Sandwich Structures: Theory and Responses 3030718948, 9783030718947

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Terry John Hause

Sandwich Structures: Theory and Responses

Sandwich Structures: Theory and Responses

Terry John Hause

Sandwich Structures: Theory and Responses

Terry John Hause U.S. Army Warren, MI, USA

ISBN 978-3-030-71894-7 ISBN 978-3-030-71895-4 https://doi.org/10.1007/978-3-030-71895-4

(eBook)

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

This book is in honor of Dr. Liviu Librescu who sacrificed his life, on April 16th, 2007, during the Virginia Tech Shooting, to save the students in his class by holding the door shut while the other students escaped through the window. Dr. Liviu Librescu was a great mentor, colleague, and friend. May he never be forgotten for the selfless act he carried out.

Preface

Over the past several decades sandwich structures have increased in their use and applications. Just to name a few, sandwich structures have applications in supersonic/hypersonic aircraft, Cryogenic tanks, Flight and launch vehicles, trains, automobiles, naval ships, and several components found in satellites. With all of these applications it is imperative that a good understanding of these structures is in hand both from a theoretical and practical standpoint. In the future the applications will increase for the use of these structures. With the consideration of all of these applications a great deal of knowledge with regard to their behavior or response under complex loading conditions is required. Currently, there is need for a comprehensive book on sandwich structures that provides a comprehensive theoretical base upon which to build. Very few if any books on sandwich structures provide this background from a consistent standpoint. There is a plethora of texts on plates, shells, and beams. A comprehensive theory that provides a solid foundation in the subject matter for the practicing engineer, instructor, or scientist is needed. In general, the majority of technical information on sandwich structures is found in technical journals. Unfortunately, the theory varies among the various researchers. This is another reason to present a consistent understandable theory in text format. For these reasons, this text is being written. This text will expose scientists and engineers to the most advanced state-of-theart theory and response behavior. This text will serve mainly as a theoretical reference in conjunction with various applications. The scientist and practicing engineer will be exposed to several theoretical aspects and their solution techniques. These techniques can serve as a basis for other types of structures such as composite structures, monocoque plates and shells, thin-walled beams, etc. Also, this text will serve to better understand how certain variables can affect the design of components which are of the sandwich-type construction. Additional benefits for the interested reader are it will provide a theoretical tool which will provide an understanding of the response and instability of advanced sandwich structures exposed to complex loading conditions. It will discuss and present both the incompressible and compressible core theories. The idea of vii

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Preface

integrating functionally graded materials into these structures are presented. A better understanding of how to alleviate unwanted detrimental effects due to various complex loading conditions by the use of the structural tailoring technique is promoted. Many aspects discussed and presented in this book is of benefit to the interested reader. In addition, it will provide a basic solid mechanics foundation which can be applied to other types of structures. Chapter 2 presents the theoretical foundation for sandwich plates and shells with a transversely incompressible core. The theory is very comprehensive covering a broad spectrum of various specialized cases concerning both the linear and nonlinear theories with the inclusion of the geometrical imperfections. Also considered are the external loadings on the structure pertaining to the compressive/tensile edge loadings as well as the transverse pressure. The governing equations are developed via Hamilton’s principle which is an enormously powerful theoretical tool which can be used for a whole host of solid mechanics problems. Chapter two sets the tone for Chaps. 3, 4, 5 and 6. Chapters 3, 4, 5 and 6 contain applications of the theory developed in Chap. 2 such as the buckling problem, post buckling, free vibration, and the dynamic response problem. Chapter 3 considers the linear governing equations developed in Chap. 2 as it pertains to the buckling problem. These equations are then solved via an immensely powerful solution technique referred to as the Extended Galerkin Method (EGM). Several validations are made to validate the theory with various other numerical results which clearly reveal the response of the structure under various material and geometrical properties. The nonlinear governing equations from Chap. 2 are applied to the post buckling problem in Chap. 4 where three cases are considered. The first case considers cross-ply laminated sandwich plates and shells. The second case considers angle-ply laminated sandwich plates, and finally angle-ply sandwich shells are considered. The governing equations from Chap. 2 are highlighted then the details of the solution technique are presented which involves the EGM and Newton’s method to arrive at the post buckling response. Several results are presented considering the effect of the geometric imperfections, the effect of the curvature, the effect of the ply angle, the transverse pressure, biaxial edge loading, the end -shortening of the edges, etc. Chapter 5 addresses the free vibration problem considering the linear governing equations from chapter two with the inclusion of the inertia term. The governing equations are solved via the EGM with validations and results which follow. The results consider the anisotropy of the face sheets, the stacking sequence of the face sheet lamina, the fiber orientation of the face sheets, the orthotropic properties of the core, in addition to other important geometrical and material properties. All of these results are highlighted with respect to the effects on the eigenfrequencies of the structure. Chapter 6 addresses the dynamic response of sandwich plates and shells, where in contrast to the free vibration problem, time-dependent external loadings are considered for the dynamic response problem. The governing equations from Chap. 2 as it applies to the dynamic response problem are solved via closed-form techniques such as the Laplace Transform and the EGM. To generate the results several external time-dependent loads are considered such as the sonic boom, triangular pulse,

Preface

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Heaviside step function, rectangular pulse, the sine pulse, a tangential traveling air blast, and the Friedlander in-Air explosive pulse, and a case for underwater blast loading. The highlighted results consider various geometrical and material parameters and their effect on the dynamic response of the structure. Chapters 2, 3, 4, 5 and 6 are devoted to the sandwich structure with a transversely incompressible core. In Chaps. 7 and 8 attention is given to the theory of sandwich plates and shells with a transversely compressible core which captures local and global wrinkling. Two very comprehensive detailed theories are addressed where the first theory considers that the core transverse displacement is modeled with a first order power series whereas the second theory considers that the core transverse displacement is modeled with a second order power series. The latter is considered in Chap. 8 while the former is addressed in Chap. 7. Both theories are identical in their approach to the development of the governing equations where Hamilton’s principle is utilized. Theory two requires more computational effort to produce the governing equations, as a result of an assumed higher order transverse core displacement. At the conclusion of both Chaps. 7 and 8 there is an application of the theory to highlight a solution technique to solving these governing equations. In Chap. 9, a nonlinear theoretical foundation considering the first order shear deformation theory for functionally graded sandwich plates and shells is presented. Two types of functionally graded sandwich structures are considered. The first type considers the that the face sheets are functionally graded while the core is homogeneous. The second case considers the opposite scenario where the face sheets are homogeneous, and the core is functionally graded. The theory employs Hamilton’s principle while considering the tangential and rotatory inertias. It is hoped that the material covered in this book sets a foundation upon which to build from. It should be noted that although very comprehensive and detailed theories have been presented which have enormous applications, this material was not meant to cover every single case or application in which these equations could be applied. There are still several areas that need to be addressed both for the incompressible and compressible core case. The theoretical tools have been provided in this text with the idea that the many other applications for sandwich structures can now be explored. Warren, MI, USA

Terry John Hause

Acknowledgements

I would like to thank God for the inspiration, motivation, and opportunity afforded me to write this book. I would also like to thank my dear wife Marilou and my dear son Jourdan for their understanding and patience during the writing of this book.

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Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Preliminary Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Composition of the Sandwich Structure . . . . . . . . . . . . . . . . . . . 1.3 Failure Modes of Sandwich Structures . . . . . . . . . . . . . . . . . . . . 1.4 Contents of the Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

Theory of Sandwich Plates and Shells with an Transversely Incompressible Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Preliminaries and Basic Assumption . . . . . . . . . . . . . . . . . . . . . 2.3 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Displacement Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Strain–Displacement Relationships . . . . . . . . . . . . . . . . . 2.3.3 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Stress and Moment Resultants . . . . . . . . . . . . . . . . . . . . 2.4 Hamilton’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Hamilton’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Strain Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Work Done by External Loads . . . . . . . . . . . . . . . . . . . . 2.5 Equations of Motion – Nonlinear Formulation . . . . . . . . . . . . . . 2.5.1 The Mixed Formulation for Sandwich Plate and Shells . . . 2.5.2 Displacement Formulation for Sandwich Plates and Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Displacement Formulation for Sandwich Plates . . . . . . . . 2.6 Equations of Motion – Linear Formulation . . . . . . . . . . . . . . . . . 2.6.1 Displacement Formulation for Sandwich Shells . . . . . . . . 2.6.2 Displacement Formulation for Sandwich Plates . . . . . . . .

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1 1 2 3 3 5

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7 7 8 9 9 11 18 21 25 25 26 29 30 32 32

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37 46 49 49 50

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2.7 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 52 53

3

Buckling of Sandwich Plates and Shells . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Preliminaries and Basic Assumptions . . . . . . . . . . . . . . . . . . . . . . 3.3 Buckling of Flat Sandwich Panels . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Governing System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Solution Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Validation of the Theoretical Results . . . . . . . . . . . . . . . . . 3.3.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Buckling of Doubly Curved Sandwich Panels . . . . . . . . . . . . . . . . 3.4.1 Governing System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Solution Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Validation of the Theoretical Results . . . . . . . . . . . . . . . . . 3.4.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Stress/Strain Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 55 55 56 56 58 61 62 65 65 68 71 72 75 77 78

4

Post-Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Preliminaries and Basic Assumptions . . . . . . . . . . . . . . . . . . . . . 4.3 Cross-Ply Laminated Sandwich Shells . . . . . . . . . . . . . . . . . . . . 4.3.1 Governing System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Solution Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Numerical Results and Discussion . . . . . . . . . . . . . . . . . 4.4 Angle-Ply Laminated Sandwich Plates . . . . . . . . . . . . . . . . . . . . 4.4.1 Governing System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Solution Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Numerical Results and Discussion . . . . . . . . . . . . . . . . . 4.5 Angle-Ply Laminated Sandwich Shells . . . . . . . . . . . . . . . . . . . . 4.5.1 Governing System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Solution Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Numerical Results and Discussion . . . . . . . . . . . . . . . . . 4.6 Immovability of the Edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 End-Shortening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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79 79 80 80 80 82 90 92 92 96 102 105 105 110 119 125 126 127 127

5

Free Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Preliminaries and Basic Assumptions . . . . . . . . . . . . . . . . . . . . . 5.3 Flat Sandwich Panels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Governing System . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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129 129 130 130 130

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5.3.2 Solution Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Validation of the Theoretical Structural Model . . . . . . . . . . 5.3.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Doubly Curved Sandwich Panels . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Governing System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Solution Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Validation of the Theoretical Model . . . . . . . . . . . . . . . . . 5.4.4 Present Results and Discussion . . . . . . . . . . . . . . . . . . . . . 5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

132 135 135 140 140 143 147 148 154 154

6

Dynamic Response to Time-Dependent External Excitations . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Preliminaries and Basic Assumptions . . . . . . . . . . . . . . . . . . . . . 6.3 Flat Sandwich Panels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Governing System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Solution Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Doubly Curved Sandwich Panels . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Governing System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Solution Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Explosive Pressure Pulses and Numerical Results . . . . . . . . . . . . 6.5.1 Sonic Boom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Rectangular Pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.3 Heaviside Pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.4 Sine Pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.5 Tangential Blast Pulse . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.6 Friedlander Explosive Blast Pulse . . . . . . . . . . . . . . . . . . 6.5.7 Underwater Shock Pulse . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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155 155 156 156 156 157 160 160 162 164 165 167 170 170 173 178 181 182 182

7

Theory of Sandwich Plates and Shells with an Transversely Compressible Core – Theory One . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Preliminaries and Basic Assumptions . . . . . . . . . . . . . . . . . . . . . 7.3 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Displacement Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Nonlinear Strain–Displacement Equations . . . . . . . . . . . . 7.3.3 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Hamilton’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Strain Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Work Done by External Loads . . . . . . . . . . . . . . . . . . . . 7.4.3 Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . .

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183 183 184 185 185 188 192 193 193 199 203 204 204 208

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7.6 7.7 7.8

8

9

The Stress Resultants, Stress Couples, and Stiffnesses . . . . . . . . . Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Application – Dynamic Response of Flat Sandwich Panels . . . . . 7.8.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8.3 Solution Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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209 210 210 210 211 213 221 221

Theory of Sandwich Plates and Shells with an Transversely Compressible Core – Theory Two . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Preliminaries and Basic Assumptions . . . . . . . . . . . . . . . . . . . . . 8.3 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Displacement Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Nonlinear Strain–Displacement Equations . . . . . . . . . . . . 8.3.3 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Hamilton’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Strain Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Work Done by External Loads . . . . . . . . . . . . . . . . . . . . 8.4.3 Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Governing System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 The Stress Resultants, Stress Couples, and Stiffnesses . . . . . . . . . 8.7 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 Application – Buckling/Post-Buckling . . . . . . . . . . . . . . . . . . . . 8.8.1 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8.3 Solution Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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223 223 224 224 224 227 230 231 232 239 242 243 243 254 254 256 256 256 257 259 263

Theory of Functionally Graded Sandwich Plates and Shells . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Preliminaries and Basic Assumptions . . . . . . . . . . . . . . . . . . . . . 9.3 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Displacement Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Nonlinear Strain–Displacement Equations . . . . . . . . . . . . 9.3.3 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Hamilton’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Strain Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 Work Done by External Loads . . . . . . . . . . . . . . . . . . . . 9.4.3 Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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265 265 265 267 267 268 269 271 271 272 274

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9.5

275 275 276 279 279 280

Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1 Equations of Motion – Mixed Formulation . . . . . . . . . . . . 9.5.2 The Equations of Motion – The Displacement Formulation 9.5.3 The Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Final Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

Chapter 1

Introduction

Abstract Sandwich structures have several prominent roles in the aerospace industry such as their use in aircraft engines, the fuselage, the floor, side-panels, overhead bins, and the ceiling. These structures have many additional applications such as their use in naval ship bulkheads, deck houses, aircraft hangars, locomotive cabs, buses, satellites, automobiles, etc. With this in mind, a very intensive study of their structural performance is necessary under extreme complex loading conditions. As an introduction, a very preliminary overview of the composition of typical sandwich structures are discussed with regard to their structural elements. Secondly various applications of this type of structure are listed. A few historical applications are briefly mentioned with various types of failure modes, such as wrinkling, that sandwich structures can endure. Finally, an overview of the text is presented for the readers benefit.

1.1

Preliminary Overview

A sandwich structure is a three-layered structure composed of two facings with a thick core layer in-between. The core layer is multiple times the thickness of the facings. The facings are usually very thin and stiff which can be constructed from a variety of different kinds of materials. Some of the most common types are anisotropic laminated composites, a single-layered isotropic or orthotropic material, and functionally graded materials. Generally, the core is soft and only carries the transverse shear stresses. The other type of core is referred to as a strong core which carries both the tangential and transverse shear stresses. Typical types of core constructions are foam, honeycomb, web, and truss-type core construction. The former two are soft-core type while the latter two are of the strong-core type. Usually these type structures are exposed to in-plane and lateral loading. Sandwich structures have several applications. Overtime their use seems to keep increasing due to advanced manufacturing techniques and the introduction of new types of materials integrated into the structure, either within the facings or within the core. Some of the aeronautical applications of sandwich structures includes aircraft engines, the fuselage, the floor, side panels, overhead bins, and the ceiling. These © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 T. J. Hause, Sandwich Structures: Theory and Responses, https://doi.org/10.1007/978-3-030-71895-4_1

1

2

1 Introduction

structures have many additional applications such as use in naval ship bulkheads, deck houses, aircraft hangars, locomotive cabs, buses, satellites, automobiles, etc. According to Vinson (2005), during the last decade, sandwich structures have found their way into wind energy systems. GE Energy declares there are 6900 installations worldwide, while their growth rate is up to 20% annually. According to Vinson (2005) Germany, Spain, the US, and Denmark are the leaders in wind energy installations. These structures inherently have many benefits such as large bending stiffness, providing a smoother surface finish for aerodynamic applications, provide excellent sound and thermal insulation depending on the materials used in their construction, increased strength at elevated temperatures, increased operational time, and lightweight in construction. The application of sandwich construction has been around for several decades. The British de Havilland Mosquito bomber of World War II utilized the sandwichtype construction within the airframe. The face sheets were birch bonded to a balsa wood core. Other airplanes such as the B-58, B-70, F-111, C-5a, and many others employed this type of construction. The advantage is the high strength-to-weight ratio inherent within the structure. Various spacecraft have leveraged this type of structure such as the Apollo spacecraft, The Spacecraft LM Adapter fairings on the Centaur, and other launch vehicles. They were also used in propellant tank bulkheads.

1.2

Composition of the Sandwich Structure

Sandwich structures provide a wide range of facing and core material selections. The choice of materials depends on the application in which the structure will be utilized, the loading conditions, the availability, and the cost. As an example, in aerospace applications glass-epoxy or glass-vinyl ester are used in the facings of civil and marine structures. The core is often aluminum or Nomex honeycomb. In civil engineering applications, the core is usually a closed- or open-cell foam. Balsa core is used in ships according to Birman and Kardomateas (2018). There are four types of commonly used cores. They are foam, honeycomb, truss, and web-type core construction. Considering foam, there are several types of foam materials. The first is Polystyrene, which has a high compressive strength and resist water penetration. Then there is Phenolic, which is fire resistant, has a low density, and low mechanical properties. Next there is polyurethane. This is utilized for producing the fuselage, wing tips, and other curved parts of small aircraft. There is polystyrene which is used for airfoil shapes. Then there is Polyvinyl Chloride (PVC), which has a high compressive strength and durability and is fire resistant. Finally, there is Polymethacrylimide which is used for lightweight sandwich construction. The second type of core is the honeycomb-type structure. The cell-type structure can be round, square, or triangular. Some of the materials used in the construction of the honeycomb core are kraft paper, thermoplastics, aluminum, steel, titanium, aramid paper, fiberglass, carbon, and ceramics. Each has their own properties and

1.4 Contents of the Text

3

benefits depending on the application in which it will be used. The honeycomb-type core construction is used mainly in the aerospace industry. They are lightweight, flexible, fire retardant, have good impact resistance, and have the best strength-toweight ratios. The third and fourth types are the truss and web core which are used in civil engineering applications. Besides the traditional construction, current research shows that the latest designs in sandwich construction include new core concepts in design, incorporating nanotubes and smart materials, as well as functionally graded materials.

1.3

Failure Modes of Sandwich Structures

The structural instability of a sandwich structure can appear in any number of ways depending on the loading and construction. The first type of failure mode is Intracellular Buckling or face dimpling. This is a localized form of failure which occurs when the core is not continuous. Above the core cells, the facings buckle with the cell walls acting as edge supports. As this progresses, buckling ensues. The second type of failure is face wrinkling. This usually occurs if the core is compressible. A compressible core can carry normal or extensible straining thus stresses. This mode of failure appears as short wavelengths in the facings, which involves the normal to facings straining of the core. This wrinkling can by symmetrical or asymmetrical with respect to the mid-surface of the core before being deformed. Ultimate failure usually occurs by core crushing, tensile rupture of the core, or tensile rupture of the core-to-facing bond. The third type of failure is shear crimping. This is a general instability mode where the buckle wavelength is noticeably short due to a low transverse modulus for the core. This failure mode occurs quickly and causes failure due to shearing. The fourth and fifth types are the General Instability and the Panel Instability. With all of the applications and construction scenarios available to these type structures it is imperative that a good theoretical understanding of some of these concepts are understood. In the forthcoming chapters a comprehensive theoretical base is presented to obtain the pertinent response under various loading scenarios.

1.4

Contents of the Text

The governing equations for a sandwich plates and shells modeled with a transversely incompressible core are presented in Chap. 2. The kinematic equations such as the displacement field and the strain–displacement relationships are formulated for the case of thick doubly curved shells with the First-Order Shear Deformation theory in mind. The Green’s Strain Tensor is presented with the introduction of geometric imperfections. The constitutive equations are developed based on the generalized Hooke’s Law following with the developments of the equations of

4

1 Introduction

motion. Two forms of the equations of motion are presented. One is referred to as the mixed formulation and the other form is the displacement formulation. These equations of motion are developed via Hamilton’s Principle. As a byproduct from Hamilton’s Principle, the boundary conditions are simply supported and clamped boundary conditions. The pertinent governing equations are formulated for the general case of thick shells including the geometric imperfections, body forces and inertia terms, prescribed edge loadings, and the transverse loadings. In general, the body forces are neglected and considered negligible to the response of the structure. Chapters 3 and 4 provide the theoretical developments, the solution methodology, and in addition, some numerical results concerning the problem of buckling and post-buckling, respectively. In the case of buckling, which is the linearized counterpart of the post-buckling solution, validations are made against the theory with remarkable agreement. The presentation of the numerical results, in the case of post-buckling, are broken down into two categories, cross-ply and angle-ply laminated facings. Several issues are highlighted in both cases such as the effect of the geometric imperfections, the effect of the aspect ratio, the material directional properties in the face sheets, the effect of the panel face thickness, the effect of curvature, the snap-through-type behavior, the layup sequence of the facings (structural tailoring), the in-plane and tangential prescribed edge loadings, etc. In Chaps. 5 and 6, the free vibration and dynamic response of sandwich structures are investigated. The dynamic response considers various time-dependent external loadings for both sandwich plates and shells. The solution methodology leads to an eigenvalue problem which is conducive to a closed-form solution utilizing the Extended Galerkin Method (EGM) and the Laplace Transform Method (LTM) to determine both the frequency and the dynamic response. The effects of the panel curvature, anisotropy, structural tailoring, the orthotropy of the core, and the effects of damping are all considered in regard to the structural response. Various types of stimuli such as the sonic boom, the triangular pulse, Heaviside step function, the rectangular pulse, sine pulse, tangential traveling air blast, and the Friedlander in-air explosive pulse are considered. Chapter 7 presents the basic governing equations for the transversely compressible core model, whereby the core is considered extensible in the transverse direction. By modeling the core as extensible, the wrinkling phenomenon is captured which can lead to a local (face wrinkling) and global instability mode which is not captured by the incompressible core model. Chapter 8 is built upon Chap. 7 in that the core is modeled with a higher-order transverse displacement function to the second order, in contrast to Chap. 7 where the transverse displacement is considered linear. The theory is computationally more intensive but provides for an improved behavior of the extensibility of the core. Following the theoretical developments in Chap. 7, an application making use of the theory concerning the dynamic response is presented. A brief overview of how to apply and solve the equations is highlighted. In contrast, an application of the theory presented following the theoretical developments in Chap. 8 is discussed highlighting buckling and post-buckling. In both cases numerical results are omitted. Prominent authors such as Hohe and Librescu (2003, 2006) have generated and published several numerical results regarding this subject matter. The reader is referred to their

References

5

publications for the results. There has been a plethora of results compiled by these top researchers where a detailed analysis of both the local (wrinkling) and global bucking and post-buckling under various geometrical and loading scenarios have been presented and discussed in sufficient detail. In Chap. 9, the state-of-the-art components of functionally graded sandwich plates and shells are discussed. An introduction to functionally graded sandwich structures is provided where the governing equations are derived for a both symmetric and asymmetric cases. It is hoped that this text will provide some insight and promote a better understanding of sandwich structures under complex loading conditions for the practicing engineer, research scientist, or graduate student wanting to acquire knowledge with regard to these type structures.

References Birman, V., & Kardomateas, G. A. (2018). Review of current trends in research and applications of sandwich structures. Composites B, 142, 221–240. Hohe, J., & Librescu, L. (2003). A nonlinear theory for doubly curved anisotropic sandwich shells with transversely compressible core. International Journal of Solids and Structures, 40, 1059–1088. Hohe, J., & Librescu, L. (2006). Dynamic buckling of flat and curved sandwich panels with transversely compressible core. Composite Structures, 74, 10–24. Vinson, J. R. (2005). Sandwich structures: Past, present, and future. In O. T. Thomsen et al. (Eds.), Sandwich structures 7: Advancing with sandwich structures and materials. Dordrecht: Springer.

Chapter 2

Theory of Sandwich Plates and Shells with an Transversely Incompressible Core

Abstract An extremely robust geometrical nonlinear theory of initially imperfect doubly curved sandwich shells with an incompressible core is presented. The aspects of the theory consider dissimilar face sheets with imposed anisotropic laminated composite construction. The core is considered broad in its inherent properties by considering both the weak and strong core types. The influence of the geometric imperfections on the theoretical developments are also included which adds an additional level of complexity. The governing equations are simplified for various specialized cases for two formulations referred to as the mixed formulation and the displacement formulation. These formulations are then broken down individually each for plate and shell-type sandwich construction. Finally, the boundary conditions are briefly discussed.

2.1

Introduction

In this chapter, the sandwich structure is introduced with a very comprehensive and detailed presentation of the basic terminology, basic assumptions, and the governing nonlinear theory regarding the case for the transversely incompressible core. The derivation is provided in extreme detail which is then specialized for several different cases. The presentation of the governing equations commences with the highlights of the displacement field, the nonlinear Green–Lagrange strain displacement field with the introduction of geometrical imperfections along with the stress– strain relationships, based on the generalized Hooke’s Law. Following in a similar vein, the principles of shallow shell theory are adopted and the concept of the stress and stress couple resultants are presented. The equations of motion are derived via an energy approach known as Hamilton’s principle. The equations are built upon a very general base considering the strong core–type construction with transverse shear considered in the facings (thick facings). The equations are then reduced considering symmetry, a soft/weak core, with the adoption of the Love–Kirchhoff hypothesis where the facings are assumed thin whereby the transverse shear stresses can be neglected. Following these assumptions, the equations are further reduced

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 T. J. Hause, Sandwich Structures: Theory and Responses, https://doi.org/10.1007/978-3-030-71895-4_2

7

8

2 Theory of Sandwich Plates and Shells with an Transversely Incompressible Core

Fig. 2.1 The geometry of a doubly curved sandwich panel

considering the linear theory for both plates and shells with applications of these equations in mind. This chapter serves as a precursor to Chaps. 3, 4, 5, and 6.

2.2

Preliminaries and Basic Assumption

Referring to Fig. 2.1, the middle surface of the core becomes the global mid-surface of the structure which is referred to an orthogonal curvilinear coordinate system xi (i ¼ 1,2,3). The coordinate x3 is considered positive when measured in the downward normal direction. The thickness of the core is 2 h which is uniform throughout. The thicknesses of the bottom and top facings are h0 and h00 , respectively. This implies that H ( h0 + 2 h + h00 ) is the total thickness of the structure. For the identification purposes unless otherwise noted, quantities with single primes refer to the bottom face and those with double primes refer to the top face. These primes may be placed on the right or left of the respective quantity. The geometrically nonlinear theory of doubly curved sandwich panels is based on a series of assumptions which are as follows: The face sheets are constructed of a number of orthotropic material layers, the axes of orthotropy of the individual plies being not necessarily coincident with the geometrical axes xα (α ¼ 1,2) of the structure. 1. The thickness of the core is much larger than those of the face sheets, that is, 2h  h= , h== 2. The core material features orthotropic properties, the axes of orthotropy being parallel to the geometrical axes xα 3. The cases of both the weak and strong core type sandwich structures are considered. In the weak-core case, the core is capable of carrying transverse shear

2.3 Basic Equations

4. 5. 6. 7.

9

stresses only. Whereas in the strong core case, the core can carry both tangential and transverse shear stresses. A perfect bonding between the face sheets and between the faces and the core exists. All three layers, the facings and the core, are incompressible in the transverse direction. The geometrical nonlinearities in the von Kármán sense with the geometrical imperfections are included in the sandwich structure model. The principles of shallow shell theory apply.

2.3 2.3.1

Basic Equations Displacement Field

In line with the typical plate and shell theory, a power series expansion is utilized for the displacement field. For the case of the sandwich structure, with three different layers, each layer assumes its own displacement field. The First-Order Shear Deformation Theory will be assumed for each layer which is based on the Mindlin– Reissner Theory. Each layer is treated as a separate monocoque shell. It is assumed that the normal to the mid-surface remains straight but not necessarily normal after deformation. The distance measured from each layer’s respective mid-surface is measured with respect to the global mid-surface. The displacement field for a point in each layer is then given by
 Bottom Face h  x3  h þ h=    = V 1 ð x 1 , x 2 , x 3 Þ ¼ = V 1 ð x 1 , x 2 Þ þ x 3  a= ψ 1 ð x 1 , x 2 Þ    = = V 2 ð x 1 , x 2 , x 3 Þ ¼ = V 2 ð x 1 , x 2 Þ þ x 3  a= ψ 2 ð x 1 , x 2 Þ =

=

ð2:1a  cÞ

V 3 ð x1 , x2 , x3 Þ ¼ = v3 ð x1 , x2 Þ


 Core h  x3  h 

V 1 ðx1 , x2 , x3 Þ ¼ V 1 ðx1 , x2 Þ þ x3 ψ 1 ðx1 , x2 Þ 

V 2 ðx1 , x2 , x3 Þ ¼ V 2 ðx1 , x2 Þ þ x3 ψ 2 ðx1 , x2 Þ V 3 ðx1 , x2 , x3 Þ ¼ v3 ðx1 , x2 Þ
 Top Face h  h==  x3  h

ð2:2a  cÞ

10

2 Theory of Sandwich Plates and Shells with an Transversely Incompressible Core

   == V 1 ðx1 , x2 , x3 Þ ¼ == V 1 ðx1 , x2 Þ þ x3 þ a== ψ 1 ðx1 , x2 Þ    == == V 2 ðx1 , x2 , x3 Þ ¼ == V 2 ðx1 , x2 Þ þ x3 þ a== ψ 2 ðx1 , x2 Þ ==

== 

ð2:3a  cÞ

V 3 ðx1 , x2 , x3 Þ ¼ == v3 ðx1 , x2 Þ 

where = V α , V α , == V α (α ¼ 1, 2) are the tangential displacements for points which lie on the mid-surface and = ψ α , ψ α , == ψ α are the shear angles for points on the mid-surface. In addition, (a= ¼ h þ h= =2 ) and (a== ¼ h þ h== =2 ) which are the distances from the global mid-surface to the mid-surface of the bottom and top faces, respectively. At the interfaces between both facings and the core, continuity must exist regarding the displacements. For this reason, the kinematic continuity conditions must be fulfilled at these interfaces (see Librescu et al. (1997), Hause et al. (1998, 2000), Librescu (1975)). The kinematic continuity conditions can be expressed mathematically as
 Bottom face sheet/core interface x3 ¼ h V 1 ¼ = V 1 and V 2 ¼ = V 2

ð2.4a, bÞ


 Top face sheet/core interface x3 ¼ h V 1 ¼ == V 1 and V 2 ¼ == V 2

ð2.5a, bÞ

Because the core is considered incompressible, =

V 3 ¼ V 3 ¼ == V 3 ¼ v3 :

ð2:6Þ

When Eqs. (2.4a, b)–(2.6) are used in conjunction with Eqs. (2.1a–c)–(2.3a–c), the following 3D displacement relationships result fulfilling the kinematic continuity conditions.
 Bottom Face h  x3  h þ h=   = V 1 ð x 1 , x 2 , x 3 Þ ¼ ξ 1 ð x 1 , x 2 Þ þ η 1 ð x 1 , x 2 Þ þ x 3  a= ψ 1 ð x 1 , x 2 Þ   = = V 2 ð x 1 , x 2 , x 3 Þ ¼ ξ 2 ð x 1 , x 2 Þ þ η 2 ð x 1 , x 2 Þ þ x 3  a= ψ 2 ð x 1 , x 2 Þ =

=

V 3 ðx1 , x2 , x3 Þ ¼ v3 ðx1 , x2 Þ


 Core h  x3  h

ð2:7a  cÞ

2.3 Basic Equations

11

V 1 ðx1 , x2 , x3 Þ ¼ξ1 ðx1 , x2 Þ 

1 4

h

=

=

==

hψ 1 ðx1 , x2 Þ  == hψ 1 ðx1 x2 Þ

i

h io
n 1 = == þ x3 =h η1 ðx1 , x2 Þ  = hψ 1 ðx1 , x2 Þ þ == hψ 1 ðx1 , x2 Þ 4 h i 1 = == V 2 ðx1 , x2 , x3 Þ ¼ ξ2 ðx1 , x2 Þ  = hψ 2 ðx1 , x2 Þ  == hψ 2 ðx1 , x2 Þ 4 n h io
 1 = == þ x3 =h η2 ðx1 , x2 Þ  = hψ 2 ðx1 , x2 Þ þ == hψ 2 ðx1 , x2 Þ 4 V 3 ð x1 , x2 , x3 Þ ¼ v3 ð x1 , x2 Þ

ð2:8a  cÞ


 Top Face h  h==  x3  h   == V 1 ðx1 , x2 , x3 Þ ¼ ξ1 ðx1 , x2 Þ  η1 ðx1 , x2 Þ þ x3 þ a== ψ 1 ðx1 , x2 Þ   == == V 2 ðx1 , x2 , x3 Þ ¼ ξ2 ðx1 , x2 Þ  η2 ðx1 , x2 Þ þ x3 þ a== ψ 2 ðx1 , x2 Þ ==

==

ð2:9a  cÞ

V 3 ð x1 , x2 , x3 Þ ¼ v3 ð x1 , x2 Þ

Within the above displacement equations, ξ1(x1, x2), ξ2(x1, x2), η1(x1, x2), η2(x1, x2) are defined as: ξ 1 ð x1 , x2 Þ ¼

=

=









= V 1 þ == V 1 V þ == V 2 , ξ2 ðx1 , x2 Þ ¼ 2 2 2 





ð2.10a, bÞ



= V  == V 1 V  == V 2 , η2 ðx1 , x2 Þ ¼ 2 η1 ðx1 , x2 Þ ¼ 1 2 2

ð2.11a, bÞ

With these newly defined 2D tangential displacement measures in hand, the problem is reduced from 15 displacement quantities to 9 which are: ξα ðx1 , x2 Þ, ηα ðx1 , x2 Þ, v3 ðx1 , x2 Þ,

2.3.2

=

ψ α ðx1 , x2 Þ,

==

ψ α ðx1 , x2 Þ,

ðα ¼ 1, 2Þ

Strain–Displacement Relationships

The strain–displacement relationships at a point in the structure are given by the Green–Lagrange strain tensor for a shallow shell. With the assumption that the transverse displacements are much larger than the in-plane or tangential displacements, in addition to the geometric imperfections, only the nonlinear terms, associated with the transverse displacements, are retained in the Green–Lagrange strain tensor (Librescu and Chang (1993)). These expressions are given by

12

2 Theory of Sandwich Plates and Shells with an Transversely Incompressible Core

e11

 2 ∂V 1 V 3 1 ∂V 3 ¼  þ ∂x1 R1 2 ∂x1

ð2:12Þ

e22

 2 ∂V 2 V 3 1 ∂V 3 ¼  þ ∂x2 R2 2 ∂x2

ð2:13Þ

∂V 2 ∂V 1 ∂V 3 ∂V 3 þ þ ∂x1 ∂x2 ∂x1 ∂x2

ð2:14Þ

γ 12 ¼

γ 13 ¼ 2e13 ¼

∂V 1 ∂V 3 ∂V 3 ∂V 3 þ þ ∂x3 ∂x1 ∂x1 ∂x3

ð2:15Þ

∂V 2 ∂V 3 ∂V 3 ∂V 3 þ þ ∂x3 ∂x2 ∂x2 ∂x3  2 ∂V 3 1 ∂V 3 ¼ þ ∂x3 2 ∂x3

γ 23 ¼ 2e23 ¼ e33

ð2:16Þ ð2:17Þ 

With the introduction of a stress free initial geometric imperfection V 3 , in Eqs. (2.12)–(2.17), The Green–Lagrange strain tensor for finite deformations with imperfections becomes as follows: e11 ¼

 2  ∂V 1 V 3 1 ∂V 3 ∂V 3 ∂V 3  þ þ ∂x1 R1 2 ∂x1 ∂x1 ∂x1

ð2:18Þ

e22 ¼

 2  ∂V 2 V 3 1 ∂V 3 ∂V 3 ∂V 3  þ þ ∂x2 R2 2 ∂x2 ∂x2 ∂x2

ð2:19Þ



2e12



∂V 2 ∂V 1 ∂V 3 ∂V 3 ∂V 3 ∂V 3 ∂V 3 ∂V 3 ¼ þ þ þ þ ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x2 ∂x1 

2e13 ¼

ð2:20Þ



∂V 1 ∂V 3 ∂V 3 ∂V 3 ∂V 3 ∂V 3 ∂V 3 ∂V 3 þ þ þ þ ∂x3 ∂x1 ∂x1 ∂x3 ∂x1 ∂x3 ∂x3 ∂x1

ð2:21Þ

2.3 Basic Equations

13 

2e23 ¼



∂V 2 ∂V 3 ∂V 3 ∂V 3 ∂V 3 ∂V 3 ∂V 3 ∂V 3 þ þ þ þ ∂x3 ∂x2 ∂x2 ∂x3 ∂x2 ∂x3 ∂x3 ∂x2

e33

 2  ∂V 3 1 ∂V 3 ∂V 3 ∂V 3 ¼ þ þ ∂x3 2 ∂x3 ∂x3 ∂x3

ð2:22Þ

ð2:23Þ

substituting Eqs. (2.7a–c)–(2.9a–c) into Eqs. (2.18)–(2.23) one obtains the 3D strain quantities in terms of the 2D strain measures for each respective layer of the sandwich structure. These 3D strain quantities are given as
 Bottom Facings h  x3  h þ h=   = e11 ¼ = ε11 þ x3  a= κ 11   = = e22 ¼ = ε22 þ x3  a= κ 22   = 2= e12 ¼ = γ 12 þ x3  a= κ12 =

=

ð2:24a  fÞ

=

2 e13 ¼ γ 13 2= e23 ¼ = γ 23 =

e33 ¼ 0


 Core h  x3  h e11 ¼ ε11 þ x3 κ11 e22 ¼ ε22 þ x3 κ22 2e12 ¼ ε12 þ x3 κ 12 2e13 ¼ γ 13

ð2:25a  fÞ

2e23 ¼ γ 23 e33 ¼ 0
 Top Face h  h==  x3  h   == e11 ¼ == ε11 þ x3 þ a== κ11   == == e22 ¼ == ε22 þ x3 þ a== κ22   == == 2e12 ¼ == γ 12 þ x3 þ a== κ12 ==

==

2e13 ¼ == γ 13

==

2e23 ¼ == γ 23

==

e33 ¼ 0

ð2:26a  fÞ

14

2 Theory of Sandwich Plates and Shells with an Transversely Incompressible Core =

==

=

==

=

==

In the above equations, εij , εij , εi3 , εi3 , γ ij , γ ij and
εi3 ði ¼ 1, 2, 3Þ represent = == the 2D tangential and transverse strain measures, respectively. While κij and κij represent the bending strains. Their expressions are given as
 Bottom Face h  x3  h þ h=  2 ∂ξ1 ∂η1 1 ∂v3 ∂v þ þ þ 3 ∂x1 ∂x1 2 ∂x1 ∂x1  2 ∂η ∂ξ 1 ∂v3 ∂v ¼ 2þ 2þ þ 3 ∂x2 ∂x2 2 ∂x2 ∂x2

ε11 ¼

∂v3 v3  ∂x1 R1

=

ε22

∂v3 v3  ∂x2 R2



=

γ 12



=

ð2:27Þ



ð2:28Þ 

∂v ∂v ∂η ∂η ∂ξ ∂ξ ∂v ∂v ∂v ∂v ¼ 1þ 2þ 1þ 2þ 3 3þ 3 3þ 3 3 ∂x2 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2

=

ð2:29Þ

=

γ 13 ¼ = ψ 1 þ

∂v3 ∂x1

ð2:30Þ

=

γ 23 ¼ = ψ 2 þ

∂v3 ∂x2

ð2:31Þ

=

κ11 ¼

∂= ψ 1 ∂x1

ð2:32Þ

=

κ22 ¼

∂= ψ 2 ∂x2

ð2:33Þ

∂= ψ 1 ∂= ψ 2 þ ∂x2 ∂x1

ð2:34Þ

κ12 ¼


 Core h  x3  h

ε11

=

==

∂ψ 1 ∂ξ 1 = ∂ψ 1 h ¼ 1  h== ∂x1 4 ∂x1 ∂x1

! þ

 2  ∂v ∂v 1 ∂v3 v þ 3 3 3 2 ∂x1 ∂x1 ∂x1 R1

ð2:35Þ

!  2  = == ∂v3 ∂v3 v3 ∂ψ ∂ξ2 1 = ∂ψ 2 1 ∂v3 == 2 ε22 ¼  h þ  ð2:36Þ h þ 2 ∂x2 ∂x2 4 ∂x2 ∂x2 ∂x2 ∂x2 R2 " ! !# = = == == ∂ψ 2 ∂ξ1 ∂ξ2 1 = ∂ψ 1 ∂ψ 2 ∂v == ∂ψ 1 γ 12 ¼ þ  h þ þ h þ 3 ∂x2 ∂x1 4 ∂x2 ∂x1 ∂x2 ∂x1 ∂x1 





∂v3 ∂v3 ∂v3 ∂v3 ∂v3 þ þ ∂x2 ∂x1 ∂x2 ∂x1 ∂x2

ð2:37Þ

2.3 Basic Equations

15

n  o 1 1 ∂v = == η1  h= ψ 1 þ h== ψ 1 þ 3 4 ∂x1 h n  o 1 1 ∂v = == γ 23 ¼ η2  h= ψ 2 þ h== ψ 2 þ 3 4 ∂x2 h ( !) = == 1 ∂η1 1 = ∂ψ 1 == ∂ψ 1 h κ 11 ¼  þh ∂x1 ∂x1 h ∂x1 4 γ 13 ¼

κ 22

κ12

ð2:38Þ ð2:39Þ ð2:40Þ

( !) = == 1 ∂η2 1 = ∂ψ 2 == ∂ψ 2 h ¼  þh ∂x2 ∂x2 h ∂x2 4

ð2:41Þ

( " ! ! #) = = == == ∂ψ 2 1 ∂η1 ∂η2 1 = ∂ψ 1 ∂ψ 2 == ∂ψ 1 ¼ þ  h þ þ þh ∂x2 ∂x1 ∂x12 ∂x1 h ∂x2 ∂x1 4

ð2:42Þ


 Top Face h  h==  x3  h

==

ε11 ¼

 2  ∂ξ1 ∂η1 1 ∂v3 ∂v ∂v v  þ þ 3 3 3 ∂x1 ∂x1 2 ∂x1 ∂x1 ∂x1 R1

ð2:43Þ

==

ε22 ¼

 2  ∂ξ2 ∂η2 1 ∂v3 ∂v ∂v v  þ þ 3 3 3 ∂x2 ∂x2 2 ∂x2 ∂x2 ∂x2 R2

ð2:44Þ



==

γ 12 ¼



∂ξ1 ∂ξ2 ∂η1 ∂η2 ∂v3 ∂v3 ∂v3 ∂v3 ∂v3 ∂v3 þ   þ þ þ ∂x2 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2

==

ð2:45Þ

==

γ 13 ¼ == ψ 1 þ

∂v3 ∂x1

ð2:46Þ

==

γ 23 ¼ == ψ 2 þ

∂v3 ∂x2

ð2:47Þ

==

κ11 ¼

∂== ψ 1 ∂x1

ð2:48Þ

==

κ22 ¼

∂== ψ 2 ∂x2

ð2:49Þ

∂== ψ 1 ∂== ψ 2 þ ∂x2 ∂x1

ð2:50Þ

κ 12 ¼

16

2 Theory of Sandwich Plates and Shells with an Transversely Incompressible Core

Up to this point it has been assumed that the facings are thick enough to contain transverse shear stresses of significant value. If the facings are regarded as being inherently thin the transverse shear stresses can be neglected. This is based on the Love–Kirchhoff assumption. By setting the transverse shear stresses to zero in Eqs. (2.30), (2.31), (2.46), and Eq. (2.47) the shear angles become =

ψ 1 ¼ == ψ 1 ¼ 

∂v3 ∂v and = ψ 2 ¼ == ψ 2 ¼  3 : ∂x1 ∂x2

ð2.51a, bÞ

In addition, if the sandwich structure is globally symmetric (with respect to the global mid-surface) and both of the facings share the same thickness. This implies that
 h= ¼ h== ¼ h and a= ¼ a== ¼ a  h þ h=2 :

ð2.52a, bÞ

With the Love–Kirchhoff assumption and assuming local and global symmetry, the strain-displacement Eqs. (2.27)–(2.50) become
 Bottom Face h  x3  h þ h=

=

=

 2 ∂ξ1 ∂η1 1 ∂v3 ∂v þ þ þ 3 ∂x1 ∂x1 2 ∂x1 ∂x1  2 ∂η ∂ξ 1 ∂v3 ∂v ¼ 2þ 2þ þ 3 ∂x2 ∂x2 2 ∂x2 ∂x2

∂v3 v3  ∂x1 R1

ε22

∂v3 v3  ∂x2 R2



=

γ 12



ε11 ¼

ð2:53Þ



ð2:54Þ 

∂v ∂v ∂η ∂η ∂ξ ∂ξ ∂v ∂v ∂v ∂v ¼ 1þ 2þ 1þ 2þ 3 3þ 3 3þ 3 3 ∂x2 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2

=


 Core h  x3  h

ð2:55Þ

=

γ 13 ¼ 0

ð2:56Þ

=

γ 23 ¼ 0

ð2:57Þ

2

=

κ11 ¼ 

∂ v3 ∂x21

=

κ22 ¼ 

∂ v3 ∂x22

ð2:58Þ

2

ð2:59Þ

2

κ12 ¼ 2

∂ v3 ∂x1 ∂x2

ð2:60Þ

2.3 Basic Equations

17

γ 12 ¼

ε11 ¼

 2 ∂v∘ ∂v3 ∂ξ1 1 ∂v3 þ þ 3 ∂x1 2 ∂x1 ∂x1 ∂x1

ð2:61Þ

ε22 ¼

 2 ∂v∘ ∂v3 ∂ξ2 1 ∂v3 þ þ 3 ∂x2 2 ∂x2 ∂x2 ∂x2

ð2:62Þ

∂ξ1 ∂ξ2 ∂v3 ∂v3 ∂v∘3 ∂v3 ∂v3 ∂v∘3 þ þ þ þ ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2

ð2:63Þ

κ 12

κ11

 2 1 ∂η1 h ∂ v3 ¼ þ h ∂x1 2 ∂x21

ð2:64Þ

κ22

 2 1 ∂η2 h ∂ v3 ¼ þ h ∂x2 2 ∂x22

ð2:65Þ

 2 1 ∂η1 ∂η2 ∂ v3 ¼ þ þh ∂x1 ∂x2 h ∂x2 ∂x1

ð2:66Þ

 1 h ∂v3 ∂v η1 þ þ 3 2 ∂x1 ∂x1 h

ð2:67Þ

 1 h ∂v3 ∂v ¼ η2 þ þ 3 2 ∂x2 ∂x2 h

ð2:68Þ

γ 13 ¼

γ 23


 Top Face h  h==  x3  h

==

==

 2 ∂ξ1 ∂η1 1 ∂v3 ∂v  þ þ 3 ∂x1 ∂x1 2 ∂x1 ∂x1  2 ∂η ∂ξ 1 ∂v3 ∂v ¼ 2 2þ þ 3 ∂x2 ∂x2 2 ∂x2 ∂x2



ε11 ¼

∂v3 v3  ∂x1 R1

ε22

∂v3 v3  ∂x2 R2

ð2:69Þ



ð2:70Þ

18

2 Theory of Sandwich Plates and Shells with an Transversely Incompressible Core



==

γ 12 ¼

==

2.3.3



∂ξ1 ∂ξ2 ∂η1 ∂η2 ∂v3 ∂v3 ∂v3 ∂v3 ∂v3 ∂v3 þ   þ þ þ ∂x2 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2

ð2:71Þ

==

γ 13 ¼ 0

ð2:72Þ

==

γ 23 ¼ 0

ð2:73Þ

2

==

κ11 ¼ 

∂ v3 ∂x21

==

κ22 ¼ 

∂ v3 ∂x22

ð2:74Þ

2

ð2:75Þ

2

κ 12 ¼ 2

∂ v3 ∂x1 ∂x2

ð2:76Þ

Constitutive Equations

Within the 3D geometrically nonlinear elasticity theory, the constitutive equations are described by the linear relationship between the second Piola–Kirchhoff stress and Lagrange strain tensor. As a result, for an anisotropic material featuring monoclinic symmetry Hooke’s Law for the kth layer of a composite lamina can be described by (see Reddy 2004 and Jones 1999) 9 9 9 8 8 2 38 b 12 Q b 16 > b 11 Q > > Q λ11 > = = = < σ 11 > < e11 >

> > > > > ; ; : b 16 Q b 26 Q b 66 : γ 12 ;k : b σ 12 k Q λ12 k k 9 8 b > = < β11 >  b β22 ΔM > > ; : b β12 k " #  b b 45 2e23 σ 23 Q 2 Q44 ¼K b 45 Q b 55 σ 13 k 2e13 k Q

ð2:77aÞ

ð2:77bÞ

k

b ij , b where, Q λij , b μij are termed the reduced elastic moduli, the reduced thermal moduli, and the reduced moisture moduli, respectively. The expressions for these quantities are provided as

2.3 Basic Equations

19


 b k ¼ Qk cos 4 θk þ 2 Qk þ 2Qk sin 2 θk cos 2 θk þ Qk sin 4 θk Q 11 11 12 66 22
k 
 k b ¼ Q þ Qk  4Qk sin 2 θk cos 2 θk þ Qk sin 4 θk þ cos 4 θk Q 12 11 22 66 12
 b k ¼ Qk sin 4 θk þ 2 Qk þ 2Qk sin 2 θk cos 2 θk þ Qk cos 4 θk Q 22 11 12 66 22
k 
k  k k k 3 k b ¼ Q  Q  2Q sin θk cos θk þ Q  Q þ 2Qk sin 3 θk cos θk Q 16 11 12 66 12 22 66

 b k ¼ Qk  Qk  2Qk sin 3 θk cos θk þ Qk  Qk þ 2Qk sin θk cos 3 θk Q 26 11 12 66 12 22 66

 b k ¼ Qk þ Qk  2Qk  2Qk sin 2 θk cos 2 θk þ Qk sin 4 θk þ cos 4 θk Q 66 11 22 12 66 66 b k ¼ Qk cos 2 θk þ Qk sin 2 θk Q 44 44 55
k  k k b Q45 ¼ Q55  Q44 cos θk sin θk b k ¼ Qk cos 2 θk þ Qk sin 2 θk Q 55 55 44 k k k b bk b b k αk þ Q bk b λ11 ¼ Q 11 α11 þ Q12 b 16 α12 22 k k k b bk b b k αk þ Q bk b λ22 ¼ Q 22 α22 þ Q12 b 26 α12 22 k k k b bk b b k αk þ Q bk b λ12 ¼ Q 16 α11 þ Q26 b 66 α12 22 k

k

k

b b b b b b b μk11 ¼ Q 11 β 11 þ Q12 β22 þ Q16 β12 k

k

k

b b b b b b b μk22 ¼ Q 22 β 22 þ Q12 β11 þ Q26 β12 k

k

k

k

k

k

k bk b b k bk b k bk b μk12 ¼ Q 16 β 11 þ Q26 β22 þ Q66 β12

b αk11 ¼ αk11 cos 2 θk þ αk22 sin 2 θk b αk22 ¼ αk22 cos 2 θk þ αk11 sin 2 θk
 b αk12 ¼ αk11  αk22 cos θk sin θk k b β11 ¼ βk11 cos 2 θk þ βk22 sin 2 θk k b β22 ¼ βk22 cos 2 θk þ βk11 sin 2 θk
 k b β12 ¼ βk11  βk22 cos θk sin θk

ð2:78a  uÞ where the material stiffnesses are Qk11 ¼

E k1 , 1  νk12 νk21

Q22 ¼

E k2 νk21 E k1 νk12 E k2 , Q ¼ ¼ , 12 1  νk12 νk21 1  νk12 νk21 1  νk12 νk21

Qk44 ¼ Gk23 ,

Qk55 ¼ Gk13 ,

Qk66 ¼ Gk12 ð2:79a  fÞ

20

2 Theory of Sandwich Plates and Shells with an Transversely Incompressible Core

While Ei, νij, Gij, Gi3 (i, j ¼ 1, 2) are the Young’s Modulus, Poisson’s Ratio, Shear Modulus of the facings, and the core shear moduli, respectively; While βij, αij are the moisture expansion coefficient and the coefficient of thermal expansion, respectively. Assigning the constitutive equations to each layer of the sandwich panel results in
 Bottom Face h  x3  h þ h= 9 8 2 3 = = 9 9 8 > > b b 11 = Q b 12 = Q b 16 8 = = > > λ Q 11 σ e > > > > > > = < < 11 = 6 7 < 11 = 6= 7 = = = = = b b 12 Q b 22 Q b 26 7 ¼6 Q  σ 22 e22 λ22 > ΔT > > > 4 5> > ; ; > := > : = > = ; σ 12 k γ 12 k : =b > b 16 = Q b 26 = Q b 66 Q λ 12 k k 9 8 = > μ11 > > > = < b =  b μ22 ΔM > > > ; := > b μ12 k (

=

σ 23 = σ 13

2

) k

=

b 44 Q ¼ =K24 = b 45 Q

3( ) = b 45 Q 2 e 23 5 2= e13 k b Q

ð2:80aÞ

=

55

ð2:80bÞ

k


 Core h  x3  h 9 2 8 Q11 > = < σ 11 > 6 σ 22 ¼ 4 Q12 > > ; : σ 12 0

Q22 0

9 8 9 9 8 38 > > = > = = < e11 > < λ11 > < μ11 > 7 0 5 e22  λ22 ΔT  μ22 ΔM > > > > > ; > ; ; : : : γ 12 μ12 Q66 λ12





Q12

σ 23 σ 13

0

" ¼K


 Top Face h  h==  x3  h

2

Q44 0

0 Q55

# k

2e23

ð2:81aÞ



2e13

9 8 2 3 == == == == 9 9 8 8 > > b b b b == == > > λ Q Q Q 11 12 16 7 > > > = = = < 11 > < σ 11 > < e11 > 6 6 7 == == == == == b b 12 b 22 b 26 7 == e22 ¼6 Q  ΔT σ 22 Q Q λ 22 > > > > > 4 5> > ; ; > : == : == > > == σ 12 k γ 12 k : ==b ; b 16 == Q b 26 == Q b 66 Q λ12 k k 9 8 == > μ11 > > > = < b ==  b μ22 ΔM > > > ; : == > b μ12 k

ð2:81bÞ

ð2:82aÞ

2.3 Basic Equations

(

2.3.4

21 ==

σ 23 == σ 13

2

) k

3( ) == b 45 Q 2 e 23 5 == 2== e13 k b Q ==

==

b 44 Q ¼ == K 2 4 == b 45 Q

55

ð2:82bÞ

k

Stress and Moment Resultants

By definition and consistent with shallow shell theory, the stress resultants for each layer of the sandwich structure are defined below. Unless otherwise stated, (α, β ¼ 1, 2).
 Bottom Face h  x3  h þ h= n= Z n o X = = N αβ , M αβ ¼ k¼1

ðx3 Þk



=

ðx3 Þk1

σ αβ

n= Z n o X = N α3 ¼

n  o 1, x3  a= dx3 k



ð x3 Þ k

=

 dx3

ð2:83bÞ

σ αβ f1, x3 gdx3

ð2:84aÞ

σ α3 dx3

ð2:84bÞ

n  o 1, x3 þ a== dx3

ð2:85aÞ

ðx3 Þk1

k¼1

σ α3

ð2:83aÞ

k


 Core h  x3  h

 N αβ , M αβ ¼

Z

h

h

Z N α3 ¼

h

h


 Top Face h  h==  x3  h n

== == N αβ , M αβ

o

¼

n== Z X k¼1

ðx3 Þk



==

ðx3 Þk1

σ αβ

n== Z n o X == N α3 ¼ k¼1

k

ðx3 Þk

ðx3 Þk1



==

σ α3

 k

dx3

ð2:85bÞ

22

2 Theory of Sandwich Plates and Shells with an Transversely Incompressible Core

It should be noted that (x3)k and (x3)k  1 denote the distances from the global mid-surface to the upper and bottom interfaces of the kth layer, respectively. These definitions are similar to the ones defined in Librescu (1970, 1975). Substituting Eqs. (2.80a) and (2.80b) into Eqs. (2.83a) and (2.83b) gives the stress and stress couple resultants in terms of the strain measures for the bottom face of the sandwich panel as
 Bottom Face h  x3  h þ h= 9 2 8 = = > N 11 > A11 = A12 = A16 = E11 > > > > 6 > > > = = > > N > A12 = A22 = A26 = E12 > 6 > > > 6 > = 22 > = < 6 = = = = N 12 6 A16 A26 A66 E16 ¼ 6 = = = = = > 6 E 11 E 12 E16 F 11 > M 11 > > > > 6 > > > > 6 =E = > > M > > 4 12 = E 22 = E26 = F 12 22 > > > > ; := = M 12 E 16 = E 26 = E66 = F 16 9 8 9 8 = T > = m > > N 11 > > N 11 > > > > > > > > > > > = T > = m > > > > > N 22 > N 22 > > > > > > > > > > = > = < =NT > < =Nm > 12  = 12  T > > > > =Mm > > > > > M 11 > > 11 > > > > > > > > > T m = = > > > > > M 22 > > M 22 > > > > > > > > ; := T ; := m > M 12 M 12 (

= =

)

N 23

" =

¼ K

N 13

2

= =

A44 A45

= = = = = =

9 38 > = ε11 > E16 > > > > 7> > > > = = > 7 > ε E26 7> 22 > > > > > < 7 = E66 7 = γ 12 = 7 = > = κ11 > F 16 7 > > > 7> > > > =κ > 7 = > > F 26 5> > 22 > > > > ; : = = κ12 F 66 =

E 12 E 22 E 26 F 12 F 22 F 26

ð2:86aÞ

= =

#( A45 A55

=

γ 23

=

γ 13

) ð2:86bÞ

where the stiffnesses are defined as n

n= Z o X A=ωρ , B=ωρ , D=ωρ ¼

ðx3 Þk

ðx3 Þk1

k¼1

 = 
 b Q 1, x3 , x23 dx3 ωρ k

ðω, ρ ¼ 1, 2, 6Þ ð2:87aÞ

= AIJ

¼

n= Z X k¼1

ðx3 Þk

ðx3 Þk1

 = b Q dx3 IJ k

ðI, J ¼ 4, 5Þ

ð2:87bÞ

2.3 Basic Equations

23

 2 F =ωρ ¼ D=ωρ  2a= B=ωρ þ a= A=ωρ

E =ωρ ¼ B=ωρ  a= A=ωρ ,

ð2.87c, dÞ

and the thermal stress and stress couple resultants are defined as =

N Tαβ

n= Z X

¼

ðx3 Þk

ðx3 Þk1

k¼1

n Z X =

=

=

M Tαβ

Nm αβ

¼

¼

Mm αβ

¼

k



ðx3 Þk

k¼1

ðx3 Þk1

n= Z X

ð x3 Þ k

k¼1 =

=  bλαβ ΔTdx3

k¼1

k

=

ðx3 Þk1

n= Z X

ðx3 Þk

ðx3 Þk1

ð2.88a, bÞ

=  b x3  a λαβ ΔTdx3 =



b βαβ

 k

ΔMdx3

x3  a

=

=

b βαβ

 k

ð2.89a, bÞ ΔMdx3


 Core h  x3  h Substituting Eqs. (2.81a) and (2.81b) into Eqs. (2.84a) and (2.84b) gives the stress and stress couple resultants for the strong core layer in terms of the strain measures which are determined to be 9 8 2 Q11 > = < N 11 > 6 N 22 ¼ 2h4 Q12 > > ; : N 12 0

Q12 Q22 0

9 8 2 Q11 > = < M 11 > 2 36 M 22 ¼ h 4 Q12 > > ; 3 : M 12 0

Q12 Q22

(

)

N 23 N 13

0

9 8 T 9 8 m 9 38 > = > = = > < ε11 > < N 11 > < N 11 > 7 m T 0 5 ε22  N 22  N 22 > > > > ; > ; ; > : : : γ 12 Q66 0 0

ð2:90aÞ

9 8 T 9 8 m 9 38 0 > = > = > = < κ11 > < M 11 > < M 11 > 7 m 0 5 κ22  M T22  M 22 > > > > ; > ; > ; : : : κ12 Q66 0 0

ð2:90bÞ

0

" ¼ 2hK

2

Q44

0

0

Q55

#

γ 23 γ 13

ð2:90cÞ

where Qij (i, j ¼ 1, 2, 6) are given by Eqs. (2.79a–f). Finally, the thermal stress and stress couple resultants in Eqs. (2.90a) and (2.90b) are expressed as

24

2 Theory of Sandwich Plates and Shells with an Transversely Incompressible Core



 Z h T T N αβ , M αβ ¼ ð1, x3 Þλαβ ΔTdx3

ð2:91aÞ

h



m m N αβ , M αβ



Z ¼

h

h

ð1, x3 Þμαβ ΔMdx3

ð2:91bÞ


 Top Face h  h==  x3  h Substituting Eqs. (2.82a) and (2.82b) into Eqs. (2.85a) and (2.85b) gives the stress and stress couple resultants in terms of the strain measures for the top face as 9 2 8 3 == == > N 11 > A11 == A12 == A16 == E11 == E12 == E 16 > > > > 7 > 6 == > > == > > 6 A12 == A22 == A26 == E12 == E22 == E 26 7 N 22 > > > > > 7 6 > > == < == == N 12 = 6 A26 == A66 == E16 == E26 == E 66 7 7 6 A16 ¼ 6 == 7 == == == == == == > > 6 M E E E F F F > 6 > 11 > 11 12 16 11 12 16 7 > 7 > > > > 7 6 == E == == == == == == > > M E E F F F > > 5 4 22 12 22 26 12 22 26 > > > > ; : == == == == == == == M 12 E 16 E26 E 66 F 16 F 26 F 66 9 8 9 8 9 8 == T == m > == > > ε11 > N 11 > > > > > > N 11 > > > > > > > > > > > == > > > > > > > > > T == == > > > > > > ε N Nm 22 > > > > 22 > 22 > > > > > > > > > > > > = < == γ = < == N T = < == N m > 12   == 12  == 12 == m > > > > κ11 > M T11 > M 11 > > > > > > > > > > > > > > > > > > > > > > > == T > == m > == > > > > κ > > > > > > 22 > > M 22 > > M 22 > > > > > > > > ; : == ; : == T ; : == m > κ12 M 12 M 12 (

==

N 23 == N 13

)

" ¼

==

K

2

==

A44 == A45

==

#(

A45 == A55

==

γ 23 == γ 13

ð2:92aÞ

) ð2:92bÞ

where the stiffnesses for the top face are defined as n

== == A== ωρ , Bωρ , Dωρ

o

¼

n== Z X k¼1

ðx3 Þk

ðx3 Þk1

 == 
 b Q 1, x3 , x23 dx3 ωρ k

ðω, ρ ¼ 1, 2, 6Þ ð2:93aÞ

2.4 Hamilton’s Principle

== AIJ

25

¼

n== Z X

ðx3 Þk

ðx3 Þk1

k¼1

 ==  b Q dx3 IJ

ðI, J ¼ 4, 5Þ

k

 2 == == == == F == A== ωρ ¼ Dωρ þ 2a Bωρ þ a ωρ

== == == E == ωρ ¼ Bωρ þ a Aωρ ,

ð2:93bÞ

ð2:93cÞ

and the thermal stress and stress couple resultants are defined as ==

n== Z X

N Tαβ ¼

ðx3 Þk1

k¼1

n Z X ==

==

==

M Tαβ ¼

Nm αβ ¼ Mm αβ

¼

==  b λαβ ΔTdx3 k



ð x3 Þ k

k¼1

ðx3 Þk1

n== Z X

ðx3 Þk

k¼1 ==

ðx3 Þk

==

ðx3 Þk



ðx3 Þk1

ð2.94a, bÞ

k

ðx3 Þk1

n== Z X k¼1

==  b x3 þ a== λαβ ΔTdx3 b βαβ

 k

x3 þ a

ΔMdx3

==

==

b βαβ

ð2.95c, dÞ

 k

ΔMdx3

In the above equations, K = , K, K == are known as the shear correction factors.

2.4 2.4.1

Hamilton’s Principle Hamilton’s Equation

To derive the equations of motion, Hamilton’s variation principle will be adopted. The advantage of using this method is that, as a byproduct, the boundary conditions appear within the developments. Hamilton’s equation is given by Z δJ ¼ δ

t1

ðU  W  T Þdt ¼ 0

ð2:96Þ

t0

where t0, t1 are two arbitrary instants of time. U denotes the strain energy within a deformable body, W represents the work due to surface tractions, the work due to edge loads, and the work due to body forces. T denotes the kinetic energy of the 3D body.

26

2 Theory of Sandwich Plates and Shells with an Transversely Incompressible Core

2.4.2

Strain Energy

The strain energy stored in a 3D elastic body is expressed mathematically by U¼

1 2

Z τ

σ ij eij dτ

ð2:97Þ

dτ implies a volume element (dτ ¼ dσdx3) where dσ represents the planar area of an element. The total strain energy in a sandwich structure is a summation of the strain energies in the upper facings, the bottom facings, and the core. A variation in the total energy, considering three separate layers of the sandwich panel, is expressed as 1 δU ¼ 2

Z "Z σ

hþh=

= = σ ij δeij

h

Z þ

h h

Z σ ij δeij þ

h hh

# == == σ ij δeij þ ==

dx3 dσ

ði, j ¼ 1, 2, 3Þ ð2:98Þ

An alternative form which involves the temperature terms will become useful later on and is expressed as 1 δU ¼ δ 2

Z (Z σ

hþh=  h

Z  = = b = e= e=  2b Q λ Te þ dx 3 αβ αβωρ αβ ωρ αβ

h hh=



b == e== e== Q αβωρ αβ ωρ

) Z h   == == b b 2b λαβ Teαβ dx3 þ Qαβωρ eαβ eωρ  2λαβ Teαβ þ Qα3ω3 eα3 eω3 dx3 dσ h

ð2:99Þ Equation (2.99) is valid for both the weak- and strong-core model. For the weakcore model, the underlined terms should be discarded. The Greek indices assume the values 1,2 and the summation convention is held over a repeated index. The terms   Qαβ33 Q33ωρ b Qαβωρ  Qαβωρ  Q3333

  Qαβ33 b λ and λαβ  λαβ  Q3333 33

ð2:100Þ

are referred to as the modified elastic moduli and the thermal compliance expansion coefficients, respectively (see Librescu, 1975). It is also assumed that Qijmn and λij are temperature independent. Expanding Eq. (2.98) based on Einstein’s summation convention gives

2.4 Hamilton’s Principle

Z (Z δU ¼ Z


h

þ


h

Z þ


hþh= 
h

A

27

 = = = = = = = = = = σ 11 δe11 þ σ 22 δe22 þ σ 12 δγ 12 þ σ 13 δγ 13 þ σ 23 δγ 23 dx3

ðσ
11 δ
e11 þ σ
22 δ
e22 þ σ
12 δ
γ 12 þ σ
13 δ
γ 13 þ σ
23 δ
γ 23 Þdx3 þ 


h 
hh==

== == σ 11 δe11

þ

== == σ 22 δe22

þ

== == σ 12 δγ 12

þ

== == σ 13 δγ 13

þ

== == σ 23 δγ 23

)  dx3 dA ð2:101Þ

where σ ij are the components of the second Piola–Kirchhoff stress tensor and A is the area of the sandwich structure. Substituting Eqs. (2.24a–f)–(2.26a–f) into Eq. (2.101) results in Z (Z δU ¼ A

hþh= n

        = = = = = = σ 11 δε11 þ x3  a= δκ 11 þ σ 22 δε22 þ x3  a= δκ 22 þ

h

Z h     o = = = = = = = σ 12 δγ 12 þ x3  a= δκ 12 þ σ 13 δγ 13 þ σ 23 δγ 23 dx3 þ fσ 11 ðδε11 þ h

x3 δκ11 Þ þ σ 22 ðδε22 þ x3 δκ22 Þ þ σ 12 ðδε12 þ x3 δκ12 Þ þ σ 13 δγ 13 þ Z h      == == == == == σ 13 δγ 13 gdx3 þ σ 11 δε11 þ x3 þ a== δκ 11 þ σ 22 δε22 þ ðx3 þ hh==       o o == == == == == == == == a== δκ 22 þ σ 12 δγ 12 þ x3 þ a== δκ 12 þ σ 13 δγ 13 þ σ 23 δγ 23 dx3 dA ð2:102Þ Using the definitions of the local stress resultants and stress couples from Eqs. (2.83a, b)–(2.85a, b) allows the variation in the strain energy to be expressed as δU ¼

Z h A

=

=

=

=

=

=

=

=

=

=

=

=

N 11 δε11 þ M 11 δκ 11 þ N 22 δε22 þ M 22 δκ 22 þ N 12 δγ 12 þ M 12 δκ 12 þ

=

=

=

=

N 13 δγ 13 þ N 23 δγ 23 þ N 11 δε11 þ M 11 δκ 11 þ N 22 δε22 þ M 22 δκ22 þ N 12 δε12 þ ==

==

==

==

==

==

==

==

M 12 δκ12 þ N 13 δγ 13 þ N 23 δγ 23 þ N 11 δε11 þ M 11 δκ 11 þ N 22 δε22 þ M 22 δκ 22 þ i == == == == == == == == N 12 δγ 12 þ M 12 δκ 12 þ N 13 δγ 13 þ N 23 δγ 23 dA ð2:103Þ Substituting the strain–displacement equations, Eqs. (2.53)–(2.76) into Eq. (2.103) integrating by parts, simplifying, and gathering coefficients of like virtual displacements, results in

28

2 Theory of Sandwich Plates and Shells with an Transversely Incompressible Core

     ∂N 11 ∂N 12 ∂N 22 ∂N 12 ∂L11 ∂L12 N 13 δη1 δξ1  δξ2  δU ¼  þ þ þ  ∂x1 ∂x2 ∂x2 ∂x1 ∂x1 ∂x2 h 0 0     ∂L22 ∂L12 N 23 ∂P11 ∂P12 h= = =  þ  þ þ N 13  N 13 δψ 1 þ δη2 þ ∂x2 ∂x1 ∂x1 ∂x2 h 4h     == ∂P22 ∂P12 h= ∂S11 ∂S12 = = == h == þ þ N 23  N 23 δψ 2  þ  N 13 þ N 13 δψ 1 ∂x2 ∂x1 ∂x1 ∂x2 4h 4h      == ∂S22 ∂S12 ∂N 11 ∂N 12 ∂v3 ∂v∘3 == h == þ  N 23 þ N 23 δψ 2  þ þ þ ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x1 4h ! !    2 2 2 2 ∂ v∘3 ∂N 22 ∂N 12 ∂v3 ∂v∘3 ∂ v3 ∂ v∘3 ∂ v3 þ N 11 þ 2N 12 þ þ þ þ þ ∂x2 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x21 ∂x21 ! ) + 2 2 ∂ v3 ∂ v∘3 N 11 N 22 ∂N 13 ∂N 23 N 22 þ þ   þ δv3 dx1 dx2 þ R1 R2 ∂x1 ∂x2 ∂x22 ∂x22 Z l2 = = == N 11 δξ1 þ N 12 δξ2 þ L11 δη1 þ L12 δη2  P11 δψ 1  P12 δψ 2 þ S11 δψ 1 þ 0       ∂v3 ∂v∘3 ∂v3 ∂v∘3  l1 == þ N 12 þ N 13 δv3  dx2 þ þ S12 δψ 2 þ N 11 0 ∂x1 ∂x1 ∂x2 ∂x2 Z l1 = = == N 22 δξ2 þ N 12 δξ1 þ L22 δη2 þ L12 δη1  P22 δψ 2  P12 δψ 1 þ S22 δψ 2 þ 0       ∂v3 ∂v∘3 ∂v3 ∂v∘3  l2 == þ þ S12 δψ 1 þ N 22 þ N 12 þ N 23 δv3  dx1 0 ∂x2 ∂x2 ∂x1 ∂x1 Z

l2 Z l1



ð2:104Þ where the global stress and stress couple resultants appearing in this equation are defined as =

==

=

==

N 11 ¼ N 11 þ N 11 þ N 11 ,

=

==

=

==

N 22 ¼ N 22 þ N 22 þ N 22 ,

=

==

N 12 ¼ N 12 þ N 12 þ N 12

N 13 ¼ N 13 þ N 13 þ N 13 , N 23 ¼ N 23 þ N 23 þ N 23     = == = == L11 ¼ N 11  N 11 þ M 11 =h, L22 ¼ N 22  N 22 þ M 22 =h   = == L12 ¼ h N 12  N 12 þ M 12 =h  
 
  = = P11 ¼ h= =4 N 11 þ M 11 =h  M 11 , P22 ¼ h= =4 N 22 þ M 22 =h  M 22  
 = P12 ¼ h= =4 N 12 þ M 12 =h  M 12  
 
  == == S11 ¼ h== =4 N 11  M 11 =h þ M 11 , S22 ¼ h== =4 N 22  M 22 =h þ M 22  
 == S12 ¼ h== =4 N 12  M 12 =h þ M 12

ð2:105a  nÞ

2.4 Hamilton’s Principle

2.4.3

29

Kinetic Energy

The kinetic energy of elastic body by definition is of the form 1 T¼ 2

"

Z Z Z ρ x3 x2 x1

∂V 1 ∂t

2



∂V 2 þ ∂t

2  2 # ∂V 3 dx1 dx2 dx3 ∂t

ð2:106Þ

where ρ is the mass density per unit area. In general, the tangential velocities can be neglected due to the small tangential displacements as a function of time. As a result, this assumption is adopted and only the transverse inertia will be considered within the expression for the kinetic energy. Considering that the kinetic energy of a sandwich structure is the summation of the kinetic energies of each of the individual layers, the variation in the kinetic energy becomes Z

t1

δTdt ¼ 

t0

σ

t0

Z þ

"Z

t1 Z

Z

hþh=

=

h

== h == ∂ V 3 ρ ∂t 2 hh== 2

=

2

ρ

∂ V3 = δV 3 dx3 þ ∂t 2 #

Z

h h

2

ρ

∂ V3 δV 3 dx3 ∂t 2

==

δV 3 dσdt

ð2:107Þ

Substituting in the expressions for the transverse displacements, from Eqs. (2.7c), (2.8c), and (2.9c), gives Z

t1

t1 Z

Z δTdt ¼ 

t0

"Z

σ

t0

hþh=

2

=

ρ

h

∂ v3 δv3 þ ∂t 2

Z

h h

2

ρ

∂ v3 δv3 ∂t 2

 2 h ∂ v þ ρ== 23 δv3 dσdt ∂t hh== Z

ð2:108Þ

Simplifying Eq. (2.108) by combining the coefficients of the transverse variational displacement gives Z

t1

Z δTdt ¼ 

t0

t1 Z

2

σ

t0

m0

∂ v3 δv3 dσdt ∂t 2

ð2:109Þ

where, the inertia quantity, m0 is defined as Z m0 ¼ h

hþh=

=

ρðkÞ dx3 þ

Z

h h

Z ρdx3 þ

h hh

==

==

ρðkÞ dx3

ð2:110Þ

30

2 Theory of Sandwich Plates and Shells with an Transversely Incompressible Core

2.4.4

Work Done by External Loads

The total work consists of the work due to edge loads, surface tractions, and body forces. Surface tractions includes such loadings as lateral or transverse pressure loading. Body forces are forces which act through the body such as gravity, electrical forces, or magnetic forces. Edge loads are tangential, vertical, or normal loadings on the edge of the structure. The total work due to these loadings is the sum of the work applied to all three layers, the top face, the core, and the bottom face. This is expressed mathematically as W total ¼ W body

forces

þ W edge

loads

þ W surface

tractions

ð2:111Þ

• Work due to body forces Z (Z δW b ¼

σ

hþh=

ρ

=

Z

= = H i δV i dx3

þ

h h

h

Z ρH i δV i dx3 þ

)

h hh==

ρ

==

== == H i δV i dx3

dσ

ð2:112Þ where ρ is the mass density and Hi is the body force vector. The body forces are neglected such that gravitational, electrical, and magnetic forces are irrelevant. • Work due to surface tractions l2 Z l1

Z δW st ¼ 0

q3 ðx1 , x2 ÞδV 3 dx1 dx2

ð2:113Þ

0

• Work due to edge loads The work due to edge loads along the boundaries is expressed fundamentally as Z l1 (Z hþh=   = = = = = = δW el ¼ e σ 22 δV 2 þ e σ 21 δV 1 þ e σ 23 δV 3 dx3 0

h

Z þ

h


 e σ 22 δV 2 þ e σ 21 δV 1 þ e σ 23 δV 3 dx3

h

Z þ

h

hh==

Z þ

l2

)   == == == == == == e σ 22 δV 2 þ e σ 21 δV 1 þ e σ 23 δV 3 dx3 dx1

(Z

0

h

hþh= 

= = e σ 11 δV 1

þ

= = e σ 12 δV 2

þ

= = e σ 13 δV 3

ð2:114Þ

 dx3

Z h  e þ σ 11 δV 1 þ e σ 12 δV 2 þ e σ 13 δV 3 dx3 þ Z

h

h

hh==

)   == == == == == == e σ 11 δV 1 þ e σ 12 δV 2 þ e σ 13 δV 3 dx3 dx2

2.4 Hamilton’s Principle

31

(Note: Quantities with a tilde on top implies on the boundary.) Substituting in the expressions for the displacement quantities = V i , V i , and //Vi from Eqs. (2.7a–c)– (2.9a–c) while using the definition of stress and stress couples resultants defined earlier in Eqs. (2.83a, b)–(2.85a, b) results in Z

l1 h =

δW el ¼ 0

e 22 δξ2 þ = N e 22 δη2 þ = N e 21 δξ1 þ = N e 21 δη1 þ = N e 23 δv3 þ = M e 22 δψ =2 þ = M e 21 δψ =1 N

  e e = = e h δψ = e δξ þ N e δξ þ M 21 δη þ M 22 δη  N e h þM þN 22 2 21 1 21 21 1 2 1 4 h h 4h       == == = = == == e h M e h þM e h M e h δψ ==  N e h δψ = þ N e h δψ == þ N 21 21 22 22 22 22 1 2 2 4 4 4 4h 4h 4h e δv þ == N e 22 δξ2  == N e 22 δη2 == N e 21 δξ1  == N e 21 δη1 þ == N e 23 δv3 þ == M e 22 δψ == þN 23 3 2 i == == e 21 δψ 1 dx1 þ M Z

l2 h =

þ

e 11 δξ1 þ = N e 11 δη1 þ = N e 12 δξ2 þ = N e 12 δη2 þ = N e 13 δv3 N

0

e e = e δξ þ N e δξ þ M 12 δη þ M 11 δη e 11 δψ =1 þ = M e 12 δψ =2 þ N þM 11 1 12 2 2 1 h h       = = == == = = e h δψ = þ N e h δψ ==  N e h δψ = e h þM e h M e h þM  N 12 12 12 12 11 11 2 2 1 4 4 4 4h 4h 4h   == == e h δψ == þ N e h M e δv þ == N e 11 δξ1  == N e 11 δη1 þ N 11 11 13 3 1 4 4h i == == == e 12 δξ2  == N e 12 δη2 þ == N e 13 δv3 þ == M e 11 δψ == e N 1 þ M 12 δψ 2 dx2 þ

ð2:115Þ Simplifying and combining like terms gives the total variation in the work due to edge loads shown as Z δW el ¼

l1 

== e 12 δξ1 þ N e 22 δξ2 þ e e12 δψ =1  P e22 δψ =2 þ e N L12 δη1 þ e L22 δη2  P S12 δψ 1 þ == e e S22 δψ 2 þ N 23 δv3 dx1 Z l2  == e 11 δξ1 þ N e 12 δξ2 þ e e11 δψ =1  P e12 δψ =2 þ e N þ L11 δη1 þ e L12 δη2  P S11 δψ 1 þ 0 == e e 13 δv3 dx2 S12 δψ 2 þ N 0

ð2:116Þ where the boundary global stress resultants and stress couples are given by

32

2 Theory of Sandwich Plates and Shells with an Transversely Incompressible Core

e þN e =11 þ N e == e 11 ¼ N N 11 11 ,

e þN e 22 ¼ N e =22 þ N e == N 22 22 ,

e þN e 12 ¼ N e =12 þ N e == N 12 12

e þN e þN e =13 þ N e == e 23 ¼ N e =23 þ N e == e 13 ¼ N N N 13 23 13 , 23  =    == = == e e =h e e e e e e L11 ¼ N 11  N 11 þ M 11 =h, L22 ¼ N 22  N 22 þ M 22  =  e e 12  N e == e L12 ¼ N 12 þ M 12 =h       e þM e =h  M e þM e =h  M e =11 , P e22 ¼ h= =4 N e =22 e11 ¼ h= =4 N P 11 11 22 22    e þM e =h  M e =12 e12 ¼ h= =4 N P 12 12

   e M e =h þ M e e == S11 ¼ h== =4 N 11 11 11 ,    e M e =h þ M e e == S12 ¼ h== =4 N 12 12 12

   e M e =h þ M e e == S22 ¼ h== =4 N 22 22 22

ð2:117a  nÞ

2.5

Equations of Motion – Nonlinear Formulation

2.5.1

The Mixed Formulation for Sandwich Plate and Shells

The mixed formulation which is expressed in terms of the transversal displacement and the stress and stress couple resultants can now easily be obtained by substituting the expressions for δU, δT, δW from Eqs. (2.104), (2.109), (2.113), and (2.116) into Eq. (2.96) this results in Hamilton’s Equation being expressed in a more useful form as       ∂N 11 ∂N 12 ∂N 22 ∂N 12 ∂L11 ∂L12 N 13  þ þ þ  δη1  δξ1  δξ2  ∂x1 ∂x2 ∂x2 ∂x1 ∂x1 ∂x2 h t0 0 0      ∂L22 ∂L12 N 23 ∂P11 ∂P12 h= ∂P22 ∂P12 = = δη2 þ þ  þ þ N 13  N 13 δψ 1 þ þ þ ∂x2 ∂x1 ∂x1 ∂x2 ∂x2 ∂x1 h 4h     h= ∂S11 ∂S12 h== ∂S22 ∂S12 = = == == == þ þ N 13  N 13 δψ 1  þ þ N 23  N 23  N 23 δψ 2  ∂x ∂x ∂x2 ∂x1 4h 4h 1 2 !     2 2 h== ∂N 11 ∂N 12 ∂v3 ∂v∘3 ∂ v3 ∂ v∘3 == þ þ N 11 N 23 δψ 2 þ þ þ þ ∂x1 ∂x2 ∂x1 ∂x1 ∂x21 ∂x21 4h ! !  2 2 2 2 ∂ v∘3 ∂ v3 ∂ v3 ∂ v∘3 N N ∂N 13 ∂N 23 þ2N 12 þ þ  þ N 22 þ 11 þ 22  R1 R2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x22 ∂x22  2 ∂ v m0 23  q3 δv3 dx1 dx2 dt ∂t Z t1 Z l2 



 e 11 δξ1 þ N 12  N e 12 δξ2 þ L11  e N 11  N L11 δη1 þ L12  e L12 δη2 þ Z t1Z l2Z

t0

0

l1 

2.5 Equations of Motion – Nonlinear Formulation

33

   

 == == e11 δψ =1  P12  P e12 δψ =2 þ S11  e  P11  P S11 δψ 1 þ S12  e S12 δψ 2       ∂v3 ∂v∘3 ∂v3 ∂v∘3 e þ N 11 þ þ þ N 12 þ N 13  N 13 δv3 dx2 dt þ ∂x1 ∂x1 ∂x2 ∂x2 Z t1 Z l1 



 e 22 δξ2 þ N 12  N e 12 δξ1 þ L22  e þ N 22  N L22 δη2 þ L12  e L12 δη1 t0

0

    
 == == e22 δψ =2  P12  P e12 δψ =1 þ S22  e  P22  P S22 δψ 2 þ S12  e S12 δψ 1       ∂v3 ∂v∘3 ∂v3 ∂v∘3 e 23 δv3 dx1 dt ¼ 0 þ N 22 þ þ ð2:118Þ þ N 12 þN 23  N ∂x2 ∂x2 ∂x1 ∂x1


The above equation can only be satisfied if each of the triple and double integrals are zero. Since the coefficients of the variational displacements are arbitrary the integrals are only satisfied if the coefficients are set equal to zero. Setting the coefficients to zero gives nine equations of motion and nine prescribed boundary conditions along each edge which are listed below as δξ1 :

∂N 11 ∂N 12 þ ¼0 ∂x1 ∂x2

ð2:119Þ

δξ2 :

∂N 22 ∂N 12 þ ¼0 ∂x2 ∂x1

ð2:120Þ

δη1 :

∂L11 ∂L12 N 13 þ  ¼0 ∂x1 ∂x2 h

ð2:121Þ

δη2 :

∂L22 ∂L12 N 23 þ  ¼0 ∂x2 ∂x1 h

ð2:122Þ

=

  ∂P11 ∂P12 = þ þ N 13  h= =4h N 13 ¼ 0 ∂x1 ∂x2

ð2:123Þ

δψ 2 :

=

  ∂P22 ∂P12 = þ þ N 23  h= =4h N 23 ¼ 0 ∂x2 ∂x1

ð2:124Þ

==

  ∂S11 ∂S12 == þ þ N 13  h== =4h N 13 ¼ 0 ∂x1 ∂x2

ð2:125Þ

δψ 1 :

δψ 1 :

34

2 Theory of Sandwich Plates and Shells with an Transversely Incompressible Core

  ∂S22 ∂S12 == þ þ N 23  h== =4h N 23 ¼ 0 ∂x2 ∂x1

==

δψ 2 :

2

δv3 : N 11

∘

2 ∂ v3 ∂ v3 1 þ þ ∂x21 ∂x21 R1 ∘

2

þN 22

!

