Dimension Real(kind=8) Dimension

953

E35 CoA

,E42 ,E23 ,E46 E35(nC,nC),E42(nC,nC),E23(nC,nC),E46(nC,nC) ,CoB CoA(3,3),CoB(3,3)

! DATA zero/0.D0/,one/1.0D0/,two/2.0D0/,three/3.0D0/,onem/-1.0D0/ !---------------------------------------------- Get EBlocks for dir 2 E31 = zero E32 = zero E35 = zero E42 = zero E46 = zero ! ========================================= call FormEBlocks(nC,E31,E32,E35,E42,E46,2,Iout) ! ========================================= !--------------------------------------------------------------- Form EMatrix do 40 i = 1,3 do 40 j = 1,3 EMat(i+6 ,j ) = E31(i,j) EMat(i+6 ,j+3 ) = E32(i,j) EMat(i+6 ,j+12) = E35(i,j) EMat(i+9 ,j+3 ) = E42(i,j) EMat(i+9 ,j+15) = E46(i,j) 40 continue !--------------------------------------------------return End Subroutine FormEBlocks(nC,E11,E12,E13,E22,E24,iCurDir,Iout) ! -------------------------------------------------------------------------! Inputs ! C0i(nC) = initial Curvature vector for dir iCurDir ! nC = 3 ! AMat = Rotation Matrix ! W = matrix ! R = matrix ! AA = a Vector for dir iCurDir ! CK = Kc Vector for dir iCurDir ! ! Outputs ! E11(nC,nC) = E11 block ! E12(nC,nC) = E12 block ! E13(nC,nC) = E13 block ! E22(nC,nC) = E22 block ! E24(nC,nC) = E24 block ! ---------------------------------------------------Implicit Real(kind=8) (a-h,o-z) ! include 'ThetaConstants.h' include 'BeamParams.h' =============================== ! Real(kind=8) E11 ,E12 ,E13 ,E22 Dimension E11(nC,nC),E12(nC,nC),E13(nC,nC),E22(nC,nC)

Computation of Nonlinear Structures

954

Real(kind=8) Dimension Real(kind=8) Dimension

TAMat W

TAMat(nC,nC) ,R ,E24 ,sAA ,sKc W(nC,nC),R(nC,nC),E24(nC,nC),sAA(nC,nC),sKc(nC,nC)

! DATA zero/0.D0/,one/1.0D0/,two/2.0D0/,three/3.0D0/ ------------------------------------------------------------------W = WMat R = RMat !---------------------------------------------------------------AAMatTran TAMat = TRANSPOSE(AMat) !----------------------------------------------------------------------E11Block call FormSkew(C0,nC,E11,Iout) !skew[k0] E11 = MATMUL(TAMat, E11) !AMatTransposeE11 !----------------------------------------------------------------------E12Block call FormSkew(AA,nC,sAA,Iout) !skew[a] E12 = MATMUL(sAA, W) ![a]W E12 = MATMUL(TAMat, E12) !AMatTransposeE12 !----------------------------------------------------------------------E13Block E13 = TAMat !AMatTransposeE13 !----------------------------------------------------------------------E22Block call FormSkew(CK,nC,sKc,Iout) !skew[Kc] E22 = R + MATMUL(sKc,W) !R + [Kc]W E22 = MATMUL(TAMat, E22) !AMatTransposeE22 !----------------------------------------------------------------------E24Block E24 = W E24 = MATMUL(TAMat, E24) !AMatTransposeE24 !----------------------------------------return end !

Subroutine FormGMatrix(G,nG,nC,nDof,Iout) ! --------------------------------------------------------------------! Inputs ! HD(nG) = {dis,rot,disPrimed_1,rotPrimed_1,disPrimed_2,rotPrimed_2} ! nG = 18 ! FM(nC) = {force1,moment1,force2,moment2} ! nF = 12 ! C01(nC) = initial Curvature vector = k01 ! C02(nC) = initial Curvature vector = k02 ! nC = 3 (= nDim) ! ! Outputs ! G22(nC,nC) = GThetaBarDelThetaBlock ! ------------------------------------------------------------Implicit Real(kind=8) (a-h,o-z) !====================================== include 'LoadStepIter.h' include 'CurParams.h' include 'ThetaConstants.h' include 'BeamParams.h' include 'ShellParams.h' !=======================================

Nonlinear Shell

Real(kind=8) Dimension Real(kind=8) Dimension !

955

G G(nG,nG) G12 ,G22 ,G23 ,G24 G12(3,3),G22(3,3),G23(3,3),G24(3,3)

============================================ DATA zero/0.D0/,one/1.0D0/,two/2.0D0/,three/3.0D0/

!

===========

G = zero !-------------------------------------------------------------- Form GBLocks_Dir_1 G12 = zero G22 = zero G23 = zero G24 = zero ! --------------C0 = C01 Ro = RR Rp = Rp1 WMat = WMat1 RMat = RMat1 CK = CK1 AA = AA1 FMt = FMt1 ! --------------------------------------call FormGBlocks(nDof,nC,G12,G22,G23,G24,1,Iout) !----------------------------------------------------- Fill GMatrix for Dir_1 ! ---------------------------------- Upper Triangle with Diagonals do 10 i = 1,3 do 10 j = 1,3 G(i,j+3) = G12(i,j) G(i+3,j+3) = G22(i,j) G(i+3,j+6) = G23(i,j) G(i+3,j+9) = G24(i,j) 10 continue !------------------------------------------------------ Form GBLocks_Dir_2 G12 = zero G22 = zero G23 = zero G24 = zero ! ------------C0 = C02 Ro = RR Rp = Rp2 WMat = WMat2 RMat = RMat2 CK = CK2 AA = AA2 FMt = FMt2 ! ------------------------------------call FormGBlocks(nDof,nC,G12,G22,G23,G24,2,Iout) !--------------------------------------------------------- Fill GMatrix for Dir_2 ! --------------------------- Upper Triangle with Diagonals do 40 i = 1,3 do 40 j = 1,3

Computation of Nonlinear Structures

956

G(i ,j+ 3) = G(i ,j+ 3) + G12(i,j) G(i+3,j+ 3) = G(i+3,j+ 3) + G22(i,j) G(i+3,j+12) = G(i+3,j+12) + G23(i,j) G(i+3,j+15) = G(i+3,j+15) + G24(i,j) 40 continue !-------------------------------------------- Symmetry:Lower Triangle do 50 i = 1,17 do 50 j = i+1,18 G(j,i) = G(i,j) 50 continue !-----------------------------------------return End

! ! ! ! ! ! ! ! ! !

Subroutine FormGBlocks(nF,nC,G12,G22,G23,G24,iCurDir,Iout) -----------------------------------------------------Inputs FM(nC) = {forcei,momenti} for dir iCurDir nF = 6 C0(nC) = initial Curvature vector for dir iCurDir nC = 3 Outputs G22(nC,nC) = GThetaBarDelThetaBlock ---------------------------------------------------Implicit Real(kind=8) (a-h,o-z)

! include 'ThetaConstants.h' include 'BeamParams.h' !

=============================== Real(kind=8) G12 ,G22 ,G23 Dimension Real(kind=8) Dimension Real(kind=8) Dimension Real(kind=8) Dimension Real(kind=8) Dimension Real(kind=8) Dimension

Fo BB W FA WMR

,G24 G12(nC,nC),G22(nC,nC),G23(nC,nC),G24(nC,nC) ,XM Fo(nC),XM(nC) ,CC BB(nC),CC(nC) ,R ,SM ,SF ,SC W(nC,nC),R(nC,nC),SM(nC,nC),SF(nC,nC),SC(nC,nC) ,XMK ,TA ,ATA FA(nC,nC),XMK(nC,nC),TA(nC,nC),ATA(nC,nC) ,TMT ,SMt WMR(nC,nC),TMT(nC,nC),SMt(nC,nC)

! DATA nV/3/ DATA zero/0.D0/,one/1.0D0/,two/2.0D0/,three/3.0D0/ ! -----------------------------------------------------W = WMat R = RMat ! ======== ! istart = (iCurdir-1)*6 ! ! call VectorExtract_Vector(HD,nG,Ro,nC,4,Iout) !rotation ! call VectorExtract_Vector(HD,nG,Rp,nC,istart+10,Iout)!rotationPrimed

