Foundations of Elastoplasticity: Subloading Surface Model [4 ed.] 3030931374, 9783030931377

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Koichi Hashiguchi

Foundations of Elastoplasticity: Subloading Surface Model Fourth Edition

Foundations of Elastoplasticity: Subloading Surface Model

Koichi Hashiguchi

Foundations of Elastoplasticity: Subloading Surface Model Fourth Edition

123

Koichi Hashiguchi MSC Software Japan Ltd. Shinjuku-Ku, Tokyo, Japan

ISBN 978-3-030-93137-7 ISBN 978-3-030-93138-4 https://doi.org/10.1007/978-3-030-93138-4

(eBook)

1st–3rd editions: © Springer International Publishing AG 2009, 2014, 2017 4th edition: © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 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

Preface

The elastoplasticity theory has developed drastically during the last half century. Then, the elastoplastic constitutive equation was formulated so as to describe not only the monotonic but also the cyclic loading behaviors accurately. Further, the rigorous formulations of the viscoplastic constitutive equation for the rate-dependent deformation behavior, the multiplicative hyperelastic-based plastic constitutive equation for the finite deformation, the multiplicative hyperelastic-based crystal plasticity and the constitutive equation for friction have been attained. This book is the standard text book for elastoplasticity which is explained comprehensively covering the monotonic to the cyclic loading behaviors. Further, the viscoplastic constitutive equation, the constitutive equation of elastoplastic-damage, the multiplicative hyperelastic-based plastic constitutive equation, the crystal plasticity and the constitutive equation for friction are explained thoroughly. Prior to these descriptions, the comprehensive explanations on vector-tensor analysis and continuum mechanics will be provided as a foundation for elastoplasticity theory, covering the underlying physical concepts, mathematical operations and theories, e.g. convective time-derivative, integration of convective rate variable, pull-back and push-forward operations, objectivity, work-conjugacy and multiplicative decomposition. The following theories/models, which provide the fundamental framework of the constitutive equations for irreversible (plastic) deformation/sliding of solids, have been proposed by the author and thus they are explained concisely and comprehensively in this book, which can never be described by the other plasticity models, e.g. the Chaboche, the Mroz and the Dafalias models assuming the purely-elastic domain within which the stress lies always. 1. The loading criterion, i.e. the judgment whether the plastic strain rate is induced is formulated based on the rational physical background (Chap. 8 Sect. 8.5), which holds in the general material exhibiting not only the hardening but also the softening behaviors. 2. The continuity condition and the smoothness condition are formulated (Chap. 9 Sect. 9.1), which must be satisfied in constitutive equations.

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3. The subloading surface concept underling the cyclic plasticity is introduced (Chap. 9 Sect. 9.2), which insists that the plastic deformation is not induced suddenly at the moment when the stress reaches the yield surface as has been postulated in the other models but it develops continuously as the stress approaches the yield surface, renamed the normal-yield surface. Here, the approaching-degree of the stress to the normal-yield surface is represented by the ratio (called the normal-yield ratio) of the size of the subloading surface to that of the normal-yield surface, while the subloading surface passes through the current stress point and is similar to the normal-yield surface. Therefore, the smooth elastic-plastic transition leading to the continuous variation of the tangent stiffness modulus is described always. Consequently, the yield judgment whether the stress reaches the yield surface is not required in constitutive equations based on the subloading surface concept. This concept is inevitable to describe the cyclic loading behavior under a general stress amplitude. 4. The explicit constitutive equation based on the subloading surface concept is formulated (Chaps. 9, 11 and 12), by which the cyclic loading behavior under the general stress amplitude is described accurately. On the other hand, the other models, i.e. the cylindrical yield surface model (Chaboche and Ohno-Wang), the multi-surface model (Mroz) and the two(bounding)-surface model (Dafalias-Yoshida) which adopt the yield surface enclosing a purely-elastic domain inside which the stress lies always are incapable of describing the cyclic loading behavior under the stress amplitude inside the yield surface (Chap. 10). Therefore, these models other than the subloading surface model are inapplicable to the mechanical design of solids and structures subjected to the general cyclic loading. Here, it should be noted that there exists the limitation in the downsizing of the elastic domain in the multi-surface and the two-surface models, because the outward-normal direction of the yield surface fluctuates busily by the downsizing of the yield surface, so that the direction of the plastic strain rate based on the normality-rule becomes unstable. Worst of all, the yield surface is limited to the large Mises yield surface in the cylindrical yield surface (Chaboche) model falling within the framework of the conventional plasticity. Soon or later these models will have to disappear from the history of plasticity. 5. The subloading-overstress model is formulated (Chap. 14), which is capable of describing the elastoplastic deformation in the quasi-static loading and the viscoplastic deformation in the general rate ranging from the quasi-static to the impact loading during the monotonic and the cyclic loading with a general stress amplitude. On the other hand, the past overstress (Bingham-Prager-Perzyna) model is incapable of describing the cyclic loading behavior induced by the variation of stress inside the yield surface and the loading behavior at high rate, predicting the elastic response under the impact loading. 6. The tangential-inelastic strain rate induced by the rate of stress tangential to the subloading surface was formulated (Chap. 9 Sect. 9.7 and Chap. 11 Sect. 11.10), which is inevitable in the analysis of the non-proportional deformation resulting in the instability and localization. On the other hand, the continuity

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condition is violated if the tangential-inelastic strain rate is incorporated into the other models because they assume the purely-elastic domain. 7. The constitutive equations of various materials, i.e. metals, soils, polymers, etc. are formulated (Chaps. 12, 13 and 18) by virtue of the high generality of the subloading surface model. In contrast, the Chaboche (Ohno) model published and praised mainly in Int. J. Plasticity for the last nearly half a century is limited to the description of only Mises metal without the inherent (e.g. orthotropic) anisotropy and the pressure-dependence, etc. and thus it is the typical ad hoc model only for the limited particular deformation and material, i.e. metals. 8. The subloading-damage model is formulated (Chap. 15), while the formulation by the subloading surface model is of crucial importance because the damage progresses by the slight plastic deformation induced by the variation of stress inside the yield surface. 9. The complete multiplicative-subloading hyperelastic-based (visco)plasticity for the exact descriptions of the finite elastic and (visco)plastic deformations under the monotonic/cyclic loadings is formulated (Chap. 17), which cannot be formulated by the other plasticity models. 10. The subloading-multiplicative hyperelastic-based crystal (visco)plasticity model is formulated (Chap. 21), by which the finite elastic and visco(plastic) deformations of crystalline solids can be described exactly. On the other hand, the other crystal plasticity models fall within the framework of the hypoelastic-based plasticity so that their formulations are of quite complicated forms, requiring the cumbersome stress integration procedure, and the elastic deformation described by them is limited to be infinitesimal. 11. The subloading-friction model is formulated (Chap. 22), which is capable of describing the smooth elastic-plastic transition, the reduction of friction-coefficient from the static to the kinetic friction, the recovery to the static friction during the cease of sliding, the saturation of the tangential contact stress with the increase of the normal contact stress and the dry and the fluid (lubricated) frictions at the general rate of sliding from the static to the impact sliding. Consequently, the subloading-overstress model is to be the only model which is capable of describing the rate-independent (elastoplastic) and rate-dependent (viscoplastic) deformation/sliding behaviors of solids under the monotonic and the cyclic loading processes in the general stress/strain amplitude and the general rate ranging from the static to the impact loading. Then, all the constitutive models for the irreversible deformation/sliding behaviors of solids will have to be unified to this model in the near future. The subloading surface model will be engraved as the governing law of the irreversible phenomena in the history of solid mechanics. However, what a lot of effort and time have been wasted repeatedly in order to be converged to the subloading surface model. The Chaboche model, the multi surface (Mroz) model, the two surface (Dafalias) model, the creep model, the two scale damage model, the cap soil model, the rate-and-state model, etc. are the typical huge wastes.

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The subloading surface model for metals has been implemented to the commercial FEM software “Marc” marketed from MSC Software Ltd. as the standard uploaded (ready-made) program by the name “Hashiguchi model” after the 2017 version, so that Marc users can apply these models to their deformation analyses. Further, the performance that the material parameters can be determined automatically by inputting the stress-strain curve is installed after the 2019 version, so that even the users unfamiliar to the elastoplasticity theory can receive the high benefits of the subloading surface model. These implementations have been highly supported by Dr. Motohatu Tateishi under the active encouragement by President Takehiko Kato, MSC Software Ltd., Japan. The subloading surface model for soils and the subloading-friction model will be also installed to the Marc as the standard uploaded programs soon. The computer programs of the subloading surface model for metals and soils and the subloading-friction models are given in Appendix L so as to use these models, capturing clearly the formulations and the calculation processes. The detail of the implicit numerical calculation method will not be provided, since it has been described in detail in the former monographs (Hashiguchi 2017, 2020). The author wishes to express cordial thanks to his colleagues at Kyushu University, who have discussed and collaborated with the author over a long period of time: Prof. Masami Ueno (currently Emeritus Prof., Univ. Ryukyus) in particular, and Dr. Takashi Okayasu (currently Prof. Kyushu Univ.), Dr. Seiichiro Tsutsumi (currently Associate Prof., Osaka Univ.), Dr. Toshiyuki Ozaki, Kyushu Electric Eng. Consult. Inc., Dr. Shingo Ozaki (currently Prof. Yokohama National Univ.) and Dr. Tatsuya Mase, Tokyo Electric Power Services Co., Ltd (currently Prof. Tezukayama Univ.). In addition, Emeritus Prof. Tadatsugu Tanaka, Univ. of Tokyo, Dr. Jnnji Yoshida, Yamanashi Univ. and Dr. Shogo Sannomaru, Mazuda Motor Corporation are appreciated for their valuable discussions and advices. Further, the author would like to express his sincere gratitude to Emeritus Prof. Akira Asaoka and his colleagues at Nagoya University: Prof. Masaki Nakano and Prof. Toshihiro Noda who have appreciated and used the author’s model widely in their analyses and who have offered discussion continuously on deformation of geomaterials. In addition, the author thanks Emeritus Prof. Teruo Nakai and Prof. Feng Zhang, the Nagoya Inst. Tech., for their valuable comments. Furthermore, the author is thankful to Dr. Kazuo Okamura, Dr. Noriyuki Suzuki, Dr. Ryoichi Higuchi and Dr. Masahiro Ogawa, Nippon Steel Corporation, Dr. Atsuro Iga, Dr. Masanori Oka and Mr. Takuya Anjiki, Yanmar Co. Ltd. for the collaborations on constitutive relations of metals and the numerical calculations. He is also grateful to Mr. Takehiko Kato (President) and Dr. Motoharu Tateishi, MSC Software, Ltd., Japan for the standard implementation of the Hashiguchi (subloading surface) model to the commercial FEM software Marc.

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The heartfelt gratitude of the author is dedicated to Prof. Yuki Yamakawa, Tohoku University, for a lot of valuable advices for the improvements of the descriptions through the intensive reading the manuscript and a close collaboration with detailed discussions on elastoplasticity theory, particularly on the finite strain theory based on the multiplicative elastoplasticity. In particular, the author expresses his sincere gratitude to emeritus Prof. Genki Yagawa, University of Tokyo, for encouraging always the author with undeserved high appreciation of research contributions, and thus the author was stimulated to the publication of this book. Finally, the author would like to acknowledge the enthusiastic supports by Dr. Thomas Ditzinger (Editorial director), Mr. Holger Schöpe (Book editorial service team), Springer, Heidelberg and Ms. Sindhu Sundararajan (Project manager at Scientific Publishing Services, Chennai) for the publication of this book. Tokyo, Japan Feb 23, 2022

Koichi Hashiguchi Technical Adviser, MSC Software Ltd. (Emeritus Professor, Kyushu University, Japan)

Contents

1

Mathematical Preliminaries: Vector and Tensor Analysis . . . . . 1.1 Conventions and Symbols . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Summation Convention . . . . . . . . . . . . . . . . . . . 1.1.2 Kronecker’s Delta and Permutation Symbol . . . . 1.1.3 Matrix and Determinant . . . . . . . . . . . . . . . . . . 1.2 Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Definition of Vector . . . . . . . . . . . . . . . . . . . . . 1.2.2 Operations of Vectors . . . . . . . . . . . . . . . . . . . . 1.2.3 Coordinate Transformation of Vector . . . . . . . . . 1.3 Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Definition of Tensor . . . . . . . . . . . . . . . . . . . . . 1.3.2 Quotient Law . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Notations of Tensors . . . . . . . . . . . . . . . . . . . . . 1.3.4 Orthogonal Tensor . . . . . . . . . . . . . . . . . . . . . . 1.4 Operations of Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Notations in Tensor Operations . . . . . . . . . . . . . 1.4.2 Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Various Tensors . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . 1.6 Calculations of Eigenvalues and Eigenvectors . . . . . . . . . . 1.6.1 Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.2 Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Eigenvalues and Eigenvectors of Skew-Symmetric Tensor . 1.8 Cayley-Hamilton Theorem . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Scalar Triple Products with Invariants . . . . . . . . . . . . . . . 1.10 Positive Definite Tensor . . . . . . . . . . . . . . . . . . . . . . . . . 1.11 Polar Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.12 Isotropic Tensor-Valued Tensor Function . . . . . . . . . . . . . 1.13 Representation of Tensor in Principal Space . . . . . . . . . . .

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1.14 1.15 1.16 1.17 1.18

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Two-Dimensional State . . . . . . . . . . . . . . . . . . . . Tensor Functions . . . . . . . . . . . . . . . . . . . . . . . . . Partial Differential Calculi . . . . . . . . . . . . . . . . . . Differentiation and Integration in Tensor Field . . . Representation in General Coordinate System . . . . 1.18.1 Primary and Reciprocal Base Vectors . . 1.18.2 Metric Tensor and Base Vector Algebra . 1.18.3 Tensor Representations . . . . . . . . . . . . .

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Description of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Motion of Material Point . . . . . . . . . . . . . . . . . . . . . . . 2.2 Time-Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Variations and Rates of Geometrical Elements . . . . . . . 2.3.1 Deformation Gradient and Variations of Line, Surface and Volume Elements . . . . . . . . . . . . 2.3.2 Velocity Gradient and Rates of Line, Surface and Volume Elements . . . . . . . . . . . . . . . . . . 2.4 Material-Time Derivative of Volume Integration . . . . . .

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Description of Tensor (Rate) in Convected Coordinate System . 3.1 Reference and Current Primary and Reciprocal Base Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Description of Deformation Gradient Tensor by Embedded Base Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Pull-Back and Push-Forward Operations . . . . . . . . . . . . . . 3.4 Convected Time-Derivative . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 General Convected Derivative . . . . . . . . . . . . . . 3.4.2 Corotational Rate . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 On Adoption of Convected Rate Tensor in Hypoelastic Constitutive Equation . . . . . . . . . 3.4.4 Time-Integration of Convected Rate Tensor . . . .

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Deformation/Rotation Tensors . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Deformation Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Strain Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Volumetric and Isochoric Parts of Deformation Gradient Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Strain Rate and Spin Tensors . . . . . . . . . . . . . . . . . . . . . 4.5 Logarithmic (True) and Infinitesimal (Nominal) Strains . .

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Stress 5.1 5.2 5.3 5.4

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Tensors and Conservation Laws . . . . . . . Stress Tensor . . . . . . . . . . . . . . . . . . . . . Conservation Law of Mass . . . . . . . . . . . Conservation Law of Linear Momentum . . Conservation Law of Angular Momentum

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Contents

5.5 5.6 5.7 5.8 5.9 5.10

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Equilibrium Equation . . . . . . . . . . . . . . . . . . . . Equilibrium Equation of Angular Moment . . . . . Virtual Work Principle . . . . . . . . . . . . . . . . . . . Conservation Law of Energy . . . . . . . . . . . . . . . Work Conjugacy . . . . . . . . . . . . . . . . . . . . . . . . Various Simple Deformations . . . . . . . . . . . . . . 5.10.1 Uniaxial Loading . . . . . . . . . . . . . . . . 5.10.2 Simple Shear . . . . . . . . . . . . . . . . . . . 5.10.3 Combination of Tension and Distortion

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Objectivity and Objective (Rate) Tensors . . . . . . . . . . . . . . . . 6.1 Objectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Influence of Rigid-Body Rotation on Various Mechanical Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Material-Time Derivative of Tensor . . . . . . . . . . . . . . . . 6.4 Objectivity of Convective Time-Derivative and Corotational Rate . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Various Objective Stress Rate Tensors . . . . . . . . . . . . . . 6.6 Jaumann Rate with Plastic Spin . . . . . . . . . . . . . . . . . . . 6.7 Time-Derivative of Scalar-Valued Tensor Function . . . . . Elastic 7.1 7.2 7.3

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Constitutive Equations . . . . . . . . . . . . . . . Definition of Hyperelasticity . . . . . . . . . . . Hyperelastic Equations . . . . . . . . . . . . . . . Explicit Hyperelastic Models . . . . . . . . . . . 7.3.1 St.Venant-Kirchhoff Model . . . . . 7.3.2 Neo-Hookean Model . . . . . . . . . . 7.3.3 Mooney Model . . . . . . . . . . . . . . 7.3.4 Ogden Model . . . . . . . . . . . . . . . Rate Forms of Hyperelastic Equation . . . . . Infinitesimal Strain-Based Elastic Equation . Cauchy Elasticity . . . . . . . . . . . . . . . . . . . Hypoelasticity . . . . . . . . . . . . . . . . . . . . . .

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Elastoplastic Constitutive Equations . . . . . . . . . . . . . . . . . . . . 8.1 Fundamental Requirements for Elastoplastic Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Classification of Elastoplastic Constitutive Equations . . . . 8.2.1 Infinitesimal Hyperelastic-Based Plasticity . . . . 8.2.2 Hypoelastic-Based Plasticity . . . . . . . . . . . . . . 8.2.3 Multiplicative Hyperelastic-Based Plasticity . . . 8.3 Conventional Plastic Constitutive Equation . . . . . . . . . . . 8.4 Constitutive Equation of Metals . . . . . . . . . . . . . . . . . . . 8.5 Formulation of General Loading Criterion . . . . . . . . . . . 8.6 Physical Backgrounds of Associated Flow Rule . . . . . . .

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8.6.1

8.7

8.8 8.9 8.10 9

Positiveness of Second-Order Plastic Work Rate: Prager’s Interpretation . . . . . . . . . . . . . . . . . . . . . 8.6.2 Principle of Maximum Plastic Work . . . . . . . . . . 8.6.3 Positiveness of Work Done During Stress Cycle: Drucker’s Interpretation . . . . . . . . . . . . . . . . . . . 8.6.4 Positiveness of Second-Order Plastic Relaxation Work Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.5 Comparison of Interpretations for Associated Flow Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.1 Definition of Isotropy . . . . . . . . . . . . . . . . . . . . . 8.7.2 Elastoplastic Constitutive Equation with Kinematic Hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.3 Kinematic Hardening Rules . . . . . . . . . . . . . . . . . Plastic Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Physical Interpretation of Nonlinear Kinematic Hardening Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Limitations of Conventional Elastoplasticity . . . . . . . . . . . .

Unconventional Elastoplasticity Model: Subloading Surface Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Mechanical Requirements . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Continuity Condition in the Small . . . . . . . . . . . . 9.1.2 Continuity Condition in the Large: Smoothness Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Subloading Surface (Hashiguchi) Model . . . . . . . . . . . . . . . 9.3 Distinguished Advantages of Subloading Surface Model . . . 9.4 Numerical Performance of Subloading Surface Model . . . . . 9.5 On Bounding Surface Model with Radial-Mapping: Misuse of Subloading Surface Concept . . . . . . . . . . . . . . . . . . . . . 9.6 Incorporation of Kinematic Hardening . . . . . . . . . . . . . . . . 9.7 Incorporation of Tangential-Inelastic Strain Rate . . . . . . . . . 9.8 Limitation of Initial Subloading Surface Model . . . . . . . . . .

10 Classification of Plasticity Models: Critical Reviews and Assessments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Cyclic Loading Behavior . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Classification and Assessment of Plasticity Models . . . . . . 10.3 Plasticity Models with Elastic Domain . . . . . . . . . . . . . . . 10.3.1 Common Drawbacks in Models with ElasticDomain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Cylindrical Yield Surface (Chaboche) Model: Ad Hoc. Primitive Conventional Model Limited to Simple Metal Behavior . . . . . . . . . . . . . . . . .

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10.3.3

10.4 10.5

Multi-surface (Mroz) Model: Incapable of Describing Mechanical Ratchetting . . . . . . . 10.3.4 Two Surface (Dafalias) Model: Incapable of Describing Plastic Strain Rate in Unloading Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extended Subloading Surface (Hashiguchi) Model: Capable of Describing General Loading Behavior . . . . . . Overall Assessment of Plasticity Models . . . . . . . . . . . .

11 Extended Subloading Surface Model . . . . . . . . . . . . . . . . . . . . 11.1 Normal-Yield and Subloading Surfaces . . . . . . . . . . . . . 11.2 Evolution Rule of Elastic-Core . . . . . . . . . . . . . . . . . . . 11.3 Plastic Strain Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Stain Rate Versus Stress Rate Relations . . . . . . . . . . . . . 11.5 Calculation of Normal-Yield Ratio in Unloading Process . 11.6 Improvement of Inverse and Reloading Responses . . . . . 11.7 Loading Criterion for Large Loading Increment . . . . . . . 11.7.1 Exact Judgment of Loading . . . . . . . . . . . . . . . 11.7.2 Initial Value of Normal-Yield Ratio in Plastic Corrector Step . . . . . . . . . . . . . . . . . . . . . . . . 11.8 Plastic Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.9 Incorporation of Tangential-Inelastic Strain Rate . . . . . . .

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293 293 296 302 303 304 305 308 309

. . . . 312 . . . . 315 . . . . 315

12 Constitutive Equations of Metals . . . . . . . . . . . . . . . . . . . . . . . 12.1 Yield Surface, Isotropic, Kinematic Hardening and Elastic-Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Cyclic Stagnation of Isotropic Hardening . . . . . . . . . . . . . 12.3 Calculation of Normal-Yield Ratio in Unloading Process . . 12.4 Implicit Stress-Integration . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Material Parameters and Comparisons with Test Data . . . . 12.5.1 Material Parameters . . . . . . . . . . . . . . . . . . . . . 12.5.2 Comparisons with Test Data . . . . . . . . . . . . . . . 12.6 Analyses of Engineering Phenomena . . . . . . . . . . . . . . . . 12.7 Orthotropic Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7.1 Representation of Isotropic Mises Yield Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7.2 Plane Strain State . . . . . . . . . . . . . . . . . . . . . . . 12.8 Subloading Surface Model with Orthotropic Anisotropy . . 12.8.1 Subloading Surface with Orthotropic Anisotropy 12.8.2 Plastic Strain Rate . . . . . . . . . . . . . . . . . . . . . . 12.8.3 Normal-Yield Ratio . . . . . . . . . . . . . . . . . . . . . 12.8.4 Elastic-Core Yield Ratio . . . . . . . . . . . . . . . . . . 12.8.5 Cyclic Stagnation of Isotropic Hardening . . . . . .

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319 321 326 327 329 329 331 342 346

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356 359 360 360 361 362 363 363

xvi

13 Constitutive Equations of Soils . . . . . . . . . . . . . . . . . . . . . . . 13.1 Isotropic Consolidation Characteristics . . . . . . . . . . . . . 13.2 Yield Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1 Yield Functions . . . . . . . . . . . . . . . . . . . . . . 13.2.2 Critical State Surface Taken Account of Third Deviatoric Invariant . . . . . . . . . . . . . . . . . . . 13.3 Subloading Surface Model for Soils . . . . . . . . . . . . . . . 13.4 Extension of Material Functions . . . . . . . . . . . . . . . . . . 13.4.1 Yield Surface with Tensile Strength . . . . . . . . 13.4.2 Rotational Hardening . . . . . . . . . . . . . . . . . . 13.5 Extended Subloading Surface Model . . . . . . . . . . . . . . 13.5.1 Superyield, Normal-Yield and Subloading Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5.2 Evolution Rules of Internal Variables . . . . . . . 13.5.3 Plastic Strain Rate . . . . . . . . . . . . . . . . . . . . 13.5.4 Yield Stress Function . . . . . . . . . . . . . . . . . . 13.5.5 Partial Derivatives of Subloading Surface Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5.6 Calculation of Normal-Yield Ratio . . . . . . . . . 13.6 Simulations of Test Results . . . . . . . . . . . . . . . . . . . . . 13.7 Numerical Analysis of Footing Settlement Problem . . . . 13.8 Hyperelastic Equation of Soils . . . . . . . . . . . . . . . . . . .

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365 365 374 374

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377 383 393 393 397 401

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401 405 407 409

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409 413 419 423 430

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433 433 435 438 440

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442 442 445 447 449

14 Viscoplastic Constitutive Equations with Subloading Surface Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Rate-Dependent Deformation of Solids . . . . . . . . . . . . . . . . 14.2 History of Viscoplastic Constitutive Equations . . . . . . . . . . 14.3 Irrationality of Creep Model . . . . . . . . . . . . . . . . . . . . . . . 14.4 Mechanical Response of Past Overstress Model . . . . . . . . . 14.5 Subloading Overstress Model: Extension to Description of General Rate of Deformation . . . . . . . . . . . . . . . . . . . . . 14.5.1 Static and Limit Subloading Surfaces . . . . . . . . . . 14.5.2 Viscoplastic Strain Rate . . . . . . . . . . . . . . . . . . . 14.5.3 Strain Rate Versus Stress Rate Relation . . . . . . . . 14.6 Comparison with Test Data . . . . . . . . . . . . . . . . . . . . . . . . 14.6.1 Dynamic Loading Process Inducing Elastic-Viscoplastic Deformation . . . . . . . . . . . . . 14.6.2 Quasi-static Loading Process Inducing Elastoplastic Deformation Behaviors . . . . . . . . . . 14.7 Temperature Dependence of Elasto-Viscoplastic Deformation Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . 449 . . 451 . . 454

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xvii

15 Continuum Damage Model with Subloading Surface Concept 15.1 Basic Hypothesis of Strain Equivalence in Constitutive Equation with Brittle Damage . . . . . . . . . . . . . . . . . . . . 15.2 Hyperelastic Equation in Undamaged Variables . . . . . . . 15.3 Hyperelastic Equations with Damage . . . . . . . . . . . . . . . 15.3.1 Bilateral Damage . . . . . . . . . . . . . . . . . . . . . . 15.3.2 Unilateral Damage . . . . . . . . . . . . . . . . . . . . . 15.4 Evolution of Damage Variable . . . . . . . . . . . . . . . . . . . . 15.4.1 Bilateral Damage . . . . . . . . . . . . . . . . . . . . . . 15.4.2 Unilateral Damage . . . . . . . . . . . . . . . . . . . . . 15.5 Elastoplastic-Damage Model with Subloading Surface Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5.1 Plastic Strain Rate . . . . . . . . . . . . . . . . . . . . . 15.5.2 Stress(Rate) Versus Strain(Rate) Relation and Stress Integration . . . . . . . . . . . . . . . . . . . 15.6 Anisotropic (Orthotropic) Damage Tensor . . . . . . . . . . . 15.7 Subloading-Overstress Damage Model . . . . . . . . . . . . . . 15.7.1 Bilateral Damage . . . . . . . . . . . . . . . . . . . . . . 15.7.2 Unilateral Damage . . . . . . . . . . . . . . . . . . . . . 15.8 Subloading-Gurson Model for Ductile Damage . . . . . . . . 15.9 High Cycle Fatigue: Redundancy of Two-Scale Damage Model and Necessity of Subloading-Damage Surface Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Subloading Phase-Transformation Model . . . . . . . . . . . . . . . 16.1 Constitutive Equation . . . . . . . . . . . . . . . . . . . . . . . . . 16.1.1 Elastic Strain Increment . . . . . . . . . . . . . . . . 16.1.2 Plastic Strain Increment Based on Subloading Surface Model . . . . . . . . . . . . . . . . . . . . . . . 16.2 Thermal and Transformation Strain Increments . . . . . . . 16.2.1 Heat-Transformation Strain Increment . . . . . . 16.2.2 Transformation-Plastic Strain Increment . . . . . 16.3 Stress Rate Versus Strain Rate Relation . . . . . . . . . . . . 17 Multiplicative Hyperelastic-Based Plasticity with Subloading Surface Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1 Exact Elastic–Plastic Decomposition of Deformation Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1.1 Necessity of Multiplicative Decomposition of Deformation Gradient Tensor . . . . . . . . . . 17.1.2 Embedded Base Vectors in Intermediate Configuration . . . . . . . . . . . . . . . . . . . . . . . .

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458 458 460 460 462 472 472 473

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477 481 484 485 485 486

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494 496 496 497 498

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Contents

17.2

17.3 17.4 17.5 17.6 17.7 17.8 17.9 17.10 17.11

17.12 17.13 17.14

Deformation Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.1 Elastic and Plastic Right Cauchy-Green Deformation Tensor . . . . . . . . . . . . . . . . . . . . . . 17.2.2 Strain Rate and Spin Tensors . . . . . . . . . . . . . . . On Limitation of Hypoelastic-Based Plasticity . . . . . . . . . . Further Multiplicative Decomposition of Plastic Deformation Gradient Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Formulation and Calculation in Intermediate Configuration: Isoclinic Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stress Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Internal Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Normal-Yield, Subloading and Elastic-Core Surfaces . . . . . Plastic Flow Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plastic Strain Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Material Functions for Metals and Soils . . . . . . . . . . . . . . . 17.11.1 Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.11.2 Soils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculation Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . Isotropic Hardening Stagnation . . . . . . . . . . . . . . . . . . . . . Subloading-Overstress Model . . . . . . . . . . . . . . . . . . . . . . . 17.14.1 Constitutive Equation . . . . . . . . . . . . . . . . . . . . . 17.14.2 Calculation Procedure . . . . . . . . . . . . . . . . . . . . .

. . 505 . . 505 . . 506 . . 509 . . 510 . . . . . . . . . . . . . .

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516 517 519 522 524 527 529 529 531 536 538 542 542 548

18 Viscoelastic-Viscoplastic Model of Polymers . . . . . . . . . . . . . 18.1 Viscoelastic Rheological Model . . . . . . . . . . . . . . . . . . 18.2 Viscoelastic Deformation with Elastic Strain Energy Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2.1 Elastic Strain Free-Energy Function . . . . . . . . 18.2.2 Second Piola–Kirchhoff Stress Tensor . . . . . . 18.3 Viscoelastic-Damage Model: Subloading-Mullins Effect 18.4 Viscoplastic Constitutive Equation in Glassy State . . . .

. . . . . 549 . . . . . 549 . . . . .

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551 551 552 557 561

19 Corotational Rate Tensors . . . . . . . . . . 19.1 Hypoelasticity . . . . . . . . . . . . . . . 19.1.1 Zaremba-Jaumann Rate . 19.1.2 Green-Naghdi Rate . . . . 19.2 Kinematic Hardening Material . . . 19.2.1 Zaremba-Jaumann Rate . 19.2.2 Green-Naghdi Rate . . . . 19.3 Plastic Spin . . . . . . . . . . . . . . . . .

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565 565 566 567 569 571 571 573

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Contents

20 Localization of Deformation . . . . . . . . . . . . . . . . . . 20.1 Element Test . . . . . . . . . . . . . . . . . . . . . . . . . 20.2 Gradient Theory . . . . . . . . . . . . . . . . . . . . . . 20.3 Shear-Band Embedded Model: Smeared Crack 20.4 Necessary Condition for Shear Band Inception

xix

....... ....... ....... Model . . .......

21 Hypoelastic- and Multiplicative Hyperelastic-Based Crystal Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1 Description of Strain Rate and Spin by Crystal Lattice Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Resolved Shear Stress (Rate) . . . . . . . . . . . . . . . . . . . . 21.3 Stress Rate Versus Plastic Shear Strain Rate Relation . . 21.4 Conventional Crystal Plasticity Model . . . . . . . . . . . . . 21.4.1 Yield Condition and Flow Rule . . . . . . . . . . . 21.4.2 Evolution of Isotropic Hardening . . . . . . . . . . 21.4.3 Evolution of Kinematic Hardening . . . . . . . . . 21.4.4 Stress Rate Versus Strain Rate Relation . . . . . 21.5 Subloading Crystal Plasticity Model . . . . . . . . . . . . . . . 21.6 Subloading-Overstress Crystal Plasticity Model . . . . . . . 21.7 Extension to Description of Cyclic Loading Behavior . . 21.8 Uniqueness of Slip Rate Mode . . . . . . . . . . . . . . . . . . . 21.9 Various Schemes for Calculation of Shear Strain Rates . 21.9.1 Singular Value Decomposition . . . . . . . . . . . 21.9.2 Regularized Schmid Law . . . . . . . . . . . . . . . 21.9.3 On Creep-Type Crystal Plasticity Model . . . .

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583 583 584 588 589

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596 601 604 605 605 606 607 609 613 619 623 629 632 633 636 637

22 Constitutive Equation for Friction: Subloading-Friction Model . 22.1 History of Constitutive Equation for Friction . . . . . . . . . . . 22.2 Sliding Displacement and Contact Traction . . . . . . . . . . . . . 22.3 Hyperelastic Sliding Behavior . . . . . . . . . . . . . . . . . . . . . . 22.4 Elastoplastic Sliding Velocity . . . . . . . . . . . . . . . . . . . . . . . 22.4.1 Sliding Normal-Yield and Subloading Surfaces . . 22.4.2 Evolution Rule of Sliding Hardening Function . . . 22.4.3 Evolution Rule of Sliding Normal-Yield Ratio . . . 22.4.4 Elastoplastic Sliding Velocity . . . . . . . . . . . . . . . 22.5 Loading Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.6 Calculation of Normal Sliding-Yield Ratio . . . . . . . . . . . . . 22.7 Fundamental Mechanical Behavior of Subloading-Friction Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.7.1 Relation of Tangential Contact Stress Rate and Sliding Velocity . . . . . . . . . . . . . . . . . . . . . . 22.7.2 Numerical Experiments and Comparisons with Test Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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641 641 643 646 647 647 648 651 654 659 661

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Contents

22.8 22.9

Stick–Slip Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . Friction Condition with Saturation of Tangential Contact Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.10 Subloading-Overstress Friction Model . . . . . . . . . . . . . . 22.10.1 Subloading-Overstress (Viscoelastic) Friction Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.10.2 Numerical Experiments . . . . . . . . . . . . . . . . . . 22.10.3 Comparison with Test Data . . . . . . . . . . . . . . . 22.11 Extension to Rotational and Orthotropic Anisotropy . . . .

. . . . 669 . . . . 672 . . . . 683 . . . .

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684 689 692 694

Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703 Appendix A: Projection of Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709 Appendix B: Logarithmic Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711 Appendix C: Matrix Representation of Tensor Relations . . . . . . . . . . . . 715 Appendix D: Euler’s Theorem for Homogeneous Function . . . . . . . . . . . 721 Appendix E: Outward-Normal Tensor of Surface . . . . . . . . . . . . . . . . . . 723 Appendix F: Relationships of Material Constants in ln v
ln p and e
ln p Linear Relations . . . . . . . . . . . . . . . . . . . . . . . . 725 Appendix G: Derivative in Critical State . . . . . . . . . . . . . . . . . . . . . . . . . 727 Appendix H: Convexity of Two-Dimensional Curve . . . . . . . . . . . . . . . . . 729 Appendix I: Normal Tensor to Subloading Surface with Anisotropic Damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733 Appendix J: Tensor Exponential Map for Time-Integration of First-Order Linear Differential Equation . . . . . . . . . . . . 737 Appendix K: Eyring Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739 Appendix L: Computer Programs of Subloading Surface Models . . . . . 741 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 841

Chapter 1

Mathematical Preliminaries: Vector and Tensor Analysis

Physical quantities appearing in continuum mechanics are mathematically expressed by tensors because they possess not only magnitudes but also directions in multi-dimensional space. Therefore, their relations are described by tensor relations. Before the explanation on the main theme of this book, i.e. the elastoplasticity theory, the mathematical properties of tensors and the mathematical rules on tensor operations are explained on the level necessary to understand the continuum mechanics mainly related to the elastoplasticity theory. The readers will able to understand the formulations almost completely, since the comprehensive explanations and proofs for almost all of the concepts and the equations are given without any logical jumps.

1.1

Conventions and Symbols

Some basic conventions and symbols appearing in the tensor analysis are described in this section.

1.1.1

Summation Convention

We first introduce the Cartesian summation convention. Repeated suffix in a term is summed over numbers that the suffix can take. For instance, ur v r ¼

3 P

ur vr ; Tir vr ¼

r¼1

Trr ¼

3 P

Trr

3 P r¼1

9 > Tir vr > = > > ;

ð1:1Þ

r¼1

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 K. Hashiguchi, Foundations of Elastoplasticity: Subloading Surface Model, https://doi.org/10.1007/978-3-030-93138-4_1

1

2

1

Mathematical Preliminaries: Vector and Tensor Analysis

where the range of suffixes is 1; 2; 3. Because of ur vr ¼ us vs ; Tir vr ¼ Tis vs ; Trr ¼ Tss a letter of the repeated suffix is arbitrary. It is therefore called as the dummy index. The convention described above is also called Einstein’s summation convention. Hereinafter, a repeated index obeys this convention unless otherwise specified by the additional remark “(no sum)”.

1.1.2

Kronecker’s Delta and Permutation Symbol

The symbol dij ði; j ¼ 1; 2; 3Þ defined in the following equation is called the Kronecker’s delta. dij ¼

 1: i ¼ j 0: i ¼ 6 j

ð1:2Þ

for which one has dir drj ¼ dij ;

dii ¼ 3

ð1:3Þ

Furthermore, the symbol eijk defined by the following equation is called the alternating (or permutation) symbol or Eddington’s epsilon or Levi-Citiva “e” tensor.

eijk

8

> a  a ¼     a ¼ 1 > > > ½a b c  > > > > < b  c a  b ¼     b ¼ 0 ½a b c  > > > > > > ½b  c c  a a  b  ½a b c 2 1 > >  > ða  bÞ c ¼ ¼ ¼    :   3   3   ½a b c  ½a b c  ½a b c 

ð1:64Þ

noting Eq. (1.59) with Eq. (1.54). Further, it follows from Eqs. (1.52) and (1.59) that a  a þ b  b þ c  c ¼ 0

ð1:65Þ

noting a  a þ b  b  þ c  c  ¼

a  ðb  cÞ þ b  ðc  aÞ þ c  ða  bÞ ½abc

(5) Tensor product and component Based on the vectors vð1Þ ; vð2Þ ;   ; vðmÞ , one can make the m-th order tensor as follows: ð2Þ    ðmÞ vð1Þ  vð2Þ     vðmÞ ¼ vð1Þ vpm ep1  ep2     epm p 1 vp 2

ð1:66Þ

For two vectors, one has the second-order tensor u  v ¼ ui ei vj ej ¼ ui vj ei ej

ð1:67Þ

which is expressed in the matrix form 2

u1 v1 ½u  v ¼ 4 u2 v1 u3 v 1

u1 v 2 u2 v 2 u3 v 2

3 u1 v 3 u2 v 3 5 u3 v 3

ð1:68Þ

As described above, one can make a tensor from two vectors. After the scalar product u  v and the vector product u  v for the two vectors u and v, one calls

14

1

Mathematical Preliminaries: Vector and Tensor Analysis

u  v as the tensor (cross) product or dyad which means “one set by two”. Particularly, it holds for three arbitrary vectors u, v and w that ðu  vÞw ¼ ðui ei  vj ej Þwk ek ¼ ui ei ðvj wk djk Þ ¼ ui ei ðvj wj Þ ¼ uðv  wÞ and thus the following expression holds. u  v ¼ uðv 

ð1:69Þ

It follows from Eq. (1.8) that detðu  vÞ ¼ eijk u1 vi u2 vj u3 vk ¼ u1 u2 u3 eijk vi vj vk ¼ u1 u2 u3 ½vvv leading to detðu  vÞ ¼ 0

ð1:70Þ

Equation (1.62) is rewritten in terms of the tensor product as follows: v ¼ gv

ð1:71Þ

where g ¼ a  a þ b  b þ c  c ¼ a  a þ b  b þ c  c ð¼gT Þ

ð1:72Þ

g is regarded as the generalized identity tensor and is called the metric tensor transforming the vector to itself in the general coordinate system. In other words, the identity tensor in the general coordinate system is defined by incorporating the reciprocal base vectors ða ; b ; c Þ for the primary base vectors ða; b; cÞ. In particular, g is reduced in the normalized rectangular coordinate system ða ¼ a ¼ e1 ; b ¼ b ¼ e2 ; c ¼ c ¼ e3 Þ as follows: I ¼ dij ei  ej ¼ ei  ei ð¼IT Þ;

dij ¼ ei  ej

ð1:73Þ

which possesses the components of the Kronecker’s delta dij and is called the unit tensor or the identity tensor transforming the vector to itself. Incidentally, the permutation tensor is defined by e ¼ eijk ei  ej  ek ;

eijk ¼ ½ei ej ek 

ð1:74Þ

noting Eq. (1.48). The following relation holds. e :ðu  vÞ ¼ u  v

ð1:75Þ

1.2 Vector

15

noting ðeijk ei  ej  ek Þ:ður er  vs es Þ ¼ eijk ei ur vs djr dks ¼ eijk ei uj vk ¼ uj ej  vk ek

1.2.3

Coordinate Transformation of Vector

Adopt the other normalized orthogonal coordinate system fO  xi g with the base fei g in addition to the normalized orthogonal coordinate system fO  xi g with the base fei g (Fig. 1.1). Noting v ¼ vj ej ¼ ðv  ej Þej in general, the following relations hold for the base vectors. ei ¼ ðei  ej Þej ; ei ¼ Qri er ;

ei ¼ ðei  ej Þej

ð1:76Þ

ei ¼ Qir er

ð1:77Þ

where the coordinate transformation operator Qij is defined by Qij  cosðangle between ei and ej Þ ¼ ei  ej

ð1:78Þ

Moreover, because of Qir Qjr ¼ ðei  er Þðej  er Þ ¼ ei  ðej  er Þer ¼ ei  ej Qri Qrj ¼ ðer  ei Þðer  ej Þ ¼ ðei  er Þer  ej ¼ ei  ej



It follows that Qir Qjr ¼ Qri Qrj ¼ dij

ð1:79Þ

It is assumed for a while that the relative (parallel and rotational) motion does not exist between the above-described coordinate systems, and that their origins mutually coincide. Then, denoting the component on the base ei by ðÞ , the x2

x*2

v

v2

v*2

e2

v*1

e*2 e*1 0

e1

x*1

θ v1

x1

Fig. 1.1 Coordinate transformation of vector illustrated in the two-dimensional state

16

1

Mathematical Preliminaries: Vector and Tensor Analysis

coordinate transformation rule, i.e. the transformation rule of the components of v in these coordinate systems is given by vi ¼ Qij vj

ð1:80Þ

noting vr er  ei ¼ vj ej  ei and based on v ¼ vj ej ¼ vr er

ð1:81Þ

Furthermore, noting Qri vr ¼ Qri Qrs vs ¼ dis vs , the inverse relation of Eq. (1.80) is given as vi ¼ Qji vj

ð1:82Þ

Equations (1.80) and (1.82) are expressed in matrix form as 38 9 8 9 2 Q11 Q12 Q13 > < v1 > = < v1 = 6 7  v2 ¼ 4 Q21 Q22 Q23 5 v2 ; > : ; : > ; v3 Q31 Q32 Q33 v3

8 9 2 38 9 Q11 Q21 Q31 < v = > < v1 > = 6 7 1 v2 ¼ 4 Q12 Q22 Q32 5 v2 > : ; : > ; v3 v3 Q13 Q23 Q33

ð1:83Þ

which are often expressed simply as fv g ¼ ½Qfvg;

fvg ¼ ½QT fv g

ð1:84Þ

Needless to say, an equation including ð Þ and ð Þ does not describe the relation between different vectors, but describes the relations between components when a certain vector is described by two different coordinate systems. As known from the following equation, the magnitude of vector is not influenced by the coordinate transformation, whilst it is the basic property of the scalar quantity. kv k ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi Qir vr Qis vs ¼ Qir Qis vr vs ¼ drs vr vs ¼ vr vr ¼ kvk

The relations described above are shown below. 

e  e ½Q ¼ 1 1 e2  e1 

e1  e2 e2  e2

cos h sin h ½Q½Q ¼  sin h cos h T



  e ¼ 1 1 e2 1



cos h sin h

  e

1 2 ¼ cos h sin h  sin h cos h e2 2

  sin h 1 ¼ cos h 0

0 ¼ dij ¼ ½I 1

1.2 Vector

17



 ei ¼ ei  e1 e1 þ ei  e2 e2 ¼ Qi1 e1 þ Qi2 e2 ; e1 ¼ cos he1 þ sin he2 e2 ¼  sin he1 þ cos he2





 ei ¼ ei  e1 e1 þ ei  e2 e2 ¼ Q1i e1 þ Q2i e2 v ¼ v1 e1 þ v2 e2 ¼ v1 e1 þ v2 e2 

v1 v2



 ¼

cos h sin h  sin h cos h



 v1 ; v2



v1 v2



 ¼

cos h sin h

 sin h cos h



v1 v2



  where h designates the angle that the base ei rotates in the anti-clock direction from the base fei g. Choosing the position vector x as the vector v, it follows from Eqs. (1.80) and (1.82) that xi ¼ Qir xr

)

xi ¼ Qri xr

ð1:85Þ

from which one has 9 @xi @Qir xr @xr > ¼ ¼ Qir ¼ Qir djr ¼ Qij > = @xj @xj @xj > @xj @Qrj xr @x ; ¼ ¼ Qrj r ¼ Qrj dir ¼ Qij >   @xi @xi @xi

ð1:86Þ

Consequently, Qij can be also described as Qij ¼

1.3

@xi @xj ¼  @xj @xi

ð1:87Þ

Tensor

The vector described in the foregoing possesses the direction in first order but there exit quantities possessing the direction in high order. They are collectively called the tensor. The general definition and mathematical properties of tensor are described in this section.

18

1

1.3.1

Mathematical Preliminaries: Vector and Tensor Analysis

Definition of Tensor

Let the set of nm functions be described as Tðp1 ; p2 ;   ; pm Þ in the coordinate system fO  xi g with the origin O and the axes xi ði ¼ 1; 2;  ;nÞ in the n-dimensional space, where each of the indices p1 ; p2 ;   ; pm takes the number 1; 2;  ;n. This set of functions is defined as the m th-order tensor in the n-dimension, if the set of functions is observed in the other coordinate system fOxi g with the origin O and the axes xi as follows: T  ðp1 ; p2 ;    pm Þ ¼ Qp1 q1 Qp2 q2    Qpm qm Tðq1 ; q2 ;    qm Þ

ð1:88Þ

or T  ðp1 ; p2 ;    pm Þ ¼

@xp1 @xp2 @xq1 @xq2



@xpm Tðq1 ; q2 ;    qm Þ @xqm

ð1:89Þ

provided that only the directions of axes are different but the origin is common and the relative motion does not exist. Here, Eq. (1.87) is used. Then, designating Tðp1 ; p2 ;    pm Þ by the symbol Tp1 p2 pm for the simplicity of notation, Eqs. (1.88) or (1.89) is expressed as 

Tp1 p2

¼ Qp1 q1 Qp2 q2    Qpm qm Tq1 q2

pm





ð1:90Þ

qm

or Tp1 p2



pm

¼

@xp1 @xp2 @xpm  Tq q @xq1 @xq2 @xqm 1 2



ð1:91Þ

qm

Noting that Qp1 r1 Qp2 r2    Qpm rm Tp1 p2



pm

¼ Qp1 r1 Qp2 r2    Qpm rm Qp1 q1 Qp2 q2    Qpm qm Tq1 q2



qm

¼ ðQp1 r1 Qp1 q1 ÞðQp2 r2 Qp2 q2 Þ    ðQpm rm Qpm qm ÞTq1 q2 ¼ dr1 q1 dr2 q2





qm

drm qm Tq1 q2 qm

the inverse relation of Eq. (1.90) is given by Tr1 r2



rm

¼ Qp1 r1 Qp2 r2    Qpm rm Tp1 p2



pm

ð1:92Þ

While the transformation rule of the first-order tensor, i.e. vector is given by Eqs. (1.80) and (1.82), the transformation rule of the second-order tensor is given by Tij ¼ Qir Qjs Trs ;

Tij ¼ Qri Qjs Trs

ð1:93Þ

The transformation between the coordinate systems without relative motion is considered above in the definition of the tensor, whereas the transformation in the

1.3 Tensor

19

form of (1.90) or (1.92) is called as the objective transformation. A tensor that obeys the objective transformation even between the coordinate systems with the relative motion is called an objective tensor.

1.3.2

Quotient Law

One has a convenient law, called the quotient law, which is used to judge whether or not a quantity is a tensor as will be explained below. Quotient law: “If a set of functions Tðp1 ; p2 ;   ; pm Þ becomes Bpl þ 1 pl þ 2 pm (m-l-th order tensor lacking the suffices p1 pl ) by multiplying it by Ap1 p2 pl (l-th order tensor ðl  mÞ), the set is an m-th order tensor”. (Proof) The proof can be achieved by showing that the quantity Tðp1 ; p2 ;   ; pm Þ is the m-th order tensor when it holds that 



Tðp1 ; p2 ;   ; pm ÞAp1 p2



pl

¼ Bpl þ 1 pl þ 2



ð1:94Þ

pm

which is described in the coordinate system fO  xi g as follows: T  ðp1 ; p2 ;   ; pm ÞAp1 p2



pm

¼ Bpl þ 1 pl þ 2



ð1:95Þ

pm

Here, the following relation holds. Bpl þ 1 pl þ 2



pm

¼ Qpl þ 1 rl þ 1 Qpl þ 2 rl þ 2   Qpm rm Brl þ 1 rl þ 2



rm

¼ Qpl þ 1 rl þ 1 Qpl þ 2 rl þ 2   Qpm rm Tðr1 ; r2 ;   ; rm ÞAr1 r2 rl ¼ Qpl þ 1 rl þ 1 Qpl þ 2 rl þ 2   Qpm rm Tðr1 ; r2 ;    rm Þ Qp1 r1 Qp2 r2    Qpl rl Ap1 q2 pl |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 



lþ1 m

1 l

ð1:96Þ Substituting Eq. (1.95) into Eq. (1.96) yields 

 T  ðp1 ; p2 ;   ; pm Þ  Qp1 r1 Qp2 r2   Qpm rm T ðr1 ; r2 ;   ; rm Þ Ap1 p2



pl

¼0

ð1:97Þ

from which it holds that T  ðp1 ; p2 ;   ; pm Þ ¼ Qp1 r1 Qp2 r2   Qpm rm T ðr1 ; r2 ;   ; rm Þ

ð1:98Þ

Therefore, taking account of the definition of tensor in Eq. (1.88) into Eq. (1.98), the quantity Tðp1 ; p2 ;   ; pm Þ is the m-th order tensor. ðQ.E.DÞ According to the proof presented above, Eq. (1.94) can be written as

20

1

Tp1 p2



Mathematical Preliminaries: Vector and Tensor Analysis pm Ap1 p2 pl

¼ Bpl þ 1 pl þ 2



ð1:99Þ

pm

For instance, if the quantity Tði; jÞ transforms the first-order tensor, i.e. vector vi to the vector ui by the operation Tði; jÞvj ¼ui , one can regard Tði; jÞ as the second-order tensor. Eventually, in order to prove that a certain quantity is a tensor, one needs only to show that it obeys the tensor transformation rule (1.90) or that the multiplication of a tensor to the quantity leads to a tensor by the quotient rule. Tensors fulfill linearity as follows: Tp1 p2 Tp1 p2

 

pm ðGp1 p2  pl pm ðaAp1 p2 

þ Hp1 p2 pl Þ ¼ Tp1 p2 pl Þ ¼ aTp1 p2 pm Ap1 p2 



pm Gp1 p2  pl





þ Tp1 p2



pm Hp1 p2  pl



pl

where a is an arbitrary scalar. Therefore, the tensor plays the role to transform linearly a tensor to the other tensor and thus it is called the linear transformation. The operation that lowers the order of tensor by multiplying the other tensor is called the contraction.

1.3.3

Notations of Tensors

When we express the tensor T as T ¼ Tp1 p2



pm ep1  ep2   

 ep m

ð1:100Þ

in a similar form to the case of vector in Eq. (1.25), Eq. (1.100) is called the component notation with bases, defining ep1  ep2     epm as the base of m-th order tensor. The transformation of T between the bases in Eq. (1.77) leads Eq. (1.100) to T ¼ Tp1 p2



 pm Qr1 p1 er1

 Qr2 p2 er2     Qrm pm erm

¼ Qr1 p1 Qr2 p2    Qrm pm Tp1 p2 ¼

Tr1 r2  rm er1



er2



  

 p m er 1

 er2      erm

erm

The following various notations are used for tensors. Indicial (or component) notation: Tp1 p2 pm . Component notation with base: Tp1 p2 pm ep1 ep2      epm . Symbolic (or direct) notation: T. Matrix notation: Eq. (1.6) for second-order tensor as an example. 



ð1:101Þ

1.3 Tensor

21

The matrix notation holds only for a vector or a second-order tensor or for a fourth-order tensor if it is formally expressed by two suffixes. For instance, the stress–strain relation can be expressed in matrix notation by expressing the stress and the strain of second-order tensors as a form of vector and the stiffness coefficient of fourth-order tensor as a form of second-order tensor. Various contractions exist in the operation of higher-order tensors. and thus the symbolic notation is not useful in general. For instance, which of the following does ST mean: Sijk Tjk ; Sijk Tkj ; Sijk Tij ; Sijk Tji ; Sijk Tjl ; Sijk Tkl ; Sijk Til ? In other words, the application of symbolic notation is limited to the multiplication between low order tensors. On the other hand, component notation with bases holds always without defining special rule. Introducing the notation ðQ½½TÞp1 p2 pm  Qp1 q1 Qp2 q2    Qpm qm T q1 q2

T  Q ½½T p1 p2 pm  Qq1 p1 Qq2 p2    Qqm pm Tq1 q2







qm



ð1:102Þ

qm

Equations (1.90) and (1.92) can be expressed by the symbolic notation as follows: T ¼ Q½½T

ð1:103Þ

T ¼ QT ½½T 

In particular, transformations of the vector and the second-order tensor are expressed as follows: vi ¼ Qir vr ¼ vr Qir ;

vi ¼ Qri vr ¼ vr Qri

v ¼ Qv ¼ vQT ; v ¼ QT v ¼ v Q Tij ¼ Qir Qjs Trs ;

ð1:104Þ

Tij ¼ Qri Qsj Trs

T ¼ QTQ ; T ¼ Q T Q T

T

ð1:105Þ

noting Tv ¼ vTT ðTij vj ¼ vj Tij Þ in general for Eq. (1.104), where TT ððTT Þij ¼ Tji Þ is the transposed tensor defined exactly in Sect 1.4.3. The component of vector is expressed by the direct notation in Eq. (1.26). Here, consider the component of second-order tensor in the direct notation. The second-order tensor T is expressed from Eq. (1.100) as T ¼ Tij ei  ej

ð1:106Þ

22

1

Mathematical Preliminaries: Vector and Tensor Analysis

from which, noting Eq. (1.69), it follows that ei  Tej ¼ ei  Trs er  es ej ¼ Trs dir dsj and thus the component of T in the direct notation is given as Tij ¼ ei  Tej

ð1:107Þ

As known from Eq. (1.107), the orthogonal projection of the vector Tej to the base vector ei is the component of the tensor T. Especially, Tii (no sum) and Tij ði 6¼ jÞ are called the normal and the shear components, respectively. Further, noting the relation ðTu  TvÞij ¼ Tir ur Tjs us ¼ Tir ur us Tjs ¼ ðTðu  vÞTT Þij , one has Tu  Tv ¼ Tu  vTT

1.3.4

ð1:108Þ

Orthogonal Tensor

The coordinate transformation operator Qij which appeared in Sects. 1.2.3 and 1.3.3 plays an important role in the coordinate transformation. The component notation with bases is obtained from Qij ei  ej ¼ ei  ðei  ej Þej ð ¼ ei  ei Þ ¼ ðei  er Þer  ei ¼ Qri er  ei

ð1:109Þ

as follows: Q ¼ Qij ei  ej ¼ Qij ei  ej

ð1:110Þ

Furthermore, considering Eq. (1.77), the direct notation of Q is given by Q ¼ ei  ei

ð1:111Þ

Because of ei ¼ er dir ¼ er  er ei ei ¼ er dir ¼ er  er ei

)

it follows that ei ¼ Qei ¼ Qri er ;

ei ¼ QT ei ¼ Qir er

ð1:112Þ

1.3 Tensor

23

Furthermore, changing Eq. (1.79) to the direct notation or noting the relation (

QQT ¼ ei  ei ej  ej ¼ ei dij  ej ¼ ei  ei QT Q ¼ ei  ei ej  ej ¼ ei dij  ej ¼ ei  ei

obtained from Eq. (1.111), it holds that QQT ¼ QT Q ¼ I

ð1:113Þ

where I possesses the components of the Kronecker’s delta, i.e. ðIÞij ¼ dij

ð1:114Þ

which transforms vector and tensor to original vector and tensor as Iv ¼ vI ¼ v and IT ¼ TI ¼ T, and thus it is called the identity tensor. The tensor satisfying Eq. (1.113) is defined as the orthogonal tensor in general, the properties of which are described below. It follows from Eq. (1.113) that QT ¼ Q1

ð1:115Þ

det Q ¼ det QT ¼ 1

ð1:116Þ

Moreover, one has

for the right-handed system, noting detðQQT Þ ¼ detQ detQT ¼ ðdet QÞ2 ¼ det I ¼ 1 which is derived from Eq. (1.113) with Eqs. (1.9), (1.17). Further, from Eq. (1.113) one obtains ðQIÞQT ¼ ðQIÞT Making the determinant of this equation and noting Eqs. (1.12), (1.17) and (1.116), it holds that detðQIÞ ¼  detðQIÞ ! detðQIÞ ¼ 0

ð1:117Þ

Then, one of the principal values of the orthogonal tensor is unity in order to satisfy ðQ1  1ÞðQ2  1ÞðQ3  1Þ ¼ 0 as known from the fact which will be described in Sect. 1.5.

24

1

1.4

Mathematical Preliminaries: Vector and Tensor Analysis

Operations of Tensors

Various basic operations of tensors are shown collectively in this section, which are used for the representations of constitutive relations in the later chapters.

1.4.1

Notations in Tensor Operations

The following notations are used throughout this book for the vectors a; b; c; d and v, the second-order tensors A; B and T, the third-order tensor N and fourth-order tensor T. 8 < a  b ¼ trða  bÞ for ai bi a  b ¼ e:ða  bÞ for eijk aj bk ð1:118Þ : a  b for ai bj with a  b  c for ai bj cj 

Tv for Tij vj ; vT for vi Tij v  T for ðv  TÞij ¼ eikl vk Tlj ; T  v for ðT  vÞij ¼ ejkl Tik vl

ð1:119Þ

v  T ¼ vk ek  Tlj el  ej ¼ vk Tlj ekli ei  ej ¼ ekli vk Tlj ei  ej ¼ eikl vk Tlj ei  ej

!

T  v ¼ Tik ei  ek  vl el ¼ Tik vl ei  eklj ej ¼ eklj Tik vl ei  ej ¼ ejkl Tik vl ei  ej 8 AB for Air Brj > > > A : B ¼ trðABT Þ ¼ trðAT BÞ for A B > > ij ij > > > T : I ¼ I : T ¼ trT for T d ¼ T > ij ij ii > < A  B for Aij Bkl T :ða  bÞ¼ a  Tb for Tij ai bj > > > > ðT  aÞb ¼ ða  bÞT for Tij ar br ; ða  TÞb ¼ a  ðTbÞ for ai Tjr br > > > > > ða  bÞ:ðc  dÞ ¼ ða  cÞðb  dÞ for ai ci bj dj > : ða  b  cÞv ¼ ðc  vÞa  b for ai bj cr vr 

N T for Nijk Tkl ; T N for Tij Njkl N : T for Nijk Tjk ; T : N for Tij Nijk

ð1:120Þ

ð1:121Þ ð1:122Þ

ð1:123Þ

1.4 Operations of Tensors



25

pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi jjvjj ¼ v  v for p viffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vi ffi pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi jjTjj ¼ T : T ¼ trðTTT Þ for Tij Tij @f 2 ðTÞ @f 2 ðTÞ for @T  @T @Tij @Tkl

9 ðTvÞi ¼ Tir vr ; ðTSÞij ¼ Tir Srj ; T : S ¼ trðTST Þ ¼ Tij Sij = ðRvÞij ¼ Rijr vr ; ðRTÞijk ¼ Rijr Trk ; ðR : TÞi ¼ Rirs Trs ; ðN : TÞij ¼ Nijrs Trs ; ðT : NÞij ¼ Trs Nrsij ; ðN : PÞijkl ¼ Nijrs Prskl

ð1:124Þ

ð1:125Þ

ð1:126Þ

where v; ðT; SÞ, R and ðN; PÞ designate the vector, the second-order, the third-order and the fourth-order tensors, respectively.

1.4.2

Trace

An operation taking the sum of the components having the same suffixes, i.e. the sum of diagonal components in the matrix notation is called the trace and is expressed as tr Tð¼T : IÞ ¼ Trs drs ¼ Trr ¼ T11 þ T22 þ T33

ð1:127Þ

trðABÞð¼A : BT Þ ¼ Air Bri ¼ A11 B11 þ A12 B21 þ A13 B31 þ A21 B12 þ A22 B22 þ A23 B32 þ A31 B13 þ A32 B23 þ A33 B33

ð1:128Þ

The following relations hold for the trace. tr ðA þ SÞ¼ tr A þ tr B; trðaTÞ ¼ atr T; tr ðABÞ ¼ tr ðBAÞ; trðu  vÞ ¼ u  v ð1:129Þ

1.4.3

Various Tensors

Various basic tensors used widely in tensor operations are defined in this subsection. (1) Transposed tensor The tensor TT satisfying the following equation for any arbitrary vectors a and b is defined as the transposed tensor of a tensor T. a  Tb ¼ b  TT a

ð1:130Þ

26

1

Mathematical Preliminaries: Vector and Tensor Analysis

Noting a  ðu  vÞb ¼ b  ðv  uÞa it follows from Eq. (1.130) that ðu  vÞT ¼ v  u

ð1:131Þ

Further, comparing the equation ei  ðTrs er  es Þej ¼ Tij ¼ ej  ðTrs es  er Þei with Eq. (1.130), one has ðTrs er  es ÞT ¼ Trs es  er ¼ Tsr er  es

ð1:132Þ

ðTT Þij ¼ ðTÞji

ð1:133Þ

ðA þ BÞT ¼ AT þ BT ð½ðA þ BÞT ij ¼ Ajr þ Bri Þ

ð1:134Þ

i.e.

The following relations hold.

It holds from ½ðABÞT ij ¼ Ajr Bri that ðABÞT ¼ BT AT tr TT ¼ tr T;

tr ðABÞ ¼ A : BT ¼ AT : B

ð1:135Þ ð1:136Þ

The magnitude of tensor is defined as the square root of the sum of the squares of each components and thus it is expressed using Eq. (1.133) as kTk ¼

ffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Tij Tij ¼ T : T ¼ tr ðTTT Þð¼ trT2 for T¼TT Þ

ð1:137Þ

The tensor whose magnitude is unity is called the unit tensor. It follows by use of Eq. (1.133) that Tv ¼ vTT ; Tij vj ¼vj Tij Tu  v ¼ u  TT v ¼ u  vT;

ðTij uj Þvi ¼ ui ðvj Tji Þ

Au  Bv ¼ u  AT Bv (

Tu  v ¼ uTT  v; ðTir ur Þvi ¼ ður Tir Þvi u  Tv ¼ u  vTT ; ui ðTjr vr Þ ¼ ui ðvr Tjr Þ

ð1:138Þ ð1:139Þ ð1:140Þ ð1:141Þ

1.4 Operations of Tensors

27

(2) Determinant The determinant is studied already in Sect. 1.1 but it will be studied in more detail in this subsection. The determinant is defined in Eqs. (1.8)–(1.10) already as follows:   9  T11 T12 T13  8   < epqr T1p T2q T3r =   or det T ¼ jTij j ¼  T21 T22 T23  ¼ ¼ 1 e e T T T ð1:142Þ   : ; 3! abc pqr ap bq cr epqr Tp1 Tq2 Tr3  T31 T32 T33  We obtain the following equations (

detðTÞ ¼  det T;

detðsTÞ ¼s3 det T;

detð ABÞ ¼ det A det B;

det T¼ det TT

detðTn Þ ¼ ð det TÞn ;

detðexp TÞ¼ expðtrTÞ ð1:143Þ

detða  bÞ ¼ 0

ð1:144Þ

by virtue of 8 detðTÞ ¼ eijk ðT1i ÞðT2j ÞðT3k Þ ¼ eijk T1i T2j T3k > > > > > < detðsTÞ ¼ eijk sT1i sT2j sT3k ¼ s3 eijk T1i T2j T3k 1 1 1 > det T ¼ eabc epqr Tap Tbq Tcr ¼ epqr eabc Tpa Tqb Trc ¼ eabc epqr Tpa Tqb Trc > > > 3! 3! 3! > : detðABÞ ¼ epqr ðA1a Bap ÞðA2b Bbq ÞðA3c Bcr Þ ¼ epqr A1a A2b A3c Bap Bbq Bcr ¼ A1a A2b A3c eabc det B

ð1:145Þ     expðT11 Þ 0 0    ¼ expðT11 þ T22 þ T33 Þ ð1:146Þ detðexp TÞ ¼  0 0 expðT22 Þ   0 0 expðT33 Þ  detða  bÞ ¼ eijk ða1 bi Þða2 bj Þða3 bk Þ ¼ a1 a2 a3 eijk bi bj bk ¼ ða1 a2 a3 Þðb  bÞ  b ð1:147Þ noting Eqs. (1.8), (1.10), (1.17) and (1.45). The following equations hold for the cofactor, noting Eq. (1.13). 8 cofðsTÞ ¼ s2 cofT > > > > < ðcofTÞT ¼ cofðTT Þ; ðcofTÞ1 ¼ cofðT1 Þ; ðcofTÞT ¼ cofðTT Þ cofðABÞ ¼ cofðAÞcofðBÞ > > > trðcofTÞ ¼ 12 ðtr2 T  trT2 Þ ¼ II > : detðcofTÞ¼ 1

ð1:148Þ

28

1

Mathematical Preliminaries: Vector and Tensor Analysis

noting 1 1 1 trðcofTÞ ¼ eabc eaqr Tbq Tcr ¼ ðdbq dcr  dbr dcq ÞTbq Tcr ¼ ðTqq Trr  Trq Tqp Þ 2 2 2 II is the second principal invariant as will be defined in Sect. 1.5. The vector product in Eq. (1.34) and the scalar triple product in Eq. (1.36) are described by the determinant as follows:          e 1 e2 e3  u u  u u  u u     2 3  3 1  1 2   uv ¼  e1 þ  e2 þ  e3 ¼ u1 u2 u3   v 2 v3   v3 v1   v1 v2     v1 v2 v3  vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi      u u u2 u3 2  u3 u1 2  u1 u2 2 t jjuvjj ¼   þ  þ   v2 v3   v3 v1   v1 v2       u 1 u2 u3   u  e 1 u  e 2 u  e 3          ½uvw ¼  v1 v2 v3 ¼ v  e1 v  e2 v  e3       w 1 w 2 w 3   w  e1 w  e2 w  e3     a  c b  c   ða  bÞ  ðc  dÞ¼  a  d b  d      a1 a2 a3  p1 q2 r3   ai pi         ½abc½pqr ¼  b1 b2 b3  p2 q2 r3 ¼ bi pi      c1 c2 c3  p3 q2 r3   ci pi

   ai qi ai ri   a  p a  q a  r     bi qi bi ri ¼ b  p b  q b  r     ci qi ci ri   c  p c  q c  r 

ð1:149Þ

ð1:150Þ

ð1:151Þ

ð1:152Þ

ð1:153Þ

noting Eq. (1.143)4 . The following equation is derived as the special case of Eq. (1.153) for the three vectors v1 ; v2 ; v3 .    v 1  v1 v 1  v2 v1  v3    ð1:154Þ v2  v2 v2  v3  v2 ¼ ½v1 v2 v3 2 ¼   sym: v3  v 3  where v ¼ ½v1 v2 v3  ¼ eijk ðv1 Þi ðv2 Þj ðv2 Þk ð¼eijk ðv1  ei Þðv2  ej Þðv3  ek ÞÞ

ð1:155Þ

1.4 Operations of Tensors

29

is the volume of the parallelepiped formed by the line-elements v1 ; v2 ; v3 . By introducing the symbol vij  vi  vj

ð1:156Þ

   v11 v12 v13      v22 v23 ¼ detðvij Þ v2 ¼     sym: v33 

ð1:157Þ

Equation (1.154) can be written as

It follows from Eq. (1.153) that detT ¼ ½Te1 Te2 Te3  ¼ Te1  ðTe2  Te2 Þ

ð1:158Þ

noting     T11 T12 T13   ðe1  Te1 Þ ðe1  Te2 Þ       detT ¼  T21 T22 T23 ¼ ðe2  Te1 Þ ðe2  Te2 Þ     T31 T32 T33   ðe3  Te1 Þ ðe3  Te2 Þ

 ðe1  Te3 Þ   ðe2  Te3 Þ   ðe3  Te3 Þ 

¼ ½e1 e2 e3 ½Te1 Te2 Te3  ¼ ½Te1 Te2 Te3  (3) Inverse tensor The tensor T1 fulfilling the following relation is defined as the inverse tensor of the tensor T. TT1 ¼ I;

Tir ðT1 Þrj ¼ dij

ð1:159Þ

Equation (1.14) is expressed in the direct notation as follows: ðdetTÞI ¼ TðcofTÞT ;

ðdetTÞdij ¼ Tip ðcofTÞjp

from which the cofactor is given as follows: cofT ¼ ðdetTÞTT ; (cofTÞij ¼

1 eipq ejrs Tpq Trs ¼ ðdetTÞTji1 2!

ð1:160Þ

and then the inverse tensor is given as follows: T1 ¼

ðcofTÞT ; detT

Tij1 ¼

ðcofTÞji detT

ð1:161Þ

30

1

Mathematical Preliminaries: Vector and Tensor Analysis

Equation (1.161) is represented for the 2  2 and 3  3 matrices as follows:

 2 3    T11 T12 T13  " #  T22 T33 T23 T32 T32 T13  T33 T12 T12 T23  T13 T22 T T   

T T 22 21 11 12     6 7 = T1 ¼ ; T1 ¼4 T23 T31  T21 T33 T33 T11  T31 T13 T13 T21  T11 T23 5= T21 T22 T23    T12 T11  T21 T22  T21 T32  T22 T31 T31 T12  T32 T11 T11 T22  T12 T21  T31 T32 T33 

ð1:162Þ Then, det T 6¼ 0 is required in order that T1 exists, so that the tensor fulfilling this condition is called the non-singular (or invertible) tensor. The partial derivative of Eq. (1.18) is rewritten by Eq. (1.160) as @ det T ¼ ðdet TÞTT @T

ð1:163Þ

The derivation of Eq. (1.163) starting from the definition of the total differential equation has been often described in some literatures (cf. Leigh, 1968; Hashiguchi and Yamakawa, 2012) but it needs cumbersome manipulations. Compared with it, the derivation shown above would be concise and straightforward. The following relations hold for the inverse tensor. ðTT Þ1 ¼ ðT1 ÞT ð  TT Þ;

ðABÞ1 ¼ B1 A1

ð1:164Þ

ðA1 þ B1 Þ1 ¼ ½B1 ðB þ AÞA1 1 ¼ AðB þ AÞ1 B ¼ BðA þ BÞ1 A ð1:165Þ detTdetðT1 Þ ¼ detð TT1 Þ ¼1

ð1:166Þ

detðT1 Þ ¼ ðdetTÞ1

ð1:167Þ

because of ððTT1 ÞT ¼Þ ðT1 ÞT TT ¼ I ¼ ðTT Þ1 TT ABðABÞ1 ¼ I ! BðABÞ1 ¼ A1 Now, when we regard the transformation of the vector v to the vector u by the tensor T, i.e. Tv ¼ u; Tij vj ¼ ui

ð1:168Þ

as the simultaneous equation in which the components of v are the unknown numbers, solution exists for u 6¼ 0 if detT 6¼ 0 and is given by v ¼ T1 u, noting Eq. (1.161), as

1.4 Operations of Tensors

31

v¼

ðcofTÞT u; detT

vi ¼

ðcofTÞji detT

ð1:169Þ

uj

Here, T must be the non-singular tensor fulfilling det T 6¼ 0 in order that the non-trivial solution v 6¼ 0 exists for u 6¼ 0. On the other hand, T must be the singular tensor fulfilling det T ¼ 0 in order that the solution v 6¼ 0 exists for u ¼ 0. (4) Symmetric and skew-symmetric tensors Tensors TS and TA fulfilling the following relations are defined as the symmetric and the skew-(or anti-)symmetric tensor, respectively. TST ¼ TS ;

TjiS ¼ TijS

ð1:170Þ

TjiA ¼ TijA

ð1:171Þ

and TAT ¼ TA ;

An arbitrary tensor T is uniquely decomposed into the symmetric and the skew (anti)-symmetric tensors. T ¼ TS þ TA 2

3

2

T11 T11 T12 T13 6 T12 þ T21 7 6 6 6 4 T21 T22 T23 5 ¼ 6 2 4 T31 T32 T33 T13 þ T31 2

T12 þ T21 2 T22 T23 þ T32 2

ð1:172Þ

T13 þ T31 3 2 0 7 6 2 6 T23 þ T32 7 7 þ 6  T12  T21 7 6 2 2 5 4 T13  T31 T33  2

1 TS ¼ sym½T ¼ ðT þ TT Þ; 2

T12  T21 2 0 T23  T32  2

1 TA ¼ ant½T ¼ ðT  TT Þ 2

sym½A þ B ¼ sym½A þ sym½B; ant½A þ B ¼ ant½A þ ant½B trðA0 BÞ ¼ trðAB0 Þ ¼ trðA0 B0 Þ

T13  T31 3 7 2 T23  T32 7 7 7 2 5 0

ð1:173Þ ð1:174Þ ð1:175Þ

while the components of TS and TA are often denoted by TðijÞ and T½ij , respectively. Equation (1.172) is called the Cartesian decomposition, following the decomposition of a complex number to a real and an imaginary parts. It holds that trTS ¼ trT;

tr TA ¼ 0;

ðTA Þii ¼ 0

tr( AS BA Þ ¼0

ðno sumÞ

ð1:176Þ ð1:177Þ

32

1

Mathematical Preliminaries: Vector and Tensor Analysis

det TA ¼ 0 (

u  TS v ¼ v  TS u u  ðTA vÞ¼ v  ðTA uÞ v  ðTA vÞ ¼ 0

ð1:178Þ ð1:179Þ ð1:180Þ

noting Eq. (1.130). (5) Spherical and deviatoric parts The tensor T is decomposed as follows: T ¼ Tm þ T0

ð1:181Þ

9 1 1 Tm  Tm I; Tm  ðtr TÞ¼ Tii = 3 3 ; T0  TTm I ðtrT0 ¼ 0Þ

ð1:182Þ

while Tm and T0 are called the spherical (or mean) part and the deviatoric part of the tensor T. Noting Eq. (1.177), the skew-symmetric tensor T0A of the deviatoric tensor T0 is given by T0 A ¼ TA

ð1:183Þ

Then, the symmetric part of the deviatoric tensor is given by T0 S ¼ T0 TA ¼ TTm ITA

ð1:184Þ

T ¼ Tm I þ T0S þ TA

ð1:185Þ

from which one has

The decomposition of T into the mean component Tm I, the deviatoric symmetric component T0 S and the skew-symmetric component TA is called triple decomposition. The following holds: trðA0 BÞ ¼ trðAB0 Þ ¼ trðA0 B0 Þ

ð1:186Þ

(6) Axial vector The skew-symmetric tensor TA has three independent components in the three-dimensional state. Therefore, the vector tA having the following components is called the axial vector.

1.4 Operations of Tensors

33

   A 1 tiA ¼  eirs TrsA t1A t2A t3A ¼ T23 2

A  T31

A  T12



ð1:187Þ

Inversely from Eq. (1.187) it is obtained that 2 TijA ¼ 4

TijA ¼ eijr trA ;

0

t3A 0 ant:

3 t2A t1A 5 0

ð1:188Þ

Furthermore, noting Eq. (1.36) and the relation TirA vr ¼ eirs tsA vr ¼ eirs trA vs

ð1:189Þ

the following relation holds. TA v ¼ tA  v

ð1:190Þ

The relation of TA and tA is shown in Fig. 1.2 in the case that tA is the angular velocity vector and v is the position vector of particle. The quantity in Eq. (1.190) designates the peripheral velocity vector, while TA is called the spin tensor which induces the peripheral velocity by undergoing the multiplication of the position vector.

tA

|| v || sin

n



TAv = t A v = ( || t A |||| v ||sin ) n

v

0 Fig. 1.2 Anti-symmetric tensor and axial vector

34

1

Mathematical Preliminaries: Vector and Tensor Analysis

(7) Fourth-order transformation tensors The fourth-order tracing identity tensor T for a second-order identity tensor I is defined by T ≡ I ⊗ I = δ ijδ kl ei ⊗ e j ⊗ ek ⊗ el = ei ⊗ ei ⊗ e j ⊗ e j ð1:191Þ leading to T : T = T : T = (trT)I = 3Tm I for an arbitrary second-order tensor T. The fourth-order identity tensor I and the fourth-order transposing tensor I for a second-order tensor are defined by I ≡ δ ik δ jl ei ⊗ e j ⊗ ek ⊗ el = ei ⊗ e j ⊗ ei ⊗ e j I ≡ δ il δ jk ei ⊗ e j ⊗ ek ⊗ el = ei ⊗ e j ⊗ e j ⊗ ei

leading

I :T = T: I = T, I :T = T: I = TT ,

to

I :I = I :I = I

ð1:192Þ

and

∂T / ∂T = I , ∂T / ∂T = I .. T

The symmetrizing tensor S and the skew-(or anti-)symmetrizing tensor A are defined by S ≡ 1 (δ ik δ jl + δ ilδ jk )ei ⊗ e j ⊗ ek ⊗ el = 1 (I + I ) 2 2

A ≡ 1 (δ ik δ jl − δ ilδ jk )ei ⊗ e j ⊗ ek ⊗ el = 1 (I − I ) 2 2

ð1:193Þ

leading to S :T = T S , A :T = TA . The deviatoric projection tensor I ' is defined by I ' ≡ (δ ik δ jl − 1 δ ijδ kl )ei ⊗ e j ⊗ ek ⊗ el = I − 1 T 3 3

ð1:194Þ

leading to I ' :T = T : I ' = T' . The following relations hold from Eqs. (1.191)(1.194). ⎫ ⎪ ⎪ ⎪ T :I = I , T : S = T, T :A = O, T : I ' = O, ⎪ , (T : T)ijkl = δ ijTrrkl , I : T = T ⎬ ⎪ (S : T)ijkl = (Tijkl + T jikl ) / 2 , (A : T)ijkl = (Tijkl −T jikl ) / 2 , ⎪ ⎪ ⎪ (I ' : T)ijkl = Tijkl − 1 δ ijTrrkl 3 ⎭

T : T = 3 T, S : S = S , A : A = O, I ' : I ' = I ' ,

ð1:195Þ

where T is an arbitrary fourth-order tensor and O is the fourth-order zero tensor. Further, the following fourth-order tensors are defined by the four types of tensor products of the second-order tensors A; B; C (cf. e.g. del Piero 1979; Steinmann et al. 1997; Kintzel and Bazar 2006; Wang and Dui 2008).

1.4 Operations of Tensors

35

9 ðA  BÞijkl ¼ Aij Bkl with A  B : C ¼ AðB : CÞ ððA  B : CÞij ¼ Aij ðBkl Ckl ÞÞ and > > > > > C : A  B ¼ ðC : AÞBððC : A  BÞkl ¼ Cij Aij Bkl Þ > > > > > T > ðABÞijkl ¼ Aik Bjl with AB : C ¼ ACB ððAB : CÞij ¼ Aik Bjl Ckl ¼ Aik Ckl Bjl Þ and > > > > > T > C : AB ¼ A CBððC : ABÞkl ¼ Cij Aik Bjl ¼ Aik Cij Bjl Þ > > > > > T T > ðABÞijkl ¼ Ail Bjk with AB : C ¼ AC B ððAB : CÞij ¼ Ail Bjk Ckl ¼ Ail Ckl Bjk Þ and > = C : AB ¼ BT CT AððC : ABÞkl ¼ Cij Ail Bjk ¼ Bjk Cij Ail Þ > > > > > e ijkl ¼ Aik Blj with A B e : C ¼ ACBððA B e : CÞij ¼ Aik Blj Ckl ¼ Aik Ckl Blj Þ and > ðA BÞ > > > > T T > e kl ¼ Cij Aik Blj ¼ Aik Cij Blj Þ > e ¼ A CB ððC : A BÞ C : A B > > > > T > ðA  BÞijkl ¼ Ail Bkj with A  B : C ¼ AC BððA  B : CÞij ¼ Ail Bkj Ckl ¼ Ail Ckl Bkj Þ and > > >



> > > > C : A  B ¼ BCT AððC : A  BÞ ¼ Cij Ail Bkj ¼ Bkj Cij Ail Þ ;



kl

ð1:196Þ from which it follows that 

I

I



ð1:197Þ 

1.5



Eigenvalues and Eigenvectors

The application of the tensor T to an arbitrary vector v, i.e. Tv imposes v an extension or a contraction and a rotation in general. However, it imposes only an extension or a contraction without a rotation for the vector possessing the particular direction. Examine this fact in the following. Here, consider the following relation for the unit vector p. Tp ¼ Tp ;

ðTrj er  ej Þp ¼ Tp ! ei



Tij pj ¼ Tpi

ðTrj er  ej Þp ¼ ei



ð1:198Þ

Tp ! dir Trj pj ¼ Tei ! Tij pj ¼ Tpi



36

1

Mathematical Preliminaries: Vector and Tensor Analysis

i.e. ðTTIÞp ¼ 0; ðTij Tdij Þpj ¼ 0

ð1:199Þ

where the unit vector p is called the eigenvector (or principal or characteristic or proper vector) and the scalar T is called the eigenvalue (or principal or characteristic or proper value) when Eq. (1.198) holds. The eigenvector and the eigenvalue are called the eigenpair. Here, note the fact that the eigenvector is transformed to its scalar multiplication by a linear transformation, i.e. the multiplication of tensor. The necessary and sufficient condition that the simultaneous Eq. (1.199) has a non-zero solution of p is given by detðT  TIÞ ¼ 0 ;

  Tij  Tdij  ¼ 0

ð1:200Þ

noting u ¼ 0 in the solution of Eq. (1.169) for the simultaneous Eq. (1.168). Equation (1.200) is called the characteristic equation of the tensor, which is regarded as the cubic equation of T. Unit vectors P1 ; P2 ; P3 are derived for each of solutions T1 ; T2 ; T3 from Eq. (1.199). The mathematical verification of Eq. (1.200) is omitted even in the well-known books (e.g. Fung 1965; Ogden 1984; Bonet and Wood 1997; Holzapfel 2000; de Souza Neto et al., 2008, etc.). Then, it will be explained in detail without fear of redundancy in the following. Besides, the necessary and sufficient condition in Eq. (1.200) for the existence of the non-zero solution of p in Eq. (1.198) is verified by the other way: If the inverse tensor of TTI exists, Eq. (1.199) leads to Ip ¼ p ¼ 0 by multiplying ðTTIÞ1 to Eq. (1.199). In other words, the inverse tensor ðTTIÞ1 must not exist (must be a singular tensor) in order that the non-zero solution of p exists. The condition that the inverse tensor ðTTIÞ1 does not exist is nothing but (1.200), noting Eq. (1.169). Therefore, the simultaneous equation (1.199) for p becomes indeterminate. Then, the vector p cannot be determined only by this equation, while the eigenvalues T are already determined. Thus, an additional property for the magnitude of p must be added in order to determine p. Usually, the incidental condition that p is the unit vector, i.e. ||P|| = 1 is added in order to determined uniquely. The fact that Eq. (1.200) is the necessary and sufficient conditions that non-zero (non-trivial) solution(s) of P exist(s) in Eq. (1.199) is verified below by the elementary method. (Note) Consider the simultaneous equation for the unknown values x1 ; x2 ; x3 : 9 a11 x1 þ a12 x2 þ a13 x3 ¼ c1 > = a21 x1 þ a22 x2 þ a23 x3 ¼ c2 a31 x1 þ a32 x2 þ a33 x3 ¼ c3

> ;

ðaÞ

1.5 Eigenvalues and Eigenvectors

37

where aij ði;j ¼ 1; 2; 3Þ are the known coefficients. Equation (a) is described in the forms: (

Ax ¼ c: direct tensor notation ½Afxg ¼ fcg: matrix form

ðbÞ

setting 2

a11 6 A ¼ 4 a21

a12 a22

3 a13 7 a23 5;

a31

a32

a33

8 9 > = < x1 > x ¼ x2 ; > ; : > x3

8 9 > = < c1 > c ¼ c2 > ; : > c3

Solutions of Eq. (a) are given by    c1 a12 a13     c2 a22 a33     c3 a32 a33  ; x1 ¼    a11 a12 a13   a21 a22 a23     a31 a32 a33 

   a11 c1 a13     a21 c2 a23     a31 c3 a33  ; x2 ¼    a11 a12 a13   a21 a22 a23     a31 a32 a33 

   a11 a12 c1     a21 a22 c2     a31 a32 c3   x3 ¼    a11 a12 a13   a21 a22 a23     a31 a32 a33 

In the case of c1 ¼ c2 ¼ c3 ¼ 0 as seen in Eq. (1.200), only the trivial solutions x1 ¼ x2 ¼ x3 ¼ 0 can exist for det A 6¼ 0. Then, the following equation must hold in order that nontrivial solution exits.     a11 a12 a13   T11 T      det A ¼  a21 a22 a23  ¼ 0 ! detðTTIÞ ¼  T21    T31  a31 a32 a33 

T12 T22 T T32

 T13  T23  ¼ 0 T33 T 

In what follows, it is proven that the eigenvalues are real and the eigenvectors are mutually orthogonal in the second-order real symmetric tensor. The complex conjugate relation for Eq. (1.198) is generally described as follows: Tp ¼ Tp designating the conjugate quantities by the over bar ðÞ. The multiplication of p and p to Eqs. (1.198) and (1.201) leads to p  Tp ¼ Tp  p;

p  Tp ¼ Tp  p

ð1:201Þ

38

1

Mathematical Preliminaries: Vector and Tensor Analysis

from which we have p  Tpp  Tp ¼ ðTTÞp  p The left-hand side of this equation is zero because of p  Tpp  Tp ¼ p  ðTTT Þ p ¼ 0. Then, we have T ¼ T so that T must be a real number. Further, it follows from Eq. (1.198) that Tpa ¼ Ta pa (no sum)

ð1:202Þ

for the eigenvectors pa ða ¼ 1; 2; 3Þ of T. By making the scalar products of Eq. (1.202) and the eigenvectors, we have pb  Tpa ¼ Ta pa  pb

)

pa  Tpb ¼ Tb pb  pa

ðno sum)

Subtracting the lower equation from the upper equation, one has pb  Tpa pa  Tpb ¼ ðTa  Tb Þpa  pb which reads: ðTa  Tb Þpa  pb ¼ 0 (no sum)

ð1:203Þ

noting pb  Tpa pa  Tpb ¼ pb  Tpa TT pa  pb ¼ pb  ðTTT Þpa ¼ 0. The following facts can be concluded from Eq. (1.203). (1) If three principal values are all different to each other, there exist the three principal directions which are perpendicular to each other. (2) If two of three principal values are same, all directions in the plane perpendicular to the principal direction for the other principal value are the principal directions for the same principal value. (3) If all three principal values are same, all directions in the space are the principal directions. Based on the result described above, denoting the eigenvectors by pJ and the corresponding eigenvalues as TJ , one can write TpJ ¼ TJ pJ ðno sum)

ð1:204Þ

In addition, noting that the shear component on the coordinate system with the base vector feP g is zero, i.e. TPQ ¼ 0 ðP 6¼ QÞ

ð1:205Þ

1.5 Eigenvalues and Eigenvectors

39

the symmetric tensor T possessing orthogonal principal directions is expressed by T¼

3 P

TJ pJ  pJ

ð1:206Þ

J¼1

which is called the spectral representation. Then, the tensor function of tensor T is described in general as fðTÞ 

3 X

f ðTJ ÞpJ  pJ

ð1:207Þ

J¼1

and then one has ½fðTÞT ¼ fðTT Þ

ð1:208Þ

The spectral representation of the asymmetric tensor can be referred to Istov (2015). ~ having the eigenvectors p ~J has the same eigenvalues TJ as those If the tensor T of the tensor T possessing the eigenvectors pJ , it holds that ~ pJ ¼ TJ p ~J ðno sumÞ T~

ð1:209Þ

where the orthogonal tensor Q between the eigenvectors of these tensors is given by ~J  pK ; Q ¼ pJ  p ~J ; pJ ¼ Q~ ~J ¼ QT pJ : QJK ¼ p pJ ; p

ð1:210Þ

Applying QT to Eq. (1.204), one has QT TpJ ¼ QT TJ pJ ¼ TJ QT pJ

ðno sumÞ

from which, considering Eqs. (1.209) and (1.210), it holds that ~J ¼ T ~P ~ J ¼ TJ P ~J QT TQP

ðno sumÞ

ð1:211Þ

Then, one obtains the relation ~ ¼ QT TQ T

ð1:212Þ

As presented above, tensors having identical eigenvalues can be related by the orthogonal tensor; they are called the similar tensor mutually. The coordinate transformation rule (1.105) of a certain tensor and the relation (1.212) of two tensors having identical eigenvalues but different eigenvectors, are of mutually opposite forms.

40

1

Mathematical Preliminaries: Vector and Tensor Analysis

If the function f of tensor A; B;    is observed to be identical independent of observers, i.e. if it fulfills the relation f ðA;B;  Þ ¼ f ðQ½½A;Q½½B;  Þ

ð1:213Þ

using the symbol in Eq. (1.102), f is called the isotropic scalar-valued tensor function, which is none other than the invariant. In particular, if the isotropic scalar-valued tensor function f ðTÞ of single second-order tensor T fulfills f ðTÞ ¼ f ðQTQT Þ

ð1:214Þ

f ðTÞ can be expressed by three principal values in the three-dimensional case, including them in symmetric form so as to be identical even if they are exchanged to each other. Then, there exist three independent invariants for a single tensor. Their explicit forms are presented below. The expansion of the characteristic Eq. (1.200) of T leads to   T11 T    T21   T31

T12 T22 T T32

 T13   T23   T33 T 

¼ ðT11 TÞðT22 TÞðT33 TÞ þ T12 T23 T31 þ T21 T32 T13 ðT11 TÞT23 T32 ðT22 TÞT31 T13 ðT33 TÞT12 T21 ¼ T3 þ ðT11 þ T22 þ T33 ÞT2 ðT11 T22 þ T22 T33 þ T33 T11 ÞT þ T11 T22 T33 þ T12 T23 T31 þ T21 T32 T13 þ ðT12 T21 þ T23 T32 þ T31 T13 ÞT  T11 T23 T32  T22 T31 T13  T33 T12 T21 ¼ T3 þ ðT11 þ T22 þ T33 ÞT2 ðT11 T22 þ T22 T33 þ T33 T11 T12 T21  T23 T32  T31 T13 ÞT þ T11 T22 T33 þ T12 T23 T31 þ T21 T32 T13  T11 T23 T32  T22 T31 T13  T33 T12 T21 ¼ 0

ð1:215Þ from which the characteristic equation is given as T 3 IT 2 þ IITIII ¼ 0

ð1:216Þ

I  T11 þ T22 þ T33 ¼ Tii ¼ trT ¼ T1 þ T2 þ T3

ð1:217Þ

where

       T11 T12   T22 T23   T33 T13       1  2 II   þ þ  ¼ ðT T  Trs Tsr Þ¼ 12 ½ðtrTÞ  trT2   T21 T22   T32 T33   T31 T11  2 rr ss ¼ T11 T22 þ T22 T33 þ T33 T11 T12 T21  T23 T32  T31 T13 ¼ T1 T2 þ T2 T3 þ T3 T1

ð1:218Þ

1.5 Eigenvalues and Eigenvectors

41

   T11 T12 T 13      1 1 1 III   T21 T22 T23  ¼ detT ¼ erst Tr1 Ts2 Tt3 ¼ ðtrTÞ3  trTtrT2 þ trT3 6 2 3    T31 T32 T33  ¼ T11 T22 T33 þ T12 T23 T31 þ T21 T32 T13  T11 T23 T32  T22 T31 T13  T33 T12 T21 ¼ T1 T2 T3

ð1:219Þ I, II, III designate the change of the total length, the total surface area and the volume, respectively, of the unit cube in the initial state, if T1, T2, T3 are the principal stretches in Eq. (4.19). The last expressions in III is derived from ðtrT3  I trT2 þ II trT  3III ¼Þ trT3  ðtrTÞtrT2 þ

1 ½ðtrTÞ2  trT2 trT  3III¼ 0 2

which is obtained by taking the trace of Eq. (1.250) in the Cayley–Hamilton theorem which will be described in Sect. 1.8 and by substituting Eq. (1.218) for II. On the other hand, the characteristic Eq. (1.215) is expressed using the three principal values T1 ; T2 ; T3 as follows:   T1 T   0   0

0 T2 T 0

 0  0  ¼ ðTT1 ÞðTT2 ÞðTT3 Þ ¼ 0 T3 T 

ð1:220Þ

i.e. T 3  ðT1 þ T2 þ T3 ÞT 2 þ ðT1 T2 þ T1 T2 þ T3 T1 ÞT  T1 T2 T3 ¼ 0 Comparing Eqs. (1.216) and (1.220), coefficients I, II and III are described as I ¼ T1 þ T2 þ T3 II ¼ T1 T2 þ T2 T3 þ T3 T1 III ¼ T1 T2 T3

ð1:221Þ

Equation (1.221) can also be derived by substituting T11 ¼ T1 ; T22 ¼ T2 ; T33 ¼ T3 , T12 ¼ T23 ¼ T31 ¼ 0 in Eqs. (1.217)–(1.219), while hereinafter principal value is denoted by only one suffix. Since I, II and III are the symmetric functions of principal values, they are the invariants and are called the principal invariants. The following invariants are called the moments. I  I ¼ trT;

II  trT2 ;

III  trT3

ð1:222Þ

42

1

Mathematical Preliminaries: Vector and Tensor Analysis

The principal invariants are described in terms of these moments from Eqs. (1.217)–(1.219) as follows: I¼I 2

II ¼ 12 ðI IIÞ

ð1:223Þ

1 3 1 1 III ¼ I  I II þ III 6 2 3 Next, consider the deviatoric tensor T0 . The characteristic equation of T0 is given by replacing T to T0 in Eq. (1.216) as follows: T 0 3 þ II0 T0 III0 ¼ 0

ð1:224Þ

0 0 0 I0  tr T0 ¼ T11 þ T22 þ T33 ¼0

ð1:225Þ

where

1 1 1 II0 ¼  Trs0 Tsr0 ¼  tr T02 ¼  jjT0 jj2 2 2 2  1 02 02 02 02 02 02 ¼  ðT11 þ T22 þ T33 Þ þ T12 þ T23 þ T31 2 1 02 02 02 þ T23 þ T31 g ¼  f ½ðT11  T22 Þ2 þ ðT22  T33 Þ2 þ ðT33  T11 Þ2  þ T12 6 1 1 ¼  ðT102 þ T202 þ T302 Þ¼  ½ðT1  T2 Þ2 þ ðT2  T3 Þ2 þ ðT3  T1 Þ2  2 6 0 0 0 0 0 0 0 0 0 0 0 0 ¼  ðT11 T22 þT22 T33 þT33 T11 T12 T21  T23 T32  T31 T13 Þ¼  ðT10 T20 þT20 T30 þT30 T10 Þ ð1:226Þ  0 0 0  T T T   11 12 13   0 0 0  1 1 0 III0   T21 T22 T23  ¼ det T0 ¼ eijk Ti10 Tj20 Tk3 ¼ trT03 ¼ Tij0 Tjk0 Tki0 3 3  0 0 0  T T T  31 32 33 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ¼ T11 T22 T33 þ T12 T23 T31 þ T21 T32 T13  T11 T23 T32  T22 T31 T13  T33 T12 T21 ¼ T10 T20 T30

ð1:227Þ noting Eqs. (1.218) and (1.219).