2

þ 2N 12

2 ∂ v3 ∂ v3 1 þ þ ∂x22 ∂x22 R2

!

∘

2 ∂ v3 ∂ v3 þ ∂x1 ∂x2 ∂x1 ∂x2

ð2:126Þ

!

2

þ

∂N 13 ∂N 23 ∂v ∂ v þ þ q3  c 3 ¼ m0 23 ∂x1 ∂x2 ∂t ∂t ð2:127Þ

(Note: In Eq. (2.127) the damping term has been manually added.) q3 denotes the distributed transversal load. The associated boundary conditions along the edges xn ¼ const(n ¼ 1, 2) become e nn N nn ¼ N e N nt ¼ N nt Lnn ¼ e Lnn ent Lnt ¼ L

or

ξn ¼ e ξn e ξt ¼ ξt

or or

ηn ¼ e ηn

or

ηt ¼ e ηt

enn Pnn ¼ P

or

e =n ψ =n ¼ ψ

ent Pnt ¼ P Snn ¼ e Snn

or

e =t ψt ¼ ψ

or

e == ψ == n n ¼ ψ

=

e == Snt ¼ e Snt or ψt ¼ ψ t     ∂v3 ∂v3 ∂v3 ∂v3 e n3 N nt þ þ þ N nn þ N n3 ¼ N ∂xt ∂xt ∂xn ∂xn ==

or

v3 ¼ ev3 ð2:128a  iÞ

The subscripts n and t are used to imply the normal and tangential in-plane directions to an edge and hence, n ¼ 1 when t ¼ 2, and vice versa. There are nine boundary conditions which means the governing equations are of the eighteenth order. There are a few special cases of these equations presented next. Special Cases • The discarding of the transverse shear effects in the facings When the facings are considered thin, the transverse shear stresses become negligible. Therefore, the Love–Kirchhoff assumption is adopted for the facings. Using the variational principle Eq. (2.96) in conjunction with Eqs. (2.51a, b) the equations of motion become δξ1 :

∂N 11 ∂N 12 þ ¼0 ∂x1 ∂x2

ð2:129Þ

2.5 Equations of Motion – Nonlinear Formulation

∂N 22 ∂N 12 þ ¼0 ∂x2 ∂x1

ð2:130Þ

δη1 :

∂L11 ∂L12 N 13 þ  ¼0 ∂x1 ∂x2 h

ð2:131Þ

δη2 :

∂L22 ∂L12 N 23 þ  ¼0 ∂x2 ∂x1 h

ð2:132Þ

δξ2 :

δv3 :

35

! ! 2 ∘ 2 ∘ 2 2 ∂ v3 ∂ v3 ∂ v 3 1 ∂ v3 N 11 þ þ Þ þ 2N 12 þ ∂x1 ∂x2 ∂x1 ∂x2 ∂x21 ∂x21 R1 !  2 2 ∘ 2 2
2

11 ∂ v3 ∂ v3 1 ∂ N ∂ N ∂ N 12 22 þ N 22 þ þ þ 2 þ  C 2 ∂x1 ∂x2 ∂x22 ∂x22 R2 ∂x21 ∂x22    2 2 2
13 ∂N
23 C ∂N ∂ M 11 ∂ M 12 ∂ M 22 þ 1 þ
1 þ þ 2 þ þ q3 þ ∂x1 ∂x2 ∂x1 ∂x2 ∂x21 ∂x22 h 2

c

∂v3 ∂ v ¼ m0 23 ∂t ∂t

ð2:133Þ

The associated boundary conditions along the edges xn ¼ const(n ¼ 1, 2) become e nn N nn ¼ N e nt N nt ¼ N enn Lnn ¼ L

or or

Lnt ¼ e Lnt

or

or

ξn ¼ e ξn e ξt ¼ ξt ηn ¼ e ηn ηt ¼ e ηt

∂v3 ∂ev3 e M e nn C 2 N nn  M nn ¼ C 2 N or ¼ nn ∂xn ∂xn       ∂v3 ∂v3 ∂v3 ∂v3 ∂M nn ∂M nt N nt þ þ þ2 or v3 ¼ ev3 þ N nn þ ∂xt ∂xt ∂xn ∂xn ∂xn ∂xt     e e nt e ∂N nn ∂N nt C1 ∂N ∂M nt C 2 þ2 N n3 ¼ C2 þ þ N n3 þ 1þ ∂xn ∂xt ∂x ∂x h t t ð2:134a  fÞ =

==

=

For this case, the local stress and stress couple resultants, N 13 ¼ N 13 ¼ N 23 ¼ == N 23 ¼ 0. This implies that the global stress resultants, N13 and N23 are reduced to N 13 ¼ N 13

ð2:135aÞ

N 23 ¼ N 23

ð2:136bÞ

while the global stress couple resultants M11, M22, M12 are defined as

36

2 Theory of Sandwich Plates and Shells with an Transversely Incompressible Core


 = == M 11 ¼ M 11 þ M 11  C 1 =h M 11
 = == M 22 ¼ M 22 þ M 22  C 1 =h M 22
 = == M 12 ¼ M 12 þ M 12  C 1 =h M 12

ð2:137a  cÞ

where   C 1 ¼ h= þ h== =4,

  C2 ¼ h=  h== =4

ð2.138a, bÞ

For this case there are six boundary conditions required at each edge. This reduces the governing equations to the twelfth order. • Weak (soft) core with symmetric facings In this case the core is only capable of carrying the transverse shear stresses. As a result, N αβ and M αβ become immaterial. Because of this, the equations of motion, the stress and stress couple resultants, and the associated boundary conditions can be further simplified. Also with symmetric facings with respect to the global mid-surface (mid-surface of the core), C1 ¼ h/2, C2 ¼ 0. The equations of motion simplify to ∂N 11 ∂N 12 þ ¼0 ∂x1 ∂x2

ð2:139Þ

δξ2 :

∂N 22 ∂N 12 þ ¼0 ∂x2 ∂x1

ð2:140Þ

δη1 :

∂L11 ∂L12 N 13 þ  ¼0 ∂x1 ∂x2 h

ð2:141Þ

δη2 :

∂L22 ∂L12 N 23 þ  ¼0 ∂x2 ∂x1 h

ð2:142Þ

! 2 ∘ 2 ∂ v3 ∂ v3 þ þ 2N 12 ∂x1 ∂x2 ∂x1 ∂x2 !    2 ∘ 2 2
13 ∂N
23 ∂ v3 ∂ v3 1 h ∂N ∂ M 11 þ þ þ þ 1 þ þ ∂x1 ∂x2 ∂x22 ∂x22 R2 ∂x21 2
h 2

δv3 : N 11

δξ1 :

∘

2 ∂ v3 ∂ v 3 1 þ þ ∂x21 ∂x21 R1

þ N 22 2

þ2

2

!

2

∂ M 12 ∂ M 22 ∂v ∂ v þ þ q3  c 3  m0 23 ¼ 0 2 ∂x1 ∂x2 ∂t ∂t ∂x2 ð2:143Þ

2.5 Equations of Motion – Nonlinear Formulation

37

The associated boundary conditions along the edges xn ¼ const(n ¼ 1, 2) simplify to e nn N nn ¼ N e N nt ¼ N nt

ξn ¼ e ξn e ξt ¼ ξt

or or or

Lnn ¼ e Lnn ent Lnt ¼ L

ηn ¼ e ηn

ηt ¼ e ηt ∂v3 ∂ev3 e nn M nn ¼ M or ¼ ∂xn ∂xn     ∂v3 ∂v3 ∂v3 ∂v3 ∂M nn ∂M nt N nt þ þ þ2 þ N nn þ ∂xt ∂xt ∂xn ∂xn ∂xn ∂xt   e h ∂M nt e þ 1þ N n3 ¼ þ N n3 ∂xt 2h or

or

v3 ¼ ev3

ð2:144a  fÞ

The global stress and stress couple resultants reduce to =

==

=

==

=

==

N 11 ¼ N 11 þ N 11 , N 22 ¼ N 22 þ N 22 , N 13 ¼ N 13 , N 23 ¼ N 23 L11 ¼ N 11  N 11 ,

=

==

L22 ¼ N 22  N 22 ,

M 11 ¼ = M 11 þ == M 11 ,

=

==

N 12 ¼ N 12 þ N 12 =

==

L12 ¼ N 12  N 12

M 22 ¼ = M 22 þ == M 22 ,

M 12 ¼ = M 12 þ == M 12 ð2:145a  kÞ

2.5.2

Displacement Formulation for Sandwich Plates and Shells

• Strong Core Formulation The displacement formulation can be developed by replacement of equations Eqs. (2.99), (2.109), (2.113), (2.116), (2.24a–f)–(2.26a–f), and (2.53)–(2.76) into Eq. (2.96) carrying out the integration with respect to x3, integrating by parts wherever possible and invoking the arbitrary character of the variations δη1, δη2, δξ1, δξ2, and δv3 by setting the variation of these five coefficients to zero. The result is five equations of motion and a set of boundary conditions as by-products. This governing system is valid for double curved sandwich shells with a strong core and symmetric facings neglecting the transverse shear in the facings. It is assumed that symmetry exists both locally and globally. This system of equations is provided as

38

2 Theory of Sandwich Plates and Shells with an Transversely Incompressible Core

( δξ1 : Λ11 (

)  2  2 2 2 ∂ ξ1 ∂v3 ∂ v3 ∂ v3 ∂v3 ∂v3 ∂ v3 1 ∂v3 þ þ þ þ  ∂x21 ∂x1 ∂x21 ∂x21 ∂x1 ∂x1 ∂x21 R1 ∂x1 2



2 2 2 2 ∂ ξ1 ∂ ξ2 ∂v ∂ v3 ∂v3 ∂ v3 ∂v3 ∂ v3 þ þ 3 þ þ 2 ∂x1 ∂x22 ∂x2 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ) (   2  2 2 ∂v3 ∂2 v3 ∂v3 ∂2 v3 ∂v3 ∂ v3 ∂ ξ2 ∂v ∂ v3 þ Λ12 þ þ þ 3 þ 2 2 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ) (  2  2 2 ∂v ∂2 v3 ∂v ∂ v3 1 ∂v3 ∂ ξ1 ∂ ξ2  A16 2 þ þ 3  þ 3 ∂x2 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 R2 ∂x1 ∂x1 ∂x2 ∂x21

Λ66

2





2



2 2 ∂v ∂2 v3 ∂v3 ∂ v3 ∂v ∂ v3 ∂v ∂ v ∂v ∂ v3 þ2 3 þ2 3 þ 3 23 þ 3 þ ∂x1 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x21 ) ( )   2  2 2 2 ∂v3 ∂2 v3 1 ∂v3 ∂ ξ2 ∂v3 ∂ v3 ∂v3 ∂ v3 ∂v3 ∂ v3 1 ∂v3  A26  þ þ þ  ∂x2 ∂x21 R1 ∂x2 ∂x22 ∂x2 ∂x22 ∂x2 ∂x22 ∂x2 ∂x22 R2 ∂x2

þ2



∂N T11 ∂N T12  ¼0 ∂x1 ∂x2

ð2:146Þ (

2





2 2 2 ∂ ξ2 ∂v3 ∂ v3 ∂ v3 ∂v3 ∂v3 ∂ v3 1 ∂v3 þ þ þ  2 2 2 ∂x2 ∂x2 ∂x2 ∂x2 ∂x2 ∂x2 ∂x22 R2 ∂x2

)

(

2

2

∂ ξ2 ∂ ξ1 þ ∂x21 ∂x1 ∂x2 )   2  2  2 2 ∂v3 ∂2 v3 ∂v3 ∂2 v3 ∂v3 ∂ v3 ∂v3 ∂ v3 ∂v3 ∂ v3 ∂v3 ∂ v3 þ þ þ þ þ þ þ ∂x1 ∂x1 ∂x2 ∂x2 ∂x21 ∂x1 ∂x1 ∂x2 ∂x2 ∂x21 ∂x1 ∂x1 ∂x2 ∂x2 ∂x21 ( ) (  2  2 2 2 ∂v3 ∂2 v3 ∂ ξ1 ∂v3 ∂ v3 ∂v3 ∂ v3 1 ∂v3 ∂ ξ2 þ þ þ  A26 2 Λ12 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 R1 ∂x2 ∂x1 ∂x2

δξ2 : Λ22

2

þ







þ Λ66

2



2 2 2 ∂v ∂2 v3 ∂ ξ1 ∂v ∂ v3 ∂v ∂ v3 ∂v ∂ v3 ∂v3 ∂ v3 þ2 3 þ2 3 þ2 3 þ 3 þ þ 2 ∂x2 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x1 ∂x22 ∂x1 ∂x22 ∂x2 ) ( )   2  2 2 2 ∂v3 ∂2 v3 1 ∂v3 ∂ ξ1 ∂v3 ∂ v3 ∂v3 ∂ v3 ∂v3 ∂ v3 1 ∂v3  þ þ þ   A16 ∂x1 ∂x22 R2 ∂x1 ∂x21 ∂x1 ∂x21 ∂x1 ∂x21 ∂x1 ∂x21 R1 ∂x1

∂N T22 ∂N T12  ¼0 ∂x2 ∂x1

ð2:147Þ  2 2 2 2 3 ∂ η1 hhQ11 ∂3 v3 b ∂ η1 ∂ η2 2hhQ66 ∂ v3 b b 12 ∂ η2 þ δη1 : Λ11 þ þ Λ66 þ þΛ 2 3 2 2 3 ∂x1 3 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2   2 2 2 3 b T11 ∂ η2 ∂ η1 ∂ η2 hhQ12 ∂ v3 ∂v3 1 ∂N þ þ A16 þ2  d 1 η1 þ a þ A26  3 ∂x1 ∂x22 ∂x1 ∂x2 ∂x1 ∂x21 ∂x22 h ∂x1 b 12 1 ∂N  ¼0 h ∂x2 T

ð2:148Þ

2.5 Equations of Motion – Nonlinear Formulation

39

 2 2 2 2 3 3 b 22 ∂ η2 þ hhQ22 ∂ v3 þ Λ b 66 ∂ η2 þ ∂ η1 þ 2hhQ66 ∂ v3 þ Λ b 12 ∂ η1 δη2 : Λ 2 3 2 2 3 ∂x2 3 ∂x1 ∂x2 ∂x1 ∂x2 ∂x2 ∂x2 ∂x1 ∂x2   2 2 2 3 b T22 ∂ η1 ∂ η2 ∂ η1 hhQ12 ∂ v3 ∂v3 1 ∂N þ A16  þ þ A26 þ2  d 2 η2 þ a 3 ∂x21 ∂x2 ∂x1 ∂x2 ∂x2 ∂x22 ∂x21 h ∂x2 b T12 1 ∂N  ¼0 h ∂x1

ð2:149Þ (

! !  2 2  2  2  2 ∂ξ1 ∂ v3 ∂ v3 1 ∂v3 ∂ v3 ∂ v3 ∂v3 ∂v3 þ þ δv3 : Λ11 þ þ 2 ∂x1 ∂x1 ∂x21 ∂x1 ∂x1 ∂x21 ∂x21 ∂x21 ! " ! #)  2  2  2  2 2 ∂ v3 ∂ v3 1 ∂ξ1 1 ∂v3 ∂v3 ∂v3 ∂ v3 ∂ v3 v3 þ þ þ  v3 þ þ  R1 ∂x1 2 ∂x1 R1 ∂x1 ∂x1 ∂x21 ∂x21 ∂x21 ∂x21 ( ! !  3 2  2  4 2 2 hhQ11 ∂ η1 h ∂ v3 ∂ξ1 ∂ v3 ∂ v3 ∂ξ2 ∂ v3 ∂ v3 þ þ þ  þ Λ12 þ 3 ∂x1 ∂x22 ∂x2 ∂x21 ∂x22 ∂x21 ∂x31 2 ∂x41 ! ! !  2 2  2 2  2  2  2  2 1 ∂v3 ∂ v3 ∂ v3 1 ∂v3 ∂ v3 ∂ v3 ∂v3 ∂v3 ∂ v3 ∂ v3 þ þ þ þ þ þ 2 ∂x1 2 ∂x2 ∂x1 ∂x1 ∂x22 ∂x22 ∂x22 ∂x21 ∂x21 ∂x22 ! " ! #  2   2  2  2 2 ∂v3 ∂v3 ∂ v3 ∂ v3 1 ∂ξ2 1 ∂v3 ∂v3 ∂v3 ∂ v3 ∂ v3 v3 þ þ þ v3 þ þ þ  R1 ∂x2 2 ∂x2 R2 ∂x2 ∂x2 ∂x21 ∂x2 ∂x2 ∂x21 ∂x22 ∂x22 " ! #)  2  2  2 1 ∂ξ1 1 ∂v3 ∂v3 ∂v3 ∂ v3 ∂ v3 v3 þ  v3 þ þ þ  R2 ∂x1 2 ∂x1 R1 ∂x1 ∂x1 ∂x21 ∂x21 ( !  3  2 2  3 4 2 ∂ η2 hhQ12 ∂ η1 ∂ v3 ∂ξ2 ∂ v3 ∂ v3 1 ∂v3 þ h þ  þ þ Λ22 2 ∂x2 3 ∂x2 ∂x22 ∂x1 ∂x22 ∂x21 ∂x2 ∂x21 ∂x22 ∂x22 ! ! "  2   2  2  2  2 ∂ v3 ∂ v3 ∂v3 ∂v3 ∂ v3 ∂ v3 1 ∂ξ2 1 ∂v3 ∂v3 ∂v3 þ þ þ þ þ þ R2 ∂x2 2 ∂x2 ∂x2 ∂x2 ∂x22 ∂x22 ∂x2 ∂x2 ∂x22 ∂x22 ! #)  3 2  2 4 ∂ v3 ∂ v 3 v3 hhQ22 ∂ η2 h ∂ v3 v3 þ þ   R2 3 ∂x22 ∂x22 ∂x32 2 ∂x42 ( ! ! 2  2  2 2 ∂ v3 ∂ v3 ∂ξ1 ∂ξ2 ∂ v3 ∂ v3 ∂v ∂v þ þ þ2Λ66 þ þ 3 3 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ! !    2  2  2 2 2 ∂ v3 ∂v ∂v ∂ v3 ∂ v3 ∂ v3 ∂v ∂v ∂ v3 þ þ þ þ 3 3 þ 3 3 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2

40

2 Theory of Sandwich Plates and Shells with an Transversely Incompressible Core

2



∂ v3 ∂x1 ∂x2

!)

( !  3 2  3 4 2 ∂ η2 2hhQ66 ∂ η1 ∂ v3 ∂ξ1 ∂ v3 ∂ v3 þ þ h 2 2 þ A16 þ  3 ∂x2 ∂x21 ∂x1 ∂x22 ∂x21 ∂x2 ∂x1 ∂x2 ∂x21 2



2 ∂ξ ∂ v3 ∂ v3 þ 2 þ ∂x1 ∂x21 ∂x21 

∂v ∂v þ 3 3 ∂x1 ∂x2

!

2



2 ∂ v3 ∂ v3 þ ∂x21 ∂x21

!





2

∂v ∂v ∂2 v3 ∂ v3 þ þ 3 3 ∂x1 ∂x2 ∂x21 ∂x21

!

! 2  2 ∂ v3 ∂ξ1 ∂ v3 þ þ2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 ! !  2  2  2 2 ∂ v3 ∂ v3 ∂ v3 ∂v3 ∂v3 ∂ v3 þ þ þ2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 2



2 ∂ v3 ∂ v3 þ ∂x21 ∂x21

 2 ∂v3 þ ∂x1

∂v ∂v þ 3 3 ∂x1 ∂x2

!

" !#)   2  2 ∂ v3 1 ∂ξ1 ∂ξ2 ∂v3 ∂v3 ∂v3 ∂v3 ∂v3 ∂v3 ∂ v3 þ þ þ þ þ þ  2v3 R1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ( þA26



2

2 ∂ξ2 ∂ v3 ∂ v3 þ ∂x1 ∂x22 ∂x22





2

∂v ∂v ∂2 v3 ∂ v3 þ 3 3 þ ∂x2 ∂x1 ∂x22 ∂x22 2

!

!



2 ∂ v3 ∂ξ ∂ v3 þ2 2 þ ∂x2 ∂x1 ∂x2 ∂x1 ∂x2

2



2 ∂ξ ∂ v3 ∂ v3 þ þ 1 ∂x2 ∂x22 ∂x22 

!

2



2 ∂v ∂v ∂ v3 ∂ v3 þ þ 3 3 ∂x2 ∂x1 ∂x22 ∂x22

!



2

2 ∂v ∂v ∂ v3 ∂ v3 þ þ 3 3 ∂x2 ∂x1 ∂x22 ∂x22

!

!

!  2 2  2 ∂ v3 ∂v3 ∂ v3 þ þ ∂x2 ∂x2 ∂x1 ∂x2 ∂x1

!   2  2 ∂ v3 ∂v3 ∂v3 ∂ v3 þ2 þ ∂x2 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2   2   1 ∂ξ2 ∂ξ1 ∂v3 ∂v3 ∂v3 ∂v3 ∂v3 ∂v3 ∂ v3 þ þ þ þ þ 2v3 þ R2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 4

F 11

4

4

2



∂ v3 ∂x1 ∂x2

4

∂ v3 ∂ v ∂ v3 ∂ v  2ðF 12 þ 2F 66 Þ 2 3 2  F 22  4F 16 3 3 4 4 ∂x1 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2

    2 4 2 2 ∂ v3 h 1 ∂η1 C 1 ∂ v3 4F 26 þ 2hK G13 1 þ þ 1þ þ 2hK G23 ∂x1 ∂x32 2h h ∂x1 h ∂x21 !     2 2  2 ∂ v h 1 ∂η2 C 1 ∂ v3 ∂ v 1 3 3 1þ þ 1þ þ þ  N T11 ∂x21 ∂x21 R1 2h h ∂x2 h ∂x22

!#)

2.5 Equations of Motion – Nonlinear Formulation

2

N T12

N m 11

!

2



N T22



2 ∂ v3 ∂ v3 1 þ þ ∂x22 ∂x22 R2

!

! !  2 2  2  2 ∂ v3 ∂ v3 ∂ v3 1 ∂ v3 m þ þ  N 12 þ ∂x1 ∂x2 ∂x1 ∂x2 ∂x21 ∂x21 R1 2

N m 22



2 ∂ v3 ∂ v3 þ ∂x1 ∂x2 ∂x1 ∂x2

41



2 ∂ v3 ∂ v3 1 þ þ ∂x22 ∂x22 R2

!

2

þC

∂v3 ∂ v þ mo 23 ¼ q3 ∂t ∂t

ð2:150Þ

It should be noted that the terms containing the squares of the geometrical imperfections can be discarded. The associated boundary conditions along the edges xn ¼ const(n ¼ 1, 2) are e nn N nn ¼ N e nt N nt ¼ N

or

Lnn ¼ e Lnn e Lnt ¼ Lnt

or or

or

ξn ¼ e ξn ξt ¼ e ξt

ηn ¼ e ηn ηt ¼ e ηt ∂v ∂ev 3 e nn M nn ¼ M or ¼ 3 ∂xn ∂xn     ∂v3 ∂v3 ∂v3 ∂v3 ∂M nn ∂M nt N nt þ þ þ2 þ N nn þ ∂xt ∂xt ∂xn ∂xn ∂xn ∂xt e nt e
 ∂M þ a=h N n3 ¼ þ N n3 ∂xt

or

v3 ¼ ev3

ð2:151a  fÞ In the above equations of motion, the stiffness coefficients are defined as Λ11 ¼ A11 þ 2hQ11 ,

Λ22 ¼ A22 þ 2hQ22 ,

Λ66 ¼ A66 þ 2hQ66
 b 11 ¼ A11 þ 2h=3 Q11 , Λ
 b 66 ¼ A66 þ 2h=3 Q66 Λ

Λ12 ¼ A12 þ 2hQ12


 b 22 ¼ A22 þ 2h=3 Q22 , Λ


 b 12 ¼ A12 þ 2h=3 Q12 Λ ð2:152a  hÞ

While the global mechanical stiffness measures Aωρ, Fωρ are defined as Aωρ ¼ = Aωρ þ == Aωρ ,

F ωρ ¼ = F ωρ þ == F ωρ

ðω, ρ ¼ 1, 2, 6Þ ð2.153a, bÞ

42

2 Theory of Sandwich Plates and Shells with an Transversely Incompressible Core

if the face sheets are symmetric with respect to both their local and global mid-surfaces then, =

Aωρ ¼ == Aωρ ,

=

F ωρ ¼ == F ωρ

ð2.154c, dÞ

The global thermal stress and stress couple resultants appearing in the equations are defined as N T11 ¼ = N T11 þ N 11 þ == N T11 , N T22 ¼ = N T22 þ N 22 þ == N T22 , N T12 ¼ = N T12 þ == N T12       b T11 ¼ h = N T11  == N T11 , N b T22 ¼ h = N T22  == N T22 , N b T12 ¼ h = N T12  == N T12 N

 b T11 ¼ = M T11 þ == M T11  h=2h M T11 , M b T22 ¼ = M T22 þ == M T22  h=2h M T22 M T

T

b T12 ¼ = M T12 þ == M T12 M ð2:155a  iÞ • Weak/Soft-Core Formulation Eqs. (2.146)–(2.150) can be reduced somewhat if one considers the weak core model by discarding all of the underlined terms appearing in these equations, as well as in the stiffness coefficients. For consideration of the weak core with symmetric facings both locally and globally the governing equations of motion can be modified to appear as (

) (  2  2 2 2 2 2 ∂ ξ1 ∂v3 ∂ v3 ∂ v3 ∂v3 ∂v3 ∂ v3 ∂ ξ1 ∂ ξ2 1 ∂v3 δξ1 : A11 þ þ þ  þ A þ 66 ∂x21 ∂x1 ∂x21 ∂x21 ∂x1 ∂x1 ∂x21 R1 ∂x1 ∂x22 ∂x1 ∂x2 )   2  2  2 2 ∂v3 ∂2 v3 ∂v3 ∂2 v3 ∂v3 ∂ v3 ∂v3 ∂ v3 ∂v3 ∂ v3 ∂v3 ∂ v3 þ þ þ þ þ þ þ ∂x2 ∂x1 ∂x2 ∂x1 ∂x22 ∂x2 ∂x1 ∂x2 ∂x1 ∂x22 ∂x2 ∂x1 ∂x2 ∂x1 ∂x22 ( ) (  2  2 2 2 ∂v3 ∂2 v3 ∂ ξ2 ∂ ξ1 ∂v3 ∂ v3 ∂v3 ∂ v3 1 ∂v3 A12 þ þ þ  A16 2 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 R2 ∂x1 ∂x1 ∂x2 2





2



2 2 2 ∂v ∂2 v3 ∂ ξ2 ∂v ∂ v3 ∂v ∂ v3 ∂v ∂ v3 ∂v3 ∂ v3 þ2 3 þ2 3 þ2 3 þ 3 þ þ 2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x2 ∂x21 ∂x2 ∂x21 ∂x1 ) ( )   2  2 2 2 ∂v3 ∂2 v3 1 ∂v3 ∂ ξ2 ∂v3 ∂ v3 ∂v3 ∂ v3 ∂v3 ∂ v3 1 ∂v3  þ þ þ   A26 ∂x2 ∂x21 R1 ∂x2 ∂x22 ∂x2 ∂x22 ∂x2 ∂x22 ∂x2 ∂x22 R2 ∂x2

þ



∂N T11 ∂N T12  ¼0 ∂x1 ∂x2

ð2:156Þ

2.5 Equations of Motion – Nonlinear Formulation

(

2



43 

2 2 2 ∂ ξ2 ∂v3 ∂ v3 ∂ v3 ∂v3 ∂v3 ∂ v3 1 ∂v3 þ þ þ  ∂x22 ∂x2 ∂x22 ∂x22 ∂x2 ∂x2 ∂x22 R2 ∂x2

)

(

2

2

∂ ξ2 ∂ ξ1 þ ∂x21 ∂x1 ∂x2 )   2  2  2 2 ∂v3 ∂2 v3 ∂v3 ∂2 v3 ∂v3 ∂ v3 ∂v3 ∂ v3 ∂v3 ∂ v3 ∂v3 ∂ v3 þ þ þ þ þ þ þ ∂x1 ∂x1 ∂x2 ∂x2 ∂x21 ∂x1 ∂x1 ∂x2 ∂x2 ∂x21 ∂x1 ∂x1 ∂x2 ∂x2 ∂x21 ( ) (  2  2 2 2 ∂v3 ∂2 v3 ∂ ξ1 ∂v3 ∂ v3 ∂v3 ∂ v3 1 ∂v3 ∂ ξ2 A12 þ þ þ  A26 2 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 R1 ∂x2 ∂x1 ∂x2

δξ2 : A22

2



þ A66



2



2 2 2 ∂v3 ∂2 v3 ∂ ξ1 ∂v3 ∂ v3 ∂v3 ∂ v3 ∂v3 ∂ v3 ∂v3 ∂ v3 þ 2 þ 2 þ 2 þ þ þ ∂x2 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x1 ∂x22 ∂x1 ∂x22 ∂x22 ) ( )   2  2 2 2 ∂v3 ∂2 v3 1 ∂v3 ∂ ξ1 ∂v3 ∂ v3 ∂v3 ∂ v3 ∂v3 ∂ v3 1 ∂v3  þ þ þ   A16 ∂x1 ∂x22 R2 ∂x1 ∂x21 ∂x1 ∂x21 ∂x1 ∂x21 ∂x1 ∂x21 R1 ∂x1

þ



∂N T22 ∂N T12  ¼0 ∂x2 ∂x1 2



2



2

2



2



2



ð2:157Þ 2 2 ∂ η2 ∂ η1 þ 2 ∂x1 ∂x2 ∂x21

2



ð2:158Þ 2 2 ∂ η1 ∂ η2 þ2 ∂x1 ∂x2 ∂x22

∂ η1 ∂ η1 ∂ η2 ∂ η2 þ A66 þ þ A16 þ A12 ∂x1 ∂x2 ∂x21 ∂x22 ∂x1 ∂x2  2 b T11 1 ∂N b T12 ∂ η2 ∂v3 1 ∂N  d η þ a  ¼0 þA26  1 1 ∂x1 ∂x22 h ∂x1 h ∂x2

δη1 : A11

2

∂ η2 ∂ η2 ∂ η1 ∂ η1 þ A66 þ þ A26 þ A12 ∂x1 ∂x2 ∂x22 ∂x22 ∂x1 ∂x2  2 b T22 1 ∂N b T12 ∂ η1 ∂v3 1 ∂N  d η þ a  ¼0 þA16  2 2 ∂x2 ∂x21 h ∂x2 h ∂x1

δη2 : A22

ð2:159Þ !    2 2  2 2 2 ∂ξ1 ∂ v3 ∂ v3 1 ∂v3 ∂ v3 ∂ v3 ∂v3 ∂v3 þ þ δv3 : A11 þ þ 2 ∂x1 ∂x1 ∂x21 ∂x1 ∂x1 ∂x21 ∂x21 ∂x21 ! " ! #)  2  2  2  2 2 ∂ v3 ∂ v3 1 ∂ξ1 1 ∂v3 ∂v3 ∂v3 ∂ v3 ∂ v 3 v3 þ þ  þ þ  v3 þ R1 ∂x1 2 ∂x1 R1 ∂x1 ∂x1 ∂x21 ∂x21 ∂x21 ∂x21 ( ! ! !  2  2  2 2  2  2  2 ∂ξ1 ∂ v3 ∂ v3 ∂ξ2 ∂ v3 ∂ v3 1 ∂v3 ∂ v3 ∂ v3 þ þ þ þA12 þ þ 2 ∂x1 ∂x1 ∂x22 ∂x22 ∂x2 ∂x21 ∂x21 ∂x22 ∂x22 ! ! !  2 2    2 2  2  2  2 1 ∂v3 ∂ v3 ∂ v3 ∂v3 ∂v3 ∂ v3 ∂ v3 ∂v3 ∂v3 ∂ v3 ∂ v3 þ þ þ þ þ þ 2 ∂x2 ∂x1 ∂x1 ∂x22 ∂x2 ∂x2 ∂x21 ∂x21 ∂x21 ∂x22 ∂x21 (

! 

44

2 Theory of Sandwich Plates and Shells with an Transversely Incompressible Core

" ! #  2  2  2 1 ∂ξ2 1 ∂v3 ∂v3 ∂v3 ∂ v3 ∂ v3 v3 þ þ þ  v3 þ  R1 ∂x2 2 ∂x2 R2 ∂x2 ∂x2 ∂x22 ∂x22 " ! #)  2  2  2 1 ∂ξ1 1 ∂v3 ∂v3 ∂v3 ∂ v3 ∂ v3 v3 þ þ  þ  v3 þ R2 ∂x1 2 ∂x1 R1 ∂x1 ∂x1 ∂x21 ∂x21 ( þA22

!    ! !   2 2  2  2  2 2 2 ∂ξ2 ∂ v3 ∂ v3 1 ∂v3 ∂ v3 ∂ v3 ∂v3 ∂v3 ∂ v3 ∂ v3 þ þ þ þ þ 2 ∂x2 ∂x2 ∂x22 ∂x2 ∂x2 ∂x22 ∂x22 ∂x22 ∂x22 ∂x22

" ! #)  2  2  2 1 ∂ξ2 1 ∂v3 ∂v3 ∂v3 ∂ v3 ∂ v3 v3 þ þ  v3 þ þ  R2 ∂x2 2 ∂x2 R2 ∂x2 ∂x2 ∂x22 ∂x22 ( þ2A66

! !  2 2  2  2 ∂ v3 ∂ v3 ∂ξ1 ∂ v3 ∂ξ2 ∂ v3 þ þ þ ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 

2

2 ∂ v3 ∂v ∂v ∂ v3 þ 3 3 þ ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2

!

!   2  ∂v3 ∂v3 ∂2 v3 ∂ v3 þ þ ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2

!) ( !   2  2  2 2 ∂ v3 ∂v3 ∂v3 ∂ v3 ∂ξ1 ∂ v3 ∂ v3 þ þ A16 þ þ ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x2 ∂x21 ∂x21 !  2 2  ∂ξ2 ∂ v3 ∂ v3 ∂v ∂v þ þ þ 3 3 ∂x1 ∂x21 ∂x21 ∂x1 ∂x2 

2



2 ∂v ∂v ∂ v3 ∂ v3 þ 3 3 þ ∂x1 ∂x2 ∂x21 ∂x21

!

2



2 ∂ v3 ∂ v3 þ ∂x21 ∂x21

!



2



∂v ∂v ∂2 v3 ∂ v3 þ þ 3 3 ∂x1 ∂x2 ∂x21 ∂x21 2



2 ∂ v3 ∂ξ ∂ v3 þ þ2 1 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2

!

!

! !  2   2  2  2 2 ∂ v3 ∂ v3 ∂v3 ∂ v3 ∂v3 ∂v3 ∂ v3 þ þ2 þ þ ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 " !#)   2  2 ∂ v3 1 ∂ξ1 ∂ξ2 ∂v3 ∂v3 ∂v3 ∂v3 ∂v3 ∂v3 ∂ v3 þ þ þ þ þ þ  2v3 R1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ( þA26

2



2 ∂ξ2 ∂ v3 ∂ v3 þ ∂x1 ∂x22 ∂x22

!

2



2 ∂ξ ∂ v3 ∂ v3 þ þ 1 ∂x2 ∂x22 ∂x22

!

2



2 ∂v ∂v ∂ v3 ∂ v3 þ þ 3 3 ∂x2 ∂x1 ∂x22 ∂x22

!

2.5 Equations of Motion – Nonlinear Formulation

45

! ! !   2  2  2  2 2 ∂v3 ∂v3 ∂2 v3 ∂ v3 ∂ v3 ∂v3 ∂v3 ∂ v3 ∂ v3 ∂ξ2 ∂ v3 þ þ þ þ þ þ2 ∂x2 ∂x1 ∂x22 ∂x2 ∂x1 ∂x22 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x22 ∂x22 ! !  2   2  2  2 2 ∂ v3 ∂ v3 ∂v3 ∂ v3 ∂v3 ∂v3 ∂ v3 þ þ þ þ2 ∂x2 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 " !#)  2   2  ∂ v3 1 ∂ξ2 ∂ξ1 ∂v3 ∂v3 ∂v3 ∂v3 ∂v3 ∂v3 ∂ v3 þ þ þ þ þ  2v3 þ R2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 4

F 11

4

4

4

4

∂ v3 ∂ v ∂ v3 ∂ v ∂ v3  2ðF 12 þ2F 66 Þ 2 3 2  F 22  4F 16 3 3  4F 26 ∂x41 ∂x1 ∂x2 ∂x42 ∂x1 ∂x2 ∂x1 ∂x32

     2   2 h 1 ∂η1 C 1 ∂ v3 h þ2hK G13 1 þ þ 1þ  þ 2hK G23 1 þ 2h 2h h ∂x1 h ∂x21 2



! !   2  2 2  2  2 ∂ v3 1 ∂η2 C 1 ∂ v3 ∂ v3 ∂ v3 1 ∂ v3 T T þ 1þ þ þ N 11 N 12 þ ∂x1 ∂x2 ∂x1 ∂x2 ∂x21 ∂x21 R1 h ∂x2 h ∂x22

N T22

! ! !  2 2  2  2  2 2 ∂ v ∂ v ∂ v3 ∂ v3 1 ∂ v 1 ∂ v 3 3 3 3 þ þ þ þ  Nm  Nm þ 11 12 ∂x1 ∂x2 ∂x1 ∂x2 ∂x22 ∂x22 R2 ∂x21 ∂x21 R1 2

N m 22



2 ∂ v3 ∂ v3 1 þ þ ∂x22 ∂x22 R2

!

2

þC

∂v3 ∂ v þ mo 23 ¼ q3 ∂t ∂t

ð2:160Þ

The associated boundary conditions along the edges xn ¼ const(n ¼ 1, 2) are ξn ¼ e ξn e ξt ¼ ξt or ηn ¼ e ηn Lnt ¼ e Lnt or ηt ¼ e ηt ∂v3 ∂ev3 e nn M nn ¼ M or ¼ ∂xn ∂xn     ∂v3 ∂v3 ∂v3 ∂v3 ∂M nn ∂M nt N nt þ þ þ2 þ N nn þ ∂xt ∂xt ∂xn ∂xn ∂xn ∂xt e
 ∂M nt e þ a=h N n3 ¼ þ N n3 ∂xt e nn N nn ¼ N e N nt ¼ N nt Lnn ¼ e Lnn

or or

or

v3 ¼ ev3

ð2:161a  fÞ

46

2 Theory of Sandwich Plates and Shells with an Transversely Incompressible Core

2.5.3

Displacement Formulation for Sandwich Plates

Considering the weak core model and discarding the terms with curvatures in Eqs. (2.156)–(2.160) results in the nonlinear equations of motion for flat sandwich panels given as (

2





2 2 2 ∂ ξ1 ∂v3 ∂ v3 ∂ v3 ∂v3 ∂v3 ∂ v3 þ þ þ 2 2 2 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂x21

)

(

2

2

∂ ξ1 ∂ ξ2 þ þ ∂x22 ∂x1 ∂x2 )   2  2  2 2 ∂v3 ∂2 v3 ∂v3 ∂2 v3 ∂v3 ∂ v3 ∂v3 ∂ v3 ∂v3 ∂ v3 ∂v3 ∂ v3 þ þ þ þ þ þ ∂x2 ∂x1 ∂x2 ∂x1 ∂x22 ∂x2 ∂x1 ∂x2 ∂x1 ∂x22 ∂x2 ∂x1 ∂x2 ∂x1 ∂x22 ( ) (  2  2 2 2 ∂v3 ∂2 v3 ∂ ξ2 ∂v3 ∂ v3 ∂v3 ∂ v3 ∂ ξ1  A16 2 þ þ þ A12 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2

δξ1 : A11

2



þ A66



2



2 2 2 ∂v ∂2 v3 ∂ ξ2 ∂v ∂ v3 ∂v ∂ v3 ∂v ∂ v3 ∂v3 ∂ v3 þ2 3 þ2 3 þ2 3 þ 3 þ þ 2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x2 ∂x21 ∂x2 ∂x21 ∂x1 ) ( )   2  2 2 2 ∂v3 ∂2 v3 ∂N T11 ∂N T12 ∂ ξ2 ∂v3 ∂ v3 ∂v3 ∂ v3 ∂v3 ∂ v3  A  þ þ þ  ¼0 26 ∂x2 ∂x21 ∂x1 ∂x2 ∂x22 ∂x2 ∂x22 ∂x2 ∂x22 ∂x2 ∂x22

þ

ð2:162Þ (

2





2 2 2 ∂ ξ2 ∂v3 ∂ v3 ∂ v3 ∂v3 ∂v3 ∂ v3 þ þ þ 2 2 2 ∂x2 ∂x2 ∂x2 ∂x2 ∂x2 ∂x2 ∂x22

)

(

2

2

∂ ξ2 ∂ ξ1 þ þ ∂x21 ∂x1 ∂x2 )   2  2  2 2 ∂v3 ∂2 v3 ∂v3 ∂2 v3 ∂v3 ∂ v3 ∂v3 ∂ v3 ∂v3 ∂ v3 ∂v3 ∂ v3 þ þ þ þ þ þ ∂x1 ∂x1 ∂x2 ∂x2 ∂x21 ∂x1 ∂x1 ∂x2 ∂x2 ∂x21 ∂x1 ∂x1 ∂x2 ∂x2 ∂x21 ( ) (  2  2 2 2 ∂v3 ∂2 v3 ∂ ξ1 ∂ ξ2 ∂v3 ∂ v3 ∂v3 ∂ v3 A12 þ þ þ  A26 2 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2

δξ2 : A22

2





þ A66

2



2 2 2 ∂v ∂2 v3 ∂ ξ1 ∂v ∂ v3 ∂v ∂ v3 ∂v ∂ v3 ∂v3 ∂ v3 þ2 3 þ2 3 þ2 3 þ 3 þ þ 2 ∂x2 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x1 ∂x22 ∂x1 ∂x22 ∂x2 ) ( )   2  2 2 2 ∂v3 ∂2 v3 ∂ ξ1 ∂v3 ∂ v3 ∂v3 ∂ v3 ∂v3 ∂ v3 þ þ þ  A16 ∂x1 ∂x22 ∂x21 ∂x1 ∂x21 ∂x1 ∂x21 ∂x1 ∂x21

þ



∂N T22 ∂N T12  ¼0 ∂x2 ∂x1

ð2:163Þ

2.5 Equations of Motion – Nonlinear Formulation

δη1 :

47

 2  2 2 2 2 2 ∂ η1 ∂ η1 ∂ η2 ∂ η2 ∂ η2 ∂ η1 þ A þ A þ þ A þ 2 66 12 16 ∂x1 ∂x2 ∂x1 ∂x2 ∂x21 ∂x22 ∂x1 ∂x2 ∂x21  T T 2 b 11 1 ∂N b 12 ∂ η2 ∂v 1 ∂N þA26  d 1 η1 þ a 3   ¼0 2 ∂x ∂x ∂x ∂x2 h h 1 1 2 A11

ð2:164Þ δη2 :

 2  2 2 2 2 2 ∂ η2 ∂ η2 ∂ η1 ∂ η1 ∂ η1 ∂ η2 A22 þ A66 þ þ A26 þ2 þ A12 ∂x1 ∂x2 ∂x1 ∂x2 ∂x22 ∂x22 ∂x1 ∂x2 ∂x22  T T 2 b 22 1 ∂N b 12 ∂ η1 ∂v 1 ∂N  d 2 η2 þ a 3   ¼0 þA16 2 ∂x2 ∂x1 h ∂x2 h ∂x1

(

2



!



2

2



!

ð2:165Þ 

2 2 ∂ξ1 ∂ v3 ∂ v3 1 ∂v3 ∂ v3 ∂ v3 ∂v ∂v þ þ þ þ 3 3 2 ∂x1 ∂x1 ∂x21 ∂x1 ∂x1 ∂x21 ∂x21 ∂x21 ( !) ! !  2 2  2  2  2 2 ∂ v3 ∂ v3 ∂ξ1 ∂ v3 ∂ v3 ∂ξ2 ∂ v3 ∂ v3 þ þ þ þ A12 þ ∂x1 ∂x22 ∂x22 ∂x2 ∂x21 ∂x21 ∂x21 ∂x21 ! ! !  2 2  2 2  2  2  2  2 1 ∂v3 ∂ v3 ∂ v3 1 ∂v3 ∂ v3 ∂ v3 ∂v3 ∂v3 ∂ v3 ∂ v3 þ þ þ þ þ þ 2 ∂x1 2 ∂x2 ∂x1 ∂x1 ∂x22 ∂x22 ∂x22 ∂x21 ∂x21 ∂x22 !) ( ! !  2 2   2 2  2  2  2 ∂v3 ∂v3 ∂ v3 ∂ v3 ∂ξ2 ∂ v3 ∂ v3 1 ∂v3 ∂ v3 ∂ v3 þ þ þ þ þA22 þ 2 ∂x2 ∂x2 ∂x2 ∂x21 ∂x21 ∂x2 ∂x22 ∂x22 ∂x22 ∂x22 ( !) !  2   2 2  2  ∂ v3 ∂v3 ∂v3 ∂ v3 ∂ v3 ∂ξ1 ∂ v3 ∂ξ þ þ þ þ 2A66 þ 2 ∂x2 ∂x2 ∂x22 ∂x22 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ! ! !  2   2  2  2  2 ∂ v3 ∂ v3 ∂v3 ∂v3 ∂2 v3 ∂ v3 ∂ v3 ∂v3 ∂v3 ∂ v3 þ þ þ þ þ ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 !) ( ! !  2  2  2  2  2 2 ∂ v3 ∂v3 ∂v3 ∂ v3 ∂ξ1 ∂ v3 ∂ v3 ∂ξ2 ∂ v3 ∂ v3 þ þ þ þ þA16 þ ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x2 ∂x21 ∂x1 ∂x21 ∂x21 ∂x21 ! ! !   2  2  2  2 2 ∂v3 ∂v3 ∂2 v3 ∂ v3 ∂v3 ∂v3 ∂ v3 ∂ v3 ∂v3 ∂v3 ∂ v3 ∂ v3 þ þ þ þ þ þ ∂x1 ∂x2 ∂x21 ∂x1 ∂x2 ∂x21 ∂x1 ∂x2 ∂x21 ∂x21 ∂x21 ∂x21 !   ! 2  2  2 2 2 ∂ v3 ∂ v3 ∂ξ1 ∂ v3 ∂v3 ∂ v3 þ þ þ þ2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 !) ! (  2   2  2  2 ∂ v3 ∂v3 ∂v3 ∂ v3 ∂ξ2 ∂ v3 ∂ v3 þ2 þ þ A26 þ ∂x1 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x22 ∂x22

δv3 :

A11

48

2 Theory of Sandwich Plates and Shells with an Transversely Incompressible Core



2

2 ∂ξ ∂ v3 ∂ v3 þ 1 þ ∂x2 ∂x22 ∂x22

!



2

∂v ∂v þ 3 3 ∂x2 ∂x1 

2 ∂v ∂v ∂ v3 ∂ v3 þ 3 3 þ ∂x2 ∂x1 ∂x22 ∂x22 2



2 ∂ v3 ∂ v3 þ ∂x2 ∂x1 ∂x2 ∂x1

!

!



!



2



∂v ∂v ∂2 v3 ∂ v3 þ þ 3 3 ∂x2 ∂x1 ∂x22 ∂x22 2



2 ∂ v3 ∂ξ ∂ v3 þ þ2 2 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2

! þ

!

 2 ∂v3 ∂x2

!)   2  2 4 ∂ v3 ∂v3 ∂v3 ∂ v3 ∂ v3 þ þ2  F 11 ∂x2 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x41

4

2ðF 12 þ2F 66 Þ

2

2 ∂ v3 ∂ v3 þ ∂x22 ∂x22

4

4

4

∂ v3 ∂ v3 ∂ v ∂ v3  F 22  4F 16 3 3  4F 26 ∂x21 ∂x22 ∂x42 ∂x1 ∂x2 ∂x1 ∂x32

    2   2 h 1 ∂η1 C 1 ∂ v3 h 1 ∂η2 þ2hK G13 1 þ þ 1þ þ 2hK G23 1 þ 2 ∂x 2h h 1 2h h ∂x2 h ∂x1 2

! !   2  2 2  2  2 ∂ v3 C 1 ∂ v3 ∂ v3 ∂ v3 ∂ v3 T T þ þ 1þ  N 11  N 12 þ ∂x1 ∂x2 ∂x1 ∂x2 ∂x21 ∂x21 h ∂x22 2



!

2



!

N T22

2 ∂ v3 ∂ v3 þ ∂x22 ∂x22

N m 22

2 ∂ v3 ∂ v3 þ ∂x22 ∂x22

! !  2 2  2  2 ∂ v ∂ v ∂ v ∂ v 3 3 3 3 þ  Nm þ  Nm 11 12 ∂x1 ∂x2 ∂x1 ∂x2 ∂x21 ∂x21 2

þC

∂v3 ∂ v þ mo 23 ¼ q3 ∂t ∂t

ð2:166Þ

The associated boundary conditions along the edges xn ¼ const(n ¼ 1, 2) are e nn N nn ¼ N e N nt ¼ N nt Lnn ¼ e Lnn

or

Lnt ¼ e Lnt

or

or or

ξn ¼ e ξn e ξt ¼ ξt ηn ¼ e ηn

ηt ¼ e ηt ∂v3 ∂ev3 e nn M nn ¼ M or ¼ ∂xn ∂xn     ∂v3 ∂v3 ∂v3 ∂v3 ∂M nn ∂M nt N nt þ þ þ2 þ N nn þ ∂xt ∂xt ∂xn ∂xn ∂xn ∂xt e
 ∂M nt e þ a=h N n3 ¼ þ N n3 ∂xt

or

v3 ¼ ev3

ð2:167a  fÞ

2.6 Equations of Motion – Linear Formulation

2.6

49

Equations of Motion – Linear Formulation

2.6.1

Displacement Formulation for Sandwich Shells

Considering Eqs. (2.156)–(2.160), the following linear equations of motion for sandwich shells can be obtained by discarding all of the nonlinear terms and neglecting the thermal and moisture terms. This results in δξ1 :

δξ2 :

 2   2   2  2 ∂ ξ1 1 ∂v3 ∂ ξ1 ∂ ξ2 ∂ ξ2 1 ∂v3  þ  þ A þ A  66 12 ∂x1 ∂x2 R2 ∂x2 ∂x21 R1 ∂x1 ∂x22 ∂x1 ∂x2    2  2 2 ∂ ξ1 ∂ ξ ∂ ξ2 1 ∂v3 1 ∂v3 þ 22    A26 ¼0 A16 2 ∂x1 ∂x2 ∂x1 R1 ∂x1 ∂x22 R2 ∂x2

A11

ð2:168Þ  2   2    2 2 ∂ ξ2 1 ∂v3 ∂ ξ2 ∂ ξ1 ∂ ξ1 1 ∂v3 A22  þ  þ A66 þ A12  ∂x1 ∂x2 R1 ∂x1 ∂x22 R2 ∂x2 ∂x21 ∂x1 ∂x2     2 2 2 ∂ ξ2 ∂ ξ1 1 ∂v3 ∂ ξ1 1 ∂v3   þ  A ¼0 A26 2 16 2 2 R R ∂x1 ∂x2 ∂x2 ∂x1 2 ∂x2 1 ∂x1 ð2:169Þ  2  2 2 ∂ η1 ∂ η1 ∂ η2 ∂ η2 A11 þ A þ þ A 66 12 ∂x1 ∂x2 ∂x21 ∂x22 ∂x1 ∂x2  2    2 2 ∂ η2 ∂ η1 ∂ η2 ∂v3 þA16 þ2  d 1 η1 þ a þ A26 ¼0 ∂x1 ∂x2 ∂x1 ∂x21 ∂x22  2  2 2 2 ∂ η2 ∂ η2 ∂ η1 ∂ η1 A22 þ A þ þ A 66 12 ∂x1 ∂x2 ∂x22 ∂x22 ∂x1 ∂x2  2    2 2 ∂ η1 ∂ η2 ∂ η1 ∂v3 þA26 þ 2  d η þ a þ A ¼0 16 2 2 ∂x1 ∂x2 ∂x2 ∂x22 ∂x21 2

δη1 :

δη2 :

ð2:170Þ

ð2:171Þ

       A11 A12 ∂ξ1 A16 A26 ∂ξ1 A22 A12 ∂ξ2 A26 A16 ∂ξ2 δv3 :  þ  þ  þ  þ Þ R1 R2 ∂x1 R1 R2 ∂x2 R2 R1 ∂x2 R2 R1 ∂x1       2 2 4 ∂η1 ∂η2 A11 A22 2A12 ∂ v ∂ v ∂ v3 v3  d 1 a  þ 2 þ þ a 23  d2 a þ a 23 þ F 11 2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x41 R1 R2 R1 R2 

4

þ F 22

4

4

4

2

∂ v3 ∂ v ∂ v ∂ v3 ∂ v3 þ 2ðF 12 þ 2F 66 Þ 2 3 2 þ 4F 16 3 3 þ 4F 26 þ N 011 þ ∂x42 ∂x1 ∂x2 ∂x21 ∂x1 ∂x2 ∂x1 ∂x32 2

þ 2N 012

2

2

∂ v3 ∂ v3 ∂v ∂ v þ N 022  c 3  m0 23 ¼ q3 ∂x1 ∂x2 ∂t ∂t ∂x22

ð2:172Þ

50

2 Theory of Sandwich Plates and Shells with an Transversely Incompressible Core

where N 011 , N 022 , and N 012 are the prescribed edge loads. The associated boundary conditions are along the edges xn ¼ const(n ¼ 1, 2) e nn N nn ¼ N e nt N nt ¼ N Lnn ¼ e Lnn ent Lnt ¼ L

or

ξn ¼ e ξn e ξt ¼ ξt

or

ηn ¼ e ηn

or

ηt ¼ e ηt

or

∂v3 ∂ev3 ¼ ∂xn ∂xn     ∂v3 ∂v3 ∂v3 ∂v3 ∂M nn ∂M nt N nt þ þ þ2 þ N nn þ ∂xt ∂xt ∂xn ∂xn ∂xn ∂xt e nn M nn ¼ M

or

or

v3 ¼ ev3

e nt e
 ∂M þ a=h N n3 ¼ þ N n3 ∂xt ð2:173a  fÞ

2.6.2

Displacement Formulation for Sandwich Plates

The governing equations of motion for flat sandwich plates can be obtained by setting the curvatures to zero in Eqs. (2.168)–(2.172). This results in decoupling occurring between stretching and bending, leaving the first two equations of motion Eqs. (2.168) and (2.169) decoupled from the last three. As a result, the system of equations, Eqs. (2.168)–(2.172) are reduced to

δη1 :

δη2 :

 2  2 2 2 ∂ η1 ∂ η1 ∂ η2 ∂ η2 A11 þ A þ þ A 66 12 ∂x1 ∂x2 ∂x21 ∂x22 ∂x1 ∂x2  2    2 2 ∂ η2 ∂ η1 ∂ η2 ∂v3 þA16 þ 2  d η þ a þ A ¼0 26 1 1 ∂x1 ∂x2 ∂x1 ∂x21 ∂x22

ð2:174Þ

 2  2 2 2 ∂ η2 ∂ η2 ∂ η1 ∂ η1 A22 þ A þ þ A 66 12 ∂x1 ∂x2 ∂x22 ∂x22 ∂x1 ∂x2  2    2 2 ∂ η1 ∂ η2 ∂ η1 ∂v3 þA26 þ 2  d η þ a þ A ¼0 16 2 2 ∂x1 ∂x2 ∂x2 ∂x22 ∂x21

ð2:175Þ

2.7 Boundary Conditions

51



δv3 :

 2  2  4 4 ∂η1 ∂η2 ∂ v3 ∂ v3 ∂ v3 ∂ v3 d1 a þ a 2  d2 a þ a 2 þ F 11 þ F 22 ∂x1 ∂x2 ∂x1 ∂x2 ∂x41 ∂x42 4

þ 2ðF 12 þ 2F 66 Þ 2

þ 2N 012

4

4

2

∂ v3 ∂ v ∂ v3 ∂ v3 þ 4F 16 3 3 þ 4F 26 þ N 011 þ ∂x21 ∂x22 ∂x21 ∂x1 ∂x2 ∂x1 ∂x32 2

2

∂ v3 ∂ v3 ∂v ∂ v þ N 022  c 3  m0 23 ¼ q3 ∂x1 ∂x2 ∂t ∂t ∂x22 ð2:176Þ

Because of the decoupling, the associated boundary conditions are reduced to the following four boundary conditions along the edges xn ¼ const(n ¼ 1, 2) enn Lnn ¼ L Lnt ¼ e Lnt

ηn ¼ e ηn ηt ¼ e ηt ∂v3 ∂ev3 e nn M nn ¼ M or ¼ ∂xn ∂xn     ∂v3 ∂v3 ∂v3 ∂v3 ∂M nn ∂M nt N nt þ þ þ2 þ N nn þ ∂xt ∂xt ∂xn ∂xn ∂xn ∂xt e
 ∂M nt e þ a=h N n3 ¼ þ N n3 ∂xt or or

or

v3 ¼ ev3

ð2:177a  dÞ

2.7

Boundary Conditions

Two types of boundary conditions that will be considered here are simply supported and clamped. Referring to the boundary conditions in Eqs. (2.151a–f) and (2.161a– f), the following determination is made. For Simply Supported edge conditions of the sandwich shell there are two cases. Case I: The edges xn ¼ 0, Ln is loaded in compression and freely movable. In this case, along these edges, the following conditions need to be fulfilled. e nn , N nt ¼ 0, ηn ¼ 0, ηt ¼ 0, M nn ¼ 0, v3 ¼ 0 N nn ¼ N

ð2:178Þ

Case II: The edges xn ¼ constant is unloaded and immovable. For this case, the following conditions have to be fulfilled. ξn ¼ 0, N nt ¼ 0, ηn ¼ 0, ηt ¼ 0, M nn ¼ 0, v3 ¼ 0 For Clamped boundary conditions, along the edges xn¼ 0, Ln,

ð2:179Þ

52

2 Theory of Sandwich Plates and Shells with an Transversely Incompressible Core

ξn ¼ 0, ξt ¼ 0, ηn ¼ 0, ηt ¼ 0,

∂v3 , v3 ¼ 0: ∂xn

ð2:180Þ

n ¼ normal. The subscripts n and t are used to designate the normal and tangential in-plane directions to an edge and hence, n ¼ 1 when t ¼ 2 and vice versa. For the case expressing the immovability of the edges ξn ¼ 0, it is implied that this is fulfilled in an average sense which is expressed mathematically as Z

l2 Z l1

0

0

∂ξn dxn dxt ¼ 0 ∂xn

ð2:181Þ

Analogously, the static boundary conditions, Eqs. (2.178a, b) and (2.179b) are also fulfilled in an average sense expressed as Z

Lt

e nn Ln N nn dxt ¼ N

ð2:182Þ

0

 N nt dxt ¼ 0,

0

n ¼ 1, 2 t ¼ 2, 1

 X

ð2:183Þ

n, t

e nn denotes the compressive edge load on the edges xn ¼ 0, Ln. The sign where N means no summation on n, t.

2.8

P



Lt



Z

Summary

In this chapter, the governing equations for the case of the core being incompressible have been derived. First the derivations started out with the most general case considering the strong core with the transverse stresses in the facings. Hamilton’s principle was then leveraged for the theoretical developments. The equations of motion were derived for both the mixed formulation and the displacement formulation. Later on, special cases were considered which were the discarding of the transverse shear stresses in the facings, adopting the Love–Kirchhoff assumptions, and the case of the weak core neglecting the in-plane stresses. Only the transverse stresses were considered in the weak core case. Finally, various types of boundary conditions were discussed. These governing equations will be utilized in later chapters on buckling, post-buckling, and the dynamic response where several solution methodologies will be applied to these equations. Several references to the equations in this chapter and the next few chapters will be forthcoming.

References

53

References Hause, T., Librescu, L., & Johnson, T. F. (1998). Thermomechanical load-carrying capacity of sandwich flat panels. Journal of Thermal Stresses, 21(6), 627–653. Hause, T., Johnson, T. F., & Librescu, L. (2000). Effect of face-sheet anisotropy on buckling and post buckling of flat sandwich panels. Journal of Spacecraft and Rockets, 37(3), 331–341. Jones, R. M. (1999). Mechanics of composite materials (2nd ed.). New York/London: Taylor and Francis. Librescu, L. (1970). On a geometrically non-linear theory of elastic anisotropic sandwich type plates. Revue Roumaine des Sciences techniques-Mecanique Appliquee, 15(2), 323–339. Librescu, L. (1975). Elastostatics and kinetics of anisotropic and heterogeneous shell-type structures. Leyden: Noordhoff International Publishing. Librescu, L., & Chang, M. J. (1993). Effects of geometric imperfections on vibration of compressed shear deformable composite curved panels. Acta Mechanica, 96, 203–224. Librescu, L., Hause, T., & Camarda, C. J. (1997). Geometrically nonlinear theory of initially imperfect sandwich plates and shells incorporating non-classical effects. AIAA Journal, 35(8), 1393–1403. Reddy, J. N. (2004). Mechanics of laminated composite plates and shells-theory and analysis (2nd ed.). Boca Raton: CRC Press.

Chapter 3

Buckling of Sandwich Plates and Shells

Abstract Application of the governing linear equations of equilibrium and the associated boundary conditions as applied to the stability of sandwich plates and shells with anisotropic laminated composite facings and a weak core is addressed while the facings are considered symmetric with respect to both their local and global mid-surfaces. Both sandwich plates and shells are considered with implications of various structural configurations. Such structural configurations consider the structural tailoring of the face sheets, the panel face thickness, the aspect ratio, etc. on the buckling load. Finally, validations are made with results found within the existing literature.

3.1

Introduction

This chapter is concerned with the static or stability behavior of anisotropic laminated composite sandwich plates and shells. Both flat and curved sandwich panels are discussed and treated separately. This study concerns symmetric laminated facings both locally and globally in conjunction with a weak or soft core. In addition, the facings are assumed thin so that the Kirchhoff–Love assumptions apply. Simply supported boundary conditions with all four edges freely movable are also considered. The chosen solution methodology is devoted to the extended Galerkin method, while the influence of a number of kinematical and physical parameters, on the loadcarrying capacity of the structure, are investigated. Finally, validations are presented to highlight the accuracy of the theory.

3.2

Preliminaries and Basic Assumptions

The governing linear theory for both flat and doubly curved sandwich panels is based on a number of previously mentioned assumptions in Chap. 2 but are repeated here for reference. These assumptions are relisted as

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 T. J. Hause, Sandwich Structures: Theory and Responses, https://doi.org/10.1007/978-3-030-71895-4_3

55

56

3 Buckling of Sandwich Plates and Shells

1. The face sheets are constructed of a number of orthotropic material layers, the axes of orthotropy of the individual plies being not necessarily coincident with the geometrical axes xα (α ¼1, 2) of the structure. 2. The thickness of the core is much larger than those of the face sheets, that is, 2h  h= , h== . 3. The core material features orthotropic properties, the axes of orthotropy being parallel to the geometrical axes xα. 4. The cases of the weak-core type are considered. 5. A perfect bonding between the face sheets and between the faces and the core exists. 6. All three layers, the facings and the core, are incompressible in the transverse direction. 7. The global middle surface of the structure is selected to coincide with that of the core layer which is referred to a curvilinear and orthogonal coordinate system, xα(α ¼ 1, 2). The transverse normal coordinate x3 is considered positive when measured in the direction of the downward normal. 8. The principles of shallow shell theory apply in regard to doubly curved sandwich panels.

3.3

Buckling of Flat Sandwich Panels

3.3.1

Governing System

This section is concerned with the stability of sandwich plates at the critical load. The solution methodology begins with the linear governing system of partial differential equations for sandwich plates considering a weak core and the transverse shear effects neglected in the facings. The pertinent equations are a subset of the governing equations for shells. These equations were developed in Chap. 2 as Eqs. (2.174)– (2.176) with the reduced set of boundary conditions, Eqs. (2.177a–d). These governing equations are • Equations of Equilibrium:
2 
2  2 2 2 2 ∂ η1 ∂ η1 ∂ η2 ∂ η2 ∂ η2 ∂ η1 A11 þ A66 þ þ A16 þ2 þ A12 ∂x1 ∂x2 ∂x1 ∂x2 ∂x21 ∂x22 ∂x1 ∂x2 ∂x21
 2 ∂ η2 ∂v þA26  d 1 η1 þ a 3 ¼ 0 ∂x1 ∂x22 ð3:1Þ

3.3 Buckling of Flat Sandwich Panels

57


2 
2  2 2 2 2 ∂ η2 ∂ η2 ∂ η1 ∂ η1 ∂ η1 ∂ η2 A22 þ A66 þ þ A26 þ2 þ A12 ∂x1 ∂x2 ∂x1 ∂x2 ∂x22 ∂x22 ∂x1 ∂x2 ∂x22
 2 ∂ η1 ∂v þA16  d 2 η2 þ a 3 ¼ 0 2 ∂x2 ∂x1 ð3:2Þ

2  2  4 4 ∂η1 ∂η2 ∂ v3 ∂ v3 ∂ v3 ∂ v3 d 1 a þ a 2  d2 a þ a 2 þ F 11 þ F 22 4 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x42 4

þ2ðF 12 þ2F 66 Þ 2

þ2N 012

4

4

2

∂ v3 ∂ v ∂ v3 ∂ v3 þ 4F 16 3 3 þ 4F 26 þ N 011 2 2 3 ∂x1 ∂x2 ∂x21 ∂x1 ∂x2 ∂x1 ∂x2

ð3:3Þ

2

∂ v3 ∂ v3 þ N 022 ¼0 ∂x1 ∂x2 ∂x22

where N 011 , N 022 , and N 012 are the prescribed edge loadings in both the x1 and x2 directions. It should be noted that the damping, inertia, and transversal load terms have been neglected. Because of the decoupling between stretching and bending, the associated boundary conditions are reduced to the following four boundary conditions along the edges xn ¼ 0, Ln which are • Boundary conditions: Lnn ¼ e Lnn

or

ηn ¼ e ηn

ð3:4Þ

Lnt ¼ e Lnt

or

ηt ¼ e ηt

ð3:5Þ

∂v3 ∂ev3 ¼ ∂xn ∂xn

ð3:6Þ

e nn M nn ¼ M

or

∂v3 ∂v ∂M nn ∂M nt þ N nn 3 þ þ2 ∂xt ∂xn ∂xn ∂xt e nt e   ∂M þ a=h N n3 ¼ þ N n3 ∂xt N nt

or

v3 ¼ ev3 ð3:7Þ

considering simply supported boundary conditions, the following conditions apply Along the edges x1 ¼ 0, L1 η1 ¼ 0, η2 ¼ 0, M 11 ¼ 0, v3 ¼ 0

ð3:8a  dÞ

Along the edges x2 ¼ 0, L2 η2 ¼ 0, η1 ¼ 0, M 22 ¼ 0, v3 ¼ 0

ð3:9a  dÞ

58

3 Buckling of Sandwich Plates and Shells

In terms of displacements, the third boundary conditions are expressed as 2

2

2

M 11 ¼ F 11

∂ v3 ∂ v3 ∂ v3 þ F 12 þ 2F 16 ¼0 2 2 ∂x ∂x1 ∂x2 1 ∂x2

M 22 ¼ F 22

∂ v3 ∂ v3 ∂ v3 þ F 12 þ 2F 26 ¼0 ∂x1 ∂x2 ∂x22 ∂x21

2

2

ð3:10aÞ

2

ð3:10bÞ

These expressions will become useful later on when the displacement functions have been determined and thus the fulfillment of these particular boundary conditions will become readily apparent.