Nonlinear Shell

957

call VectorExtract_Vector(FMt,nF,Fo,nC,1,Iout)!Force call VectorExtract_Vector(FMt,nF,XM,nC,4,Iout) !Moment !---------------------------------------------------------------G22Block ! ---------------------------------------------------- Translational ! ----------------------------------------- FA = WTran(Fxa + axF -2(F.a)I)W call FormF_a_PartMatrix(Fo,AA,FA,3,1,Iout) !Fxa + axF -2(F.a)I ! ! call Form_WMatrix(Ro,W,nC,c1,c2,c1b,c2b,Iout) FA = MATMUL(FA, W) !(Fxa + axF -2(F.a)I)W FA = MATMUL(TRANSPOSE(W), FA) !WTran(Fxa + axF -2(F.a)I)W ! call CrossProduct(Fo,AA,BB,Iout) ! b = F x a call DotProduct(Ro,BB,dot,nV,Iout) ! theta dot b C2TdotB = c2*dot ! c2*(theta dot b) ! ---------------------------------------.5*FA + c2(theta dot b)I FA = 0.5D0*FA FA(1,1) = FA(1,1) + C2TdotB FA(2,2) = FA(2,2) + C2TdotB FA(3,3) = FA(3,3) + C2TdotB ! ------------------------------------------------------- Rotational ! ---------------------------------- XMK = WTran(Mxkc + kcxM -2(M.kc)I)W call FormF_a_PartMatrix(XM,CK,XMK,3,2,Iout) !Mxkc + kcxM -2(M.kc)I ! XMK = MATMUL(XMK, W) !(Mxkc + kcxM -2(M.kc)I)W !------------------------------------WTran(Mxkc + kcxM -2(M.kc)I)W XMK = MATMUL(TRANSPOSE(W), XMK) call CrossProduct(XM,CK,CC,Iout) ! c = M x kc call DotProduct(Ro,CC,dot,nV,Iout) ! theta dot c C2TdotC = c2*dot ! c2*(theta dot c) ! --------------------------------------------.5*XMK + c2(theta dot c)I XMK = 0.5D0*XMK XMK(1,1) = XMK(1,1) + C2TdotC XMK(2,2) = XMK(2,2) + C2TdotC XMK(3,3) = XMK(3,3) + C2TdotC ! --------------------------------------------------AlfaCxRot + RotxAlfaC call FormThetaAlfaTrans_Matrix(Ro,CC,TA,nC,c1,c2,c1b,c2b,Iout) !RotxAlfa ATA = 0.5D0*(TA + TRANSPOSE(TA)) ! --------------------------------------------------AlfaBxRot + RotxAlfaB call FormThetaAlfaTrans_Matrix(Ro,BB,TA,nC,c1,c2,c1b,c2b,Iout) !RotxAlfa ATA = ATA + 0.5D0*(TA + TRANSPOSE(TA)) ! ---------------------------------------- WTranMR+RTranMTranW call FormSkew(XM,nC,SM,Iout) !skew[M] ! call FormR_Matrix(Ro,Rp,R,nC,c1,c2,c1b,c2b,Iout)!R matrix WMR = MATMUL(TRANSPOSE(W),SM) !WTran[M] WMR = MATMUL(WMR,R) !WTran[M]R WMR = WMR + TRANSPOSE(WMR) ! --------------------------------------------- G22Block G22 = FA + XMK + ATA + WMR !----------------------------------------------------G12Block call FormSkew(Fo,nC,SF,Iout) !skew[F] call FormSkew(C0,nC,SC,Iout) !skew[k0] ! G12 = MATMUL(SC,SF) ![k0][Ft]

Computation of Nonlinear Structures

958

G12 = MATMUL(G12,W) ![k0][Ft]W !----------------------------------------------------- G23Block G23 = MATMUL(TRANSPOSE(W),SF) !WTran[Ft] !------------------------------------------------------G24Block TMT = MATMUL(TRANSPOSE(W),SM)!WTran[Mt] TMT = MATMUL(TMT,W) !WTran[Mt]W call FormS_Matrix(Ro,XM,SMt,nC,c1,c2,c1b,c2b,Iout) !S(Mt) Matrix ! G24 = TMT + SMt !----------------------------------------------------return end

11.20.7

Incremental Equation Solution and Updates

Now, let us recall the geometrically exact linearized finite element equation in unknown incremental generalized displacement, Δq̂ d , and the incremental load factor, Δ𝜆, as: ̂ 0 , d̂ 0 ) ̂ KcT Δq̂ d + p(𝜆) Δ𝜆 = −R(𝜆

(11.723)

It may be recalled that the linearized equation (11.723) has been formulated to be solved by a Newton-type iterative solution procedure for the configuration path evaluation. The configuration path is sought for a set of discrete load level parameters: 𝜆 = 𝜆0 , 𝜆1 , 𝜆2 , ⋯. Generally, a nonlinear configuration path may consist of both snap-through and snap-back buckling phenomena. A most suitable procedure for capturing these unstable situations is provided by embedding the arc length constraint algorithm in a Newton-type incremental/iterative method. At each step of an iterative solution procedure, both d, the displacement vector, and R, the rotation tensor, need to be updated for use in the next step of iteration, and for the determination of the shell configuration path, (d, R), of interest. For the displacement vector, being a member of a vector space, the updating is additively straightforward as given by: d(i+1) = d(i) + Δd(i)

(11.724)

where Δd(i) , d(i) , d(i+1) are the incremental displacement vector obtained as a solution of the linearized incremental equation (11.723), the known starting displacement vector at iteration, i, and the updated displacement vector, respectively. An additive update for the rotation tensor, being a member of a manifold, that is, a curved space (SO(3), the special orthogonal group), is not allowed. However, we can locally parameterize the rotation tensor by θ, the Rodrigues rotation vector that belongs to the tangent spaces; a tangent space being a vector space, we can have an additive updating scheme for the rotation vector, without leaving the space, similar to that of the displacement update of the reference surface as: θ(i+1) = θ(i) + Δθ(i)

(11.725)

where Δθ(i) , θ(i) , θ(i+1) are the incremental rotation vector obtained as a solution of the linearized incremental equation (11.723), the known starting rotation vector at iteration, i, and the updated rotation vector, respectively. For most structural problems of interest, the additive updating

Nonlinear Shell

959

procedure suffices, resulting in extremely accurate approximations close to the exact solution for moderately large rotations. Remark

r Recalling our section on the rotation tensor, the updating scheme given by equation (11.725) will not work for ‖𝜃‖ = (2n + 1)𝜋, singular at those θ’s.

11.20.8

n = 0, 1, ⋯, as the Rodrigues rotation tensor becomes

Singularity-free Update for the Configuration

The update for the translational configuration, that is, the displacement of the shell reference surface, is done as shown earlier by equation (11.724). Here, we are mainly concerned with a singularity-free rotational configuration update. For details on the main ideas about quaternion, and so on, and the numerical algorithm, we refer to Section 10.20 on the numerical implementation for beams.

11.20.8.1

Update for the Rotation Configuration

With the above conversions from the rotation matrix to the quaternion and back, in place, let Δθ(i) be the incremental Rodrigues rotation vector. We will have in storage the current quaternion, Q(i) , for the iteration step i; we can retrieve and convert it to the rotation matrix, R(i) . Then, we can generate the quaternion, ΔQ(i) , corresponding to Δθ(i) by replacing θ by Δθ(i) , and similarly, the norm; we can construct the incremental rotation matrix, ΔR(i) from the quaternion ΔQ(i) . Now, we get the updated rotation matrix, R(i+1) , as R(i+1) = ΔR(i) R(i) . Finally, by application of the algorithm for the conversion of a rotation matrix to the corresponding quaternion, we convert R(i+1) to Q(i+1) and store it in the database.