1.5 Eigenvalues and Eigenvectors

43

(Note) 1 02 02 ðT þ T 02 þ T33 Þ 2 11( 22  2  2  2 ) 1 1 1 1 T11  ðT11 þ T22 þ T33 Þ þ T22  ðT11 þ T22 þ T33 Þ þ T33  ðT11 þ T22 þ T33 Þ ¼ 2 3 3 3   1 2 1 2 2 2 T11 ¼  T11 ðT11 þ T22 þ T33 Þ þ ðT11 þ T22 þ T33 Þ2 þ T22  T22 ðT11 þ T22 þ T33 Þ 2 3 9 3   1 2 1 2  T33 ðT11 þ T22 þ T33 Þ þ ðT11 þ T22 þ T33 Þ2 þ ðT11 þ T22 þ T33 Þ2 þ T33 9 3 9  1 2 2 3 2 2 2 þ T33  ðT11 þ T22 þ T33 Þ þ ðT11 þ T22 þ T33 Þ2 ¼ T11 þ T22 2 3 9  1 2 1 2 2 2 ¼ T11 þ T22 þ T33  ðT11 þ T22 þ T33 Þ 2 3  1 2 1 2 2 2 2 2 þ T22 þ T33 þ 2T11 T22 þ 2T22 T33 þ 2T33 T11 Þ ¼ T11 þ T22 þ T33  ðT11 2 3  1 2 2 2 2 ðT11 þ T22 þ T33  T11 T22  T22 T33  T33 T11 Þ ¼ 2 3   1 1 ½ðT11  T22 Þ2 þ ðT22  T33 Þ2 þ ðT33  T11 Þ2  ¼ 2 3 1 ¼ ½ðT11  T22 Þ2 þ ðT22  T33 Þ2 þ ðT33  T11 Þ2  6

Then, the following expressions in terms of the principal values hold from Eqs. (1.225)–(1.227) or from Eq. (1.221). I0 ¼ 0 II0 ¼  ðT10 T20 þ T20 T30 þ T03 T10 Þ 03 03 III0 ¼ T01 T02 T03 ¼ 13 ðT03 1 þ T2 þ T3 Þ

1.6

ð1:228Þ

Calculations of Eigenvalues and Eigenvectors

The second-order symmetric tensor can be represented by Eq. (1.206) as the spectral representation in the eigendirections. To express the tensor in the eigendirections, one must calculate the eigenvalues and the eigenvectors of the tensor. The solutions for them (cf. Hoger and Carlson 1984; Carlson and Hoger 1986) are shown in this section.

44

1.6.1

1

Mathematical Preliminaries: Vector and Tensor Analysis

Eigenvalues

In order to obtain eigenvalues, one try to calculate the deviatoric components from the characteristic Eq. (1.224) which is the cubic equation having the coefficients as the functions of invariants. Now, infer the form rffiffiffiffiffiffiffiffi 4 II0 cos w T ¼ 3 0

ð1:229Þ

for the eigenvalues of deviatoric part of tensor T. The substituting Eq. (1.229) into Eq. (1.224), we have  4II0 3=2 3 which is reduced to

cos3 w  II0

 4II0 1=2 3

cos wIII0 ¼ 0

pffiffiffi 4 pffiffiffi III0 3=2 cos 3wIII0 ¼ 0 3 3

ð1:230Þ

ð1:231Þ

using the trigonometric formula 1 cos3 w ¼ ðcos 3w þ 3 cos wÞ 4

ð1:232Þ

It is obtained from (1.231) that pffiffiffi 3 3III0 cos 3w¼ 2II0 3=2

ð1:233Þ

Noting that the cosine is the periodic function with the period 2p, the angle w is expressed by the following equation with a natural number J in general. w¼

pffiffiffi i 1 h 1  3 3III0   2pJ cos 0 3=2 3 2II

ð1:234Þ

Substituting Eq. (1.234) into Eq. (1.229), one has Tp0

1 ¼ 3

rffiffiffiffiffiffiffi n1h  3pffiffi3ffiIII0  io 4II0 cos  2pJ cos1 3 3 2II0 3=2

ð1:235Þ

and adding the isotropic component I=3, the eigenvalues of T are given as follows:

1.6 Calculations of Eigenvalues and Eigenvectors

1 TP ¼ Iþ 3

1.6.2

45

! rffiffiffiffiffiffiffi n1h  pffiffiffi 0  io 4II0 1 3 3III cos  2pJ cos 3 3 2 II0 3=2

ð1:236Þ

Eigenvectors

Equation (1.206) can be expressed as follows: 3 X

T¼

ð1:237Þ

TP EP

P¼1

while the tensor EJ is called the eigenprojection of T, which is defined by EP  eP  eP (no sum)

ð1:238Þ

fulfilling 3 X

EP ð¼e1  e1 þ e2  e2 þ e3  e3 Þ ¼ I

ð1:239Þ

P¼1

(

EP for P¼Q

EP E Q ¼

O

for P 6¼ Q

EP : EQ ¼ dPQ

9 > > = > > ;

ð1:240Þ

It holds that

9 > > EP ¼ ð TQ EQ ÞeP  eP ¼ ð TQ eQ  eQ ÞeP  eP ¼ TP eP  eP > > > = Q¼1 Q¼1 3 X

E P T ¼ e P  eP ð

3 X

3 X

TQ EQ Þ ¼ eP  eP ð

Q¼1

3 X Q¼1

> > > TQ eQ  eQ Þ ¼ eP  eP TP > > ;

ðno sum for PÞ

and thus one has TEP ¼ EP T ¼ TP EP (no sum) On the other hand, it holds from Eq. (1.239) that TTQ I ¼

3 X P¼1

TP EP  TQ

3 X P¼1

EP

ð1:241Þ

46

1

Mathematical Preliminaries: Vector and Tensor Analysis

and thus it is obtained that T  TQ I¼

3 X

ðTP TQ ÞEP

ð1:242Þ

P¼1

from which one has 3 Y Q6¼h Q¼1

ðTTQ IÞ ¼

3 X 3 Y

ðTP TQ ÞEQ ¼

3 hY

Q6¼h P¼1 Q¼1

i ðTh TQ Þ Eh

ð1:243Þ

Q6¼h Q¼1

from which the following Sylvester’s formula is obtained. Eh ¼

3 TT I Q Q T T Q Q 6¼ h h

ð1:244Þ

Q¼1

For instance, E2 ðh¼ 2Þ in the popular case of n ¼ 3 is obtained from Eq. (1.244) as follows: Q ðTTP IÞ ðT  T1 IÞðT  T3 IÞ P¼1;3 ¼ E2 ¼ Q ðT2 TP Þ ðT2 T 1 ÞðT2 T 3 Þ P¼1;3

1.7

Eigenvalues and Eigenvectors of Skew-Symmetric Tensor

The characteristic equation of skew-symmetric tensor is given by substituting TA into Eq. (1.216) as follows: T 3  tr TA T 2 þ

1 2 A ðtr T  tr TA2 ÞTdet TA ¼ 0 2

ð1:245Þ

Noting tr TA ¼ det TA ¼ 0 in Eqs. (1.176) and (1.178), Eq. (1.245) leads to ð2T 2  trTA 2 Þ

T ¼0 2

ð1:246Þ

from which the eigenvalues are given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T ¼ i jtr TA 2 j=2 ¼ ijjtA jj

and 0

ð1:247Þ

1.7 Eigenvalues and Eigenvectors of Skew-Symmetric Tensor

47

noting 02

0 t3A A2 @ 4 trT ¼ tr 0 ant:

32 0 t2A t1A 54 0

t3A 0 ant:

31 t2A t1A 5A¼ 2ðt1A 2 þ t2A 2 þ t3A 2 Þ\0 0

obtained from Eq. (1.188), where tiA is the axial vectors defined in Eq. (1.187). It is known from Eq. (1.247) that one principal value is zero and two principal values are pure-imaginary numbers without a real part. Here, if one of the principal direction e3 is chosen to the direction of the zero A A A principal value of TA leading to T13 ¼ t23 ¼ 0, it follows by denoting T12 ¼ A t12  x that TA ¼ QTA QT 3 2 32 32 sinh 0 cos h sinh 0 0 x 0 cos h 7 6 76 76 ¼4 sinh cos h 0 54 x 0 0 54 sinh cos h 0 5 0 0 1 0 0 0 0 0 0 2 3 2 2 xsinh cos h þ xsinh cos h x sin h þ x cos h 0 6 7 ¼4 xsinh cos h  xsinh cos h 0 5  x cos2 h  xsin2 h 2

0 6 ¼4 x 0

0 3

x 0

0 7 0 5¼ TA

0

0

0

ð1:248Þ

0

meaning that the components do not change in the coordinate transformation. It is caused from the fact that the independent component of skew-symmetric tensor is only one when the one of bases in the coordinate system is chosen to the principal direction of skew-symmetric tensor.

1.8

Cayley-Hamilton Theorem

Denoting the eigenvector of the tensor T by e, it follows from Eq. (1.198) that Tr e ¼ T r e

ð1:249Þ

by the repeated applications of Eq. (1.198), i.e. T r e ¼ T r1 Te ¼ T r1 Te ¼ T T r2 Te ¼ TT r2 Te ¼ T2 T r2 e ¼ T2 T r3 Te ¼    ¼ Tr e. Equation (1.249) means that the eigenvalues of the tensor Tr are given by TJr where TJ ðJ¼ 1; 2; 3Þ are the

48

1

Mathematical Preliminaries: Vector and Tensor Analysis

eigenvalues of T, and the eigenvectors of the tensor Tr coincide with those of T. Tensors having an identical set of principal directions are called to be coaxial or said to fulfill the coaxiality. Then, the linear associative function fðTÞ of T is coaxial with T and the principal values of fðTÞ are given by f ðTJ Þ. The multiplication of the eigenvector e to the characteristic Eq. (1.216) leads to ðT3  I T2 þ II T  III IÞe ¼ 0 noting Eq. (1.249). Because of e 6¼ 0, the following Cayley-Hamilton theorem holds. T3  I T2 þ II T  III I ¼ O

ð1:250Þ

It follows from the Cayley-Hamilton theorem that T4 ¼ ðI T2  II T þ IIIIÞT ¼ I T3  II T2 þ III T ¼ IðIT2  II T þ III IÞ  II T2 þ III T ¼ (I2 IIÞT2  ðIII  IIIÞT þ III I

ð1:251Þ III T1 ¼ T2  I T þ II I

ð1:252Þ

It is concluded that the power of the tensor T is expressed by the linear associative of T2 ; T; I with coefficients consisting of the principal invariants. The Cayley-Hamilton theorem can be also derived from the characteristic equation as follows: Eq. (1.216) holds for the three principal values T1 , T2 and T3 , and thus the following equation must holds. 2

T13  IT12 þ II T1  III 6 0 4

0 0

0

T33  IT32 þ II T3  III

0 2

T13 0 0

3

2

3

0 T23  IT22 þ IIT2  III

T12 0 0

3

2

T1 0 0

3

2

1 0 0

3

7 5

2

0 0 0

3

6 6 6 7 7 7 6 7 6 7 3 2 6 6 7 7 7 6 7 6 7 ¼6 4 0 T2 0 5  I4 0 T2 0 5 þ II4 0 T2 0 5  III4 0 1 0 5 ¼ 4 0 0 0 5 0 0 T3 0 0 1 0 0 0 0 0 T33 0 0 T32 ð1:253Þ the tensor expression of which is nothing but Eq. (1.250).

1.8 Cayley-Hamilton Theorem

1.9

49

Scalar Triple Products with Invariants

The following formulae of the scalar triple products related to the principal invariants hold. ½Tu v w þ ½u Tv w þ ½u vT w ¼ trT½uvw ¼ I½uvw ½u Tv Tw þ ½Tu v Tw þ ½Tu Tv w ¼ II½uvw

ð1:254Þ

½Tu Tv Tw ¼ detT½uvw ¼ III½uvw where I; II; III are the principal invariants of T in Eqs. (1.217)–(1.219), i.e. I  trT;

II  ðtr2 T  trT2 Þ=2; III  detT

which will be delineated in the next section. The proof for Eq. (1.254) is given below (cf. Chadwick 1976; Kyoya 2008). Proof of Eq. (1.254)1: We have the relation ½Tu; v; w þ ½u; Tv; w þ ½u; v; Tw ¼ ui vj wk ð½Tei ; ej ; ek  þ ½ei ; Tej ; ek  þ ½ei ; ej ; Tek Þ ¼ eijk ui vj wk ð½Te1 ; e2 ; e3  þ ½e1 ; Te2 ; e3  þ ½e1 ; e2 ; Te3 Þ

ð1:255Þ

¼ ½uvwð½Te1 ; e2 ; e3  þ ½e1 ; Te2 ; e3  þ ½e1 ; e2 ; Te3 Þ making use of Eq. (1.49) with Eq. (1.47) and noting ½Tei ; ej ; ek  þ ½ei ; Tej ; ek  þ ½ei ; ej ; Tek  ¼ eijk ð½Te1 ; e2 ; e3  þ ½e1 ; Te2 ; e3  þ ½e1 ; e2 ; Te3 Þ

ð1:256Þ Further, the inside of bracket in the last equation in Eq. (1.255) is described as follows: ½Te1 ; e2 ; e3  þ ½e1 ; Te2 ; e3  þ ½e1 ; e2 ; Te3  ¼ ½Tr1 er ; e2 ; e3  þ ½e1 ; Tr2 er ; e3  þ ½e1 ; e2 ; Tr3 er  ¼ Tr1 ½er e2 e3  þ Tr2 ½e1 er e3  þ Tr3 ½e1 e2 er  ¼ T11 ½e1 e2 e3  þ T22 ½e1 e2 e3  þ T33 ½e1 e2 e3  ¼ T11 þ T22 þ T33 ¼ trT ¼ I

ð1:257Þ Equation (1.254)1 is obtained by substituting Eq. (1.257) into Eq. (1.255). Proof of Eq. (1.254)2: In the similar way as Eq. (1.255), we have first

50

1

Mathematical Preliminaries: Vector and Tensor Analysis

½u; Tv; Tw þ ½Tu; v; Tw þ ½Tu; Tv; w ¼ eijk ui vj wk ð½e1 ; Te2 ; Te3  þ ½Te1 ; e2 ; Te3  þ ½Te1 ; Te2 ; e3 Þ

ð1:258Þ

¼ ½uvwð½e1 ; Te2 ; Te3  þ ½Te1 ; e2 ; Te3  þ ½Te1 ; Te2 ; e3 Þ Here, applying Eqs. (1.47) and (1.256), the inside of bracket in this equation is described as follows: ½e1 ; Te2 ; Te3  þ ½Te1 ; e2 ; Te3  þ ½Te1 ; Te2 ; e3  ¼ Tr2 Ts3 ½e1 er es  þ Tr1 Ts3 ½er e2 es  þ Tr1 Ts2 ½er es e3  ¼ Tr2 Ts3 e1rs þ Tr1 Ts3 er2s þ Tr1 Ts2 ers3 ¼ ðT22 T33  T23 T32 Þ þ ðT11 T33  T13 T31 Þ þ ðT11 T22  T12 T12 Þ ¼ T11 T22 þ T22 T33 þ T33 T11  ðT12 T12 þ T23 T32 þ T31 T13 Þ

ð1:259Þ

2 2 2 ¼ fT11 þ T22 þ T33 þ 2ðT11 T22 þ T22 T33 þ T33 T11 Þg=2 2 2 2  fT11 þ T22 þ T33 þ 2ðT12 T12 þ T23 T32 þ T31 T13 Þg=2 ¼ ðTrr Tss Trs Tsr Þ=2

leading to ½e1 ;Te2 ;Te3  þ ½Te1 ; e2 ; Te3  þ ½Te1 ; Te2 ; e3 ¼ ½ðtrTÞ2  tr2 T=2

ð1:260Þ

Equation (1.254)2 is obtained by substituting Eq. (1.260) into Eq. (1.258). Proof of Eq. (1.254)3: Changing the representations of the tree vectors to the component-based representations and then applying Eqs. (1.47) and (1.256), we have ½Tu; Tv; Tw ¼ eijk ui vj wk ½Te1 ;Te2 ;Te3  ¼ ½uvw½Te1 ;Te2 ;Te3 

ð1:261Þ

Equation (1.254)3 is obtained by substituting Eq. (1.158) into Eq. (1.261). Tep ¼ Tij ei  ej ep ¼ Tip ei jTe1 Te2 Te3 j ¼ jTi1 ei Tj2 ej Tk3 ek j ¼ Ti1 Tj2 Tk3 jei ej ek j ¼ Ti1 Tj2 Tk3 eijk je1 e2 e3 j ¼ det T with the aid of Eq. (1.47). It follows from Eq. (1.254)3 for T ¼ Q that Qu  ðQv  QwÞ ¼ det½Qu  ðv  wÞ

ð1:262Þ

1.9 Scalar Triple Products with Invariants

51

Then, the scalar product u  ðv  wÞ is transformed to the identical and the opposite handed system for det½Q ¼ 1 and det½Q ¼ 1, respectively, in the orthogonal transformation. Further, it follows from Eq. (1.254)3 that eijk det T ¼ ½Tei ; Tej ; Tek ; det T ¼ ½Te1 ;Te2 ;Te3 

1.10

ð1:263Þ

Positive Definite Tensor

The second-order symmetric tensor P fulfilling Pv  v [ 0

ð1:264Þ

for an arbitrary vector vð6¼ 0Þ is called the positive-definite tensor. Here, denoting the principal value and the corresponding eigen values of P as PJ and eJ , respectively, it follows that PeJ  eJ ¼ PJ eJ  eJ ¼ PJ jjeJ jj2 [ 0 ðno sum)

ð1:265Þ

Therefore, the principal values of the positive-definite symmetric tensor are all real and positive, i.e. PJ [ 0 and further it follows noting Eq. (1.221)3 that det P ¼ III [ 0

1.11

ð1:266Þ

Polar Decomposition

The following positive-definiteness holds for the symmetric tensors TT T and TTT . (

TT Tv  v ¼ Tv  Tv ¼ jjTvjj2 [ 0 TTT v  v ¼ TT v  TT v ¼ jjTT vjj2 [ 0

ð1:267Þ

noting Eq. (1.139). Therefore, TT T and TTT are the positive-definite tensors. Then, 2

designating their principal values by k2a and ka and their eigenvectors N a and na ða¼ 1; 2; 3Þ for TT T and TTT , respectively, the following representations in the spectral decomposition hold. 8 3 X > T > > T ¼ k2a N a  N a C  T > < a¼1

3 > X > 2 > T > ka na  na : b  TT ¼ a¼1

Further, introduce the following positive-definite tensors.

ð1:268Þ

52

1

Mathematical Preliminaries: Vector and Tensor Analysis

8 3 X > 1=2 > > U ¼ C ¼ ka N a  N a > < a¼1

3 > X > > 1=2 > ka na  na :V ¼ b ¼

ð1:269Þ

a¼1

which gives UN a ¼ ka N a ; Vna ¼ ka na (no sum)

ð1:270Þ

Further, let the following tensor be introduced. R ¼ TU 1 ;

R ¼ V 1 T

ð1:271Þ

for which one has (

RRT ¼ ðTU 1 ÞðTU1 ÞT ¼ TðU 2 Þ1 TT ¼ TT1 TT TT ¼ I R R ¼ ðV 1 TÞT ðV 1 TÞ ¼ TðV 2 Þ1 TT ¼ TT TT T1 T ¼ I T

ð1:272Þ

Therefore, R and R are the orthogonal tensors. It follows from Eq. (1.271) that T ¼ RU ¼ VR

ð1:273Þ

from which the following expressions are derived. U ¼ RT VR;

V ¼ RUR

T

ð1:274Þ

The substitution of Eq. (1.269)2 into Eq. (1.274)2 gives T

V ¼ RUR ¼ R

3 X

T

ka N a  N a R ¼

a¼1

3 X

ka RN a  RN a

a¼1

The coincidence of this equation with Eq. (1.269)2 requires the following relations. k a ¼ ka ; na ¼ RN a ;

R¼R N a ¼ RT na

ð1:275Þ ð1:276Þ

Then, it follows from Eqs. (1.269), (1.273), (1.274), (1.275) and (1.276) that

1.11

Polar Decomposition

53

T ¼ RU ¼ VR U ¼ RT VR;

V ¼ RURT

U ¼ C1=2 ¼ ðTT TÞ1=2 ¼ V¼b

T 1=2

¼ ðTT Þ

1=2

¼

3 P a¼1 3 P a¼1

R¼

3 P a¼1

T¼

3 P a¼1

na  N a ;

ka na  N a ;

ð1:277Þ ð1:278Þ

ka N a  N a ð ¼ U T Þ

ð1:279Þ

ka na  na ð ¼ V Þ T

det R ¼ 1

T1 ¼

3 1 P N a  na a¼1 ka

ð1:280Þ

ð1:281Þ

Equation (1.277) is called the polar (spectral) decomposition, and RU and VR are called the right and left polar decompositions, respectively. U and V possess the same principal values ka but different principal directions N a and na , so that they are called the similar tensors to each other. Now, the following relation holds from Eq. (1.268) that dC¼

3 X

½2ka dka N a  N a þ k2a ðdN a  N a þ N a  dN a Þ

ð1:282Þ

a¼1

from which one has N a  dCN a ¼ 2ka dka

ð1:283Þ

noting N a  dN a ¼ 0 derived from N a  N b ¼ dab . Here, substitutions of N a  dCN a ¼ dC:N a  N a by virtue of Eq. (1.120)5 and dC ¼ ð@C=@ka Þdka into the left-hand side of Eq. (1.283) leads to @C dka : N a  N a ¼ 2ka dka @ka

ð1:284Þ

1 @C dka : N a  N a ¼ 1 2ka @ka

ð1:285Þ

leading to

Here, taking account of the contraction @C=@ka :@ka =@C ¼ 1, it follows from Eq. (1.285) that

54

1

Mathematical Preliminaries: Vector and Tensor Analysis

@ka ¼ ð2ka Þ1 N a  N a @C

ð1:286Þ

Then, one has @k2a ¼ Na  Na @C 9 > @k21 = ¼ IN 3  N 3 > @C > @k23 ; ¼ N3  N3 > @C 3 @k2 X ¼ Na  Na ¼ I @C a¼1

for k1 6¼ k2 6¼ k3 6¼ k1 for k1 ¼ k2 6¼ k3

ð1:287Þ

for k1 ¼ k2 ¼ k3 ¼ k

noting @k2a @k2a @ka ¼ ¼ 2ka ð2ka Þ1 N a  N a @C @ka @C Equation (1.287) is useful for the derivations of stress in the hyperelastic equation described by the principal stretches as will be shown in Chap. 7.

1.12

Isotropic Tensor-Valued Tensor Function

If the tensor-valued function f of tensors S; T;    fulfills the following equation, it is called the isotropic function. Q½½fðS; T;    Þ ¼ fðQ½½S; Q½½T;    Þ

ð1:288Þ

where use is made of the symbol in Eq. (1.103). If f is a scalar, it is to be the invariant defined in Eq. (1.213) and if it is a tensor, it is called the isotropic tensor-valued tensor function. Now, consider the isotropic second-order tensor function B of a single second-order tensor A as the simplest case of Eq. (1.288), i.e. B ¼ fðAÞ

ð1:289Þ

fðQAQT Þ ¼ QfðAÞQT

ð1:290Þ

where f fulfills

First introducing the coordinate system with the bases e1 ; e2 ; e3 , which are the unit eigenvector of the tensor A and further adopting the another coordinate system

1.12

Isotropic Tensor-Valued Tensor Function

55

rotated 180° around the base e3 , the orthogonal tensor between the bases of these coordinate systems is given by 2

1 Q0 ¼ 4 0 0

3 0 0 1 0 5 0 1

ð1:291Þ

where Q0 fulfills Q0 ¼ QT0 resulting in the symmetric tensor and it holds that 2

1 4 0 0

0 1 0

38 0 9 8 0 9 0 > < > = > < > = 5 0 0 ¼ 0 ; > > > > 1 :1; :1;

i:e: Q0 e3 ¼ e3

ð1:292Þ

Then, it is known that e3 is one eigenvector not only of A but also of Q0 . Furthermore, denoting the principal values of A by A1 ; A2 ; A3 , it holds that 2

1 4 0 0

0 1 0

32 0 A1 0 54 0 0 1

0 A2 0

32 0 1 0 54 0 A3 0

0 1 0

3 2 0 A1 0 5¼4 0 0 1

0 A2 0

3 0 0 5; A3

i.e: Q0 AQT0 ¼ A

ð1:293Þ and thus it holds that fðQ0 AQT0 Þ ¼ fðAÞ ¼ B

ð1:294Þ

On the other hand, from Eqs. (1.289), (1.290) and (1.293) one has B ¼ fðAÞ ¼ fðQ0 AQT0 Þ ¼Q0 fðAÞQT0 ¼ Q0 BQT0

ð1:295Þ

Then, one obtains B ¼ Q0 BQT0 leading to the commutative law Q0 B ¼ BQ0

ð1:296Þ

and further, noting Eq. (1.292), the following relation is obtained. Q0 Be3 ¼ BQ0 e3 ¼ Be3

ð1:297Þ

which means that Be3 is the eigenvector of Q0 . Then, reminding that the eigenvector of Q0 is e3 , it follows that Be3 has the same direction as e3 . Then, it follows that Be3 ¼ B3 e3

ð1:298Þ

where B3 is the eigenvalue of B for the eigenvector e3 . Performing the similar manipulations also for e1 and e2 , it can be concluded that the tensor B has the same eigenvectors as the tensor A, leading to the coaxiality. Therefore, the principal

56

1

Mathematical Preliminaries: Vector and Tensor Analysis

values B1 ; B2 ; B3 of the tensor B can be represented in unique relation to the principal values A1 ; A2 ; A3 of the tensor A, i.e. ^ i ðA1 ; A2 ; A3 Þ Bi ¼ B

ði ¼ 1; 2; 3Þ

ð1:299Þ

Now, if the principal values Ai is different from each other, the most general expression of Bi is given by ^ i ðAj Þ ¼ /0 þ /1 Ai þ /2 A2i Bi ¼ B ði ¼ 1; 2; 3Þ Bi ei  ei ¼ ð/0 þ /1 Ai þ /2 A2i Þei  ei ðno sumÞ

ð1:300Þ

i.e. 9 B1 ðA1 ; A2 ; A3 Þ ¼ /0 þ /1 A1 þ /2 A21 > = B2 ðA1 ; A2 ; A3 Þ ¼ /0 þ /1 A2 þ /2 A2 ; 2 > B3 ðA1 ; A2 ; A3 Þ ¼ /0 þ /1 A3 þ /2 A23 ;

8 9 2 > 1 = < B1 > i:e: B2 ¼ 4 1 > ; : > 1 B3

A1 A2 A3

38 / 9 A21 > = < 0> 25 /1 A2 > > A23 : / ; 2

ð1:301Þ where /0 ; /1 ; /2 are scalar functions of A1 ; A2 ; A3 . The rightness of Eq. (1.300) or (1.301) can be verified by the reason that /0 ; /1 ; /2 can be determined if A1 ; A2 ; A3 and B1 ; B2 ; B3 are given, because the following Vandermonde’s determinant for the above simultaneous equation of the unknown variables /0 ; /1 ; /2 is not zero for mutually different values of A1 ; A2 ; A3 , i.e.   1 A1   1 A2   1 A3 leading to the   B0   B1   B2 /0 ¼  1 1  1

 A21  A22  ¼ ðA1 A2 ÞðA2 A3 ÞðA3 A1 Þ 6¼ 0 A23 

ð1:302Þ

solutions  A1 A21  A2 A22  A3 A23   ; A1 A21  A2 A22  A3 A23 

 1  1  1 /1 ¼  1 1  1

B1 B2 B3 A1 A2 A3

 A21  A22  A23  ; A21  A22  A2  3

 1  1  1 /2 ¼  1 1  1

A1 A2 A3 A1 A2 A3

 B1  B2  B3   A21  A22  A2 

ð1:303Þ

3

as known from the description in Sect. 1.5. While Eq. (1.301) is regarded as a representation of the relation of the tensors A and B in their common principal coordinate system, it is expressed in the direct notation of tensor as

1.12

Isotropic Tensor-Valued Tensor Function

57

B ¼ /0 I þ /1 A þ /2 A2

ð1:304Þ

This fact can also be verified simply using Cayley-Hamilton theorem (Sect. 1.8) for the special case that f is a polynomial equation of the power of A in Eq. (1.289). However, for the case in which f is the general function of A, one must depend on the above-mentioned proof. It follows analogously to the above-mentioned method that B ¼ /0 I þ / 1 A

ð1:305Þ

when the two of principal values are same, and ð1:306Þ

B ¼ /0 I

when all principal values are same. In the particular case in which f is the linear function of the tensor A, Eq. (1.304) is reduced to B ¼ aðtr AÞI þ bA

ð1:307Þ

where a and b are the constants. Equation (1.307) is rewritten as B ¼ N: A

ð1:308Þ

1 N  aT þ bS; Nijkl  adij dkl þ bðdik djl þ dil djk Þ 2

ð1:309Þ

where

While the second-order isotropic tensor-valued tensor function of single tensor is considered above, the representation theorem of the second-order isotropic tensorvalued symmetric tensor function f S and anti-symmetric tensor function f A of the two tensors A and B are represented as follows (Rivlin 1955; Truesdell and Noll 1965). 9 f S ðA; BÞ ¼ u0 I þ u1 A þ u2 B þ u3 A2 þ u4 B2 þ u5 ðAB þ BAÞ > > > > 2 2 2 2 2 2 2 2 þ u ðA B þ BA Þ þ u ðAB þ B AÞ þ u ðA B þ B A Þ = 6

7

8

f ðA; BÞ ¼ g1 ðABBAÞ þ g2 ðA BBA Þ þ g3 ðAB2 B2 AÞ > > > > ; 2 2 2 2 þ g4 ðABA A BAÞ þ g5 ðBAB B ABÞ A

2

2

ð1:310Þ

where u0 ; u1 ;   ; u8 and g1 ; g2 ;   ; g5 are the scalar-valued isotropic functions of invariants of A and B, i.e.  tr A; tr A2 ; tr A3 ; tr B; tr B2 ; tr B3 ð1:311Þ tr ðABÞ; tr ðAB2 Þ; tr ðA2 BÞ; tr ðA2 B2 Þ

58

1

Mathematical Preliminaries: Vector and Tensor Analysis

noting that all tensors can be represented by powers of tensor lower than the second power by the Cayley-Hamilton theorem in Sect. 1.8.

1.13

Representation of Tensor in Principal Space

Second-order tensor T is described by only three independent components, i.e. the principal values, in the directions of the eigenvectors e1 ; e2 ; e3 . Designating the principal values by T1 ; T2 ; T3 , it can be represented as follows: T ¼ T1 e1 þ T2 e2 þ T3 e3

ð1:312Þ

Equation (1.312) may be called the representation of tensor in principal space (Fig. 1.3) by which the second-order tensor can be visualized in the three dimensional space. Equation (1.312) is rewritten by decomposing T into the mean and the deviatoric components as follows: 0 ÞeIII ¼ Tm þ T0 T ¼ ðTm þ TI0 Þe1 þ ðTm þ TII0 Þe2 þ ðTm þ TIII

ð1:313Þ

where pffiffiffi Tm  Tm Im ¼ 3Tm em ðTm  ðTI þ TII þ TIII Þ=3Þ

ð1:314Þ

1 Im  eI þ eII þ eIII ; em  pffiffiffi Im ðjjem jj¼1Þ 3

ð1:315Þ

0 eIII ¼ jjT0 jjt0 T0  TI0 eI þ TII0 eII þ TIII

ð1:316Þ

TIII

' TIII TI' e III

0

Space diagonal

T

em TII e II TI e I

T' t'

Deviatoric plane

TII'

Tm

Fig. 1.3 Representation of second-order tensor in principal space

1.13

Representation of Tensor in Principal Space

59

TIII

n'III θ + 2π 3

_

0

n'I

T 'I

θ T'

TI

_

T 'II

n'II

2 π −θ 3

TII

_

( −) TIII '

Fig. 1.4 Deviatoric part of tensor in deviatoric plane (p-plane)

t0  T0 =jjT0 jjðjjt0 jj¼ 1Þ

ð1:317Þ

9 0 TI0  TI  Tm ; TII0  TII  Tm ; TIII  TIII  Tm = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 02 ¼ pffiffiffi jjT0 jj ¼ TI0 2 þ TII0 2 þ TIII ðTI TII Þ2 þ ðTII TIII Þ2 þ ðTIII TI Þ2 ; 3

ð1:318Þ

Here, introduce the following unit vectors n0I , n0II and n0III which are the projections of the unit base vectors eI , eII and eIII onto the deviatoric plane as shown in Fig. 1.4. 9 1 > > n0I  pffiffiffi ð2eI  eII  eIII Þðjjn0I jj ¼ 1Þ > > > 6 > > = 1 0 0 nII  pffiffiffi ðeI þ 2eII  eIII ÞðjjnII jj ¼ 1Þ > 6 > > > > 1 > 0 0 nIII  pffiffiffi ðeI  eII þ 2eIII ÞðjjnIII jj ¼ 1Þ > ; 6

ð1:319Þ

fulfilling n0I  em ¼ n0II  em ¼ n0III  em ¼ cos ½ð1=2Þp ¼0; pffiffiffiffiffiffiffiffi n0I  eI ¼ n0II  eII ¼ n0III  eIII ¼ 2=3; n0I  n0II ¼ n0II  n0III ¼ n0III

9 > > > =

> > > ;  n0I ¼ cos ½ð2=3Þp ¼ 1=2

ð1:320Þ

60

1

Mathematical Preliminaries: Vector and Tensor Analysis

It follows from Eqs. (1.316) and (1.319) for the projections of T0 onto the directions n0I , n0II and n0III that 9 0 T I  T0  n01 ¼ jjT0 jjt0  n01 ¼ jjT0 jj cos h > > >  2  > = 0 T II  T0  n0II ¼ jjT0 jjt0  n0II ¼ jjT0 jj cos h p 3 > >  2 > > 0 0  0 0 0  0 0 T III  T nIII ¼ jjT jjt nIII ¼ jjT jj cos h þ p ; 3

ð1:321Þ

where h is the angle measured in the anti-clock wise direction from n0I to T0 in the deviatoric plane as shown in Fig. 1.4 and it is called the Lode angle. Further, one 0 has the following relation for T I for example. 0

ðT I n0I Þ  eI ¼ ½ðT0  n0I Þn0I   eI 1 1 0 ¼ ½ðTI0 eI þ TII0 eII þ TIII eIII Þ  pffiffiffi ð2eI  eII  eIII Þ pffiffiffi ð2eI  eII  eIII Þ  eI 6 6 1 0 0 0 0 0 0 0 ¼ ð2TI  TII  TIII Þ  2 ¼ T1 ð*TI þ TII þ TIII ¼ 0Þ 6

Then, we have rffiffiffi 9 2 0 > > eI ¼ jjT eI ¼ jjT jj cos h > > > 3 > > rffiffiffi = 2 2 0 2 0 0 0 0 0 jjT jj cosðh pÞ > ð1:322Þ TII ¼ ðT II nII Þ  eII ¼ jjT jj cosðh pÞnII  eII ¼ 3 3 3 > > rffiffiffi > > 2 2 0 2 > > 0 0 0 0 0 jjT jj cosðh þ pÞ ; TIII ¼ ðT III nIII Þ  eIII ¼ jjT jjðh þ pÞnIII  eIII ¼ 3 3 3 T10 ¼

0 ðT I n0I Þ 

0

jj cos h n0I 

The product of the three components is given by III0

0 T 0 TII0 TIII 2 ¼ 10 0 0 ¼ 0 3=2 jjT jj jjT jj jjT jj 3 ð2II Þ

rffiffiffi 2 2 2 1 cos h cosðh pÞ cosðh þ pÞ ¼ pffiffiffi cos 3h 3 3 3 3 6 ð1:323Þ

noting Eqs. (1.226) and (1.228)3 and cos h cosðh þ

h1 2 2 4 i pÞ cosðh pÞ ¼ cos h ðcos 2h þ cos pÞ 3 3 2 3 1h 4 4 i ¼ cos 3h þ cos h þ cosðh þ pÞ þ cosðh  pÞ 4 3 3 1h 4 i 1 ¼ cos 3h þ cos h þ 2 cos h cos p ¼ cos 3h 4 3 4

1.13

Representation of Tensor in Principal Space

61

exploiting the formula cos A cos B ¼ ½cosðA þ BÞ þ cosðABÞ=2, cos A þ cos B ¼ 2cos½ðA þ BÞ=2cos½ðABÞ=2. It follows from Eq. (1.323) that pffiffiffi pffiffiffi 3 3 III0 cos 3h ¼ ¼ 6tr t0 3 0 3=2 2 II

ð1:324Þ

noting Eqs. (1.226) and (1.227) for the third term. The different proof of this equation is shown in Ottosen and Ristinmaa (2005). Incidentally, the variable lð1  l  1Þ defined by l

0 pffiffiffi 2TII  TI  TIII 2TII0  TI0  TIII ¼ ¼ 3 tanðh  p=6Þ 0 0 TI  TIII TI  TIII

ð1:325Þ

is called the Lode variable which is reduced to ( l¼

 1 for TII ¼ TIII ðh ¼ 0 : triaxial extensionÞ þ 1 for TII ¼ TI ðh ¼ p=3 : triaxial compressionÞ

ð1:326Þ

describing the state of the intermediate principal stress. The following relation is exploited in Eq. (1.325)4. 2T2  T1  T3 T1  T3 2 cosðh  23 pÞ  cos h  cosðh þ 23 pÞ cosðh  23 pÞ  cos h þ cosðh  23 pÞ  cosðh þ 23 pÞ ¼ ¼ cos h  cosðh þ 23 pÞ 2 sinðh þ 13 pÞ sinð 13 pÞ pffiffi pffiffi 1 1 4 2 sinðh  3 pÞ sinð 3 pÞ  2 sin h sinð 3 pÞ  sinðh  13 pÞð 23Þ  sin hð 23Þ pffiffi pffiffi ¼ ¼ 2 cosðh  16 pÞð 23Þ  cosðh  16 pÞð 23Þ pffiffi sinðh  13 pÞ þ sin h 2 sinðh  16 pÞ cosð 16 pÞ 2 sinðh  16 pÞð 23Þ pffiffiffi 1 ¼ ¼ ¼ 3 tanðh  pÞ ¼ 6 cosðh  16 pÞ cosðh  16 pÞ cosðh  16 pÞ

ð1:327Þ

1.14

Two-Dimensional State

Consider the two-dimensional state in which the components related to the e3 direction in the coordinate system ðx1 ; x2 ; x3 Þ with the fixed base ðe1 ; e2 ; e3 Þ are zero, i.e. T33 ¼ T 31 ¼ T23 ¼ 0. Furthermore, introduce the coordinate system ðx1 ; x2 ; x3 Þ with the bases ðe1 ; e2 ; e3 ð¼ e3 ÞÞ which is rotated by the angle a in the counterclockwise direction around the axis x3 as shown in Fig. 1.5, where the normal components Tij ði; j ¼ 1; 2; i ¼ jÞ and the shear components Tij ði 6¼ jÞ in

62

1

x2

Mathematical Preliminaries: Vector and Tensor Analysis

x2

Tt

Material

x1* * * (T11 , T12 )

x*2 x1*

RT

(T22 , T12 ) x*2

α 0

x1

2



T2 P

2 p T1

Tm

Tn

(T11 , T12 ) x1

(T22* , T12* ) (a) Physical plane

(b) (Tn , Tt ) plane

Fig. 1.5 Mohr’s circle

the base ðe1 ; e2 Þ are designated by Tn and by Tt , respectively. The orthogonal tensor between these bases is given from Eq. (1.78) as follows: 2

3 0 05 1

cos a sina ½Q ¼4 sina cosa 0 0

ð1:328Þ

Substituting Eq. (1.328) into Eq. (1.93), one has 9  T11 ¼ T11 cos2 a þ T22 sin2 a þ 2T12 cos a sin a > =

 T22 ¼ T11 sin2 a þ T22 cos2 a  2T12 sin a cos a  T12

¼ ðT22  T11 Þ cos a sin a þ T12 ðcos a  sin aÞ 2

2

> ;

ð1:329Þ

which is rewritten as 9  T11 ¼ Tm þ T cos 2a þ T12 sin 2a > =

 T22 ¼ Tm  T cos 2a  T12 sin 2a > ;  ¼ T sin 2a þ T12 cos 2a T12

ð1:330Þ

where Tm 

T11 þ T22 ; 2

T

Furthermore, it follows from Eq. (1.330) that

T11  T22 2

ð1:331Þ

1.14

Two-Dimensional State

63   T11 þ T22 ¼ T11 þ T22

 @T11  ¼ 2T12 ; @a

ð1:332Þ

 @T22  ¼ 2T12 @a

ð1:333Þ

While the axis x3 ð¼ x3 Þ is one of the principal directions, the other principal directions exist on the plane ðx1 ; x2 Þ. Denoting the principal direction from the x1     =@a ¼ T12 ¼ 0 or @T22 =@a ¼ 0 in axis by a, it is obtained by substituting @T11 Eq. (1.333) into Eq. (1.330) that tan 2ap ¼

T12 T

ð1:334Þ

from which one obtains T ¼ RT cos 2ap ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 RT ¼ T þ T212

9 T12 ¼ RT sin 2ap = ;

ð1:335Þ

Substituting Eq. (1.335) into the upper two of Eq. (1.330) with specifying a as ap , the maximum and the minimum principal values T1 and T2 are described by T1 T2

) ¼ Tm R T

ð1:336Þ

Equation (1.336) can also be derived directly from the quadratic equation 2 ¼0 T 2  ðT11 þ T22 ÞT þ T11 T22  T12 2 which is obtained by inserting I ¼ T11 þ T22 ; II ¼ T11 T22  T12 ; III ¼ 0 ðT33 ¼ T31 ¼ T23 ¼ 0Þ in Eq. (1.216).  Furthermore, denoting a for the extremal value of T12 as as , it follows by taking  @T12 =@a ¼ 0 in Eq. (1.330) that

tan 2as ¼ 

T 2T12

ð1:337Þ

Equations (1.334) and (1.337) yield the relation as ¼ ap p=4 (tan 2ap  tan 2as ¼ 1) and thus there exist the two directions for the extremal values of T12 and they divide the two principal directions into two equal angles, i.e. p=4. The  denoted by TM is given from Eq. (1.335)2 as follows: extremal values of T12

64

1

Mathematical Preliminaries: Vector and Tensor Analysis

TM ¼ RT

ð1:338Þ

which is also expressed by Eq. (1.336) as TM ¼

T1  T2 2

ð1:339Þ

The following equation is derived from Eqs. (1.330) and (1.335)3. 2

ðTn  Tm Þ2 þ Tt2 ¼ RT

ð1:340Þ

noting the following equations obtained from Eq. (1.335)1;3 or (1.335)2;3 with the    replacement of T11 ðT22 Þ ! Tn ; T12 ! Tt . 2

2 ðTn Tm Þ2 ¼ T cos2 2a þ T12 sin2 2a þ 2TT12 sin 2a cos 2a

)

2

2 Tt2 ¼ T sin2 2a þ T12 cos2 2a  2TT12 sin 2a cos 2a

Consequently, the components on an arbitrary plane is expressed by the point on the circle with the radius RT centering at ðTm ; 0Þ in the two-dimensional plane ðTn ; Tt Þ as shown in Fig. 1.5. This circle is called the Mohr’s circle. Substituting Eq. (1.335) into Eq. (1.330), we have the expressions 9  ¼ Tm þ RT cosð2a  2ap Þ; > T11 =

 T22 ¼ Tm  RT cosð2a  2ap Þ; > ;  T12 ¼ RT cosð2a  2ap Þ

ð1:341Þ

  Therefore, T11 and T12 are shown by the values in the ordinate and abscissa axes, respectively, of the point rotated 2a counterclockwise from point T11 ; T12 on the Mohr’s circle as shown in Fig. 1.5, provided that the definition for the sign of shear component is altered to be positive when it applies to the body surface in the clockwise direction, in the Mohr’s circle. As shown in Fig. 1.5, the intersecting angle of the two straight lines drawn in parallel to the physical plane x1 and x1 stemming from the points ðT11 ; T12 Þ and   ðT11 ; T12 Þ, respectively, on the Mohr’s circle is a which is the angle of circumference of Mohr’s circle and thus the intersecting point lies on the circle. This point is called the pole. Generally speaking, the normal component Tn and the shear component Tt applying to a certain physical plane are given by the intersecting point of Mohr’s circle and the straight line drawn parallel to that physical plane from the pole P.

1.14

Two-Dimensional State

1.15

65

Tensor Functions

The following various tensor functions are defined. sin T ¼

1 X ð1Þn n 1 1 TP eP  eP ¼ ðTP  TP3 þ TP5     ÞeP  eP ð2n þ 1Þ! 3! 5! n¼0

1 X ð1Þn 2n þ 1 1 1 T ¼ T T3 þ T5     ¼ ð2n þ 1Þ! 3! 5! n¼0

cos T ¼

  1 1 TP2n eP  eP ¼ 1 TP2 þ TP4     eP  eP ð2nÞ! 2! 4!

1 X ð1Þn n¼0

1 X ð1Þn 2n 1 1 T ¼ I T2 þ T4     ¼ ð2nÞ! 2! 4! n¼0

exp T ¼

1 X 1 n 1 1 TP eP  eP ¼ ð1 þ TP þ TP2 þ TP3 þ n! 2! 3! n¼0

1 1 ¼ I þ T þ T2 þ T 3 þ 2! 3! log T ¼

1 X ð1Þn1 n¼1

ð1:342Þ

n





ð1:343Þ

ÞeP  eP

1 X 1 n T ¼ n! n¼0

ð1:344Þ

 1 1 ðTp  1Þn eP  eP ¼ ðTp  1Þ  ðTp  1Þ2 þ ðTp  1Þ3     eP  eP 2 3

1 X 1 1 ð1Þn1 ¼ ðT  IÞ  ðT  IÞ2 þ ðT  IÞ3     ¼ ðT  IÞn 2 3 n n¼1

ð1:345Þ These functions satisfy the following basic properties noting Eqs. (1.207) and (1.344). ðexp TA ÞT ¼ ðexp TAT Þ ¼ expðTA Þ

ð1:346Þ

ðexp TÞT ¼ expðTT Þ

ð1:347Þ

expðnTÞ ¼ ðexp TÞn

ð1:348Þ

expðA þ BÞ ¼ exp A exp B

ð1:349Þ

det½expðA þ BÞ ¼ expðtrAÞ expðtrBÞ

ð1:350Þ

ðexp TA Þðexp TA ÞT ¼ ðexp TA Þðexp TAT Þ ¼ expðTA TA Þ ¼ I

ð1:351Þ

66

1

Mathematical Preliminaries: Vector and Tensor Analysis

noting 1 X 1 ðA þ BÞn n! n¼0

¼

1 1 1 X X 1 n X 1 n 1 n n1 n A Þð AÞ ðA þ nAn1 B þ    þ nAB þ B Þ ¼ ð n! n! n! n¼0 n¼0 n¼0

det½exp T ¼ ð exp TÞ1 ðexp TÞ2 ðexp TÞ3 ¼ exp T1 exp T2 exp T3 ¼ expðT1 þ T2 þ T3 Þ for Eqs. (1.349) and (1.350). For tensor functions of diagonalizable tensors, one can obtain alternative closed-form expressions for the tensor functions and their derivatives in the spectral representations. Numerical computation of the exponential and logarithmic functions of a tensor, as well as their derivatives, can be referred to de Souza Neto (2001), Ortiz et al. (2001), Itskov and Aksel (2002), Itskov (2003, 2019), etc.

1.16

Partial Differential Calculi

Partial derivatives of symmetric tensors appearing often in elastoplasticity are shown this section. (1) Power functions n @Tn X ~ þ Tn1 I ~ ~ n1 þ TT ~ n2 þ    þ Tn2 T ~ nk ¼ IT Tk1 T ¼ @T k¼1

 @Tn  @T

ijkl

¼

n X

ðTk1 Þik ðTnk Þlj

ð1:352Þ

k¼1

Examples for low-order tensors are shown below. @Tij ¼ dik djl @Tkl @ðTir Trj Þ ¼ dik drl Trj þ Tir drk djl ¼dik Tlj þ Tik djl @Tkl @ðTir Trs Tsj Þ ¼ dik drl Trs Tsj þ Tir drk dsl Tsj þ Tir Trs dsk djl ¼dik Tls Tsj þ Tik Tlj þ Tir Trk djl @Tkl

ð1:353Þ

1.16

Partial Differential Calculi

67

using the symbol in Eq. (1.196)3 . (2) Invariants @ðTrs drs Þ ¼ dir djs drs ¼ dij @Tij @ðTrs Tsr Þ ¼ dri dsj Tsr þ Trs dsi drj ¼ 2Tji @Tij @ðTrs Tst Ttr Þ ¼ dri dsj Tst Ttr þ Trs dsi dtj Ttr þ Trs Tst dti drj ¼Tjt Tti þ Tri Tjr þ Tjs Tsi ¼ 3Tjt Tti @Tij

@Tij0 ¼ I 0ijkl @Tkl 

@Tij0 @ðTij  Tm dij Þ 1 ¼ ¼ dik djl  dij dkl @Tkl @Tkl 3

@trT @I @I ¼ ¼ ¼ I; @T @T @T

@trT2 @ II ¼ ¼ 2TT ; @T @T



@trT3 @ III ¼ ¼ 3TT 2 ð1:354Þ @T @T

2 @II @ 12 ððtrTÞ  trT2 Þ ¼ I I  TT ¼ @T @T @III @ detT ¼ ¼ ðdetTÞTT ¼ II I  I TT þ T2T ¼ ðII I  IT þ T2 ÞT @T @T ð1:355Þ

noting Eq. (1.163) and h1 1 1 3i 3 2 @III @ 6 ðtrTÞ  2 trTtrT þ 3 trT 3 1 1 1 ¼ ðtrTÞ2 I  ðtrT2 ÞI  ðtrTÞ2TT þ 3TT2 ¼ @T @T 6 2 2 3 1 2 2 T T2 ¼ ½ðtrTÞ  trT I  ðtrTÞT þ T 2

Then, the following equations are derived.

and

pffiffiffiffiffiffiffiffiffiffi @ detT 1 pffiffiffiffiffiffiffiffiffiffi T ¼ detTT @T 2

ð1:356Þ

pffiffiffiffiffiffiffiffiffiffi @ ln detT 1 1 ¼ T 2 @T

ð1:357Þ

! pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi @ ln det T 1 @ det T 1 1 ¼ pffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi ðdet TÞT1 @T det T @T det T 2 det T

68

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Mathematical Preliminaries: Vector and Tensor Analysis

Further one has pffiffiffiffiffiffiffiffiffiffiffi @ Trs Trs 1 Tij @jjTjj T ¼ ðTrs Trs Þ1=2 ðdri dsj Tsr þ Trs dri dsj Þ ¼ pffiffiffiffiffiffiffiffiffiffiffi ; ¼ @Tij 2 @T jjTjj Trs Trs



@



ð1:358Þ

ð1:359Þ

pffiffiffiffiffiffiffiffiffiffiffi Trs0 Trs0 1 0 0 1=2 @ðTrs0 Trs0 Þ 1 0 0 1=2 ¼ ðTrs Trs Þ ¼ ðTrs Trs Þ 2dir djs Trs0 ¼ tij0 ; 0 @Tij0 2 2 @Tij ð1:360Þ

(3) Deviatoric tensors

@tij0 @Tkl0

¼

Tij0 @ pffiffiffiffiffiffiffiffiffiffiffiffiffi 0 T0 Tpq pq @Tkl0

dik djl ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffi Tkl0 0 T 0  T 0 pffiffiffiffiffiffiffiffiffiffiffiffiffi Tpq pq ij 0 T0 Tpq pq 0 T0 Tpq pq

1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi ðdik djl  tij0 tkl0 Þ; 0 T0 Tpq pq



ð1:361Þ

@tij0 @tij0 @Trs0 1 1 0 ¼ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi Þðdrk dsl  drs dkl Þ ðdir djs  tij0 trs 0 T0 @Tkl @Trs0 @Tkl 3 Tpq pq 1 1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi ðdik djl  dij dkl  tij0 tkl0 Þ; 0 T0 3 Tpq pq



 cos 3h 

 pffiffiffi 0 3 6trt

ð1:362Þ ð1:363Þ

pffiffiffi 0 0 0 pffiffiffi pffiffiffi tqr trp @ cos 3h @ 6tpq 0 0 ¼ ¼ 3 6dip djq tqr trp ¼ 3 6tir0 trj0 ; 0 0 @tij @tij pffiffiffi @ cos 3h ¼ 3 6t 0 2 0 @t

ð1:364Þ

1.16

Partial Differential Calculi

69

0 pffiffiffi 0 0 1 @ cos 3h @ cos 3h @trs 0 0 ¼ ¼ 3 6trp tps 0 ðdri dsj  trs tij Þ 0 @Tij0 @Tij0 @trs jjT jj 3 pffiffiffi ¼ 0 ð 6tit0 ttj0  cos 3htij0 Þ jjT jj

@ cos 3h 3 pffiffiffi ¼  0 ð 6t0  cos 3ht0 Þ 0 @T jjT jj

ð1:365Þ

0 @ cos 3h @ cos 3h @trs ¼ 0 @Tij @Tij @trs   pffiffiffi 1 1 0 0 d ¼ 3 6trt0 tts0 pffiffiffiffiffiffiffiffiffiffiffiffiffi d  d  t t d ir js ij rs ij rs 0 T0 3 Tpq pq   pffiffiffi 1 1 0 0 0 0 0 0 0 0 t t  t d  t t t t t ¼ 3 6 pffiffiffiffiffiffiffiffiffiffiffiffiffi ij ir rj rs st tr ij ; 0 T0 3 rt tr Tpq pq

pffiffiffi   @ cos 3h 3 6 0 2 1 1 0 2 0 ¼ 0 t  pffiffiffi cos 3ht  jjt jj I @T jjT jj 3 6

ð1:366Þ

(4) Various tensor functions Differentiating Tir1 Trs ¼ dis , one has @ðTir1 Trs Þ @Tir1 @T 1 ¼ Trs þ Tir1 drk dsl ¼ 0 ! ir Trs Tsj1 þ Tir1 drk dsl Tsj1 ¼ 0 @Tkl @Tkl @Tkl 





d ij ¼ ðTir1 Trs Þ ¼ T

1 ir



Trs þ Tir1 T

rs

¼0

which leads to



ð1:367Þ

Further, one has 1 n 1 n  @ exp T  X @ exp T X 1X 1X ~ nk ; ¼ Tk1 T ¼ ðTk1 Þik ðTnk Þlj ijkl @T n! k¼1 @T n! k¼1 n¼1 n¼1

ð1:368Þ

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Mathematical Preliminaries: Vector and Tensor Analysis

noting 1 2 1 3 @ exp T @ðI þ T þ 2! T þ 3! T þ Þ ¼ @T @T due to Eq. (1.344) with Eq. (1.352). One should notice @½f ðAÞBij  @f ðAÞ @Bij ¼ Bij þ f ðAÞ @Tkl @Tkl @Tkl @½f ðAÞB @f ðAÞ @B  @f ðAÞ @B  ¼B þ f ðAÞ  B þ f ðAÞ 6¼ @T @T @T @T @T For symmetric tensors, the fourth-order identity tensor  replaced to the fourth-order symmetrizing tensor  equations.

1.17

ð1:369Þ can be in the above

Differentiation and Integration in Tensor Field

Scalar s, vector v, and tensor T are called the scalar field, the vector field, and the tensor field, respectively when they are functions of the position vector x. Their differentiation and integration in fields are shown below, in which the following operator, called the nabra or Hamilton operator, is often used. $

@ @ er ¼ @xr @x

ð1:370Þ

(1) Gradient Scalar field : grads ¼ $s ¼

@s er @xr

8 @ @vi > > ej ¼ ei  ej : rear (right) form < v  $ ¼ vi ei  @xj @xj Vector field : gradv ¼ @ @vj > > :$  v ¼ ei  vj ej ¼ ei  ej : front (left) form @xi @xi

ð1:371Þ

ð1:372Þ

1.17

Differentiation and Integration in Tensor Field

71

Second-order tensor field: 8 @ @Tij > > ek ¼ ei  ej  ek : rear (right) form < T  $ ¼ Tij ei  ej  @xk @xk grad T ¼ > @ @Tjk > :$ T ¼ ei  Tjk ej  ek ¼ ei  ej  ek : front (left) form @xi @xi ð1:373Þ (2) Divergence Vector field: divv ¼ $  vð ¼ v  $Þ ¼ vi ei 

@ @vi ej ¼ @xi @xj

ð1:374Þ

Second-order tensor field: 8 @ @Tir > > ek ¼ ei : rear (right) form < T$ ¼ Tij ei  ej @xr @xk divT ¼ > @ @Tri > : $T ¼ ei Tjk ej  ek ¼ ei : front (left) form @xr @xi

ð1:375Þ

(3) Rotation (or curl) Vector field:

rotv ¼

8 @ @vi > > < v  $ ¼ vi ei  @x ej ¼ eijk @x ek : rear (right) form j

j

> > : $  v ¼ @ ei  vj ej ¼ eijk @vj ek : front (left) form @xi @xi

ð1:376Þ

noting Eq. (1.35). Second-order tensor field: 8 @ > > T  $ ¼ Tij ei  ej  ek > > > @xk > > > > > @Tij @Tij > > ¼ ei  ðej  ek Þ ¼ ejkr ei  er : rear (right) form < @xk @xk rotT ¼ > @ > > ei  Tjk ej  ek $T¼ > > @x > i > > > > @Tjk @Tjk > > ¼ ðei  ej Þ  ek ¼ eijr er  ek : front (left) form : @xi @xi ð1:377Þ

72

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Mathematical Preliminaries: Vector and Tensor Analysis

The symbol $ is regarded as a vector, and the scalar product of itself, i.e. D  r2  $  $ ¼

@ @ @2 er  es ¼ @xr @xr @xr @xs

ð1:378Þ

has the meaning of r2 ð Þ  divðgradð ÞÞ. The symbol D is called the Laplacian or Laplace operator, which is often used for scalar or vector fields as @2s @xr @xr

ð1:379Þ

@ 2 vs es @xr @xr

ð1:380Þ

Ds ¼ Dv ¼

The following relations hold between the above-mentioned operators. 9 gradðsvÞ ¼ v  grads þ sgradv; > > > > divðsvÞ ¼ sdivv þ v  grads; > > > > divðu  vÞ ¼ v  rotu  u  rotv; > > = rotðu  vÞ ¼ ðgraduÞvðgradvÞu þ ðdivvÞuðdivuÞv; ð1:381Þ gradðu  vÞ ¼ ðgradvÞu þ ðgraduÞv þ u  rotv þ v  rotu; > > > T > divðsTÞ ¼ T grads þ sdivT; > > > > > divðTvÞ ¼ v  divT þ trðTgradvÞ > ; T divðABÞ ¼ B  divA þ divA  B (4) Gauss’ divergence theorem Consider the physical quantity TðxÞ in the zone surrounded by a smooth surface inside a material. Then, suppose the slender prism cut by the four planes perpendicular to the x2 -axis and x3 -axis in infinitesimal intervals from a zone inside the material. The following equation holds for the prism possessing the infinitesimal volume @v ¼ dx1 dx2 dx3 . Z

@T dv ¼ @x 1 @v

Z

@T xþ dx1 dx2 dx3 ¼ ½Tx11 dx2 dx3 ¼ ðT þ T  Þdx2 dx3 @v @x1

ð1:382Þ

where ðÞ þ and ðÞ designate the values of physical quantity at the maximum and the minimum x1 -coordinates, respectively. The neighborhood of the surface cut by the prism is magnified in Fig. 1.6. Consider the infinitesimal rectangular surface PQRS of the prism exposed at the surface in the maximum x1 -coordinate and the infinitesimal rectangular section PQ R S cut by the plane passing through the point P and perpendicular to the x1 



!

!

axis by the prism. Then, denoting QQ ¼ dxQ ; SS ¼ dxS , the vectors PQ; PS are given by

1.17

Differentiation and Integration in Tensor Field

73

S*

R* dx

dx2

1

R

dx s S

e3

n

dx3

e2

Q*

0

dx Q

Q

P

e1

Fig. 1.6 Infinitesimal square pillar cut from a zone in material

! PQ ¼ dx2 e2 þ dxQ e1 ;

!

PS ¼ dx3 e3 þ dxS e1

ð1:383Þ

and thus it holds that !

!

n þ da þ ¼ PQ  PS ¼ ðdx2 e2 þ dxQ e1 Þ  ðdx3 e3 þ dxS e1 Þ ¼ dx2 dx3 e2  e3 þ dxQ dx3 e1  e3 þ dxS dx2 e2  e1

ð1:384Þ

¼ dx2 dx3 e1  dxQ dx3 e2  dxS dx2 e3 Comparing the components in the base e1 on the both sides in Eq. (1.384), one has n1þ da þ ¼ dx2 dx3

ð1:385Þ

In a similar manner for the surface of the prism exposed on surface in the minimum x1 -coordinate, one has  n 1 da ¼ dx2 dx3

ð1:386Þ

The general expression of projected area is given in Appendix A. Adopting Eqs. (1.385) and (1.386) in Eq. (1.382), it holds for the prism that Z @T  þ þ þ    dv ¼ T þ n1þ da þ T  ðn ð1:387Þ 1 da Þ ¼ T n1 da þ T n1 da @v @x1 Then, the following equation holds for the whole zone. Z v

@T dv ¼ @x1

Z a

Tn1 da

ð1:388Þ

In a similar manner also for the x2 - and x3 -directions, the following Gauss’ divergence theorem holds.

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1

Mathematical Preliminaries: Vector and Tensor Analysis

Z v

@T dv ¼ @xi

Z Tni da

ð1:389Þ

a

The following equations for the scalar s, the vector v and the tensor T hold from Eq. (1.389).

1.18

Ð Ð Ð Ð @s v @x dv ¼ a sni da; v $sdv ¼ a snda i

ð1:390Þ

Ð @vi Ð Ð Ð v @x dv ¼ a vi ni da; v $  vdv ¼ a v  nda i

ð1:391Þ

Ð @Tij Ð Ð Ð v @x dv ¼ a Tij ni da; v $Tdv ¼ a Tnda i

ð1:392Þ

Representation in General Coordinate System

The physical meanings of finite strain (rate) tensors are captured clearly by their representations in the embedded (convected) coordinate system, since the components in the primary bases are kept constant. The embedded coordinate system changes to a curvilinear coordinate system even if it is the rectangular coordinate system at the initial state of deformation process. Then, mathematics in the general curvilinear coordinate system with the primary and the reciprocal base vectors, the representations and the coordinate transformations of vector and tensor, the metric tensor, the scalar, the vector and the tensor products, etc. are addressed in this section.

1.18.1

Primary and Reciprocal Base Vectors

The following reciprocal base vectors g1 ; g2 ; g3 are defined by the primary base vectors g1 ; g2 ; g3 as follows: gi ¼ 12 eijk

gj  gk ; vg

i:e:

g1 ¼

g2  g3 2 g3  g1 3 g1  g2 ;g ¼ ;g ¼ ð1:393Þ ½g1 g2 g3  ½g1 g2 g3  ½g1 g2 g3 

noting Eq. (1.59) and e1jk gj  gk ¼ 2g2  g3 for example, where vg is the volume of parallelepiped made up of the primary vectors g1 ; g2 ; g3 in the right hand coordinate system, i.e.

1.18

Representation in General Coordinate System

vg  ½g1 g2 g3 

75

ð1:394Þ

noting Eq. (1.47). The inverse expression of Eq. (1.393) is given by 1 gi ¼ eijk vg g j  gk ; 2

i:e:

g1 ¼

g2  g3 g3  g1 g1  g2 ; g ; g ¼ ¼ 2 3 ½g1 g2 g3  ½g1 g2 g3  ½g1 g2 g3  ð1:395Þ

noting Eq. (1.59), where it is satisfied noting Eq. (1.63)3 that ½g1 g2 g3 ½g1 g2 g3  ¼ 1

ð1:396Þ

1 ¼ ½g1 g2 g3  vg

ð1:397Þ

leading to

Here, note that the primary and the reciprocal basis in Eqs. (1.393) and (1.395) satisfy the following relations. gi  g j ¼ dij ; gi  gi ¼ 3

ð1:398Þ

gi  gi ¼0

ð1:399Þ

which are equivalent to Eqs. (1.63)1; 2 and (1.65). Now, we consider the general curvilinear coordinate systems fhi g with the primary base fgi g and its reciprocal base fgi g. The infinitesimal line-element dx at the material point vector x is described as follows: ð1:400Þ dx ¼ dxi ei ¼ dhi gi where dxi ¼ dx  ei ; dhi ¼dx  gi

ð1:401Þ

by virtue of Eq. (1.62). It follows from Eq. (1.400) with Eq. (1.398) that dx  dx ¼ dxi dxi ¼ dhi dh j gi  gj

ð1:402Þ

Further, one has the expressions dx ¼

@x @x dxi ¼ i dhi @xi @h

ð1:403Þ

76

1

Mathematical Preliminaries: Vector and Tensor Analysis

Then, the following expressions are derived from Eqs. (1.400) and (1.403). ei ¼

@x ; @xi

gi ¼

@x @hi

ð1:404Þ

The reciprocal base vector gk is always perpendicular to the coordinate plane formed by the primary base vectors ðgi ; gj Þðk 6¼ i; k 6¼ jÞ. Note, however, that fgi g and fhi g are merely defined momentarily from the primary base vector fgig. Therefore, a continuous coordinate system fhi g with the reciprocal vector base fgi g does not exist in actuality when the coordinate system fhi g with the primary base fgig is a curvilinear coordinate system.

1.18.2

Metric Tensor and Base Vector Algebra

Now, we introduce the scalar quantities gij  gi  gj ; gij  gi  g j ¼ dij ; gij  gi  gj ¼ dij ; gij  gi  g j

ð1:405Þ

noting Eqs. (1.398), where the symmetries gij ¼ gji ; gij ¼ gji ; gij ¼ gij hold. Here, note that gij and gij are necessary to express the length of the line-element and the angle between line-elements which are most basic geometrical ingredients. Then, they are called the metric tensors. It follows from Eq. (1.405) that gir grj ¼ dij ;

½gir ½grj  ¼ ½gir grj  ¼ ½dij 

ð1:406Þ

noting gir grj ¼ ðgi  gr Þðgr  g j Þ ¼ gi  g j ¼ dij or gir grj ¼

@hr @h j @h j ¼ i ¼ dij @hi @hr @h

Hence, we have gij ¼ g1 ij ;

½gij  ¼ ½gij 1

ð1:407Þ

1.18

Representation in General Coordinate System

77

from which it follows that detðgij Þ detðgij Þ ¼1 (no sum); detðgij Þ ¼ 1= detðgij Þ

ð1:408Þ

and g  detðgij Þ ¼ ½g1 g2 g3 2 ¼ v2g ;

detðgij Þ ¼ 1=g ¼ 1=v2g

ð1:409Þ

noting Eqs. (1.17) or (1.143) 4 , (1.154) and (1.157). Equation (1.107) means that the matrices with the components gij and gij are inverse tensors to each other, while this property is based on their non-singularities, i.e., ½gij  6¼ 0 and ½gij  6¼ 0. By use of the notation in Eq. (1.405), the following transformation rules between the primary and the reciprocal base vectors hold. gi ¼ gir gr ;

gi ¼ gir gr

ð1:410Þ

It is known from Eq. (1.410) that gij and gij play the role of the shifter which makes the index go up and down, respectively. The generalized identity tensor in the general coordinate system is given from Eq. (1.72) with Eq. (1.405) as follows: gð¼ ei  ei Þ ¼ gi  gi ¼ gi  gi ¼ gij gi  gj ¼ gij gi  g j ð¼ gT Þ

ð1:411Þ

fulfilling pffiffiffi jjgjj ¼ 3; detg ¼ 1; g ¼ gT ¼ g1

ð1:412Þ

The fact that the tensor g is the generalized identity tensor is confirmed noting Eq. (1.72) as follows: ( gv ¼ ( gT ¼

gi  gi v ¼ ðv  gi Þgi ¼ vi gi gi  gi v ¼ ðv  gi Þgi ¼ vi gi

) ¼v

gi  gi Trs gr  gs ¼ dir Trs gi  gs ¼ Tis gi  gs 





gi  gi Tr s gr  gs ¼ dri Tr s gi  gs ¼ Ti s gi  gs

ð1:413Þ

) ¼T

ð1:414Þ

The components of g shown in Eq. (1.411) are necessary for the description of the geometrical elements, e.g., the length of line-element and the angle between line elements in the general space as will be shown in Subsection 1.18.3. Then, it is called the metric tensor. Further, Eqs. (1.393) and (1.395) are written as

78

1

Mathematical Preliminaries: Vector and Tensor Analysis

1 gi ¼ eijk gj  gk ; 2

1 gi ¼ eijk g j  gk 2

ð1:415Þ

and one has the following equations from Eqs. (1.394) and (1.397) ½gi gj gk  ¼ eijk ;

½gi g j gk  ¼ eijk

ð1:416Þ

eijk vg

ð1:417Þ

eijk ¼ gir gjs gkt erst

ð1:418Þ

introducing the symbols eijk  eijk vg ; eijk  which satisfy the following transformation rules. eijk ¼ gir gjs gkt erst ; noting eijk ¼ vg detðgpq Þeijk ¼ vg gir gjs gkt erst ¼ gir gjs gkt vg erst vg vg eijk ¼

1 1 1 detðgpq Þeijk ¼ gir gjs gkt erst ¼ gir gjs gkt erst vg vg vg

with the aid of Eqs. (1.15) and (1.409). The following relations hold by virtue of Eq. (1.21). 8 ijk > < e eirs ¼ ejki ersi ¼ djr dks  djs dkr eijk eijs ¼ 2dks > : ijk e eijk ¼ 3! ¼ 6

ð1:419Þ

Further, the vector products of the both primary bases or reciprocal bases are given by gi  gj ¼ eijk gk ; gi  g j ¼ eijk gk

ð1:420Þ

noting 8 1 1 1 > < gi  gj ¼ ðgi  gj gj  gi Þ ¼ ðdir djs dis djr Þgr  gs ¼ ekij ekrs gr  gs ¼ ekij gk 2 2 2 > : gi  g j ¼ 1 ðgi  g j g j  gi Þ ¼ 1 ðd d d d Þgr  gs ¼ 1 ekij e gr  gs ¼ ekij g ir js is jr krs k 2 2 2

by virtue of Eqs. (1.39), (1.415) and (1.419).

1.18

Representation in General Coordinate System

1.18.3

79

Tensor Representations

The vector is described by Eq. (1.62) as v ¼ ðv  gi Þgi ¼ ðv  gi Þgi ð¼ gi  gi v ¼ gi  gi vÞ

ð1:421Þ

v ¼ vi g i ¼ v i g i

ð1:422Þ

leading to

with v i ¼ v  gi ;

vi ¼v  gi

ð1:423Þ

where vi and vi are called the contravriant component and the covariant component, respectively. The relation of vi and vi are given as vi ¼gir vr ; vi ¼gir vr

ð1:424Þ

noting v i ¼ v r gr  gi ;

vi ¼ v r gr  gi

The metric tensor component plays the role of shifter as known from Eq. (1.424). The scalar, the vector and the tensor products are given by Eqs. (1.420) and (1.422) as u  v ¼ ui vi ¼ ui vi ¼ gij ui v j ¼ gij ui vj jjvjj ¼

qffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi v  v ¼ vi vi ¼ gij vi v j ¼ gij vi vj

ð1:425Þ ð1:426Þ

u  v ¼ eijk ui v j gk ¼ eijk ui vj gk

ð1:427Þ

ðu  vÞ  w ¼ eijk ui v j wk ¼ eijk ui vj wk

ð1:428Þ

u  v ¼ ui v j gi  gj ¼ ui vj gi  g j ¼ ui v j gi  gj ¼ ui vj gi  g j

ð1:429Þ

The square of the length of an infinitesimal line-element vector dx is expressed using the metric tensor by Eq. (1.426) as follows: jjdxjj2 ¼ dx  dx ¼ gij dhi dh j ¼ gij dhi dhj ¼ dhi dhi

ð1:430Þ

80

1

Mathematical Preliminaries: Vector and Tensor Analysis

noting dx  dx ¼ dhi gi  dh j gj ¼ dhi gi  dhj g j ¼ dhi gi  dhj g j The infinitesimal surface vector dak of the surface formed by the line-elements dhi gi (no sum) and dh j gj (no sum) along the primary base fgi g is given from Eq. (1.420) by dak ¼ dhi gi  dh j gj ¼ eijk dhi dh j gk

ð1:431Þ

(no sum)

where i; j; k are different from each other and are ordered in an even permutation, which is called the surface element vector. The volume dV of the infinitesimal parallelepiped formed by the sides along the primary base fgi g is given by dv ¼ ½dh1 g1 dh2 g2 dh3 g3  leading to dv ¼ ½g1 g2 g3 dh1 dh2 dh3 ¼ vg dh1 dh2 dh3

ð1:432Þ

Following the tensor product of two vectors in Eq. (1.429), the second-order tensor T is defined as follows: 

T ¼ T ij gi  gj ¼ Tij gi  g j ¼ Ti j gi  gj ¼ Tij gi  g j

ð1:433Þ

where the components of T are derived from as follows: T ij ¼ gi  Tg j ; Tji ¼ gi  Tgj ; Tij ¼ gi  Tg j ; Tij ¼ gi  Tgj

ð1:434Þ 

exploiting Eq. (1.398)1 . Here, we cannot write Tji because of Tij 6¼ Ti j . Then, the relations between the components in the different variance are given by substituting Eq. (1.433) into Eq. (1.434) as follows: (



i rj g ¼ gir Tr j ¼ gir Trs gsj ; T ij ¼ Tr 

Ti j ¼ gir T rj ¼ gir Trs gsj ¼ Tir grj ;



Tij ¼ T ir grj ¼ gir Tr s gsj ¼ gir Trj 

Tij ¼ gir T rs gsj ¼ gir Trj ¼ Ti r grj 

ð1:435Þ

Analogously to the vector described in the foregoing, T ij , Tij , Ti j and Tij are called the contravariant, the contravariant-covariant, the covariant-contravariant and the covariant components, respectively. The scalar product of second-order tensors A and B is expressed in the components as follows:

1.18

Representation in General Coordinate System

81

j i ij ij rs js A : B ¼ Aij Bij ¼ Aij Bj i ¼ Ai Bj ¼ Aij B ¼ gir gjs A B ¼ gir g Aij Brs

ð1:436Þ

noting 8 ij ðA gi  gj Þ:ðBrs gr  gs Þ ¼ ðgi  gr Þðgj  gs ÞAij Brs > > > > i j s r j i s > >  r  > > ðAj gi  g Þ:ðBr g  gs Þ ¼ ðgi g Þðg gs ÞAj Br > > < ðAj gi  g Þ:ðBr g  gs Þ ¼ ðgi  g Þðg  gs ÞAj Br s r j r j i i s i j rs i s > ðAij g  g Þ:ðB gr  gs Þ ¼ ðg  gr Þðgj  g ÞAij Brs > > > > > > ðAij gi  gj Þ:ðBrs gr  gs Þ ¼ gir gjs Aij Brs > > > : ðAij gi  g j Þ:ðBrs gr  gs Þ ¼ gir gjs Aij Brs : The magnitude of tensor T is given by setting A ¼ B ¼ T in Eq. (1.436) as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi  jjTjj ¼ T : T ¼ T ij Tij ¼ T ij Ti j ¼ gir gjs T ij T rs

ð1:437Þ

The definition of the trace trT  T : I in the Cartesian space is extended as the scalar product with the metric tensor in the Riemann space, leading to the sum of the diagonal components of the mixed variant form, i.e. 

trT  T : g ¼ g : T ¼ T ii ¼ Tii ¼ Ti i ¼ Tii

ð1:438Þ

noting 8 i ðg  gi Þ:ðT rs gr  gs Þ ¼ T ii > > > > < ðgi  g Þ:ðT r g  gs Þ ¼ T i s r i i s r i > ðgi  g Þ:ðTr g  gs Þ ¼ Tii > > > : ðgi  gi Þ:ðTrs gr  gs Þ ¼ Tii where use is made of Eq. (1.120)7 . The following relations hold. 8 i g  ðT rs gr  gs Þg j ¼ T ij ¼ g j  ðT rs gs  gr Þgi > > > > < gi  ðT r g  gs Þg ¼ T i ¼ g  ðT r g  gr Þgi j j s r j s s > gi  ðTrs gr  gs Þg j ¼ Tij ¼ g j  ðTrs gs  gr Þgi > > > : i g  ðTrs gr  gs Þg j ¼ Trs ¼ g j  ðTrs gs  gr Þgi Comparing these relations with Eq. (1.130), the transposed tensor TT is given by

82

1

Mathematical Preliminaries: Vector and Tensor Analysis



TT ¼T ji gi  gj ¼ Tj i gi  g j ¼ Tji gi  gj ¼ Tji gi  g j

ð1:439Þ

Therefore, the transposed tensor is given by exchange of the front and the rear base vectors. However, it is not obtained by the exchange of front and rear suffices in components, differing from the tensor expressions in the Cartesian coordinate system, as known from ( T ¼ T



Tj i g i  g j

)

( 6¼

Tji gi  gj

Tji gi  g j 

Tj i g i  g j

ð1:440Þ

If T is the symmetric tensor, it follows from Eqs. (1.433) and (1.439) that 



T ij ¼ T ji ; Tij ¼ Tj i ðTi j ¼ Tji Þ;

Tij ¼ Tji

ð1:441Þ

However, the indices cannot be exchanged between the contravariant and the covariant ones, i.e. 



Tij 6¼ Tji ; Ti j 6¼ Tj i

ð1:442Þ

Therefore, the mixed components Tij and Tji possess the nine independent values even in the symmetric tensor, leading to the inconvenience of analysis.

Chapter 2

Description of Motion

The tensor analysis providing the mathematical foundation for the continuum mechanics was described in Chap. 1. Basic concepts and quantities for continuum mechanics will be studied in the four chapters up to Chap. 6. The appropriate descriptions of motion of basic material elements are of importance in the formulation of constitutive relations. The variations of some basic material elements during the deformation will be studied in this chapter prior to the explanation of the deformation/rotation measures in the subsequent chapter.

2.1

Motion of Material Point

A material body is assembly of material particles (or material elements). The map of positions of material particles in a space is referred to as the configuration. Here, the configurations in the initial time t ¼ t0 and the current time t are called the initial (or Lagrangian) configuration and the current (or Eulerian) configuration, respectively. Deformation is described by the change of configuration from a particular configuration which is called the reference configuration. Here, the reference configuration can be chosen at arbitrary intermediate time sðt0  s  tÞ which is called the reference time. The position vectors of material particle in the initial and the current configurations are designated by X and xðtÞ, respectively. Here, X is fixed and thus it can be regarded as a label of each material particle. The motion of material point during the time t0 ! t is described as x ¼ vðX; tÞ;

X ¼ v1 ðx; tÞ

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 K. Hashiguchi, Foundations of Elastoplasticity: Subloading Surface Model, https://doi.org/10.1007/978-3-030-93138-4_2

ð2:1Þ

83

84

2

Description of Motion

Incidentally, the motion of material point during the time t0 ! s is described as xðsÞ ¼ vðX; sÞ;

X ¼ v1 ðxðsÞ; sÞ

ð2:2Þ

The fact that a material point does not overlap or separate by the motion of material requires the existence of the one-to-one correspondence between X and x (x is uniquely determined by X and vice versa) so that x1 ðX1 ; X2 ; X3 Þ; x2 ðX1 ; X2 ; X3 Þ and x3 ðX1 ; X2 ; X3 Þ must be mutually independent. Now, let the mathematical requirement for this fact be derived by the reductive absurdity. x1 ; x2 ; x3 are not mutually independent if they satisfy the constraint f ðx1 ðX1 ; X2 ; X3 Þ; x2 ðX1 ; X2 ; X3 Þ; x3 ðX1 ; X2 ; X3 ÞÞ ¼ 0

ð2:3Þ

from which it follows that 9 2 @x @f @x1 @f @x2 @f @x3 1 > þ þ ¼ 0> > > 6 @X1 @x1 @X1 @x2 @X1 @x3 @X1 > = 6 @x @f @x1 @f @x2 @f @x3 6 1 þ þ ¼ 0 ; i.e:6 > 6 @X2 @x1 @X2 @x2 @X2 @x3 @X2 > > 4 @x > @f @x1 @f @x2 @f @x3 > 1 ; þ þ ¼0 @X3 @x1 @X3 @x2 @X3 @x3 @X3

@x2 @X1 @x2 @X2 @x2 @X3

9 8 9 8 @x3 3> @f > > 0 > > > > > > > > > > > > @X1 7 @x1 > > > > > = < = < 7 @x3 7 @f ¼ 0 7 > > @X2 7> > > > > @x2 > > > > > > > > @x3 5> @f > > > ; ; : > : 0 @X3 @x3 ð2:4Þ

The equation J¼0

ð2:5Þ

must hold in order that @f =@x1 ; @f =@x2 ; @f =@x3 are determined uniquely on account of Eq. (1.169) regarding TiJ ¼ @xi =@XJ and vi ¼ @f =@xi , where J is defined by   @xi @x1 @x2 @x3 ¼ eIJK J  det @XJ @XI @XJ @XK

ð2:6Þ

and is called the functional determinant or Jacobian. In contrast, in order that they are mutually independent, it must hold that J 6¼ 0

ð2:7Þ

being free from the constraint in Eq. (2.3). The transformation between x and X is called the admissible transformation, if f1 ; f2 ; f3 in x1 ¼ f1 ðX1 ; X2 ; X3 Þ; x2 ¼ f2 ðX1 ; X2 ; X3 Þ, x3 ¼ f3 ðX1 X2 X3 Þ are single-valued and continuous functions, so that the Jacobian is not zero as shown in Eq. (2.7). Further, if the Jacobian is positive, a right-hand coordinate system is transformed to other right-hand one, and it is called the positive transformation. Inversely, if the Jacobian is negative, a

2.1 Motion of Material Point

85

right-hand coordinate system is transformed to a left-hand one, and it is called the negative transformation. Admissible and positive transformation with J [ 0 is assumed throughout this book. Physical quantity, say T, in the body changes generally with the position and the time. Physical quantity at current time is described TðX; tÞ in terms of the reference configuration X and the current time t. This type of description of mechanical state is called the Lagrangian (or material) description. On the other hand, the physical quantity at current time is described Tðx; tÞ in terms of the current configuration x and the current time t. This type of description of physical quantity is called the Eulerian description or spatial description. Further, the physical quantity at current time can be described in terms of the configuration xðsÞ at arbitrary reference time s and the current time t as   T v1 ðxðsÞ; sÞ; t ¼ s TðxðsÞ; tÞ

ð2:8Þ

() designating to choose the reference time s as s [ t0 . Equation (2.8) is called the relative description. Specifically, t TðxðtÞ; tÞ is called the updated Lagrangian description, where the reference configuration is taken as the current configuration, choosing the reference time as the current time, i.e. s ¼ t. In contrast, the description TðX; tÞ will be called the total Lagrangian description. s

2.2

Time-Derivatives

The time-derivative of the tensor in the spatial description @Tðx; tÞ @t

ð2:9Þ

describes the rate of the physical quantity at a certain spatial point and thus it is called the spatial-time (or local) derivative. In many cases of fluid mechanics, a motion and its history of individual particle from the initial state is immaterial and thus the spatial-time derivative is often adopted. In contrast, the time-derivative of the tensor in the material description @TðX; tÞ @t

ð2:10Þ

describes the rate of the physical quantity in a certain material particle and thus is called the material-time derivative. It is denoted by the symbol 

T

@TðX; tÞ DT @TðX; tÞ or  @t Dt @t

ð2:11Þ

86

2

Description of Motion

In solid mechanics, the rate of deformation and its history of individual material particle is required and thus the material-time derivative is used usually. The material-time derivative in Eq. (2.11) and the spatial-time derivative in Eq. (2.9) are related by 

T

 @Tðx; tÞ @Tðx; tÞ @Tij ðx; tÞ @Tij ðx; tÞ þ v ; Tij  þ vk @t @x @t @xk

ð2:12Þ

where v  @x=@t is the velocity vector of material particle. The first term in the right-hand side signifies the non-steady (or local time derivative) term describing the change with time at fixed spatial point and the second term signifies the steady (or convective) term describing the change due to the movement of material, which results from the existence of a spatial gradient of the physical quantity T. Rate-type constitutive equations for the irreversible deformation of solids, e.g. the viscoelastic, the elastoplastic and the viscoplastic deformation, must be described by the material-time derivative pursuing a material particle because they must describe the relation of physical quantities in each material particle. Here, it should be noticed that the material-time derivative of physical quantity describes the rate observed by moving in parallel with material particle as known from Eq. (2.11) which concerns only with the position vector of material particle and the time. Then, the objective time-derivative based on the rate of physical quantity observed by the coordinate system deforming/rotating with a material must be used for constitutive equations of solids as will be described in Chap. 6.

2.3

Variations and Rates of Geometrical Elements

Variations of line, surface and volume elements and their rates during the deformation are described in this section.

2.3.1

Deformation Gradient and Variations of Line, Surface and Volume Elements

Variations of line, surface and volume elements are described below. Relation of the current infinitesimal line element dx to the initial infinitesimal line element dX is represented as

2.3 Variations and Rates of Geometrical Elements

87

dx ¼ FdX; dX ¼ F1 dx

ð2:13Þ

where F

@x @X

ð2:14Þ

which is referred to as the deformation gradient tensor, for which it follows that 8 @x @xi > > ei  ej ¼ F¼ > > @X @Xj > >  T   > > > T @x T @xi @xi @xj > > ¼ ei  ej ¼ ej  ei ¼ ei  ej

> F1 ¼ ei  ej ¼ > > > @xj @x > >  T   > > @X T @Xi @Xj > > : FT ¼ ¼ ei  ej ¼ ei  ej @xj @xi @x

ð2:15Þ

noting FF1 ¼

@xi @Xr @xi @Xj ei  ej er  es ¼ ei  es ¼ dis ei  es ¼ I @Xj @xs @Xj @xs

Equation (2.15) is represented in the fixed coordinate system, while it is represented concisely by the embedded base vectors in the initial and the current configurations in the embedded coordinate system as will be described in the next chapter. The following expression holds by introducing the polar decomposition for the deformation gradient tensor in Eq. (1.273) into Eq. (2.13). dx ¼ RUdX ¼ VRdX;

dX ¼ U1 RT dx ¼ RT V1 dx

ð2:16Þ

The current position vector is related to the reference position vector by use of the displacement vector u as x ¼ X þ u ¼ ðXi þ ui Þei

ð2:17Þ

Then, the deformation gradient is represented as follows: @ðX þ uÞ @u F¼ ¼ Iþ ¼ @X @X



 @ui ei  ej dij þ @Xj

ð2:18Þ

88

2

Description of Motion

from which it follows that dF ¼

@du @dui ¼ ei  ej @Xj @X

ð2:19Þ

The current and the reference infinitesimal volume elements dv and dV formed by the infinitesimal line-elements dxa; dxb; dxc and dXa; dXb; dXc are related by       dv ¼ dxa dxb dxc ¼ FdXa FdXb FdXc ¼ det F dXa dXb dXc ¼ ðdet FÞdV ð2:20Þ by virtue of Eq. (1.254)3 . Then, the ratio of the current to the reference infinitesimal volume elements, i.e., Jacobian is given by J  det F ¼

m q0 ¼ q V

ð2:21Þ

where q0 and q are the mass densities in the reference and the current configurations, respectively. The following equalities hold for the infinitesimal volume elements, designating the infinitesimal reference and current surface element vectors as dA ¼ dXa  dXb and da ¼ dxa  dxb , respectively, noting Eq. (1.139).  dv ¼

da  dxc JdV ¼ JdA  dXc ¼ JdA  F1 dxc ¼ JFT dA  dxc

ð2:22Þ

where the area vectors da and dA in the current and the reference configurations are given by da ¼ dan;

dA ¼ dAN

ð2:23Þ

designating the infinitesimal area and the unit outward-normal as ðda; nÞ and ðdA; NÞ in the current and the reference configurations, respectively. The following relations are obtained from Eq. (2.22). da ¼ JFT dA ¼ ðcofFÞdA;

dA ¼ FT da=J ¼ ðcofFÞ1 da

ð2:24Þ

noting cofF ¼ JFT

ð2:25Þ

2.3 Variations and Rates of Geometrical Elements

89

by virtue of Eq. (1.160), leading to da ¼ FT N  nJdA;

dA ¼ FT n  Nda=J

ð2:26Þ

n ¼ FT NJdA=da;

N ¼ FT nda=ðJdAÞ

ð2:27Þ

Equation (2.26) is referred to as the Nanson’s formula. Here, the following Euler’s formula holds for the cofactor. rX ðcofFÞ ¼

@cofF ¼ 0; @X

@ðcofFÞap @Xp

¼

@ 12 eabc epqr Fbq Fcr ¼0 @Xp

ð2:28Þ

noting Eq. (1.13).

2.3.2

Velocity Gradient and Rates of Line, Surface and Volume Elements

Rates of line, surface and volume elements are described below. Differentiating Eq. (2.13), we have 

ðdxÞ ¼ F dx;

ðdxÞ ¼ dv ¼

@v dx ¼ ldx @x

ð2:29Þ

where l  gradv ¼ v  rx ¼

@v  1 ¼ FF @x

ð2:30Þ

noting 



@v @ x @ x @X ¼ ¼ @x @x @X @x

ð2:31Þ

The tensor l is called the velocity gradient tensor. The following expression holds        noting FF1 ¼ F F1 þ F F1 ¼ l þ F F1 ¼ O with the caution ðF1 Þ ¼ 



F1 F F1 6¼ ðFÞ1 ¼ F1 ½ðF1 Þ 1 F1 that   F F1 ¼ l

ð2:32Þ

90

2

Description of Motion

It follows from Eqs. (2.21) and (2.30) that 

trl ¼ divv ¼ J =J

ð2:33Þ



J ¼ Jtrl 

ð2:34Þ



ðdvÞ ¼ ð J =JÞdv ¼ J dV ¼ ðtrlÞdv

ð2:35Þ

noting 

 J ¼ ðdet FÞ ¼



 

@ det F  : F ¼ ðdet FÞFT : F ¼ ðdet FÞtr F1 F ¼ ðdet FÞtr F F1 ¼ Jtrl @F

or 

trl ¼

@vr @v1 @v2 @v3 @v1 @x2 @x3 þ @x1 @v2 @x3 þ @x1 @x2 @v3 ð@x1 @x2 @x3 Þ v ¼ þ þ ¼ ¼ ¼ @xr @x1 @x2 @x3 @x1 @x2 @x3 @x1 @x2 @x3 v

with Eqs. (1.163). Further, it follows from Eq. (2.25) that ðcofFÞ ¼ ~lcofF

ð2:36Þ

~l  ðtrlÞI  lT

ð2:37Þ

where

noting 



ðcofFÞ ¼ J F

T





þF

T



  J J J ¼ cofF þ FT FT cofF ¼ cofF  FT FT cofF J J

with the use of Eq. (1.160). ~l is referred to as the surface strain rate tensor. Further, differentiating Eq. (2.24) and noting Eq. (2.36), we have ðdaÞ ¼ ðcofFÞ dA ¼ ~lðcofFÞdA which is rewritten as ðdaÞ ¼ elda in the spatial description.

ð2:38Þ

2.3 Variations and Rates of Geometrical Elements

91

Now, we express the surface vectors as follows: da  nda;

dA  NdA

ð2:39Þ

where da and dA are the infinitesimal areas, and n and N are the unit normal vectors of the current and the reference infinitesimal surface vectors, respectively.  Here, noting n  n ¼ 0 from n  n ¼ 1 for the unit vector n, it follows that h i  ðdaÞ ¼ n  nðdaÞ ¼ n  ðndaÞ  n da ¼ n  ðdaÞ

ð2:40Þ

Substituting Eq. (2.38) into Eq. (2.40), one obtains the rate of the current infinitesimal area as follows: ðdaÞ ¼ n  elda

ð2:41Þ

ðdaÞ ¼ ðtrl  n  dnÞda

ð2:42Þ

d  sym½l

ð2:43Þ

or

where

noting n  wn ¼ 0 because of Eq. (1.180), where w  ant½l

ð2:44Þ

Here, d and w are called the strain rate tensor and the continuum spin tensor, respectively. Further, the rate of the unit normal of the current surface element is given from Eqs. (2.37), (2.39), (2.41) and (2.42) as follows:      n ¼ ðn  dnÞI  lT n ¼ ðn  lnÞI  lT n

ð2:45Þ

noting 

n da ¼ ðndaÞ  nðdaÞ   ¼ ðtrlÞI  lT nda  nðtrl  n  dnÞda ¼ lT nda þ n  dnda The formula for the physical quantities on the geometrical variation of material formulated in this section will be often used in the subsequent chapters.

92

2

2.4

Description of Motion

Material-Time Derivative of Volume Integration

Supposing that the zone of material occupying the volume v at the current moment ðt ¼ tÞ changes to occupy the volume v þ dv after the infinitesimal time R ðt ¼ t þ dtÞ, the material-time-derivative of the volume integration v Tðx; tÞdv of the physical quantity Tðx; tÞ involved in the volume is given by the following equation. Z

 Tðx; tÞdv

1 dt!0 dt

¼ lim

v

1 dt!0 dt

Z



Z v þ dv

Tðx; t þ dtÞdv 

Z

Tðx; tÞdv v



Z fTðx; t þ dtÞ  Tðx; tÞgdv þ

¼ lim

Tðx; t þ dtÞdv

ð2:46Þ

dv

v

The integration of the first term in the right-hand side in Eq. (2.46) is transformed as 1 lim dt!0 dt

Z

Z ½Tðx; t þ dtÞ  Tðx; tÞdv ¼

v

v

@Tðx; tÞ dv @t

ð2:47Þ

On the other hand, the second term in Eq. (2.46) describes the influence caused by the change of volume during the infinitesimal time increment. Here, the volume increment dv is given by subtracting the volume flowing out from the boundary of the zone from the volume flowing into the boundary, which is the sum of dvð¼ v  ndadtÞ over the whole boundary surface (Fig. 2.1). Therefore, substituting the Gauss’ divergence theorem in Eq. (1.389) and ignoring the second-order infinitesimal quantity, the integration of the second term in the right-hand side of Eq. (2.46) is given by

Fig. 2.1 Translation of a zone in material

v t n

a da

n

v

v v

2.4 Material-Time Derivative of Volume Integration

93

Z 1 Tðx; t þ dtÞdv ffi lim Tðx; tÞdv dt!0 dt dv dv Z Z 1 Tðx; tÞvr nr dadt ¼ Tðx; tÞvr nr da ¼ lim dt!0 dt a a Z Z Z @Tðx; tÞvr @Tðx; tÞ @vr dv ¼ vr dv þ Tðx; tÞ dv ¼ @xr @xr @xr v v v

1 lim dt!0 dt

Z

The sum of the first term in the right-hand side in this equation and Eq. (2.47) is equal to the material-time derivative of Tðx; tÞ by virtue of Eq. (2.12). Then, Eq. (2.46) is given by

R



 v

Tðx; tÞdv

¼

R

  @vr dv ¼ T ðx; tÞ þ Tðx; tÞ v @xr

R



v

½T ðx; tÞ þ Tðx; tÞdiv v dv ð2:48Þ

which is called the Reynolds’ transportation theorem. Equation (2.48) can be obtained also by the following simple manner. Z

 Tðx; tÞdv

Z



¼

TðX; tÞJdV

v

Z ¼

V

_ ðTðX; tÞJ þ TðX; tÞ  JÞdV

v

 Z   @vr dv ¼ Tðx; tÞ þ Tðx; tÞ @xr v where V is the initial volume, while Eq. (2.34) is used. For the physical quantity T kept constant in a volume element, say a mass, Eq. (2.48) leads to Z  ðT ðx; tÞ þ Tðx; tÞdivvÞdv ¼ 0 ð2:49Þ v

The local (weak) form of Eq. (2.49) is given as 

T ðx; tÞ þ Tðx; tÞdivv ¼ 0

ð2:50Þ

Chapter 3

Description of Tensor (Rate) in Convected Coordinate System

Material involved in a unit mass in the current state (configuration) changes by the deformation. Therefore, constitutive equation describing the inherent property of material cannot be formulated by tensor variables in the current configuration, i.e. the spatial, i.e. Eulerian tensors but must be formulated by tensor variables in the reference (initial) configuration, i.e. the material, i.e. Lagragian tensors. Needless to say, however, variables which one can actually observe are the one in the current configuration. Therefore, the transformation of tensor variables from the current to the reference configuration, called the pull-back operation, and it inverse transformation, called the push-forward operation, are required in the formulation of constitutive equation and the deformation analysis. These operations can be clearly interpreted by the description of tensors in the convected (embedded) coordinate system in which the coordinate axes are caved inside the material itself. The description of tensors in the convected coordinate system and the pull-back, push-forward operations are described comprehensively in this chapter. The convected derivatives, the corotational rates and the objectivity of rate tensors are also described in detail.