3.3.2

Solution Methodology

The solution methodology chosen here is the extended Galerkin method (EGM). The advantage of using this method relies on the fact that the unfulfilled boundary conditions are satisfied in an average sense. As a starting point, the transverse displacement can be represented as v3 ðx1 , x2 , t Þ ¼ wmn sin λm x1 sin μn x2

ð3:11Þ

where λm ¼ mπ/L1 and μn ¼ nπ/L2. This completely satisfies the condition that v3 ¼ 0 along all four edges. But does not satisfy the conditions for M11 ¼ 0, M22 ¼ 0 along the edges x1 ¼ 0, L1, x2 ¼ 0, L2, respectively. These will be satisfied later on, in an average sense, through the use of the EGM. The first two equations of equilibrium Eqs. (3.1) and (3.2) can be satisfied by assuming η1 and η2 as follows 

η 1 ð x1 , x2 Þ η 2 ð x1 , x2 Þ



(

) 1Þ H ðmn cos λm x1 sin μn x2 ¼ 1Þ I ðmn ( ) 2Þ H ðmn sin λm x1 cos μn x2 þ 2Þ I ðmn

ð3.12a, bÞ

1Þ 2Þ ð1Þ 2Þ , H ðmn , I mn , and I ðmn are undetermined coefficients. Substituting the where H ðmn expressions for η1 and η2 back into Eqs. (3.1) and (3.2) and comparing coefficients of like trigonometric functions gives the following matrix equation in terms of the unknown coefficients as

3.3 Buckling of Flat Sandwich Panels

2 6 6 6 6 6 4

ðm,nÞ

ðm,nÞ

J 11

59

3 8 ð1Þ 9 8 ð1Þ 9 K mn > > H mn > > > > > > 7 > > > ðm,nÞ 7> = < < ð 2 Þ 2Þ = J 13 7 H mn K ðmn ¼ 7 ðm,nÞ > I ð1Þ > > K ð3Þ > > > > J 34 7 > > > > 5> ; ; : mn : mn ð 2 Þ ð4Þ ðm,nÞ I K mn mn J 33

ðm,nÞ

J 12

ðm,nÞ

J 13

ðm,nÞ

J 14

ðm,nÞ

J 11

J 14

ðm,nÞ

J 33 Symm:

ð3:13Þ

where ðm,nÞ

  ¼  λ2m A11 þ μ2n A66 þ d1 ,

ðm,nÞ

ðm,nÞ

J 12 ¼ 2λm μn A16 , J 13 ¼ λ2m A16  μ2n A26   ðm,nÞ ðm,nÞ ðm,nÞ J 14 ¼ λm μn ðA12 þ A66 Þ, J 33 ¼  μ2n A22 þ λ2m A66 þ d2 , J 34 ¼ 2λm μn A26 J 11

1Þ K ðmn ¼ d1 aλm wmn ,

2Þ K ðmn ¼ 0,

3Þ K ðmn ¼ 0,

3Þ K ðmn ¼ d 2 aμn wmn

ð3:14a  jÞ αÞ αÞ From Eq. (3.13), using Cramer’s rule, the coefficients H ðmn , I ðmn ðα ¼ 1, 2Þ can be expressed as

  ð1Þ ð2Þ ð1Þ ð2Þ 1Þ 2Þ ð1Þ ð2Þ e mn , eI mn , eI mn wmn , e mn , H , H ðmn , I mn , I mn ¼ H H ðmn

ð3:15Þ

where 1Þ detðJ 1 Þ e ðmn H ¼ , detðJ Þ

¼

2Þ detðJ 2 Þ e ðmn ¼ H , detðJ Þ

1Þ detðJ 3 Þ eI ðmn ¼ , detðJ Þ

detðJ 4 Þ detðJ Þ

2Þ eI ðmn

ð3:16a  dÞ

While, 0

1Þ K ðmn

ðm:nÞ

J 12

B B K ð2Þ J ðm,nÞ B 11 J 1 ¼ B mn B K ð3Þ J ðm,nÞ @ mn 14 ðm,nÞ ð4Þ K mn J 13 0 ðm,nÞ 1Þ K ðmn J B ð11m:nÞ BJ 2Þ K ðmn B ¼ B 12 B J ðm,nÞ K ð3Þ @ 13 mn ðm,nÞ 4Þ J 14 K ðmn

ðm,nÞ

J 13

ðm,nÞ

J 14

ðm,nÞ

J 33

ðm,nÞ

J 34

ðm,nÞ

J 13

ðm,nÞ

J 14

ðm,nÞ

J 33

ðm,nÞ

J 34

ðm,nÞ

J 14

1

C C C C, ðm,nÞ J 34 C A ðm,nÞ J 33 1 ðm,nÞ J 14 C ðm,nÞ J 13 C C C ðm,nÞ C J 34 A ðm,nÞ J 33 ðm,nÞ

J 13

J2

ð3.17a, bÞ

60

3 Buckling of Sandwich Plates and Shells

0

ðm,nÞ

J 11

B ðm:nÞ BJ B J 3 ¼ B 12 B J ðm,nÞ @ 13 ðm,nÞ J 14 0 ðm,nÞ J B ð11m:nÞ BJ B ¼ B 12 B J ðm,nÞ @ 13 ðm,nÞ J 14 0 B B B J¼B B @

ðm:nÞ

1Þ K ðmn

J 14

ðm,nÞ

2Þ K ðmn

J 13

ðm,nÞ

3Þ K ðmn

ðm,nÞ

4Þ K ðmn

J 12

J 11 J 14 J 13

ðm:nÞ

J 12

ðm,nÞ

J 11

ðm,nÞ

J 14

ðm,nÞ

J 13

ðm,nÞ

J 11

ðm,nÞ

1

C C C C, ðm,nÞ J 34 C A ðm,nÞ J 33 1 1Þ K ðmn C 2Þ C K ðmn C C ð3Þ C K mn A 4Þ K ðmn ðm,nÞ

ðm,nÞ

J 13

ðm,nÞ

J 14

ðm,nÞ

J 33

ðm,nÞ

J 34

ðm,nÞ

J 12

ðm,nÞ

J 11

ðm,nÞ

J 13

ðm,nÞ

J 14

ðm,nÞ

J 33 Symm:

J4

ð3.17c, dÞ

ðm,nÞ

J 14

1

C C C C ðm,nÞ C J 34 A ðm,nÞ J 33 ðm,nÞ

J 13

ð3:17eÞ

Although the first two equations of equilibrium are satisfied by the assumed functional forms for η1 and η2, the boundary conditions for ηn and ηt along the edges xn ¼ 0, Ln are not satisfied. Again, these will be satisfied in an average sense through the use of the EGM. At this juncture, all of the displacement quantities v3, η1, η2 are known. In addition, there remains only the fifth equation of equilibrium unfulfilled along with the first, second, and third boundary conditions. To solve for the critical load, the unfulfilled fifth equation of motion and the unfulfilled boundary conditions will be retained in the energy functional with the appropriate integrations carried out. In the end, this will supply the critical load for the stability problem in terms of the geometrical and material parameters. Retaining these unfulfilled expressions in the Hamilton’s energy functional results in Z

t 1 Z l 2 Z l 1  t0

0

0

d1 a



2  2  4 4 ∂η1 ∂η2 ∂ v ∂ v ∂ v3 ∂ v3 þ a 23 d 2 a þ a 23 þF 11 þ F 22 ∂x1 ∂x2 ∂x1 ∂x2 ∂x41 ∂x42 4

4

4

2

∂ v3 ∂ v ∂ v3 ∂ v3 þ 4F 16 3 3 þ 4F 26 þ N 011 þ ∂x21 ∂x22 ∂x21 ∂x1 ∂x2 ∂x1 ∂x32   Z t1 Z l2 2 2 ∂ v3 ∂ v3 þ N 022 dx dx þ2N 012 δv dt þ L11 δη1 þ L12 δη2 þ 3 1 2 ∂x1 ∂x2 ∂x22 t0 0

      Z t1 Z l1 ∂v3  l1 ∂v3  l2 M 11 δ dx2 dt þ dx1 dt L22 δη2 þ L12 δη1 þ M 22 δ ∂x1  0 ∂x2  0 t0 0 ¼0 ð3:18Þ þ 2ðF 12 þ 2F 66 Þ

3.3 Buckling of Flat Sandwich Panels

61

Substituting in the expressions for v3, η1, η2 and carrying out the indicated integrations and solving for the prescribed edge loads results in the buckling solution for angle-ply laminated sandwich structures which results in a linear algebraic equation as  ð1Þ  2Þ e mn þ aλm þ μn ad 2 eI ðmn N 011 λ2m þ N 022 μ2n ¼ λm ad1 H þ aμn þ F 11 λ2m þ F 22 μ2n þ 4F 66 λ2m μ2n þ 2F 12 λ2m μ2n ð3:19Þ Where the shear loading term, N 012 has been discarded. Nondimensionalizing Eq. (3.19) results in   n4 ϕ4 F 22 2m2 n2 ϕ2 ðF 12 þ 2F 66 Þ a2 L21  2 þ þ 2 m d1 þ K x m 2 π 2 þ LR n 2 ϕ 2 ¼ m 4 þ F 11 F 11 π F 11 i  aL3 h 1Þ ð2Þ e ðmn þn2 ϕ2 d 2 þ 3 1 md 1 H þnϕd2eI mn π F 11 ð3:20Þ Where the nondimensional parameters are defined as Kx ¼

L21 N 011 N 022 L , L ¼ , ϕ¼ 1 R 4 0 L2 π F 11 N 11

ð3:21Þ

1Þ ð2Þ e ðmn and eI mn should also be It should be mentioned that the coefficients H nondimensionalized which is not shown here.

3.3.3

Validation of the Theoretical Results

Before addressing the present results some validations are displayed which compare the results from the present theory with the experimental counterparts (Alexandrov et al. 1960). Table 3.1 lists the geometrical and material properties, in addition to the buckling response, for a three-layered flat sandwich panel with isotropic facings (Dura Aluminum, ν ¼ 0.3, E ¼ 6.96  105 kg/cm2) and a transversely isotropic core (Penoplast). Comparing the present theory with the experimental results reasonable agreement is seen keeping in mind that with the inclusion of geometric imperfections, within the theory, would provide more exact agreement. Results in Table 3.2 display the material and geometrical properties and the buckling response for a three-layered flat sandwich panel with isotropic facings (Dura Aluminum) and an orthotropic core. In both cases, simply supported boundary conditions are assumed. As with the results in Table 3.1, the results in Table 3.2

62

3 Buckling of Sandwich Plates and Shells

Table 3.1 Comparisons of theoretical and buckling predictions for a flat sandwich panel with a transversely isotropic core with isotropic faces (Validation No. 1) Case 1 2 3 4 5 6

L1(cm) 60 60 40 40 80 80

L2(cm) 40 40 60 60 60 60

h (cm) 0.05 0.10 0.10 0.10 0.05 0.05

h ðcmÞ 0.425 0.650 0.700 1.400 0.450 0.450

G ðkg=cm Þ 99.4 149.6 117.1 96.5 73.5 74.1 2

e 11 L2  103 kg N Present A. et al. 3.79 3.60 9.03 8.25 11.30 12.30 17.40 16.00 4.21 4.00 4.24 4.10

% Error +5.28 +9.45 -8.13 +8.75 +5.25 +3.41

A. et al. ! Alexandrov et al. (1960) Table 3.2 Comparisons of theoretical and buckling predictions for a flat sandwich panel with an orthotropic core and isotropic faces (Validation No. 2) e 11 L2  103 kg N Case 1 2 3 4

L1(cm) 60 60 60 80

L2(cm) 40 40 40 60

hf (cm) 0.05 0.25 0.25 0.10

h ðcmÞ 0.45 1.25 1.15 0.95

a

G13 140.4 390.0 337.0 138.1

a

G23 100.8 103.0 97.0 78.6

Present 5.29 47.22 38.20 17.34

A. et al. 5.85 46.57 36.50 15.25

% Error -9.57 +1.11 +4.66 +13.7

ν12 0.28 0.06 0.3 0.32

α1 (1/K) 11.34 1.62 – –

α2 (1/K) 36.9 1.674 – –

a

Units are in (kg/cm2) A. et al. ! Alexandrov et al. (1960)

Table 3.3 Face sheet material properties Type F1 F2 F3 F4

Material HS Graph. Ep. IM7/977-2 SCS-6/Ti-15-3 CFRP

E1(N/mm2) 1.8375 0.0812 0.19404 2.324

E2(N/mm2) 0.105 0.0763 0.12663 0.13552

G12 (N/mm) 0.0735 0.0098 0.05705 0.05327

Note: Multiply E1, E2, G12  105

reveal a slight over agreement which again is most likely due to the imperfections within the test specimen.

3.3.4

Results and Discussion

For the following numerical results, unless specified otherwise, the material properties for the face sheets and the core are listed in Tables 3.3 and 3.4 by type. Some additional validations are made in Fig. 3.1 where the effect of the panel face thickness on the critical buckling load for a flat sandwich panel of a fixed stacking sequence in the facings is shown for specific values of a (the distance between the

3.3 Buckling of Flat Sandwich Panels

63

Table 3.4 Core material properties Type C1 C2 C3

Core type Titanium honeycomb Aluminum honeycomb Aluminum honeycomb

G13 (N/mm2) 0.0145  10 148.239 99.4

G23 (N/mm2) 0.0066  105 91.7847 60.2

Fig. 3.1 The compressive buckling load vs. panel face thickness for fixed values of the distance between the mid-surface of the core and the mid-surface of the upper and local face sheets

mid-surface of the core and the mid-surface of the facings). The results in Fig. 3.1 are of the F4:C1 type. It can be seen that as the face thickness increases, the critical buckling load increases. Overlaid on the plot are data points from Pearce and Webber (1972) who considered the same results. Almost perfect agreement is seen. In Fig. 3.2, the effect of the ply angle on the buckling load for various aspect ratios of a four-layered flat sandwich panel are depicted for a F3:C1 type. All of the trends appear to be more flat than curved. For lower aspect ratios, the critical buckling load is higher across the entire ply angle spectrum. For higher aspect ratios, the critical buckling loads are lower. In this case, data taken from Ko and Jackson (1991, 1993) for the same configuration are overlaid on the present results and perfect agreement is seen. Figures 3.3 and 3.4 which are of the F1:C1 type display the effect of the directional material properties on the critical buckling load for a single-layered and a three-layered flat sandwich panel for the given layup. The buckling loads in both cases appear to peak around 45 degrees then drop off up until 90 degrees. The three-layered sandwich panel can sustain larger buckling loads

64

3 Buckling of Sandwich Plates and Shells

Fig. 3.2 The effect of the fiber orientation of the face sheet lamina on the buckling load of the flat sandwich panel

Fig. 3.3 The effect of the fiber orientation of the face sheet lamina on the buckling load of singlelayered flat sandwich panel

as compared to its single-layered counterpart. The peak critical loads at their determined ply angle are recorded in Table 3.5 for later reference in Chap. 5 concerning free vibration where it is found that at the peak buckling load the eigenfrequencies vanish.

3.4 Buckling of Doubly Curved Sandwich Panels

65

Fig. 3.4 The effect of the fiber orientation of the face sheet lamina on the buckling load of threelayered flat sandwich panel Table 3.5 Maximum buckling loads and associated ply angles for two different layups of the face sheets for a flat sandwich panel ψa 0.9 Layup [θ/c/θ] (2.021) [47.7] [θ/  θ/θ/c]s (2.089) [47.7]  a e 11  103 Sequence ( ) [ ]  N

3.4 3.4.1

1.0 1.2 (1.634) [44.1] (1.193) [37.8] (1.682) [44.1] (1.222) [38.6]  N=mm ½θdeg

1.4 (0.975) [31.5] (0.994) [31.5]

1.6 (0.878) [18.9] (0.889) [19.8]

Buckling of Doubly Curved Sandwich Panels Governing System

This section is concerned with the stability of doubly curved sandwich shells. In contrast to the previous section covering flat sandwich plates, the governing equations are coupled between stretching and bending. There are two additional equations of equilibrium which exhibits this coupling along with two additional boundary conditions prescribed along each edge. The governing equations which apply to the case of doubly curved sandwich shells are Eqs. (2.168)–(2.172) in conjunction with the boundary conditions, Eqs. (2.173a–f). These are presented as

66

3 Buckling of Sandwich Plates and Shells

• Equations of Equilibrium:



2 
2  2 2 ∂ ξ1 1 ∂v3 ∂ ξ1 ∂ ξ2 ∂ ξ2 1 ∂v3  þ  þ A þ A 66 12 ∂x1 ∂x2 R2 ∂x1 ∂x21 R1 ∂x1 ∂x22 ∂x1 ∂x2

 2 2 2 ∂ ξ1 ∂ ξ2 1 ∂v3 ∂ ξ2 1 ∂v3 A16 2  þ 2   A26 ¼0 ∂x1 ∂x2 ∂x1 R1 ∂x2 ∂x22 R2 ∂x2

A11

ð3:22Þ

 2 2 2 2 ∂ ξ2 1 ∂v3 ∂ ξ2 ∂ ξ1 ∂ ξ1 1 ∂v3  þ  þ A þ A 66 12 2 2 R R ∂x ∂x ∂x ∂x ∂x ∂x2 ∂x1 2 1 ∂x2 2 1 2 1 2

2  2 2 ∂ ξ2 ∂ ξ1 ∂ ξ1 1 ∂v3 1 ∂v3 A26 2   þ  A16 ¼0 ∂x1 ∂x2 ∂x22 R2 ∂x1 ∂x21 R1 ∂x1


A22



ð3:23Þ
2 
2  2 2 2 2 ∂ η1 ∂ η1 ∂ η2 ∂ η2 ∂ η2 ∂ η1 þ A66 þ þ A16 þ2 A11 þ A12 ∂x1 ∂x2 ∂x1 ∂x2 ∂x21 ∂x22 ∂x1 ∂x2 ∂x21
 2 ∂ η2 ∂v þA26  d1 η1 þ a 3 ¼ 0 ∂x1 ∂x22 ð3:24Þ
2 
2  2 2 2 2 ∂ η2 ∂ η2 ∂ η1 ∂ η1 ∂ η1 ∂ η2 þ A66 þ þ A26 þ2 A22 þ A12 ∂x1 ∂x2 ∂x1 ∂x2 ∂x22 ∂x22 ∂x1 ∂x2 ∂x22
 2 ∂ η1 ∂v þA16  d2 η2 þ a 3 ¼ 0 ∂x2 ∂x21 ð3:25Þ



 A11 A12 ∂ξ1 A16 A26 ∂ξ1 A22 A12 ∂ξ2 A26 A16 ∂ξ2 þ  þ  þ  þ   R1 R2 ∂x1 R1 R2 ∂x2 R2 R1 ∂x2 R2 R1 ∂x1


 2 2 4 ∂η1 ∂η2 A A22 2A12 ∂ v3 ∂ v3 ∂ v3 þ þ  d a þ a a þ a þ  11 v  d þ F 3 1 2 11 ∂x1 ∂x2 ∂x21 ∂x22 ∂x41 R21 R22 R1 R2 4

F 22

4

4

4

2

∂ v3 ∂ v ∂ v ∂ v3 ∂ v þ 2ðF 12 þ 2F 66 Þ 2 3 2 þ 4F 16 3 3 þ 4F 26 þ N 011 23 þ ∂x42 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x32 2

2N 012

2

∂ v3 ∂ v þ N 022 23 ¼ 0 ∂x1 ∂x2 ∂x2 ð3:26Þ

3.4 Buckling of Doubly Curved Sandwich Panels

67

• Boundary conditions: The associated boundary conditions along the edges xn ¼ 0, Ln are e nn N nn ¼ N

or

ξn ¼ e ξn

ð3:27Þ

e nt N nt ¼ N

or

ξt ¼ eξt

ð3:28Þ

Lnn ¼ e Lnn

or

ηn ¼ e ηn

ð3:29Þ

Lnt ¼ e Lnt

or

ηt ¼ e ηt

ð3:30Þ

∂v3 ∂ev3 ¼ ∂xn ∂xn

ð3:31Þ

e nn M nn ¼ M

or

∂v3 ∂v ∂M nn ∂M nt þ N nn 3 þ þ2 ∂xt ∂xn ∂xn ∂xt e   ∂M nt e þ a=h N n3 ¼ þ N n3 ∂xt N nt

v3 ¼ ev3

or

ð3:32Þ

In the case of simply supported boundary conditions freely movable on all four edges at x1 ¼ 0, L1 N 11 ¼ N 12 ¼ η1 ¼ η2 ¼ M 11 ¼ v3 ¼ 0

ð3:33a  fÞ

N 22 ¼ N 12 ¼ η1 ¼ η2 ¼ M 22 ¼ v3 ¼ 0

ð3:34a  fÞ

at x2 ¼ 0, L2

In terms of displacements, the first, second, and fifth boundary conditions from Eqs. (3.33) and (3.34) can be written as N 11 ¼ A11 ¼ 0,
N 12 ¼ A66



 ∂ξ1 ∂ξ ∂ξ2 ∂ξ1 A11 A12 þ A12 2 þ A16 þ þ  v R1 R2 3 ∂x1 ∂x2 ∂x1 ∂x2 ð1⇄2Þ

∂ξ2 ∂ξ1 þ ∂x1 ∂x2

 þ A26 2

ð3:35aÞ
 ∂ξ2 ∂ξ A16 A26 þ A16 1  þ v ¼0 R1 R2 3 ∂x2 ∂x1 2

2

M 11 ¼ F 11

∂ v3 ∂ v3 ∂ v3 þ F 12 þ 2F 16 ¼0 ∂x1 ∂x2 ∂x21 ∂x22

M 22 ¼ F 22

∂ v3 ∂ v3 ∂ v3 þ F 12 þ 2F 26 ¼0 ∂x1 ∂x2 ∂x22 ∂x21

2

2

ð3:35bÞ

ð3:36aÞ

2

ð3:36bÞ

68

3 Buckling of Sandwich Plates and Shells

3.4.2

Solution Methodology

Following the same procedure as was carried out for flat sandwich plates, v3(x1, x2, t) can be assumed in the following form v3 ðx1 , x2 , t Þ ¼ wmn sin λm x1 sin μn x2

ð3:37Þ

v3(x1, x2, t) identically fulfills the sixth boundary conditions provided in Eqs. (3.33f) and (3.34f). The next step is to fulfill the first two equations of equilibrium, Eqs. (3.22) and (3.23). To achieve this, ξ1 and ξ2 can be assumed in the following form 

ξ1 ðx1 , x2 , t Þ



ξ2 ðx1 , x2 , t Þ

(

) 1Þ F ðmn cos λm x1 sin μn x2 ¼ 1Þ Gðmn ( ) 2Þ F ðmn þ sin λm x1 cos μn x2 2Þ Gðmn

ð3.38a, bÞ

1Þ 2Þ 1Þ 2Þ where F ðmn , F ðmn , Gðmn , Gðmn are constants to be determined. Substituting the expressions for v3, ξ1, ξ2 into Eqs. (3.22) and (3.23) and comparing coefficients of the same trigonometric functions gives the following matrix equation in terms of 1Þ 2Þ 1Þ 2Þ the unknown constants F ðmn , F ðmn , Gðmn , Gðmn which is expressed as

2 6 6 6 6 6 4

ðm,nÞ

L11

ðm,nÞ

L12

ðm,nÞ

L11

ðm,nÞ

L13

ðm,nÞ

L14

ðm,nÞ

L33 Symm:

3

8 ð1Þ 9 8 ð1Þ 9 M mn > > > > > > F mn > > 7 > > > ðm,nÞ 7> = < < ð 2 Þ 2Þ = L13 7 F mn M ðmn ¼ 7 ð3Þ ðm,nÞ > Gð1Þ > > > > M mn > L34 7 mn > > > 5> ; : ð2Þ ; > : ð4Þ > ðm,nÞ Gmn M mn L ðm,nÞ

L14

ð3:39Þ

33

where ðm,nÞ

  ¼  λ2m A11 þ μ2n A66 ,

ðm,nÞ

ðm,nÞ

¼ 2λm μn A16 , L13 ¼ λ2m A16 þ μ2n A26   ðm,nÞ ðm,nÞ ðm,nÞ L14 ¼ λm μn ðA12 þ A66 Þ, L33 ¼  μ2n A22 þ λ2m A66 , L34 ¼ 2λm μn A26

 A A A A 1Þ 2Þ M ðmn ¼ μn 16 þ 26 wmn , M ðmn ¼ λm 11 þ 12 wmn R1 R2 R1 R2

 A A A A 3Þ 4Þ M ðmn ¼ μn 12 þ 22 wmn , M ðmn ¼ λm 16 þ 26 wmn R1 R2 R1 R2 L11

L12

ð3:40a  jÞ αÞ αÞ From Eq. (3.39), using Cramer’s rule, the coefficients F ðmn , Gðmn be expressed as

ðα ¼ 1, 2Þ can

3.4 Buckling of Doubly Curved Sandwich Panels



69

 ð1Þ 2Þ 1Þ 2Þ 1Þ 2Þ e ð1Þ , G e ð2Þ wmn e ðmn e mn , F F ðmn , F ðmn , Gðmn , Gðmn , G ¼ F mn mn

ð3:41Þ

where 1Þ detðL1 Þ e ðmn F ¼ , detðLÞ

¼

2Þ detðL2 Þ e ðmn ¼ F , detðLÞ

e ð1Þ ¼ detðL3 Þ , G mn detðLÞ

e ð2Þ G mn

detðL4 Þ detðLÞ

ð3:42a  dÞ

while 0

ðm:nÞ

1Þ F ðmn

L12

B B ð2Þ B F mn L1 ¼ B B ð1Þ B Gmn @ 0

ðm,nÞ

L11

ðm,nÞ

L14

ðm,nÞ

2Þ Gðmn

L13

ðm,nÞ

L11

B B ðm:nÞ B L12 L3 ¼ B B ðm,nÞ B L13 @ ðm,nÞ

0 B B B L¼B B B @

L14

ðm,nÞ

L11

ðm,nÞ

ðm,nÞ

L13

L14

1

C C C C, ðm,nÞ C L34 C A

ðm,nÞ

ðm,nÞ

L14

L13

ðm,nÞ

L33

ðm,nÞ

L33

ðm,nÞ

1

1Þ F ðmn

L14

ðm,nÞ

2Þ F ðmn

L13

ðm,nÞ

1Þ Gðmn

C C C C, ðm,nÞ C L34 C A

ðm,nÞ

2Þ Gðmn

L33

L11 L14 L13

ðm:nÞ

L12

ðm,nÞ

L11

ðm,nÞ ðm,nÞ

ðm,nÞ

ðm,nÞ

L13

ðm,nÞ

L14

ðm,nÞ

L33

ðm,nÞ

L11

B B ðm:nÞ B L12 L2 ¼ B B ðm,nÞ B L13 @

ðm,nÞ

L34

ðm:nÞ

L12

0

ðm,nÞ

L14

0

L14

ðm,nÞ

L11

B B ðm:nÞ B L12 L4 ¼ B B ðm,nÞ B L13 @ ðm,nÞ

1

L14

ðm,nÞ

1Þ F ðmn

L13

2Þ F ðmn

L14

1Þ Gðmn

L33

2Þ Gðmn

L34

ðm:nÞ

L12

ðm,nÞ

L11

ðm,nÞ

L14

ðm,nÞ

L13

ðm,nÞ

L14

C C C C ðm,nÞ C L34 C A

ðm,nÞ

ðm,nÞ

L13

ðm,nÞ ðm,nÞ

ðm,nÞ

L33

ðm,nÞ

L13

ðm,nÞ

L14

ðm,nÞ

L33

ðm,nÞ

L34

1

1Þ F ðmn

1

C 2Þ C F ðmn C C C 1Þ C Gðmn A 2Þ Gðmn

C C C C ðm,nÞ C L34 C A ðm,nÞ

L13

ðm,nÞ

Symm:

L33

ð3:43a  eÞ Following in a similar manner, as is customary, the third and fourth equations of equilibrium can be satisfied by assuming η1 and η2 in the usual form as 

η 1 ð x1 , x2 Þ η 2 ð x1 , x2 Þ



(

) 1Þ H ðmn cos λm x1 sin μn x2 ¼ 1Þ I ðmn ( ) 2Þ H ðmn þ sin λm x1 cos μn x2 2Þ I ðmn

ð3.44a, bÞ

70

3 Buckling of Sandwich Plates and Shells

Substituting the expressions for η1 and η2 back into Eqs. (3.24) and (3.25) and comparing coefficients of like trigonometric functions gives the same result as was found for the buckling of flat sandwich panels. Therefore, no further development is necessary. The first four equations of equilibrium, Eqs. (3.22)–(3.25) and the boundary conditions, Eqs. (3.33f) and Eqs. (3.34f) are identically fulfilled. There remains the fifth equation of equilibrium and the remaining unfulfilled boundary conditions, Eqs. (3.26), (3.33a–e), and (3.34a–e). The identical procedure that was carried out for flat plates will be duplicated here for the case of doubly curved sandwich shells through the use of the extended Galerkin method. The unfulfilled quantities are retained in the energy functional and thus by performing the necessary operations will result in fulfilling the last equation of motion and the remaining boundary conditions in an average sense. Inserting these unfilled expressions back into Hamilton’s equation gives

   A11 A12 ∂ξ1 A16 A26 ∂ξ1 A22 A12 ∂ξ2 þ  þ þ þ   R1 R2 ∂x1 R1 R2 ∂x2 R2 R1 ∂x2 t0 0 0


  
2 ∂η1 ∂η2 A26 A16 ∂ξ2 A11 A22 2A12 ∂ v3 þ  þ þ  d a þ a a þ v  d 3 1 2 2 2 2 R2 R1 ∂x1 ∂x1 ∂x2 ∂x1 R1 R2 R1 R2
  4 2 4 4 4 ∂ v ∂ v3 ∂ v3 ∂ v3 ∂ v a 23 þ F 11 þ F þ 2 F þ 2F þ 4F 16 3 3 þ 22 12 66 4 4 2 2 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2   4 2 2 2 ∂ v3 ∂ v3 ∂ v3 ∂ v3 þ N 011 þ 2N 012 þ N 022 4F 26 δv3 dx1 dx2 dtþ 3 2 ∂x1 ∂x2 ∂x1 ∂x22 ∂x1 ∂x2
  Z t 1 Z l2   l1 ∂v3  dx2 dtþ N 11 δξ1 þ N 12 δξ2 þ L11 δη1 þ L12 δη2 þ M 11 δ  ∂x1 t0 0 0
  Z t 1 Z l 1  l  ∂v3  2 dx2 dt ¼ 0 N 12 δξ1 þ N 22 δξ2 þ L12 δη1 þ L22 δη2 þ M 22 δ  ∂x2 t0 0 0 Z

t 1 Z l2 Z l1 




ð3:45Þ Substituting in the expressions for η1, η2, ξ1, ξ2, v3 into Eq. (3.45) and carrying out the indicated operations results in an algebraic equation which governs the stability of doubly curved sandwich structures with symmetric anisotropic laminated face sheets. This algebraic equation is presented as  1Þ λ2m N 011 þ μ2n N 022 ¼ λ4m F 11 þ 2λ2m μ2n ðF 12 þ 2F 66 Þ þ μ4n F 22 þ λm ad1 H ðmn þ aλm

  A A A A 2Þ 2Þ 1Þ þμn ad 2 I ðmn þ aμn þ λm 11 þ 12 F ðmn þ μn 12 þ 22 Gðmn R1 R2 R1 R2


 A A A A A11 A12 A22 1Þ 2Þ þμn 16 þ 26 F ðmn þ λm 16 þ 26 Gðmn þ þ 2 þ R1 R2 R1 R2 R1 R2 R22 R21 ð3:46Þ

3.4 Buckling of Doubly Curved Sandwich Panels

71

where the shear load, N 012 has been discarded. Eq. (3.46) can be nondimensionalized as   n4 ϕ4 F 22 2m2 n2 ϕ2 ðF 12 þ 2F 66 Þ K x m2 ψ 21 þ LR n2 ϕ2 ψ 22 ¼ m4 þ þ F 11 F 11 i  aL31 h a2 L21  2 L2 h ð1Þ 2Þ 1Þ 2 2 e ðmn e mn þnϕd 2eI ðmn þ 3 1 mðψ 1 A11 þ ϕψ 2 A12 ÞF þ 2 m d1 þ n ϕ d2 þ 3 md 1 H þ π F 11 π F 11 π F 11  i 2Þ 1Þ 2Þ e ðmn e ðmn e ðmn þ nϕF þ mG ðψ 1 A16 þ ϕψ 2 A26 Þ nϕðψ 1 A12 þ ϕψ 2 A22 ÞG þ

 L21  2 ψ A þ ψ 22 ϕ2 A22 þ 2A12 ψ 1 ψ 2 ϕ 4 π F 11 1 11

ð3:47Þ With Eq. (3.47) in hand, the critical load can be determined for various geometrical and material parameters to study their effects on the stability of the doubly curved sandwich panel. Also, the effect of the structural tailoring can be determined as to its beneficial structural behavior.

3.4.3

Validation of the Theoretical Results

As was presented in the section for flat plates, validations are made first for a circular cylindrical panel composed of isotropic facings (Dura Aluminum, ν ¼ 0.3, E ¼ 6.96  105 kg/cm2 ) and a transversely isotropic core of penoplast (see Karavanov 1960) as shown in Table 3.6. The panel geometrical properties are L1 ¼ 60 cm, L2 ¼ 40 cm, R2 ¼ 100 cm, hf ¼ 0.1 cm (facing thickness). Herein, the results again reveal overpredictions which are most likely due to imperfections Table 3.6 Comparisons of theoretical and experimental buckling predictions for a circular cylindrical panel with a transversely isotropic core of penoplast Case 1 2 3 4 5 6 7 8 9 10

h (cm) 0.750 0.750 0.475 0.200 0.200 0.225 0.500 0.475 0.500 0.700

K.! Karavanov (1960)

G ðkg=cm Þ 81.3 84 150 127 566 92 32.6 40 141 104 2

(N11)cr 0.365 0.373 0.977 2.726 6.708 2.023 0.431 0.5 0.879 0.468

e 11 L2  103 kg N Present K. 9.114 8.2 9.3 7.8 10.49 8.9 6.643 6.28 16.35 14.6 5.965 5.0 5.087 4.4 5.369 4.62 10.36 9.25 10.26 8.55

% Error +11.4 +19.23 +17.865 +5.78 +11.99 +19.3 +15.61 +16.212 +12.00 +20.00

72

3 Buckling of Sandwich Plates and Shells

within the test specimens and other miscellaneous considerations. A concept known as the knockdown factor which utilizes a numerical factor to knockdown the theoretical results to the experimental ones within a reasonable amount would most likely bring the results within agreement. The knockdown factor in practice should be no greater than 20%.

3.4.4

Results and Discussion

For the following numerical results, the material properties for the core and face sheets are displayed in Tables 3.7 and 3.8 by type below. Additionally, unless specified otherwise, L1 ¼ 609.6 mm, h ¼ 12:7 mm. hf (Face thickness) is given in the figure caption. Herein, results are presented which show the effect of certain geometrical properties on the buckling strength of curved sandwich panels. Figure 3.5 depicts the effect of the ply angle for single-layered facings on the buckling strength of a cylindrical sandwich panel for various aspect ratios. The material characteristics for this case are of the F1:C1 type. The results show that as the aspect ratio of the panel becomes smaller the critical buckling load becomes larger and peaks around 45 degrees and drops off afterwards. Cylindrical sandwich panels with larger aspect ratios have a much lower critical buckling load. In Fig. 3.6 considering the F1:C1 type highlights the effect of the ply angle for various aspect ratios on the critical buckling load. It is apparent that at smaller aspect ratios the panel can carry larger compressive edge loading which peaks around 45 degrees. At the larger aspect ratios, the load carrying capacity of the panel is diminished in comparison. Figure 3.7 depicts the effect of the panel aspect ratio on the critical buckling load of a cylindrical sandwich panel for various fiber orientation angles in the face sheets. These configurations are of the material type F2: C1. It is seen that as the aspect ratio becomes smaller the critical buckling load increases. As the aspect ratio increases the critical load drops significantly up to a point at which the trend seems to flatten out. As it can be seen that the 45-degree fiber Table 3.7 Face sheet material properties Type F1 F2

Material HS Graph. Ep IM7/977-2

E1(N/mm2) 1.8375 0.0812

E2(N/mm2) 0.105 0.0763

G12(N/mm2) 0.0735 0.0098

ν12 0.28 0.06

α1 (1/K) 11.34 1.62

α2 (1/K) 36.9 1.674

Note: Multiply E1, E2, G12  105

Table 3.8 Core material properties Type C1

Core type Titanium Honeycomb

G13 (N/mm2) 0.0145  105

G23 (N/mm2) 0.0066  105

3.4 Buckling of Doubly Curved Sandwich Panels

73

Fig. 3.5 The effect of the fiber orientation of the face sheet lamina on the buckling load for a cylindrical sandwich panel with single-layered facings (hf ¼ 0.508 mm)

Fig. 3.6 The effect of the fiber orientation of the face sheet lamina on the buckling load for a cylindrical sandwich panel with single-layered facings (hf ¼ 1.524 mm)

orientation has the larger load carrying capacity of the panel. There also seems to be no advantage between the 0-degree and the 90-degree fiber orientation from a loadcarrying capacity standpoint.

74

3 Buckling of Sandwich Plates and Shells

Fig. 3.7 The effect of the panel aspect ratio on the buckling load for a cylindrical sandwich panel with various ply angles in the facings (hf ¼ 0.635 mm)

Fig. 3.8 The effect of the panel aspect ratio on the buckling load for a cylindrical sandwich panel with various curvatures (hf ¼ 0.635 mm)

Figure 3.8 which depicts results for the material type F2:C1 reveals that for the chosen stacking sequence in the face sheets that at appreciable small aspect ratios the critical load seems to be the same for all three degrees of curvature. As the aspect

3.5 Stress/Strain Theory

75

ratio increases, the larger load-carrying capacity of the structure appears to occur in the panel with the larger curvature.

3.5

Stress/Strain Theory

The stress and strain relationships were introduced in Chap. 2 without any discussion on their use. Now that the buckling bifurcation load can be determined it is important to know what stresses or strains are produced in the structure from this loading. The first step is to quantify the strains. Once these are known the next step is determining the stress at any location within the structure. It will be assumed that we are only interested in the mechanical stresses and that the thermal and moisture stresses will not be considered. Also, full symmetry is assumed for both the top and bottom facings. The facings are also assumed thin and the Kirchhoff–Love assumptions hold. With this in mind, considering the bottom facing, Eq. (2.86a) is simplified and decoupled to account for the full symmetry in the structure as n

o h in o N = ¼ A= ε= n o h in o M = ¼ F= κ=

ð3:48aÞ ð3:48bÞ

Because of symmetry, [E/] ¼ 0. Let h

= 

A

i

h i1 ¼ A=

h and

= 

i

F

h i1 ¼ F=

ð3.49a, bÞ

With some simple algebraic matrix manipulation, Eqs. (3.48a, b) can be expressed as n o h in o ε= ¼ = A N =

ð3:50Þ

n o h in o κ= ¼ = F  M =

ð3:51Þ

and

In terms of the geometrical coordinate system, the strain relationships are given by Eq. (2.24a–f) as n o n o  n o e= ¼ ε = þ x 3  a= κ =

ð3:52Þ

76

3 Buckling of Sandwich Plates and Shells

Substituting Eqs. (3.50) and (3.51) into the above expression gives n o h in o  h in o e = ¼ = A  N = þ x 3  a= = F  M =

ð3:53Þ

With edge loading and applied moments on the structure, 9 8 e 11 > =

e 22 N= ¼ N > ; :e > N 12

and

9 8 e 11 > =

e 22 M= ¼ M > ; :e > M 12

ð3.54a, bÞ

e ij , M e ij ði ¼ 1, 2Þ are the applied edge and couple loadings on the strucwhere N ture. If just uniaxial compressive edge loading is applied in the x1 direction then e 22 ¼ N e 12 ¼ 0, and Mij ¼ 0. Eq. (3.53) can be written as N 9 8 9 9 8 8 = > > e 11 > e 11 > e > > 11 = =  =
> > > ; ; > :e > :e > ; : = > N 12 M 12 2γ 12 G

ð3:55Þ

The above expressions for the strain relationships as a function of the edge loading is only valid for the geometrical coordinate system. To obtain the strains in the material coordinate system, a coordinate transformation needs to be applied. It can be shown, the details of which are not provided here, that the strains in the material coordinate system can be expressed by 9 9 8 8 = = > > > > e e > > > = < 11 = < 11 > = ¼ ½T ðθÞT e=22 e22 > > > > > > ; ; : = > : = > 2γ 12 M 2γ 12 G

ð3:56Þ

where 2 6 T ðθ Þ ¼ 4

cos 2 ðθÞ sin 2 ðθÞ

sin 2 ðθÞ cos 2 ðθÞ

 sin ðθÞ cos ðθÞ

sin ðθÞ cos ðθÞ

3 2 cos ðθÞ sin ðθÞ 7 2 cos ðθÞ sin ðθÞ 5 cos 2 ðθÞ  sin 2 ðθÞ

ð3:57Þ

where θ is the fiber orientation angle of each ply. The next step is to determine the stresses within the structure. For the bottom face, in the geometrical coordinate system, the stresses can be obtained from Eq. (2.80a) as

3.6 Summary

77

2 = 9 8 b = Q > = < σ 11 > 6 11 6 = b ¼ 6 =Q σ 22 > 4 12 ; := > = σ 12 G b 16 Q

=

b 12 Q

=

b 22 Q

=

b 26 Q

3 9 b 16 8 = Q = < e11 > 7> 7 = = b e 7 Q26 5> 22 > := ; = γ 12 G b 66 Q =

ð3:58Þ

Substituting Eq. (3.55) into Eq. (3.58) gives 9 9 8 8 = e 11 > > = =

h = ih i> = b = A ¼ Q σ 22 0 > > > ; ; : := > σ 12 G 0

ð3:59Þ

Within this expression the noncontributing elements and or terms have been dropped. In terms of the material coordinate system, the stresses can be obtained from the following expression: 9 9 8 8 = = > > = = < σ 11 > < σ 11 > = ¼ T ðθÞ = σ 22 σ 22 > > ; ; := > := > σ 12 M σ 12 G

ð3:60Þ

The same approach can be applied for the strains and stresses in the top face. A quicker approach would be to just replace a/ with a// and single primes with double primes in the above expressions.

3.6

Summary

The theory governing the buckling response of both flat and curved sandwich panels with a weak (soft) core have been presented and discussed in detail. The theory accounted for the directional properties of the face sheets, the orthotropy of the core all within a geometrical linear context, while the principles of shallow shell theory were adhered to. As an addendum a section on the deformation of sandwich panels was presented covering the stress–strain behavior of sandwich panels. Following the theory, results were presented for both flat and curved sandwich panels separately. The effects of the ply angle, aspect ratio, the panel face thickness, and the effects of the curvature on the critical load were discussed. Several validations were made for both flat and curved sandwich panels where excellent agreement was seen. These results play an important role in the design of aerospace vehicles and structures where lightweight materials and strength are a necessity.

78

3 Buckling of Sandwich Plates and Shells

References Alexandrov, A. I., Briuker, L. E., Kurshin, L. M., & Prusakov, A. P. (1960). Research on three layered panels (in Russian). Moskow: Oboronghiz. Karavanov, V. F. (1960). Stability of three-layer shallow cylindrical panels with soft core. Izvestia Vishish Ukebnich Zavedenia-Aviatsionaia Technika, 2, 50–60. (in Russian). Ko, W. L., & Jackson, R. H. (1991). Combined load buckling behavior of metal matrix composite sandwich panels under different thermal environments. NASA TM-4321. Ko, W. L., & Jackson, R. H. (1993). Compressive and shear buckling analysis of metal matrix composite sandwich panels under different thermal environments. Composite Structures, 25, 227–239. Pearce, T. R. A., & Webber, J. P. H. (1972). Buckling of sandwich panels with laminated face plates. Aeronautical Quarterly, 23, 148–160.

Chapter 4

Post-Buckling

Abstract Sandwich structures are capable of sustaining loading beyond the buckling bifurcation point. With this in mind, a comprehensive geometrically nonlinear theoretical treatment of sandwich plates and shells is considered to provide insight into their structural performance within this loading region. Within this comprehensive theoretical treatment, there are few structural and material conditions considered. The face sheets are considered symmetric with respect to their local and global mid-surfaces while constructed from anisotropic laminated composites. The facings are assumed imperfect and thin thereby neglecting the transverse shear stresses in the facings. The core is assumed to be of the weak-type construction. With these theoretical aspects, two cases of laminated facings are considered, cross-ply and angle-ply. Two formulations, the mixed formulation and the displacement formulation, are presented within the theoretical developments. To arrive at the postbuckling solution, the extended Galerkin method and Newton’s method are adopted. Finally, a number of kinematical and physical parameters are considered in regard to the load-carrying capacity of these sandwich panels. Several results are presented which provides sufficient insight to the behavior of these structures.

4.1

Introduction

This chapter is concerned with the post-buckling behavior of anisotropic laminated composite sandwich plates and shells. The governing nonlinear equations developed in Chap. 2 are applied to three cases for post-buckling solution. The first case considers cross-ply laminated facings which lend themselves to the stress potential solution technique. The second case considers angle-ply laminated sandwich plates. While the third case considers angle-ply laminated sandwich shells. The theory considers that the facings are symmetrically laminated both locally and globally in conjunction with a weak (soft) core. In addition, the facings are assumed thin so that the Kirchhoff–Love assumptions apply where the transverse shear stress can be neglected. Simply supported boundary conditions are considered with all four edges freely movable. The solution methodologies involve both the extended Galerkin method and Newton’s method for nonlinear polynomials. Finally, the influence of a © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 T. J. Hause, Sandwich Structures: Theory and Responses, https://doi.org/10.1007/978-3-030-71895-4_4

79

80

4 Post-Buckling

number of kinematical and physical parameters, as well as the tangential boundary conditions on the load-carrying capacity of the structure are investigated with several results presented.

4.2

Preliminaries and Basic Assumptions

The governing nonlinear theory for the postcritical loading of doubly curved sandwich panels is based on a number of previously mentioned assumptions which were presented in Chaps. 2 and 3. The reader is referred to those chapters for an overview. Those preliminaries and basic assumptions apply here as well.

4.3 4.3.1

Cross-Ply Laminated Sandwich Shells Governing System

This section is concerned with the stability of sandwich plates and shells both at the critical load and above. This solution methodology begins with the nonlinear governing system of partial differential equations for sandwich shells for the case of a weak core and the transverse shear effects neglected in the facings. The applicable equations for this case are given by Eqs. (2.139)–(2.143) along with the boundary conditions Eqs. (2.144a–f). The governing equations are • Equations of Equilibrium ∂N 11 ∂N 12 þ ¼0 ∂x1 ∂x2

ð4:1aÞ

∂N 22 ∂N 12 þ ¼0 ∂x2 ∂x1

ð4:1bÞ

∂L11 ∂L12 N 13 þ  ¼0 ∂x1 ∂x2 h

ð4:1cÞ

∂L22 ∂L12 N 23 þ  ¼0 ð4:1dÞ ∂x2 ∂x1 h ! ! ! 2  2  2  2 2 2 ∂ v3 ∂ v3 ∂ v3 1 ∂ v3 ∂ v3 ∂ v3 1 N 11 þ þ þ þ þ þ 2N 12 þ N 22 ∂x1 ∂x2 ∂x1 ∂x2 ∂x21 ∂x21 R1 ∂x22 ∂x22 R2    2 2 2 h ∂N 13 ∂N 23 ∂ M 11 ∂ M 12 ∂ M 22 þ þ þ2 þ ¼ q3 þ 1þ 2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x22 2h ð4:1eÞ

4.3 Cross-Ply Laminated Sandwich Shells

81

• Boundary Conditions The associated boundary conditions along the edges xn ¼ const(n ¼ 1, 2) are ξn ¼ e ξn ξt ¼ e ξt or ηn ¼ e ηn e Lnt ¼ Lnt or ηt ¼ e ηt ∂v ∂ev 3 e nn M nn ¼ M or ¼ 3 ∂xn ∂xn     ∂v3 ∂v3 ∂v3 ∂v3 ∂M nn ∂M nt N nt þ þ þ2 þ N nn þ ∂xt ∂xt ∂xn ∂xn ∂xn ∂xt   e nt e h ∂M þ 1þ N n3 ¼ þ N n3 ∂xt 2h e nn N nn ¼ N e nt N nt ¼ N enn Lnn ¼ L

or or

or

v3 ¼ ev3

ð4:2a  fÞ In Eq. (4.1e), the transverse inertia and damping terms have been eliminated due to the fact that they are not part of the post stability problem. Generally, it is common to consider two types of simply supported boundary conditions, Type A and Type B. In this text only Type A will be considered. For Type A all four edges are freely movable where the following conditions require fulfillment. Along the edges xn ¼ 0, Ln e nn , N nt ¼ 0, ηn ¼ 0, ηt ¼ 0, M nn ¼ 0, v3 ¼ 0 N nn ¼ N

ð4:3a  fÞ

For Type B, the unloaded edges are immovable where the following conditions apply. Along the edges xn ¼ 0, Ln ξn ¼ N nt ¼ ηn ¼ ηt ¼ M nn ¼ v3 ¼ 0

ð4:4a  fÞ

The boundary condition (4.4a) is fulfilled in an average sense through enforcing the following condition. Z 0

Ln Z Lt  0

 ∂ξn dxn dxt ¼ 0 ∂xn

ð4:5Þ

xt ¼ 0, Lt denote the edges parallel to the direction of the uniaxial compressive e nn rendering the edges e nn . From this condition, the fictious edge load N edge load N e nn is determined it immovable can be determined. Once the fictitious edge load N needs to be incorporated into the post-buckling equation. Also, boundary conditions (4.3a, b) and (4.3b) are fulfilled in an average sense through the following conditions.

82

4 Post-Buckling Lt

e nn Ln , N nn dxt ¼ N

0

Z

Lt



X N nt dxt ¼ 0 ,

0

n ¼ 1, 2 t ¼ 2, 1

n, t

 ð4.6a, bÞ



Z

e nn are the compressive edge loads along the edges xn ¼ 0, Ln. Herein, N

4.3.2

Solution Methodology

The solution methodology chosen here will use a stress potential approach which will require a compatibility equation. To fulfill the complete governing system of equations, focus will begin with the first two equations of equilibrium, Eqs. (4.1a and 4.1b). These first two equations of equilibrium can be satisfied with a stress potential representation for the stress resultants N11, N22 and N12 as 2

N 11 ¼

∂ φ , ∂x22

2

N 22 ¼

∂ φ , ∂x21

2

N 12 ¼ 

∂ φ ∂x1 ∂x2

ð4:7a  cÞ

Equations (4.1a and 4.1b) are now identically fulfilled in terms of the Airy’s potential function. Next attention will be given to the third and fourth equations of equilibrium. In Eqs. (4.1c and 4.1d), h can be factored out of both equations. As a result, Eqs. (4.1c and 4.1d) become ∂L11 ∂L12 þ  N 13 ¼ 0 ∂x1 ∂x2

ð4:8aÞ

∂L22 ∂L12 þ  N 23 ¼ 0 ∂x2 ∂x1

ð4:8bÞ

where       = == = == = == ð4:9a  cÞ L11 ¼ h N 11  N 11 , L22 ¼ h N 22  N 22 , L12 ¼ h N 12  N 12 Expressing Eqs. (4.1c and 4.1d) in terms of displacements and setting A16 ¼ A26 ¼ 0 results in A11

 2    2 2 2 ∂ η1 ∂ η1 ∂ η2 ∂ η2 ∂v3 þ A þ  d η þ a þ A ¼ 0 ð4:10aÞ 66 12 1 1 ∂x1 ∂x2 ∂x1 ∂x21 ∂x22 ∂x1 ∂x2

 2    2 2 2 ∂ η2 ∂ η2 ∂ η1 ∂ η1 ∂v3 þ A66 þ  d2 η2 þ a A22 þ A12 ¼ 0 ð4:10bÞ ∂x1 ∂x2 ∂x2 ∂x22 ∂x22 ∂x1 ∂x2

4.3 Cross-Ply Laminated Sandwich Shells

83

The above equations can be satisfied by assuming η1 and η2 as follows 1Þ η1 ðx1 , x2 Þ ¼ Bðmn cos λm x1 sin μn x2

ð4:11aÞ

1Þ sin λm x1 cos μn x2 η2 ðx1 , x2 Þ ¼ Cðmn

ð4:11bÞ

Substituting the expressions for η1 and η2 into Eqs. (4.10a and 4.10b) and comparing coefficients of like trigonometric functions gives the expressions for (1) (1) Bmn and Cmn as ð1Þ

1Þ emn wmn , ¼B Bðmn

ð1Þ

1Þ e wmn C ðmn ¼C mn

ð4:12aÞ

where 1Þ eðmn B ¼



a ½ðA12 þ A66 Þd2  A22 d1 λm μ2n  d1 A66 λ3m  d1 d2 λm b mn Δ

ð4:12bÞ

e ð1Þ ¼ C mn



a ½ðA12 þ A66 Þd1  A11 d2 λ2m μn  d 2 A66 μ3n  d 1 d 2 μn b mn Δ

ð4:12cÞ

and  b mn ¼A11 A66 λ4 þ ðd 2 A11 þ d 1 A66 Þλ2 þ A11 A22  2A12 A66  A2 λ2 μ2 þ Δ m m 12 m n ðd 1 A22 þ d 2 A66 Þμ2n þ A22 A66 μ4n þ d1 d2 ð4:12dÞ Up to this point the first four equations of equilibrium are satisfied. The fifth equation of equilibrium will now be addressed which can be expressed in the following form as ! 2 2 2 ∂ ϕ ∂ v3 ∂ v∘3 1 þ þ þ 2 ∂x1 ∂x22 ∂x22 R2 ! 2 2 2 4 4 4 ∂ v∘3 ∂ ϕ ∂ v3 ∂ v3 ∂ v3 ∂ v3 2 þ  F  4F  F 11 22 16 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x41 ∂x42 ∂x31 ∂x2 ð4:13Þ ! 2 4 4 ∂ v3 ∂ v 2aK 4F 26  2ðF 12 þ 2F 66 Þ 2 3 2 þ ∂x1 ∂x2 ∂x∂x32 h       2 2 ∂η1 ∂η2 ∂ v ∂ v G13 þ a 23 þ G23 þ a 23 ¼0 ∂x1 ∂x2 ∂x1 ∂x2 2 2 ∂ ϕ ∂ v3 ∂ v∘3 1 þ þ 2 ∂x2 ∂x21 ∂x21 R1 2

!

84

4 Post-Buckling

The above equation has three unknowns ϕ, v3 and v∘3 . Based on Seide (1974) and Simitses (1986), v∘3 can be represented as v∘3 ¼ w∘mn sin λm x1 sin μn x2

ð4:14Þ

where w∘mn are the modal amplitudes of the initial geometric imperfection shape. Eq. (4.13) is in terms of two unknowns at this point. These quantities are ϕ and v3. To ensure single-valued displacements, one more partial differential equation in terms of these two unknown quantities is needed. This will come from a compatibility equation. With the use of Eqs. (2.53)–(2.55) and (2.69)–(2.71), a compatibility equation can be derived by eliminating the in-displacements which is determined to be  2 2 2 2 2 2 2 2 2 ∂ γ 12 ∂ ε11 ∂ ε22 2 ∂ v3 2 ∂ v3 ∂ v3 ∂ v ∂ v þ  þ þ  2 þ 2 23 23 2 2 2 2 ∂x1 ∂x2 R1 ∂x2 R2 ∂x1 ∂x1 ∂x2 ∂x2 ∂x1 ∂x1 ∂x2 2 2 ∂ v∘3 ∂2 v3 ∂ v3 ∂ v∘3 ∂ v3 ∂ v∘3  4 þ 2 ¼0 ∂x1 ∂x2 ∂x1 ∂x2 ∂x21 ∂x22 ∂x21 ∂x22 2

þ2

2

2

ð4:15Þ where =

==

=

ε11 ¼ ε11 þ ε11 ,

==

ε22 ¼ ε22 þ ε22 ,

=

==

γ 12 ¼ γ 12 þ γ 12

ð4:16Þ

To express ε11, ε22, and γ 12 in terms of ϕ, the constitutive equations will be utilized. With the use of Eqs. (2.86a) and (2.92a) and neglecting the thermal and moisture terms, the following matrix relationship can be established assuming identical symmetric laminated facings. 

N M



¼

A E

E F

  ε κ

where N ¼ fN11 , N22 , N12 gT

M ¼ fM11 , M22 , M12 gT

ð4:17Þ

4.3 Cross-Ply Laminated Sandwich Shells

2 6 A ¼6 4 2

A12

A11

Symm: F 11

6 F ¼6 4

3

2

7 A26 7 5,

6 E¼6 4

A16

A22

A66 F 12

85

F 16

E 12

E 11

E 22 Symm:

3

E 16

3

7 E 26 7 5, E 66

7 F 26 7 5

F 22 Symm:

F 66 ε ¼ fε11 , ε22 , γ 12 gT κ ¼ fκ11 , κ 22 , κ12 gT

Performing a partial inversion of this matrix equation provides 

ε



M

¼

A

E

 ðE  ÞT

F



N

 ð4:18Þ

κ

where A ¼ A1 ,

E  ¼ A1 E,

 ðE  ÞT ¼ EA1 ,

F  ¼ F  EA1 E ð4:19Þ

For symmetric laminated facings [E] ¼ 0. With the use of Eqs. (4.7a–c), Eq. (4.18) can be expressed in terms of ϕ as ε11 ¼ A11 γ 12 ¼

2

2

∂ φ ∂ φ þ A12 2 , 2 ∂x2 ∂x1

A66

ε22 ¼ A12

2

∂ φ ∂x1 ∂x2

2

2

∂ φ ∂ φ þ A22 2 , 2 ∂x2 ∂x1

ð4:20Þ

Substituting these expressions into Eq. (4.15) gives the compatibility equation in the required form as 4 4 2 2   ∂2 ϕ ∂ ϕ ∂ ϕ ∂ ϕ  ∂ ϕ    þ A  2A  2A þ A þ 2A 11 16 26 66 12 ∂x41 ∂x42 ∂x21 ∂x22 ∂x1 ∂x32 ∂x31 ∂x2  2 2 2 2 2 2 2 ∂ v∘3 ∂2 v3 2 ∂ v3 2 ∂ v3 ∂ v3 ∂ v3 ∂ v3 þ þ  2 þ 2 þ 2 R1 ∂x22 R2 ∂x21 ∂x1 ∂x2 ∂x21 ∂x22 ∂x21 ∂x22

A22

2 2 ∂ v3 ∂ v∘3 ∂ v ∂ v∘3 þ 2 23 ¼0 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x22 2

4

2

ð4:21Þ

86

4 Post-Buckling

The governing system of equations for the problem at hand are Eqs. (4.13) and (4.21) along with the appropriate boundary conditions. For simply supported boundary conditions of Type A, v3 can be assumed as v3 ¼ wmn sin λm x1 sin μn x2

ð4:22Þ

This representation satisfies both the fifth and sixth boundary conditions (Mnn ¼ v3 ¼ 0) leaving the remaining ones unfulfilled. The only unknown at this point is ϕ. This quantity is assumed in the following form (see Librescu 1965, 1975; Librescu and Chang 1992) ϕðx1 , x2 Þ ¼ ϕ1 ðx1 , x2 Þ 

1 e 2 e 2 e 12 x1 x2 N x þ N 22 x1  2N 2 11 2

ð4:23Þ

e 11 , N e 22 , and N e 12 represent the average compressive and shear edge Here N loadings, while ϕ1 is the particular solution of Eq. (4.21). Substituting Eqs. (4.14), (4.22), and (4.23) into Eq. (4.21) and comparing coefficients of like trigonometric functions provides ϕ1 in the form 1Þ 2Þ 3Þ ϕ1 ðx1 , x2 Þ ¼ Aðmn cos 2λm x1 þ Aðmn cos 2μn x2 þ Aðmn sin λm x1 sin μn x2

ð4:24Þ

Herein,  1Þ eð1Þ w2 þ 2wmn w∘ , Aðmn ¼A mn mn mn

 2Þ eð2Þ w2 þ 2wmn w∘ , Aðmn ¼A mn mn mn

ð 3Þ

3Þ e wmn Aðmn ¼A mn

ð4:25aÞ where e ð1Þ ¼ A mn

eð2Þ ¼ A mn  eð3Þ A mn and

¼

μ2n R1





λ2

þ Rm2

A11 A22  A212 μ2n 32A11 λ2m

ð4:25bÞ

A11 A22  A212 λ2m 32A22 μ2n

ð4:25cÞ



A11 A22 A66  A66 A212 e mn Δ

ð4:25dÞ

4.3 Cross-Ply Laminated Sandwich Shells

87

 e mn ¼ A11 A66 λ4 þ A11 A22  2A12 A66  A2 λ2 μ2 þ A22 A66 μ4 Δ m 12 m n n

ð4:25eÞ

The particular solution ϕ1 satisfies the following conditions.  Z L1 2  L  2 ∂ ϕ1  L1 ∂ ϕ1  2  dx1 ¼ 0,  dx1 ¼ 0, 2  ∂x2 0 ∂x21  0 0 0   Z L2 2 Z L1 2   ∂ ϕ1  L 1 ∂ ϕ1  L2  dx2 ¼ 0,  dx1 ¼ 0 0 ∂x1 ∂x2  0 0 ∂x1 ∂x2  0

Z

L2

ð4:26a  dÞ

e 11 , N e 22 acquire the meaning of average in-plane compressive This reveals that N edge loads expressed by Z

L2 0

2  ∂ ϕ  e 11 L2 , dx2 ¼ N ∂x22  x1 ¼ 0, L1

Z 0

L1

2  ∂ ϕ  e 22 L1 dx1 ¼ N ∂x21  x2 ¼ 0, L2

ð4.26e, fÞ Finally, ξ1 and ξ2 can be determined from the expressions of stress resultants N11 and N22 expressed in terms of displacements which have the following form N 11

N 22

N 12 ¼ 

( )  2 2 ∂v∘3 ∂v3 v3 ∂ ϕ ∂ξ1 1 ∂v3 ¼ ¼ A11 þ þ  ∂x1 2 ∂x1 ∂x1 ∂x1 R1 ∂x22 ( )  2 ∂v∘3 ∂v3 v3 ∂ξ2 1 ∂v3 þ A12 þ þ  ∂x2 2 ∂x2 ∂x2 ∂x2 R2

ð4:27aÞ

( )  2 2 ∂v∘3 ∂v3 v3 ∂ ϕ ∂ξ2 1 ∂v3 ¼ 2 ¼ A22 þ þ  ∂x2 2 ∂x2 ∂x2 ∂x2 R2 ∂x1 ( )  2 ∂v∘3 ∂v3 v3 ∂ξ1 1 ∂v3 þ A12 þ þ  ∂x1 2 ∂x1 ∂x1 ∂x1 R1

ð4:27bÞ

  2 ∂ ϕ ∂ξ1 ∂ξ2 ∂v3 ∂v3 ∂v∘3 ∂v3 ∂v3 ∂v∘3 ¼ A66 þ þ þ þ ð4:27cÞ ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2

Equations (4.27a, 4.27b, and 4.27c) considered in conjunction with Eqs. (4.14), (4.22), (4.23), and (4.24) allow ξ1 and ξ2 to be determined as

88

4 Post-Buckling 1Þ 2Þ 3Þ ξ1 ðx1 , x2 Þ ¼ Dðmn x1 þ Dðmn sin 2λm x1 þ Dðmn sin 2λm x1 cos 2μn x2   4Þ 5Þ e 6Þ e N 11 þ Dðmn N 22 x1 þ Dðmn cos λm x1 sin μn x2 þ Dðmn

1Þ 2Þ 3Þ x2 þ Eðmn sin 2μn x2 þ E ðmn cos 2λm x1 sin 2μn x2 ξ2 ðx1 , x2 Þ ¼E ðmn   4Þ 5Þ e 6Þ e þ E ðmn sin λm x1 cos μn x2 þ E ðmn N 11 þ E ðmn N 22 x2

ð4:28Þ

ð4.3-29Þ

1Þ 6Þ 1Þ 6Þ  Dðmn and E ðmn  E ðmn are expressed as where the constants Dðmn

   ðiÞ ðiÞ  iÞ iÞ e mn w2mn þ 2wmn w∘mn , e mn , E Dðmn , E ðmn ¼ D 4Þ Dðmn

4Þ e ðmn ¼D wmn ,

4Þ E ðmn

ði ¼ 1, 2, 3Þ

4Þ e ðmn ¼E wmn

ð4:30a  cÞ

while λ2 A12 μ2n  A11 λ2m 1Þ 2Þ 3Þ λ e ðmn e ðmn e ðmn D ¼ m, D ¼ , D ¼ m 8 16A11 λm 16  3 2 2 A12 A66 þ A12  A11 A22 λm μn  A11 A66 λm A22 A66 λm μ2n  A12 A66 λ3m 4Þ e ðmn D ¼ þ b mn b mn R1 Δ R2 Δ A22 A12 5Þ 6Þ Dðmn ¼ , Dðmn ¼ A11 A22  A212 A11 A22  A212 μ2 A12 λ2m  A22 μ2n 1Þ 2Þ 3Þ μ e ðmn e ðmn e ðmn E ¼ n, E ¼ , E ¼ n 8 16A22 μn 16  A12 A66 þ A212  A11 A22 λ2m μn  A22 A66 μ3n A11 A66 λ2m μn  A12 A66 μ3n ð4Þ e þ E mn ¼ b mn b mn R2 Δ R1 Δ A12 A11 5Þ 6Þ E ðmn ¼ , E ðmn ¼ A11 A22  A212 A11 A22  A212 ð4:31a  lÞ and  b mn ¼ A11 A66 λ4 þ A11 A22  2A12 A66  A2 λ2 μ2 þ A22 A66 μ4 Δ m 12 m n n

ð4:32Þ

At this juncture, all of the displacement quantities v3 , v∘3 , ξ1 , ξ2 , η1 , η2 , ϕ, ϕ1 are known. In addition, there remains only the fifth equation of equilibrium unfulfilled and the unfulfilled boundary conditions. To solve for the postcritical solution, the EGM will be leveraged (see Fulton 1961). The unfulfilled fifth equation of motion and the unfulfilled boundary conditions will be retained in the energy functional with the appropriate integrations carried out. In the end, this will supply the postcritical solution

4.3 Cross-Ply Laminated Sandwich Shells

89

in terms of the geometrical and material parameters. Retaining these unfulfilled expressions in the Hamilton’s energy functional results in ! ! 2 2 2 2 2 2 ∂ ϕ ∂ v3 ∂ v∘3 1 ∂ ϕ ∂ v3 ∂ v∘3 1 þ þ þ þ þ 2 ∂x22 ∂x21 ∂x1 ∂x22 ∂x21 R1 ∂x22 R2 t0 0 0 !  2 2 2 4 4 4 ∂ v∘3 ∂ ϕ ∂ v3 ∂ v3 ∂ v3 ∂ v3 2 þ  F  2 ð F þ 2F Þ  F 11 22 12 66 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x41 ∂x42 ∂x21 ∂x22 !    2 2  2  ∂η1 ∂η2 2aK ∂ v ∂ v þ G13 þ a 23 þG23 þ a 23 δv3 dx1 dx2 dtþ ∂x1 ∂x2 ∂x1 ∂x2 h +   Z t 1 *Z l 2  2 2 ∂ ϕ e ∂ φ  l1 þ N 11 δξ1  δξ2 þ L11 δη1 þ L12 δη2  dx2 dt þ 0 ∂x1 ∂x2 ∂x22 t0 0  + 

   Z t1 Z l1  l2 2 2 ∂ ϕ e ∂ φ  þ N 22 δξ2  δξ1 þ L22 δη2 þ L12 δη1  dx1 dt 0 ∂x1 ∂x2 ∂x21 t0 0 ¼0

Z

t1

*Z

l2 Z l1

(

ð4:33Þ Substituting in the expressions for v3 , v∘3 , ξ1 , ξ2 , η1 , η2 , ϕ, ϕ1 and carrying out the indicated integrations results in the governing post-buckling solution for cross-ply laminated sandwich structures resulting in a nonlinear algebraic equation expressed in terms of the modal amplitudes as  2   5Þ 4Þ Pðmn wmn þ 2wmn w∘mn wmn þ w∘mn þ Pðmn wmn þ w∘mn wmn ð4:34Þ  2  3Þ 2Þ 1Þ 0Þ þPðmn wmn þ 2wmn w∘mn þ Pðmn wmn þ w∘mn þ Pðmn wmn þ Pðmn þ qmn ¼ 0 Where  ð 1Þ  5Þ e þA eð2Þ ¼ 2λ2m μ2n A Pðmn mn mn 4Þ Pðmn

16Δm n ¼ 3L1 L2

" 

eð2Þ D e ð4Þ μn A mn mn



eð1Þ E e ð4Þ λm A mn mn

ð4:35Þ ð2Þ

ð1Þ

e e μ A λm A mn þ n mn þ λm R1 μn R2

#

  ð3Þ 8Δm λm μn 16Δm 3Þ 2Þ 3Þ 3Þ n λm e ð2Þ e e ðmn e ðmn e ðmn A ¼  n  þE þE Pðmn Dmn þ D mn 3L1 L2 3L1 L2  2Þ Pðmn ¼  λ2m N 11 þ μ2n N 22

ð4:36Þ

ð4:37Þ ð4:38Þ

90

4 Post-Buckling

h ð3Þ 1Þ e þ F 11 λ4 þ F 22 μ4 þ 2ðF 12 þ 2F 66 Þλ2 μ2 Pðmn ¼ μ2n =R1 þ λ2m =R2 A mn m n m n  ð1Þ   ð1Þ i e þ aμn emn þ aλm þ d 2 aμn C þd1 aλm B mn

ð4:39Þ

  e 11 N e 22 4Δm N n ¼ þ λm μn L1 L2 R1 R2

ð4:40Þ

m n Δm n ¼ ½ð1Þ  1½ð1Þ  1

ð4:41Þ

0Þ Pðmn

where

4.3.3

Numerical Results and Discussion

In what follows are a few post-buckling results which are presented for the case of cross-ply laminated face sheets. Applicable to these results are the material properties for the face sheets and the core which are given in Tables 4.1 and 4.2. The geometrical properties are provided as, L1 ¼ L2 ¼ 609.6 m, hf ¼ 0.635 mm, h ¼ 12:7 mm. Figure 4.1 shows the effect of curvature on the post-buckling response of a cylindrical sandwich panel with the given layup in the face sheets. It is apparent that as the curvature increases there is more of a tendency for snap through–type behavior beyond the buckling bifurcation point. For the case of a flat plate, the equilibrium paths are given by the standard Euler behavior. In Fig. 4.2, the effect of the geometric imperfections on the post-buckling behavior of a cylindrical sandwich panel for the given layup is shown. The case where the imperfection is zero is considered the ideal post-buckling behavior; where the existence of an geometric imperfection is considered the real behavior that the load–deflection interaction would follow. It can be seen that the larger the geometric imperfections are, the farther the ideal load–deflection interaction is from the ideal Table 4.1 Face sheet material properties Type F1

Material HS Graph. Ep.

E1(N/mm2) 1.8375

E2(N/mm2) 0.105

G12 (N/mm2) 0.0735

ν12 0.28

α1 (1/K) 11.34

α2 (1/K) 36.9

Note: Multiply E1, E2, G12  105 Table 4.2 Core material properties Type C1

Core type Titanium honeycomb

G13 (N/mm2) 0.0145  105

G23 (N/mm2) 0.0066  105

4.3 Cross-Ply Laminated Sandwich Shells

91

Fig. 4.1 The effect of the curvature of a cylindrical sandwich panel under uniaxial compressive edge loading for the depicted cross-ply stacking sequence

Fig. 4.2 The effect of imperfections on a cylindrical sandwich panel under uniaxial compressive edge loading for the displayed cross-ply stacking sequence

Euler behavior. Also, for the given fixed curvature, a minor snap through–type behavior is seen beyond the buckling bifurcation point for the ideal case. Figure 4.3 shows the effect of the combination of transverse pressure loading and biaxial edge loading on the post-buckling response of a cylindrical sandwich panel for the given layup and curvature. The case where the transverse pressure and biaxial edge loading is nonexistent, the load–deflection interact is the seen to be the standard Euler-type post-buckling behavior for a cylindrical panel.

92

4 Post-Buckling

Fig. 4.3 The effect of biaxial edge loading and transverse pressure on a cylindrical sandwich panel under uniaxial compressive edge loading of cross-ply face-sheets

With the inclusion of the transverse pressure, in addition to the biaxial edge loading, the behavior is similar to the behavior seen with the presence of geometric imperfections. The transverse pressure behaves like a geometric imperfection while the biaxial edge loading determines if the imperfection type behavior due to the transverse pressure is negative or positive. Various combinations of the effect of transverse pressure and biaxial edge loading are displayed for both positive and negative loading scenarios. It is seen in general that the combination of a positive transverse pressure and a compressive biaxial edge loading (positive) that the behavior resembles a positive geometric imperfection hugging the ideal load–deflection curve. For a combination of negative transverse pressure and negative biaxial edge loading (tension), it behaves more like a negative geometric imperfection. In Fig. 4.4, the effect of the face sheet thickness on the post-buckling response is seen. The trends appear to be identical for the entire range of face sheet thicknesses. With this in mind, the higher face sheet thickness provides a higher buckling bifurcation point with a minor increase in the snap through–type behavior.