11.20.9

Newton-type Iteration and Arc Length Constraint

Recalling Section 8.5 on Newton-type iteration with arc length constraint, a method can be devised to solve problems and perform stability or buckling analysis with quadratic convergence. For proportional loading, the coefficient of proportionality for the loading, 𝜆, is gradually incremented through a Newton-type iterative method to trace the configuration path of the system. However, a straightforward application of the method fails near, at or beyond the stability limit point. For details of the main ideas about Newton-type method and arc length constraint, and so on, and the numerical algorithm, see Section 10.20 on numerical implementation for beams. Recall from Section 8.5 on Newton-type iteration with arc length constraint that there are generally two fundamentally different scenarios

11.20.10

Numerical Examples

Now that we have some implementation details, we would like to apply these to various problems of interest (that have become industry benchmarks), examine them and interpret the accuracy, robustness and numerical efficiency of the obtained results and compare them with the published solutions by other methods, whenever available. It is important to note that in all of the following

960

Computation of Nonlinear Structures

Figure 11.22 Example 1: Flat plate curling problem.

examples full Gauss quadrature points (see Section 10.20 on numerical implementation for beams) were selected with an absolute minimum number of mega c-type shell elements to show that locking is a non-issue and that there is no need for any ad hoc so-called “under integration” and/or “reduced integration”; this was only possible by c-type finite elements that are backed by a balanced exact theoretical formulation and an associated numerical implementation with Bernstein–Bezier polynomials as the basis functions. Example 1: A 2D straight thin plate coiling to circle A straight cantilevered plate, reported in the literature of unit cross-sectional depth is subjected to a moment loading at the free end. The plate is modeled as a single degenerate 3D quintic in X-Y plane with Z-directional constraints. The theoretical solution for a beam results in a complete 2𝜋EI EI , where the tip moment, M = , with L, as the length of a circular plate of radius, r = M L beam. The elasticity modulus, E, and the shear modulus, G, are 106 and 0.5 × 106 , respectively; the length is taken as L = 10.0, the depth and width as 1.0 with Poisson’s ratio, 𝜇 = 0. Finally, the problem is solved subparametrically, with the geometry modeled linearly and the displacement modeled by a c-type quintic shell element. The deformed shapes for various tip moment loading are shown in Figure 11.22. Note that compared to the reported analyses that have used eight elements, a single mega quintic element suffices, and thus points towards the huge reduction in subsequent quality assurance efforts in practical endeavors.

Example 2: Geometrically exact nonlinear curved plate as deep arch To exemplify unification of geometry modeling and finite element analysis in both formulation and subsequent numerical solution, a geometrically exact nonlinear deep arch problem with shear deformation and extensibility is considered here. The deformed shapes at various load levels are shown in Figures 11.23(a) and 11.23(b) for I where, I, is the moment of inertia, A two Huddleston compressibility parameters, c = A(2bd) represents cross-sectional area, and for b, d. One model, as in Figure 11.23(a), is analyzed with c = 2.55x10−6 , that is, the flexibility is so low that it renders the arch relatively incompressible. The other, as shown in Figure 11.23(b), has c = .01, that is, it has very high I, and thus, making the arch substantially compressible. It may be noted that it is solved with only one c-type quintic element to accurately model the problem, while it takes 40 Q-type shell elements of Palazotto and Dennis (1992), to seriously underestimate the limit point and track the wrong equilibrium path.

Nonlinear Shell

961

(a)

(b)

Figure 11.23 (a) Example 2: Nonlinear deep shell with Huddleston parameter, c = 0.01. (b) Example 2: Nonlinear deep shell with Huddleston parameter, c = 2.55 × 10−6 .

The Q-type degenerate shell element, with Poisson’s ratio vanishing, of Palazotto and Dennis is characterized by Reddy-type (Palazotto and Dennis, 1992) quadratic transverse shear (Palazotto and Dennis, 1992) and quadratic in-plane displacement approximations resulting in 36 degrees of freedom. For flexible bodies, that is, beams, plates and shells, undergoing extreme geometric nonlinearity through large rotations and displacement, it seems unwise to compromise theoretical formulation prematurely for numerical manipulations early in the process.

962

Computation of Nonlinear Structures

Figure 11.24 Example 3: A cantilevered plate under end torque.

Example 3: A cantilevered plate under end torque In order to demonstrate the versatility of the c-type formulation for large rotation and displacement, a flat cantilevered plate is subjected to end torque; the elasticity modulus, E, is 12 × 106 and the Poisson’s ratio, 𝜇 = 0.3 with the length taken as L = 1.0, and the depth and width as 0.1 and 0.25, respectively. Figure 11.24 shows the deformed shape of the plate at 90◦ twist; it may be noted that it took only one quintic c-type plate element to solve the problem.

Example 4: A Shallow cylindrical shell snap-through Here we analyze the snap-through phenomenon of a shallow cylindrical shell. The material and geometric properties are shown in Figure 11.25(a); only a quadrant modeled by a single c-type quintic shell element with appropriate boundary conditions is used for the analysis. Two cases R with different thicknesses are studied with = 200, 400. In contrast with Figure 11.25(b) for h R R the relatively thicker case with = 200, it is seen from Figure 11.25(c) that for larger = 400, h h that is, for the thinner shell, a “kick-in” phenomenon is present as the sides deform more than the center; it is captured quite accurately as the deformed shapes for increasing load parameter for the cases are shown in Figure 11.25(d) and Figure 11.25(e), respectively. Finally, the result due to one c-type element is compared in Figure 11.25(f) with other published work and we mention that locking is still a non-issue for c-type finite element method of analysis.

Example 5: A pinched hemi-spherical shell with an apical hole A hemisphere of radius, R = 10, and thickness, h = .04, with an apical hole of 18◦ is analyzed for very large rotation and displacement with near inextensibility due to two inward and two outward forces in mutually perpendicular directions, as shown in Figure 11.26(a. Only a quadrant of the hemisphere is considered with meridian symmetric and latitudinal free boundary conditions as shown in Figure 11.26(b). The material properties are: elasticity modulus, E = 6.825 × 107 , and the Poisson’s ratio, 𝜇 = 0.3. In order to handle this problem with the largest possible c-type shell elements, it is absolutely necessary that the geometry of the quadrant is accurately modeled; for example, the cubic non-rational Bernstein–Bezier element are not acceptable. The rational quadratic Bernstein–Bezier elements turned out to be the perfect choice; just a 3 × 3 set of these elements produced the accurate result; a quadrant with 2 × 2 set of rational elements is shown in Figure 11.26(c).

Nonlinear Shell

963

(a)

(b)

(c)

(d)

(e)

(f)

Figure 11.25 (a) Example 4: Shallow shell with one c-type quintic element for a quadrant. (b) Example 4: Shallow shell load deformation at load point for R/h = 200. (c) Example 4: Shallow shell load deformation at load point for R/h = 400. (d) Example 4: Shallow shell deformation shapes for R/h = 200. (e) Example 4: Shallow shell deformation shapes for R/h = 400. (f) Example 4: Number of elements comparison.

Computation of Nonlinear Structures

964

(a)

(b)

(c)

(d)

(e)

(f)

Figure 11.26 (a) Example 5: Hemispherical shell with hole geometry. (b) Example 5: Hemispherical shell with hole geometry. (c) Example 5: Hemispherical shell: rational quadratic Bezier mesh for a quadrant. (d) Example 5: Hemispherical shell: load deformation curves under load points for 2 × 2 quintics for a quadrant. (e) Example 5: Hemispherical shell: deformed shape. (f) Example 5: Hemispherical shell: number of elements comparison.

Nonlinear Shell

965

For the linear range, the displacement inward or outward at the load application point is symmetric and was found to produce the exact value of 0.093. However, in the nonlinear regime, the symmetry disappears with the inward displacements much more than the outward ones for the same load level; load–deformation relationships at load points up to 70 of the radius are shown in Figure 11.26(d); the corresponding deformation shape is shown in Figure 11.26(e). Finally, the comparison of element numbers with published results is shown in Figure 11.26(f).