3.1

Reference and Current Primary and Reciprocal Base Vectors


 Suppose the coordinate system Hi with the primary base fGi g ði ¼ 1; 2; 3Þ, possessing the origin P0 at the fixed material point, which are embedded inside the
 material itself at the reference time t0 , changes to the coordinate system hi with the primary base fgi ðtÞg at the current time t under the deformation. Here, it should be noted that hi ¼ Hi holds even during the deformation since they are the components in the embedded coordinate system, although the primary base fgi ðtÞg © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 K. Hashiguchi, Foundations of Elastoplasticity: Subloading Surface Model, https://doi.org/10.1007/978-3-030-93138-4_3

95

96

3

Description of Tensor (Rate) in Convected Coordinate System



 changes. The coordinate system Hi (and hi ðtÞ ) is called the convective (convected or embedded) coordinate system and the primary base fGi g (and fgi ðtÞg) is called the convective (convected or embedded) base. On the other hand, the reciprocal bases in the reference configuration and in the current configuration
 are denoted by Gi and fgi ðtÞg, respectively, and their components are designated as fHi g and fhi ðtÞg, respectively. Here, it should be noted that the reciprocal bases and coordinates must be reformed at every moment with a deformation in order to keep the mathematical relation in Eq. (1.393) to the primary base so that they cannot be embedded. Here, the covariant components change in general as known from dhi ðtÞ ¼ gij ðtÞdh j ¼ gij ðtÞdH j 6¼ Gij dH j ¼ dHi noting Eq. (1.424) because of gij ðtÞ 6¼ Gij . Consequently, it follows that hi ¼ H i ;

dhi ¼ dHi ðhi ðtÞ 6¼ Hi ;

dhi ðtÞ 6¼ dHi Þ

ð3:1Þ

The following relations hold for the primary and the reciprocal base vectors by virtue of Eq. (1.405)2,3. Gi hG j ¼ dij ; Gi ¼ Gir Gr ;

gi ðtÞhg j ðtÞ ¼ dij

Gi ¼ Gir Gr

gi ðtÞ ¼ gir ðtÞgr ðtÞ;

gi ðtÞ ¼ gir ðtÞgr ðtÞ

ð3:2Þ ð3:3Þ

where Gij ¼ Gi hGj ;

Gij ¼ Gi hG j

gij ðtÞ ¼ gi ðtÞhgj ðtÞ;

gij ðtÞ ¼ gi ðtÞhgJ ðtÞ

ð3:4Þ

The following tensors are nothing else the metric tensors defined by Eq. (1.411). (

  G  Gi  Gi ¼ Gi  Gj ¼ Gij Gi  Gj ¼ Gij Gi  G j ¼ GT gðtÞ  gi ðtÞ  gi ðtÞ ¼ gi ðtÞ  gi ðtÞ ¼ gij ðtÞgi ðtÞ  gj ðtÞ ¼ gij ðtÞgi ðtÞ  g j ðtÞð¼ gT ðtÞÞ

ð3:5Þ The vector and the tensor based in the reference and the current configuration are called the Lagrangian and the Eulerian vector and tensor, respectively. In principle, the Lagrangian and the Eulerian vector and tensor are denoted by the uppercase and the lowercase letters, respectively, throughout this book. Further, the tensor based in both of the reference and the current configurations is called the Lagrangian– Eulerian or Eulerian–Lagrangian two-point tensor and denoted by the uppercase letter throughout this book.

3.2 Description of Deformation Gradient Tensor by Embedded Base Vectors

3.2

97

Description of Deformation Gradient Tensor by Embedded Base Vectors

The infinitesimal line-element dX in the reference configuration and dxðtÞ in the current configuration are described as follows: 

  dX ¼ dHi Gi ¼ dhi Gi ¼ dHi Gi 6¼ dhi Gi dxðtÞ ¼ dhi gi ðtÞ ¼ dHi gi ðtÞð¼ dhi gi ðtÞ 6¼ dHi gi ðtÞÞ

ð3:6Þ

where 

dHi ¼ dXhGi dhi ¼ dxðtÞhgi ðtÞ

ð3:7Þ

noting Eq. (1.423). The relation of the primary base vectors gi ðtÞ and Gi is given by replacing dxðtÞ and dX, respectively, in Eq. (2.13) as follows: gi ðtÞ ¼ FðtÞGi Gi ¼ F1 ðtÞgi ðtÞ

ð3:8Þ

or noting 8 @xðtÞ @xðtÞ @xðtÞ @X > > ¼ ¼ < gi ðtÞ ¼ @X @Hi @Hi @hi @X @X @X @xðtÞ > > ¼ ¼ : Gi ¼ @Hi @hi @xðtÞ @hi by of virtue of Eq. (1.404). Then, it follows from Eq. (3.8) that   @hi @hi FðtÞ ¼ gi ðtÞ  Gi ¼ Fi j gi ðtÞ  G j ; Fi j ¼ dij ¼ ¼ @H j @h j    i  i @Hi @hi F1 ðtÞ ¼ Gi  gi ðtÞ ¼ F1 j Gi  g j ðtÞ; F1 j ¼ dij ¼ ¼ @h j @h j noting  (

FðtÞ ¼ FðtÞG ¼ FðtÞGi  Gi F1 ðtÞ ¼ F1 ðtÞg ¼ F1 ðtÞgi ðtÞ  gi ðtÞ

F i ¼ gi ðtÞhFðtÞGj ¼ gi ðtÞhðgr ðtÞ  Gr ÞGj ¼ dir drj  j1 i F j ðtÞ ¼ Gi hF1 ðtÞgj ðtÞ ¼ Gi hðGr  gr ðtÞÞgj ðtÞ ¼ dir dtj

ð3:9Þ

98

3

Description of Tensor (Rate) in Convected Coordinate System

with Eqs. (3.1) and (3.8). The fact that the deformation gradient is the Eulerian– Lagrangian two point tensor based in the initial and the current base vectors can be understood by the representation in the embedded coordinate system, i.e. Equation (3.9), although this fact is hidden in the rectangular coordinate system. Further, it is noteworthy that the deformation gradient tensor is regarded as the identity tensor possessing the Kronecker’s delta dij in the convected coordinate system with both the covariant and the contravariant bases as known from Eq. (3.9), although it is represented with the components @xi =@Xj in the fixed coordinate system as seen in Eq. (2.15). Here, it follows from Eq. (3.9) that  i   i FT ðtÞ ¼ Gi  gi ðtÞ ¼ FT j G j  gi ðtÞ; FT j ¼ dij  j  j FT ðtÞ ¼ gi ðtÞ  Gi ¼ FT i gi ðtÞ  Gj ; FT i ¼ dij

ð3:10Þ

from which one has gi ðtÞ ¼ FT ðtÞGi

ð3:11Þ

Gi ¼ FT ðtÞgi ðtÞ for the reciprocal base vectors, noting (

  gi ðtÞ ¼ g j ðtÞdij ¼ g j ðtÞ  Gj Gi ¼ FT ðtÞGi   Gi ¼ G j dij ¼ G j  gj ðtÞ gi ðtÞ ¼ FT ðtÞgi ðtÞ

The symbol ðtÞ designating the time-dependence will be eliminated for the sake of simplicity in the following. The rates of the current base vectors are given in terms of the velocity gradient tensor l in Eq. (2.30) from Eqs. (3.8) and (3.11) as follows: h



gi ¼ F Gi ¼ lgi    gi ¼ FT Gi ¼ lT gi 



F ¼ g Gi ¼ lF;



ð3:12Þ



FT ¼ Gi  gi ¼ FT lT

ð3:13Þ

i  1

F ¼ Gi 



gi

1

¼ F l;



F

T

¼



gi

T

Gi ¼ l F T

3.2 Description of Deformation Gradient Tensor by Embedded Base Vectors

99

noting 8 h > 1 > < F F FGi ¼ lgi   T 

T  T  i  T  T T i  T  T i > > F G ¼ F F F G ¼ F F g ¼ F F1 gi ¼  F F1 gi ¼ lT gi :

Further, it follows from Eqs. (3.9) and (3.12) that l ¼ gi  gi 

ð3:14Þ



lT ¼ gi  gi noting 

lgi  gi ¼ gi  gi

3.3

Pull-Back and Push-Forward Operations

The Eulerian tensor defined by the current base fgi g or fgi g is transformed to the
 Lagrangian tensor by replacing these bases to the reference base fGi g or Gi , keeping the conponents. This operation is called the pull-back and its inverse operation is called the push-forward. These operations can be performed also by the multiplication of the deformation gradient tensor because the current and the reference base vectors are related by the deformation gradient tensor as shown in Eqs. (3.8) and (3.11). The Eulerian vector v is expressed by the following contravariant and the covariant forms in the current configuration by Eq. (1.422), i.e. v ¼ v i gi ¼ v i gi

ð3:15Þ

The pull-back transformations from the Eulerian vector to the Lagrangian vectors are performed by replacing the current base fgi g or fgi g to the reference base fGi g
 or Gi and designated as follows: v G  vi Gi ;

v

G

 vi G i

ð3:16Þ

where the over arrow directed to the left ð Þ is added (the right arrow ð!Þ for the push-forward operation described later). Further, the uppercase index G is added in order to stipulate the replacement of the current base to the reference base (the lowercase index g for the push-forward operation described later) and it is put in a

100

3

Description of Tensor (Rate) in Convected Coordinate System

lower or upper position of suffix of the transposed base vector. These symbols for the pull-back and the push-forward operations have been introduced by Hashiguchi (2011). Equation (3.16) is rewritten as v G ¼ vi F1 gi ¼ F1 vi gi ;

v

G

¼ v i F T gi ¼ F T v i gi

using Eqs. (3.8) and (3.11), and thus the following direct notations are obtained. v G ¼ vi Gi ¼ F1 v;

 vG ¼ v i G i ¼ F T v

ð3:17Þ

which are called the contravariant and the covariant pull-back operation, respectively. In particular, for the pure rotation with F ¼ R leading to Gi ¼ RT gi ; Gi ¼ RT gi , the two relations in Eq. (3.17) unified to the single relation.    R v R ¼ vi RT gi ¼ V ¼ vi RT gi ¼ RT v

ð3:18Þ

in which the distinction between the covariant and the contravariant transformation diminishes. In the inverse to the above, the Lagrangian vector defined on the reference base
 fGi g or Gi , i.e. V ¼ V i Gi ;

V ¼ Vi Gi

ð3:19Þ

!g V ¼ Vi gi

ð3:20Þ

is transformed to the Eulerian vectors ! V g ¼ V i gi ;


 by the push-forward operations replacing the reference base fGi g or Gi to the ! !g current base fgi g or fgi g, where V g and V are called the contravariant and the covariant push-forward operation, respectively. Here, because of ! V g ¼ V i FGi ;

!g V ¼ Vi FT Gi

noting Eq. (3.11), the following direct notations hold. ! V g ¼ V i gi ¼ FV;

!g V ¼ Vi gi ¼ FT V

ð3:21Þ

Further, for the pure rotation with F ¼ R leading to gi ¼ RGi ; gi ¼ RGi , it follows from Eq. (3.21) that

3.3 Pull-Back and Push-Forward Operations

101

 !R   !  V R ¼ V i RGi ¼ V ¼ Vi RGi ¼ RV

ð3:22Þ

Next, the second-order tensor t in the current configuration is represented by the following four types from Eq. (1.433). i t ¼ tij gi  gj ¼ tj gi  g j ¼ tij gi  gj ¼ tij gi  g j

ð3:23Þ

Here, when tij ¼ tji ; ti j ¼ tji ; tij ¼ tji ; tij ¼ tji , t is the symmetric tensor and thus t ¼ tT holds by virtue of Eq. (1.441). Here, note that the primary base vector gi and that of the the component vi obey the ordinary coordinate-transformation

rule as

derivative @f =@hi of scalar function f , i.e. @f =@hi ¼ @f =@h

j

j

@h =@hi , while gi

and vi obey the opposite transformation rule as explained in detail in Hashiguchi and Yamakawa (2012) and Hashiguchi (2020). Then, gi and vi are called the covariant base vector and the covariant component, respectively, and gi and vi are called the contravariant base vector and the contravariant component, respectively. The Eulerian tensor in Eq. (3.23) is replaced to the Lagrangian tensors by the pull-back operations replacing both of the current base vectors to the reference base vectors as follows: 8 > t > > > < t > t > > > : t

1 1 ij T ij ij 1 GG  t Gi  Gj ¼ t F gi  F gj ¼ F t gi  gj F G j T j 1 i i i 1 j G  tj Gi  G ¼ tj F gi  F g ¼ F tj gi  g F j i j T i 1 G gj ¼ FT tij gi  gj FT G  ti G  Gj ¼ ti F g  F i j T i T j T i j GG  tij G  G ¼ tij F g  F g ¼ F tij g  g F

leading to t

T

¼ tij Gi  Gj ¼ F1 tFT ¼ t GG ;

 i T T GT tG ¼ t G  G ¼ F tF ¼ 6 t j G i G ;

GG



T i ¼ tj Gi  G j ¼ F1 tF 6¼ t G G

GG t ¼ tij Gi  G j ¼ FT tF ¼ t GG T

t

G G

ð3:24Þ where t gg , t gg t gg and t gg are called the contravariant, the contravariantcovariant, the covariant-contravariant and the covariant push-forward operation, respectively. For the pull-back operation by which only one of the two current base vectors is exchanged to the reference base vector, the hat symbol (^) is added to the unchanged one as follows:

102

3

Description of Tensor (Rate) in Convected Coordinate System

8 9 9 ij ii 1 ij ij 1 > = = t ¼ t G  g ¼ t F g  g t ¼ t g  G ¼ t g  F g G^ g i g ^ j > j r J i i j G > 1 > > ¼ F ¼ tFT t; > ^ g ^ g > ; j i j i i j i 1 1 ; > < t G ¼ tj Gi  g ¼ tj F gi  gj t G ¼ ti g  Gj ¼ ti g  F gj 9 9 G^ g > > i j T i j > t ^gG ¼ tij gi  G j ¼ tij gi  FT g j = ¼ tij G  g ¼ tij F g  g = > t > > ¼ FT t; ¼ tF > > > j i i T j; : t G ¼ tj Gi  g ¼ tj FT gi  g > ; t G ^ g ¼ tj gi  G ¼ tj gi  F g j j i i ^ g

ð3:25Þ Incidentally, when only the rotation is given to both base vectors, the pull-backed tensor are described by the following unique equation in which the distinction of covariant, contravariant and mixed-variant operations vanishes: ^

T t

R

^

¼ RT tR ¼ TT

ð3:26Þ

^

which is derived by setting F ¼ R in Eq. (3.24). T is called the corotational Lagrangian tensor. On the other hand, the Lagrangian tensor defined by the reference bases
 fGi g; Gi is represented in the four types T ¼ T ij Gi  Gj ¼ Tji Gi  G j ¼ Tij Gi  Gj ¼ Tij Gi  G j

ð3:27Þ

The following push-forwarded Eulerian tensors are obtained by replacing the reference base vectors to the current base vectors. 8! > T gg  T ij gi  gj ¼ T ij FGi  FGj ¼ FT ij Gi  Gj FT > > > !g < T g ¼ T ji gi  g j ¼ T ji FGi  FT G j ¼ FTij Gi  G j F1 g >! T g ¼ Tij gi  gj ¼ Tij FT Gi  FGj ¼ FT Tij Gi  Gj FT > > > : !gg T ¼ Tij gi  g j ¼ Tij FT Gi  FT Gj ¼ FT Tij Gi  G j F1 leading to !

! T T gg ¼ T ij gi  gj ¼ FTFT ¼ T gg ;

!

! gT T g ¼ Tji gi  g j ¼ FTF1 6¼ T g !ggT

!g !gT !gg T g ¼ Tij gi  gj ¼ FT TFT ð6¼ T g Þ; T ¼ Tij gi  g j ¼ FT TF1 ¼ T

ð3:28Þ !gg ! !g !g where T gg , T g , T g and T are called the contravariant, the contravariantcovariant, the covariant-contravariant and the covariant push-forward operation, respectively.

3.3 Pull-Back and Push-Forward Operations

103

For the push-forward operation in which only one of the two reference base vectors is changed to the current base vector, it is denoted by adding the hat symbol (^) to the unchanged one as follows:

ð3:29Þ Incidentally, when only the rotation is given to both base vectors, the push-forwarded tensor is represented by the following unified equation in which the distinction of covariant, contravariant and mixed-variant operations vanishes: _

!R t  T ¼ RTRT ¼ t T

_

ð3:30Þ

which is called the corotational Eulerian tensor. The metric tensors in the reference and the current configurations are related in the pull-back and the push-forward operations as follows: (

  G Gi  Gi ¼ F1 gi  FT gi ¼ F1 gF ¼ g G G¼   G Gi  Gi ¼ FT gi  F1 gi ¼ FT gFT ¼ g G 8 g < g  gi ¼ FG  FT Gi  ¼ FGF1 ¼ ! Gg i i g¼ g : gi  g ¼ FT Gi  FG  ¼ FT GFT ¼ ! G i

i

g

9 > > > > > > = > > > > > > ;

ð3:31Þ

The tensors in the current configuration and in the referent configuration are referred to as the Eulerian tensor and the Lagrangian tensor, respectively. It is shown that we have the two and the four types of transformations from the Eulerian to the Lagrangian vector and second-order tensor or its inverse transformations, respectively, since the vector and the second-order tensor possesses one and two base vectors, respectively. In general, there exist n  2 types of transformations for the n-th order tensor. It should be noticed from the physical point of view that practically the Eulerian tensor is defined first and thereafter the Lagrangian tensors are produced from the Eulerian tensor by the pull-back operation. It will be shown for stress tensors in Sect. 5.1.

104

3

Description of Tensor (Rate) in Convected Coordinate System

A material involved in the unit volume in the current configuration changes but that in the reference configuration does not change during a deformation. Therefore, a constitutive equation describing the inherent property of material must be formulated by the tensors based in the reference configuration, which are given by the pulled-back of the tensors in the current configuration and the deformation analysis must be performed by the constitutive equation in the reference configuration. Note, however, that the boundary condition is given in the current configuration and thus first the variables must be pulled-back to the reference configuration. Therefore, the deformation analysis must be performed by the constitutive equation in the reference configuration and after that the calculated result is push-forwarded to the current configuration, by which the actual states of stress and deformation are captured. Gurtin et al. (2010) proposed the symbols specifying the pull-back operations P½t  FT tF; P½t  F1 tFT and the push-forward operations P1 ½t  FT tF1 ; P1 ½t  FtFT However, the covariant-contravariant mixed operations cannot be specified by these symbols. On the other hand, the symbols with the arrow and superscript or subscript shown in this section are capable of specifying the four types of the pull-back operations and the four thypes of the push-forward operations, providing the whole informations for these operations in the concise forms.

3.4

Convected Time-Derivative

As will be described in detail in Chap. 6, the constitutive relation of material does not depend on the rigid body rotation of material. Here, the material-time derivatives of the stress and the tensor-valued internal variables observed by a fixed coordinate system are influenced by the rigid-body rotation of material. Therefore, the material-time derivatives in Eq. (2.12) described in a fixed coordinate system cannot be used in the description of constitutive equation. On the other hand, the rates of stress and internal variables observed by the embedded coordinate system, which deforms and rotates with the material itself, is irrelevant to the rigid body rotation. Therefore, the constitutive equation must be represented in terms of the embedded derivatives of variables. The physical meaning and the mathematical expression of the time-derivatives of vector- and tensor-valued variables observed by the embedded coordinate system will be explained in this section, which is called the embedded time-derivative or the convective time-derivative or the convected time-derivative.

3.4 Convected Time-Derivative

105

3.4.1 General Convected Derivative The material-time derivative of the second-order tensor t based in the current configuration in Eq. (1.433) is described by 8  ij     t gi  gj ¼ tij gi  gj þ t ij gi gj þ tij gi  g > > > j > > >

 > j  <  i i j j i j i  tj gi  g ¼ t j gi  g þ tj gi g þ tj gi  g t¼ >  j i   > > ti g  gj ¼ tij gi  gj þ tij gi  gj þ ti gi  gj > > > >    :  tij gi  g j ¼ tij gi  g j þ tij gi  g j þ tij gi  g j

ð3:32Þ

from which the convective derivatives, i.e. the rate of the tensor t observed by the observer moving with the embedded coordinate system are given by 8 ij      > t gi  gj ¼ t tij gi gj  tij gi  gj ¼ t tij lgi  gj  tij gi  gj lT ¼ t lt  tlT > > > >     > > i  < ti j gi  g j ¼ t tj gi gj  ti j gi  gj ¼ t ti j lgi  gj þ ti j gi  g j l ¼ t lt  tl       > > > ti jgi  gj ¼ t tij gi gj  tij gi  gj ¼ t þ ti lT gi  gj  ti j gi  gj lT ¼ t lT t  tlT > > > >     :  t_ij gi  g j ¼ t tij gi  g j  tij gi  g j ¼ t þ tij lT gi  g j þ tij gi  g j l ¼ t lT t  tl

ð3:33Þ noting Eq. (3.12). The following four types of convected derivatives can be defined from Eq. (3.33).

o

o

o

o

o

o

o

o

ð3:34Þ

106

3

Description of Tensor (Rate) in Convected Coordinate System

       noting F F1 ¼ l; FT FT ¼ FT FT ¼ lT because of FF1 ¼ F F1 þ gg    g g F F1 ¼ O. The four types of the convected time-derivatives tgg , tg , tg and t are the push-forward of the material-time derivative of the pull-backed tensor by the contravariant, the contravariant-covariant, the covariant-contravariant and the covariant type operation, respectively. Namely, the convective time-derivative are attained by. 1. first pulling-back the tensor in the current configuration to the reference configuration, 2. then taking its time-differentiation, 3. finally returning (pushing-forward) it to the current configuration, There are the four types of the pull-back operations in the second-order tensor which possesses the two base vectors. Here, the variant type of the pull-back and the push-forward operation is same to each other. Incidentally, the particular operators, e.g. the identity tensor I, R are also used in addition to the original operator F. The convective time-derivative has been called often the Lie derivative and denoted by the symbol Lv ðf Þ defined by Lv ðf Þ  v

D  1  v f Dt

where f is scalar, vector or tensor variable, and v and v1 designate the push-forward and the pull-back operation, respectively (cf. e.g., Truesdell and Noll 1965; Marsden and Hughes 1983; Simo and Hughes 1998; Holzpfel 2000; Bonet and Wood 2008; de Souza Neto et al. 2008; Belytschko et al. 2014). However, the symbol with the double arrows and the two indices shown in this section would express the operations and modes (covariant or contravariant) of the convected time-derivatives clearly and intuitively. It was introduced by Hashiguchi (2011). However, the hypoelastic-based constitutive equation represented by the above-mentioned convective time-derivative will be disused sooner or later by the adoption of the multiplicative-based constitutive equation as will be explained in detail in Chap. 17.

3.4.2 Corotational Rate An irreversible deformation exhibits the loading-path dependency. Therefore, a constitutive equation for an irreversible deformation must be described by the rate-form. Therein, pertinent rates of stress and internal variables must be used in the constitutive equation, which satisfy the objectivity described the next

3.4 Convected Time-Derivative

107

subsection. Here, consider the particular cases in which the deformation and its rate are negligible compared with the rotation, i.e. U ffi I or d ffi O: For the case that the deformation is small compared with the rotation, i.e. F ffi R, all the four kinds of derivatives in Eq. (3.34) reduce to the single equation: oR

o

o• g

og

o gg

t ≡ t gg = t g = t • g = t

→

)• )• R • • • = R (R tR) R = R T RT ≡ T = t − R RT t + t R RT •

T

T

 tR

ð3:35Þ

is zero which is referred to the Green-Naghdi rate (Green and Naghdi 1965). when the quantity RT tR, which is observed by the coordinate system moving and rotating with a material line-element, is kept constant. The relation of the base vectors are given from Eq. (3.8) under F ¼ R as follows: 8 < gi ¼ RGi ; gi ¼ RGi G ¼ RT gi ; Gi ¼ RT gi : i R ¼ gi  Gi ¼ gi  Gi

ð3:36Þ

Then, the rate of the current base vector is given as follows: 

gi ¼ XR gi 



ð3:37Þ



noting gi ¼ R Gi ¼ R RT gi , where 

XR  R RT

ð3:38Þ

is referred to the relative spin. It designates the spin of the normalized convected covariant base vector gi =kgi k under the rigid-body rotation in which ðgi =kgi kÞ ¼ XR ðgi =kgi kÞ holds because of kgi k ¼ const. and kðgi =kgi kÞk ¼ 1. The physical meaning of the relative spin based on the name capped with the adjective “relative” will be given in Sect. 4.4. The Green-Naghdi rate in Eq. (3.35) is represented in terms of the relative spin as follows: 

 t R ¼ t XR t þ tXR ¼ t RT



ð3:39Þ

Further, in the case that the strain rate d in Eq. (2.43) is infinitesimal compared with the continuum spin w in Eq. (2.44), leading to l ¼ w, Eq. (3.34) is reduced to 

 t w  t wt þ tw ¼ t wT



ð3:40Þ

108

3

Description of Tensor (Rate) in Convected Coordinate System

which is called the Zaremba-Jaumann rate (Zaremba 1903; Jaumann 1911). Here, 

tw ignoring the strain rate d is related to the convected derivative in Eq. (3.34) as follows: o

o

o

o

g

o

gg • g t w = t gg + t d +dt = t g − td +dt = t • g + t d − dt = t −td −dt

ð3:41Þ

resulting in o o o o g og o g o g og og t w = 1 (t gg + t gg ) = 1 (t •g + t • g ) = 1 [t •g + (t •g )T ] = 1 [t • g + (t • g )T ] 2 2 2 2

ð3:42Þ

The corotational rates in Eqs. (3.39) and (3.40) depend only on the geometrical change of material which can be observed from the outside appearance of material. However, the physically meaningful rotation is the rotation of the substructure of material which is induced by the elastic distortion and the rigid-body rotation but is not induced by the plastic deformation related merely to the mutual slips between material particles, and thus the rotation of the substructure of material cannot be known only from the outside appearance of material. Based on this physical interpretation, it has been pointed out by Mandel (1971, 1973), Kratochvil (1971), etc. that the rotation of substructure is suppressed by the plastic deformation such that the substructure spin wS is given by the subtraction of the plastic spin wp (Dafalias 1984, 1985a, 1998) from the continuum spin w, i.e. 



t  t ws t þ tws

ð3:43Þ

wS ¼ w  wp ¼ we

ð3:44Þ

with

The explicit equation of the plastic spin wp will be shown in the subsequent chapters. The oscillations of the stress and the kinematic hardening variable are suppressed by the corotational rate with the modified spin tensor in Eq. (3.44).

3.4.3 On Adoption of Convected Rate Tensor in Hypoelastic Constitutive Equation The influence of the rigid-body rotation occurs in the deformation analysis by the constitutive equation described in terms of the material time-derivative denoted by the symbol ð Þ. The influence can be reduced greatly by exploiting the convected (embedded) time-derivative which is the variation of the tensor observed from the coordinate system deforming/rotating with the material itself. However, there exist

3.4 Convected Time-Derivative

109

the four types of the convected time-derivative. Unfortunately, which one is rational in them cannot be concluded. In addition, the time-integration of them by the forward Euler method leads to an erroneous result as far as the calculation by infinitesimal increments is not performed. Instead, the time-integration must be executed by the pull-back, the time-integration and the push-forward operation in the three steps as will be explained in the next subsection.

3.4.4 Time-Integration of Convected Rate Tensor The exact time-integration of the convective rate variable, e. g. stress rate and rate of internal variables is of crucial importance in the deformation analysis by the hypoelastic-based constitutive equation. The forward-Euler calculation method of the stress and internal variables in the hypoelastic constitutive relation has to be done in infinitesimally small incremental steps, otherwise large error will be induced. Then, an efficient calculation method has been developed and installed as the standard-function in the commercial FEM software, e.g. Mark and Abaqus, etc., while a detailed theoretical background is not written in their user-manuals. In what follows, it will be explained systematically from the general aspect of the convected rate tensors and the corotational tensors, although it has been delineated in fragments taking account of only the material rotation by some literatures (e.g. Hughes and Winget 1980; Pinsky et al. 1983; Rubinstein and Atluri 1983; Flagnan and Taylor 1987; Simo and Hughes 1998; de Souza Neto et al. 2008). (a) Pull-back and Push-forward time-integration method The rigid-body rotation of material occurs finitely, which is larger compared with the deformation in many cases as known from the screw, the turbine, the whleel, etc., while needless to say the rigid-body rotation is irrelevant to the material property, i.e. the constitutive relation. Therefore, the variation of tensor variables must be executed by an infinitesimal increment in the current configuration in general. However, the time-integration of tensor variables can be executed in finite increments by the calculation with the three steps, i.e. the pull-back operation to the reference configuration, the incremental calculation by the constitutive equation in the reference configuration and the push-forward operation to the current configuration as described in Sect. 3.4.1. 

The convected rate t in Eq. (3.34) can be represented collectively in terms of the second-order tensors F l and F r which play the roles of the pull-back and push-forward operations, noting that the operators for the pull-back and the push-forward are the inverse relation to each other, as follows: 1

 t ¼ F l F l tF r F r1



ð3:45Þ

110

3

Description of Tensor (Rate) in Convected Coordinate System

leading to ð3:46Þ where the second-order tensors F l and F r are given for the four types of the 

convective time-derivative t in Eq. (3.34) as follows: ⎧ → • ⎪ l o −T r • for t gg ≡ ← t gg = F(F −1tF −T )• FT = t − lt − tl T ⎪  = F,  = F ⎪ → ⎪ • •g g • −1 −1 ⎪  l = F,  r = F for to g• ≡ ← t g = F (F tF)• F = t − lt + tl ⎪⎪ ⎨ → ⎪ l •g og ← r − ⎪  = F −T ,  = F T for t • g ≡ t • g = F −T (FT tF −T )• FT = t• + l T t −t l T ⎪ ⎪ → gg ⎪ • gg ← ⎪  l = F −T ,  r = F for to ≡ t = F −T (FT tF)• F −1 = t• + l T t + t l ⎪⎩

ð3:47Þ

We consider the four examples of the typical convective rates. 

(1) The Oldroyd rate is based on the contravariant rate tgg in Eq. (3.34)1 , i.e. (3.47)1 : F l ¼ F; F r ¼ FT 

(2) The Cotter-Rivlin rate is based on the covariant rate tgg in Eq. (3.34)4 , i.e. (3.47)4 : F l ¼ FT ; F r ¼ F (3) The Green-Naghdi rate in Eq. (3.39) is derived by setting Fl ¼ Fr ¼ R (4) The Jaumann rate in Eq. (3.40) is given by 1   ¼ F r F r1 ¼ w F l F l i.e.



F l ¼ wF i ;



F r ¼ wF r

ð3:48Þ

ð3:49Þ ð3:50Þ . Then, denoting F l

noting

and F r collectively by F , i.e. F l ¼ F r ¼ F , Eq. (3.50) is represented in the unified form as follows: 

F ¼ wF

ð3:51Þ

3.4 Convected Time-Derivative

111

The incremental relation of Eq. (3.51) is given by   F n þ 1 ¼ exp wn þ 1=2 Dt F n

ð3:52Þ

noting Eq. (A.57) with h ¼ 1=2.



In what follows, we consider the time-integration of the convected rate tensor t for the all cases in Eq. (3.47). Firstly, the pull-back of t in the step n þ 1 to the reference configuration is given by t



nþ1



r ¼ t n þ tn þ 1=2 Dt ¼ F nl1 tn F rn þ F nl1 þ 1=2 tn þ 1=2 F n þ 1=2 Dt

ð3:53Þ



noting Eq. (3.45), where t is given by the constitutive relation in the current configuration. Then, the updated tensor tn þ 1 in the current configuration is calculated by the push-forward of Eq. (3.53) as follows: tn þ 1 ¼ F ln þ 1 t

r1 n þ 1F n þ 1



 r l1 r ¼ F ln þ 1 F l1 t F þ F t F Dt F r1 n n þ 1=2 n n n þ 1=2 n þ 1=2 nþ1

ð3:54Þ leading to 

tn þ 1 ¼ F lD tn F rD þ F ld tn þ 1=2 F rd Dt

ð3:55Þ

where (

F lD  F ln þ 1 F nl1 ; F rD  F rn F r1 nþ1 r r r1 F ld  F ln þ 1 F nl1 ; F  F þ 1=2 d n þ 1=2 F n þ 1

ð3:56Þ

The variables in Eq. (3.56) are represented by F noting Eq. (3.47) as follows:

112

3

8 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > :

Description of Tensor (Rate) in Convected Coordinate System

r T T F lD ¼ Fn þ 1 F1 n ; F D ¼ Fn Fn þ 1

9 =

T ; F ld ¼ Fn þ 1 Fn þ 1=2 ; F rd ¼ FT n þ 1=2 Fn þ 1

F lD ¼ Fn þ 1 F1 n ;

for contravariant: Oldroyd rate

9 = F rD ¼ Fn F1 nþ1

; F rd ¼ Fn þ 1=2 F1 nþ1 9 T = F lD ¼ F1 F rD ¼ FT n þ 1 Fn ; n Fn þ 1

F ld ¼ Fn þ 1 F1 n þ 1=2 ;

for contravariant-covariant

for covariant-contravariant T ; F rd ¼ FT n þ 1=2 Fn þ 1 9 T = F lD ¼ FT F rD ¼ Fn F1 n þ 1 Fn ; nþ1 for covariant: Cotter-Rivlin rate T ; F ld ¼ FT F rd ¼ Fn þ 1=2 F1 n þ 1 Fn þ 1=2 ; nþ1 9 F lD ¼ Rn þ 1 RTn ; F rD ¼ Rn RTn þ 1 = for Green-Naghdi rate F ld ¼ Rn þ 1 RTn þ 1=2 ; F rd ¼ Rn þ 1=2 RTn þ 1 ;    ) F lD ¼ exp wn þ 1=2 Dt ; F rD ¼ F lT D ¼ exp wn þ 1=2 Dt     for Jaumann rate F ld ¼ exp wn þ 1=2 Dt=2 ; F rd ¼ F lT d ¼ exp wn þ 1=2 Dt=2 F ld ¼ F1 n þ 1 Fn þ 1=2 ;

ð3:57Þ noting )     F lD ¼ exp wn þ 1=2 Dt F n F 1 ¼ exp w Dt ; n þ 1=2   n1   F n ¼ exp wn þ 1=2 Dt F rD ¼ F n exp wn þ 1=2 Dt

ð3:58Þ

     1   ) F ld ¼ exp wn þ 1=2 Dt F n exp wn þ 1=2 Dt=2 F n ¼ exp wn þ 1=2 Dt=2      1   F rd ¼ exp wn þ 1=2 Dt=2 F n exp wn þ 1=2 Dt F n ¼ exp wn þ 1=2 Dt=2 ð3:59Þ with Eq. (3.52). The following approximate equations for the Jaumann rate in Eq. (3.57) were proposed by Hughes and Winget (1980) taking only the first and second-order terms in the Taylor expansion in Eq. (1.344). 

F TD þ tn þ 1=2 Dt tn þ 1 ¼ f F D tn F n f

ð3:60Þ

f F D  I þ wn þ 1=2 Dt

ð3:61Þ

where

3.4 Convected Time-Derivative

113

noting (

    exp wn þ 1=2 Dt ffi I þ wn þ 1=2 Dt; exp wn þ 1=2 Dt ffi I  wn þ 1=2 Dt     exp wn þ 1=2 Dt=2 ffi I þ wn þ 1=2 Dt=2 ffi I; exp wn þ 1=2 Dt=2 ffi I  wn þ 1=2 Dt=2 ffi I

ð3:62Þ Only the influence caused by the rigid-body rotation is excluded as seen in the first term in the right-hand side in Eq. (3.60). Equation (3.60) is described for the Cauchy stress by

rn þ 1 ¼ f F D rn F n f F TD þ E : dn þ 1=2  dpn þ 1=2 Dt

ð3:63Þ

and for the kinematic hardening variable a described in Chap. 8 by  F D an F n f F TD þ ck dpn þ 1=2  an þ 1 ¼ f

1 bk Fn þ 1=2

    p  dn þ 1=2 an þ 1=2 Dt

ð3:64Þ

(b) Numerical examples The numerically calculated results by the above-mentioned Jaumann rate and the Green-Naghdi rate in the simple shear deformation of the hypoelastic material are shown in Figs. 3.1 and 3.2. These calculations have been executed by Prof. Yuki Yamakawa (Tohoku university) and they are published here by his courtesy. The calculation by the forward-Euler method deviates from the exact calculation curve with the decrease of the number of calculation, i.e. the increase of strain increment as shown in Figs. 3.1a and 3.2a for the Jaumann rate and the

(a) Forward-Euler method

Fig. 3.1 Numerical time-integrations of Jaumann rate

(b) Pull-back and push-forward method

114

3

Description of Tensor (Rate) in Convected Coordinate System

(a) Forward-Euler method

(b) Pull-back and push-forward method

Fig. 3.2 Numerical time-integrations of Green-Naghdi rate

Green-Naghdi rate, respectively. Here, the exact calculation for the Jaumann rate is shown by the sinusoidal curve, because the material rotates constantly by the constant spin w if the plastic spin is not incorporated as described in Sect. 3.4.2. On the other hand, the exact calculations can be executed by the pullback-push forward calculation method in Eq. (3.55) with Eq. (3.57) independent of the shear strain increment as shown in Figs. 3.1b and 3.2b. There does not exist the definite answer to the question which convected rate tensor is best. The Jaumann rate is practically accepted widely because it based on the continuum spin as the anti-symmetric part of the velocity gradient tensor which is the most basic quantity describing the rate of deformation and the incorporation of the plastic spin tensor is physically acceptable. However, the hypoelastic equation holds only for the infinitesimal elastic deformation as will be described in Sect. 17.3. Such defect in the hypoelasticty can be excluded in the multiplicative hyperelastic-based plastic constitutive equation described in Chap. 17.

Chapter 4

Deformation/Rotation Tensors

There are various deformation/rotation (rate) measures, from which the most suitable measure must be adopted in the description of constitutive equation, depending on the purpose, e.g. the description of infinitesimal deformation, finite deformation, hyper- or hypo-elastic deformation, (visco)plastic deformation, etc. In this chapter, definitions and physical meanings of main deformation/rotation (rate) measures used in continuum mechanics will be explained based on the deformation gradient tensor from which all the deformation/rotation (rate) measures are deduced. In particular, various strains, strain rates and spins will be introduced as the deformation/rotation (rate).

4.1

Deformation Tensors

Applying the polar decomposition in Sect. 1.11 to the deformation gradient F, we have F ¼ RU ¼ VR ;

FiA ¼ RiR U RA ¼ Vir RrA

ð4:1Þ

U ¼ RT F ¼ ðFT FÞ1=2 ðU2 ¼ FT FÞ ðUT ¼ UÞ

ð4:2Þ

V ¼ FRT ¼ ðFFT Þ1=2 ðV2 ¼ FFT Þ ðVT ¼ VÞ

ð4:3Þ

where

R ¼ FU1 ¼ FðFT FÞ1=2 ; R ¼ V1 F ¼ ðFFT Þ1=2 F ðdetR ¼ 1Þ V ¼ RURT ; U ¼ RT VR

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 K. Hashiguchi, Foundations of Elastoplasticity: Subloading Surface Model, https://doi.org/10.1007/978-3-030-93138-4_4

ð4:4Þ ð4:5Þ

115

116

4

Deformation/Rotation Tensors

U and V are the symmetric tensors describing the pure deformation which possess the different principal directions depending on the orthogonal tensor R describing the rotation. They are the similar tensors to each other, since they are related by Eq. (4.5) through the orthogonal tensor R as was described in Sect. 1.11. Therefore, they possess the same principal stretches, say ka ð[ 0Þ ða ¼ 1; 2; 3Þ. Denoting the bases for the principal directions of U and V by fNa ðtÞg and fnðaÞ ðtÞg, respectively, they can be written as U¼

3 P a¼1

ka NðaÞ ðtÞ  NðaÞ ðtÞ; V ¼

3 P a¼1

ka nðaÞ ðtÞ  nðaÞ ðtÞ

ð4:6Þ

where the relation of NðaÞ ðtÞ and nðaÞ ðtÞ is given from Eq. (1.276) as follows: nðaÞ ðtÞ ¼ RðtÞNðaÞ ðtÞ; NðaÞ ðtÞ ¼ RT ðtÞnðaÞ ðtÞ

ð4:7Þ

with RðtÞ ¼

3 P

nðaÞ ðtÞ  NðaÞ ðtÞ

ð4:8Þ

a¼1

NðaÞ ðtÞ and nðaÞ ðtÞ are called the Lagrangian triad and the Eulerian triad, respectively. Substituting Eqs. (4.6) and (4.8) into Eq. (4.1), F and its inverse tensor are described by FðtÞ ¼

3 P a¼1

ka ðtÞnðaÞ ðtÞ  NðaÞ ðtÞ; F1 ðtÞ ¼

3 P 1 NðaÞ ðtÞ  nðaÞ ðtÞ a¼1 ka ðtÞ

ð4:9Þ

Let the mechanical meanings of U; V and R be examined below. The variation of infinitesimal line-element is given by the polar decomposition F ¼ RU as follows: dx ¼ FdX ¼ RUd X ¼ R

3 X b¼1

kb NðbÞ  NðbÞ

3 X a¼1

dXa NðaÞ ¼ R

3 X

ka dXa NðaÞ

a¼1

ð4:10Þ Equation (4.10) means that the infinitesimal line-elements dXa NðaÞ (no sum) in the principal directions NðaÞ are first stretched by ka times to ka dXa NðaÞ (no sum) and then undergoes the rotation R as shown in Fig. 4.1. On the other hand, the change of the infinitesimal line-element by the polar decomposition VR is described as

4.1 Deformation Tensors

117

RL

RE

RL

R

RE

R

L

Fig. 4.1 Polar decomposition of deformation gradient

dx ¼ FdX ¼ VRdX ¼

3 X

kb nðbÞ  nðbÞ R

b¼1

¼

3 X a¼1

ka dXa nðaÞ ¼

3 X

dXa NðaÞ ¼

a¼1 3 X

3 X b¼1

kb nðbÞ  nðbÞ

3 X

dXa nðaÞ

a¼1

ka RdXa NðaÞ

a¼1

ð4:11Þ

118

4

Deformation/Rotation Tensors

Equation (4.11) means that the infinitesimal line-elements dXa NðaÞ (no sum) in the principal directions NðaÞ first becomes dXa nðaÞ (no sum) by rotation R and then are stretched by ka times to ka dXa nðaÞ (no sum) (see Fig. 4.1). As described above, U and V designates the deformation and R the rotation. ka is called the principal stretch, and U and V are called the right and left stretch tensor, respectively. Letting RL and RE designate the rotations of the Lagrangian triad fNðaÞ g and the Eulerian triad fnðaÞ g, respectively, from the fixed base fea gða ¼ 1; 2; 3Þ, they are given by RL 

3 P a¼1

NðaÞ  ea ; RE 

3 P a¼1

nðaÞ  ea

ð4:12Þ

where the following relations hold. NðaÞ ¼ RL ea ; nðaÞ ¼ RE ea

ð4:13Þ

RE ¼ RRL

ð4:14Þ

Considering the particle P and the adjacent particles P0 and P00 , we designate their position vectors before and after the deformation by X; X þ dX; X þ dX and x; x þ dx; x þ dx, respectively. Then, noting (1.139), one has 

dx  dx ¼ FdX  FdX ¼ FT FdX  dX ¼ CdX  dX  1 dX  dX ¼ F1 dx  F1 dx ¼ FT F1 dx  dx ¼ FFT dx  dx ¼ b1 dx  dx ð4:15Þ

where   C  U2 ¼ FT F ¼ FT g F ¼ g GG ¼ gij Gi  G j ¼ CT   C1 ¼ F1 FT ¼ F1 g FT ¼ g ¼ gij Gi  Gj ¼ CT

ð4:16Þ

  ! b  V2 ¼ FFT ¼ FGFT ¼ G gg ¼ Gij gi  gj ¼ bT   !gg b1 ¼ FT F1 ¼ FT GF1 ¼ G ¼ Gij gi  g j ¼ bT

ð4:17Þ

GG

as ascertained by 

FT F ¼ Gi  gi gj  Gj ¼ gij Gi  Gj FFT ¼ gi  Gi Gj  gj ¼ Gij gi  gj

4.1 Deformation Tensors

119

so that C and b are related to each other as follows: (

C ¼ FT bFT ¼ b G

G 

ð4:18Þ

G

b ¼ FT CT ¼ C  G

C and b are referred to as the right and left Cauchy-Green deformation tensor, respectively. In accordance with Eq. (4.6) they are described by C¼

3 P a¼1

k2a NðaÞ  NðaÞ ; b ¼

3 P a¼1

k2a nðaÞ  nðaÞ

ð4:19Þ

where ka is called the principal stretch. The principal values k2a are obtained by the solutions T of the characteristic equation T 3  Ic T 2  IIc T þ IIIc ¼ 0

ð4:20Þ

based on Eq. (1.216), where Ic  trC; IIc 

i 1h 1 1 1 ðtrCÞ2  trC2 ; IIIc  ðtrCÞ3  trCtrC2 þ trC3 2 6 2 3

ð4:21Þ

The principal values and directions are calculated by the method described in Sect. 1.6. The similar equations hold for b instead of C. Using the relative description (2.8), the relative deformation gradient tensor in the reference configuration xðsÞ is defined as s FðtÞ

¼

@xðtÞ @xðsÞ

ð4:22Þ

which is related to the deformation gradient FðtÞ ð 0 FðtÞÞ as FðtÞ ¼

@xðtÞ @xðtÞ @xðsÞ ¼ ¼ s FðtÞFðsÞ @X @xðsÞ @X

ð4:23Þ

and is further expressed in the polar decomposition as s FðtÞ

¼ s RðtÞs UðtÞ ¼ s VðtÞs RðtÞ

ð4:24Þ

while s CðtÞ and s bðtÞ are defined by s CðtÞ

¼ ðs FðtÞÞ T s FðtÞ ¼ s U2 ðtÞ

s bðtÞ

¼ s FðtÞ ð s FðtÞÞT ¼ s V2 ðtÞ

) ð4:25Þ

which are called the relative right and the left Cauchy-Green deformation tensors.

120

4.2

4

Deformation/Rotation Tensors

Strain Tensors

Consider the scalar quantity which changes only by the pure deformation but is independent of the rotation. Subtracting the upper equation from the lower equation in Eq. (4.15), one has ( 2EdX  dX ð¼ 2EAB dXA dXB Þ dx  dx  dX  dX ¼ ð4:26Þ 2edx  dx ð¼ 2eij dxi dxj Þ where  1 1 1 1  @x @x E  ðC  IÞ ¼ ðU2  IÞ ¼ ðFT F  IÞ ¼ ð ÞT ð Þ  I 2 2 2 2 @X @X   1 1 @xk @xk EAB  ðFkA FkB  dAB Þ ¼  dAB 2 2 @XA @XB  T 1 1 1 1 @X @X  e  ðIb1 Þ ¼ ðIV2 Þ ¼ ðI  FT F1 Þ ¼ I  2 2 2 2 @x @x   1 1 @XK @XK eij  ½dij  ðF1 ÞKi ðF1 ÞKj  ¼ dij  @xi @xj 2 2

9 > > > > > > > > > > > > > = > > > > > > > > > > > > > ;

ð4:27Þ

which are defined by C and b describing the pure deformations, noting dx  dx ¼ FdX  FdX ¼ FT FdX  dX and dX  dX ¼ F1 dx  F1 dx ¼ FT F1 dx  dx¼ ðFFT Þ1 dx  dx. Applying the quotient law described in Sect. 1.3.2 to Eq. (4.26), it is confirmed that E and e are the second-order tensors. If a deformation is not induced, the triangle PP0 P00 keeps the same shape and thus the left-hand side in Eq. (4.26) is zero so that E and e are zero independent of rotation. Conversely, if E 6¼ O; e 6¼ O, the left-hand side in Eq. (4.26) is not zero so that the shape of the triangle is not same as that in the initial, resulting in a deformation. Therefore, E and e are the quantities describing the deformation independent of rigid-body rotation and called the Green strain tensor and the Almansi strain tensor, respectively. Using the displacement vector u ¼ x  X ¼ ui ei ;

@u ¼FI @X

ð4:28Þ

they are expressed by "  9  T  # 1 @u @u @u @u 1 @uA @uB @uK @uK > > > þ Þð ; EAB ¼ E¼ þ þ þ > = 2 @X @X @X @X 2 @XB @XA @XA @XB > " #    T  T   > > 1 @u @u @u @u 1 @ui @uj @uk @uk > > > e¼  þ  þ ; eij ¼ ; @xi @xi @xj 2 @x @x @x @x 2 @xj ð4:29Þ

4.2 Strain Tensors

121

The following relation exists between them, noting Eq. (3.24)4 for the pull-back operation and Eq. (3.28)4 for the push-forward operation. 9 GG > T = E ¼ F eF ¼ e ; EAB ¼ FiA FjB eeij ð4:30Þ !gg > T 1 e ¼ FT EF1 ¼ E ; eij ¼ FiA FjB EAB ; The Green strain tensor E and the Almansi strain tensor e are called also the Lagrangian strain tensor and the Eulerian strain tensor, respectively, since the former and the latter are represented by the reference and the current variables. Now, consider the symmetric part of the displacement gradient tensor u which is the eliminations of the third terms, i.e. the second-order infinitesimal terms in the brackets E and e in Eq. (4.29), i.e.  e

@u @X

s

"  T #    1 @u @u 1 1 @uA @uB T þ ¼ þ ¼ F þ F  I ; eAB  2 @X @X 2 2 @XB @XA ð4:31Þ

or "    s  T #  @u 1 @u @u 1 1 @ui @uj þ e ¼ þ ¼ I  F1 þ FT ; eij  @xi @x 2 @x @x 2 2 @xj ð4:32Þ which describe roughly deformation, depending not only on U or V but also on the rotation tensor R, noting F ¼ RU ¼ VR. Incidentally, the distinction between the infinitesimal line elements dX in the initial state and dx in the current state becomes meaningless. Then, e is called the infinitesimal (nominal) strain tensor, while the Green and the Almansi strain tensors are called the finite (true) strain tensor. Consequently, the distinction of the Lagrange and the Eulerian description, i.e. the difference of Eqs. (4.31) and (4.32) vanishes because of dx ffi dX in an infinitesimal deformation. Hereinafter, unless otherwise noted, the infinitesimal strain tensor refer to Eq. (4.31) in the Langrange description. The defects of the infinitesimal (nominal) strain tensor will be explained in Sect. 4.5. In what follows, the geometrical interpretation of E and e will be given. Considering the case that the two infinitesimal line-elements PP0 and PP00 coincide to each other, i.e. dX ¼ dX; dx ¼ dx and denoting their direction vectors in the initial and current configurations by NðjjNjj ¼ 1Þ and nðjjnjj ¼ 1Þ leading to dX ¼ jjdXjjN and dx ¼ jjdxjjn, it follows from Eq. (4.26) that

122

4

( 2

2

jjdxjj  jjdXjj ¼

Deformation/Rotation Tensors

2EN  NjjdXjj2 2en  njjdxjj2

ð4:33Þ

By selecting the X1 -axis and the x1 -axis in the reference and the initial stage, respectively, of this line-element, ðN1 ; N2 ; N3 Þ ¼ ð1; 0; 0Þ and ðn1 ; n2 ; n3 Þ ¼ ð1; 0; 0Þ hold leading to EN  N ¼ E11 and En  n ¼ e11 and thus we have " #! 9  > 1 jjdxjj2  jjdXjj2 1 jjdxjj 2 > > > ¼ 1 E11 ¼ > 2 = 2 2 jjdXjj jjdXjj " # ! ð4:34Þ   > > 1 jjdxjj2  jjdXjj2 1 jjdXjj 2 > > e11 ¼ ¼ 1 > ; 2 2 jjdxjj jjdxjj2 Therefore, E11 and e11 describes the half of the change in the square of the line-element vector. Equation (4.34) becomes E11 e11

9 1 ðjjdxjj  jjdXjjÞðjjdxjj þ jjdXjjÞ > > ¼ > = 2 jjdXjj2 1 ðjjdxjj  jjdXjjÞðjjdxjj þ jjdXjjÞ > > > ¼ ; 2 jjdXjj2

ð4:35Þ

E11 and e11 are simplified to the following equation in the case of the infinitesimal deformation fulfilling jjdxjj ffi jjdXjj. 9 jjdxjj  jjdXjj > ¼ e11 > = jjdXjj > jjdxjj  jjdXjj ; ¼ e11 > ffi jjdxjj

E11 ffi e11

ð4:36Þ

which coincides with the infinitesimal strain e in Eq. (4.31) or (4.32). On the other hand, denoting the direction vectors of the two distinct infinitesimal line-element PP0 and PP00 as N0 and N00 , respectively, in the initial state and the angles contained by them as h, it holds from Eq. (4.26) that jjdxjjjjdxjj cos h  jjdXjjjjdXjj cos h0 ¼ 2EN0  N00 jjdXjjjjdXjj

ð4:37Þ

i.e. jjdxjj jjdxjj cos h  cos h0 ¼ 2EN0  N00 ¼ 2Eij Nj0 Nj00 jjdXjj jjdXjj

ð4:38Þ

where h0 is the initial value of h. Here, assuming that the infinitesimal line-elements PP0 and PP00 were mutually perpendicular before a deformation, i.e. h0 ¼ p=2 leading to cos h0 ¼ 0; and making their directions coincide to the X1 -and X2 -axes,

4.2 Strain Tensors

123

i.e. ðN10 ; N20 ; N30 Þ ¼ ð1; 0; 0Þ; ðN100 ; N200 ; N300 Þ ¼ ð0; 1; 0Þ leading to Eij Ni0 Nj00 ¼ E12 , it follows that E12 ¼

1 jjdxjj jjdxjj 1 jjdxjj jjdxjj cos h ¼ sinðp=2  hÞ 2 jjdXjj jjdXjj 2 jjdXjj jjdXjj

ð4:39Þ

Further, for the particular deformation in which the lengths of the line-elements PP′ and PP″ do not change, i.e., jjdxjj=jjdXjj; jjdxjj=jjdXjj, Eq. (4.39) leads to E12 ¼ ðp=2  hÞ=2

ð4:40Þ

Consequently, E12 describes half of the decrease in the angle contained by the two line-elements which were perpendicular before deformation. In addition to the Lagrangian and Eulerian strain tensors defined above, we can define various strain tensors in terms of U or V, fulfilling the condition that they are zero when U ¼ V ¼ I as follows (Seth 1964; Hill 1968): EðmÞ

8 < 1 ðU2m  IÞ for m 6¼ 0 2m ¼ fðUÞ ¼ : ln U for m ¼ 0

ð4:41Þ

eðmÞ

8 < 1 ðV2m  IÞ for m 6¼ 0 2m ¼ fðVÞ ¼ : ln V for m ¼ 0

ð4:42Þ

where 2m is the integer (positive or negative). The Green strain tensor is obtained by choosing m ¼ 1 in Eq. (4.41), i.e. E ¼ Eð1Þ and the Almansi strain tensor is obtained by choosing m ¼ 1 in Eq. (4.42), e ¼ eð1Þ . The Biot strain tensor (Biot 1965) is given by choosing m ¼ 1=2 in Eq. (4.41), i.e. BUI

ð4:43Þ

The generalized strain tensors in Eqs. (4.41) and (4.42) are mutually related by virtue of Eq. (4.5) as follows. EðmÞ ¼ RT eðmÞ R

ð4:44Þ

The strain tensors in Eqs. (4.41) and (4.42) are coaxial with U and V, respectively, and their principal values are given by the function of the principal stretch ka ða ¼ 1; 2; 3Þ as follows: 8 < 1 ðk2m  1Þ for m 6¼ 0 2m a f ðka Þ ¼ : ln ka for m ¼ 0

ð4:45Þ

124

4

Deformation/Rotation Tensors

The function f ðka Þ fulfills f ð1Þ ¼ 0; f 0 ð1Þ ¼ 1

ð4:46Þ

f 0 ðsÞ [ 0

ð4:47Þ

and

where s is an arbitrary positive scalar quantity. The function f ðka Þ for several values of m is shown in Fig. 4.2. (Note) Eq. (4.45)2 for m ¼ 0 is derived as follows: lim

1

m!0 m

ðkm a  1Þ ¼ lim

m!0

expðm ln ka Þ  1 expðm ln ka Þ ln ka ¼ ln ka ðno sumÞ ¼ lim m!0 1 m

by the aid of l’H^ ospital’s rule. Further, adopt the second-order tensor function fðUÞ which is coaxial with the right stretch tensor U and has the principal values f ðka Þ. Therefore, we can define the general strain tensor in the spectral decomposition as follows: 3 P

EðmÞ ¼

a¼1

f ðka ÞNðaÞ  NðaÞ ¼

3 P a¼1

2m 1 2m ðka

 1ÞNðaÞ  NðaÞ

ð4:48Þ

In addition, for the left stretch tensor V, we can define the following strain tensor.

f(

m =1

)

m =1/2

1.5

1

m=0 m = 1/2 m= 1

0.5

0

0.5

1

2

0.5

1

Fig. 4.2 Relation of principal strain to principal stretch in generalized strain tensor

4.2 Strain Tensors

125

eðmÞ ¼

3 P

f ðka ÞnðaÞ  nðaÞ ¼

a¼1

¼

3 X

3 P a¼1

2m 1 2m ðka

 1ÞnðaÞ  nðaÞ

f ðka ÞRNðaÞ  RNðaÞ ¼ RfðUÞRT

ð4:49Þ

a¼1

noting Eq. (1.108). In the particular case of m ¼ 0, noting ka [ 0, the strains defined by the following equation are called the logarithmic or Hencky strain tensor. Lagrangian-logarithmic strain tensor: Eð0Þ ¼

3 P a¼1

Eulerian-logarithmic strain tensor: eð0Þ ¼

3 P a¼1

lnka NðaÞ  NðaÞ  ln U ¼ 12 ln C

lnka nðaÞ  nðaÞ  ln V ¼ 12 ln b

ð4:50Þ where ka ¼ Ua ¼ Va ¼

pffiffiffiffiffiffi pffiffiffiffiffi Ca ¼ ba

ð4:51Þ

Ua ; Ca ; Va and ba being the principal values of U; C; V and b, respectively. When the principal directions of U and V are fixed, the following equations hold in these directions. ka ¼ 

ð0Þ



ðE Þa ¼ ð e

ð0Þ

 Þa ¼

@xa @Xa

@xa ðno sumÞ @Xa

  

@xa @Xa



ð4:52Þ



¼

@ xa ¼ ðln ka Þ  ¼ daa @xa

ðno sumÞ ð4:53Þ

where ln ka in Eq. (4.52) is the logarithmic (or Hencky or natural or true) strain and daa (no sum) is the principal component of the strain rate tensor defined in the next section. It follows from Eq. (4.50) that 8 3 3 3 X X > 1X ð0Þ > > ¼ ln k ¼ ln C ¼ ln Ua ¼ lnðU1 U2 U3 Þ trE a a > < 2 a¼1 a¼1 a¼1 3 3 3 > X X > 1X > ð0Þ > ¼ ln k ¼ ln b ¼ ln Va ¼ lnðV1 V2 V3 Þ tre : a a 2 a¼1 a¼1 a¼1

ð4:54Þ

126

4

Deformation/Rotation Tensors

which is nothing but the logarithmic volumetric strain trEð0Þ ¼ treð0Þ ¼

3 P a¼1

4.3

ln

@xa v ¼ ln J ¼ ln ¼ ev @Xa V

ð4:55Þ

Volumetric and Isochoric Parts of Deformation Gradient Tensor

The deformation gradient tensor F is defined as the ratio of the length of the infinitesimal-line element in the current configuration to that of the reference configuration. Therefore, it may not be additively decomposed but it is obliged to be multiplicatively decomposed. The multiplicative decomposition of the deformation gradient into the volumetric part Fvol and the volume preserving, i.e. isochoric (distortional) part F will be described in the following, where F ¼ gFvol F; Fvol  ðdetFÞ1=3 g ¼ FTvol ; F  ðdetFÞ1=3 F

ð4:56Þ

noting F ¼ gF ¼ ½ðdetFÞ1=3 g ½ðdetFÞ1=3 F ¼ Fvol F F ¼ gi  Gi ¼ gi  dij Gj ¼ gi  gi gj  Gj ¼ gF based on Eq. (3.9), where it holds that detFvol ¼ detF ¼ J; detF ¼ 1

ð4:57Þ

noting detFvol ¼ det½ðdetFÞ1=3 g ¼ ½ðdetFÞ1=3 3 detg detF ¼ det½ðdetFÞ1=3 F ¼ ½ðdetFÞ1=3 3 detF ¼ ðdetFÞ1 detF by Eq. (1.143)2 with s ¼ ðdet FÞ1=3 . Then, the tensor F for the isochoric part is called a unimodular tensor. Then, let the following decomposition of the right Cauchy-Green deformation tensor be defined.

4.3 Volumetric and Isochoric Parts of Deformation Gradient Tensor

127

C ¼ FT Cvol F Cvol  FTvol Fvol ¼ ðdetCÞ1=3 g;

ð4:58Þ

C  FT F ¼ ðdetCÞ1=3 C satisfying detCvol ¼ detC ¼ J 2 ; detC ¼ 1

ð4:59Þ

F ¼ g; trC ¼ 3; C0 ¼ O for F ¼ Fvol

ð4:60Þ

F ¼ ðdetFvol Þ1=3 Fvol ¼ ðdetFvol Þ1=3 ðdetFvol Þ1=3 g ¼ g

ð4:61Þ

by Eq. (1.143)2 . One has

noting

ð4:62Þ

The following partial derivatives hold for the volumetric and the isochoric tensors.

ð4:63Þ

noting @Cvol @ðdetCÞ1=3 g 1 @detC ¼ ¼ g  ðdetCÞ2=3 @C @C 3 @C 1 1 2=3 ðdetCÞC1 ¼ g  ðdetCÞ1=3 C1 ¼ g  ðdetCÞ 3 3

128

4

@C ¼ @C

h i @ ðdetCÞ1=3 C @C

Deformation/Rotation Tensors

 1 ¼ C   ðdetCÞ4=3 ðdetCÞC1 þ ðdetCÞ1=3 S 3

pffiffiffiffiffiffiffiffiffiffi @ detC 1 @detC 1 ¼ pffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffi ðdetCÞC1 @C @C 2 detC 2 detC pffiffiffiffiffiffiffiffiffiffi @ ln detC 1 1 ¼ C @C 2

h i 1=3 trC @trC @ ðdetCÞ 1 ¼  ðdetCÞ4=3 ðdetCÞC1 trC þ ðdetCÞ1=3 G ¼ @C @C 3 by virtue of Eqs. (1.355), (1.357), (1.369) and (4.58). The expression in terms of the principal stretches ka ða ¼ 1; 2; 3Þ will be shown below. trF ¼ k1 þ k2 þ k3 ; detF ¼ k1 k2 k3

ð4:64Þ

trC ¼ k21 þ k22 þ k23 ; detC ¼ ðk1 k2 k3 Þ2

ð4:65Þ

The principal stretch ka is described in terms of the volumetric part kvol and the isochoric part ka as follows: ka ¼ kvol ka

ð4:66Þ

kvol  ðk1 k2 k3 Þ1=3 ¼ J 1=3 ; ka  ka =J 1=3

ð4:67Þ

where

trF ¼ k1 þ k2 þ k3 ¼ ðk1 þ k2 þ k3 Þ=J 1=3 ; detF ¼ k1 k2 k3 ¼ k1 k2 k3 Þ=J ¼ 1 ð4:68Þ trC ¼ k21 þ k22 þ k23 ¼ ðk21 þ k22 þ k23 Þ=J 3=3 ; detC ¼ ðk1 k2 k3 Þ2 ¼ 1

ð4:69Þ

The invariants of C are defined by 9 IC  trC ¼ k21 þ k22 þ k23 > > > = 1 2 2 2 2 2 2 2 2 IIC  ðtr C  trC Þ ¼ k1 k2 þ k2 k3 þ k3 k1 > 2 > > ; 2 2 2 IIIC  detC ¼ k1 k2 k3

ð4:70Þ

4.3 Volumetric and Isochoric Parts of Deformation Gradient Tensor

with

9 @IC @ trC > > ¼ ¼ I > > @C @C > > = 2 1 2 @ 2 ðtr C  tr C Þ @IIC ¼ ¼ IC I  C > @C @C > > > > @IIIC @detC 2 1 > ¼ ¼ IIC G  IC C þ C ¼ IIIC C ; @C @C

129

ð4:71Þ

noting Eqs. (1.354) and (1.355). The invariants of the isochoric part are given by 9 > IC  trC ¼ k21 þ k22 þ k23 > > = 1 2 2 2 2 2 2 2 2 IIC  ðtr C  trC Þ ¼ k1 k2 þ k2 k3 þ k3 k1 > 2 > > ; 2 2 2 IIIC  detC ¼ k1 k2 k3

ð4:72Þ

noting tr2 C ¼ ðk21 þ k22 þ k23 Þ2 ; trC2 ¼ k41 þ k42 þ k43 , and their derivatives are given by 9 @IC @trC > > ¼ ¼I > > > @C @C > > > > = 2 1 2 @IIC @ 2 ðtr C  trC Þ ð4:73Þ ¼ ¼ IC I  C > @C @C > > > > > > @IIIC @detC 2 1 > ; ¼ ¼ II C G  IC C þ C ¼ III C C > @C @C

4.4

Strain Rate and Spin Tensors

The idealized deformation process in which the deformation is uniquely determined by the state of stress independent of the loading path is called the elastic deformation process. To describe it, it is required to introduce the strain tensor describing the deformation from the initial state and relate it to the stress. Here, since the superposition rule does not hold in the strain tensor, the null stress state is chosen usually as the reference state of strain. On the other hand, the deformation is not determined uniquely by the state of stress depending on the loading path and thus it cannot be related to the stress in the irreversible deformation process, e.g. the viscoelastic, the plastic and the viscoplastic loading processes. Therefore, it is obligatory to relate the infinitesimal changes of stress and deformation and to integrate them along the loading path in order to capture the current states of stress and deformation.

130

4

Deformation/Rotation Tensors

Here, introduce the velocity gradient tensor in Eq. (2.30), i.e. l 

@v @vi ; lij   @ j vi @xj @x

ð4:74Þ





Noting F ¼ @ x =@X ¼ @v=@X ðdv ¼ F dXÞ and the chain rule of derivative, Eq. (4.74) can be rewritten as Eq. (2.30), i.e l¼

@v @X  1  @vi @XA ¼ F F ðF ¼ lFÞ; lij ¼ @XA @xj @X @x ðdxÞ ¼ ldx

ð4:75Þ ð4:76Þ

noting ðdxÞ ¼ dv ¼

@v dx @x

ðdxÞ , i.e. dv describes the rate of the infinitesimal line element, i.e. the relative velocity between the velocities of material particles in both sides of infinitesimal line element dx. Here, we can choose the time s ð  tÞ to be arbitrary, resulting in 

l ¼ s FðtÞs F1 ðtÞ because the velocity gradient tensor l is substantially independent of the reference infinitesimal line element dX but dependent only on rates of deformation and rotation relative to a deformed configuration. Now, choosing the current state for the reference state leading to t F1 ðtÞ ¼ I, the velocity gradient tensor l can be expressed in the updated Lagrangian description as follows: 

l ¼ t FðtÞ

ð4:77Þ

Further, taking the time-derivative of Eq. (4.24) and noting t RðtÞ ¼ t UðtÞ ¼ ¼ I, it follows that

t VðtÞ



t F ðtÞ









¼ t UðtÞ þ t RðtÞ ¼ tV þ t RðtÞ

ð4:78Þ

Decomposing l additively into the symmetric and the skew-symmetric parts and noting Eqs. (4.75)–(4.78), it is obtained that l ¼ dþw

ð4:79Þ

4.4 Strain Rate and Spin Tensors

131

where 9 "  T #   > 1 @v @v > d  12 ðl þ l Þ ¼ þ ¼ t UðtÞ ¼ t VðtÞ > > > = 2 @x @x T

> > > > > ;

  1 @vi @vj dij  þ 2 @xj @xi 9 "  T #  > 1 @v @v > w  12 ðl  l Þ ¼  ¼ t RðtÞ > > > = 2 @x @x

ð4:80Þ

T



wij 

1 @vi @vj  2 @xj @xi

> > > > > ;



ð4:81Þ

where d is called the strain rate tensor or deformation rate tensor or stretching and w is called the (continuum) rotation rate tensor or continuum spin tensor. Here, note that d is not a time-derivative of any strain tensor but is defined independently as the rate variable although it is called the strain rate tensor. In addition, note that the time-integration of d cannot lead to any deformation measure in general, because it concerns with different material line-elements which rotate with material. Only the time-integration of axial component of d coincides with the axial component of the Hencky strain in Eq. (4.50) if the axial direction is fixed. Substituting Eqs. (4.1) and (4.75) into Eqs. (4.80) and (4.81), d and w are described by U; R as follows: 9  1  1 1  1 T  T > 1 T d ¼ ½ F F þ ðF F Þ  ¼ fðRUÞ ðRUÞ þ ðRUÞ ½ðRUÞ  g > > > > 2 2 > > >   1 > 1 1 T > = ¼ RðU U þ U U ÞR 2  1  1 > > w ¼ ½F F1  ðF F1 ÞT  ¼ fðRUÞ  ðRUÞ1  ðRUÞT ½ðRUÞ T g > > > 2 2 > > >    > 1 > T 1 1 T ; ¼ RR þ RðUU  U U ÞR 2

ð4:82Þ

Consequently, we obtain 

~ RT d ¼ Rsym½U

9 > =

 > ~ RT ; w ¼ X þ Rant½U R

ð4:83Þ

132

4

Deformation/Rotation Tensors

where 



e  U U1 U 

XR  R R T

ð4:84Þ ð4:85Þ

XR is the relative spin appeared in Eq. (3.38). The following relation holds in the pure rotation ðFð¼ @x=@XÞ ¼ R; dx ¼ RdXÞ. 



ðdxÞ ¼ Rd X ¼ R RT dx ¼ XR dx ¼ x dx

ð4:86Þ

Therefore, XR designates the spin of the line element during the pure rotation, while x is the axial vector to XR . Further, d and w are described by V; R as follows: 9 1 > 1 T T >   d ¼ fðVRÞ ðVRÞ þ ðVRÞ ½ðVRÞ  g > > > 2 > > >    1  1 1 T T T T 1 T T > = ¼ ðV V þ V V Þ þ ðV R R V  V R R V Þ > 2 2 1 > > > w ¼ fðVRÞ ðVRÞ1  ðVRÞT ½ðVRÞ T g > > 2 > > >     1 1 T 1 T T T 1 T T > ; ¼ ðV R R V þ V R R V Þ þ ðV V  V V Þ > 2 2 and thus

9  R > ~ ~ d ¼ sym½V þ sym½X  =  > ~ R  þ ant½V ~ ; w ¼ ant½X

ð4:87Þ

ð4:88Þ

where 



~  V V1 V

ð4:89Þ

~ R  VXR V1 X

ð4:90Þ

Equations (4.83) and (4.88) mean that d is not a pure rate of deformation and w is not a pure rate of rotation. It follows from Eq. (4.12) that  L

XL  R RLT ¼

3 X a;b¼1

ðNðaÞ  ea Þ ðNðbÞ  eb ÞT

4.4 Strain Rate and Spin Tensors

¼

3 X

133



ðN ðaÞ  eðaÞ þ NðaÞ  eðaÞ Þ eðbÞ  NðbÞ

a;b¼1

¼

3 X



ðaÞ

N

 NðaÞ

ð4:91Þ

a¼1

and thus one has 

ðaÞ

N

¼ XL NðaÞ

ð4:92Þ

 ðaÞ

noting e ¼ 0 since feðaÞ g is the fixed base. Therefore, XL describes the spin of the Lagrangian principal triad fNðaÞ g of the right stretch tensor U and is called the Lagrangian spin tensor. The components of XL in the Lagrangian triad fNðaÞ g are described as  ðbÞ

XLab ¼ NðaÞ  XL NðbÞ ¼ NðaÞ  N

ð4:93Þ

On the other hand, it follows from Eq. (4.12) that  E

XE  R RET ¼

3 X

 T nðaÞ  eðaÞ  nðbÞ  eðbÞ

a;b¼1

¼

3 X  nðaÞ  nðaÞ

ð4:94Þ

a¼1

and thus one has  ðaÞ

n

¼ XE nðaÞ

ð4:95Þ

Therefore, XE describes the spin of the Eulerian principal triad fnðaÞ g of the right stretch tensor V and is called the Eulerian spin tensor. The components of XE in the Eulerian triad fnðaÞ g are described as 

XEab ¼ nðaÞ  XE nðbÞ ¼ nðaÞ  n ðbÞ

ð4:96Þ

It follows from Eq. (4.9) that 

F ¼

3 X a¼1

(

 )  a a a ðaÞ k a n  N þ ka n  N þ n  N 

a

a

ð4:97Þ

134

4

Deformation/Rotation Tensors

which is rewritten by Eqs. (4.92) and (4.95) as 

F ¼

3 X



ka nðaÞ  NðaÞ þ XE F  FXL

ð4:98Þ

a¼1

or by Eqs. (4.93) and (4.96), Eq. (4.97) leads to 

F ¼

3 X



ka nðaÞ  NðaÞ þ

a¼1

3 X

 kb XEab  ka XLab nðaÞ  NðbÞ

ð4:99Þ

a;b¼1

Here, it holds that 

R RT ¼

3  X nðaÞ  NðaÞ



T nðbÞ  NðbÞ

a;b¼1

¼

 3  X  ðaÞ  NðbÞ  nðbÞ n ðaÞ  NðaÞ þ nðaÞ  N a;b¼1

    3 X  ðcÞ  ðaÞ ðaÞ ðaÞ ðaÞ ðcÞ ðbÞ ðbÞ N n ¼ n  n þn  N N N a;b;c¼1

¼

  ðcÞ   3 X   ðaÞ n  nðaÞ  nðaÞ  NðaÞ N  NðcÞ NðbÞ  nðbÞ a;b;c¼1

and thus the following relations hold. XR ¼ XE  RXL RT ; XE ¼ XR þ RXL RT ;

  XL ¼ R T XE  XR R

ð4:100Þ

The velocity gradient is described noting Eq. (4.9) as l¼

3 X a¼1

¼

! ðaÞ

ka n

N

ðaÞ

3 X 1 ðbÞ N  nðbÞ k b¼1 b

 3 3  X  ðaÞ X 1 ðbÞ  ðaÞ ka nðaÞ  NðaÞ þ ka n  NðaÞ þ ka nðaÞ  N N  nðbÞ k b a¼1 b¼1

0 1   3 X ka  @ka nðaÞ  nðaÞ þ nðaÞ  nðaÞ A þ ¼ N ðaÞ  NðbÞ nðaÞ  nðbÞ ka k a¼1 a;b¼1 b 3 X

ð4:101Þ

4.4 Strain Rate and Spin Tensors

135

which is rewritten using Eqs. (4.93) and (4.96) as 

l¼

3 X ka a¼1

ka

n

ðaÞ

ðaÞ

n

 3  X ka L E þ Xa;b  Xab nðaÞ  nðbÞ kb a;b¼1

ð4:102Þ

from which the strain rate and the continuum spin are represented as 

3 X k

d¼

a¼1

w¼

a ðaÞ

ka

3 X a;b¼1

 nðaÞ þ

3 k2  k2 X a b

XLab nðaÞ  nðbÞ

ð4:103Þ

! 2 2 k þ k a b XEab  XL nðaÞ  nðbÞ ða 6¼ bÞ 2ka kb ab

ð4:104Þ

n

a;b1

2ka kb

noting 

d¼

3 X ka a¼1

2 sym

ka

ðaÞ

n

" 3 X

n

ðaÞ

 3   k  1X ka ðaÞ ðbÞ b ðbÞ ðaÞ nðaÞ  nðbÞ þ N N N N þ ka 2 a;b¼1 kb



n ðaÞ  nðaÞ  ¼

a¼1

w¼

3 1X 2 a¼1

þ

 3 X   n ðaÞ  nðaÞ þ nðaÞ  nðaÞ ¼

3 X

a¼1

a¼1

! nðaÞ  nðaÞ



¼I¼O

! 



n ðaÞ  nðaÞ  nðaÞ  n ðaÞ     3  1X ka  ðaÞ kb  ðbÞ ðaÞ nðaÞ  nðbÞ N  NðbÞ  N N ka 2 a;b¼1 kb

From the relation dab ¼

k2a  k2b L X ða 6¼ bÞ 2ka kb ab

ð4:105Þ

obtained from the anti-symmetric part in Eq. (4.103), the Lagrangian spin is written as XLab ¼

2ka kb dab ða 6¼ bÞ k2b  k2a

ð4:106Þ

The Eulerian spin is given from Eq. (4.104) as follows: XEa;b ¼ wab 

k2a þ k2b k2a  k2b

dab ða 6¼ bÞ

ð4:107Þ

136

4 







Deformation/Rotation Tensors  ðaÞ

~ ¼V¼V ~ ¼ O; N In the rigid-body rotation (U ¼ U Eqs. (4.79), (4.83), (4.91) and (4.100) that l ¼ w ¼ XR ¼ X E ;

¼ O), it follows from

XL ¼ O

ð4:108Þ

In what follows, we consider the physical meanings of d and w. The relative velocity of the particle points P and P0 , the current position vectors of which are x and x þ dx, respectively, is given by dv ¼ ldx

ð4:109Þ

from Eq. (4.76) and it is additively decomposed as dv ¼ dvd þ dvw

ð4:110Þ

dvd  ddx

ð4:111Þ

dvw  wdx

ð4:112Þ

where

The following equation is obtained for the infinitesimal line-element dx ¼ dxi ei ðno sumÞ. dj€i ¼

dvdj (no sum) dxi

ð4:113Þ

noting dji ¼ ej  dei ¼ ej  d

dx dvd dvdj ¼ ej  ¼ dxi dxi dxi

ð4:114Þ

with the aid of Eqs. (1.107) and (4.111). Therefore, dji is the ej -component of the relative velocity dvd of the unit line element ðdxi ¼ 1Þ in the ei -direction. Consequently, the infinitesimal line-element dx ¼ dxi ei ðno sumÞ rotates in the velocity given by the tangential component dji ðj 6¼ iÞ of the strain rate d. On the other hand, denoting the axial vector described in Eqs. (1.187) and (1.188) _ for the skew-symmetric tensor w by x, it holds that 1 _ xi ¼  eirs wrs ; 2

_

wij ¼ eijr xr

and thus Eq. (4.112) is rewritten by Eq. (1.190) as

ð4:115Þ

4.4 Strain Rate and Spin Tensors

137

 _ _ dvwi ¼ wis dxs ¼ eisr xr dxs ¼ eirs xr dxs

_

dvw ¼ x dx;

ð4:116Þ

Therefore, the arbitrary line-element dx rotates in the peripheral velocity dvw _ and angular velocity x, termed often the spin vector, by the continuum spin w, _ whereas 2x is called the vorticity. Noting Eqs. (4.110)–(4.116), it follows for the material line element dx1 e1 that dv ¼ lij ei  ej dx1 e1 ¼ li1 dx1 ei ¼ ðd21 þ w21 Þdx1 e2 þ d11 dx1 e1 ¼ ðd12  w12 Þdx1 e2 þ d11 dx1 e1 ¼ -3 dx1 e2 þ d11 dx1 e1

ð4:117Þ

as shown in Fig. 4.3, where we set ^ 3 þ d12 -3  w12 þ d12 ¼ x

ð4:118Þ _

-3 designates the clock-wise angular velocity of the line-element. x 3 denotes the average angular velocity of the line-elements in the plane, which coincides with the angular velocity of the line element in the principal directions of strain rate fulfilling d12 ¼ 0. By choosing dn ; dt for Tn ; Tt described in Sect. 1.14, the relation of the rate of elongation and the rate of rotation is shown in Fig. 4.4. It is depicted by the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 centering in circle of relative velocity with the radius ½ðd11 þ d22 Þ=22 þ d12 _

ððd11 þ d22 Þ=2; x3 Þ in the two-dimensional plane ðdn ; -3 Þ. The parallelepiped in the _ principal directions of the strain rate d rotates by the angular velocity x as shown in Fig. 4.5. The rate of the scalar product of the vectors dx and dx of the infinitesimal elements connecting the three points P; P0 ; P00 with the position vectors x; x þ dx; x þ dx, respectively, is given noting Eq. (1.139) as follows:

(

dv w = 3 dx1e2 = w12 dx1e2

x2 dv (

3

dv d

d12

e2

0

e1

dx = dx1e1

Fig. 4.3 Extension and rotation of the line-element

dv2d = d12dx1e 2

x1 dv1d = d11dx1e1

138

4 3

=

3

Deformation/Rotation Tensors

dt = w12 dt (d 11,

3

3

d12)

= w12

(d 22 ,

3

0

d12 )

d2

d1

d11 d 22 2

dn

Fig. 4.4 Circle of relative velocity

(= )

d 3 dx3 x3

d 2 dx2 x2

d1dx1

x1 Fig. 4.5 Deformation and rotation for principal directions of strain rate

ðdx  dxÞ  ¼ dv  dx þ dx  dv @v @v dx  dx þ dx  dx ¼ ¼ @x @x

("

 T # ) @v @v þ dx  dx @x @x

leading to ðdx  dxÞ ¼ 2ddx  dx

ð4:119Þ

If the vicinity of the particle P undergoes the rigid-body rotation, the quantity in Eq. (4.119) for an arbitrary scalar quantity dx  dx is zero and thus d ¼ O has to

4.4 Strain Rate and Spin Tensors

139

hold. Inversely, if d ¼ O, the quantity in Eq. (4.119) for the scalar quantity dx  dx of arbitrary line-element vectors becomes zero and thus it can be stated that the vicinity of the particle P does not undergo a deformation. Then, d ¼ O is the necessary and the sufficient condition for the situation that a deformation is not induced, allowing only a rigid-body rotation. Denoting the lengths of the line-elements PP0 and PP00 as ds and ds, respectively, and the angle contained by them as h, it holds that ðdx  dxÞ ¼ ðdsdscos hÞ  ¼

ðdsÞ ðdsÞ þ ds ds



  cos h  hsin h dsds

ð4:120Þ

Further, denoting the unit vectors in the directions of the line-elements PP0 and PP as l and m, respectively, and noting dx ¼ lds, dx ¼ mds, it holds from Eqs. (4.119) and (4.120) that 00



ðdsÞ ðdsÞ þ ds ds



   cos h  hsin h ¼ 2dl  v ¼ 2dij li vj

ð4:121Þ

If the particles P0 and P00 chosen in same direction ðh ¼ 0Þ, it follows from Eq. (4.121) that ðdsÞ ¼ dl  l ds

ð4:122Þ

The left-hand side of Eq. (4.122) designates the rate of elongation of the line-element. Therefore, the rate of elongation is given by the normal component of d in the relevant direction, noting Eq. (1.107). On the other hand, choosing the line-element PP00 to be perpendicular to the line element PP0 ðh ¼ p=2Þ, it follows from Eq. (4.121) that h  ¼ 2dl  m

ðl  m ¼ 0Þ

ð4:123Þ

The left-hand side of Eq. (4.123) designates the decreasing rate of the angle contained by the two line-elements mutually perpendicular instantaneously and is called the shear strain rate.   Next, the relations of the rate E of Green strain tensor E and the rate e of the Almansi strain tensor e to the strain rate tensor d are formulated below. The material-time derivative of Eq. (4.26) is given by 

ðdx  dxÞ ¼ 2 E dX  dX

ð4:124Þ

140

4

It follows from Eqs. (4.119) and (4.124) that

Deformation/Rotation Tensors

: 

ddx  dx ¼ dFdX  FdX ¼ E dX  dX

ð4:125Þ

from which, noting Eq. (1.139), we have the relation of Green strain tensor to the strain rate tensor as follows: 



1

E ¼ FT dF; d ¼ FT E F

ð4:126Þ

which is obtained also from

ð4:127Þ

Next, the time-differentiation of Eq. (4.27)2 leads to ð4:128Þ 

Here, it follows from ðFF1 Þ ¼ FðF1 Þ þ F F1 ¼ O with Eq. (4.75) that 

F1



¼ F1 l

ð4:129Þ

which is derived also by      1  @F1 @F1 @ @X @ @X þ v¼ F ¼ þ v @t @x @t @x @x @x      @ @X @ @X @X @v @ @X @X @X @v ¼ þ v  ¼ þ v  @x @t @x @x @x @x @x @t @x @x @x ¼

@  @X @v @X @v X  ¼ @x @x @x @x @x

ð4:130Þ

noting that the inside of the bracket ½  in Eq. (4.130) is the material-time derivative of the initial position vector X and thus it is zero.

4.4 Strain Rate and Spin Tensors

141

Substituting Eq. (4.129) into Eq. (4.128), one has 

i 1 h 1 T 1 F l F þ FT F1 l 2      1 1 1 T 1 T 1 T 1 I IF F I IF F l ¼l þ 2 2 2 2     1 1 ¼ lT Ie þ Ie l 2 2

e ¼

ð4:131Þ

from which one has the relation of the rate of the Almansi strain tensor to the strain rate tensor: 

e ¼ d  lT e  el

ð4:132Þ

Equation (4.132) is rewritten as 

e ¼d

      1  1  l þ lT  l  lT e  e l þ lT þ l  lT 2 2

and thus we obtain

w

e ¼ d  de  ed

ð4:133Þ

where



ew  e  we þ ew

ð4:134Þ

is called the Zaremba-Jaumann rate of Almansi strain tensor, while the Zaremba-Jaumann rate will be explained in Sect. 4.4. 





E ¼ e ¼ e w ¼ d holds in the initial state ðF ¼ I; E ¼ e ¼ OÞ and thus all the strain rates mutually coincide by Eqs. (4.126), (4.127), (4.132) and (4.134).

4.5

Logarithmic (True) and Infinitesimal (Nominal) Strains

In what follows, we introduce the following notation. 

e  d; de  ddt

ð4:135Þ

142

4

Deformation/Rotation Tensors

If the direction of the material line-element always coincides with the xi -axis, the principal strain rate in this direction is given by 

di ¼ e i ¼

     @vi @ui @ui @ui @ui (no sum) ¼ ¼ ¼ = 1þ @xi @xi @ ðXi þ ui Þ @Xi @Xi

ð4:136Þ

The time-integration of Eq. (4.136) leads to   @ui @xi ð0Þ ð0Þ ¼ ln ei ¼ ln 1 þ ¼ ln ki ¼ Ei ¼ ei ¼ lnð1 þ ei Þ (no sum) @Xi @Xi ð4:137Þ Therefore, the time-integration ei of principal strain rate di does not coincide with the principal infinitesimal strain ei in Eq. (4.32). Setting @Xi ! l0 ; @xi ! l; @ui ! l  l0 , where l0 and l are the lengths of the line-element in the initial and the current states, respectively, it follows that 9 l > ¼ ln ð 1 þ e Þ i = l0 l  l0 > ; ei ¼ 0 l ei ¼ ln

ð4:138Þ

Note that the logarithimic axial strain and the infinitesimal strain are denoted by the italic letter ei and the roman letter ei , respectively, while it needs the caution to distinguish them. Consequently, the time-integration of principal component of strain rate tensor d and the Hencky strain tensor Eð0Þ and eð0Þ coincide with ei which is called the logarithmic (or natural) strain, provided that their principal directions are fixed. On the other hand, the principal value of infinitesimal strain tensor e does not coincide with them and it is called the nominal strain. It follows from Eq. (4.138) that ei ¼ 1ne 2 ðffi 0:693Þ for l ¼ 2l0 and ei ¼ 1 for l ¼ 0 ei ¼ þ 1 for l ¼ 2l0 and ei ¼ 1 for l ¼ 0

) ð4:139Þ

Therefore, the magnitude of nominal strain in the deformation that the material length becomes zero, i.e. the material diminishes is identical with that in the deformation that the material length becomes only twice. As a practical example, about 5% error is induced in the nominal strain for 10% elongation as known from ei =ei ¼ 0:1= lnð1:1Þ ¼ 1:049 for l ¼ 1:1 l0 . This property would cause the inconvenience for the adoption in constitutive equation for the wide range of deformation.

4.5 Logarithmic (True) and Infinitesimal (Nominal) Strains

143

Further, one has Z

ln l0

dl ¼ l

Z

l1 l0

dl þ l

Z

l2 l1

dl þ  þ l

Z

ln

dl ln1 l

i.e. ln

ln l1 l2 ln ¼ ln þ ln þ    þ ln l0 l0 l1 ln1

On the other hand, one sees ln  l0 l1  l0 l2  l1 ln  ln1 ¼ 6 þ þ    þ l0 l0 l1 ln1 Thus, it follows for the superposition of strains that n e0i n ¼ e0i 1 þ e1i 2 þ    þ en1 i

) ð4:140Þ

n e0i n 6¼ e0i 1 þ e0i 2 þ    þ en1 i

while eai b and eai b designate the longitudinal strain in the xi -direction when the length of the line-element changes from la to lb , provided that the principal direction of strains are fixed. Consequently, the superposition rule holds in the logarithmic strain but it does not hold in the nominal strain. Furthermore, one has ln

l1 l2 l3 l1 l2 l3 ¼ ln 0 þ ln 0 þ ln 0 l01 l02 l03 l1 l2 l3

l1 l2 l3  l01 l02 l03 l1  l01 l2  l02 l3  l03 6¼ þ þ l01 l02 l03 l01 l02 l03 where l1 ; l2 ; l3 are the lengths of line-elements in the directions of three fixed principal strains. Thus, it follows for the sum of the principal strains that ev ¼

ln Vv

9 > > ¼ ln J ¼ ln F ¼ ei ¼ lnð1 þ ev Þ > > = i¼1

3 vV X 6¼ ei ev ¼ V i¼1

3 P

> > > > ;

ð4:141Þ

where V and v are the initial and the current volumes, respectively, of material. Therefore, the sum of logarithmic strains in orthogonal directions coincides with the logarithmic volumetric strain but the sum of nominal strains in orthogonal

144

4

Deformation/Rotation Tensors

directions does not coincide with the nominal volumetric strain. Equation (2.21) is exploited in Eq. (4.141). Further, it follows from Eqs. (4.136) and (4.141) that 

ev ¼ dv ¼ trd ¼







v v @vr  J ¼ ¼ 6¼ e v ¼ @xr v J V

ð4:142Þ

Therefore, the volumetric strain rate trd coincides with the material-time derivative of the logarithmic volumetric strain em but it does not coincide with that of the nominal volumetric strain em . Consequently, the nominal strain is applicable only to the description of infinitesimal deformation, while the logarithmic strain is applicable to constitutive equations for finite deformation.

Chapter 5

Stress Tensors and Conservation Laws

Conservation laws of mass, momentum, angular momentum, etc. must be fulfilled during a deformation. These laws are described first in detail. Then, the Cauchy stress tensor is defined and further, based on it, various stress tensors are derived from the Cauchy stress tensor. Introducing the stress tensor, the equilibrium equations of force and moment are formulated from the conservation laws. The virtual work principle required for the analyses of boundary value problems are also described. Further, the work-conjugate pairs of the stresses and the deformation rate tensors are explained, which must be adopted in the formulation of constitutive equations.

5.1

Stress Tensor

When the infinitesimal force vector df applies to the surface with the infinitesimal area da and the unit normal vector n, the stress vector, i.e. traction t is given as t


df da

ð5:1Þ

Now, introduce the following second-order tensor r fulfilling the relation t ¼ rn; ti ¼ rij nj

ð5:2Þ

by the quotient law described in Sect. 1.3.2. The following relation holds from Eqs. (5.1) and (5.2) df ¼ rnda; dfi ¼ rij nj da

ð5:3Þ

Equation (5.2) is called the Cauchy’s fundamental theorem or Cauchy’s stress principle. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 K. Hashiguchi, Foundations of Elastoplasticity: Subloading Surface Model, https://doi.org/10.1007/978-3-030-93138-4_5

145

146

5

Stress Tensors and Conservation Laws

The components of the tensor r are given by Eq. (1.107) as rij ¼ ei  rej ; r ¼ rij ei  ej

ð5:4Þ

As will be verified by the equilibrium of the angular moment in Sect. 5.6 that r is the symmetric tensor, and thus one has rij ¼ rei  ej

ð5:5Þ

from Eq. (5.4). Here, when we choose ei to the unit normal vector n of the surface on which t applies, the following equation holds by substituting Eq. (5.2) with n ¼ ei , i.e. t ¼ rn ¼ rei into Eq. (5.5). rij ¼ ti  ej

ð5:6Þ

where the expression ti specifies the stress vector applying to the unit surface possessing the unit normal vector ei ð¼ nÞ. Therefore, rij can be interpreted as the component in the direction ej of the stress vector ti applying to the unit surface element having the outward-normal vector ei . The tensor r is called the Cauchy stress tensor. The six independent components of r exist for the orthonormal surface base vectors ðe1 ; e2 ; e3 Þ by the symmetry, which is described in the matrix form as 2 4

r11

3 r12 r13 r22 r23 5 Sym: r33

The Cauchy’s fundamental theorem and the components of the Cauchy stress tensor are illustrated in Fig. 5.1. The physical meaning of the stress tensor in the general coordinate system is explained comprehensively in Hashiguchi (2020). Various stress tensors are defined from the Cauchy stress tensor described above. Some of them, which are often used in continuum mechanics, are presented below. The tensor s defined by the following equation is called the Kirchhoff stress tensor. s ¼ Jr

ð5:7Þ

The vector T defined by the following equation is called the nominal stress vector. T


df dA

ð5:8Þ

5.1 Stress Tensor

147

n = e3 33

df = tda

n

t3

da 31

σda

13

32

23

t2

t1 22 12

11

n = e2

21

n = e1 t = σn

ij

(a) Cauchy’s fundamental theorem

ti e j

ji

(b) Components of Cauchy stress

Fig. 5.1 Cauchy’s fundamental theorem and components of Cauchy stress tensor

The stress tensor P, which is related to T by the following equation, is called the first Piola-Kirchhoff stress tensor which is the Eulerian-Lagrangian two-point tensor. T
PN ðTi
PiA NA Þ

ð5:9Þ

where N is the unit outward-normal vector on the surface in the reference configuration. Here, substituting Eqs. (2.24) and (5.9) into Eq. (5.8), we have 1 1 df
PNdA ¼ PFT nda; dfi ¼ PiA NA dA ¼ PiA FrA nr da J J

ð5:10Þ

On the other hand, the substitution of Eq. (5.2) into Eq. (5.1) yields df ¼ rnda

ð5:11Þ

It follows from Eqs. (5.10) and (5.11) that 1 df ¼ rnda ¼ PFT nda J

ð5:12Þ

The relation of r and P is given from Eq. (5.12) as follows: 9 1 1 1 1 r ¼ FPT ð¼ PFT Þ; rij ¼ FiA PjA ¼ PiA FjA = J J J J ; T T T P ¼ JrF ¼ sF ð6¼ P Þ; PiA ¼ Jrir ðF1 ÞAr   ^g P ¼ Pij gi  Gj
sij gi  Gj ¼ s ^gG ¼ s G ¼ sFT

ð5:13Þ

148

5

df = (σda) n

df = (P dA)N N

n

F

dA P (= FS)

Stress Tensors and Conservation Laws

σ da

dX dx = F dX

dv

dV

Fig. 5.2 Cauchy stress and first Piola-Kirchhoff stress

Equation (5.12) is illustrated in Fig. 5.2. The stress tensor P is called the first Piola-Kirchhoff stress tensor, noting PN ¼ df=dA applied to the reference unit area vector. Incidentally, it is the pull-back of the Cauchy stress tensor only from the right in Eq. (3.25)2 and thus it is the contravariant two-point tensor with the covariant current and reference base vectors. The first Piola-Kirchhoff stress is called also the nominal stress while the Cauchy stress is called the true stress, since the former designates the force per the initial (reference) unit surface while the latter designates the force per the current unit surface. It is used when the rate form of the equilibrium equation in the current configuration is derived as will be shown in Sect. 5.5. Further, the stress tensor S defined by the following equation is called the second Piola-Kirchhoff stress tensor which is the Lagrangian tensor. T G ¼ SN

ð5:14Þ

where TG


F1 df ¼ F1 T dA

ð5:15Þ

is the contravariant pull-back of the nominal stress vector T. Using Eq. (2.24) into these equations, one has the following expression. 1 df
FSNdA ¼ FSFT nda J

ð5:16Þ

Comparing Eqs. (5.11) and (5.16), it follows that 1 1 r ¼ FSFT ; rij ¼ FiA SAB FBj J J 1 1 S ¼ F P ¼ JF rF T ¼ F1 sF T ð¼ST Þ;

9 = SAB ¼ JðF1 ÞAi rij ðF 1 ÞBj

;

ð5:17Þ

5.1 Stress Tensor

149

    S ¼ Sij Gi  Gj ¼F1 P ¼ F1 Pij gi  Gj ¼ sij Gi  Gj ¼ F1 sFT ¼ s GG

S is the full pull-back of the Cauchy stress tensor, and thus is based in the covariant reference base vectors. (Note) F1 df ¼ ð@XA =@xi ÞeA  ei dfj ej ¼ ð@XA =@xi Þdfi eA is the pull-back of df from the current to the reference configurations. One can pull-back df also by FT df as shown in Eq. (3.17). However, the stress tensor R ¼ FT rFT =J obtained by setting FT df=dA ¼ RN instead of Eq. (5.14) does not satisfy the symmetry, i.e. R 6¼ RT and thus it is inconvenient for the practical use of deformation analysis. The substitution of the decomposition of the Cauchy stress into the deviatoric and the spherical part r ¼ r0 þ rm I; rm
ð1=3Þtrr

ð5:18Þ

to Eq. (5.17)2 for the second Piola–Kirchhoff stress leads to S ¼ JF1 rFT ¼ JF1 r0 FT þ JF1 rm IFT i.e. S ¼ S0 þ JC1 rm

ð5:19Þ

S0 ¼ JF1 r0 FT

ð5:20Þ

trS0 6¼ 0; S0 :Cð¼ tr½ðJF1 rFT ÞðFT FÞT  ¼ Jtrr0 Þ ¼ 0

ð5:21Þ

where

Here, note that

Now, introduce the Mandel stress M ¼ CS ¼ JFT rFT ¼ FT sFT ð6¼ MT Þ

ð5:22Þ

 G  M ¼ Mij Gi  Gj
si j Gi  Gj ¼ FT sFT ¼ s G i.e. M ¼ M0 þ M m I

ð5:23Þ

150

5

Stress Tensors and Conservation Laws

where M0 ¼ JFT r0 FT ;

Mm ¼ Jrm

ð5:24Þ

Therefore, one has trM0 ¼ 0

ð5:25Þ

The Mandel stress is adopted in the multiplicative hyperelastic-based plasticity for the finite deformation as will be described in Chap. 17. The following stress is called the covariant convected stress tensor. cr


FT rF ð¼ c rT Þ;

cs


FT sF ð¼ c sT Þ

ð5:26Þ

The relations between the above-mentioned stress tensors as the transformations between the Lagrangian and the Eulerian tensors will be described in Sect. 5.9. The relations of various stress tensors defined above are summarized in Table 5.1. (Note) t


df df ;T
; da dA

TG


F1 df ¼ F1 T dA

t ¼ rn; Jt ¼ sn; T ¼ PN; T G ¼ SN

Table 5.1 Relations of various stress tensors Names, notations Cauchy r Kirchhoff s 1st Piola-Kirchhoff (Nominal) P 2nd Piola-Kirchhoff S Mandel M

rð¼ rT Þ

sð¼ sT Þ 1 s J

Jr

  P ¼ 6 PT

  S ¼ ST

  M ¼ 6 MT

1 T FP J PFT

1 FSFT J FSFT

1 T F MFT J FT MFT

FS

FT M

JrFT

sFT

JF1 rFT

F1 sFT

F1 P

JFT rFT

FT sFT

FT P

C1 M CS

5.1 Stress Tensor

5.2

151

Conservation Law of Mass

Denoting the field of material density as qðx; tÞ, the mass in a current volume v is R given as m ¼ v qðx; tÞdv which is kept constant because the mass neither flow into the volume element nor flow out from it. Therefore, the following conservation law of mass must hold. Z





m¼

¼0

qðx; tÞdv

ð5:27Þ

v

from which, noting Eq. (2.50), one has the continuity equation. 



q þ qdivv ¼ 0 ;

q þq

@vr ¼0 @xr

ð5:28Þ

Further, setting Tðx; tÞ
q/, where / is a physical quantity per unit mass, one has Z

 Z Z   @vr  dv ¼ q/ þ q/ þ q/ q /dv @xr v v

 q/dv

¼

v

ð5:29Þ

noting the Reynolds’ transportation theorem in Eq. (2.48) and Eq. (5.28).