4.4 4.4.1

Angle-Ply Laminated Sandwich Plates Governing System

In the previous section, the Airy’s stress potential method was limited to sandwich panels with cross-ply laminated facings because of the global stiffness terms, A16, A26. When these terms are not zero, the stress potential method is not feasible and

4.4 Angle-Ply Laminated Sandwich Plates

93

Fig. 4.4 The effect of the face sheet thickness on the compressive uniaxial edge loading of a cylindrical sandwich panel for a cross-ply stacking sequence

resorting to the displacement formulation is the most favorable approach. This will be the approach for the formulation governing angle-ply laminated flat sandwich panels in this section and for sandwich shells in Sect. 4.5. The governing nonlinear equations for sandwich plates can be obtained from Eqs. (2.162)–(2.166) as • Equations of Motion ( A11

2





2 2 2 ∂ ξ1 ∂v3 ∂ v3 ∂ v3 ∂v3 ∂v3 ∂ v3 þ þ þ ∂x21 ∂x1 ∂x21 ∂x21 ∂x1 ∂x1 ∂x21

)

 þ A66

2

2

∂ ξ1 ∂ ξ2 þ þ ∂x22 ∂x1 ∂x2

)   2  2  2 2 ∂v3 ∂2 v3 ∂v3 ∂2 v3 ∂v3 ∂ v3 ∂v3 ∂ v3 ∂v3 ∂ v3 ∂v3 ∂ v3 þ þ þ þ þ þ ∂x2 ∂x1 ∂x2 ∂x1 ∂x22 ∂x2 ∂x1 ∂x2 ∂x1 ∂x22 ∂x2 ∂x1 ∂x2 ∂x1 ∂x22 ( ) (  2  2 2 2 ∂v3 ∂2 v3 ∂ ξ2 ∂ ξ1 ∂v3 ∂ v3 ∂v3 ∂ v3 þ þ þ A12  A16 2 þ ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 2





2



2 2 2 ∂v3 ∂2 v3 ∂ ξ2 ∂v3 ∂ v3 ∂v3 ∂ v3 ∂v3 ∂ v3 ∂v3 ∂ v3 þ 2 þ 2 þ 2 þ þ þ ∂x1 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x2 ∂x21 ∂x2 ∂x21 ∂x21 ) ( )   2  2 2 2 ∂v3 ∂2 v3 ∂ ξ2 ∂v3 ∂ v3 ∂v3 ∂ v3 ∂v3 ∂ v3 þ þ þ  A26 ¼0 ∂x2 ∂x21 ∂x22 ∂x2 ∂x22 ∂x2 ∂x22 ∂x2 ∂x22

ð4:42aÞ

94

4 Post-Buckling

( A22

2





2 2 2 ∂ ξ2 ∂v3 ∂ v3 ∂ v3 ∂v3 ∂v3 ∂ v3 þ þ þ 2 2 2 ∂x2 ∂x2 ∂x2 ∂x2 ∂x2 ∂x2 ∂x22

)

( þ A66

2

2

∂ ξ2 ∂ ξ1 þ þ ∂x21 ∂x1 ∂x2

)   2  2  2 2 ∂v3 ∂2 v3 ∂v3 ∂2 v3 ∂v3 ∂ v3 ∂v3 ∂ v3 ∂v3 ∂ v3 ∂v3 ∂ v3 þ þ þ þ þ þ ∂x1 ∂x1 ∂x2 ∂x2 ∂x21 ∂x1 ∂x1 ∂x2 ∂x2 ∂x21 ∂x1 ∂x1 ∂x2 ∂x2 ∂x21 ( ) (  2  2 2 2 ∂v3 ∂2 v3 ∂ ξ1 ∂ ξ2 ∂v3 ∂ v3 ∂v3 ∂ v3 þ þ þ A12  A26 2 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 2





2



2 2 2 ∂v ∂2 v3 ∂ ξ1 ∂v ∂ v3 ∂v ∂ v3 ∂v ∂ v ∂v ∂ v3 þ þ2 3 þ2 3 þ2 3 þ 3 23 þ 3 þ 2 ∂x2 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x22 ∂x2 ) ( )   2  2 2 2 ∂v3 ∂2 v3 ∂ ξ1 ∂v3 ∂ v3 ∂v3 ∂ v3 ∂v3 ∂ v3 þ þ þ  A16 ¼0 ∂x1 ∂x22 ∂x21 ∂x1 ∂x21 ∂x1 ∂x21 ∂x1 ∂x21

ð4:42bÞ 2



2

2



2



2

2

∂ η1 ∂ η1 ∂ η2 ∂ η2 ∂ η2 ∂ η1 þ A66 þ þ A16 þ2 þ A12 A11 2 2 2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1   2 ∂ η2 ∂v þA26  d 1 η1 þ a 3 ¼ 0 ∂x1 ∂x22



ð4:42cÞ  2   2  2 2 2 2 ∂ η2 ∂ η2 ∂ η1 ∂ η1 ∂ η1 ∂ η2 þ A66 þ þ A26 þ2 A22 þ A12 ∂x1 ∂x2 ∂x1 ∂x2 ∂x22 ∂x22 ∂x1 ∂x2 ∂x22   2 ∂ η1 ∂v þA16  d2 η2 þ a 3 ¼ 0 ∂x2 ∂x21 ð4:42dÞ (

! ! !)  2 2  2  2  2  2 2 ∂ξ1 ∂ v3 ∂ v3 1 ∂v3 ∂ v3 ∂ v3 ∂v3 ∂v3 ∂ v3 ∂ v3 þ þ A11 þ þ þ 2 ∂x1 ∂x1 ∂x21 ∂x1 ∂x1 ∂x21 ∂x21 ∂x21 ∂x21 ∂x21 ! ! ! (  2  2  2 2  2  2  2 ∂ξ1 ∂ v3 ∂ v3 ∂ξ2 ∂ v3 ∂ v3 1 ∂v3 ∂ v3 ∂ v3 þ A12 þ þ þ þ þ 2 ∂x1 ∂x1 ∂x22 ∂x22 ∂x2 ∂x21 ∂x21 ∂x22 ∂x22 ! ! !)  2 2    2 2  2  2  2 1 ∂v3 ∂ v3 ∂ v3 ∂v3 ∂v3 ∂ v3 ∂ v3 ∂v3 ∂v3 ∂ v3 ∂ v3 þ þ þ þ þ þ 2 ∂x2 ∂x1 ∂x1 ∂x22 ∂x2 ∂x2 ∂x21 ∂x21 ∂x21 ∂x22 ∂x21

4.4 Angle-Ply Laminated Sandwich Plates

95

(

! ! !)  2 2  2  2  2  2 2 ∂ξ2 ∂ v3 ∂ v3 1 ∂v3 ∂ v3 ∂ v3 ∂v3 ∂v3 ∂ v3 ∂ v3 þA22 þ þ þ þ þ 2 ∂x2 ∂x2 ∂x22 ∂x2 ∂x2 ∂x22 ∂x22 ∂x22 ∂x22 ∂x22 ( ! ! 2  2  2 2 2 ∂ v3 ∂ v3 ∂ξ1 ∂ v3 ∂ξ2 ∂ v3 ∂v ∂v ∂ v3 þ2A66 þ þ þ þ 3 3 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 2



∂ v3 þ ∂x1 ∂x2

!



2



2 ∂v ∂v ∂ v3 ∂ v3 þ þ 3 3 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2

!



∂v ∂v þ 3 3 ∂x1 ∂x2

2



2 ∂ v3 ∂ v3 þ ∂x1 ∂x2 ∂x1 ∂x2

!)

(

! ! ! 2  2  2  2 2 2 ∂ξ1 ∂ v3 ∂ v3 ∂ξ2 ∂ v3 ∂ v3 ∂v3 ∂v3 ∂ v3 ∂ v3 þA16 þ þ þ þ þ ∂x2 ∂x21 ∂x1 ∂x21 ∂x21 ∂x1 ∂x2 ∂x21 ∂x21 ∂x21 ! ! !   2  2  2  2 2 ∂v3 ∂v3 ∂2 v3 ∂ v3 ∂ v3 ∂v3 ∂v3 ∂ v3 ∂ v3 ∂ξ1 ∂ v3 þ þ þ þ þ þ2 ∂x1 ∂x2 ∂x21 ∂x1 ∂x2 ∂x21 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 ∂x21 ∂x21 !  2  2  2 ∂ v3 ∂v3 ∂ v3 ∂v3 ∂v3 þ þ þ2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x1 ( þA26 

þ

∂ξ2 ∂x1

∂v3 ∂v3 ∂x2 ∂x1

2



2 ∂ v3 ∂ v3 þ ∂x1 ∂x2 ∂x1 ∂x2

!)

! ! ! 2  2  2  2 2 2 ∂ v3 ∂ v3 ∂ξ1 ∂ v3 ∂ v3 ∂v3 ∂v3 ∂ v3 ∂ v3 þ þ þ þ þ ∂x2 ∂x22 ∂x2 ∂x1 ∂x22 ∂x22 ∂x22 ∂x22 ∂x22 ! ! !  2  2  2  2 2 2 ∂ v3 ∂ξ2 ∂ v3 ∂ v3 ∂ v3 ∂v3 ∂v3 ∂ v3 ∂ v3 þ þ þ þ þ2 ∂x2 ∂x1 ∂x22 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x22 ∂x22 ∂x22

!  2  2  2 ∂ v3 ∂v3 ∂ v3 ∂v ∂v þ þ þ2 3 3 ∂x2 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x2 4

4

2



2 ∂ v3 ∂ v3 þ ∂x1 ∂x2 ∂x1 ∂x2 4

!)

4

 F 11

∂ v3 ∂x41

4

∂ v3 ∂ v3 ∂ v ∂ v3  F 22  4F 16 3 3  4F 26 ∂x21 ∂x22 ∂x42 ∂x1 ∂x2 ∂x1 ∂x32     2  2 h 1 ∂η1 C 1 ∂ v3 þ2hK G13 1 þ þ 1þ 2h h ∂x1 h ∂x21     2  2 h 1 ∂η2 C ∂ v3 þ2hK G23 1 þ þ 1þ 1  q3 ¼ 0 2h h ∂x2 h ∂x22 2ðF 12 þ2F 66 Þ

ð4:42eÞ

• Boundary Conditions and the associated boundary conditions along the edges xn ¼ const(n ¼ 1, 2) are

96

4 Post-Buckling

e nn N nn ¼ N e nt N nt ¼ N

or

Lnn ¼ e Lnn e Lnt ¼ Lnt

or or

or

ξn ¼ e ξn ξt ¼ e ξt

ηn ¼ e ηn ηt ¼ e ηt ∂v ∂ev 3 e nn M nn ¼ M or ¼ 3 ∂xn ∂xn     ∂v3 ∂v3 ∂v3 ∂v3 ∂M nn ∂M nt N nt þ þ þ2 þ N nn þ ∂xt ∂xt ∂xn ∂xn ∂xn ∂xt e nt e  ∂M þ a=h N n3 ¼ þ N n3 ∂xt

v3 ¼ ev3

or

ð4:43a  fÞ Considering simply supported boundary conditions, of Type A, they are assumed as Along the edges xn ¼ 0, Ln e nn , N nn ¼ N ¼0

4.4.2

N nt ¼ 0,

ηn ¼ 0,

ηt ¼ 0,

M nn ¼ 0,

v3

ð4:44a  fÞ

Solution Methodology

Equations (4.42a–e) and (4.44a–f) constitutes the governing system of equations for the post-buckling solution for flat sandwich panels with angle-ply laminated facings and a weak incompressible core. The method of approach will be similar as in the previous section but without the stress potential to aid in the solution. It should also be mentioned that the inertia and thermal terms have been discarded within the equations of motion which are now referred to as the equations of equilibrium. The transversal deflection and the geometric imperfection representations are the same as in the previous section for the same reasons. The boundary conditions remain the same and the assumptions for the transversal displacement and the peak geometric imperfection remain unchanged. Again, these are represented by the following expressions 

v3 v∘3



 ¼

wmn w∘mn

 sin λm x1 sin μn x2

ð4.45a, bÞ

Unlike the linear theory the first two equations of equilibrium are coupled with the last three. Therefore, there are five equations of motion to contend with. All five equations of equilibrium need to be fulfilled as well as the boundary conditions.

4.4 Angle-Ply Laminated Sandwich Plates

97

Keeping the EGM in mind and knowing whatever is not identically fulfilled is fulfilled in an average sense allows ξ1 and ξ2 to be assumed in the following form 9 8 < ξ1 ðx1 , x2 Þ = :

ξ2 ðx1 , x2 Þ

;

¼

9 8 1Þ = < F ðmn :

1Þ Gðmn

;

8 9 4Þ = < F ðmn

sin 2λm x1 þ

9 8 2Þ = < F ðmn :

2Þ Gðmn

sin 2μn x2 þ

8 9 5Þ = < F ðmn

:

3Þ Gðmn

;

sin λm x1 cos μn x2 þ

sin 2λm x1 cos 2μn x2 þ 5Þ ; Gðmn 8 9 8 9 8 9 6Þ = 7Þ = 8Þ = < F ðmn < F ðmn < F ðmn cos 2λm x1 sin 2μn x2 þ x þ x : ð6Þ ; : ð7Þ ; 1 : ð8Þ ; 2 Gmn Gmn Gmn :

4Þ Gðmn

;

cos λm x1 sin μn x2 þ

;

9 8 3Þ = < F ðmn

:

ð4.46a, bÞ Substituting the expressions for v3 , v∘3 , ξ1 , and ξ2, from Eqs. (4.45a, b) and (4.46a, b), into Eqs. (4.42a and 4.42b) and comparing coefficients of like 1Þ 6Þ 1Þ 6Þ  F ðmn and Gðmn  Gðmn as trigonometric functions provides F ðmn 

  ðiÞ ðiÞ ð5Þ ð6Þ  iÞ iÞ 5Þ 6Þ e ,F e e mn , G e mn , G F ðmn , Gðmn , F ðmn , Gðmn ¼ F w2mn þ 2wmn w∘mn , ði ¼ 1, 2Þ mn mn

ð4:47aÞ 6Þ 5Þ F ðmnjÞ ¼ GðmnjÞ ¼ F ðmn ¼ Gðmn ¼ 0,

ð j ¼ 3, 4Þ

ð4:47cÞ

where, A216  A11 A66 λ2m þ ðA12 A66  A16 A26 Þμ2n 2Þ ðA16 A22  A26 A12 Þλ2m e ðmn   , F ¼ 16λm A11 A66  A216 16μn A22 A66  A226  2 A26  A22 A66 μ2n þ ðA12 A66  A16 A26 Þλ2m ð1Þ ð2Þ ðA26 A11  A16 A12 Þμ2n e e  , Gmn ¼  Gmn ¼ 16λm A11 A66  A216 16μn A22 A66  A226 5Þ λ e ð6Þ ¼ μn e ðmn ¼ m, G F mn 16 16 ð4:48a  fÞ 1Þ e ðmn ¼ F



7Þ 8Þ 7Þ 8Þ The constants, F ðmn , F ðmn , Gðmn , Gðmn are four constants which remain undetermined. These are determined from enforcing the in-plane static boundary conditions which are expressed mathematically as

0

Lt

e nn Lt N nn dxt ¼ N



n ¼ 1, 2 t ¼ 2, 1



X n, t



Z

ð4:49Þ

98

4 Post-Buckling

Z

Lt

N nt dxt ¼ 0

ð4:50Þ

0

These conditions fulfill the boundary conditions, Eqs. (4.44a, b) in an average sense. In order to utilize these conditions the stress resultants N11, N22, and N12 need to be expressed in terms of displacements. It should be recalled that for the case of a weak core =

==

N 11 ¼ N 11 þ N 11 ,

=

==

N 22 ¼ N 22 þ N 22 ,

=

==

N 12 ¼ N 12 þ N 12

ð4:51a  cÞ

Making use of Eqs. (2.86a) and (2.92a) coupled with the strain–displacement relationships, Eqs. (2.53)–(2.55) and Eqs. (2.69)–(2.71), while discarding the terms with the curvatures gives the global stress resultants in terms of displacements ξ1, ξ2, v3, and v∘3 as ( ) )  2  2 ∂v∘3 ∂v3 ∂v∘3 ∂v3 ∂ξ1 1 ∂v3 ∂ξ2 1 ∂v3 N 11 ¼ A11 þ þ þ þ  þ A12 ∂x1 2 ∂x1 ∂x1 ∂x1 ∂x2 2 ∂x2 ∂x2 ∂x2    ∂ξ1 ∂ξ2 ∂v3 ∂v3 ∂v3 ∂v3 ∂v3 ∂v3 þ A16 þ þ þ þ ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 (

ð4:52aÞ (

N 22

) ( )  2  2 ∂v∘3 ∂v3 ∂v∘3 ∂v3 ∂ξ2 1 ∂v3 ∂ξ1 1 ∂v3 ¼ A22 þ þ þ þ þ A12 ∂x2 2 ∂x2 ∂x2 ∂x2 ∂x1 2 ∂x1 ∂x1 ∂x1    ∂ξ2 ∂ξ1 ∂v3 ∂v3 ∂v3 ∂v3 ∂v3 ∂v3 þ A26 þ þ þ þ ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ð4:52bÞ (

N 12

) ( )  2  2 ∂v∘3 ∂v3 ∂v∘3 ∂v3 ∂ξ1 1 ∂v3 ∂ξ2 1 ∂v3 þ A26 ¼ A16 þ þ þ þ ∂x1 2 ∂x1 ∂x1 ∂x1 ∂x2 2 ∂x2 ∂x2 ∂x2   ∂ξ1 ∂ξ2 ∂v3 ∂v3 ∂v∘3 ∂v3 ∂v3 ∂v∘3 þ A66 þ þ þ þ ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ð4:52cÞ

Substituting the expressions for v3 , v∘3 , ξ1 , and ξ2 into Eqs. (4.52a, 4.52b, and 4.52c) then substituting the result into the requirements for the in-plane static boundary conditions, Eqs. (4.49) and (4.50) and carrying out the indicated opera7Þ 8Þ 7Þ 8Þ tions provides the constants F ðmn , F ðmn , Gðmn , Gðmn as

4.4 Angle-Ply Laminated Sandwich Plates

99

A2  A22 A66 λ2m  2 e 11 wmn þ 2wmn w∘mn þ 26 N 8 Ω A A  A16 A26 e N 22 þ 12 66 Ω A A  A16 A26 μ2  e 11 ¼  n w2mn þ 2wmn w∘mn þ 12 66 N 8 Ω A2  A11 A66 e þ 16 N 22 Ω 7Þ F ðmn ¼

8Þ Gðmn

ð4:53aÞ

ð4:53bÞ

where Ω ¼ A11 A22 A66  A212 A66 þ 2A12 A16 A26  A11 A226  A22 A216

ð4:53cÞ

7Þ 8Þ F ðmn and Gðmn are arbitrary and have no effect on the final post-buckling solution. With the first two equilibrium equations fulfilled, in addition to the boundary condition Eq. (4.44f), attention is now given to the third and fourth equilibrium Eqs. (4.42c and 4.42d). It can be seen that these two governing equations have no curvature terms in them, as a result, the curvature plays no part on the fulfillment of these two equations. Except for the consideration of cross-ply laminated sandwich plates and shells where the A16 and A26 terms are zero the expressions for η1 and η2 will remain the same for other pre- and postcritical stability problems. The expressions for these displacement functions have already been determined in Chap. 3 (see Eqs. 3.12a, b) – (3.17a–e). Therefore, the first four equations of equilibrium are fulfilled with the boundary condition (4.44f). The fifth equation of equilibrium and the unfulfilled boundary conditions will be satisfied in an average sense through the application of the EGM. Retaining the fifth equilibrium equation along with the unfulfilled boundary conditions within the energy functional gives

Z

t1

*Z

l1 Z l2

"

( A11

t0

0

0 

∂v ∂v þ 3 3 ∂x1 ∂x1



( þ A22

2



2 ∂ v3 ∂ v3 þ ∂x21 ∂x21

 2 1 ∂v3 þ 2 ∂x1 ∂v ∂v þ 3 3 ∂x1 ∂x1

2



2 ∂ξ1 ∂ v3 ∂ v3 þ ∂x1 ∂x21 ∂x21

2

þ A12 

2 ∂ v3 ∂ v3 þ ∂x22 ∂x22 2

(

!)



2 ∂ v3 ∂ v3 þ ∂x22 ∂x22

!

!

!

∂ξ1 ∂x1

!  2 2 2  1 ∂v3 ∂ v3 ∂ v3 þ þ 2 ∂x1 ∂x21 ∂x21 

2

2 ∂ v3 ∂ v3 þ ∂x22 ∂x22

!

2



2 ∂ξ ∂ v3 ∂ v3 þ þ 2 ∂x2 ∂x21 ∂x21

!

!  2 2 2  1 ∂v3 ∂ v3 ∂ v3 þ þ 2 ∂x2 ∂x21 ∂x21 

∂v ∂v þ 3 3 ∂x2 ∂x2

2



2 ∂ v3 ∂ v3 þ ∂x21 ∂x21

!)

! ! !)  2 2  2  2  2  2 2 ∂ξ2 ∂ v3 ∂ v3 1 ∂v3 ∂ v3 ∂ v3 ∂v3 ∂v3 ∂ v3 ∂ v3 þ þ þ þ þ 2 ∂x2 ∂x2 ∂x22 ∂x2 ∂x2 ∂x22 ∂x22 ∂x22 ∂x22 ∂x22

100

4 Post-Buckling

( þ2A66

2



2 ∂ v3 ∂ v3 þ ∂x1 ∂x2 ∂x1 ∂x2

∂ξ1 ∂x2

2



2 ∂ v3 ∂ v3 þ ∂x1 ∂x2 ∂x1 ∂x2 2

!



2 ∂ v3 ∂ v3 þ ∂x1 ∂x2 ∂x1 ∂x2



( þ A16

2



2 ∂ v3 ∂ξ ∂ v3 þ þ 2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 2

∂v ∂v þ 3 3 ∂x1 ∂x2

!)

!



2 ∂ v3 ∂ v3 þ ∂x1 ∂x2 ∂x1 ∂x2 2



2 ∂ξ1 ∂ v3 ∂ v3 þ ∂x2 ∂x21 ∂x21

!

!

! þ

∂v3 ∂v3 ∂x1 ∂x2



þ

∂v3 ∂v3 ∂x1 ∂x2 2



2 ∂ v3 ∂ v3 þ ∂x21 ∂x21

∂ξ þ 2 ∂x1

!

! !   2  2  2 ∂v3 ∂v3 ∂2 v3 ∂ v3 ∂v3 ∂v3 ∂ v3 ∂ v3 ∂v3 ∂v3 þ þ þ þ þ ∂x1 ∂x2 ∂x21 ∂x1 ∂x2 ∂x21 ∂x1 ∂x2 ∂x21 ∂x21 ! !   ! 2  2  2  2 2 2 2 ∂ v3 ∂ v3 ∂ξ1 ∂ v3 ∂ v3 ∂ v3 ∂v3 ∂ v3 þ þ þ þ2 þ ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 ∂x21 ∂x21 

∂v ∂v þ2 3 3 ∂x1 ∂x1

2

2



2 ∂ξ ∂ v3 ∂ v3 þ 1 þ ∂x2 ∂x22 ∂x22 

∂v ∂v þ 3 3 ∂x2 ∂x1



2 ∂ v3 ∂ v3 þ ∂x1 ∂x2 ∂x1 ∂x2

!

!)

∂v ∂v þ 3 3 ∂x2 ∂x1

2



2 ∂ v3 ∂ v 3 þ ∂x22 ∂x22

! þ2

∂ξ2 ∂x2

( þ A26

4

2



2 ∂ v3 ∂ v3 þ ∂x22 ∂x22

!

2



2 ∂ v3 ∂ v3 þ ∂x22 ∂x22

!



2



∂v ∂v ∂2 v3 ∂ v3 þ þ 3 3 ∂x2 ∂x1 ∂x22 ∂x22 ! 2  2 ∂ v3 ∂ v3 þ ∂x1 ∂x2 ∂x1 ∂x2

!  2  2  2 ∂ v3 ∂v3 ∂ v3 ∂v ∂v þ2 3 3 þ þ ∂x2 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x2 4

∂ξ2 ∂x1

2



2 ∂ v3 ∂ v3 þ ∂x1 ∂x2 ∂x1 ∂x2 4

!)

!

4

 F 11

∂ v3 ∂x41

4

∂ v3 ∂ v3 ∂ v ∂ v3  F 22  4F 16 3 3  4F 26 ∂x21 ∂x22 ∂x42 ∂x1 ∂x2 ∂x1 ∂x32     2  2 h 1 ∂η1 C ∂ v3 þ2hK G13 1 þ þ 1þ 1 2h h ∂x1 h ∂x21      2   2 h 1 ∂η2 C 1 ∂ v3 þ2hK G23 1 þ þ 1þ  q3 ¼ 0 δv3 dx1 dx2 dtþ 2h h ∂x2 h ∂x22    Z t1Z l2 l  e 11 δξ1 þ N 12 δξ2 þ L11 δη1 þ L12 δη2 þ M 11 δ ∂v3  1 dx2 dt N 11 þ N þ ∂x1  0 t0 0    Z t1Z l1  ∂v3  l2 e dx1 dt þ N 22 þ N 22 δξ2 þ N 12 δξ1 þ L22 δη2 þ L12 δη1 þ M 22 δ ∂x2  0 t0 0 2ðF 12 þ2F 66 Þ

¼0

ð4:54Þ

4.4 Angle-Ply Laminated Sandwich Plates

101

Substituting the expressions for v3 , v∘3 , ξ1 , ξ2 , η1 , η2 into Eq. (4.54) and carrying out the indicated integrations results in the governing post-buckling solution for angle-ply laminated flat sandwich panels which results in a nonlinear algebraic equation expressed in terms of the modal amplitudes as  2   5Þ 2Þ 1Þ Pðmn wmn þ 2wmn w∘mn wmn þ w∘mn þ Pðmn wmn þ w∘mn þ Pðmn wmn þ qmn ¼ 0 ð4:55Þ Where  ð1Þ  3 ð2Þ  3 ð1Þ 5Þ e  λ A16 þ λm μ2 A26 G e e mn  μn A22 þ λ2m μn A12 G Pðmn ¼  λ3m A11 þ λm μ2n A12 F mn mn m n  3 ð2Þ 2  μn A26 þ λm μn A16 F mn

ð4:56Þ  2Þ e 11 þ μ2n N e 22 Pðmn ¼  λ2m N

ð4:57Þ

 ð1Þ  1Þ e mn þ aλm þ Pðmn ¼F 11 λ4m þ 2ðF 12 þ 2F 66 Þλ2m μ2n þ F 22 μ4n þ d 1 aλm H  ð2Þ  d2 aμn eI mn þ aμn

ð4:58Þ

Equation (4.55) is put in a more conducive form for generating results as      ∘ 2 5Þ 3 5Þ ∘ 5Þ 2Þ 1Þ 2Þ ∘ Pðmn wmn þ 3Pðmn wmn w2mn þ 2Pðmn wmn þ Pðmn þ Pðmn wmn þ qmn ¼ 0 wmn þ Pðmn

ð4:59Þ which can be further expressed as L1 w3mn þ L2 w2mn þ L3 wmn þ L4 ¼ 0

ð4:60Þ

where 5Þ L1 ¼ Pðmn

ð4:61aÞ

5Þ ∘ wmn L2 ¼ 3Pðmn  2 5Þ 2Þ 1Þ w∘mn þ Pðmn þ Pðmn L3 ¼ 2Pðmn

ð4:61bÞ

2Þ ∘ L4 ¼ Pðmn wmn þ qmn

ð4:61dÞ

ð4:61cÞ

102

4 Post-Buckling

Equation (4.60) is easily solved via Newton’s method for the various equilibrium configurations of the structure. Usually, a more convenient form is to introduce the following Parameter δ as δ ¼ wmn þ w∘mn ) wmn ¼ δ  w∘mn

ð4:62Þ

Substituting Eq. (4.62) into Eq. (4.60) gives  3  2  L1 δ  w∘mn þ L2 δ  w∘mn þ L3 δ  w∘mn þ L4 ¼ 0

ð4:63Þ

Expanding, combining like terms, and simplifying gives    2   3 L1 δ3mn þ L2  3L1 w∘mn δ2mn þ L3  2L2 w∘mn  L1 w∘mn δmn þ L4  L1 w∘mn þ  2 L2 w∘mn  L3 w∘mn ¼ 0 ð4:64Þ Solving Eq. (4.64) via Newton’s method gives the compressive uniaxial load vs δmn (amplitude of deflection plus amplitude of imperfection).

4.4.3

Numerical Results and Discussion

A few results of the applied theory of sandwich plates as it pertains to post-buckling are presented next. For the following numerical illustrations which follow, Tables 4.3 and 4.4 contain the material and geometrical properties for both the face sheets and the core. Figure 4.5 depicts the effect of the geometrical imperfections on the uniaxial compressive edge loading of a flat sandwich panel versus amplitude of deflection for a fixed layup of the facing. The entire map of the stability paths is shown. For the case of zero imperfection, there is a bifurcation point which shows several directions Table 4.3 The geometrical and material properties for the face sheets t ak (mm) 0.005

L1 (mm) 609.6 (24in)

L2 (mm) 609.6 (24in)

E1 (GPa) 180.987

E2 (GPa) 10.342

G12 (GPa) 7.239

ν12 0.28

t ak implies the ply thickness

Table 4.4 The geometrical material properties for the core h in (mm) 12.7 (0.5in)

G13 (MPa) 1.437

G23 (MPa) 0.651

4.4 Angle-Ply Laminated Sandwich Plates

103

Fig. 4.5 The effect of geometric imperfections on the compressive uniaxial edge loading vs. amplitude of deflection for a flat sandwich panel

Fig. 4.6 The counterpart of Fig. 4.5 in the compressive edge load-end shortening plane for various geometric imperfections of a flat sandwich panel

in which the loading can occur. This bifurcation point is the point at which the panel starts to buckle. The results also reveal that with an imperfection present, the bifurcation disappears and there is an initial deflection with the start of compressive loading. As the imperfection increases, the initial deflection increases, and the direction of loading appears to proceed farther away from the bifurcation point. Figure 4.6 is the

104

4 Post-Buckling

Fig. 4.7 The effect of the ply angle on the compressive edge loading for a single-layered flat sandwich panel

counterpart of Fig. 4.5 revealing the end-shorting of panel. As the compressive loading increases, the edges displace inward. This amount of displacement is depicted for various amounts of imperfections inherent within the structure. With no imperfection there is a discontinuity within the trend line. This discontinuity is the buckling bifurcation point for the panel which matches with the buckling point in Fig. 4.5. Beyond the bifurcation point, the panel continues to displace transversely as the compressive loading increases until the panel fails. Also, as the imperfection increases, the end-shortening increases for a fixed compressive edge loading. Figures 4.7 and 4.8 highlight the influence played by the various ply angles on the uniaxial compressive strength of a flat sandwich panel for two different layups. In each case, it appears that the ply angle of 45 degrees contains the higher buckling bifurcation point whereas at a ply angle of zero degrees the panel has a larger loadcarrying capacity from a load–deflection standpoint. In Fig. 4.9, the effect of biaxial edge loading is depicted. When both compression and tension are present simultaneously the effect seems to increase the point at which the structure buckles beyond which the loading increases until the panel fails. In comparison, when both edges are loaded in compression the effect seems to diminish the point at which the structure buckles. Figure 4.10 illustrates the effect of the ply angle for the given layup of a flat sandwich panel on the transverse pressure deflection interaction. It clearly reveals that the structure has a larger load-carrying capacity at a ply angle of 45 degrees as compared with the other ply angle configurations. It can also be seen that 0- and 90-degree ply angle configurations have the least load-carrying capacity.

4.5 Angle-Ply Laminated Sandwich Shells

105

Fig. 4.8 The effect of the ply angle on the compressive edge loading for the given layup in the facings

Fig. 4.9 The effect of the biaxial edge loading, for the given fixed layup, on the uniaxial compressive edge load

4.5 4.5.1

Angle-Ply Laminated Sandwich Shells Governing System

In contrast to the previous section, this section considers the curvature terms included in the governing equations. The solution methodology is exactly the

106

4 Post-Buckling

Fig. 4.10 The effect of the transverse pressure on the deflection of a flat sandwich panel for various ply angles

same as in the previous section with only an additional level of computational effort. The governing equations can be obtained from Eqs. (2.156)–(2.160) and (2.161a–f) which are given as • Equations of Equilibrium (

2





2 2 2 ∂ ξ1 ∂v3 ∂ v3 ∂ v3 ∂v3 ∂v3 ∂ v3 1 ∂v3 þ þ þ  ∂x21 ∂x1 ∂x21 ∂x21 ∂x1 ∂x1 ∂x21 R1 ∂x1

)

(

2

2

∂ ξ1 ∂ ξ2 þ ∂x22 ∂x1 ∂x2 )   2  2  2 2 ∂v3 ∂2 v3 ∂v3 ∂2 v3 ∂v3 ∂ v3 ∂v3 ∂ v3 ∂v3 ∂ v3 ∂v3 ∂ v3 þ þ þ þ þ þ ∂x2 ∂x1 ∂x2 ∂x1 ∂x22 ∂x2 ∂x1 ∂x2 ∂x1 ∂x22 ∂x2 ∂x1 ∂x2 ∂x1 ∂x22 ( )  2  2 2 ∂v3 ∂2 v3 ∂ ξ2 ∂v3 ∂ v3 ∂v3 ∂ v3 1 ∂v3 þA12 þ þ þ  ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 R2 ∂x2 (  2  2 2 2 ∂v3 ∂2 v3 ∂ ξ1 ∂ ξ2 ∂v3 ∂ v3 ∂v3 ∂ v3 A16 2 þ 2 þ 2 þ 2 þ ∂x1 ∂x2 ∂x21 ∂x1 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ) 2  2 2 ∂v3 ∂ v3 ∂v3 ∂ v3 ∂ v3 1 ∂v3 þ þ þ  2 2 ∂x2 ∂x1 ∂x2 ∂x1 ∂x21 R1 ∂x1 ( )  2  2 2 2 ∂ ξ2 ∂v3 ∂ v3 ∂v3 ∂ v3 ∂v3 ∂ v3 1 ∂v3 A26 þ þ þ  ¼0 ∂x22 ∂x2 ∂x22 ∂x2 ∂x22 ∂x2 ∂x22 R2 ∂x2 A11

þ A66

ð4:65aÞ

4.5 Angle-Ply Laminated Sandwich Shells

(

2

107





2 2 2 ∂ ξ2 ∂v3 ∂ v3 ∂ v3 ∂v3 ∂v3 ∂ v3 1 ∂v3 þ þ þ  2 2 2 ∂x2 ∂x2 ∂x2 ∂x2 ∂x2 ∂x2 ∂x22 R2 ∂x2

)

(

2

2

∂ ξ2 ∂ ξ1 þ þ ∂x21 ∂x1 ∂x2 )   2  2  2 2 ∂v3 ∂2 v3 ∂v3 ∂2 v3 ∂v3 ∂ v3 ∂v3 ∂ v3 ∂v3 ∂ v3 ∂v3 ∂ v3 þ þ þ þ þ þ ∂x1 ∂x1 ∂x2 ∂x2 ∂x21 ∂x1 ∂x1 ∂x2 ∂x2 ∂x21 ∂x1 ∂x1 ∂x2 ∂x2 ∂x21 ( ) (  2  2 2 2 ∂v3 ∂2 v3 ∂ ξ1 ∂v3 ∂ v3 ∂v3 ∂ v3 1 ∂v3 ∂ ξ2 A26 2 þ þ þ  A12 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 R1 ∂x2 ∂x1 ∂x2 A22

2





þ A66

2



2 2 2 ∂v3 ∂2 v3 ∂ ξ1 ∂v3 ∂ v3 ∂v3 ∂ v3 ∂v3 ∂ v ∂v3 ∂ v3 þ 2 þ 2 þ 2 þ þ þ ∂x2 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x1 ∂x22 ∂x1 ∂x22 ∂x22 ) ( )   2  2 2 2 ∂v3 ∂2 v3 1 ∂v3 ∂ ξ1 ∂v3 ∂ v3 ∂v3 ∂ v3 ∂v3 ∂ v3 1 ∂v3 A16 ¼0  þ þ þ  ∂x1 ∂x22 R2 ∂x2 ∂x21 ∂x1 ∂x21 ∂x1 ∂x21 ∂x1 ∂x21 R1 ∂x1

þ

ð4:65bÞ  2   2  2 2 2 2 ∂ η1 ∂ η1 ∂ η2 ∂ η2 ∂ η2 ∂ η1 þ A66 þ þ A16 þ2 A11 þ A12 ∂x1 ∂x2 ∂x1 ∂x2 ∂x21 ∂x22 ∂x1 ∂x2 ∂x21   2 ∂ η2 ∂v þA26  d1 η1 þ a 3 ¼ 0 2 ∂x1 ∂x2 ð4:65cÞ  2   2  2 2 2 2 ∂ η2 ∂ η2 ∂ η1 ∂ η1 ∂ η1 ∂ η2 þ A þ þ A þ 2 þ A 66 12 26 ∂x1 ∂x2 ∂x1 ∂x2 ∂x22 ∂x22 ∂x1 ∂x2 ∂x22   2 ∂ η1 ∂v þA16  d2 η2 þ a 3 ¼ 0 ∂x2 ∂x21 A22

ð4:65dÞ (

! ! !  2 2  2  2  2  2 2 ∂ξ1 ∂ v3 ∂ v3 1 ∂v3 ∂ v3 ∂ v3 ∂v3 ∂v3 ∂ v3 ∂ v3 A11 þ þ þ þ þ 2 ∂x1 ∂x1 ∂x21 ∂x1 ∂x1 ∂x21 ∂x21 ∂x21 ∂x21 ∂x21 " ! #) (  2  2  2 2 ∂ξ1 ∂ v3 1 ∂ξ1 1 ∂v3 ∂v3 ∂v3 ∂ v3 ∂ v3 v3 þ  v3 þ þ þ  þ A12 2 2 R1 ∂x1 2 ∂x1 R1 ∂x1 ∂x1 ∂x1 ∂x22 ∂x1 ∂x1 ! ! !  2  2 2  2  2  2 2 ∂ v3 ∂ξ2 ∂ v3 ∂ v3 1 ∂v3 ∂ v3 ∂ v3 1 ∂v3 þ þ þ 2 þ þ þ 2 ∂x1 2 ∂x2 ∂x2 ∂x21 ∂x2 ∂x21 ∂x22 ∂x22 ! ! !   2  2  2  2 2 2 ∂ v3 ∂ v3 ∂v3 ∂v3 ∂ v3 ∂ v3 ∂v3 ∂v3 ∂ v3 ∂ v3 þ þ þ þ þ ∂x1 ∂x1 ∂x22 ∂x2 ∂x2 ∂x21 ∂x21 ∂x22 ∂x21 ∂x21

108

4 Post-Buckling

" ! #  2  2  2 1 ∂ξ2 1 ∂v3 ∂v3 ∂v3 ∂ v3 ∂ v3 v3 þ þ þ v3 þ  R1 ∂x2 2 ∂x2 R2 ∂x2 ∂x2 ∂x22 ∂x22 " ! #)  2  2  2 1 ∂ξ1 1 ∂v3 ∂v3 ∂v3 ∂ v3 ∂ v3 v3 þ  v3 þ þ þ  R2 ∂x1 2 ∂x1 R1 ∂x1 ∂x1 ∂x21 ∂x21 ( þA22

! !  2 2  2  2  2 ∂ξ2 ∂ v3 ∂ v3 1 ∂v3 ∂ v3 ∂ v3 ∂v3 ∂v3 þ þ þ þ 2 ∂x2 ∂x2 ∂x22 ∂x2 ∂x2 ∂x22 ∂x22 ∂x22

2



2 ∂ v3 ∂ v3 þ ∂x22 ∂x22

" ! #)  2  2  2 1 ∂ξ2 1 ∂v3 ∂v3 ∂v3 ∂ v3 ∂ v3 v3 þ  v3 þ þ þ  R2 ∂x2 2 ∂x2 R2 ∂x2 ∂x2 ∂x22 ∂x22 ( þ2A66

2



2

2 ∂ v3 ∂ v3 þ ∂x1 ∂x2 ∂x1 ∂x2 2



∂ v3 ∂x1 ∂x2

!)

( þ A16

2



2 ∂ v3 ∂ v3 þ ∂x21 ∂x21

!

!

2

2



!



∂v ∂v þ 3 3 ∂x2 ∂x1

!

!

!

þ

∂v3 ∂v3 ∂x1 ∂x2 

∂v ∂v þ 3 3 ∂x1 ∂x2

2

∂ v3 þ ∂x1 ∂x2

!  2 2  ∂ξ2 ∂ v3 ∂ v3 ∂v ∂v þ 2 þ 3 3 þ 2 ∂x1 ∂x1 ∂x1 ∂x1 ∂x2 

2





!

!





2

∂v ∂v þ 3 3 ∂x1 ∂x2

2 ∂ v3 ∂ v3 þ ∂x21 ∂x21

!

!  2  2  2 ∂ v3 ∂v3 ∂ v3 ∂v ∂v þ þ þ2 3 3 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x1

"   1 ∂ξ1 ∂ξ2 ∂v3 ∂v3 ∂v3 ∂v3 ∂v3 ∂v3 þ þ þ þ þ R1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2

2



!





2 ∂ v3 ∂ v3 þ ∂x1 ∂x2 ∂x1 ∂x2 2



∂v ∂v ∂2 v3 ∂ v3 þ þ 3 3 ∂x1 ∂x2 ∂x21 ∂x21

2 ∂ v3 ∂ v3 þ ∂x1 ∂x2 ∂x1 ∂x2

2 ∂ v3 ∂ v3 þ ∂x22 ∂x22

2

2 ∂ v3 ∂ v3 þ ∂x1 ∂x2 ∂x1 ∂x2

2 ∂ξ1 ∂ v3 ∂ v3 þ ∂x2 ∂x21 ∂x21



2

2 ∂ v3 ∂ v3 þ ∂x1 ∂x2 ∂x1 ∂x2

∂ξ þ 2 ∂x1



2



!

∂v ∂v þ 3 3 ∂x1 ∂x2

2 ∂ v3 ∂ξ ∂ v3 þ þ2 1 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2

2v3



2 ∂ v3 ∂ v3 þ ∂x1 ∂x2 ∂x1 ∂x2

∂ξ1 ∂x2

!#)

( þ A26

2

2



2 ∂v ∂v ∂ v3 ∂ v3 þ þ 3 3 ∂x2 ∂x1 ∂x22 ∂x22 2



2 ∂ v3 ∂ v 3 þ ∂x22 ∂x22

!



2 ∂ξ2 ∂ v3 ∂ v3 þ ∂x1 ∂x22 ∂x22

!



∂v ∂v þ 3 3 ∂x2 ∂x1 2



2 ∂ v3 ∂ξ ∂ v3 þ þ2 2 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2

! þ

∂ξ1 ∂x2 2



2 ∂ v3 ∂ v3 þ ∂x22 ∂x22

! þ

!

!

4.5 Angle-Ply Laminated Sandwich Shells

109

! !  2   2  2  2 2 ∂ v3 ∂ v3 ∂v3 ∂ v3 ∂v3 ∂v3 ∂ v3 þ2 þ þ ∂x2 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 " !#)  2   2  ∂ v3 1 ∂ξ2 ∂ξ1 ∂v3 ∂v3 ∂v3 ∂v3 ∂v3 ∂v3 ∂ v3 þ þ þ þ  2v3 þ þ R2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 4

F 11

4

4

4

4

∂ v3 ∂ v ∂ v3 ∂ v ∂ v3  2ðF 12 þ2F 66 Þ 2 3 2  F 22  4F 16 3 3  4F 26 ∂x41 ∂x1 ∂x2 ∂x42 ∂x1 ∂x2 ∂x1 ∂x32

     2    2 h 1 ∂η1 C 1 ∂ v3 h þ2hK G13 1 þ þ 1þ  þ 2hK G23 1 þ 2h 2h h ∂x1 h ∂x21 !    2  2  2 1 ∂η2 C 1 ∂ v3 ∂ v3 ∂ v3 1 T þ 1þ þ þ  N 11 ∂x21 ∂x21 R1 h ∂x2 h ∂x22 2

 N T12

N m 11

2

!

2



N T22



2 ∂ v3 ∂ v3 1 þ þ ∂x22 ∂x22 R2

!

! !  2 2  2  2 ∂ v3 ∂ v3 ∂ v3 1 ∂ v3 m þ þ  N 12 þ ∂x1 ∂x2 ∂x1 ∂x2 ∂x21 ∂x21 R1 2

N m 22



2 ∂ v3 ∂ v3 þ ∂x1 ∂x2 ∂x1 ∂x2



2 ∂ v3 ∂ v3 1 þ þ ∂x22 ∂x22 R2

!  q3 ¼ 0

ð4:65eÞ

• Boundary Conditions and the associated boundary conditions along the edges xn ¼ const(n ¼ 1, 2) are e nn N nn ¼ N e N nt ¼ N nt Lnn ¼ e Lnn

or or

Lnt ¼ e Lnt

or

or

ξn ¼ e ξn e ξt ¼ ξt ηn ¼ e ηn

ηt ¼ e ηt ∂v3 ∂ev3 e nn M nn ¼ M or ¼ ∂xn ∂xn     ∂v3 ∂v3 ∂v3 ∂v3 ∂M nn ∂M nt N nt þ þ þ2 þ N nn þ ∂xt ∂xt ∂xn ∂xn ∂xn ∂xt e  ∂M nt e þ a=h N n3 ¼ þ N n3 ∂xt

or

v3 ¼ ev3

ð4:66a  fÞ

110

4 Post-Buckling

As was assumed previously, simply supported boundary conditions of Type A are assumed and restated as: Along the edges xn ¼ 0, Ln e nn , N nn ¼ N

N nt ¼ 0,

ηn ¼ 0,

ηt ¼ 0,

M nn ¼ 0,

v3 ¼ 0

ð4:67a  fÞ

4.5.2

Solution Methodology

Equations (4.65a–e) and (4.67a–f) constitute the governing system of equations concerning the post-buckling problem for doubly curved sandwich panels with angle-ply laminated facings and a weak incompressible core. The method of approach will be the same as in the previous section. As previously the inertia and thermal terms have been discarded. The assumed forms for transversal deflection and the geometric imperfection representations are represented by the following expressions 

v3 v∘3



 ¼

wmn w∘mn

 sin λm x1 sin μn x2

ð4.68a, bÞ

additionally, the first two equations of equilibrium, Eqs. (4.65a and 4.65b) can be fulfilled by assuming ξ1 and ξ2 in the following form which is identical to the previously assumed form in the previous section. The difference will appear in the expressions for the coefficients. (

ξ1 ðx1 , x2 Þ ξ2 ðx1 , x2 Þ

(

) ¼

(

1Þ F ðmn 1Þ Gðmn 4Þ F ðmn

)

( sin 2λm x1 þ

)

2Þ F ðmn 2Þ Gðmn

( sin 2μn x2 þ

(

5Þ F ðmn

)

3Þ F ðmn 3Þ Gðmn

) sin λm x1 cos μn x2 þ

sin 2λm x1 cos 2μn x2 þ 5Þ Gðmn ( ð6Þ ) ( ð7Þ ) ( ð8Þ ) F mn F mn F mn cos 2λm x1 sin 2μn x2 þ x1 þ x2 ð6Þ ð7Þ 8Þ Gmn Gmn Gðmn 4Þ Gðmn

cos λm x1 sin μn x2 þ

)

ð4.69a, bÞ Substituting the expressions for v3 , v∘3 , ξ1 , and ξ2, from Eqs. (4.68a, b) and (4.69a, b), into Eqs. (4.65a and 4.65b) and comparing coefficients of like trigono1Þ 6Þ 1Þ 6Þ metric functions provides F ðmn  F ðmn and Gðmn  Gðmn as

4.5 Angle-Ply Laminated Sandwich Shells



111

  ðiÞ ðiÞ ð5Þ ð6Þ  iÞ iÞ 5Þ 6Þ e mn , F e mn w2mn þ 2wmn w∘mn , e mn , G e mn , G F ðmn , Gðmn , F ðmn , Gðmn ¼ F



F ðmnjÞ , GðmnjÞ





 e ð jÞ wmn , e ðmnjÞ , G ¼ F mn

ði ¼ 1, 2Þ

ð4:70aÞ ð j ¼ 3, 4Þ

ð4:70bÞ

6Þ 5Þ F ðmn ¼ Gðmn ¼0

ð4:70cÞ

where A216  A11 A66 λ2m þ ðA12 A66  A16 A26 Þμ2n ðA16 A22  A26 A12 Þλ2m 2Þ e ðmn   , F ¼ ¼ 16λm A11 A66  A216 16μn A22 A66  A226  2 A26  A22 A66 μ2n þ ðA12 A66  A16 A26 Þλ2m ð1Þ ð2Þ ðA26 A11  A16 A12 Þμ2n e e  , Gmn ¼  Gmn ¼ 16λm A11 A66  A216 16μn A22 A66  A226 5Þ λ e ð6Þ ¼ μn e ðmn F ¼ m, G mn 16 16 ð4:71a  fÞ 1Þ e ðmn F



3Þ 4Þ 3Þ 4Þ The constants F ðmn , F ðmn , Gðmn , Gðmn are determined from the following matrix equation.

2 6 6 6 6 6 4

ðm,nÞ

R11

ðm,nÞ

R12

ðm,nÞ

R11

ðm,nÞ

R13

ðm,nÞ

R14

ðm,nÞ

R33 Symm:

3

8 ð3Þ 9 8 ð1Þ 9 Smn > > > > > > F mn > > 7 > > > ðm,nÞ 7> = < < 2Þ = ð 4 Þ R13 7 F mn Sðmn ¼ 7 ð3Þ ðm,nÞ > Gð3Þ > > > > > mn > > Smn > R34 7 5> ; : ð4Þ ; > : ð4Þ > ðm,nÞ G S mn mn R ðm,nÞ

R14

ð4:72Þ

33

where ðm,nÞ

 ¼  λ2m A11 þ μ2n A66 ,

ðm,nÞ

ðm,nÞ

¼ 2λm μn A16 , R13 ¼ λ2m A16 þ μ2n A26  ðm,nÞ ðm,nÞ ðm,nÞ R14 ¼ λm μn ðA12 þ A66 Þ, R33 ¼  μ2n A22 þ λ2m A66 , R34 ¼ 2λm μn A26     A A A A 1Þ 2Þ Sðmn ¼ μn 16 þ 26 wmn , Sðmn ¼ λm 11 þ 12 wmn R1 R2 R1 R2     A12 A22 A16 A26 ð3Þ ð2Þ Smn ¼ μn þ Smn ¼ λm þ wmn , wmn R1 R2 R1 R2 R11

R12

ð4:73a  jÞ From Eq. (4.72), Using Cramer’s rule, the coefficients F ðmnjÞ , GðmnjÞ be expressed as

ð j ¼ 3, 4Þ can

112

4 Post-Buckling



  ð3Þ  4Þ 3Þ 4Þ 3Þ 4Þ e ð3Þ , G e ð4Þ wmn e ðmn e mn , F F ðmn , F ðmn , Gðmn , Gðmn , G ¼ F mn mn

ð4:74Þ

where e ð3Þ ¼ detðR1 Þ , F mn detðRÞ ¼

e ð4Þ ¼ detðR2 Þ , F mn detðRÞ

e ð3Þ ¼ detðR3 Þ , G mn detðRÞ

e ð4Þ G mn

detðR4 Þ detðRÞ

ð4:75a  dÞ

and 0

3Þ F ðmn

B B F ð4Þ B R1 ¼ B mn B Gð3Þ @ mn 4Þ Gðmn

ðm:nÞ

ðm,nÞ

L12

L13

ðm,nÞ

ðm,nÞ

L11

L14

ðm,nÞ

ðm,nÞ

L14

L33

ðm,nÞ

ðm,nÞ

L13

L34

ðm,nÞ

L14

0

1

C C C C, ðm,nÞ L34 C A ðm,nÞ L33 ðm,nÞ

L13

ðm,nÞ

L11

B ðm:nÞ BL B R2 ¼ B 12 B Lðm,nÞ @ 13 ðm,nÞ L14

ðm,nÞ

3Þ F ðmn

L13

4Þ F ðmn

L14

3Þ Gðmn

L33

4Þ Gðmn

L34

ðm,nÞ ðm,nÞ ðm,nÞ

ðm,nÞ

L14

1

C C C C ðm,nÞ L34 C A ðm,nÞ L33 ðm,nÞ

L13

ð4.76a, bÞ 0

ðm,nÞ

L11

B ðm:nÞ BL B R3 ¼ B 12 B Lðm,nÞ @ 13 ðm,nÞ L14

ðm:nÞ

3Þ F ðmn

L14

ðm,nÞ

4Þ F ðmn

L13

ðm,nÞ

3Þ Gðmn

ðm,nÞ

4Þ Gðmn

L12

L11 L14 L13

ðm,nÞ

0

1

C C C C, ðm,nÞ L34 C A ðm,nÞ L33 ðm,nÞ

ðm,nÞ

L11

B ðm:nÞ BL B R4 ¼ B 12 B Lðm,nÞ @ 13 ðm,nÞ L14

ðm:nÞ

L12

ðm,nÞ

L11

ðm,nÞ

L14

ðm,nÞ

L13

ðm,nÞ

L13

ðm,nÞ

L14

ðm,nÞ

L33

ðm,nÞ

L34

3Þ F ðmn

1

C 4Þ C F ðmn C C 3Þ C Gðmn A ð4Þ Gmn

ð4.76c, dÞ 0 B B B R¼B B @

ðm,nÞ

L11

ðm:nÞ

ðm,nÞ

L12

L13

ðm,nÞ

ðm,nÞ

L11

L14

ðm,nÞ

L33 Symm:

ðm,nÞ

L14

1

C C C C ðm,nÞ C L34 A ðm,nÞ L33 ðm,nÞ

L13

ð4:76eÞ

7Þ 8Þ 7Þ 8Þ Finally, F ðmn , F ðmn , Gðmn , Gðmn are four constants which remain undetermined. These are determined as before from enforcing the in-plane static boundary conditions which are expressed mathematically as

0

Lt

e nn Lt N nn dxt ¼ N



n ¼ 1, 2 t ¼ 2, 1



X n, t



Z

ð4:77Þ

4.5 Angle-Ply Laminated Sandwich Shells

Z

Lt

113

N nt dxt ¼ 0

ð4:78Þ

0

These conditions fulfill the boundary conditions, Eqs. (4.67a, b) in an average sense. To apply these conditions, N11, N22, and N12 are expressed in terms of displacements by Making use of Eqs. (2.86a) and (2.92a) coupled with the strain– displacement relationships, Eqs. (2.53)–(2.55) and Eqs. (2.69)–(2.71) gives the global stress resultants in terms of displacements as (

N 11

)  2 ∂v∘3 ∂v3 v3 ∂ξ1 1 ∂v3 ¼ A11 þ þ  ∂x1 2 ∂x1 ∂x1 ∂x1 R1 ( )  2 ∂v∘3 ∂v3 v3 ∂ξ2 1 ∂v3 þ A12 þ þ  ∂x2 2 ∂x2 ∂x2 ∂x2 R2    ∂ξ1 ∂ξ2 ∂v3 ∂v3 ∂v3 ∂v3 ∂v3 ∂v3 þ A16 þ þ þ þ ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 (

N 22

)  2 ∂v∘3 ∂v3 v3 ∂ξ2 1 ∂v3 ¼ A22 þ þ  ∂x2 2 ∂x2 ∂x2 ∂x2 R2 ( )  2 ∂v∘3 ∂v3 v3 ∂ξ1 1 ∂v3 þA12 þ þ  ∂x1 2 ∂x1 ∂x1 ∂x1 R1    ∂ξ2 ∂ξ1 ∂v3 ∂v3 ∂v3 ∂v3 ∂v3 ∂v3 þA26 þ þ þ þ ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1

ð4:79aÞ

(

N 12

)  2 ∂v∘3 ∂v3 v3 ∂ξ1 1 ∂v3 ¼ A16 þ þ  ∂x1 2 ∂x1 ∂x1 ∂x1 R1 ( )  2 ∂v∘3 ∂v3 v3 ∂ξ2 1 ∂v3 þA26 þ þ  ∂x2 2 ∂x2 ∂x2 ∂x2 R2   ∂ξ1 ∂ξ2 ∂v3 ∂v3 ∂v∘3 ∂v3 ∂v3 ∂v∘3 þA66 þ þ þ þ ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2

ð4:79bÞ

ð4:79cÞ

Substituting the expressions for v3 , v∘3 , ξ1 , and ξ2 into Eqs. (4.79a, 4.79b, and 4.79c) then substituting the result into the requirements for the in-plane static boundary conditions, Eqs. (4.77) and (4.78) and carrying out the indicated opera7Þ 8Þ 7Þ 8Þ tions provides the relationships for the constants F ðmn , F ðmn , Gðmn , Gðmn as

114

4 Post-Buckling

A2  A22 A66 λ2m  2 e 11 wmn þ 2wmn w∘mn þ 26 N 8 Ω A A  A16 A26 e þ 12 66 N 22 Ω A A  A16 A26 μ2  e 11 ¼  n w2mn þ 2wmn w∘mn þ 12 66 N 8 Ω A2  A11 A66 e þ 16 N 22 Ω 7Þ F ðmn ¼

8Þ Gðmn

ð4:80aÞ

ð4:80bÞ

where Ω ¼ A11 A22 A66  A212 A66 þ 2A12 A16 A26  A11 A226  A22 A216

ð4:80cÞ

7Þ 8Þ F ðmn and Gðmn are arbitrary and have no effect on the final post-buckling solution. These expressions are identical in the case of sandwich plates. With the first two equilibrium equations fulfilled, the third and fourth equations of equilibrium will be addressed in a similar manner to determine the expressions for η1 and η2 which were addressed in Chap. 3 (see Chap. 3, Eqs. (3.12a, b)-(3.17a–e)). Continuing in this manner with the first four equations of equilibrium and the sixth boundary condition fulfilled, the remaining unfulfilled equilibrium equation and boundary conditions will be retained in Hamilton’s energy functional resulting in

! !  2 2  2  2  2 ∂ξ1 ∂ v3 ∂ v3 1 ∂v3 ∂ v3 ∂ v3 ∂v3 ∂v3 A11 þ þ þ þ 2 ∂x1 ∂x1 ∂x21 ∂x1 ∂x1 ∂x21 ∂x21 ∂x21 t0 0 0 ! " ! #)  2  2  2  2 2 ∂ v3 ∂ v3 1 ∂ξ1 1 ∂v3 ∂v3 ∂v3 ∂ v3 ∂ v3 v3 þ þ  v3 þ þ þ  R1 ∂x1 2 ∂x1 R1 ∂x1 ∂x1 ∂x21 ∂x21 ∂x21 ∂x21 ( ! ! !  2  2 2  2  2  2 2 ∂ξ1 ∂ v3 ∂ v3 ∂ξ2 ∂ v3 ∂ v3 1 ∂v3 ∂ v3 ∂ v3 þ þ þ þ A12 þ þ 2 ∂x1 ∂x1 ∂x22 ∂x22 ∂x2 ∂x21 ∂x21 ∂x22 ∂x22 ! ! !  2 2   2  2  2  2 2 1 ∂v3 ∂ v3 ∂ v3 ∂v3 ∂v3 ∂ v3 ∂ v3 ∂v3 ∂v3 ∂ v3 ∂ v3 þ þ þ þ þ þ 2 ∂x2 ∂x1 ∂x1 ∂x22 ∂x2 ∂x2 ∂x21 ∂x21 ∂x21 ∂x22 ∂x21 ! # "  2  2  2 1 ∂ξ2 1 ∂v3 ∂v3 ∂v3 ∂ v3 ∂ v3 v3  þ þ v3 þ þ R1 ∂x2 2 ∂x2 R2 ∂x2 ∂x2 ∂x22 ∂x22 " ! #)  2  2  2 1 ∂ξ1 1 ∂v3 ∂v3 ∂v3 ∂ v3 ∂ v3 v3 þ  v3 þ þ þ  R2 ∂x1 2 ∂x1 R1 ∂x1 ∂x1 ∂x21 ∂x21 ( ! ! !  2 2  2  2  2  2 2 ∂ξ2 ∂ v3 ∂ v3 1 ∂v3 ∂ v3 ∂ v3 ∂v3 ∂v3 ∂ v3 ∂ v3 þ þ þ A22 þ þ þ 2 ∂x2 ∂x2 ∂x22 ∂x2 ∂x2 ∂x22 ∂x22 ∂x22 ∂x22 ∂x22

Z

t1

*Z

l1 Z l2

"

(

4.5 Angle-Ply Laminated Sandwich Shells

115

" ! #)  2  2  2 1 ∂ξ2 1 ∂v3 ∂v3 ∂v3 ∂ v3 ∂ v3 v3 þ þ  v3 þ þ  R2 ∂x2 2 ∂x2 R2 ∂x2 ∂x2 ∂x22 ∂x22 ! ! ( 2  2  2 2 ∂ v3 ∂ v3 ∂ξ1 ∂ v3 ∂ξ2 ∂ v3 ∂v ∂v þ þ 3 3 þ2A66 þ þ ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 

2

2 ∂ v3 ∂ v3 þ ∂x1 ∂x2 ∂x1 ∂x2 2



!

2 ∂ v3 ∂ v3 þ ∂x1 ∂x2 ∂x1 ∂x2



∂v ∂v þ 3 3 ∂x1 ∂x2

!)

( þ A16

2



2 ∂ v3 ∂ v3 þ ∂x1 ∂x2 ∂x1 ∂x2 2



2 ∂ξ1 ∂ v3 ∂ v3 þ ∂x2 ∂x21 ∂x21

!

!



þ

∂v3 ∂v3 ∂x1 ∂x2 2



2 ∂ v3 ∂ v3 þ ∂x21 ∂x21

∂ξ þ 2 ∂x1

!

! ! !   2  2  2  2 2 ∂v3 ∂v3 ∂2 v3 ∂ v3 ∂v3 ∂v3 ∂ v3 ∂ v3 ∂v3 ∂v3 ∂ v3 ∂ v3 þ þ þ þ þ þ ∂x1 ∂x2 ∂x21 ∂x1 ∂x2 ∂x21 ∂x1 ∂x2 ∂x21 ∂x21 ∂x21 ∂x21 !   !  2  2  2 2 2 ∂ v3 ∂ v3 ∂ξ1 ∂ v3 ∂v3 ∂ v3 ∂v3 ∂v3 þ2 þ þ þ þ2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x1 

2



2 ∂ v3 ∂ v3 þ ∂x1 ∂x2 ∂x1 ∂x2

!

"   1 ∂ξ1 ∂ξ2 ∂v3 ∂v3 ∂v3 ∂v3 ∂v3 ∂v3 þ þ þ þ þ R1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2

! ! 2  2  2 2 ∂ξ2 ∂ v3 ∂ v3 ∂ξ1 ∂ v3 ∂ v3 2v3 þ þ þ A26 þ ∂x1 ∂x22 ∂x2 ∂x22 ∂x22 ∂x22 !  ! !  2  2  2  2 2 2 ∂v3 ∂v3 ∂ v3 ∂ v3 ∂v3 ∂v3 ∂ v3 ∂ v3 ∂v3 ∂v3 ∂ v3 ∂ v3 þ þ þ þ þ ∂x2 ∂x1 ∂x22 ∂x2 ∂x1 ∂x22 ∂x22 ∂x2 ∂x1 ∂x22 ∂x22 ∂x22 !   !  2  2  2 2 2 ∂ v3 ∂ v3 ∂ξ2 ∂ v3 ∂v3 ∂ v3 ∂v ∂v þ2 þ þ þ þ2 3 3 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x2 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x2 2



2 ∂ v3 ∂ v3 þ ∂x1 ∂x2 ∂x1 ∂x2

2



2 ∂ v3 ∂ v3 þ ∂x1 ∂x2 ∂x1 ∂x2

!

2

2v3

!#)

(

"   1 ∂ξ2 ∂ξ1 ∂v3 ∂v3 ∂v3 ∂v3 ∂v3 ∂v3 þ þ þ þ þ R2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x1 

2 ∂ v3 ∂ v3 þ ∂x1 ∂x2 ∂x1 ∂x2

!#)

4

 F 11

4

4

∂ v3 ∂ v ∂ v3  2ðF 12 þ2F 66 Þ 2 3 2  F 22 ∂x41 ∂x1 ∂x2 ∂x42

    2  4 4 2 ∂ v3 ∂ v3 h 1 ∂η1 C 1 ∂ v3 4F 16 3  4F 26 þ 2hK G13 1 þ þ 1þ ∂x1 ∂x2 ∂x1 ∂x32 2h h ∂x1 h ∂x21     2    2 h 1 ∂η2 C 1 ∂ v3 þ2hK G23 1 þ þ 1þ  q3 δv3 dx1 dx2 dt 2h h ∂x2 h ∂x22

116

4 Post-Buckling

   ∂v3  l1 e dx2 dt þ N 11 þ N 11 δξ1 þ N 12 δξ2 þ L11 δη1 þ L12 δη2 þ M 11 δ ∂x1  0 t0 0    Z t1Z l1 l  e 22 δξ2 þ N 12 δξ1 þ L22 δη2 þ L12 δη1 þ M 22 δ ∂v3  2 dx1 dt þ N 22 þ N ∂x2  0 t0 0 Z t1Z

l2 

¼0

ð4:81Þ

Substituting the expressions for v3 , v∘3 , ξ1 , ξ2 , η1 , η2 , into Eq. (4.81) and carrying out the indicated integrations results in the governing post-buckling solution for angle-ply laminated doubly curved sandwich structures resulting in a nonlinear algebraic equation expressed in terms of the modal amplitudes as  2    2 5Þ 4Þ 3Þ Pðmn wmn þ 2wmn w∘mn wmn þ w∘mn þ Pðmn wmn þ w∘mn wmn þ Pðmn wmn þ 2wmn w∘mn þ  2Þ 1Þ 0Þ wmn þ w∘mn þ Pðmn wmn þ Pðmn þ qmn ¼ 0 Pðmn

ð4:82Þ

where   1Þ 2Þ e ð1Þ þ N 54 G e ð2Þ e ðmn e ðmn þ N 52 F þ N 53 G Pmn ¼  N 51 F mn mn 4Þ Pðmn ¼

ð4:83Þ

  16Δm 3Þ 4Þ n e ð3Þ þ N 44 G e ð4Þ þ e ðmn e ðmn N 40 þ N 41 F þ N 42 F þ N 43 G mn mn 9λm μn L1 L2   ð1Þ   ð2Þ  ð6Þ 16Δm 2Þ ð5Þ n e ð1Þ þ N 47 F e e e ðmn e e G N 45 F þN 46 G þ F þ G þ N þ 48 mn mn mn mn mn 3λm μn L1 L2 ð4:84Þ

3Þ Pðmn ¼

  8Δm 1Þ 2Þ n e ð1Þ þ N 34 G e ð2Þ e ðmn e ðmn N 31 F þ N 32 F þ N 33 G mn mn 3λm μn L1 L2   m 4Δn 3Þ 4Þ e ð3Þ þ N 38 G e ð 4Þ e ðmn e ðmn þ N 35 F þ N 36 F þ N 37 G mn mn λm μn L1 L2

ð4:85Þ

 2Þ Pðmn ¼  λ2m N 11 þ μ2n N 22

ð4:86Þ

     3Þ A11 A12 e ð4Þ A A e ð3Þ þ μn A16 þ A26 F e ðmn þ F mn þ μn 12 þ 22 G mn R1 R2 R1 R2 R1 R2     A16 A26 e ð4Þ A11 A22 2A12 þλm þ þ 2 þ Gmn þ þ F 11 λ4m þ 2ðF 12 þ 2F 66 Þλ2m μ2n R1 R2 R21 R2 R1 R2  ð1Þ   2Þ  e mn þ aλm þ d 2 aμn eI ðmn þF 22 μ4n þ d 1 aλm H þ aμn

1Þ Pðmn ¼ λm



ð4:87Þ 0Þ ¼ Pðmn

 4Δm n e22 N e11 N e 11 þ P e 22 P λm μn L1 L2

ð4:88Þ

4.5 Angle-Ply Laminated Sandwich Shells

117

    2   A26  A22 A66 A11 A12 A12 A22 A12 A66  A16 A66 e P11 ¼ þ þ þ  R1 R2 Ω R1 R2 Ω     2 2  A11 A226  A11 A22 A66 A12 A66  A12 A16 A26 4Þ e ðmn λm 1 þ þ þ F Ω Ω   A A2  A A A  ð3Þ A22 A12 A66  A22 A16 A26 12 26 12 22 66 e þ G μn þ mn Ω Ω        A A2  A A A A26 A12 A66  A16 A226 ð3Þ 16 26 16 22 66 e ð4Þ þ μn F e λm G þ mn mn Ω Ω ð4:89Þ e22 ¼ P

    2   A16  A11 A66 A11 A12 A12 A66  A16 A26 A12 A22 þ  þ þ R1 R2 Ω R1 R2 Ω   A A2  A A A  ð4Þ A11 A12 A66  A11 A16 A26 12 16 12 11 66 e mn þ F λm þ Ω Ω     2  A22 A216  A22 A11 A66 A12 A66  A12 A16 A26 e ð3Þ þ G μn 1 þ þ mn Ω Ω    2    2 ð4Þ A16 A12 A66  A16 A26 A16 A26  A11 A26 A66 ð3Þ e e λm Gmn þ μn F mn þ Ω Ω ð4:90Þ

and N 51 ¼ λ3m A11 þ λm μ2n A12 ,

N 52 ¼ λ2m μn A16 þ μ3n A26 ,

N 53 ¼ λ3m A16 þ λm μ2n A26

N 54 ¼ μ3n A22 þ λ2m μn A12

  3Þ 4Þ e ð4Þ þ λm μn F e ð3Þ A66 e ðmn e ðmn N 47 ¼ N 48 ¼ λ2m A16 F þ λm μn A26 G þ λ2m G mn mn   ð3Þ ð4Þ 2 e ð3Þ e ð4Þ þ λ2 A12 F e e F G N 46 ¼ λm μn A22 G þ λ μ þ λ m n mn m m mn A26 mn mn   3Þ 4Þ e ð4Þ þ λm μn F e ð3Þ A16 e ðmn e ðmn N 45 ¼ λ2m A11 F þ λm μn A12 G þ λ2m G mn mn 3 N 44 ¼ λ3m A16 þ λm μ2n A26 , 2 N 42 N 40 N 38

1 N 43 ¼ μ3n A22 þ λ2m μn A12 þ λ2m μn A66 2 1 3 3 2 2 3 ¼ λm A11 þ λm μn A12 þ λm μn A66 , N 41 ¼ μn A26 þ λ2m μn A16 2 2     A A A A ¼ λ2m 11 þ 12 þ μ2n 22 þ 12 R1 R2 R2 R1 (   ð2Þ   5Þ 2Þ 1 e þG e ð6Þ þ 2μn A66 F e ðmn e ðmn 2λm A16 F ¼ λm þ 2μn A26 G  λ2m A16 mn mn 3 ) 2 2 ð1Þ A16 λm A μ ð1Þ 26 n e e mn  2λm A66 Gmn  2λm A16 F  8 8

118

N 37

N 36

N 35

N 34 N 32 N 31

4 Post-Buckling

   ð1Þ   ð5Þ 1 e ð6Þ þ 2λm A12 F e ð1Þ  μ2 A22 e e G ¼ μn þ F A 2μn A22 G þ 2λ m 26 mn mn mn mn n 3  A22 μ2n A12 λ2m ð2Þ e ð2Þ  2μn A26 F e þ þ 2μn A22 G mn mn 8 8    ð2Þ   5Þ ð2Þ 1 2 e þG e ð6Þ þ 2μn A16 F e ðmn e 2λm A11 F ¼ λm þ 2μn A12 G  λ A 11 mn mn mn m 3  ð 1Þ A11 λ2m A12 μ2n ð1Þ e e 2λm A11 F mn  2λm A16 Gmn þ þ 8 8  ð1Þ  o 5Þ μ n e ð6Þ þ 2λm A66 G e ð1Þ  μ2 A26 e ðmn e mn þ F ¼ n 2λm A16 F þ 2μn A26 G mn mn n 3    A16 λ2m A26 μ2n ð2Þ e ð2Þ þ A66 F e þ þ 2μn A26 G μn mn mn 8 8   A12 A22 ¼ μn þ , N 22 ¼ N 12 ¼ η1 ¼ η2 ¼ M 22 ¼ v3 ¼ 0, R1 R2   A A ¼ μn 16 þ 26 R1 R2     A11 A12 A16 A26 ¼ λm þ þ , N 33 ¼ λm ð4:91a  tÞ R1 R2 R1 R2

To determine the post-buckling behavior, Eq. (4.82) is put in a more suitable form which is conducive to generating results. Algebraically Eq. (4.82) can be put in the following form.     ∘ 2 5Þ 3 5Þ ∘ 4Þ 3Þ 5Þ 4Þ ∘ 3Þ ∘ wmn þ 3Pðmn wmn þ Pðmn þ Pðmn wmn þ Pðmn wmn þ 2Pðmn wmn þ w2mn þ 2Pðmn Pðmn  2Þ 1Þ 2Þ ∘ 0Þ Pðmn þ Pðmn wmn þ Pðmn þ qmn ¼ 0 wmn þ Pðmn ð4:92Þ which can be further expressed as L1 w3mn þ L2 w2mn þ L3 wmn þ L4 ¼ 0

ð4:93Þ

where 5Þ L1 ¼ Pðmn

ð4:94aÞ

4.5 Angle-Ply Laminated Sandwich Shells 5Þ ∘ 4Þ 3Þ L2 ¼ 3Pðmn wmn þ Pðmn þ Pðmn

 ∘ 2 5Þ 4Þ ∘ 3Þ ∘ 2Þ 1Þ L3 ¼ 2Pðmn wmn þ Pðmn wmn þ 2Pðmn wmn þ Pðmn þ Pðmn 2Þ ∘ 0Þ wmn þ Pðmn þ qmn L4 ¼ Pðmn

119

ð4:94bÞ ð4:94cÞ ð4:94dÞ

Equation (4.93) can be solved via Newton’s method to solve for the various equilibrium configurations of an anisotropic laminated doubly curved sandwich panel with an incompressible (weak/soft) core. A more convenient form is to introduce the following Parameter δas δ ¼ wmn þ w∘mn ) wmn ¼ δ  w∘mn

ð4:95Þ

Substituting Eq. (4.95) into Eq. (4.93) gives  3  2  L1 δ  w∘mn þ L2 δ  w∘mn þ L3 δ  w∘mn þ L4 ¼ 0

ð4:96Þ

Expanding, combining like terms, and simplifying gives    2   3 L1 δ3mn þ L2  3L1 w∘mn δ2mn þ L3  2L2 w∘mn  L1 w∘mn δmn þ L4  L1 w∘mn þ  2 L2 w∘mn  L3 w∘mn ¼ 0 ð4:97Þ Solving Eq. (4.97) via Newton’s method gives the compressive uniaxial load vs δmn (amplitude of deflection plus amplitude of imperfection).

4.5.3

Numerical Results and Discussion

With the theory in hand, several results are presented to better understand the effect of both the geometrical and material parameters on the post-buckling behavior of sandwich panels. For all of the numerical simulations, unless otherwise specified, Tables 4.5 and 4.6 contain the geometrical and material properties for the face sheets and the core for each of the figures. In addition, unless otherwise specified for the following numerical illustrations (m ¼ n ¼ 1). Figure 4.11 depicts the effect of the geometric imperfections on the post-buckling behavior of a cylindrical sandwich panel. This figure reveals all of the stable and unstable directional loading paths that are theoretically possible for both the primary and secondary branches. The effect of the geometric imperfections shows the realistic direction the loading would take for compressive edge loading. This figure also shows that the larger imperfections whether negative or positive cause the

120

4 Post-Buckling

Table 4.5 The geometrical and material properties for the face sheets hf (mm) 0.02

L1 (mm) 609.6 (24in)

L2 (mm) 609.6 (24in)

E1 (GPa) 180.987

E2 (GPa) 10.342

G12 (GPa) 7.239

ν12 0.28

Table 4.6 The geometrical material properties for the core hin (mm) 12.7 (0.5in)

G13 (MPa) 1.437

G23 (MPa) 0.651

Fig. 4.11 The effect of geometric imperfections on the compressive uniaxial edge loading vs. amplitude of deflection for a four-layered symmetrical cylindrical sandwich panel

loading path proceed away from the ideal or perfect loading path shown as a solid line. The smaller the geometric imperfection, the closer the loading path is to the ideal or perfect one. Figure 4.12 displays the counterpart of Fig. 4.11 in the compressive load-end shortening plane where the stable and unstable paths are displayed, while the presence of geometric imperfections shows the realistic behavior of the structure under a compressive edge loading. In Fig. 4.13, the effect of curvature on the postbuckling behavior of a cylindrical sandwich is shown. The typical response is seen where an increase in curvature results in a marginally higher buckling bifurcation point with an additional possibility of a snap-through-type behavior. Figure 4.14 presents the effect of the curvature on the end-shortening of a cylindrical sandwich panel of the given stacking sequence in the facings. The typical one-to-one or linear behavior between the compressive edge loading and the end-shortening is seen up to the buckling bifurcation point beyond which varying amounts of snap-through behavior are seen depending on the amount of curvature

4.5 Angle-Ply Laminated Sandwich Shells

121

Fig. 4.12 The counterpart of Fig. 4.11 in the compressive edge load-end shortening plane for various geometric imperfections of a cylindrical sandwich panel

Fig. 4.13 The effect of the curvature on the compressive uniaxial edge loading vs. amplitude of deflection for the given layup of a cylindrical sandwich panel

present in the structure. Figure 4.15 highlights the effect of the ply angle for the given layup on the uniaxial compressive edge loading for a cylindrical sandwich panel. Similar behavior is seen as in the case of a flat panel. The buckling bifurcation rises with an increase of ply angle up to 45 degrees. Additionally, at a ply angle of 45 degrees there seems to be a diminishing capacity to sustain the compressive edge loading in contrast to the 0-degree layup where the load-carrying capacity of the

122

4 Post-Buckling

Fig. 4.14 The effect of imperfections on the end-shortening of a cylindrical sandwich panel for various curvatures under uniaxial compressive edge loading

Fig. 4.15 The effect of the ply angle on the compressive uniaxial edge loading vs. amplitude of deflection for a cylindrical sandwich panel

structure seems to rise beyond the buckling bifurcation point. In Fig. 4.16, the effects of the geometric imperfections on a doubly curved sandwich panel are shown. The results reveal that for the case of a doubly curved sandwich panel, there is extraordinarily little effect on the ideal compressive edge load-displacement interaction of the structure. There is only a marginal effect with an increase in the amount of geometric imperfections. Figure 4.17 depicts the effects of the various magnitudes of curvature on the amplitude of deflection for a doubly curved sandwich panel for the

4.5 Angle-Ply Laminated Sandwich Shells

123

Fig. 4.16 The effect of geometric imperfections on the compressive uniaxial edge loading vs. amplitude of deflection for a doubly curved sandwich panel

Fig. 4.17 The effect of the curvature on the compressive uniaxial edge loading vs. amplitude of deflection for a doubly curved sandwich panel

given layup. Considering the stable equilibrium paths, a marginal increase in the load-carrying capacity of the structure seems to follow an increase in curvature. Figure 4.18 illustrates the effect of the various ply angles for the given layup on the uniaxial compressive edge loading of a doubly curved sandwich panel. Along the stable loading direction, it is apparent that the ply angle configuration of 45 degrees provides the larger load-carrying capacity of the panel. In Fig. 4.19, the effect of the transverse pressure on the amplitude of deflection with an applied fixed-edge loading

124

4 Post-Buckling

Fig. 4.18 The effect of the ply angle on the compressive uniaxial edge loading vs. amplitude of deflection for a doubly curved sandwich panel

Fig. 4.19 Transverse pressure vs amplitude of deflection for various amounts of compressive uniaxial edge loading for cylindrical sandwich panel

as a percentage of the critical load for a cylindrical sandwich panel is displayed. With the transverse pressure loading present, as the compressive edge loading reaches within the vicinity of the critical load, the implications for the load-carrying capacity of the structure become more catastrophic. In Fig. 4.20, given a fixed compressive edge load, the implications of the transverse pressure on the amplitude of deflection are displayed. It appears that the construct with the larger curvature, from a structural standpoint, benefits from a larger load-carrying capacity.