Index Adjoint Map and Left and Right Translations, 126 Admissible Virtual Space, 91, 125, 524, 560–561, 564, 722, 788, 790, 793 affine combinations, 166 Affine Space, 33–34, 41, 43, 136–137, 139, 155, 248, 280, 352–353, 355–356 Affine Transformation Identity, 21–22, 39, 47, 60–61, 63–64, 67–68, 70, 81, 90, 98–100, 104–106, 116, 125–126, 137, 255, 259, 268, 331, 347, 362–363, 367, 372, 375, 377, 380, 398, 538, 550, 559, 562, 564–565, 568, 575–576, 581, 599, 609, 616, 622, 628, 648, 655, 663, 668, 670, 730, 733, 788, 790, 792–793, 797–798, 805–806, 811, 825, 834, 852, 862, 871–872, 916 Rotation, 1, 6, 9–15, 17, 22, 27, 33, 36, 39, 44, 64, 74, 90–94, 97–101, 104–108, 110, 112, 114–116, 118–121, 123–127, 131, 137, 165–166, 325, 334, 336–337, 353, 356, 359–360, 488, 490, 496, 498–499, 501, 504, 508, 525–527, 531, 533, 541–542, 555–556, 558–566, 572, 574, 580–582, 584–585, 590, 596–597, 599, 601–602, 605–611, 613, 616, 618, 623, 626–628, 643, 647, 679, 684–685, 688, 693–694, 698, 701, 707, 709–711, 715, 722–724, 745, 750, 767, 782, 784, 786–788, 790–794, 800, 803–804, 810–812, 815, 819, 830, 832, 834, 836–837,

839–842, 845, 850–854, 857–858, 861–862, 880, 885, 906, 913–914, 925, 934, 950–953 Scaling, 37, 137, 161, 164, 166, 249 Translation, 33–34, 42–43, 75, 77, 92, 127, 136–137, 155, 164, 166–167, 280, 334, 355–356, 373, 397, 421, 533, 565, 793 Algorithm For Newton Method of Iteration, 441 Algorithm: Numerical Iteration, 445 Allowable Parametric Representation, 37, 140, 249 An Important Lemma, 89, 605, 839 Angle between Vectors, 259 Angular Momentum, 523–524, 537–538, 543, 547–549, 553–554, 556, 558, 575, 585, 591–592, 594, 608, 622, 640, 647–648, 722, 748, 750, 755–756, 759–760, 762, 769–770, 776, 778, 780–783, 787, 805, 815, 822, 828, 842 Angular Velocity Tensor, 125, 127, 526, 542, 550, 565, 571, 723, 794, 800 Arc Length Constraint, 368, 438–439, 441–445, 643, 707, 714, 879 Arc Length Parameterization, 140, 270, 272, 534, 735, 773 augmented interpolation transformation matrix, 679, 691, 906, 925 Axiom #1: Conservation of Mass, 338 Axiom #2: Balance of Momentum, 339 Axiom #3: Balance of Moment of Momentum, 342 Axiom #4: Conservation of Energy Energy Conjugated Stress-strain, 348

Computation of Nonlinear Structures: Extremely Large Elements for Frames, Plates and Shells, First Edition. Debabrata Ray. © 2016 John Wiley & Sons, Ltd. Published 2016 by John Wiley & Sons, Ltd.

968

B-Rep, 8 B-spline, 8, 14–16, 135–136, 150, 154, 163, 184–185, 190–191, 193–194, 198, 218–219, 222, 224, 228, 230–231, 234–235, 237, 239, 242–243, 245, 247–248, 280, 296, 299–301, 304, 306, 367, 412, 414, 429–432, 434, 460, 474–476, 478, 487–488, 492, 509, 726 B-spline Curve Construction, 163 Balance Equations, 2D For Linear Beam, 595 For Nonlinear Beam, 595 Balance Principles, 74, 325–326, 337–338, 343, 351, 399, 523–524, 722 Banach Space, 25 Barycentric Combination, 33, 35, 137, 164, 166–167, 169, 190, 201, 206–207, 213, 306, 313 Basis Functions, 2–6, 14, 23, 135, 148–149, 151–155, 157, 163, 167, 170, 174, 186–187, 189–190, 193–194, 201, 206, 247, 280, 286, 293, 307, 309, 321, 367–368, 374, 376, 380, 403–404, 406–407, 422–423, 433–435, 511, 513, 517, 521, 687–688, 691, 715, 930, 932, 934, 943 Basis Functions:, 433 Beam 1D Direct Engineering Representation, 532 Beam 3D Continuum Representation Beam Body: Interior and Boundary Surfaces, 533 Beam Angular Momentum Balance Equations, 553 Beam Balance Equations of Motion: Dynamic, 554 Beam Geometry: Definition and Assumptions Beam: Undeformed Geometry, 531 Bernstein basis function, 149, 153 Bernstein Basis Functions: Properties Degree Elevation, 154, 164, 176, 178–179, 198, 292, 367, 406, 411–412, 423, 498, 514, 517–518 End Point Interpolation, 151, 156, 164, 169, 410 First Derivative, 5, 28–29, 154, 172, 174, 202, 286, 691 Integral Property, 154

Index

linear precision, 150–151, 198, 285, 367, 406, 411–412, 459, 514, 517 Non-negativity and Unique Maximum, 153, 309 Partition of Unity, 151, 153, 169, 190, 307, 309 Power Basis Connection, 153 Product Rule, 153, 562, 592, 790, 822, 848 Recursive Property, 153 Subdivision Property, 153, 201 Symmetry Property, 10, 12, 154, 170, 181, 605, 839, 921 Bernstein polynomials, 8, 14, 148–151, 154–157, 164, 169–170, 174, 280, 286, 292–293, 307, 314–315, 406, 411, 413, 422–423, 428, 432, 434, 436, 486, 490, 498, 513–514, 517, 520–522 Bernstein-Bezier curves, 136, 148, 154, 157, 198, 228, 406, 411, 459, 474 Bernstein-Bezier Generalized Displacement Element, 934 Bernstein-Bezier Geometric Element, 932 Bezier control vector, 8, 367, 413, 472, 489 Bezier form of B-spline curves, 154 Bezier form of B-spline surfaces, 280 Bezier-Bernstein Curves Affine Invariance, 164–165, 285 Affine Reparameterization Invariance, 164–166 global parameter, 165–167, 182, 430–431 local parameter, 165–167, 182 Cubic Bezier Curves, 182, 218, 475 Bezier-Bernstein-de Casteljau, 135–136, 154, 247–248 bound vectors, 10 bound vectors, 34, 51, 255 Boundary or End Conditions, 230 boundary representation, 8, 148, 176, 367, 407 C++ Code Snippets for Geometric Properties, 940 C++ Code Snippets for Material and Geometric Stiffness Matrices, 948 C++ Code: B-spline to Bezier, 243 C++ Code: Generating Missing Control Points, 304 c-type Curved Beam Element, 493–494, 496 c-type Deep Beam: Plane Stress Element, 498

Index

c-type FE Formulation: Dynamic Loading, 673, 891 c-type FE Implementation and Examples: Quasi-static Loading c-type Beam Element Library, 687 Code Fragment for E Matrix, 693 Code Fragment for G Matrix, 697 Code Fragment for Tangent Stiffness Matrix, 711 Non-rational Elements, 688 Rational Elements, 688 c-type Finite Element Discretization, 687 c-type Finite Element Formulation, 460, 529, 574, 677, 684, 725, 803, 901, 913, 921 c-type Finite Element Library, 435 c-type Finite Element Space, 434 c-type Shell Element Library Non-rational Tensor-product Quadrilateral Elements, 930 c-type Shell Elements and State Control Vectors, 904, 923 c-type Solutions: Locking Problems Remedy for shear locking, 511 Remedy for transverse shear locking, 520 c-type Truss and Bar Element, 460 CAGD, 8, 15, 137, 148, 152, 154, 195, 205, 406 Cartesian Product Bernstein-Bezier surfaces, 248, 280 Cartesian product patches, 280, 292 Cartesian Product Surfaces Control Net Generation, 296 Cauchy Reciprocal Theorem, 340, 345 Cayley-Hamilton theorem, 66, 357 Centripetal Parameterization, 230 characteristic polynomial, 66, 68, 70, 91, 106, 118, 357, 662 Classes of Functions, 23 Classical Rayleigh-Ritz-Galerkin Method, 403 Classification of Conics, 210 Component Force and Moment Vectors, 583–584, 774, 815, 819 Component or Operational Vector Form, 555, 580, 782, 810 composite Bezier curve, 181–182, 299 Composite Bezier Form Quadratic and Cubic B-splines, 248, 300 Computational Derivatives and Variations, 596, 830