5.3

Conservation Law of Linear Momentum

R The linear momentum in a current volume v is given by v qvdv. On the other hand, R the traction applied to the surface of the region is given as a tda and the body force R applied to the region is given by v qbdv, denoting the body force per unit mass as b. The rate of momentum has to be equivalent to the sum of the traction and the body force applied to the region. Therefore, the Euler’s first law of motion (or conservation law of momentum) is given as Z

 qvdv

Z

Z

¼

tda þ

v

qbdv

a

v

or R



v

by virtue of Eq. (5.29).

q v dv ¼

R a

tda þ

R v

qbdv

ð5:30Þ

152

5

5.4

Stress Tensors and Conservation Laws

Conservation Law of Angular Momentum

R The angular momentum in a current volume v is given as v qðx  vÞdv. On the other hand, since the angular momentum caused by the tractionR and the angular momentum caused by the body force are described by a ðx  tÞda and R v qðx  bÞdv, respectively, the Euler’s second law of motion, i.e. conservation law of angular momentum is described as Z

 qx  vdv

Z

Z

¼

x  tda þ

v

qx  bdv

a

Z

 qeijk xj vk dv

v

Z

Z

¼

eijk xj tk da þ

v

qeijk xj bk dv

a

v

which is reduced to R

R



v

qx  v dv ¼

Z

a

Z



k

R v

qx  bdv

Z

qeijk xj v dv ¼ v

x  tda þ

eijk xj tk da þ a

ð5:31Þ

qeijk xj bk dv v





noting ðx  vÞ ¼ v  v þ x  v ¼ x  v and Eq. (5.29).

5.5

Equilibrium Equation

Substituting Eq. (5.2) into Eq. (5.30) for the conservation law of momentum and noting Eq. (5.45), the following equation is obtained. Z



Z

Z

q v dv ¼ v

rT nda þ a

Z



Z

q vi dv ¼

qbdv; v

v

Z rir nr da þ

a

qbi dv

ð5:32Þ

v

The first term in the right-hand side of Eq. (5.32) is given by the Gauss’ divergence theorem in Eq. (1.392) as Z Z @rir rir nr da ¼ dv a v @xr By this equation the local form of Eq. (5.32) is given as 

rx r þ qb ¼ q v ;

@rij  þ qbi ¼ q vi @xj

ð5:33Þ

5.5 Equilibrium Equation

153

This equation is called the Cauchy’s first law of motion, i.e. the equilibrium equation. On the other hand, substituting Eqs. (2.21) and (5.12) into Eq. (5.32), one has Z Z Z  q0 v dV ¼ PNdA þ q0 bdV ð5:34Þ V

A

V

which is rewritten by the Gauss’ divergence theorem as follows: Z

Z



Z

q0 v dV ¼ V

PrX dV þ V

ð5:35Þ

q0 bdV V

where rX
ð@=@XA ÞeA ¼ @=@X. The local form of this equation is given as 

rX P þ q0 b ¼ q0 v ;

@PiA  þ q 0 bi ¼ q 0 v i @XA

ð5:36Þ

The equilibrium equation in a rate form is required in constitutive equations for irreversible deformation including elastoplastic deformation. The time-differentiation of Eq. (5.36) engenders the following rate-type (or incremental-type) equilibrium  equation, provided that the acceleration does not change, i.e. v ¼ 0. 





P rX þ q0 b ¼ 0 ;

 @ PiA þ q0 bi ¼ 0 @XA

ð5:37Þ

In order to describe Eq. (5.37) by the Cauchy stress, specifying 

D
1J PFT Pr



D 1  6¼P r T ¼ FPT J

 ð5:38Þ

and noting Eq. (5.13)2 , we have

By substituting

(due to Eq. (4.75)) and Eqs. (4.79) and (2.34) to this equation, we obtain D

P D



w

r ¼ r þ rtrl  rlT ¼ r þ rtrd  rd þ wr w

ð5:39Þ 

Therein, P r is referred to as the nominal stress rate, whereas r
r  w r þ r w is the Zaremba-Jaumann stress rate, nothing Eq. (3.40).

154

5

Stress Tensors and Conservation Laws

It follows that 

D

@P r 1 @ P ¼ @x J @X

ð5:40Þ

      D  @ FT =J @p r @ P FT =J 1@P T 1@ P ¼ F þP ¼ ¼ @x @x @x J @x J @X

ð5:41Þ

by virtue of

0

1      @ FjA =J 1 @PiA @xj 1 @PiA C B ij 1 @Pi;A ¼ Fj;A þ PiA ¼ ¼ @ A @xj @xj J @xj J @xj @XA J @XA D

@P r



with   @ FT =J ¼O @x

ð5:42Þ

where O is the third-order zero tensor, noting "      T # @ FT =J 1 @FT 1 @FT T @J T @ det F @F JF JF ¼ 2 ¼ 2 @x @x @x J @x J @F @x " #   1 @FT @F T J  FT JFT ¼O ¼ 2 @x J @x Substitution of Eq. (5.40) into Eq. (5.37) yields the rate-type equilibrium defined in the current configuration: 

D

P r rx þ q b ¼ 0 ;

D

 @P rij þ qbi ¼ 0 @xj

ð5:43Þ 

This equation is derived also by the following manner. From Eq. (5.32) with v ¼ 0, one has Z

 Z  Z  T q vdv ¼ r nda þ qbdv Z v  Za  Zv   T q0 v dV ¼ r nda þ q0 bdV V a V |fflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflffl} 

0

5.5 Equilibrium Equation

155

is Z

Z

rT ðndaÞ þ

r_ T nda þ

0¼

Z



q0 b dV a V Za Z Z 

¼ r_ T nda þ rT ðtrlÞI  lT nda þ q b dv a v Z Za   T  T T T q b dv r_ þ r trd  r l $x dv þ ¼ v

v

which results in Eq. (5.43), noting Eqs. (1.392), (2.38) and r ¼ rT which will be verified in the next section. The equilibrium equation in the rate form, e.g. Eq. (5.43) is not needed necessarily by checking the equilibrium for the current variables pushing-forwarded to the ones in the reference configuration calculated by the constitutive equation, noting that the boundary condition is given by the stress tensor in the current configuration.

5.6

Equilibrium Equation of Angular Moment

Substituting Eq. (5.2) into Eq. (5.31) of the conservation law of angular momentum and noting Eq. (5.45), one has Z

Z



Z

qeijk xj vk dv ¼ v

eijk xj rkr nr da þ a

qeijk xj bk dv

ð5:44Þ

v

Because the first term in the right-hand side of this equation is rewritten as Z

Z eijk xj rkr nr da ¼ a

a

@xj rkr eijk dv ¼ @xr

 Z  @rkr dv eijk rkj þ eijk xj @xr v

Equation (5.44) leads to Z  eijk rkj þ eijk xj v

  @rkr  dv ¼ 0 þ qbk  qvk @xr

Noting the equilibrium Eq. (5.33) to this equation, it holds that eijk rkj ¼ 0 (e.g. e123 r23 þ e132 r32 ¼ r23  r32 ¼ 0) from which we have the symmetry of Cauchy stress tensor, i.e. r ¼ rT ;

rij ¼ rji

ð5:45Þ

156

5.7

5

Stress Tensors and Conservation Laws

Virtual Work Principle

Giving the scalar product of the arbitrary velocity increment dv to the equilibrium Eq. (5.33), we find 

dv  rx r þ qdv  b ¼ qdv v

ð5:46Þ

Noting 

divðdvrÞ ¼ r : graddv þ dvdivr r : graddv ¼ r : dl ¼ r : dd

ð5:47Þ

Equation (5.46) leads to divðdvrÞ  r : dd þ qdv b ¼ qd

  1 vv 2

i.e.  r : dd ¼ divðdvrÞ þ þ qdv b  qd

1 vv 2

 ð5:48Þ

which is the local form of the virtual work equation in the current configuration. Now, consider the integration of Eq. (5.48) over the current volume. The integration of the first term in the right hand side in Eq. (5.48) yields Z

Z

Z

div ðdvrÞdv ¼

Z

ðdvrÞ  nda ¼

v

a

dv  ðrnÞda ¼ a

dv  tda

ð5:49Þ

a

by virtue of Eqs. (1.391), (5.2) and (5.45). Further, noting Eq. (5.49), the integration of Eq. (5.48) leads to Z

Z r : dddv ¼ v

Z t  dvda þ

a



Z qb  dvdv 

v

qd v

1 vv 2

 dv

ð5:50Þ

Incidentally, applying the velocity v instead of dv, one has the following equation analogously to Eq. (5.50). Z

Z r : ddv ¼ v

t  vda þ a

Z

Z qb  vdv  v

v

1 qv  vdv 2

 ð5:51Þ

The quantity in the left-hand side means the stress power, while the first, the second and the third terms in the right-hand side mean the powers done by the surface force (traction), the body force and the rate of the kinetic energy. Therefore, r : d designates the stress power done in the unit current volume.

5.8 Conservation Law of Energy

5.8

157

Conservation Law of Energy

The internal energy U is defined by Z U


ð5:52Þ

qedv v

where e is the internal energy per unit mass. The heat input caused by the internal heat source r per unit current unit mass and the heat flux vector q per unit area is given by Z Z Z Z qrdv  qnda ¼ qrdv  divqdv ð5:53Þ Q
v

a

v

v

The stress power is given by Z Pr ¼

r : ddv

ð5:54Þ

v

The following energy equilibrium equation must hold. 

U ¼ Pr þ Q

ð5:55Þ

i.e. Z

 qedv

Z

Z

¼

r : ddv þ

v

v

Z qrdv 

v

divqdv

ð5:56Þ

v

from which the conservation law of energy in the local form is given as 

qe ¼ r : d þ qr  divq

5.9

ð5:57Þ

Work Conjugacy

The work rate done for the unit volume in the current configuration is given by  r : d ¼ rij dij which is the basic variable governing the conservation law of energy in Eq. (5.57). Designating the infinitesimal volumes in a specific region of material in the reference (initial) and the current configurations as dV and dvð¼JdVÞ, respec tively, the work rate w0 done per the unit reference volume is given from Eqs. (4.27), (4.75), (4.80), (4.126), (5.7), (5.13) and (5.17) as follows:

158

5

Stress Tensors and Conservation Laws



w0 ¼ r : ddv=dV ¼ trðrdÞJ ¼ trðslÞ ¼ trðsdÞ  T 8 h i   T  T > > ðlFÞ ¼ tr PF tr sF > > > > > >   <  1 T  T 

tr F sF dF ¼ trðSE Þ ¼ trðSC =2Þ F ¼ h i      > T > > ¼ tr MLT tr FT sFT F1 lF > > > > >  : trðBBÞ leading to 









w0 ¼ Jr : d ¼ s : d ¼ P : F ¼ S : E ¼ S : C =2 ¼ M : L ¼ B : B

ð5:58Þ

where L ¼ F1 lF ¼ ~lG

ð5:59Þ

B
ðSU þ USÞ

ð5:60Þ

G

which is called the Biot stress tensor. Equation (5.58)8 is derived also as follows:        1 1 trðS EÞ ¼ tr S ðU U þ U UÞ ¼ trðSU U þ US UÞ 2 2 i  1 h ¼ tr ðSU þ USÞ U ¼ trðB B Þ 2

ð5:61Þ

noting Eqs. (4.27)1 , (4.43) and (5.60). The work-conjugate pairs are described noting Eq. (3.24) as 8  G ^g j i T > P ¼ sF ¼ s ¼ s g  G ; F ¼ IF ¼ l ^g ¼ lij gi  G j > j i  G <  S ¼ F1 sFT ¼ s GG ¼ sij Gi  Gj ; E ¼ FT dF ¼ d GG ¼ dij Gi  G j > > G : G M ¼ FT sFT ¼ s G ¼ si j Gi  Gj ; L ¼ F1 IF ¼ l G ¼ lij Gi  G j leading to 8  j i j r i s j i r s > > > P : F ¼si g  Gj :ls gr  G ¼ si ls dr dj ¼ si lj < 

S : E ¼sij Gi  Gj :drs Gr  Gs ¼ sij drs dri dsj ¼ sij dij ¼ sij lij > > > : M : L¼si j Gi  Gj :lrs Gr  Gs ¼si j lrs dir dsj ¼si j lij

5.9 Work Conjugacy

159

Therefore, the work-conjugate pair tensors possess the opposite (covariant and contravariant) variants to each other so that their scalar product is represented only by their components, while, needless to say, all tensors can be represented in any variants. The work rate reflecting the constitutive property is not concerned with a current unit volume but is concerned with a reference unit volume, noting that the mass in the current unit volume is variable but the mass in the reference unit volume is invariable. All the solids are composed of the solid particles, e.g. the crystals in metals and the soil particles in soils and thus they exhibit the hyperelastic deformation behavior in a low stress level and the inelastic deformations due to the mutual slips or the change of mutual positions between solid particles is induced in addition to the elastic deformation when a stress increases. Then, the stress tensor and the strain (rate) tensor used in a constitutive equation for solids must be the pair causing the work rate done to the unit reference volume. This requirement was proposed by Hill (1968, 1978) and Macvean (1968) through Truesdell (1952), Ziegler and Macvean (1967). This concept is called the work-conjugacy and the pair of the stress and the strain (rate) fulfilling the work-conjugacy is called the work-conjugate pair. The elastic constitutive equation describing the relation between the stress and the deformation is formulated by the second Piola-Kirchhoff stress tensor S and the right Cauchy-Green deformation tensor C. On the other hand, the plastic constitutive equation is formulated by the Mandel stress tensor M and the plastic part of the velocity gradient tensor L as will be shown Chap. 17. Here, note that the rate of C cannot be decomposed into the elastic and the plastic parts, while L can be decomposed into these parts exactly within the framework of the multiplicative decomposition of the deformation gradient tenor. Now, consider the unit reference orthogonal cell with the side vectors ðN1 ; N2 ; N3 Þ ðjjNi jj ¼ 1Þ which changes to the current cell with the side vectors ðn1 ; n2 ; n3 Þ by the relation ni ¼ FNi due to dx ¼ FdX as shown in Fig. 5.3. By taking account of the force f i ¼ PNi due to Eqs. (5.8) and (5.9) with dA ¼ 1 and (5.58)4 into the relation, it follows that

f 2 = PN 2 N2 f1 = PN1

N2 N3

N1 f 3 = PN3 (|| Ni || = 1)

f2

F

n2 n2

f1

N1

n1

n3 f3

n1

(||ni || 1)

Fig. 5.3 Current cell deformed from reference orthogonal unit cell to which First Piola-Kirchhoff stress applies

160

5







P : F ¼ trðPFT Þ ¼ ðF T PÞii ¼

Stress Tensors and Conservation Laws 3 X



Ni  F T PNi

i¼1

¼

3 X

PNi  ðFNi Þ ¼

i¼1

3 X





f i  ni ¼ w0

ð5:62Þ

i¼1 

Therefore, we can confirm the fact that P : F designates the work rate (power) done to the current cell which was the orthogonal unit cell in the reference state. The 



equation P : C ¼ w0 can be derived by a similar way. The pairs of stresses and strain rates (or rates of deformation gradient) shown in Eq. (5.58) are called the work-conjugate pair. Stress and strain rate tensors in the work-conjugate pair have to be used for the formulation of constitutive equation. The stress vs. strain relations based on the typical work-conjugate pairs in the uniaxial loading process will be shown below. 

(1) P F pair The relation df ¼ PdA in Eq. (5.10) is reduced to f ¼ PA, where the uniaxial components of f, P and A are designated by f , P and A, respectively. The time-integration of the work-conjugate deformation measure to the first Piola-Kirchhoff stress P is the deformation gradient Fð¼ @x=@XÞ which is 



reduced to F ¼ l=L ¼ 1 þ u=L ¼ 1 þ e leading to F ¼ e. Consequently, designating the Young’s modulus E, the following relation holds. P ¼ Ee

ð5:aÞ



(2) S E pair The relation df ¼ FSdA in Eq. (5.16) is reduced to f ¼ FSA¼ ð1 þ eÞSA leading to S ¼ ðf =AÞ=ð1 þ eÞ ¼ P=ð1 þ eÞ ¼ Ee=ð1 þ eÞ, where the uniaxial component of S is designated by S. The work-conjugate deformation measure: Green strain E to the second Piola-Kirchhoff stress S is represented as pffiffiffiffiffiffiffiffiffiffiffiffiffiffi E ¼ u=L þ ðu=LÞ2 =2 ¼ e þ e2 =2 leading to e ¼ 1 þ 1 þ 2E. Consequently, noting Eq. (5.a), one has the relation ! 1 S ¼ E 1  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2E

ð5:bÞ

(3) rd pair The relation df ¼ sda in Eq. (5.3) leads to df ¼ rda under the volume-preserving deformation J ¼ 1, which is reduced to r ¼ f =a ¼ f =ðAL=lÞ¼ ðf =AÞðl=LÞ¼ ðf =AÞ½ðL þ uÞ=L ¼ Pð1 þ eÞ ¼ Eeð1 þ eÞ in the uniaxial loading process. The work-conjugate deformation rate measure to the Cauchy stress tensor r is the strain rate tensor d as shown in Eq. (5.58). The

5.9 Work Conjugacy

161 

time-integration of the axial component l =l of d is identical to the axial component e of the Eulerian logarithmic strain eð0Þ in Eq. (4.50), i.e. e ¼ lnðl=LÞ¼ lnð1 þ u=LÞ ¼ lnð1 þ Þ leading to e ¼ exp e  1 in the uniaxial loading process. Consequently, noting Eq. (5.a), one has the relation r ¼ Eðexp e  1Þ exp e

ð5:cÞ

The stress versus strain relations in Eqs. (5.a), (5.b) and (5.c) are shown in Table 5.2 and depicted choosing the Young’s modulus E ¼ 100; 000N=mm2 in Fig. 5.4 referring to Ishikawa (2015). It is known from this figure that the stress versus strain relations differ significantly from each other for the strain greater than 0.1. Table 5.2 Various stress vs. strain relations by various work-conjugate pairs in uniaxial loading Classification of deformation

Strains and stresses

Stress versus strain relations

Infinitesimal deformation

Nominal strain 1st Piola-Kirchhoff (nominal) stress Green strain

e ¼ u=L F P¼ A

Finite deformations

Lagrangian (reference) description

Eulerian (current) description

2nd Piola-Kirchhoff stress Logarithmic (true) strain Cauchy (true) stress

E¼ S¼

u þ L ð1 þ

1 u 2 1 ð Þ ð¼ e þ e2 Þ 2 L 2 u 1 Þ F P L ð¼ Þ A 1þe

l u e ¼ ln ¼ lnð1 þ Þ ð¼ lnð1 þ eÞÞ L L F F u r ¼ ¼ ð1 þ Þð¼ Pð1 þ eÞÞ a A L

E(exp e 1)exp e

P

S

E E (1

1 ) 1 2E

Fig. 5.4 Various work-conjugate stress versus strain relations premised on linear relation of P ¼ Ee in uniaxial loading ðE ¼ 100; 000 N/mm2 Þ

162

5.10

5

Stress Tensors and Conservation Laws

Various Simple Deformations

Let various strain (rate) and stress (rate) described in the foregoing be shown explicitly and let their relation be described for various simple deformations. These deformations are often observed in experiments for measurement of material properties. Homogeneous and isotropic deformation is assumed therein.

5.10.1

Uniaxial Loading

For a cylindrical specimen with the initial length L and the initial radius R, suppose that the length and the radius changes to l and r (Khan and Huang 1995). Choosing the X1 -axis to the axial direction of cylinder, it holds that x 1 ¼ k1 X 1 ; x 2 ¼ k2 X 2 ; x 3 ¼ k3 X 3

ð5:63Þ

where k1 ¼

l r ; k2 ¼ k 3 ¼ L R

ð5:64Þ

from which one has 2

k1 6 F ¼U ¼ V ¼ 4 0 0

0 k2 0

3 0 7 0 5; k2

2

F1

k1 1 6 ¼4 0 0

0 k1 2 0

R ¼I

3 0 7 0 5; k1 2

9 > > > 2> J ¼ detF ¼ k1 k2 = > > > > ;

ð5:65Þ The aforementioned measures of deformation (rate) are given as 2

C ¼ U2 ;

k21 T 4 F F¼ 0 0

0 k22 0

3 2 2 0 k1 0 5; b1 ¼ V2 ¼ FT F1 ¼ 4 0 k22 0

0 k2 2 0

3 0 0 5 k2 2 ð5:66Þ

2

2 1 1 4 k1  1 E ¼ ðC  IÞ ¼ 0 2 2 0

0 k22  1 0

3 0 0 5 k22  1

5.10

Various Simple Deformations

163

2 2   1 1 4 1  k1 1 ¼ e¼ Ib 0 2 2 0 2

Eð0Þ ¼ eð0Þ



l ¼ F F1

ln k1 ¼4 0 0

0 2 ll1 6 ¼4 0 0

3 2 lnðl=LÞ 0 0 5¼4 0 0 ln k2

0 ln k2 0

2 k1 0 6  6 ¼¼ 4 0 k2

0 1  k2 2 0

3 0 0 5 1  k2 2

0 lnðr=RÞ 0

ð5:67Þ

3 0 0 5 ð5:68Þ lnðr=RÞ

32 3 3 2  1 k k 0 0 1 k1 0 0 1 76 1 7  7 6 1 0 5¼6 k2 k1 0 7 07 4 0 54 0 k2 5 2  1  0 0 k 1 2 0 k2 0 0 k2 k2 3 0 0  ð0Þ  ð0Þ  7 1 rr 0 5¼d¼E ¼e 0



rr 1

0

ð5:69Þ w¼O

ð5:70Þ

The infinitesimal strain in Eq. (4.31) is given by 2

ðl  LÞ=L e¼4 0 0

0 ðr  RÞ=R 0

3 2 0 0 k1  1 5¼4 0 k2  1 0 ðr  RÞ=R 0 0

3 0 0 5¼UI k2  1

ð5:71Þ 2 l =L  6 e¼4 0 0

3 0 r =R 0 

0 7 0 5  r =R

ð5:72Þ

Denoting the axial load as F, various stresses are shown as follows: 2

F 6 pr2 r¼4 0 0

3

2

F 0 0 2 7 6 6 k2 pR2 0 05 ¼ 4 0 0 0 0

3

0 0 0

2

F 07 6 2 7 ¼ 6 k2 A0 05 4 0 0 0

3 0 0 0

07 7 05 0

ð5:73Þ

164

5

2

3

F 2 6 k2 A 0 s ¼ Jr ¼ k1 k22 6 4 0 0 2

k1 1

P ¼ JF1 r ¼ k1 k22 4 0 0

0 k1 2 0

0 0 0

2F 07 k1 A0 7¼6 4 0 05 0 0

2 3 F 0 6 2 k2 A0 0 56 4 0 1 k2 0

S ¼ JF1 rFT

5.10.2

Stress Tensors and Conservation Laws

2 F 6 ¼ 4 k10A0 0

0 0 0

3 0 0 0

0

7 05 0

3 2 F 07 6 A 7¼4 0 0 05 0 0

ð5:74Þ 3 0 0

7 0 0 5 ð5:75Þ 0 0

3 0 0

7 0 05 0 0

ð5:76Þ

Simple Shear

Consider the simple shear in which the shear deformation is induced in parallel to the x1 -axis as shown in Fig. 5.5. x1 ¼ X1 þ cX2 ; x2 ¼ X2 ; x3 ¼ X3

ð5:77Þ

where c is the engineering shear strain. Denoting the shear angle by h, it holds that c ¼ 2tanh;

2 cos h ¼ pffiffiffiffiffiffiffiffiffiffiffiffi ; 4 þ c2

/2

/2

1

X 2 = x2 e2

X

X2 x

e1 0 Fig. 5.5 Simple shear

X1

x1

c sin h ¼ pffiffiffiffiffiffiffiffiffiffiffiffi 4 þ c2

ð5:78Þ

5.10

Various Simple Deformations

165 



  2c   ¼ c cos2  h; h h¼ c ¼ 2 h sec2  2 4 þ c2

ð5:79Þ

It holds in this situation that 2

1 F ¼ 40 0

2

3 0 0 5; 1

c 1 0

F1

3 0 0 5; 1

c 1 0

1 ¼ 40 0

J ¼ detF ¼ 1

ð5:80Þ

where the inverse tensor F1 is derived using Eq. (1.161). The components in the third line and those in the third row are zero except for unity in the third line and the third row in all tensors appearing hereinafter for the simple shear deformation. Then, for simplicity, let them be expressed by the matrix with two lines and two rows.

 c ; 1

1 F¼ 0

F

1

1 ¼ 0

c 1

 ð5:81Þ

from which it is obtained that 

l ¼ FF

1

d¼



¼ 0 c 0 0 0 1

 1 c ; 0 2



0 0

 c ¼ 0 1 0

w¼

0 1



c 0

 ð5:82Þ

 1 c 0 2

ð5:83Þ

Further, it follows noting Eq. (1.161) that

     1 c 1 þ c2 1 2 T 2 1 T C ¼U ¼F F ¼ C ¼U ¼F F ¼ 2 ; c 1þ c c



2 1 1 2 c 2 þ c c U1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffi U ¼ pffiffiffiffiffiffiffiffiffiffiffiffi 2 ; c 2 4 þ c2 c 2 þ c 4 þ c2



    1 1 þ c2 c b ¼ V2 ¼ FFT ¼ ; b1 ¼ V2 ¼ FT F1 ¼ c c 1



 1 1 2 c 2 þ c2 c ; V1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffi V ¼ pffiffiffiffiffiffiffiffiffiffiffiffi 2 c 2 4 þ c2 4 þ c2 c c þ 2

9 c > > 1 = > > ;

ð5:84Þ 9 c > > 1 þ c2 = > > ; ð5:85Þ

  b1 ¼ V2 ¼ FT F1 ¼

1 c

c 1 þ c2

 ð5:86Þ

166

5

1 2 R ¼ FU ¼V F ¼ pffiffiffiffiffiffiffiffiffiffiffiffi 2 c 4þc 1

 c cos h ¼ 2  sin h

1

 sin h R¼h  cos h 



Stress Tensors and Conservation Laws

sin h cos h

  2c c cos h ¼ 3=2 2 2  sin h ð4 þ c Þ

2 c

 ð5:87Þ

 ð5:88Þ



  2c 1 0 1 ¼ 0 4 þ c2 1 0



  1 0 1 1 0 c 1 c 1 E ¼ ðC  GÞ ¼ ;e ¼ g  b ¼ 2 2 c c2 2 2 c c2

  1 1 0 c T e ¼ FþF  I ¼ 2 2 c 0 



0 X ¼ RR ¼ h 1 R

ð5:89Þ

T

ð5:90Þ ð5:91Þ

noting FU1 ¼





R¼h



1 1 c 2 þ c2 pffiffiffiffiffiffiffiffiffiffiffiffi 0 1 c 4 þ c2 



2

c 2

c ffi  pffiffiffiffiffiffiffiffi 2



2 ffi pffiffiffiffiffiffiffiffi

2c 4 4þc  sin h cos h 5 ¼ c ffi 2 ffi  pffiffiffiffiffiffiffiffi  cos h  sin h 4 þ c2  pffiffiffiffiffiffiffiffi 2 2 4þc



 sin h cos h RR ¼ h  cos h  sin h 

3

4 þ c2

T



cos h sin h

4þc

 sin h cos h



Various stress tensors are described using the notation s ¼ r12 as follows:

r¼P¼ P ¼ JrFT ¼

r11 s

s r22



1 c

r11 s

s r22



 0 r11  cs ¼ s  cr22 1

ð5:92Þ s r22





 1 c r11  cs s S ¼ JF1 rFT ¼ F1 P ¼ 0 1  s  cr22 r22

r11  c2 r22  2cs s  cr22 ¼ s  cr22 r22

ð5:93Þ

ð5:94Þ

5.10

Various Simple Deformations

5.10.3

167

Combination of Tension and Distortion

Consider a thin cylindrical specimen subjected to the combination of tension and distortion described by the following equation in the polar coordinate system r ¼ aR; h ¼ H þ xZ;

z ¼ kZ

ð5:95Þ

where ðR; H; ZÞ signifies the initial configuration, and a, x and k denote the proportionality factors depending on the deformation, while x is described by the relative distortion angle / between both ends as follows: x
/=L

ð5:96Þ

L being the length of the specimen. The explanation in the following is referred to Khan and Huang (1995). Variables describing the deformation are given as follows: 2

@r 6 @R FrZ 6 7 6 6 r@h 7 FhZ 5 ¼ 6 6 @R 6 FzZ 4 @z @R

@r R@H FrR FrH 6 r@h F¼6 4 FhR FhH R@H FzR FzH @z R@H 2 3 @aR @aR @aR 6 7 2 @R R@H @Z a 6 7 6 7 6 r@ðH þ xZÞ r@ðH þ xZÞ r@ðH þ xZÞ 7 6 6 ¼6 7 ¼ 40 6 7 @R R@H @Z 6 7 0 4 5 @kZ @kZ @kZ @R R@H @Z 2

3

3 @r @Z 7 7 7 r@h 7 7 @Z 7 7 @z 5 @Z

0 a 0

0

3

7 xaR 7 5

ð5:97Þ

k

from which we have 21 F1

6a 6 6 ¼6 60 6 4 0

3 0

0

7 7 1 xR 7 7  a k 7 7 5 1 0 k

ð5:98Þ

168

5

Stress Tensors and Conservation Laws

and 2

a V ¼ 40 0

3 0 ksin/ 5 k cos /

ð5:99Þ

3 0 5 a sin / axR sin / þ k cos /

ð5:100Þ

0 a cos / þ axR sin / k sin /

2

a U ¼ 40 0

0 a cos / a sin /

where kþa cos / ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ða þ kÞ2 þ ðaxRÞ2

axR sin / ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða þ kÞ2 þ ðaxRÞ2

ð5:101Þ

R being the mean radius ðR ffi RÞ of the thin cylindrical specimen in the initial state. Further, one has 2

a2 C ¼ FT F ¼ 4 0 0

3 0 5 xa2 R 2 k2 þ x 2 a 2 R

0 a2 xa2 R

2

b1 ¼ FT F1

1 6 a2 6 6 ¼6 60 6 4 0



l ¼ FF1

ð5:102Þ 3

0 1 a2 xR  ak

0

7 7 xR 7 7  ak 7 7 2 x2 R 15 þ 2 k2 k

2 a 3  xZ 0 6 a 7 6 7   6 7 a xaR 6 7 ¼ 6xZ a k 7 6 7 4  5 k 0 0 k

ð5:103Þ

ð5:104Þ

from which we have 2 aa 6 6 0 d¼6 6 4 0

0 a a  x aR 2k

3 0 x aR 7 7 7; 2k 7  5 k k 

2

0

6 6xZ w¼6 6 4 0



xZ 0 

x aR  2k

3 0 x aR 7 7 2k 7 7 5 0 

ð5:105Þ

5.10

Various Simple Deformations

169

Stresses in various definitions are described by the following equations, designating the normal stress rzz and the shear stress rrh applied to the traverse section of the cylinder by r and s, respectively. 2

3 0 0 0 s5 s r

0 r ¼ 40 0

ð5:106Þ

It holds from Eqs. (5.97), (5.98) and (5.106) that 2

3

1 6a 6 6 P ¼ JF1 r ¼ a2 k6 0 6 4 0 2

0 1 a 0

0 0 ¼ 4 0 a2 xsR 0 a2 s 2

S ¼ PFT

0 ¼ 40 0 2

0 6 6 ¼ 60 4 0

0 a2 xsR a2 s

0

72 7 0 xR 74 7 0  k 7 1 5 0 k 0

0 0 s

3 0 s5 r

3 0 aks  a2 xrR 5 a2 r

2 1 36 0 6a 6 2 aks  a xrR 56 0 6 2 ar 4 0

2

a2 x2 rR 2axsR þ k a2 xrR as  k

ð5:107Þ 3 0 1 a xR  k 3

0 2 7 7 as  a xrR k 7 5 2 a k

07 7 7 07 7 15 k

ð5:108Þ

Chapter 6

Objectivity and Objective (Rate) Tensors

Constitutive property of material is independent of observers. Therefore, constitutive equation has to be described by variables obeying the common objective transformation rule described in Sect. 1.3.1. State variables, e.g. stress, strain and back stress tensors in the same configuration obey the common coordinate transformation rule. However, the material-time derivatives of tensors in the current configuration do not obey the objective transformation rule, since they are influenced by the rigid-body rotation. Then, instead of the material-time derivative of tensors, particular time-derivatives of tensors obeying the objective transformation rule, i.e. the convective time-derivative have to be adopted in constitutive equations. The consideration on the fulfillment of objectivity is of great importance for the hypoelastic-based constitutive equation formulated in the current configuration which is influenced directly by the rigid-body rotation. Then, the objectivity and the formulation of constitutive relations fulfilling the objectivity will be explained in this chapter.

6.1

Objectivity

Physical quantities except for scalar ones are observed to be different depending on the state, e.g. position, direction, velocity of observers. On the other hand, mechanical property of material is observed identically independent of the state of observers. In particular, it is observed identically independent of the rigid-body rotation of material. Therefore, a constitutive equation describing material property must be expressed in a common form independent of coordinate systems. Then, it must be described so as not to be influenced by the rigid-body rotation of material. This fact was not so obvious in the olden time and was advocated by Oldroyd (1950) in the middle of the last century. It is referred to as the principle of material-frame indifference (Oldroyd 1950) or principle of objectivity or simply © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 K. Hashiguchi, Foundations of Elastoplasticity: Subloading Surface Model, https://doi.org/10.1007/978-3-030-93138-4_6

171

172

6

Objectivity and Objective (Rate) Tensors

objectivity. This would be regarded as one of the starting points of the modern continuum mechanics which is called sometimes as the rational mechanics (Truesdell and Toupin 1960; Truesdell and Noll 1965) in order to describe the finite elastoplastic deformation and has been studied over the last half century prior to the recent formulation by the hyperelastic-based plasticity which will be comprehensively described in Chap. 17. Here, note that components of tensor describing mechanical state of material, e.g. stress, strain and anisotropic internal variables are observed to be changed by the fixed coordinate system if the material rotates, even when the components are observed to be unchanged by the coordinate system rotating concurrently with the material itself. Therefore, the material-time derivative of tensor describing mechanical state is observed to be non-zero, by the fixed coordinate system when the material rotates even when it is observed to be zero by the observer rotating concurrently with the material itself. It is caused by the fact that the material-time derivative of tensor designates the rate of tensor observed by the coordinate system moving in parallel with material but without rotation. Then, the material-time derivative cannot be adopted for the description of constitutive equations in a current rate form. Machine elements are often subjected not only to deformation but also to rigid-body rotation, as seen in metal forming, gears, wheels, etc. Soils near the side edges of footings, at the bottom ends of piles, etc. undergo a large rigid-body rotation. Therefore, formulations of constitutive equations which is not influenced by the rigid-body rotation are of great importance in practical engineering problems.

6.2

Influence of Rigid-Body Rotation on Various Mechanical Quantities

In order to check whether or not a constitutive equation is formulated so as to satisfy the objectivity principle, it is expedient to examine the influence of rigid-body rotation on the tensor variables used in constitutive equations. Instead, one may examine how the components of these variables are
observed  by the coordinate systems with the fixed base fei g and the rotating base ei ðtÞ which are related as ei ðtÞ ¼ QT ðtÞei ;


ei ð0Þ ¼ ei ;





ei ðtÞ ¼ QT ðtÞei

ð6:1Þ



provided that the rotating base ei ðtÞ coincides with the fixed base fei g at the beginning of deforming/rotation ðt ¼ 0Þ, noting Eq. (1.111), where one has QðtÞ ¼ ei  ei ðtÞ;

Qð0Þ ¼ I

These bases are illustrated in Fig. 6.1 for the two-dimensional state.

ð6:2Þ

6.2 Influence of Rigid-Body Rotation on Various Mechanical Quantities

173


 Fig. 6.1 Coordinate systems with the fixed base fei g and the rotating base ei ðtÞ which coincides with the base fei g at the beginning of deformation/rotation (illustrated in two-dimensional state for e3 ¼ e3 )

The components of the initial infinitesimal line element
 dX  in the initial state (t ¼ 0) is observed to be identical by these bases, since ei ðtÞ coincides with fei g in the initial state. On the other hand, the components of the current infinitesimal line-element dxðtÞ is observed to be different by these bases as the rotating base
 ei ðtÞ differs from the fixed base fei g for t [ 0. Here, noting dxðtÞ ¼ FðtÞdX, we have dX ¼ dX




  ei ð0Þ ¼ fei g ;

dx ðtÞ ¼ QðtÞdxðtÞ ¼ QðtÞFðtÞdX

ð6:3Þ

and dx ðtÞ ¼ F ðtÞdX ð0Þ ¼ F ðtÞdX

ð6:4Þ

from which it follows that F ðtÞ ¼ QðtÞFðtÞ

ð6:5Þ

It is known from Eq. (6.5) that the deformation gradient FðtÞ is the second-order tensor but it obeys the transformation rule of the first-order tensor. This is based on the fact that the deformation gradient is the two-point tensor as specified in Eq. (3.9). Substituting Eq. (6.5) into Eqs. (4.2)–(4.5), (4.16), (4.17) and (4.27), the following relations are obtained for various quantities describing a deformation.

174

6

U ¼ U;

ðF TF ¼ ðQFÞT QF ¼ FT QT QF ¼ FT FÞ

C ¼ C

V ¼ QVQT ;

Objectivity and Objective (Rate) Tensors

b ¼ QbQT

ð6:6Þ

ðF FT ¼ QFðQFÞT ¼ QFFT QT Þ

ð6:7Þ

ðF ¼ QF ¼ QRU ¼ QRU ¼ R U Þ

R ¼ QR

ð6:8Þ

E ¼ E ð ¼ FT eFÞ; e ¼ QeQT

ð6:9Þ

Noting the relation 









F F1 ¼ ðQFÞ ðQFÞ1 ¼ ðQF þ QFÞF1 Q1 ¼ QðFF1  QT QÞQT it holds for the velocity gradient in Eq. (4.75) that l ¼ Qðl  XÞQT ¼ QlQT þ X

ð6:10Þ

where X and X are defined by 



X  QT Q;

X  Qri Qrj ei  ej

X  QQT ;

X  Qir Qjr ei  ej



)



ð6:11Þ

and they are related by X ¼ QXQT ;

X ¼ QT XQ

ð6:12Þ



where Q is given by 



Q ¼ er  er

ð6:13Þ



from Eq. (6.2) because of ei ¼ 0. Substituting Eqs. (6.2) and (6.13) into Eq. (6.11), we have 



X ¼ er  er ;

X ¼ ðei  ej Þei  ej

ð6:14Þ

from which it follows that 

ei ¼ Xei

ð6:15Þ


 It is known from Eq. (6.15) that X is the spin of the base ei . The substitution of Eq. (6.10) into Eqs. (4.80) and (4.81) yields the following transformation rules.

6.2 Influence of Rigid-Body Rotation on Various Mechanical Quantities

175

d ¼ QdQT

ð6:16Þ

w ¼ QðwXÞQT ¼ QwQT þ X

ð6:17Þ

The relative spin in Eq. (4.85), i.e. 

XR  R RT

ð6:18Þ

obeys the transformation identical to that of w as follows: 

XR ¼ QðXR  XÞQT ¼ QXR QT þ X

ð6:19Þ

noting 





R RT ¼ ðQRÞ ðQRÞT ¼ ðQR þ QRÞRT QT 







¼ QRRT QT þ QQT ¼ QXR QT þ QQT QQT ¼ QXR QT  QQT QQT We obtain the following conclusions for the influence of rigid-body rotation from Eqs. (6.6)–(6.19). (1) The right Cauchy-Green deformation tensor C and the Green strain tensor E are based in the reference configuration and thus they are observed to be unchangeable, i.e. invariant, obeying the transformation rule of scalar quantities independent of the rigid-body rotation. On the other hand, the left Cauchy-Green deformation tensor b and the Almansi strain tensor e are based in the current configuration and thus obey the transformation rule of second-order tensor. (2) The strain rate tensor d obeys the transformation rule of the second-order tensor. On the other hand, the velocity gradient tensor l, the continuum spin tensor w and the relative spin XR are directly subjected to the influence of rate of rigid-body rotation, lacking the objectivity. The following transformations hold for stress tensors described in Chap. 5. r ¼ QrQT

ðtt ¼ Qtt ¼ Qrn ¼ QrQT Qn ¼ r n Þ s ¼ QsQT



 T

ðP ¼ Jr F

ð6:20Þ ð6:21Þ

P ¼ QP ¼ JQrQ ðQFÞ ¼ JQrQT QT FT ¼ QJrFT ¼ QPÞ T

T

ð6:22Þ

176

6



Objectivity and Objective (Rate) Tensors

S ¼ S ¼ ðQFÞ1 ðQsQT ÞðQFÞT S ¼F s F  ¼ F1 QT QsQT QT FT ¼ F1 sFT ¼ S 

1  T

ð6:23Þ

Then, the Cauchy stress tensor r and the Kirchhoff stress tensor s obeys the transformation rule of the second-order tensor. On the other hand, the first Piola-Kirchhoff stress tensor P obeys the transformation rule of the first-order tensor so that it is the two-point tensor. The second Piola-Kirchhoff stress tensor S is the invariant under the superposition of rigid-body rotation. The consideration of objectivity is of great importance in the formulation of hypoelastic-based plastic constitutive equations since the rates of stress and anisotropic internal state variables in the current configuration are influenced directly by the rigid-body rotation as described above. Then, the time-derivatives of state variables will be further considered in the subsequent sections.

6.3

Material-Time Derivative of Tensor

The material-time derivatives of state variables is the rates of them observed by the coordinate system moving in parallel with material particle as explained in Sect. 2.2. However, it will be mathematically verified in this section that the material-time derivative of tensor does not obey the objective transformation rule and thus it cannot be used in constitutive equations. Consider the transformation of the material-time derivative of a state variable obeying the objective transformation in Eqs. (1.90) and (1.92). The material-time derivative of the tensor t in the current configuration reads (Hashiguchi 2007a): 



t p1 p2



pm

¼ Qp1 q1 Qp2 q2





Qpm qm tq1 q2



qm

þ Qp1 q1 Qp2 q2  Qpm qm tq1 q2



þ Qp1 q1 Qp2 q2    Qpm qm tq1 q2



qm

þ





qm

þ Qp1 q1 Qp2 q2  Qpm qm tq1 q2



qm

ð6:24Þ 



tp1 p2



pm

¼ Qq1 p1 Qq2 p2    Qqm pm tq1 q2





qm

þ Qq1 p1 Qq2 p2    Qqm pm tq1 q2

qm

þ







þ Qq1 p1 Qq2 p2    Qqm pm tq1 q2





qm

þ Qq1 p1 Qq2 p2    Qqm pm tq1 q2



qm

ð6:25Þ 







Noting the relation Qpi qi ¼ dpi s Qsqi ¼ Qpi t Qst Qsqi ¼ Qpi t Qst Qsqi ¼ Qpi t Xtqi and replacing t ! qi , qi ! ri , then Eqs. (6.28) and (6.29) can be rewritten as follows:

6.3 Material-Time Derivative of Tensor 

tp1 p2





pm

¼ Qp1 q1 Qp2 q2    Qpm qm ðtq1 q2  Xq1 r1 tr1 q2



qm



qm

 Xq2 r2 tq1 r2



qm





 Xq1 rm tq1 q2





 Xqm rm tq1 q2



rm Þ

ð6:26Þ





t p1 p2

177



pm

¼ Qq1 p1 Qq2 p2    Qqm pm ðtq1 q2 

Xq1 r1 tr1 q2    qm



   qm  Xq2 r2 tq1 r2    qm



rm Þ

ð6:27Þ

It is known from Eqs. (6.26) and (6.27) that the material-time derivative does not 

obey the objective transformation rule, noting that the components tp1 p2 



pm

in the

t p1 p2    pm

in the fixed coordinate system is not zero even when the components coordinate system rotating with the material is zero. Equations (6.26) and (6.27) are expressed for the vector v and the second-order tensor t in symbolic notation as follows: 







v ¼ Qðv  XvÞ; v ¼ QT ðv  Xv Þ 





ð6:28Þ



t ¼ Qðt  Xt þ tXÞQT ; t ¼ QT ðt  Xt þ t XÞQ

ð6:29Þ

Consequently, the material-time derivative cannot be adopted in constitutive equations. In order to see the irrationality for using the material-time derivative of tensor, consider the hypoelastic constitutive equation in which the material-time derivative of Cauchy stress tensor is related linearly to the strain rate tensor as follows: 

r¼H:d

ð6:30Þ

where the tangent modulus tensor H (fourth-order tensor) is the function of stress and anisotropic internal variables in general. It follows from this equation with Eq. (6.29) that 

d ¼ H1 : QT ðr  Xr þ r XÞQ 



ð6¼ H1 : QT r Q ¼ H1 : QT ðQrQT Þ Q ¼ H1 : rÞ

ð6:31Þ

This leads to the irrational result that the deformation is induced, i.e. d 6¼ O even if the stress observed by the coordinate system rotating ðX ¼ QXQT 6¼ OÞ with the  material itself does not change, i.e. r ¼ O. This is caused by the non-objectivity of material-time derivative of tensor, while the strain rate d is not a material-time derivative of tensor but is the original tensor defined so as to obey the objective transformation by excluding the continuum spin tensor w from the velocity gradient tensor l. In the next section, the objective time-derivative of tensor will be introduced which is based on the rate of tensor observed by the coordinate system deforming/rotating with material itself, satisfying the objectivity.

178

6

Objectivity and Objective (Rate) Tensors

The Eulerian tensor changes even when the state of physical quantity observed from the material itself does not change under a material rotation, since the base vectors change even by a rotation of the material. On the other hand, the Lagrangian tensor pulled-back to the reference configuration with the fixed base vectors does not change in the material rotation and thus it called the rotation-free tensor, while the Lagrangian tensor inherits the components in the Eulerian tensor. Then, the constitutive relation described by the Lagrangian tensors is used in the deformation analysis. The Eulerian base vectors are calculated by the push-forward operation of the Lagrangian (reference) base vectors through the deformation gradient tensor, which is required to capture the Eulerian tensor in the current configuration from the Lagrangian tensor and vice versa. The physical and geometrical interpretations for the relations between the above-mentioned Eulerian and Lagrangian tensors can be referred to Hashiguchi and Yamakawa (2012).

6.4

Objectivity of Convective Time-Derivative and Corotational Rate

The convected derivatives in Eq. (3.34) satisfy the objectivity obviously because they are based on the rates of tensor observed by a material itself. In addition, this fact can be mathematically confirmed as shown below for Eq. (3.34)1 as an example, noting Eq. (6.10). ! 

t

gg

¼ F ðF1 t FT Þ FT ¼ QF½ðQFÞ1 QtQT ðQFÞT  ðQFÞT ¼ QF½F1 QT QtQT QFT  FT QT ¼ Q½FðF1 tFT Þ FT QT

ð6:32Þ

or ! 

t

gg



¼ t  l t  t lT ¼ ðQtQT Þ  Qðl  XÞQT QtQT  QtQT ½Qðl  XÞQT T 





¼ QtQT þ QtQT þ QtQT  Qðl  XÞtQT  QtQT QðlT  XT ÞQT 







T



¼ QtQT þ Q tQT þ QtQT  Qðl  Q QÞtQT  QtQT QðlT  QT QÞQT 







T



¼ QtQT þ QtQT þ QtQT  QltQT  QQT Q tQT  QtlT QT  QtQT QQT 

¼ Qðt  lt  tlT ÞQT ð6:33Þ Hereinafter, we consider the general corotational rate tensor and show its objectivity in the following.

6.4 Objectivity of Convective Time-Derivative and Corotational Rate

179

The general corotational rate tensor is represented as  t ¼ R ðRT ½½tÞ



ð6:34Þ

by the symbolic notation defined in Eq. (1.103). Equation (6.34) is expressed in the component description as follows (Hashiguchi 2003):  t p1 p2  pn

¼ Rp1 q1 Rp2 q2  Rpn qn ðRs1 q1 Rs2 q2  Rsn qn ts1 s2





sn Þ



¼ Rp1 q1 Rp2 q2  Rpn qn ðRs1 q1 Rs2 q2  Rsn qn ts1 s2 

þ Rs1 q1 Rs2 q2  Rsn qn ts1 s2 þ

sn

þ Rs1 q1 Rs2 q2  Rsn qn ts1 s2

sn











þ Rs1 q1 Rs2 q2  Rsn qn ts1 s2





sn

sn Þ 

¼ Rp1 q1 Rs1 q1 Rp2 q2 Rs2 q2  Rpn qn Rsn qn ts1 s2



sn



þ Rp1 q1 Rs1 q1 Rp2 q2 Rs2 q2  Rpn qn Rsn qn ts1 s2



sn



sn



þ Rp1 q1 Rs1 q1 Rp2 q2 Rs2 q2  Rpn qn Rsn qn ts1 s2 þ  

þ Rp1 q1 Rs1 q1 Rp2 q2 Rs2 q2  Rpn qn Rsn qn ts1 s2 

¼ dp1 s1 dp2 s2  dpn sn ts1 s2



¼ t p1 p2



pn

 Xp1 s1 ts1 p2



sn

 Xp1 s1 dp2 s2  dpn sn ts1 s2

sn

þ dp1 s1 ðXp2 s2 Þ  dpn sn ts1 s2 



pn



sn

þ



 Xp2 s2 tp1 s2





sn

þ dp1 s1 dp2 s2  ðXpn sn Þts1 s2 pn





X p n s n tp 1 p 2





sn

sn

ð6:35Þ 





noting Eq. (6.18) with Rp1 q1 Rs1 q1 ¼ ðRRT Þp1 s1 ¼ ðRRT Þp1 s1 ¼ Xp1 s1 . The objectivity of the corotational rate is verified as follows:  t p1 p2    pn

¼ Rp1 q1 Rp2 q2  Rpn qn ðRs1 q1 Rs2 q2  Rsn qn ts1 s2



sn Þ



¼ Qp1 t1 Rt1 q1 Qp2 t2 Rt2 q2  Qpn tn Rtn qn ½Qs1 r1 Rrn qn Qs2 r2 Rr2 q2  Qsn rn Rrn qn ðQs1 u1 Qs2 u2  Qsn un tu1 u2 ¼ Qp1 t1 Qp2 t2  Qpn tn Rt1 q1 Rt2 q2  Rtn qn ½Rrn qn Rr2 q2  Rrn qn ðQs1 r1 Qs1 u1 Qs2 r2 Qs2 u2  Qsn rn Qsn un tu1 u2



un Þ







un Þ

¼ Qp1 t1 Qp2 t2  Qpn tn Rt1 q1 Rt2 q2  Rtn qn ½Rrn qn Rr2 q2  Rrn qn ðdr1 u1 dr2 u2  drn un tu1 u2 un Þ ¼ Qp1 t1 Qp2 t2  Qpn tn Rt1 q1 Rt2 q2  Rtn qn ½Rr1 q1 Rr2 q2  Rrn qn tr1 r2 



¼ Qp1 t1 Qp2 t2  Qpn tn t t1 t2



tn



rn 



ð6:36Þ

180

6

Objectivity and Objective (Rate) Tensors

i.e. 



t ¼ Q½½t

ð6:37Þ

The fourth-order tensor as the special case of Eq. (6.36) was used to the anisotropic tensor for the directional distortional yield surface by Feigenbaum and Dafalias (2014) referring to Hashiguchi (2003).

6.5

Various Objective Stress Rate Tensors

Various rates of the Cauchy stress r and the Kirchhoff stress s ð¼ JrÞ can be obtained from the aforementioned convected and corotational time-derivatives as will be shown in this section. Corotational time-derivative with a spin tensor is designated by the symbol ð Þ as shown in the last section. On the other and, the time-derivatives other than corotational time derivatives are designated by the symbol ðD Þ. (a) Contravariant convected rates Based on Eq. (3.34)1, the contravariant convected rate of the Cauchy stress r is given by !  gg

D

rOl  r





D



¼ FðF1 rFT Þ FT ¼ FS=J FT ¼ r  lr  rlT ð¼ rOl Þ T

ð6:38Þ

which is termed the Oldroyd rate of Cauchy stress (Oldroyd 1950). Likewise, it holds for the Kirchhoff stress that DOl

s

!  gg

 s



D



¼ FðF1 sFT Þ FT ¼ FSFT ¼ s  ls  slT ð¼ sOl Þ T

ð6:39Þ

which is termed the Oldroyd rate of Kirchhoff stress. Further, D

rTr

J

1 DOl

s

¼J

1

! 



gg

D

ðJrÞ ¼ J 1 FðF1 ðJrÞFT Þ FT ¼ J 1 FS FT ¼ rOl þ rtrd 



¼ r  lr  rlT þ rtrdð¼ rTrT Þ ð6:40Þ is termed the Truesdell rate of Cauchy stress.

6.5 Various Objective Stress Rate Tensors

181

(b) Covariant-contravariant convected rates The covariant-contravariant convected rate of the Kirchhoff stress s is given from Eq. (3.34)3 as ð6:41Þ t t The particular case of the rate in Eq. (6.41) is given by D

! 





D

T T  T T T P s  s ^g ¼ ðsF Þ F ¼ P F ¼ s  sl ð6¼ P s Þ g

ð6:42Þ

which is termed the relative 1st Piola-Kirchhoff stress rate. The following stress rate is defined as the nominal stress rate in Eqs. (5.38) and (5.39). D Pr



1   D T P s ¼ r  rl þ rtrd ð6¼P rT Þ J

ð6:43Þ

which is the nominal stress rate and used for the equilibrium equation of rate-form in the current configuration as described in Eq. (5.43). (c) Covariant convected rates The covariant convected rate of Cauchy stress is given from Eq. (3.34)4 as ! 

D



D

rCR  r gg ¼ FT ðFT rFÞ F1 ¼ r þ lT r þ rl ð¼ rCR Þ T

ð6:44Þ

which is termed the Cotter-Rivlin rate of Cauchy stress (Cotter and Rivlin 1995). Likewise, the covariant convected rate of Kirchhoff stress is given by DCR

s

! 



D

 s gg ¼ FT ðFT sFÞ F1 ¼ s þ lT s þ sl ð¼ sCR Þ T

ð6:45Þ

(d) Corotational rates The following stress rate based on Eq. (3.39) is termed the Green-Naghdi rate of Cauchy stress (Green and Naghdi 1965). 

! 





rR  R r ¼ RðRT rRÞ RT ¼ r  XR r þ rXR ð¼ rRT Þ

ð6:46Þ

Similarly, the Green-Naghdi rate of Kirchhoff stress is given by 

! 





sR  R s ¼ RðRT sRÞ RT ¼ s  XR s þ sXR ð¼ sRT Þ

ð6:47Þ

182

6

Objectivity and Objective (Rate) Tensors

The stress rate based on Eq. (3.40) is given by 





T

rw  r  wr þ rwð¼ rw Þ

ð6:48Þ

which is termed the Zaremba-Jaumann rate of Cauchy stress (Zaremba 1903; Jaumann 1911). Likewise, it follows for the Kirchhoff stress that 





T

sw  s  ws þ sw ð¼ sw Þ

ð6:49Þ

The stress rate tensors described above are collectively shown below. 





Oldroyd rate of Cauchy stress : rOl  r lr  rlT ð¼ rOl Þ 



T





Truesdell rate of Cauchy stress : rTr  rOl þ r trd ¼ r  lr  rlT þ r tr d ð¼ rTR Þ Covariant-contravariant D

T

D



convected rate of the Kirchhoff stress : t  s þ lT s  slT ð6¼ t T Þ 1 D  D D Nominal stress rate : P r  P s ¼ r  rlT þ r tr d ð6¼ PrT Þ J D

D



Cotter-Rivlin rate of Cauchy stress : rCR  r þ lT r þ rl ð¼ rCRT Þ 





T

Green-Naghdi rate of Cauchy stress : rR  r  XR r þ rXR ð¼ rR Þ 





T

Zaremba-Jaumann rate of Cauchy stres : rw  r  wr þ rw ð¼ rw Þ

ð6:50Þ Here, it should be noted that a corotational rate must be used also for evolution equations of all internal variables in addition to the stress. The above-mentioned rate tensors are used for hypoelastic constitutive equations. In particular, the hypoelastic constitutive equation in terms of the Oldroyd rate is derived directly from the hyperelastic equation as will be described in Sect. 7.4. However, it should be noticed that the hypoelastic-based plastic constitutive equation is limited to the infinitesimal elastic deformation as will be revealed in Sect. 17.3. Incidentally, the particular spin, called the logarithmic spin, by which the corotational rate of the Eulerian-logarithmic (Hencky) strain ln V in Eq. (4.50)2 coincides with the strain rate d, was proposed and the hyperelastic constitutive equation was derived from the hypoelastic constitutive equation in terms of the logarithmic rate of Cauchy stress and the strain rate d by Xiao and his colleagues (Xiao 1995; Xiao et al. 1997, 1999) as explained in Appendix B. However, it would be limited to the isotropy.

6.6 Jaumann Rate with Plastic Spin

6.6

183

Jaumann Rate with Plastic Spin

The physically meaningful rotation of material is the rotation of the substructure of material (material fibers) which cannot be observed from the outside appearance of material. It is not caused by the plastic deformation which causes only the parallel slip of material particles. Therefore, the spin of substructure xs is given by subtracting the plastic spin wp from the continuum spin w, resulting in xs ¼ wwp . Here, the plastic spin wp is determined depending on the constitutive property of plastic anisotropy, while the continuum spin w is determined only by the geometrical variation of outside appearance. Consequently, xs ¼ we holds, i.e. 

xs ¼ we ¼ Re Re ¼ w  wp T

ð6:51Þ

(Mandel 1971, 1985a; Kratochvil 1971; Dafalias 1983; Loret 1983). The explicit formulation of the plastic spin wp will be given in Sect. 8.8.

6.7

Time-Derivative of Scalar-Valued Tensor Function

Scalar-valued tensor functions of stress and internal variables appear often in continuum mechanics as seen in the strain energy function and the yield function. Then, the time-derivative of scalar-valued tensor function is required in order to derive the rate-type relation of variables, e.g. the consistency condition of yield condition. The time-derivative of scalar function is independent of rigid-body rotation and thus it can be given primarily by its material-time derivative. Here, it should be noticed that the internal variables are formulated by the objective time-derivatives and thus the consistency condition must be transformed to the objective time-derivative. It can be proved that the material-time derivative of scalar-valued tensor function is transformed only to its corotational time-derivative. This fact would seem physically obvious but it must be proved mathematically. To this end, its mathematically exact proof for scalar valued function of general tensor will be given below, referring to the previous studies by Dafalias (1985a, 1998, 2011) for vector and second-order tensor and Hashiguchi (2007b) for general tensor. The corotational rate of the general tensor is defined based on Eq. (3.34) for the second-order tenor as follows: hh _ ii _ t ¼ R ðRT ½½tÞ



ð6:52Þ

_

denoting the general orthogonal tensor by R, where use is made of the symbol ½½  _

for general objective transformation in Eq. (1.103) with the replacement Q ! RT .

184

6 

Objectivity and Objective (Rate) Tensors

 _

_

Here, noting f ðtÞ ¼ f ðRT ½½tÞ because of the requirement f ðtÞ ¼ f ðRT ½½tÞ for scalar variable, one has 

 _



_

f ðtÞ ¼ f ðR ½½tÞ ¼ T

@f ðRT ½½tÞ _

@ðR ½½t Þ h h ii @f ðtÞ _ _ T ¼  R ðR ½½tÞ @t



f ðtw1 w2 Þ ¼ 

T

_

_

_

_

_

_



 ðR ½½tÞ ¼ R T

@f ðRu1 w1 Ru2 w2  tu1 u2 Þ 

T

_

@f ðtÞ @t



_

 ðRT ½½tÞ ð6:53Þ

_

ðRv1 p1 Rv2 p2  tv1 v2



Þ

@ðRs1 p1 Rs2 p2  ts1 s2 Þ _ _ @f ðtw1 w2 Þ _ _ ¼ Rs1 p1 Rs2 p2  ðRv1 p1 Rv2 p2  tv1 v2 Þ @ts1 s2 

ð6:54Þ







where the symbol  designates the full contraction between derivative components in order between derivative components, i.e. t  s ¼ tp1 p2 pm sp1 p2 pm . The derivation of Eq. (6.53) is shown for vector and second-order tensor as follows: 



_

 _

f ðvÞ ¼ f ðRT vÞ ¼

@f ðRT vÞ _

@ðRT vÞ

_

ðR

T

_

vÞ ¼ RT



@f ðvÞ _ T  @f ðvÞ _ _ T  ðR vÞ ¼ RðR vÞ @v @v ð6:55Þ



 _

_

_

f ðtÞ ¼ f ðR tRÞ ¼ T

_

@f ðRT tRÞ _

_

@ðRT tRÞ

_

_

_

: ðRT tRÞ ¼ RT

@f ðtÞ _ T _ T _  R :ðR tRÞ @t

ð6:56Þ

@f ðtÞ _ _ T _  _ T : RðR tRÞ R ¼ @t Equations (6.53), (6.55) and (6.56) can be satisfied by the corotational rate in Eq. (6.52) amongst objective rates. Then, we have the following relation. 

f ðtÞ ¼ 

f ðtq1 q2

@f ðtÞ  @f ðtÞ  t ¼ t @t @t @f ðtq1 q2 qm Þ  tp1 p2 qm Þ ¼ @tp1 p2 pm 







pm

¼

@f ðtq1 q2 @tp1 p2





ð6:57Þ

qm Þ 

pm

t p1 p2



pm

which is described for vector and second-order tensor as follows: 

f ðvÞ ¼

@f ðvÞ   v; @v



f ðvr Þ ¼

@f ðvr Þ  vi @vi

ð6:58Þ

6.7 Time-Derivative of Scalar-Valued Tensor Function 

f ðtÞ ¼

@f ðtÞ  : t; @t



f ðtrs Þ ¼

185

@f ðtrs Þ  t ij @tij

ð6:59Þ

It follows from Eq. (6.54) that 



_ _ _ _ _ _ @f ðtw1 w2 w3 Þ _ _ _ Rs1 p1 Rs2 p2 Rs3 p3  ðRv1 p1 Rv2 p2 Rv3 p3  þ Rp1 v1 Rp2 v2 Rp3 v3  þ @ts1 s2 s3 



Þtv1 v2 v3









_ _ @f ðtw1 w2 w3 Þ _ _ ðRs1 p1 Rv1 p1 ds2 v2 ds3 v3  þ ds1 v1 Rs2 p2 Rv2 p2 ds3 v3  þ @ts1 s2 s3 @f ðtw1 w2 w3 Þ ¼ ðxv1 s1 tv1 s2 s3  þ xs2 v2 ts1 v2 s3  þ  Þ ¼ 0 @ts1 s2 s3

¼





Þtv1 v2 v3





ð6:60Þ









which is reduced for vector and second-order tensor as follows: 

@f ðvÞ @f ðvu Þ @f ðvu Þ @f ðvÞ Þx ¼ 0 ð6:61Þ  xv ¼ xri vi ¼ vi xri ¼ 0; i.e. tr ðv  @v @vr @vr @v tr½ð

@f ðtÞ T @f ðtÞ t  tT Þx ¼ 0 @t @t

ð6:62Þ

The fulfillments of Eqs. (6.61) and (6.62) require for the tensors in the brackets ð Þ to be zero or symmetric tensor, while Dafalias (1998) has required for the latter to be zero tensor. The fulfillment of Eq. (6.61) is easily known for f ðvÞ ¼ svv because of @f ðvÞ=@v ¼ 2sv with Eq. (1.176)3, and that of Eq. (6.62) for the second-order symmetric tensor t because of ð@f ðtÞ=@tÞtT  tT ð@f ðtÞ=@tÞ ¼ O. Equation (6.57) is extended for plural variables as follows: 

f ðt1 ; t2 ;



@f ðt1 ; t2 ; @t1 @f ðt1 ; t2 ; ¼ @t1

Þ¼

@f ðt1 ; t2 ;  Þ   t2 þ  @t2   Þ  @f ðt1 ; t2 ;  Þ  t1 þ t2 þ  ¼ f ðt1 ; t2 ; @t2 

Þ



 t1 þ



Þ ð6:63Þ

which is shown for the function of two tensors in Belytschko et al. (2014). Here, it should be noted that the mathematical property does not hold for each term, i.e. @f ðt1 ; t2 ;  ; tm Þ  @f ðt1 ; t2 ;  ; tm Þ  ti 6¼ ti @ti @ti

(no sum)

ð6:64Þ

Scalar-valued functions must be independent of rigid-body rotation so that their material-time derivative possess a unique value which coincides with their corotational time-derivative as can be confirmed by the above-mentioned proof.

186

6

Objectivity and Objective (Rate) Tensors

However, they do not lead to the other convected time-derivatives which depend on the rate of deformation, i.e. velocity gradient. Therefore, corotational timederivatives can be adopted in the time-derivatives of scalar functions in constitutive relations, e.g. a strain energy function and a yield function of tensors in the current configuration but convected time-derivatives other than corotational rates cannot be adopted in them. The most popular scalar-valued tensor functions are the principal invariants of tensor. Principal invariants of tensor are described by three independent principal invariants of tensor. Then, the material-time derivatives of the principal invariants are transformed to the corotational time-derivatives merely by replacing all the material time-derivatives of tensor to the corotational time-derivatives of tensor as will be written below. It follows from Eqs. (6.59) and (6.63) that 



9 > > > > =



I ¼ ðtrtÞ ¼ ðtrtÞ ¼ I : t ¼ tr t 





II ¼ ðtrt2 Þ ¼ ðtrt2 Þ ¼ 2tT : t ¼ 2trðt tÞ 





III ¼ ðtrt3 Þ ¼ ðtrt3 Þ ¼ 3t2 : t ¼ 3trðt2 tÞ T

> > > > ; 9 > > > > =





I ¼ ðtrtÞ ¼ tr t o    1n ðtrtÞ2  trt2 ¼ ðtrtÞtr t trðt tÞ II ¼ 2 





III ¼ ðdettÞ ¼ ðdettÞtT : t ¼ ðdettÞtrðt1 tÞ

> > > > ;

ð6:65Þ

ð6:66Þ

for the principal invariants in Eqs. (1.222) and (1.217)–(1.219), noting Eqs. (1.354) and (1.355). Further, it holds that 







ðt1 : t2 Þ ¼ t1 : t2 þ t1 : t2 ¼ trðtT2 t1 Þ þ trðtT1 t2 Þ

ð6:67Þ

for the two tensor variables. If t1 and t2 are commutative (possessing same principal directions) leading to t1 t2 ¼ t2 t1 ¼ t1 tT2 ¼ tT2 t1 , one has 



t1 : t2 ¼ t1 : t2 ;





t1 : t2 ¼ t 1 : t2

ð6:68Þ

noting t1 : ðt2 x  xt2 Þ ¼ tr½t1 ðt2 xÞT   tr½t1 ðxt2 ÞT  ¼ trðtT2 t1 xÞ þ trðt1 tT2 xÞ ¼ 0 ð6:69Þ

6.7 Time-Derivative of Scalar-Valued Tensor Function

187

All the equations in Eqs. (6.65)–(6.67) hold for arbitrary corotational tensors as proved here, although they are written explicitly for the Zaremba-Jaumann rate in some literatures (e.g. Prager 1961b, Belytschko et al. 2014).

Chapter 7

Elastic Constitutive Equations

Elastic deformation is induced by the reversible deformation of material particles themselves without a mutual slip between them. They therefore exhibit higher stiffness compared to the stiffness associated with plastic deformation. Elastic constitutive equations are classifiable into the three types depending on the exactness in the description of reversibility, i.e. the hyperelasticity (or Green elasticity) possessing the strain energy function, the Cauchy elasticity possessing the one-to-one correspondence between stress and strain and the hypoelasticity possessing the linear relation between stress rate and strain rate. As preparation for the study of elastoplasticity in the subsequent chapters, they will be explained previously in this chapter.

7.1

Definition of Hyperelasticity

The material in which a work done per unit reference volume during a certain deformation state to other deformation state is invariant independent of a deformation path is defined as a hyperelastic material or Green elastic material. Therefore, the hyperelastic material possesses the following mechanical properties. (1) An energy is not dissipated/produced during a stress or strain cycle. (2) There exists the one-to-one correspondence between the stress and the deformation. Therefore, the current stress is determined only by the current deformation, and vice versa independently of loading path.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 K. Hashiguchi, Foundations of Elastoplasticity: Subloading Surface Model, https://doi.org/10.1007/978-3-030-93138-4_7

189

190

7.2

7

Elastic Constitutive Equations

Hyperelastic Equations

R   The work done per unit reference volume is designated as w0 ¼ w0 dt where w0 is described by the scalar product of the work-conjugate pair of stress and deformation rate in Eq. (5.58). Now, let the right Cauchy-Green deformation tensor   C ¼ U2 in Eq. (4.16) be adopted as the deformation measure which designates the pure deformation independent of the rigid-body rotation R, although the deformation gradient tensor F is influenced by the rigid-body rotation superimposed on the deformed configuration. Therefore, the single-valued scalar function wðCÞ for C must exist and the following relation must hold. Z w0 ¼

C

Z dw0 ¼ wðCÞ  wðC0 Þ ¼

C0

C

C0



@wðCÞ : dC @C

 ð7:1Þ

where wðCÞ is called the elastic strain (or potential) energy function or stored energy function or Helmholtz free energy function. (Note) Substituting DU ¼ Q þ W ¼ TDS þ W (U: internal energy, T: absolute temperature, Q: heat, W: work, S: entropy) into the Helmholtz free energy function F ¼ U  TS ! DF ¼ DU  TDS under the isothermal process (DT ¼ 0), one has W ¼ DF. Therefore, the work increment done by the external force is identical to the increment of the Helmholtz free energy. Then, it follows that dw0 ¼

@wðCÞ @wðCÞ : dC ¼ 2 : dC=2 @C @C

ð7:2Þ

On the other hand, the work increment done to the unit reference volume is given by dw0 ¼ S : dC=2 in Eq. (5.58). Therefore, the second Piola-Kirchhoff stress is given by S¼2

@wðCÞ @C

ð7:3Þ

where the one-to-one correspondence between the stress S and the deformation variable C exists. Here, note that the function wðCÞ must satisfy the complete integrability condition @Sij @Skl ¼ @Ckl @Cij

ð7:4Þ

which is imposed by the exchangeability of the order of partial derivatives, i.e.

7.2 Hyperelastic Equations

191

@2w @2w ¼ @Cij @Ckl @Ckl @Cij

ð7:5Þ

which holds for an arbitrary scalar continuous function wðCÞ. Now, the strain energy function is chosen to be positive and zero in the reference configuration for convenience, i.e. wðCÞ  0;

wðIÞ ¼ 0

ð7:6Þ

leading to S¼

@w  ¼O @C C¼I

ð7:7Þ

Then, the strain energy function wðCÞ takes the minimal-minimum (global minimum) stationary value in the reference configuration C ¼ I (no deformation) and the stress S is zero in the reference configuration C ¼ I (no deformation) as shown in Fig. 7.1. In the above, the hyperelastic constitutive equation is defined in terms of the right Cauchy-Green deformation tensor C which is independent of the rotation of material but depends only on the pure deformation. On the other hand, it is defined first in terms of the deformation gradient tensor F which depends not only on the deformation also on the rotation of material and after that it is reduced to the function of the right Cauchy-Green deformation tensor C or the Green strain tensor E in order to fulfill the objectivity by Holzapfel (2000), Bonet and Wood (2008), de Souza Neto et al. (2008), Gurtin et al. (2010), which would be roundabout explanation compared with the explanation in terms of the right Cauchy-Green deformation tensor C as mentioned above.

S

ψ (C) C=I

C ij

C =I

Fig. 7.1 Rough illustrations of strain energy function and Piola-Kirchhoff stress

C ij

192

7

Elastic Constitutive Equations

On the other hand, the constitutive equation which describes the one-to-one correspondence of stress and strain but does not possess an elastic strain energy function and thus does not fulfill the integrability condition described by the partial derivative of the stress tensor by the strain tensor, resulting in the energy production or dissipation during a loading cycle, is defined as the Cauchy elasticity. Furthermore, a linear relation between a stress rate and a strain rate which does not lead to the one-to-one correspondence between a stress and a strain and in which the strain energy function does not exist and thus the energy is produced or dissipated during a loading cycle is called the hypoelasticity (Truesdell 1955). Here, note that the elastoplastic constitutive equation must be formulated in a rate form because it exhibits the loading-path dependency. Concurrently, the hyperelastic-based plastic constitutive equation describing both the elastic and the plastic deformations exactly must be formulated in terms of the work-conjugate pair 

inducing the work rate identical to S : C=2. Here, note that the elastoplastic constative equation described in the intermediate configuration based on the isoclinic concept must be described in terms of the Mandel stress tensor M and the velocity gradient tensor L in the intermediate configuration which are work-conjugate pair fulfilling 



s : l ¼ S : C=2 ¼ M : L which is reduced to the hyperelastic equation w0 ¼ 



e

wðCe Þ ¼ s : le ¼ S : Ce=2 ¼ M : L in the elastic deformation process as will be described in Sect. 17.6. Now, substituting Eq. (7.3) into the stress tensors listed in Table 5.1, we obtain various expressions of the hyperelasticity as follows: 1 @wðCÞ T 1 @wðEÞ T @wðCÞ T @wðEÞ T F ¼ F F ; s ¼ 2F F ¼F F r¼2 F J @C J @E @C @E @wðCÞ @wðEÞ ¼F ; P ¼ 2F @C @E @wðCÞ @wðEÞ @wðCÞ @wðEÞ S¼2 ¼ ; M ¼ 2C ¼ ð2E þ IÞ @C @E @C @E

ð7:8Þ

The tangent stiffness modulus tensor CS  2

@S @ 2 wðCÞ ¼4 @C @C  @C

ð7:9Þ

in the hyperelastic relation in terms of the second Piola-Kirchhoff stress tensor S satisfies both the minor symmetry and the major symmetry, i.e. ð7:10Þ

7.2 Hyperelastic Equations

193

Furthermore, noting @ @ @CPQ @ @FrP FrQ @ ¼ ¼ ¼ ðdri dPA FrQ þ FrP dri dQA Þ @FiA @CPQ @FiA @CPQ @FiA @CPQ @ @ @ ¼ FiQ þ FiP ¼ 2FiP @CAQ @CPA @CPA

ð7:11Þ

@ @ @bpq @ @FpR FqR @ ¼ ¼ ¼ ðdip dAR FqR þ FpR diq dAR Þ @FiA @bpq @FiA @bpq @FiA @bpq @ @ @ FqA þ FpA ¼ 2 FqA ¼ @biq @bpi @biq

ð7:12Þ

one has

and

9 @ @ @ > > ¼ 2F ¼F > > @F @C @E > > > > @ 1 1 @ 1 @ > > > > ¼ F ¼ @C 2 @F 2 @E =   > @ > > F ¼ F1 > > @b > > > > > @ @ > > 1 @ ; ¼F ¼2 @E @F @C

ð7:13Þ

9 @ @ > ¼2 F > > > > @F @b > = @ 1 @ 1 ¼ F > @b 2 @F > > > @ @ 1 > > ; ¼ F @e @F

ð7:14Þ

by which Eq. (7.8) is described in terms of the deformation measures in the current configuration as follows: 1 @wðbÞ @wðbÞ b; s ¼ 2 b J @b @b @wðbÞ P¼2 F; @b @wðbÞ @wðbÞ F; M ¼ 2FT F S ¼ 2F1 @b @b

r¼2

ð7:15Þ

194

7

Elastic Constitutive Equations

noting @wðCÞ 1 @wðCÞ @wðbÞ ¼ 2 F1 ¼ 2F1 F @C 2 @F @b @wðbÞ T @wðbÞ FF ¼ 2 b s ¼ FSFT ¼ 2FF1 @b @b @wðbÞ @wðbÞ P ¼ FS ¼ 2FF1 F¼2 F @b @b @wðbÞ @wðbÞ F ¼ 2FT F M ¼ CS ¼ 2FT FF1 @b @b S¼2

Noting @ 1 @ 1 ¼ V @b 2 @V

ð7:16Þ

@ @ @b @ @V2 @ ¼2 V ¼ ¼ @V @b @V @b @V @b

ð7:17Þ

derived from

the stress tensors in Eq. (7.15) are expressed in terms of V as follows: @wðVÞ @wðVÞ ; s¼V @V @V @wðVÞ 1 V F; P¼ @V @wðVÞ 1 @wðVÞ 1 V F; M ¼ FT V S ¼ F1 @V @V r ¼ 1J V

ð7:18Þ

If the strain energy function w is represented by the principal stretches ka ða ¼ 1; 2; 3Þ, the principal values of stress tensors are given as follows (Holzapfel, 2000) @w @w ; s a ¼ ka @ka @ka @w ða ¼ 1; 2; 3Þ P a ¼ Ma ¼ @ka @w Sa ¼ k1a @ka ra ¼ 1J ka

ð7:19Þ

by Eq. (7.18) with the replacements of F and V to ka . These principal stresses are expressed by the principal values ra of the Cauchy stress as follows: Sa ¼ J

1 1 ra ; Pa ¼ Ma ¼ J ra ka k2a

ða ¼ 1; 2; 3Þ

ð7:20Þ

7.2 Hyperelastic Equations

195

The stress-strain relation of the isotropic material is given from Eq. (1.304) by S ¼ /E0 I þ /E1 E þ /E2 E2

ð7:21Þ

where /E0 ; /E1 ; /E2 are the functions of invariants of E, while it is not necessary hyperelastic equation. By eliminating the third term, Eq. (7.21) is reduced to the isotropic linear hyperelastic equation: S ¼ aðtrEÞI þ 2bE

ð7:22Þ

possessing the elastic strain energy function 1 wðEÞ ¼ aðtrEÞ2 þ bE2 2

ð7:23Þ

where a and b are the material constants. Now, consider the strain energy function in terms of the invariants of C, i.e. wðCÞ ¼ wðIC ; IIC ; IIIC Þ

ð7:24Þ

The substituting Eq. (7.24) into Eq. (7.8), it follows that  1 @wðCÞ @wðCÞ @wðCÞ 2 Iþ ðIC I  CÞ þ ðIIC I  IC C þ C Þ FT r ¼ 2 pffiffiffiffiffiffiffiffi F @IC @IIC @IIIC IIIC ð7:25Þ noting Eq. (4.71). It follows by virtue of Eq. (7.3) that    @wðCÞ : C¼0 S2 @C

ð7:26Þ

On the other hand, the incompressibility requires      1 J ¼ Jtrd ¼ JtrðFT E F1 Þ ¼ JtrðC1 EÞ ¼ JC1 : E ¼ JC1 : C ¼ 0 ð7:27Þ 2

The following relation must hold from Eqs. (7.26) and (7.27) for the incompressible material. S2

@wðCÞ 1 ¼ rm JC1 @C 2

ð7:28Þ

where rm is the hydrostatic stress and can be determined by the additional equation based on the incompressibility condition J ¼ 1. Eventually, the constitutive equation of the incompressible material is given from Eq. (7.28) by

196

S¼2

7

Elastic Constitutive Equations

@wðCÞ 1 þ rm JC1 @C 2

ð7:29Þ

which is represented by the other stress tensors M ¼ CS ¼ 2C

@wðCÞ þ rm JI @C

ð7:30Þ

1 1 @wðCÞ T r ¼ FSFT ¼ 2 F F þ rm I J J @C

7.3

ð7:31Þ

Explicit Hyperelastic Models

Various explicit hyperelastic constitutive equations will be shown in this section.

7.3.1

St.Venant-Kirchhoff Model

The simplest hyperelastic material model is the St. Venant-Kirchhoff model which is just an extension of the geometrically linear elastic material model to the geometrically nonlinear regime. It possesses the following strain energy function. 1 wðEÞ ¼ kðtrEÞ2 þ ltrE2 ; 2

1 wðCÞ ¼ kftr½ðC  IÞ=2g2 þ ltr½ðC  IÞ=22 2 ð7:32Þ 0

where the material constants k and l correspond to the Lame constants in the infinitesimal linear elasticity theory. Equation (7.32) is identical to Eq. (7.23) with the replacement of a ! k and b ! l. The substitution of Eq. (7.32) into Eq. (7.8) reads: S ¼ kðtrEÞI þ 2lE ¼ kftr½ðC  IÞ=2gI þ lðC  IÞ

ð7:33Þ

It is followed that

 where



ð7:34Þ

ð7:35Þ

7.3 Explicit Hyperelastic Models

197

Further, the Mandel stress in Eq. (5.22) is given for Eq. (7.33) as M ¼ ktr½ðC  IÞ=2C þ lCðC  IÞ

ð7:36Þ

The Kirchhoff stress is given from Eq. (7.33) as s ¼ ktr½ðb  IÞ=2b þ lbðb  IÞ

ð7:37Þ

noting s ¼ FSFT ¼ F½kftr½ðC  IÞ=2gg þ lðC  IÞFT   1 1 ¼ ktr ðFT F  IÞ FFT þ 2lF ðFT FIÞ FT 2 2 The multiplications of C to S, C and I from the left lead to M, CC and C. On the other hand, the contravariant push-forward operation, i.e. the multiplications of F from the left and FT from the right to S, C and I lead to FSFT ¼ s, FCFT ¼ FðFT FÞFT ¼ bb and FIFT ¼ b. Therefore, the hyperelastic equations in terms of ðM; CC; CÞ and ðs; bb; bÞ possess the identical form. It is unfortunate that the St. Venant-Kirchhoff equation is practically useless except for the small deformation range.

7.3.2

Neo-Hookean Model

The neo-Hookean elastic constitutive equation is derived by Rivlin (1948) based on the statistical thermodynamics of cross-linked polymer chains, which is usable for incompressible plastics and rubber-like substances. It is given as 1 1 wðCÞ ¼ lðIC  3Þ ¼ lðk21 þ k22 þ k23  3Þ 2 2

ð7:38Þ

noting Eq. (4.70)1 , where l is the martial constant. The substitution of Eq. (7.38) into Eq. (7.8) with Eqs. (4.71) and (5.22) leads to S ¼ lI

ð7:39Þ

M ¼ lC

ð7:40Þ

s ¼ lb

ð7:41Þ

The neo-Hookean model can be derived through the statistical thermodynamics of cross-linked polymer chains. The model is also inadequate for biaxial states of stress and incompressible materials and has been superseded by the Mooney-Rivlin model and the others described in the following.

198

7

Elastic Constitutive Equations

The hyperelastic constitutive models other than the Neo-Hookean model have been formulated by the statistical thermodynamics of cross-linked polymer. The well-known models are the Arruda-Boyce model (Arruda and Boyce 1993), the Gent model (Gent 1996) and the micro-sphere model (Miehe et al. 2004). The neo-Hookean model was extended to the following various compressible hyperelastic models. (a) Modified Neo-Hookean Model (1) The following strain energy function for the Neo-Hookean model is extended to the compressible material as follows: 1 1 wðCÞ ¼ lðIc  3Þ  l ln J þ kðln JÞ2 2 2

ð7:42Þ

where k and l are the material constants, which is adopted for the multiplicative elastoplasticity model by Ulm and Celigoj (2021). It follows from Eq. (7.42) that  1 1 1 1 S ¼ 2 lI  l C1 þ k2ðln JÞ C1 2 2 2 2 leading to S ¼ lðI  C1 Þ þ kðln JÞC1

ð7:43Þ

pffiffiffiffiffiffiffiffiffiffiffi 8 @J @ det C 1 1 > > ¼ ¼ JC < @C @C 2 pffiffiffiffiffiffiffiffiffiffiffi > @ ln J @ ln det C 1 > : ¼ ¼ C1 @C @C 2

ð7:44Þ

noting

with the aid of Eq. (4.71). Further, the Mandel stress M is given by M ¼ lðC  IÞ þ kðln JÞI

ð7:45Þ

Equation (7.43) is described by the Kirchhoff stress as s ¼ FSFT ¼ F½lðI  C1 Þ þ kðln JÞC1 FT ¼ F½lðI  F1 FT Þ þ kðln JÞF1 FT FT leading to s ¼ lðb  IÞ þ kðln JÞI

ð7:46Þ

7.3 Explicit Hyperelastic Models

199

(b) Modified Neo-Hookean Model (2) The following strain energy function is adopted by Simo and Pister (1984), Ciarlet (1988), etc. 1 1 1 wðCÞ ¼ kðJ 2  1Þ  ð k þ lÞ ln J þ lðIc  3Þ 4 2 2

ð7:47Þ

from which, noting Eq. (7.44), we have 1 S ¼ kðJ 2  1ÞC1 þ lðI  C1 Þ 2

ð7:48Þ

1 M ¼ kðJ 2  1ÞI þ lðC  IÞ 2

ð7:49Þ

1 s ¼ kðJ 2  1ÞI þ lðb  IÞ 2

ð7:50Þ

Equation (7.47) is reduced to the Neo-Hookean model in Eq. (7.38) for the incompressible material ðJ ¼ 1Þ. (c) Modified Neo-Hookean Elasticity (3) The following strain energy function is used for the finite elastoplastic constitutive relation by Vladimirov et al. (2008, 2010). 1 l wðCÞ ¼ KðJ 2  1  2lnJÞ þ ðtrC  3  2lnJÞ 4 2

ð7:51Þ

where K and l are material constants, from which we have K 2 ðJ  1ÞC1 þ lðI  C1 Þ 2

ð7:52Þ

M¼

K 2 ðJ  1ÞI þ lðC  IÞ 2

ð7:53Þ

s¼

K 2 ðJ  1ÞI þ lðb  IÞ 2

ð7:54Þ

S¼

noting Eq. (7.44). Equation (7.51) will be adopted for the formulation of the finite elastoplastic constitutive equation in Sect. 17.11.1 for the multiplicativehyperelastic-based plasticity for metals. (d) Modified Neo-Hookean Elasticity (4) The following the strain energy function is assumed by Simo and Pister (1984), Ciarlet (1988), Simo and Miehe (1992), etc., which has the separated form into two terms related to the volume change J and the isochoric deformation C.

200

7

Elastic Constitutive Equations

  K J2  1 l wðCÞ ¼  lnJÞ þ ðtrC  3 2 2 2

ð7:55Þ

where K and l are the material constants, from which we have  1 1 2 1 2=3 1 G  ðtrCÞC S ¼ KðJ  1ÞC þ lJ 2 3

ð7:56Þ

1 1 M ¼ KðJ 2  1ÞG þ lJ 2=3 C' ¼ KðJ 2  1ÞG þ lC' 2 2

ð7:57Þ

1 s ¼ KðJ 2  1Þg þ lb' 2

ð7:58Þ

exploiting Eq. (4.63). Except for the St. Venant-Kirchhoff elastic constitutive equation in Sect. 7.3.1, the hyperelastic equations are described in complex forms in terms of the second Piola-Kirchhoff stress tensor S but it can be described concisely in terms of the Mandel stress tensor M.