4.6 Immovability of the Edges

125

Fig. 4.20 The effect of curvature on the transverse pressure vs. amplitude of deflection for a fixed amount of compressive edge loading of cylindrical sandwich panel

4.6

Immovability of the Edges

So far only simply supported boundary conditions with freely movable edges have been considered. Some situations arise where all four edges are not necessarily movable such as two edges freely movable, and two edges immovable while under compressive edge loading. In such situations, the immovable edges are prevented from moving. To prevent these edges from moving some type of fictious edge loading is present to prevent these edges from moving. For the case that two edges are immovable along the edges (x1 ¼ 0, L1) fulfillment of the following e 11 necessary to prevent the immovcondition can provide the fictitious edge load N able edges from moving. Z 0

L2 Z L1

ð∂ξ1 =∂x1 Þdx1 dx2 ¼ 0

ð4:98Þ

0

Substituting Eq. (4.69a) into Eq. (4.98) carrying out the indicated operations results in the fictitious edge load. Once this is determined it can be substituted into the post-buckling solution to determine the response for two immovable and two freely movable edges. In the case, where two edges are immovable along the edges (x2 ¼ 0, L2) the following condition needs to be fulfilled.

126

4 Post-Buckling

Z

L2 Z L1

0

ð∂ξ2 =∂x2 Þdx1 dx2 ¼ 0

ð4:99Þ

0

e 22 can be determined and then substituted into By which the fictitious edge load N the post-buckling solution

4.7

End-Shortening

As a result of uniaxial or biaxial compression, the edges in both the x1 and x2 direction have a tendency to displace inward or outward depending on the loaded edges. This amount of edge displacement can be determined from the following expressions. For displacement in the x1 direction, the end-shortening, Δ1 is determined from Δ1 ¼ 

1 L1 L2

Z 0

L2 Z L1  0

 ∂ξ1 dx1 dx2 ∂x1

ð4:100Þ

Substituting in the expression for ξ1 from Eq. (4.69a) and integrating gives a quantifiable expression for Δ1 as Δ1 ¼

e3 Δm F λ2m  2 w wmn þ 2wmn w∘mn þ n 8 μn L1 L2 mn  2  A16  A11 A66 A12 A66  A16 A26 e  N 11 þ LR Ω Ω

ð4:101Þ

For displacement in the x2 direction, the end-shortening is determined from 1 Δ2 ¼  L1 L2

Z 0

L2 Z L1  0

 ∂ξ2 dx1 dx2 ∂x2

ð4:102Þ

Substituting in the expression for ξ2 from Eq. (4.69b) and integrating gives   e Δm μ2n  2 A12 A66  A16 A26 A226  A22 A66 ∘ n G3 e w þ 2wmn wmn þ þ LR Δ2 ¼ w  N 11 8 mn λm L1 L2 mn Ω Ω

ð4:103Þ

References

4.8

127

Summary

A comprehensive theoretical base governing the post-buckling response of flat and doubly curved sandwich panels has been presented in sufficient detail considering the geometric imperfections and the anisotropy of the face sheets. The theory considered three parts. The first part was concerned with the special case of flat and doubly curved sandwich panels with cross-ply laminated facings, while the second and third part was concerned with flat and doubly curved sandwich panels with symmetrically laminated angle ply face sheets, respectively. In contrast to the latter parts, a solution technique referred to as the stress potential method was used to solve the governing equations in conjunction with the extended Galerkin method and Newton’s method. Following the theoretical developments, several results were presented considering the effects of curvature, geometric imperfections, panel face thickness, material directional properties (fiber orientation angles), biaxial edge loading, the transverse pressure and various stacking sequences and or layups. The results showed that all of these geometrical and material considerations play an important role in the post-buckling response of flat and doubly curved sandwich panels.

References Fulton, R. E. (1961). Non-linear equations for a shallow unsymmetric sandwich shell of double curvature. In Developments in mechanics, proceedings of 7th midwestern mechanics conference (pp. 365–380). New York: Plenum. Librescu, L. (1965). Aeroelasticity stability of orthotropic heterogeneous thin panels in the vicinity of the flutter critical panel. Journal de Mecanique, 4(1), 51–76. Librescu, L. (1975). Elastostatics and kinetics of anisotropic and heterogeneous shell-type structures. Leyden: Noordhoff International Publishing. Librescu, L., & Chang, M. J. (1992). Post-buckling and imperfection sensitivity of shear deformable composite doubly curved panels. International Journal of Solids and Structures, 29(9), 1065–1083. Seide, P. (1974). A reexamination of Koiter’s theory of initial post-buckling behavior and imperfection sensitivity of structures. In C. Y. Fung & E. E. Sechler (Eds.), Thin shell structures: Theory, experiment, and design (pp. 59–80). Englewood Cliffs: Prentice-Hall. Simitses, G. J. (1986). Buckling and post-buckling of imperfect cylindrical shells: A review. Applied Mechanics Reviews, 39(10), 1517–1524.

Chapter 5

Free Vibration

Abstract An advanced model of sandwich plates and shells considering anisotropic laminated composite face sheets with a weak orthotropic core, in the context of determining the eigenfrequencies, is presented. Within this advanced model, adoption of the shallow shell theory is adhered to. The influence of several geometrical and material characteristics of the sandwich panel are considered with regard to the eigenfrequencies. The influence of these characteristics such as the panel geometry, the ply angle and stacking sequence of the face sheets, the orthotropicity ratio of the core, the panel face thickness, the aspect ratio, etc. are all determined to have an effect on the eigenfrequencies of the sandwich panel. Due to the nature of the governing equations, a closed form solution is obtainable with reasonable results. Several validations are made with several prominent author’s results found within the literature. With this is hand, appropriate conclusions are made.

5.1

Introduction

This chapter is concerned with the free vibration behavior of doubly curved sandwich panels. This study is carried out in the framework of an advanced sandwich model, utilizing the linear equations from Chap. 2, which will allow for a closedform solution despite the intricacy of the governing equations. The anisotropy of the face sheets, the stacking sequence, the fiber orientation, orthotropic properties of the core as well as other important geometrical and material properties of both the facings and the core are analyzed with respect to the effects on the eigenfrequencies of the structure. Numerical solutions and validations are presented which show the effects of the above-mentioned geometrical and material effects on the eigenfrequencies of the structure in a detailed optimized fashion through the use of the structural tailoring technique. Finally, this analysis is carried out in two parts. The first part is concerned with flat sandwich panels, while the second part addresses the analysis of doubly curved sandwich shells.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 T. J. Hause, Sandwich Structures: Theory and Responses, https://doi.org/10.1007/978-3-030-71895-4_5

129

130

5 Free Vibration

5.2

Preliminaries and Basic Assumptions

In consideration of the free or natural vibration of sandwich structures, the incompressible core case will be considered for both the plate and shell configuration. With this in mind, the following basic assumptions are adopted: 1. The face sheets are orthotropic layers not necessarily coincident with the geometrical axes. 2. The core features orthotropic properties in the transverse direction and is considered the weak-type and much larger in thickness than the facings. 3. Perfect bonding between the face sheets and the facings and the core are assumed. 4. The shallow shell theory is assumed. 5. The transverse shear effects in the facings are discarded. 6. The face sheets are symmetric with respect to their local and global mid-surfaces. 7. The tangential and rotatory inertia terms are neglected. (Librescu 1987)

5.3

Flat Sandwich Panels

5.3.1

Governing System

The governing equations which apply to the free vibration problem of flat sandwich panels were developed in Chap. 2. The equations of interest are based on the linear theory for plates with symmetric laminated facings, a weak orthotropic core, and with the transverse shear effects discarded in the facings. Under these conditions, Eqs. (2.174)–(2.176) apply along with the set of simply supported boundary conditions with freely movable edges, from Eqs. (2.177a–d). This set of governing equations are the linearized counterpart of Eqs. (2.156)–(2.160). Also, as mentioned in Chap. 2, for flat panels, setting the curvatures to zero decouples the first two equations of motion from the last three. The last three equations of motion govern the bending problem separate from the stretching problem. This implies that, in the case of flat sandwich plates, the eigenfrequencies are only dependent on the bending problem. The governing system of equations for flat sandwich panels are given as • Equations of Motion
2 
2  2 2 2 2 ∂ η1 ∂ η1 ∂ η2 ∂ η2 ∂ η2 ∂ η1 þ A þ þ A þ 2 þ A 66 12 16 ∂x1 ∂x2 ∂x1 ∂x2 ∂x21 ∂x22 ∂x1 ∂x2 ∂x21
 2 ∂ η2 ∂v þA26  d 1 η1 þ a 3 ¼ 0 ∂x1 ∂x22 A11

ð5:1aÞ

5.3 Flat Sandwich Panels

131


2 
2  2 2 2 2 ∂ η2 ∂ η2 ∂ η1 ∂ η1 ∂ η1 ∂ η2 þ A þ þ A þ 2 þ A 66 12 26 ∂x1 ∂x2 ∂x1 ∂x2 ∂x22 ∂x22 ∂x1 ∂x2 ∂x22
 2 ∂ η1 ∂v þA16  d 2 η2 þ a 3 ¼ 0 ∂x2 ∂x21

A22


d1 a

2 ∂η1 ∂ v þ a 23 ∂x1 ∂x1




  d2 a

4

þ2ðF 12 þ2F 66 Þ

4

þ F 11

4

4

∂ v3 ∂ v3 þ F 22 ∂x41 ∂x42

4

2

∂ v3 ∂ v ∂ v3 ∂ v3 þ 4F 16 3 3 þ 4F 26 þ N 011 2 2 3 ∂x1 ∂x2 ∂x21 ∂x1 ∂x2 ∂x1 ∂x2

2

þ2N 012

2 ∂η2 ∂ v þ a 23 ∂x2 ∂x2

ð5:1bÞ



2

ð5:1cÞ

2

∂ v3 ∂ v3 ∂ v þ N 022  m0 23 ¼ 0 ∂x1 ∂x2 ∂t ∂x22

• Boundary Conditions The boundary conditions are reduced from six to only four required along each edge as Along the edges xn ¼ 0, Ln enn Lnn ¼ L e Lnt ¼ Lnt

ηn ¼ e ηn ηt ¼ e ηt ∂v ∂ev 3 e nn M nn ¼ M or ¼ 3 ∂xn ∂xn ∂v ∂v ∂M nn ∂M nt N nt 3 þ N nn 3 þ þ2 ∂xt ∂xn ∂xn ∂xt e   ∂M nt e þ a=h N n3 ¼ þ N n3 ∂xt or or

ð5:2a  dÞ or

v3 ¼ ev3

Because there are four boundary conditions prescribed along each edge, the governing system of equations should be of the eighth order. For simply supported boundary conditions freely movable on all edges at x1 ¼ 0, L1 η1 ¼ η2 ¼ M 11 ¼ v3 ¼ 0

ð5:3a  dÞ

η1 ¼ η2 ¼ M 22 ¼ v3 ¼ 0

ð5:4a  dÞ

at x2 ¼ 0, L2

132

5 Free Vibration

5.3.2

Solution Methodology

The methodology applied in Librescu et al. (1997) and Hause et al. (1998, 2000) will be utilized here. Due to the complexity of the governing equations, an approximate solution methodology such as the extended Galerkin method (EGM) will be adopted. The goal is to satisfy both the equations of motion and the boundary conditions in the Galerkin sense. To satisfy the kinematic boundary conditions, Eqs. (5.3d) and (5.4d), v3(x1, x2, t) can be expressed as v3 ðx1 , x2 , t Þ ¼ ðwmn sin λm x1 sin μn x2 Þ exp ðiωmn t Þ

ð5:5Þ

pffiffiffiffiffiffiffi where ωmn are the undamped natural frequencies; i ¼ 1; λm ¼ mπ/L1, μn ¼ nπ/ L2; L1 and L2 are the panel length and width, respectively. With the first two equations of motion, Eqs. (5.1a and 5.1b), η1 and η2 are assumed in the following form   1Þ 2Þ cos λm x1 sin μn x2 þ H ðmn sin λm x1 cos μn x2 exp ðiωmn t Þ η1 ðx1 , x2 , t Þ ¼ H ðmn ð5:6aÞ   1Þ 2Þ cos λm x1 sin μn x2 þ I ðmn sin λm x1 cos μn x2 exp ðiωmn t Þ ð5:6bÞ η2 ðx1 , x2 , t Þ ¼ I ðmn 1Þ 2Þ 1Þ 2Þ , H ðmn , I ðmn , I ðmn are undetermined coefficients and. Substituting η1 and where H ðmn η2 into Eqs. (5.1a and 5.1b) and comparing the coefficients of the same trigonometric functions results in a system of equations in matrix form as

2 6 6 6 6 6 4

ðm,nÞ

U 11

ðm,nÞ

U 12

ðm,nÞ

U 11

ðm,nÞ

U 13

ðm,nÞ

U 14

ðm,nÞ

U 33 Symm:

38 ðm,nÞ 9 8 9 ðm,nÞ > > H1 > > > > V > > > > 1 > > 7> ðm,nÞ > > > > ðm,nÞ 7> = = < < U 13 7 H 2 0 ¼ 7 ðm,nÞ > ðm,nÞ > > > > > I1 > 0 > U 34 7 > > > 5> > > > > ; > : ðm,nÞ > ðm,nÞ : ðm,nÞ ; V 2 I2 U 33 ðm,nÞ

U 14

ð5:7Þ

ðm,nÞ

where the elements U ij of the matrix are expressions in terms of the geometrical and mechanical properties of the structure which are given as d1 L21 ðm,nÞ ðm,nÞ , U 12 ¼ 2A16 mnϕ, U 13 ¼ A16 m2 þ A26 n2 ϕ2 π2 d 2 L2 ðm,nÞ ðm,nÞ ðm,nÞ U 14 ¼ ðA12 þ A66 Þmnϕ, U 33 ¼ A22 n2 ϕ2 þ A66 m2 þ 2 1 , U 34 ¼ 2A26 mnϕ π ð5:8a  fÞ ðm,nÞ

U 11

¼ A11 m2 þ A66 n2 ϕ2 þ

5.3 Flat Sandwich Panels

133

while ðm,nÞ

V1

¼

ðd1 amL1 Þ wmn , π

ðm,nÞ

V2

¼

ðd 2 anϕL1 Þ wmn π

ð5.9a, bÞ

αÞ From Eq. (5.7), using Cramer’s Rule, the expressions for H ðmn , can be expressed as

~ ðαÞ H ðαÞ mn ¼ H mn wmn ,

ðαÞ

~ I ðαÞ mn ¼ I mn wmn ,

=

X

αÞ I ðmn (α ¼ 1, 2)

ð5.10a, bÞ

m, n

where e ð1Þ ¼ detðU 1 Þ , H mn detðU Þ

e ð2Þ ¼ detðU 2 Þ , H mn detðU Þ

e ð1Þ ¼ detðU 3 Þ , I mn detðU Þ

e ð2Þ ¼ detðU 4 Þ I mn detðU Þ ð5:11a  dÞ

while 0 B B B U1 ¼ B B B @

ðm,nÞ

V1

U 11

0

U 14

V2

ðm,nÞ

U 11

B B ðm,nÞ B U 12 U2 ¼ B B ðm,nÞ B U 13 @ ðm,nÞ

U 14 0

ðm,nÞ

0

ðm,nÞ

0

ðm:nÞ

U 12

ðm,nÞ

U 11

B B ðm,nÞ B U 12 U3 ¼ B B ðm,nÞ B U 13 @ ðm,nÞ U 14 0 ðm,nÞ U 11

B B ðm,nÞ B U 12 U4 ¼ B B ðm,nÞ B U 13 @ ðm,nÞ

U 14

ðm,nÞ ðm,nÞ

U 13

ðm:nÞ

V1

ðm,nÞ

U 13

ðm,nÞ

U 14

ðm,nÞ

U 33

ðm,nÞ

U 34

ðm,nÞ

U 13

ðm,nÞ

0

U 14

0

U 33

ðm,nÞ

V2

ðm:nÞ

U 12

ðm,nÞ

U 11

ðm,nÞ

U 14

ðm,nÞ U 13 ðm:nÞ U 12 ðm,nÞ

U 11

ðm,nÞ

U 14

ðm,nÞ

U 13

ðm,nÞ ðm,nÞ

U 34

ðm,nÞ

V1

0 0 ðm,nÞ V2 ðm,nÞ U 13 ðm,nÞ

U 14

ðm,nÞ

U 33

ðm,nÞ

U 34

ðm,nÞ

U 14

1

C C C C, ðm,nÞ C U 34 C A ðm,nÞ

U 13

ðm,nÞ

U 33

ðm,nÞ

U 14

1

ð5.12a, bÞ

C C C C ðm,nÞ C U 34 C A ðm,nÞ

U 13

ðm,nÞ

U 33

ðm,nÞ

U 14

1

C C C C, ðm,nÞ C U 34 C A ðm,nÞ

U 13

ðm,nÞ

U 33

ðm,nÞ

V1

0 0 ðm,nÞ

V2

1 C C C C C C A

ð5.12c, dÞ

134

5 Free Vibration

0 B B B U¼B B @

ðm,nÞ

U 11

ðm,nÞ

U 12

ðm,nÞ

U 11

ðm,nÞ

U 13

ðm,nÞ

U 14

ðm,nÞ

U 33 Symm:

ðm,nÞ

U 14

1

C C C C ðm,nÞ U 34 C A ðm,nÞ U 33 ðm,nÞ

U 13

ð5:12eÞ

At this point, Eqs. (5.1a and 5.1b) are identically fulfilled. There remains the third equation of motion currently unfulfilled. This will be fulfilled in an average sense through the use of the already familiar extended Galerkin method. In addition, the boundary conditions, Eqs. (5.3d) and (5.4d) are fulfilled with the remaining ones unfulfilled. Retaining the unfulfilled third equation of motion and the remaining unfulfilled boundary conditions from Eqs. (5.3a–c) and (5.4a–c) in the energy functional (Hamilton’s equation) results in Z

t 1 Z l2 Z l1

t0

0

0





2  2  4 4 ∂η1 ∂η2 ∂ v3 ∂ v3 ∂ v ∂ v d1 a þ a 2  d2 a þ a 2 þ F 11 43 þ F 22 43 þ ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 4

4

4

2

2

∂ v3 ∂ v ∂ v3 ∂ v ∂ v þ 4F 16 3 3 þ 4F 26 þ N 011 23 þ N 022 23  ∂x21 ∂x22 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x32 +
 Z t1 *Z l2 2  ∂ v3 ∂v3  l1 m0 2 δv3 dx1 dx2 dt þ L11 δη1 þ L12 δη2 þ M 11 δ  dx dtþ ∂t ∂x1  0 t0 0 +
 Z t1 *Z l1 ∂v3  l2 L12 δη1 þ L22 δη2 þ M 22 δ  dx dt ¼ 0 ∂x2  0 t0 0 2ðF 12 þ2F 66 Þ

ð5:13Þ Substituting the expressions for η1, η2, and v3 into Eq. (5.13) and carrying out the necessary integrations and simplifying realizing the independent character of the variations results in an eigenvalue problem which gives the expression for the eigenfrequencies as ω2mn ¼

 1 2Þ d aλ H ð1Þ þ d2 aμn I ðmn þ d1 a2 λ2m þ d2 a2 μ2n þ F 11 λ2m þ F 22 μ2n þ 4F 66 λ2m μ2n þ m0 1 m mn  2F 12 λ2m μ2n  N 011 λ2m  N 022 μ2n ð5:14Þ

Where the expression for the undamped natural frequency can be arranged in dimensionless form as

 aL3 1Þ n4 ϕ4 F 22 2m2 n2 ϕ2 ðF 12 þ 2F 66 Þ a2 L21  2 e ðmn þ þ 2 m d 1 þ n2 ϕ2 d 2 þ 3 1 md 1 H F 11 F 11 π F 11 π F 11    ð2Þ e þnϕd 2 I mn þ K x m2 þ LR n2 ϕ2

Ω2mn ¼ m4 þ

ð5:15Þ

5.3 Flat Sandwich Panels ð1Þ

135

ð2Þ

e mn and eI mn are nondimensionalized also which are not provided here). (Note: H The nondimensional parameters are defined as Ω2mn ¼

5.3.3

m0 L41 ω2mn L2 N 0 N0 L , K x ¼ 14 11 , LR ¼ 22 , ϕ¼ 1 4 L2 π F 11 π F 11 N 011

ð5:15a  dÞ

Validation of the Theoretical Structural Model

Prior to the presentation of the numerical results validations are made to assess the accuracy of the theory. The case considered for the validations considers a threelayered flat sandwich panel whose material characteristics are listed in Table 5.1. The facings are aluminum while the core consists of an aluminum honeycomb–type construction. The length of the edges of the panel are L1 ¼ 1.828 m and L2 ¼ 1.219 m. In Table 5.2, the eigenfrequencies from three different sources are compared with the present theoretical results. Excellent agreement is seen. Some additional sources are listed in Table 5.3 for the same geometrical and material characteristics as in Table 5.1. Again, remarkable agreement is seen. Hohe et al. (2006) considered the transverse compressibility of the core. With this additional consideration, exceptional agreement is seen revealing that the compressibility of the core, for a flat sandwich panel, has extraordinarily little effect on the eigenfrequencies.

5.3.4

Results and Discussion

In consideration of the following numerical results, the material properties for the face sheets and the core are listed in Tables 5.4 and 5.5. Unless otherwise specified, the face sheets are of the F1 type and the core is of the C1 type while the following geometrical properties are given as L1 ¼ 0.6096 m, h ¼ 0:0127 m. Figure 5.1 shows the effects of the compressive edge loading on the eigenfrequencies Ω1=2 mn for various aspect ratios. It is known that when the Table 5.1 Material and geometrical properties

Upper face Bottom face Core

Thickness (mm) 0.4064

Elastic modulus (GPa) 68.95

Poisson’s ratio 0.33

Mass density N  s2  m4 2768.93

G12 25.92

G13 –

G23 –

0.4064

68.95

0.33

2768.93

25.92

–

–

6.35

0

0

121.83

–

0.0517

0.134

Shear modulus (GPa)

136

5 Free Vibration

Table 5.2 Eigenfrequency comparisons (validation no. 1) m 1

2

3

4

n 1 23.5 23.4 – – 45.1 44.8 45.0 45.0 80.7 80.3 78.0 80.0 130.0 129.3 133.0 129.0

2 71.0 70.7 69.0 71.0 92.1 91.5 92.0 91.0 126.8 126.1 129.0 126.0 174.9 173.9 177.0 174.0

3 146.5 145.8 152.0 146.0 166.7 165.9 169.0 165.0 200.1 199.1 199.0 195.0 – – – –

4 245.3 244.4 246.0 244.0 264.5 263.5 262.0 263.0 – – – – – – – –

5 362.5 361.7 381.0 360.0 – – – – – – – – – – – –

Present study Hohe et al. (2006) Raville and Ueng (1967) (exp) Raville and Ueng (1967) (numerical) Present study Hohe et al. (2006) Raville and Ueng (1967) (exp) Raville and Ueng (1967) (numerical) Present study Hohe et al. (2006) Raville and Ueng (1967) (exp) Raville and Ueng (1967) (numerical) Present study Hohe et al. (2006) Raville and Ueng (1967) (exp) Raville and Ueng (1967) (numerical)

Table 5.3 Eigenfrequency comparisons (validation no. 2) ωij(Hz) ω1(¼ω11) ω2(¼ω21) ω3(¼ω12) ω4(¼ω31) ω5(¼ω22) ω6(¼ω32)

Experiment (R. and U.) 45 69 78 92 125

Exact (R. and U.) 23 44 71 80 91 126

FEM (M. and S.) 23 44 70 80 90 125

SFPM (Z. and L.) 23.3 44.48 71.36 78.81 91.90 125.16

Present 23.40 44.64 71.51 79.27 92.2 125.97

R. and U. ! Raville and Ueng (1967); M. and S. ! Monforton and Schmidt (1968); Z. and L. ! Zhou and Li (1996) Table 5.4 Material properties for the face sheets Type F1 F2

Material HS Graph. Ep. IM7/977-2

E1 (GPa) 180.99 79.98

E2(GPa) 10.34 75.15

G12(GPa) 7.24 9.65

ν12 0.28 0.06

Table 5.5 Core material properties Type C1

Core type Titanium honeycomb

G13 ðGPaÞ 1.44

G23 ðGPaÞ 0.651

5.3 Flat Sandwich Panels

137

Fig. 5.1 The effect of the compressive edge load on the eigenfrequencies Ω1/2 of a flat sandwich panel for various aspect ratios Table 5.6 Comparison of the critical buckling load Layup [θ/  θ/θ/core/θ/  θ/θ] θ (deg) 47 44.1 38.6 31.5 19.8

ϕ(L1/L2) 1.11 1 0.833 0.714 0.625

L1(mm) 609.6 609.6 609.6 609.6 609.6

 Kcr 7110.28 4913.96 2734.11 1664.60 1056.09

 N ∘11 cr ðN=mÞ (Chap. 3) 2.08  106 1.67  106 1.22  106 0.99  106 0.88  106



 N ∘11 cr ðN=mmÞ (Present) 2.089  106 1.682  106 1.222  106 0.994  106 0.889  106

eigenfrequency vanishes, the critical load has been reached. At this point the buckling load is identified. It can be seen that at smaller aspect ratios the buckling loads are higher. In Table 5.6, comparisons are made between the direct buckling approach as was presented in Chap. 3 and the present results where the buckling load is determined from the point at which the eigenfrequencies vanish. Excellent agreement is made between the two approaches. In Fig. 5.2, the effect of the panel face thickness on the eigenfrequency Ω1/2 for various distances a is shown. It is revealed that the eigenfrequency decays as the panel face thickness increases. Also, as the distance a increases for a fixed face thickness, the eigenfrequencies become larger. In Fig. 5.3, the effect of the fiber orientation of the face sheets on the eigenfrequency Ω1/2 for various aspect ratios is depicted. At a larger aspect ratio, the range between the eigenfrequencies Ω1/2 between 0- and 90-degree ply angles is larger than at a small aspect ratio. The trend seems to increase for the larger aspect ratio where it appears flat for the small

138

5 Free Vibration

Fig. 5.2 The effect of the face thickness on the eigenfrequencies Ω1/2 of a flat sandwich panel for various distances from the mid-surface of the core to the mid-surface of the facings

Fig. 5.3 The effect of the ply angle on the eigenfrequencies Ω1/2 of a flat sandwich panel for various aspect ratios

aspect ratio of around 0.25 or smaller. In Fig. 5.4, the effect of the ply angle on the eigenfrequency Ω1/2 at various mode numbers. It is evident that at the higher modes, the eigenfrequencies Ω1/2 are increased. In addition, for each mode the highest frequency appears to be around a ply angle of 45 degrees.

5.3 Flat Sandwich Panels

139

Fig. 5.4 The effect of ply angle on various mode eigenfrequencies Ω1/2 of a flat sandwich panel

Fig. 5.5 The effect of the orthotropicity ratio on the eigenfrequencies Ω1/2 of a flat sandwich panel for various ply angles

In Fig. 5.5, the effect of the orthotropicity ratio on the eigenfrequencies Ω1/2 is displayed for various ply angles. Above 45 degrees the orthotropicity ratio seems to have a trend which slightly increases resulting in higher eigenfrequencies. In a similar sense, the effect of the core orthotropy ratio is to increase the eigenfrequency Ω1/2 with an increase in the ply angle as shown in Fig. 5.6. Larger ratios provide

140

5 Free Vibration

1/2 Fig. 5.6 The effect of the ply angle on  the eigenfrequencies Ω of a flat sandwich panel for various orthotropy ratios η  G13 =G23 of the core

smaller eigenfrequencies, but the trends appear to be the same across the spectrum with an increase in ply angle.

5.4 5.4.1

Doubly Curved Sandwich Panels Governing System

The governing equations which apply to the free vibration problem of doubly curved sandwich panels are an extension of the governing equations for the flat panels. In this case the governing equations are not decoupled, and six boundary conditions are required along each edge requiring the system of equations to be of the twelfth order. The equations of interest are based on the linear theory for sandwich shells assuming symmetric anisotropic laminated facings, a weak orthotropic core with the transverse shear effects discarded in the facings. Under these conditions, Eqs. (2.168)–(2.172) apply along with the corresponding set of simply supported boundary conditions Eqs. (2.173a–f). This set of governing equations are general in the sense that they can apply to a whole host of static and dynamic problems of doubly curved shallow sandwich shells. The governing equations for double-curved sandwich shells are

5.4 Doubly Curved Sandwich Panels

141

• Equations of Motion



2 
2  2 2 ∂ ξ1 1 ∂v3 ∂ ξ1 ∂ ξ2 ∂ ξ2 1 ∂v3  þ  þ A þ A 66 12 ∂x1 ∂x2 R2 ∂x2 ∂x21 R1 ∂x1 ∂x22 ∂x1 ∂x2

 2 2 2 ∂ ξ1 ∂ ξ2 1 ∂v3 ∂ ξ2 1 ∂v3 A16 2  þ 2   A26 ¼0 ∂x1 ∂x2 ∂x1 R1 ∂x1 ∂x22 R2 ∂x2

A11

ð5:16aÞ



2 
2  2 2 ∂ ξ2 ∂ ξ2 ∂ ξ1 ∂ ξ1 1 ∂v3 1 ∂v3  þ  þ A þ A 66 12 ∂x1 ∂x2 R1 ∂x1 ∂x22 R2 ∂x2 ∂x21 ∂x1 ∂x2

 2 2 2 ∂ ξ2 ∂ ξ1 1 ∂v3 ∂ ξ1 1 ∂v3 A26 2  þ 2   A16 ¼0 ∂x1 ∂x2 ∂x2 R2 ∂x2 ∂x21 R1 ∂x1 A22

ð5:16bÞ
2 
2  2 2 2 2 ∂ η1 ∂ η1 ∂ η2 ∂ η2 ∂ η2 ∂ η1 þ A þ þ A þ 2 þ A 66 12 16 ∂x1 ∂x2 ∂x1 ∂x2 ∂x21 ∂x22 ∂x1 ∂x2 ∂x21
 2 ∂ η2 ∂v þA26  d 1 η1 þ a 3 ¼ 0 ∂x1 ∂x22 A11

ð5:16cÞ
2  2 2 2 ∂ η2 ∂ η2 ∂ η1 ∂ η1 þ A þ þ A12 66 2 2 ∂x1 ∂x2 ∂x2 ∂x2 ∂x1 ∂x2
2 
 2 2 ∂ η1 ∂ η2 ∂ η1 ∂v3 þA26 þ2  d2 η2 þ a þ A16 ¼0 ∂x1 ∂x2 ∂x2 ∂x22 ∂x21 A22

ð5:16dÞ




    A11 A12 ∂ξ1 A16 A26 ∂ξ1 A22 A12 ∂ξ2 A26 A16 ∂ξ2 þ  þ  þ  þ R1 R2 ∂x1 R1 R2 ∂x2 R2 R1 ∂x2 R2 R1 ∂x1


   2 2 ∂η1 ∂η2 A11 A22 2A12 ∂ v3 ∂ v3  þ 2 þ þ a 2  d2 a þa 2 v3  d 1 a ∂x1 ∂x2 ∂x1 ∂x2 R21 R2 R1 R2




4

4

4

4

4

þ F 11

∂ v3 ∂ v3 ∂ v ∂ v ∂ v3 þ F 22 þ 2ðF 12 þ 2F 66 Þ 2 3 2 þ 4F 16 3 3 þ 4F 26 ∂x41 ∂x42 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x32

þ N 011

∂ v3 ∂ v3 ∂ v3 ∂v ∂ v þ 2N 012 þ N 022  c 3  m0 23 ¼ 0 ∂x1 ∂x2 ∂t ∂t ∂x21 ∂x22

2

2

2

2

ð5:16eÞ

142

5 Free Vibration

• Boundary Conditions It should be noted that for the free vibration problem, the thermal, the moisture, and the transversal loading terms have been discarded. The boundary conditions are Along the edges xn ¼ 0, Ln e nn N nn ¼ N e nt N nt ¼ N enn Lnn ¼ L e Lnt ¼ Lnt

ξn ¼ e ξn ξt ¼ e ξt

or or

ηn ¼ e ηn ηt ¼ e ηt ∂v ∂ev 3 e nn M nn ¼ M or ¼ 3 ∂xn ∂xn ∂v3 ∂v3 ∂M nn ∂M nt N nt þ N nn þ þ2 ∂xt ∂xn ∂xn ∂xt e   ∂M nt e þ a=h N n3 ¼ þ N n3 ∂xt or or

ð5:17a  fÞ or

v3 ¼ ev3

For simply supported boundary conditions freely movable on all edges at x1 ¼ 0, L1 N 11 ¼ N 12 ¼ η1 ¼ η2 ¼ M 11 ¼ v3 ¼ 0

ð5:18a  fÞ

N 22 ¼ N 12 ¼ η1 ¼ η2 ¼ M 22 ¼ v3 ¼ 0

ð5:19a  fÞ

at x2 ¼ 0, L2

It will be helpful to determine, further in the developments, if the boundary conditions are satisfied by expressing the boundary conditions in terms of displacements. In terms of displacements, the first, second, and fifth boundary conditions can be written as N 11 ¼ A11



 ∂ξ1 ∂ξ ∂ξ2 ∂ξ1 A11 A12 þ A12 2 þ A16 þ þ  v R1 R2 3 ∂x1 ∂x2 ∂x1 ∂x2

¼ 0,
N 12 ¼ A66

ð1⇄2Þ ∂ξ2 ∂ξ1 þ ∂x1 ∂x2

 þ A26

ð5:20Þ
 ∂ξ2 ∂ξ A16 A26 þ A16 1  þ v ¼0 R1 R2 3 ∂x2 ∂x1

ð5:21Þ

5.4 Doubly Curved Sandwich Panels 2

143 2

2

M 11 ¼ F 11

∂ v3 ∂ v3 ∂ v3 þ F 12 þ 2F 16 ¼0 ∂x1 ∂x2 ∂x21 ∂x22

M 22 ¼ F 22

∂ v3 ∂ v3 ∂ v3 þ F 12 þ 2F 26 ¼0 ∂x1 ∂x2 ∂x22 ∂x21

2

2

ð5:22Þ

2

ð5:23Þ

where the global stiffness quantities Aij, Fij(i, j ¼ 1, 2, 6) were defined in Chap. 2. The sign (1 ⇄ 2) in Eq. (5.20) indicates that N22 which is not explicitly supplied can be obtained by replacing the subscript 1 with 2 and vice versa.

5.4.2

Solution Methodology

The methodology applied in Librescu et al. (1997) and Hause et al. (1998, 2000) will be utilized here. To begin the process, attention is given to the first two equations of motion, Eqs. (5.16a and 5.16b). here ξ1, ξ2 are assumed in the following form   1Þ 2Þ cos λm x1 sin μn x2 þ F ðmn sin λm x1 cos μn x2 exp ½ðiωmn  αmn Þt  ξ1 ðx1 , x2 , t Þ ¼ F ðmn ð5:24Þ   1Þ 2Þ cos λm x1 sin μn x2 þ Gðmn sin λm x1 cos μn x2 exp ½ðiωmn  αmn Þt  ξ2 ðx1 , x2 , t Þ ¼ Gðmn ð5:25Þ 1Þ 2Þ 1Þ 2Þ , F ðmn , Gðmn , Gðmn are arbitrary constants to be determined; while where F ðmn λm ¼ mπ/L1 and μn ¼ nπ/L2. v3(x1, x2, t) can be expressed as

v3 ðx1 , x2 , t Þ ¼ ðwmn sin λm x1 sin μn x2 Þ exp ½ðiωmn  αmn Þt 

ð5:26Þ

ωmn are the undamped natural frequencies and αmn are constants which provide a measure of damping. Substituting Eqs. (5.24 and 5.25) into Eqs. (5.16a and 5.16b) and identifying the coefficients of the like trigonometric functions provides the 1Þ 2Þ 1Þ 2Þ coefficients F ðmn , F ðmn , Gðmn , Gðmn as 

  ð1Þ  2Þ 1Þ 2Þ 1Þ 2Þ e ð1Þ , G e ð2Þ wmn e ðmn e mn , F F ðmn , F ðmn , Gðmn , Gðmn , G ¼ F mn mn

ð5:27Þ

1Þ 2Þ e ð1Þ , G e ð2Þ are solutions to the following matrix e ðmn e ðmn , F , G The coefficients F mn mn equation

144

5 Free Vibration

2 6 6 6 6 6 4

ðm,nÞ

Y 11

ðm,nÞ

Y 12

ðm,nÞ

Y 11

ðm,nÞ

Y 13

ðm,nÞ

Y 14

ðm,nÞ

Y 33 Symm:

38 ð1Þ 9 8 ð1Þ 9 > e mn > > F > > > > > > > > > > Z mn > 7 ð2Þ > ðm,nÞ > = > =

e Y 13 7 7 mn mn ¼ 7 ð3Þ ðm,nÞ > ð1Þ > e > > > > >G > Z mn > Y 34 7 > > 5> mn > ; > : > ð4Þ > > ð2Þ > ðm,nÞ : Z ; mn Y 33 e Gmn ðm,nÞ

Y 14

ð5:28Þ

where ðm,nÞ

Y 11

¼ A11 m2 þ A66 n2 ϕ2 ,

ðm,nÞ

Y 14

1Þ Z ðmn 3Þ Z ðmn

ðm,nÞ

¼ 2A16 mnϕ,

Y 12

ðm,nÞ

Y 13

ðm,nÞ

¼ A16 m2 þ A26 n2 ϕ2 ðm,nÞ

¼ ðA12 þ A66 Þmnϕ, Y 33 ¼ A22 n2 ϕ2 þ A66 m2 , Y 34 ¼ 2A26 mnϕ m nϕ 2Þ ¼  ðψ 1 A16 þ ψ 2 ϕA12 Þ, Z ðmn ¼  ðψ 1 A16 þ ψ 2 ϕA26 Þ n π m nϕ ð4Þ ¼  ðψ 1 A16 þ ψ 2 ϕA26 Þ, Z mn ¼  ðψ 1 A12 þ ψ 2 ϕA22 Þ π π ð5:29a  jÞ

In the above expressions, ϕ ¼ L1/L2, ψ 1 ¼ L1/R1, ψ 2 ¼ L2/R2 where ϕ is the 1Þ 2Þ e ð1Þ , G e ð2Þ can be evale ðmn e ðmn aspect ratio. From the above matrix equation, F , F , G mn mn uated, using Cramer’s rule such that 1Þ F ðmn

detðY1 Þ , detðYÞ

2Þ F ðmn

detðY2 Þ , detðYÞ

1Þ Gðmn

detðY3 Þ , detðYÞ

2Þ Gðmn

detðY4 Þ detðYÞ

ð5:30a  dÞ

where 0

1Þ Z ðmn

B B ð2Þ B Z mn Y1 ¼ B B ð3Þ B Z mn @ 0

4Þ Z ðmn ðm,nÞ

Y 11

B B ðm,nÞ B Y 12 Y2 ¼ B B ðm,nÞ B Y 13 @ ðm,nÞ

Y 14

ðm,nÞ

Y 12

ðm,nÞ

Y 11

ðm,nÞ

Y 14

ðm,nÞ

Y 13

ðm,nÞ

Y 13

ðm,nÞ

Y 14

ðm,nÞ

Y 33

ðm,nÞ

Y 34

ðm,nÞ

1Þ Z ðmn

Y 13

2Þ Z ðmn

Y 14

3Þ Z ðmn

Y 33

4Þ Z ðmn

Y 34

ðm,nÞ ðm,nÞ ðm,nÞ

ðm,nÞ

Y 14

1

C C C C, ðm,nÞ C Y 34 C A ðm,nÞ

Y 13

ðm,nÞ

Y 33

ðm,nÞ

Y 14

1

C C C C ðm,nÞ C Y 34 C A ðm,nÞ

Y 13

ðm,nÞ

Y 33

ð5.31a, bÞ

5.4 Doubly Curved Sandwich Panels

0

ðm,nÞ

Y 11

B B ðm,nÞ B Y 12 Y3 ¼ B B ðm,nÞ B Y 13 @ ðm,nÞ

0

Y 14

ðm,nÞ

Y 11

B B ðm,nÞ B Y 12 Y4 ¼ B B ðm,nÞ B Y 13 @ ðm,nÞ

Y 14

0 B B B Y¼B B @

145 1Þ Z ðmn

Y 14

ðm,nÞ

2Þ Z ðmn

Y 13

ðm,nÞ

3Þ Z ðmn

C C C C, ðm,nÞ C Y 34 C A

ðm,nÞ

4Þ Z ðmn

Y 33

Y 11 Y 14 Y 13

ðm,nÞ

Y 12

ðm,nÞ

Y 11

ðm,nÞ

Y 14

ðm,nÞ

Y 13

ðm,nÞ

Y 11

ðm,nÞ

Y 13

ðm,nÞ

Y 14

ðm,nÞ

Y 33

ðm,nÞ

Y 34

ðm,nÞ

Y 12

ðm,nÞ

Y 11

ðm,nÞ

1

ðm,nÞ

Y 12

ðm,nÞ

ðm,nÞ 1Þ Z ðmn

C 2Þ C Z ðmn C C C 3Þ C Z ðmn A 4Þ Z ðmn

ðm,nÞ

Y 13

ðm,nÞ

Y 14

ðm,nÞ

Y 33 Symm:

ð5.31c, dÞ

1

ðm,nÞ

Y 14

1

C C C C ðm,nÞ C Y 34 A ðm,nÞ Y 33 ðm,nÞ

Y 13

ð5:31eÞ

The expressions for ξ1, ξ2, and v3 identically fulfill the first two equations of motion. A similar procedure to identically fulfill the third and fourth equation of motion is followed. With this in mind η1 and η2 are assumed in the following form   1Þ 2Þ cos λm x1 sin μn x2 þ H ðmn sin λm x1 cos μn x2 exp ½ðiωmn  αmn Þt  η1 ðx1 , x2 , t Þ ¼ H ðmn ð5:32aÞ   1Þ 2Þ cos λm x1 sin μn x2 þ I ðmn sin λm x1 cos μn x2 exp ½ðiωmn  αmn Þt  η2 ðx1 , x2 , t Þ ¼ I ðmn ð5:32bÞ 1Þ 2Þ 1Þ 2Þ , H ðmn , I ðmn , I ðmn are undetermined coefficients. Substituting η1 and η2 where H ðmn into Eqs. (5.16c and 5.16d) and comparing the coefficients of the same trigonometric functions yields the unknown coefficients. For doubly curved sandwich shells these coefficients are given in Eqs. (5.10a–d)–(5.12a–e). This leaves Eqs. (5.16c and 5.16d) identically fulfilled. There remains the fifth equilibrium equation currently unfulfilled. As previously, this will be fulfilled in an average sense through the use of the EGM. In addition, the boundary conditions, Eqs. (5.18f) and (5.19f) are fulfilled with the remaining ones unfulfilled. Retaining the unfulfilled fifth equation of motion and the remaining unfulfilled boundary conditions from Eqs. (5.18a–e) and (5.19a–e) in the energy functional (Hamilton’s equation) results in

146

5 Free Vibration



 A11 A12 ∂ξ1 A16 A26 ∂ξ1 A22 A12 ∂ξ2  þ  þ  þ  R1 R2 ∂x1 R1 R2 ∂x2 R2 R1 ∂x2 t0 0 0



2  2  ∂η1 ∂η2 A26 A16 ∂ξ2 A11 A22 2A12 ∂ v3 ∂ v3 þ þ þ þ a 2  d2 a þa 2 v  d1 a  R2 R1 ∂x1 R21 R22 R1 R2 3 ∂x1 ∂x2 ∂x1 ∂x2 Z

t1 Z l2 Z l1

4




4

4

4

4

∂ v3 ∂ v ∂ v ∂ v ∂ v3 þ F 22 43 þ 2ðF 12 þ 2F 66 Þ 2 3 2 þ 4F 16 3 3 þ 4F 26 ∂x41 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x32  2 2 2 ∂v ∂ v ∂v ∂ v þN 011 23 þ N 022 23 þ C 3 þ m0 23 δv3 dx1 dx2 dtþ ∂t ∂t ∂x1 ∂x2
 l2 + Z t1 Z l2 ∂v3  N 11 δξ1 þ N 12 δξ2 þ L11 δη1 þ L12 δη2 þ M 11 δ  dx2 dtþ 0 ∂x 1 t0 0 þF 11

Z

t1 Z l1

t0

0




∂v3 N 22 δξ2 þ N 12 δξ1 þ L22 δη2 þ L12 δη1 þ M 22 δ ∂x2

 l1 +   dx1 dt ¼ 0 0

ð5:33Þ Substituting the expressions for ξ1, ξ2, η1, η2, and v3 into Eq. (5.33) and carrying out the necessary integrations and simplifying realizing the independent character of the variations gives the following characteristic equation S2mn þ 2Δmn ωmn Smn þ ω2mn ¼ 0

ð5:34aÞ

where ω2mn ¼ K mn =m0

and

Δmn ¼ C=2m0 ωmn

ð5:34bÞ

Denote the undamped natural frequencies squared and the modal damping, respectively. The expression for the undamped natural frequency is given by ω2mn ¼


    1 1Þ 2Þ λ4m F 11 þ 2λ2m μ2n ðF 12 þ 2F 66 Þ þ μ4n F 22 þ λm ad 1 H ðmn þ aλm þ μn ad 2 I ðmn þ aμn m0



 A A A A A A A 1Þ 2Þ 2Þ þλm 11 þ 12 F ðmn þ μn 12 þ 22 Gðmn þ μn 16 þ 26 F ðmn þ λm 16 þ R1 R2 R1 R2 R1 R2 R1 
  A26 A11 A A 1Þ þ þ 2 12 þ 22 Gðmn  λ2m N 011  μ2n N 022 R2 R1 R2 R22 R21

ð5:35Þ The expression for undamped natural frequencies can be arranged in dimensionless form as

5.4 Doubly Curved Sandwich Panels

147

 n4 ϕ4 F 22 2m2 n2 ϕ2 ðF 12 þ 2F 66 Þ a2 L21  2 þ þ 2 m d 1 þ n2 ϕ 2 d 2 F 11 F 11 π F 11 i aL3 h L2 h 1Þ ð2Þ 1Þ e ðmn e ðmn þ 3 1 md 1 H þnϕd2eI mn þ 3 1 mðψ 1 A11 þ ϕψ 2 A12 ÞF π F 11 π F 11 i   2Þ e ð2Þ þ nϕF e ð1Þ ðψ 1 A16 þ ϕψ 2 A26 Þ e ðmn þnϕðψ 1 A12 þ ϕψ 2 A22 ÞG þ mG mn mn

Ω2mn ¼ m4 þ

þ

   L21  2 ψ A þ ψ 22 ϕ2 A22 þ 2A12 ψ 1 ψ 2 ϕ  K x m2 ψ 21 þ LR n2 ϕ2 ψ 22 π 4 F 11 1 11 ð5:36Þ

where the nondimensional parameters Ω2mn , K x , LR are defined by Eq. (5.15a–d).

5.4.3

Validation of the Theoretical Model

For the case of cylindrical shells, validations are made with Rahmani et al. (2010) who considered the free vibration of cylindrical sandwich shells with a flexible core using a higher order theoretical model. Although the current theoretical model does not incorporate the theoretical aspects of the flexible core, particularly good agreement is found between the current or present model and that of Rahmani et al. (2010). This shows, as far as the natural frequency is concerned, at least for the fundamental mode, that the compressibility of the core has only a minor effect on the fundamental natural frequency. For the following validations, Table 5.7 specifies the material properties for this particular case while Tables 5.8 and Table 5.9 contain the comparisons between the present model and that of Rahmani et al. (2010) of the fundamental frequencies of a cylindrical sandwich shell. Table 5.8 compares the Table 5.7 Material and geometrical properties

Facings Core

Elastic modulus (GPa) E1 E2 24.51 7.77 –

Poisson’s ratio 0.078 0

Table 5.8 Natural frequencies of circular  no. 1 H=L2 ¼ 1 m, 2h=H ¼ 0:88, L1 ¼ 1 m R. et al. Present % Error

Frequency (Hz) ω1 ω11

R. et al. ! Rahmani et al. (2010)

Mass density (kg/m3) 1800 130

cylindrical

R2 ¼ 1 m, L2 ¼ 1 m 234.77 252.034 6.8

Shear modulus (GPa) G12 G13 G23 3.34 – – – 0.05 0.05

sandwich

shell:

validation

R2 ¼ 2 m, L2 ¼ 1 m 211.92 217.21 2.4

148

5 Free Vibration

Table 5.9 Natural frequencies of circular cylindrical  no. 2 H=L2 ¼ 1 m, 2h=H ¼ 0:88, L1 ¼ 1 m, R2 ¼ 1 m R. et al. Present % Error

Frequency (Hz) ω1 ω1

L1 ¼ 1 m 211.92 217.21 2.4

L1 ¼ 2 m 141.35 146.88 3.8

sandwich

L1 ¼ 3 m 128.18 133.72 4.1

shell:

L1 ¼ 5 m 122.33 127.77 4.3

validation L1 ¼ 10 m 120.27 125.64 4.3

R. et al. ! Rahmani et al. (2010)

Table 5.10 Material characteristics of the face sheets Type F1 F2

Material HS Graph. Ep. IM7/977-2

E1(GPa) 180.99 79.98

E2(GPa) 10.34 75.15

G12(GPa) 7.24 9.65

ν12 0.28 0.06

Table 5.11 Material characteristics of the core Type C1

G13 ðGPaÞ 1.44

Core type Titanium honeycomb

G23 ðGPaÞ 0.651

Table 5.12 Comparisons of eigenfrequencies and buckling loads θ 47 44.1 38.6 31.5 19.8

ψ 0.9 1.0 1.2 1.4 1.6

Plate (see Fig. 5.1) N ∘11 ðN=mÞ Kcr 7110.28 2.08  106 4913.96 1.67  106 2734.11 1.22  106 1664.60 0.99  106 1056.09 0.88  106

pffiffiffiffiffiffiffiffi Ω11 16.28 14.84 12.82 11.32 10.10

Shell (see Fig. 5.7) Kcr N ∘11 ðN=mÞ 8946.94 2.61  106 6945.93 2.37  106 4674.68 2.08  106 2289.93 1.36  106 1256.84 1.05  106

pffiffiffiffiffiffiffiffi Ω11 18.7631 17.5176 14.6559 12.2611 10.5534

fundamental frequencies (m ¼ n ¼ 1) for different radius of curvatures while Table 5.9 compares the fundamental frequencies for various lengths (L1) of the side/edge of the sandwich panel. In both cases the facings consist of a [0/90/0/core/0/ 90/0] layup. Remarkably close agreement is seen between the present results and that of Rahmani et al. (2010) with discrepancies under 7%.

5.4.4

Present Results and Discussion

The material properties of the core and face sheets for the numerical results that follow are listed in Tables 5.10 and 5.11. Table 5.12 displays the comparison between the critical buckling loads for a flat and doubly curved sandwich panel where the eigenfrequency vanishes. Also, in comparison are the eigenfrequencies at

5.4 Doubly Curved Sandwich Panels

149

Fig. 5.7 The effect of the compressive edge load on the eigenfrequencies Ω1/2 of a cylindrical sandwich panel for various aspect ratios

zero compressive edge load. The comparison indicates that at the exclusion of the compressive edge load, higher eigenfrequencies are inherent for curved sandwich panel construction over flat sandwich panels. Figure 5.7 illustrates the vanishing of the eigenfrequencies at the critical buckling load for a cylindrical sandwich panel (L1/R1 ¼ L2/R2 ¼ 0.4) for various characteristic ply angles. At the larger ply angles, the structure benefits from its capacity to carry a larger compressive edge load prior to buckling. In Fig. 5.8, higher eigenfrequencies appear to be the norm at larger length-to-curvature ratios. There also appears to be a continual increase in the eigenfrequencies over the span from the 0- to 90-degree ply angles per length-tocurvature ratio. There also appears to be only a marginal difference between the eigenfrequencies, for the various curvature ratios, at the two extremes of 0- and 90-degrees. As illustrated in Fig. 5.8, similar behavior is seen in Figure 5.9 with the increase of the length-to-curvature ratio. In addition, as the core thickness increases, providing a higher thickness ratio, the eigenfrequencies increase. An increase in the core thickness-to-face thickness ratio seems to act as a catalyst to boosting the eigenfrequencies. Figure 5.10 highlights the effect of the orthotropicity ratio on the normalized eigenfrequencies for characteristic ply angles of a single-layered cylindrical sandwich panel. It appears that for ply angles less than 45 degrees there seems to be a marginal continuous increase in the eigenfrequencies over the orthotropicity ratio span of between 0 and 1. At the 45-degree level, the effect of the orthotropicity ratio seems to be flat or null. Above 45 degrees, as the orthotropicity ratio increases, the eigenfrequencies trend marginally downward. Figure 5.11 conveys the effect of the material directional properties on the frequency ratio for characteristic curvature

150

5 Free Vibration

Fig. 5.8 The effect of the ply angle on the eigenfrequencies Ω1/2 of a cylindrical sandwich panel for various length-to-curvature ratios

Fig. 5.9 The effect of the ply angle on the eigenfrequencies Ω1/2 of a cylindrical sandwich panel for various normalized curvatures and core-to-face thickness ratios

5.4 Doubly Curved Sandwich Panels

151

Fig. 5.10 The effect of the orthotropicity ratio on the eigenfrequencies Ω1/2 of a cylindrical sandwich panel

Fig. 5.11 The effect of the ply angle on the ratio of the shell-to-plate eigenfrequencies Ω1/2 of a cylindrical sandwich panel for various normalized curvatures

ratios. The larger curvature ratio exhibits higher frequency ratios which is consistently larger than one. This implies that the eigenfrequencies for a curved sandwich panel are higher than for flat sandwich panel. Additionally, evident, for a fixed

152

5 Free Vibration

Fig. 5.12 The effect of the thickness ratio on the ratio of the shell-to-plate eigenfrequencies Ω1/2 of a cylindrical sandwich panel for various ply angles

curvature ratio, is the increase in the frequency ratio up to 45 degrees followed by a decrease until a ply angle of 90 degrees. In Fig. 5.12 the effect of the core-to-face sheet thickness ratio on the frequency ratio for fixed ply angles concerning a single-layered curved sandwich panel is highlighted. At small core thickness-to- face thickness ratios there appears to be a much larger spread between the values of the frequency ratio for the various fixed ply angles as compared to the larger ones. At the extreme of the larger thickness ratios there is only a negligible difference in the values of the frequency ratio between the fixed ply angle constructs. It is in the vicinity of the larger thickness ratios that the flat and curved sandwich panels exhibit almost identical eigenfrequency values. It can also be seen that as the thickness ratio increases the frequency ratio decays exponentially. Figure 5.13 displays the effect of the normalized curvature on the frequency ratio of a curved sandwich panel for characteristic ply angles. It is seen that an increase in the curvature ratio results in higher frequency ratios for all ply angles. The larger the curvature ratio, the larger the margin between the frequency of the flat sandwich panel and a curved sandwich panel. Figure 5.14 exhibits the effect of the aspect ratio on the frequency ratio for fixed values of ply angles. There appears to be a continual increase in the frequency ratio for all cases considered up to a peak value followed by a decrease which asymptotically approaches a frequency ratio value of 1. Additionally, the larger peak values pertaining to the frequency ratio seem to rest with the fiber orientations of 30, 45, and 60 degrees.

5.4 Doubly Curved Sandwich Panels

153

Fig. 5.13 The effect of the normalized curvature on the ratio of the shell-to-plate eigenfrequencies Ω1/2 of a cylindrical sandwich panel for various ply angles

Fig. 5.14 The effect of the panel aspect ratio on the ratio of the shell-to-plate eigenfrequencies Ω1/2 of a cylindrical sandwich panel for various curvatures

154

5.5

5 Free Vibration

Summary

A theoretical model applied to the free vibration of flat and curved sandwich panels with anisotropic laminated face sheets and a weak/soft core has been presented. Flat panels were considered in the first part which then followed with a discussion of the free vibration of curved sandwich panels in the second part. In both cases, very intricate analytical solution methodologies were applied. The theory was then validated against other cases found in the literature. Remarkable close agreement was seen. Finally numerical results were presented which included the effects of the orthotropicity ratios of both the core and the face sheets, fiber orientation angles within the face sheets, the aspect ratio of the panel, the curvature ratio, the core thickness-to face thickness ratio, and other additional geometrical parameters. Also, the concept of the eigenfrequencies vanishing at the critical buckling load was discussed. Additionally, all of these effects on the frequency ratio were addressed. It was found that the eigen-frequencies of a sandwich shell are higher than for a sandwich plate.

References Hause, T., Librescu, L., & Johnson, T. F. (1998). Thermomechanical load-carrying capacity of sandwich flat panels. Journal of Thermal Stresses, 21(6), 627–653. Hause, T., Johnson, T. F., & Librescu, L. (2000). Effect of face-sheet anisotropy on buckling and post buckling of flat sandwich panels. Journal of Spacecraft and Rockets, 37(3), 331–341. Hohe, J., Librescu, L., & Oh, S. Y. (2006). Dynamic buckling of flat and curved sandwich panels with transversely compressible core. Composite Structures, 74(1), 10–24. Librescu, L. (1987). Refined geometrically non-linear theories of anisotropic laminated shells. Quarterly of Applied Mathematics, 45(1), 1–22. Librescu, L., Hause, T., & Camarda, C. J. (1997). Geometrically nonlinear theory of initially imperfect sandwich plates and shells incorporating non-classical effects. AIAA Journal, 35(8), 1393–1403. Monforton, G. R., & Schmidt, L. A., Jr. (1968). Finite element analysis of sandwich plates and cylindrical shells with laminate faces. Proceedings of the conference on matrix methods on structural mechanics, TR-68-150, Air Force Flight Dynamics Lab., Wright Patterson Air Force Base, OH, pp. 573–616. Rahmani, O., Khalili, S. M. R., & Malekzadeh, K. (2010). Free vibration response of composite sandwich cylindrical shell with flexible core. Composite Structures, 92, 1269–1281. Raville, M. E., & Ueng, C. E. S. (1967). Determination of natural frequencies of vibration of a sandwich plate. Experimental Mechanics, 7(4), 490–493. Zhou, H. B., & Li, G. Y. (1996). Free vibration analysis of sandwich plates with laminated faces using spline finite point method. Computers & Structures, 59(2), 257–263.

Chapter 6

Dynamic Response to Time-Dependent External Excitations

Abstract An advanced theoretical model pertaining to sandwich structures characterized by anisotropic laminated facings and a weak orthotropic core is addressed, in regard to the dynamic response problem. In this regard, several time-dependent external pressure pulses are considered which lend themself to a closed form solution, for both the eigenfrequencies and the dynamic response problem, utilizing the Laplace transform and the extended Galerkin method. Finally, a detailed analysis considering the influence of several parameters is conducted, with numerical results covering a broad spectrum of responses to various material or geometrical configurations presented with a thorough discussion of the results.

6.1

Introduction

This chapter is concerned with the dynamic response behavior of doubly curved sandwich panels experiencing forced vibration. The first part of the chapter is concerned with the dynamic response of sandwich plates followed by doubly curved sandwich shells. Solution methodologies are applied which allows for a closed form solution for both the eigenfrequencies and the dynamic response. Such closed form solutions involve the Laplace Transform and the extended Galerkin method. The dynamic response considers various time-dependent external loadings for both plates and shells with anisotropic laminated face sheets. The implications of the panel curvature, anisotropy and stacking sequence of the face sheets, the orthotropy of the core, and the structural damping are considered as to their effects on the structural response to time-dependent stimuli, of the structure. Various types of timedependent loads are considered such as the sonic boom, triangular pulse, Heaviside step function, rectangular pulse, the sine pulse, a tangential traveling air blast, and the Friedlander in-Air explosive pulse, numerical results are presented which covers a broad spectrum of responses to various material or geometrical configurations.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 T. J. Hause, Sandwich Structures: Theory and Responses, https://doi.org/10.1007/978-3-030-71895-4_6

155

156

6.2

6 Dynamic Response to Time-Dependent External Excitations

Preliminaries and Basic Assumptions

In consideration of the dynamic response of sandwich structures, the incompressible core case will be considered for both the plate and shell configuration. With this in mind the following basic assumptions are adopted: 1. The face sheets are orthotropic layers not necessarily coincident with the geometrical axes. 2. The core features orthotropic properties in the transverse direction and is considered the weak-type and much larger in thickness than the facings. 3. Perfect bonding between the face sheets and the facings and the core are assumed. 4. The shallow shell theory is assumed. 5. The transverse shear effects in the facings are discarded. 6. The face sheets are symmetric with respect to their local and global mid-surfaces. 7. The linearized counterpart of the equations of motion, Eqs. (2.156)–(2.160) are adopted. 8. The tangential and rotatory inertia terms are neglected (Librescu 1987).

6.3 6.3.1

Flat Sandwich Panels Governing System

In this section, the governing equations which are considered for the dynamic response problem of flat sandwich panels are based on the linear theory for flat sandwich panels assuming symmetric laminated facings, a weak orthotropic core, and with the transverse shear effects discarded in the facings. With these considerations in mind, Eqs. (2.174)–(2.176) are applicable along with the set of simply supported boundary conditions expressed in Eqs. (2.177a–d). As mentioned in Chap. 2, for flat panels, setting the curvatures to zero decouples the first two equations of motion from the last three. The last three equations of motion govern the bending problem separate from the stretching problem. This implies that, in the case of flat sandwich plates, the dynamic response problem is only dependent on the bending problem. This chapter will be concerned with both flat and curved sandwich panels. The dynamic response problem is an extension of the free vibration problem in that the transverse pressure loading term p3 6¼ 0. The same governing equations apply except with the transverse pressure loading term p3 included. This allows for the ability to include several time-dependent loading scenarios into the equations. Neglecting the thermal terms, the governing equations of motion are

6.3 Flat Sandwich Panels

157

• Equations of Motion
2 
2  2 2 2 2 2 ∂ η1 ∂ η1 ∂ η2 ∂ η2 ∂ η2 ∂ η1 ∂ η2 þ A þ þ A þ 2 þ A þ A 66 12 16 26 ∂x1 ∂x2 ∂x1 ∂x2 ∂x2 ∂x2 ∂x1 ∂x2 ∂x21 ∂x22
1  2 ∂v d1 η1 þ a 3 ¼ 0 ∂x1 A11

ð6:1aÞ


2

2

2

∂ η2 ∂ η2 ∂ η1 þ A66 þ ∂x22 ∂x22 ∂x1 ∂x2
 ∂v3 d2 η2 þ a ¼0 ∂x2



A22




2

þ A12

2

2

∂ η1 ∂ η1 ∂ η2 þ A26 þ2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x22



2

þ A16

∂ η1 ∂x21

ð6:1bÞ
d1 a

2 ∂η1 ∂ v þ a 23 ∂x1 ∂x1 4

2F 66 Þ

 d2 a

2 ∂η2 ∂ v þ a 23 ∂x2 ∂x2

4



4

þ F 11

4

2

4

∂ v3 ∂ v3 þ F 22 þ 2ðF 12 þ ∂x41 ∂x42 2

∂ v3 ∂ v ∂ v3 ∂ v3 ∂ v3 þ 4F 16 3 3 þ 4F 26 þ N 011 þ 2N 012 þ 2 2 3 2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 2

N 022






2

∂ v3 ∂v ∂ v  c 3  m0 23 ¼ P3 ðtÞ ∂t ∂t ∂x22 ð6:1cÞ

This leaves three equations in terms of three unknowns, η1, η2, v3. These unknown variables need to be determined in such a manner as to satisfy the corresponding equations of motion, while at the same time satisfying the boundary conditions. The governing equations of motion and the boundary conditions constitute the governing system for the dynamic response of plates. For simply supported boundary the following conditions hold. at x1 ¼ 0, L1 η1 ¼ η2 ¼ M 11 ¼ v3 ¼ 0

ð6:2a  dÞ

η1 ¼ η2 ¼ M 22 ¼ v3 ¼ 0

ð6:3a  dÞ

at x2 ¼ 0, L2

6.3.2

Solution Methodology

To identically satisfy the fourth and fifth boundary conditions, Eq. (6.2-2f) and Eq. (6.2-3f), the transverse displacement v3(t) is assumed in the following form

158

6 Dynamic Response to Time-Dependent External Excitations

v3 ðx1 , x2 , t Þ ¼ wmn ðt Þ sin λm x1 sin μn x2

ð6:4Þ

where λm ¼ mπ/L1, μn ¼ nπ/L2 and wmn(t) is the amplitude of deflection as a function of time. Regarding η1, η2, they are assumed in the following form.   1Þ 2Þ cos λm x1 sin μn x2 þ H ðmn sin λm x1 cos μn x2 η1 ðx1 , x2 , t Þ ¼ wmn ðt Þ H ðmn   1Þ 2Þ cos λm x1 sin μn x2 þ I ðmn sin λm x1 cos μn x2 η2 ðx1 , x2 , t Þ ¼ wmn ðt Þ I ðmn

ð6:5Þ ð6:6Þ

1Þ 2Þ ð1Þ 2Þ , H ðmn , I mn and I ðmn are undetermined coefficients. These coefficients are where, H ðmn easily determined by the use of Eqs. (6.5)–(6.6) in Eqs. (6.1a, b) and comparing coefficients of like trigonometric functions. These coefficients are identical to the ones previously determined in earlier sections. The first two equations of motion, Eqs. (6.1a, b) and the boundary conditions, Eqs. (6.2d) and (6.3d) are identically fulfilled. There remains the third equation of motion and the remaining unfulfilled boundary conditions. These will be retained in the energy functional and thus by performing the necessary operations will result in fulfilling the last equation of motion and the remaining boundary conditions in an average sense. Inserting these unfilled expressions back into Hamilton’s equation from gives t 1 Z l2 Z l1 

Z

d 1 a

t0

0

0 4

2F 66 Þ




2 ∂η1 ∂v þ a 23 ∂x1 ∂x1 4




 d2a 4

2 ∂η2 ∂ v þ a 23 ∂x2 ∂x2 2



4

þ F 11

4

∂ v3 ∂ v3 þ 2ðF 12 þ þ F 22 ∂x41 ∂x42 2

2

∂ v3 ∂ v ∂ v3 ∂ v3 ∂ v3 ∂ v3 þ 4F 16 3 3 þ 4F 26 þ N 011 þ 2N 012 þ N 022 þ ∂x1 ∂x2 ∂x21 ∂x22 ∂x21 ∂x22 ∂x1 ∂x2 ∂x1 ∂x32

 2 ∂v3 ∂ v þ m0 23  P3 ðt Þ δv3 dx1 dx2 dtþ ∂t ∂t +


Z t1 *Z l2  ∂v3

l1 L11 δη1 þ L12 δη2 þ M 11 δ

dx dtþ

0 ∂x1 t0 0 +


Z t1 *Z l1  ∂v3

l2 L12 δη1 þ L22 δη2 þ M 22 δ

dx dt ¼ 0

0 ∂x2 t0 0

C

ð6:7Þ

Substituting in the expressions for L11, L22, L12 along with δη1, δη2, δv3 into Eq. (6.7) and carrying out the indicated operations, results in a second-order differential equation which governs the dynamic response of sandwich plate structures. This governing differential equation is presented as € mn þ 2Δmn ωmn w_ mn þ ω2mn wmn ¼ F mn ðt Þ w where the expression for the natural undamped frequencies, ω2mn is given by

ð6:8Þ

6.3 Flat Sandwich Panels

ω2mn ¼

159

 1 2Þ d aλ H ð1Þ þ d2 aμn I ðmn þ d1 a2 λ2m þ d2 a2 μ2n þ F 11 λ2m þ F 22 μ2n þ 4F 66 λ2m μ2n þ m0 1 m mn  2F 12 λ2m μ2n  N 011 λ2m  N 022 μ2n ð6:9Þ

Or in dimensionless form as a2 L21  F 22 n4 ϕ4 2ðF 12 þ 2F 66 Þm2 n2 ϕ2 þ þ d m 2 þ d 2 n2 ϕ2 þ F 11 F 11 F 11 π 2 1   aL31  1Þ 2Þ þ d1 mH ðmn þ d2 nϕI ðmn þ K x m2 þ LR n2 ϕ2 3 F 11 π ð6:10Þ

Ω2mn ¼ m4 þ

The nondimensional parameters ϕ, K x , LR , Ω2mn were defined previously in Chap. 5. Typically, the undamped natural frequency is expressed as ωmn ¼

rffiffiffiffiffiffiffiffi K mn m0

ð6:11Þ

Leaving Kmn determined as 1Þ 2Þ K mn ¼ d1 aλm H ðmn þ d 2 aμn I ðmn þ d 1 a2 λ2m þ d 2 a2 μ2n þ F 11 λ2m þ F 22 μ2n þ 4F 66 λ2m μ2n þ 2F 12 λ2m μ2n  N 011 λ2m  N 022 μ2n

ð6:12Þ In addition, Δmn ¼ C/2m0ωmn which denotes the modal viscous damping ratio, while the expression for the generalized load, Fmn(t) is given by F mn ðt Þ ¼

16δm,2s1 δn,2q1 p ðt Þ m0 ð2s  1Þð2q  1Þπ 2 3

ð6:13Þ

where,  δm,2s1 ¼

1

m odd

0

m even

ðs ¼ 0, 1, 2, :: . . .Þ

ð6:14Þ

The same holds true for δn, 2q  1. From Eq. (6.13), p3(t) can represent any one of a number of different types of external time-dependent loadings. This will be discussed in sect. 6.5.

160

6 Dynamic Response to Time-Dependent External Excitations

6.4

Doubly Curved Sandwich Panels

6.4.1

Governing System

For doubly curved sandwich shells there is no decoupling present among the governing equations of motion. For this case coupling exists between stretching and bending. With this in mind, there are five coupled equations of motion which constitute the governing system of equations, in addition to the boundary conditions which needs to be fulfilled. For the dynamic response problems simply supported boundary conditions which are freely movable in both the tangential and normal directions will be assumed. There are five unknown displacement functions which are ξ1, ξ2, η1, η2, v3 that need to be determined which satisfy both the chosen boundary conditions and the equations of motion. With this in mind the governing system of equations are • Equations of Motion.
2 
2 
2  2 ∂ ξ1 1 ∂v3 ∂ ξ1 ∂ ξ2 ∂ ξ2 1 ∂v3 A11  þ  þ A66 þ A12 ∂x1 ∂x2 R2 ∂x2 ∂x21 R1 ∂x1 ∂x22 ∂x1 ∂x2

2  2 2 ∂ ξ1 ∂ ξ 1 ∂v3 ∂ ξ2 1 ∂v3  A16 2  þ 22   A26 ¼0 ∂x1 ∂x2 ∂x1 R1 ∂x1 ∂x22 R2 ∂x2 ð6:15aÞ



2 
2  2 2 ∂ ξ2 1 ∂v3 ∂ ξ2 ∂ ξ1 ∂ ξ1 1 ∂v3 A22  þ  þ A66 þ A12 ∂x1 ∂x2 R1 ∂x1 ∂x22 R2 ∂x2 ∂x21 ∂x1 ∂x2

2  2 2 ∂ ξ2 ∂ ξ 1 ∂v3 ∂ ξ1 1 ∂v3 A26 2  þ 21   A16 ¼0 ∂x1 ∂x2 ∂x2 R2 ∂x2 ∂x21 R1 ∂x1 ð6:15bÞ
2 
2  2 2 2 2 ∂ η1 ∂ η1 ∂ η2 ∂ η2 ∂ η2 ∂ η1 þ A66 þ þ A16 þ2 A11 þ A12 ∂x1 ∂x2 ∂x1 ∂x2 ∂x21 ∂x22 ∂x1 ∂x2 ∂x21
 2 ∂ η2 ∂v þA26  d 1 η1 þ a 3 ¼ 0 ∂x1 ∂x22 ð6:15cÞ
2 
2  2 2 2 ∂ η2 ∂ η2 ∂ η1 ∂ η1 ∂ η1 ∂ η2 þ A þ þ A þ 2 A22 þ A 66 12 26 ∂x1 ∂x2 ∂x1 ∂x2 ∂x22 ∂x22 ∂x1 ∂x2 ∂x22
 2 ∂ η1 ∂v þA16  d 2 η2 þ a 3 ¼ 0 2 ∂x2 ∂x1 2

ð6:15dÞ

6.4 Doubly Curved Sandwich Panels

161





    A11 A12 ∂ξ1 A16 A26 ∂ξ1 A22 A12 ∂ξ2 A26 A16 ∂ξ2  þ  þ  þ  þ R1 R2 ∂x1 R1 R2 ∂x2 R2 R1 ∂x2 R2 R1 ∂x1


   2 2 4 ∂η1 ∂η2 A A22 2A12 ∂ v3 ∂ v3 ∂ v3  11 þ þ d a þ a a þ a v  d þ F 3 1 2 11 ∂x1 ∂x2 ∂x21 ∂x22 ∂x41 R21 R22 R1 R2 4

þ F 22

4

4

4

2

∂ v3 ∂ v ∂ v ∂ v3 ∂ v3 þ 2ðF 12 þ 2F 66 Þ 2 3 2 þ 4F 16 3 3 þ 4F 26 þ N 011 þ ∂x42 ∂x1 ∂x2 ∂x21 ∂x1 ∂x2 ∂x1 ∂x32 2

þ 2N 012

2

2

∂ v3 ∂ v3 ∂v ∂ v þ N 022  c 3  m0 23 ¼ P3 ðtÞ 2 ∂x1 ∂x2 ∂t ∂t ∂x2 ð6:15eÞ

• Boundary Conditions. Along the edges xn ¼ 0, Ln e nn N nn ¼ N e nt N nt ¼ N Lnn ¼ e Lnn e Lnt ¼ Lnt

or or or

ξn ¼ e ξn e ξt ¼ ξt

ηn ¼ e ηn or ηt ¼ e ηt ∂v ∂ev 3 e nn M nn ¼ M or ¼ 3 ∂xn ∂xn

  ∂v3 ∂v3 ∂v3 ∂v3 ∂M nn ∂M nt N nt þ þ þ2 þ N nn þ ∂xt ∂xt ∂xn ∂xn ∂xn ∂xt e nt e  ∂M þ a=h N n3 ¼ þ N n3 ∂xt

or

v3 ¼ ev3

ð6:16a  fÞ For simply supported boundary conditions freely movable on all edges at x1 ¼ 0, L1 N 11 ¼ N 12 ¼ η1 ¼ η2 ¼ M 11 ¼ v3 ¼ 0

ð6:17a  fÞ

N 22 ¼ N 12 ¼ η1 ¼ η2 ¼ M 22 ¼ v3 ¼ 0

ð6:18a  fÞ

at x2 ¼ 0, L2

In terms of displacements, the first, second, and fifth boundary conditions can be written as

162

6 Dynamic Response to Time-Dependent External Excitations

N 11 ¼ A11

N 12



 ∂ξ1 ∂ξ ∂ξ2 ∂ξ1 A11 A12 þ A12 2 þ A16 þ þ  v R1 R2 3 ∂x1 ∂x2 ∂x1 ∂x2

¼ 0, ð1⇄2Þ

 ∂ξ2 ∂ξ1 ∂ξ2 ∂ξ1 A16 A26 ¼ A66 þ þ A16  þ þ A26 v ¼0 R1 R2 3 ∂x1 ∂x2 ∂x2 ∂x1

2

∂ v3 ∂ v3 ∂ v3 þ F 12 þ 2F 16 ¼0 ∂x1 ∂x2 ∂x21 ∂x22

M 22 ¼ F 22

∂ v3 ∂ v3 ∂ v3 þ F 12 þ 2F 26 ¼0 ∂x1 ∂x2 ∂x22 ∂x21

2

ð6:20Þ

2

M 11 ¼ F 11

2

6.4.2

2

ð6:19Þ

ð6:21Þ

2

ð6:22Þ

Solution Methodology

The transverse displacement can be assumed in the following form v3 ðx1 , x2 , t Þ ¼ wmn ðt Þ sin λm x1 sin μn x2

ð6:23Þ

v3(x1, x2, t) identically fulfills the sixth boundary conditions provided in Eqs. (6.17f) and (6.18f). To fulfill the first two equations of motion, ξ1 and ξ2 can be assumed in the following form   1Þ 2Þ cos λm x1 sin μn x2 þ F ðmn sin λm x1 cos μn x2 ξ1 ðx1 , x2 , t Þ ¼ wmn ðt Þ F ðmn

ð6:24aÞ

  1Þ 2Þ ξ2 ðx1 , x2 , t Þ ¼ wmn ðt Þ Gðmn cos λm x1 sin μn x2 þ Gðmn sin λm x1 cos μn x2

ð6:24bÞ

1Þ 2Þ 1Þ 2Þ Where F ðmn , F ðmn , Gðmn , Gðmn are coefficients that have been previously determined and provided in Eqs. (5.27) – (5.31a–e). Following in a similar manner, η1 and η2 can be assumed in the following form.