969

computational element centrifugal stiffness matrix, 911 computational element symmetric mass matrix, 681 Computational Equations of Motion, 590 Computational Real Strains Real Translational and Bending Strains, 864 computational tangent stiffness matrix, 701 Computational Virtual Work Equations External virtual work, 397, 402, 419, 557, 613, 622–623, 845, 857, 886 Inertial virtual work, 402, 529, 557, 613, 622, 643, 725, 845, 880 Internal virtual work, 397, 402, 419, 500, 525, 527, 529, 557–558, 560, 613, 618, 620, 622, 627, 631–632, 723–725, 788, 845, 847, 861, 864–866, 882 computer-aided geometric design, 8, 15, 21, 137 Configuration Space, 9, 91–92, 125, 331, 524–526, 528, 539, 542–543, 555, 558–561, 564, 570, 607, 626, 722–724, 749, 782–783, 787–788, 793, 799, 806, 841, 861, 898, 901, 918, 921 conforming, 6, 8–9, 434, 511, 516 conics, 135–136, 154, 198, 205–207, 209–210, 247, 280, 687 Conics as Implicit Functions, 205 Conics as Quadratic Rational Bezier, 207 conics of revolution, 247 conservative system, 10, 13, 470, 606, 610, 618, 840, 898, 901, 921 constitutive or material tensor, 363 Constitutive Theory: Hyperelastic Stress-Strain, 351 Construction of S-form from W-form, 381 Construction of W-form from S-form, 380 constructive solid modeling, 367, 407 Continuum Approach – 3D to 1D, 539 Contravariant Base Vectors, 38–39, 46–47, 53–54, 58, 60, 62, 75, 91, 94, 251–253, 257, 328, 343, 580, 592, 732–733, 805, 810–811, 822 contravariant components, 47, 53, 55, 57–58, 60–61, 63, 67, 77, 85, 87, 252–254, 257, 336, 584, 757, 815 convected, 327, 329, 558, 575, 786, 804

Index

970

Conventional Finite Element Methods Bubble Mode, 7, 367, 509 Drilling Freedom, 7, 509 h-type, 2–4, 6–8, 406, 433, 438, 509, 511, 522 Incomplete Strain States, 5 Induced Anisotropy, 5 interpolation, 2–3, 6–9, 15, 23, 123–124, 135, 151, 155–156, 164, 166–167, 169, 182, 185, 187, 189, 193, 198, 218, 228, 230–233, 235, 237, 245, 247, 280, 282, 285, 307, 313–314, 410–411, 429, 486, 493, 509, 511, 514, 517, 522, 679, 685, 687–688, 691, 904, 906, 914, 923, 925, 930 Locking Problems, 8, 14, 460, 494, 509, 511 Shear and Membrane Locking, 7, 9 non-conforming, 6, 8, 511, 516 p-type, 3–5, 406, 433, 438, 509, 511, 522 Patch Test, 6–7 reduced or selective integration, 7, 407, 459 selective under-integration, 8, 509 Convex Hull, 23, 164, 169–170, 198, 201, 206, 285 Convex Set, 23 Coons’ Patch, 297–299, 303–304 Coordinate Frame Rotation, 566, 794 coordinate systems 4D homogeneous coordinate system, 139 Cartesian, 5, 10, 13, 23, 30, 36–37, 39, 42, 55, 71, 74–76, 136–137, 142–144, 248–250, 280, 285–286, 290, 292–293, 296, 298–299, 304, 306, 319, 321, 327–329, 336, 367, 408, 435, 498, 508, 774, 815, 942, 946 Base or Coordinate Vectors for Cartesian Coordinate Systems, 36 curvilinear, 10, 13, 16, 22–23, 36, 42, 75–76, 327–329, 331, 336–338, 352, 488–489, 524–526, 532, 537, 539, 542, 580, 582, 584, 722–723, 727–728, 733, 774, 778, 810, 812, 815, 930 spinning coordinate system, 12, 327, 568, 571, 580, 620, 786, 796, 800, 810, 847 Covariant Base Vectors, 38–39, 45, 47, 53, 58, 60–61, 74, 77, 91, 94, 251–253, 257,

261, 328, 343, 543, 580, 732, 737, 748, 810–811 covariant components, 53–54, 56–58, 60, 63, 66–67, 77, 83, 88–89, 252–253, 257, 345, 582, 812, 869 covariant derivative, 11, 13, 32, 76–77, 83, 85, 87, 91, 125–126, 331, 346–347, 545, 560–562, 588, 590–591, 644, 762, 772, 778, 791, 818–819, 822, 882 Covariant Linearization of Virtual Work Geometric stiffness functional, 652, 660, 672–674, 685–686, 888, 892, 898, 901–902, 914–915, 918, 921 linearized dynamic virtual work equations, 652, 673, 892 linearized quasi-static virtual work equations, 685, 914 Material stiffness functional, 652, 656, 672–674, 685–686, 888, 892, 897–898, 902, 914–915, 917–918 residual force functional, 644, 652, 673–674, 676, 685–686, 888, 892, 914–915, 917 unbalanced force functional, 652, 673–674, 685–686, 888, 892, 914–917 covariant metric tensor, 39 covariant Riemann curvature tensor, 279 Curvature, 3, 7, 9–10, 13, 16, 90–91, 125–127, 136, 145–147, 173, 180, 187, 189, 202, 248, 261, 263, 265–270, 272, 277, 279, 337, 433, 470, 474, 488, 496, 524–526, 560–562, 565, 568, 571–572, 574, 588, 591, 597–599, 601, 605, 610, 620, 628, 693–694, 697–698, 703–704, 706, 713, 722–723, 728, 730, 733, 735, 738–739, 746, 763, 772–773, 778, 787–788, 790, 794, 797, 800, 803, 810, 818, 825, 832, 834, 836, 839, 847, 852–853, 855, 862, 867, 869, 897, 901, 904, 907, 917, 921, 923, 925–926, 929, 938, 941, 947–948, 950–954 curved beam, 9–10, 12, 15–17, 327, 438, 460, 486–487, 492–494, 496, 498, 500, 522, 531, 687, 715–718 Curvilinear (Convected) Coordinate System, 327, 329 Darboux’s vector, 146, 588 de Boor-Cox, 8, 135–136, 195, 247–248

Index

de Casteljau Algorithm linear interpolations, 161, 166, 282, 307, 314 Decomposition of Tensors Polar Decomposition, 72, 91, 334, 359–360 Spherical and Deviatoric Decomposition, 72 Deformation Geometry, 326 deformation gradient, 91, 325, 327, 329, 331–334, 337, 339, 347, 349, 351, 356, 358–362, 575, 632–633, 805, 866–867, 869, 872 deformation gradient tensor Tangent Map, 331–332 deformation map, 327–329, 331–332, 336, 338, 524, 541, 545, 547, 574–575, 722, 749–751, 755, 766–767, 804–805 Deformation Power, 524, 574, 576, 722, 785, 803, 806 Deformed Geometry, 325, 523, 534, 722, 772–773 Derivative of a Curve, 140 Derivatives of Bezier Curves End Point First Derivatives, 172, 182, 288 End Point Higher Derivatives, 173 First Derivative, 5, 28–29, 154, 172, 174, 202, 286, 691 General Derivatives, 172, 174 Derivatives, Angular Velocity and Variations, 125 determinant, 37, 39, 51, 58, 66, 68, 70–71, 93–94, 98, 148, 205, 249, 254–255, 265, 279, 343, 417, 445, 485, 488, 507, 562, 677, 687, 691, 703–704, 735, 776, 790, 872, 926, 938 Differential Area Element, 332 direct stiffness, 11, 406, 478, 529, 677, 687, 726 Divergence of Vector Functions, 84–85 Divergence Operator, 84–85, 87, 755 Dual Base vectors, 38 Dynamic Balance Equations, 537, 543, 590, 592, 607, 748, 750, 763, 770, 778, 781, 825, 841 Dynamic Momentum Balance Equation, 547 Effective angular momentum, 548–549, 554, 608, 640 Effective External Applied Force, 547

971

Effective External Applied Moment, 553–554 Einstein summation convention, 23, 39, 47, 51, 63, 137, 345, 347, 551 Elastic Materials Homogeneous Elastic Material, 359 Isotropic, Homogeneous, 358, 360, 362, 882 Principle of Determinism, 326, 358 Principle of Local Action, 326, 351, 358, 632, 866 End Point First Derivatives (Tangents), 288 energy conjugate, 326, 338, 349, 359, 399, 557–558, 560, 632, 785–786, 788, 866 Energy inner product, 27–28, 376–378, 382–385, 392, 421 energy space Essential Boundary Conditions: Dirichlet Type, 379 Natural Boundary Conditions: Neumann Type, 379 Solution Set, 372–373, 378, 381, 407, 421 Equivalence of S-form and W-form, 380 Equivalence of S-form, W-form and V-form, 382 Euclid’s geometry fifth or the parallel postulate, 42 Euclidean spaces, 22, 28, 39, 42, 51 Euclidean to Riemannian Space, 36 Euler Theorem, 100 Eulerian, 325, 327, 336, 338, 345, 367, 399 External applied power, 557, 785 Extremum Values of Curvature Gaussian Curvature, 269–270, 279 Mean Curvature, 269–270 Finite Difference Methods, 371 finite element method c-type finite element method, 1, 7, 9, 14–15, 91, 148, 325–326, 337, 367–368, 371, 373, 375, 377, 379, 381, 383, 385, 391, 393, 395, 397, 399, 403, 405–407, 411, 413, 417, 419, 421, 423, 427, 429, 431, 433, 435, 438–439, 441, 443, 445, 450–451, 453, 455–456, 459, 509, 522, 524, 526, 528–529, 559, 561, 626, 717, 722–724, 726, 773, 788, 860