7.3.3

Mooney Model

The strain energy function in the Mooney model (Mooney 1940) is given by wðCÞ ¼

C1 ðk21

þ k22

þ k23

 3Þ þ C2

1 1 1 þ 2 þ 23 2 k1 k2 k 3

! ð7:59Þ

where C1 and C2 are the material constants. Equation (7.59) is regarded as the particular case of the Ogden model which will be shown in the next subsection. For the incompressible case ðk1 k2 k3 ¼ 1Þ, noting Eq. (4.70), Eq. (7.59) is reduced to the Mooney-Rivlin model: wðCÞ ¼ C1 ðIc  3Þ þ C2 ðIIc  3Þ ¼ C1 ðk21 þ k22 þ k23  3Þ þ C2 ðk21 k22 þ k22 k23 þ k23 k21  3Þ

ð7:60Þ for which one has S ¼ 2C1 I þ 2C2 ðIC I  CÞ

ð7:61Þ

M ¼ 2C1 C þ 2C2 ðIC C  C2 Þ

ð7:62Þ

7.3 Explicit Hyperelastic Models

201

s ¼ 2C1 b þ 2C2 ðIC b  b2 Þ

7.3.4

ð7:63Þ

Ogden Model

The generalized hyperelastic model was proposed by Ogden (1972) referring to the generalized strain tensor in Eq. (4.48) or (4.49) as follows: w¼

N X bn

a n¼1 n

ðka1n þ ka2n þ ka3n  3Þ

ð7:64Þ

where N, bn and an are the material constants. The Ogden model in Eq. (7.64) is reduced to the Mooney model in Eq. (7.59) by choosing N ¼ 2; a1 ¼ 2; b1 ¼ C1 and a2 ¼ 2; b2 ¼ C2 . Equation (7.64) is rewritten as follows: w¼

3 X

xðka Þ

ð7:65Þ

a¼1

with xðka Þ ¼

7.4

N X b

n

a n¼1 n

ðkaan  1Þ

ð7:66Þ

Rate Forms of Hyperelastic Equation

The time-differentiation of Eq. (7.8)4 leads to 

S¼

@ 2 wðEÞ  :E @E  @E

ð7:67Þ

Substituting Eqs. (4.126) and (7.67) into Eq. (6.39), the Truesdell rate of  Ol

Kirchhoff stress s

is rewritten as

202

7 DOl

s

 ¼F

Elastic Constitutive Equations

  @ 2 wðEÞ @ 2 wðEÞ DOl dkl : ðFT dFÞ FT sij ¼ FiA FjB FkC FlD @E  @E @EAB @ECD

ð7:68Þ DOl

which is the rate of hyperelastic equation in the current configuration. Here, s related to the Zaremba-Jaumann rate of Cauchy stress in Eq. (6.48) as DOl

s

w

¼ Jðr dr  rd þ rtrdÞ

is

ð7:69Þ

The Zaremba-Jaumann rate of Cauchy stress is related to the strain rate from these equations as  2 1 @ wðEÞ T r ¼ F ðF dFÞ FT þ dr þ rd  rtrd detF @E  @E w

ð7:70Þ

which is expressed as w

~ :d r ¼E

ð7:71Þ

~ in the current configuration is where the hyperelastic tangent modulus tensor E given by ð7:72Þ

 with 1 Rijkl  ðrik djl þ ril djk þ rjk dil þ rjl dik Þ ðRijkl ¼ Rklij ¼ Rjikl ¼ Rijlk Þ 2

7.5

ð7:73Þ

Infinitesimal Strain-Based Elastic Equation

For the infinitesimal deformation, the hyperelastic constitutive equation can be given as r¼ e¼

@wðeÞ ; @e

@/ðrÞ ; @r



r¼ 

e¼

@w2 ðeÞ   : e ¼ E : e; @e  @e

@/2 ðrÞ   : r ¼ E : r; @r  @r

E

@r @ 2 wðeÞ ¼ @e @e  @e

E1 

@e @ 2 /ðrÞ ¼ @r @r  @r

ð7:74Þ ð7:75Þ

7.5 Infinitesimal Strain-Based Elastic Equation

203

where e is the infinitesimal strain in Eq. (4.32). The work w done during the elastic deformation is uniquely determined by the elastic strain energy function before and after the deformation as follows: Z W¼

e

e0

Z r : de ¼

e

e0

@wðeÞ : de ¼ ½wðeÞee0  ¼ wðeÞ  wðe0 Þ @e

ð7:76Þ

Adopting the elastic strain energy function 1 wðeÞ ¼ LðtreÞ2 þ Gtre2 2

ð7:77Þ

the stress is given by the linear relation to the elastic strain e as r ¼ Lev I þ 2Ge

ð7:78Þ

i.e. r ¼ ðL þ

2 GÞev I þ 2Ge' 3

ð7:79Þ

which is to be the general isotropic linear elastic equation in Eq. (1.307) or (1.308) with Eq. (1.309) and is referred to as the Hooke’s law, where L and G are called the 0

Lame constants. It follows by taking the trace and the deviatoric part of Eq. (7.78) that rm ¼ Kev ; r0 ¼ 2Ge0

ð7:79Þ

1 1 0 rm ; e 0 ¼ r K 2G

ð7:80Þ

ev ¼ where

2 2 K  L þ GðL ¼ K  GÞ 3 3

ð7:81Þ

rm ð ðtr rÞ=3Þ is the spherical stress. K and G are called the bulk elastic modulus and the shear elastic modulus, respectively. It follows from Eqs. (7.79) and (7.80) that r ¼ Kev I þ 2Ge' ¼ ðK  23 GÞev I þ 2Ge 1 rm I þ e ¼ 3K

1 1 1 1 r' ¼ ð  Þrm I þ r 2G 3K 2G 2G

ð7:82Þ

204

7

Elastic Constitutive Equations

Equation (7.82) are represented as r¼E:e

ð7:83Þ

using the elastic modulus tensor E given by 



 



 

ð7:84Þ



where is the fourth-order tracing tensor and is the fourth-order deviatoric projection tensor defined in Eqs. (1.191) and (1.194), respectively. The inverse relation between the two equation in Eq. (7.84) is confirmed by ( KT + 2GI ' ) : [(1/ 9) KT + (1/ 2G )I ' ] = (1/ 3)T + I ' = I , noting Eq. (1.194). It follows from Eq. (7.80) for the uniaxial loading process (rij ¼ 0 for i ¼ j 6¼ 1and i 6¼ j) that e11 ¼

1 r11 ; E

m e22 e33 e22 ¼ e33 ¼  r11 ! ¼ ¼ m e11 e11 E

ð7:85Þ

where E¼

9KG ; 3K þ G

m¼

3K  2G 2ð3K þ GÞ

ð7:86Þ

noting 9 h 1 i 1  9K þ 3G 1 1 1 r11 þ ðr11  r11 Þ ¼ r11 þ r11 ¼ r11 > > > > 9K 2G 3 9K 3G 27KG > > = 3K  2G

1  2G  3K > 1 2ð3K þ GÞ > > r11  r11 ¼ r11 ¼  ¼ r11 > > 9KG > 9K 6G 18KG ; 3K þ G ð7:87Þ

e11 ¼

e22

The inverse relations of Eq. (7.86) are given as K

E ; 3ð1  2mÞ

G

E 2ð1 þ mÞ

ðL ¼

vE Þ ð1 þ mÞð1  2mÞ

ð7:88Þ

Here, in the uniaxial loading state, E is the ratio of the axial stress to the axial strain and is called the Young’s modulus, and m is the ratio of lateral strain to axial strain and is called the Poisson’s ratio. The strain energy function in (7.77) with Eq. (7.88) is expressed noting e ¼ ½ðtreÞ=3I þ e0 is expressed as

7.5 Infinitesimal Strain-Based Elastic Equation

vE E ðtreÞ2 þ tre2 2ð1 þ mÞð1  2mÞ 2ð1 þ mÞ  vE E 1 2 2 02 ðtreÞ þ ðtre Þ þ tre ¼ 2ð1 þ mÞð1  2mÞ 2ð1 þ mÞ 3 E E 2 ðtreÞ2 þ tre0 ¼ 6ð1  2mÞ 2ð1 þ 2mÞ

205

wðeÞ ¼

ð7:89Þ

Both of the variables ðtre2 Þ for the volumetric deformation and tre0 2 ð¼ e0rs e0rs Þ for the deviatoricdeformation are positive. Then, the Poisson’s ratio must be limited in the range 1\v\1=2

ð7:90Þ

in order that wðeÞ  0 hold in any deformation, e.g. in the pure volumetric deformationðtre0 2 ¼ 0Þ and the pure deviatoric deformation ðtre ¼ 0Þ. Here, note that the lower limit ðm ¼ 1Þ and the upper limit ðm ¼ 0:5Þ in Eq. (7.90) correspond to keep the similar shape and the constant volume, respectively. The former is seen in artificial structures, e.g. honeycomb. Substituting Eq. (7.88) into Eq. (7.84), the elastic modulus tensor is also described using the Young’s modulus and the Poisson’s ratio as follows: E=

E T+ E E δ δ + E 1( , E ijkl = + δ ilδ jk ) − 1 δ ijδ kl ] 3(1 − 2ν ) 3(1 − 2ν ) ij kl 1 +ν [ 2 δ ik δ jl 1 +ν I ' 3

−1 −1 E = 1 − 2ν T + 1 +ν I ' , E ijkl = 1 − 2ν δ ij δ kl + 1 +ν [ 1 (δ ik δ jl + δ ilδ jk ) − 1 δ ijδ kl ] 3E 3E E E 2 3

ð7:91Þ

or E = E (S + ν T ), E ijkl = E [ 1 (δ ik δ jl + δ ilδ jk ) + ν δ ijδ kl ] 1 +ν 1 − 2ν 1 +ν 2 1 − 2ν −1 −1 E = 1 [(1 +ν )S −νT ], E ijkl = 1 [ 1 (1 +ν )(δ ik δ jl + δ ilδ jk ) −νδ ijδ kl ] E E 2

ð7:92Þ

noting Eqs. (1.191)–(1.195), which is confirmed by −1 E : E = E (S + ν T ) : 1 [(1 + ν )S −νT ] E 1 +ν 1 − 2ν

(1 +ν ) ) S :T − 1 ν− 22ν T :T ] = 1 [(1 +ν )S + (−ν + ν 1 +ν 1 − 2ν −ν (1 − 2ν ) +ν (1 +ν ) − 3ν 2 T =S =S + (1 +ν )(1 − 2ν )

ð7:93Þ

206

7

Elastic Constitutive Equations

noting Eqs. (1.1.91) and (1.1.92) Incidentally, by the matrix notation 2

3 T1111 T1122 T1133 T1123 T1131 T1112 6 7 6 T2211 T2222 T2233 T2223 T2231 T2212 7 6 7 6T 7 6 3311 T3322 T3333 T3323 T3331 T3312 7 T ¼ ½Tijkl  ¼ 6 7 6 T2311 T2322 T2333 T2323 T2231 T2312 7 6 7 6T 7 4 3111 T3122 T3133 T3123 T3131 T3112 5 T1211 T1222 T1233 T1223 T1231 T1212

ð7:94Þ

as explained in Appendix C, where the components ð9 9Þ are abbreviated to ð6 6Þ, Eq. (7.92) is represented as follows: 2 6 6 6 E 6 E¼ ð1 þ mÞð1  2mÞ 6 6 4

1v

2 E1

v v 1v v 1v Sym:

m m 1 Sym:

0 0 0 1þv

0

0

0

1v v 0 1v 0

0 0

0 0

0

0

1  2v

0

1

6 6 16 ¼ 6 E6 6 4

m 1

0 0 0 1  2v

3 0 0 7 0 0 7 7 0 0 7 ð7:95Þ 7 0 0 7 5 1  2v 0 1  2v 3 0 7 0 7 7 0 7 7 0 7 5 0 1þv

0 0 0 0 1þv

ð7:96Þ

which is confirmed by 2

1

E:E

6 6 6 6 E 6 ¼ ð1 þ mÞð1  2mÞ 6 6 6 4 2

1v

v

1

60 6 6 60 6 ¼6 60 6 40 0

0

0

1

0

0

1

0

0

0 0

0 0

1  2v

Sym:

ð1mÞm2 m2

6 mmð1mÞm2 6 6 6 mmð1mÞm2 1 6 ¼ ð1 þ mÞð1  2mÞ 6 60 6 40 2

v

0 3 0 0 0 0 0 07 7 7 0 0 07 7 1 0 07 7 7 0 1 05 0 0

3 2 7 6 7 6 7 6 716 7 6 7E6 7 6 7 6 5 4

1

m

m

0

0

0

1

m 1

0 0

0 0

0 0

1þv

0

0

1þv

0

Sym:

3 7 7 7 7 7 7 7 7 5

mð1mÞ þ mm2

1  2v mð1mÞm2 þ m

0

0

1þv 0

ð1mÞm2 m2

mð1mÞm2 þ m

0

0

0

mmð1mÞm2

ð1mÞm2 m2

0

0

0

0

0

ð1 þ mÞð12mÞ 0

0 0

0 0

0 0

ð1 þ mÞð12mÞ 0

3

0 0 ð1 þ mÞð12mÞ

7 7 7 7 7 7 7 7 5

1

ð7:97Þ

7.5 Infinitesimal Strain-Based Elastic Equation

207

Table 7.1 Relationships between two independent elastic constants E; m

G; m

E; G

E; K

E

E

2ð1 þ mÞG

E

E

G

E 2ð1 þ mÞ E 3ð1  2mÞ

G

G EG 3ð3G  EÞ E  2G 2G GðE  2GÞ 3G  E

3EK 9K  E K

m

m

2ð1 þ mÞG 3ð1  2mÞ m

L

mE ð1 þ mÞð1  2mÞ

2Gm 1m

K

G; K 9KG 3K þ G G K

3K  E 6K 3Kð3K  EÞ 9K  E

3K  2G 2ð3K þ GÞ 2 K G 3

L; G lð3L þ 2GÞ LþG G 2 G 3 L 2ðL þ GÞ L

Lþ

Relationships between two independent elastic constants are listed in Table 7.1. The elastic strain energy function wðeÞ and the complementary energy function /ðrÞ are given for the linear elasticity as ⎧ ψ (ε) = 1 ε : E: ε = 1 E [ε ij εij + E ( ε kk ) 2 ] ⎪ 2 2 1 +ν 1 − 2ν ⎪ ⎨ ⎪φ (σ ) = 1 σ : E −1 : σ = 1 [(1+ν )σ σ −ν (σ ) 2 ] ij ij kk 2 ⎪ 2E ⎩

ð7:98Þ

from which it follows that ⎧ ∂ψ ν E 1 1 ⎪σ = ∂ε = E: ε = 1 + ν (1 − 2ν ε v I + ε ) = E[3(1 − 2ν ) ε v I + 1 + ν ε' ] ⎪ ⎨ 1 ⎪ε = ∂φ = E−1 : σ = 1 [ ν σ − ν σ I ] = E [(1 − 2ν ) σm I + (1 + ν )σ' ] ⎪ ∂σ E (1+ ) 3 m ⎩

ð7:99Þ

The rate forms of the infinitesimal elastic equation is given from Eqs. (7.80) and (7.99) as follows: 8   2 >      > > < r ¼ K ev I þ 2G e' ¼ K  3 G ev I þ 2G e   ð7:100Þ > 1  1  1 1  1   > >e¼ rm I þ r' ¼  rm I þ r : 3K 2G 3K 2G 2G 



r m ¼ K ev ;





r' ¼ 2G e'

 8  1 1  E m     > > ' r e e ev I þ e ¼ E I þ ¼ < v 3ð1  2mÞ 1þm 1 þ m 1  2m h i > > : e ¼ 1 ð1  2mÞ rm I þ ð1 þ mÞ r ' ¼ 1 ½ð1 þ mÞ r 3m rm I E E

ð7:101Þ

ð7:102Þ

208

7.6

7

Elastic Constitutive Equations

Cauchy Elasticity

The elastic material which does not have a strain energy function but has a one-to-one correspondence between the Cauchy stress and a strain is called the Cauchy elastic material. Here, the stress tensor is given by an equation of strain tensor and thus the equation includes six strain components. The equation of six strain components does not fulfill the condition of complete integration leading to the strain energy function so that it does not result in the hyperelasticity in general. Then, the work done by the stress is generally dependent on the deformation path. For that reason, an energy dissipation/production is induced during the stress or strain cycle. In the above-mentioned definition, the Cauchy elastic material is described as r ¼ fðeÞ

ð7:103Þ

in terms of the Almansi strain tensor e in Eq. (4.27) or (4.29). Equation (7.103) is reduced to the following equation by virtue of Eq. (1.304) for the isotropic material. r ¼ /e0 I þ /e1 e þ /e2 e2

ð7:104Þ

where /e0 ; /e1 ; /e2 are functions of invariants of e. Furthermore, for an isotropic linear elastic material, Eq. (7.104) is reduced r ¼ LðtreÞI þ 2Ge

ð7:105Þ

noting Eq. (7.78). Limiting to the infinitesimal strain leading to e ffi e, Eq. (7.105) results in Eq. (7.76), i.e. r ¼ LðtreÞI þ 2Ge

ð7:106Þ

Here, substituting Eq. (7.106) with Eq. (4.32) into Eq. (5.33), the Navier’s equation is obtained by replacing L and G to a and b, respectively, as follows: 

ða þ bÞ$ð$ uÞ þ bDu þ qb ¼ q v ða þ bÞ

@ 2 uj @ 2 ui  þb þ qbi ¼ q vi @xj @xi @xj @xj

noting Eqs. (1.374), (1.380) and h

1 @ui k @ a @u d þ 2b ij 2 @xj þ @xk @xj

@uj @xi

i ¼a

@ 2 uj @ 2 ui @ 2 uj þb þb @xj @xi @xj @xj @xj @xi

ð7:107Þ

7.7 Hypoelasticity

7.7

209

Hypoelasticity

The following relation, called the hypoelasticity, was proposed by Truesdell (1955) in order to take the material rotation into account. 

r ¼ HðrÞ : d

ð7:108Þ

where HðrÞ is the forth-order tensor-valued isotropic function of the stress tensor r and it is called hypoelastic response function. However, the hypoelastic equation is incapable of describing the constitutive equation exactly, since d does not mean the pure rate of deformation as shown in Eqs. (4.83) and (4.88). In addition, there does not exist a one-to-one correspondence between the stress and the deformation and the work done during a certain variation of stress depends on the loading path. If we adopt the Hookean elastic modulus tensor E for HðrÞ, Eq. (7.108) is reduced to 

r¼E:d

ð7:109Þ

which is described as follows, noting Eqs. (7.100) and (7.102). 9   2 > > K  G dv I þ 2Gd > = 3   1  1 0 1 1  1 > > rm I þ r  rm I þ r> d¼ ; 3K 2G 3K 2G 2G 

r ¼ Kdv I þ 2Gd0 ¼

  rm ¼ Kdv ; r0 ¼ 2Gd0   9 1 1 0 E m  > dv I þ d ¼ dv I þ d > r¼E = 3ð1  2mÞ 1þm 1 þ m 1  2m i> i 1h 1h     > d¼ ð1  2mÞ rm I þ ð1 þ mÞ r0 ¼ ð1 þ mÞ r 3m rm I ; E E

ð7:110Þ

ð7:111Þ

ð7:112Þ

Incidentally, the following equation in which the Jaumann rate of Cauchy stress is related nonlinearly to the strain rate is called the hypoplastic material (Kolymbas and Wu 1993) named by Dafalias (1986). 

r ¼ fðd; rÞ;



rij ¼ fij ðdkl ; rkl Þ

ð7:113Þ

pffiffiffiffiffiffiffiffiffiffiffi where fij is the nonlinear function of dkl , e.g. drs drs ð¼ jjdjjÞ. However, the hypoplastic constitutive equation would not express the real elastoplastic deformation behavior as will be described in Sect. 8.1.

210

7

Elastic Constitutive Equations

Equation (7.109) is modified to the following rational relation in terms of the 

work-conjugate pair ðs; dÞ as shown in Eq. (5.58) in Sect. 5.8. τo = E : d

ð7:114Þ

While the three major types of elastic materials are described in this chapter, the other elastic material, called the Cosserat elastic material, was advocated by Cosserat and Cosserat (1909). The couple stress is related to the rotational strain in this material. It has been applied to the prediction of localized deformation (e.g. cf. Mindlin, 1963; Muhlhaus and Vardoulaskis, 1987).

Chapter 8

Elastoplastic Constitutive Equations

Elastic deformation is induced microscopically by the elastic deformations of the material particles themselves, exhibiting a one-to-one correspondence to the stress as described in Chap. 7. However, when the stress reaches an yield stress, slippages between material particles (e.g. crystal lattice in metals and soil particles in soils) are induced, which do not disappear even if the stress is removed, leading macroscopically to the plastic deformation. Then, the one-to-one correspondence between the stress and the strain, i.e. the stress–strain relation does not hold in the elastoplastic deformation process, exhibiting the loading-path dependence. Therefore, one must formulate the elastoplastic constitutive equation as a relation between the stress rate and the strain rate. This chapter addresses the basic concept and formulation for elastoplastic constitutive equations in the conventional elastoplasticity (Drucker 1988) based on the assumption that the inside of the yield surface is a purely elastic domain as the introduction to elastoplasticity, while the formulations are given within the framework of the infinitesimal strain theory in terms of the material-time derivative. The unconventional elastoplasticity describing the plastic strain rate induced by the rate of stress inside the yield surface will be described in the subsequent chapters, which is required to describe the cyclic loading behavior, and further the exact finite strain theory based on the multiplicative decomposition of the deformation gradient tensor will be given in Chap. 17.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 K. Hashiguchi, Foundations of Elastoplasticity: Subloading Surface Model, https://doi.org/10.1007/978-3-030-93138-4_8

211

212

8.1

8

Elastoplastic Constitutive Equations

Fundamental Requirements for Elastoplastic Constitutive Equations

The elastoplastic constitutive equations must be formulated to satisfy the following requirements. (1) Decomposition of deformation measure into elastic and plastic parts The deformation of solids which are assemblies of sloid particles, e.g. crystal particles in metals and soil particles in sands and clays are induced by the deformations of material particles themselves and their mutual slips. The deformations of material particles themselves cause the macroscopic elastic deformation of the material as far as a large stress such as material particles deforms irreversibly is not applied. On the other hand, when the macroscopic stress applied to the material increases up to the yield stress, the mutual slips between the material particles are not removed even if the stress is removed (unloaded), so that they cause the plastic deformation of the material. Therefore, the decomposition of strain rate into the elastic and the plastic parts is inevitable for the formulations of the evolution rules of internal variables which is irrelevant to the elastic deformation but changes with the plastic deformation history in order to describe the irreversible change of the internal structure. Eventually, the deformation measure of material is decomposed into the elastic part and the plastic part. (2) Rate(incremental) form equation by loading-path dependence The plastic deformation induced during the change from a certain stress to the other stress depends on the stress path during which the stress traces. Therefore, it exhibit the loading path dependence. Then, the constitutive equation describing the deformation including the plastic deformation is obliged to be formulated in the rate (incremental) form. The elastic deformation reflects the variation of the internal structure of material, i.e. the elastic deformational state of material particles. Although the plastic deformation itself is irrelevant to the internal structure of material, the plastic internal variable reflects the variation of the internal structure of material which influences on the plastic deformational characteristics. The examples of the plastic internal variables are the isotopic hardening variable and the kinematic hardening variable, i. e. the backstress which will be described in the subsequent sections.

8.1 Fundamental Requirements for Elastoplastic Constitutive Equations

213

(3) Incorporation of yield surface The stress applied between the material particles needs to increase to overcome the friction resistance in order to induce their mutual slips leading to the macroscopic plastic deformation. Then, the stress which causes the plastic deformation is called the yield stress. Therefore, the yield surface must be incorporated, which is formed by connecting the yield stresses in various directions. In addition, the incorporation of the yield surface is required for the formulation of the plastic strain rate which is associated to the yield surface. Besides, the above-mentioned basic requirements for the constitutive equation of irreversible deformation are ignored in the hypoplasticity in which the rate-nonlinear terms of the strain rate are incorporated. The hypoplasticity was disused in the field of metal plasticity twenty years later after the proposition of the hypoelasticity by Truesdell (1955) but sadly it is still used in the field of soil plasticity.

8.2

Classification of Elastoplastic Constitutive Equations

The elastoplastic constitutive equations can be classified into the three types described below.

8.2.1

Infinitesimal Hyperelastic-Based Plasticity

The infinitesimal strain tensor e in Eq. (4.31) or (4.32) is additively decomposed into the elastic strain tensor ee and the plastic strain tensor ep as follows: 





e ¼ e e þ ep ; e ¼ ee þ e p

ð8:1Þ

The Cauchy stress is given by the hyperelastic relation through the strain energy function wðee Þ in Eq. (7.74) as follows: r¼

@wðee Þ @ee

ð8:2Þ

Then, the work We done during the elastic deformation process is uniquely determined by the values of the strain energy function before and after the elastic deformation based on Eq. (7.76), i.e., Z w ¼ e

ee ee0

Z r : de ¼ e

ee ee0

e @wðee Þ : dee ¼ ½wðee Þee0 ¼ wðee Þ  wðee0 Þ @ee

214

8

Elastoplastic Constitutive Equations

Further, the strain energy function wðee Þ is given in the quadratic form for the linear elasticity: 1 wðee Þ ¼ ee : E : ee 2

ð8:3Þ

r ¼ E :ee ¼ E :ðe ep Þ; ee ¼ E1 :r; e ¼ E1 :r þ ep

ð8:4Þ

leading to

which further leads to 















p

r ¼ E : ee ¼ E : ðe  e p Þ; ee ¼ E1 : r; e ¼ E1 : r þ e

ð8:5Þ

for the materials possessing the constant elastic modulus tensor, i.e. E ¼ const The stress r is calculated by substituting the plastic strain ep calculated by the plastic constitutive equation into Eq. (8.4). On the other hand, it can be calculated also by the time-integration by Eq. (8.5). The accuracy and the efficiency in the numerical calculations by these two methods are almost same. This can be stated also for tensor-valued internal variables, e.g. the kinematic hardening variables and the elastic-core which will be described in the subsequent sections and chapters. The stress rate versus elastic strain rate for the Hooke’s law is given from Eqs. (7.82) and (7.89) as follows: 9   2 > e e > ¼ K  G ev I þ 2Ge r¼ > = 3   1 1 0 1 1 1 > > ee ¼ rm I þ r ¼  rm I þ r> ; 3K 2G 3K 2G 2G Keev I þ 2Gee0

r0 ¼ 2Gee0

rm ¼ Keev ; 

ð8:6Þ



ð8:7Þ 9 e e > ev I þ e > =

1 1 e0 E  m eev I þ e ¼ 3ð1  2mÞ 1þm 1 þ m 1  2m 1 1 ee ¼ ½ð1  2mÞrm I þ ð1 þ mÞr0  ¼ ½ð1 þ mÞr  3mrm I E E r¼E

> > ;

ð8:8Þ

where 1 eev  tree ; ee0  ee  eev I 3

ð8:9Þ

The infinitesimal hyperelastic-based plasticity is limited to the infinitesimal elastic and plastic deformation without a rotation of material. Elastoplastic constitutive equations will be described in the infinitesimal hyperelastic-based plasticity

8.2 Classification of Elastoplastic Constitutive Equations

215

in the subsequent sections, since it is quite simple to be applied to various engineering problems in practice and it can be transformed directly to the multiplicative hyperelastic-based plasticity in Chap. 17. Further, Eq. (8.5) will be used for soils, etc. even if the elastic modulus tensor E is not con-stant but it is a function of stress r as will be described in Chap. 13, in which a strain energy function in Eq. (8.3) does not hold usually so that Eqs. (8.4) and (8.5) cannot lead to the hyperelastic equation in that case.

8.2.2

Hypoelastic-Based Plasticity

A constitutive equation describing the inherent deformation properties of materials which is not influenced by the rigid-body rotation of materials. However, the influence of the rigid-body rotation is not excluded in the infinitesimal strain formulation. Then, the hypoelasticity was proposed by Truesdell (1955) as the modification of the infinitesimal strain formulation so as to exclude the influence of the rigid-body rotation as described in Sect. 7.7. The strain rate d in Eq. (4.62) is additively decomposed into the elastic strain rate de and the plastic strain rate dp , i.e. d ¼ de þ dp

ð8:10Þ

The corotational stress rate is given by 



r ¼ E : de ¼ E : ðd dp Þ; de ¼ E1 : r

ð8:11Þ

based on Eq. (7.109). The stress rate versus elastic strain rate is given for the Hooke’s law from Eqs. (7.110) as follows: )   r ¼ Kdve I þ 2Gde0 ¼ K  23 G dve I þ 2Gde 1 1  1 0 1  1  rm I þ 2G de ¼ 3K rm I þ 2G r ¼ 3K  2G r 

rm ¼ Kdve ;

0 e0 r ¼ 2Gd

 9 1 1 e0 E  m > dve I þ d ¼ dve I þ de > > = 3ð1  2mÞ 1þm 1 þ m 1  2m > > 1 1 0    ; de ¼ ½ð1  2mÞ rm I þ ð1 þ mÞ r  ¼ ½ð1 þ mÞ r 3m rm I > E E 

ð8:12Þ ð8:13Þ



r¼E

ð8:14Þ

216

8

Elastoplastic Constitutive Equations

where dve  trde ;

1 de0  de  dve I 3

ð8:15Þ

The convected time-derivative of internal variables, e.g. the kinematic hardening must be also given by the convected time-derivative explained in Sect. 3.4. The hypoelastic-based plastic constitutive equation is obtained from the infinitesimal strain-based elastoplasticity by replacing 

 

e

p



r ! r; e ! d; e ! de ; e ! dp ; e ! d The hypoelastic-based plasticity is capable of describing the finite plastic deformation and rotation. It has been studied widely by numerous researchers represented by Rodney Hill and James R. Rice after the proposition of hypoelasticity (Truesdell 1955). However, the hypoelastic-based framework possesses the following various defects. (1) The accurate description of elastic deformation is limited to be infinitesimal, (2) The energy dissipation and its accumulation are caused even within a purely elastic range during a cyclic loading (cf. Kojic and Bathe 1987; Brepols 2014), (3) A cumbersome time-integration scheme is required for the calculation of the stress and internal variables, and then a careful treatment is necessary in the numerical time-integration scheme to guarantee the objectivity of the constitutive relation as was described in detail in Sect. 3.4.4. (4) There exists the arbitrariness or the non-uniqueness regarding the choice of a corotational rate (cf. Peric 1992). Then, the hypoelastic-based plasticity will be disused by the skip from the infinitesimal hyperelastic-based plasticity to the multiplicative hyperelastic-based plasticity in the near feature.

8.2.3

Multiplicative Hyperelastic-Based Plasticity

The exact description of the finite deformation and rotation can be described by the multiplicative hyperelastic-based plasticity, in which the deformation gradient tensor is multiplicatively decomposed into the elastic and the plastic parts as will be explained in detail in Chap. 14. The primary purpose of this monograph is the explanation of the exact finite strain theory within the framework of the multiplicative hyperelastic-based plasticity. The multiplicative hyperelastic-based plasticity cannot be formulated from the hypoelastic-based plasticity but can be derived as the extension of the infinitesimal hyperelastic-based plasticity. Therefore, the formulation of the infinitesimal hyperelastic-based plasticity will be described in detail prior to the extensions.

8.3 Conventional Plastic Constitutive Equation

8.3

217

Conventional Plastic Constitutive Equation

The conventional plastic constitutive equation incorporating the yield surface enclosing a purely-elastic domain is described in this section. Now, consider first the following isotropic yield condition incorporating the isotropic hardening/softening. f ðrÞ ¼ FðHÞ

ð8:16Þ

where Fð  0Þ is the function of the isotropic hardening variable H and is called the hardening function which describes the isotropic hardening or softening, i.e. the expansion or contraction of yield surface. Here, we choose the yield stress function f ðrÞð  0Þ in Eq. (8.16) to be the homogeneous function of r in degree-one. Therefore, it follows that f ðjsjrÞ ¼ jsjf ðrÞ

ð8:17Þ

@f ðrÞ : r ¼ f ðrÞ ¼ FðHÞ @r

ð8:18Þ

for an arbitrary scalar s and

for the sake of Euler’s theorem for homogeneous function in degree-one (see Appendix D). Then, it follows from Eq. (8.18) that @f ðrÞ

@f ðrÞ

: r @f ðrÞ n : r





=

1=

¼ @r

¼ f ðrÞ @r @r F

ð8:19Þ

where n is the normalized outward-normal of the yield surface (see Appendix E). @f ðrÞ n @r



@f ðrÞ



@r ðknk ¼ 1Þ

ð8:20Þ

The material-time derivative of Eq. (8.16) leads to the consistency condition: @f ðrÞ  0  : r F H ¼ 0 @r

ð8:21Þ

which is ttransformed to the following normalized equation by multiplying Eq. (8.19). 

0



n : r F H

n:r ¼0 F

ð8:22Þ

218

8

Elastoplastic Constitutive Equations

where   p   0 p p p F  dF=dH; H r; H; e ¼ fHd r; H; e jje k ke k 

ð8:23Þ

p

H has to be the homogeneous function of e in degree-one for the rate-independent deformation behavior because it evolves only in the plastic loading process  ðe p 6¼ OÞ and possesses the dimension of time in minus one, while, needless to say,  it is a nonlinear equation of the components eijp in general as seen in metals, i.e. ffiffiffiffiffiffiffiffiffiffi ffi q pffiffiffiffiffiffiffiffi  p pffiffiffiffiffiffiffiffi  p  p  H ¼ 2=3 jje k ¼ 2=3 ers ers described in Sect. 8.4. Now, assume the associated flow rule in which the plastic potential function is given by the yield function: 

p

  p k ¼ ke k [ 0



e ¼ kn

ð8:24Þ



where k is the magnitude of plastic strain rate, called often plastic positive proportionality factor or plastic multiplier. The expression of the flow rule in Eq. (8.24) possesses the clear physical meaning as definitely divided into the 

magnitude k and the pure direction tensor n. Then, the magnitude of plastic strain 



p

rate, i.e. ke k appearing often in the hardening variable can be replaced to k so that the physical interpretation of plastic constitutive relation can be captured clearly. 

On the other hand, the magnitude cannot be represented only by k in the flow rule 



with the expression e p ¼ k @f ðrÞ=@r except for the particular yield function given by the magnitude of stress jjrjj leading to jj@f ðrÞ=@rjj ¼ jj@jjrjj=@rjj ¼ 1. The physical backgrounds of the associated flow rule will be explained in Sect. 8.6. Substituting the plastic flow rule in Eq. (8.24) into the consistency condition (8.22), one has 

n :r

0

F  k fHn ðr; H; nÞn : r ¼ 0 F

ð8:25Þ

which leads to 



n : r  k Mp ¼ 0

ð8:26Þ

where M p is called the plastic modulus and is given by 0

Mp 

F fHn ðr; H; nÞn : r F

ð8:27Þ

8.3 Conventional Plastic Constitutive Equation

219

with 



fHn ðr; H; nÞ ¼ H = k

ð8:28Þ

noting Eq. (8.23) with Eq. (8.24). Here, the yield surface in Eq. (8.16) retains the similar shape and orientation with respect to the origin of stress space by virtue of homogeneity of function f ðrÞ. It follows from Eq. (8.26) that 



n : r p n : r ; e ¼ p n Mp M



k¼

ð8:29Þ

Substituting Eqs. (8.11) and (8.29) into Eq. (8.5), the strain rate is given by 



e ¼ E1 : r þ

  n :r n  n  1 n ¼ E þ :r Mp Mp

ð8:30Þ

The scalar product of n : E to Eq. (8.30) leads to     n :r n :r n :r  p p n : E : e ¼ n :r þ n : E : n ¼ ðM þ n : E : nÞ p ¼ ðM þ n : E : nÞ p Mp M M 



¼ ðM p þ n : E : nÞ k

from which the magnitude of plastic strain rate described in terms of strain rate, 



denoted by K instead of k, in the flow rule (8.24) is described as follows: 



K¼



n : E :e n : E :e p ; e ¼ p n p M þn : E : n M þn : E : n

ð8:31Þ

Incidentally, Eq. (8.31)1 can be also derived directly from the following relation obtained by substituting Eqs. (8.1) and (8.5) into the consistency condition (8.26). 



e











p

n : r  K M p ¼ n : E : e  K M p ¼ n : E :ðe  e Þ  K M p 





¼ n : E :ðe  K nÞ  K M p ¼ n : E : e ðn : E : n þ M p Þ K ¼ 0 ð8:32Þ The stress rate is given from Eq. (8.5) with Eq. (8.31) as follows: 

e



p



p





r ¼ r þ r ¼ E :ðe  e Þ ¼ E : e 

n : E :e E:n¼ Mp þ n : E : n

 E

 E:nn:E  :e Mp þ n : E : n

220

8

e

p

Elastoplastic Constitutive Equations

n pq  pqkl ε• kl  ijrsn rs M p + nab abcd n cd

p

σ• ij = σ• ije + σ• ij =  ijkl (ε• kl −ε• kl ) =  ij kl ε• kl −

ð8:33Þ

where e



r  E : e;





p



p

r  E : e ¼ Kpr : e

ð8:34Þ

E ijrs nrs npq E pqkl p :n n K pr ≡ E p ⊗ : E , K ijklr ≡ M + n: E: n M p + nab E abcd ncd

ð8:35Þ

e

r is called the elastic stress rate since it is the stress rate calculated supposing that p a purely elastic deformation is induced, and r is called the plastic relaxation stress pr rate. The fourth-order tensor K is called the plastic relaxation stiffness modulus tensor. Furthermore, using the elastoplastic stiffness modulus tensor

 ijrs n n pq  pqkl ep pr : n ⊗ n :  ,  ijkl =  ijkl − p rs ≡  ijkl −  ijkl ep ≡  −  pr =  − p M + nab  abcd ncd M + n: : n ð8:36Þ the stress rate can be described as 



r ¼ Kep : e

ð8:37Þ

Here, note that E possesses not only the minor symmetry but also the major symmetry E ijkl = E klij providing ( E : n)ij = ( n : E)ij = E ijkl nkl, so that the symmetries of E : n  n : E ¼ ½ðE : nÞ  ðn : EÞT and thus Kep ¼ KepT hold. Here, consider the non-associated flow rule 

p



e ¼ k m ðjjmjj ¼ 1; m 6¼ nÞ

ð8:38Þ

where m is the normalized second-order function of stress and internal variables. 

The magnitude of plastic strain rate K is given instead of Eq. (8.31) as follows: 



n : E :e n : re ¼ p K¼ p M þn : E : m M þn : E : m 

! ð8:39Þ

noting Eq. (8.34), Then, the elastoplastic stiffness modulus tensor is given by Kep ¼ EKpr ¼ E

E:mn:E Mp þ n : E : m

ð8:40Þ

8.3 Conventional Plastic Constitutive Equation

221

Therefore, the plastic relaxation modulus tensor and the elastoplastic stiffness modulus tensor are not the symmetric tensors, i.e. Kp 6¼ KpT ; Kep 6¼ KepT in the non-associated flow rule, which causes the complexity and the inefficiency in numerical calculation.

8.4

Constitutive Equation of Metals

The general form of the elastoplastic constitutive equations for isotropic materials are described above. The constitutive equation of metals is shown below, which has made an important contribution to the development of elastoplasticity. The following von Mises yield condition (von Mises, 1923) with the associated flow rule can be assumed for metals. rffiffiffi pffiffiffiffiffiffiffi 3 0 jjr jj ¼ 3J2 2 qffiffi qffiffi p ¼ 23jj e jj; fHn ¼ 23

f ðrÞ ¼ req ; req  (

H ¼ eeqp 

Rqffiffi2

p

3jj e





jjdt; H ¼ e

eqp

ð8:41Þ

0

FðHÞ ¼ F0 f1 þ sr ½1  expðcH HÞg; F  dF=dH ¼ sr cH F0 expðcH HÞ ð8:42Þ where F0 ; sr and cH are the material constants. The hardening function F in Eq. (8.42) increases from the initial value F0 with the equivalent plastic strain eeqp and saturates when it reaches the maximum value ð1 þ sr ÞF0 as shown in Fig. 8.1.

F( Fs

eqp)

(1 sr ) F0 1

F0

F ' = sr cH F0 exp( cH

0 Fig. 8.1 Isotropic hardening function in the uniaxial loading process

eqp)

eqp

222

8

Elastoplastic Constitutive Equations

In the monotonic uniaxial loading with the assumption of the plastically-pressure p p p p p independence (rij ¼ 0 except for i ¼ j ¼ 1; e2 ¼ e3 ¼  e1 =2 ðtr e ¼ 0Þ, eij ¼ 0 ði 6¼ jÞ), it holds that 8 pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < req  3=2jjr0 jj ¼ 3=2 ðr1  r1 =3Þ2 þ 2ð0  r1 =3Þ2 ¼ r1 ð8:43Þ ffi  R pffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R p  p : eeqp  R pffiffiffiffiffiffiffi 2 2=3jjep0 jj dt ¼ 2=3 ep2 e1 dt ¼ ep1 1 þ 2ð e1 =2Þ dt ¼

Then, req and eeqp coincide with the axial stress and the axial plastic strain in the uniaxial loading and thus are called the equivalent stress and the equivalent (or accumulated) plastic strain, respectively. Substituting Eq. (8.42) into Eqs. (8.30) and using the relations 9 rffiffiffi @f ðrÞ 3 r0 r0 0 > > ; n ¼ 0 ; n : r ¼ jjr jj > ¼ > > jjr jj @r 2 jjr0 jj > > > r ffiffi ffi > rffiffiffi 0  > 0 > 3 r 3r :r 3  = 0  eq : r n : r ¼ ¼ r ¼ 0 eq ð8:44Þ 2 jjr jj 2 r 2 > > > p  p0 > > e ¼e > > rffiffiffi > 0 > > F 2 2 0 > 0 p ; M ¼ jjr jj ¼ F F 3 3 the constitutive equation of the isotropic Mises material is given as follows: qffiffi 

n :r  e ¼ E1 : r þ 2 0 n ¼ E1 : r þ 3F 



2  eq 3r 2 0 3F

0



eq

r 31r  ¼ E1 : r þ r0 0 jjr jj 2 F 0 req

ð8:45Þ

which is called the Prandtl-Reuss equation. The plastic work rate of this material is described as 

p



r:e ¼ r:k

 r0 p  eqp  eqp ¼ k jjr0 jj ¼ jjr0 jjjj e jj ¼ req e ¼ Fðeeqp Þ e 0 jjr jj

ð8:46Þ

which is the product of the hardening function and the rate of the equivalent plastic strain and thus the hardening attributable to the plastic work is called the work hardening, too. The traction t acting on the octahedral plane is expressed by   1 1 1 t ¼ r em ¼ ðr1 e1  e1 þ r2 e2  e2 þ r3 e3  e3 Þ pffiffiffi e1 þ pffiffiffi e2 þ pffiffiffi e3 3 3 3 1 ¼ pffiffiffi ðr1 e1 þ r2 e2 þ r3 e3 Þ ð8:47Þ 3

8.4 Constitutive Equation of Metals

223

on the principal base, noting Eq. (5.2) with n ¼ em and Eq. (1.135). Then, the normal component roct and the tangential component s oct of the traction t are given as   1 1 1 1 roct  t  em ¼ pffiffiffi ðr1 e1 þ r2 e2 þ r3 e3 Þ  pffiffiffi e1 þ pffiffiffi e2 þ pffiffiffi e3 3 3 3 3 1 ¼ ðr1 þ r2 þ r3 Þ ¼ rm 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi soct  jjt jj2  r2oct rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 1 ðr þ r22 þ r23 Þ  ðr1 þ r2 þ r3 Þ2 ¼ 3 1 9 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ¼ fr þ r22 þ r23  ðr1 r2 þ r2 r3 þ r3 r1 Þg 9 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffipffiffiffiffiffi pffiffiffi 2 eq 1 2 r ðr1  r2 Þ2 þ ðr2  r3 Þ2 þ ðr3  r1 Þ2 ¼ J2 ¼ ¼ 3 3 3

ð8:48Þ

ð8:49Þ

where soct is called the octahedral shear stress and J2 is given by choosing the deviatoric Cauchy tensor r0 as the second invariant of the deviatoric tensor T0 in Eq. (1.182) as follows: 1 1 1 1 2 J2  kr0 k ¼ tr r0 2 ¼ r0rs r0sr ¼ req 2 2 2 2 3 1 02 0 2 0 2 0 2 ¼ ðr11 þ r22 þ r33 Þ þ r12 þ r0232 þ r0312 2 1 1 ¼ ðr012 þ r022 þ r032 Þ ¼ fðr1  r2 Þ2 þ ðr2  r3 Þ2 þ ðr3  r1 Þ2 g 2 6

ð8:50Þ

It is interpreted from Eqs. (8.49) and (8.50) for the Mises yield condition that the yielding is induced when the octahedral shear stress reaches a certain value. Equations (8.48) and (8.49) are also derived as the components in the directions pffiffiffi Im ð¼ 3em Þ and t0 ð¼ T0 =jjT0 jjÞ, i.e. Tm and jjT0 jj of Tm and T0 , respectively, regarding T as r in Eqs. (1.314) and (1.318).

8.5

Formulation of General Loading Criterion

The judgment of whether or not the plastic strain rate is induced is required for the elastoplastic deformation analysis. The criterion for this judgment is called the loading criterion. In what follows, it is formulated (Hashiguchi 2000), which is not limited to the associated flow rule but applicable even to the non-associated flow rule described at the end of Sect. 8.3.

224

8

Elastoplastic Constitutive Equations

1. In the loading (plastic deformation) process, it is required noting Eqs. (8.29) and (8.39) that 9   > n :r > = k ¼ p [0 p M for e 6¼ O ð8:51Þ   > n : E :e > ; [0 K¼ p M þn : E : m 

p

2. In the unloading (elastic deformation) process e ¼ O, it holds that 

n :r 0

ð8:52Þ

n :r [ 0 for M p \0 Mp

ð8:53Þ

and thus





k¼ 





e



e



On the other hand, substituting e ¼ e leading to n : E : e ¼ n : E : e ¼ n : r into 

Eq. (8.31), it holds for K that 

n :r \0 ð8:54Þ Mp þ n : E : m in this process. Here, note that the elastic modulus tensor E is the positive definite tensor fulfilling n : E : n [ 0 for an arbitrary n as defined in Eq. (1.264) for the second-order tensor, noting  (ε e)  ε e: E: ε e  0. Here, we may assume 

K¼

n : E : m jM p j, provided that m is not far different from n. Eventually, it follows that M p þ n : E : m [ 0:

ð8:55Þ

Also, note that the infinite rate of plastic relaxation is induced so that the stress decreases in the infinite rate if the denominator becomes zero, i.e. M p þ n : E : m ! 0 as known from Eq. (8.33) illustrated in Fig. 8.2 for the unip e axial loading process. Then, in the unloading process ðe ¼ O; e 6¼ OÞ, the following inequalities hold from Eqs. (8.53) and (8.54), depending on the sign of the plastic modulus M p leading to the hardening, perfectly-plastic and softening state. 8  p > > < k \0 and K \0 for M [ 0   k ! Indeterminate and K \0 for M p ¼ 0 > >  : k [ 0 and K \0 for M p \0

ð8:56Þ

8.5 Formulation of Loading Criterion





225





Fig. 8.2 Signs of k and K in uniaxial loading process ðr e ¼ E :
Þ 

Therefore, the sign of k at the moment of unloading from the state M p 0 can be 

positive or indeterminate in the unloading process. On the other hand, K is definitely negative in the unloading process. Thus, the distinction between a loading 

and an unloading process cannot be judged by the sign of k but it can be done by 

that of K. Consequently, the loading criterion is given as either

p  6 O for f ðrÞ ¼ FðHÞ and K [ 0 e ¼ p e ¼ O for others

ð8:57Þ

or 

p



e ¼ 6 O for f ðrÞ ¼ FðHÞ and n : E : e [ 0 p e ¼ O for others

ð8:58Þ

in lieu of Eq. (8.55). Limiting to the hardening process with M p [ 0, Eq. (8.58) leads to

p  6 O for f ðrÞ ¼ FðHÞ and n : r [ 0 e ¼ p e ¼ O for others

ð8:59Þ

However, the loading criterion in Eq. (8.59) is inapplicable to the softening state observed in loose soils, rocks, concrete and damage materials, etc. The plastic loading is interpreted as the process that the plastic relaxation stress  p rate r is induced, noting Eq. (8.34).

226

8

Elastoplastic Constitutive Equations

dσ e (n: d σ e = n: : dε > 0): loading

n

σ dσ e (n : d σ e = n : : unloading

0

: dε < 0)

dσ e (n : d σ e = n : : dε = 0) : neutral loading ij

Yield surface f ( : (ε ε p )) = F ( H ) Fig. 8.3 Loading criterion by the direction of the elastic stress increment dre  E : de. in strain space 



The loading criterion in Eq. (8.58) is rewritten by the elastic strain rate r  E : e in Eq. (8.34) as

p e e ¼ 6 O for f ðrÞ ¼ FðHÞ and n : r [ 0 p e ¼ O for others

ð8:60Þ

Equation (8.60) is interpreted as follows: The loading, the neutral loading and the 

unloading are defined as processes that the elastic stress rate re is directed outward, tangential and inward direction, respectively, of the yield surface as shown in Fig. 8.3.   In elastoplastic deformation analysis, suppose to calculate first r and e by either of the elastic or the elastoplastic constitutive equation. Then, check the sign of  n : E : e. If the sign conflicts with the loading criterion, it is required to recalculate them using another constitutive equation. Here, it would be efficient to calculate first by the elastoplastic constitutive equation since the monotonic loading process in which the elastoplastic deformation process continues is seen often in practical engineering problems.  The loading judgment by the direction of strain rate, i.e. the sign of n : E : e (or  n : r for the hardening state) is not required for the numerical calculation exploiting the return-mapping projection for hyperelastic-based constitutive equations.

8.6 Physical Backgrounds of Associated Flow Rule

8.6

227

Physical Backgrounds of Associated Flow Rule

The associated flow rule holds for a wide range of materials, which insists “The direction of the plastic strain rate is the outward-normal of the yield surface in the coordinate space where the each directions of the plastic strain rate components p eij are taken same to the each directions of the stress components rij ”. The adjective “associated” is added because the flow rule is associated to the yield surface. On the other hand, the flow rule which is not associated to the yield surface is called the non-associated flow rule which is observed in highly frictional (pressure-dependent yielding) materials. Several physical interpretations for the associated flow rule are described in this section.

8.6.1

Positiveness of Second-Order Plastic Work Rate: Prager’s Interpretation

Prager (1949) reported that the associated flow rule must hold to fulfill the positivity of the second-order plastic work rate, i.e. 



p

r:e 0

ð8:61Þ

The direction of the plastic strain rate must be outward-normal to the yield surface for any stress rate directing outwards the yield surface, assuming that the direction of plastic strain rate depends on the state of stress but independent of the rate of stress as shown in Fig. 8.4. However, Eq. (8.61) holds only for hardening materials in which the stress rate is directed outwards the yield surface but does not hold for softening materials. εp

σ

σy

0

ij ,

Yield surface

Fig. 8.4 Prager’s positive plastic work rate for hardening materials

p

ij

228

8.6.2

8

Elastoplastic Constitutive Equations

Principle of Maximum Plastic Work

The postulate, called the principle of maximum plastic work, was proposed by Mises for rigid-plastic materials and Hill (1948b, 1950) and Mandel (1964) for elastoplastic materials. It insists that the plastic work rate done by the actual stress ry on the yield surface is greater than a plastic work rate done by any mechanically-admissible stress r inside the convex yield surface, leading to 





ry : e [ r : e ; i:e:ðry  r Þ: e [ 0 p

p

p

ð8:62Þ

as depicted in Fig. 8.5. Then, the plastic strain rate must be directed to the outward-normal of convex yield surface, so that the associated flow rule must hold in order to satisfy the principle of maximum plastic work under the assumption that p the direction of plastic strain rate e depends only on the state of stress but it is independent of the rate of stress. This fact was discussed for the finite strain theory by Lubliner (1984).

8.6.3

Positiveness of Work Done During Stress Cycle: Drucker’s Interpretation

Drucker (1951) proposed the background as will be described in the following. Drucker (1951) postulated “the work done during the stress cycle by the external agency is positive”. It is described mathematically as follows: Z

tðr0 Þ t0 ðr0 Þ



ðr  r0 Þ : e dt  0

ð8:63Þ

εp

σy

σy σ*

0

σ* ij ,

Yield surface

Fig. 8.5 Principle of maximum plastic work

p

ij

8.6 Physical Backgrounds of Associated Flow Rule

229

y

(σ y σ 0 ) : ε pdt

0

d

0

p

0 Fig. 8.6 Positive work done by external agency in Drucker’s (1951) postulate (Illustration in uniaxial loading process)

where r0 stands for the initial stress at the initial time t0 ðr0 Þ, and tðr0 Þ designates the time that the stress returns to the initial stress. The following inequality is obtained from Eq. (8.63) under the assumption that the inside of the yield surface is the purely-elastic domain (see Fig. 8.6). 

p

ðry  r0 Þ: e  0

ð8:64Þ 

where ry designates the stress on the yield surface in which the plastic strain rate e is induced. The followings must hold in order to fulfill Eq. (8.64).

p

(1) The plastic strain rate is directed outward-normal of the yield surface in the coordinate system where the components of the stress and the corresponding components of the plastic strain rate are taken to the same directions. Then, the associated flow rule must hold, provided that the direction of plastic strain rate is determined solely by the current stress and internal variable but independent of stress rate. (2) In this occasion the yield surface has to be the convex surface (see Fig. 8.7). The result (1) is called the associated flow rule or the normality rule and the result (2) is called the convexity of yield surface.

8.6.4

Positiveness of Second-Order Plastic Relaxation Work Rate

Ilyushin (1961) postulated that “the work done during the strain cycle is positive”. Limiting to the infinitesimal deformation process. It leads to the postulate “the second-order work increment d 2 w is not larger than the second-order elastic stress

230

8

εp

Yield surface

σy

σ0 σ0

σ0

σ0

εp

σ0

σy

Elastic region

0

Elastoplastic Constitutive Equations

σ0

(σ y σ 0 ) : ε p 0 ij ,

p

ij

Concave yield surface: Violation of Drucker (1951)’s postulate.

Normality rule Fig. 8.7 Associated flow (normality) rule and convexity of yield surface based on the Drucker’s (1951) postulate

work increment d 2 wes calculated by presuming that the strain increment is induced elastically” or “the second-order plastic relaxation work increment d 2 wpr , i.e. the work increment done during the infinitesimal strain cycle is not negative” (Hill 1968; Petryk 1991, 1997; Hashiguchi 1993a). The second-order work increment d 2 w is shown by Dabf, the second-order elastic stress work increment d 2 wes by Dadf and the second-order plastic relaxation work increment d 2 wpr by Dabc noting Dadb ¼ Dabc in Fig. 8.8. It is described mathematically as follows: d 2 w ¼ d 2 wes  d 2 wpr ; d 2 w d 2 wes ; d 2 wpr  0

ð8:65Þ

where 9 1 1 > d 2 w  dr : de ¼ dee : E : de > > > 2 2 = 1 e 1 2 es d w  dr : de ¼ de : E : de > 2 2 > > > 1 p 1 p 1 2 pr d w   dr : de ¼ de : E : de ¼ k dtm : E : de ; 2 2 2

ð8:66Þ

with dre  E : de; drp  E : dep

ð8:67Þ

dre and drp are called the elastic stress increment and the plastic relaxation stress  e  p increment, respectively, following r and r in Eq. (8.34).

8.6 Physical Backgrounds of Associated Flow Rule

231

d

d d

p

E

e

1

d

a

d 2wpr

e

b f

1

E

c

0

d

p

d

e

d

(a) Hardening process ( M p

0) d

d

e

d

E

p

1

a

f e

d

b

d 2wpr 1 E

c

0

d

d d

(b) Softening process ( M p

e

p

0)

Fig. 8.8 Positiveness of work done during strain cycle, i.e. second-order plastic work rate ðde ¼ dee þ dep Þ (Illustration in uniaxial loading)

232

8

Elastoplastic Constitutive Equations

It should be noted that the associated flow rule with m ¼ n in Eq. (8.38) must hold in order that Eq. (8.65) conforms to the loading condition in Eq. (8.58), 3

although Eq. (8.58) was not restricted to the associated flow rule. The above-mentioned positiveness of second-order plastic relaxation work rate is violated in the non-associated flow which causes i) the physically unacceptable deformation in which the tangent stiffness modulus is larger than the elastic modulus and ii) the asymmetry of the tangent stiffness modulus tensor, iii) the difficulty in the formulation of the variational principle and iv) the necessity for the incorporation of additional material constant(s) as revealed by Hashiguchi (1991).

8.6.5

Comparison of Interpretations for Associated Flow Rule

Prager’s (1949) interpretation of the associated flow rule is concerned only with hardening materials as described previously. Hill’s (1948b, 1950) principle of maximum plastic work is not limited to the hardening behavior but does not provide a clear physical background. On the other hand, the interpretations of the positivity of work done by the external agency, i.e. the additional stress during a stress cycle by Drucker (1951) and of the positivity of the second-order plastic relaxation work rate during a strain cycle by Ilyushin (1961) are based on the conceivable physical postulates of the dissipation energy of materials unlimited to the hardening behavior. Here, Drucker’s (1951) postulate is related to the stress cycle but Ilyushin’s (1961) postulate of the second-order plastic relaxation work rate is related with the infinitesimal strain cycle. Now, compare below the pertinence of these postulates. (1) The strain cycle can be realized always. However, the stress cycle cannot be made in a softening state in which the stress cannot be returned to the initial state if the plastic strain rate is induced. It is based on the fact that any deformation can be given but a stress cannot be given arbitrarily to materials since strength of materials is limited. (2) Limiting to the infinitesimal cycles, consider the stress and the strain cycles. The second-order work increment done during the infinitesimal stress cycle is given by ð1=2Þdr : dep (Dabe in Fig. 8.8a). On the other hand, the additional work increment ð1=2Þdep : E : dep (Daec in Fig. 8.8a) must be done to close the strain cycle, whilst ð1=2Þdep : E : dep  0 holds because of the positive-definiteness of the elastic modulus tensor E. Therefore, the work done during the infinitesimal stress cycle is far smaller than the work during the infinitesimal strain cycle. In other words, Drucker’s (1951) postulate holds for the materials fulfilling a more restricted condition, i.e., more particular materials than the materials fulfilling the positivity of the second-order plastic relaxation work rate. (3) Strain (increment) in any definition is determined uniquely by the displacement (increment) induced in the material. Therefore, the configuration of material

8.6 Physical Backgrounds of Associated Flow Rule

233

returns to the initial configuration only if the strain returns to the initial value. In other words, if a cycle of strain in a certain definition closes, cycle of strain in any other definition (Lagrangian, Almansi, logarithmic, nominal and infinitesimal strains for instance) also closes. On the other hand, the stress is defined by the force per unit area and thus it is influenced by the deformation. The configuration in the end of stress cycle differs from the initial configuration depending on the loading path chosen during the cycle and on the definition of stress (Cauchy, Kirchhoff, nominal and second Piola-Kirchhoff stresses for instance). Then, even if a cycle of stress in a certain definition closes, the cycle in the stress in the other definition does not close. Eventually, the positivity of the strain cycle possesses the objectivity, but that of the stress cycle is not objective. (4) The assumption that the interior of the yield surface is the purely elastic domain is adopted in Drucker’s interpretation. On the other hand, this assumption is not required by the postulate of the positivity of second-order plastic relaxation work rate, which holds on the quite natural premise that the purely elastic deformation is induced at the moment of reverse loading. Eventually, it can be stated that postulate of the positivity of second-order plastic relaxation work rate is more general than the Drucker’s postulate. However, even the former is based on the premise that the direction of the plastic strain rate is dependent on the normal component but independent of the tangential component of stress rate to the yield surface. It is observed in the test result that the inelastic deformation is induced even by the tangential component. The inelastic strain rate induced by the component of stress rate tangential to yield surface will be described in Sect. 9.7.

8.7

Anisotropy

The plastic strain rate described in Sect. 8.3 concerns the yield condition with the function of stress invariants and scalar-valued internal variables. Therefore, it is limited to the materials exhibiting the isotropic responses in the plastic deformation behavior. In what follows, first the isotropy in constitutive equation is defined. Then, the plastic strain rate extended to the anisotropy will be explained in this section.

8.7.1

Definition of Isotropy

An isotropic material is defined as one exhibiting identical mechanical response that is independent of the chosen direction of material element or of the coordinate

234

8

Elastoplastic Constitutive Equations

system by which the response is observed. Here, the input/output variables are the stress rate and the strain rate in the irreversible deformation. The rate-type constitutive equation is described in general as follows: 



fðr; r; Hi ; eÞ ¼ O

ð8:68Þ

where Hi ði ¼ 1; 2; 3;   Þ denotes collectively scalar-valued or tensor-valued internal state variables. When the following equation holds by giving coordinate transformations only for stress (rate) and strain rate tensors in the tensor-valued function f, it can be stated that Eq. (8.68) describes the constitutive equation of isotropic material. 







fðQ r QT ; Q r QT ; Hi ; Q e QT Þ ¼ Qfðr; r; Hi ; eÞQT

ð8:69Þ

In the plastic constitutive equation formulated incorporating the yield and/or plastic potential function, the isotropy holds if the yield and/or plastic potential function is given by the function of stress invariants and scalar internal variables. Then, designating these scalar-valued functions by f , it must fulfill the following equation. f ðQ r QT ; Hi Þ ¼ f ðr; Hi Þ

ð8:70Þ

In contrast, the anisotropic plastic constitutive equation is described by incorporating the yield and/or plastic potential function including tensor-valued internal variable in addition to the stress invariants and scalar internal variables. Then, Eqs. (8.69) and (8.70) do not hold in anisotropic constitutive equations.

8.7.2

Elastoplastic Constitutive Equation with Kinematic Hardening

It is well-known through the experimental observation that if the monotonic loading proceeds towards a certain direction in the stress space, the hardening develops in that direction but the yield stress lowers in the opposite direction. From the microscopic viewpoint, this phenomenon is induced by the staticallyindeterminable deformation of internal structure and is called the Bauschinger effect. To reflect this effect in the elastoplastic constitutive equation, the translation or the rotation of the yield surface is adopted widely. The translation of the yield surface, called the kinematic hardening, is realized by introducing the back-stress translating towards the loading direction and replacing the stress tensor with the relative tensor given by subtracting the back-stress tensor from the stress tensor. On the other hand, soils, which is the assembly of particles with weak adhesion among them, can bear a far larger compression stress than the tensile stress. Therefore, they exhibit a strong frictional property that the deviatoric yield stress increases with the pressure, while the yield surface only slightly includes the tensile stress regime near

8.7 Anisotropy

235

the origin of the stress space. Therefore, once the yield surface translates leaving the origin, it can never come back to include the origin again because the yield surface contracts with the plastic volume expansion leading to the softening. Therefore, the kinematic hardening cannot be applied but the rotation of the yield surface around the origin of stress space, i.e. the rotational hardening, is suitable to soils as will be described in Chap. 13. Now, let the yield condition (8.16) be extended to describe the anisotropy by introducing the internal variables of second-order tensor a as follows: f ð^ rÞ ¼ FðHÞ

ð8:71Þ

^  ra r

ð8:72Þ

where

að¼ a0 Þ being the back-stress (kinematic hardening variable) proposed by Prager (1956) in order to describe the induced anisotropy, the evolution rule of which depends on stress and internal variables as will be described in the next section. Note that an anisotropic hardening variable is deviatoric in general. Here, it is ^ in the homogeneous degree-one assumed that f in Eq. (8.71) is the function of r fulfilling f ðjsj^ rÞ ¼ jsjf ð^ rÞ. Then, the yield surface (8.71) maintains the similar shape and orientation with respect to the stress point r ¼ a. The material time-derivative of Eq. (8.71) leads to the consistency condition. @f ð^ rÞ  @f ð^ rÞ  0  :r : a F H ¼ 0 @^ r @^ r

ð8:73Þ

)

 p  p

  p p F; e = e e

H ¼ fHe ðr; F; e Þ ¼ fH^n r;

 p  p

  p p a ¼ f ke ðr; F; a; e Þ ¼ f ke r; F; a; e = e e

ð8:74Þ

where



p

noting that they are homogeneous functions of e in degree-one since they are p induced only in the plastic loading process e 6¼ O and the first-order time-differential quantities. The associated flow rule in Eq. (8.24) is extended as

 p    p ^ ðk [ 0; jj^ njj ¼ 1; e ¼ kÞ e ¼kn

ð8:75Þ



@f ð^ rÞ

@f ð^ rÞ



^ = n @r @r

ð8:76Þ

with

236

8

Elastoplastic Constitutive Equations

noting @f ðb r Þ=@r ¼ @f ðb r Þ=@ b r Substituting 9 @f ðb rÞ > > :b r ¼ f ðb rÞ ¼ F > = @b r @f ðb r Þ





:b r @f ðb

@f ðb

n :b > r Þ

r> > @r

n

r Þ ¼ ^ ; ^ = ¼ 1=

@r



f ðb rÞ @r F

ð8:77Þ

based on the Euler’s homogeneous function of tensor variable (see Appendix D), Eq. (8.73) leads to 

^:r ^ 0  n F H¼0 F



^n : r  ^n : a 

The substitution of the associated flow rule in Eq. (8.75) into this equation leads to 



^n : r  k M p ¼ 0

ð8:78Þ

where 

F0 fH^n b M  ^n : r þ f k^n F

 ð8:79Þ

p





^ Þ ¼ H = k; fH^n ðr; H; n





^Þ ¼ a = k f k^n ðr; F; a; n

ð8:80Þ

It follows from Eq. (8.78) that 

^ :r n ; Mp



k¼





p

e ¼

^ :r n ^ n Mp

ð8:81Þ

and 





e ¼ E1 : r þ

^ :r n ^¼ n Mp

  ^n ^  n :r E1 þ Mp

ð8:82Þ

from which it follows that 

^ : E :e n ^ : E :n ^ Mp þ n    ^n ^:E ^ : E :e E :n n    ^ ¼ E p E :n :e r ¼ E :e p ^ : E :n ^ ^ : E :n ^ M þn M þn 

K¼

ð8:83Þ ð8:84Þ

8.7 Anisotropy

237

The loading criterion is given as follows (Hashiguchi 1994, 2000) 



p



)

^ : E :e [0 6 O for f ð^ rÞ ¼ FðHÞ and K [ 0 or n e ¼ p e ¼ O for others

8.7.3

ð8:85Þ

Kinematic Hardening Rules

The evolution rule of the back-stress for the plastically-incompressible metals was given by Prager (1956) as follows (see Fig. 8.9): 



a ¼ ck e

p

ð8:86Þ

where ck is the material constant with the dimension of stress. Equation (8.86) is described in the uniaxial loading process by 



p

aa ¼ c k e a

ð8:87Þ

where ð Þa designates the axial component. If ck is extended to be a monotonic decreasing function of the equivalent plastic strain, the nonlinear behavior is described in the initial monotonic loading process but the peculiar behavior is described in the inverse loading process as shown in Fig. 8.9b. It should be noted that this extension is not accepted, resulting in the worsening, although it has been recommended in some literatures (e.g. de Souza Neto et al. 2008). Shield and Ziegler (1958) and Ziegler (1959) proposed the following modification of Prager (1956)’s linear kinematic hardening rule in Eq. (8.86), insisting that Preger’s rule possesses the mathematical inconvenience such that the components of back-stress tensor are induced also in the directions in which the components of stress tensor are zero.

a

a

ck

1 0

(a) ck : constant

1

p

a

0

ck

p

a

(b) ck : monotonic-decreasing function of plastic strain

Fig. 8.9 Rate-linear kinematic hardening rules illustrated in one-dimensional state

238

8

Elastoplastic Constitutive Equations

0  p

 ^ e

a ¼ cz r

ð8:88Þ



b 0i ¼ 0 which is seen in a where cz is the material constant, noting that ai ¼ 0 for r plane stress condition. The directions of the translations of the back-stresses in Eqs. (8.86) and (8.88) are identical to each other for the Mises yield surface whose 0 0 ^¼b r jj holds, while section cut by the deviatoric plane is circle for which n r =jjb they are obviously different from each other for the Tresca yield surface for instance. It would be merely the disturbance in the history of plasticity, although it has been introduced in a lot of literatures (e.g. Chakrabarty 1987; Duszek and Perzyna 1991; Khan and Huang 1995; Lubarda 2002; Asaro and Lubarda 2006). The following nonlinear-kinematic hardening rule of metals was formulated by Hashiguchi (2000).      1 p 1  p ^ ke ka ¼ k f k^n ; f k^n ¼ ck n a a ¼ ck e  bk F bk F

ð8:89Þ

pffiffiffiffiffiffiffiffi where ck and bk ð  3=2Þ are the material constants, while the first and the second terms in Eq. (8.89) are called the hardening part and the dynamic recovery part, respectively. On the other hand, the original Armstrong-Frederick nonlinear kinematic hardening rule is given as   1 p p a ¼ ck e  ke ka bk 

from which Eq. (8.89) is modified as the saturation value of the movement of the yield surface depend on the size of the yield surface F by the modifications bk ! bk F . The pertinence ofEq. (8.89) for the evolution rule of the kinematic hardening can be perceived by the physical interpretation that the saturation of the kinematic hardening would decrease with the isotropic softening and further will be confirmed in Subsection 15.5.1 as the saturation of the kinematic hardening decreases with the damage evolution. This pertinent property is not provided in the Armstrong-Frederick rule. Now, Eq. (8.89) is reduced to the following equation for the uniaxial loading process of the Mises metals exhibiting the plastically-incompressibility, i.e. p p el ¼  ea =2. ! rffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   1 3 1 p p2 p2 p e a þ 2 e l aa ¼ c k 1 aa e a aa ¼ c k e a  bk F 2 bk F   p p  : ea [ 0; þ : ea \0 

leading to

ð8:90Þ

8.7 Anisotropy

239 a

2b F 3 k

ck (1

Limit kinematic hardening surface

3 a 2 bk F

)

1

ck

0

ck (1

3 a 2 bk F

1

) ck 1

1

1

2ck

1

p

a

2ck

Limit kinematic hardening surface

2b F 3 k

Fig. 8.10 Nonlinear kinematic hardening rule illustrated for uniaxial loading process

) 8 p pffiffiffiffiffiffiffiffi 0 for ea [ 0 > > > in aa ¼ 2=3bk F > p > > > < 2ck for ea \0  p aa = ea ¼ ck in aa ¼ 0 > ) > p > > pffiffiffiffiffiffiffiffi 2ck for ea [ 0 > > > in aa ¼  2=3bk F : p 0 for ea \0 



p

ð8:91Þ

p

noting el ¼ ea =2, where ðÞl designates the lateral component. Equation (8.90) is 

time-integrated under the isotropic nonhardening state F ¼ 0 as follows:  

 

 

ð8:92Þ

where A is the saturation value of aa , i.e. A

pffiffiffiffiffiffiffiffi 2=3bk F

ð8:93Þ

The relation of the axial components aa of the back-stress versus the axial plastic strain epa in the uniaxial loading is illustrated in Fig. 8.10 in which the kinematic hardening gradually saturates in the monotonic loading process but the abrupt increase of the kinematic hardening rate occurs at the moment of reverse loading. pffiffiffiffiffiffiffiffi The kinematic hardening saturates at 2=3bk F. The Armstrong-Frederick kinematic hardening rule have been worsened to various complex forms by Chaboche (1991), Hassan and Kyriakides (1991), Ohno and Wang (1993), Dafalias et al. (2008), Hassan et al. (2008), Dafalias and Feigenbaum (2011), etc. Here, it should be noted that the kinetic hardening is important for plastically-incompressible metals but it is irrelevant to the plastically-incompressible

240

8

Elastoplastic Constitutive Equations

materials, e.g. soils, rocks, concrete, ceramics, etc. for which the rotational hardening is dominant as will be described in Chap. 13. Further, the directional distortional hardening has been studied hitherto, which leads to the distortion of the shape of the yield surface such that a region of high curvature develops in the direction of loading and flattening develops in the opposite direction by Kurtyka and Zyczkowski (1996), Voyiadjis and Forooze (1990), Feigenbaum and Dafsalias (2007, 2014), Shi et al. (2017), etc.

8.8

Plastic Spin

The infinitesimal hyperelastic-based plastic constitutive equation is formulated so far. In order to transform it to the hypoelastic-based plastic constitutive equation, the plastic spin in Eq. (3.44) must be incorporated. It is given specifically by the following equation in terms of the plastic-work conjugate pair ðs; dp Þ as would be suggested from Eq. (5.58) in Sect. 5.58 and shown specifically in Sect. 17.6. wp ¼ gp ant½s dp 

ð8:94Þ

where gp is the material parameter. Equation (8.94) is based on the non-coaxiality between the stress and the plastic strain rate, which is generally accompanied with the anisotropy. Obviously, the plastic spin is not induced in isotropic materials 

^Þ are co-axial and thus they are fullfilling a ¼ O holds, for which s and dp ð¼ kn commutable. Equation (8.94) is the modification of the proposition wp ¼ gp ant½r dp  by Zbib and Aifantis (1988) in which the work-conjugacy is ignored. It will be extended to the multiplica-tive hyperelastic-based plasticity in Sect. 17.9.

8.9

Physical Interpretation of Nonlinear Kinematic Hardening Rule

The strain e is additively decomposed into the elastic strain ee as the storage part and the plastic strain ep as the pseudo dissipative part, and further the plastic strain ep is additively decomposed into the storage part epks which induces the kinematic hardening and the dissipative part epkd , i.e. ep ¼ epks þ epkd

ð8:95Þ

The stress r and the kinematic hardening variable a which describe the variation of substructures in material are given by the partial derivatives of the potential functions of the elastic strain and the storage parts of the plastic strain so that the following relations hold.

8.9 Physical Interpretation of Nonlinear Kinematic Hardening Rule

r¼

@we ðee Þ ; @ee

a¼

@wk ðepks Þ @epks

241

ð8:96Þ

Now, assuming 1 1 we ðee Þ ¼ ee : E : ee ; wk ðepks Þ ¼ ck epks : epks 2 2

ð8:97Þ

it follows from Eq. (8.96) that a ¼ ck epks ¼ ck ðep epkd Þ

r ¼ E : ee ¼ E :ðeep Þ;

ð8:98Þ

The time-differentiation of Eq. (8.98) reads: 



p



p

p

r ¼ E :ðe  e Þ; a ¼ ck ðe  ekd Þ

ð8:99Þ

Then, the rates of the storage and the dissipative parts for the stress and the kinematic hardening are given as Storage part ⎧ 644 7448 ⎪ p • • ⎪σ =  : ( ε − ), ε• { ⎪ Dissipative part ⎪ Storage part ⎨ 6447448 ⎪ ⎪α• = ck ( ε• p − 1 || ε• p||α ) bk F ⎪ 14243 ⎪ Dissipative part ⎩

ð8:100Þ

from Eqs. (8.5) and (8.89). The dissipative parts of the strain and the plastic strain satisfy the positivity of the dissipation energy, i.e. • ⎧ •p ⎪σ : ε = (σ : nˆ )λ ≥ 0 ⎨ • p ⎪ α : ε• kd = (α : α )λ / (b k F ) ≥ 0 ⎩

8.10

ð8:101Þ

Limitations of Conventional Elastoplasticity

The conventional elastoplasticity described in this chapter is premised on the assumption that the interior of yield surface is a purely elastic domain. Therefore, the relation of stress rate versus strain rate is predicted to change abruptly at the moment when the stress reaches the yield surface. Therefore, the smooth stress– strain curve observed in real materials is not predicted as shown in Fig. 8.11. This results in the defects: a determination of offset value, i.e. plastic strain at yield point

242

8

Elastoplastic Constitutive Equations

Experiment Elastic state Elastoplastic state

Prediction by conventional plasticity Unrealistic prediction: Excessively high peak stress

Roughly real prediction

0

0 Hardening

Softening

Fig. 8.11 Prediction of monotonic loading behavior by conventional plasticity

is required, which is influenced by an arbitrariness since a plastic deformation develops gradually and thus a stress–strain curve is smooth usually in real materials, and a peak stress value is predicted to be excessively high in a softening behavior observed in frictional materials, e.g. soils and rocks and concrete. Further, only an elastic deformation is predicted for the cyclic loading of stress below the yield stress. In real materials, however, plastic deformation is accumulated even for stress cycles lower than the yield stress and the strain is accumulated resulting in the failure as will be described in subsequent chapters. Thus, the conventional plasticity possesses the above-mentioned various defects in the application to the mechanical design of machines and structures in engineering practice.

Chapter 9

Unconventional Elastoplasticity Model: Subloading Surface Model

Elastoplastic constitutive equations with the yield surface enclosing the elastic domain possess many limitations in the description of elastoplastic deformation, as explained in the last chapter. They are called the conventional model in the Drucker’s (1988) classification of plasticity models. Various unconventional elastoplasticity models have been proposed, which are intended to describe the plastic strain rate induced by the rate of stress inside the yield surface. Among them, the subloading surface model is the only pertinent model fulfilling the mechanical requirements for unconventional models. These mechanical requirements are first described and then the subloading surface model is explained in detail.

9.1

Mechanical Requirements

There exist various mechanical requirements, e.g., the thermodynamic restriction and the objectivity for constitutive equations. Among them, the continuity and the smoothness conditions are violated in many elastoplasticity models, while their importance for formulation of constitutive equations has not been sufficiently recognized to date. Before formulation of the plastic strain rate, these conditions will be explained below (Hashiguchi 1993a, b, 1997, 2000).

9.1.1

Continuity Condition in the Small

It is observed in experiments that “stress rate changes continuously for a continuous change of strain rate”. This fact is called the continuity condition in the small and is expressed mathematically as follows (Hashiguchi 1993a, b, 1997, 2000).

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 K. Hashiguchi, Foundations of Elastoplasticity: Subloading Surface Model, https://doi.org/10.1007/978-3-030-93138-4_9

243

244

9

Unconventional Elastoplasticity Model: Subloading Surface Model














lim rðr; Hi ; e þ d eÞ rðr; Hi ; eÞ ! O


ð9:1Þ

d e!O

where Hi ði ¼ 1; 2; 3;


Þ denotes collectively scalar-valued or tensor-valued internal state variables. In addition, dðÞ stands for an infinitesimal variation. The response of the stress rate to the input of strain rate in the current state of stress and

internal variables is designated by rðr; Hi ; eÞ. Uniqueness of solution is not guaranteed in constitutive equations violating the continuity condition, predicting different stresses or deformations for identical input loading. The violation of this condition is schematically shown in Fig. 9.1. Ordinary elastoplastic constitutive equations in which the plastic strain rate is derived obeying the consistency condition, fulfill the continuity condition. As described later, however, no elastoplastic constitutive equation fulfills it except for the subloading surface model when they are extended to describe the tangential inelastic strain rate. The concept of the continuity condition was first advocated by Prager (1949). However, a mathematical expression of this condition was not given. The condition was defined as the continuity of strain rate to the input of stress rate by Prager (1949) inversely to the definition given above. However, identical stress rate directing inwards the yield surface can induce different strain rates in loading and unloading states for softening materials. Also, identical stress rate along the yield surface can induce different strain rates in a perfectly-plastic material as illustrated in Fig. 9.2. Besides, it is noteworthy that a stress rate cannot be given arbitrarily since there exists a limitation in strength of materials although a strain rate can be given arbitrarily. For that reason, the Prager’s (1949) notion does not hold in the general loading state including softening and the perfectly plastic states.

jump

jump

Input : ε•

Fig. 9.1 Violation of continuity condition in the small

Response : σ•

9.1 Mechanical Requirements

245

σ Input: dσ ( > <
[ 0 for R\1 R ¼ 0 for R ¼ 1 > > : ð\0 for R [ 1Þ 
e
¼ 0 for e ¼O R
6 O \0 for ee ¼




for ep 6¼ O




for ep ¼ O

ð9:7Þ

ð9:8Þ

Here, the rate of the normal-yield ratio evolves with the plastic strain rate, obeying Eq. (9.7). On the other hand, when only the elastic strain rate is induced, the updated value of R needs to be calculated by solving Eq. (9.6) for R. Note that in the calculation, the stress is updated obeying the elastic constitutive equation and internal variables are fixed since no plastic strain evolves during this process. Then, it follows that











R ¼ UðRÞjj ep jj ¼ UðRÞ k for ep 6¼ O

Fig. 9.5 Plastic strain rate based on the subloading surface concept

ð9:9Þ

9.2 Subloading Surface (Hashiguchi) Model

R¼

f ðrÞ

for ee 6¼ O and ep ¼ O F

249

ð9:10Þ

where UðRÞ is the monotonically-decreasing function of R fulfilling the conditions (see Fig. 9.6a). 8 ! þ 1 for R ¼ 0 ðelastic stateÞ > > < [ 0 for R\1 ðsubyield stateÞ UðRÞ ¼ 0 for R ¼ 1 ðnormal-yield stateÞ > > : \0 for R [ 1 ðover normal-yield stateÞ

ð9:11Þ

The explicit form of the function UðRÞ satisfying Eq. (9.11) is given by p  UðRÞ ¼ u cot R 2

ð9:12Þ

where u is the dimensionless material parameter regulating the increase of the normal-yield ratio R for a certain plastic strain increment, noting dR=jjdep jj ¼ UðRÞ ¼ u cot½ðp=2ÞR. Therefore, the stress vs. strain curve is smoother for a smaller value of the parameter u, while the conventional elastoplastic behavior is realized for u ! 1. However, note that there exist a lot of materials including usual metals in which the plastic strain rate is hardly induced blow a certain value of the normal-yield ratio. Then, let Eq. (9.11) be modified to the following equation, by which the plastic strain rate is not induced until the normal-yield ratio R reaches a certain value of the material parameter Re ð\1Þ (see Fig. 9.6b).

Fig. 9.6 Function UðRÞ in rate of normal-yield ratio R which is attract to unity in plastic loading process

250

9

Unconventional Elastoplasticity Model: Subloading Surface Model

8 ! þ 1 for 0  R  Re ðelastic stateÞ > > < [ 0 for Re \R\1 ðsubyield stateÞ UðRÞ ¼ 0 for R ¼ 1 ðnormal-yield stateÞ > > : \0 for R [ 1 ðover normal-yield stateÞ

ð9:13Þ

The material parameter Re is interpreted to be the ratio of the (half) stress amplitude rfl at the fatigue (or endurance) limit, i.e. the fatigue limit stress to the yield stress ry under the zero value of average stress r, i.e. Re ¼ rfl =ry jr¼0 . Fatigue limit is observed in steels, titanium, etc. but it is not observed in other materials involving non-ferrous metals. Note here that the incorporation of the material parameter Re does not mean the incorporation of the yield surface enclosing a purely-elastic domain as known from the fact: The plastic strain rate is predicted for the cyclic loading with a small stress amplitude under a high average stress by the subloading surface model with the incorporation of Re but it cannot be predicted if the yield surface enclosing a purely-elastic domain is incorporated as seen in the cyclic kinematic hardening model, i.e. the cylindrical yield surface, the multi-surface, the two-surface models which will be described in Sect. 10.3. Equation (9.12) is modified to satisfy Eq. (9.13) as follows:   p hR  Re i UðRÞ ¼ u cot 2 1  Re

ð9:14Þ

where h i is the Macaulay’s bracket defined by hsi ¼ ðs þ jsjÞ=2, i.e. s\0:hsi ¼ 0 and s  0: hsi ¼ s (s: arbitrary scalar variable). Equation (9.14) conforms to the fulfillment of the smoothness condition since the function UðRÞ decreases continuously from the infinite value UðRe Þ ð! 1Þ. If u is fixed to be constant, Eq. (9.9) with Eq. (9.14) can be integrated analytically as 9      2 p R0  Re p ep  ep0 > 1 > exp  u þ Re > R ¼ ð1  Re Þ cos cos > > p 2 1  Re >  2 1  Re > = p R0 Re cos for R0  Re 2 1  Re 2 1  Re > >  > ln  ep  ep0 ¼ > > p R  Re p u > > cos ; 2 1  Re ð9:15Þ R
where ep  jj ep jjdt and ep0 is the initial value of ep , whilst one must set R0 ¼ Re for R0 \Re . However, the judgment whether of R\Re or R  Re is required in Eq. (9.14), although the yield judgment is not required. The time-differentiation of Eq. (9.6) of the subloading surface leads to

9.2 Subloading Surface (Hashiguchi) Model

251



@f ðrÞ
: r R F  R F ¼ 0 @r

ð9:16Þ

The following relation holds by virtue of the homogeneity of the function f ðrÞ in degree-one. @f ðrÞ : r ¼ f ðrÞ ¼ RF @r

ð9:17Þ

which yields @f ðrÞ :r @r ¼ RF ; n :r ¼ @f ðrÞ @f ðrÞ @r @r

1 ¼ n :r @f ðrÞ RF @r

ð9:18Þ

where @f ðrÞ @f ðrÞ = n @r @r

ð9:19Þ

Equation (9.16) with Eq. (9.18) results in " n:



! # F R r þ r ¼0 F R


ð9:20Þ

Now, adopt the associated flow rule





ep ¼ k n







ðk ¼ jj ep jj [ 0Þ

ð9:21Þ





The evolution rule of the normal-yield ratio is described as the equation R ¼ UðRÞ k instead of Eq. (9.9) for the expression of the flow rule in Eq. (9.21). Note, however,





that the equation R ¼ UðRÞ k does not hold for the expression of the flow rule





ep ¼ k @f ðrÞ=@r because of jj@f ðrÞ=@rjj 6¼ 1 in general. Substituting Eqs. (8.28) and (9.9) with Eq. (9.21) into Eq. (9.20), one has "

! # '


F UðRÞ
fHn ðr; H; nÞ k þ k r ¼ 0 ð H ¼ fHn ðr; H; nÞ kÞ n : r F R


ð9:22Þ

252

9

Unconventional Elastoplasticity Model: Subloading Surface Model

It follows from Eqs. (9.21) and (9.22) that





k¼

n: r p ; M


p

e ¼




n: r p n M

ð9:23Þ

where   F' UðRÞ fHn ðr; H; nÞ þ n :r M  F R p

ð9:24Þ

which is reduced to the plastic modulus of the conventional elastoplasticity in p Eq. (8.27) in which note that M p is replaced to M in Eq. (9.24), i.e. p

M ¼

F' fHn ðr; H; nÞn : r ¼ M p F

ð9:25Þ

in the normal-yield state ðR ¼ 1 ! UðRÞ ¼ 0Þ. The strain rate is described by extending Eq. (8.30) to the subloading surface model as follows: • −1 n ⊗ n • = ( + ε• = −1 : σ• + n : σ p ): σ p n

M

M

ð9:26Þ




from which the magnitude of plastic strain k in terms of strain rate, denoted by the


symbol K, is derived as follows: •

Λ=

n :  : ε• n :  : ε• •p n , = ε p M p + n:: n M + n : : n

ð9:27Þ

The stress rate is given by extending Eq. (8.33) as follows: σ• = : ε• −

n :  : ε•  :n ⊗ n :  ) : ε•  : n = ( − p M + n:: n M + n : : n p

ð9:28Þ

The loading criterion is given by •

ε• p ≠ O : R > R e and Λ > 0 or n:  : ε• > 0 ε• p = O : others

ð9:29Þ

where the judgment whether or not the stress reaches the yield surface is not required since the plastic strain rate is induced continuously as the stress approaches the normal-yield surface.

9.2 Subloading Surface (Hashiguchi) Model

253

1

1 Conventional plasticity model (u

)

Subloading surface model

σ R 1

0

1

0

2

Subloading surface f (σˆ ) = R F (H )

3

Normal-yield surface

ˆ ) = F (H ) f (σ

Fig. 9.7 Smooth stress–strain curve predicted by the subloading surface

The stress vs. strain curve by the subloading surface model is illustrated for the simplest case of the perfectly-plastic material in Fig. 9.7.

9.3

Distinguished Advantages of Subloading Surface Model

This model possesses the following distinguished abilities. (1) Smooth transition from elastic to plastic state is described, which is observed in real material behavior. Then, we don’t need suffer from the determination of an offset value (plastic strain value in yield point). In contrast, the determination is required in all of the other elastoplastic models since they assume a surface enclosing a purely-elastic domain leading to the abrupt elastic–plastic transition, while the determination is accompanied with an arbitrariness. The influences of the material parameter u on the function UðRÞ and the stress–strain curve are depicted in Figs. 9.8 and 9.9, respectively. The larger the material parameter u, the more rapidly the normal-yield ratio R increases causing the more rapid increase of stress, i.e. approaching the behavior of the conventional plasticity. (2) Plastic strain rate can be described even for low stress level and for cyclic loading process under small stress amplitudes since a purely-elastic domain is not assumed.

254

9

Unconventional Elastoplasticity Model: Subloading Surface Model

U ( R)

Smaller value of u

0

1

R

Fig. 9.8 Influence of u on function UðRÞ

(3) The yield-judgment whether or not the stress reaches the yield surface is unnecessary since the plastic strain rate develops continuously as the stress approaches the normal-yield surface. In contrast, the yield judgment is required in all of the other elastoplastic models since they assume a surface enclosing a purely-elastic domain. (4) The stress is automatically pulled back to the normal-yield surface when it goes


over the surface in numerical calculation because of R \0 for R [ 1 from Eq. (9.7) with Eq. (9.11)4 as seen in Fig. 9.10. In contrast, the particular operation to pull back the stress is required in all of the other models because they assume a surface enclosing a purely-elastic domain. The above-mentioned stress controlling function involved in the subloading surface model to pull-back the stress automatically to the yield surface in the plastic deformation process is quite beneficial for the explicit method in the numerical



0 Fig. 9.9 Influence of u on stress–strain curve



9.3 Distinguished Advantages of Subloading Surface Model

255

Fig. 9.10 Stress is automatically controlled to be attracted to the normal-yield surface in the subloading surface model

stress integration process. Besides, it is particularly important for the softening state in which the tangent stiffness modulus changes intensely deviating from the linear isotropic hardening curve. For the normal-yield state R ¼ 1 ðU ¼ 0Þ, the plastic strain rate in Eq. (9.23) with Eq. (9.24) is reduced to Eq. (8.29) with Eq. (8.27) for the conventional plasticity, i.e.
p

e ¼




n: r

F' fHn ðr; H; nÞn : r F

n

For u ! 1 leading to the sudden decrease of the function U from U ! 1 p for R\1 to U ¼ 0 for R ¼ 1 in Eq. (9.11), the plastic modulus M in Eq. (9.24) drops suddenly from the infinite value to the value M p in Eq. (9.25) so that the present model behavior is reduced to the conventional elastoplasticity model behavior by choosing a large value of the material parameter u, thereby exhibiting an sudden transition from the elastic to plastic state. On the other hand, as u becomes smaller, a gentler transition from the elastic to plastic state is described. Therefore, u plays the role to alleviate the sudden transition from the elastic to plastic state. It follows from Eqs. (8.29) and (9.23) in the plastic loading process, fulfilling





k  0 or k  0, that

256

9

Unconventional Elastoplasticity Model: Subloading Surface Model

9

> M p [ 0 ! n : r [ 0; F [ 0 : normal hardening =

p M ¼ 0 ! n : r ¼ 0; F ¼ 0 : normal nonhardening >
;
M p \0 ! n : r \0; F \0 : normal softening

ð9:30Þ

for the conventional model and 9
p > M [ 0 ! n : r [ 0 : subloading hardening =
p M ¼ 0 ! n : r ¼ 0 : subloading nonhardening >
; p M \0 ! n : r \0 : subloading softening

ð9:31Þ

for the subloading surface model. Here, it should be noted that the signs of M p or p




M and n : r coincide with each other in both models but they do not necessarily


coincide with the sign of F in the subloading surface model. The distinguished advantages of the subloading surface model in the descriptions of irreversible mechanical phenomena can be obtained by the simple modification of existing computer program for the conventional elastoplasticity model to add only one material parameter u for the evolution rule of the normal-yield ratio without any expense.

9.4

Numerical Performance of Subloading Surface Model

The stress controlling function of the subloading surface model is described in Sect. 7.3. This fact will be shown below by the numerical calculation for the response of the uniaxial loading behavior, adopting the simplest subloading surface model for the isotropic Mises material with the evolution rule of the normal-yield ratio in Eq. (9.9) with Eq. (9.12). The response of the conventional elastoplastic constitutive model is also shown for the comparison. The relations of the axial stress ra and the normal-yield ratio R versus the axial strain ea are depicted in Fig. 9.11. The responses adopting the linear isotropic hardening F ¼ F0 þ hc eep (hc : material constant) are depicted in Fig. 9.11a and those for the nonlinear isotropic hardening in Eq. (8.42) are shown in Fig. 9.11b. The two levels of axial strain increment dea ¼ 0:0006 and 0:0055 are input in the numarical calculations. Here, any special stress controlling algorithm to pull it back to the yield surface is not introduced. The material parameters are chosen as follows:

9.4 Numerical Performance of Subloading Surface Model

257

Material constants: Youg’s modulus : E ¼ 100000 MPa;  Linear isotropic : hc ¼ 7000 MPa Hardening Nonlinar isotropic : sr ¼ 0:8; cH ¼ 50; Evolution of normal-yield ratio : u ¼ 200: Initial values: Hardening function: F0 ¼ 500 MPa; Stress : r0 ¼ O MPa The nonsmooth curves bent at the yield stress are expressed by the conventional model. Moreover, the stress deviates from the exact curve of concventional elastoplasticity. The deviation becomes large with the increases in the nonlinearity of hardening and in the increase of input strain increment. On the other hand, the stress is automatically attracted to the normal-yield surface in the subloading subloading surface model even for the quite large strain increment dea ¼ 0:0055 ð0:55%Þ. The zigzag lines tracing the exact curve are calculated such that the stress rises up when the it lies below the normal-yield surface but it drops 1000 0.0055

d a = 0.0006

800

600

a

0.0055

(MPa)

400

d a = 0.0006

200

0

2.0

1.50

d a = 0.0006 d a = 0.0055

1.0 d a = 0.0055 d a = 0.0006

R

0.50

0

0.02

0.04

a

0.06

0.08

(a) Linear isotropic hardening Exact curve of conventional elastoplasticity Calculated by the conventional elastoplastic model Calculated by the subloading surface model

Fig. 9.11 Numerical accuracies of the conventional elastoplastic model and the subloading surface model: Uniaxial loading behavior of Mises material with isotropic hardening

258

9

Unconventional Elastoplasticity Model: Subloading Surface Model 1000

0.0055 d a = 0.0006

800

a (MPa)

600 0.0055 400

d a = 0.0006

200

0

2.0

1.50

0.0055

0.0006

1.0 0.0055 d a = 0.0006

R

0.50

0

0.02

0.04

a

0.06

0.08

(b) Nonlinear isotropic hardening

Fig. 9.11 (continued)

down immediately if it goes over the normal-yield surface, obeying the evolution


rule of normal-yield ratio in Eq. (9.9) with Eq. (9.12), i.e. R [ 0 for R\1 and


p

R \0 for R [ 1. The plastic modulus M lowers than that in the conventional one and further it can be negative at the over normal-yield state R [ 1 leading to U\0,


p

while n : r \0 (subloading softening defined in Eq. (9.31)) holds for M \0


because of k [ 0 as known from Eqs. (9.23)–(9.25). The amplitude of zigzag decreases gradulally in the monotonic loading process, while, needless to say, the amplitude is smaller for a smaller input increment of strain. Eventually, the subloading surface model posseses the distinguished high ability for numerical calculation as verified also quantitatively in these concrete examples, which has not been attained in any other elastoplastic constitutive equations assumng a purely-elastic domain as will be described in Chap. 10. The drastic improvement of the conventional eladtoplasticity model by the subloading surface model is realized in Fig. 9.12. Neddless to say, this model is essntial for predicting the fatigue phenomenon in which a plastic deformation is induced during the cyclic loading behavior with a small stress amplitude. Besides, the two scale model (Lemaitre 1990) for the damage phenomenon in a high cycle fatigue, which requires the complex and irrigorous formulation, can be disused by adopting the subloading surface model as will be described in Sect. Sect. 15.9.

9.4 Numerical Performance of Subloading Surface Model

259

Conventional plasticity model 



Excessively high peak stress

Pull-back of stress to yield surface is necessary

 Yield judgement is necessary

Elastic deformation

0 Hardening state



Elastic deformation

0

 Softening state





Yield judgement is unnecessary

0

Hardening state





Improvement

Subloading surface model 

Offset value

0

Stress is pulled-back automatically to yield surface

 Softening state

Improvement of monotonic loading behaviors



0

Improvement of accuracy and efficiency in numerical calculation

Fig. 9.12 Improvement of conventional elastoplasticity model by subloading surface model

These improvements can be attained merely by adding only one matrial constant u


in the evolution rule of the normal-yield ratio R. Then, the cost and the time required for the industrial design and development can be reduced tremendously even if we input irresponsible large value for the material constant u, noting that the deformation behavior based on the conventional elastoplastic constitutive equation is calculated for u ! 1.

9.5

On Bounding Surface Model with Radial-Mapping: Misuse of Subloading Surface Concept

The bounding surface model with a radial mapping by Dafalias and Herrmann (1980), Dafalias (1986), Dafalias and Talebat (2016), etc. possesses the primitive mathematical formulation which is basically different from the subloading surface model, although Dafalias stated: “It appears that the first time a radial mapping formulation was proposed, it was in reference to granular materials by Hashiguchi and Ueno (1977)” which is the original sentence in Dafalias (1986, p. 980). However, it has been adopted to the description of deformation behavior of soils as SANICLAY and SANISAND models (Dafalias et al. 2006; Dafalias and Talebat 2016, etc.). First note that the coined word “bounding surface” is nothing but the usual word “yield surface”. Therefore, this coined word must be disused in order to avoid the disturbance/confusion. Accordingly, let the “bounding surface model with a radial mapping” be called merely the “radial mapping model” hereinafter.

260

9

Unconventional Elastoplasticity Model: Subloading Surface Model

Note the following fundamental difference of the radial mapping model from the subloading surface model. (1) Any loading surface passing through the current stress is not incorporated, (2) Then, the strain rare cannot be formulated by rigorously based on the consistency condition and then it is formulated irrationally by the interpolation method. Therefore, the radial mapping model possesses the following crucial drawbacks and limitations which are not involved in the subloading surface model. (a) It is not guaranteed that the stress approaches the yield surface in the plastic loading process in the general loading state. (b) The cyclic loading behavior cannot be described leading to the open hysteresis loop in the unloading–reloading process and thus resulting in the excessively large plastic strain accumulation, while the subloading surface model has been extended to describe the cyclic loading behavior accurately as will be described in Chap. 11. Nevertheless, the bounding surface model with radial-mapping is used for the description of soil deformation, although it is never used for the description of metal deformation. The similar situation is seen in the hypoplasticity (Kolymbas and Wu 1993) in Eq. (7.113). The constitutive modelling in terms of the rate-nonlinear equation of the strain rate is performed in the hypoplasticity by extending the hypoelasticity in terms of the rate-linear equation of strain rate (Treuesdell 1955), while the fundamental requirements, i.e. the decomposition of the strain rate into the elastic and the plastic parts and the yield surface are ignored in the hypoplasticity. Unfortunately, the hypoplasticity is still now studied for soils, although it was studied once but faded out soon for metals ten years after the proposition of the hypoelasticity (Treuesdell 1956). This difference between metals and soils would be caused by the facts: The prediction of deformation behavior is required a high accuracy for metals but a quite low accuracy for soils. The radial mapping model and the hypopolasticity will disappear in the near future by the sound development of soil mechanics. Eventually, the ones using the bounding surface model with radial mapping for soils, i.e. the SANICLAY and the SANISAND models should dispose them and instead they should notice the subloading surface model for the rigorous deformation analyses and the sound development of soil plasticity.

9.6

Incorporation of Kinematic Hardening

The subloading surface based on the yield surface in Eq. (8.71) with the kinematic hardening is described as f ðb r Þ ¼ RFðHÞ The material-time derivative of Eq. (9.32) leads to

ð9:32Þ

9.6 Incorporation of Kinematic Hardening

261



@f ðb r Þ
@f ðb rÞ
:r : a RF H  R F ¼ 0 @b r @b r

ð9:33Þ

Substituting the associated flow rule








^ ep ¼ k n




ðk ¼ jj ep jj  0Þ

ð9:34Þ

Equation (9.33) is rewritten substituting Eq. (8.74) as

@f ðb r Þ
@f ðb rÞ
^Þ  RF k fH^n ðr; H; n ^Þ  U k F ¼ 0 : r : k f k^n ðr; a; F; n @b r @b r

ð9:35Þ

Noting 9 > > > > > > > > > > =

@f ð^ rÞ :^ r ¼ f ð^ rÞ ¼ RF @^ r @f ð^ rÞ @f ð^ rÞ n ^ ¼ @^ @^ r r

ð9:36Þ

> > > > rÞ > @f ð^ > > :^ r @f ð^ > ^ r Þ @f ð^ r Þ n :^ r > @^ r > = ¼ ¼ 1= ; f ð^ rÞ @^ @^ r r RF Equation (9.35) is rewritten as





^: r ^ ^' Þ  n ^: r ^ n n : k f k^n ðr; a; F; n


 F' U
^Þ þ k ¼ 0 k fH^n ðr; H; n F R

ð9:37Þ

from which one has





^ : r
p n ^: r n ^ k¼ p ; e ¼ p n M M


ð9:38Þ

where p

^: M n

h

F'

U ^ F fH^n ðr; H; nÞ þ R



'

^Þ b r þ f k^n ðr; a; F; n

i

ð9:39Þ

The loading criterion is given by ⎧• p • • ⎪ε ≠ O for R > R e and Λ > 0 or nˆ :  : ε > 0 ⎨ ⎪ε• p = O for others ⎩

ð9:40Þ

262

9.7

9

Unconventional Elastoplasticity Model: Subloading Surface Model

Incorporation of Tangential-Inelastic Strain Rate

As presented in Eqs. (8.29), (8.81) and (9.38), the inelastic strain rate in the traditional constitutive equation has the following limitations. (i) The inelastic strain rate depends solely on the stress rate component normal to the yield surface, called the normal stress rate, but is independent of the component tangential to the yield surface, called the tangential stress rate, since it is derived merely based on the consistency condition. (ii) The direction of inelastic strain rate is determined solely by the current state of stress and internal variables but it is independent of the stress rate. On the other hand, it has been verified by experiments that an inelastic strain rate induced by the deviatoric part of the tangential stress rate, called the deviatoric tangential stress rate, influences considerably on a deformation in the non-proportional loading process deviating from the proportional loading path normal to the yield surface, which is called the tangential inelastic strain rate. Here, the spherical part of the tangential stress rate does not induce an inelastic strain rate, as Rudnicki and Rice (1975) verified based on the fissure model. In addition, the tangential inelastic strain rate is induced considerably in the plastic instability phenomena with the strain localization induced by the generation of the shear band and it influences on the macroscopic deformation and strength characteristics. To remedy these insufficiencies of the traditional plastic constitutive equation, various models have been proposed to date as follows: (1) Intersection of plural yield surfaces: Various models assuming the intersection of plural yield surfaces have been proposed (Batdorf and Budiansky 1949; Koiter 1953; Bland 1957; Mandel 1965; Hill 1966; Sewell 1973, 1974). The Koiter’s (1953) model was adopted by Sewell (1973, 1974), but it is indicated that the applicability of the model is limited to the inception of uniaxial loading. Models in this category cannot describe the latent hardening pertinently and are not readily applicable to general loading processes (cf. Christoffersen and Hutchinson (1979)). (2) Corner theory: The singularity of outward-normal of the yield surface is introduced by assuming the conical corner or vertex at the stress point on the yield surface. Therefore, the direction of plastic strain rate can take a wide range surrounded by the outward-normal of the yield surface (Christoffersen and Hutchinson 1979; Ito 1979; Gotoh 1985; Goya and Ito, 1991; Petryk and Thermann, 1977). There exist the two kinds of models: One kind is based on the assumption of an imaginary infinitesimal vertex and the other subsumes a finite projecting cone. The evolution rule of the cone cannot be formulated and the reloading from the cone surface after partial unloading cannot be described pertinently in the latter models. It was described by Hecker (1976) and Ikegami (1979) that the yield surface projects towards the loading direction generally but the formation of the so-called vertex is doubtful.