  1Þ 2Þ cos λm x1 sin μn x2 þ H ðmn sin λm x1 cos μn x2 η1 ðx1 , x2 , t Þ ¼ wmn ðt Þ H ðmn   1Þ 2Þ cos λm x1 sin μn x2 þ I ðmn sin λm x1 cos μn x2 η2 ðx1 , x2 , t Þ ¼ wmn ðt Þ I ðmn

ð6:25aÞ ð6:25bÞ

6.4 Doubly Curved Sandwich Panels

163

1Þ 2Þ ð1Þ 2Þ where, the coefficients H ðmn , H ðmn , I mn , and I ðmn have previously been determined in earlier sections. The first four equations of motion, Eqs. (6.15a–d) and the boundary conditions, Eqs. (6.18f) and Eqs. (6.18f) are identically fulfilled. There remains the fifth equation of motion and the remaining unfulfilled boundary conditions. The identical procedure as was carried out for flat plates will be duplicated here for the case of doubly curved sandwich shells. The unfulfilled quantities will be retained in the energy functional and thus by performing the necessary operations will result in fulfilling the last equation of motion and the remaining boundary conditions in an average sense. Inserting these unfilled expressions back into Hamilton’s equation gives





 A A ∂ξ1 A16 A26 ∂ξ1 A22 A12 ∂ξ2 A26 A16 ∂ξ2  11 þ 12  þ þ þ  þ R1 R2 ∂x1 R1 R2 ∂x2 R2 R1 ∂x2 R2 R1 ∂x1 t0 0 0


2  2  4 4 ∂η1 ∂η2 A A 2A ∂ v ∂ v ∂ v3 ∂ v3  11 þ 22 þ 12 v3  d1 a þ a 23  d2 a þ a 23 þ F 11 þ F 22 2 2 4 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x42 R1 R2 R1 R2

Z

t1 Z l2 Z l1 

4

4

4

2

2

2

∂ v3 ∂ v ∂ v ∂ v3 ∂ v3 ∂ v3 þ 4F 16 3 3 1 þ 4F 26 1 3 3 þ N 11 þ þ2N 012 þ N 22 ∂x1 ∂x2 ∂x21 ∂x22 ∂x21 ∂x22 ∂x1 ∂x2 ∂x1 ∂x2  2 ∂v ∂ v þC 3 þ m0 23  P3 ðt Þ δv3 dx1 dx2 dtþ ∂t ∂t +


Z t1 *Z l2  ∂v3

l1 N 11 δξ1 þ N 12 δξ2 þ L11 δη1 þ L12 δη2 þ M 11 δ

dx2 dtþ

0 ∂x1 t0 0 * +


Z t1 Z l1  ∂v3

l2 N 22 δξ2 þ N 12 δξ1 þ L22 δη2 þ L12 δη1 þ M 22 δ

dx1 dt ¼ 0

0 ∂x2 t0 0 þ2ðF 12 þ 2F 66 Þ

ð6:26Þ Substituting in the expressions for ξ1, ξ2, η1, η2, v3 into Eq. (6.26) and carrying out the indicated operations results in a second-order differential equation which governs the dynamic response of shallow sandwich shell structures. This governing differential equation is expressed as € mn þ 2Δmn ωmn w_ mn þ ω2mn wmn ¼ F mn ðt Þ w

ð6:27Þ

It can be seen that this is the same result as was shown in Eq. (6.8) where the undamped natural frequency squared ω2mn is expressed by ω2mn ¼


    1 1Þ 2Þ λ4 F þ 2λ2m μ2n ðF 12 þ 2F 66 Þ þ μ4n F 22 þ λm ad1 H ðmn þ aλm þ μn ad2 I ðmn þ aμn m0 m 11



A A A A A A A 1Þ 2Þ 2Þ þ μn 12 þ 22 Gðmn þ μn 16 þ 26 F ðmn þ λm 16 þ þλm 11 þ 12 F ðmn R1 R2 R1 R2 R1 R2 R1 
  A26 A11 A12 A22 ð1Þ 2 0 2 0 Gmn þ  λm N 11  μn N 22 þ2 þ R2 R1 R2 R22 R21

ð6:28Þ

164

6 Dynamic Response to Time-Dependent External Excitations

Or in dimensionless form as a2 L21  F 22 n4 ϕ4 2ðF 12 þ 2F 66 Þm2 n2 ϕ2 þ þ d m 2 þ d 2 n2 ϕ2 þ F 11 F 11 F 11 π 2 1  aL31  L2 h 1Þ 2Þ 1Þ d1 mH ðmn þ d2 nϕI ðmn þ þ 1 3 mðψ 1 A11 þ ψ 2 ϕA12 ÞF ðmn 3 F 11 π F 11 π i   2Þ 1Þ nϕðψ 1 A12 þ ψ 2 ϕA22 ÞG2mn þ nϕF ðmn þ mGðmn ðψ 1 A16 þ ψ 2 ϕA26 Þ þ

Ω2mn ¼ m4 þ

 L21  2 ψ A þ ψ 22 ϕ2 A22 þ 2A12 ψ 1 ψ 2 ϕ  K x m2 ψ 21 þ LR n2 ψ 22 ϕ2 F 11 π 4 1 11 ð6:29Þ Where the nondimensional parameters ψ 1 , ψ 2 , ϕ, K x , LR , Ω2mn were defined previously in Chap. 5. Equation (6.27) can be solved for various pressure pulses via the Laplace transform method. First a discussion and understanding of the various pressure pulses needs to be ascertained. In the next section various types of pressure pulses are discussed.

6.5

Explosive Pressure Pulses and Numerical Results

The dynamic response of structures is an especially important and essential area for understanding the effects of rapid-type, time-dependent loadings on the structural response and how to design the structure to withstand such large stresses generated within the structure under such loadings. Some examples would include aircraft exposed to shockwaves or space vehicles exposed to blast pulses. These types of external stimuli can be very damaging to the structure. It is therefore imperative to gain an understanding of the structural response within these kinds of environments to design against catastrophic failure. Several numerical simulations are presented for each of the various types of pressure pulses which are discussed next. Unless otherwise specified, Tables 6.1 and 6.2 contain the material and geometrical characteristics for the results which follow. The first type of external pulse is the Sonic Boom. This is expressed as follows.

Table 6.1 Material properties for the face sheets and the core Face sheets E1(GPa) 207 Core G13 ðGPaÞ 0.1027

E2(GPa) 5.17

G12(GPa) 2.55

ρ (kg/m3) 1588.22

G23 ðGPaÞ 0.0621

– –

ρ ðkg=m3 Þ 16

ν12 0.25

6.5 Explosive Pressure Pulses and Numerical Results

165

Table 6.2 Geometrical properties Ply Thickness, tk(m) 0.0005

6.5.1

hf (m) 0.002

L1 (m) 0.420

L2 (m) 0.420

2h ðmÞ 0.0250

Δmn 0.05

Sonic Boom ( p3 ð t Þ ¼

 p0 1  t=t p

0 < t < rt p

0

t < 0, t > rtp

ð6:30Þ

p0 denotes the peak reflected pressure in excess of the ambient, tp denotes the positive phase duration of the pulse measured from the time of impact of the structure, and r denotes the shock pulse length factor. If r ¼ 1, the sonic boom becomes a triangular pulse, if r ¼ 2 a symmetric sonic boom is generated, for r 6¼ 2 a nonsymmetric N-Shaped pulse results. If r ¼ 1, tp!1 the N-shaped pulse degenerates into a step pulse. p3(t) can be expressed (see Marzocca et al. 2001; Librescu and Nosier 1990) in terms of the Heaviside function as
p3 ð t Þ ¼ p0

   t  1 H ðt Þ  H t  rt p tp

ð6:31Þ

Substituting this expression into Eq. (6.27) gives € mn þ 2Δmn ωmn w_ mn þ w

ω2mn wmn


 e   F t  ¼ 1 H ðt Þ  H t  rt p m0 tp

ð6:32Þ

where, e ¼ 16p0 δm,2s1 δn,2q1 F ð2s  1Þð2q  1Þπ 2

ð6:33Þ

The Laplace Transform Method in time is used to solve Eq. (6.32), assuming zero initial conditions. The Laplace transform operator is defined as Z L fg ¼

1

fgest dt

ð6:34Þ

0

And is applied to Eq. (6.32) where s is the transform variable. For the case of the sonic boom type pulse, applying the Laplace transform to Eq. (6.32) results in

166

6 Dynamic Response to Time-Dependent External Excitations

  e ð1  r Þertp s ertp s F 1 1   W mn ðsÞ ¼  2  þ ð6:35Þ s t p s2 m0 s þ 2Δmn ωmn s þ ω2mn s t p s2 where the following zero initial conditions were assumed. wmn ð0Þ ¼ w_ mn ðt Þ ¼ 0

ð6:36Þ

Taking the inverse Laplace Transform of Eq. (6.35), to arrive back in the time domain, one obtains 

2  e 2Δmn þ Δmn ωmn t p  1 2Δmn t 2Δmn Δmn ωmn t F 1 þ  1 þ cos Ω t   e  mn t p ωmn t p t p ωmn Ωmn t p m0 ω2mn 
  2Δ t 2Δ eΔmn ωmn t sin Ωmn t  1 þ mn   ð1  r Þ þ mn eΔmn ωmn ðtrtp Þ cos Ωmn t  rt p  t p ωmn t p t p ωmn  
2    2Δmn þ ð1  r ÞΔmn ωmn t p  1 Δmn ωmn ðtrtp Þ  sin Ωmn t  rt p H t  rt p e Ωmn t p

wmn ðt Þ ¼

ð6:37Þ In Eq. (6.37), Ωmn ¼ ωmn

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  Δ2mn

ð6:38Þ

denotes the damped natural frequency. Figure 6.1 illustrates the effect of curvature on the deflection–time response of a doubly curved sandwich panel due to a sonic

Fig. 6.1 The effects of panel curvature on the dynamic response of a sandwich panel due to a sonic boom (r ¼ 2, tp ¼ 0.005, P0 ¼ 5 MPa)

6.5 Explosive Pressure Pulses and Numerical Results

167

Fig. 6.2 The effects of the ply angle on the dynamic response due to a sonic boom (r ¼ 2, tp ¼ 0.005, P0 ¼ 5 MPa, L1/R1 ¼ L2/R2 ¼ 0.4)

boom for a given fixed layup. It appears that the larger curvature ratios are less detrimental providing smaller amplitudes of oscillations and a more rapid decaying response. Figure 6.2 shows the response due to a sonic boom for a doubly curved sandwich panel for various ply angles under a specific stacking sequence in the facings. It clearly shows that the ply angle plays an important role in the amplitude and decay of the response in both the free and forced regimes. The results show that a ply angle of θ ¼ 45 deg appears to be the most beneficial from an amplitude of oscillation standpoint. In Fig. 6.3, the response to a sonic boom for a doubly curved sandwich panel for various amounts of damping is depicted. It is apparent that an increase in the amount of damping leads to a more rapid decay in the response.

6.5.2

Rectangular Pulse  p3 ðt Þ ¼

p0 0

0 < t < tp t > tp

In terms of the Heaviside step function, p3(t) is expressed as

ð6:39Þ

168

6 Dynamic Response to Time-Dependent External Excitations

Fig. 6.3 The implications of damping on the dynamic response due to a sonic boom (r ¼ 2, tp ¼ 0.005, P0 ¼ 5 MPa, L1/R1 ¼ L2/R2 ¼ 0.4)

   p3 ð t Þ ¼ p0 H ð t Þ  H t  t p

ð6:40Þ

Substituting Eq. (6.40) into Eq. (6.27) results in € mn þ 2Δmn ωmn w_ mn þ ω2mn wmn ¼ w

e    F H ðt Þ  H t  t p m0

ð6:41Þ

Taking the Laplace Transform and assuming zero initial conditions, the response in the Laplace domain is W mn ðsÞ ¼

e mn 1 F  ð1  etp s Þ m0 s s2 þ 2Δmn ωmn s þ ω2mn

ð6:42Þ

Arriving back in the time domain by taking the inverse Laplace transform gives 8 >
> > b μ11 > 11 > > > > > > > > 7 = = < < B t C t t 7 t B C 7 b b μ22 ΔM ð7:36Þ Q26 7 @ γ 22 A  λ22 ΔT  b > > > > > > > 5 > ; > : t > > t t t > ; : γ b μ12 k b 66 bλ 12 k Q 12 k k bt Q 11

3

0

γ t11

1

b ij for i, j ¼ (1, 2, 6) are the where k represents the kth lamina in the facing and Q transformed plane-stress reduced stiffness measures (Reddy (2004) and Jones (1999)). These were given in Chap. 2.
 Bottom face sheets t2c  x3  t2c þ t bf 2 1 bb Q τb11 6 11 B b C 6 @ τ22 A ¼ 6 4 τb12 k Sym 0

b b11 Q b b22 Q

9 8 8 b 9 1 > bλb > b > > μ11 > γ > > > 11 11 = =
> 5 > ; > : b > b > b b > ; : γ b μ12 k 12 b 66 bλ k Q 12 k k b b11 Q

3

0

The stress–strain relationships for the orthotropic weak (soft) core with the geometrical and material axes coincident are expressed as   Core  t2c  x3  t2c 0

1 2 c τc33 Q33 B c C 6 @ τ23 A ¼ 4 0 τc13

0

0 Qc44 0

30 c 1 γ 33 0 7B c C 0 5@ γ 23 A Qc55 γ c13

ð7:38Þ

The core transverse and normal moduli are given as Qc33 ¼ Ec ,

Qc44 ¼ Gc23 ,

Qc55 ¼ Gc13

ð7:39a  cÞ

7.4 Hamilton’s Principle

193

where Ec is the Young’s modulus and Gc13 , Gc23 are the shear moduli for the core which allows the core stresses to be expressed as τc33 ¼ E c γ c33 ,

7.4

τc13 ¼ G13 γ c13 ,

τc23 ¼ G23 γ c23

ð7:40a  cÞ

Hamilton’s Principle

An energy approach using Hamilton’s Principle is used to derive the equations of motion, as was shown in Chap. 2. Letting U represent the strain energy, W represent the work done by external forces, and T represent the kinetic energy, Hamilton’s Principle (see Soedel (2004)) is expressed as Z

t1

δJ ¼

ðδU  δW  δT Þdt ¼ 0

ð7:41Þ

t0

7.4.1

Strain Energy

Utilizing Eq. (2.98) from Chap. 2 and assuming a weak compressible core where the core carries only the transverse shear stresses, the variation in the strain energy is given by ! Z þt c Z tc þtb Z Z t c 2 2 2 f τtαβ δγ tαβ dx3 þ τci3 δγ ci3 dx3 þ τbαβ δγ bαβ dx3 dA ð7:42Þ δU ¼ A

t c t 2 t f

tc 2

tc 2

where τij are the tensorial components of the second Piola–Kirchhoff stress tensor, while A is attributed to the planar area of the sandwich shell. The strain energy in the tensorial form is expanded out and shown as Z (Z δU ¼

tc 2

A

Z

þ þ

tc 2

t2c

Z

tc b 2 þt f



 τb11 δγ b11 þ τb22 δγ b22 þ 2τb12 δγ b12 dx3 þ

 c c  2τ13 δγ 13 þ 2τc23 δγ c23 þ τc33 δγ c33 dx3 þ

t2c

t2c t tf



τt11 δγ t11

þ

τt22 δγ t22

þ

2τt12 δγ t12



ð7:43Þ

) dx3 dA

Substituting in the expressions for the strain relationships, Eqs. (7.14a–c)– (7.16a–c) results in

7 Theory of Sandwich Plates and Shells with an Transversely Compressible. . .

194

Z (Z δU ¼

tc b 2 þt f

"

tc 2

A

þτb22



δγ a22



τb11

δγ d22



þ

! t c þ t bf x3  δκ a22  2

þ2τb12 δγ a12  δγ d12 þ Z þ Z þ

tc 2



t2c

δγ d11

þ

! t c þ t bf x3  δκ a11  2

δγ a11

! ! t c þ t bf d x3  δκ 11 2

! ! t c þ t bf d x3  δκ 22 2

! ! !# b t c þ t bf t þ t c f x3  δκ a12  x3  δκ d12 dx3 2 2

      2τc13 δγ c 13 þ x3 δκ c13 þ 2τc23 δγ c 23 þ x3 δκ c23 þ τc33 δγ c33 þx3 δκ c33 dx3

t2c t2c t tf



      t c þ t tf t c þ t tf τt11 δγ a11 þ δγ d11 þ x3 þ δκ a11 þ x3 þ δκ d11 2 2 

     t c þ t tf t c þ t tf a d t þ þ x3 þ δκ 22 þ x3 þ δκ 22 þ 2τ12 δγ a12 2 2        t c þ t tf t c þ t tf þδγ d12 þ x3 þ δκ a12 þ x3 þ x3 þ δκ a12 δκ d12 dx3 dA 2 2 

þτt22

δγ a22

δγ d22

ð7:44Þ which can be expressed in the following form as Z δU ¼ A



N b11 δγ a11  N b11 δγ d11 þ M b11 δκ a11  M b11 δκ d11 þ N b22 δγ a22  N b22 δγ d22

þM b22 δκ a22  M b22 δκ d22 þ 2N b12 δγ a12  2N b12 δγ d12 þ 2M b12 δκ a12  2M b12 δκ d12 þ2N c13 δγ c13 þ 2M c13 δκ c13 þ 2N c23 δγ c23 þ 2M c23 δκ c23 þ N c33 δγ c33 þ M c33 δκ c33 þN t11 δγ a11 þ N t11 δγ d11 þ M t11 δκ a11 þ M t11 δκ d11 þ N t22 δγ a22 þ N t22 δγ d22 þ M t22 δκ a22 þM t22 δκ d22 þ 2N t12 δγ a12 þ 2N t12 δγ d12 þ 2M t12 δκ a12 þ 2M t12 δκ d12 dA ð7:45Þ where the local stress resultants N tαβ , N bαβ , N ci3 M tαβ , M bαβ , M ci3 are defined as n o Z t t N αβ :M αβ ¼

t2c

t2c t tf

τtαβ

and the stress couples

  t c þ t tf 1, x3 þ dx3 2

ð7:46aÞ

7.4 Hamilton’s Principle

195

n o Z N bαβ :M bαβ ¼

tc b 2 þt f

tc 2



N ci3 , M ci3



( τbαβ

Z ¼

t c þ t bf 1, x3  2

tc 2

t2c

!) dx3

τci3 ð1, x3 Þdx3

ð7:46bÞ

ð7:46cÞ

Combining like terms gives Z



δU ¼ A

       N t11 þ N b11 δγ a11 þ N t11  N b11 δγ d11 þ M t11 þ M b11 δκ a11 þ M t11  M b11 δκ d11



       þ N t22 þ N b22 δγ a22 þ N t22  N b22 δγ d22 þ M t22 þ M b22 δκ a22 þ M t22  M b22 δκ d22         þ2 N t12 þ N b12 δγ a12 þ2 N t12  N b12 δγ d12 þ2 M t12 þ M b12 δκ a12 þ2 M t12  M b12 δκ d12  þN c33 δγ c33 þ M c33 δκ c33 þ 2N c13 δγ c13 þ 2M c13 δκ c13 þ 2N c23 δγ c23 þ2M c23 δκ c23 dA

ð7:47Þ Utilizing the definition of global stress resultants and global stress couples, as defined below, in Eqs. (7.49a and 7.49b), allows the variation in the strain energy to be written as Z δU ¼ A

 a a 2N 11 δγ 11 þ 2N d11 δγ d11 þ 2M a11 δκ a11 þ 2M d11 δκ d11 þ 2N a22 δγ a22 þ 2N d22 δγ d22 þ

2M a22 δκ a22 þ 2M d22 δκ d22 þ 4N a12 δγ a12 þ 4N d12 δγ d12 þ 4M a12 δκ a12 þ 4M d12 δκ d12 þ N c33 δγ c33 þ M c33 δκ c33 þ 2N c13 δγ c13 þ 2M c13 δκ c13 þ 2N c23 δγ c23 þ 2M c23 δκ c23 dA ð7:48Þ where the global stress resultants and global stress couples are defined as
 n

o 1 N tαβ þ N bαβ , M tαβ þ M bαβ N aαβ , M aαβ ¼ 2

ð7:49aÞ


 n

o 1 N tαβ  N bαβ , M tαβ  M bαβ N dαβ , M dαβ ¼ 2

ð7:49bÞ

Substituting in the strain–displacement equations, Eqs. (7.18)–(7.35) into Eq. (7.48) results in

7 Theory of Sandwich Plates and Shells with an Transversely Compressible. . .

196

Z " δU ¼ A

2N a11

(         ∂ δua1 ∂ δua3 ∂u∘ a3 ∂ud3 ∂ δud3 δua3 ∂ua3 ∂ δua3  þ þ þ R1 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1

) (         ∂ δua3 ∂u∘ d3 ∂ δud1 δud3 ∂ud3 δ ua3 ∂ua3 ∂ δud3 d þ  þ þ 2N 11 þ R1 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 )     2 a 2 d  ∂ δua3 ∂u∘ d3 ∂ δud3 ∂u∘ a3 a ∂ δu3 d ∂ δu3 þ þ  2M 11  2M 11 ∂x1 ∂x1 ∂x1 ∂x1 ∂x21 ∂x21 þ2N a22

(         ∂ δua2 ∂ δua3 ∂u∘ a3 ∂ud3 ∂ δud3 δua3 ∂ua3 ∂ δua3  þ þ þ R2 ∂x2 ∂x2 ∂x2 ∂x2 ∂x2 ∂x2 ∂x2

) (         ∂ δua3 ∂u∘ d3 ∂ δud2 δud3 ∂ud3 ∂ δua3 ∂ua3 ∂ δud3 d þ  þ þ þ 2N 22 þ R2 ∂x2 ∂x2 ∂x2 ∂x2 ∂x2 ∂x2 ∂x2 )     2 a 2 d  ∂ δua3 ∂u∘ d3 ∂ δud3 ∂u∘ a3 a ∂ δu3 d ∂ δu3 þ þ  2M 22  2M 22 ∂x2 ∂x2 ∂x2 ∂x2 ∂x22 ∂x22 þ4N a12

(           ∂ δua1 ∂ δua2 ∂ δua3 ∂ua3 ∂ua3 ∂ δua3 ∂ δud3 ∂ud3 þ þ þ þþ ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2

         ) ∂ δua3 ∂u∘ a3 ∂u∘ a3 ∂ δua3 ∂ δud3 ∂u∘ a3 ∂u∘ d3 ∂ δud3 ∂ud3 ∂ δud3 þ þ þ þ þ ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 þ4N d12

(           ∂ δud1 ∂ δud2 ∂ δua3 ∂ud3 ∂ua3 ∂ δud3 ∂ δua3 ∂ud3 þ þ þ þ ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 ∂x2 ∂x1

)          2 ∂ δua3 ∂u∘ d3 ∂u∘ a3 ∂ δud3 ∂ δua3 ∂u∘ d3 ∂ δua3 ∂ua3 ∂ δud3 þ þ þ þ  4M a12 ∂x1 ∂x2 ∂x1 ∂x2 ∂x2 ∂x1 ∂x1 ∂x2 ∂x2 ∂x1 

2 ∂ δud3 2 4 4∘ þ N c33  δud3 þ 2 ud3 δud3 þ 2 ud3 δud3 tc tc tc ∂x1 ∂x2 ( !!   !   t b t b ∂ δua3 1 d 1 1 tf þtf 1 t f  t f ∂ δud3 c þ2N 13  δu1 þ þ  þ tc 2 tc tc 4 4 ∂x1 ∂x1

4M d12

)  a  a ∘ 1 ∂ua3 d 1 d ∂ δu3 1 ∘ a ∂ δu3 1 ∂ua3 d  δu  u  u3  δu t c ∂x1 3 t c 3 ∂x1 tc t c ∂x1 3 ∂x1 ( þ2M c13

      4 c 1 ∂ δud3 2 ∂ud3 d 2 d ∂ δud3 2 ∘ d ∂ δud3  þ δu þ u þ u δΦ t 2c 1 t c ∂x1 t 2c ∂x1 3 t 2c 3 ∂x1 t 2c 3 ∂x1

7.4 Hamilton’s Principle ∘

2 ∂ud þ 2 3 δud3 t c ∂x1

197

(

) þ 2N c23

1 1 1 þ  δud2 þ tc 2 tc

t tf þ t bf 4

!!   ∂ δua3 ∂x2

!       ∂ δud3 1 ∂ua3 d 1 d ∂ δua3 1 ∘ ∂ δua3  δu3  u3  ua3 t c ∂x2 tc tc ∂x2 ∂x2 ∂x2 ) (     ∘ 1 ∂ua3 d 4 c 1 ∂ δud3 2 ∂ud3 d 2 d ∂ δud3 c  δu þ 2M 23 2 δΦ2  þ 2 δu þ u t c ∂x2 3 t c ∂x2 tc t c ∂x2 3 t 2c 3 ∂x2 )#   ∘ 2 ∘ d ∂ δud3 2 ∂ud3 d þ 2 u3 þ 2 δu dA ð7:50Þ tc t c ∂x2 3 ∂x2 1 þ tc

t tf  t bf 4

Integrating each appropriate term by parts, combining coefficients of like variational displacements and simplifying gives the variation in the strain energy as  a   a  ∂N 11 ∂N a12 ∂N 22 ∂N a12 þ þ δua1  2 δua2 ∂x1 ∂x2 ∂x2 ∂x1 0 0  a   a  ∂N 11 ∂N a12 N c13 ∂N 22 ∂N a12 N c23 2 þ þ þ þ δud1  2 δud2 tc tc ∂x1 ∂x2 ∂x2 ∂x1 ( !     2 a 2 ∘a ∂ u ∂ u 8 c 8 1 3 þ 2 M 13 δΦc1 þ 2 M c23 δΦc2 þ 2N a11 þ 23 þ R1 tc tc ∂x21 ∂x1 Z

δU ¼

l2 Z l1 

2

2∘

2

4N a12

∂ ua3 ∂ ua3 þ ∂x1 ∂x2 ∂x1 ∂x2

2

!  2N a22 2

2

2 ∘a

2

∂ ua3 ∂ u3 1 þ þ R2 ∂x22 ∂x22 2∘

∂ ua3 ∂ ua3 ∂ M a12 ∂ M a22 4 2  2N d11 þ 2 ∂x1 ∂x2 ∂x2 ∂x21 ∂x21 2

2N d22

2∘

∂ ua3 ∂ ua3 þ ∂x22 ∂x22

! 2  tc

!

!

2

2 2

 4N d12

t b tc t f þ t f þ  ud3  ∘ud3 2 4

∂ M a11 ∂x21 2∘

∂ ud3 ∂ ud3 þ ∂x1 ∂x2 ∂x1 ∂x2

!

∂N c13 ∂N c23 þ ∂x1 ∂x2

!

!

     a  ∂u3 ∘∂ua3 4 ∂ud3 ∘∂ud3 4 ∂ud3 ∘∂ud3 c c þ þ þ þ N 23 þ N 13  2 t c ∂x2 t c ∂x1 ∂x2 ∂x1 ∂x1 ∂x1      a    ∂ua3 ∘∂ua3 ∂ud3 ∘∂ud3 ∂N 11 ∂N a12 ∂N a22 ∂N a12  þ þ þ þ 2 2 ∂x2 ∂x1 ∂x1 ∂x1 ∂x1 ∂x2 ∂x2 ∂x2 )  d   d  d  ∂u3 ∘∂ud3 ∂N 11 ∂N d12 ∂N 22 ∂N d12  þ þ 2 þ δua3 ∂x1 ∂x2 ∂x2 ∂x2 ∂x2 ∂x1

7 Theory of Sandwich Plates and Shells with an Transversely Compressible. . .

198

(

! ! 2 a 2 ∘a 2 a 2 ∘a ∂ u ∂ u ∂ u ∂ u 1 3 3 3 3 þ 2N d11 þ þ þ  4N d12 R1 ∂x1 ∂x2 ∂x1 ∂x2 ∂x21 ∂x21 ! 2 2∘ 2 2 2 ∂ ua3 ∂ ua3 ∂ M d11 ∂ M d12 ∂ M d22 1 d 2N 22 þ þ  4  2  2 R2 ∂x1 ∂x2 ∂x22 ∂x22 ∂x21 ∂x22 ! ! ! 2 2∘ 2 2∘ 2 d 2∘ ∂ ud3 ∂ ud3 ∂ ud3 ∂ ud3 ∂ ud3 a a ∂ u3 þ þ þ  4N 12 2N 22 ∂x1 ∂x2 ∂x1 ∂x2 ∂x21 ∂x21 ∂x22 ∂x22 !  t b 
 t  t ∂N c13 ∂N c23 4 tc 4 f f d d c  þ  u3  ∘u3 N 33 þ þ tc 2 tc 16 ∂x1 ∂x2

2N a11

∘

∂ud3 ∂ud3 2 þ ∂x1 ∂x1 ∘

∂ua3 ∂ua3 2 þ ∂x1 ∂x1

!

∂N a11 ∂N a12 þ ∂x1 ∂x2



∘

∂ud3 ∂ud3 þ 2 ∂x2 ∂x2

!  ∂N a22 ∂N a12 þ ∂x2 ∂x1

! !  ∘ ∂ua3 ∂ua3 ∂N d11 ∂N d12 N c13 þ þ þ 2 tc ∂x1 ∂x2 ∂x2 ∂x2

) +  d 
∂M c ∂M c  ∂N 22 ∂N d12 N c23 4 tc 13 23  þ þ  ud3  ∘ud3 þ þ 2 δud3 dx1 dx2 tc tc 2 ∂x2 ∂x1 ∂x1 ∂x2 Z

l2

þ

*

( 2N a11 δua1

0

 þ2N a12

þ

2N a12 δua2

∘

∂ua3 ∂ua3 þ ∂x2 ∂x2

þ

!

2N d11 δud2

þ ∘

þ

2N d11

2N d12 δud2

∂ud3 ∂ud3 þ ∂x1 ∂x1

þ

∘

2N a11

!

∂ua3 ∂ua3 þ ∂x1 ∂x1 ∘

þ

∂ud3 ∂ud3 þ ∂x2 ∂x2

2N d12

!

!

" # ) t b ∂M a11 ∂M a12 2 t c t f þ t f þ2 þ4 þ þ  ud3  ∘ud3 N c13 δua3 tc 2 4 ∂x1 ∂x2 ( þ

∘

2N a11

∂ud3 ∂ud3 þ ∂x1 ∂x1 ∘

þ2N d12

∂ua3 ∂ua3 þ ∂x2 ∂x2

!

!

∘

þ

2N d11

∂ua3 ∂ua3 þ ∂x1 ∂x1

!

∂M d11 ∂M d12 1 þ4  þ2 tc ∂x1 ∂x2

∘

þ

2N a12

∂ud3 ∂ud3 þ ∂x2 ∂x2

!

! t tf  t bf N c13 4

)  a  d 
  l1 ∂u3 ∂u3 2 tc d d c d a d  dx2  2  u3  ∘u3 M 13 δu3 þ 2M 11 δ þ 2M 11 δ 0 tc 2 ∂x1 ∂x1 Z

l1

þ 0

*

( 2N a22 δua2

þ

2N a21 δua1

þ

2N d22 δud1

þ

2N d21 δud1

þ

∘

2N a22

∂ua3 ∂ua3 þ ∂x2 ∂x2

!

7.4 Hamilton’s Principle

 þ2N a21

199

∘

∂ua3 ∂ua3 ∂x1 ∂x1

!

∘

þ

∂ud3 ∂ud3 þ ∂x2 ∂x2

2N d22

!

∘

þ

∂ud3 ∂ud3 þ ∂x1 ∂x1

2N d21

!

" # ) t b ∂M a22 ∂M a21 2 t c t f þ t f d d þ2 þ4 þ þ  u3  ∘u3 N c23 δua3 tc 2 4 ∂x2 ∂x1 (

∘

∂ud3 ∂ud3 þ ∂x2 ∂x2

þ 2N a22

∘

þ2N d21

∂ua3 ∂ua3 þ ∂x1 ∂x1

!

!

∘

∂ua3 ∂ua3 þ ∂x2 ∂x2

þ 2N d22

!

∘

þ 2N a21

∂ud3 ∂ud3 þ ∂x1 ∂x1

!

! t tf  t bf N c23 4

∂M d22 ∂M d21 1 þ4  þ2 tc ∂x2 ∂x1

)  a  d 
 ∂u3 ∂u3  l2 2 tc d d c d a d dx1  u3  ∘u3 M 23 δu3 þ2M 22 δ  2 þ2M 22 δ tc 2 ∂x2 ∂x2  0 ð7:51Þ

7.4.2

Work Done by External Loads

The total work considered in this text that is performed on an elastic body is the summation of the work due to body forces, edge loads, surface tractions, and damping. This is expressed mathematically as W total ¼ W body

forces

þ W edge

loads

þ W surface

tractions

þ W Damping

ð7:52Þ

• Work due to body forces Z (Z δW b ¼

σ

hþh=

ρ

=

= = H i δV i dx3

Z þ

h

h h

Z ρH i δV i dx3 þ

h hh==

) ρ

==

== == H i δV i dx3

dσ ð7:53Þ

where ρ is the mass density and Hi is the body force vector. It should be noted that the body forces will not be included in the following developments. Typical body forces are gravity, electrical and magnetic forces which will be neglected. • Work due to edge loads W edge

loads

¼ W tedge

loads

þ W cedge

loads

þ W bedge

loads

ð7:54Þ

7 Theory of Sandwich Plates and Shells with an Transversely Compressible. . .

200

By definition, the work due to edge loads starts with the basic expression below. Considering the contribution from each layer, the total work due to edge loads is expressed as Z

Z

δW el ¼

t c 2

t c t 2 t f

x1

Z

þ Z

Z þ

 t t  bτ22 δv2 þ bτt21 δvt1 dx3 þ

tc b 2 þt f

tc 2 tc 2

tc t 2 t f

x2

Z þ

tc 2

tc 2

Z

tc 2 t c 2

bτc23 δvc3

!
 b b bτ22 δvb2 þ bτ21 δvb1 dx3 dx1  t t  bτ11 δv1 þ bτt12 δvt2 dx3

bτc13 δvc3

Z þ

tc b 2 þt f tc 2

ð7:55Þ

!
 b b b b bτ11 δv1 þ bτ12 δv2 dx3 dx2

Substituting in the expressions for the displacements, Eqs. (7.6a–c)–(7.8a–c) while considering the expressions for the local stress resultants and stress couples, defined earlier in Eqs. (7.46a–c), give Z 

 a  d t t t t ∂u3 ∂u3 a d b b b b t21 δud1 þ b b t21 δua1 þ N δW el ¼ N 22 δu2 þ N 22 δu2  M 22 δ  M 22 δ þN ∂x2 ∂x2 x1  a  d t t c c ∂u ∂u 3 3 b 23 δud3 þ N b b22 δua2  N b b22 δud2 b 21 δ b 23 δua3  2 M b 21 δ M þN M tc ∂x1 ∂x1  a  d  a b b b b 22 δ ∂u3 þ M b b21 δud1  M b b21 δ ∂u3 þ b 22 δ ∂u3 þ N b 21 δua1  N M ∂x2 ∂x2 ∂x1  d  b b21 δ ∂u3 dx1 þM ∂x1  a  d Z  b t11 δua1 þ N b t11 δud1  M b t11 δ ∂u3  M b t12 δud2 þ b t11 δ ∂u3 þ N b t12 δua2 þ N N ∂x ∂x 1 1 x2  a  d t t ∂u3 ∂u3 b c δud þ N b b11 δua1  N b b11 δud1 b b c13 δua3  2 M b  M 12 δ þN M 12 δ t c 13 3 ∂x2 ∂x2  a  d  a b b b b b ∂u3 ∂u3 ∂u3 a d b b b b b M 11 δ þ M 11 δ þ N 12 δu2  N 12 δu2  M 12 δ þ ∂x1 ∂x1 ∂x2  d  b b12 δ ∂u3 dx2 þM ∂x2 ð7:56Þ In the above equation, Eq. (7.56), gathering coefficients of identical variational displacements utilizing the definition of global stress and couple resultants from Eqs. (7.49a and 7.49b) while simplifying gives

7.4 Hamilton’s Principle

201

! b a21 c ∂M b 23 δua3 W el ¼ þ þ þ þ 2 þN ∂x1 x1 !  a  d  b c23 b d21 M a d ∂u3 ∂u3 ∂M d b b δu3  2M 22 δ þ 2   2M 22 δ dx1 t ∂x1 ∂x2 ∂x2 c ! Z b a12 a d a d c ∂ M a d a d b 11 δu1 þ 2N b 11 δu1 þ 2N b 12 δu2 þ 2N b 12 δu2 þ 2 b 13 δua3 þ 2N þN ∂x 2 x2 !  a  d  b c13 b d12 M a d ∂u3 ∂u3 ∂M d b b þ 2  δu3  2M 11 δ 2M 11 δ dx2 t ∂x2 ∂x1 ∂x1 c Z

b a22 δua2 2N

b d22 δud2 2N

b a21 δua1 2N

b d21 δud1 2N

ð7:57Þ where
a  n
t 
t o b αβ , M b αβ þ N b aαβ ¼ 1 N b bαβ , M b bαβ b αβ þ M N 2

ð7:58Þ


d  n
t 
t o b αβ  N b dαβ ¼ 1 N b bαβ , M b bαβ b αβ , M b αβ  M N 2

ð7:59Þ

• Work due to surface tractions The expression for the work done by lateral vertical loading such as an external pressure is by definition given as Z
 b W st ¼ qb3 δvb3 dA qt3 δvt3 þ b ð7:60Þ A

Substituting Eqs. (7.6c) and (7.7c) into Eq. (7.60) gives W st ¼

Z
    b qt3 δua3 þ δud3 þ b qb3 δua3  δud3 dA

ð7:61Þ

A

Simplifying results in Z n
Z

 o  b b W st ¼ qb3 δua3 þ b qt3  b qb3 δud3 dA ¼ 2 qd3 δud3 dA ð7:62Þ qt3 þ b qa3 δua3 þ b A

A

where b qb3 qt3 þ b 2 t b b  qb3 q b qd3 ¼ 3 2 b qa3 ¼

ð7.63a, bÞ

7 Theory of Sandwich Plates and Shells with an Transversely Compressible. . .

202

• Work due to damping By definition, the work due to damping for the transverse direction is given by Z Wd ¼ 



A

 C t v_ t3 δvt3 þ C c v_ c3 δvc3 þ C b v_ b3 δvb3 dA

ð7:64Þ

where Ct, Cc and Cb are the structural damping coefficients per unit area of the facings and the core. Substituting in vt3 , vc3 and vt3 from Eqs. (7.6c), (7.7c), and (7.8c) gives Z  t a      C u_ 3 þ u_ d3 δua3 þ δud3 þ C c u_ a3 δua3 þ C b u_ a3  u_ d3 δua3  δud3 dA Wd ¼  A

ð7:65Þ It should be mentioned that damping is assumed to be constant throughout the thickness of the core. As a result, simplifying and gathering like variational terms gives Z  t      Wd ¼  C þ C b þ C c u_ a3 δua3 þ C t  Cb u_ a3 δud3 þ Ct  C b u_ d3 δua3 þ 

A

 C þ Cb u_ d3 δud3 dA t

ð7:66Þ which can be written as    Z   d a  d Cc a a d d a a d Wd ¼ 2 C þ u_ 3 þ C u_ 3 δu3 þ C u_ 3 þ C u_ 3 δu3 dA 2 A ð7:67Þ where Ca ¼

Ct þ Cb 2

Ct  Cb Cd ¼ 2

ð7.68a, bÞ

Ca and Cd are the average and half difference of the damping coefficients with regard to the top and bottom facings.

7.4 Hamilton’s Principle

7.4.3

203

Kinetic Energy

By definition, the kinetic energy is given by 1 T¼ 2

" 2  2  2 # ∂V 1 ∂V 2 ∂V 3 ρ þ dx1 dx2 dx3 ∂t ∂t ∂t x1

Z Z Z x3 x2

ð7:69Þ

Taking a variation in the expression for the kinetic energy and integrating with respect to time gives Z

t1

t1 Z

Z δTdt ¼ 

t0

σ

t0

Z þ

t2c t2c tt f

"Z

tc b 2 þt f tc 2

Z
b  € b2 δV b2 þ V € b3 δV b3 dx3 þ € 1 δV b1 þ V ρb V

#
t  € t2 δV t2 þ V € t3 δV t3 dx3 dσdt € 1 δV t1 þ V ρt V

tc 2

t2c

 c c  € c2 δV c2 þ V € c3 δV c3 dx3 € 1 δV 1 þ V ρc V

ð7:70Þ If the in-plane inertias are neglected and only the transverse inertias are considered, the kinetic energy can be expressed as Z

t1

δTdt ¼ 

t0

"Z

t1 Z

Z t0

A

tc b 2 þt f

tc 2

€ b3 δV b3 dx3 ρb V

Z þ

tc 2

t2c

€ c3 δV c3 dx3 ρc V

Z þ

t2c t2c t tf

# € t3 δV t3 dx3 ρt V

dAdt

ð7:71Þ Using the expressions for the displacements for each respective layer, from Eqs. (7.6c), (7.7c), and (7.8c) give Z

t1 t0

Z (Z

  2 a d δTdt ¼  ρ þ þ ρ € u3  x 3 € u3 dx3 þ tc t0 A t2c t tf t2c !   Z tc þtb 2 f   a  2 a d b a d d  δu3  x3 δu3 dx3 þ ρ €u3  € u3 δu3  δu3 dx3 dAdt tc tc Z

t1

t2c

t



€ua3

€ud3



δua3

δud3



Z

tc 2

c

2

ð7:72Þ Integrating and simplifying give Z t0

t1



 mc a a a €u3 δu3 þ md € δTdt ¼ 2 m þ ua3 δud3 þ md € ud3 δua3 2 t0 A    mc d d €u3 δu3 dAdt þ ma þ 3 Z

t1 Z

ð7:73Þ

7 Theory of Sandwich Plates and Shells with an Transversely Compressible. . .

204

where ma ¼

mtf þ mbf , 2

md ¼

mtf  mbf 2

ð7.74a, bÞ

and Z m c ¼ ρc t c ,

7.5

mtf ¼

t2c

t2c t tf

Z ρtðkÞ dx3 ,

mbf ¼

tc 2

tc b 2 þt f

ρbðkÞ dx3

ð7:75a  cÞ

Governing Equations

7.5.1

Equations of Motion

Substituting δU, δW, δT back into Hamilton’s equation gives  a   a  ∂N 11 ∂N a12 ∂N 22 ∂N a12 þ þ δua1  2 δua2 ∂x1 ∂x2 ∂x2 ∂x1 t0 0 0  a   a    ∂N 11 ∂N a12 N c13 ∂N 22 ∂N a12 N c23 8 2 þ þ þ þ δud1  2 δud2 þ 2 M c13 δΦc1 tc tc tc ∂x1 ∂x2 ∂x2 ∂x1 ( ! !   ∘ 2 2 2 2∘ ∂ ua3 ∂ ua3 ∂ ua3 ∂ ua3 8 c 1 c a a þ 2 M 23 δΦ2 þ 2N 11 þ þ þ  4N 12 tc ∂x1 ∂x2 ∂x1 ∂x2 ∂x21 ∂x21 R1 ! 2 a 2 ∘a 2 2 2 ∂ u ∂ u ∂ M a11 ∂ M a12 ∂ M a22 1 3 3 2N a22 þ þ  4  2  2 ∂x1 ∂x2 ∂x22 ∂x22 R2 ∂x21 ∂x22 ! ! ! 2 2∘ 2 2∘ 2 2∘ ∂ ua3 ∂ ua3 ∂ ud3 ∂ ud3 ∂ ua3 ∂ ua3 d d d 2N 11 þ þ þ  4N 12  2N 22 ∂x1 ∂x2 ∂x1 ∂x2 ∂x21 ∂x21 ∂x22 ∂x22 ! !  ∘ t b ∂N c13 ∂N c23 2 tc t f þ t f 4 ∂ud3 ∂ud3 ∘d d  þ þ þ  u3  u3 þ N c23 tc 2 t c ∂x2 ∂x2 4 ∂x1 ∂x2 ! !  ∘ ∘ ∂ua3 ∂ua3 ∂N a11 ∂N a12 4 ∂ud3 ∂ud3 c þ þ þ þ N 13  2 t c ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂x2 ! !  d  a   ∘ ∘ ∂ua3 ∂ua3 ∂ud3 ∂ud3 ∂N 22 ∂N a12 ∂N 11 ∂N d12 2 þ þ þ þ 2 ∂x2 ∂x2 ∂x2 ∂x1 ∂x1 ∂x1 ∂x1 ∂x2

Z

t 1 Z l 2 Z l 1

2

7.5 Governing Equations

!

205

   ∂N d22 ∂N d12 Cc a a a u_ 3  2C d u_ d3 þ  2b q3  2 C þ 2 ∂x2 ∂x1 ) ( ! 2 2∘
 ∂ ua3 ∂ ua3 mc a 1 a d d a d €u  2m €u3 δu3 þ 2N 11 2 m þ þ þ 2 3 ∂x21 ∂x21 R1 ! ! 2 2∘ 2 2∘ 2 2 ∂ ua3 ∂ ua3 ∂ ua3 ∂ ua3 ∂ M d11 ∂ M d12 1 d d 4N 12 þ þ þ 4  2N 22 2 2 2 2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x2 ∂x2 R2 ∂x1 ! ! 2 2∘ 2 2∘ 2 ∂ ud3 ∂ ud3 ∂ ud3 ∂ ud3 ∂ M d22 a a 2  2N þ þ  4N 11 12 ∂x1 ∂x2 ∂x1 ∂x2 ∂x22 ∂x21 ∂x21 ! !  2 2∘ t b 
 ∂ ud3 ∂ ud3 ∂N c13 ∂N c23 4 tc 4 tf tf ∘d a d c 2N 22 þ  u þ þ  u N  3 3 33 tc 2 tc 16 ∂x1 ∂x2 ∂x22 ∂x22 ! !  a   a  ∘ 2∘ ∂ud3 ∂ ud3 ∂ud3 ∂ud3 ∂N 11 ∂N a12 ∂N 22 ∂N a12 2 þ þ þ þ 2 ∂x1 ∂x1 ∂x1 ∂x2 ∂x2 ∂x2 ∂x2 ∂x1 ! !     ∘ ∘ ∂ua3 ∂ua3 ∂N d11 ∂N d12 N c13 ∂ua3 ∂ua3 ∂N d22 ∂N d12 N c23 2 þ þ þ þ þ 2 þ tc tc ∂x1 ∂x1 ∂x1 ∂x2 ∂x2 ∂x2 ∂x2 ∂x1  
 ∂M c ∂M c 4 t 13 23 þ 2 c  ud3  ∘ud3 þ  2b qd3  2C d u_ a3  2C a u_ d3 tc 2 ∂x1 ∂x2 ) # +
 m 2 ma þ c €ud3  2md €ua3 δud3 dx1 dx2 dt 6 * ( ! Z t1 Z l2 ∘a a ∂u ∂u 3 þ 2N a11 δua1 þ 2N a12 δua2 þ 2N d11 δud2 þ 2N d12 δud2 þ 2N a11 þ 3 ∂x1 ∂x1 t0 0 ! ! !  a ∘ ∘ ∘ ∂u3 ∂ua3 ∂ud3 ∂ud3 ∂ud3 ∂ud3 ∂M a11 a d d þ2N 12 þ þ þ þ 2N 11 þ 2N 12 þ2 ∂x2 ∂x2 ∂x1 ∂x1 ∂x2 ∂x2 ∂x1 " # ) t b ∂M a12 2 t c t f þ t f ∘ þ4 þ þ  ud3  ud3 N c13 δua3 tc 2 4 ∂x2 ( ! ! ! ∘ ∘ ∘ ∂ud3 ∂ud3 ∂ua3 ∂ua3 ∂ud3 ∂ud3 a d a þ 2N 11 þ þ þ þ 2N 11 þ 2N 12 ∂x1 ∂x1 ∂x1 ∂x1 ∂x2 ∂x2 ! ! ∘ t b ∂ua3 ∂ua3 ∂M d11 ∂M d12 1 t f  t f d þ2N 12 N c13 þ þ þ4  þ2 tc 4 ∂x2 ∂x2 ∂x1 ∂x2 )  a  d 
  l1 ∂u3 ∂u3 2 tc d d c  dx2 dt  2  u3  ∘u3 M 13 δud3 2M a11 δ þ 2M d11 δ 0 tc 2 ∂x1 ∂x1 ∘

∂ud3 ∂ud3 2 þ ∂x2 ∂x2

7 Theory of Sandwich Plates and Shells with an Transversely Compressible. . .

206

! ∘a a ∂u ∂u 3 þ 2N a22 δua2 þ 2N a21 δua1 þ 2N d22 δud1 þ 2N d21 δud1 þ 2N a22 þ 3 ∂x2 ∂x2 t0 0 ! ! !  a ∘ ∘ ∘ ∂u3 ∂ua3 ∂ud3 ∂ua3 ∂ud3 ∂ud3 ∂M a22 a d d þ2N 21 þ þ þ þ 2N 22 þ 2N 21 þ2 ∂x1 ∂x1 ∂x2 ∂x2 ∂x1 ∂x1 ∂x2 " # ) t b ∂M a21 2 t c t f þ t f þ4 þ þ  ud3  ∘ud3 N c23 δua3 tc 2 4 ∂x1 ( ! ! ! ∘ ∘ ∘ ∂ud3 ∂ud3 ∂ua3 ∂ua3 ∂ud3 ∂ud3 a d a þ 2N 22 þ þ þ þ 2N 22 þ 2N 21 þ 2N d21 ∂x2 ∂x2 ∂x2 ∂x2 ∂x1 ∂x1 ! ! ) ∘ t b
 ∂ua3 ∂ua3 ∂M d22 ∂M d21 1 t f  t f 2 tc ∘d c d c þ þ4  N 23  2  u3  u3 M 23 δud3 þ2 tc 4 tc 2 ∂x1 ∂x1 ∂x2 ∂x1  a  d   l2 ∂u3 ∂u3 a d  dx1 dt ¼ 0 þ2M 22 δ ð7:76Þ þ 2M 22 δ 0 ∂x2 ∂x2 Z

t 1 Z l1

*

(

This equation can only be satisfied by setting the coefficients of the variational displacements equal to zero since the variational displacements are arbitrary. Setting the coefficients of δua1 , δua2 , δud1 , δud2 , δΦc1 , δΦc2 , δua3 , and δud3 to zero provides the equations of motion along with the corresponding boundary conditions. These are provided as δua1 :

∂N a11 ∂N a12 þ ¼0 ∂x1 ∂x2

ð7:77Þ

δua2 :

∂N a12 ∂N a22 þ ¼0 ∂x1 ∂x2

ð7:78Þ

δud1 :

∂N d11 ∂N d12 N c13 þ þ ¼0 tc ∂x1 ∂x2

ð7:79Þ

δud2 :

∂N d12 ∂N d22 N c23 þ þ ¼0 tc ∂x1 ∂x2

ð7:80Þ

δΦc1 :

M c13 ¼ 0

ð7:81Þ

δΦc2 :

M c23 ¼ 0

ð7:82Þ

7.5 Governing Equations

δua3 :

207

! ! 2 2∘ 2 2∘ ∂ ua3 ∂ ua3 ∂ ua3 ∂ ua3 1 a þ þ þ þ þ N 22 ∂x1 ∂x2 ∂x1 ∂x2 ∂x22 ∂x22 R2 ! ! 2 2∘ 2 2∘ 2 2 2 ∂ ud3 ∂ ud3 ∂ ud3 ∂ ud3 ∂ M a11 ∂ M a12 ∂ M a22 d d þ þ2 þ þ N 11 þ þ þ 2N 12 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x21 ∂x22 ∂x21 ∂x21 ! !  2 2∘ t b ∂ ud3 ∂ ud3 ∂N c13 ∂N c23 1 tc t f þ t f ∘d d d þ þN 22 þ  u3  u3 þ þ 4 tc 2 ∂x1 ∂x2 ∂x22 ∂x22 ! !   ∘ ∘ 2 ∂ud3 ∂ud3 2 ∂ud3 ∂ud3 Cc a a c c a u_ 3 þ C d u_ d3 þ þ q3 þ C þ N 23  N 13 þ b  2 t c ∂x2 ∂x2 t c ∂x1 ∂x1
 m þ ma þ c €ua3 þ md €ud3 ¼ 0 2 ð7:83Þ 2

N a11

2∘

∂ ua3 ∂ ua3 1 þ þ 2 2 R ∂x1 ∂x1 1

δud3 :

!

2N a12

! ! 2 2∘ 2 2∘ ∂ ua3 ∂ ua3 ∂ ua3 ∂ ua3 1 d þ þ þ þ þ N 22 ∂x1 ∂x2 ∂x1 ∂x2 ∂x22 ∂x22 R2 ! ! 2 2∘ 2 2∘ 2 2 2 ∂ ud3 ∂ ud3 ∂ ud3 ∂ ud3 ∂ M d11 ∂ M d12 ∂ M d22 a a þ2 þ þ N 11 þ þ þ 2N 12 þ ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x21 ∂x22 ∂x21 ∂x21 ! !  2 d 2 ∘d t b 
 t  t ∂ u ∂ u ∂N c13 ∂N c23 2 t 1 ∘ f f c a d c d 3 3 þN 22  u3  u3 N 33  þ þ þ 8t c tc 2 3 ∂x1 ∂x2 ∂x22 ∂x22 2

N d11

2∘

∂ ua3 ∂ ua3 1 þ þ R1 ∂x21 ∂x21

!

2N d12


 m þb qd3 þ C d u_ a3 þ C a u_ d3 þ ma þ c €ud3 þ md €ua3 ¼ 0 6

ð7:84Þ The corresponding boundary conditions become uan ¼ b uan

or

b nn N ann ¼ N

a

ð7:85Þ

uat uat ¼ b

or

b ant N ant ¼ N

ð7:86Þ

udn ¼ b udn

or

b dnn N dnn ¼ N

ð7:87Þ

7 Theory of Sandwich Plates and Shells with an Transversely Compressible. . .

208

udt ¼ b udt

ua3

or

b dnt N dnt ¼ N

ð7:88Þ

! ! ! ∘ ∘ ∘ ∂ua3 ∂ua3 ∂ua3 ∂ua3 ∂ud3 ∂ud3 a a ¼ or þ þ þ N nn þ N nt þ N dnn þ ∂xn ∂xn ∂xt ∂xt ∂xn ∂xn ! ! ∘ ∂ud3 ∂ud3 ∂M ant 1 t c t tf þ t bf ∂M ann ∘d d d þ þ2 þ þ  u3  u3 N c13 N nt þ tc 2 4 ∂xt ∂xt ∂xn ∂xt b ua3

b nt 1 c ∂M b þ N 2 n3 ∂xt a

¼

ð7:89Þ

ud3

! ! ! ∘ ∘ ∘ ∂ud3 ∂ud3 ∂ud3 ∂ud3 ∂ua3 ∂ua3 a a ¼ or þ þ þ N nn þ N nt þ N dnn þ ∂xn ∂xn ∂xt ∂xt ∂xn ∂xn ! ! ∘ ∂ua3 ∂ua3 ∂M dnt 2 Φc3 t tf  t bf ∂M dnn d N cn3 ¼ N nt þ þ þ2   tc 3 8 ∂xt ∂xt ∂xn ∂xt b ud3

b b nt M ∂M  n3 tc ∂xt d

c

ð7:90Þ

7.5.2

∂ua3 ∂b ua ¼ 3 ∂xn ∂xn

or

b ann M ann ¼ M

ð7:91Þ

∂ud3 ∂b ud ¼ 3 ∂xn ∂xn

or

b nn M dnn ¼ M

ð7:92Þ

d

Boundary Conditions

Where n and t are the normal and tangential directions to the boundary, when n ¼ 1, t ¼ 2 and when n ¼ 2, t ¼ 1. For the case of simply supported boundary conditions the following conditions must be satisfied. Along the edges xn ¼ 0, Ln N ann ¼ N dnn ¼ N ant ¼ N dnt ¼ M ann ¼ M dnn ¼ ua3 ¼ ud3 ¼ 0

ð7:93Þ

For the case of clamped boundary conditions, the following conditions must be satisfied.

7.6 The Stress Resultants, Stress Couples, and Stiffnesses

uan ¼ uat ¼ udn ¼ udt ¼ ua3 ¼ ud3 ¼

7.6

209

∂ua3 ∂ud3 ¼ ¼0 ∂xn ∂xn

ð7:94Þ

The Stress Resultants, Stress Couples, and Stiffnesses

The local stress resultants and stress couples can be expressed in terms of the strain measures by substituting Eqs. (7.36)–(7.38) into Eqs. (7.46a–c). For the top face, the matrix form is 1 0 t N t11 A11 B Nt C B B 22 C B B t C B B N 12 C B C B B B Mt C ¼ B B 11 C B B t C B @ M 22 A @ 0

M t12

At12 At22

At16 At26 At66

Bt11 Bt12 Bt16

Bt12 Bt22 Bt26

Dt11

Dt12 Dt22

Sym

10 t 1 0 t 1T 0 t 1m N 11 N 11 γ 11 Bt16 B γt C B N t C B Nt C Bt26 C CB 22 C B 22 C B 22 C CB C B C B t C BN C Bt66 CB 2γ t12 C B N t12 C CB CB C  B 12 C B B Mt C t C t CB t C D16 CB κ 11 C B M 11 C B 11 C B t C B t C B t C t C @ M 22 A D26 A@ κ 22 A @ M 22 A t t t 2κ 12 M 12 M t12 D66

ð7:95Þ where the local stiffnesses Atij , Btij , Dtij are defined as
 Z Atij , Btij , Dtij ¼

t2c

t2c t tf

bt Q ij

(     ) t c þ t tf t c þ t tf 2 dx3 1, x3 þ , x3 þ 2 2

ð7:96Þ

For the bottom face, the local stress resultants and couples in matrix form become 1 0 b N b11 A11 B Nb C B B 22 C B B b C B B N 12 C B C B B B Mb C ¼ B B 11 C B B b C B @ M 22 A @ M b12 Sym 0

Ab12 Ab22

Ab16 Ab26

Bb11 Bb12

Bb12 Bb22

Ab66

Bb16 Db11

Bb26 Db12 Db22

10 b 1 0 b 1T 0 b 1m γ 11 N 11 N 11 Bb16 B B b C b C b CB b C γ N N B26 CB 22 C B 22 C B 22 C B b C B b C B b C b C C B C C B B B66 C CB 2γ 12 C  B N 12 C  B N 12 C B κb C B M b C B Mb C Db16 C CB 11 C B 11 C B 11 C CB C B C B b C @ M 22 A Db26 A@ κ b22 A @ M b22 A b b b 2κ 12 M 12 M b12 D66

ð7:97Þ where the local stiffnesses Abij , Bbij , Dbij are defined as




Abij , Bbij , Dbij ¼

Z

tc b 2 þt f tc 2

8 < b

! !2 9 t c þ t bf t c þ t bf = b 1, x3  Q dx , x3  ij 2 2 : ; 3

ð7:98Þ

210

7 Theory of Sandwich Plates and Shells with an Transversely Compressible. . .

If the structural tailoring of the faces is such that h thei stacking sequence is ðt,bÞ symmetric with respect to its local mid-surface, then Bij ¼ 0. The stress resultants and stress couples for the core are expressed as 0

N c33

1

0

Ac33

B c C B @ N 23 A ¼ @ 0 0 N c13 0 c 1 0 c D33 M 33 B c C B M ¼ @ 23 A @ 0 0 M c13

0 Ac44 0 0 Dc44 0

0

10

γ c33

1

CB C 0 A@ 2γ c23 A Ac55 2γ c13 10 c 1 κ33 0 CB C 0 A@ 2κc23 A Dc55 2κc13

ð7:99aÞ

ð7:99bÞ

where the core stiffnesses are given by 

Acii , Dcii



Z ¼

tc 2

t2c

  Qcii 1, x23 dx3 ,

i ¼ ð3, 4, 5Þ

ð7:100Þ

Note: No summation with respect to the repeated indices is assumed. The core transverse and normal moduli are given in Eqs. (7.39a–c).

7.7

Comments

To-date only a few applications have been addressed utilizing these state-of-the-art theoretical developments. Such areas include buckling, dynamic buckling, post-buckling, and the transient dynamic response problem (Hohe and Librescu (2003, 2006) and Hause (2012)). As an example, on how to apply these equations, the dynamic response problem is presented in the next section. This example is not meant to be an exhaustive treatment of possible applications of this theory nor is it meant to be an exhaustive treatment of the dynamic response. Several results have been compiled by Hohe and Librescu (2003, 2006), where they considered the buckling, post-buckling, and the transient dynamic buckling problem. The reader is referred to these authors to see additional applications of this theory as well as the results.

7.8 7.8.1

Application – Dynamic Response of Flat Sandwich Panels Preliminaries

With the theoretical foundation in hand some basic assumptions are in order toward the dynamic response of flat sandwich panels. The following assumptions are made. The tangential deformations are assumed negligible in contrast to the transverse

7.8 Application – Dynamic Response of Flat Sandwich Panels

211

direction where the deformations can be appreciable. The face sheets are assumed incompressible, while the core can exhibit extensibility in the transverse direction. Likewise, the tangential and rotary inertias are assumed to be negligible, whereas the transverse inertia is retained. With this in mind, the following additional basic assumptions are adopted: 1. The face sheets are orthotropic layers not necessarily coincident with the geometrical axes. 2. The core features orthotropic properties in the transverse direction and is considered the weak-type and much larger in thickness than the facings. 3. Perfect bonding between the face sheets and the facings and the core are assumed. 4. The transverse shear effects in the facings are discarded. 5. The face sheets are symmetric with respect to their local and global mid-surfaces. 6. All time-dependent external pressures are uniformly distributed over the panel face.

7.8.2

Governing Equations

The equations of motion, Eqs. (7.77)–(7.84), are provided below which will be referred to for the solution methodology. These equations have been modified for the case of symmetric facings with respect to the global and their respective local mid-surfaces. For this specialized case of symmetry, the following relationships hold.
 t b t b 2t tf ¼ 2t bf t  t t þ t f f f f td ¼ ¼ 0, t a ¼ ¼ ¼ t f , at ¼ ab ¼ a 2 2 2 tc þ t f ¼ 2

ð7:101Þ

In addition, the transverse pressure, b qb3 has been discarded. As a result of these relationships, the nonlinear equations of motion become δua1 :

∂N a11 ∂N a12 þ ¼0 ∂x1 ∂x2

ð7:102aÞ

δua2 :

∂N a12 ∂N a22 þ ¼0 ∂x1 ∂x2

ð7:102bÞ

∂N d11 ∂N d12 N c13 þ þ ¼0 tc ∂x1 ∂x2

ð7:102cÞ

δud1 :

7 Theory of Sandwich Plates and Shells with an Transversely Compressible. . .

212

δud2 :

∂N d12 ∂N d22 N c23 þ þ ¼0 tc ∂x1 ∂x2

ð7:102dÞ

δΦc1 : M c13 ¼ 0

ð7:102eÞ

δΦc2 : M c23 ¼ 0

ð7:102fÞ

δua3 : 2

2

2

2

2

2 ∂ ua3 a ∂ ua3 a ∂ M a11 ∂ M a12 ∂ M a22 ∂ u3 a N þ 2 N 12 þ N 22 þ þ2 þ þ 11 2 2 2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x22  2 2 2
∂N c ∂ ua3 d ∂ ua3 d ∂ ua3 d ∂N c23 1 tc þ t f d 13 N þ2 N þ N þ þ  u3 2 ∂x1 ∂x2 12 ∂x1 ∂x2 ∂x21 11 ∂x22 22 t c   d d  2 ∂u3 c 2 ∂u3 c mc a  f €u3  C þ C c u_ a3 þ b  N  N  mf þ qt3 ¼ 0 t c ∂x1 13 t c ∂x2 23 2

ð7:102gÞ δud3 : 2

2

2

2

2

2

2 ∂ ua3 d ∂ ua3 d ∂ M d11 ∂ M d12 ∂ M d22 ∂ ud3 a ∂ u3 d N 11 þ 2 N 12 þ N 22 þ þ2 þ þ N 2 2 2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x22 ∂x21 11   2 2
 ∂ ud3 a ∂ ud3 a 2 tc mc d d c f € þ2 N 12 þ N þ  m þ qt3 ¼ 0  u u3  C f u_ d3 þ b N 3 33 3 ∂x1 ∂x2 ∂x22 22 t c 2

ð7:102hÞ Simply supported boundary conditions will be assumed. These are specified as Along the edges x1 ¼ (0, L1) N a11 ¼ N d11 ¼ N a12 ¼ N d12 ¼ M a11 ¼ M d11 ¼ ua3 ¼ ud3 ¼ 0

ð7:103a  hÞ

Along the edges x2 ¼ (0, L2) N a22 ¼ N d22 ¼ N a21 ¼ N d21 ¼ M a22 ¼ M d22 ¼ ua3 ¼ ud3 ¼ 0

ð7:104a  hÞ

Equations (7.8-2a–h)–(7.104a–h) constitute the governing system to determine the dynamic response for flat sandwich panels with symmetric thin facings and a thick orthotropic weak transversely compressible core.

7.8 Application – Dynamic Response of Flat Sandwich Panels

7.8.3

213

Solution Methodology

As a beginning to the solution process for the dynamic response, ua3 , ud3 can be assumed in the following form as ua3 ¼ wamn ðt Þ sin λm x1 sin μn x2

ð7:105aÞ

ud3 ¼ wdmn ðt Þ sin λm x1 sin μn x2

ð7:105bÞ

where λm ¼ mπ/L1, μn ¼ nπ/L2, m and n are the number of half sine waves in the respective directions, whereas wamn ðt Þ and wdmn ðt Þ denote the modal amplitudes as a function of time of the transverse displacement functions. It should be noted that Hohe and Librescu considered the mode numbers and modal amplitudes as independent of one another. For the current case they are not considered to be independent. For the case where they are assumed independent, they would take the following form     ua3 ¼ wamn sin λam x1 sin μan x2

 ud3 ¼ wdpq sin λdp x1 sin μdq x2 where λam ¼

mπ nπ , μan ¼ , L1 L2

λdp ¼

pπ qπ , μdq ¼ L1 L2

With the current representation, the transverse geometric boundary conditions are identically fulfilled. The transverse pressure is represented by b qt3 ðx1 , x2 , t Þ ¼ qmn ðt Þ sin λm x1 sin μn x2

ð7:106Þ

Integrating both sides of Eq. (7.106) over the plate area such that 4 qmn ðt Þ ¼ L1 L2

Z 0

L2 Z L1

qt ðx1 , x2 , t Þ sin λm x1 sin μn x2 dx1 dx2

ð7:107Þ

0

gives qmn ðt Þ ¼

16qt mnπ 2

ð7:108Þ

Equations (7.8-2a, b) can be fulfilled by assuming a stress potential for N a11 , N a22 , N a12 in the following form.

7 Theory of Sandwich Plates and Shells with an Transversely Compressible. . .

214

2

N a11 ¼

2

∂ φ , ∂x22

N a22 ¼

2

∂ φ , ∂x21

N a12 ¼ 

∂ φ ∂x1 ∂x2

ð7:109a  cÞ

Another variable φ has been thrown into the governing system of equations. This variable will be determined from a compatibility equation. Eliminating the tangential strains from the in-plane displacements ua1 and ua2 results in 2

2

2

∂ γ a11 ∂ γ a12 ∂ γ a22 2 þ ¼ 2 ∂x1 ∂x2 ∂x2 ∂x21



∂ua3 ∂x1 ∂x2

2



∂ud3 þ ∂x1 ∂x2

2

2



2

2

2

∂ ua3 ∂ ua3 ∂ ud3 ∂ ud3  ∂x21 ∂x22 ∂x21 ∂x22

ð7:110Þ Since symmetry exists in the facings locally and globally the stress resultants from Eqs. (7.95) and (7.97) can be expressed in decoupled form as 0

N a11

1

2

Aa11

B a C 6 @ N 22 A ¼ 4 Sym N a12

Aa12 Aa22

30 a 1 Aa16 γ 11 a 7B a C A26 5@ γ 22 A Aa66 2γ a12

ð7:111Þ

By performing a matrix inversion of Eq. (7.111) utilizing Eqs. (7.109a–c) and substituting γ a11, γ a22, and γ a12 into Eq. (7.110) results in the compatibility equation in terms of the variables φ, ua3 , and ud3 . This appears as  ∂ φ ∂ φ   ∂ φ ∂ φ ∂ φ þ 2A12 þ A66 þ A22 4  2A16  2A26 3 4 2 2 3 ∂x2 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2  2  2 2 a 2 a 2 d 2 d a d ∂u3 ∂u3 ∂ u3 ∂ u3 ∂ u3 ∂ u3 ¼ þ   ∂x1 ∂x2 ∂x1 ∂x2 ∂x21 ∂x22 ∂x21 ∂x22 A11

4

4

4

4

4

ð7:112Þ where, [A] ¼ [A]1. Since ua3 and ud3 are known, φ can be determined by assuming the following form in terms of unknown coefficients then substituting into Eq. (7.112) and solving for the unknown coefficients. This assumed form is expressed in terms of trigonometric functions as  2  2  2 φðx1 , x2 , t Þ ¼ wamn C1 cos 2λm x1 þ wamn C 2 cos 2μn x2 þ wdmn C3 cos 2λm x1 þ  d 2 wmn C 4 cos 2μn x2 ð7:113Þ After substituting Eq. (7.113) into Eq. (7.112) and comparing coefficients, C1  C4 are determined to be

7.8 Application – Dynamic Response of Flat Sandwich Panels

215


2 
2
2 f f f f f f f f f f f μ2n A66 A11 A22  A12 A16 A26  A11 A26  A22 A16 þ 2A12

C1 ¼ C3 ¼
2  f f f 32λ2m A11 A66  A16 ð7:114Þ


2 
2
2 f f f f f f f f f f f λ2m A66 A11 A22  A12 A16 A26  A11 A26  A22 A16 þ 2A12

C2 ¼ C4 ¼
2  f f f 2 32μn A22 A66  A26 ð7:115Þ Thus far, φ, ua3 , ud3 are known. Next ua1 and ua2 will be determined from the stress resultants expressed in terms of displacements. From Eq. (7.111), N a11 and N a22 expressed in terms of displacements are N a11

(  2  2 ) 2 ∂ua1 1 ∂ua3 ∂ φ 1 ∂ud3 a ¼ 2 ¼ A11 þ þ 2 ∂x1 ∂x1 2 ∂x1 ∂x2 (  a 2  d 2 ) a ∂u ∂u3 ∂u 1 1 3 2 þAa12 þ þ 2 ∂x2 ∂x2 2 ∂x2

∂ua1 ∂ua2 ∂ua3 ∂ua3 ∂ud3 ∂ud3 þ þ þ ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 (  a 2  d 2 ) 2 a ∂u ∂u3 ∂u ∂ φ 1 1 3 2 ¼ 2 ¼ Aa22 þ þ 2 ∂x2 ∂x2 2 ∂x2 ∂x1 þAa16

N a22

( þAa12

þAa26

ð7.116a, bÞ

 2  2 ) ∂ua1 1 ∂ua3 1 ∂ud3 þ þ 2 ∂x1 ∂x1 2 ∂x1

∂ua1 ∂ua2 ∂ua3 ∂ua3 ∂ud3 ∂ud3 þ þ þ ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2



Equations (7.8-16a, 7.116a, b) are two coupled inhomogeneous partial differential equations in terms of two unknowns ua1 and ua2 which can be assumed in the following form

7 Theory of Sandwich Plates and Shells with an Transversely Compressible. . .

216




ua1 ðx1 , x2 , t Þ ¼



wamn wamn




wamn wamn

2


  2  2  2  þ wdmn D1 x1 þ wamn þ wdmn D2 sin 2λm x1 þ

2


  2  2  2  þ wdmn D3 sin 2μn x2 þ wamn þ wdmn D4 sin 2λm x1 cos 2μn x2

2

 2  þ wdmn D5 cos 2λm x1 sin 2μn x2


ua2 ðx1 , x2 , t Þ ¼


wamn

wamn

2


  2  2  2  þ wdmn E 1 x2 þ wamn þ wdmn E 2 sin 2μn x2 þ

2


  2  2  2  þ wdmn E3 sin 2λm x1 þ wamn þ wdmn E 4 cos 2λm x1 sin 2μn x2

2

 2  þ wdmn E5 sin 2λm x1 cos 2μn x2 ð7.117a, bÞ

Substituting the expressions for ua1 and ua2 into Eqs. (7.116a, b) and comparing coefficients of like trigonometric functions give D1  D5 and E1  E5. These coefficients are provided as the solution to the following matrix equations. 

N1

N2

N3

N4



D1



E1

 ¼

S1 S2



 ,

N4

N5

N6

N7



D2 E3



 ¼

S3 S4

 ,



N8

N9

N 10

N 11



D3 E2



 ¼

S5



S6

ð7:118a  cÞ 0

N 12 B N 13 B B @ N 16 N 17

N 13 N 12

N 14 N 15

N 17 N 16

N 18 N 19

10 1 0 1 D4 S7 N 15 B C B C N 14 C C B D 5 C B S8 C CB C ¼ B C N 19 A@ E4 A @ S9 A N 18 E5 S10

ð7:119Þ

where f N 1 ¼ A11 ,

f N 2 ¼ A12 ,

f N 3 ¼ A22

f N 4 ¼ 2λm A11 ,

f N 5 ¼ 2λm A16 ,

f N 6 ¼ 2λm A12 ,

f N 7 ¼ 2λm A26

f N 8 ¼ 2μn A16 ,

f N 9 ¼ 2μn A12 ,

f N 10 ¼ 2μn A26 ,

f N 11 ¼ 2μn A22

f N 12 ¼ 2λm A11 ,

f N 13 ¼ 2μn A16 ,

f N 14 ¼ 2μn A12 ,

f N 15 ¼ 2λm A16 ,

f N 16 ¼ 2λm A12

f N 17 ¼ 2μn A26 ,

f N 18 ¼ 2μn A22 ,

f N 19 ¼ 2λm A26

ð7:120a  sÞ

7.8 Application – Dynamic Response of Flat Sandwich Panels

217

f f f λ2m A11 þ μ2n A12 λ2 A f þ μ2n A22 , S2 ¼  m 12 8 8 2 f 2 f 2 f 2 f μ A  λm A11 μ A  λm A12 S3 ¼ n 12 , S4 ¼ n 22  4λ2m C 1 8 8 f f λ2 A f  μ2n A12 λ2 A f  μ2n A22 S5 ¼  m 11 S6 ¼  m 12  4μ2n C 2 , 8 8 f f 2 f 2 f λm μn A16 λ A þ μn A12 λ2 A f þ μ2n A22 S7 ¼ m 11 , S8 ¼  , S9 ¼ m 12 8 4 8 f λm μn A26 S10 ¼  4 ð7:121a  jÞ

S1 ¼ 

The third and fourth equations of motion are now addressed neglecting the thermal terms. These equations in terms of the displacement quantities are expressed as 2

2

Aa11

2

∂ ud1 ∂ud3 ∂ ua3 ∂ua3 ∂ ud3 þ þ ∂x1 ∂x21 ∂x1 ∂x21 ∂x21 2

! þ

2

∂ud ∂ ua3 ∂ua ∂ ud3 ∂ ud2 þ 3 þ 3 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2

2

2

2

2

Aa12

2

2

2

!

∂ua ∂ ud3 ∂ud ∂ ua3 ∂ ud3 ∂ua3 ∂ ua3 ∂ud3 ∂ ud2 ∂ ud1 þ2 þ2 3 þ2 3 þ þ 2 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x21 ∂x2 ∂x21 ∂x2 ! 2 a 2 2 d 2 2 d a 2 d ∂ u ∂u ∂u ∂ u ∂ua3 ∂ ud3 ∂ u ∂ ud1 ∂ ud2 a 3 3 3 3 2 þ A þ þ þ þ þAa26 66 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x22 ∂x22 ∂x2 ∂x2 ∂x22 ∂x22 !

2 2 2   a ∂ud3 ∂ ua3 ∂ua3 ∂ ud3 ∂ud3 ∂ ua3 Ac55 d d ∂u3 þ þ þ þ 2u þ t þ t  2u c f 1 3 t2c ∂x2 ∂x1 ∂x2 ∂x1 ∂x22 ∂x1 ∂x22 ∂x1

!

þAa16

¼0 2

Aa22

2

2

∂ ud2 ∂ud3 ∂ ua3 ∂ua3 ∂ ud3 þ þ ∂x2 ∂x22 ∂x2 ∂x22 ∂x22

!