972

First Fundamental form Riemannian Metric, 39, 252–253 flat-spaces, 10 For Linear Circular Beam, 595 Forward Difference Operator, 32, 172, 174, 178, 202 Frame Indifference, 326, 351–352, 359–360, 632, 866 free vectors, 10 Frenet Frame bi-normal vector, 143–145 osculating plane, 144, 147–148, 263, 485 Principal Normal Vector, 143–144 Rectifying Plane, 144–145 Unit Tangent Vector, 142, 144, 147 Frenet-Serret Derivatives, 136, 145 Gateaux or Directional Derivative, 28–29 Gauss-Legendre Quadrature Formulas Composite Formulas, 448 Higher Dimensional Gauss-Legendre, 452 Interpolatory Quadratures, 446 Linear Transformation of Interval, 448, 450 Newton-Cotes Formulas, 446, 448 Tabular Examples of Gauss-Legendre Nodes and Weights, 453 Gauss-Weingarten Formulas Christoffel symbol, 77, 272, 745 first kind, 272–273 second kind, 33, 74, 77, 272, 277 Gauss Formula, 272 Normal Coordinate System, 272, 580, 584, 728, 810–811, 815, 871 Weingarten Formula, 248, 272, 275, 733 Geometric stiffness matrix, 12–13, 528, 655, 667, 672, 681, 707, 725, 897, 901, 917, 921, 948 Geometric Stiffness Matrix with Distributed Moment Loading, 672 global basis function, 368 Gradient of Tensor Functions, 84 Gradient of Vector Functions, 83 Gram-Schmidt Orthogonalization, 14, 143–144 Green’s identity or the divergence theorem, 372, 375, 380

Index

Groups Abelian, 22 Lie, 2, 10, 12, 22, 91–92, 98–100, 104–105, 115, 118, 120, 125–126, 265–266, 561–562, 564–566, 568, 571, 598, 601–602, 605–606, 609, 616, 648, 709, 790, 792–793, 796–798, 800, 825, 832, 836, 839–840, 852 Heat Conduction and Potential Flow Problems, 428 Hellinger-Reissner Variational Theorem, 391 Hessian Symmetry Lemma, 83 Higher-order Tensor Component, 55 Hilbert Space, 25, 33, 71 Horner’s Algorithm, 157 Hyperelastic Material Property Principle of Equivalence, 632, 866 Thin, Linear, Homogeneous, Isotropic, 635, 869 Hyperelastic, Isotropic, Homogeneous Material, 362 Important Computational Identities, 127 incremental generalized displacement, 528–529, 656, 679, 687, 701, 707, 725–726, 879, 897, 904, 916–917, 929 incremental strain-displacement, 656, 898, 918 Inertial power, 785 Inner Product Space, 25, 27–28, 33, 405, 421 Integral Transforms: Green-Gauss Theorems, 87 Integration by Parts, 28, 87, 367, 372, 375–377, 380–381, 383–385, 557 Inter-element Continuity inner control points, 184 junction control point, 184 Inter-element Continuity (IEC): Bezier to B-spline, 429 Internal deformation power, 785 internal energy, 338, 348–349, 631, 866 internal energy density, 348 Iso-curves, 250–251, 266, 306, 343, 728, 730, 773 Isotropic Tensor Functions, 326, 356–357 Jacobian determinant, 39, 677, 687 Jacobian matrix, 37, 51, 249, 255, 417, 691

Index

Kinematic and Configuration Space, 555, 559, 783 Kinematic Parameters displacement vector field, 91, 125, 524–526, 559, 561, 564, 580, 687, 722–723, 787, 790, 793, 804, 810 rotational tensor field, 91, 125, 524, 526, 559, 574, 722, 724, 787–788, 804 Kinetic Energy Balance, 349, 351 knot sequence, 182, 186, 190, 192–193, 234–235, 245 knot vector, 182, 192, 194 Knots of Multiplicity, 187, 190, 235 Kronecker delta tensor, 38, 252 Lagrange equations of motion, 346 Lagrangian, 2–3, 7, 16, 135, 148, 247, 325–329, 331, 333–343, 345–347, 349, 351, 353, 355–357, 359, 361, 363, 367, 399, 543, 550–551, 738, 746, 748 Lagrangian inertia tensor, 551 Lame constants, 362–363 Least Square Form (L–form), 385 Length of a Curve, 259 Length of a Vector on a Surface, 257 Linear Balance Equations, 3D, 595 Linear Elasto-dynamic Problems, 427 Linear Interpolation, 123–124, 166–167, 198, 237, 282, 307, 313–314, 685, 914 Linear Interpolation or Approximation on 3-Sphere, 123, 914 Linear Momentum, 537, 545, 547, 549, 554, 556, 585, 591, 594, 608, 640, 751, 753–755, 759, 767, 776, 780, 783, 815, 828 Linear Transformations Linear Functionals, 14, 26–27, 42, 44–45, 248 Linear Operators, 26–27, 42, 44, 174, 248 Orthogonal transformation, 27, 33, 64, 91–93, 97, 100, 118, 125, 136, 353, 358 Linear Vector Space, 22, 26–27, 41, 44, 46, 62, 75, 126, 135–136, 155, 247–248, 280, 560, 563, 628, 792, 862 Linear, Hyperelastic, Isotropic, Homogeneous Material, 362 Lines of Curvature, 91, 125, 266–270, 272, 337, 723, 728, 730, 733, 735, 738–739,

973

746, 763, 772–773, 778, 787, 803, 810, 836, 852, 867, 869, 897, 901, 904, 907, 917, 921, 923, 925–926, 929, 938, 941, 948 local basis functions, 3, 368, 404, 687 Local Gauss Frame, 261 Love-Kirchhoff, 11, 542, 767 Love-Kirchhoff-type, 11, 542, 767 Mainardi-Codazzi conditions, 787 material coordinate system, 327–328, 335, 338–339, 558, 568, 574, 786, 796, 803 Material Curvature Tensor, 127 Material stiffness matrix, 527–528, 655, 674, 686, 693, 707, 725, 918, 926 material symmetry, 351, 360, 632, 866 Material Tensor, 361, 363 Matrix Form of a Bezier Curve, 157, 164 Matrix Representation, 46–47, 49, 529, 556, 585, 592, 597, 725, 733, 784, 822, 830, 832 mega-element, 531 mesh generation, 1, 8–10, 13, 133, 135, 145, 148, 155–156, 163–164, 170, 173, 180–182, 189, 247, 280–282, 286, 290, 292, 312, 407–408, 417, 423, 434, 460, 486, 498, 507, 717, 719, 923, 932 metric tensor, 39, 44, 56, 58, 60, 85, 87, 91, 253–254, 259, 267–268, 273, 485, 730, 735, 778, 940, 946 Meusnier’s Theorem Normal Section of a Surface, 263 Minimal Support Property, 193 Minimum Potential Energy Principle, 189, 368, 395, 399, 419, 470 Minimum Potential Energy Theorem, 391 Nanson’s Formula, 332 New Finite Element Method: c-type Coerced Isoparametry, 406, 411–412, 423 Element (Local) Level Formulation, 414 Finite Element Basis Functions, 422 Finite Element Mesh, 407, 422, 429 Finite Element Trial and Virtual Subspace, 422 Root Element, 4, 6, 282, 285, 408, 410, 417, 422, 433, 435, 488, 500, 507–508, 513, 517, 520, 677, 687–688, 904, 923, 930