9.7 Incorporation of Tangential-Inelastic Strain Rate

263

(3) Hypoplasticity: This term was first used by Dafalias (1986) in the analogy to the term hypoelasticity introduced by Truesdell (1955) described in Sect. 5.3. Models in this category are classified into the two kind of models in which the direction of plastic





strain rate depends on the direction of the stress rate r =jj r jj (Mroz, 1966; Wang et al. 1970; Dafalias and Popov 1977; Hughes and Shakib 1986; Hashiguchi 1993a) and the models in which the direction of the plastic strain rate depends on





the direction of strain rate e =jj e jj (Hill 1959; Simo 1987b; Hashiguchi 1997; Kuroda and Tvergaard 2001). The singularity in the field of direction of plastic strain rate is introduced in the algebraic ways into these models, although it is done geometrically in the models described in (1) and (2). However, the magnitude of the plastic strain rate is derived from the consistency condition. Therefore, the plastic strain rate diminishes when the stress rate is directed tangentially to the yield surface, as in the traditional constitutive equations without the vertex. The constitutive equations described in (1)–(3) possess the following problems. (i) A formulation of pertinent model which fulfills the consistency condition and is applicable to the general loading process is difficult. (ii) The stress rate vs. strain rate relation becomes nonlinear. Therefore, the inverse expression cannot be derived, which renders deformation analysis to be difficult. Differently from the above-mentioned models, the following linear relation between the stress rate vs. strain rate with the tangential-inelastic strain rate, called the J2 -deformation theory, has been formulated by Budiansky (1959) and later Rudnicki and Rice (1975) by extending Eq. (6.73) with the isotropic Mises yield condition as follows: ð9:41Þ which can be rewritten as

ð9:42Þ

where the rate-linearity is retained. On the other hand, Hencky’s deformation theory (Hencky 1924) is described as ð9:43Þ

264

9

Unconventional Elastoplasticity Model: Subloading Surface Model

The time-derivative of Eq. (9.43) leads to ð9:44Þ Comparing Eq. (9.42) with Eq. (9.44), choosing Fðreq Þ so as to fulfill F ' ðeeqp ðreq ÞÞ ¼

3 1 2 /ðreq Þ þ /'ðreq Þreq

ð9:45Þ

and it is known that the J2 deformation theory coincides with Hencky’s deformation theory (9.43). However, it possesses crucial limitations as described in below. In what follows, let the tangential inelastic strain rate induced by the stress rate tangential to the loading surface be incorporated into the subloading surface model in the following (Hashiguchi 1998, 2005; Hashiguchi and Tsutsumi 2003; Hashiguchi and Protasov 2004; Khojastepour and Hashiguchi 2004a, b; Khojastepour et al. 2006). Here, note that the tangential inelastic strain rate would not be induced by the mean stress rate as would be inferred from the example for the fact that inelastic volumetric change would not be induced in metals (see Fig. 9.13). In what follows, let the inelastic strain rate induced by the deviatoric stress rate component tangential to the subloading surface be formulated for the general material unlimited to the Mises metal. Firstly, assume that the strain rate is com
posed of the tangential-inelastic strain rate et in addition to the elastic strain rate and the plastic strain rate as follows (Hashiguchi 1998, 2013):











e ¼ ee þ e p þ et

ð9:46Þ


Further, assume that the tangential-inelastic strain rate et is induced by the
tangential component of the stress rate r to the subloading surface in the deviatoric
stress space, which is denoted by rt (Fig. 9.14) defined by ð9:47Þ

ð9:48Þ T't ≡  ' − n' ⊗ n' , T t'ijkl ≡  'ijkl − n 'ijn 'kl

ð9:49Þ

9.7 Incorporation of Tangential-Inelastic Strain Rate

265

Fig. 9.13 Example showing the fact that tangential-inelastic strain rate is not induced by mean stress rate since inelastic volumetric change is not induced in metals

σ 'n

σ' 0 Subloading surface

n' σ ' σ 't  ij'

f (σ ) = R F ( H )

Normal-yield surface f (σ) = F ( H )

Fig. 9.14 Normal and tangential stress rates for subloading surface model in deviatoric stress plane

fulfilling

ð9:50Þ

by virtue of ð9:51Þ

266

9

Unconventional Elastoplasticity Model: Subloading Surface Model

The fourth-order tensor  is the deviatoric projection tensor in Eq. (1.194). The fourth-order tensor T't is the deviatoric-tangential projection tensor which transforms an arbitrary second-order tensor to the tangential part to the subloading surface in the deviatoric stress space and the second-order tensor subjected to this
projection is designated by ðÞ't, i.e. t't  T't: t leading further to T't: t't ¼ t't. Then, r't is


the deviatoric-tangential projection tensor of the stress rate r, which is called the deviatoric-tangential stress rate. Note that the tangential-inelastic strain rate does not affect the yield surface because it is relevant to the tangential component but irrelevant to the normal component to the yield surface.


Now, assume that the tangential-inelastic strain rate et is related linearly to the
tangential-deviatoric stress rate r't in the normal-yield state (R ¼ 1) as follows: • −1 ε t =  : σ• 't

ð9:52Þ

Let Eq. (9.52) be extended for the sub-yield state as follows: • −1 ε t = T ( R)  : σ• 't

ð9:53Þ

In this equation TðRÞ was given by (Hashiguchi, 1998, 2013, 2017, 2020) as follows: TðRÞ ¼ ~cR~n

ð9:54Þ

which is the monotonically-increasing function of the normal-yield ratio R, where ~c and ~ n are the material constants, always fulfilling the continuity and the smoothness conditions in Eqs. (9.1) and (9.2). However, it increases unlimitedly with R and thus it would be physically unacceptable to the high value of the normal-yield ratio, i.e. high over-yield state R [[ 1 observed in the viscoplastic deformation. The function TðRÞ without this effect would be given by the following equation.

ð9:55Þ

9.7 Incorporation of Tangential-Inelastic Strain Rate

267

where ~c and ~ n are the material constants. The following properties are furnished in Eq. (9.55) 1) The material parameter ~c exhibits the maximum value of the function TðRÞ. 2) The function TðRÞ approaches the maximum ~c earlier for a larger value of the material constant ~ n. 3) The relation of the function TðRÞ vs. the normal-yield ration R exhibits the inflection point in a smaller value of R for a larger value of the material constant ~n. In particulier, the inflection point appears at R = 1 for ~n ¼ 1 as shown in Fig. 9.15. Adding the tangential-inelastic strain rate in Eq. (9.53) to Eq. (9.26), the strain rate is given by n: σ• • ε =  −1 : σ• + p n + T ( R)  −1 : σ• 't M ð9:56Þ n n −1 ⊗ −1 • ( R ) =  +  : T't : σ p +T

(

)

M

In what follows, we assume the elastic modulus tensor fulfilling

T't : : n = O

ð9:57Þ

R T ( R)

c

1

d 2T ( R ) 0 dR 2

0 c (1 e)e

d 2T ( R) 0 dR 2

c (1 e)e

c

T ( R)

Fig. 9.15 Function TðRÞ for ~ n¼1

1

R

268

9

Unconventional Elastoplasticity Model: Subloading Surface Model

which holds in the Hooke’s law in Eq. (5.35) for example. Taking account of Eq. (9.57), it follows from Eq. (9.56) that • • T't :  : ε = [1 + T ( R)] σ't

ð9:58Þ

1 • T't :  : ε 1 + T ( R)

ð9:59Þ

leading to σ• 't =

Substituting Eq. (9.59) into Eq. (9.53), one obtains •

εt =

T ( R) −1 •  : T't :  : ε 1 + T ( R)

ð9:60Þ

The expression Eq. (9.27) for the plastic multiplier in terms of strain rate is obtained from Eq. (9.56), noting Eq. (9.47)1. Then, the stress rate is given by substituting Eq. (9.46) with Eq. (9.27) and Eq. (9.60) into the relation based on the infinitesimal linear elasticity in Eq. (7.83) as follows:

σ• =  : (ε• − ε• p− ε• t ) =  : ε• −

n:  : ε•  :n − T ( R) T' :  : ε• t 1 + T ( R) M + n:  : n p

: n ⊗ n:  − T ( R) T' :  =  − p t M + n:  : n 1 + T ( R)

(

):ε

ð9:61Þ

•

Equations (9.56) and (9.61) are given for the Hooke’s law in Eq. (7.84) as follows: n n T ( R) • −1 ð9:62Þ T't : σ ε• =  + ⊗p +

(

M

2G

)

T ( R) n n: T't : ε• σ• =  − p: ⊗  − 2G + n: :n 1 + T ( R) M

(

)

ð9:63Þ

Here, it is known that the bulk modulus K is irrelevant to the tangential-inelastic strain rate which is induced only by the deviatoric part of stress rate. The tangential-inelastic strain rate is essentially tangential to the loading surface (yield and subloading surface) and thus it is always induced as long as the stress rate tangential to the loading surface is imposed. Therefore, the loading criterion for the tangential inelastic strain rate is not required and thus the loading criterion for the constitutive equation with the tangential-inelasticity is given by the equation identical to the loading criterion for the ordinary elastoplastic constitutive equation without the tangential-inelastic strain rate, i.e. Eq. (9.40).

9.7 Incorporation of Tangential-Inelastic Strain Rate

269

Equations (9.56), (9.60) and (9.63) were altered by Fincato and Tsutsumi (2019) and Tsutsumi et al. (2019) as

(

n n −1 ε• =  + ⊗p + M

)

T ( R) T ( R) • • σ• 't T't : σ , ε t = 2G [1 − T ( R)] 2G [1 − T ( R)]

ð9:64Þ

n n: σ• =  − p: ⊗  − 2GT ( R)T't : ε• M + n:  : n

(

)

ð9:65Þ
t

This alteration cannot be allowed because the tangential-inelastic strain rate e is induced infinitely for TðRÞ ! 1, i.e. R ! c1=n for Eq. (9.54) in which the sinn gularity of the tangential inelastic strain rate field is induced. It contradicts to the original physical meaning of the tangential-inelastic strain rate induced by the stress rate tangential to the subloading surface, which is ignored in the associated flow rule. It cannot be applied to the over normal-yield state ðR [ 1Þ occurring in the overstress state which will be described in Chap. 14. 1

σ 0 2

Yield surface

3

Input : σ

1

1

ε t : suddenly induced.

ε t : gradually develops as the subloading surface expands.

0 2

Subloading surface

0 3

2

Yield surface

3

Normal-yield surface Subloading surface model: Hashiguchi (2005), fulfilling continuity and smoothness conditions.

Conventional plasticity model: Rudnicki and Rice model (1975), violating continuity and smoothness conditions.

Fig. 9.16 Incorporation of tangential inelastic strain rate illustrated for von Mises yield surface

270

9

Unconventional Elastoplasticity Model: Subloading Surface Model
t

The tangential-inelastic strain rate e develops gradually as the current stress approaches the normal-yield surface, i.e. the subloading surface expands fulfilling the continuity and the smoothness condition in the subloading surface model as shown in Fig. 9.16 for the isotropic Mises material. The validity of Eq. (9.62) or (9.63) has been verified by Hashiguchi and Protasov (2004) for metals and Hashiguchi and Tsutsumi (2001, 2003, 2007) and Tsutsumi and Hashiguchi (2005) and for geomaterials. On the other hand, if the tangential-inelastic strain rate is incorporated into the plasticity model assuming the yield surface enclosing a purely-elastic domain, both of the continuity and the smoothness conditions in Eqs. (9.1) and (9.2) are violated, since the tangential-inelastic strain rate is induced suddenly at the moment when the stress reaches the yield surface as illustrated for the J2 -deformation model of Rudnicki and Rice (1975) in Fig. 9.16.

9.8

Limitation of Initial Subloading Surface Model

The conventional elastoplastic constitutive equation is incapable of predicting the plastic strain accumulation because the conventional yield surface is assumed to enclose the purely-elastic domain as shown in Fig. 9.17a. On the other hand, an excessively large strain accumulation is described during the cyclic loading process because only elastic deformation is induced in the unloading process by the initial subloading surface model described in this chapter as shown in Fig. 9.17b. The cyclic plasticity models proposed to describe the cyclic loading behavior will be described in the subsequent chapters.

Elastoplastic state Elastic state





0



(a) Conventional elastoplastic model

0

 (b) Initial subloading surface model

Fig. 9.17 Stress versus strain curve predicted by conventional elastoplastic model and initial subloading surface model in cyclic uniaxial loading

Chapter 10

Classification of Plasticity Models: Critical Reviews and Assessments

Accurate description of plastic deformation induced during a cyclic loading process is required for the mechanical design of machinery subjected to vibration and buildings and soil structures subjected to earthquakes since the middle of the last century. Elastoplastic constitutive model formulated for this aim is called the cyclic plasticity model. Substantially, the key of the pertinence in cyclic plasticity model is how to describe appropriately a small plastic strain rate induced by the rate of stress inside the yield surface. Therefore, a quite delicate formulation of plastic strain rate developing gradually as the stress approaches the yield surface is required to this end. Here, needless to say, the continuity and the smoothness conditions described in Sect. 8.1 would have to be fulfilled in a cyclic plasticity model. Various cyclic plasticity models have been proposed to date, while most of them violate the continuity and/or the smoothness conditions unfortunately. Then, the beginners for the cyclic plasticity model would be perplexed as to which model is most pertinent and should be chosen for their study and analyses. In this chapter, the basic mathematical structures of plasticity models proposed hitherto will be discussed and assessed comprehensively. In this context, explicit evolution rules of particular internal variables such as the isotropic hardening, the kinematic hardening rule for metals and the rotational hardening for soils, etc. which differ depending on the kinds of materials will not be discussed in detail. On the other hand, the kinematic hardening rules which are concerned only to the metals are discussed still now extensively by J.-L. Chaboche, S. Hassan, S. Kyrakides, N. Ohno, Y. F. Dafalias, etc., where unfortunately the basic mathematical structures of plasticity models themselves influencing basically the cyclic loading behavior have not been discussed enough.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 K. Hashiguchi, Foundations of Elastoplasticity: Subloading Surface Model, https://doi.org/10.1007/978-3-030-93138-4_10

271

272

10.1

10

Classification of Plasticity Models: Critical Reviews …

Cyclic Loading Behavior

The actual stress versus strain curve during the cyclic loading under a constant stress amplitude is shown by the blue colored line in Fig. 10.1, showing the accumulation of strain, i.e. the mechanical ratchetting. Here, note that the slight compressive plastic strain is induced in the unloading process to the stress-free state by the following mechanical reason: The solids are composed of material particles, e.g. the crystal particles in metals and soil particles in soils. Now, suppose the uniaxial loading process of the solid. The material particles are elongated to the tensional direction and the mutual slips are induced so that the tensional plastic strain is induced in the loading process. On the other hand, the elongated material particles return to the initial shapes which cause the slight mutual slips to the compressive direction so that the slight compressive plastic strain is induced in the unloading process to the stress free state. Further, the tensional elastic deformation and then the tensional plastic strain rate are induced in the reloading process. Consequently, the closed hysteresis loop is induced as shown in Fig. 10.1. However, the stress versus strain curve in which an merely an elastic deformation is repeated during the cyclic loading is predicted by the conventional elastoplastic model as shown by the red-colored line in Fig. 10.1. On the other hand, the excessively large strain accumulation is predicted by the initial subloading surface model described in Chap. 9 as was shown in Fig. 9.17, because only the elastic deformation is induced in the unloading process. Consequently, the cyclic loading behavior cannot be predicted realistically by the conventional elastoplastic model and the initial subloading surface model. Then, various plasticity models described in the next section have been proposed hitherto.

Fig. 10.1 Cyclic loading (mechanical ratchetting) behavior

10.2

Classification and Assessment of Plasticity Models

10.2

273

Classification and Assessment of Plasticity Models

The plasticity models, i.e. the cylindrical yield surface (Chaboche) model (Chaboche et al. 1979), the multi-surface (Mroz) model (Mroz 1967; Iwan 1967), the two-surface (Dafalias) model (Dafalias and Popov 1975; Krieg 1975) and the subloading surface (Hashiguchi) model (Hashiguchi 1978, 1980, 1989; Hashiguchi and Ueno 1977) proposed to date are classified from some aspects (Fig. 10.2). (1) Classification from conventional and unconventional models The plasticity models were classified into the conventional plasticity models and the unconventional plasticity models by Drucker (1988). The inside of the yield surface is assumed to be the purely-elastic domain inside which the stress lies always in the former. On the other hand, the latter aim at the description of the plastic deformation induced by the rate of stress inside the yield surface. The plasticity models proposed so far are classified as follows (Figure 10.3): Conventional plasticity model: Cylindrical yield surface (Chaboche) model, The Chaboche model is the conventional elastoplasticity model possessing the cylindrical yield surface which obeys the isotropic and the kinematic hardenings in the principal stress space. Therefore, needless to say, it is incapable of describing the cyclic loading behavior for the stress amplitude smaller than the size of the cylindrical surface. Moreover, it is incapable of describing the cyclic loading behavior appropriately even for the stress amplitude larger than the size of yield surface as will be described in detail in Sect. 10.3.2. Therefore, the Chaboche model is the primitive model incapable of the cyclic loading behavior, although it has been used and studied widely against the progress of time. First of all, it would be quite reckless to describe the cyclic loading behavior only by making the conventional yield surface expand (isotropic hardening) and translate (kinematic hardening). Unconventional plasticity models: Multi-surface (Mroz) model, Two-surface (Dafalias) model and the subloading surface (Hashiguchi) model. The cyclic plasticity model would have to belong to the unconventional model. In this aspect, the Chaboche model cannot be classified into the cyclic plasticity model. (2) Classification from existence or removal of elastic-domain Existence of elastic domain: Cylindrical yield surface (Chaboche) model, Multi-surface (Mroz) model and Two-surface (Dafalias) model, Removal of elastic-domain: Subloading surface (Hashiguchi) model (3) Classification from geometrical variation of elastic domain or loading surface The cyclic plasticity models are classified from the aspect of the geometrical variation of elastic domain or loading surface as follows:

Fig. 10.2 Classification of plasticity models

Chaboche et al. (1979) Ohno-Wang (1993)

Two-surface model Dafalias-Popov (1975) Yoshida-Uemori (2002)

Multi-surface model Mroz (1966, 1967)

Plural surface models with small elastic-domain

Hashiguchi-Ueno (1977) Hashiguchi (1980, 1989)

Subloading surface model

Unconventional plasticity models

10

Cylindrical yield surface model

Conventional plasticity model

Plasticity models

274 Classification of Plasticity Models: Critical Reviews …

Classification and Assessment of Plasticity Models

Fig. 10.3 Conventional and unconventional plasticity models

10.2 275

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Translation of elastic domain: Cylindrical yield surface (Chaboche) model, Multi-surface (Mroz) model and Two-surface (Dafalias) model Expansion of loading surface: Subloading surface (Hashiguchi) model The critical reviews for each of the above-mentioned plasticity models will be given in the subsequent sections.

10.3

Plasticity Models with Elastic Domain

The models other than the subloading surface model, i.e. the cylindrical yield surface model (Chaboche et al. 1979; Ohno and Wang 1993), the multi-surface model (Mroz 1967) and the two-surface model (Dafalias and Popov 1975; Yoshida and Uemori 2002a) assume the existence of the purely elastic domain inside which the stress lies always. The basic mathematical structures and the mechanical properties of these models will be examined in this section.

10.3.1

Common Drawbacks in Models with Elastic-Domain

The plasticity models assuming the elastic-domain inside which the stress lies always possess the following common crucial drawbacks. (1) The plastic strain rate cannot be described for the rate of stress inside the elastic-domain surface, so that the plastic strain accumulation, i.e. mechanical ratchetting phenomenon cannot be predicted at all. In facts, the original proposers (Chaboche, Morz, Dafalias) and their followers (Ohno and Yoshida in Japan unfortunately) have never shown a simulation of the mechanical ratchetting behavior in the tension (positive) or compression (negative) one side, i.e. the pulsating loading except for the physically unacceptable method by Yoshida’s group (e.g. Yoshida and Uemori 2002b, 2003; Yoshida and Amaishi 2020) as will be explained in Sect. 10.3.3. (2) The tangent stiffness modulus changes discontinuously resulting in the violation of the smoothness condition in Eq. (9.2) at the moment when the stress reaches the elastic-domain surface, (3) The judgement whether the stress reaches the elastic-domain surface is necessary, (4) Then, the numerical calculation of cyclic loading behavior is the time-consuming. Here, note that an excessive downsizing of the elastic-domain would results in the instability of its outward-normal, so that the direction of the plastic strain rate based on the normality-rule (associated flow rule) changes unstably as shown in Fig. 10.4. Therefore, the cyclic plasticity models other than the subloading surface model are inapplicable to the description of the cyclic loading behavior under the small stress amplitude.

10.3

Plasticity Models with Elastic Domain

277 ε p : stable

Yield surface

σ ε p : unstable σ Downsized yield surface or elastic domain

0

ij

Fig. 10.4 Downsizing of elastic-domain causes the instability of direction of plastic strain rate even for slight change of stress.

Needless to say, the cylindrical yield surface model is incapable of describing the cyclic loading (mechanical ratchetting) behavior at all, since it is the typical conventional elastoplasticity model as will be described in detail in Sect. 10.3.2. Further, the multi-surface (Mroz) model and the two-surface (Dafalias) model are also incapable of describing that behavior as will be described in Sects. 10.3.3 and 10.3.4. The plasticity models with an elastic domain, i.e. the cylindrical elastic-domain model, the multi-surface model and the two surface model described in this section have not been propagated by the original proposers themselves after the recent ten years. The proposers of these models themselves (Chaboche, Mroz, Dafalias) would have recognized the crucial defects of their own-models so that they have not commented on their models in the recent ten years, although these models are still studied by their followers (Ohno, Yoshida, etc.) unfortunately yet at present.

10.3.2

Cylindrical Yield Surface (Chaboche) Model: Ad Hoc. Primitive Conventional Model Limited to Simple Metal Behavior

The conventional plasticity model limited to the cylindrical yield surface with the plural kinematic hardening rules was proposed by Chaboche et al. (1979) and improved by Chaboche and Rousselier (1983), Chaboche (1989), Ohno and Wang (1993), etc. It may be called the cylindrical yield surface model which is called the

278

Classification of Plasticity Models: Critical Reviews …

10

multikinematic model by Chaboche (2008) himself. Strictly speaking, however, it may not be called a cyclic plasticity model, since it is incapable of describing the plastic strain rate by the rate of stress inside the yield surface. (a) Chaboche model The following yield surface with the isotropic hardening and the Armstrong-Frederick (1966) kinematic hardenings was introduced by Chaboche et al. (1979). rffiffiffi 3 0 jj^ r jj ¼ F0 þ F 2

ð10:1Þ

where the isotropic hardening function F evolves by the following equation. 



F ¼ cðF s  FÞe epq

ð10:2Þ

c is the material constant and F s is the material constant describing the saturation value of F. Equation (10.2) leads to F ¼ F s þ ðF 0  F s Þ exp½cðeeqp  eeqp 0 Þ which coincides with Eq. (8.42) by setting Fs ¼ ð1 þ sr ÞF0 and c ¼ cH . The kinematic hardening rule was given by the superposition of the several non-linear kinematic hardening rules of Armstrong and Frederick (1966) in Eq. (8.89) as follows: 

a¼

n X



ai

ð10:3Þ

i¼1

where  ai

rffiffiffi 2  bi ai jjep k ðno sumÞ ¼ Ai e  3 p

ð10:4Þ

Ai and bi ði ¼ 1; 2;   ; nÞ are the material constants, while n is chosen usually 4  8. Equation (10.3) is integrated for the uniaxial loading process as follows: aa ¼

n X Ai i¼1

bi

½1  expðbi epa Þ

ð10:5Þ

for epa [ 0 under the initial condition aa ¼ 0 and epa ¼ 0. The uniaxial loading behavior of Chaboche model is illustrated in Fig. 10.5, where the isotropic hardening is not incorporated by setting c ¼ 0 for simplicity.

10.3

Plasticity Models with Elastic Domain

279

Fig. 10.5 Chaboche model (after Lamaitre and Chaboche 1990)

The mechanical ratchetting is overestimated by use of the Armstrong-Frederic nonlinear kinematic hardening equation in its original form, because the kinematic hardening is suppressed but it induced excessively in the opposite direction in the reverse loading by the dynamic recovery part in the Armstrong-Frederic nonlinear kinematic hardening rule. Then, Chaboche (1991) modified Eq. (10.4) by incorporating the threshold for activation of the dynamic recovery part into the fourth kinematic equation, while Eq. (10.4) is used for the first to the third kinematic equations, as follows:  a4

rffiffiffi   2 a  a4 jjep jj b4 1  ¼ A4 e  3 jja4 jj p

ð10:6Þ

where a is the material constant designating the threshold value of the activation of the dynamic recovery which is zero for jja4 jj\a. (b) Ohno-Wang model Based on the opinions: (1) By introducing the threshold for the activation of the dynamic recovery the excessive mechanical ratchetting is suppressed as was proposed by Chaboche (1991), (2) The rate of the dynamic recovery depends on direction of the backstress rate in the multi-dimensional deformation. Ohno and Wang (1993) have modified the Armstrong-Frederic kinematic hardening rule by incorporating the idea of the threshold value by Chaboche (1991) as follows:

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Classification of Plasticity Models: Critical Reviews … 

a¼

n X



ai

ð10:7Þ

i¼1  ai

    2 p  p ai ai ¼ hi e  H½fi  e : ai r i 3

ðno sumÞ

ð10:8Þ

where hi and ri ði ¼ 1; 2;   ; nÞ are the material constants. The dynamic recovery activates when the following condition is satisfied in each evolution rule. fi  a2i  ri2 where

ð10:9Þ

rffiffiffi 3 ai  jjai jj 2

ð10:10Þ

H½  is the Heaviside step function, i.e., H½s ¼ 1 for s  0, H½s ¼ 0 for s\0. Then, the kinematic hardening proceeds when fi  a2i  ri2 ¼ 0 is satisfied and the plastic strain rate is induced directing outward of the surface described by Eq. (10.9) but it ceases until fi  a2i  ri2 ¼ 0 ðH½fi  ¼ 1Þ is satisfied. The fact that Eq. (10.8) is the possible candidate for the rate of the kinematic hardening fulfilling the time-derivative of Eq. (10.9) is as ascertained by 



fi ¼ 2ai ai ¼ 2ai

rffiffiffi rffiffiffi 3 3 ai  : ai ðjjai jjÞ ¼ 2ai 2 2 jjai jj

rffiffiffi     3 ai 2  ai ai ¼ 2ai :hi ep  H½fi  ep : ai ri 2 jjai jj 3 rffiffiffi      a 3 ai 2  ai ai ai i :hi k  k : ¼ 2ai jjai jj ai ai 2 jjai jj 3 jjai jj 0 1 * + rffiffiffi  a  a 3 ai 2 a a i i i i B C qffiffi ¼ 2ai :hi @ k  k : qffiffi A¼0 jjai jj 2 jjai jj 3 jjai jj 3 3 jja jj jja jj 2

i

2

i

The piecewise-linear relation of the backstress versus plastic strain is depicted for the proportional loading process by Eq. (10.7) with Eq. (10.8). Further, Eq. (10.8) was modified by replacing the Heaviside step function to the power function proposed by Henshall et al. (1987) as follows: 

ai ¼ hi



 mi    2 p ai ai ai  e  ep : ri ai r i 3

ðno sumÞ

ð10:11Þ

10.3

Plasticity Models with Elastic Domain

281

where mi ði ¼ 1; 2;   ; nÞ are the material constants. On account of this modification, the backstress versus plastic strain curve in the monotonic loading process becomes smooth. Here, note that Eq. (10.11) for mi ! 1 is reduced to Eq. (10.8). On the other hand, Eq. (10.11) for mi ¼ 0 is reduced to the Armstrong-Frederick nonlinear kinematic hardening rule. Further, the following model was proposed by Abdel-Karim and Ohno (2000). 

a i ¼ hi

    2 p ai ai   e  li ai jjep jj  H½fi  ep : ai r i 3

ðno sumÞ

ð10:12Þ

which comprises both of the Armstrong-Frederick’s (1966) and Ohno and Wang’s (1993) equations, li being the material constants. The defect of the original Armstrong-Frederick nonlinear kinematic hardening rule has been revealed such that the overshooting of the stress vs strain curve is predicted in the reloading process after the small reverse loading (cf. Dafalias, 2008), when this model is applied to the prediction of the cyclic loading behavior by the conventional plasticity model with the yield surface enclosing the purely-elastic domain obeying the isotropic and the kinematic hardenings. The overshooting of the stress-strain curve resulting in the overestimation of the mechanical ratchetting is the unavoidable nature of this model, because the reloading curve translates in parallel to the preceding monotonic curve in its underside keeping the almost constant interval which is caused by the dynamic recovery part in the kinematic hardening rule. Nevertheless, various modifications of the Armstrong-Frederick nonlinear kinematic hardening model have been proposed by Henshall et al. (1987), Hassan et al. (2008), Dafalias et al. (2008), Dafalias and Feigenbaum (2011), Okorokov et al. (2019), etc. in addition to Chaboche et al. (1979), Chaboche (1991), Ohno and Wang (1993) and Abdel-Karim and Ohno (2000) described above. Then, the formulation has become more and more complicated. Therein, the modifications of the Armstrong-Frederick kinematic hardening model have been repeated through its superposition by Chaboche et al. (1979) in the first stage, the incorporation of the threshold term (Chaboche, 1991) or the power function (Henshall et al., 1987), the multiplication (Dafalias, 2008), etc. Unfortunately, however, they adopt only the cylindrical yield surface (Chaboche-Ohno) without taking care about the afore-mentioned defect of the basic primitive structure of this model as a cyclic plasticity model, although they are concerned only with the large stress amplitude surrounding the yield surface. In facts, the Chaboche-Ohno model has been irrespective to the general cyclic loading behavior including the partial unloading-reloading, the pulsating loading behaviors, etc. because it falls within the framework of the conventional plasticity. Nevertheless, an appropriate modification of Chaboche-Ohno model could not be found and will not found forever even for the description of the cyclic loading behavior with large stress amplitude contrary to the expectation of these researchers represented by Chaboche. The reason for the impossibility is caused by the fact that they are concerned only with the unloading behavior through the dynamic recovery part. However, the rational formulation of the cyclic loading behavior cannot be

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attained by the modification of the unloading behavior in the kinematic hardening rule but it can be attained by the modification of the reloading behavior as will be shown actually in Sect. 11.6 for the extended subloading surface model. Eventually, it should be concluded that the conventional plasticity model including the Chaboche model is basically incapable of describing the cyclic loading behavior even for a large stress amplitude, although a half century has been wasted after Chaboche et al. (1979). First of all, the superposition of equations started by Chaboche et al. (1979) itself results in containing many material parameters whose physical meanings are unclear, exposing the physical and mathematical immaturity as a constitutive equation. The Armstrong-Frederick model itself is physically and mathematically reasonable for metals by the slight modification in Eq. (8.89). In fact, the general cyclic loading behavior of metals can be described accurately by adopting it in the extended subloading surface model as will be shown in Chaps. 11 and 12. Besides, the kinematic hardening is relevant to the plastically-incompressible metals but it is irrelevant to the plastically-compressible materials, e.g. soils, rocks, concrete, ceramics, glass, etc. for which the rotational hardening is dominant as will be described in Chap. 13 for soils. The cyclic plasticity model should be studied seriously by adopting the unconventional yield surface and formulating the plastic strain rate induced by the rate of stress inside the yield surface. Besides, the improvement for avoiding the undershooting curve cannot be attained by the modification of the unloading property as the dynamic recovery part but can be attained by the improvement of the reloading behavior in the unconventional plasticity model as will be described in Sect. 11.6. Consequently, it should be recognized that 1. The cyclic plasticity formulation can never be realized by adopting the conventional plasticity model enclosing the purely-elastic domain. 2. It is impossible to describe the cyclic loading behavior only by the modification of the kinematic hardening rule. 3. Only the cyclic loading behavior under the large stress amplitude in positive and negative stress sides covering the yield surface has been shown in the Chaboche model and its related models. It can be described realistically even by the initial subloading surface model in Sect. 9.2 as shown by Hashiguchi (1981). However, the description of the cyclic loading behavior under the half stress amplitude in positive or negative one side inducing the mechanical ratchetting is of crucial imporatance for the engineering practice. 4. The rational formulation for the cyclic plastic model can be realized only by the extended subloading surface model described in detail from the subsequent chapters in which the plastic strain rate is induced by the rate of stress inside the yield surface and reloading behavior is modified so as to return promptly to the preceding stress vs. strain curve, keeping the kinematic hardening rule in Eq. (8.89) as it is. On the other and, the various nonsense modifications of the unloading behavior have been repeated through the alteration of the dynamic recovery part in the Armstrong-Frederick kinematic hardening equation.

10.3

Plasticity Models with Elastic Domain

283

Unfortunately, however, the Chaboche model was officially implemented in the commercial software Marc and Abaqus, LS-DYNA etc. so that it is used widely by metal engineers because it can be understood even by the beginners of elastoplasticity theory possessing the elementary knowledge only of the Mises yield condition with the kinematic hardening. However, it should be recognized that the mechanical designs of solids and structures subjected to the general cyclic loading ehaviour by this model would result in dangerous accidents because this model is incapable of describing the cyclic loading behavior pertinently even for the stress amplitude larger than the yield surface, while, needless to say, it cannot describe the cyclic loading behavior for the stress amplitude smaller than the yield surface at all.

10.3.3

Multi-surface (Mroz) Model: Incapable of Describing Mechanical Ratchetting

Mroz (1966, 1967, 1976) and Iwan (1967) proposed the multi-surface (Mroz) model based on the following basic assumptions. (a) The elastic-domain surface and plural encircled subyield surfaces are incorporated inside the yield surface, while the ratios of the sizes of these surfaces to the yield surface are kept constant throughout a deformation. (b) The elastic-domain surface and the subyield surfaces are pushed out by the current stress point so that plural surfaces contact at a point. (c) Plastic modulus is prescribed by the size of the subyield surface on which the current stress lies, while it is smaller for a more outer subyield surface. The uniaxial deformation ehaviour of the multi-surface model is illustrated in Fig. 10.6 for the simple material without the isotropic and the kinematic hardenings. However, this model possesses the following defects in addition to the crucial defects caused by the incorporation of the elastic-domain described in Sect. 10.3.1. (1) Plural subyield surfaces contact at the current stress point and there exist plural plastic moduli at that contact point so that the singular point in the field of plastic modulus is induced therein. Numerical calculation becomes unstable for the cyclic loading behavior in the vicinity of contact point. (2) It is physically impertinent that the elastic-domain surface contacts with the conventional yield surface in the fully-plastic state. (3) Stress transfers to a larger subyield surface by moving the half of the difference of the radii of subyield surfaces in the initial loading process as shown in Fig. 10.6b. On the other hand, it transfers to a larger subyield surface by moving the difference of the diameters of subyield surfaces in the unloading-reverse loading process as shown in Fig. 10.6d. Therefore, the Masing rule (Masing 1926) meaning that the curvature of stress–strain curve in the unloading-reverse loading decreases to a half of the curvature of initial loading curve is described exactly and simply as shown in Fig. 10.6d. By virtue

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Classification of Plasticity Models: Critical Reviews …

Fig. 10.6 One-dimensional loading ehaviour predicted by multi-surface model

of this mechanical feature, this model has been used widely. However, the variation of curvature observed in real material behavior is not so large as described by the Masing rule. (4) The stress–strain curve converges to the fixed loop for the nonhardening yield surface as shown in Fig. 10.7a. The hysteresis loop of the stress-strain curve becomes thinner with increase of the cycles for the hardening yield surface as shown in Fig. 10.7b. Therefore, the accumulation of plastic strain during a pulsating stress loading, called the mechanical ratcheting, cannot be described at all by this model. Therefore, the deformations of machinery and structures subjected to cyclic loading are predicted to be unrealistically small by the multi-surface model, resulting in a risky mechanical design of solids and structures.

10.3.4

Two Surface (Dafalias) Model: Incapable of Describing Plastic Strain Rate in Unloading Process

Dafalias and Popov (1975, 1976) and Krieg (1975) proposed the two surface model based on the following assumptions:

10.3

Plasticity Models with Elastic Domain

285

Fig. 10.7 Prediction of cyclic loading behavior by the multi surface model under a constant stress amplitude: Plastic strain accumulation cannot be described at all

(a) The elastic domain is incorporated inside the conventional yield surface which is renamed as the “bounding surface” by Dafalias and Popov (1975) and “limit surface” by Krieg (1975). The bounding surface is merely the usual yield surface. Therefore, this coined word must be disused in order to avoid the disturbance/confusion. (b) The ratio of the size of the elastic-domain surface to that of the yield surface is kept constant throughout the deformation. (c) The elastic-domain is pushed out by the current stress point and translates toward the conjugate point on the yield surface, while the outward-normal at conjugate point on the yield surface is identical to that at the current stress point on the elastic-domain surface. (d) The plastic modulus depends on the distance from the current stress on the elastic-domain surface to the conjugate stress on the yield surface. Here, it is required that the elastic-domain surface must translate so as not to intersect with the yield surface because the direction of plastic strain rate becomes indeterminate if they intersects. The rigorous translation rule was derived by Hashiguchi (1981, 1988). This model has been adopted widely for the prediction of deformation behavior of metals (cf. e.g. Dafalias and Popov 1976; McDowell 1985, 1989; Ohno and Kachi 1986; Ellyin 1989; Hassan and Kyriakides 1992; Yoshida and Uemori 2002a, 2002b, 2003; Yoshida and Amaishi 2003).

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Fig. 10.8 Uniaxial loading behavior predicted by two surface model

The uniaxial loading behavior of the two surface model is illustrated in Fig. 10.8 for the simple material without the isotropic and the kinematic hardenings. However, this model possesses the following defects in addition to the crucial defects caused by the incorporation of the elastic-domain described in Sect. 10.3.1. (1) The plastic modulus depends on the distance from the current stress to the conjugate stress irrespective of the loading process, i.e. the initial, the reverse and the reloading processes and thus the curvature of stress–strain curves are identical irrespective of these processes. Then, the Masing effect cannot be described at all contrary to the multi-surface model. (2) Only the elastic deformation without a plastic strain rate is induced in unloading process, although the plastic deformation in the opposite direction to the preceding deformation is induced in real material behavior as seen in Fig. 10.1. Consequently, the open hysteresis loop is depicted in the unloading–reloading process. (3) The excessively large mechanical ratcheting phenomenon is depicted as shown in Fig. 10.9. However, the mechanical ratcheting cannot be described at all if the inner yield surface (purely elastic domain) is assumed to be larger than the half of the outer-yield (bonding) surface. Therefore, the finitely different results

10.3

Plasticity Models with Elastic Domain

287

Bounding surface Constant stress amplitude

Elastic state

p

0

(a) Small inner yield surface: Excessively large strain accumulation

Bounding surface Constant stress amplitude

Elastic state

0

p

(b) Large inner yield surface: No strain accumulation

Fig. 10.9 Prediction of cyclic loading behavior by the two-surface model under a constant stress amplitude: excessively large strain accumulation

of the mechanical ratcheting are described by the selection of the size of the inner yield surface. Besides, the spring-back phenomenon can never be predicted by this model, since the description of the plastic deformation induced in the unloading process is of crucial importance for the prediction of this phenomenon. Nevertheless, Yoshida et al. (Yoshida and Uemori, 2002b, 2003; Yoshida and Amaishi, 2020) insist that they can predict the spring-back phenomenon by the two surface model by making the Young’s modulus decrease asymptotically to the saturation value with the plastic equivalent strain or by using the chord modulus defined by the inclination of the straight line connecting the beginning and the unloaded points of the unloading stress-strain curve, which decreases with the plastic deformation. In fact, however, the exact Young’s modulus represented by the inclination of the stress vs. strain curve at the moment just after the reverse loading in the uniaxial loading is kept constant as far as a large deformation causing the damage is not induced as known from the elementary knowledge of the continuum damage mechanics. Besides, once the damage begins, the damage develops increasingly to the failure in the monotonic loading process. In fact, however, the plastic deformation is induced during the unloading process, so that the spring-back phenomenon must be analyzed by an elastoplastic constitutive equation which is capable of describing the plastic

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deformation induced in the unloading loading process. Besides, the mechanical ratcheting phenomenon, i.e. the strain accumulation during the cyclic loading cannot be described by such easy-going method altering the elastic deformation behavior which woud result in the risky mechanical design. The spring-back phenomenon can be analyzed rigorously by the extended subloading surface model which will be explained in the next section and Sect. 12.6. Unfortunately, however, the two surface model worsened by the Yoshida’s group has been installed as the standard program in the commercial software Ls-Dyna, PAM-STSAMP and JSOL-JSTAMP. Further, the similar irrational methods have been reported by Sun and Wagoner (2011), Wagoner (2013), Chen et al. (2016), Okorokov et al. (2019), etc. One should make effort seriously to describe the plastic deformation induced in the unloading process in order to predict rationally the spring-back phenomenon. Vanishing Elastic Domain Model The modification of the two surface model was proposed by Dafalias and Popov (1977) in which the inner yield surface is contracted to a point, naming it “vanishing elastic domain”. However, this modification results in the serious defect that the direction of the plastic strain rate depends on the direction of the stress rate and thus it becomes unstable deviating from the associated flow rule. In other words, it is the quite peculiar model ignoring the physical background of the associated flow rule described in Sect. 8.6. Bounding Surface Model with Radial-Mapping The bounding surface model with radial-mapping by Dafalias and Harmann (1980) is basically different from the above-mentioned two-surface model and it is basically different also from the subloading surface model contrary to the Dafalias’ statement (Dafalias 1986) as was explained exhaustively in Sect. 9.5. The term “bounding surface” should be excluded from the formal technical terms, since it is merely a yield surface itself so that all the plasticity models using the yield surface are included in the bounding surface model if this term is accepted. As studied deeper, the inherent defects become clearer in the Chaboche, the multi and the two surface models.

10.4

Extended Subloading Surface (Hashiguchi) Model: Capable of Describing General Loading Behavior

The subloading surface model formulated in Chap. 9, called the initial subloading surface model hereinafter, is incapable of describing cyclic loading behavior appropriately, predicting an open hysteresis loop in an unloading-reloading process and thus overestimating a mechanical ratcheting phenomenon. The insufficiency is

10.4

Extended Subloading Surface (Hashiguchi) …

289

caused by the fact that the similarity-center of the normal-yield and the subloading surfaces is fixed at the back-stress and thus a purely-elastic deformation is described in the unloading process, resulting in the open hysteresis loop. Then, the insufficiency was remedied by making the similarity-center translate with the plastic deformation (Hashiguchi 1985b, 1986, 1989) as will be described in the following, while it is called the extended sublading surface model. The uniaxial loading behavior is depicted in Fig. 10.10 for the simple material behavior without a variation of the normal-yield surface. The similarity-center goes up following the stress by the plastic strain rate in the initial loading process as seen in Fig. 10.10a and b. The subloading surface shrinks and thus only elastic strain rate is 1

1

σ

Normal-yield surface

1

1

c α σ

σ

c

σ = c =α c

p

0

0

3

1 



σ

c

σ c



α

Subloading surface: expands

3

c

(c = O)

σ

σ



0

0



α

p

σ c 

0

Contracts p ( ε = O)

α p

0

3 2 (c) Beginning instant of unloading process

Expands



α Contracts p ( ε = O)



σ σ

σ c



α

α α

2 3 (e) Reloading process until reaching similarity-center 1

1

σ

c

c

(c = O)

p

0

1

1





α c α σ σ

0

2 (b) Initial loading process

3 1

1

α

p

0

Expands 0

2 (d) Reverse loading process

(a) Beginning of initial loading

1

σ

p

0

2



c α

2 (f) Reloading process

0

3

1

1

σ

σ  c 



α

0

Expands

c α p

0 2

3

Expansion of (f): Closed hysteresis loop is depicted in unloading-reloading process.

Fig. 10.10 Prediction of uniaxial loading behavior by extended subloading surface model: a initial state, b initial loading process, c unloading process until similarity-center, d unloading-inverse loading process after passing similarity-center, e reloading process until reaching similarity-center and f reloading process.

290

10

Classification of Plasticity Models: Critical Reviews …

Fig. 10.11 Modification of subloading surface model to describe cyclic loading behavior.

induced until the stress goes down to the similarity-center in the unloading process as seen in Fig. 10.10c. After that the subloading surface begins to expand and thus the plastic strain rate in the compression is induced in the unloading-inverse loading process whilst the similarity-center goes down following the stress by the compressive plastic strain rate as seen in Fig. 10.10d. Further, only the elastic strain rate is induced until the stress goes up to the similarity-center in the reloading process from the complete unloading as seen in Fig. 10.10e. After that the subloading surface begins to expand and thus the plastic strain rate is induced whilst the similarity-center goes up following the stress by the plastic strain rate as seen in Fig. 10.10f. The expanded figure of Fig. 10.10f is shown in the lowest part of Fig. 10.10. Consequently, the closed hysteresis loop is depicted realistically as shown in this figure. Besides, the spring-back phenomenon can be described rigorously only by the extended subloading model, while it can never be analyzed by such irrational model as the Yoshida et al’s model based on the two surface model (Yoshida and Uemori, 2002b, 2003; Yoshida and Amaishi, 2020) as was explained in Sect. 10.3.4. The extended subloading surface model would describe the cyclic loading behavior realistically as illustratively shown in Fig. 10.11, resolving all drawbacks in the cyclic plasticity models based on the kinematic hardening concept, while the continuity and the smoothness conditions in Eqs. (9.1) and (9.2) are satisfied only in this model. Then, it has been applied to the descriptions of rate-independent (elastoplastic) and rate-dependent (elasto-viscoplastic) deformation behavior of not only metals but also geomaterials and further the friction phenomena between solids and the crystal plasticity as will be described in detail in the subsequent chapters.

10.5

Overall Assessment of Plasticity Models

The overall assessment of cyclic plasticity models is summarized in Table 10.1. All the serious drawbacks involved in the other plasticity models with elastic domain, i.e. the cylindrical yield surface model Chaboche-Ohno), the multi-surface model

Table 10.1 Classification and over-all evaluation of plasticity models

10.5 Overall Assessment of Plasticity Models 291

292

10

Classification of Plasticity Models: Critical Reviews …

(Mroz) and the two surface model (Dafalias-Yoshida) are resolved in the extended subloading surface model. Eventually, it can be concluded definitely that the rigorous plasticity model is the extended subloading surface model which possesses the capability of describing the general rate-independent and—dependent monotonic and the cyclic loading behavior for any small stress amplitude and the finite deformation behavior by the multiplicative elastic-plastic decomposition. The detailed formulations for the extended subloading surface model will be described in the subsequent chapters.

Chapter 11

Extended Subloading Surface Model

As was assessed in detail in Chap. 10, it can be concluded that only the extended subloading surface model is capable of describing the general loading behavior of materials appropriately. The explicit formulation of the extended model is described in detail in this chapter. The central point of the extension in the formulation from the initial subloading surface model in Chap. 9 is the translation of the similarity-center of the subloading and the normal-yield surfaces, so that the cyclic loading behavior with the closed hysteresis loop is described realistically. The extended model will be applied to the description of the elastoplastic deformations of various materials, e.g. metals, soils and the various phenomena, e.g. damage, phase transformation and friction and some of their validities will be verified by comparisons with test data of metals and soils in the subsequent chapters.

11.1

Normal-Yield and Subloading Surfaces

The normal-yield surface with the isotropic and the kinematic hardening is described as f ð^ rÞ ¼ FðHÞ

ð11:1Þ

as shown in Eq. (8.71) already. The extended subloading surface associated with the normal-yield surface in Eq. (11.1) is given as follows (see Fig. 11.1). f ðrÞ ¼ RFðHÞ

ð11:2Þ

where Rð0  R  1Þ is the normal-yield ratio and

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 K. Hashiguchi, Foundations of Elastoplasticity: Subloading Surface Model, https://doi.org/10.1007/978-3-030-93138-4_11

293

294

11

Extended Subloading Surface Model

n 

εp

σy

σ

σ 

ˆ ) = F (H ) f (σ Subloading surface

nˆ c

σ

σ

Rσ

Normal-yield surface

f (σ) = RF (H )

c

cˆ

α α

Elastic-core surface

f (cˆ) = R cF (H )

 ij

0

3

n

σy

α

α

ˆ ' || = F ( H ) 3 / 2||σ



σ

cˆ

(  / R) σ

Normal-yield surface

εp

σ Rσ

f (cˆ ) =  F ( H )

σˆ y  σ y  α  σ / R σˆ   σ  α   σˆ y

(a) General material



Limit elastic-core surface

Subloading surface

σ σ

ˆ' ||  RF ( H ) 3 / 2 || σ'  Rc

c

Elastic-core surface

0

1

ˆ' || = R cF ( H ) 3 / 2 || c

2 (b) Mises material in deviatoric stress plane

Fig. 11.1 Normal-yield, subloading and elastic-core surfaces

Limit elastic-core surface ˆ' || =  F ( H ) 3 / 2 || c

11.1

Normal-Yield and Subloading Surfaces

295

rra

ð11:3Þ

a stands for the conjugate (similar) point in the subloading surface to the point a representing the back-stress tensor for kinematic hardening in the normal-yield surface. Here, note that the purely elastic behavior is induced when the stress coincides with the similarity-center of the normal-yield and the subloading surfaces and thus the normal-yield ratio becomes zero ðR ¼ 0Þ. Then, let the similarity-center be called the elastic-core and let it be denoted by the bold letter c. Here, the following relation holds by virtue of the similarity (see Fig. 11.1). a  c ¼ Rða  cÞ

ð11:4Þ

a ¼ c  R^c

ð11:5Þ

which yields

_

r ¼ r þ R^c

ð11:6Þ

where ^c  c  a

) ð11:7Þ

_

rrc

Besides, ry in Fig. 11.1 is the conjugate stress on the normal-yield surface to the current stress r on the subloading surface, fulfilling ra ¼ Rðry aÞ. The time-derivative of a is described from Eq. (11.5) as 







a ¼ R a þ ð1  RÞ c  R ^c

ð11:8Þ

All the relations in Eqs. (11.4)–(11.6) hold by virtue of the similarity of the subloading surface to the normal-yield surface. The subloading surface in Eq. (11.2) is rewritten by Eq. (11.6) as follows: _

f ðr þ R^cÞ ¼ RFðHÞ

ð11:9Þ

Adopt the associated flow rule for the subloading surface:        e p ¼ k nðk ¼ e p  [ 0Þ

ð11:10Þ

296

11

Extended Subloading Surface Model

where n

@f ðrÞ @r

  @f ðrÞ    @r 

ðknk ¼ 1Þ

ð11:11Þ

The rate of the isotropic hardening variable is described by using the function fHn ^ to n as given from Eqs. (8.23) and (8.28) with Eq. (11.10) and the replacement of n follows: 





p H ðr; H; e Þ ¼ fHn ðr; H; nÞ k

ð11:12Þ

The rate of the kinematic hardening variable is described by using the function ^ to f kn given from Eqs. (8.89) and (8.100) with Eq. (11.10) and the replacement of n n as follows: Storage part

zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{   1      p a ¼ ck ðe p  e a Þ ¼ k f kn ; bk F |fflfflfflfflfflffl ffl{zfflfflfflfflfflfflffl}

 f kn ¼ ck



1 a n bk F

ð11:13Þ

Diddipative part

11.2

Evolution Rule of Elastic-Core

The rigorous evolution rule of the elastic-core is formulated in this section, which plays the central role in the description of the cyclic loading behavior leading to the closed hysteresis loop as shown in Fig. 10.1. Now, the following facts should be noticed. (1) In the physical view-point, the elastic-core at which a purely-elastic deformation is induced must not lie on the normal-yield surface at which a fully-plastic deformation is induced. In addition, it should be noted that a smooth elastic– plastic transition leading to the continuous variation of the tangent stiffness modulus tensor is not described violating the continuity condition in the large, i.e. the smoothness condition in Eq. (9.5), if the elastic-core lies on the normal-yield surface. On the other hand, the other plasticity models, i.e. the cylindrical superposed kinematic hardening model (Chaboche et al. 1979; Ohno and Wang, 1993), the multi surface model (Mroz, 1967; Iwan, 1967) and the two surface model (Dafalias and Popov, 1975; Krieg, 1975; Yoshida and Uremori, 2003) violate the smoothness condition, since the purely-elastic domain contacts to the yield surface directly. (2) In the mathematical view-point, the subloading surface is not determined uniquely if the stress coincides with the elastic-core lying on the normal-yield surface, noting the fact that R is indeterminate as known from the relation

11.2

Evolution Rule of Elastic-Core

297

0 ¼ R0 which is induced by substituting r ¼ c ¼ ry into rc ¼ Rðry cÞ based on the similarity of the subloading surface to the normal-yield surface. Consequently, the elastic-core is not allowed to approach to the normal-yield surface unlimitedly. Now, let the following elastic-core surface be introduced, which always passes through the elastic-core c and maintains a similarity to the normal-yield surface with respect to the kinematic-hardening variable a. ð11:14Þ where ℜ c designates the ratio of the size of the elastic-core surface to the normal-yield surface (see Fig. 11.1) so that let it be called the elastic-core yield ratio. Then, let it be postulated that the elastic-core can never reach the normal-yield surface designating the fully-plastic stress state so that the elastic-core does not go over the following limit elastic-core surface. f ð^cÞ ¼ vFðHÞ

ð11:15Þ

where vð\1Þ is the material constant designating the maximum value of ℜ c and the following inequality must be satisfied. f (c ˆ ) ≤ χ F ( H ), i.e. ℜ c ≤ χ

ð11:16Þ

The material-time derivative of Eq. (11.16) at the limit state that c lies on the limit elastic-core surface in Eq. (11.15) yields: ð11:17Þ Here, noting ð11:18Þ on account of the Euler’s homogeneous function f ð^cÞ in degree-one for the variable ^c, the substitution of Eq. (11.18) to Eq. (11.17) leads to ð11:19Þ The inequality (11.16) is called the enclosing condition of elastic-core and Eq. (11.19) is its rate form.

298

11

Extended Subloading Surface Model

Now, assume the equation (Hashiguchi 2018b)       r    F     c  a  ^c ¼ ce e p ðrv cÞ ¼ ce e p  v ^c F R

ð11:20Þ

where ce is the dimensionless material constant and rv is the conjugate stress on the limit elastic-core surfaces to the current stress r on the subloading surface, i.e. rv 

v r þ a; R

rv  c 

v r  ^c R

ð11:21Þ

which is based on the relation rv  a ¼ vðry  aÞ ¼ v

ra r ¼v R R

ð11:22Þ

where ry is the conjugate stress on the normal-yield surface (Fig. 11.2). Equation (11.20) means that the elastic-core translates to approach the conjugate 



stress rv in the nonhardening state: F ¼ 0 and a ¼ O. It follows from Eq. (11.19) with Eq. (11.20) that

σy

(nˆc : (σ  c ) 0 )

σ

σ 

Rσ

nˆ c

σ c

σ

Normal-yield surface

ˆ ) = F (H ) f (σ Subloading surface f (σ) = RF (H )

cˆ

α

Limit elastic-core surface

α 0

f (cˆ ) =  F ( H )

 ij

σˆ y 



 σy α  σ /R σˆ  



 σ α   σˆ y  (  / R)σ Fig. 11.2 Translation of elastic-core when it lies on the limit surface ℜ c = χ fulfilling the enclosing condition in Eq. (11.19) because of the convexity of the surface

11.2

Evolution Rule of Elastic-Core

299

ð11:23Þ

as far as the normal-yield surface satisfies the convexity condition (Appendix H), noting that @f ð^cÞ=@c which is the outward-normal of the elastic-core surface at the current elastic-core c and rv c makes an obtuse angle when c lies on the limit ^c is the norelastic-core surface, while rv lies on the limit elastic-core surface. n malized outward-normal of the elastic-core surface (see Fig. 11.2), i.e. @f ð^cÞ ^c  n @c

,  @f ð^cÞ   ^c k¼ 1Þ  @c  ðkn

ð11:24Þ

Therefore, Eq. (11.20) satisfies the enclosing condition of the elastic-core so that the elastic-core can never go out from the limit elastic-core surface. Then, the evolution rule of the elastic-core is given from Eq. (11.20) by      v   F   c ¼ ce ep  r^c þ a þ ^c F R

ð11:25Þ

The second and the third terms in the right-hand side designate the influences by the kinematic and the isotropic hardening. These terms cannot be excluded in general as known from the fact that the quantity in the left-hand term in Eq. (11.19) leads to the following equation if these terms are ignored. "  !  #     v   @f ð^cÞ @f ð^cÞ   F F  p  p r^c  a  ^c ¼ : ce e  : ce e ðrv cÞ a  ^c F F @^c R @^c    @f ð^cÞ   @f ð^cÞ F   ¼ ce :ðrv cÞep   : a  f ð^cÞ F @^c @^c 

which is not necessarily negative for the softening process F \0 and thus does not fulfill the enclosing condition of the elastic-core (Eq. 11.19) in general as observed in soils exhibiting the softening (see Chap. 13). The translation of the elastic-core would have to be influenced by the variation of the isotropic hardening/softening and the kinematic hardening of the normal-yield surface in general, since its movement is limited to the interior of the normal-yield surface which expands and translates with the plastic deformation. This property is different from the evolution rule of the kinematic hardening which causes the movement of the normal-yield surface. Equation (11.25) with Eqs. (11.12) and (11.13) leads to

300

11

Extended Subloading Surface Model

     v   F' f     1        p Hn  p  c ¼ ce ep  r^c þ ck ep  e a þ e ^c ¼ f cn k ð11:26Þ F R bk F where  f cn  ce

   v 1 F' fHn ^c a þ r^c þ ck n F R bk F

ð11:27Þ

Now, considering the uniaxial loading behavior of the simple material with the pffiffiffiffiffiffiffiffi p p  p0 p plastic incompressibility e ¼ e leading jjje k ¼ 3=2jea j without the isotropic 

hardening F ¼ 0 and the kinematic hardening a ¼ O and designating the compop p nents of c and e in the axial direction by ca and ea , respectively, it follows noting p

p

rva ¼ vF ðupper: ea [ 0; lower: ea \0Þ that 

ca ¼ c e

pffiffiffiffiffiffiffiffi  p 3=2ð ea ÞðvF  ca Þ





ðupper : eap [ 0; lower : eap \0Þ

leading to 

ca ¼

pffiffiffiffiffiffiffiffi  3=2ce ðvF  ca Þ eap





ðupper : eap [ 0; lower : eap \0Þ

ð11:28Þ

from which one has 8 pffiffiffiffiffiffiffiffi > ¼0 > < 3=2ce vF for ca ( ca pffiffiffiffiffiffiffiffi  3=2ce ðvF  ca Þ ¼ pffiffiffi vF and eap \0 p ¼ > ea >  : 6ce vF for ca ¼ vF and eap [ 0 

ð11:29Þ

The time-integration of Eq. (11.28) is given as follows: pffiffiffiffiffiffiffiffi vF  ca ¼ exp½ 3=2ce ðepa  epa0 Þ vF  ca0 i.e. ca ¼ vF  ðvF  ca0 Þ exp½ 

pffiffiffiffiffiffiffiffi 3=2ce ðepa  epa0 Þ 

ðupper: eap [ 0; lower : eap \0Þ

ð11:30Þ

where ca0 and epa0 are the initial values of ca and epa , respectively. The relation of ca versus epa is shown for the incompressible material without hardening in Fig. 11.3 in which the nonlinear evolution rule is depicted actually. The plastic strain ep be additively decomposed into the storage part epcs and the dissipative part epcd for the elastic-core c, i.e.

11.2

Evolution Rule of Elastic-Core

301

Fig. 11.3 Translation of elastic-core in uniaxial loading for nonhardening Mises material



ep ¼ epcs þ epcd ;





p p ep ¼ ecs þ ecd

ð11:31Þ

Then, the elastic-core c is formulated using the elastic strain energy function wc ðepcs Þ as follows: c¼

@wc ðepcs Þ @epcs

ð11:32Þ

Here, let wc ðepcs Þ be given by the quadratic equation 1 wc ðepcs Þ ¼ cs epcs : epcs 2

ð11:33Þ

Then, it follows that 







p p c ¼ cs epcs ¼ cs ðep epcd Þ; c ¼ cs ecs ¼ cs ðep  ecd Þ;

ð11:34Þ

where cs is the material constant with the dimension of stress. The storage and the dissipative parts in the rate of the elastic-core in Eq. (11.26) are expressed formally noting Eq. (11.34) as follows:

ð11:35Þ

302

11

Extended Subloading Surface Model

Here, note that c e || ε p || ( / R ) σ / c s in the elastic-core itself, ck ε p / c s in the p

kinematic hardening and ( F' f Hn / F )|| ε || cˆ / c s in the isotropic hardening contribute p

to the evolution of the elastic-core. On the other hand, c e || ε ||cˆ / c s in the elastic-core itself and ck [1/ (bk F )] || ε p || α / c s in the kinematic hardening suppress the evolution of p the elastic-core. In other words, c e || ε ||cˆ / c s and ck [1/ (bk F )] || ε p || α / c s lead to the dissipative part in the rate of the elastic-core. Equations (11.13) and (11.35) will be used in the formulation of the dissipative part of the plastic strain rate for the rates of the kinematic hardening variable and the elastic-core in the multiplicative hyperelastic-based plasticity in Sect. 17.9.

11.3

Plastic Strain Rate

The material-time derivative of Eq. (11.2) leads to the consistency condition of the subloading surface as follows:  @f ðrÞ  @f ðrÞ   :r :aRF  RF ¼ 0 @r @r

ð11:36Þ

@f ðrÞ :r ¼ f ðrÞ ¼ RF @r

ð11:37Þ

Here, one has

based on the homogeneous function f ðrÞ of r in degree-one by the Euler’s theorem. Then, it follows that @f ðrÞ :r @r  ¼  RF  ; n :r ¼  @f ðrÞ @f ðrÞ      @r   @r 

1 n :r   @f ðrÞ ¼ RF    @r 

ð11:38Þ

The substitution of Eq. (11.38) into Eq. (11.36) leads to " 

n : r n :

#  !  F R þ rþ a ¼ 0 F R

ð11:39Þ

The substitution of Eq. (11.8) into Eq. (11.39) leads to " #    F R n : r n : r þ R a þ ð1  RÞ c þ ðr  R^cÞ ¼ 0 F R 

ð11:40Þ

11.3

Plastic Strain Rate

303

Noting the relation _

r  R^c ¼ r  a  ðc  aÞ ¼ r

ð11:41Þ

it follows from Eq. (11.40) that     F R_ n: r n : r þ R a þ ð1  RÞ c þ r ¼ 0 F R





ð11:42Þ

The substitutions of Eq. (11.10) with Eqs. (9.9), (11.12), (11.13) and (11.26) into Eq. (11.42) leads to   F'  U_ n : r n : k fHn r þ R k f kn þ ð1  RÞ k f cn þ k r ¼ 0 F R 







ð11:43Þ p

from which the plastic multiplier k and the plastic strain rate e are given as follows: 



k¼



n : r p n : r p ; e ¼ p n M M

ð11:44Þ

where 

F' U fHn r þ Rf kn þ ð1  RÞf cn þ _ M ¼ n: r F R



p

ð11:45Þ

The plastic modulus in Eq. (11.45) is reduced to the one for the conventional model in the normal-yield state by setting R ¼ 1 leading to U ¼ 0.

11.4

Stain Rate Versus Stress Rate Relations

The strain rate is given by substituting Eqs. (8.5) and (11.44)2 into Eq. (8.1) as follows: 





e ¼ E1 : r þ

n :r p n¼ M

  n :n  :r E1 þ p M

ð11:46Þ

from which the magnitude of plastic strain rate described in terms of the strain rate, 



denoted by K instead of k, in the flow rule of Eq. (11.47) is given as follows:

304

11 

K¼

Extended Subloading Surface Model





n:E: e n:E: e p ;e ¼ p n p M þ n:E:n M þ n:E:n

ð11:47Þ

The stress rate is given from Eq. (8.5) with Eq. (11.47) as follows: ð11:48Þ The loading criterion is given as follows (Hashiguchi 2000, 2013b): 





ep ¼ 6 O for R [ Re and K [ 0 or n : E : e [ 0  p e ¼ O for others

ð11:49Þ

p

premising M þ n : E : n [ 0 based on the same physical background to that for Eq. (8.55) described in Sect. 8.5, where the judgment whether or not the stress reaches the yield surface is not required since the plastic strain rate is induced continuously as the stress approaches the normal-yield surface. Equation (11.49) is applicable not only to the hardening state but also to the perfectly-plastic and softening state (Hashiguchi 2000, 2013a).

11.5

Calculation of Normal-Yield Ratio in Unloading Process

The normal-yield ratio R is calculated by the time-integration of Eq. (9.9) in the  plastic loading process ep 6¼ O. On the other hand, it can be calculated efficiently for large incremental steps by using the analytical solution in Eq. (9.15) under u ¼ const: during an incremental step. The former is used for the forward-Euler method premising on the infinitesimal incremental steps and the latter is used in the return-mapping method for large incremental steps. On the other hand, R is calculated by solving Eq. (11.9), i.e. the following   equation in the elastic loading process ee 6¼ O; ep ¼ O. _

f ðr þ R^cÞ ¼ RFðHÞ

ð11:50Þ

by substituting the updated values of r; a; c; F. Equation (11.50) is expressed by the quadratic equation and thus the normal-yield ratio can be expressed analytically for the Mises material as will be shown in Chap. 12. For soil plasticity, however, Eq. (11.50) is expressed by higher-order nonlinear equation and thus the normal-yield ratio must be calculated numerically for soils as will be described in Chap. 13.

11.5

11.6

Calculation of Normal-Yield Ratio in Unloading Process

305

Improvement of Inverse and Reloading Responses

The unique relation ep  ep0 ¼ f ðR  R0 Þ holds as shown in Eq. (9.15) under the initial condition ep ¼ ep0 : R ¼ R0 in the monotonic loading process if U in Eq. (9.9) is the function of only the normal-yield ratio R. Therefore, ep induced during a certain change of R in the monotonic loading process is identical irrespective of the difference of loading processes, e.g. the initial loading, the reloading and the inverse loading and of the proportional and non-proportional loadings. This property causes the description that the returning of the reloading stress–strain curve to the previous loading curve is unrealistically gentle as shown in Fig. 11.4a. Therefore, it engenders the impertinent prediction of cyclic loading behavior, i.e. the prediction of the unrealistically large plastic strain accumulation during the

Fig. 11.4 The defect of past subloading surface model: Unrealistically gentle returning to preceding loading curve

306

11

Extended Subloading Surface Model

cyclic loading process as shown in the upper part of Fig. 11.4b. This insufficiency in the past formulation of the subloading surface model was criticized intensely by Dafalias, stating “the predictions reported in Hashiguchi (1980) for the uniaxial loading of metals were quite unrealistic, basically due to the strong undershooting phenomenon” (Dafalias 1986, p. 980). However, the insufficiency in the description of deformation behavior by the past formulation of the subloading surface model would not originate from the intrinsic nature of this model contrary to the criticism by Dafalias (1986), although the identical defect is involved in the Chaboche model as the unavoidable (intrinclic) nature as described in Sect. 10.3.2. In what follows, the past formulation of subloading surface model will be modified so as to remedy the insufficiency in the description of reloading behavior. First, note the following facts: (1) The difference between the curvatures in the loading and the inverse loading curves becomes larger as the plastic deformation proceeds continuously. This fact is known as the Masing rule (Masing 1926). (2) The similarity-center corresponding to the most elastic stress state approaches the normal-yield surface as the plastic deformation proceeds continuously, and the approaching degree of the similarity-center to the normal-yield surface is expressed by the elastic-core yield ratio ℜ c ( = f (cˆ ) / F ) in Eq. (11.14) as described in Sect. 11.2. (3) The transition from the elastic to plastic state is more abrupt, i.e. the curvature of stress–strain curve is greater for a larger value of the material parameter u in the function UðRÞ in Eq. (9.14), as described in Sect. 9.2. The faster and slower rising up of the stress vs. strain curve can be described by setting a larger value and a smaller value, respectively, to the material parameter u. (4) The larger difference between the values of u in the reloading and the inverse loading states can be described by the larger value of ℜ c. ^c (5) The direction n of plastic strain rate is close to that of the outward-normal n (Eq. 11.24) of the elastic-core surface in the reloading process but it is far from ^c in the unloading process. The degree how near the or even opposite to that of n reloading process can be expressed by the following scalar product of these unit tensors: ^c : n Cn  n

ð1  Cn  1Þ

ð11:51Þ

^c involved in the variable C n are shown in where the unit normal tensors n and n Fig. 11.5. Eventually, let the material parameter u in the function U(R) for the evolution rule of the normal-yield ratio R in Eq. (9.14) be extended to the following equation for the extended subloading surface model in which the elastic-core, i.e. the similarity-center c moves with the plastic deformation, introducing the variables ℜ c and Cn .

11.6

Improvement of Inverse and Reloading Responses

n

nˆ c

307

Normal-yield surface

ˆ )  F (H ) f (σ σ

c α α

σ σ

Subloading surface f (σ )  RF ( H ) Elastic-core surface f (cˆ ) / F ( H ) c

 ij

^c : n ð1  Cn  1Þ for improvement of reloading behavior Fig. 11.5 Scalar variable Cn  n

⎧u exp(uc χ ) (largest) for ℜ c = χ and Cn = 1 ⎪ u → u exp(ucℜ c Cn ) = ⎨ (medium) for ℜ c = 0 or Cn = 0 u ⎪u exp(−u χ ) (smallest) for ℜ = χ and C = −1 n c c ⎩

ð11:52Þ

leading to the replacement 〈 R − Re 〉 ) → U ( R, ℜc , Cn ) = u exp(ucℜc Cn ) cot(π2 〈 R1 −−RRee〉 ) U ( R) = u cot(π 2 1 − Re

ð11:53Þ 

where uc is the material constant. Then, the plastic positive proportionality factor k is re-placed to the following equation by Eq. (11.53), noting Eq. (9.9) with Eq. (9.14).

ð11:54Þ By the above-mentioned modification, the phenomenon that the reloading curve after a partial unloading returns rapidly to the preceding loading curve and the curvature of inverse loading curve decreases can be described realistically as shown in Fig. 11.6a. Besides, the excessive plastic strain accumulation for the cyclic loading process in the neighborhood of yield surface is suppressed as shown in the lower part of Fig. 11.6b. The above-mentioned improvement of the reloading behavior is inevitably required for the description of the accurate cyclic loading behavior. On the other hand, the improvement of the cyclic loading behavior can never be attained by the modification of the dynamic recovery part of the Armstrong-Frederick kinematic hardening rule as has been repeated after Chaboche et al. (1979), since it is relevant to the reverse loading behavior but irrelevant to the reloading behavior.

308

11

uc 0



Extended Subloading Surface Model

R =1

uc = 0

uc 0

c

uc = 0

c

uc 0

0 uc = 0 uc 0

uc = 0

c

p

R =1

(a) Reloading after partial-unloading and reverse loading



R =1

Elastic-core

0

p

(b) Cyclic loading under small stress amplitude near normal-yield state Fig. 11.6 Stress-plastic strain curve predicted by modified evolution rule of normal-yield ratio: Rapid recovery to preceding monotonic loading curve

11.7

Loading Criterion for Large Loading Increment

The exact loading criterion will be given in this section, which is required for the numerical calculation with large loading increments involving not only the monotonic but also the inverse loading.

11.7

Loading Criterion for Large Loading Increment

11.7.1

309

Exact Judgment of Loading

The following loading criterion for the return-mapping method was incorporated by Hashiguchi (2013, 2017a) and has been used for the FEM analyses by Hashiguchi (2013), Anjiki et al. (2016), Yamakawa et al. (2010a, b), Iguchi (2019), Fincato and Tsutsumi (2017, 2018, 2019a, b, 2020a, b) and Tsutsumi et al. (2019). (

trial trial trial Depn þ 1 ¼ O; rFinal n þ 1 ¼ rn þ 1 for f ðrn þ 1  an Þ  Rn FðHn Þ  0 or f ðrn þ 1  an Þ  Re FðHn Þ  0 trial Depn þ 1 6¼ O; rFinal n þ 1 6¼ rn þ 1 for other

ð11:55Þ which persists that the plastic strain rate is induced only when the subloading surface just after the elastic trial step is larger than subloading surface at the beginning of the elastic trial step, i.e. the previous step. This standard loading criterion gives the correct judgment for conventional plasticity models. However, when this loading criterion is applied to the subloading surface model, it may give incorrect judgment due to the following facts. (1) The subloading surface in the elastic-trial step cannot be described by f ðrtrial nþ1  trial trial trial an Þ  Rn FðHn Þ ¼ 0 but can be described by f ðrn þ 1  an þ 1 Þ  Rn þ 1 FðHn Þ ¼ trial 0 with atrial cn exactly, noting that only the internal variables Hn , n þ 1 ¼ cn  Rn þ 1^ an and cn are fixed in the elastic trial step. (2) Any elastic trial steps inducing the shrink of the subloading surface are judged as the elastic loading process by the past (incorrect) loading criterion. In fact, however, the plastic strain rate must be induced when once the subloading surface contracts but then it expands over the elastic domain in the elastic trial step even if the subloading surface after the elastic trial step is smaller than the subloading surface in the beginning of the elastic trial step. Needless to say, such incorrect loading judgment would result in the accumulation of plastic strain rate during the cyclic loading for a constant or a decreasing amplitude cannot be described at all. Then, the exact loading criterion will be formulated in the following. The following facts should be noticed for the formulation of the loading criterion required at the beginning of the plastic corrector step, referring to Fig. 11.7, where the elastic trial step for the initial subloading surface model ðc ¼ aÞ and for the extended subloading surface model ðc 6¼ aÞ are shown in Fig. 11.7a and b, respectively (The subscript n and n þ 1 is added to the variables at the end of the step n and the step n þ 1, respectively). “The plastic strain increment Depn þ 1 is not induced if the elastic trial stress rtrial nþ1 stays inside the elastic response region, i.e. f ðrtrial n þ 1 Þ  Re FðHn Þ\0 in the step n or trial if the stress increment Drtrial n þ 1 makes an obtuse angle with the outward-normal nn þ 1 of the subloading surface in the elastic trial step n þ 1. Otherwise, it is induced.”

310

11

Extended Subloading Surface Model

Fig. 11.7 Correct loading criterion when elastic trial stress increment is directed inward of subloading surface at step n in return-mapping method for subloading surface model

11.7

Loading Criterion for Large Loading Increment

311

Then, the correct loading criterion in the return-mapping method for the subloading surface model is given as follows: Loading criterion in returnmapping for subloading surface model trial trial trial trial trial Depn þ 1 ¼ O and rFinal n þ 1 ¼ rn þ 1 for Rn þ 1  Rn or Rn þ 1  Re or nn þ 1 :Drn þ 1  0 p Final trial Den þ 1 6¼ O and rn þ 1 6¼ rn þ 1 for others ð11:56Þ where trial etrial rtrial n þ 1 ¼ rn þ Drn þ 1 ¼ rn þ E :Den þ 1

ð11:57Þ

  @f ðrtrial n þ 1 Þ @f ðrtrial Þ nþ1  ntrial   nþ1 @rtrial  @rtrial  nþ1 nþ1

ð11:58Þ

trial trial trial trial rtrial cn Þ n þ 1 ¼ rn þ 1  a n þ 1 ¼ rn þ 1  ðcn  Rn þ 1^

ð11:59Þ

where

noting Eq. (11.5). Rtrial n þ 1 is calculated from the following equation of the subloading surface in the elastic trial step. trial f ðrtrial n þ 1 Þ ¼ Rn þ 1 FðHn Þ

ð11:60Þ

Equation (11.60) is the nonlinear equation of the normal-yield ratio Rtrial n þ 1 and thus must be calculated by the numerical method, e.g. the Newton–Raphson Rtrial nþ1 method in general. However, the function f ðrtrial n þ 1 Þ is given for the Mises material by f ðrtrial n þ 1Þ ¼

pffiffiffiffiffiffiffiffi trial pffiffiffiffiffiffiffiffi  trial' trial ' 3=2jjrn þ'1  a n þ 1 k ¼ 3=2rtrial cn Þ'  n þ 1  ðcn  Rn þ 1^

ð11:61Þ

Rtrial n þ 1 is calculated analytically by solving the quadratic equation derived by substituting Eq. (11.61) into Eq. (11.60) as follows:

Rtrial nþ1 ¼

' ' ðrtrial c'n þ n þ 1  cn Þ: ^

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2ffi ' ' ' '  ½ðrtrial c'n 2 þ ð23 ½FðHn Þ2  ^c'n  Þrtrial n þ 1  cn Þ: ^ n þ 1  cn  ' 2 2 2 ^  3 ½FðHn Þ  cn

ð11:62Þ The detailed derivation of Eq. (11.62) can be referred to Sect. 12.3. Further, ntrial nþ1 is given for the Mises material as follows:

312

11

Extended Subloading Surface Model

trial ' ' ' rtrial cn Þ n þ 1  ðcn  Rn þ 1^   ntrial  nþ1 trial trial ' ' r  ðc'  R ^c' Þ nþ1

11.7.2

n

ð11:63Þ

nþ1 n

Initial Value of Normal-Yield Ratio in Plastic Corrector Step

As known from the correct loading criterion described in the latest section, the initial value of normal-yield ratio must be determined at the beginning of the plastic-corrector step. It has been formulated and derived by Hashiguchi (2018, 2020) as explained below. Noting the loading criterion described in the foregoing, the plastic corrector step must be started from (1) the subloading surface coinciding with the elastic domain surface, when the elastic-trial stress increment intersects with the elastic domain surface, for which the normal-yield ratio is given by Re . (2) the subloading surface in the transitional state at which the subloading surface once contracts and then begins to expand in the elastic-trial step, while the elastic-trial stress increment Drtrial n þ 1 contacts tangentially to the subloading surface, for which the normal-yield ratio is designated as R0n þ 1 . Then, designating the stress at which the stress increment vector Drtrial n þ 1 becomes tangential to the subloading surface which changes from the contraction to the expansion by r0n þ 1 , the following relations must be satisfied at the contact point, i.e. the loading-start stress r0n þ 1 . f ðr0n þ 1 Þ ¼ R0n þ 1 FðHn Þ n0n þ 1 : Drtrial nþ1 ¼ 0

ð11:64Þ

The first equation is required for the subloading surface to pass through the contact point r0n þ 1 and the second equation is required for the elastic-trial increment Drtrial nþ1 to contact tangentially to the subloading surface at the contraction–expansion transition. r0n þ 1  sDrtrial n þ 1 þ rn ð0  s  1Þ

ð11:65Þ

a0n þ 1 ¼ cn  R0n þ 1^cn

ð11:66Þ _

0 r0n þ 1  r0n þ 1  a0n þ 1 ¼ sDrtrial cn n þ 1 þ rn þ Rn þ 1^

ð11:67Þ

11.7

Loading Criterion for Large Loading Increment

n0n þ 1

@f ðr0n þ 1 Þ  @r0n þ 1

313

,  @f ðr0n þ 1 Þ    @r0  nþ1

ð11:68Þ

sð0  s  1Þ is the unknown scalar parameter which must be determined so as to satisfy Eq. (11.64) at the stress r0n þ 1 . The two unknown variables s and R0n þ 1 are calculated by solving Eq. (11.64) and then all the variables r0n þ 1 , a0n þ 1 , r0n þ 1 in the subloading surface passing through the contact point and the stress increment Drtrial n þ 1 at the elastic trial step are calculated. Equation (11.64) is regarded as the simultaneous quadratic equation of the unknown scalar variables R0n þ 1 and s so that the numerical calculation is required for their solutions in general. In what follows, the analytical equation of R0n þ 1 in terms of the known variables will be derived for the Mises material. Equation (11.64) is explicitly described for the Mises material with f ðr0n' þ 1 Þ ¼ pffiffiffiffiffiffiffiffi 0    3=2rn' þ 1  leading to n0n þ 1 ¼ r0n' þ 1 =r0n' þ 1  as follows: pffiffiffiffiffiffiffiffi 0  3=2rn' þ 1  ¼ R0n þ 1 FðHn Þ 0' rn þ 1 :Drtrial nþ1 ¼ 0

ð11:69Þ

i.e. 8 pffiffiffiffiffiffiffiffi  _' 0 '  ' þr < 3=2 ^ þ R c sDrtrial  ¼ R0n þ 1 FðHn Þ n n n þ 1 nþ1 :

_

0 ' ' ðsDrtrial c'n Þ:Drtrial nþ1 ¼ 0 n þ 1 þ rn þ Rn þ 1^

ð11:70Þ

The upper equation in Eq. (11.70) is expressed as _

_

0 0 ' ' ' ' ð3=2ÞðsDrtrial c'n Þ:ðsDrtrial c'n Þ n þ 1 þ rn þ Rn þ 1^ n þ 1 þ rn þ Rn þ 1^

¼ ðR0n þ 1 FðHn ÞÞ2 leading to _

trial' trial' 0 ' ' c'n Þ s2 Drtrial n þ 1 : Drn þ 1 þ 2sDrn þ 1 : ðrn þ Rn þ 1^ _

_

þ ðrn' þ R0n þ 1^c'n Þ : ðrn' þ R0n þ 1^c'n Þ  ð2=3ÞðR0n þ 1 FðHn ÞÞ2 ¼ 0 from which we have _

0 ' ' c'n Þ þ s ¼ ½Drtrial n þ 1 :ðrn þ Rn þ 1^ _

_' pffi trial' 0 ' ' f½Drtrial c'n Þ2  Drtrial n þ 1 :ðrn þ Rn þ 1^ n þ 1 :Drn þ 1

_

trial' ' ½ðrn' þ R0n þ 1^c'n Þ:ðrn' þ R0n þ 1^c'n Þ  ð2=3ÞðR0n þ 1 FðHn ÞÞ2 g=ðDrtrial n þ 1 :Drn þ 1 Þ

ð11:71Þ

314

11

Extended Subloading Surface Model

The substitution of Eq. (11.71) into the second equation in Eq. (11.70)2 leads to _'

0 ' Drtrial c'n Þ þ 1 :ðrn þ Rn þ 1^ qnffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ' _ _' trial' _' 0 0 ' ' þ ½Drtrial c'n Þ2  Drtrial c'n Þ:ðrn þ R0n þ 1^c'n Þ  ð2=3ÞðR0n þ 1 FðHn ÞÞ2  n þ 1 :ðrn þ Rn þ 1^ n þ 1 :Drn þ 1 ½ðrn þ Rn þ 1^ ' _

0 ' þ Drtrial c'n Þ ¼ 0 n þ 1 :ðrn þ Rn þ 1^

i.e. _

0 ' ' c'n Þ2 ½Drtrial n þ 1 :ðrn þ Rn þ 1^ _

_

trial' 0 ' '  Drtrial c'n Þ:ðrn' þ R0n þ 1^c'n Þ  ð2=3ÞðR0n þ 1 FðHn ÞÞ2  ¼ 0 n þ 1 :Drn þ 1 ½ðrn þ Rn þ 1^

ð11:72Þ which is the quadratic equation of R0n þ 1 . Equation (11.72) is rewritten as trial' 02 ' ' ' c' Þ2  ðDrtrial' :Drtrial' Þð^ ' cn Þ þ ð2=3ÞðFðHn ÞÞ2 ðDrtrial ½ðDrtrial n n þ 1 :^ nþ1 n þ 1 cn :^ n þ 1 :Drn þ 1 ÞRn þ 1 _

_

trial' ' trial' ' ' c' ÞR0 ' ' þ 2½ðDrtrial cn Þ  ðDrtrial n nþ1 n þ 1 :rn ÞðDrn þ 1 :^ n þ 1 :Drn þ 1 Þðrn :^ _

_

_

trial' trial' ' 2 ' ' ' þ ðDrtrial n þ 1 :rn Þ  ðDrn þ 1 :Drn þ 1 Þðrn :rn Þ ¼ 0

The solution of R0n þ 1 in Eq. (11.72) is given by R0n þ 1

¼

B 

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B2  AC A

ð11:73Þ

where 8 > A  S2ca  Sss ½^c'n :^c'n  ð2=3ÞðFðHn ÞÞ2  > < _ B  Ssc Sca  Sss rn' :^c'n > > _ _ : C  S2sc  Sss rn' : rn'

ð11:74Þ

with trial' trial' ' ' Sss  Drtrial cn ; n þ 1 :Drn þ 1 ; Sca  Drn þ 1 :^

_

' ' Ssc  Drtrial n þ 1 :rn

ð11:75Þ

One must choose the solution satisfying 0  R0n þ 1  1 in Eq. (11.73). Here, we must set R0n þ 1 ¼ Re if R0n þ 1  Re in the calculated result. It is enough to calculate the initial normal-yield ratio R0n þ 1 , while it is not necessary to calculate the scalar number s and the stress r0n þ 1 for the return-mapping calculation. The correct loading criterion for the return-mapping method in the initial subloading surface model is given by setting c ¼ a in the above-mentioned formulations, referring to Fig. 11.7a.

11.8

11.8

Plastic Spin

315

Plastic Spin

The plastic spin in Eq. (3.44) for the hypoelastic-based plasticity is given by extending Eq. (8.94) to the extended subloading surface model as follows: 

wp ¼ gp ant½r dp  ¼ gp k xp ;

11.9

xp ¼ ant½r n

ð11:76Þ

Incorporation of Tangential-Inelastic Strain Rate

The tangential-inelastic strain rate described in Sect. 9.7 is extended for the extended subloading surface model as shown in this section. '

The tangential-deviatoric stress rt (Fig. 11.8) is defined by ð11:77Þ

ð11:78Þ ð11:79Þ fulfilling ð11:80Þ

by virtue of

'

The fourth-order tensor Tt is the tangential-deviatoric projection tensor which transforms an arbitrary second-order tensor to the tangential part to the subloading surface in the deviatoric stress space and the second-order tensor subjected to this ' ' projection is designated as ð Þ't , i.e. t't  Tt : t leading further to Tt : t't ¼ t't .

316

11

Extended Subloading Surface Model

n'

σ '



σ 'n

σ' σ'



σ'

σ 't

ε t =

T (R )  2G σ't

c'

α' α'

Subloading surface f (σ) = RF ( H )

 'ij

0

Normal-yield surface f (σˆ ) = F ( H )

Fig. 11.8 Tangential-deviatoric stress rate and tangential-inelastic strain rate in the deviatoric stress plane

All the relations in Eqs. (9.53), (9.62) and (9.63) hold under the replacements of p ð Þ't ! ð Þ't ; M p ! M . Consequently, it follows that t

  n n TðRÞ '  :r E1 þ þ T p 2G t M

ð11:82Þ



r¼



ð11:81Þ

e¼ 

'

e ¼ TðRÞE1 : rt ¼ TðRÞE1 : T't : r



 E:n n:E TðRÞ  '  2G E p Tt : e 1 þ TðRÞ M þn : E : n

ð11:83Þ

The loading criterion in Eq. (11.49) which was given for the equation without the tangential-inelastic strain rate holds as it is, as was described in Sect. 9.7. The importance for the introduction of the tangential-inelastic strain rate will be verified by comparison with test data on a metal in Sect. 12.5. The subloading surface model has been applied to metals (Hashiguchi 1980; 1989; Hashiguchi and Yoshimaru 1995; Hashiguchi and Tsutsumi 2001; Hashiguchi and Protasov 2004; Khojastehpor et al. 2006; Tsutsumi et al. 2001, 2003, 2005, 2007; Hashiguchi et al. 2012; Fincato and Tsutsumi 2017, 2018, 2019, 2020a, b; 2021; Anjiki et al. 2020; Anjiki and Hashiguchi 2021, Liu et al. 2022) and soils (Hashiguchi and Ueno 1977; Hashiguchi 1978; Topolnicki 1990; Kohgo et al. 1993; Asaoka et al. 1997; Hashiguchi and Chen 1998; Chowdhury et al. 1999;

11.9

Incorporation of Tangential-Inelastic Strain Rate

317

Hashiguchi et al. 2002; Khojastehpor and Hashiguchi 2004a, b; Khojastehpor et al. 2006; Nakai and Hinokio 2004; Hashiguchi and Tsutsumi 2001, 2003, 2005, 2007; Hashiguchi and Mase 2007, 2011; Wongsaroj et al. 2007; Yuanming et al. 2009, 2016; Maranha et al. 2016; Hashiguchi et al. 2021; Yamada et al. 2022; Lu et al. 2022), rocks (Xiong et al. 2018), asphalt (Darabai et al. 2012), etc. Consequently, its capability has been verified widely. The explicit constitutive equations for metals and soils will be described in the subsequent chapters.

Chapter 12

Constitutive Equations of Metals

The plasticity theory has been highly developed through the prediction of deformation of metals up to date. The reason for this would lie in the fact that, among various materials exhibiting plastic deformation, metals are used most widely as engineering materials and exhibit the simplest plastic deformation behavior without a pressure dependence, a plastic compressibility, a dependence on the third invariant of deviatoric stress and a softening. Nevertheless, metals exhibit various particular aspects, e.g., the kinematic hardening and the stagnation of isotropic hardening in a cyclic loading. Explicit constitutive equations of metals will be delineated in this chapter, which are based on the extended subloading surface model described in the preceding chapters. The constitutive equation detailed in this chapter has been implemented to the commercial FEM software “Marc” marketed from MSC Software Ltd. as the standard uploaded (ready-made) program by the name “Hashiguchi model” after the 2017 version, so that Marc users can apply these models to their deformation analyses. Further, the performance that the material parameters can be determined automatically by inputting the stress–strain curve is installed after the 2019 version, so that even the users unfamiliar to the elastoplasticity theory can execute an accurate analysis by the subloading surface model. Incidentally, the computer program is shown in Appendix L(a).

12.1

Yield Surface, Isotropic, Kinematic Hardening and Elastic-Core

The yield function for the Mises yield condition is extended to incorporate kine^0 in Eq. (8.41) as follows: matic hardening by replacing r0 to r rffiffiffi 3 0 ^k f ð^ rÞ ¼ kr 2 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 K. Hashiguchi, Foundations of Elastoplasticity: Subloading Surface Model, https://doi.org/10.1007/978-3-030-93138-4_12

ð12:1Þ

319

320

12

Constitutive Equations of Metals

for which the subloading function f ðrÞ for Eq. (12.1) is given by ð12:2Þ

Further, the elastic-core function in Eq. (11.14) for Eq. (12.1) is given by rffiffiffi rffiffiffi 3 0 @f ð^cÞ 3 ^c0 ¼ f ð^cÞ ¼ k^c k ; 2 @^c 2 k^c0 k

ð12:3Þ

It follows from Eqs. (11.14) and (11.24) for Eq. (12.3) that ð12:4Þ

The isotropic hardening function is given by Eq. (8.42), i.e. FðHÞ ¼ F0 f1 þ Sr ½1  expðcH H Þg:

ð12:5Þ

F 0 ¼ F0 Sr cH expðcH H Þ

ð12:6Þ ð12:7Þ

The rates of the nonlinear kinematic hardening and the elastic-core are given by Eqs. (11.13) and (11.26) as ð12:8Þ

ð12:9Þ

The plastic modulus is given by substituting Eqs. (12.2), (12.3), (12.7), (12.8) and (12.9) into Eq. (11.45) as follows:

12.1

Yield Surface, Isotropic, Kinematic Hardening and Elastic-Core

321

(rffiffiffi   2 F0 r0 1 a r þ ck R  3F kr0 k bk F " )   pffiffiffiffiffiffiffiffi 0 # v  2=3F r0 1 U_ ^c þ r þ ð1  RÞ ce r  ^c þ ck  a þ F R R kr0 k bk F

r0 M  : kr0 k p

ð12:10Þ

12.2

Cyclic Stagnation of Isotropic Hardening

As a plastic deformation induced by the mutual slips of solid particles proceeds, the mutual slips becomes difficult due to the closer packaging of solid particles, the accumulation of obstacles, etc. resulting in the isotropic hardening. However, if the loading direction is reversed, the solid particles are released from the close packaging, the obstacles, etc. so that the mutual slips recover so that the isotropic hardening stagnates temporarily. This phenomenon considerably affects the cyclic loading behavior in which the reverse loading is repeated. To describe this phenomenon, the concept of the cyclic stagnation of isotropic hardening, i.e. the existence of nonhardening region was proposed by Chaboche et al. (1979; see also Chaboche, 1989) and studied also by Ohno’s group (Ohno 1982; Ohno and Kachi 1986; Ohno et al. 2021). The concept insists that isotropic hardening does not proceed when the plastic strain given by the time-integration of plastic strain rate lies inside a certain region, called the nonhardening region, in the plastic strain space. The non-hardening region expands and translates when the plastic strain lies on the boundary of the region and the plastic strain rate is induced directing outwards the region. It is similar to the notion of the yield surface based on the assumption that the plastic strain rate is induced only when the stress lies on that surface, while the plastic strain and the rate of isotropic hardening variable for the nonhardening region correspond to the stress and the plastic strain rate, respectively, for the yield surface. The extended formulation for the isotropic hardening stagnation by the notion of the subloading surface concept will be shown in this section (Hashiguchi 2009; 2013a). Assuming that the isotropic hardening stagnates when the plastic strain ep lies inside a certain region, let the following surface, called the normal-isotropic hardening surface, be introduced. ~ gð~ep Þ ¼ K

ð12:11Þ

322

12

n~ p

p

ρ ε~ p

Constitutive Equations of Metals

Normal-isotropic hardening surface g (ε~ p ) = K~ Sub-isotropic hardening surface ~ ~ g (ε~ p ) = RK

ρ p

0

ij

Fig. 12.1 Normal- and sub-isotropic hardening surfaces

where ~ep  ep  q

ð12:12Þ

~ and q designate the size and the center, respectively, of the normal-isotropic K hardening surface, the evolution rules of which will be formulated later. Furthermore, we introduce the surface, called the subloading-isotropic hardening surface, which always passes through the current plastic strain ep and which has a similar shape and a same orientation to the normal-isotropic hardening surface (see Fig. 12.1). Here, the plastic strain ep is regarded as an internal variable. The same situation can be reminded in the Prager’s lineal kinematic hardening rule in Eq. (8.86). The subloading-isotropic stagnation surface is expressed by the following equation. ~K ~ gð~ep Þ ¼ R

ð12:13Þ

  ~ 0R ~  1 is the ratio of the size of subloading-isotropic hardening surwhere R face to that of the normal-isotropic hardening surface. It plays the role as the measure for the approaching degree of the plastic strain to the normal-isotropic ~ is referred to as the normal-isotropic hardening ratio. It hardening surface. Then, R ~ in terms of the known values of ep , q ~ ¼ gð~ep Þ=K is calculable from the equation R ~ and K.