2

þ

Aa12

2

2

∂ud ∂ ua3 ∂ua ∂ ud3 ∂ ud1 þ 3 þ 3 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2

!

! 2 2 2 2 2 2 ∂ua3 ∂ ud3 ∂ud3 ∂ ua3 ∂ ud3 ∂ua3 ∂ ua3 ∂ud3 ∂ ud1 ∂ ud2 þ2 þ2 þ2 þ þ ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x22 ∂x22 ∂x1 ∂x22 ∂x1 ! 2 a 2 2 d 2 2 d a 2 d ∂ u ∂u ∂u ∂ u ∂ua3 ∂ ud3 ∂ u ∂ ud2 ∂ ud1 a 3 3 3 3 1 þ A þ þ þ þ þAa16 66 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x21 ∂x21 ∂x1 ∂x1 ∂x21 ∂x21 !

2 2 2 h
 i a ∂ud3 ∂ ua3 ∂ua3 ∂ ud3 ∂ud3 ∂ ua3 Ac55 d a d ∂u3 þ þ þ þ 2u þ t þ t  2u c 2 f 3 t2c ∂x1 ∂x1 ∂x2 ∂x2 ∂x21 ∂x2 ∂x21 ∂x2 þAa26

¼0

ð7.122a, bÞ

7 Theory of Sandwich Plates and Shells with an Transversely Compressible. . .

218

It can be seen that Eqs. (7.121a, b) are two couple partial differential equations in terms of two unknowns ud1 and ud2 . These displacement quantities can be assumed in the following form following the same procedure as before by inserting the unknown functions into the two coupled partial differential equations, Eqs. (7.122a, b), followed by comparing coefficients of like trigonometric functions. ud1 and ud2 are assumed as follows ud1 ðx1 , x2 , t Þ ¼ wamn wdmn A1 sin 2λm x1 þ wamn wdmn A2 sin 2μn x2 þwamn wdmn A3 sin 2λm x1 cos 2μn x2 þ wamn wdmn A4 cos 2λm x1 sin 2μn x2 þwamn A5 cos λm x1 sin μn x2 þ wamn A6 sin λm x1 cos μn x2 ud2 ðx1 , x2 , t Þ ¼ wamn wdmn B1 sin 2μn x2 þ wamn wdmn B2 sin 2λm x1 þwamn wdmn B3 cos 2λm x1 sin 2μn x2 þ wamn wdmn B4 sin 2λm x1 cos 2μn x2 þwamn B5 sin λm x1 cos μn x2 þ wamn B6 cos λm x1 sin μn x2 ð7.123a, bÞ Following the described procedure just mentioned provides the constants, A1  A6 and B1  B6 which are presented as the solution to the following matrix equations. 

0 B B B B B @

M7

M8

M9

M 11

M 10 M 13

Sym 1

0

R5 C B B R6 C C B ¼B C, B R7 C A @ R8 where

0 B B B B B @

M1 M2

M2 M3

M 10



10

A1 B2

A3



 ¼

R1 R2

 ð7:124Þ

1

CB C B C M 12 C CB B3 C CB C B C M8 C A@ A4 A M 11 B4

M 14

M 15

M 16

M 18

M 17 M 14

Sym

10

1

1

0

A5 R9 C CB C B C B C B M 19 C CB B5 C B R10 C C CB C ¼ B C B C B M 15 C A@ A6 A @ 0 A M 17

M 18

B6

0

ð7:125Þ

7.8 Application – Dynamic Response of Flat Sandwich Panels

2Gc13 , tc 2Gc f M 4 ¼ 4μ2n A66 þ 13 , tc

219

2Gc23 tc 2Gc23 2 f 2 f M 5 ¼ 4μn A26 , M 6 ¼ 4μn A22 þ t  c 
 Gc13 f f f 2 2 2 f 2 f M 7 ¼ 8λm μn A16 , M 8 ¼ 4 λm A16 þ μn A26 , M 9 ¼ 4 λm A11 þ μn A66 þ 2tc  
 Gc f f f f f þ A66 , M 12 ¼ 4 λ2m A66 þ μ2n A22 þ 23 , M 11 ¼ 8λm μn A26 M 10 ¼ 4λm μn A12 2t c
 c G f f f f f , M 14 ¼ λ2m A11 þ μ2n A66 þ 13 , M 15 ¼ λm μn A12 þ A66 M 13 ¼ 8λm μn A16 tc 2Gc f f f f f , M 17 ¼ λ2m A16 þ μ2n A26 , M 18 ¼ λ2m A66 þ μ2n A22 þ 23 M 16 ¼ 2λm μn A16 tc f M 1 ¼ 4λ2m A11 þ

f M 2 ¼ 4λ2m A16 ,

f M 3 ¼ 4λ2m A66 þ

f M 19 ¼ 2λm μn A26

ð7:126a  sÞ  
 Gc λm 2 f λ f f f þ 13 ,  μ2n A26 λm A11  μ2n A12 R2 ¼ m λ2m A16 , 2 tc 2    Gc μ
μ f f f f  λ2m A16  λ2m A12 þ 23 , R4 ¼ n μ2n A22 R3 ¼ n μ2n A26 2 2 tc 
 μn
2 f λ f f f m 3λm A16 þ μ2n A26 λ2m A16 þ 3μ2n A26 R6 ¼  , R5 ¼  2  2
 Gc  λ f f f þ μ2n A12 þ 2A66 þ 13 R7 ¼  m λ2m A11 2 tc  
 Gc13 2aλm Gc13 μn 2 f f f 2 R8 ¼  μn A22 þ λm A12 þ 2A66 þ , R9 ¼  2 tc tc c 2aμn G23 R10 ¼  tc ð7:127a  jÞ R1 ¼

Addressing the fifth and sixth equations of motion, Eqs. (7.102e, f), by substituting the core deformation components κci3 ði ¼ 1, 2, 3Þ into Eqs. (7.99b) then substituting into the respective equations of motion provides two simultaneous equations in terms of Φcα ðα ¼ 1, 2Þ. The result is given as Φc1 ¼

t c ∂ud3 1 d ∂ud3  u , 4 ∂x1 2 3 ∂x1

Φc2 ¼

t c ∂ud3 1 d ∂ud3  u 4 ∂x2 2 3 ∂x2

ð7.128a, bÞ

Up to this point the entire governingsystem for the  dynamic response has been c a d , u , Φ , φ to a seven-parameter system reduced from a nine-parameter system u α i i  a d  ui , ui , φ by eliminating Φcα from the governing system. At this juncture, the first six equations of motion are satisfied, and all of the displacement functions are known. Also in agreement with this fulfillment are the transverse boundary

7 Theory of Sandwich Plates and Shells with an Transversely Compressible. . .

220

conditions and the natural boundary conditions M ann and M dnn . The remaining natural boundary conditions with respect to the stress resultants N ann , N ant , N dnn , N dnt will be satisfied through the use of the extended-Galerkin method in an average sense. The only unknowns at this point are the modal amplitudes wamn ðt Þ and wdmn ðt Þ which will be determined through the use of the extended-Galerkin method. By expressing the last two equations of motion, Eqs. (7.102g, h), and the unfulfilled boundary conditions in terms of displacements and retaining these expressions in Hamilton’s energy functional carrying out the indicated operations and collecting identical variational coefficients of the modal amplitudes wamn ðt Þ and wdmn ðt Þ noting that the variations of δwamn and δwdmn are arbitrary and independent from each other and that the corresponding coefficients must vanish results in two nonlinear coupled second-order ordinary differential equations in terms of the modal amplitudes. These are solved via fourth-order Runge–Kutta numerical procedure for a system of differential equations. This system of differential equations is presented as  2  3 q ∂ wamn ∂wamn þ Ca10 wamn þ C a11 wamn wdmn þ Ca12 wamn wdmn þ C a30 wamn ¼ mn þC 2 2 ∂t ∂t 2

m1

 2  3  2 ∂ wdmn ∂wdmn þ Cd01 wdmn þ C d02 wdmn þ C d03 wdmn þ C d20 wamn þC 2 ∂t ∂t   q 2 þC d21 wamn wdmn ¼ mn 2 ð7.129a, bÞ 2

m2

These coupled differential equations can be used in conjunction with any transient dynamical transverse pressure pulse qmn. The coefficients C10  C12, C30, C01  C03, C20, C21 which depend on the material and geometric properties are given as
 2aμn Gcyz 2aλm Gcxz f f f f þ μ4n D22 Ca10 ¼ λ4m D11 þ 2λ2m μ2n D12 þ 2D66 þ ðaλm  A3 Þ þ ðaμn  B3 Þ tc tc


 32 f f f f A5 þ λm μn A66 þ 2A12 ðλm B5 þ μn A5 Þ þ 2μ3n A22 B5 þ Ca11 ¼  2 2λ3m A11 9π
 2a 2 c f f λ2m A16 ð3μn A6 þ 2λm B6 Þ þ μ2n A26 ð3λm B6 þ 2μn A6 Þ  λm Gxz þ μ2n Gcyz þ tc h i 64a c c þ 2 λm Gxz ðA2  3A1 Þ þ μn Gyz ðB2  3B1 Þ 9π t c f λ3m A11 μ3 A f ½3λm þ 8ðA3  2A1 Þ þ n 22 ½3μn þ 8ðB3  2B1 Þ þ 16 16 f λ2 A f λm μn A12 ½4λm ðB3  2B1 Þ þ 4μn ðA3  2A1 Þ þ 3λm μn þ m 16 ½μn ð3A4  2A2 Þþ 8 2 f f μ2n A26 λm μn A66 λm ðB4  2B2 Þ þ ½μn ðA4  2A2 Þ þ λm ð3B4  2B2 Þ þ ½4ðμn A3 þ 2 4 λm B3 Þ  λm μ n

Ca12 ¼ 2λ2m μ2n ðC 1 þ C2 Þ þ

References

221

C a30 ¼ 2λ2m μ2n ðC1 þ C 2 Þ
 f f f f þ μ4n D22 C d01 ¼ λ4m D11 þ 2λ2m μ2n D12 þ 2D66 þ 2E c C d02 ¼ 

128Ec 3π 2 t c

C d03 ¼ 2λ2m μ2n ðC1 þ C 2 Þ þ C d20

C d21

9E c 4t 2c

" f λm μn A66 64 3 f f f ¼  2 λm A11 A5 þ λm μn A12 ðλm B5 þ μn A5 Þ þ μ3n A22 B5 þ ðλm B5 þ μn A5 Þþ 2 9π # f f λ2m A16 μ2n A26 ð3μn A6 þ 2λm B6 Þ þ ð2μn A6 þ 3λm B6 Þ 2 2

  h i 1 3λ 3 f f ¼ λ3m A11 A3  2A1 þ m þ λm μn A12 μn ðA3  2A1 Þ þ λm ðB3  2B1 Þ þ λm μn þ 8 2 4

 3 1 f f B3  2B1 þ μn þ 2λm μn A66 μ n A3 þ λm B3  λm μ n μ3n A22 8 4 f f ½λm ðB4  2B2 Þ þ μn ð3A4  2A2 Þ þ μ2n A26 ½λm ð3B4  2B2 Þ þ μn ðA4  2A2 Þ λ2m A16  þ 4λ2m μ2n ðC 1 þ C 2 Þ ð7:130a  iÞ

7.9

Comments

A very comprehensive theoretical treatment of doubly curved sandwich shells considering a transversely compressible core has been presented. In addition, an application was introduced to demonstrate how the equations can be applied toward solving the many types of typical structural response problems in engineering mechanics. The presented application is not meant to be an exhaustive treatment of the typical response problems such as buckling, post-buckling, free vibration, dynamic response, thermal buckling, etc. As a matter of fact, there are many response problems that yet need to be addressed regarding these theoretical equations. The researcher and/or scientist is encouraged to further advance the field with these equations. There are a series of results by Hohe and Librescu (2003, 2006) that the reader is referred to.

References Amabili, M. (2004). Nonlinear vibrations and stability of shells and plates. Cambridge University Press.

222

7 Theory of Sandwich Plates and Shells with an Transversely Compressible. . .

Hohe, J., & Librescu, L. (2003). A nonlinear theory for doubly curved anisotropic sandwich shells with transversely compressible core. International Journal of Solids and Structures, 40, 1059–1088. Hohe, J., & Librescu, L. (2006). Dynamic buckling of flat and curved sandwich panels with transversely compressible core. Composite Structures, 74, 10–24. Jones, R. M. (1999). Mechanics of composite materials (2nd ed.). New York, London: Taylor and Francis. Reddy, J. N. (2004). Mechanics of laminated composite plates and shells-theory and analysis (2nd ed.). Boca Raton: CRC Press. Soedel, W. (2004). Vibrations of shells and plates (3rd ed.). New York: Marcel Dekker, Inc. Hause, T. (2012). Elastic structural response of anisotropic sandwich plates with a first-order compressible core impacted by a Friedlander-type shock loading. Composite Structures, 94, 1634–1645.

Chapter 8

Theory of Sandwich Plates and Shells with an Transversely Compressible Core – Theory Two

Abstract A further comprehensive detailed nonlinear theoretical model of asymmetric anisotropic doubly curved laminated composite sandwich plates and shells considering the case for the transversely compressible core in sufficient detail, thereby capturing the wrinkling phenomenon of the sandwich panel, is presented. The theory assumes a transversal displacement of the core in terms of a second-order power series expansion, thus theory two. The governing equations are developed by way of an energy approach known as Hamilton’s principle. Finally, an application of the governing equations with regard to buckling and postbuckling is provided to demonstrate the solution approach to these equations through the use of the extended Galerkin method.

8.1

Introduction

This chapter considers many of the same theoretical aspects of a sandwich panel as was presented in Chap. 7. The principal difference resides with the transverse displacement representation for the core. In Chap. 7, a linear or first-order power series was assumed for the transverse displacement function, whereas in the present case a second-order polynomial is assumed for the transverse displacement thereby capturing the local and global instability modes on a higher-order basis. Some of the theoretical considerations include the Kirchhoff assumptions in the facings, a weak core, geometric imperfections, anisotropic laminated face sheets, and large displacements in the transverse direction. The governing equations are derived via an energy approach using Hamilton’s principle. The result is 11 equations of motion and 9 boundary conditions prescribed along each edge. This is in comparison with 8 equations of motion and 8 prescribed boundary conditions required along each edge from the first theory (Chap. 7). At the conclusion of the chapter, an application of the buckling and post-buckling response is presented demonstrating how to apply and solve these theoretical equations.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 T. J. Hause, Sandwich Structures: Theory and Responses, https://doi.org/10.1007/978-3-030-71895-4_8

223

224

8.2

8 Theory of Sandwich Plates and Shells with an Transversely Compressible. . .

Preliminaries and Basic Assumptions

The geometrically nonlinear theory of doubly curved sandwich panels with a transversely compressible core considering a higher-order representation for the core transverse displacement function contains the same theoretical assumptions as was considered in the previous chapter along with the same terminology. Without repeating the same preliminaries and basic assumptions the reader is referred back to the Chap. 7.

8.3 8.3.1

Basic Equations Displacement Field

The displacement field for the top and bottom facings are given as
 Bottom face sheets t2c
x3
t2c þ t bf

þ

! t c þ t bf x3  ψ b1 2

ð8:1aÞ

vb2 ¼ ub2 þ

! t c þ t bf x3  ψ b2 2

ð8:1bÞ

vb1

¼

ub1

vb3 ¼ ub3


Top face sheets  t2c  t tf
x3
 t2c

ð8:1cÞ



  t c þ t tf þ x3 þ ψ t1 2

ð8:2aÞ

  t c þ t tf vt2 ¼ ut2 þ x3 þ ψ t2 2

ð8:2bÞ

vt3 ¼ ut3

ð8:2cÞ

vt1

¼

ut1

8.3 Basic Equations

225

where, in the above equations, ψ b1 , ψ b2 , ψ t1 , ψ t2 represent the shear angles, while tand b represent the association with the top and bottom facings, respectively. In contrast to Chap. 3 a third-order power series is assumed for the core tangential displacements while a second-order polynomial is assumed for the core transverse displacements as shown below. This is referred to as the {3, 2}-order theory (Barut et al. (2001).   Core  t2c
x3
t2c vc1 ¼ uc1 þ x3 ψ c1 þ x23 γ c1 þ x33 θ1

ð8:3aÞ

vc2 ¼ uc2 þ x3 ψ c2 þ x23 γ c2 þ x33 θ2

ð8:3bÞ

vc3 ¼ uc3 þ x3 ψ c3 þ x23 γ 3

ð8:3cÞ

The underlined term in Eq. (8.3c) is a second-order term added to capture a more enhanced behavior of the compressible core. This is in contrast to Chap. 3 where the expression for the transverse displacement was only carried to the first order (linear). Next the interfacial continuity conditions at the interfaces need to be satisfied. This was shown and presented in Chap. 3. Applying the interfacial continuity conditions and at the same time adopting the Love–Kirchhoff assumptions results in the following displacement field for the facings and the core.
 Top face sheets  t2c  t tf
x3
 t2c     t c þ t tf ∂ua3 t c þ t tf ∂ud3 vt1 ¼ ua1 þ ud1  x3 þ  x3 þ 2 2 ∂x1 ∂x1

ð8:6aÞ

    t c þ t tf ∂ua3 t c þ t tf ∂ud3 vt2 ¼ ua2 þ ud2  x3 þ  x3 þ 2 2 ∂x2 ∂x2

ð8:6bÞ

vt3 ¼ ua3 þ ud3

ð8:6cÞ


Bottom face sheets

tc 2


x3
t2c þ t bf

vb1 ¼ ua1  ud1 



t c þ t bf x3  2

!

∂ua3 þ ∂x1

t c þ t bf x3  2

!

∂ud3 ∂x1

ð8:7aÞ

8 Theory of Sandwich Plates and Shells with an Transversely Compressible. . .

226

vb2

¼

ua2



ud2

t c þ t bf x3  2



!

∂ua3 þ ∂x2

t c þ t bf x3  2

!

vb3 ¼ ua3  ud3

∂ud3 ∂x2

ð8:7bÞ ð8:7cÞ

  Core  t2c
x3
t2c

vc1

! ! ! t tf  t bf ∂ua3 t tf þ t bf ∂ud3 2x3 d 1 t tf þ t bf ∂ua ¼    u1 þ x3 3 tc 4 4 tc 4 ∂x1 ∂x1 ∂x1 !  2   2  t b ∂ud 4x3 4x3 1 tf tf c þ  1 Φ þ  1 x3 Ωc1 x3 3 þ 1 tc 2 t 2c t 2c ∂x1 ua1

ð8:8aÞ vc2

! ! ! t tf  t bf ∂ua3 t tf þ t bf ∂ud3 2x3 d 1 t tf þ t bf ∂ua ¼    u2 þ x3 3 tc 4 4 tc 2 ∂x2 ∂x2 ∂x2 !  2   2  t b ∂ud 4x3 4x3 1 tf tf c þ  1 Φ þ  1 x3 Ωc2 x3 3 þ 2 tc 2 t 2c t 2c ∂x2 ua2

ð8:8bÞ

vc3 ¼ ua3 

 2  4x3 2x3 d u3 þ  1 Φc3 tc t 2c

ð8:8cÞ

In the above displacement equations, the average and half-difference displacement function which were introduced in Chap. 3 are uai ¼

 1 t ui þ ubi , 2

udi ¼

 1 t ui  ubi 2

ð8.9a, bÞ

ψ aα , ψ dα represent the rotation angles; while the core quantities Φc1 , Φc2 , Φc3 , Ωc1 , and Ωc2 represent the warping functions. In the following section, a stress-free initial geometric imperfection is introduced which remains constant during deformation. These geometric imperfections as before are introduced (see Hohe and Librescu (2003)) into the Green–Lagrange strain tensor as vt3 ¼ ua3 þ ud3

∘

∘

∘

ð8:10Þ

∘

∘

∘

ð8:11Þ

∘

∘

2x3 ∘ d u tc 3

ð8:12Þ

vb3 ¼ ua3  ud3 vc3 ¼ ua3 

8.3 Basic Equations

8.3.2

227

Nonlinear Strain–Displacement Equations

The Green–Lagrange strain tensor with the geometric imperfections, from Eqs. (2.18)–(2.23), in conjunction with the von Kármán assumptions is (Librescu and Chang 1993, Amabili 2004) γ 11 ¼

 2 ∂v1 v3 1 ∂v3 ∂v ∂v∘  þ þ 3 3 ∂x1 R1 2 ∂x1 ∂x1 ∂x1

ð8:13aÞ

γ 22 ¼

 2 ∂v2 v3 1 ∂v3 ∂v ∂v∘  þ þ 3 3 ∂x2 R2 2 ∂x2 ∂x2 ∂x2

ð8:13bÞ

  1 ∂v2 ∂v1 1 ∂v3 ∂v3 1 ∂v3 ∂v∘3 1 ∂v3 ∂v∘3 ¼ þ þ þ þ 2 ∂x1 ∂x2 2 ∂x1 ∂x2 2 ∂x1 ∂x2 2 ∂x2 ∂x1

ð8:13cÞ

γ 13 ¼

  1 ∂v1 ∂v3 1 ∂v3 ∂v3 1 ∂v3 ∂v∘3 1 ∂v3 ∂v∘3 þ þ þ þ 2 ∂x3 ∂x1 2 ∂x1 ∂x3 2 ∂x1 ∂x3 2 ∂x3 ∂x1

ð8:13dÞ

γ 23 ¼

  1 ∂v2 ∂v3 1 ∂v3 ∂v3 1 ∂v3 ∂v∘3 1 ∂v3 ∂v∘3 þ þ þ þ 2 ∂x3 ∂x2 2 ∂x2 ∂x3 2 ∂x2 ∂x3 2 ∂x3 ∂x2

ð8:13eÞ

γ 12

γ 33 ¼

 2 ∂v3 1 ∂v3 ∂v ∂v∘ þ þ 3 3 ∂x3 2 ∂x3 ∂x3 ∂x3

ð8:13fÞ

Substituting the displacement equations, Eqs. (8.6a–c), (8.7a–c), and (8.8a–c) along with the introduction of the geometric imperfections, Eqs. (8.10)–(8.12) into the nonlinear strain–displacement relationships Eqs. (8.13a–f) gives for each layer
 Top face sheets  t2c  t tf
x3
 t2c

γ t11 γ t22 γ t12

    t c þ t tf a t c þ t tf d ¼ þ þ x3 þ κ11 þ x3 þ κ 11 2 2     t c þ t tf a t c þ t tf d ¼ γ a22 þ γ d22 þ x3 þ κ22 þ x3 þ κ 22 2 2     t c þ t tf a t c þ t tf d ¼ γ a12 þ γ d12 þ x3 þ κ12 þ x3 þ κ 12 2 2 γ a11

γ d11

ð8:14a  cÞ

8 Theory of Sandwich Plates and Shells with an Transversely Compressible. . .

228

  Core  t2c
x3
t2c 

γ c13 γ c23 γ c33

   t 2c t 2c 2 ¼ γ 13 þ þ  η þ x3  xυ 4 13 4 3 13     t2 t2 ¼ γ c 23 þ x3 κ c23 þ x23  c η23 þ x23  c x3 υ23 4 4     2 tc t 2c c c 2 2 ¼ γ 33 þ x3 κ 33 þ x3  η þ x3  xυ 4 33 4 3 33 x3 κ c13

c


Bottom face sheets

γ b11

¼

γ a11



tc 2

γ d11

x23


x3
t2c þ t bf

þ

γ b22 ¼ γ a22  γ d22 þ γ b12 ¼ γ a12  γ d12 þ

ð8.15a-cÞ



! t c þ t bf a x3  κ11  2 ! t c þ t bf a κ22  x3  2 ! t c þ t bf a x3  κ12  2

! t c þ t bf d x3  κ11 2 ! t c þ t bf d κ22 x3  2 ! t c þ t bf d x3  κ12 2

ð8.16a-cÞ

where γ aαβ ¼


 1 t γ αβ þ γ bαβ , 2

γ dαβ ¼


 1 t γ αβ  γ bαβ 2

ð8:17aÞ

κ aαβ ¼


 1 t κ αβ þ κbαβ , 2

κ dαβ ¼


 1 t καβ  κ bαβ 2

ð8:17bÞ

ða,d Þ

In the above strain displacement equations, γ ij

are the in-plane average and the ða,dÞ

half difference of the tangential strains of the top and bottom facings, while καβ are the average and half difference bending strains of the top and bottom facings. For the core, γ ci3 and κ ci3 are the tangential and bending strains, respectively, while ηci3 and υci3 are higher-order strains. The expressions for these strains are given as γ a11 ¼

 2  2 ∘ ∘ ∂ua ∂ua3 1 ∂ud3 ∂ua ∂ud3 ∂ua1 ua3 1 ∂ua3  þ þ 3 þ þ 3 ∂x1 R1 2 ∂x1 ∂x1 ∂x1 2 ∂x1 ∂x1 ∂x1

ð8:18Þ

γ a22 ¼

 2  2 ∘ ∘ ∂ua ∂ua3 1 ∂ud3 ∂ua ∂ud3 ∂ua2 ua3 1 ∂ua3  þ þ 3 þ þ 3 ∂x2 R2 2 ∂x2 ∂x2 ∂x2 2 ∂x2 ∂x2 ∂x2

ð8:19Þ

8.3 Basic Equations

229 ∘

γ a12 ¼

∘

1 ∂ua1 1 ∂ua2 1 ∂ua3 ∂ua3 1 ∂ud3 ∂ud3 1 ∂ua3 ∂ua3 1 ∂ua3 ∂ua3 þ þ þ þ þ 2 ∂x2 2 ∂x1 2 ∂x1 ∂x2 2 ∂x1 ∂x2 2 ∂x1 ∂x2 2 ∂x1 ∂x2 ∘

∘

1 ∂ud3 ∂ud3 1 ∂ud3 ∂ud3 þ þ 2 ∂x1 ∂x2 2 ∂x1 ∂x2 ∘

∘

γ d11 ¼

∂ud1 ud3 ∂ua3 ∂ud3 ∂ua3 ∂ud3 ∂ua3 ∂ud3  þ þ þ ∂x1 R1 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1

γ d22 ¼

∂ud2 ud3 ∂ua3 ∂ud3 ∂ua3 ∂ud3 ∂ua3 ∂ud3  þ þ þ ∂x2 R2 ∂x2 ∂x2 ∂x2 ∂x2 ∂x2 ∂x2

∘

ð8:21Þ

∘

∘

γ d12 ¼

ð8:22Þ

∘

1 ∂ud1 1 ∂ud2 1 ∂ua3 ∂ud3 1 ∂ua3 ∂ud3 1 ∂ua3 ∂ud3 1 ∂ua3 ∂ud3 þ þ þ þ þ 2 ∂x2 2 ∂x1 2 ∂x1 ∂x2 2 ∂x2 ∂x1 2 ∂x1 ∂x2 2 ∂x1 ∂x2 ∘

ð8:20Þ

∘

1 ∂ua3 ∂ud3 1 ∂ua3 ∂ud3 þ þ 2 ∂x2 ∂x1 2 ∂x2 ∂x1

ð8:23Þ

2

κ a11 ¼ 

∂ ua3 ∂x21

κ a22 ¼ 

∂ ua3 ∂x22

ð8:24Þ

2

ð8:25Þ

2

κ a12 ¼ 

∂ ua3 ∂x1 ∂x2

ð8:26Þ

2

κd11 ¼ 

∂ ud3 ∂x21

κd22 ¼ 

∂ ud3 ∂x22

ð8:27Þ

2

ð8:28Þ

2

κ d12 ¼ 

∂ ud3 ∂x1 ∂x2

ð8:29Þ

2 2  2 4  d 8  2 γ 33 ¼  ud3 þ 2 ud3 þ 2 ud3 u∘3 þ 2 Φc3 tc tc tc tc

γ c13

ud ¼ 1þ tc

t b 1 1 tf þtf þ 2 tc 4 ∘

!!

∂ua3 1 þ ∂x1 t c ∘

t tf  t bf 4

!

ð8:30Þ

∂ud3 ∂x1 ∘

∂ud Ωc ud ∂ua ua ∂ua ud ∂ua 2Φc ∂ud 2 þ 1  3 3  3 3  3 3  3 3  Φc3 3 tc t c ∂x1 t c ∂x1 t c ∂x1 t c ∂x1 t c ∂x1

ð8:31Þ

8 Theory of Sandwich Plates and Shells with an Transversely Compressible. . .

230

γ c23

ud ¼ 2þ tc

t b 1 1 tf þtf þ 2 tc 4

∘

!!

∂ua3 1 þ ∂x2 t c

∘

t tf  t bf 4

!

∂ud3 Ωc2 ud3 ∂ua3 þ  tc t c ∂x2 ∂x2

∘

ua ∂ua ud ∂ua 2Φc ∂ud 2 ∂ud  3 3  3 3  3 3  Φc3 3 t c ∂x2 t c ∂x2 t c ∂x2 t c ∂x2 ð8:32Þ κc33 ¼

8 c 16 d c 16 c  ∘ d Φ  u Φ  Φ u t 2c 3 t 3c 3 3 t 3c 3 3

ð8:33Þ ∘

κc13 ¼

∘

4 c 1 ∂ud3 4 c ∂ua3 2 d ∂ud3 2  ∘ d ∂ud3 2 d ∂ud3 4 c ∂ua3 Φ  þ Φ þ u þ u þ u þ Φ t 2c 1 t c ∂x1 t 2c 3 ∂x1 t 2c 3 ∂x1 t 2c 3 ∂x1 t 2c 3 ∂x1 t 2c 3 ∂x1

ð8:34Þ ∘

κc23 ¼

∘

4 c 1 ∂ud3 4 c ∂ua3 2 d ∂ud3 2  ∘ d ∂ud3 2 d ∂ud3 4 c ∂ua3 Φ  þ Φ þ u þ u þ u þ Φ t 2c 2 t c ∂x2 t 2c 3 ∂x2 t 2c 3 ∂x2 t 2c 3 ∂x2 t 2c 3 ∂x2 t 2c 3 ∂x2

ð8:35Þ ηc33 ¼

32  c 2 Φ3 t 4c

ð8:36Þ

∘

ηc13 ¼

6 c 8 c ∂ud3 8 c ∂ud3 2 ∂Φc3 4 d ∂Φc3 4  ∘ d ∂Φc3 Ω  Φ  Φ þ  u  u ð8:37Þ t 3c 1 t 3c 3 ∂x1 t 3c 3 ∂x1 t 2c ∂x1 t 3c 3 ∂x1 t 3c 3 ∂x1

ηc23 ¼

6 c 8 c ∂ud3 8 c ∂ud3 2 ∂Φc3 4 d ∂Φc3 4  ∘ d ∂Φc3 Ω  Φ  Φ þ  u  u ð8:38Þ t 3c 2 t 3c 3 ∂x2 t 3c 3 ∂x2 t 2c ∂x2 t 3c 3 ∂x2 t 3c 3 ∂x2

∘

υc33 ¼ 0

8.3.3

ð8:39Þ

υc13 ¼

16 c ∂Φc3 Φ t 4c 3 ∂x1

ð8:40Þ

υc23 ¼

16 c ∂Φc3 Φ t 4c 3 ∂x2

ð8:41Þ

Constitutive Equations

The top and bottom facings are considered to be constructed from unidirectional fiber-reinforced anisotropic laminated composites. The stress–strain relationships per lamina (see Reddy (2004) and Jones (1999)) of the facings is repeated here as

8.4 Hamilton’s Principle

231


 Top face sheets  t2c  t tf
x3
 t2c 0

τt11

1

2

b t11 Q

6 B t C @ τ22 A ¼ 6 4 τt12 k Sym

b t11 Q bt Q 22

8 t 9 30 8 t 9 1 bλ > > γ t11 μ11 > > > 11 > = =
> > ; > : t > γ t12 k : bλt ; b μ12 k bt Q b t11 Q

66

12

k

k

b ij for i, j ¼ (1, 2, 6) are the where k represents the kth lamina in the facing and Q transformed plane-stress reduced stiffness measures. These were given in Chap. 2.
 Bottom face sheets t2c
x3
t2c þ t bf 0

τb11

1

2

6 B b C 6 @ τ22 A ¼ 6 4 τb12 k

bb Q 11

bb Q 11 bb Q 22

Sym

9 8 8 b 9 > bλb > > > μ11 > > > > 11 = =
> > ; > : b > b > > b b γ 12 k : b ; b μ12 k b Q λ12 k 66 k bb Q 11

3

0

γ b11

1

The stress–strain relationships for the orthotropic core with the geometrical and material axes coincident are expressed as   Core  t2c
x3
t2c 0

1 2 c Q33 τc33 B c C 6 @ τ23 A ¼ 4 0 τc13

0

0 Qc44 0

30 c 1 γ 33 0 7B c C 0 5@ γ 23 A Qc55 γ c13

ð8:44Þ

The core transverse and normal moduli are given as Qc33 ¼ E c ,

Qc44 ¼ Gc23 ,

Qc55 ¼ Gc13

ð8:45Þ

where Ec is the young’s modulus for the core, and Gc13 , Gc23 are the shear moduli for the core.

8.4

Hamilton’s Principle

An energy approach using Hamilton’s Principle is used to derive the equations of motion. Letting U represent the strain energy, W represent the work done by external forces, and T represent the kinetic energy, Hamilton’s Principle (see Soedel 2004) is expressed as ð t1 ð8:46Þ δJ ¼ ðδU  δW  δT Þdt ¼ 0 t0

8 Theory of Sandwich Plates and Shells with an Transversely Compressible. . .

232

8.4.1

Strain Energy

Assuming a weak compressible core where the core carries only the transverse shear stresses, the variation in the strain energy is given by ð (ð tc þtb 2 f  δU ¼ τb11 δγ b11 þ τb22 δγ b22 þ 2τb12 δγ b12 dx3 tc 2

A

ð tc þ þ



2

t2c

 2τc13 δγ c13 þ 2τc23 δγ c23 þ τc33 δγ c33 dx3

ð t c 2

t2c t tf





ð8:47Þ )

τt11 δγ t11 þ τt22 δγ t22 þ 2τt12 δγ t12 dx3 dA

where τij are the tensorial components of the second Piola–Kirchhoff stress tensor, while A is attributed to the planar area of the sandwich shell. Substituting in the expressions for the strain relationships, Eqs. (8.14a–c)–(8.16a–c) results in ! ! ! b b t þ t t þ t c c f f τb11 δγ a11  δγ d11 þ x3  δU ¼ δκ a11  x3  δκ d11 tc 2 2 A 2 ! ! !  a t c þ t bf t c þ t bf b d a d þ τ22 δγ 22  δγ 22 þ x3  δκ 22  x3  δκ 22 2 2 ! ! !# t c þ t bf t c þ t bf b a d a d þ2τ12 δγ 12  δγ 12 þ x3  δκ 12  x3  δκ 12 dx3 2 2 ð (ð tc þtb " 2

f

     t 2c c c 2 c δη33 þ 2τc13 δγ c13 þ x3 δκ c13 þ δγ 33 þ x3 δκ 33 þ x3  tc 4 2         t2 t2 t2 þ x23  c δηc13 þ x23  c x3 δυc13 þ 2τc23 δγ c23 þ x3 δκ c23 þ x23  c δηc23 4 4 4       tc ð 2 t c þ t tf t2 þ x23  c x3 δυc23 dx3 þ τt11 δγ a11 þ δγ d11 þ x3 þ δκ a11 þ 4 2 t2c t tf ! ! ! b t c þ t bf t þ t c f þ x3 þ δκ d11 þ τt22 δγ a22 þ δγ d22 þ x3 þ δκ a22 2 2 ! ! !  a t c þ t bf t c þ t bf d t d þ x3 þ δκ 22 þ 2τ12 δγ 12 þ δγ 12 þ x3 þ δκ a12 2 2 ! !# ) t c þ t bf d δκ 12 dx3 dA þ x3 þ 2 ð tc  2

τc33

ð8:48Þ

8.4 Hamilton’s Principle

233

which can be written in terms of the local stress resultants and couples defined below in Eq. (8.50)–(8.52) as ð



δU ¼ A

N t11 δγ a11 þ N t11 δγ d11 þ M t11 δκ a11 þ M t11 δκ d11 þ N t22 δγ a22 þ N t22 δγ d22

þM t22 δκ a22 þ M t22 δκ d22 þ 2N t12 δγ a12 þ 2N t12 δγ d12 þ 2M t12 δκ a12 þ 2M t12 δκ d12 þN c33 δγ c33 þ M c33 δκ c33 þ Lc33 δηc33 þ 2N c13 δγ c13 þ 2M c13 δκ c13 þ 2Lc13 δηc13 þ2K c13 δυc13 þ 2N c23 δγ c23 þ 2M c23 δκ c23 þ 2Lc23 δηc23 þ 2K c23 δυc23 þ N b11 δγ a11 N b11 δγ d11 þ M b11 δκ a11  M b11 δκ d11 þ N b22 δγ a22  N b22 δγ d22 þ M b22 δκ a22  M b22 δκ d22 þ 2N b12 δγ a12  2N b12 δγ d12 þ 2M b12 δκ a12  2M b12 δκ d12 dA ð8:49Þ where the local stress resultants and stress couples are defined as   t c þ t tf τtαβ 1, x3 þ dx3 2

ð8:50Þ

( !) n o ð t2c þtbf t c þ t bf b b b N αβ :M αβ ¼ ταβ 1, x3  dx3 tc 2

ð8:51Þ

n

N tαβ :M tαβ

o

¼

ð t c 2

t2c t tf

2



N ci3 , M ci3 , Lci3 , K ci3





ð tc ¼

2

t2c

τci3

     t 2c t 2c 2 2 1, x3 , x3  , x3  x dx3 4 4 3

ð8:52Þ

Combining like terms which contain the same variational displacements results in ð δU ¼ A

 t        N 11 þ N b11 δγ a11 þ N t11  N b11 δγ d11 þ M t11 þ M b11 δκ a11 þ M t11  M b11 δκd11 þ

 t        N 22 þ N b22 δγ a22 þ N t22  N b22 δγ d22 þ M t22 þ M b22 δκ a22 þ M t22  M b22 δκd22 þ         2 N t12 þ N b12 δγ a12 þ 2 N t12  N b12 δγ d12 þ 2 M t12 þ M b12 δκ a12 þ 2 M t12  M b12 δκ d12 þN c33 δγ c33 þ M c33 δκc33 þ Lc33 δηc33 þ 2N c13 δγ c13 þ 2M c13 δκc13 þ 2Lc13 δηc13 þ 2K c13 δυc13 þ  2N c23 δγ c23 þ 2M c23 δκ c23 þ 2Lc23 δηc23 þ 2K c23 δυc23 dA

ð8:53Þ Utilizing the definition of global stress resultants and global stress couples as defined below in Eqs. (8.55a and 8.55b), provides the variation in the strain energy as

8 Theory of Sandwich Plates and Shells with an Transversely Compressible. . .

234

ð



δU ¼ A

2N a11 δγ a11 þ 2N d11 δγ d11 þ 2M a11 δκ a11 þ 2M d11 δκ d11 þ 2N a22 δγ a22 þ 2N d22 δγ d22 þ

2M a22 δκ a22 þ 2M d22 δκ d22 þ 4N a12 δγ a12 þ 4N d12 δγ d12 þ 4M a12 δκ a12 þ 4M d12 δκ d12 þ N c33 δγ c33 þ M c33 δκ c33 þ Lc33 δηc33 þ 2N c13 δγ c13 þ 2M c13 δκ c13 þ 2Lc13 δηc13 þ 2K c13 δυc13 þ  2N c23 δγ c23 þ 2M c23 δκc23 þ 2Lc23 δηc23 þ 2K c23 δυc23 dA

ð8:54Þ where the global stress resultants and global stress couples are defined as
 n

o 1 N tαβ þ N bαβ , M tαβ þ M bαβ N aαβ , M aαβ ¼ 2

ð8:55aÞ


 n

o 1 N tαβ  N bαβ , M tαβ  M bαβ N dαβ , M dαβ ¼ 2

ð8:55bÞ

Substituting in the strain–displacement equations, Eqs. (8.18)–(8.40) into Eq. (8.54) results in ð " δU ¼

       a ∂ δu1 ∂ δua3 ∂u∘ a3 ∂ud3 ∂ δud3 δua ∂ua ∂ δua3  3þ 3 þ þ R1 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 A )        d ∂ δua3 ∂u∘ d3 ∂ δu1 δud3 ∂ud3 δ ua3 ∂ua3 ∂ δud3 d þ  þ þ þ 2N 11 R1 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 2N a11

    2 a 2 d  ∂ δua3 ∂u∘ d3 ∂ δud3 ∂u∘ a3 a ∂ δu3 d ∂ δu3 þ  2M 11  2M 11 þ ∂x1 ∂x1 ∂x1 ∂x1 ∂x21 ∂x21         ∂ δua2 ∂ δua3 ∂u∘ a3 ∂ud3 ∂ δud3 δua3 ∂ua3 ∂ δua3 a þ2N 22  þ þ þ R2 ∂x2 ∂x2 ∂x2 ∂x2 ∂x2 ∂x2 ∂x2 )        d ∂ δua3 ∂u∘ d3 ∂ δu2 δud ∂ud ∂ δua3 ∂ua ∂ δud3 þ  3þ 3 þ 3 þ þ 2N d22 R2 ∂x2 ∂x2 ∂x2 ∂x2 ∂x2 ∂x2 ∂x2 )     2 a 2 d  ∂ δua3 ∂u∘ d3 ∂ δud3 ∂u∘ a3 a ∂ δu3 d ∂ δu3 þ  2M 22 þ  2M 22 ∂x2 ∂x2 ∂x2 ∂x2 ∂x22 ∂x22          a ∂ δu1 ∂ δua2 ∂ δua3 ∂ua3 ∂ua3 ∂ δua3 ∂ δud3 ∂ud3 þ4N a12 þ þ þ þþ ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2  d  a ∘a      ) ∂ δu3 ∂u3 ∂u∘ a3 ∂ δua3 ∂ δud3 ∂u∘ a3 ∂u∘ d3 ∂ δud3 ∂ud3 ∂ δu3 þ þ þ þ þ ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2

8.4 Hamilton’s Principle

235

         d ∂ δu1 ∂ δud2 ∂ δua3 ∂ud3 ∂ua3 ∂ δud3 ∂ δua3 ∂ud3 þ þ þ þ ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 ∂x2 ∂x1 )         ∂ δua3 ∂u∘ d3 ∂ δud3 ∂u∘ a3 ∂u∘ d3 ∂ δua3 ∂ua ∂ δud3 þ 3 þ þ þ ∂x1 ∂x2 ∂x2 ∂x1 ∂x1 ∂x2 ∂x2 ∂x1 þ4N d12

  2 2 ∂ δua3 ∂ δud3 2 4  4M d12 þ N c33  δud3 þ 2 ud3 δud3 þ tc tc ∂x1 ∂x2 ∂x1 ∂x2

4 ∘ d d 16 c c 8 16 16 c þ 2 u3 δu3 þ 2 Φ3 δΦ3 þ M 33 2 δΦc3  3 Φc3 δud3  3 ud3 δΦc3 tc tc tc tc tc 4M a12

16 ∘ d  3 u3 δΦc3 tc

þ

Lc33

16 c c Φ δΦ t 4c 3 3

þ

2N c13

1  δud1 þ tc

1 1 þ 2 tc

t tf þ t bf 4

!!

!       t b ∂ δua3 1 t f  t f ∂ δud3 1 c 1 ∂ua3 d 1 d ∂ δua3 þ þ δΩ1  δu  u tc tc t c ∂x1 3 t c 3 ∂x1 4 ∂x1 ∂x1 )     ∘ ∘ 1 ∘ a ∂ δua3 1 ∂ua3 d 2 ∂ud3 c 2 c ∂ δud3 2 ∂ud3 c  u3  δu  δΦ  Φ  δΦ tc t c ∂x1 3 t c ∂x1 3 t c 3 ∂x1 t c ∂x1 3 ∂x1     4 c 1 ∂ δud3 4 ∂ua3 c 4 c ∂ δua3 2 ∂ud3 d  þ δΦ þ Φ þ δu δΦ t 2c 1 t c ∂x1 t 2c ∂x1 3 t 2c 3 ∂x1 t 2c ∂x1 3 )     ∘ ∘ 2 d ∂ δud3 2 ∘ d ∂ δud3 2 ∂ud3 d 4 ∂ua3 c 6 þ 2 u3 þ 2 u3 þ 2 δu3 þ 2 δΦ3 þ 2Lc13 3 δΩc1 tc tc t c ∂x1 t c ∂x1 tc ∂x1 ∂x1

þ2M c13

   c ∘ 8 ∂ud3 c 8 c ∂ δud3 8 ∂ud3 c 2 ∂ δΦ3 4 ∂Φc3 d  3 δΦ3  3 Φ3  3 δΦ3 þ 2  3 δu t c ∂x1 tc t c ∂x1 t c ∂x1 t c ∂x1 3 ∂x1  c  c   c  4 d ∂ δΦ3 4 ∘ d ∂ δΦ3 16 ∂Φc3 c 16 c ∂ δΦ3 c  3 u3  3 u3 δΦ3 þ 4 Φ3 þ 2K 13 4 tc tc t c ∂x1 tc ∂x1 ∂x1 ∂x1 ! ! !     t b t b ∂ δua3 1 d 1 1 tf þtf 1 t f  t f ∂ δud3 c þ þ þ2N 23  δu2 þ tc 2 tc tc 4 4 ∂x2 ∂x2     ∘ 1 c 1 ∂ua3 d 1 d ∂ δua3 1 ∘ a ∂ δua3 1 ∂ua3 d þ δΩ2  δu3  u3  u3  δu tc t c ∂x2 tc tc t c ∂x2 3 ∂x2 ∂x2 )     ∘ 2 ∂ud3 c 2 c ∂ δud3 2 ∂ud3 c 4 c 1 ∂ δud3 c  δΦ  Φ  δΦ þ 2M 23 2 δΦ2  t c ∂x2 3 t c 3 ∂x2 t c ∂x2 3 t c ∂x2 tc

8 Theory of Sandwich Plates and Shells with an Transversely Compressible. . .

236

      4 ∂ua3 c 4 c ∂ δua3 2 ∂ud3 d 2 d ∂ δud3 2 ∘ d ∂ δud3 þ 2 δΦ þ Φ þ 2 δu þ u þ 2 u3 t c ∂x2 3 t 2c 3 ∂x2 t c ∂x2 3 t 2c 3 ∂x2 tc ∂x2 )   ∘ ∘ ∂ δud3 2 ∂ud 4 ∂ua 6 8 ∂ud 8 þ 2 3 δud3 þ 2 3 δΦc3 þ 2Lc23 3 δΩc2  3 3 δΦc3  3 Φc3 t c ∂x2 t c ∂x2 tc t c ∂x2 tc ∂x2  c  c  c  ∘ 8 ∂ud3 c 2 ∂ δΦ3 4 ∂Φc3 d 4 d ∂ δΦ3 4 ∘ d ∂ δΦ3  3 δΦ þ  3 δu  u  3 u3 t c ∂x2 3 t 2c ∂x2 t c ∂x2 3 t 3c 3 ∂x2 tc ∂x2   c 16 ∂Φc3 c 16 c ∂ δΦ3 þ2K c23 4 δΦ3 þ 4 Φ3 dA ð8:56Þ t c ∂x2 tc ∂x2 Integrating each appropriate term by parts, combining coefficients of identical variational displacements, and simplifying gives the variation in the strain energy as δU ¼

ð l2 ð l1 



  a  ∂N a11 ∂N a12 ∂N 22 ∂N a12 þ þ δua1  2 δua2 ∂x ∂x ∂x ∂x 1 2 2 1 0 0  a   a  ∂N 11 ∂N a12 N c13 ∂N 22 ∂N a12 N c23 2 þ þ þ þ δud1  2 δud2 tc tc ∂x1 ∂x2 ∂x2 ∂x1      c  N 8 8 6 þ 2 M c13 δΦc1 þ 2 M c23 δΦc2 þ 2 13 þ 3 Lc13 δΩc1 tc tc tc tc ( !  c  2 2 ∘a N 23 6 c ∂ ua3 ∂ u 1 c a þ2 þ 3 L23 δΩ2 þ 2N 11 þ þ tc tc ∂x21 ∂x21 R1 ! ! 2 2 ∘a 2 2 ∘a 2 ∂ ua3 ∂ u3 ∂ ua3 ∂ u3 ∂ M a11 1 a a þ þ þ 4N 12  2N 22 2 2 2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x2 ∂x2 R2 ∂x21 2

! ! 2 2 ∘a 2 2 ∘d 2 2 ∂ ua3 ∂ u3 ∂ ud3 ∂ u3 ∂ M a12 ∂ M a22 d d 4 2  2N 11 þ þ  4N 12 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x22 ∂x21 ∂x21 ! !  2 a 2 ∘a t b t þ t d ∂ u ∂ u ∂N c13 ∂N c23 2 t ∘ f f c d d 3 3 þ þ  u3  u3 2N 22 þþ  tc 2 4 ∂x1 ∂x2 ∂x22 ∂x22 ! ∘d 4 ∂ud3 ∂u3 4 þ þ N c23 þ t c ∂x2 ∂x2 tc

! ! ∘d ∘a ∂ud3 ∂u3 ∂ua3 ∂u3 c þ þ N 13  2 ∂x1 ∂x1 ∂x1 ∂x1

! !  a   ∘a  ∘d ∂ua3 ∂u3 ∂ud3 ∂u3 ∂N 11 ∂N a12 ∂N a22 ∂N a12 þ þ þ þ 2 2 ∂x1 ∂x2 ∂x2 ∂x2 ∂x2 ∂x1 ∂x1 ∂x1

8.4 Hamilton’s Principle



∂N d11 ∂N d12 þ ∂x1 ∂x2



237



∘d

∂ud3 ∂u þ þ 3 2 ∂x2 ∂x2

!  ∂N d22 ∂N d12 þ ∂x2 ∂x1

 

4 c ∂M c13 ∂M c23 4 ∂Φc3 c 4 ∂Φc3 c Φ þ M  M  δua3 þ t 2c 3 ∂x1 t 2c ∂x1 13 t 2c ∂x1 13 ∂x2 ( ! ! 2 a 2 ∘a 2 a 2 ∘a ∂ u ∂ u ∂ u ∂ u 1 3 3 3 3 2N d11 þ þ þ  4N d12 ∂x1 ∂x2 ∂x1 ∂x2 ∂x21 ∂x21 R1 

2 ∘a

2

2N d22

∂ ua3 ∂ u3 1 þ þ 2 2 R ∂x2 ∂x2 2

2N a11

∂ ud3 ∂ u3 þ ∂x21 ∂x21

2 ∘d

2

2 ∘d

2

2N a22 

∂ ud3 ∂ u3 þ ∂x22 ∂x22

∂N c13 ∂N c23 þ ∂x1 ∂x2

!

2

2

!

 þþ

4N a12

2

2 ∘d

2



!

2

∂ M d11 ∂ M d12 ∂ M d22  4  2 ∂x1 ∂x2 ∂x21 ∂x22 ∂ ud3 ∂ u3 þ ∂x1 ∂x2 ∂x1 ∂x2

!


 4 tc 4 ∘d  ud3  u3 N c33 þ  tc 2 tc



Φc3

∘d

∘d

!

∘a

!

∂ua3 ∂u3 þ 2 ∂x2 ∂x2 ∂M c23 þ ∂x2

) þ

!

∂N a11 ∂N a12 þ ∂x1 ∂x2



!   ∘a  ∂ua3 ∂u3 ∂N d11 ∂N d12 N c13 ∂N a22 ∂N a12 þ þ þ 2 þ tc ∂x2 ∂x1 ∂x1 ∂x1 ∂x1 ∂x2 ∂N d22 ∂N d12 N c23 þ þ tc ∂x2 ∂x1

( δud3

!

 c  ∂L13 ∂Lc23 4 ∂Φc3 c 4 ∂Φc3 c 16 N 13 þ N 23 þ 3 Φc3 þ t c ∂x1 t c ∂x2 tc ∂x1 ∂x2

∂ud3 ∂u3 8 ∂Φc3 c 8 ∂Φc3 c 16 þ 3 L13 þ 3 L23 þ 3 Φc3 M c33  2 þ t c ∂x1 t c ∂x2 tc ∂x1 ∂x1 ∂ud3 ∂u3 2 þ ∂x2 ∂x2

t tf  t bf þ 16

4  tc




∂M c 4 tc ∘d d 13 þ  u3  u3 þ 2 tc 2 ∂x1

! ∘d ∂ud3 ∂u3 4 þ N c13  tc ∂x1 ∂x1

! ∘d ∂ud3 ∂u3 þ N c23 ∂x2 ∂x2

!  
∘d  8 ∂Lc13 ∂Lc23 t c 8 c ∂ud3 ∂u3 ∘d d  3 þ þ  u3  u3  3 L13 2 t c ∂x1 tc ∂x2 ∂x1 ∂x1 ∘d

∂ud3 ∂u3 8  3 Lc23 þ tc ∂x2 ∂x2

! þ


 16 c c 16 t c 64 c c ∘d d c Φ N þ  u  u 3 3 M 33 þ 4 Φ3 L33 t 2c 3 33 t 3c 2 tc

8 Theory of Sandwich Plates and Shells with an Transversely Compressible. . .

238

!   ∘a 32 c ∂K c13 ∂K c23 8 c ∂ua3 ∂u3  4 Φ3 þ þ þ 2 M 13 tc tc ∂x1 ∂x2 ∂x1 ∂x1 !) +  ∘a 8 c ∂ua3 ∂u3 þ 2 M 23 þ δΦc3 dx1 dx2 þ tc ∂x2 ∂x2 ð l2 * 0

( 2N a11 δua1

þ

2N a12 δua2 ∘

∂ua3 ∂ua3 þ ∂x2 ∂x2

þ2N a12

þ

2N d11 δud2

!

þ

2N d12 δud2 ∘

þ

2N d11

∂ud3 ∂ud3 þ ∂x1 ∂x1

þ

∘

2N a11

!

∂ua3 ∂ua3 þ ∂x1 ∂x1 ∘

þ

∂ud3 ∂ud3 þ ∂x2 ∂x2

2N d12

!

! þ2

þ ∂M a11 ∂x1

) t b i ∂M a12 2 ht c t f þ t f 8 c c ∘d d c þ4 þ þ  u3  u3 N 13 þ þ 2 M 13 Φ3 δua3 tc 2 4 tc ∂x2 ( þ

∘

2N a11

∂ud3 ∂ud3 þ ∂x1 ∂x1 ∘

∂ua3 ∂ua3 þ ∂x2 ∂x2

þ2N d12

!

!

∘

þ

2N d11

∂ua3 ∂ua3 þ ∂x1 ∂x1

!

∘

þ

2N a12

∂ud3 ∂ud3 þ ∂x2 ∂x2

!

! t b t  t ∂M d11 ∂M d12 8 c c 2 f f þ4  3 L13 Φ3  Φc3  N c13 þ2 tc 8 tc ∂x1 ∂x2

)



  2 tc 8 tc 32 c c ∘d ∘d d c d d c  2  u3  u3 M 13 δu3 þ 3  u3  u3 L13 þ 4 Φ3 K 13 δΦc3 tc 2 tc 2 tc þ2M a11 δ ð l1 * 0

 a  d   l1 ∂u3 ∂u3  dx2 þ þ 2M d11 δ 0 ∂x1 ∂x1 (

2N a22 δua2

þ

2N a21 δua1 ∘

þ2N a21

∂ua3 ∂ua3 þ ∂x1 ∂x1

þ

2N d22 δud1

!

þ

2N d21 δud1 ∘

þ

þ2N d22

∂ud3 ∂ud3 þ ∂x2 ∂x2

þ

∘

2N a22

!

∂ua3 ∂ua3 þ ∂x2 ∂x2 ∘

þ

2N d21

∂ud3 ∂ud3 þ ∂x1 ∂x1

!

) t b i ∂M a22 ∂M a21 2 ht c t f þ t f 8 c c ∘d d c þ  u3  u3 N 23 þ 2 M 23 Φ3 δua3 þ4 þ þ2 tc 2 4 tc ∂x2 ∂x1 ( þ 2N a22

∘

∂ud3 ∂ud3 þ ∂x2 ∂x2

!

∘

þ 2N d22

∂ua3 ∂ua3 þ ∂x2 ∂x2

!

∘

þ 2N a21

∂ud3 ∂ud3 þ ∂x1 ∂x1

!

!

8.4 Hamilton’s Principle

239

∘

∂ua3 ∂ua3 þ ∂x1 ∂x1

þ2N d21

!

! t tf  t bf ∂M d22 ∂M d21 8 c c 2 c þ4  3 L23 Φ3  Φ3  N c23 þ2 tc 8 tc ∂x2 ∂x1

)



  2 tc 8 tc 32 c c ∘d ∘d d c d d c  2  u3  u3 M 23 δu3 þ 3  u3  u3 L23 þ 4 Φ3 K 23 δΦc3 tc 2 tc 2 tc þ2M a22 δ

8.4.2

 a  d   l2 ∂u3 ∂u3 d  dx1 þ þ 2M 22 δ 0 ∂x2 ∂x2

ð8:57Þ

Work Done by External Loads

The total work consists of the work due to body forces, edge loads, surface tractions, and damping. The work due to each of these is presented mathematically as • Work due to body forces

δW b ¼

ð (ð hþh= σ

ρ

=

= = H i δV i dx3

þ

h

ðh h

ρH i δV i dx3 þ

ð h hh==

) ρ

==

== == H i δV i dx3

dσ ð8:58Þ

where ρ is the mass density and Hi is the body force vector. The body forces will not be included in the following developments. Typical body forces are gravity, electrical and magnetic forces which will be neglected. • Work due to edge loads W edge

loads

¼ W tedge

loads

þ W cedge

loads

þ W bedge

loads

ð8:59Þ

Considering the contribution from each layer, the total work due to edge loads along each boundary is the total summation of the work of each layer. The work due to edge loads is by definition ð

ð tc 2

δW el ¼ x1

t c t 2 t f

ð ð tc 2 þ x2

t c t 2 t f

 t t  bτ22 δv2 þ bτt21 δvt1 dx3 þ

ð tc

 t t  bτ11 δv1 þ bτt12 δvt2 dx3 þ

2

t c 2

bτc23 δvc3 dx3 þ

ð tc 2

t c 2

! ð tc þtb
 2 f b b b b bτ22 δv2 þ bτ21 δv1 dx3 dx1 tc 2

! ð tc þtb
 2 f c b b c b b bτ13 δv3 dx3 þ bτ11 δv1 þ bτ12 δv2 dx3 dx2 tc 2

ð8:60Þ

8 Theory of Sandwich Plates and Shells with an Transversely Compressible. . .

240

Substituting in the pertinent displacement equations, Eq. (8.6a–c)–(8.8a–c) into Eq. (8.60) using the expressions for the local stress resultants and stress couples defined earlier in Eqs. (8.50)–(8.52) gives ð 

 a  d b t22 δua2 þ N b t22 δud2  M b t22 δ ∂u3  M b t21 δud1 þ b t22 δ ∂u3 þ N b t21 δua1 þ N N ∂x ∂x 2 2 x1  a  d c b c δud þ 4 b b b22 δua2 b t21 δ ∂u3 þ N b c23 δua3  2 M b t21 δ ∂u3  M L δΦc þ N M t c 23 3 t 2c 23 3 ∂x1 ∂x1  a  d  a b b21 δ ∂u3 þ b b22 δud2  M b b22 δ ∂u3  M b b21 δud1  M b b22 δ ∂u3 þ N b b21 δua1  N N ∂x2 ∂x2 ∂x1  d ! b b21 δ ∂u3 M dx1 þ ∂x1  a  d ð  t t t t ∂u3 ∂u3 a d b b b b t12 δud2 þ b b t12 δua2 þ N N 11 δu1 þ N 11 δu1  M 11 δ  M 11 δ þN ∂x ∂x 1 1 x2  a  d t t c c ∂u ∂u 3 3 b c δud þ 4 b b b11 δua1 b 12 δ b 13 δua3  2 M b 12 δ L δΦc þ N M þN M t c 13 3 t 2c 13 3 ∂x2 ∂x2  a  d  a b b11 δud1  M b b11 δ ∂u3  M b b12 δud2  M b b12 δ ∂u3 þ b b11 δ ∂u3 þ N b b12 δua2  N N ∂x1 ∂x1 ∂x2  d ! b b12 δ ∂u3 M dx2 ∂x2

δW el ¼

ð8:61Þ Integrating the underlined terms, simplifying, and gathering like terms results in ! b a21 c ∂M b W el ¼ 2 þ N 23 δua3 þ ∂x1 x1 !  a  d

b c23 b d21 M c ∂M c b d22 δ ∂u3 þ 4 b b a22 δ ∂u3 þ 2M L dx1 þ  δΦ 2 δud3 þ 2M tc t 2c 23 3 ∂x1 ∂x2 ∂x2 ! ð n b a12 a d a d c ∂ M a d a d b 11 δu1 þ 2N b 11 δu1 þ 2N b 12 δu2 þ 2N b 12 δu2 þ 2 b 13 δua3 þ 2N þN ∂x 2 x2 !  a  d

b c13 b d12 M a d ∂u3 ∂u3 ∂M 4 bc d c b b þ2M 11 δ þ 2 L13 δΦ3 dx2  2 δu3 þ 2M 11 δ tc tc ∂x2 ∂x1 ∂x1 ð n

b a22 δua2 þ 2N b d22 δud2 þ 2N b a21 δua1 þ 2N b d21 δud1 þ 2N

ð8:62Þ where the defined global stress resultants and stress couples from Eqs. (8.55a and 8.55b) have been utilized.

8.4 Hamilton’s Principle

241

• Work due to surface tractions The expression for the work done by lateral vertical loading such as an external pressure is by definition given as ð
 b W st ¼ qb3 δvb3 dA qt3 δvt3 þ b

ð8:63Þ

A

Substituting Eqs. (8.6c) and (8.7c) into Eq. (8.63) gives ð
    b W st ¼ qt3 δua3 þ δud3 þ b qb3 δua3  δud3 dA

ð8:64Þ

A

Simplifying results in ð n
ð

 o  t b t b a d b b q3 δu3 þ b q3  b q3 δu3 dA ¼ 2 qd3 δud3 dA ð8:65Þ q3 þ b qa3 δua3 þ b W st ¼ A

A

where b qa3 ¼

b qb3 qt3 þ b 2

ð8:66Þ

b qd3 ¼

b qb3 qt3  b 2

ð8:67Þ

• Work due to damping By definition, the work due to damping for the transverse direction is given by ð Wd ¼ A



 C t v_ t3 δvt3 þ C c v_ c3 δvc3 þ C b v_ b3 δvb3 dA

ð8:68Þ

where Ct, Cc and Cb are the structural damping coefficients per unit area of the facings and the core. Substituting in Eqs. (8.6)–(8.8) gives ð Wd ¼ A

 t a      C u_ 3 þ u_ d3 δua3 þ δud3 þ C c u_ a3 δua3 þ C b u_ a3  u_ d3 δua3  δud3 dA ð8:69Þ

Assuming that the damping is constant throughout the thickness of the core, simplifying, and gathering like variational terms gives

8 Theory of Sandwich Plates and Shells with an Transversely Compressible. . .

242

Wd ¼

ð h 

     Ct þ C b þ Cc u_ a3 δua3 þ C t  Cb u_ a3 δud3 þ C t  Cb u_ d3 δua3 þ A ð8:70Þ i  t  C þ C b u_ d3 δud3 dA

which can be written as ð  Wd ¼ 2

Ca þ A

    Cc a u_ 3 þ Cd u_ d3 δua3 þ Cd u_ a3 þ Ca u_ d3 δud3 dA 2

ð8:71Þ

where Ca ¼

Ct þ Cb , 2

Cd ¼

Ct  Cb 2

ð8:72Þ

This expression is substituted into energy function Eq. (8.46).

8.4.3

Kinetic Energy

By definition, the kinetic energy is given by 1 T¼ 2

"

ð ð ð ρ x3 x2 x1

∂V 1 ∂t

2

 2  2 # ∂V 2 ∂V 3 þ dx1 dx2 dx3 ∂t ∂t

ð8:73Þ

Taking a variation in the expression for the kinetic energy and integrating with respect to time gives ð t1 t0

δTdt ¼

ð t1 ð "ð tc þtb 2

t0 σ

tc 2

ð tc þ þ

f

2

t2c


 € b2 δV b2 þ V € b3 δV b3 dx3 € b1 δV b1 þ V ρb V

 c c  € c2 δV c2 þ V € c3 δV c3 dx3 € 1 δV 1 þ V ρc V

ð t c 2

t2c t tf

ð8:74Þ

#  t t t t t t € € € ρ V 1 δV 1 þ V 2 δV 2 þ V 3 δV 3 dx3 dσdt t




If the in-plane inertias are neglected and only the transverse inertias are considered, the kinetic energy can be expressed as

8.5 Governing System

ð t1

δTdt ¼

t0

243

ð t1 ð "ð tc þtb 2

tc 2

t0 A

f

€ b3 δV b3 dx3 ρb V

ð tc þ

2

t2c

€ c3 δV c3 dx3 ρc V

þ

ð t c 2

t2c t tf

# € t3 δV t3 dx3 ρt V

dAdt

ð8:75Þ Using the expressions for the displacements for each respective layer, from Eqs. (8.6)–(8.8), gives ð t1

δTdt ¼

t0

ð t 1 ð (ð  t c

 2    4x3 2 €c ρc u€a3  x3 €ud3 þ  1 Φ 3 tc t 2c t0 A t2c ttf t2c !  2    ð tc þtb 2 f    4x3 2  δua3  x3 δud3 þ  1 δΦc3 dx3 þ ρb €ua3  €ud3 δua3  δud3 dx3 dAdt tc tc t 2c 2 2

 a   €3 þ € ρt u ud3 δua3 þ δud3 dx3 þ

ð tc 2

ð8:76Þ Integrating and simplifying gives ð t 1 ð 
n
  o m 2m € c m d δTdt ¼ 2 ma þ c €ua3 þ 2md €ud3  c Φ u þ δua3 þ 2 ma þ c € 3 2 3 6 3 t0 t0 A

 2mc a 6mc € c €u þ þ2md €ua3 δud3  Φ δΦc3 dAdt 3 3 15 3

ð t1

ð8:77Þ where ma ¼

mtf þ mbf , 2

md ¼

mtf  mbf 2

ð8.78a, bÞ

and mtf ¼

8.5 8.5.1

ð t c 2

t2c t tf

ρtðkÞ dx3 ,

mbf ¼

ð tc þtb 2

tc 2

f

ρbðkÞ dx3 ,

m c ¼ ρc t c

ð8:79a  cÞ

Governing System Equations of Motion

Substituting δU, δW, and the δT back into Hamilton’s Equation, Eq. (8.46) gives

8 Theory of Sandwich Plates and Shells with an Transversely Compressible. . .

244

ð t1 ð l2 ð l1  t0

 a   a   a ∂N 11 ∂N a12 ∂N 22 ∂N a12 ∂N 11 ∂N a12 a a 2 þ þ þ δu1  2 δu2  2 ∂x1 ∂x2 ∂x2 ∂x1 ∂x1 ∂x2 0 0   a      Nc ∂N 22 ∂N a12 N c23 8 8 þ 13 δud1  2 þ þ δud2 þ 2 M c13 δΦc1 þ 2 M c23 δΦc2 tc tc tc tc ∂x2 ∂x1 ( !  c   c  2 2 ∘a N 13 6 c N 23 6 c ∂ ua3 ∂ u3 1 c c a þ2 þ 3 L13 δΩ1 þ 2 þ 3 L23 δΩ2 þ 2N 11 þ þ tc tc tc tc ∂x21 ∂x21 R1 2 ∘a

2

4N a12

∂ ua3 ∂ u3 þ ∂x1 ∂x2 ∂x1 ∂x2

!

2

 2N a22

2 ∘a

∂ ua3 ∂ u3 1 þ þ 2 2 R ∂x2 ∂x2 2

!

2

2

∂ M a11 ∂x21

! ! 2 a 2 ∘a 2 d 2 ∘d 2 2 ∂ u ∂ u ∂ u ∂ u ∂ M a12 ∂ M a22 3 3 3 3 4 2  2N d11 þ þ  4N d12 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x22 ∂x21 ∂x21 ! !  2 2 ∘a t b ∂ ua3 ∂ u3 ∂N c13 ∂N c23 2 tc t f þ t f ∘d d d 2N 22 þ  u þ  u þ  3 3 tc 2 4 ∂x1 ∂x2 ∂x22 ∂x22 4 þ tc

! ! ! ∘d ∘d ∘a ∂ud3 ∂u3 ∂ua3 ∂u3 4 ∂ud3 ∂u3 c c þ þ þ N 23 þ N 13  2 t c ∂x1 ∂x1 ∂x2 ∂x2 ∂x1 ∂x1

 a   a  a   d  ∂u3 ∘∂ua3 ∂u3 ∘∂ud3 ∂N 11 ∂N a12 ∂N 22 ∂N a12 þ þ þ þ 2 2 ∂x1 ∂x2 ∂x2 ∂x2 ∂x2 ∂x1 ∂x1 ∂x1   d     ∂ud3 ∘∂ud3 ∂N 11 ∂N d12 ∂N d22 ∂N d12 þ þþ 2 þ ∂x1 ∂x2 ∂x2 ∂x2 ∂x2 ∂x1   ∂M c13 ∂M c23 4 4 ∂Φc3 c 4 ∂Φc3 c þ M 13  2 M  2b qa3  2 Φc3  2 tc t c ∂x1 t c ∂x1 13 ∂x1 ∂x2 )
 mc a 2mc € c a c a d d a d d €u  2m € 2ðC þ C =2Þu_ 3  2C u_ 3  2 m þ Φ δua3 þ u3 þ 2 3 3 3 (

! ! 2 a 2 ∘a 2 a 2 ∘a ∂ u ∂ u ∂ u ∂ u 1 3 3 3 3 2N d11 þ þ þ  4N d12 ∂x1 ∂x2 ∂x1 ∂x2 ∂x21 ∂x21 R1 ! 2 2 ∘a 2 2 2 ∂ ua3 ∂ u3 ∂ M d11 ∂ M d12 ∂ M d22 1 d 2N 22 þ þ  4  2  2 ∂x1 ∂x2 ∂x22 ∂x22 R2 ∂x21 ∂x22 2

2N a11

2 ∘d

∂ ud3 ∂ u3 þ ∂x21 ∂x21

!

2



4N a12

2 ∘d

∂ ud3 ∂ u3 þ ∂x1 ∂x2 ∂x1 ∂x2

!

2



2N a22

2 ∘d

∂ ud3 ∂ u3 þ ∂x22 ∂x22

! 
 t tf  t bf ∂N c13 ∂N c23 4 tc 4 ∘d d c c  Φ3 þ þ  u3  u3 N 33 þ þ tc 2 tc 16 ∂x1 ∂x2

!

8.5 Governing System

245

  4 ∂Φc3 c 4 ∂Φc3 c 16 c ∂Lc13 ∂Lc23 8 ∂Φc3 c 8 ∂Φc3 c þ N 13 þ N 23 þ 3 Φ3 þ L13 þ 3 L þ 3 t c ∂x1 t c ∂x2 tc t c ∂x1 t c ∂x2 23 ∂x1 ∂x2 ! !  ∘d  ∘d ∂ud3 ∂u3 ∂ud3 ∂u3 ∂N a11 ∂N a12 16 c c þ 3 Φ3 M 33  2 þ þ þ 2 tc ∂x1 ∂x1 ∂x1 ∂x2 ∂x2 ∂x2 ! !  a   ∘a  ∘a ∂ua3 ∂u3 ∂ua3 ∂u3 ∂N 22 ∂N a12 ∂N d11 ∂N d12 N c13 þ þ þ þ 2 þ 2 tc ∂x2 ∂x1 ∂x1 ∂x1 ∂x1 ∂x2 ∂x2 ∂x2  d  
∂M c ∂M c23 ∂N 22 ∂N d12 N c23 4 t ∘d 13 þ þþ þ þ 2 c  ud3  u3  2b qd3  2C d u_ a3 tc tc 2 ∂x2 ∂x1 ∂x1 ∂x2 ) ( ! ∘d
 mc d 4 ∂ud3 ∂u3 a d a d a d €u  2m €u3 δu3 þ  N c13 2C u_ 3  2 m þ þ t c ∂x1 ∂x1 6 3 ! !  
∘d 4 ∂ud3 ∂u3 8 ∂Lc13 ∂Lc23 t c ∘d c d  þ þ  u3  u3 N 23  3 t c ∂x2 ∂x2 2 t c ∂x1 ∂x2 ∘d

∂ud3 ∂u3 8  3 Lc13 þ tc ∂x1 ∂x1

!

∘d

∂ud3 ∂u3 8 þ  3 Lc23 tc ∂x2 ∂x2

! þ

16 c c Φ N t 2c 3 33

 
 16 t c 64 c c 32 c ∂K c13 ∂K c23 8 ∘d d c  u þ Φ L  Φ þ  u M þ 2 M c13 3 3 33 t 3c 2 t 4c 3 33 t 4c 3 ∂x1 tc ∂x2 ! ! ) + ∘a ∘a ∂ua3 ∂u3 8 c ∂ua3 ∂u3 2mc a 6mc € c c €u þ þ þ Φ δΦ3 dx1 dx2 dt þ 2 M 23 þ 3 3 15 3 tc ∂x1 ∂x1 ∂x2 ∂x2

þ

ð t 1 ð l2 D


 b a11 δua1 þ 2 N a12  N b a12 δua2 þ 2 N d11  N b d11 δud2 þ 2 N a11  N t0

0

! ! ∘ ∘ ∂ua3 ∂ua3 ∂ua3 ∂ua3 a þ2  þ þ þ þ 2N 12 ∂x1 ∂x1 ∂x2 ∂x2 ! ! ∘d ∘d d d ∂u ∂u ∂u ∂u ∂M a11 ∂M a12 3 3 þ2N d11 þ 3 þ 2N d12 þ 3 þ2 þ4 ∂x1 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ) t b h i b a12 c ∂M 2 tc t f þ t f 8 c c ∘d d c b 13 δua3 þ þ N þ  u3  u3 N 13 þ 2 M 13 Φ3  2 tc 2 4 tc ∂x2 ( ! ! ! ∘ ∘ ∘ ∂ud3 ∂ud3 ∂ua3 ∂ua3 ∂ud3 ∂ud3 a d a þ 2N 11 þ þ þ þ 2N 11 þ 2N 12 ∂x1 ∂x1 ∂x1 ∂x1 ∂x2 ∂x2


N d12

b d12 N



(

δud2

2N a11

8 Theory of Sandwich Plates and Shells with an Transversely Compressible. . .

246

∘

þ2N d12

∂ua3 ∂ua3 þ ∂x2 ∂x2

!

! t tf  t bf ∂M d11 ∂M d12 8 c c 2 c þ4  3 L13 Φ3  Φ3  N c13 þ2 tc 8 tc ∂x1 ∂x2

)
 b c13 b d12 M ∂M 2 tc ∘d d c  2 þ  u3  u3 M 13  2 δud3 tc tc 2 ∂x2



 ∂ua  a 8 tc 32 c c 4 bc ∘d d c c a 3 b  u3  u3 L13 þ 4 Φ3 K 13  2 L13 δΦ3 þ 2 M 11  M 11 δ þ 3 tc 2 tc tc ∂x1 ð t 1 ð l1 D

  d   b d11 δ ∂u3 dx2 dt þ b a22 δua2 2 N a22  N þ2 M d11  M ∂x1 t0 0


 a d b 21 δua1 þ 2 N d22  N b 22 δud1 þ 2 N d21  N b d21 δud1 þ þ2 N a21  N (

∘

2N a22

∂ua3 ∂ua3 þ ∂x2 ∂x2

!

∘

þ

2N a21

∂ua3 ∂ua3 þ ∂x1 ∂x1

!

∘

þ

2N d22

∂ud3 ∂ud3 þ ∂x2 ∂x2

!