974

New Moving or Spinning Coordinate System, 786 Newton Iteration, 438 Newton’s Method, 438–439, 442 Newton-type iteration, 11, 368, 439, 643, 714, 880, 901, 921 Non-conservative Systems, 12 non-rational, 136, 139, 154, 163, 198–199, 201, 205–207, 222, 224, 228, 230, 232, 243, 248, 280–281, 283, 286, 296, 299–301, 306, 322, 508–509, 687–688, 930, 932 non-rational Bernstein-Bezier surfaces, 280 Non-rational Bezier Surfaces hyperbolic paraboloid surface, 282–283 Normal Vector for a Degenerate Patch, 290 Surface Normal Vectors, 290 Non-rational Curves, 154, 198 Non-symmetric Arc Length Constraint Method, 441 Norm, 5, 25–26, 33, 70–71, 81, 98, 115–116, 195, 376, 385, 399, 404–405, 421, 442, 486, 496, 711, 872 normal curvature, 266–267 Normal Plane, 142–143 NURB, 235 Objective Functions, 355 Objective Tensors, 351, 353 objective vector, 355 One-dimensional Strain Energy, 525 Open Curve Interpolation, 218 Operators, 17, 26–29, 42, 44, 83, 104–105, 135, 151, 170, 172, 174, 247–248, 286, 371, 385 Orthogonal Transformation and the Rotation Tensor determinant of an orthogonal transformation, 93 Parametric Mapping isoparametry, 6, 406, 411–413, 417, 423, 688 subparametric, 6, 368, 407, 411, 435–436, 904, 923 superparametric, 6 Pascal triangle, 5

Index

Patch by Extrusion of One Boundary Curve, 296 Lofting of Two Boundary Curves, 296 Revolution of a Curve, 299 Permutation symbol, 33, 71 Piola Transformation, 332, 346, 359 Points on a Surface Elliptical Point, 265 Hyperbolic Point, 266 Parabolic Point, 266, 270 Position or radius vector, 137 Potential Energy Functional, 368, 392, 394–395, 419, 470, 488–489 Projective Map and Cross Ratio Invariance, 198 Properties of Metric or Fundamental Tensor, 253 Pseudo-vector Representation of Tensors, 50 Quadratic Rational Bezier Curves and Conics, 206 quadrature rules, 414, 438 Quasi-static Proportional Loading, 439, 529, 622, 639, 643, 725, 877, 879–880 Quotient Rule and Invariance, 57 Radius of Curvature, 146–147, 261, 267, 869 Ratio Invariance, 198 Rational B-spline Curves, 135, 230 rational B-spline curves, 135, 230 Rational Bernstein-Bezier curves, 154 rational Bernstein-Bezier surfaces, 280 rational Bezier, 135–136, 163, 198–199, 201–210, 213, 232–234, 248, 280–281, 292–293, 295–296, 321–322, 486, 488 Rational Bezier Curves, 163, 198–199, 201, 205–206, 210, 280, 292, 321 Rational Bezier Surfaces, 280, 292, 295 Real Coordinate Space, 22–23, 33, 41, 43–44, 136, 248 Real Parametric Curves, 139 Real Parametric Surface, 248–249 Real Strain and Strain Rates Real Strain Field, 527, 572, 724, 800 Recursion Relation, 190, 315 Recursive Algorithm: de Boor-Cox spline, 195 Reduced Internal Energy, 631, 866 Reference Configuration, 325, 353, 356, 360, 362, 574, 803, 805

Index

Regular Curve, 140, 250 regular points, 250 Regular Reparameterization, 140 Reissner-Mindlin-type, 542, 574, 750, 767 represent a curve implicit form, 135, 205, 247 parametric form, 135, 150, 247, 406, 459 Representation of the 3D Rotation Matrix Cayley’s Representation, 100 Cayley’s Theorem, 100–101, 105–107, 564, 596, 609, 616, 792, 830, 851 Instantaneous Rotation, 560, 596, 608, 788, 830, 851 Euler–Rodrigues Parameters, 112 4D Euler-Rodrigues Vector, 112 Hamilton’s Quaternions, 115 Four-dimensional Quaternions, 115 Quaternion Properties, 116 inverse quaternion, 115–116 multiplication rule, 116, 118 pure or vector quaternion, 116 unit quaternion, 115–116, 118 Hamilton-Rodrigues Quaternion, 115, 118–121, 685 4D Hamilton-Rodrigues Quaternion, 119 Rodrigues Parameters, 107, 112, 114, 560, 597, 609, 616, 709, 788, 852 3D Rotation Vector, 108 Geometry of the Rotation Matrix and the Rodrigues Vector, 110 Rodrigues Vector, 92, 100, 107–108, 110, 112, 114, 119–120, 127, 560, 564, 580, 596–597, 599, 608–609, 616, 788, 792, 810, 830, 851–852 Rotation Matrix, 91–92, 94, 97–101, 104–108, 110, 112, 114, 116, 118–121, 125, 127, 137, 560, 562–564, 596–597, 607–609, 616, 647, 709–711, 788, 790–792, 810, 830, 841, 850–851 Residual Error Form: (R-Form), 371 Riemann curvature tensor mixed Riemann curvature tensor, 279 Riemannian geometry, 39, 253 Riemannian spaces, 36, 39, 44 Rigid Body Dynamics, 549 Rigid Body Rotation, 6, 91–92, 94, 97, 100, 334, 336, 533

975

rotating coordinate system, 327, 574, 803 Rotation Matrix: Components of a Rotation Tensor, 94 Rotation Tensor, 11, 14, 39, 74, 90–94, 97, 99–101, 105, 107, 115, 119, 121, 123, 125–127, 131, 325, 334, 337, 353, 356, 359–360, 525–526, 533, 541–542, 556, 558–564, 566, 574, 580–581, 584–585, 596–597, 601, 605, 608, 626, 643, 684, 707, 723–724, 750, 784, 786–788, 790–791, 793–794, 803, 810–811, 815, 830, 832, 836, 839, 841, 851, 861, 880 Rotational Variation, 126–127, 565, 793, 850 Schoenberg, 8, 17, 135, 148, 185, 188, 193, 247 Schoenberg-deBoor B-spline forms, 247 Schoenberg-Whitney Theorem, 188 screw motion, 92 Second Fundamental form Surface Curvatures, 261, 947 Second-Order Tensor, 44, 46–47, 50, 54–58, 60–63, 71–72, 80–81, 83–84, 87–89, 94, 253, 334, 345, 570 Second-order Tensor Component Contravariant Components of a Second-order Tensor, 55 Covariant Components of a Second-order, 54 Mixed Components of a Second-order Tensor, 54–55 Shear, 7–12, 16–17, 137, 185, 326–327, 337, 363, 395, 438, 460, 475, 478, 485–488, 493–497, 499, 502, 509, 511–512, 516, 520, 522, 529, 531–532, 539, 541–542, 555, 572, 575, 583, 595, 628, 633, 656, 715–716, 726, 746, 750, 767, 774, 782, 804, 814, 828, 850, 862, 864, 867 Shell analysis derived approach, 11 direct approach, 11, 524, 722 Shell Geometry: Definition and Assumptions 3D Shell-like Body, 722, 727, 750, 754, 766, 782, 803–804, 842, 866 Optimal Normal Coordinate System, 728, 871 Reference Surface Oriented Differential Area, 738 Shell Boundary, 735–738, 751, 756–758

976

Shell Geometry: Definition and Assumptions (Continued) Shell Coordinates, 736 Shell Interior Differential Volume, 739 Shell Optimal Normal Coordinate System, 728 Shell Reference Surface, 91, 125, 609, 722–723, 728, 730, 732, 735–738, 745–746, 748, 750–751, 756, 763, 773–774, 776, 778, 782, 784–787, 798, 800, 803–804, 810, 815, 819, 828, 830, 836, 841, 844, 848, 852, 861, 866, 872, 923 Shell Shifter Tensor, 732 Shell Triads of Covariant Base Vectors, 737 Shell Undeformed Geometry, 727 Shoulder Point and Parameterization, 209 simple knot, 190 singular points, 250 Singularity-free Update, 959 Sobolev Space, 14, 25–26, 367, 372–373, 376 spatial coordinate system, 327, 558, 795 Special Orthogonal Group, SO(3) Adjoint Map, 104, 106, 126–127, 564–565, 793 Exponential Map, 104–106, 126, 562–564, 609, 616, 790–793, 852 Lie algebra, 22, 99–100, 105, 125–126, 564–565, 609, 616, 792–793, 852 Matrix Lie Group, SO(3), 98 Tangent Operators of SO (3), 104 tangent space at identity, so (3), 100 Special Tensors Fourth-order Identity Tensor, 61, 81 Inverse Tensors, 63 Metric or Fundamental Tensor, 58, 253 Orthogonal and Rotation Tensors, 64 Second-order Identity Tensor, 39, 60–61, 63 Skew-symmetric Tensors, 66–68, 72, 565, 794, 825 Axial Vectors, 67–68, 91, 100, 105, 107, 568, 598, 605, 610, 628, 648, 772, 797–798, 818, 825, 832, 839, 853, 862 Eigenproperties of Skew-symmetric Tensors, 67 Important Identities of Skew-symmetric Tensors, 68