12.2

Cyclic Stagnation of Isotropic Hardening

323

The consistency condition of the subloading isotropic hardening surface is given by  @gð~e p Þ  p @gð~e p Þ  ~ ~ ~ ~ K: :e  : q ¼ RK þ R p p @~e @~e

ð12:14Þ

Let the following postulates be adopted for the formulations of the evolution ~ and q. rules of K ~ and q evolve when the plastic strain rate e p is induced directing outwards the (1) K subloading-isotropic hardening surface, fulfilling @gð~ep Þ  p : e [ 0: @~ep

ð12:15Þ

~ and q increase as the plastic strain approaches the (2) The rates of K normal-isotropic hardening surface, i.e. as the normal-isotropic hardening ratio ~ increases. Therefore, they are monotonic-increasing function of the K ~ normal-isotropic hardening ratio R. p (3) The plastic strain e is assumed to exist inside the normal-isotropic hardening surface. Therefore, it must hold that 

~¼1 ~ ¼ 0 for R R

ð12:16Þ

(4) The consistency condition in Eq. (12.14) is reduced to the following relation which must be fulfilled when the plastic strain just lies on the normal-isotropic hardening surface.  @gð~ep Þ  p @gð~ep Þ  ~ : e  : q ¼ K @~ep @~ep

for

~¼1 R

ð12:17Þ

Then, we assume the following equations so as to fulfill all these postulates. ð12:18Þ ð12:19Þ where 0  C  1 and fð  1Þ are the material constants and ð12:20Þ

324

12

Constitutive Equations of Metals

~

R

1 ~ K

g (ε~ p ) ~

εp

: εp

n~ : ε p 0 0

1

ε p O R~

Fig. 12.2 Evolution of normal-isotropic hardening ratio: Plastic strain is attracted to the normal-isotropic hardening surface

~ and q into Substituting Eqs. (12.18) and (12.19) for the evolution rules of K Eq. (12.14), the rate of the normal-isotropic hardening ratio is given by

ð12:21Þ

~ fulfilling which is the monotonically-decreasing function of R 8 > 1 @gð~ep Þ  p > > ~¼0 h ¼ : e ið[ 0Þ for R > > ~ > p K > @~ e > > >  < p ~ \ 1 h@gð~e Þ : e p ið[ 0Þ for R\1 ~ R > ~ p > K > @~ e > > > ~¼1 > ¼ 0 for R > > > : ~[1 \0 for R

ð12:22Þ

12.2

Cyclic Stagnation of Isotropic Hardening

325

~ increases as shown in Fig. 12.2. Therefore, the normal-isotropic hardening ratio R when the plastic strain moves to the outward of the sub-isotropic hardening surface but it decreases such that the normal-isotropic hardening surface encloses the plastic strain when the plastic strain tends to go out from the normal-isotropic hardening 

~ [ 1 as shown in ~ \0 for R surface encloses by virtue of the inequality R Eq. (12.22). In other words, the normal-isotropic hardening surface is automatically controlled such that the plastic strain does not go out from that surface. Furthermore, needless to say, the judgment of whether the plastic strain reaches the normal-isotropic hardening surface is not necessary in the present formulation consistently based on the subloading concept in all aspects. In contrast, the judgment whether the plastic strain or the back-stress reaches the isotropic hardening (stagnation) surface is required in the other models (Chaboche et al., 1979; Chaboche, 1991; Ohno, 1982; Yoshida and Uemori, 2002b). It is assumed that the isotropic hardening variable H evolves under the following conditions. 

(1) The isotropic hardening is induced when the plastic strain rate ep is induced directing outwards the sub-isotropic hardening surface, i.e. 

H

n



~ : ep [ 0 [ 0 for n ¼ 0 for others:

ð12:23Þ

(2) The isotropic hardening rate increases as the plastic strain approaches the normal-isotropic hardening surface, i.e. as the normal-isotropic hardening ratio  ~ increases. Then, H is the monotonically-increasing function of R. ~ Here, in R order that the isotropic hardening develops continuously, its rate must be zero,  ~ ¼ 0, i.e. when the plastic strain lies just on the center of the i.e. H ¼ 0 for R normal-isotropic hardening surface because the rate is zero during the process in which the plastic strain moves towards the inside of the sub-isotropic stagnation surface. (3) The isotropic hardening rule of Eq. (12.7) in the monotonic loading process holds ~ ¼ 1Þ and when the plastic strain lies on the normal-isotropic hardening surface ðR the plastic strain rate is induced in the outward-direction of that surface. Eventually, let the following evolution rule of isotropic hardening be assumed by extending Eq. (12.7). ð12:24Þ

326

12

Constitutive Equations of Metals

where t ð = 1Þ is the material constant and ~fHn  R ~ t h~ ifHn n:n

ð12:25Þ

The three material constants C, 1 and t in the isotropic hardening stagnation can be fixed as C = 0.5, 1 = 3 and t = 3 actually. Employing the extended isotropic hardening rule in Eqs. (12.24) instead of Eqs. (11.12) into Eq. (11.45), the plastic modulus is modified as follows: (

ð12:26Þ

in general and (

ð12:27Þ

for metals with Eq. (12.7). The function gð~ep Þ is given in the simplest form as follows: gð~ep Þ ¼ jj~ep jj;

ð12:28Þ

~ep ~ep @gð~ep Þ ~ ; n ¼ ¼ jj~ep jj jj~ep jj @~ep

ð12:29Þ

The Yoshida’s group (Yoshida and Uemori, 2002a, b; Yoshida and Amaishi, 2020) proposed the isotropic hardening stagnation by incorporating the Armstrong-Frederick (1966)’s nonlinear kinematic hardening variable (back-stress) instead of the plastic strain. However, it is to be physical worsening and the mathematical complication, since the physical mechanism of the isotropic hardening is different from that of the kinematic hardening.

12.3

Calculation of Normal-Yield Ratio in Unloading Process

The normal-yield ratio R must be calculated from the equation of the subloading  surface in the unloading process (ep ¼ O). It can be calculated directly by R ¼ f ð^ rÞ=F in the initial subloading surface model. However, it has to be calculated by solving the equation of the subloading surface in the extended subloading surface model as described below. Substituting Eq. (11.6) into Eq. (12.2), the extended subloading surface is described as follows:

12.3

Calculation of Normal-Yield Ratio Unloading Process

327

rffiffiffi 3 _0 jjr þ R^c0 jj ¼ RFðHÞ 2

ð12:30Þ

2 _ trðr0 þ R^c0 Þ2 ¼ R2 F 2 3

ð12:31Þ

i.e.

The normal-yield ratio R is derived from the quadratic Eq. (12.31) as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2 2 _ _ 2 2 0 0 0 0 0 F  jj^c jj jjr0 jj2 r : ^c þ ðr : ^c Þ þ 3 R¼ 2 2 F  jj^c0 jj2 3 _

12.4

ð12:32Þ

Implicit Stress-Integration

The implicit stress-integration method for the extended subloading surface model without the isotropic hardening stagnation and the tangential-inelastic strain rate was given by Anjiki et al. (2016, 2019), while the generalized loading criterion (Hashiguchi, 2018a) described in Sect. 11.7 was incorporated in Anjiki et al. (2019) (see Hashiguchi (2020) in detail). Further, it was given by Iguchi et al. (2019) and Ttutsumi et al. (2019). However, the tangential-inelastic strain rate is ignored and the inefficient calculation by the double loop is used in the former. The isotropic hardening stagnation is ignored and the inefficient return-mapping by the cutting-plane projection is adopted in the latter. The formulation of the complete implicit calculation method by the closed point projection for the extended subloading surface model incorporating all the functions including the isotropic hardening stagnation and the tangential-inelastic strain rate is required. The simultaneous equation for the evolution rules of the variables to this aim is given in the following (Hashiguchi, 2021a).

328

12

Constitutive Equations of Metals

ð12:33Þ

where eetnal n þ 1 is the elastic strain in the elastic trial step and Fn þ 1 ¼ F0 ½1 þ Sr f1  expðcH Hn þ 1 Þg n þ 1 ¼ n

0n þ 1 r jj r0n þ 1 jj

_

0n þ 1 ¼ r0n þ 1 þ Rn þ 1^c0n þ 1 ; r

n þ 1 D epn þ 1 ¼ epn þ n kn þ 1

ð12:34Þ ð12:35Þ ð12:36Þ

ð12:37Þ

12.4

Implicit Stress-Integration

329

Tn þ 1 ¼ ~cf½1  expðRn þ 1 Þge

ð12:38Þ

~ep ~n þ 1 ¼ np þ 1 ; ~epn þ 1 ¼ epn þ 1  qn þ 1 n ~e

ð12:39Þ

nþ1

ð12:40Þ ð12:41Þ ^c n þ 1 : n n þ 1 Cn n þ 1  n

ð12:42Þ

u n +1 = u exp(ucℜ cn +1 Cn n +1)

ð12:43Þ

~ n þ 1, R ~ n þ 1 and The eight unknown variables rn þ 1 , H n þ 1 , an þ 1 , cn þ 1 , qn þ 1 , K Dkn þ 1 are involved in the simultaneous Eq. (12.33) composed of the eight equations, where en þ 1 and en are the known input variables.

12.5

Material Parameters and Comparisons with Test Data

Material parameters will be collectively shown and the capability of the subloading surface model to describe various loading behavior will be verified by the comparisons with test data in this section.

12.5.1

Material Parameters

Material parameters are shown collectively below for three versions of the subloading surface model. (i) The initial subloading surface model, which is the improvement of the conventional elastoplasticity model only with the isotropic and the kinematic hardenings, contains the following 7 material constants and 2 initial values. Material constants: Elastic moduli: E;m ( isotropic : sr ; cH Hardening kinematic : ck ; bk Evolution of normal - yieldratio : u

330

12

Constitutive Equations of Metals

Initial values:

Normal-yield surface

size : F0 center: a0

The computer program is shown in Appendix L(a) (i). (ii) The extended subloading surface model, in which the tangential-inelastic strain rate and the isotropic hardening stagnation are ignored, contains the following 12 material constants and 3 initial values. Material constants: Elastic moduli: E;m ( isotropic : isotropic : sr ; cH Hardening kinematic : ck ; bk Evolution of normal - yield ratio : u; uc ; Re ð\1Þ Translationon of elastic-core : ce ; v ( \1) Initial values: ( Normal - yield surface

size : F 0 center : a0

Elastic-core: c0 The computer program is shown in Appendix L(a) (ii). (iii) The general subloading surface model, which possesses all the behaviors involving the tangential-inelastic strain rate and the isotropic stagnation, contains the following 16 material constants and 5 initial values at most, while the full version of computer program is shown in Appendix L(a) (iii). Material constants: Elastic moduli: E;m ( isotropic : sr ; cH Hardening kinematic : ck ; bk Evolution of normal - yield ratio : u; uc ; Re ð\1Þ Translationon of elastic - core : ce ; v ( \1) Tangential inelasticity : ~c; ~ nð  1Þ Stagnation of isotropic hardening : C (0  C  1); 1 ([ 1); t

12.5

Material Parameters and Comparisons with Test Data

Initial values: Normal-yield surface

(

331

size : F 0 center : a0

Elastic-core: c0 Normal - isotropic hardening surface

(

~0 size : K center : q0

The determination of these material parameters is explained below in brief. (1) Young’s modulus E and Poisson’s ratio m are determined from the slope and the ratio of lateral to axial strains in the initial part of stress–strain curve. (2) sr , cH and F0 for the isotropic hardening and ck , bk and a0 for the kinematic hardening are determined from stress–strain curves in the initial and the inverse loadings. (3) u, uc and Re ð\1Þ for the evolution of the normal-yield ratio are determined from the stress–strain curve in the subyield state, i.e. the elastic–plastic transitional state. (4) ce , v and c0 for the elastic-core are determined from the stress–strain curves in cyclic loading. (5) ~c and ~ n for the tangential-inelastic strain rate are determined by the difference of the strain in the non-proportional loading from that in the proportional loading. ~ 0 and q0 for the isotropic-hardening stagnation are determined from (6) C, 1, t, K the stress–strain curves in cyclic loading under a constant strain amplitude. All of these material parameters except for ~c and n~ for the tangential-inelastic strain rate can be determined only by the stress–strain curves in the uniaxial loading for initial isotropic materials. One can put a0 ¼ c0 ¼ q0 ¼ O for the initial isotropy, ~ 0 ¼ ~ep0 ~ 0 by K which is assumed in all the subsequent simulations. We calculate K ~ 0 ¼ 1 by inputting a small value of ep0 and q0 ¼ O in order that the leading to R isotropic hardening rule in Eq. (12.7) without the isotropic stagnation holds in the initial loading process for all the present simulations. Needless to say, the tangential inelasticity is irrelevant to the proportional loading. The efficient and accurate determination of the material parameters in the subloading surface model for metals can be referred to Liu et al. (2022) by adopting the artificial neural network.

12.5.2

Comparisons with Test Data

The capability of the present model for describing the deformation behavior of metals is verified through comparisons with several basic test data in this section, referring to Hashiguchi and Yamakawa (2012) and Hashiguchi and Ueno (2017). Capability of

332

12

Constitutive Equations of Metals

unconventional plasticity model aimed at describing plastic strain rate induced by a rate of stress inside yield surface must be evaluated by a degree in which cyclic loading behavior can be described appropriately. Then, various cyclic loading test data in uniaxial loading are first simulated and thereafter a circular strain path test datum is simulated to verify capability for describing non-proportional loading behavior. The cyclic loading behavior under the stress amplitude to both positive and negative sides can be predicted to some extent by any models, including even the conventional plasticity model. On the other hand, the prediction of the cyclic loading behavior under the stress amplitude in positive or negative one side, i.e. the pulsating loading inducing the so-called mechanical ratcheting effect requires a high ability for the description of plastic strain rate induced by the rate of stress inside the yield surface. Furthermore, it is noteworthy that we often encounter the pulsating loading phenomena in the boundary-value problems in engineering practice, e.g. railways and gears. The comparison with the test data for the 1070 steel under the cyclic loading of axial stress between 0 and þ 830 MPa after Jiang and Zhang (2008) is depicted in Fig. 12.3, where the material parameters are selected as follows: Material constants: Elastic moduli : E ¼ 160; 000 MPa; m ¼ 0:3; ( isotropic hardening : sr ¼ 0:58; cH ¼ 170; Hardening kinematic hardening : ck ¼ 5; 000 MPa; bk ¼ 0:5; Evolution of normal - yield ratio: u ¼ 200; uc ¼ 6; Re ¼ 0:5 Translationon of elastic - core: ce ¼ 1; 000; v ¼ 0:7; Stagnation of isotropic hardening: C ¼ 0:5; 1 ¼ 5; t ¼ 0:1; Initial values: Isotropic hardening function : F0 ¼ 507 MPa: The relation of the axial stress and the axial components of back-stress and similarity-center versus the axial strain and the relation of the axial strain versus the number of cycles are depicted in Fig. 12.3a, where the axial components are designated by ðÞa . The accumulation of axial strain is simulated closely by the present model. The calculation is controlled automatically such that the stress and the back-stress are attracted to the normal-yield and the normal-isotropic hardening surfaces, respectively, as known from the variations of the normal-yield ratio R and ~ depicted in Fig. 12.3b. Accumulation of the normal-isotropic hardening ratio R axial strain is overestimated as depicted in Fig. 12.3c if the reloading behavior is not improved by setting uc ¼ 0 ignoring the Masing effect. Despite of the improvement for reloading behavior, however, hysteresis loops are simulated as

12.5

Material Parameters and Comparisons with Test Data

333

Test result (Jiang and Zhang, 2008)

Model simulation

900

3.0

800

2.5

700

a

(MPa)

a

600

(%)

2.0

sa

500

1.5

400

1.0

300 200

a

0.5

100

0.0

0 0.0

0.5

1.0

1.5

2.0

2.5

3.0 a

3.5

0

(%)

100

200

(a) 1.0

R

1.0

R

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0 0.0

300 400 500 600 Number of cycles

0.0

0.5

1.0

1.5

2.0

2.5

3.0 a

0.0

3.5 (%)

0.5

1.0

1.5

2.0

2.5

3.0 a

(b)

3.5 (%)

3.5 900

3.0

800

2.5

700 a

600

sa

(MPa) 500

a

(%)

2.0

1.5

400

1.0

300

a

200

0.5

100

0 0.0

0.0 0.5

1.0

1.5

2.0

2.5

3.0 a

(%)

3.5

0

(c)

100

200

300 400 500 Number of cycles

600

Fig. 12.3 Uniaxial cyclic loading behavior under the pulsating loading between 0 and 830 MPa of 1070 steel (Test data after Jiang and Zhang (2008)): a Test result and simulation result and simulation without stagnation of isotropic hardening, b Variations of normal-yield ratio and normal-isotropic hardening ratio and c Test result and simulation without improvement of reloading behavior

narrower than those in the test result in order to fit the strain accumulation in the test result. A further improvement is desirable for this insufficiency. Next, examine the uniaxial cyclic loading behavior under the constant stress amplitude to both positive and negative sides with different magnitudes. Comparison with the test data for the 304L steel under the cyclic loading of axial stress between þ 250 and 150 MPa after Hassan et al. (2008) is depicted in Fig. 12.4 where the material parameters are selected as shown below.

334

12

Constitutive Equations of Metals

Test result (Hassan et al., 2008) a

Model simulation

(MPa)

300

10 20

2.5

200

2.0 1.5

sa

100

1.0

a

0

0.5

1.0

1.5

2.0

2.5 a

0.5

(%)

0.0

-100

0.0

40

20

60

100

120

140

Number of cycles

(a)

-200

R

(%)

a

40 6080100cycles

1.0

1.0

R

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0

0.5

1.0

1.5

2.0 a

0.5

2.5

(%)

1.0

1.5

2.0

2.5 a

(b)

10 20 40 60 80 100cycles

a

(%)

(%)

㻞㻚㻡㻌 㻞㻚㻜㻌 㻝㻚㻡㻌

a (MPa)

㻝㻚㻜㻌 a

㻜㻚㻡㻌

(%)

㻜㻚㻜㻌 㻜

㻞㻜

㻠㻜

(c)

㻢㻜

㻤㻜

㻝㻜㻜

Number of cycles

Fig. 12.4 Uniaxial cyclic loading behavior under the constant stress amplitude between –150 and 250 MPa of 304L steel (Test data after Hassan et al. (2008)): a Test result and simulation, b Variations of normal-yield ratio and normal-isotropic hardening ratio and c Simulation by modified Chaboche model (cf. Hassan et al. (2008))

Material constants: Elastic moduli : E ¼ 200; 000 MPa; m ¼ 0:3; ( isotropic : sr ¼ 0:3; cH ¼ 30; Hardening kinematic : ck ¼ 130 MPa; bk ¼ 0:9; Evolution of normal - yield ratio : u ¼ 2; uc ¼ 10; Re ¼ 0:5 Translationon of elastic - core : ce ¼ 143; v ¼ 0:7; Stagnation of isotropic hardening : C ¼ 0:5; 1 ¼ 15; t ¼ 1;

12.5

Material Parameters and Comparisons with Test Data

335

Initial values: Isotropic hardening function: F0 ¼ 232 MPa: The relation of the axial stress and the axial components of back-stress and similarity-center vs. the axial strain and the relation of the axial strain vs. the number of cycles are depicted in Fig. 12.4a. Both the accumulation of strain and the hysteresis loops are simulated closely by the present model. The calculation is controlled automatically such that the stress and the back-stress are attracted to the normal-yield and the normal-isotropic hardening surfaces, respectively, as known from the variations of the normal-yield ratio R and the normal-isotropic hardening ~ depicted in Fig. 12.4b. The relations of the axial stress versus the axial strain ratio R and the relation of the axial strain vs. the number of cycles simulated using the modified Chaboche model (Chaboche 1991) are also depicted in Fig. 12.4c in which the strain is simulated as larger than the test result and the hysteresis loops are simulated as narrower than the test data. The prediction of this steel deformation behavior will be improved by incorporating the rate-dependence. Further, we examine the uniaxial cyclic loading behavior for constant symmetric stress amplitude to both positive and negative sides. Comparison with test data of the 304 steel under the cyclic loading of axial stress between +182 and −182 MPa under the constant hoop stress 80 MPa after Xia and Ellyin (1994) is depicted in Fig. 12.5 where the material parameters are selected as follows: Material constants: Elastic moduli : E ¼ 190; 000 MPa; m ¼ 0:3; ( isotropic : sr ¼ 1:3; cH ¼ 100; Hardening kinematic : ck ¼ 25 MPa; bk ¼ 0:3; Evolution of normal - yield ratio : u ¼ 200; uc ¼ 6; Re ¼ 0:5; Translationon of elastic - core : ce ¼ 1; 429; v ¼ 0:7; Stagnation of isotropic hardening : C ¼ 0:5; 1 ¼ 8; t ¼ 5; Initial values: Isotropic hardening function: F0 ¼ 212 MPa: The relation of the axial stress and the axial components of back-stress and similarity-center versus the axial strain and the circumferential strain el with the number of cycles are shown in Fig. 12.5a, while the back-stress is induced quite slightly so that it is invisible in this figure. The simulations for the accumulation of

336

12

Constitutive Equations of Metals Model simulation

Test result ( Xia and Ellyin, 1994 ) N=20, 15 1 to 10

a

(MPa) a



(MPa)

N=1 to 10

N=15, 20

 

















l

 





a







(%)

(a)























R



















a

N=20, 15

N=1 to 10





(%)



(MPa)

a





(MPa)

200 150 100 50 0 50 100 150 200 0.0

200 a



0.0

(c)



a

a

200

100 0.2



(b)

0

0.4 (%)

R

 

100

0.6

(%)





 







N=1 to 10

0.2





(%)

N=15, 20

0.4

0.6 l

(%)

Fig. 12.5 Uniaxial cyclic loading behavior under the constant stress amplitude between 182 and +182 MPa of 304L steel (Test data after Xia and Ellyin (1994)): a Test result and simulation, b Variations of normal-yield ratio and normal-isotropic hardening ratio and c Simulation by Xia and Ellyin (1994)

axial strain and the hysteresis loops agree well with the test result, except for the prediction of hysteresis loops as narrower than the test result in the initial stage. Here, the axial strain and the lateral strain are accumulated to the compression side and the extension side, respectively, by the application of the hoop stress 80 MPa. The calculation is automatically controlled such that the stress and the back-stress are attracted to the normal-yield and the normal-isotropic hardening surfaces, respectively, as known from the variations of the normal-yield ratio R and the ~ depicted in Fig. 12.5b. The relations of the axial normal-isotropic hardening ratio R stress vs. the axial and lateral strains and the relation with the number of cycles simulated by Xia and Ellyin (1994; cf. also Ellyin 1997) are also depicted in Fig. 12.5c where both the axial and the circumferential strains are overestimated.

12.5

Material Parameters and Comparisons with Test Data

337 Model simulation

Test result (Xia and Ellyin, 1994) a

(MPa)

(MPa)

a

400

400

sa

300

sa

300 

200

200 a

a

-3.0 -2.0

-1.0

0 0.0

1.0

2.0

3.0 a

-200

(%)

-3.0 -2.0

0 -1.0 0.0 -200

-400

(a)

1.0

1.0

0.6

0.6

0.4

0.4

0.2

0.2

0.0 -1 .5 -1 .0 -0 .5 0

a

a

(%)

(b) R

0.8

0.8

-2

3.0

-300

-400 R

-2 .5

2.0



 -300

-3 .5 -3

1.0

0 .5

1

1 .5

2

2 .5

3

3 .5

-3 .5 -3 -2 .5

-2

0.0 -1 .5 -1 .0 -0 .5 0

(%)

a

0 .5

1

1 .5

2

2 .5

3

3 .5

(%)

(c) a

(MPa)

a

a

(d)

(MPa)

(%)

a

(%)

(e)

Fig. 12.6 Uniaxial cyclic loading behavior under the constant strain amplitude the 5 levels increasing strain amplitudes of 316 steel (Test data after Chaboche et al. (1979)): a Test result and simulation by present model, b Test result and simulation without stagnation of isotropic hardening, c Variations of normal-yield ratio and normal-isotropic hardening, d Simulation by Chaboche (2008) and e Simulation by Ellyin and Xia (1989)

Furthermore, examine the uniaxial cyclic loading behavior under the constant symmetric strain amplitudes to both positive and negative sides. Comparison with the test data of the 316 steel under the cyclic loading with the increasing axial strain amplitudes 1:0; 1.5,  2.0,  2.5,  3.0% after Chaboche et al. (1979) is depicted in Fig. 12.6 where the material parameters are selected as follows:

338

12

Constitutive Equations of Metals

Material constants: Elastic moduli : E ¼ 170; 000 MPa; m ¼ 0:3; ( isotropic : sr ¼ 0:85; cH ¼ 5; Hardening kinematic : ck ¼ 2; 000 MPa; bk ¼ 0:5; Evolution of normal - yield ratio: u ¼ 100; uc ¼ 3; Re ¼ 0:5; Translationon of elastic - core : ce ¼ 283; v ¼ 0:7; Stagnation of isotropic hardening : C ¼ 0:5; 1 ¼ 5; t ¼ 1; Initial values: Isotropic hardening function : F0 ¼ 320 MPa: The relation of the axial stress and the axial components of back-stress and similarity-center versus the axial strain are shown in Fig. 12.6a. The hysteresis loops and the stagnation of isotropic hardening are simulated closely by the present model. On the other hand, the calculated result without the cyclic stagnation of isotropic hardening overestimates the hardening behavior as shown in Fig. 12.6b. The calculation is controlled automatically such that the stress and the back-stress are attracted to the normal-yield and the normal-isotropic hardening surfaces, respectively, as known from the variations of the normal-yield ratio R and the ~ depicted in Fig. 12.6c. The relations of the axial normal-isotropic hardening ratio R stress and the axial strain simulated by Chaboche (1991) and Ellyin and Xia (1989) are depicted in Fig. 12.6d, e, respectively. The strain in the initial stage is simulated as larger than the test result by the former and the curves predicted by the latter is not smooth but piece-wise linear. Finally, we examine the non-proportional loading behavior by the comparison with the test data of the austenitic 17–12 Mo SPH carbon stainless steel after Delobelle et al. (1995). The specimen is subjected to the approximately circular strain path in the strain plane ðez ; ezh Þ by the inputs of the axial strain ez ¼ 0:004 sinðh  p=2Þ and the axial-circumferential shear strain ezh ¼ 0:0036 sin h in the sinusoidal waves under the constant circumferential normal stress rh ¼ 50 MPa during 40 cycles after the uniaxial straining to ez ¼ 0:004 as depicted in Fig. 12.7. The simulation are performed by using the material parameters selected as follows:

12.5

Material Parameters and Comparisons with Test Data z

339

z

0.004

0

r 0

0.0036

z

Fig. 12.7 Circular strain path loading given by the axial strain and the axial-circumferential engineering shear strain

Material constants: Elastic moduli : E ¼ 170; 000 MPa; m ¼ 0:3ðG ¼ 65; 385 MPaÞ; ( isotropic : sr ¼ 1:7; cH ¼ 40; Hardening kinematic : ck ¼ 200 MPa; bk ¼ 0:9; Evolution of normal-yield ratio : u ¼ 800; uc ¼ 3; Re ¼ 0:5; Translationon of elastic - core : ce ¼ 283; v ¼ 0:7; ~ ¼ 3; Tan gential inelasticity : ~c ¼ 0:6; n Stagnation of isotropic hardening : C ¼ 0:5; 1 ¼ 5; t ¼ 1; Initial values: Isotropic hardening function: F0 ¼ 240 MPa: The strain path ðez ; eh Þ (eh : circumferential normal strain) and the stress path pffiffiffi  rz ; 3rzh (rah : axial-circumferential shear stress) are shown for the test result and the model simulation in Fig. 12.8a(i) and (ii), respectively. The simulation of the stress path and the accumulation of lateral strain are in good agreement with the test result. The stress and the plastic strain are attracted to the normal-yield and the normal-isotropic hardening surfaces, respectively, as known from the variations of ~ depicted in the normal-yield ratio R and the normal-isotropic hardening ratio R Fig. 12.8a(ii). The circular strain path in this test produces the spiral stress path approaching the circular stress path along the Mises yield surface which is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2ffi expressed by the circle r2z þ 3rzh ¼ F in the two-dimensional stress plane  pffiffiffi  rz ; 3rzh . The tangential strain is added in the axial strain ez and the axial-circumferential strain ezh but it is not added to eh . Therefore, the variation of eh for the variation of ez is predicted to be larger if the tangential-inelastic strain added to ez is ignored as shown in Fig. 12.8b. 

340

12

Constitutive Equations of Metals

(%)

(%) 1.8

1.8 1.6 1.4 1.2 1.0 0.8 0.4 0.2 0.0 0.5

1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2

0.4

0.3

0.2

0.1 0

0.1 z (%)

0.2

0.3

0.4

0.5

0.0 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 -0.2

0 .1

0 .2

0 .3

0 .4

0 .5

(%)

z

3 z (MPa) 600 400 200

0 -600

-400

0

-200

200

600

400

z (MPa)

-200

-400 -600

(i) R 1.0 0.8 0.6 0.4 0.2 0.0 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

0 .1

0 .2

0 .3

0 .4 z (%)

0 .1

0 .2

0 .3

~

0.5

R 1.0 0.8 0.6 0.4 0.2 0.0 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

(a)

(ii)

z

0 .4

0.5

(%)

Fig. 12.8 a Circular strain path loading given by the axial strain and the axial-circumferential engineering shear strain during 40 cycles after the uniaxial loading of austenitic 17–12 Mo SPH carbon stainless steel (Test data after Delobelle et al. (1995)): (i) Test result, (ii) Model simulation. b (i) Test result, (ii) Model simulation without tangential-inelasticity

12.5

Material Parameters and Comparisons with Test Data

341 (%) 2.4

(%)

2.2 2.0 1.8

1.8 1.6 1.4 1.2 1.0 0.8 0.4 0.2 0.0 0.5

1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2

0.4

0.3

0.2

0.1 0

0.1

0.2

0.3

0.4

0.5

-0.5

-0.4

-0.3

-0.2

0.0 -0.1 -0.20.0

z (%)

0 .1

0 .2

0 .3

0 .4

200

400

600

0 .1

0 .2

0 .3

0 .4

z

(%)

z

(%)

0 .5

(%)

z

3 z (MPa) 600 400 200

-600

-400

0 -200

0

z (MPa)

-200

-400 -600

(i) 

R

     -0 .5 -0 .4 -0 .3 -0 .2 -0 .1 0 .0

~



R

0.5

     -0 .5 -0 .4 -0 .3 -0 .2 -0 .1 0 .0

(b) Fig. 12.8 (continued)

(ii)

0 .1

0 .2

0 .3

0 .4

0.5

342

12

12.6

Constitutive Equations of Metals

Analyses of Engineering Phenomena

The metal forming analyses are of importance in the industrial production. The analyses of the two typical phenomena, i.e. the springbuck analysis and the residual stress analysis by use of the subloading surface model will be described in this section. (1) Springback analysis The high tensile (strength) steel sheets and aluminum sheets exhibiting far larger springback than ordinary mild steel sheets are widely used in automobile industries. First, it should be noticed that not only elastic but also slight plastic deformations are induced in the spring-back process induced during the drawing out of the punch from the dies. Therefore, an appropriate unconventional plastic constitutive equation must be adopted for the prediction of this phenomenon, which is capable of describing the plastic deformation in the unloading process accurately. Needless to say, it cannot be described by the constitutive models which adopt the yield surface enclosing a purely-elastic domain, i.e. the conventional plasticity model represented by the cylindrical yield surface (Chaboche-Ohno) model, and the two-surface (Dafalias-Yoshida) models. In particular, it can never be described by the two-surface model at all contrary to the repeated propagations by Yoshida et al. (Yoshida and Uemori 2002a, b, 2003; Yoshida and Amaishi 2020) and the standard installation into the commercial software LS-DYNA, PAM-STAMP and JSOL-JSTAMP, since a plastic strain rate in the unloading process cannot be described as was described in Sect. 10.3.3. The similar irrational analyses by the two-surface model are performed by Wagoner et al. (2013), etc. In their analysis, the plastic deformation during the unloading process is ignored. Instead, the Young’s modulus E is replaced by the chord-Young’s modulus E which is calculated by connecting the beginning point of the unloading and the unloaded point to the stress-free state and the chord-Young’s modulus is assumed to decrease only by the equivalent plastic strain induced during the preceding monotonic loading process as shown in Fig. 12.9.

where E, E and Emin are the cord modulus, its initial value (true Young’s modulus) and the final converged value, respectively, and n is the material constant. However, their formulation is irrational against the facts observed in real material behavior described as follow: (1) The innegligible plastic deformation is induced during the unloading process. (2) The real Young’s modulus does not change in the range of the usual sheet forming process. Microscopically, only the elastic deformation induced by the elastic deformations of mate-rial particles themselves is induced in the

12.6

Analyses of Engineering Phenomena

343

Real unloading-reloading curve with plastic deformation Real (true) elastic curve induced in beginning state of reverse loading, which is constant in deformation for range of springback Chord modulus incorporated in Yoshida’s group

E 1

E 1

Emin

Cyclic loading under constant stress amplitude Accumulation of plastic strain cannot be described at all by Yoshida-Uemori model leading to dangerous design !!!

1

0

Fig. 12.9 Chord Young’s modulus used in ad hoc. Yoshida and Uemori (2003) model which is incapable of describing strain accumulation under constant stress amplitude, where E is the true Young’s modulus which is constant, E is the chord Young’s modulus and E min is its minimum value

initiation of the reverse loading process. In fact, the inclination of the stress vs. strain curve at the moment of the reverse loading process does not change as observed in the test data for the uniaxial loading. Besides, in fact, during a far larger deformation process, the decrease of the Young’s modulus is induced by the damage of materials by the generation and growth of minute cracks and thus the Young’s modulus decreases acceleratingly once it decreases in the monotonic loading process. The Yoshida’s group ignores this primitive knowledge in the continuum damage mechanics. Therefore, the ad hoc. method adopted in their analysis must not be used in the springback analysis and, needless to say, in the general loading process, since the plastic deformation during the pulsating loading process is ignored resulting in the dangerous engineering design. The pertinent calculation result of the springback is shown below, which was analyzed by Dr. Motoharu Tateishi (MSC Software, Ltd., Japan) by implementing the subloading surface model to the commercial software Marc (MSC Software, Ltd.). The calculation results of the shapes of the sheet after the springback are shown in Fig. 12.10, choosing the die diameter 5 mm and using the following values of material parameters. Conventional model (u

)

Subloading surface model

Fig. 12.10 Springback analysis

344

12

Constitutive Equations of Metals

Material constants: Elastic moduli : E ¼ 205; 000 MPa; m ¼ 0:3; ( isotropic : h1 ¼ 0:5; h2 ¼ 15; Hardening kinematic : ck ¼ 3000 MPa; bk ¼ 0:5; Evolution of normal - yield ratio : u ¼ 200; uc ¼ 3:5; Re ¼ 0:5; n ¼ 1; Translationon of elastic - core : ce ¼ 400; v ¼ 0:7; Stagnation of isotropic hardening : C ¼ 0:5; 1 ¼ 20; t ¼ 1; Initial values: Isotropic hardening function : F0 ¼ 400 MPa: The simulation by the conventional elastoplastic model predicting the springback only by the elastic deformation is also shown in Fig. 12.10, which is calculated by setting u ! 1 in the subloading surface model and almost identical to the simulations by the Chaboche model and the two surface model without resorting to the irrational method as done in the Yoshida-Uemori model (Yoshida and Uemori 2002a, b, 2003; Yoshida and Amaishi 2020), since the plastic deformation cannot be described in the unloading process as was explained in Sect. 10.3. The enough springback is predicted, which is caused by the plastic deformation in the stress-releasing process by virtue of the advantage of the subloading surface model describing the plastic strain rate due to the rate of stress inside the yield surface. In contrast, the springback is predicted slightly by the conventional elastoplastic model which is realized by using the large value for the material constant in the evolution of the normal-yield ratio u ¼ 100; 000. Then, the importance is recognized for the introduction of the rigorous elastoplastic model, i.e. the subloading surface model capable of describing the plastic strain rate in the stress-reducing process appropriately. Hereinafter, it is desirable that the prediction of springback behavior will be executed by the pertinent analysis exploiting the subloading surface model, aiming at the epochal improvement of the prediction of the springback behavior in industries. (2) Residual stress analysis (Higuchi and Okamura 2016) The prediction of the residual stress is of importance in the metal forming process. The estimations of residual stress change due to cyclic loading by the conventional elastoplasticity model and the subloading surface model are shown below, which was examined by Higuchi and Okamura (2016). A four-point cyclic bending test was conducted to examine the change of the residual stress in the cyclic loading process. A specimen with a width of 13 mm, a thickness of 13 mm and a length of 100 mm was cut from a seamless steel pipe P110. The specimen was loaded under a four-point bend configuration with the intervals of 20 mm between the inner rollers and 80 mm between the outer rollers as shown in Fig. 12.11. At first, the specimen was

12.6

Analyses of Engineering Phenomena

345

Fig. 12.11 Configuration of the four-point bending test

㻱㼤㼜㼑㼞㼕㼙㼑㼚㼠 㻿㼡㼎㼘㼛㼍㼐㼕㼚㼓㻌㼟㼡㼞㼒㼍㼏㼑㻌㼙㼛㼐㼑㼘 㻯㼔㼍㼎㼛㼏㼔㼑㻌㼙㼛㼐㼑㼘

Fig. 12.12 Comparison of stress–strain curves between experiment and simulations in uniaxial loading

plastically deformed by static bending load corresponding to the maximum bending stress of 900 MPa. After the unloading of the static bending load, the compressive residual stress was generated in the side of outer rollers and the tensile residual stress in the side of inner rollers. The distribution of the residual stress was measured by the X-ray stress measurement method. Then, the specimen was turned upside down, so that the side of outer rollers was in the tensile residual stress state. Sinusoidal waveform load between 5 and 100% of 900 MPa was applied 20 times to the specimen. The maximum bending stress was 500 MPa. The distributions of the residual stress after cyclic loading were measured by the X-ray stress measurement method. The simulations by the Chaboche model described in Subsect. 10.3.2 and the subloading surface model described in Chap. 12 are executed, while the commercial software Abaqus is used in the simulation by the Chaboche model. First, material constants of these two models were determined so as to fit to the

346

12

Distance from upper surface

(a) Chaboche model

Constitutive Equations of Metals

Distance from upper surface

(b) Subloading surface model

Fig. 12.13 Comparison of distributions of residual stresses before and after cyclic loading between experiment and simulations

test data of uniaxial cyclic loading of a round bar specimen of P110 as shown in Fig. 12.12. The test data can be simulated accurately by the subloading surface model. On the other hand, the simulation by the Chaboche model is not in agreement with the test data. Especially, it was difficult to simulate appropriately the smooth elastic–plastic transition in the reverse loading processes by the Chaboche model. Measured distributions of the residual stresses after the initial loading and the 20 times cyclic loading are simulated by the two models as shown in Fig. 12.13. The horizontal axis denotes the distance from upper to lower surface at center of the specimen. The initial residual stress distributions are simulated well by both of these models. As for the residual stress distribution after cyclic loading, however, the decreases of the residual stresses near the upper and the lower surfaces of the specimen is accurately simulated by the subloading surface model, whereas they are not simulated by the Chaboche model. This would be caused by the fact that the subloading surface model is capable of describing the cyclic loading behavior more accurately than the other constitutive models.

12.7

Orthotropic Anisotropy

The kinematic hardening incorporated in the foregoing is regarded to be the induced anisotropy. On the other hand, various inherent anisotropies are induced in the manufacturing process of metals. The typical inherent anisotropy is the orthotropic anisotropy formulated by Hill (1948).

12.7

Orthotropic Anisotropy

347

Now, consider the general yield function in the quadratic form shown as follows:   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f rij ¼ 12 Cijkl rij rkl

ð12:44Þ

where Cijkl is the fourth-order anisotropic tensor having eighty-one components fulfilling the symmetries Cijkl ¼ Cijlk ¼ Cjikl ¼ Cjilk ¼ Cklji ¼ Cklji ¼ Clkij ¼ Clkij

ð12:45Þ

by the minor symmetries Cijkl ¼ Cijlk ¼ Cjikl based on the symmetry of the stress tensor rij ¼ rji and the major symmetries Cijkl ¼ Cklij based on Cijkl rij rkl ¼ Cklij rkl rij ¼ Cklij rij rkl . Then, the independent components are reduced to twenty-one leading to Cijkl rij rkl ¼ C1111 r211 þ 2C1122 r11 r22 þ 2C1133 r11 r33 þ 2C1112 r11 r12 þ 2C1123 r11 r23 þ 2C1131 r11 r31 þ C2222 r222 þ 2C2233 r22 r33 þ 2C2212 r22 r12 þ 2C2223 r22 r23 þ 2C2231 r22 r31 þ C3333 r233 þ 2C3312 r33 r12 þ 2C3323 r33 r23 þ 2C3331 r33 r31 þ C1212 r212 þ 2C1223 r12 r23 þ 2C1231 r12 r31 þ C2323 r223 þ 2C2331 r23 r31 þ C3131 r231

ð12:46Þ which is the general form of yield function in the quadratic form. Here, assuming the plastic incompressibility, it holds that   2    @ 2f =@rpq dpq ¼ @Cijkl rij rkl =@rpq dpq ¼ Cijkl dpi dqj rkl dpq þ Cijkl rij dpk dql dpq ¼ Cppkl rkl þ Cijpp rij ¼ Cppkl rkl þ Cppij rij ¼ 2Cppkl rkl ¼ 0 This relation must hold for any rij and thus one obtains

which leads to

Cppkl ¼ Cijqq ¼ 0

ð12:47Þ

9 C1111 þ C2211 þ C3311 ¼ 0 = C1122 þ C2222 þ C3322 ¼ 0 ; C1133 þ C2233 þ C3333 ¼ 0

ð12:48Þ

9 9 C1112 þ C2212 þ C3312 ¼ 0 = C3312 ¼ ðC1112 þ C2212 Þ = C1123 þ C2223 þ C3323 ¼ 0 ! C1123 ¼ ðC2223 þ C3323 Þ ; ; C1131 þ C2231 þ C3331 ¼ 0 C2231 ¼ ðC1131 þ C3331 Þ

ð12:49Þ

348

12

Constitutive Equations of Metals

The substitution of Eq. (12.49) into Eq. (12.46) gives the expression Cijkl rij rkl ¼ C1111 r211 þ 2C1122 r11 r22 þ 2C1133 r11 r33 þ 2C1112 r11 r12  2ðC2223 þ C3323 Þr11 r23 þ 2C1131 r11 r31 þ C2222 r222 þ 2C2233 r22 r33 þ 2C2212 r22 r12 þ 2C2223 r22 r23  2ðC1131 þ C3331 Þr22 r31 þ C3333 r233  2ðC1112 þ C2212 Þr33 r12 þ 2C3323 r33 r23 þ 2C3331 r33 r31 þ C1212 r212

þ 2C1223 r12 r23 þ 2C1231 r12 r31 þ C2323 r223 þ 2C2331 r23 r31 þ C3131 r231

ð12:50Þ Further, noting Eq. (12.48), the terms in the form Ciijj rii rjj (no sum) are written as C1111 r211 þ C2222 r222 þ C3333 r233 þ 2C1122 r11 r22 þ 2C2233 r22 r33 þ 2C1133 r11 r33 ¼ C1111 r211 þ C2222 r222 þ C3333 r233  C1122 ðr11  r22 Þ2 þ C1122 r211 þ C1122 r222  C2233 ðr22  r33 Þ2 þ C2233 r222 þ C2233 r233  C1133 ðr33  r11 Þ2 þ C1133 r233 þ C2233 r211 ¼ ðC1111 þ C1122 þ C2233 Þr211 þ ðC1122 þ C2222 þ C2233 Þr222 þ ðC1133 þ C2233 þ C3333 Þr233  C1122 ðr11  r22 Þ2 C2233 ðr22  r33 Þ2 C1133 ðr33  r11 Þ2 ¼ C1122 ðr11  r22 Þ2 C2233 ðr22  r33 Þ2 C1133 ðr33  r11 Þ2

ð12:51Þ Then, by setting a1  C1122 ; a2  C2233 a3  C1133 a4  2C1112 ; a5  2C2212 ; a6  C2223 a7  2C3323 a8  2C3331 ; a9  2C1131 a10  2C1223 ; a11  2C2331 a12  2C1231 a13  C1212 ; a14  C2323 a15  C3131

9 > > > > = > > > > ;

ð12:52Þ

and substituting Eqs. (12.50) and (12.51) with Eq. (12.52) into Eq. (12.50) reads: Cijkl rij rkl ¼ a1 ðr11  r22 Þ2 þ a2 ðr22  r33 Þ2 þ a3 ðr33  r11 Þ2 þ ½a4 ðr33  r11 Þ þ a5 ðr33  r22 Þr12 þ ½a6 ðr11  r22 Þ þ a7 ðr22  r33 Þr23 þ ½a8 ðr22  r33 Þ þ a9 ðr22  r11 Þr31 þ a10 r12 r23 þ a11 r23 r31 þ a12 r31 r12

12.7

Orthotropic Anisotropy

349

þ a13 r212 þ a14 r223 þ a15 r231

ð12:53Þ

Equation (12.53) is the general yield function for the plastically-incompressible materials in the quadratic form. Now, assume orthotropic anisotropy. Then, if we describe the yield surface by _ the coordinate axes selected to the principal axes fe j g of orthotropic anisotropy, the yield function is independent of the sign of shear stress components in this coordinate system. Therefore, it must hold that a4 ¼ a5 ¼ a6 ¼ a7 ¼ a8 ¼ a9 ¼ a10 ¼ a11 ¼ a12 ¼ 0 Here, replacing the symbols ai as F ¼ a1 ; G ¼ a2 ; H ¼ a3 ; L ¼ a13 =2; M ¼ a14 =2; H ¼ a15 =2 used by Hill (1948), Eq. (12.53) leads to the Hill’s yield condition with orthotropic anisotropy:

1 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi F ðr11  r22 Þ2 þ Gðr22  r33 Þ2 þ H ðr33  r11 Þ2 þ 6 Lr212 þ Mr223 þ Nr231 ¼ FðHÞ

ð12:54Þ Here, note that for the isotropic material all the material parameters are unity, i.e. F ¼ G ¼ H ¼ L ¼ M ¼ N ¼ 1 holds and thus Eq. (12.54) is reduced to   pffiffiffiffiffiffiffiffi f rij ¼ 3=2kr0 k which is the equivalent stress. While Eq. (12.54) is the ^

expression on the principal axis fe i g of orthotropic anisotropy, it is rewritten by the following equation stipulating this fact. 1 2

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2  2  2 ffi ^ ^ ^ ^ ^ ^ ^ ^ 2 2 F r11  r22 þ G r22  r33 þ H r33  r11 þ r Lr12 þ M r23 þ Nr31 ¼ FðHÞ

ð12:55Þ or  2  2  2  2  ^ ^ ^ ^ ^ ^  r  r  ^12 þ Mr  ^ 223 þ Nr  ^231 ¼ 1  r þG þH þ 6 Lr F 11  r22 22  r33 33  r11

ð12:56Þ

350

12

where

H

H

Constitutive Equations of Metals

G

9 > > 2> =

F

;G  ;F  ½2FðHÞ2 ½2FðHÞ2 ½2FðHÞ > L L L > > L ;M  ;N  2 2 2; ½2FðHÞ ½2FðHÞ ½2FðHÞ ^

^

^

^

^

^

^

^

^

ð12:57Þ

^

^

^

Denoting r11 ; r22 ; r33 ; r12 ; r23 ; r31 in the yield state by ry1 ; ry2 ; ry3 ; ry12 ; ry23 ; ry31 , respectively, when only each stress applies, one has 9 ^ ^ y2 ^ y2 = ðH þ FÞry2 ¼ 1; ðG þ FÞr ¼ 1; ðG þ HÞr ¼ 1 1 2 3 ð12:58Þ ^ ^ ^ 6Ls y2 ¼ 1; 6Ms y2 ¼ 1; 6Ns y2 ¼ 1 ; 12

23

31

from which the material parameters are given by ! 1 1 1 1  F¼ þ ^y2  ^y2 ^ y2 2 r r2 r3 ! 1  ¼1 1 þ 1  1 G ^ y2 ^ ^ 2 r ry2 ry2 2 3 1 ! 1 1 1 1  ¼ þ ^y2  ^y2 H ^ y2 2 r r1 r2 3 1 1 ¼  ¼ ¼ 1 L ;M ;N ^y2 ^y2 ^ 6s 12 6s23 6s y2 23

9 > > > > > > > > > > > > > > =

ð12:59Þ

> > > > > > > > > > > > > > ;

Equation (12.55) is reduced under the uniaxial loading in the sheet metal forming as follows: 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi^ ^2 ¼1 ðF þ HÞr11 ¼ FðHÞ or ðF þ HÞr11 2

ð12:60Þ

Further, under the plane stress condition observed in the sheet metal forming it ^ ^ ^ holds that r23 ¼ r31 ¼ r33 ¼ 0 and thus Eqs. (12.55) and (12.56) are reduced to 1 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^2 ^ ^ ^2 ^2  2Fr11 r22 þ ðF þ GÞr22 þ 6Lr12 ðF þ HÞr11 ¼ FðHÞ

ð12:61Þ

i.e. ^

^

^

^

^

2  2Fr r þ ðF þ GÞr 2 þ 6Lr 2 ¼ 1 ðF þ HÞr11 11 22 22 12

ð12:62Þ

12.7

Orthotropic Anisotropy

351

Equation (12.62) is described in the state that the principal stress directions coincide with the orthogonal anisotropy axes as follows: ^

^ ^

^

ðF þ HÞr12  2Fr1 r2 þ ðF þ GÞr22 ¼ 1

ð12:63Þ

The plastic strain rate is given for Eq. (12.63) as follows:

ð12:64Þ

noting the plastic incompressibility. The orthogonal anisotropy is induced seriously in the rolling process for the ^ ^ ^ sheet metal forming. Choosing the axes e1 ; e 2 ; e3 to the rolling, the traverse, and the thickness directions, respectively, the following Lankford R-value is adopted widely in order to evaluate the intensity of the orthotropy. 



Reh ¼ eph =ep3

ð12:65Þ



where eph ð\0Þ is the lateral strain rate measured from the uniaxial tension test of the test specimen cut out at the angle h measured counterclockwise from the rolling  direction. The plastic strain rate ep3 ð\0Þ in the thickness direction is calculated from the axial and the lateral strain rates by the assumption of plastic incompressibility (see Fig. 12.14). Reh ¼ 1 means the isotropy. A small Reh -value means that the produced sheet metal is easily thinned resulting in an easy failure. It follows from Eq. (12.64) that 9  ep2 F > ^ ^ e > R0 ¼  p ¼ for r2 ¼ r3 ¼ 0 > > > H = e3 ð12:66Þ p > > > e F ^ ^ > Re90 ¼  1p ¼ for r1 ¼ r3 ¼ 0 > ; e3 G Substituting Eq. (12.66) into Eq. (12.63), one has ^

^

^ ^

ðH þ Re0 HÞr12 þ ðG þ Re0 HÞr22  2Re0 Hr1 r2 ¼ 1

352

12

Constitutive Equations of Metals

d 3p

dp

e3 e2

Rolling direction

e1

Fig. 12.14 Uniaxial tension test for Lankford R-value in metal formed by rolling process

leading to ^

r12 þ

G þ Re0 H ^ 2 2Re0 ^ ^ 1 r  r1 r2 ¼ 2 e e 1 þ R0 ð1 þ R0 ÞH ð1 þ Re0 ÞH

ð12:67Þ

Substituting Eq. (12.59) into Eq. (12.66), it follows that 9 > >  ¼ ð1  > > r3 r2 r1 = 1 1 1 > > > ð1 þ Re90 Þ  ð1  Re90 Þ ¼ ð1 þ Re90 Þ > ^ y2 ^ y2 ^ y2 ; r3 r2 r1 1 ð1 þ Re0 Þ ^ y2

1 ð1 þ Re0 Þ ^ y2

1 Re0 Þ ^ y2

By solving this equation, one has 1 ^

r2y2 1 ^

r3y2

¼

9 Re0 ð1 þ Re90 Þ 1 > > > ^ y2 > = ð1 þ Re0 ÞRe90 r 1

Re0 þ Re90 1 > > > ¼ > ^ y2 ; ð1 þ Re0 ÞRe90 r 1

Substituting Eq. (12.68) into Eq. (12.59), it follows that

ð12:68Þ

12.7

Orthotropic Anisotropy

353

9 Re0 1 > > G¼ > ^ y2 > = ð1 þ Re0 ÞRe90 r 1

1 1 > > > H¼ > ^ y2 ; 1 þ Re0 r

ð12:69Þ

1

from which we have 9 G þ Re0 H Re0 ð1 þ Re90 Þ > > > ¼ = ð1 þ Re0 ÞH ð1 þ Re0 ÞRe90 > 1 ^ > > ¼ r1y2 ; ð1 þ Re0 ÞH

ð12:70Þ

Substituting Eq. (12.70) into Eq. (12.67), it follows that

^

r12 þ

Re0 ð1 þ Re90 Þ ^ 2 2Re0 ^ ^ ^ r  r1 r2 ¼ r1y2 ð1 þ Re0 ÞRe90 2 1 þ Re0

ð12:71Þ

Equation (12.62) is rewritten as 

 1 1 1 2 2 ^2 ^ ^ ðG þ HÞ þ ðG þ H þ 4FÞ  ðG  HÞ r11 þ ðG þ HÞ  ðG þ H þ 4FÞ r11 r22 4 4 2 4 4

 1 1 1 ^2 ^2 þ ðG þ HÞ þ ðG þ H þ 4FÞ þ ðG  HÞ r22 þ 6Lr12 ¼1 4 4 2 which is arranged as follows: 1 1 ^ ^ ^ ^ ðG þ HÞðr11 þ r22 Þ2 þ ðG þ H þ 4FÞðr11  r22 Þ2 4 4 1 ^2 ^2 ^2  ðG  HÞðr11  r22 Þ þ 6Lr12 ¼ 1 2

ð12:72Þ

Denoting the angle measured in the counterclockwise direction from the principal axes of anisotropy to the principal stress as a and substituting the relations. ^

^

^

^

r11 þ r22 ¼ r1 þ r2 ; r11  r22 ¼ ðr1  r2 Þ cos 2a; ^ 2r12 ¼ ðr1  r2 Þ sin 2a

ð12:73Þ

354

12

Constitutive Equations of Metals

into Eq. (12.72), one has 1 1 ðG þ HÞðr1 þ r2 Þ2 þ ðG þ H þ 4FÞðr1  r2 Þ2 cos2 2a 4 4 1 3  ðG  HÞðr21  r22 Þ cos 2a þ Lðr1  r2 Þ2 sin2 2a ¼ 1 2 2 which is rewritten as ðr1 þ r2 Þ2  2aðr21  r22 Þ cos 2a þ bðr1  r2 Þ2 cos2 2a þ 6

L 4 ðr1  r2 Þ2 ¼ GþH GþH

ð12:74Þ where a

GH G þ H þ 4F  6L ;b GþH GþH

ð12:75Þ

Here, denoting the yielding strength in the equi-two axis tension as r and that of the pure shear as s, it follows from Eq. (12.62) that r  ðH þ GÞ1=2 ; s  ð6LÞ1=2

ð12:76Þ

The substitution of Eq. (12.76) into Eq. (12.74) leads to ðr1 þ r2 Þ2 þ

r2 ðr1  r2 Þ2  2aðr21  r22 Þ cos 2a þ bðr1  r2 Þ2 cos2 2a ¼ ð2rÞ2 s ð12:77Þ

Equation (12.77) is extended to the following equation for the in-plane isotropy with the material constant m ð  1Þ. jr1 þ r2 jm þ

rm jr1  r2 jm ¼ ð2rÞm s

ð12:78Þ

Hill (1990) proposed the following extended orthotropic yield condition from Eqs. (12.77) and (12.78). jr1 þ r2 jm þ

 r m

jr1  r2 jm h i þ jr21 þ r22 jðm=2Þ1 2aðr21  r22 Þ þ bðr1  r2 Þ2 cos 2a cos 2a ¼ ð2rÞm s

ð12:79Þ

12.7

Orthotropic Anisotropy

355

Equation (12.79) includes the five material constants, i.e. the yield stress r; s and the dimensionless number a; b; m. It is reduced to Eq. (12.77) for m ¼ 2 and to Eq. (12.78) for a ¼ b ¼ 0 (or a ¼ p=4). By use of Eqs. (12.73), (12.79) is rewritten in the anisotropic axes as follows:

m=2  r m

^ ^ ^ ^ ^2 m jðr11  r22 Þ2 þ 4r12

r11 þ r22 j þ

s

ðm=2Þ1 h i

^ ^ ^2 ^2 ^2 ^ ^ þ ðr11 þ r22 Þ2 þ 4r12 2aðr11  r22 Þ þ bðr11  r22 Þ2 ¼ ð2rÞm

ð12:80Þ Generally, the yield surface is described in the principal axes of anisotropy as follows: ^

f ðrij Þ ¼ FðHÞ

ð12:81Þ

where ^

^

^^

^

^

r ¼ R T rR; rij ¼ Rri Rsj rrs ^



^



^

Rij ðtÞ  ei  ej ðtÞ ¼ cos ei ; ej ðtÞ

ð12:82Þ 

ð12:83Þ

^

ei ðtÞ are the base vectors taken to the directions of the principal axes of the orthotropic anisotropy. Needless to say, Eq. (12.81) is not a general tensor ^ expression but is merely the expression by the components. The variation of ei is ^ calculated using the following equation with the initial value of e i0 . Z  ^ ^ ^ eidt ð12:84Þ e i ¼ ei0 þ  ^

where ei is given by    ^ ^ ^ e i¼ xa ei xa ¼ ei ei

 ^

ð12:85Þ

 ^

Here, the stress rate eij is given by  ^

 ^

^

^



^

^



rij ¼ rij ¼ Rri Rsj rrs ¼ Rri Rsj ðrrs xarp rps þ rrp xaps Þ ^

ð12:86Þ

noting Eqs. (1.105) and (3.43) with Q ¼ R T . Various anisotropic yield surfaces in the plane stress state are proposed by the Barlat’s group (e.g. Barlat et al. 2007), Yoshida et al. (2015), etc.

356

12.7.1

12

Constitutive Equations of Metals

Representation of Isotropic Mises Yield Condition

The isotropic yield function described by Eq. (8.41) can be expressed in the following various forms. rffiffiffi rffiffiffi 3 0 3pffiffiffiffiffiffiffiffiffiffiffi r0rs r0rs f ðrÞ ¼ r ¼ kr k ¼ 2 2 rffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  02  3 02 02 02 02 r02 ¼ 11 þ r22 þ r33 þ 2 r12 þ r23 þ r31 2 rffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   1 ðr11  r22 Þ2 þ ðr22  r33 Þ2 þ ðr33  r11 Þ2 þ 6 r212 þ r223 þ r231 ¼ 2 rffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 02 02 r02 ¼ 1 þ r2 þ r3 2 rffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ¼ ðr1  r2 Þ2 þ ðr2  r3 Þ2 þ ðr3  r1 Þ2 ¼ F 2 ð12:87Þ eq

The combined test of the tensile stress rð¼ r11 Þ and the distortional stress sð¼ r12 Þ for a thin wall cylinder specimen is widely adopted for metal. In this case Eq. (12.87) is rewritten as pffiffiffi r2 þ ð 3sÞ2 ¼ F 2

ð12:88Þ

pffiffiffi Then, the Mises yield condition is shown by a circle of radius F in the ðr; 3sÞ plane. The visualization of the stress state can be realized in the space of three and less dimension. The stress state can be represented completely in the principal stress space when principal stress directions are fixed to materials and only the principal stress values change. In general, however, one must use the six-dimensional space or memorize the variation of the principal stress direction if the directions change. However, in the cases for which the number of independent variable components is less than three, such as the tension-distortion test described above and the plane stress and strain tests, the state of stress can be represented in the three and less dimensional stress space. The Ilyushin’s isotropic stress space (Ilyushin 1963) is convenient to depict the Mises yield surface, which depends only on the deviatoric stress, as explained below.

12.7

Orthotropic Anisotropy

357

The deviatoric stress tensor includes the five independent variables ðtrr0 ¼ 0Þ and thus the Mises yield surface in Eq. (12.87) is described by the independent components as follows: f ðrÞ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  0  02 0 0 02 02 3r02 11 þ 3r22 þ 3r11 r22 þ 3 r12 þ r23 þ r31

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s   2  0  3 0 2 1 0 0 02 02 þ 3 r11 þ r22 þ 3 r12 þ r23 þ r31 ¼ F r ¼ 2 11 2 and thus it can be rewritten as S21 þ S22 þ S23 þ S24 þ S25 ¼ F 2

ð12:89Þ

in the five-dimensional space with the axes   pffiffiffi 1 pffiffiffi pffiffiffi pffiffiffi 3 S1 ¼ r011 ; S2 ¼ 3 r011 þ r022 ; S3 ¼ 3r012 ; S4 ¼ 3r023 ; S5 ¼ 3r031 2 2 ð12:90Þ Equation (12.89) exhibits the five-dimensional spherical super surface. Further, consider the expression of the Mises yield surface for the plane stress and strain conditions in the following.

12.7.1.1

Plane Stress State

The plane stress state fulfilling r3j ¼ 0 can be described in the three-dimensional space ðr11 ; r22 ; r12 Þ and thus the Mises yield condition (12.87) is described by the following equation. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r211  r11 r22 þ r222 þ 3r212 ¼ F

ð12:91Þ

On the other hand, Eq. (12.91) can be described in the two-dimensional principal stress plane as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r21  r1 r2 þ r22 ¼ F

ð12:92Þ

which is the section of the Mises yield condition cut by the plane r3 ¼ 0 and exhibits Mises’s ellipse in the principal stress plane ðr1 ; r2 Þ as shown in Fig. 12.15. It follows from the third equation of Eq. (1.322) that

358

Ⅱ

(1

3

( =120 ) F

(

12

Constitutive Equations of Metals

F,

2 F ) ( = 90 ) 3 ( F , F ) ( = 60 )

1 F , 1 ) ( =150 ) F 3 3

( 23 F , 1 F ) ( = 30 ) 3

( =180 ) F F ( = 0)

0 ( = 210 ) (

2 F, 3

1 ) F 3

(1

3

( = 240 ) ( F ,

F)

(

1 F, 3

F,

Ⅰ

1 ) ( = 330 ) F 3

F ( = 300 ) 2 ) ( = 270 ) F 3

Fig. 12.15 Mises yield surface in the plane stress condition (Thin curve describes Hill’s orthotropic Mises yield condition)

rffiffiffi     2 0 2 2 2 rm ¼  kr kF cos h þ p ¼  F cos h þ p 3 3 3 3

ð12:93Þ

pffiffiffiffiffiffiffiffi because of rm þ r03 ¼ 0 leading to rm ¼ r03 with kr0 k ¼ 2=3F. Substituting pffiffiffiffiffiffiffiffi Eq. (12.93) with kr0 k ¼ 2=3F into Eq. (1.322), one obtains  2 r1 ¼  F cos h þ 3  2 r2 ¼  F cos h þ 3

 2 p þ 3  2 p þ 3

9  2 2 p > > F cos h ¼ pffiffiffi F sin h þ = 3 3 3  2 2 2 > ; F cos h  p ¼ pffiffiffi F sin h > 3 3 3

ð12:94Þ

from which the coordinates of main points on the Mises’s ellipse are calculated as shown in Fig. 12.15. The thin curve shows the Hill’s orthotropic Mises yield surface in Eq. (12.61), which is rotated the principal axes of ellipse with the changes of its long and short radii from the isotropic Mises yield surface. Next, consider the Ilyushin’s isotropic stress space in which the variables in Eq. (12.90) are used. Here, in the present case fulfilling r3j ¼ 0 leading to S4 ¼ S5 ¼ 0 the Mises yield surface is represented by the sphere in the ðS1 ; S2 ; S3 Þ space, while it holds that

12.7

Orthotropic Anisotropy

359

S2

F

φ

−F

0

F

S1

−F Fig. 12.16 Mises yield surface in plane stress state without shear stress

9 > S1 ¼ ð3=2Þr011 ¼ ð3=2Þ½r11  ðr11 þ r22 Þ=3 ¼ r11  r22 =2 > = pffiffiffi  0 0 S2 ¼ 3 ð1=2Þr11 þ r22 > pffiffiffi pffiffiffi > ¼ 3fð1=2Þ½r11  ðr11 þ r22 Þ=3 þ ½r22  ðr11 þ r22 Þ=3g ¼ ð 3=2Þr22 ; ð12:95Þ Furthermore, in the case fulfilling r12 ¼ 0, the Mises yield surface is represented by the circle in the ðS1 ; S2 ; S3 Þ plane (Fig. 12.16). Here, setting S1 ¼ F cos /; S2 ¼ F sin /

ð12:96Þ

and substituting them into Eq. (12.95), it holds that  2 p r11 ¼ pffiffiffi F sin / þ ; 3 3

12.7.2

2 r22 ¼ pffiffiffi F sin / 3

ð12:97Þ

Plane Strain State

If the elastic strain rate can be ignored compared with the plastic strain rate in the 



plane strain state, the following relation holds by substituting ep33 ¼ k r033 ¼ 0 into r0rr ¼ 0.

360

12

Constitutive Equations of Metals

1 r33 ¼ ðr11 þ r22 Þ 2

ð12:98Þ

Then, the Mises yield surface is described from Eq. (12.87)3 by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi r11  r22 2 3 þ r212 ¼ F 2

ð12:99Þ

which is represented by the Mohr’s circle in the plane of the normal and the shear stresses.

12.8

Subloading Surface Model with Orthotropic Anisotropy

Explicit equations required for applying the subloading surface model to the deformation analysis will be given in this section.

12.8.1

Subloading Surface with Orthotropic Anisotropy

The subloading surface for the yield condition in Eq. (12.54) is given by rffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Fðr11  r22 Þ2 þ Gðr22  r33 Þ2 þ Hðr33  r11 Þ2 þ 2ðLr212 þ Mr223 þ Nr231 Þ ¼ RFðeeqp Þ 2

ð12:100Þ where _

rij ¼ rij þ R^cij

ð12:101Þ

(_ rij  rij  cij ^cij  cij  aij

ð12:102Þ

12.8

Subloading Surface Model with Orthotropic Anisotropy

12.8.2

361

Plastic Strain Rate

It follows from Eq. (12.100) that 8 @f 1 > > 22 Þ þ Hð 33 Þ ½Fð r11  r ¼ r11  r > > @ r 2f ðrÞ > 11 > > > @f > 1 > > 11 Þ þ Fð 33 Þ ½Gð r22  r ¼ r22  r > > > @ r 2f ðrÞ 22 > > > > 1 > @f > > 11 Þ þ Gð 22 Þ ½Hð r33  r ¼ r33  r < @ r33 2f ðrÞ > @f 1 > > L r12 ¼ > > @ r12 2f ðrÞ > > > > > @f 1 > > M r23 ¼ > > > @ r11 2f ðrÞ > > > > @f 1 > > : N r31 ¼ @ r31 2f ðrÞ

ð12:103Þ

|| ∂∂σf || =

ð12:104Þ

and 1 Ξ 2 f (σ )

where Ξ ≡

{[ F (σ 11 − σ 22 ) + H (σ 11 − σ 33 )]2 + [ F (σ 22 − σ 33 ) + G (σ 22 − σ 11 )]2 +[ H (σ 33 − σ 11 ) + G (σ 33 − σ 22 )]2 + 4[( Lσ 12 ) 2 + ( M σ 23 ) 2 + ( Nσ 31 ) 2 ]}

ð12:105Þ The components of the normalized outward-normal of the subloading surface are given by ⎧n11 = [ F (σ 11 − σ 22 ) + H (σ 11 − σ 33 )] / Ξ ⎪ ⎪n22 = [G (σ 22 − σ 11 ) + F (σ 33 − σ 11 )] / Ξ ⎪ ⎪n33 = [ H (σ 33 − σ 11 ) + G (σ 33 − σ 22 )] / Ξ ⎨ ⎪n12 = 2 Lσ 12 / Ξ ⎪n = 2M σ Ξ 23 / ⎪ 23 ⎪n = 2 Nσ 31 / Ξ ⎩ 31

ð12:106Þ

362

12

Constitutive Equations of Metals

Therefore, the plastic strain rate is given by the associated flow rule of the subloading surface as follows: ⎧ p • (σ − σ ) ⎪d11 = λ [ F 11 22 + H (σ 11 − σ 33 )] / Ξ ⎪ • ⎪d 22p = λ [G (σ 22 − σ 11 ) + F (σ 33 − σ 11 )] / Ξ ⎪ ⎪ p • (σ − σ ) ⎪d33 = λ [ H 33 11 + G (σ 33 − σ 22 )] / Ξ ⎨ ⎪d p = 2 λ• Lσ / Ξ 12 ⎪ 12 • ⎪ p ⎪d 23 = 2 λ M σ 23 / Ξ ⎪ • ⎪d p = 2 λ Nσ 31 / Ξ ⎩ 31

12.8.3

ð12:107Þ

Normal-Yield Ratio

The normal-yield ratio is calculated by the evolution rule in the plastic loading process. However, it must be calculated by the state variables, i.e. the stress, the isotropic hardening function, the kinematic hardening variable (back-stress), the elastic-core (similarity-ratio). Let the calculation method of the normal-yield ratio in the unloading process be given in the following. Equation (12.100) for the subloading surface is represented in the quadratic equation of R as follows: fFð^c11 ^c22 Þ2 þ Gð^c22  ^c33 Þ2 þ Hð^c33  ^c11 Þ2 þ 2ðL^c212 þ M^c223 þ N^c223 Þ  2½Fðeeqp Þ2 gR2 _

_

_

_

_

_

þ 2½Fðr11  r22 Þð^c11  ^c22 Þ þ Gðr22  r33 Þð^c22  ^c33 Þ þ Hðr33  r11 Þð^c33  ^c11 Þ _

_

_

þ 2ðLr12^c12 þ Mr23^c23 þ Nr31^c31 ÞR _

_

_

_

_

_

_2

_2

_2

þ Fðr11  r22 Þ2 þ Gðr22  r33 Þ2 þ Hðr33  r11 Þ2 þ 2ðLr12 þ Mr23 þ Nr31 Þ ¼ 0

ð12:108Þ R can be calculated by the following equation which is derived from Eq. (12.108). pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b  b2  ac R¼ ð12:109Þ a

12.8

Subloading Surface Model with Orthotropic Anisotropy

363

where 8 a  Fð^c11  ^c22 Þ2 þ Gð^c22  ^c33 Þ2 þ Hð^c33  ^c11 Þ2 þ 2ðL^c212 þ M^c223 þ N^c223 Þ  2½Fðeeqp Þ2 > > > > _ _ _ _ _ _ < b  Fðr c11  ^c22 Þ þ Gðr22  r33 Þð^c22  ^c33 Þ þ Hðr33  r11 Þð^c33  ^c11 Þ 11  r22 Þð^ > þ 2ðL^c12 þ M^c23 þ N^c31 Þ > > > : _ _ _ _ _ _ _ _ _ c  Fðr11  r22 Þ2 þ Gðr22  r33 Þ2 þ Hðr33  r11 Þ2 þ 2ðLr212 þ Mr223 þ Nr231 Þ  2½Fðeeqp Þ2

ð12:110Þ

12.8.4

Elastic-Core Yield Ratio

The elastic-core (similarity) yield ratio designating the approaching degree of the elastic-core to the normal-yield surface is given by

ð12:111Þ

12.8.5

Cyclic Stagnation of Isotropic Hardening

The cyclic stagnation of the isotropic hardening described in Sect. 12.2 will be extended to the orthotropic anisotropy in the following. The normal-isotropic hardening surface in Eq. (12.28) is extended to the orthotropy as follows: rffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 gð~e Þ ¼ ep2 ep2 Fð~ep11  ~ep22 Þ2 þ Gð~ep22  ~ep33 Þ2 þ Hð~ep33  ~ep11 Þ2 þ 2ðL~ep2 12 þ M~ 23 þ N~ 31 Þ 2 ð12:112Þ p

364

12

Constitutive Equations of Metals

for which one has

ð12:113Þ

ð12:114Þ

leading to p p p p ⎧n%11 = [ F (ε% 11 − ε% 22 ) + H (ε% 11 − ε% 33 )] / Ζ ⎪ p p p p ⎪n%22 = [G ( ε% 22 − ε% 33 ) + F ( ε% 22 − ε% 11 )] / Ζ p p p p ⎪ ⎪n%33 = [ H (ε% 33 − ε%11 ) + G ( ε% 33 − ε% 22 )] / Ζ ⎨ p ⎪n%12 = 2 L ε% 12 / Ζ ⎪n% = 2M ε% p 23 / Ζ ⎪ 23 ⎪⎩n% 31 = 2 N ε% p31 / Ζ

ð12:115Þ

where Ζ ≡

{ [ F (ε%11p − ε%22p ) + H (ε%11p − ε% 33p )]2 + [G ( ε% 22p − ε% 33p ) + F (ε% 22p − ε% 11p )]2 + [ H ( ε% 33 − ε% 11 ) + G ( ε%33 − ε% 22 )]2 + 4[(L ε%12 ) 2 + (M ε% 23 ) 2 + ( N ε% 31) 2]} p

p

p

p

p

p

p

ð12:116Þ

Chapter 13

Constitutive Equations of Soils

The general importance of incorporating the subloading surface model, e.g. (1) the capability of the smooth elastic-plastic transition, (2) the unnecessity of the yield judgement in the loading criterion, (3) the automatic controlling function to pull-back the stress to the yield surface in the numerical calculation with large strain increment are explained in the former sections. The deformation in the elastic-plastic transition is quite small as the 0.2% plastic strain represented by the offset-value in metals which is rather brittle material. On the other hand, the plastic strain in the elastic-plastic transition is tremendously large as several percent plastic strain in soils. (4) the large plastic deformation in the elastic-plastic transition can be described by the suloading surface model, while it cannot be described by the conventional plasticity model with the yield surface enclosing the elastic domain represented by the Cam-clay model at all. Further, soils exhibit the remarkable softening, while only the hardening is induced in metals. The smooth peak-up and down stress vs. strain is described by the subloading surface model, although the abruptly rising-up and -down stress-strain curve is described by the conventional plasticity model. Therefore, the adoption of the subloading surface model is crucially important for the formulation of the plastic constitutive equation of soils. The elastoplastic constitutive equation of soils is formulated based on the subloading surface model in this chapter.

13.1

Isotropic Consolidation Characteristics

The isotropic consolidation characteristics is the one of the fundamentals in the constitutive equations of soils, which provides the base of isotropic hardening of soils.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 K. Hashiguchi, Foundations of Elastoplasticity: Subloading Surface Model, https://doi.org/10.1007/978-3-030-93138-4_13

365

366

13 Constitutive Equations of Soils

(a) Linear relation between double-logarithm of volume and pressure Denoting the initial and the current volume as V and v, respectively, and the volume at the unloaded configuration to the stress-free state as V, the logarithmic volumetric strain ev in Eq. (4.141) is additively decomposed into the elastic logarithmic volumetric strain eev and the plastic logarithmic volumetric strain epv as follows: ev ¼ eev þ epv ðln J ¼ ln J e þ ln J p Þ

ð13:1Þ

where ev  ln J ¼ ln ¼ ln



  v vV v V v ¼ ln ¼ ln þ ln ; eev  ln J e ¼ ln ; epv  ln J p V V VV V V

V V ð13:2Þ

where J is the Jacobian representing the volume change as defined in Eq. (2.21) and, J e and J p are the elastic and the plastic parts, i.e. J  det F ¼ det

@x v v V ¼ ; J e  det Fe ¼ ; J p  det Fp ¼ @X V V V

ð13:3Þ

which lead to the multiplicative decomposition ð13:4Þ

J ¼ JeJp

Fe and Fp are the elastic and the plastic part of the deformation gradient tensor F in the multiplicative decomposition F ¼ Fe Fp as will be defined in Chap. 17. It follows from Eqs. (13.1) and (13.2) that 

e

p

ev ¼ ev þ ev

ð13:5Þ

where

ð13:6Þ As known from Eq. (13.6), the rate of the logarithmic volumetric strain is the trace of the strain rate measure d used in the hypoelasticity described in Eq. (7.109). Now, the ln v  ln p linear relation (p  ðtrrÞ=3: pressure) for the isotropic consolidation characteristics of soils was proposed by Hashiguchi (1947, 1955, 1958a, 1977, 2008), Hashiguchi and Ueno (1977) and later its conformity to test data was shown by Butterfield (1979). It is shown by the following equations and shown schematically in the upper part of Fig. 13.1.

13.1

Isotropic Consolidation Characteristics

367

Normal  consolidation line: ðElastoplastic stateÞ Swelling and recompression line: ðElastic stateÞ

vy py 9 > ¼ ~k ln > = Vy py0 > v p > ln ¼ ~ j ln > ; p0 > V

ln

ð13:7Þ

where ðV; p0 Þ, ðVy ; py0 Þ and ðvy ; py Þ are the volumes and the pressures in the initial state, the initial yield state and the current yield state, respectively. py ð [ 0Þ is the so-called preconsolidated pressure. In addition, V is the volume in the unloaded state from the yielded state ðvy ; py Þ to the initial pressure p0 , while the unloaded v

ln v

V

Vy

Normal-consolidation line

Idealization

1

V

V

1

∼

Swelling line

∼

v vy

v vy 0

Swelling line

V Vy

p0

py0

p

py

(a) Isotropic consolidation of soils

p

1

p0

p

py0

py

ln p (b) Idealization by double-logarithmic linear relation of volume and pressure. Extension to negative pressure region

v V Vy

ln v

Swelling line

V Vy

Normal-consolidation line 1 ∼

V V

v vy

1

Swelling line

∼

v vy

1 (c) Isotropic consolidation of soils with negative pressure strength

(d) Extended double-logarithmic linear relation.

Fig. 13.1 Linear relation between double-logarithms of volume and pressure for isotropic consolidation of soils

368

13 Constitutive Equations of Soils

state corresponds to the intermediate configuration described in Sect. 17.1. Further, ~ ~ are the material constants prescribing the slopes of normal-consolidation k and j line and swelling line, respectively, in the ðln v; ln pÞ plane, while in the range of infinitesimal deformation they approximately coincide to the values of k and j in the e  ln p linear relation (e: void ratio) used in the Cam-clay model as will be described subsequently in this section. Equation (13.7) is extended to the following equation so as to be applicable to the negative pressure range as shown in the lower part in Fig. 13.1, 9 Normal  consolidation line: ln vy ¼ ~k ln py þ pe > > > Vy py0 þ pe = ðElastoplastic stateÞ v p þ pe > Swelling line: ln ¼ ~ j ln > > p 0 þ pe ; V ðElastic stateÞ

ð13:8Þ

where pe ð  0Þ is the material constant prescribing the negative pressure for which the volume becomes infinite, i.e. v ! 1 for p ! pe . The logarithmic volumetric strain in Eq. (4.141) and its elastic and plastic parts are given from Eq. (13.8) as follows:
 v v V v Vy vy V ¼ eev þ epv ¼ ln þ ln ¼ ln þ ln þ ln þ ln V Vy V V vy V V
 p þ pe py0 þ pe ~ py þ pe p0 þ pe ~ ln ¼ ~ j ln þ ~ j ln  k ln j p0 þ pe p0 þ pe py0 þ pe py þ pe p þ pe p þ p y e ~Þ ln  ð~ kj ¼ ~ j ln p0 þ pe py0 þ pe

ev ¼ ln

ð13:9Þ

leading to v p þ pe j ln eev ¼ ln  ¼ ~ p0 þ pe V  V py þ pe py ~Þ ln ~Þ ln kj epv ¼ ln ¼ ð~ ffi ð~ kj py0 þ pe py0 V

9 > > = > > ;

ð13:10Þ

noting pe  py . Here, choosing py to coincide with the isotropic hardening function F, i.e. py ¼ FðHÞ; py0 ¼ F0 ðH0 Þ

ð13:11Þ

Equation (13.10) leads to p þ pe 9 > = p0 þ pe F ; ~Þ ln > epv ¼ ð~ kj F0 eev ¼ ~ j ln

ð13:12Þ

13.1

Isotropic Consolidation Characteristics

369

It follows from Eq. (13.12)2 that


 epv ~ ~ kj

Fðepv Þ ¼ F0 exp

ð13:13Þ

which is represented in terms of the general symbol H for the isotropic hardening as
FðHÞ ¼ F0 exp

H



~ ~ kj

ð13:14Þ

The strain rate is given from Eq. (13.12) as follows: 9    > p v > F > ~Þ > ev ¼ ¼ ~ j  ð~ kj > v p0 þ pe F> > >  >  =  p v V  j eev ¼  ¼ ~ > > v V p0 þ pe > >  >  > > V p > F > ~ ; ~Þ ev ¼ ¼ ðk  j F V 

ð13:15Þ

Adopting Eq. (13.15) and limiting to the infinitesimal strain with the Hooke’s law in Sect. 7.5 hereinafter, let the elastic bulk modulus K and the elastic shear modulus G in Eq. (7.84) of the elastic modulus tensor E in Eq. (7.84) with Eq. (7.101) be assumed as follows: 

rm



p

1 K ¼  ¼  ¼ ðp þ pe Þ; e e ~ j ev ev


 0 1 jjr jj p þ pe n G¼ ¼ G0 p0 þ pe 2 jje e0 jj

ð13:16Þ

where the facts that the shear modulus G depends on the pressure are taken into account. G0 is the reference value of G at p ¼ p0 and nð  1; n ffi 0:5 for common soils) is the material constant. The shear modulus in Eq. (13.16) increases depending on the pressure and its dependence can be described by the power function as verified by Tatsuoka et al. (1978) through the comparison to various test data. In addition, it is formulated to be applicable up to the negative pressure range. The hyperelastic relation based on Eq. (13.16) will be described in Sect. 17.11.2. In the case that there is no information for the determination of G0 , i.e. G-value in p ¼ p0 , we may calculate it by G0 ¼

3 1  2m 3 1  2m p0 þ pe ; K0 ¼ ~ j 2 1þm 2 1þm

with an appropriate value of the Poisson’s ratio m, noting Table 7.1. (b) Linear relation between void ratio and logarithm of pressure

ð13:17Þ

370

13 Constitutive Equations of Soils

The following e  ln p linear relation (e: void ratio) for the isotropic consolidation has been widely adopted for constitutive equation of soils after the Cam-clay models (Roscoe and Burland 1968; Schofield and Wroth 1968), although it possesses a lot of serious deficiencies described later in detail but unfortunately these deficiencies have not been recognized definitely. Normal  consolidation line: ey  ey0 ¼ k ln Swelling line: e  e ¼ j ln

p p0

py 9 > = py0 > ;

ð13:18Þ

where the material constants k and j are the slopes of normal-consolidation and swelling lines, respectively, in the (e; ln p) plane as shown in Fig. 13.2, where ðe0 ; p0 Þ, ðey0 ; py0 Þ and ðey ; py Þ are the void ratios and the pressures in the initial, the initial yield and the current yield states, respectively. In addition, e is the void ratio in the unloaded state to the initial pressure p0 . However, the e  ln p linear relation has the following physical impertinence. 1. The change of void ratio induced during a change of pressure from a certain pressure to other certain pressure (e.g. p0 to py0 in Fig. 13.2) along the swelling line is identical in spite of the plastic decrease of void ratio by the increase of a pre-consolidation pressure py . This defect is caused from the fact that the void ratio itself is adopted in vertical axis in the e  ln p linear relation, although the logarithm of volume is adopted the ln v  ln p linear relation. 2. The void ratio becomes negative if the pressure becomes large as py [ py0 expðey0 =kÞ or p [ p0 expðe=jÞ, noting Eq. (13.18). 3. The void ratio becomes infinitely large when the pressure approaches zero because of e ! 1 due to  ln p ! þ 1 for p ! 0 in Eq. (13.18)2 . This defect causes the serious problem in the deformation analysis in which pressure Fig. 13.2 Linear relation between void ratio and logarithm of pressure

13.1

Isotropic Consolidation Characteristics

371

decreases as seen in the footing-settlement analysis: Pressure in soils in the periphery of footing decreases to zero in non-cohesive soils and even to negative in cohesive soils and the cyclic mobility analysis for liquefaction in which an accurate prediction of deformation under a quite low pressure is required. 4. The pressure is not related to the volume but to the void ratio or specific volume (void ratio plus unity), although the volumetric strain is not defined by the change of void ratio but by the change of volume. Note here that the ratio of the current volume to the initial volume can be transformed to the ratio of the current specific volume to the initial specific volume as v=V ¼ ð1 þ eÞvs ðpÞ=½ð1 þ e0 Þvs ðp0 Þ ¼ ð1 þ eÞ=ð1 þ e0 Þ under the assumption of the incompressibility of soil particles themselves, i.e. vs ¼ const: where vs designates the volume occupied by the soil particles. 5. The e  ln p linear relation is not formulated by the logarithm of ratio of volumes but it is given by the difference of void ratios. Therefore, it does not fit to the logarithmic strain but it fits to the nominal strain, so that the nominal strain is obliged to be adopted in the derivation of volumetric strain from the e  ln p linear relation. However, the nominal volumetric strain does not coincide with 

the time-integration of trace of the strain rate e used in the elastoplastic constitutive equations in the subsequent chapters as known from Eq. (4.141) and it is impertinent to the description of large deformation, because (a) the strain is merely minus one even when the material vanishes completely by extreme volumetric compression as known from Eq. (4.141)2 , (b) it does not satisfy the superposition rule as known in Eq. (4.140)2 , (c) the sum of three longitudinal strains in the orthogonal directions does not coincide with the volumetric strain as shown in Eq. (4.141)2 . Because of the deficiency in 5, the e  ln p linear relation is obliged to be described by the nominal volumetric strain ev in Eq. (4.141)2 which is limited to the description of infinitesimal deformation as was delineated in Sect. 4.5. The nominal volumetric strain ev is additively decomposed into the elastic and the plastic parts as follows: ev ¼ eev þ epv

ð13:19Þ

where ev ¼

vV e vV p V V ; ev ¼ ; ev ¼ V V V

ð13:20Þ

Here, the italic letter e is used for the logarithmic strain but the roman letter e is used for the nominal strain applicable only to the infinitesimal deformation in order to distinguish them as was described in Sect. 4.5. It should be noted that 

the nominal strain cannot be related to the strain rate e in the exact sense as      known by ev ¼ tr e ¼ v =v 6¼ v =V ¼ ev . The following relations are used by substituting the specific volume instead of the volume into Eq. (13.20) on the approximation at the sacrifice of the above-mentioned deficiency 4.

372

13 Constitutive Equations of Soils

9 e  e0 > > > 1 þ e0 > > = e  e e ev ffi 1 þ e0 > > > e  e0 ðey0  e0 Þ þ ðey  ey0 Þ þ ðe  ey Þ > > ; epv ffi ¼ 1 þ e0 1 þ e0 ev ffi

ð13:21Þ

where the nearly equal symbol ffi is used to specify the approximation by the incompressibility of soil particles, noting that the symbols v and V in Eq. (13.20) are not the specific volumes but the volumes. Substituting Eq. (13.18) into Eq. (13.21), one has 9 j p k  j py > ev ffi  ln  ln > > > 1 þ e0 p0 1 þ e0 py0 > = j p e ev ffi  ln ð13:22Þ 1 þ e 0 p0 > > > j py0 k py j p0 k  j py > > ; epv ffi  ln  ln  ln ¼ ln 1 þ e0 p0 1 þ e0 py0 1 þ e0 py 1 þ e0 py0 from which the nominal volumetric strain rate is given by

9    e j p k  j py > > > ev ¼ ffi  ¼ > 1 þ e0 1 þ e 0 p 1 þ e0 p y > > > >    =   p e  e j  v V eev ¼ V  V ffi ¼ > 1 þ e0 1 þ e0 p > > >   >  > e k  j py p > > V ; ¼ ev ¼ V ffi 1 þ e0 1 þ e0 p y 



v V

ð13:23Þ

Equation (13.23) for the rates of volumetric strain rate and pressure is adopted widely for elastoplastic constitutive equations of soils after the Cam-clay model which is quite primitive model, while its basic concept was already clarified by Drucker and Prager (1952) and Drucker et al. (1957). Nevertheless, it causes the further deficiencies as follows: 6. It is not derived exactly but approximately from the e  ln p linear relation. 7. It cannot be adopted to describe finite deformation since it is derived based on the definition of nominal strain. 8. The tangent elastic bulk modulus is given by 

K

p 

eev

¼

1 þ e0 p j

ð13:24Þ

from Eq. (13.23). Here, the tangent elastic bulk modulus K depends on the initial void ratio e0 . It has a crucial physical impertinence: “The larger the initial

13.1

Isotropic Consolidation Characteristics

373

void ratio, the larger is the elastic bulk modulus. In other words, the looser the soil, the more difficult to be compressed”. Eventually, it is concluded that the e  ln p linear relation is inadequate physically and mathematically for formulation of constitutive equations for finite deformation of soils. On the other hand, all the deficiencies in the e  ln p linear relation can be excluded in the ln v  ln p linear relation. The nominal volumetric strain and its elastic and plastic parts are related to the strain rate and its elastic and plastic parts used in the elastoplastic constitutive equation as follows:

ð13:25Þ

which are of complicated forms containing the Jacobian and its elastic and plastic parts. Equation (13.25) have been used for constitutive equations under the approximation of J e ffi 1; J ffi J p by some workers (cf. Asaoka et al. 1997; Zhang et al. 2007). However, it is merely the approximation and thus the physical property cannot be remedied, so that it is limited to the description of infinitesimal deformation in spite of the complexity. Then, this modification causes no good and much harmful. A lot of constitutive equations of soils adopt the e  ln p linear relation, so that unfortunately they are applicable to the description of infinitesimal deformation. On the other hand, the ln v  ln p linear relation is applicable to the description of finite deformation. It has been adopted not only in hypoelastic-based plastic constitutive equations of soils (cf. Hashiguchi 1974, 1978; Hashiguchi and Ueno 1977; Hashiguchi and Chen 1998) but also in hyperelastic-based plastic constitutive equations of soils based on the additive decomposition of infinitesimal strain into elastic and plastic parts (e.g. Houlsby 1985; Collins and Hilder 2002; Coombs and Crouch 2011; Coombs et al. 2013). Further, it has been adopted in hyperelastic-based plastic constitutive equations of soils under the multiplicative decomposition of deformation gradient (e.g. Borja and Tamagnini 1998; Yamakawa et al. 2010, 2021) as will be described in Sect. 20.9.2. The material ~ in the ln v  ln p linear relation are approximately related to k and constants ~ k and j ~ ffi j=ð1 þ e0 Þ as described in j in the e  ln p linear relation by ~ k ffi k=ð1 þ e0 Þ; j Appendix F, while numerous test data have been accumulated for the latter.

374

13.2

13 Constitutive Equations of Soils

Yield Conditions

Yield conditions for soils are described in this section.

13.2.1

Yield Functions

Various yield conditions of soils have been proposed to date. The functions f ðrÞ of the yield condition f ðrÞ ¼ FðHÞ in Eq. (8.16) can be reduced to the following common form for soils which depend on the stress ratio and the Lode’s angle because of the frictional material. f ðrÞ ¼ pgðg=MÞ

ð13:26Þ

f ðrÞ ¼ pgðgm Þ

ð13:27Þ

or

where g

r0 ; p

g  jjgjj;

gm 

g M

ð13:28Þ

0

M is the stress ratio jjgjj in the maximum state of jjr jj in the fixed yield surface, i.e. the critical state and is called the critical state stress ratio. It is premised in Eq. (13.27) that the yield surface passes through the isotropic compression state and the null stress point. Then, the function gðgm Þ fulfill the following conditions g ¼ 1 for g ¼ 0 g ! 1 for g ! 1

 ð13:29Þ

Furthermore, note that the following equality holds in the critical state (cf. Appendix G). g0 ð1Þ ¼ gð1Þ

ð13:30Þ

which is illustrated in Fig. 13.3. Denoting the p-value in the critical state as pcr , the following equation is obtained by substituting Eq. (13.27) into Eq. (8.16) at gm ¼ 1. pcr ¼ F=gð1Þ

ð13:31Þ

13.2

Yield Conditions

375

Fig. 13.3 Function gðgm Þ in yield surface of soils

The function gðgm Þ for the yield surfaces proposed in the past are given as follows: (1) Original Cam-clay model (Schofield and Wroth 1968)


jjr0 jj=p g ¼ expðgm Þ; i:e: p exp M

 ¼F

pcr ¼ F=e

ð13:32Þ ð13:33Þ

where e is the Napier’s constant, i.e. e ¼ 2:71828. (2) Modified Cam-clay model (Burland 1965; Roscoe and Burland 1968) "

g¼

1 þ g2m ;


0  # jjr jj=p 2 i:e: p 1 þ ¼F M pcr ¼ F=2

ð13:34Þ ð13:35Þ

(3) Hashiguchi model (Hashiguchi 1972, 1985a) "
  # g2m 1 jjr0 jj=p 2 ; i:e: p exp ¼F g ¼ exp 2 2 M


pffiffiffi pcr ¼ F= e

ð13:36Þ ð13:37Þ

376

13 Constitutive Equations of Soils

The functions in (1) and (3) are unified as follows:


 
0   gnm 1 jjr jj=p n ; i:e: p exp ¼F g ¼ exp n n M

ð13:38Þ

pcr ¼ F= expð1=nÞ

ð13:39Þ

where Eqs. (13.32) and (13.36) are given for n ¼ 1 and 2, respectively. The above-mentioned three yield surfaces are shown for the axisymmetric stress states in Fig. 13.4, where 9 p   13 trr ¼  13 ðra þ 2rl Þ > >

3 > 1 3 0 >  2 ra  3 ðra þ 2rl Þ ¼  2 ra =  q  rl ra ¼ 0 ð13:40Þ 3 rl  13 ðra þ 2rl Þ ¼ 3rl > qffiffi qffiffi > > > ; jjr0 jj ¼ 2jrl  ra j ¼ 2jqj 3

3

where ra and rl are the axial and the lateral stress, respectively, in the triaxial compression/extension state, while the fact that the value of M in the axisymmetric compression is larger than that in the axisymmetric extension is taken into account in the following. Among the above-mentioned yield surfaces, only the yield surface of the modified Cam-clay model in Eq. (13.34) does not include any corner so that the singularity of the normal direction of the surface is not induced. Equation (8.16) with Eqs. (13.27) and (13.34) is rewritten as 

p  ðF=2Þ F=2

2



 jjr0 jj 2 þ ¼1 MF=2

ð13:41Þ

Axisymmetric compression

q

Hashiguchi (1972) model Modified Cam-clay model Original Cam-clay model

Mc

0

1

F Me e

F 2

F

e

F

p

Axisymmetric extension

Fig. 13.4 Various yield surfaces of soils

13.2

Yield Conditions

377

leading to jjr0 jj ¼ M

F F for p ¼ ðcritical stateÞ 2 2

Therefore, the magnitude of the deviatoric stress jjr0 jj depends on the material function M which is largest and smallest in the triaxial compression and the triaxial extension, respectively, in experiments. The pertinent equation of M will be formulated in the next subsection.