!

t b i ∂M a22 ∂M a21 2 ht c t f þ t f ∘d þ  ud3  u3 N c23 þ4 þ tc 2 4 ∂x2 ∂x1 ) ( ! ∘d d b a21 c ∂u ∂u ∂M 8 c c a a 3 3 b 23 δu3 þ 2N 22 þ 2 M 23 Φ3  2 N þ tc ∂x1 ∂x2 ∂x2 ! ! ! ∘ ∘ ∘ ∂ua3 ∂ua3 ∂ud3 ∂ud3 ∂ua3 ∂ua3 ∂M d22 d a d þ2N 22 þ þ þ þ 2N 21 þ 2N 21 þ2 ∂x2 ∂x2 ∂x1 ∂x1 ∂x1 ∂x1 ∂x2 !
 t tf  t bf ∂M d21 8 c c 2 2 t ∘d N c23  2 c  ud3  u3 M c23 þ4  3 L23 Φ3  Φc3  tc 8 tc tc 2 ∂x1 ∘

þ2N d21

∂ud3 ∂ud3 þ ∂x1 ∂x1

þ2

)


 b c23 b d21 M ∂M 8 t 32 4 c ∘d L23 δΦc3 2 þ δud3 þ 3 c  ud3  u3 Lc23 þ 4 Φc3 K c23  2 b tc tc 2 tc tc ∂x1
  a
  d  b a22 δ ∂u3 þ 2 M d22  M b d22 δ ∂u3 dx1 dt ¼ 0 þ2 M a22  M ∂x2 ∂x2

ð8:80Þ

This above equation can only be satisfied if the integrals are equal to zero each individually. This implies that, since the variational displacements are arbitrary, the coefficients of the variational displacements are zero. This results in the appearance of the equations of motion and the corresponding boundary conditions. The equations of motion appear as δua1 :

∂N a11 ∂N a12 þ ¼0 ∂x1 ∂x2

ð8:81Þ

8.5 Governing System

247

∂N a12 ∂N a22 þ ¼0 ∂x1 ∂x2

ð8:82Þ

δud1 :

∂N d11 ∂N d12 N c13 þ þ ¼0 tc ∂x1 ∂x2

ð8:83Þ

δud2 :

∂N d12 ∂N d22 N c23 þ þ ¼0 tc ∂x1 ∂x2

ð8:84Þ

δua2 :

δua3 : 2

N a11

2 ∘a

∂ ua3 ∂ u3 1 þ þ R1 ∂x21 ∂x21

δΦc1 :

M c13 ¼ 0

ð8:85Þ

δΦc2 :

M c23 ¼ 0

ð8:86Þ

δΩc1 :

N c13 þ

6 c L ¼0 t 2c 13

ð8:87Þ

δΩc2 :

N c23 þ

6 c L ¼0 t 2c 23

ð8:88Þ

!

2

þ

2N a12

2 ∘a

∂ ua3 ∂ u3 þ ∂x1 ∂x2 ∂x1 ∂x2

! þ

N a22

! 2 2 ∘a ∂ ua3 ∂ u3 1 þ þ ∂x22 ∂x22 R2 ! 2 2 ∘d ∂ ud3 ∂ u3 þ ∂x1 ∂x2 ∂x1 ∂x2

! 2 a 2 ∘a 2 2 2 ∂ u ∂ u ∂ M a11 ∂ M a12 ∂ M a22 3 3 þ2 þ þ N d11 þ þ 2N d12 þ ∂x1 ∂x2 ∂x21 ∂x22 ∂x21 ∂x21 ! !  2 2 ∘a t b ∂ ua3 ∂ u3 ∂N c13 ∂N c23 1 tc t f þ t f ∘d d d þN 22 þ þ  u3  u3 þ þ 4 tc 2 ∂x1 ∂x2 ∂x22 ∂x22  d    2 ∂u3 ∘∂ud3 2 ∂ud3 ∘∂ud3 N c23  N c13  þ þ t c ∂x2 t c ∂x1 ∂x2 ∂x1  a  a   a  a  ∂u3 ∘∂ua3 ∂u3 ∘∂ua3 ∂N 11 ∂N a12 ∂N 22 ∂N a12 þ þ þ þ þ þ ∂x1 ∂x1 ∂x1 ∂x2 ∂x2 ∂x2 ∂x2 ∂x1  d        ∂u3 ∘∂ud3 ∂ud3 ∘∂ud3 ∂N d11 ∂N d12 ∂N d22 ∂N d12 þ þ þ þ þ ∂x1 ∂x1 ∂x1 ∂x2 ∂x2 ∂x2 ∂x2 ∂x1   ∂M c13 ∂M c23 2 2 ∂Φc3 c 2 ∂Φc3 c þ 2 Φc3 þ 2 þ M 13 þ 2 M þb qa3 tc t c ∂x1 t c ∂x1 13 ∂x1 ∂x2  
 Cc a m m €c u_ 3 þ C d u_ d3 þ ma þ c €ua3 þ md €ud3  c Φ þ Ca þ ¼0 2 2 3 3

ð8:89Þ

8 Theory of Sandwich Plates and Shells with an Transversely Compressible. . .

248

The underlined terms in the ninth equation of motion vanish with the use of Eqs. (8.81), (8.82), (8.85), and (8.86). Utilizing these equations simplifies the ninth equation of motion to the following form. ! ! ! 2 2 ∘a 2 2 ∘a 2 2 ∘a ∂ ua3 ∂ u3 ∂ ua3 ∂ u3 ∂ ua3 ∂ u3 1 1 a a a þ 2N 12 þ N 22 N 11 þ þ þ þ þ ∂x1 ∂x2 ∂x1 ∂x2 ∂x21 ∂x21 R1 ∂x22 ∂x22 R2 ! ! 2 a 2 ∘a 2 d 2 ∘d 2 2 2 ∂ u ∂ u ∂ u ∂ u ∂ M a11 ∂ M a12 ∂ M a22 3 3 3 3 þ 2N d12 þ2 þ þ N d11 þ þ þ ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x21 ∂x22 ∂x21 ∂x21 ! !  2 2 ∘a t b ∂ ua3 ∂ u3 ∂N c13 ∂N c23 1 tc t f þ t f ∘d d d þ þ þ þN 22 þ  u3  u3 4 tc 2 ∂x1 ∂x2 ∂x22 ∂x22       2 ∂ud3 ∘∂ud3 2 ∂ud3 ∘∂ud3 Cc a N c23  N c13b þ þ  qa3 þ C a þ u_ 3 2 t c ∂x2 t c ∂x1 ∂x2 ∂x1
 m m €c þCd u_ d3 þ ma þ c €ua3 þ md €ud3  c Φ ¼0 2 3 3 ð8:90Þ The tenth equation of motion appears as δud3 : 2 ∘a

2

N d11

∂ ua3 ∂ u3 1 þ þ R1 ∂x21 ∂x21

!

2 ∘a

2

þ

2N d12

∂ ua3 ∂ u3 þ ∂x1 ∂x2 ∂x1 ∂x2

! þ

N d22

! 2 2 ∘a ∂ ua3 ∂ u3 1 þ þ ∂x22 ∂x22 R2 ! 2 2 ∘d ∂ ud3 ∂ u3 þ ∂x1 ∂x2 ∂x1 ∂x2

! 2 d 2 ∘d 2 2 2 ∂ u ∂ u ∂ M d11 ∂ M d12 ∂ M d22 3 3 þ2 þ þ N a11 þ þ 2N a12 þ ∂x1 ∂x2 ∂x21 ∂x22 ∂x21 ∂x21 ! !  2 2 ∘d
 t tf  t bf ∂ ud3 ∂ u3 ∂N c13 ∂N c23 2 tc 2 ∘d a d c c  u þ  u  Φ þ þ N þ þN 22 3 33 3 3 16 tc 2 tc ∂x1 ∂x2 ∂x22 ∂x22   ∂Lc13 ∂Lc23 2 ∂Φc3 c 2 ∂Φc3 c 8 4 ∂Φc3 c 4 ∂Φc3 c  3 N 13  N 23  3 Φc3 þ L13  3 L  t c ∂x1 t c ∂x2 tc t c ∂x1 t c ∂x2 23 ∂x1 ∂x2 ! !   ∘d  ∘d  ∂ud3 ∂u3 ∂ud3 ∂u3 ∂N a11 ∂N a12 ∂N a22 ∂N a12 8 c c þ  3 Φ3 M 33 þ þ þ þ þ tc ∂x1 ∂x1 ∂x1 ∂x2 ∂x2 ∂x2 ∂x2 ∂x1 ∘a

∂ua3 ∂u3 þ þ ∂x1 ∂x1 

!

∂N d11 ∂N d12 N c13 þ þ tc ∂x1 ∂x2



∘a

þ

∂ua3 ∂u3 þ ∂x2 ∂x2

!

∂N d22 ∂N d12 N c23 þ þ tc ∂x2 ∂x1





∂M c ∂M c  2 tc mc d ∘d d d d a a d a 13 23 b _ _ € þ þ q u u ua3  u u þ md €  u þ C þ C þ m þ 3 3 3 3 3 6 3 t 2c 2 ∂x1 ∂x2

¼0

ð8:91Þ

8.5 Governing System

249

The underlined terms in the tenth equation of motion vanish by using Eqs. (8.81)– (8.86). The tenth equation of motion simplifies to 2 ∘a

2

∂ ua3 ∂ u3 1 þ þ R1 ∂x21 ∂x21

N d11 2

2

!

2 ∘a

2

þ 2N d12 2

∂ ua3 ∂ u3 þ ∂x1 ∂x2 ∂x1 ∂x2 2

2 ∘d

∂ ud3 ∂ u3 ∂ M d11 ∂ M d12 ∂ M d22 þ2 þ þ N a11 þ þ 2 2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x21 ∂x21 2 ∘d

2

þN a22

∂ ud3 ∂ u3 þ ∂x22 ∂x22

!


 2 tc 2 ∘d  ud3  u3 N c33  þ tc 2 tc

!

2 ∘a

2

þ N d22

!

∂ ua3 ∂ u3 1 þ þ ∂x22 ∂x22 R2 2 ∘d

2

þ

Φc3

2N a12

∂ ud3 ∂ u3 þ ∂x1 ∂x2 ∂x1 ∂x2

t tf  t bf þ 16

!

!

!

∂N c13 ∂N c23 þ ∂x1 ∂x2



  2 ∂Φc3 c 2 ∂Φc3 c 8 c ∂Lc13 ∂Lc23 4 ∂Φc3 c 4 ∂Φc3 c  3 N 13  N 23  3 Φ3 þ L13  3 L  t c ∂x1 t c ∂x2 tc t c ∂x1 t c ∂x2 23 ∂x1 ∂x2 


 8 c c m ua3 ¼ 0 Φ3 M 33 þ b qd3 þ C d u_ a3 þ C a u_ d3 þ ma þ c €ud3 þ md € 3 6 tc

ð8:92Þ The above tenth equation of motion can be further simplified through the use of Eqs. (8.87) and (8.88) by expressing Lc13 and Lc23 in terms of N c13 and N c23 as shown below. The three double underlined terms expressed in terms of N c13 and N c23 become 1:

    8 c ∂Lc13 ∂Lc23 4 c ∂N c13 ∂N c23 Φ þ Φ þ ¼  3t c 3 ∂x1 t 3c 3 ∂x1 ∂x2 ∂x2

ð8:93Þ

2:

4 ∂Φc3 c 2 ∂Φc3 c L13 ¼  N 3 3t c ∂x1 13 t c ∂x1

ð8:94Þ

3:

4 ∂Φc3 c 2 ∂Φc3 c L23 ¼  N 3 3t c ∂x2 23 t c ∂x2

ð8:95Þ

Utilizing these expressions in terms of N c13 and N c23 in Eq. (8.92) simplifies the tenth equation of motion further while eliminating the variables Lc13 and Lc23 . The tenth equation of motion is now expressed as

8 Theory of Sandwich Plates and Shells with an Transversely Compressible. . .

250

! ! 2 2 ∘a 2 2 ∘a ∂ ua3 ∂ u3 ∂ ua3 ∂ u3 1 d þ þ þ þ þ N 22 ∂x1 ∂x2 ∂x1 ∂x2 ∂x22 ∂x22 R2 ! ! 2 2 ∘d 2 2 ∘d 2 2 2 ∂ ud3 ∂ u3 ∂ ud3 ∂ u3 ∂ M d11 ∂ M d12 ∂ M d22 a a þ þ2 þ þ N 11 þ þ þ 2N 12 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x21 ∂x22 ∂x21 ∂x21 ! !  2 d 2 ∘d
 t tf  t bf ∂ u3 ∂N c13 ∂N c23 2 tc 2 ∘d a ∂ u3 d c c þN 22  u þ  u  Φ þ þ þ N 3 33 3 3 16 tc 2 3t c ∂x1 ∂x2 ∂x22 ∂x22 2

N d11



2 ∘a

∂ ua3 ∂ u3 1 þ þ R1 ∂x21 ∂x21

!

2N d12


 4 ∂Φc3 c 4 ∂Φc3 c 8 m d u N 13  N 23  3 Φc3 M c33 þ b qd3 þ C d u_ a3 þ C a u_ d3 þ ma þ c € 6 3 3t c ∂x1 3t c ∂x2 tc

þmd €ua3 ¼ 0

ð8:96Þ The final eleventh equation of motion appears as δΦc3 :

! !  
∘d ∘d  d ∂u ∂u 4 ∂ud3 ∂u3 4 8 ∂Lc13 ∂Lc23 t c ∘d c c 3 3 N 13  N 23  3  ud3  u3 þ þ þ  t c ∂x1 ∂x1 t c ∂x2 ∂x2 2 t c ∂x1 ∂x2 ! ! ∘d ∘d
 8 c ∂ud3 ∂u3 8 c ∂ud3 ∂u3 16 16 t ∘d  3 L23 þ 2 Φc3 N c33 þ 3 c  ud3  u3 M c33  3 L13 þ þ tc tc tc tc 2 ∂x1 ∂x1 ∂x2 ∂x2 !   ∘a 64 c c 32 c ∂K c13 ∂K c23 8 c ∂ua3 ∂u3 þ 4 Φ3 L33  4 Φ3 þ þ þ 2 M 13 tc tc tc ∂x1 ∂x2 ∂x1 ∂x1 ! ∘a 8 c ∂ua3 ∂u3 2m 6m € c þ c €ua3 þ c Φ þ ¼0 þ 2 M 23 3 15 3 tc ∂x2 ∂x2

ð8:97Þ Just as was done previously, expressing Lc13 and Lc23 in terms of N c13 and N c23 with the use of Eqs. (8.87) and (8.88) for the first and second underlined terms in the above equation along with utilizing Eqs. (8.85) and (8.86) for the third and fourth underlined terms in the above equation results in the eleventh equation of motion being simplified to ! ! ∘d ∘d
 4 ∂ud3 ∂u3 4 ∂ud3 ∂u3 8 8 t ∘d c þ þ N 13 þ N c23 þ 2 Φc3 N c33  3 c  ud3  u3 M c33 3t c ∂x1 ∂x1 3t c ∂x2 ∂x2 tc tc 2  c   c 
 c ∂N 13 ∂N 23 ∂K 13 ∂K c23 32 2 tc 16 ∘d  4 Φc3 Lc33  þ  ud3  u3 þ 4 Φc3 þ 3t c 2 tc tc ∂x1 ∂x2 ∂x1 ∂x2 mc a 3mc € c  €u3  Φ ¼0 3 15 3 ð8:98Þ

8.5 Governing System

251

In summary, the final form of the eleven equations of motion are ð8:99Þ

∂N a12 ∂N a22 þ ¼0 ∂x1 ∂x2

ð8:100Þ

∂N d11 ∂N d12 N c13 þ þ ¼0 tc ∂x1 ∂x2

ð8:101Þ

∂N d12 ∂N d22 N c23 þ þ ¼0 tc ∂x1 ∂x2

ð8:102Þ

M c13 ¼ 0

ð8:103Þ

M c23 ¼ 0

ð8:104Þ

N c13 þ

6 c L ¼0 t 2c 13

ð8:105Þ

N c23 þ

6 c L ¼0 t 2c 23

ð8:106Þ

! ! 2 a 2 ∘a 2 a 2 ∘a ∂ u ∂ u ∂ u ∂ u 1 3 3 3 3 þ 2N a12 þ N a22 N a11 þ þ þ R2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x22 ∂x22 ! ! 2 2 ∘a 2 2 ∘d 2 2 2 ∂ ua3 ∂ u3 ∂ ud3 ∂ u3 ∂ M a11 ∂ M a12 ∂ M a22 d d þ þ 2N 12 þ2 þ þ N 11 þ þ ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x21 ∂x22 ∂x21 ∂x21 ! !  2 2 ∘a t b ∂ ua3 ∂ u3 ∂N c13 ∂N c23 1 tc t f þ t f ∘d d d þN 22 þ þ þ  u þ  u 3 3 4 tc 2 ∂x1 ∂x2 ∂x22 ∂x22  d      2 ∂u3 ∘∂ud3 2 ∂ud3 ∘∂ud3 Cc a c c a a u_ 3 þ C d u_ d3 q3 þ C þ þ þ  N 23  N 13b t c ∂x2 t c ∂x1 2 ∂x2 ∂x1
 m m €c þ ma þ c €ua3 þ md €ud3  c Φ ¼0 2 3 3 ð8:107Þ 2

2 ∘a

∂ ua3 ∂ u3 1 þ þ ∂x21 ∂x21 R1

!

∂N a11 ∂N a12 þ ¼0 ∂x1 ∂x2

8 Theory of Sandwich Plates and Shells with an Transversely Compressible. . .

252 2

2 ∘a

∂ ua3 ∂ u3 1 þ þ 2 2 R ∂x1 ∂x1 1

N d11 2

2

!

2 ∘a

2

þ

2N d12

∂ ua3 ∂ u3 þ ∂x1 ∂x2 ∂x1 ∂x2 2

2

2 ∘d

∂ ud3 ∂ u3 ∂ M d11 ∂ M d12 ∂ M d22 þ2 þ þ N a11 þ þ 2 2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x21 ∂x21 þN a22 

!

2 ∘a

2

N d22

∂ ua3 ∂ u3 1 þ þ 2 2 R ∂x2 ∂x2 2

þ 2N a12

∂ ud3 ∂ u3 þ ∂x1 ∂x2 ∂x1 ∂x2

þ

!

2

2 ∘d

!

!

! !  2 2 ∘d
 t tf  t bf ∂ ud3 ∂ u3 ∂N c13 ∂N c23 2 tc 2 ∘d d c c þ  u  u  Φ þ þ N þ 3 33 3 3 16 tc 2 3t c ∂x1 ∂x2 ∂x22 ∂x22


 4 ∂Φc3 c 4 ∂Φc3 c 8 m d u N 13  N 23  3 Φc3 M c33 þ b qd3 þ C d u_ a3 þ C a u_ d3 þ ma þ c € 6 3 3t c ∂x1 3t c ∂x2 tc

þmd €ua3 ¼ 0

ð8:108Þ ! ∘d
 ∂ud3 ∂u3 8 8 t ∘d þ N c23 þ 2 Φc3 N c33  3 c  ud3  u3 M c33 tc tc 2 ∂x2 ∂x2    
 ∂N c ∂N c23 32 c c 2 tc 16 c ∂K c13 ∂K c23 ∘d d 13  4 Φ3 L33  þ  u3  u3 þ 4 Φ3 þ 3t c 2 tc tc ∂x1 ∂x2 ∂x1 ∂x2

4 3t c



! ∘d ∂ud3 ∂u3 4 þ N c13 þ 3t c ∂x1 ∂x1

mc a 3mc € c €u  Φ ¼0 3 3 15 3 ð8:109Þ

Setting the boundary integrals to zero provides the associated boundary conditions along the edges xn ¼ const(n ¼ 1, 2) uan ¼ b uan

or

b ann N ann ¼ N

ð8:110Þ

uat uat ¼ b

or

b nt N ant ¼ N a

ð8:111Þ

udn ¼ b udn

or

b nn N dnn ¼ N

d

ð8:112Þ

udt ¼ b udt

or

b nt N dnt ¼ N

d

ð8:113Þ

8.5 Governing System

ua3

¼

253

! ! ! ∘ ∘ ∘ ∂ua3 ∂ua3 ∂ua3 ∂ua3 ∂ud3 ∂ud3 a a or þ þ þ N nn þ N nt þ N dnn þ ∂xn ∂xn ∂xt ∂xt ∂xn ∂xn ! ! ∘ ∂ud3 ∂ud3 ∂M ant 1 t c t tf þ t bf ∂M ann ∘d d d þ þ2 þ þ  u3  u3 N c13 N nt þ tc 2 4 ∂xt ∂xt ∂xn ∂xt

b ua3

a

b nt 1 c ∂M b ¼ þ N 2 n3 ∂xt ∘

!

∘

!

∘

!

ð8:114Þ

∂ud3 ∂ud3 ∂ud3 ∂ud3 ∂ua3 ∂ua3 þ þ þ N ann þ N ant þ N dnn þ ∂xn ∂xn ∂xt ∂xt ∂xn ∂xn ! ! ∘ ∂ua3 ∂ua3 ∂M dnt 2 Φc3 t tf  t bf ∂M dnn d þ þ2   N cn3 ¼ N nt þ tc 3 8 ∂xt ∂xt ∂xn ∂xt

ud3 ud3 ¼ b

or

b b nt M ∂M  n3 tc ∂xt d

c

ð8:115Þ

c

b Φc3 ¼ Φ 3

or


2 tc  ud3  ∘ tc 2

 48 b cn3 ud3 N cn3  4 Φc3 K cn3 ¼ N tc

ð8:116Þ

∂ua3 ∂b ua ¼ 3 ∂xn ∂xn

or

b ann M ann ¼ M

ð8:117Þ

∂ud3 ∂b ud ¼ 3 ∂xn ∂xn

or

b nn M dnn ¼ M

ð8:118Þ

d

where n and t are the normal and tangential directions to the boundary When n ¼ 1, t ¼ 2 and when n ¼ 2, t ¼ 1. It should be mentioned that the sixth and seventh boundary conditions have been simplified from their original forms just as was done for the ninth, tenth, and the eleventh equations of motion. The terms that contained Lc13 and Lc23 were expressed in terms of N c13 and N c23 with the use of the pertinent equations of motion. In addition, terms containing M c13 and M c23 were set to zero. For the case of a symmetric sandwich panel, these boundary conditions and equations of motion agree with Hohe and Librescu (2006).

8 Theory of Sandwich Plates and Shells with an Transversely Compressible. . .

254

8.5.2

Boundary Conditions

For the case of simply supported boundary conditions the following conditions must be satisfied. Along the edges xn ¼ 0, Ln N ann ¼ N dnn ¼ N ant ¼ N dnt ¼ M ann ¼ M dnn ¼ ua3 ¼ ud3 ¼ 0

ð8:119Þ

For the case of clamped boundary conditions, the following conditions must be satisfied. uan ¼ uat ¼ udn ¼ udt ¼ ua3 ¼ ud3 ¼

8.6

∂ua3 ∂ud3 ¼ ¼0 ∂xn ∂xn

ð8:120Þ

The Stress Resultants, Stress Couples, and Stiffnesses

The local stress resultants and stress couples can be expressed in terms of the strains by substituting Eqs. (8.42)–(8.44) into Eqs. (8.50)–(8.52). For the top face, the matrix form (see Hause 2012) becomes 0

N t11

1 0

At11

B t C B B N 22 C B C B B B t C B B N 12 C B C B B B t C¼B BM C B B 11 C B B t C B BM C B @ 22 A @ M t12 Sym

At12

At16

Bt11

Bt12

At22

At26

Bt12

Bt22

At66

Bt16

Bt26

Dt11

Dt12 Dt22

Bt16

10

γ t11

1 0

N t11

1T 0

N t11

1m

CB t C B t C B t C C B C B C B Bt26 C CB γ 22 C B N 22 C B N 22 C C C C C B B B B t C B t C B t C Bt66 C CB 2γ 12 C B N 12 C B N 12 C CB t C  B t C  B t C C B C B C B Dt16 C CB κ 11 C B M 11 C B M 11 C C C C C B B B B t C B t C B t C Dt26 C A@ κ 22 A @ M 22 A @ M 22 A 2κ t12 M t12 M t12 Dt66

ð8:121Þ where the local stiffnesses Atij , Btij , Dtij are defined as


Atij , Btij , Dtij



¼

ð t c 2

t2c t tf

(    ) t c þ t tf t c þ t tf 2 t b Qij 1, x3 þ , x3 þ dx3 2 2

ð8:122Þ

For the bottom face, the local stress resultants and couples in matrix form become

8.6 The Stress Resultants, Stress Couples, and Stiffnesses

0

N b11

1 0

B Nb C B B 22 C B B b C B B N 12 C B B C B B Mb C ¼ B B 11 C B B b C B @ M 22 A @ M b12

Ab11

Ab12 Ab22

Ab16 Ab26 Ab66

Bb11 Bb12 Bb16

Bb12 Bb22 Bb26

Db11

Db12 Db22

10 b 1 0 b 1T 0 b 1m γ 11 N 11 N 11 Bb16 B b CB b C B b C B26 CB γ 22 C B N 22 C B N b22 C C CB b C B b C B b C B 2γ 12 C B N 12 C B N 12 C Bb66 C CB C B C B C B b C B b C  B b C Db16 C CB κ11 C B M 11 C B M 11 C CB C B C B C Db26 A@ κb22 A @ M b22 A @ M b22 A Db66

Sym

255

2κb12

M b12

M b12

ð8:123Þ where the local stiffnesses Abij , Bbij , Dbij are defined as




Abij , Bbij , Dbij ¼

ð tc þtb 2

tc 2

f

bb Q ij

8
x3 h1 ð1Þ > V ¼ , > h2 h1 > < V ð2Þ ¼ 1, > >  N > > : V ð3Þ ¼ x3 h3 , h3 h4

x 3 2 ½ h1 , h2  x 3 2 ½ h2 , h3 

ð9:6Þ

x 3 2 ½ h3 , h4 

Here V(n)(n ¼ 1, 2, 3) denotes the volume fraction of layer n and N denotes the volume fraction index. Case (2) – Homogeneous Face-Sheets and FGM Core This case exhibits homogeneous face sheets and a functionally graded core. The face sheets are opposite in the material type. So, if one face is ceramic, the other face is metal, and vice versa. The volume fraction function can be expressed as 8 ð1Þ V ¼ 0, > > <  N 2 V ð2Þ ¼ hx33h , h2 > > : ð3Þ V ¼ 1,

x 3 2 ½ h1 , h2  x 3 2 ½ h2 , h3 

ð9:7Þ

x 3 2 ½ h3 , h4 

As in Case (2), the situation could be reversed with ceramic on the bottom face and metal on the top face (see Abdelaziz et al. 2011 and Ashraf and Alghamdi 2008)

9.3 9.3.1

Basic Equations Displacement Field

Consistent with plate and shell theory (Reddy 2004), the 3D displacement field through the wall thickness can be expressed as uð x 1 , x 2 , x 3 , t Þ ¼ u0 ð x 1 , x 2 , t Þ þ x 3 ψ 1 ð x 1 , x 2 , t Þ

ð9:8aÞ

vð x1 , x2 , x3 , t Þ ¼ v0 ð x1 , x2 , t Þ þ x3 ψ 2 ð x1 , x2 , t Þ

ð9:8bÞ

wðx1 , x2 , x3 , t Þ ¼ w0 ðx1 , x2 , t Þ

ð9:8cÞ

where u0, v0, w0 are the 2D mid-surface displacements of the core, while ψ 1, ψ 2 are the rotation angles of the mid-surface. Note: In this chapter, as compared to the theory in the past two chapters, the displacement field is unified throughout the entire structure, where before, each layer had its own defined displacement field. For functionally graded materials one equivalent displacement field will be utilized to describe the entire distribution of the displacement quantities through the thickness.

268

9.3.2

9 Theory of Functionally Graded Sandwich Plates and Shells

Nonlinear Strain–Displacement Equations

The nonlinear strain displacement relationships (Shen 2009, Reddy 2004) for doubly curved shells assuming the von Kármán assumptions across the plate thickness at distance from the mid-surface are expressed as  2 ∂u w 1 ∂w  þ ∂x1 R1 2 ∂x1  2 ∂v w 1 ∂w  þ ε22 ¼ ∂x2 R2 2 ∂x2   1 ∂v ∂u 1 ∂w ∂w ε12 ¼ þ þ 2 ∂x1 ∂x2 2 ∂x1 ∂x2   1 ∂u ∂w 1 ∂w ∂w þ ε13 ¼ þ 2 ∂x3 ∂x1 2 ∂x1 ∂x3   1 ∂v ∂w 1 ∂w ∂w þ ε23 ¼ þ 2 ∂x3 ∂x2 2 ∂x2 ∂x3 ε11 ¼

ε33

 2 ∂w 1 ∂w ¼ þ ∂x3 2 ∂x3

ð9:9aÞ ð9:9bÞ ð9:9cÞ ð9:9dÞ ð9:9eÞ ð9:9fÞ

Substituting Eqs. (9.8a–c) into Eqs. (9.9a–f) gives the 3D strain displacement equations in terms of the 2D displacement quantities of the mid-surface of the shell. These are expressed as  2 ∂ψ 1 ∂u0 1 ∂w0 w þ  0 þ x3 , R1 ∂x1 2 ∂x1 ∂x1  2 ∂ψ 2 ∂v 1 ∂w0 w ¼ 0þ  0 þ x3 R2 ∂x2 2 ∂x2 ∂x2

ε11 ¼ ε22

γ 12 ¼

  ∂ψ 1 ∂ψ 2 ∂v0 ∂u0 ∂w0 ∂w0 þ þ þ x3 þ ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1

γ 23 ¼ ψ 2 þ

∂w0 , ∂x2

γ 13 ¼ ψ 1 þ

∂w0 , ∂x1

These expressions can be put into matrix form as

ε33 ¼ 0

ð9.10a, bÞ

ð9:10cÞ ð9.10d, eÞ

9.3 Basic Equations

269

9 9 8 8 ð1Þ 9 > εð0Þ > 8 > > ε > > > > 11 11 ε > > > > > 11 > > > > > > > > > > > > ð0Þ > ð1Þ > > > > > > > > > > > > ε ε ε > > > > > 22 = = < 22 = < 22 > < γ 23 ¼ γ ð230Þ þ x3 γ ð231Þ > > > > > > > > > > > > > > > ð0Þ > ð1Þ > γ 13 > > > > > > > > > > > > > γ γ > > > > > > 13 13 > > ; > > : > > ; ; : ð0Þ > : ð1Þ > γ 12 γ 12 γ 12

ð9:11Þ

where ð0Þ ε11 ð0Þ

γ 23

ð1Þ

ε11

 2  2 ∂u0 1 ∂w0 w0 ∂v0 1 ∂w0 w ð0Þ ¼ þ  , ε22 ¼ þ  0 2 2 R R2 ∂x1 ∂x1 ∂x2 ∂x2 1 ∂w0 ∂w0 ∂v ∂u ∂w0 ∂w0 ð0Þ ¼ ψ2 þ , γ 13 ¼ ψ 1 þ , γ 012 ¼ 0 þ 0 þ ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 ∂ψ 1 ∂ψ 2 ∂ψ 1 ∂ψ 2 ð1Þ ð1Þ ¼ , ε22 ¼ , γ 12 ¼ þ ∂x1 ∂x2 ∂x2 ∂x1

ð1Þ

γ 23 ¼ 0,

ð1Þ

γ 13 ¼ 0 ð9:12a  jÞ

9.3.3

Constitutive Equations

For an elastic and isotropic body, the stress–strain relationships are governed by 9ðnÞ 2 9ðnÞ 8 3ðnÞ 8 0 Q11 Q12 > > = = < σ 11 > < ε11  αðx3 , T ÞΔT > 6 7 ε22  αðx3 , T ÞΔT σ 22 ¼ 4 Q12 Q22 0 5 > > > > ; ; : : 0 0 Q66 γ 12 τ12 

  Q44 0 ðnÞ γ 23 ðnÞ τ23 ðnÞ ¼K 0 Q55 τ13 γ 13

ð9:13Þ

ð9:14Þ

The material stiffnesses, Qij(x3), (i ¼ 1, 2, 6) are given by ðnÞ

ðnÞ

Q11 ¼ Q22 ¼ ðnÞ Q44

¼

ðnÞ Q55

¼

EðnÞ ðx3 , T Þ 1  ðνðnÞ ðT ÞÞ ðnÞ Q66

2

,

Q12 ¼

EðnÞ ðx3 , T Þ ¼ 2½1 þ νðnÞ ðT Þ

νðnÞ ðT ÞE ðnÞ ðx3 , T Þ 2

1  ðνðnÞ ðT ÞÞ

ð9:15Þ

270

9 Theory of Functionally Graded Sandwich Plates and Shells

Consistent with the standard plate and shell theory the stress resultants and stress couples are defined as 3
 X N αβ , M αβ ¼

Z

hnþ1

ðnÞ

σ αβ ð1, x3 Þdx3 ,

hn

n¼1

N α3 ¼

3 Z X n¼1

hnþ1

hn

ðα ¼ 1, 2Þ

ðnÞ

τα3 dx3

ð9:16aÞ

ð9:16bÞ

where K is the shear correction factor. With the use of Eqs. (9.13), (9.14) and (9.16a, b), the stress resultants and couples can be written in matrix form as 1

0

0

A11 N 11 B N C BA B 22 C B 12 C B B B N 12 C B A16 C B B BM C ¼ BB B 11 C B 11 C B B @ M 22 A @ B12 M 12

B16

A12 A22

A16 A26

B11 B12

B12 B22

A26 B12

A66 B16

B16 D11

B26 D12

B22

B26

D12

D22

B26

B66

D16

D26





N 23 N 13

¼K

A44

1

0

ð 0Þ

1

0 T 1 B16 B N 11 C B εð 0 Þ C B T C C B26 CB 22 C B N 22 C CB ð0Þ C B T C B66 CB γ 12 C N 12 C C B CB C ð9:17aÞ  B ð 1Þ C B C B D16 CB ε C B M T11 C C C B C 11 C B T C D26 AB B εð1Þ C @ M 22 A @ 22 A D66 M T12 ð1Þ γ 12  ð0Þ ! γ 23 ð9:17bÞ ð0Þ A55 γ ε11

13

or in compact form Eq. (9.17a) can be expressed as



½A fN g ¼ ½B fM g

½B ½D

" ð0Þ # " T # ε N ð1Þ  T M ε

ð9:18Þ

where the global stiffnesses Aij, Bij, and Dij, (i, j ¼ 1, 2, 6) are given by 3
 X Aij , Bij , Dij ¼

Z

n¼1

AIJ ¼

hnþ1

hn

3 Z X n¼1


 Qnij 1, x3 , x23 dx3 ,

hnþ1

hn

ðnÞ

QIJ dx3

ði, j ¼ 1, 2, 6Þ ðI, J ¼ 4, 5Þ

ð9:19aÞ ð9:19bÞ

9.4 Hamilton’s Principle

9.4

271

Hamilton’s Principle

As in the other chapters, an energy approach will be taken to derive the equations of motion. For convenience Hamilton’s Principle (Soedel 2004) is expressed as Z

t1

ðδU  δW  δT Þdt ¼ 0

ð9:20Þ

t0

9.4.1

Strain Energy

Assuming an inextensible core and faces, the variation in the strain energy is given by Z δU ¼

l2 Z l1

0

*Z h n 2 h2

0

ðnÞ σ 11 δε11

þ

ðnÞ σ 22 δε22

þ

ðnÞ τ12 δγ 12

þ

ðnÞ τ13 δγ 13

þ

ðnÞ τ23 δγ 23

+

o

dx3 dx1 dx2

ð9:21Þ Taking a variation in the strains from Eqs. (9.11) and substituting into Eq. (9.21) gives Z δU ¼

l2 Z l1

0

0

*Z

h 2

n

h2

    ðnÞ ð0Þ ð1Þ ðnÞ ð0Þ ð1Þ σ 11 δε11 þ x3 δε11 þ σ 22 δε22 þ x3 δε22

  o E ðnÞ ð0Þ ð1Þ ðnÞ ð0Þ ðnÞ ð0Þ þτ12 δγ 12 þ x3 δγ 12 þ τ13 δγ 13 þ τ23 δγ 23 dx3 dx1 dx2

ð9:22Þ

which can be expressed as δU ¼

Z l2Z 0

l1 n 0

ð0Þ

ð1Þ

ð0Þ

ð1Þ

ð0Þ

ð1Þ

N 11 δε11 þ M 11 δε11 þ N 22 δε22 þ M 22 δε22 þ N 12 δγ 12 þ M 12 δγ 12 þ o ð0Þ ð0Þ N 13 δγ 13 þ N 23 δγ 23 dx1 dx2 ð9:23Þ

Using the mid-surface strain–displacement relationships, Eq. (9.12a–j) in Eq. (9.23) gives the variation in the strain energy in terms of the variation in the displacement quantities which results in

272

9 Theory of Functionally Graded Sandwich Plates and Shells

∂ðδu0 Þ ∂w0 ∂ðδw0 Þ δw0 ∂ðδψ 1 Þ δU ¼ N 11 þ  þ M 11 R ∂x ∂x ∂x1 ∂x 1 1 1 1 0 0  ∂ðδv0 Þ ∂w0 ∂ðδw0 Þ δw0 ∂ðδψ 2 Þ þN 22 þ  þ M 22 R2 ∂x2 ∂x2 ∂x2 ∂x2  ∂ðδv0 Þ ∂ðδu0 Þ ∂ðδw0 Þ ∂w0 ∂w0 ∂ðδw0 Þ þN 12 þ þ þ ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2   ∂ðδψ 1 Þ ∂ðδψ 2 Þ ∂ðδw0 Þ þ þM 12 þ N 13 δψ 1 þ ∂x2 ∂x1 ∂x1   ∂ðδw0 Þ þN 23 δψ 2 þ dx1 dx2 ∂x2 Z

l2 Z l1 



ð9:24Þ

Integrating Eq. (9.24) by parts gives    Z l2 Z l1   ∂N 11 ∂N 12 ∂N 22 ∂N 12 δu0  δv0 δU ¼  þ þ ∂x1 ∂x2 ∂x2 ∂x1 0 0    ∂M 11 ∂M 12 ∂M 22 ∂M 12  þ  N 13 δψ 1  þ  N 23 δψ 2 ∂x1 ∂x2 ∂x2 ∂x1      2 ∂N 22 ∂N 12 ∂w0 ∂N 11 ∂N 12 ∂w0 ∂ w0 þ  þ  þ  N 11 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x21    2 2 ∂ w0 ∂ w0 N 11 N 22 ∂N 13 ∂N 23 2N 12  N 22    þ δw dx1 dx2 0 R1 R2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x22    Z l2  ∂w0 ∂w0  l1 þ N 11 δu0 þ N 12 δv0 þ M 11 δψ 1 þ M 12 δψ 2 þ N 11 þ N 12 þ N 13 δw0  dx2 0 ∂x1 ∂x2 0     Z l1  l2 ∂w0 ∂w0  þ N 12 δu0 þ N 22 δv0 þ M 12 δψ 1 þ M 22 δψ 2 þ N 12 þ N 22 þ N 23 δw0  dx 0 ∂x ∂x 1 2 0

ð9:25Þ

9.4.2

Work Done by External Loads

As before the external work acting on the structure is due to edge loading, lateral loading, body forces, and damping. The work due to body forces is provided below but is neglected in the theoretical developments.

9.4 Hamilton’s Principle

273

• Work due to body forces Z (Z δW b ¼

σ

hþh=

Z

= = ρ= H i δV i dx3

þ

h h

h

Z ρH i δV i dx3 þ

h

== == ρ== H i δV i dx3 == hh

) dσ ð9:26Þ

Where ρ is the mass density and Hi is the body force vector. The body forces are neglected such that gravitational, electrical, and magnetic forces are irrelevant. • Work due to surface tractions l2 Z l1

Z δW st ¼ 0

q3 ðx1 , x2 Þδw0 dx1 dx2

ð9:27Þ

0

• Work due to edge loads Z δW el ¼

l1

(Z

h 2

h2

0

Z

l2

þ 0

)   ðnÞ ðnÞ ðnÞ e σ 22 δv þ e σ 21 δu þ e σ 23 δw dx3 dx1

(Z

h 2

h2

)   ðnÞ ðnÞ ðnÞ e σ 11 δu þ e σ 12 δv þ e σ 13 δw dx3 dx2

ð9:28Þ

Substituting the displacement relationships into Eq. (9.28) gives Z

l1

δW el ¼

(Z

h 2

h2

0

Z

l2

þ

(Z

)   ð nÞ ðnÞ ðnÞ e σ 22 ðδv0 þ x3 δψ 2 Þ þ e σ 21 ðδu0 þ x3 δψ 1 Þ þ e σ 23 δw0 dx3 dx1 þ h 2

h2

0

)   ðnÞ ðnÞ ðnÞ e σ 11 ðδu0 þ x3 δψ 1 Þ þ e σ 12 ðδv0 þ x3 δψ 2 Þ þ e σ 13 δw0 dx3 dx2 ð9:29Þ

Using the definition of stress resultants and couples gives Z δW el ¼ 0

l1


 e 21 δu0 þ N e 22 δv0 þ M e 21 δψ 1 þ M e 22 δψ 2 þ N e 23 δw0 dx1 N

Z

l2


 e 12 δv0 þ M e 11 δψ 1 þ M e 12 δψ 2 þ N e 13 δw0 dx2 e 11 δu0 þ N N

0

ð9:30Þ

274

9 Theory of Functionally Graded Sandwich Plates and Shells

• Work due to damping The work due to damping for the transverse direction is given by Z Wd ¼ 

C ðaveÞ w_ 0 δw0 dA

ð9:31Þ

A

where C(a) is the average structural damping coefficient per unit area among the three layers (the facings and the core).

9.4.3

Kinetic Energy

As from the previous chapters, by definition the kinetic energy of an elastic body is given by 1 T¼ 2

" 2  2  2 # ∂V 1 ∂V 2 ∂V 3 ρ þ dx1 dx2 dx3 ∂t ∂t ∂t x1

Z Z Z x3 x2

ð9:32Þ

Taking the variation in the kinetic energy and integrating by parts with respect to time gives Z

t1

t1 Z

Z δTdt ¼ 

t0

t0

(Z

h 2

h2

A






)

€ 1 δV 1 þ V € 2 δV 2 þ V € 3 δV 3 dx3 dAdt ρ ðx3 Þ V ðnÞ

ð9:33Þ

Substituting the expressions for the displacements from Eqs. (9.8a–c) into Eq. (9.33) results in Z

t1

t0

Z δTdt ¼  t0

t1 Z A

(Z

h 2

h2

ρðnÞ ðx3 Þ½ð€u0 þ x3 ψ€ 1 Þðδu0 þ x3 δψ 1 Þð€v0 þ x3 ψ€ 2 Þ )

ð9:34Þ

€ 0 δw0 dx3 dAdt þðδv0 þ x3 δψ 2 Þ þ w Simplifying and combining like terms gives Z t0

Z t1Z n δTdt ¼  ðm0 €u0 þ m1 ψ€ 1 Þδu0 þ ðm0€v0 þ m1 ψ€ 2 Þδv0 þ ðm1 € u0 þ m2 ψ€ 1 Þδψ 1 þ t0 A o € 0 δw0 dAdt ðm1€v0 þ m2 ψ€ 2 Þδψ 2 þ m0 w

t1

ð9:35Þ where

9.5 Equations of Motion

m0 ¼

3 Z X n¼1

m2 ¼

3 Z X

ρðnÞ ðx3 Þdx3 ,

hn hnþ1 hn

n¼1

9.5

hnþ1

275

m1 ¼

3 Z X n¼1

hnþ1

ρðnÞ ðx3 Þx3 dx3 ,

hn

ð9:36Þ

ρðnÞ ðx3 Þx23 dx3

Equations of Motion Equations of Motion – Mixed Formulation

9.5.1

Substituting δU, δW, δT back into Hamilton’s Equation gives Z

t 1 Z l2 Z l1 



   ∂N 11 ∂N 12 ∂N 22 ∂N 12 € 2 δv0  þ  m0 € þ  m0€v0  m1 ψ u0  m1 ψ€ 1 δu0  ∂x1 ∂x2 ∂x2 ∂x1    ∂M 11 ∂M 12 ∂M 22 ∂M 12 þ  N 13  m1 € þ  N 23  m1€v0 u0  m2 ψ€ 1 δψ 1  ∂x1 ∂x2 ∂x2 ∂x1      2 ∂N 22 ∂N 12 ∂w0 ∂N 11 ∂N 12 ∂w0 ∂ w0 þ  þ  N 11  m2 ψ€ 2 Þδψ 2 þ  ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x21   2 2 ∂ w0 ∂ w0 N 11 N 22 ∂N 13 ∂N 23 þ q3  C ðaveÞ w_ 0 2N 12  N 22    þ R1 R2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x22  € 0 δw0 dx1 dx2 dtþ m0 w



t0

Z

0

0

t 1 Z l2 





 e 11 δu0 þ N 12  N e 12 δv0 þ M 11  M e 11 δψ 1 þ M 12  M e 12 δψ 2 þ N 11  N    l1 ∂w0 ∂w0 e 13 δw0  dx2 dtþ N 11 þ N 12 þ N 13  N 0 ∂x1 ∂x2 Z t 1 Z l1 



 e 12 δu0 þ N 22  N e 22 δv0 þ M 12  M e 12 δψ 1 þ M 22  M e 22 δψ 2 þ N 12  N t0 0    l2 ∂w0 ∂w0 e 23 δw0  dx1 dt ¼ 0 þ N 22 þ N 23  N N 12  ∂x1 ∂x2 0 t0

0

ð9:37Þ This equation can only be satisfied by setting the coefficients of the variational displacements equal to zero since the variational displacements are arbitrary. Setting the coefficients of δu0, δv0, δψ 1, δψ 2, δw0 to zero provides the equations of motion along with the boundary conditions. These are provided as δu0 :

∂N 11 ∂N 12 þ  m0 € u0  m1 ψ€ 1 ¼ 0 ∂x1 ∂x2

ð9:38aÞ

276

9 Theory of Functionally Graded Sandwich Plates and Shells

∂N 22 ∂N 12 þ  m0€v0  m1 ψ€ 2 ¼ 0 ∂x2 ∂x1

ð9:38bÞ

δψ 1 :

∂M 11 ∂M 12 þ  N 13  m1 € u0  m2 ψ€ 1 ¼ 0 ∂x1 ∂x2

ð9:38cÞ

δψ 2 :

∂M 22 ∂M 12 þ  N 23  m1€v0  m2 ψ€ 2 ¼ 0 ∂x2 ∂x1

ð9:38dÞ

δv0 :

δw0 :

  2 2 2 ∂ w0 ∂ w0 ∂ w0 N 11 N 22 ∂N 13 ∂N 23 þ 2N þ N þ þ þ þ þ C ðaÞ w_ 0 12 22 R1 R2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x21 ∂x22 €0  q ¼ 0 þm0 w

N 11

ð9:38eÞ

The boundary conditions become Along the edges x1 ¼ 0, L1 e 11 N 11 ¼ N e 12 N 12 ¼ N

or

u0 ¼ e u0

or v0 ¼ ev0 e 11 or ψ 1 ¼ ψ e1 M 11 ¼ M e e2 M 12 ¼ M 12 or ψ 2 ¼ ψ ∂w0 ∂w0 e 13 N 11 þ N 12 þ N 13 ¼ N ∂x1 ∂x2

ð9:39a  eÞ or

e0 w0 ¼ w

Along the edges x2 ¼ 0, L2 e 12 or u0 ¼ e N 12 ¼ N u0 e N 22 ¼ N 22 or v0 ¼ ev0 e 12 or ψ 1 ¼ ψ e1 M 12 ¼ M e e2 M 22 ¼ M 22 or ψ 2 ¼ ψ ∂w0 ∂w0 e 23 N 12 þ N 22 þ N 23 ¼ N ∂x1 ∂x2

9.5.2

ð9:40a  eÞ or

e0 w0 ¼ w

The Equations of Motion – The Displacement Formulation

The above equations of motion, which are expressed in terms of the mixed formulations, can be expressed in terms of displacements by using Eqs. (9.17a, b) with Eqs. (9.12a–j). With this in hand, the above equations of motion can be expressed in terms of displacements as

9.5 Equations of Motion

277



  2  2 2 2 ∂ u0 ∂w0 ∂ w0 1 ∂w0 ∂ v0 ∂w0 ∂ w0 1 ∂w0 þ A12 δu0 : A11 þ  þ  R1 ∂x1 ∂x1 ∂x21 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 R2 ∂x1 ∂x21  2  2 2 2 ∂ v0 ∂ u0 ∂ w0 ∂w0 ∂w0 ∂ w0 1 ∂w0 þA16 þ 2 þ þ 2  ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 R1 ∂x2 ∂x21 ∂x21 ∂x2   2 2 ∂ v0 ∂w0 ∂ w0 1 ∂w0 þA26 þ  R2 ∂x2 ∂x22 ∂x2 ∂x22  2  2 2 2 ∂ v0 ∂ u0 ∂ w0 ∂w0 ∂w0 ∂ w0 þA66 þ þ þ ∂x1 ∂x2 ∂x22 ∂x1 ∂x2 ∂x2 ∂x1 ∂x22   2 2 2 2 ∂ ψ1 ∂ ψ2 ∂ ψ1 ∂ ψ2 þ B12 þ B16 2 þ þB11 ∂x1 ∂x2 ∂x1 ∂x2 ∂x21 ∂x21  2  2 2 2 2 T ∂N 11 ∂N T12 ∂ ψ2 ∂ ψ1 ∂ ψ2 ∂ ψ1 ∂ u0  þ B þ  ¼ m þ m þB26 66 0 1 ∂x1 ∂x2 ∂x1 ∂x2 ∂t 2 ∂t 2 ∂x22 ∂x22

ð9:41aÞ  2   2  2 2 ∂ v0 ∂w0 ∂ w0 1 ∂w0 ∂ u0 ∂w0 ∂ w0 1 ∂w0 þ A δv0 : A22 þ  þ  12 R2 ∂x2 ∂x2 ∂x22 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 R1 ∂x2 ∂x22  2   2 2 2 2 ∂ u0 ∂w0 ∂ w0 1 ∂w0 ∂ u0 ∂ v0 ∂ w0 ∂w0 þA16 þ A þ  þ 2 þ 26 R1 ∂x1 ∂x1 ∂x21 ∂x1 ∂x2 ∂x22 ∂x21 ∂x22 ∂x1    2 2 2 2 2 ∂w ∂ w0 1 ∂w0 ∂ u0 ∂ v0 ∂ w0 ∂w0 ∂w0 ∂ w0 þ2 0 þ A66  þ þ þ ∂x2 ∂x1 ∂x2 R2 ∂x1 ∂x1 ∂x2 ∂x21 ∂x1 ∂x2 ∂x1 ∂x2 ∂x21   2 2 2 2 2 ∂ ψ2 ∂ ψ1 ∂ ψ1 ∂ ψ2 ∂ ψ1 þ B12 þ B16 þ B26 2 þ þB22 ∂x1 ∂x2 ∂x1 ∂x2 ∂x22 ∂x21 ∂x22  2  2 2 2 T T ∂N 22 ∂N 12 ∂ ψ2 ∂ ψ1 ∂ ψ2 ∂ v0  þ  ¼ m þ m þB66 0 1 ∂x1 ∂x2 ∂x2 ∂x1 ∂t 2 ∂t 2 ∂x21 ð9:41bÞ  2 2 2 2 ∂ u0 ∂w0 ∂ w0 1 ∂w0 ∂ v0 ∂w0 ∂ w0 1 ∂w0 þ B12 δψ 1 : B11 þ  þ  R1 ∂x1 ∂x1 ∂x21 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 R2 ∂x1 ∂x21  2   2 2 2 2 ∂ v0 ∂ u0 ∂ w0 ∂w0 ∂w0 ∂ w0 1 ∂w0 ∂ v0 þB16 þ B þ 2 þ þ 2  þ 26 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 R1 ∂x2 ∂x21 ∂x21 ∂x2 ∂x22   2  2 2 2 2 ∂w0 ∂ w0 1 ∂w0 ∂ v0 ∂ u0 ∂ w0 ∂w0 ∂w0 ∂ w0 þ B þ  þ þ þ 66 R2 ∂x2 ∂x2 ∂x22 ∂x1 ∂x2 ∂x22 ∂x1 ∂x2 ∂x2 ∂x1 ∂x22   2 2 2 2 2 ∂ ψ1 ∂ ψ2 ∂ ψ1 ∂ ψ2 ∂ ψ2 D11 þ D þ D þ D 2 þ 12 16 26 ∂x1 ∂x2 ∂x1 ∂x2 ∂x21 ∂x21 ∂x22  2    2 T ∂M 11 ∂M T12 ∂ ψ1 ∂ ψ2 ∂w0  A  þD66 þ ψ þ  55 1 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x22 





2

¼ m1

2 ∂ ψ1 ∂ u0 þ m2 ∂t 2 ∂t 2

ð9:41cÞ

278

9 Theory of Functionally Graded Sandwich Plates and Shells



  2  2 2 2 ∂ v0 ∂w0 ∂ w0 1 ∂w0 ∂ u0 ∂w0 ∂ w0 1 ∂w0 þ B þ  þ  12 R2 ∂x2 ∂x2 ∂x22 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 R1 ∂x2 ∂x22  2   2 2 2 2 ∂ u0 ∂w0 ∂ w0 1 ∂w0 ∂ u0 ∂ v0 ∂ w0 ∂w0 þB16 þ B þ  þ 2 þ þ 26 R1 ∂x1 ∂x1 ∂x21 ∂x1 ∂x2 ∂x22 ∂x21 ∂x22 ∂x1   2  2 2 2 2 ∂w ∂ w0 1 ∂w0 ∂ u0 ∂ v0 ∂ w0 ∂w0 ∂w0 ∂ w0 þ B66 2 0  þ þ þ ∂x2 ∂x1 ∂x2 R2 ∂x1 ∂x1 ∂x2 ∂x21 ∂x1 ∂x2 ∂x1 ∂x2 ∂x21    2 2 2 2 2 2 ∂ ψ2 ∂ ψ1 ∂ ψ1 ∂ ψ2 ∂ ψ1 ∂ ψ2 þ D66 þ D12 þ D16 þ D26 2 þ þ þD22 ∂x1 ∂x2 ∂x1 ∂x2 ∂x22 ∂x21 ∂x22 ∂x21    2 2 2 ∂M T22 ∂M T12 ∂ ψ1 ∂ ψ2 ∂w0 ∂ v  A44 ψ 2 þ   ¼ m1 20 þ m2 ∂x1 ∂x2 ∂x2 ∂x2 ∂x1 ∂t ∂t 2

δψ 2 : B22

ð9:41dÞ δw0 : ( A11 (

 2 2 2 ∂u0 ∂ w0 1 ∂w0 ∂ w0 1 þ þ 2 2 2 R ∂x1 ∂x1 ∂x1 ∂x1 1

!)  2 2 ∂u0 1 ∂w0 ∂ w0 w0 þ  w0  þ R1 ∂x1 2 ∂x1 ∂x21

 2 2  2 2  2 2 ∂u0 ∂ w0 ∂v0 ∂ w0 1 ∂w0 ∂ w0 1 ∂w0 ∂ w0 1 ∂v0 þ þ þ þ þ 2 ∂x1 2 ∂x2 R1 ∂x2 ∂x1 ∂x22 ∂x2 ∂x21 ∂x22 ∂x21 ! !)  2  2 2 2 1 ∂w0 ∂ w 0 w0 1 ∂u0 1 ∂w0 ∂ w0 w0  w0  þ  w0  þ þ R2 R1 2 ∂x2 R2 ∂x1 2 ∂x1 ∂x22 ∂x21 ( !)  2 2  2 2 2 ∂v0 ∂ w0 1 ∂w0 ∂ w0 1 ∂v0 1 ∂w0 ∂ w0 w0 þ þ þ  w0  þ A22 R2 2 ∂x2 R2 ∂x2 2 ∂x2 ∂x2 ∂x22 ∂x22 ∂x22 ( ( ) 2 2 2 2 2 ∂v0 ∂ w0 ∂u0 ∂ w0 ∂w0 ∂w0 ∂ w0 ∂v0 ∂ w0 ∂u0 ∂ w0 þ þ þ þ A16 2A66 ∂x1 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x21 ∂x2 ∂x21 A12

 2 2  2 2 ∂u0 ∂ w0 ∂w0 ∂ w0 ∂w0 ∂w0 ∂ w0 1 ∂v0 ∂u0 ∂w0 ∂w0 þ2 þ þ þ þ þ  R1 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x1 ∂x2 ∂x21 ( )  2 2 2 2 2 2 ∂ w0 ∂v0 ∂ w0 ∂u0 ∂ w0 ∂v0 ∂ w0 ∂w0 ∂ w0 þ2 þ þ þ 2w0 þ A26 ∂x1 ∂x2 ∂x1 ∂x22 ∂x2 ∂x22 ∂x2 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2  )  2 2 ∂ψ 1 ∂2 w0 ∂w0 ∂w0 ∂ w0 1 ∂v0 ∂u0 ∂w0 ∂w0 ∂ w0 þ B þ þ þ  2w þ 0 11 R2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x1 ∂x2 ∂x22 ∂x1 ∂x2 ∂x1 ∂x21 ( )   ∂ψ 1 ∂2 w0 ∂ψ 2 ∂2 w0 ∂ψ 2 ∂2 w0 1 ∂ψ 1 1 ∂ψ 2 1 ∂ψ 1 þ B12 þ þ þ þ þ B 22 R1 ∂x1 R1 ∂x2 R2 ∂x1 ∂x1 ∂x22 ∂x2 ∂x21 ∂x2 ∂x22

9.6 Final Comments

1 ∂ψ 2 R2 ∂x2



279

(   ∂ψ 1 ∂2 w0 ∂ψ 2 ∂2 w0 ∂ψ 1 ∂2 w0 ∂ψ 2 ∂2 w0 þ þ þ 2B66 þ B16 þ ∂x2 ∂x1 ∂x2 ∂x1 ∂x1 ∂x2 ∂x2 ∂x21 ∂x1 ∂x21

(  ) ∂ψ 1 ∂2 w0 ∂ψ 1 ∂2 w0 ∂ψ 2 ∂2 w0 ∂ψ 2 ∂2 w0 1 ∂ψ 1 ∂ψ 2 þ þ þ þ 2 þB26 2 ∂x1 ∂x1 ∂x2 R1 ∂x2 ∂x1 ∂x2 ∂x22 ∂x1 ∂x22 ∂x2 ∂x1 ∂x2  )     ∂ψ 1 ∂2 w0 ∂ψ 2 ∂2 w0 1 ∂ψ 1 ∂ψ 2 þ þ þ þ þ KA55 þ KA 44 R2 ∂x2 ∂x1 ∂x1 ∂x2 ∂x21 ∂x22  2   2  2 ∂ w0 1 ∂ w0 1 T ∂ w0 T N T11 þ  N þ  2N ¼0 12 22 R2 ∂x1 ∂x2 ∂x21 R1 ∂x22

9.5.3

ð9:41eÞ

The Boundary Conditions

As described in the previous chapters, n and t are the normal and tangential directions to the boundary when n ¼ 1, t ¼ 2 and when n ¼ 2, t ¼ 1. For the case of simply supported boundary conditions the following conditions must be satisfied. Along the edges xn ¼ 0, Ln N nn ¼ N nt ¼ M nn ¼ M nt ¼ w0 ¼ 0

ð9.42a, bÞ

For the case of clamped boundary conditions, the following conditions must be satisfied. N nn ¼ N nt ¼ ψ n ¼ ψ t ¼ w0 ¼ 0

9.6

ð9.43a, bÞ

Final Comments

A basic theoretical foundation concerning functionally graded doubly curved sandwich panels has been presented. Two cases were introduced. One case considered that the facings were homogeneous while the core was functionally graded, and vice versa for the second case. This chapter solely presented the theoretical equations necessary to introduce the topic. There is a plethora of results in the literature upon which the reader is referred to. The reader is referred to such authors as Shen (2009),

280

9 Theory of Functionally Graded Sandwich Plates and Shells

Abdelaziz et al. (2011), and Ashraf and Alghamdi (2008) for additional information and results on the topic.

References Abdelaziz, H. H., Atmane, H. A., Mechab, I., Boumia, L., Tounsi, A., & El Abbas, A. B. (2011). Static analysis of functionally graded sandwich plates using an efficient and simple refined theory. Chinese Journal of Aeronautics, 24, 434–448. Ashraf, Z. M., & Alghamdi, N. A. (2008). Thermoelastic bending analysis of functionally graded sandwich plates. Journal of Material Science, 43, 2574–2589. Huang, X. L., & Shen, H. S. (2004). Nonlinear vibration and dynamic response of functionally graded plates in thermal environments. International Journal of Solids and Structures, 41, 2403–2427. Reddy, J. N. (2004). Mechanics of laminated composite plates and shells-theory and analysis (2nd ed.). Boca Raton: CRC Press. Shen, H. S. (2009). Functionally graded materials-nonlinear analysis of plates and shells. Boca Raton: CRC Press. Soedel, W. (2004). Vibrations of shells and plates (3rd ed.). New York: Marcel Dekker, Inc. Touloukian, Y. S. (1967). Thermophysical properties of high temperature solid materials. New York: Macmillan.

Index

A Air-blast traveling, 175 Airy’s potential function, 82 Airy’s stress potential method, 92 Algebraic equation, 61, 70, 89, 101, 116, 260 Aluminum/Nomex honeycomb, 2 Angle-ply laminated sandwich plates boundary conditions, 97, 98 equations of motion, 96 equilibrium equations, 99 governing system, 92, 95, 96 Newton’s method, 102 numerical results and discussion, 102–104 post-buckling solution, 96, 101 strain–displacement relationships, 98 stress potential, 96 trigonometric functions, 97 Angle-ply laminated sandwich shells, 79 boundary conditions, 113 constants, 112 equilibrium equations, 110 governing system, 105, 109, 110 matrix equation, 111 Newton’s method, 119 numerical results and discussion, 119, 120, 123 post-buckling behavior, 118 problem, 110 solution, 114, 116 strain–displacement relationships, 113 trigonometric functions, 110 Angle-ply laminated sandwich structures, 61 Anisotropic face sheets, 155

Anisotropy, viii, 129 Applied fixed-edge loading, 123

B Bending strains, 14, 190, 228 Biaxial edge loading, 91, 92, 104, 105 Bifurcation, 75, 102, 104 Body forces, 30, 239 Bottom face sheet–core interface, 186 Bottom face sheets, 185, 187, 189, 192 Boundary conditions, 35, 51, 52, 246, 252, 254, 259 Buckling, 210 bifurcation, 90–92, 104, 120 doubly curved sandwich panels (see Doubly curved sandwich panels buckling) flat sandwich panels (see Flat sandwich panels buckling) post-buckling, 223, 256, 259 predictions, 62 response, 61, 77 solution, 61

C Characteristic equation, 146 Clamped boundary conditions, 51, 208, 254, 279 Compatibility equation, 82, 84, 85, 214, 259, 260 Compressible core governing equations (see Governing equations)

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 T. J. Hause, Sandwich Structures: Theory and Responses, https://doi.org/10.1007/978-3-030-71895-4

281

282 Compressible core (cont.) transversely (see Transversely compressible core Compressive edge loads, 82 Compressive uniaxial edge loading, 93, 103, 120–124 Constitutive equations, 18, 20, 192, 193, 230, 231 Contents of text, 5 Core material properties, 63, 72 Core thickness-to-face thickness, 149 Core transverse displacements, 186 Core type strong, 8 weak, 8 Cramer’s rule, 59, 68, 111, 133, 144 Critical buckling load, 63, 137 Cross-ply laminated sandwich shells Airy’s potential function, 82 boundary conditions, 86 compatibility equation, 85 constitutive equations, 84 displacements, 82, 84, 88 edge loadings, 86 equilibrium equations, 83 governing system, 80, 81 in-plane compressive edge loads, 87 matrix equation, 85 nonlinear algebraic equation, 89 numerical results and discussion, 90, 92 particular solution, 87 solution methodology, 84 stress potentials, 82 stress resultants, 87 trigonometric functions, 83 Curved sandwich panel, 152 Curvilinear and orthogonal coordinate system, 56 Cylindrical sandwich panels, 72–74, 91, 151

D Damped natural frequency, 166 Damping, 4, 34, 81, 143, 146, 159, 167, 168, 182, 199, 202, 239, 241, 272, 274 structural, 155 Deflection–time response, 170 Directional material properties, 63 Displacement field, 9–11, 185–188, 224–226, 267 Displacement formulation, 37, 41, 42, 46, 48–51, 276–279 Displacement quantities, 217

Index Doubly curved sandwich panels, 8, 55, 80 boundary conditions, 142, 161, 163 Cramer’s rule, 144 damping measures, 143 equation of motion, 145, 160 equilibrium equation, 145 governing system, 65, 67, 140, 142 Hamilton’s equation, 163 integrations, 146 nondimensional parameters, 147, 164 results and discussion, 72, 74, 148, 149, 151, 152 second-order differential equation, 163 simply supported, 160 solution methodology, 68–71 transverse displacement, 162 undetermined coefficients, 145 validations, 71, 72, 147, 148 Doubly curved sandwich shells, 221 Dynamic buckling, 210 Dynamic response doubly curved sandwich panels, 155 flat sandwich panels (see Flat sandwich panels) incompressible core, 156 pressure pulses (see Pressure pulses) sandwich panels (see Sandwich panels) sandwich structures, 156 structural damping, 155

E Edge loading, 176, 177, 239 Eigenfrequency, 137, 138, 140, 150 Einstein’s summation convention, 26 Elastic body, 269 Electrical force, 239 End-shortening, 126 Equations of equilibrium, 96 Equations of motion, 243–253, 257 Euler behavior, 90 Euler-type post-buckling behavior, 91 Explosive blast, 178, 180 Explosive pressure pulses Friedlander explosive blast pulse, 178, 180 Heaviside pulse, 170, 171 material and geometrical characteristics, 164, 165 rapid-type, 164 rectangular pulse, 169, 170 shockwaves/space vehicles, 164 sine pulse, 170, 172–174 sonic boom, 165–168

Index tangential blast pulse, 173–177 time-dependent loadings, 164 underwater shock pulse, 181–182 Extended Galerkin method (EGM), 4, 58, 70, 127, 132, 220 External pressure, 241 External stimuli, 164

F Face sheet material properties, 62, 72, 90 Face sheets, 185 Face thickness, 63, 137, 149, 150, 154 Face wrinkling, 3, 183 Failure modes, 3 Fiber orientation, 64, 65, 72, 73 Fictious edge load, 81 First-order deformation theory, 265 First-order polynomial, 186 First-Order Shear Deformation Theory (FOSDT), 3, 9, 265 Fixed compressive edge loading, 104 Flat sandwich panels compatibility equation, 214 Cramer’s Rule, 133 displacement quantities, 217 EGM, 132 face sheets, 211 face thickness, 138 fourth-order Runge–Kutta numerical procedure, 220 governing equations, 156, 157, 211, 212 governing system, 130, 131 integrations, 134 matrix, 132 modal amplitudes, 213 natural boundary conditions, 220 orthotropicity ratio, 139 partial differential equations, 218 results and discussion, 135–137 simultaneous equations, 219 solution methodology, 157–159 solution process, 213 stress potential, 213 stress resultants, 214 tangential deformations, 210 transverse boundary conditions, 220 transverse pressure, 213, 220 trigonometric functions, 132, 216 undamped natural frequency, 132, 134 unknown coefficients, 214 validation, 135, 136 Flat sandwich panels buckling governing system, 56–58 results, 62–65

283 solution methodology, 58–61 validation, 61, 62 Free vibration, 4, 64, 129, 130, 140, 142, 147, 154, 156 Frequency ratio, 149, 154 Friedlander explosive blast pulse, 178, 180 Friedlander in-air explosive pulse, 4, 155 Functionally graded sandwich clamped boundary conditions, 279 constitutive equations, 269, 270 displacement field, 267 displacement formulation, 276–279 faces sheets, 265 FGM face-sheets and homogeneous core, 266, 267 first-order deformation theory, 265 FOSDT, 265 Hamilton’s principle (see Hamilton’s principle) homogeneous face-sheets and FGM core, 267 material properties, 266 mixed formulation, 275–276 nonlinear elastic theory, 265 nonlinear strain–displacement equations, 268–269 nonlinear theory, 265 rules of mixtures, 266 shell mid-surface, 265 simply supported, 279 volume fractions, 266 Fundamental frequencies, 147

G Galerkin method, 79, 155, 262 GE Energy, 2 General Instability, 3 Geometrical and material parameters, 60, 71, 89 Geometrical and material properties, 102, 129 Geometrical imperfections, 7, 9, 41, 90, 102, 121, 183, 185, 260 Geometrically nonlinear theory, 184 Global mechanical stiffness, 41 Global mid-surfaces, 22, 42 Governing equations, 156, 157, 257, 258 boundary conditions, 208 equations of motion, 204, 206, 207 flat sandwich panels, 211, 212 stiffnesses, 209, 210 stress couples, 209, 210 stress resultants, 209, 210 Governing system boundary conditions, 254 equations of motion, 243–253

284 Gravity force, 239 Green–Lagrange, 7, 11, 12, 226, 227 Green–Lagrange strain tensor, 188 Green’s Strain Tensor, 3

H Hamilton’s energy, 60, 89, 220 Hamilton’s equation, 25, 158, 163, 204, 265 Hamilton’s principle, 7, 183 boundary conditions, 4 equation, 25, 193, 271 kinetic energy, 29, 203–204, 242, 243, 274 strain energy, 26–28, 193–199, 232–239, 271, 272 theoretical developments, 52 work done by external loads, 30, 31, 199–202, 239–242, 272, 273 Heaviside function, 165, 172 Heaviside pulse, 170, 171 Heaviside step function, viii, 4, 155, 167 Honeycomb-type core construction, 3 Honeycomb-type structure, 2 Hooke’s Law, 7

I Identical symmetric laminated facings, 84 Immovability of the edges, 125, 126 Incident pressure, 182 Incompressible core, 3, 4, 7, 96, 110, 130, 156, 184, 186 In-plane compressive edge loads, 87 In-plane static boundary conditions, 98 Integrals, 246 Intracellular Buckling/face dimpling, 3 Inverse Laplace transform, 168, 170, 172, 174, 178 Isotropic body, 269

K Kinematical and physical parameters, 80 Kinematic continuity conditions, 10 Kinetic energy, 29, 203–204, 231, 242, 243, 274 Kirchhoff–Love assumption, 55, 79 Knockdown factor, 72

L Laplace domain, 172 Laplace transform, 155, 164–166, 168, 170, 172, 174, 178, 182 Laplace Transform Method (LTM), 4

Index Linear formulation, 49–51 Load-carrying capacity, 104, 121 Love–Kirchhoff assumptions, 16, 34, 52, 185, 186, 225 Love–Kirchhoff hypothesis, 7

M Magnetic force, 239 Mass ratio, 181 Material properties, viii, 61, 62, 72, 90, 120, 148, 266 Material stiffnesses, 19 Mid-surface displacements, 188 Mid-surface strain–displacement, 271 Mindlin–Reissner theory, 9 Mixed formulation, 4, 32, 34, 275–276 Modal amplitudes, 84, 89, 101, 116, 213, 220, 260, 262 Mode eigenfrequencies, 139 Modified elastic moduli, 26 Monoclinic symmetry, 18

N Natural boundary conditions, 220 Natural frequencies, 147, 148 Newton’s method, 79, 102, 119 Nondimensionalized coefficients, 61 Nondimensionalized parameters, 61, 135, 159, 164 Nondimensionalized shear load, 71 Nonlinear elastic theory, 265 Nonlinear strain–displacement equations, 188–190, 227–230, 268–269 Normalized curvatures, 150, 153

O Orthogonal curvilinear coordinate system, 8 Orthotropicity ratio, 139, 149, 151 Orthotropy, 8

P Panel aspect ratio, 153 Panel face thickness, 62, 77, 127 Panel Instability, 3 Parameters, 55, 147, 154, 159, 164 Partial differential equations, 218 Partial inversion, 85 Particular solution, 87 Piola–Kirchhoff stress tensor, 18, 27, 193, 232 Poisson’s Ratio, 20 Polymethacrylimide, 2

Index Polystyrene, 2 Polyvinyl Chloride (PVC), 2 Positive phase duration, 165, 173, 178 Post-buckling response, 4, 127, 210 Power law, 266 Pressure pulses explosive (see Explosive pressure pulses)

R Rectangular pulse, viii, 4, 155, 169, 170 Reduced elastic moduli, 18 Reduced moisture moduli, 18 Reflected pressure, 181 Rules of mixtures, 266

S Sandwich construction, 1, 2 Sandwich linear theory, 8, 55 Sandwich panel, 46, 50 constitutive equations, 230, 231 displacement field, 224–226 doubly curved, 155 (see also Doubly curved sandwich panels) face sheets, 256 flat (see Flat sandwich panels) governing equations, 257, 258 Kirchhoff assumptions, 223 level of fidelity, 256 nonlinear strain–displacement equations, 227–230 nonlinear theory, 224 and shells, 32–34 solution methodology, 259–262 stiffnesses, 254–256 stress couples, 254–256 stress resultants, 254–256 transverse displacement function, 256 transverse displacement representation, 223 Sandwich shells, 49, 51, 183, 193, 221 Sandwich stability, 55, 56, 65, 71 Sandwich structure, 185 applications, 1, 2 composition, 2, 3 dynamic response, 4 failure modes, 3 functionally graded, 5 loading conditions, 5 three-layered structure, 1 Second-order differential equation, 158, 163 Second-order power series expansion, 183 Shallow sandwich shell theory, 7, 9, 11, 21, 77

285 Shallow shell theory principles, 185 Shear correction factors, 25, 270 Shear crimping, 3 Shear loading, 61 Shear modulus, 20 Shell mid-surface, 265 Shell theory, 270 Shell-to-plate eigenfrequencies, 151, 152 Shock loading, 181 Shockwaves/space vehicles, 164 Simply supported, 96, 156, 157, 160, 161, 212, 254, 259, 279 boundary conditions, 57, 61, 67, 81, 110, 130, 140, 208 edge conditions, 51 Sine pulse, viii, 4, 155, 170, 172–174 Single-layered flat sandwich panel, 64 Single-layered isotropic/orthotropic material, 1 Snap-through-type behavior, 4, 120 Sonic boom, viii, 4, 155, 165–169 Special cases, 34 Static boundary conditions, 52 Stiffness coefficients, 41 Stiffnesses, 22, 24, 209, 210, 254–256 Strain-displacement, 227 equations, 27, 234 nonlinear equations, 188–190, 268–269 relationships, 11–14, 16, 262 Strains energy, 26–28, 193–199, 232–239, 271, 272 measures, 13, 14, 22–24 tangential, 14 transverse, 14 Stress stress couple resultants, 7, 22–25, 27, 28, 31, 32, 35–37, 42, 82, 87, 113 stress resultants, 21, 194, 195, 200, 209, 210, 214, 215, 220, 233, 234, 240, 270, 273 Stress couples, 209, 210, 254–256 Stress-free initial geometric imperfection, 188 Stress potential, 79, 82, 92, 127, 213, 259–261 Stress resultants, 194, 195, 200, 209, 210, 214, 215, 220, 254–256, 270, 273 Stress–strain relationships, 192, 230, 269 Stress/strain theory algebraic matrix manipulation, 75 buckling bifurcation, 75 edge and couple loadings, 76 fiber orientation, 76 geometrical coordinate system, 75 Kirchhoff–Love assumption, 75

286 Stress/strain theory (cont.) material coordinate system, 77 sandwich panels, 77 Strong core formulation, 37, 41, 42 Surface tractions, 239 Symmetric anisotropic laminated facings, 140

T Tangential blast pulse, 173–177 Tangential deformations, 210 Tangential in-plane directions, 34 Tangential traveling air blast, viii, 4, 155 Tangential velocities, 29 Thermal buckling, 222 Thermal compliance expansion coefficients, 26 Thermal stress and stress couple resultants, 23, 25, 42 Thickness ratios, 152 Third-order deformation theory, 265 Time–deflection response, 175 Time-dependent loads, 155 Time domain, 166, 168, 172, 178 Time of arrival, 178 Top face sheet–core interface, 186 Top face sheets, 185, 187, 189, 192 Transient response, 220 Transverse direction, 241 Transverse displacements, 157, 162, 260, 262 Transverse inertia, 81 Transverse pressure, 30, 91, 92, 106, 123, 125, 127, 156, 211, 213, 220, 262 Transverse shear effects, 34 Transverse shear stresses, 232 Transversely compressible core applications, 210 constitutive equations, 192, 193 displacement field, 185–188 geometrical imperfections, 183 geometry, 184 Hamilton’s principle, 183 (see Hamilton’s principle) incompressible core, 184

Index mid-surface, 184 nonlinear strain–displacement equations, 188–190 notation changes, 184 power series expansion, 183 Triangular pulse, viii, 4, 155, 165 Trigonometric functions, 68, 86, 158, 214, 216, 218 2D tangential displacement measures, 11

U Undamped frequencies, 158, 159 Undamped natural frequencies, 146 Underwater shock pulse, 181–182 Unfulfilled boundary conditions, 88 Unknown coefficients, 214

V Validations, 62, 71, 136, 147–148 Variational displacements, 33 Volume fractions, 266, 267 Von-Karman, 9, 188

W Weak (soft) core, 79, 80, 98, 119 formulation, 42, 45 symmetric facings, 36 Work done by external loads, 199–202, 272, 273 work due to body forces, 30 work due to edge loads, 30, 273 work due to surface tractions, 30, 273 Work due to body forces, 199, 239, 273 Work due to damping, 202 Work due to edge loads, 199, 200 Work due to surface tractions, 201

Y Young’s modulus, 20, 193