Index

Symmetric Tensors, 66–68, 72, 357, 565, 794, 825 Eigen Properties of Symmetric Tensors, 66 Third Order Permutation Tensor, 61 Transposed Tensors, 63 spheres as rational Bezier surfaces, 295 spherical coordinates, 250, 253 Spherical Displacement, 92, 97 Spline Function Representation, 188 Spline Space, 186, 189–190, 193, 195, 434 splines B-splines (Basis Functions), 189 Cardinal splines, 189 Natural splines, 189, 195 Periodic splines, 189, 195 Physical splines, 185–186, 189 Standard Bezier Curve, 166, 169 standard form, 165, 167, 201, 207–213, 231, 233, 235, 245 Static and Dynamic Equations: Continuum Approach - 3D to 2D External applied moment, 553–554, 760, 762, 770 Internal resistive stress couple, 556, 757, 770, 783 Internal resistive stress resultant, 545, 556, 783 shell resultant stress tensor, 751 Shell total angular momentum, 759 Shell total linear momentum, 753 Stress couple tensor, 756–758 Surface angular momentum, 756, 759, 762, 769 Surface external applied force, 754 Surface linear momentum, 751, 753, 755, 767 Static and Dynamic Equations: Engineering Approach, 534 Static Balance Equations, 524, 538, 594, 721, 762–763, 781–782, 828 Strain Tensors Green-Lagrange Strain Tensor, 325, 334–335, 351, 356, 359, 361–363, 635, 869 Stress Couple, 527, 535, 548, 554, 556, 570, 574, 583–584, 631, 633, 724, 756–758, 762, 770, 774, 783, 799, 803, 815, 865, 867, 871

Index

stress power, 348–349, 361, 785 Stress Resultant, 10, 13, 17, 526, 535, 545, 547, 556, 570, 583, 633, 724, 726, 751, 753, 755, 774, 783, 799, 865, 867 Stress Tensors Piola-Kirchhoff Stress Tensors, 343 first Piola-Kirchhoff stress tensor (PK I), 345, 359 second Piola-Kirchhoff stress tensor (PK II), 345, 359 Subdivision Algorithm, 179–180, 184, 198, 201–202, 286, 293, 498, 509, 717 Surface Area, 259, 327, 340–343, 543, 739, 746, 753–755, 759–760, 762, 769–770, 784, 844, 848, 871 Surface curvature tensors, 277 Surface curvature vector, 277 surface Jacobian, 255, 732 surface traction vector, 338, 341–342 symmetrization of the geometric stiffness, 12 Symmetry of the Tangent Operator, 12–13, 16, 701 Symmetry Preserving Arc Length Constraint Iteration Method, 443 Symmetry: the Geometric Stiffness Matrix Rotational Part, 667, 897, 916 Translational Part, 897, 916 Tangent Plane, 250–252, 301, 345, 433, 751, 756–757, 784, 844, 848 Taylor series, 29–30, 143, 358, 643, 880 Tensor by Component Transformation Allowable Coordinate Transformation, 41–42, 51, 53, 255, 257 Transformation Law, 44, 53–55, 57, 257 Contravariant Components of a Vector, 53, 55, 77, 257, 336 Covariant Base Vector, 47, 53, 257, 345 Covariant Components of a Vector, 53–54, 77, 257 Tensor Calculus Covariant or Absolute Derivative, 75 Covariant or Absolute Derivatives of the Components of Vectors, 77 Differential Operators, 371 Partial Derivatives of Base Vectors, 74 Partial Derivatives of Parametric Functions of Tensors, 81

977

Partial Derivatives of Scalar Functions of Tensors, 80 Partial Derivatives of Tensor Functions of Tensors, 81 Partial Derivatives of Tensors, 74 Tensor Operations Contractions, 56, 69 double contraction, 55–56, 60–61, 64, 69–73, 81, 85, 87, 349, 848 simple contraction, 55–57, 67, 69–71, 89 Tensor Product Tensor Product (Dyadic) Notation, 45 Tensorial Description of Rotational Transformation, 94 Tensors as Linear Transformation, 44 Thin and Very Flexible Beam: Deformation Map, 541 Thin and Very Flexible Shell: Specialization Effective surface angular momentum, 769 Effective surface linear momentum, 767 Surface rotary inertia, 769 Time-dependent Dynamic Loading, 623, 857 Timoshenko, 9, 11, 17, 496, 504–505 Torsion, 147–148, 202, 595, 693–694, 697–698, 704–706, 716 Total angular momentum, 549, 759 total current “curvature” vectors, 855 Trace, 56, 60–61, 69–71, 73, 80, 91, 112, 206, 438, 623, 709–712, 714, 857, 948–949, 951 Triangular Bezier-Bernstein Surfaces Axis-Directional Tangents, 319 Derivatives and Normal to Triangular Surface, 315 Normals on a Bezier Triangle, 319 Rational Bezier Triangle and Derivatives, 321 Triangle Number, 307, 309 Triangles and Barycentric Coordinates, 306 Triangular Bernstein Polynomials, 307, 436 triangular patches or surfaces, 280 Twist, 2, 10–11, 13, 136, 145, 147–148, 161, 248, 272, 286, 288, 292–293, 301, 303–304, 306, 326, 460, 496, 500, 595, 717–718

978

Twist or Second Mixed Partial Derivative, 288, 293 Two- or Three-dimensional Scalar-valued Functions, 88 Vainberg Principle, 14, 32, 419, 626, 628, 860, 862, 864 Variational form (V-form), 381 variational formulations, 368, 371 Variational Methods, 15–17, 371–372, 386 Vector product, 33, 663, 668, 670 vector space, 22, 26–27, 34, 41, 44–47, 62, 75, 105, 126, 135–136, 155, 199, 247–248, 280, 307, 355–356, 421, 528, 560–564, 628, 647, 707, 791–792, 862 Vector triple product, 33, 91, 135, 144, 247, 550, 556–557, 560, 571, 596, 611, 613, 784–785, 788, 800, 830, 842, 844 Vector-valued Functions: Divergence or Gauss Theorem, 88 Vectors in Affine Space, 34 Vectors on a Surface, 250, 252, 277, 772 Virtual (Displacement) Work Principle for Nonlinear Systems, 399

Index

Virtual Work Principle, 11, 368, 384, 391, 397, 399, 402, 416, 419, 421, 428, 470, 486, 489, 527, 559, 561, 564, 574, 607, 613, 632, 724, 788, 793, 803, 841, 845, 866 Virtual Work Principle for Linear Systems, 397 Weak Form of the Vector Balance Equations External virtual work rate, 557 Inertial virtual work rate, 557 Internal virtual work rate, 557 Weak or Galerkin Form: (W- Form) Admissible Boundary Conditions, 373, 379, 392, 396 Collocation Method, 374 Galerkin Methods, 375 Green’s Function Method, 375 Rayleigh-Ritz-Galerkin Method, 3, 5, 368, 376, 379, 403–404, 428 test space, 368, 372–377, 379–381, 383–385 trial space, 368, 372–377, 379–381, 383–385, 421, 434 Weierstrass Approximation Theorem, 149 Weight, 207–208, 210, 212–213, 233–234, 242, 295, 299–300, 304

Figure 5.13 Definition of an inverse problem.

Figure 5.21 Convex hull.

Figure 5.23 Curve blending by degree elevation.

Computation of Nonlinear Structures: Extremely Large Elements for Frames, Plates and Shells, First Edition. Debabrata Ray. © 2016 John Wiley & Sons, Ltd. Published 2016 by John Wiley & Sons, Ltd.

Figure 5.53 Effect of boundary conditions on a Bezier curve.

Figure 5.54 Quadratic curve fitting input points.

Figure 5.55 Cubic curve fitting input points.

Figure 5.56 Exact full circle by rational Bezier.

Figure 6.13 Bilinear hyperbolic paraboloid surface.

Figure 6.17 Exact sphere as rational Bezier surface.

Figure 6.18 Patch by extrusion with Bezier control net.

Figure 6.19 Patch by lofting of two curves with Bezier control net.

Figure 6.20 Patch by lofting of three curves with Bezier control net.

Figure 6.22 Patch by revolution of a generating curve.

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