13.2.2

Critical State Surface Taken Account of Third Deviatoric Invariant

The simple equation of the critical state surface taken account of the third deviatoric invariant will be described in this subsection (Hashiguchi 2002). The Coulomb-Mohr criterion is given as follows (see Fig. 13.5):

ra  rl ra þ rl ¼ sin /c ðupper sign: Compresstion; lower sign: ExtenstionÞ 2 2 ð13:42Þ

where ra ð\0Þ and rl ð\0Þ are the axial stress and the lateral stress, respectively, and /c is the angle of the internal friction. Equation (13.42) is generalized to the three-dimensional state:

Fig. 13.5 Coulomb-Mohr yield criterion

378

13 Constitutive Equations of Soils 3  2  Y ri þ rj 2 ri þ rj sin /c ¼0  2 2 i;j¼1

ð13:43Þ

which is shown by the hexagon with the unequal radius length in the deviatoric stress plane as shown in Fig. 13.6. The expression of the Coulomb-Mohr criterion by the stress invariants is rather complicated as was given by Hashiguchi (1972). Equation (13.42) is rewritten in terms of the stress ratio Ral 

ra rl



[ 1 for triaxial compression \1 for triaxial extenssion

ð13:44Þ

as follows: sin /c ¼ 

Ral  1 ðupper: Compresstion; lower: ExtenstionÞ Ral þ 1

ð13:45Þ

i.e. Ral ¼

1  sin /c ðupper: Compresstion; lower: ExtenstionÞ 1 sin /c

ð13:46Þ

On the other hand, the ratio of the second deviatoric invariant to the pressure is given from Eq. (13.42) as follows: pffiffiffi Ral  1 jjr0 jj ðupper: compresstion; lower: extenstionÞ ¼ 6 p Ral þ 2

ð13:47Þ

noting qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 r0a 2 þ 2rl 2 ½ra  ðra þ 2rl Þ=3 2 þ 2½rl  ðra þ 2rl Þ=3 2 jjr0 jj ¼ ¼ p ðra þ 2rl Þ=3 ðra þ 2rl Þ=3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ra  ðra þ 2rl Þ=3 2 þ 2½rl  ðra þ 2rl Þ=3 2 =ðrl Þ ¼ ½ðra =rl Þ þ 2Þ=3 =ðrl Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½Ral  ðRal þ 2Þ=3 2 þ 2½1  ðRal þ 2Þ=3 2 ¼ ðRal þ 2Þ=3 pffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 6 ðRal  1Þ ¼ Ral þ 2

13.2

Yield Conditions

379

Substituting Eq. (13.46) into Eq. (13.47), we have 8 pffiffiffi > 2 6 sin /c p ffiffi ffi > 0  Mc for triaxial compresstion jjr jj 2 6 sin /c < 3  sin /c pffiffiffi ¼ ¼ > 2 6 sin /c 3 sin /c p > :  Me for triaial extension 3 þ sin /c

ð13:48Þ

Now, in order to formulate the appropriate smooth critical state surface fulfilling the convexity, we consider the conical surface given by 0

jjr jj ¼ Mðcos 3hr Þ p

ð13:49Þ

where pffiffiffi 0 cos 3hr  6trtr 3 ¼



1 for hr ¼ p=3 ðtriaxial compressionÞ 1 for hr ¼ 0 ðtriaxial extensionÞ

ð13:50Þ

0

0

tr 

r 0 ðjjtr jj ¼ 1Þ 0 jjr jj

ð13:51Þ

noting Eq. (1.324) for the Lode angle hr . Now, we will find the function Mðcos 3hr Þ in Eq. (13.49) by which a smooth convex conical surface is depicted. Then, assume the following simple equation which fulfills Eq. (13.48) in the triaxial-compression. pffiffiffi 2 6 sin /c Mðcos 3hr Þ ¼ 3  ð1  cÞ sin /c þ c sin /c cos 3hr 8 compresstion ð cos 3hr ¼ 1Þ > < Mc forptriaxial ffiffiffi

¼ Mc for c ¼ 0 2 6 sin /c > ¼ for triaxial extention ðcos 3hr ¼ 1Þ : 3  ð1  2cÞ sin /c Me for c ¼ 1

ð13:52Þ where cð0  c  1Þ is the material parameter. The variable Mðcos 3hr Þ in Eq. (13.52) coincides with Mc in the triaxial compression state but it is smaller than Me for c [ 1 and coincides with Me for c ¼ 1 in the triaxial extension state. In what follows, we will find the value of the material constant c which leads to the smallest value M in the triaxial extension state, satisfying the convexity of the conical surface. In order that the surface in Eq. (13.49) with Eq. (13.52) is convex, the curve formed by cutting it by the p plane (p constant plane) must satisfy following convexity condition (cf. Eq. (A.46) in Appendix H).

380

13 Constitutive Equations of Soils


2 1 d 1 1 þ 2 ¼ pffiffiffi ½3  ð1  cÞ sin /c þ c sin /c cos 3hr  9c sin /c cos 3hr M dhr M 2 6 sin /c 1 ¼ pffiffiffi ½3  ð1  cÞ sin /c  8c sin /c cos 3hr  0 2 6 sin /c

ð13:53Þ noting 9 !
 d 1 d 3  ð1  cÞ sin /c þ c sin /c cos 3hr 3c sin 3hr > > pffiffiffi pffiffiffi > ¼ ¼ > = dhr M dhr 2 6 sin /c 2 6
2 > d 1 9c cos 3hr > > pffiffiffi > ¼ ; 2 M dhr 2 6 In addition, Mðcos 3hr Þ takes the maximum value at hr ¼ p=3 (triaxial compression) and the minimum value at the triaxial extension at hr ¼ 0 (triaxial extension) (see Fig. 13.6), noting pffiffiffi dM 6 6c sin2 /c ¼ sin 3hr dhr ½3  ð1  cÞ sin /c þ c sin /c cos 3hr 2

Fig. 13.6 Sections of the critical state surfaces of soils in p–constant plane

ð13:54Þ

13.2

Yield Conditions

381

9 8 þ for hr ¼ p=3  0 = > > > > 0 for hr ¼ p=3 maximal for triaxial compression > > ; <  for hr ¼ p=3 þ90 ¼ >  for hr ¼ 0 = > > > > minimal for triaxial extension 0 for hr ¼ 0 > : ; þ for hr ¼ þ 0

ð13:55Þ

Therefore, the radius of the p section of the surface is the maximum and the minimum in the triaxial compression and the extension, respectively. Then, by setting hr ¼ 0 in Eq. (13.53), the following inequality must hold for the convexity of the conical surface in Eq. (13.49) with Eq. (13.52). c

3  sin /c 7 sin /c

ð13:56Þ

Equation (13.52) is reduced to the following equations for the particular values of the material constant c. 8 < Mc for pffifficffi ¼ 0 2 6 sin /c ð13:57Þ M¼ for c ¼ 1 : 3 þ sin /c cos 3hr Equation (13.57)1 leads to the Drucker-Prager yield condition and (13.57)2 was proposed by Satake (1972) and later by Gudehus (1973) and Argyris et al. (1973) as the failure criterion of soils. However, the frictional angle /c in the triaxial compression is limited to the following inequality in order to fulfill the convexity in Eq. (13.56) for c ¼ 1. 3 sin /c  ; i.e:/c  /ch  22:02 8

ð13:58Þ

Therefore, Eq. (13.57)2 is inappropriate to the material with the high frictional angle /c  22:02 . Now, let the rigorous function of Mðcos 3hr Þ be formulated, which satisfies the convexity condition in Eq. (13.53) for a whole range of frictional angle /c . Then, by choosing c to be the largest value satisfying the inequality in Eq. (13.56) for the convexity condition, i.e. c¼

3  sin /c 7 sin /c

ð13:59Þ

382

13 Constitutive Equations of Soils

M is finally determined p asffiffiffi pffiffiffi 2 6 sin /c 7 2 6 sin /c  ¼ M¼ /c /c 8 þ cos 3hr 3  sin /c 3  1  3sin sin /c þ 3sin 7 sin / 7 sin / sin /c cos 3hr c

c

i.e. pffiffiffi 7 14 6 sin /c Mðcos 3hr Þ ¼ Mc ¼ ð8 þ cos 3hr Þð3  sin /c Þ 8 þ cos 3hr

ð13:60Þ

which was proposed by Hashiguchi (2002) (see Fig. 13.6). Equation (13.60) is reduced to the following equation in the triaxial extension. pffiffiffi 7 2 6 sin /c 7 7 3 þ sin /c M¼ ¼ Mc ¼ Me for hr ¼ 0 ðtriaxial extensionÞ 9 3  sin /c 9 9 3  sin /c ð13:61Þ Therefore, M for Eq. (13.60) is smaller or larger than Me for the Coulomb-Mohr criterion in the triaxial extension state ðh ¼ 2nðp=3ÞÞ as known from the inequality 8 > < [ Me forp/ffiffiffic [ /ch M ¼ Me ¼ 2 6=9 for /c ¼ /ch ¼ 22:02 ð sin /c ¼ 3=8Þ for hr ¼ 0 ðtriaxial extensionÞ > : \Me for /c \/ch

ð13:62Þ The difference of the variable Mðhr Þ in the tri-axial compression and -extension states in most of test data is not so large as given by the Coulomb-Mohr criterion but it would be expressed approximately by Eq. (13.60). The other functions for M with some physical meanings are referred to Matsuoka and Nakai (1974), Nakai and Mihara (1984) and Matsuoka et al. (1999). However, their applications to general deformation analysis possess difficulties because they are formulated by the cubic equation of the radius of the failure surface in the deviatoric stress plane. Various modifications of the Coulomb-Mohr criterion have been proposed hitherto, e.g. Argyris et. al. (1973), William and Warnke (1974), Lin et al. (1987), Klisinski et al. (1991), Bardet (1990), Andrade and Borja (2006) but the most rigorous modification would be given by Eq. (13.49) with Eq. (13.60) only which the fulfillment of the convexity condition is verified.

13.3

13.3

Subloading Surface Model for Soils

383

Subloading Surface Model for Soils

The initial subloading surface model with the isotropic hardening is described in Chap. 9. In what follows, the simple initial subloading surface model with the modified Cam-clay yield surface will be formulated and compared with the other soil models. The following functions are adopted based on Eqs. (13.14), (13.34) and (13.60). "


0 #  g 2  jjr jj=p 2 ¼ p 1þ f ðrÞ ¼ p 1 þ M M H dF F FðHÞ ¼ F0 exp ¼ ; F0  ~j dH ~k  j ~ ~ k

9 > > > > > > > > > =

> > > H ¼ epv ; H ¼ tr ep ; fHn ð¼ H = kÞ ¼ trn > > > > > 7 > ; Mðcos 3hr Þ ¼ Mc 8 þ cos 3hr 





ð13:63Þ



p

The plastic modulus M in Eq. (9.24) is given for Eq. (13.63) as
p

M 

 trn UðRÞ n :r þ ~ R ~ kj

ð13:64Þ

and thus the plastic strain rate is described by p

e ¼





n :r n :r  n p n¼
trn UðRÞ M þ n :r ~ R ~ kj

ð13:65Þ

and thus 





n :r  n trn UðRÞ n :r þ ~ R ~ kj

e ¼ E1 : r þ


ð13:66Þ



Here, note that the subloading hardening, i.e. n : r [ 0 in Eq. (9.31) can be induced over the critical state line, fulfilling tr n  0 since the positive quantity p UðRÞ=R is contained in the plastic modulus M . The partial derivatives of the yield function in Eq. (13.63) are given as follows:

  g 2   g 2 @f ðp; g; MÞ @f ðp; g; MÞ @ pg ¼ 1þ ¼ P 1þ ¼ 2 2 ð13:67Þ ; @p @p M M @g M

384

13 Constitutive Equations of Soils

@f ðp; g; MÞ p  g 2 ¼ 2 @M M M @p @ðtrrÞ=3 1 ¼ ¼ I @r @r 3
   0 @g @jjr jj=p 1 1 0 1 0 1 0 ¼ ¼ 2 ptr þ jjr jjI ¼ tr þ gI @r @r p 3 p 3 pffiffiffi @M 14 6 sin /c M ¼ ¼ 2 @ cos 3hr 8 þ cos 3hr ð3  sin /c Þð8 þ cos 3hr Þ 3  sin /c ¼  pffiffiffi M2 14 6 sin /c pffiffiffi
 @ cos 3hr 3 6 0 2 1 1 0 ¼ 0 tr  pffiffiffi cos 3ht  jjtr jj2 I @r jjr jj 3 6

ð13:68Þ ð13:69Þ ð13:70Þ

ð13:71Þ

ð13:72Þ

@f ðrÞ @f ðp; g; MÞ @f ðp; g; MÞ @p @f ðp; g; MÞ @g ¼ ¼ þ @r @r @p @r @g @r @f ðp; g; MÞ @M @ cos 3hr þ @M @ cos 3hr @r 
  g 2  1 pg 1 0 1 Iþ2 2 tr þ gI ¼  1þ 3 M M p 3 ! pffiffiffi
  p g 2 3  sin /c 3 6 0 2 1 1 0 2 tr  pffiffiffi cos 3ht  jjtr jj2 I  pffiffiffi M2 0 jjr jj M M 3 14 6 sin /c 6
    pffiffiffi g 3  sin /c 0 2 1 g 2 g 0 1 1 0 2 pffiffiffi ¼ 1 t  pffiffiffi cos 3ht  jjtr jj I I þ 2 2 tr þ 3 6 3 M M M 7 6 sin /c r 3 6

ð13:73Þ noting the partial derivative formulae in Sect. 1.16. The predictions of the drained triaxial compression behavior of soils under the constant lateral stress by the subloading surface model in Eq. (13.66) and the conventional Cap model (Roscoe and Burland 1968; Schofield and Wroth 1968) are depicted in Fig. 13.7, where ea is the axial strain. The stress path is given by q = 3(p + rl) (rl = const.) because of p = –(ra + rl) / 3, q = rl – ra. Here, the curves of axial stress and volumetric strain vs. the axial strain in the loading from the heavily and the lightly over-consolidated states, i.e. points o and o0 , respectively are shown in this figure. In the loading from the lightly over-consolidated state o0 , the volume contraction proceeds and the axial stress increases monotonically up to the critical state. The abrupt transition from the elastic to the plastic state is predicted by the conventional Cam-clay model. On the other hand, the smooth behavior is predicted always by the subloading surface model as observed in experiments.

%

+

+

+

+

+

≥0

− t r n + U ) n:σ M p ≡( % R { λ − κ% { ≥0

%

%

+

Subloading Surface Model for Soils

Fig. 13.7 Comparison of predictions of triaxial compression behavior under constant lateral stress by the conventional Cam-clay model and the subloading surface model

%

13.3 385

386

13 Constitutive Equations of Soils

Next, consider the loading from the heavily over-consolidated state o. The elastic behavior is predicted until the stress reaches the yield surface and then the stress is predicted to decrease abruptly toward the critical state, exhibiting the intense softening by the Cam-clay model. On the other hand, the following realistic deformation behavior is predicted by the subloading surface model. (1) The first term in the plastic modulus in Eq. (13.64) decreases from the positive value to zero because of trn\0 in the process from the initial state up to the critical state line. On the other hand, the second term is always positive. Therefore, the plastic modulus is kept to be positive in this process, i.e. p ~Þ [ 0 ! 0, U=R [ 0, M [ 0 for o ! c. Both of the normal trn=ð~ kj 



hardening F [ 0 in Eq. (9.30) and the subloading hardening n : r [ 0 in Eq. (9.31) proceed in this process. (2) The first term becomes zero but the second term is positive at the point on the p ~Þ ¼ 0, U=R [ 0, M [ 0 at the point c. The critical state line, i.e. tr n=ð~ kj normal and the subloading hardenings proceed continuously rising up along this line. (3) The first term tends to be negative but the second term is positive, so that the plastic modulus is kept to be positive until the stress reaches the peak, i.e.  p ~j ~Þ\0, U=R [ 0, M [ 0 for c ! p. The normal softening F \0 trn=ðk 

but the subloading hardening n : r [ 0 proceed in this process. ~Þ ð\0Þ reaches the minimum value but the second (4) The first term trn=ð~ kj term U=Rð [ 0Þ is always positive so that the sum of them becomes zero, p canceling each other, resulting in M ¼ 0 at the peak stress point p. The normal softening proceeds continuously but the subloading hardening ceases at this 



point. The fact that ðev =j ea jÞmax : an the peak stress qmax : are induced concurrently has been recognized after (Taylor 1948). (5) The first term is negative but tends to increase toward zero while the second term is always positive but decreases continuously so that the sum of them ~Þ\0, becomes negative, exhibiting the subloading softening, i.e. tr n=ð~k  j p p U=R [ 0, M \0 and then reducing to the critical (residual) state ðM ¼ 0Þ as 

p ! c. Both of the normal softening F \0 and the subloading softening  n : r \0 proceed in this process. (6) The first and the second terms reach zero at the residual (critical) state, i.e. p ~Þ ¼ 0, U=R ¼ 0, M ¼ 0 at the final point c. Both of the normal tr n=ð~ kj and the subloading softening cease in this state. (7) The sign of the plastic volumetric strain rate is identical to that of tr n, whilst the sign of the elastic volumetric strain rate is opposite to that of the rate of pressure. The volume contraction is induced by the elastic volume contraction due to the increase of pressure in the initial stage of loading. Thereafter, the rate of plastic volume expansion tends to be larger than the rate of elastic volume contraction after passing through the critical state line, so that the rate of volume expansion

13.3

Subloading Surface Model for Soils

387

n DP Plastic potential surface assumed in non-associated Drucker-Prager model

n Normal-yield surface

σ Subloading surface

Drucker-Prager yield surface

0

Fig. 13.8 Outward-normal of subloading surface coinciding approximately with plastic potential surface assumed in Drucker-Prager model, nDP : unit normal to Drucker-Prager’s yield surface, n: unit normal to subloading surface and plastic potential surface used for Drucker-Prager’s yield surface (trn < trnDP) 

e



p proceeds. However, reaching the peak stress ðn : r ¼ M ¼ ev ¼ 0; p ¼ 0Þ at which the subloading surface expands at most and thus tr n becomes maximum 

p

leading to ev ¼ ev ¼ max: by virtue of the “associated flow rule”, the maximum 



ratio of volume expansion strain rate to axial strain rate, i.e. ev =j ea j ¼ max: are induced. This fact was indicated by Taylor (1948) based on the experimental evidence. Eventually, these typical deformation behavior in normally-consolidated and over-consolidated states can be described pertinently by the initial subloading surface model. Here, it should be emphasized that these realistic predictions are attained by the subloading surface model without resort to the non-associated flow rule. As described above, the conventional Cam-clay model predicts unrealistically high yield stress in the over-consolidated state. Then, the Cap model in which the over-consolidated side of Cam-clay yield surface is replaced by the Drucker-Prager yield surface (Drucker and Prager 1952) in Fig. 13.8 is widely used. However, the Cap model possesses various drawbacks described in the following. 1. The Cap model falls within the framework of conventional plasticity with the yield surface enclosing the purely elastic domain. Therefore, it predicts the stress–strain curve which rises up steeply (elastically) to the peak stress, and subsequently the stress decreases suddenly exhibiting a strong softening. In contrast, the subloading surface model can describe the realistic stress–strain curve with the smooth elastic–plastic transition since the plastic strain rate develops gradually as the stress approaches the yield surface. 2. The Cap model necessitates the yielding judgment, i.e. the judgment whether or not the stress lies on the yield surface in addition to the judgment on the direction of strain rate in the loading criterion as shown in Eq. (8.57). In contrast, the yielding judgment is not required in the subloading surface model since the stress lies always on the subloading surface playing the role of the loading surface as shown in Eq. (9.29).

388

13 Constitutive Equations of Soils

3. The Cap model requires the particular computer algorithm for pulling-back the stress to the yield surface when the increments of stress or strain with finite magnitudes are input in numerical calculations. Otherwise, it is obliged to perform numerical calculations with quite infinitesimal loading increments. In contrast, the subloading surface model enables numerical calculations with finite loading increments without incorporation of particular computer algorithm since it possesses an automatic controlling function to attract the stress to the yield surface in the loading process as was described in Sect. 9.3. 4. The cap model additionally adopts the plastic potential surface of the conical shape to predict a dilatancy angle lower than that predicted by applying the associated flow rule to the Drucker-Prager yield surface, avoiding an unrealistically large plastic volume expansion. Then, the formulation is obliged to adopt the non-associated flow rule which causes the serious physical drawbacks as described in Subsect. 8.6.4 (Hashiguchi et al. 1991). In contrast, the subloading surface model can use the associated flow rule, whereas the outward-normal n of the subloading surface in the current stress is approximately identical to the outward-normal nDP of the plastic potential surface adopted in the Drucker-Prager model as shown in Fig. 13.8. 5. The cap model is obliged to adopt the non-associativity for the Drucker-Prager yield surface as described in 4. Therefore, it is accompanied with the asymmetry of the elastoplastic stiffness modulus tensor Kep as shown in Eq. (8.40). This fact engenders the complexity in the formulation of variational principle and thus the difficulty in the analysis of boundary value problems. In contrast, the subloading surface model adopts the associativity leading to the symmetry of the elastoplastic stiffness modulus tensor. 6. The cap model predicts the failure surface which is determined uniquely by the Drucker-Prager yield surface itself, independent of the loading paths, because the interior of the yield surface is assumed to be a purely elastic domain and only the softening is induced when the stress reaches the Drucker-Prager yield surface. However, the surface depicted by connecting the peak stresses depends on the loading paths and its meridian section for the constant Lode angle is not straight but curved in real soils. In contrast, these facts can be described pertinently by the subloading surface model combined with the modified Cam-clay yield surface (cf. Hashiguchi et al. 2002). 7. The cap model is required the tension cut for the Drucker-Prager yield surface, which runs out sharply into the negative pressure range, when it is applied to the description of deformation in vicinity of zero pressure. In contrast, the subloading surface model does not require the tension cut because it adopts the normal-yield surface passing through the vicinity of the null stress state. 8. The cap model is accompanied with the singularity in the direction of outward-normal of the yield surface, i.e. plastic strain rate on the intersecting lines of the Drucker-Prager yield surface with the Cam-clay and the tension-cut yield surfaces. It results in unrealistic description of deformation behavior and would induce the computational difficulty in deformation analysis. In contrast,

13.3

Subloading Surface Model for Soils

389

200 Subloading surface model 150

||σ' || (kPa)

100 Drucker-Prager model (Non-AFR) Drucker-Prager model (AFR) 50

: Test data

0 0

䠉5

䠉10 a (%)

䠉15

䠉20

2.0 Drucker-Prager model (AFR) 1.5 1.0 v

(%)

0.5

Subloading surface model Drucker-Prager model (Non-AFR)

0.0 : Test data

䠉0.5 䠉1.0

0

䠉5

䠉10 a (%)

䠉15

䠉20

Fig. 13.9 Comparison of the calculated results by the Drucker-Prager and the subloading surface models with the test data (after Skempton and Brown 1961) of Weald clay for the drained triaxial compression with the constant lateral pressure

the subloading surface model adopts a single smooth normal-yield surface so that it is not accompanied with the singularity of the plastic modulus. 9. The cap model predicts the simultaneous occurrence of the peak stress and the maximum volume compression in over-consolidated clays and dense sands, in contradiction to experimental facts. In contrast, the subloading surface model provides the realistic prediction that the peak stress and the maximum ratio of volume expansion strain rate vs. axial strain rate occur simultaneously as observed in real soils: Figs. 13.9 and 13.10. 10. The cap model is required to incorporate at least two more material constants describing the inclinations of yield and plastic potential surfaces in addition to the material constants in the Cam-clay model. In contrast, the subloading surface model is required to incorporate only one more material constant u in the evolution rule of the normal-yield ratio despite the distinctively accurate description. In what follows, some comparisons of the simulations of typical triaxial test data by the Cap model and the subloading surface model are shown (Hashiguchi et al. 2002).

390

13 Constitutive Equations of Soils 4000 Subloading surface model

3000 ||σ' || (kPa) 2000

Drucker-Prager model (Non-AFR) Drucker-Prager model (AFR)

1000

: Test data

0 0

䠉5

䠉10

䠉15

䠉20

䠉25

a (%)

3 Drucker-Prager model (AFR)

2 1

v

(%)

0 Subloading surface model Drucker-Prager model (Non-AFR)

䠉1

: Test data 䠉2 0

䠉5

䠉10

䠉15

䠉20

䠉25

a (%)

Fig. 13.10 Comparison of the calculated results by the Drucker-Prager and the subloading surface models with the test data (after Stark et al. 1994) of kaolinite-silt mixtures for the drained triaxial compression with the constant lateral pressure

The simulations of the test data measured by Skempton and Brown (1961) for Weald clay subjected to the drained triaxial compression with a constant lateral stress are shown in Fig. 13.9 where the material constants and the initial value are selected as follows: ~ ~ ¼ 0:002; m ¼ 0:37; Mc ¼ 1:2; ð/c ¼ 36:17 Þ k ¼ 0:045; j My ¼ 0:574; Mp ¼ 0:071 for the Drucker  Prager yield surface; u ¼ 33 for the subloading surface model; F0 ¼ 330:0 kPa; whilst the initial stress state is r0 ¼ 67:0I kPa. Here, the Mc -value in Eq. (13.48) is given for the value of M, since the simulation is limited to the triaxial compression state. My and Mp are the inclinations of yield and the plastic potential 0 surface, respectively, of the Drucker-Prager model in the ðp; jjr jjÞ plane. The associated flow rule and the nonassociated flow rule are abbreviated as AFR and Non-AFR, respectively, in this figure. The similar simulations for the test data of kaolinite-silt mixtures measured by Stark et al. (1994) are shown in Fig. 13.10 where the material constants and the initial value are selected as follows:

13.3

Subloading Surface Model for Soils

391

~ ~ ¼ 0:006; m ¼ 0:3; Mc ¼ 1:051; k ¼ 0:1; j My ¼ 0:528; Mp ¼ 0:093 for the Drucker  Prager model; u ¼ 35 for the subloading surface model; F0 ¼ 6; 000:0 kPa; whilst the initial stress state is r0 ¼ 1275:0I kPa. The subloading surface model gives rise to the clearly better prediction than the Drucker-Prager model for both the axial stress-axial strain and the volumetric strain-axial strain curves. The curves predicted by the Drucker-Prager model are not smooth, which are formed by the three segments, i.e. the elastic, the elastoplastic and the critical state segments, whilst the former two form the concave curves of the ‘Eiffel-tower’ shape. Intense softening is induced rapidly lowering to the critical state immediately after the stress reaches the Drucker-Prager yield surface. However, note that the adoption of the non-associated flow rule in the Drucker-Prager model does not lead to the substantial improvement in simulation, whilst the subloading surface model adopting the associated flow rule gives the realistic prediction even for the volumetric strain. The parameter u is determined such that the stress–strain curve fit to the gentleness in the elastic–plastic transition. 1200 1000

||σ' || (kPa)

Drucker-Prager model Yield surface in Drucker-Prager model

800 600

Test data p0 (kPa) 104 553

400 200

Subloading surface model 0

0

200

400

600 p (kPa)

800

1000

1200

1200 1000

||σ' || (kPa)

Test data p0 (kPa) 104 553

Drucker-Prager model =0.30 =0.45

800 600 400 200

Subloading surface model 0

0

1

2

3

4

5

a (%)

Fig. 13.11 Comparison of the calculated results by the Drucker-Prager and the subloading surface models with the test data (after Bishop et al. 1965) for the undrained triaxial compression with the constant lateral pressure

392

13 Constitutive Equations of Soils

The simulations of the stress paths and the stress–strain curves to the test data measured by Bishop et al. (1965) for London clay subjected to the undrained triaxial compression are shown in Fig. 13.11 where the material constants and the initial value are selected as follows: ~ ~ ¼ 0:0063; Mc ¼ 0:82; k ¼ 0:022; j m ¼ 0:3 and m ¼ 0:45; My ¼ 0:62; Mp ¼ 0:21 for the Drucker  Prager model; m ¼ 0:3; u ¼ 70:0 for the subloading surface model; F0 ¼ 1; 700:0 kPa: The similar simulations for the test data of red clay measured by Wesley (1990) for red clay are shown in Fig. 13.12 where the material constants and the initial value are selected as follows:

400 Drucker-Prager model Yield surface of Drucker-Prager model

300

||σ' || (kPa) 200

Subloading surface model

100

0

Test data p0 (kPa) 50 100 250

100

0

400

200 p (kPa)

300

400

Drucker-Prager model =0.30 =0.43

300

||σ' || (kPa)

Test data 200

p0 (kPa) 50 100 250

100 Subloading surface model 0

0

2

4

6

8

10

a (%)

Fig. 13.12 Comparison of the calculated results by the Drucker-Prager and the subloading surface models with the test data (after Wesley 1990) for the undrained triaxial compression with the constant lateral pressure

13.3

Subloading Surface Model for Soils

393

~ ~ ¼ 0:012; Mc ¼ 1:015; k ¼ 0:035; j m ¼ 0:3 and m ¼ 0:43; My ¼ 0:767; Mp ¼ 0:24 for the Drucker  Prager model; m ¼ 0:3; u ¼ 20:0 for the subloading surface model; F0 ¼ 300:0 kPa: The test data are predicted fairly well by the subloading surface model. On the other hand, both the stress paths and the stress–strain curves predicted by the Drucker-Prager model are quite different from the test data, which are not smooth being formed by the three segments, where the Poisson’s ratio is selected two levels of m ¼ 0:30 and 0.45 in Fig. 13.11 and m ¼ 0:30 and 0.43 in Fig. 13.12. The simple (initial) subloading surface model has been widely applied to the analyses of soil deformation behavior (e.g. Hashiguchi and Ueno 1977; Hashiguchi 1978; Topolnicki 1990; Kohgo et al. 1993; Asaoka et al. 1997; Hashiguchi and Chen 1998; Chowdhury et al. 1999; Hashiguchi et al. 2002; Khojastehpor and Hashiguchi 2004a, b; Nakai and Hinokio 2004; Hashiguchi and Mase 2007; Wongsaroj et al. 2007 and many others).

13.4

Extension of Material Functions

Material functions contained in constitutive equation of soils formulated in the last section will be extended in order to describe the deformation behavior more realistically for the negative pressure range and the isotropic and anisotropic hardening behaviors.

13.4.1

Yield Surface with Tensile Strength

Consider the extended yield surface fulfilling the following conditions. (1) It includes not only positive but also negative pressure ranges. Here, note that the subloading surface is indeterminate at the null stress point when the stress reaches the null stress and thus the singular point of plastic modulus is induced since the normal-yield and the subloading surfaces pass through the null stress point which is thus to be the similarity-center of these surfaces in the initial subloading surface model described in Sect. 13.3. On the other hand, this problem is not induced in the extended subloading surface model because the similarity-center of the normal-yield and the subloading surfaces, i.e. elastic-core is not fixed at the null stress point and thus the subloading surface does not pass through the null stress point in general. The exclusion of the singularity of plastic modulus at the null stress point is of importance for the engineering design of soil structures because soils near the side edges of

394

13 Constitutive Equations of Soils

footings, soils at the pointed ends of piles, etc. are exposed to the null or further negative stress state. In addition, the incorporation of tensile yield strength is of importance for the engineering design of structures of natural soils such as soft rocks and cement-treated soils widely used recently, which have the tensile yield strength. Further, it should be noticed that the shift of the yield surface is of the importance also in the computational aspect. (2) The yield surface expands/contracts maintaining a similarity with respect to the origin of stress space so that the yield stress increases/decreases in all directions in the space. (3) For the sake of mathematical simplicity, the yield condition is described by a separate form consisting of the function of the stress, i.e. f ðrÞ and the function including the isotropic hardening variable, i.e. the isotropic hardening function FðHÞ which describes the size of the yield surface. Here, the function f ðrÞ must be a homogeneous function of the stress tensor r in order to fulfill the above-mentioned conditions (2) and (3). Equation (13.41) becomes the following equation through the translation of the yield surface to the negative pressure range by nh F ðp ! p þ nh FÞ (Hashiguchi 2007d; Hashiguchi and Mase 2007). 

p  ðð1=2Þ  nh ÞF F=2

2



2 0 jjr jj þ ¼1 MF=2

Fig. 13.13 Yield surface of soils extended to negative pressure range

ð13:74Þ

13.4

Extension of Material Functions

395

leading to ð13:740 Þ

ð1  nh Þnh F 2 þ ð1  2nh ÞpF  ðp2 þ q2 Þ ¼ 0 where 0

q

jjr jj M

ð13:75Þ

nh is the material constant, while it must fulfill nh  1=2 since the tensile yield stress is lower than the compression yield stress and further the inequality nh \pe =F is required, since the volume does not become infinite by the elastic deformation inside the yield surface, i.e. for p [  nh F. The yield surface in Eq. (13.74), i.e. ~ is given from (13.75) is depicted in Fig. 13.13 for the axisymmetric stress state. M 0 e  2nh ÞF=2 as follows: the relation jjr jj ¼ MF=2 ¼ Mð1 e ¼ M

1 M 1  2nh

ð13:76Þ

The frictional angle /c in the triaxial compression is described from Eqs. (13.61) and (13.76) as follows: /c ¼ sin1




3Mc pffiffiffi 2 6 þ Mc



¼ sin1

ec 3ð1  2nh Þ M pffiffiffi ec 2 6 þ ð1  2nh Þ M

! ð13:77Þ

e c are the values of M and M e in the axisymmmetric compression where Mc and M stress state. Equation (13.75) can be expressed in the separated form of the function f ðp; qÞ of the stress and internal variable and the hardening function F, i.e. 8 2 > < p½1 þ ðq=pÞ for nh ¼ 0 f ðp; qÞ ¼ F; f ðp; qÞ ¼ 1 > : ~ ðpq  nh pÞ for nh 6¼ 0 nh

ð13:78Þ

where ~ nh  2ð1  nh Þnh ; nh  1  2nh ; pq 

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2 þ 2~nh q2

ð13:79Þ

396

13 Constitutive Equations of Soils

The partial-derivative of the function @f ðp; qÞ=@r for nh 6¼ 0 is given as follows:
 @f ðp; qÞ 1 @pq @p  nh ¼ ~ @r @r nh @r 2

3
 16 1 @p @q @p 7 þ 4~ nh q  nh 5 ¼ 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p ~ @r @r @r nh 2 p 2 þ 2 ~ nh q2 1 3 0 2 1 6 1B p 2n~h @q7 C ¼ 4 @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  nh AI þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q 5 ~ 3 @r nh p2 þ 2 ~ nh q 2 p2 þ 2~nh q2

ð13:80Þ

where
 @p @ 1 1 ¼  trr ¼  I @r @r 3 3

ð13:81Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2 þ 2~nh q2


 1 @ðp2 þ 2~nh q2 Þ 1 @p @q ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p þ 4~nh q @r @r @r @r 2 p2 þ 2~nh q2 2 p2 þ 2~nh q2 p ffiffi ffi 
 1 2 1 0 3 6 1 1 0 0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  pI þ 4~nh q ðtr þ tr 2  pffiffiffi cos 3htr  I 3 M 8 þ cos 3hr 3 6 2 p2 þ 2~nh q2

@pq @ ¼ @r

ð13:82Þ noting pffiffiffi
 @ cos 3hr 3 6 0 2 1 1 0 ¼ 0 t  pffiffiffi cos 3htr  I @r jjr jj r 3 6
 @M @ 7 7@ cos 3hr =@r Mc ¼ Mc ¼ @r @r 8 þ cos 3hr ð8 þ cos 3hr Þ2  pffiffiffi 0 0 21 6 tr 2  p1ffiffi6 cos 3htr  13 I Mc ¼ ð8 þ cos 3hr Þjjr0 jj 8 þ cos 3hr pffiffiffi 0 0 3 6 tr 2  p1ffiffi6 cos 3htr  13 I M ¼ ð8 þ cos 3hr Þjjr0 jj

ð13:83Þ

ð13:84Þ

13.4

Extension of Material Functions

397


0  @jjr0 jj 0 M  jjr jj @M @q @ jjr jj @r ¼ ¼ @r M2 @r @r M 1 0 pffiffiffi 0 2 0 p1ffiffi cos 3ht  1 I 3 6 t  r r 3 1 0 6 ¼ 2 @t r M þ MA 8 þ cos 3hr M pffiffiffi
 1 0 3 6 1 1 0 0 Þ tr 2  pffiffiffi cos 3htr  I ¼ ðtr þ M 8 þ cos 3hr 3 6

ð13:85Þ

In the above, the yield surface of soils is formulated so as to fulfill the conditions (1)–(3) based on the modified Cam-clay model. It is difficult to derive the other yield surface fulfilling the conditions (1)–(3). For instance, consider the translation of the original Cam clay model to the negative pressure range by p ! p þ nh F.


0

jjr jj =M ðp þ nh FÞ exp p þ nh F

 ¼F

ð13:86Þ

However, a separated form into the function of stress and internal variables and the hardening function cannot be derived from this equation. On the other hand, the translation of the yield surface to the negative pressure range by the constant value Cy ðp ! p þ Cy Þ is adopted for constitutive equations for unsaturated soils (e.g. Alonso et al. 1990; Simo and Meschke 1993; Borja 2004). The modified Cam-clay model, for instance, is described by this translation as follows: 

p  ð1=2ÞF þ Cy F=2

2

 þ

2 0 jjr jj ¼1 MF=2

ð13:87Þ

In this equation, the yield surface expands/contracts from/to the fixed point r ¼ Cy I ðp ¼ Cy Þ on the hydrostatic axis and thus it does not fulfill the condition (2). The incorporation of this yield condition into the subloading surface model leads to the physical impertinence that the unloading without a plastic deformation is induced against the fact that a large plastic deformation (expansion) is induced when the stress translates towards the negative pressure direction from the origin of stress space. The computer program for soil deformation analysis based on the formulations described so far is given in the Appendix L(b).

13.4.2

Rotational Hardening

The inherent anisotropy represented in the orthotropic anisotropy described in Sect. 12.7 cannot be ignored in metals and woods. On the other hand, the induced

398

13 Constitutive Equations of Soils

Fig. 13.14 Inadequacy of kinematic hardening for description of anisotropy of soils: Stress can never return to null state

anisotropy is more dominant in soils since soils are assemblies of particles with weak cohesions between them and thus the rearrangement of soil particles is induced easily. Here, the yield surface of soils must always include the origin of stress space but does very slightly because of the weak cohesion. Besides, the remarkable softening (contraction of the yield surface) by the plastic volume expansion is induced as the stress approaches the null stress state. Then, the stress can never return to the origin of stress space, once the yield surface translates so as not to include the origin, as illustrated on the (p, q) plane for the axisymmetric stress state in Fig. 13.14. Therefore, the kinematic hardening in Eq. (8.89) is not applicable to soils. General speaking, the stress in pressure-independent materials would correspond to the stress ratio, i.e. the ratio of the deviatoric stress versus pressure in pressure-dependent, i.e. frictional materials such as soils and further the translation of yield surface, i.e. the kinematic hardening in the former would correspond to the rotation of yield surface in the latter. The description of anisotropy of soils by the rotation of the Cam-clay yield surface was proposed by Sekiguchi and Ohta (1977), 0 0 replacing the deviatoric stress r to the novel variable r  pb ðtr b ¼ 0Þ in the yield condition. This concept was called the rotational hardening in contrast to the kinematic hardening for pressure-independent materials and the second-order deviatoric dimensionless tensor b is referred to as the rotational hardening variable by Hashiguchi (1977). Then, the yield condition in Eq. (13.74) or (13.78) is extended as follows (Hashiguchi and Mase 2007):

p  ½ð1=2Þ  nh F F=2

2

2

32 _0 jjr jj 5 ¼1 þ4_ M F=2

ð13:88Þ

13.4

Extension of Material Functions

399

Fig. 13.15 Rotated yield surface in the (p, q) plane

i.e.

( _

_

f ðp; qÞ ¼ F; f ðp; qÞ ¼

_

p½1 þ ðq=pÞ2 for nh ¼ 0 1 _ ~n ðpq  nh pÞ for nh 6¼ 0

ð13:89Þ

h

where _0

r  r0  pb

ð13:90Þ

_0

_

q

jjr jj

ð13:91Þ

_

M qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi _2 _ _2 nh q pq  p þ 2~

ð13:92Þ

pffiffiffi 14 6 sin /c M ðcos 3h_ Þ ¼ r ð3  sin /c Þð8 þ cos 3h_ Þ _

ð13:93Þ

r

cos 3h_  r

_0 pffiffiffi 0 3 0 r 6trt _ ; t _  _0 r r kr k

ð13:94Þ

The yield surface in Eq. (13.88), i.e. Equation (13.89) is depicted in Fig. 13.15 for the axisymmetric stress state. The evolution rule of rotational hardening tensor b is given below (Hashiguchi and Chen 1998; Hashiguchi 2001a). The following assumptions are adopted for the formulation of the evolution rule. (1) Rotation of the yield surface is induced only by the deviatoric component of the plastic strain rate independent of the mean component. (2) The rotation ceases when the central axis of yield surface reaches the surface, called the rotational limit surface, which exhibits the conical surface having the summit at the origin of stress space. Let the rotational hardening limit surface be given by

400

13 Constitutive Equations of Soils

Normal-yield surface 3

Central axis η= β

Hydrostatic axis η=0

Rotational limit surface ||η|| = M r

2

1

Fig. 13.16 Yield surface and rotational limit surfaces (illustrated in the principal stress space)

kgk ¼ Mr

ð13:95Þ

where Mr is the stress ratio in the rotational hardening limit surface, called the rotational limit stress ratio, and let it be given following Eq. (13.60) by pffiffiffi 14 6 sin /r ð13:96Þ Mr ðcos 3h_ Þ ¼ r ð3  sin /r Þð8 þ cos 3hrÞ _

/r being the material constant, called the rotational limit angle. (3) The central axis of yield surface g ¼ b rotates towards the conjugate line g ¼ Mr t_ on the rotational limit surface, where the conjugate line is the genr

erating line of the rotational limit surface which is observed from the hydrostatic axis in the same direction observed from the central axis g ¼ b of the yield surface to the current stress (see Fig. 13.16). Based on the above-mentioned assumptions, let the following evolution rule of rotational hardening be postulated in the form in Eq. (8.89) with the replacements of a ! b; ck ! br ; bk F ! Mr as follows:
  1  p0  p0 b ¼ br e  jj e jjb Mr

ð13:97Þ

where br is the material constant. Here, it is noteworthy that the rotational hardening is not induced since the deviatoric strain rate is not induced when the stress

13.4

Extension of Material Functions

401

Fig. 13.17 Relation of axial component of rotational hardening variable versus axial plastic strain in the axisymmetric stress state

lies on the central axis of the subloading surface in the modified Cam-clay model,  p0



i.e. e ¼ O ! b ¼ O for g ¼ b as illustrated in Fig. 13.15. Here, needless to say, the deviatoric plastic strain rate is adopted in Eq. (13.97) since the anisotropic hardening is independent of plastic volumetric strain rate. Equation (13.97) is described in one-dimensional state as follows: 

ba ¼ br




 ba   p 1   ea Mr

ð13:98Þ

which is shown in Fig. 13.17.

13.5

Extended Subloading Surface Model

Elastoplastic constitutive equation for describing cyclic loading behavior of soils is formulated below by incorporating the rotational hardening instead of the kinematic hardening into the extended subloading surface model described in Chap. 11. Further, the supertield surface proposed by Asaoka et al. (2000) is incorporated in order to describe the degradation of soil skeleton structure by which the deformation behavior of soils with different void ratios may be described uniformly.

13.5.1

Superyield, Normal-Yield and Subloading Surfaces

Setting a ¼ O in Eqs. (11.1) and (11.2) and incorporating the rotational hardening b, the normal-yield and the subloading surfaces for soils are given as follows (see Fig. 13.18):

402

13 Constitutive Equations of Soils

Fig. 13.18 Rotated normal-yield, subloading, elastic-core and limit elastic-core surfaces shown in the (p, q) plane

f ðr; bÞ ¼ FðHÞ

ð13:99Þ

f ðr; bÞ ¼ RFðHÞ

ð13:100Þ

where the following relation holds by virtue of the similarity of the subloading surface to the normal-yield surface. _

r  r  að¼ Rry Þ ¼ r þ Rc

ð13:101Þ

a  ð1  RÞc ðac ¼ Rða  cÞ; a ¼ 0Þ

ð13:102Þ

with

leading to 





a ¼ ð1  RÞ c  R c

ð13:103Þ

where _

r rc

ð13:104Þ

a is the conjugate (similar) point in the subloading surface to the reference point að¼ 0Þ in the normal-yield surface as shown in the ðp; qÞ plane in Fig. 13.18. ry is the conjugate point on the normal-yield surface to the current r on the subloading surface.

13.5

Extended Subloading Surface Model

403

Now, introduce the superyield surface (Asaoka et al. 2000) (Fig. 13.19(a)): f ðr; bÞ ¼ RFðHÞ

ð13:105Þ

where Rð  1Þ is the super-normal-yield ratio designating the ratio of the size of the superyield surface to the size of the normal-yield surface, which represents the degree of the soil skeleton structure (larger in loose soils). The subloading surface in Eq. (13.100) is described by f ðr; bÞ ¼ RRFðHÞ

ð13:106Þ

where Rð  R  1Þ is the super-normal-yield ratio designating the ratio of the size of the subloading surface to the size of the superyield surface. The subloading surface model combined with the superyield surface is called the superyield-subloading surface model, abbreviated as SYS model, by Asaoka et al. (2000, 2002) which provides the primitive formulation: (1) The extended subloading surface model is not incorporated but the initial subloading surface model is incorporated, (2) The yield surface is limited to the positive pressure range and obeys the irrational rotational (rate-linear) hardening rule and (3) The isotropic hardening function is formulated based on the e  ln p linear relation limited to the infinitesimal deformation as was described in Sect. 13.1(b). Let the evolution rules of the super-normal yield ratio R based on the assumption that the super-yield surface shrinks by the deviatoric plastic strain rate and shrinks/ expands by the plastic volumetric contraction/expansion and the super-yield ratio R based on the assumption that the subloading surface approaches the super-yield surface in the plastic loading process be given by ð13:107Þ where ð13:108Þ cv ð  1Þ; u; a and u are the material constants and u is the material function which will be formulated in Eq. (13.130). The material constant cv would be larger in clays than sands. The following relation for the normal-yield ratio R in terms of the super-normal yield ratio R and the super-yield ratio R holds from Eq. (13.107) with Eq. (13.108) noting Eq. (13.106).

ð13:109Þ

404

13 Constitutive Equations of Soils

q

Subloading surface f (σ, β) = RF ( H )

Normal-yield surface f (σ, β) = F ( H )

superyield surface f (σ, β) = RF ( H )

σ

p

0

(a) Various basic surfaces in the (p, q) plane.

q

q

Stress path

Stress path

p

0 Loose sands

0

Dense sands

p

(b) Stress path under constant volume or undrained condition.

Fig. 13.19 Various basic surfaces and undrained stress path based on super-yield subloading surface model proposed by Asaoka et al. (2000)

The undrained stress paths in loose and dense sands can be predicted by the SYS model with one same set of material parameters as shown in Fig. 13.19(b) without resorting to the plastic deviatoric hardening (Nova 1977; Wilde 1977) as will be known from the plastic modulus shown in Eq. (13.129). The material-time derivative of Eq. (13.100) reads:   @f ðr; bÞ  @f ðr; bÞ  @f ðr; bÞ  :r :a þ :b ¼ RF þRF @r @r @b

ð13:110Þ

which can be rewritten as "

 
 # 1 @f ðr; bÞ  F R :b r ¼ 0 n : r n : a þ r þ r  F R RF @b 



ð13:111Þ

13.5

Extended Subloading Surface Model

405

where n

@f ðr; bÞ @r



 @f ðr; bÞ    @r ðjjnjj ¼ 1Þ

ð13:112Þ

noting the following equation based on the Euler’s theorem for the function of r in homogeneous degree-one. 1 ¼ @f ðr; bÞ @r

13.5.2

@f ðr; bÞ :r f ðr; bÞ RF @r n¼ n¼ n n:r n:r n:r

ð13:113Þ

Evolution Rules of Internal Variables

The evolution rules of the internal variables are formulated in the following. (a) Rotational hardening variable The rate of rotational hardening is given by extending Eq. (13.97) as follows: 0 1   p0    1  p0   ð13:114Þ b ¼ br @e  a e bA ¼ f bn k Mr where

pffiffiffi 14 6 sin /r M r ðcos 3hrÞ ¼ ð3  sin /r Þð8 þ cos 3hrÞ _

_

_

_0

_0 0

0

r  r pb; tr  _

r

_0

jjr jj 0

; cos 3hr 

f bn  br @n0 

_

pffiffiffi 0 3 6tr tr _

ð13:115Þ

ð13:116Þ

1 1 a

kn0 kbA

ð13:117Þ

Mr /r is the angle designating the limit of the rotation of the normal-yield surface. (b) Elastic-core To avoid the unlimited approach of the elastic-core to the normal-yield surface, first let the following surface, called the elastic-core surface, be introduced as shown in Fig. 13.18, which passes through the elastic-core c and possesses a similar shape and orientation to the normal-yield surface with respect to the null stress ðr ¼ a ¼ 0Þ

406

13 Constitutive Equations of Soils

ð13:118Þ . where the variable is the ratio of the size of the elastic-core surface to that of the normal-yield surface, called the elastic-core yield ratio. It plays the role of a measure for the approaching degree of the elastic-core to the normal-yield surface. Since the elastic-core must lie inside the normal-yield surface as described above, the elastic-core yield ratio has to be less than unity. Then, the inequality ð13:119Þ must hold, where vð\1Þ is a material constant exhibiting the maximum value of > > > < p4 1 þ

_!2

q p

3 5

> > 1 _ > > : ð p q  nh pÞ ~ nh

for nh ¼ 0

ð13:132Þ

for nh 6¼ 0

where p  ðtr rÞ=3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi _2 _ pq  p2 þ 2~ nh q

ð13:133Þ ð13:134Þ

_0

_

q

jjr jj

ð13:135Þ

_

M _

M ðcos 3hrÞ ¼ _

pffiffiffi 14 6 sin /c ð3  sin /c Þð8 þ cos 3hrÞ _

ð13:136Þ

_0

pffiffiffi r cos 3hr  6tr tr0 3 ; tr0  _0 ðjjt0_ jj ¼ 1Þ r jjr jj _

13.5.5

_

_

ð13:137Þ

Partial Derivatives of Subloading Surface Function

The partial derivatives of the function in Eq. (13.132) are shown below. 8 _ _2 > > 1 þ ðq=pÞ2 þ pð2p33 Þq for nh ¼ 0 > _ 1 @f ðp; qÞ < 0 ¼ 1 1 2p > @p @ >  nh A for nh 6¼ 0 > _ :~ nh 2 pq 8 !2 _ ð13:138Þ > > q > > 1  for n ¼ 0 > h > < p 0 1 ¼ > > 1 p > > @  nh A for nh 6¼ 0 > > _ :~ nh p q

410

13 Constitutive Equations of Soils

_

@f ðp; qÞ _

@q

¼

8 _ > q > > >2 < p

for

nh ¼ 0 ¼

_

> 1 1 4~ nh q > > > _ :~ 2 n pq

nh 6¼ 0

for

8 _ > q > > >2 < p

for

nh ¼ 0 ð13:139Þ

_

> q > > > :2_ pq

for

nh 6¼ 0

@p 1 ¼ I @r 3

ð13:140Þ

@r0 0 ¼I @r

ð13:141Þ

0

@rij @ðrij þ pdij Þ 1 1 ¼ ¼ ðdik djl þ dil djk Þ  dij dkl @rkl @rkl 2 3

!

_0

@r 1 0 ¼I þ bI @r 3

0

ð13:142Þ

1 _0 0 @rij @ðrij  pbij Þ 1 1 1 @ ¼ ¼ ðdik djl þ dil djk Þ  dij dkl þ bij dkl A @rkl 2 3 3 @rkl pffiffiffi 14 6 sin /c

_

@M _

@ cos 3hr

¼

_

ð3  sin /c Þð8 þ cos 3hrÞ2 _

3  sin /c _ 2 pffiffiffi ¼ ¼  M _ 14 6 sin /c 8 þ cos 3hr M

@tr0 _

_

¼

@ r0 0

0 _

B _0 B B @ t ij B 0 ¼ B _ B@r @ kl

¼

0 _

r ij ffi @ qffiffiffiffiffiffiffiffiffi 0 0

@ r ij 0 _

_ _

r rs r sr

_

0

¼

@ r kl

1 2 ðdik djl

_

jjrjj

0

0

_

0

0

0

ðI  tr  tr Þ _

qffiffiffiffiffiffiffiffiffiffiffi ffi 0 0 0 _ _ _ @ rrs rsr  rij _

@rkl

1

ð13:143Þ

!

ð13:144Þ

_

qffiffiffiffiffiffiffiffiffi ffi 0 0 _ _

r rs r sr 0 _

@ r kl

rrs rsr

qffiffiffiffiffiffiffiffiffiffiffi ffi 0 0 _ _ ^0ij t0 kl þ dil djk Þ rrs rsr  r _ _

0

_

0

rrs rsr

r





1

C 1 1 0 0 ffi ¼ qffiffiffiffiffiffiffiffiffiffiffi ðdik djl þ dil djk Þ  tr ij tr kl C 0 0 A _ _ 2 rrs rsr _

_

13.5

Extended Subloading Surface Model

411

@ðtrt0 r3 Þ ¼ 3t0 r2 @t0 r _

ð13:145Þ

_

_

0

1 0

0

B@tr rs tr st tr @ 0 _ @ t ij _

_

_

0

tr

0

¼ dir djs tr st tr _

_

0

_

@ cos 3hr _

0

¼

tr

_

3 _

_

0

tr

0 0 0 0 C þ tr rs tr st dit djr ¼ 3trir trrj A _

_

_

pffiffiffi 0 _ 0 ð 6tr 2  cos 3hrtr Þ _

0

jjr jj

@r

0

þ tr rs dis djt tr

_

ð13:146Þ

_

0 pffiffiffi 1 _ _ 0_ 0_ 0_ 0_ 0_ 0_ pffiffiffi @t0 r pffiffiffi 0 _ 0 _ @t0 r         r r r r r r @ 6 t t t t t @t lm mn nl lm mn nl rs rs @ sm t r nr _0 A ¼ 6 ¼ 3 6t r _ _0 _0 rs @t0 r @ rij @ rij @ rij

 pffiffiffi 0 _ 0 _  1 1 0_ 0_ sn t r nr qffiffiffiffiffiffiffiffiffiffiffiffiffi rs t r ij dri dsj þ drj dsi  t r ¼ 3 6t r _ _ 2 pq r qp r  _ _ pffiffiffi _ _ _ _ 1 in t0 r nj  t0 r sn t0 r nr t0 r rs t0 r ij ¼ 3 6 qffiffiffiffiffiffiffiffiffiffiffiffiffi t0 r 0 0 _ _ qp pq r r pffiffiffi _ _ _ 3 0 0 0    t  cos 3h r t ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r r r 6 t in nj ij pq t0 r qp t0 r _

@q

_0

@r

0

 ¼ _ t0_ þ 1

r

M

_

_

@ @q ¼ q@q ffiffiffiffiffiffiffiffiffiffiffiffiffi _0 _0 _0 @rij @ rpq rqp

3 8 þ cos 3h_

pffiffiffi 0 ð 6t0_ 2  cos 3h_ t_ Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffi _0 _0 @ rpq rqp _0

_

þ

@q _

ð13:147Þ

r r

r

r



_

@M @ cos 3h_

@ cos 3h_ _0

r

@rij @rij @M r qffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 _ _ _ @ rpq rqp @M @ cos 3h_r rij 1 q ffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ _  _0 _2 _0 _0 @ cos 3h_ @r M @ rpq rqp ij M r 2 31 pffiffiffi 0 0 14 3 ¼ _ t_ ij þ ð 6t_ ir t_ rj  cos 3h_ t_0 ij Þ5A 8 þ cos 3h_ r r r r r M r ( " _ #) _ _ _ _ @f ðp; qÞ 1 @f ðp; qÞ @f ðp; qÞ @q 1 @q Iþ  tr _0 ðI  bÞ I ¼ _ _0 @r 3 @p 3 @q @r @r _0

412

13 Constitutive Equations of Soils _

_

_

_

_

0

@f ðp; qÞ 1 @f ðp; qÞ @p @f ðp; qÞ @q @rrs ¼ þ 0 _ _ @rij 3 @p @rij @q @rrs @rij _ _ _   1 @f ðp; qÞ @p @f ðp; qÞ @q 1 1 1 ðdri dsj þ drj dsi Þ  drs dij þ brs dij ¼ þ 0 _ _ 3 @p @rij 3 3 @q @rrs 2 _ _ _   1 @f ðp; qÞ @f ðp; qÞ @q 1 1 1 dij þ ¼ d þ d d Þ  d þ d ðd d b ri sj rj si rs ij ij 0 _ _ 3 @p 3 3 rs @q @rrs 2 8 2 3 " _ #) _!2 _( _ > > 1 q q @ q 1 @ q > > 5I þ 2 >   41  0 :ðI  bÞ I > _0 > p p 3 3 _ > > _ @ r @ r > @f ðp; q Þ < for nh ¼ 0 0 1 ¼ " _ #) _ ( _ > @r > q @q 1 @q > 11@p > A >   0 :ðI  bÞ I _  nh I þ 2 _ > _0 > 3~ 3 _ nh p q > p > @ r @ r q > : for nh 6¼ 0 ð13:148Þ _

_

_

@f ðp; qÞ @f ðp; qÞ @q ¼ p _ _0 @b @q @ r _

_

_

_

0

_

ð13:149Þ

_

@f ðp; qÞ @f ðp; qÞ @q @rrs @f ðp; qÞ @q ¼ ¼ 0 0 ðpI rsij Þ _ _ _ _ @bij @q @rrs @bij @v @rrs _

¼ p

_

_

_

1

@f ðp; qÞ @q 1 @f ðp; qÞ @q A ðdrj dsj þ drj dsi Þ ¼ p 0 0 _ _ _ _ 2 @q @rrs @q @rrs

8 _ > _ @q > > > _ > 2q _0 for nh ¼ 0 @f ðp; q Þ < @r _ ¼ > @b > p _ @q > > > : 2 _ q _0 for nh 6¼ 0 pq @ r 8 _ < pc f1 þ ðqc =pc Þ2 g for nh ¼ 0 _ 1 _ f ðc; bÞ ¼ f ðpc ; qc Þ ¼ : ~ ðpqc  nh pc Þ for nh 6¼ 0 nh

ð13:150Þ

ð13:151Þ

13.5

Extended Subloading Surface Model

413

where 1 pc   tr c; 3 _

0

0

c  c þ pc I

ð13:152Þ

0

c  c  pc b

ð13:153Þ

_0

_

qc  _

M c ðcos 3h_ Þ ¼ c

jjc jj

ð13:154Þ

_

Mc

pffiffiffi 14 6 sin /c ð3  sin /c Þð8 þ cos 3h_ Þ

ð13:155Þ

c

0

_ pffiffiffi 0 3 c 0 6trt _ ; t_  _0 c c c jjc jj qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi _2 _ nh qc pqc  p2c þ 2~

cos 3h_ 

"

_

@ qc _0

@c

¼

1 _

M _c

pffiffiffi 3 0 t0_c þ 6t02 _  cos 3h_ t_ c c c 8 þ cos 3h_c

8 _ > _ @ qc > > 2q > c _0 for nh ¼ 0 @f ðc; bÞ < @c _ ¼ > pc _ @ p c @b > > p c _0 for nh 6¼ 0 > : 2 _ p qc @c @f ðc; bÞ 11 ¼ @c 3 ~nh

13.5.6

! p _

p qc

_

qc

 nh I þ 2 _ p qc

(

ð13:156Þ

ð13:157Þ # ð13:158Þ

ð13:159Þ

" _ #) 1 @ qc  ð13:160Þ 0 :ðI  bÞ I _0 3 @_ @c c _

@ qc

Calculation of Normal-Yield Ratio

The normal-yield ratio R must be calculated from the equation of the subloading surface in the unloading process. It can be calculated directly from R ¼ f ðr; bÞ=F by solving f ðr; bÞ ¼ RF in the initial subloading surface model. In the extended subloading surface model, however, it is difficult to solve Eq. (13.100) with Eq. (13.132) for R analytically. Then, we have to calculate R by the semi-analytical

414

13 Constitutive Equations of Soils

or numerical methods. The super-normal yield ratio R does not change and thus the 

superyield ratio R can be calculated by R ¼ R=R in the unloading process ðe p ¼ OÞ (a) Semi-analytical method In general, R is calculated numerically by solving the nonlinear equation obtained by substituting the current known values of r; b and c into Eq. (13.100) with _ Eq. (13.132), noting r ¼ r þ Rc in Eq. (11.6). The Newton–Raphson method would be useful for the calculation. The other numerical method is shown here. First, one has 1 _ _0 _ p ¼  trðr þ RcÞ ¼ ðrm þ Rcm Þ; r0 ¼ r þ Rc0 3

ð13:161Þ

from Eq. (11.6). Substituting Eq. (13.161) into Eq. (13.135), one has _0

_0

r ð¼ r0 pbÞ ¼ r þ Rc0 þ

1 _ _0 _ ½trðr þ RcÞ b ¼ r þ Rc0 þ ðrm þ Rcm Þb ð13:162Þ 3

_0

_

q

jjr jj _

M

_0

¼

jjr þ Rc0 þ ðrm þ Rcm Þbjj _

_

ð13:163Þ

M

Further, substituting Eq. (13.161)–(13.163) into Eq. (13.100) with Eq. (13.132) of the extended subloading surface, one has the following equation and can transform it in turn. 2

  12 3 0 _0  _  r þ Rc0 þ rm þ Rcm b 6 B C 7 _ B C 7  6 6 _ B C 7 M 6  ¼ RF for nh ¼ 0  rm þ Rcm 61 þ B C 7 _ B C 7 6 7  r þ Rc m m @ A 4 5

9 > > > > > > > > > > > > > > > > =

9> 8vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi >  32ffi u 2 > > > _0 >   _ > > u > 0 = <   > r þ Rc þ r þ Rc b   m m 2 > 1 u _ _ > t rm þ Rcm þ 2~ 4 5 > þ nh rm þ Rcm nh > _ > ~nh > > > > > ;> : M > > > ; ¼ RF or nh 6¼ 0 ð13:164Þ

13.5

Extended Subloading Surface Model

415

9 32 _0 0 _ > > jjðr þ Rc Þ þ ðr þ Rc Þbjj _ m m > 5 ¼ ðrm þ Rcm ÞRF > ð~ rm þ Rcm Þ 4 > > ^ > > M > > > = ¼ 0 for n h vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 3 u 2 0 _ 0 u > _ > u _ > > ~h 4jjr þ Rc þ ðrm þ Rcm Þbjj5 > t½ðrm þ Rcm Þ 2 þ 2n > _ > > > M > > ; _ ¼~ nh RF  nh ðrm þ Rcm Þ for nh 6¼ 0 2

2

_

2

_

_0

2

0 2

_

_0

0

M ðrm þ Rcm Þ þ jjr þ Rc jj þ 2ðrm þ Rcm Þ½ðr þ Rc Þ: b _

_

2

_

þ ðrm þ Rcm Þ2 jjbjj2 þ M ðrm þ Rcm ÞRF ¼ 0 for nh ¼ 0

9 > > > > > > =

> _0 _0 _ _ > > M ðrm þ Rcm Þ2 þ 2~ nh jjr þ Rc0 jj2 þ 4~ nh ðrm þ Rem Þ½ðr þ Rc0 Þ: b > > > _ ; _ _ 2 2 2 ~ m þ Rcm Þ jjbjj  fM ½~ þ 2nðr nh RF nh ðrm þ Rcm Þ g ¼ 0 for nh 6¼ 0 _2

9 _0 2 _0   0 0 2 2> > M rm þ 2 M rm cm R þ M þ  r  þ 2 r : c R þ kc k R > > >  _0  > > _ _ > 2 0 > þ 2 rm þ Rcm r : b þ 2 rm R þ cm R ðc : bÞ > > > >  2  > _2 > _ _ _ 2 > 2 2 2 > þ rm þ 2Rrm cm þ cm R kbk þ M rmFR þ cm FR ¼ 0 > > > > > for nh 6¼ 0 > =   _2 _2 _2 2 0 _  _2 _ ~h  r  M rm þ 2 M rm cm R þ M c2m R2 þ 2n > >  _0 _0 > > _ 2 0 0 2 > > þ 4~ n r : c R þ 2~ nh rm þ cm R r : b nh kc k R þ 4 ~ > >   2 > _ _ _ 2 > 2 0 2 2 ~ ~ > þ 4n rm R þ cm R ðc : bÞ þ 2nh rm þ 2rm cm R þ cm R kbk > > > >   > 2  > _2  > 2 2 _ _ 2 ~ ~ > M nh F  nh cm R  2nh rm nh F  nh cm R þ nh rm ¼ 0 > > > > ; for nh 6¼ 0 0h  i2 h i2 1 _ _ ~ ~ ¼ nh F  ncm R  nh rm C B nh RF  nh rm þ Rcm  2  A @ _ _2 nh F  nh cm R þ nrm ¼ ~ nh F  nh cm R2  2nh rm ~ _

2

_2

_

2

_

_

2

c2m R2

416

13 Constitutive Equations of Soils

_

M

2

0 2 2

0

þ kc k R þ 2cm ðc : bÞR þ c2m kbk2 R2 þ _0 _0 _2 _ þ 2 M rm cm R þ 2 r : c0 R þ 2cm r : b R c2m R2

_

2

2

_

2

M cm FR

2

þ 2rm ðc0 : bÞR þ M rm FR þ 2rm cm kbk2 R _0 2 _0 _2   _2 _ _2 þ M rm þ  r  þ 2rm r : b þ rm kbk2 ¼ 0 for nh ¼ 0 _

_

_

_

2

9 > > > > > > > > > > > > > > > > > > > > > > > > > =

M c2m R2 þ 2~ nh kc0 k2 R2 þ 4~ nh cm ðc0 : bÞR2 þ 2~ nh c2m kbk2 R2

> > > > > 2 > 2 ~ 0 ~ > M R ðnh F  nh cm Þ þ 2M rm cm R þ 4nh ðr : c ÞR > > > _0 _ _ 2 > 0 ~ ~ ~ > þ 4nh cm r : b R þ 4nh rm ðc : bÞR þ 4nh rm cm kbk R > > > >     > _2 _2 2 0 0 > _ _  _ _2 _ > þ 2 M nh rm ~ nh F  nh cm R þ M rm þ 2~nh  r  þ 4~nh rm r : b > > > > > 2 > _ ; 2 2 2 _ _ 2 ~ þ 2nh rm kbk  M nh rm ¼ 0 for nh 6¼ 0 _

2

_

2

_

_0

9 _2  _2 > 2 > 2 0 2 0 2 2 > M cm þ kc k þ 2cm ðc : bÞ þ cm kbk þ M cm F R > > > >  _2 > _0 _0 > > _ _ 0 0 > > þ 2 M rm cm þ 2 r : c þ 2cm r : b þ 2rm ðc : bÞ > > > >  >   _2 _2 > 2 0 > _  _ _2 2 > > ~m F R þ M rm þ  r  þ 2rm cm kbk þ M r > > > _0 > > _ _2 2 = þ 2rm r : b þ rm kbk ¼ 0 for nh ¼ 0 _2  2 > _ 2 > > R2 > M c2m þ 2~ nh k c 0 k 2 þ 4 ~ nh c m ð c 0 : b Þ þ 2 ~ nh c2m kbk2  M ~nh F  nh cm > > > h _0 _0 > > 2_ _ > 0 0 ~ ~ ~ > þ 2 M rm cm þ 2nh r : c þ 2nh cm r : b þ 2nh rm ðc : bÞ > > >  >  > _2 _ 2 > 2 _2 _ _ > 2 ~ ~ > þ 2nh rm cm kbk þ M nh rm nh F  nh cm R þ M 1  nh rm > > > > _0 2 > _0 >   _ _2 > 2 ~ ~ ~ ; þ 2nh  r  þ 4nh rm r : b þ 2nh rm kbk ¼ 0 for nh 6¼ 0 Solving this quadratic equation, the normal-yield ratio R is expressed as follows: 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B2  AC  B > > < for nh ¼ 0 A R ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~2 ~ ~ ~ > > : B  AC  B for nh ¼ 6 0 ~ A

ð13:165Þ

13.5

Extended Subloading Surface Model

417

where A

_

2

_2 þ kc k þ 2cm ðc : bÞ þ c2m kbk2 þ M cm F _0 _0 _ _ rm cm þ 2tr r c0 þ 2cm r : b þ 2rm ðc0 : bÞ

M c2m _2

B  2M

0 2

0

_

_

2

_

þ 2rm cm kbk2 þ M rm F _0 2 _0 _2   _2 _ _2 C  M rm þ  r  þ 2rm r : b þ rm kbk2 _

2

9 > > > > > > > > > > > > > > > > > > > > > > > > > > > =

~  M c2 þ 2~ A nh k c 0 k 2 þ 4 ~ n c ð c 0 : bÞ þ 2 ~ nh c2m kbk2 m > 2 h m _  > 2 > > > M ~ nh F  nh e m > > > > _0 _0 _2 > > _ _ 0 0 > ~ ~ ~ ~ B  M rm cm þ 2nh r : c þ 2nh cm r : b þ 2nh rm ðc : bÞ > > > > >  _2 > _ _ > 2 ~ ~ > þ 2nh rm em kbk þ M nh rm nh F  nh cm > > > >    > _ 2 2 > 0 0 _ _ 2 _2  _ _2 > 2 ~ ~ ~  M 1  n r þ 2~ ; n r þ 4 n r r : b þ 2 n r b C k k   h h m h m h m

ð13:166Þ

Explicit numerical calculation processes: (1) First step (beginning of calculation): Calculate the normal-yield ratio R by Eq. (13.165), substituting the trial value _ _ pffiffiffi pffiffiffi M ¼ 2 6 sin /c =3 which is the average of M ¼ 2 6 sin /c =ð3 þ sin /c Þ and pffiffiffi 2 6 sin /c =ð3  sin /c Þ. (2) Second step: Recalculate R by substituting the value 0

1 pffiffiffi pffiffiffi 14 6 sin / 14 6 sin /c c A¼ pffiffiffi M ðcos 3h_ Þ @¼ ð3  sin /c Þð8 þ cos 3h_ Þ r ð3  sin /c Þð8 þ 6trt3 Þ _

r

_

r

ð13:167Þ into Eq. (13.165), while the value of R obtained in the former step is used in Eq. (13.167). (3) Repeat the process (2) until R will reaches the convergence within a prescribed tolerance. (b) Numerical method In what follows, the numerical calculation of R by the Newton–Raphson method is described below.

418

13 Constitutive Equations of Soils

Consider the following equation of R given from Eq. (13.164)2 for the subloading surface. 3 2vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 u   u 2 16 rc _ _ 7 nh _ þ nh rm þ Rcm 5  RF ¼ 0 gðRÞ ¼ 4t rm þ Rcm þ 2~ ~ nh M ð13:168Þ where

_ 0     _ rc   r þ Rc0 þ rm þ Rcm b

ð13:169Þ

Differentiation of Eq. (13.168) leads to 2 _3 !2 31=2 2 _ @rc 2  @M M  r @gðRÞ 1 4 _ r r c @R _ c ~ c @R 42 r 5 rm þ Rcm þ 2~ nh _ 5 ¼ m þ Rcm cm þ 4nh _ _2 @R 2~nh M M M nh þ cm  F n~h

ð13:170Þ

0

g ðRÞ ¼

1 ~nh

2 _3 "
2 #1=2  @rc rc @ M _ 2 @R  _ @R _ rc M ~ ~ 4 r m þ Rcm cm þ 2nh rc _ 5 þ r m þ Rcm þ 2nh _ 2 M

M

3 nh c ~ nh m

 F5

ð13:171Þ where h_0 _ i @rc 0 ¼ v1 r þ Rc þ r þ Rc b : ð c 0 þ cm bÞ m m c @R
 @ k Tk T @T ¼ : @R kTk @R 2 _ @M 3  sin /c _ 2 4 ¼  pffiffiffi M @R 14 6 sin /c

3

kr k _0

3 pffiffiffi 6t2_  cos 3h_ t_ 5 : ðc0  btrcÞ ð13:173Þ r r

r

noting _

ð13:172Þ

_0

_

@M @M @ cos 3hr @ r ¼ : _0 @R @ cos 3hr @R @r _

_

13.5

Extended Subloading Surface Model _0

@r ¼ @R

419

h_ i _ @ ðr þ RcÞ0  trðr þ RcÞb @R

¼ c0  b tr c

ð13:174Þ

Here, noting g0 ðRn Þ ¼

gðRn þ 1 Þ  gðRn Þ 0  gðRn Þ ¼ Rn þ 1  Rn Rn þ 1  Rn

we have Rn þ 1 ¼ Rn 

gðRn Þ g0 ðRn Þ

ð13:175Þ

from which R is obtained by repeating calculation until Eq. (13.168) converges within a given tolerance. In addition, we adopt the following “reduced” Newton– Raphson iteration instead of Eq. (13.175) if the convergence is not obtained. Rn þ 1 ¼ Rn  ln

gðRn Þ g0 ðRn Þ

ð13:176Þ

Calculate first by setting ln ¼ 1. If the inequality 
    nþ1  ln  n   gðR gðR Þ Þ \ 1   4 

ð13:177Þ

is fulfilled, continue the calculation. However, if Eq. (13.177) is not fulfilled, calculate by reducing ln to the half and repeat again the calculation by reducing ln to the further half until Eq. (13.177) is fulfilled. The constitutive equation of soils based on the extended subloading surface model is described in detail so far. The tangential-inelastic strain rate has to be incorporated for the analyses of non-proportional loading behavior as shown in the analyses by Hashiguchi and Tsutsumi (2001, 2003, 2006) and Khojastehpour and Hashiguchi (2004a, b).

13.6

Simulations of Test Results

Some simulations of test data are shown below in order to show the capability of the subloading surface model to reproduce the real deformation behavior of soils (Hashiguchi and Chen 1998). The simulation of the test data (after Saada and Bianchini 1989) for Hostun sand subjected to the drained triaxial compression with a constant lateral stress, which

420

13 Constitutive Equations of Soils

Fig. 13.20 Drained behavior of Hostun sand (data from Saada and Bianchini 1989)

includes the unloading–reloading process, is shown in Fig. 13.20 where the material constants and the initial values are selected as follows: Material constants: Yield surface ðellipsoid:Þ/c ¼ 27 ; ( 8 > ~ ¼ 0:003; # ¼ 0:025; k ¼ 0:008; j < isotropic volumetric: ~ Hardening=softening deviatoric: l ¼ 0:6; / d d ¼ 25 ; > : rotational: br ¼ 10; /r ¼ 20 ; Evolution of normal  yield ratio: u1 ¼ 1:5; m1 ¼ 3:8; Translationon of elastic  core : ce = 0.05, Elastic shear modulus: G = 200 000 kPa,

13.6

Simulations of Test Results

421

Initial values: Hardening function: F0 ¼ 400 kPa; Rotational hardening variable: b0 ¼ O; Elastic  core: c0 ¼ 50I kPa; Stress: r0 ¼ 100I kPa where the hyperbolic equation UðRÞ ¼ u1 ð1=Rm1  1Þ ðu1 ; m1 : material constantsÞ is used for the evolution rule of the normal-yield ratio. Measured and calculated results are shown by the dashed and solid line, respectively. The simulation of the test data (after Saada and Bianchini 1989) for Hostun sand subjected to the drained proportional loading with b ð¼ ðr2  r3 Þ=ðr1  r3 ÞÞ ¼ 0:666 ðhr ¼ 19 090 Þ from r0 ¼ 500I kPa by the true triaxial test apparatus is shown in Fig. 13.21. The material parameters are the same as those for the above-mentioned Fig. 13.21 Drained proportional loading behavior of Hostun sand (data from Saada and Bianchini 1989). Measured and calculated results are shown by the dashed and solid line, respectively

422

13 Constitutive Equations of Soils

drained triaxial compression, while the sample was preliminarily loaded the isotropic compression from r ¼ 100I kPa to 500I kPa before the test. The simulations of the test data (after Castro 1969) for Banding sand subjected to the undrained triaxial compression with a constant lateral stress are shown in Fig. 13.22 where the material constants and the initial values are selected as follows:

Fig. 13.22 Undrained behavior of Banding sand (data from Castro 1969). Calculated results are shown by the solid lines

13.6

Simulations of Test Results

423

Material constants: Yield surfaceðellipsoid:Þ/c ¼ 26; 30; 31; 32 ; 8 8 8 ~ > > > > < k ¼ 0:025; 0:018; 0:014; 0:010; > > > > > > > > ~ ¼ 0:0067; 0:0065; 0:0060; 0:0058; > < volumetric> j > > : < isotropic # ¼ 0; 0:021; 0:058; 0:138; > Hardening=softening

> > > > l ¼ 1:00; 0:65; 0:30; 0:10; > > > > deviatoric d > : > > /d ¼ 40; 33; 30; 20 ; > > : rotational: br ¼ 10; /r ¼ 20 ;

u1 ¼ 0:1; 0:3; 0:5; 1:0; 33:0; Evolution of normal  yield ratio m1 ¼ 0:1; 0:4; 0:5; 0; 7; Translationon of elastic  core : ce = 0.049, 0.038, 0.027, 0.014;

Elastic shear modulus: G = 18 000, 23 000, 25 000, 35 000 kPa, Initial values: Hardening function: F0 = 410, 480, 520, 580 kPa; Rotational hardening variable : b0 = O, Elastic  core : c0 = - 200, - 110, - 100, - 80 I kPa, Stress: r0 =  67:0I kPa where the four values for each of the above parameters correspond to the initial relative densities Dr ¼ 0:29; 0:44; 0:47; 0:64, respectively, in this order. The different values of material parameters depending on the initial relative densities are used in this simulation because the deviatoric strain hardening (Nova 1979; Wilde 1979) in addition to the plastic volumetric strain hardening is adopted. The simulation would be performed using the unique set of material parameters by the SYS model as described in Sect. 13.5.1, while it was shown explicitly by Asaoka et al (2002) whose formulation is the immature based on the initial subloading surface model. The simulation of the cyclic mobility under the liquefaction to the test data by the subloading surface model was shown by Hashiguchi et al. (2021) although not illustrated here.

13.7

Numerical Analysis of Footing Settlement Problem

Numerical analysis of footing settlement problem will be shown in this section (Mase and Hashiguchi 2009). The prediction of peak load and post-peak behavior for the footing-settlement problem on sands having the high friction and dilatancy cannot be attained in fact by the usual implicit finite element method requiring the repeated calculations of total stiffness equation which needs quite large calculation

424

13 Constitutive Equations of Soils

time. On the other hand, it can be attained by the explicit dynamic relaxation method in which the dynamic equilibrium equation is solved directly without solving the total stiffness equation so that the calculation time is drastically reduced. The FLAC3D (Fast Lagrangian Analysis of Continua in 3 Dimensions; Cundall and Board 1988; Itasca Consulting Group 2006) based on the explicit dynamic relaxation method, i.e. the finite difference method (FDM) is adopted in the present analysis, in which the initial subloading surface model of soils with the automatic controlling function to attract the stress to the normal-yield surface is implemented as the constitutive equation. The calculation is executed by the forward Euler method without iteration calculation for convergence in this program by adopting small incremental steps so as not to influence on the calculation results, while this fact is examined prior to the calculation. The finite elements are composed of eight-node cuboidal elements. Each cuboidal element is divided into the two kinds of overlays, i.e. assembly of five tetrahedral sub-elements having different directions. Then, the deviatoric variables are analyzed using individual values in each tetrahedral sub-element. On the other hand, isotropic variables are analyzed using averaging values in five tetrahedral sub-elements in order to avoid the over-constraint problems common in finite element calculations for dilatant materials, i.e. the dilatancy locking. Test data used for numerical simulation The test data of footing settlement phenomenon on sand layers under the plane strain condition are simulated in the present analysis. The sizes of the test apparatus of type A (Tatsuoka et al. 1984) and type B (Tani 1986) have the same height 49 cm and depth 40 cm and the different widths 122 cm and 183 cm, respectively. The size of type C (Okahara et al. 1989) has the height 400 cm and the depth 350 cm and the widths 700 cm. The footings width, denoted as B, is taken 10 cm for the types A and B and 50 cm for the type C. The sand layers has been prepared carefully by the air-pluviation method for the dried Toyoura sand in order to obtain the same homogeneous layers but the test data exhibit dispersion more or less test by test despite of the laborious preparation work. Numerical analysis and comparison with test data The finite element meshes in the present analyses for the simulations of the test data are shown in Fig. 13.23. The nodal points of soil layer contacting with the footing and the bottom of soil bin are fixed to them, respectively. On the other hand, the nodal points at the side walls can move freely in the vertical direction. The right half of soil layer is analyzed in order to reduce the calculation time as has been done widely even for searching the localized deformation (cf. e.g. Sloan and Randolph 1982; Pietruszczak and Niu 1993; Stallebrass and Taylor 1997; Borja and Tamagnini 1998; Siddiquee et al. 1999; de Borst and Groen 1999; Sheng and Soloan 2000; Borja et al. 2003). First, the analysis of deformation caused by the gravity force is performed. Then, the vertical displacement of footing is imposed by incremental steps of 105  5  104 cm.

13.7

Numerical Analysis of Footing Settlement Problem

425

The material parameters in the subloading surface model are selected as F0 = 350 kPa;/c = 30 , nh = 0.001, # = 0.00003, ~ = 0:0015, j ~ = 0:00015 k m = 0:3,

/d = 29 , ld = 0.2 u = 15:0: The values of material parameters listed above are used for all the following numerical calculations because Toyoura sands having the same initial void ratio 0.66 are used in these tests. (Fig. 13.23) The comparisons of test and calculated results are shown in Fig. 13.24, where the simulation by Siddiquee et al. (1999) which is calculated by the finite element method adopting the Mohr–Coulomb model is also depicted in (c). In this figure qm is the average footing pressure, cd is the unit dry weight, Nc is the normalized footing pressure corresponding to the bearing capacity coefficient associated with frictional property of soil and S is the settlement. The qualitative trends of test results and the quantitative simulation to some extent are captured and the ultimate loads, i.e. bearing capacities are predicted well by the present analyses, although the analyses are performed for the sand with the high friction and dilatancy. Here, the post-peak behavior, i.e. the increase of load after exhibiting once the minimal value is also predicted well qualitatively. It would be provided by the adoption of the up-dated Lagrangian calculation realizing the accumulation of displacements by updating the positions of nodal points, which results in the upsurge of soils around the footing and thus the increase of footing load. However, the quantitative prediction of post-peak behavior would require the further study taking account of the tangential inelastic strain rate due to the stress rate tangential to the loading surface (Hashiguchi and Tsutsumi 2001) and the gradient effect (cf. Hashiguchi and Tsutsumi 2006) by introducing the shear-embedded model (cf. Pietruszczak and Mroz 1983; Tanaka and Kawamoto 1988; Tanaka and Sakai 1993) for example, which will be described in Sect. 20.3. The displacements of nodal points from the initiation of settlement are shown in Fig. 13.25 at the settlement 11, 15, 80 mm for Type A, B and C, respectively, which are the final stage of calculation. The Prantdl’s slip line solution with the triangle wedge, the logarithmic spiral zone and the passive Rankine zone is observed clearly in this figure (Fig. 13.25). On the other hand, the soils in the periphery of footing inevitably experience the null or further negative pressure since they are pushed upward as the footing settlement proceeds (see Fig. 13.26). It causes the singularity of plastic modulus for the normal-yield surface passing through the origin of stress space at which the normal-yield and the subloading surfaces contact with each other. This defect is improved in the present model by making the normal-yield surface translate to the

426

13 Constitutive Equations of Soils

C.L.

56cm

49cm

5cm

(a) Type A (B: 10cm) C.L.

86cm

49cm

5cm

(b) Type B (B: 10cm) C.L.

325cm

400cm

25cm

(c) Type C (B: 50cm) Fig. 13.23 Finite element meshes

13.7

Numerical Analysis of Footing Settlement Problem

427

300

300

FDM by subloading surface model Test (Tatsuoka et al., 1984): e=0.66

FDM by subloading surface model Test (Tani, 1986): e=0.669,BC2 Test (Tani, 1986): e=0.669,BC3

Test (Tatsuoka et al., 1984): e=0.66

200

200

100

100

0

0.00

0.05

0.10

0

0.15

0.00

0.05

0.10

Relative settlement S B

Relative settlement S B

(a) Type A (B: 10cm)

(b) Type B (B: 10cm)

300 FDM by subloading surface model FEM (Siddiquee et al., 1999)

Test (Okahara et al., 1989): e=0.66 Test (Okahara et al., 1989): e=0.66

200

0.15

qm : Average footing pressure : Unit dry weight S : Settlement B: Footing width

100

0

0.00

0.05

0.10

0.15

Relative settlement S B (c) Type C (B: 50cm)

Fig. 13.24 Comparisons of test and calculated results for footing settlement phenomenon

region of negative pressure as shown in Fig. 13.13, whilst the numerical difficulty can be avoided although the translation was taken quite small as 1/1000 in size of the normal-yield surface. In addition, the impertinence of infinite elastic volume expansion is avoided by shifting the isotropic consolidation characteristic into the negative range of pressure as shown in Fig. 13.1. It should be emphasized that the stable analysis cannot be executed without these improvements. The pertinent result for the footing-settlement problem on the sand with a high friction, one of the difficult problems in soil mechanics, is obtained in the present study as described above. Here, the peak, the subsequent reduction and the final

428

13 Constitutive Equations of Soils

Displacement (cm) 0.0䡚0.1 0.1䡚0.2 0.2䡚0.3 0.3䡚0.4 0.4䡚0.5 0.5䡚0.6 0.6䡚0.7 0.7䡚0.8 0.8䡚0.9 0.9䡚1.0 1.0䡚1.1

(a) Type A (B: 10cm) Displacement (cm) 0.0䡚0.2 0.2䡚0.4 0.4䡚0.6 0.6䡚0.8 0.8䡚1.0 1.0䡚1.2 1.2䡚1.4 1.4䡚1.5

(b) Type B (B: 10cm) Displacement (cm) Liv

0.00䡚1.00 1.00䡚2.00 2.00䡚3.00 3.00䡚4.00 4.00䡚5.00 5.00䡚6.00 6.00䡚7.00 7.00䡚8.00

(c) Type C (B: 50cm) Fig. 13.25 Deformed finite element meshes at final step

13.7

Numerical Analysis of Footing Settlement Problem

429

C.L.

p (kPa) 0.0001䡚0 0 䡚10 10䡚20 20䡚30 30䡚40 40䡚50 50䡚60 60䡚

I t

Fig. 13.26 Distribution of mean pressure for type A at final step

increase in footing load are predicted well qualitatively and quantitatively to some extent. The reasons for succession are summarized as follows: (1) The subloading surface model applied in the present analysis has the advantages: (i) It is furnished with the automatic controlling function to attract the stress to the yield surface, whilst all other elastoplastic constitutive models are required to incorporate a return-mapping algorithm to pull back the stress to the yield surface in the plastic deformation process: (ii) It is capable of describing the softening behavior and dilatancy characteristics quite realistically, predicting the simultaneous occurrence of the peak load and the highest dilatancy rate as was found experimentally by Taylor (1948). (iii) By virtue of the model extension to accommodate negative pressure, it has the full regularity since the normal-yield surface does not pass through the zero stress point and thus the subloading surface is always determined uniquely. In addition, the elastic property is improved such that the elastic bulk modulus does not become zero even for null stress or negative pressure inside the normal-yield surface. (2) The finite difference program FLAC3D adopted in the present study is based on the explicit-relaxation method which enables us to shorten the calculation time drastically since it is not required to solve the total stiffness matrix.

430

13 Constitutive Equations of Soils

13.8

Hyperelastic Equation of Soils

The isotropic hyperelastic constitutive equation for soils independent of the third invariant of stress and strain is given by r¼

@wðeev ; eed Þ @wðeev ; eed Þ @eev @wðeev ; eed Þ @eed ¼ þ @ee @ee @ee @eev @eed

ð13:178Þ

where ee is the infinitesimal elastic strain tensor and eev  tree ; eed  kee 0 k leading to @eev @tr ee ¼ ¼ I; @ee @ee

0

0

@eed ee ee ¼ ¼ 0 @ee kee k eed

ð13:179Þ

Equation (13.178) is rewritten by Eq. (13.179) as     0 @w eev ; eed @w eev ; eed re Iþ r¼ eed @eev @eed

ð13:180Þ

The strain energy function of soils was first proposed by Houlsby (1985) and subsequently modified by Borja and Tamagnini (1998), Tamagnini et al. (2002), in which the ln v  ln p linear relation in Eq. (13.8) under pe ¼ 0 in Sect. 13.1 is incorporated and the dependence of the shear modulus on the pressure is taken account. Mechanical and mathematical properties of this model have been studied by Callari et al. (1998), Niemunis and Cudny (1998), Houlsby et al. (2005), Amorosi et al. (2007), etc. In what follows, the hyperelastic equation of soils (Hashiguchi 2018c) is shown in the following, which possesses the identical forms for the infinitesimal and the finite strains. Firstly, we introduce the following strain energy function based on Eqs. (13.12)1 and (13.16)1. wðeev ; eed Þ

¼

pe eev


e 
e  ev e ~ðp0 þ pe Þ exp  þ G0 exp n  v ee2 þj d ~ ~ j j

ð13:181Þ

It follows for Eq. (13.181) that  e h  e i 8 e e < @wðeve;ed Þ ¼ pe  ðp0 þ pe Þ exp  ev  n G0 exp n  ev eed 2 ~ ~ ~ j j j @ev h  i e e : @wðeve;ed Þ ¼ 2G0 exp n  eev ee d ~ j @e d

ð13:182Þ

13.8

Hyperelastic Equation of Soils

431

Substituting Eq. (13.182) into Eq. (13.180) leads to
e 
e  e n e r ¼ fpe  ðp0 þ pe Þ exp  v  G0 exp n  v eed 2 gI ~ ~ ~ j j j 
e  e 0 þ 2G0 exp n  v re ~ j

ð13:183Þ

The hydrostatic and deviatoric parts in Eq. (13.183) are given by
e p þ pe e ¼ exp  v for ee 0 ¼ O ðeed ¼ 0Þ ~ j p0 þ pe i.e.


eev

p þ pe ¼ ~ j ln p0 þ pe



for ee 0 ¼ O ðeed ¼ 0Þ

ð13:184Þ

and 
e 
 ev p þ pe n e 0 e0 e ¼ 2G0 e r ¼ 2G0 exp n  ~ j p0 þ pe 0

ð13:185Þ

which conform to Eqs. (13.12)1 and (13.16)2 in the rate form, respectively, while Eq. (13.184) is used for deriving Eq. (13.185) from the second term in the right-hand side of Eq. (13.183). The general assessment of the various elastic constitutive equations for soils can be referred to Yamakawa (2021).

Chapter 14

Viscoplastic Constitutive Equations with Subloading Surface Concept

The plastic deformation depends on the rate of deformation in general. The time (or rate)-dependent inelastic deformation is called the viscoplastic deformation. The viscoplastic constitutive models are classified into the creep model and the overstress model. The viscoplastic deformation is induced depending on the ratio of the magnitude of the stress to the magnitude of the yield stress in the creep model and thus the creep model is not reduced to the elastoplastic constitutive equation in the quasi-static deformation process. On the other hand, the viscoplastic deformation is induced by the overstress from the yield surface and thus the overstress model is reduced to the elastoplastic constitutive equation in the quasi-static deformation process. Therefore, the creep model is unacceptable since it is unreasonable such that the creep model inappropriately predict a viscoplastic deformation for any low stress level and thus it cannot describe both of the rate-independent and the rate-dependent deformation behaviors by a single equation. On the other hand, the overstress model is capable of describing both of the rate-independent and the rate-dependent deformation behaviors by a single constitutive equation. However, the existing overstress model possesses the defects: (1) The plastic deformation during the cyclic loading process by the rate of stress within the yield surface cannot be described, (2) The purely elastic deformation is described for the impact loading, resulting in the inapplicability to the deformation behavior in a high strain rate. The rational overstress model is formulated by incorporating the extended subloading surface model in this chapter.

14.1

Rate-Dependent Deformation of Solids

The elastic and the inelastic deformations of solids are induced by the deformation of solid particles themselves (crystals in metals, soil particles in soils, etc.) and the mutual slips between them, respectively. Therefore, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 K. Hashiguchi, Foundations of Elastoplasticity: Subloading Surface Model, https://doi.org/10.1007/978-3-030-93138-4_14

433

434

14 Viscoplastic Constitutive Equations with Subloading Surface Concept

1. The elastic tangent modulus is high, since the elastic deformation is induced by the deformations of solid particles themselves, the rete-dependent deformation of which is described by the viscoelastic constitutive equation, 2. High stress has to be applied to solids in order that inelastic deformation is induced, which overcomes the friction resistance between solid particles. The stress inducing the inelastic rate-independent deformation is macroscopically the so-called yield stress. 3. The rate-dependent inelastic deformation is induced when an overstress from the yield surface is applied to solids, 4. The tangent modulus in the inelastic deformation process lowers from that in the elastic deformation process by the inelastic relaxation. Consequently, the rate-dependent elastic deformation induced in the stress lower than the yield stress is called the viscoelastic deformation. On the other hand, the rate-independent inelastic deformation induced by the yield stress and the rate-dependent inelastic deformation induced over the yield stress are called the plastic deformation and the viscoplastic deformation, respectively. Then, they are classified as follows: 8
Deformation of Rate-independence: Elastic constitutive equation > > > < soild particles Rate-dependence: Viscoel astic constitutive equation Deformation of soilds
> Mutual slips of Rate-independence: Plastic constitutive equation > > : solid particles Rate-dependence: Viscoplastic constitutive equation

which is illustrated in Fig. 14.1. The strain e is additively decomposed into the elastic strain ee and the viscoplastic strain rate evp , i.e. e ¼ ee þ evp ;







e ¼ ee þ evp

ð14:1Þ

Elasto-viscoplastic deformation



Elastoplastic deformation induced in the quasi-static rate

Yield stress Range of viscoelastic deformation

Elastic response

 Yield stress

Fig. 14.1 Rate-dependent deformation of solids

14.1

Rate-Dependent Deformation of Solids

435

The stress versus elastic strain relation is described by Eq. (8.4). When the strain energy function is given by the quadratic equation in Eq. (8.3), the following relations hold. r ¼ E : ee ¼ E : ðe  evp Þ; ee ¼ E1 : r; e ¼ E1 : r þ evp           r ¼ E : e e ¼ E : e  e vp ; e e ¼ E1 : r; e ¼ E : r þ e vp

ð14:2Þ ð14:3Þ

The stress r is calculated by substituting the plastic strain evp calculated by the plastic constitutive equation into Eq. (14.2). Here, let the viscoplastic strain evp be vp decomposed into the storage part evp ks and the dissipative part ekd for the kinematic vp hardening and let it be decomposed into the storage part ecs and the dissipative part evp cd for the elastic-core, i.e.

14.2

vp evp ¼ evp ks þ ekd ;

vp e vp ¼ e vp ks þ e kd







ð14:4Þ

vp evp ¼ evp cs þ ecd ;

vp e vp ¼ e vp cs þ e cd







ð14:5Þ

History of Viscoplastic Constitutive Equations

The most pertinent viscoplastic model would be the overstress model. The development of this model is reviewed in this section, while the overview of the history is portrayed simply by the one-dimensional deformation with the rheology models in Fig. 14.2. The elastic constitutive equation extended so as to describe the rate-dependence is called the viscoelastic constitutive equation and one of the typical models is the Maxwell model, in which the spring and the dashpot are connected in series.  Therefore, the strain rate e is additively decomposed into the elastic strain rate    e e ¼ E 1 r and the viscous strain rate e v ¼ l1 r, where r designates the stress, E is the elastic modulus and l is the viscous coefficient, leading to 







e ¼ e e þ e v ¼ E 1 r þ l1 r

ð14:6Þ

This model is concerned with the rate-dependent deformation at the low stress level below the yield stress. On the other hand, the elastoplastic constitutive equation for a quasi-static deformation can be schematically expressed by the Prandtl model in which the dashpot is replaced with the slider in the Maxwell model, whereas the slider begins to move in the state that the stress r reaches the yield stress ry , by which the plastic

436

14 Viscoplastic Constitutive Equations with Subloading Surface Concept

Fig. 14.2 Development of viscoplastic model

strain rate is induced (see Fig. 14.2). Then, the strain rate is additively composed of the elastic and the plastic strain rates, i.e. (  e e ¼ E 1 r for r\ry e¼ e    e þ e p ¼ E 1 r þ M p1 r for r ¼ ry 

ð14:7Þ

where M p is the plastic modulus.  Furthermore, the model which describes the rate-dependent plastic strain rate e vp induced for the state of stress over the yield stress, called the viscoplastic strain rate, was introduced by Bingham (1922), combining the above-mentioned Maxwell model and Prandtl model so as to connect the dashpot and the slider in parallel as

14.2

History of Viscoplastic Constitutive Equations

437

shown in Fig. 14.2, where l is the viscoplastic coefficient and n is the material constant, while n is chosen to be 4  8 in practice. Then, the strain rate is given by 







e ¼ e e þ evp ¼ E 1 r þ l1 \r  ry [ n

ð14:8Þ

The Bingham model for the elasto-viscoplastic deformation is the origin of the overstress model based on the concept that the viscoplastic strain rate is induced by the overstress, i.e. stress over the yield stress. The above-mentioned Bingham model for the one-dimensional deformation was extended by Hohenemser and Prager (1932) and Prager (1961) to describe the three-dimensional deformation of metals, adopting the Mises yield condition for the  slider as shown in Fig. 14.2. The viscoplastic strain rate e vp is given by  vp

e where eeqvp 

 n 1 r eq r0 1 ¼ eqvp l F ðe Þ kr0 k

ð14:9Þ

 pffiffiffiffiffiffiffiffi R   0  2=3 e vp dt is the equivalent viscoplastic strain given by 



replacing the plastic strain rate e p to the viscoplastic strain rate e vp in the plastic pffiffiffiffiffiffiffiffi  0  equivalent strain eeqp  2=3ep dt in Eq. (8.43). The viscoplastic coefficient l depends on stress, internal variables and temperature in general. Furthermore, the viscoplastic strain rate in the Prager’s overstress model was extended by Perzyna (1963, 1966) for materials having the general yield condition unlimited to the Mises yield condition as 

e vp ¼

 n 1 f ðrÞ  1 n; l FðHÞ

n

  @f ðrÞ  @f ðrÞ  = @r  @r 

ð14:10Þ

Then, substituting Eqs. (8.5) and (14.10) into Eq. (14.1), we have  n 1 f ðrÞ 1 n l FðHÞ

ð14:11Þ

 n 1 f ðrÞ 1 E:n r ¼ E : e l FðHÞ

ð14:12Þ





e ¼ E1 : r þ and thus 



where the hardening variable H evolves by      H ¼ H r; H; e vp

ð14:13Þ

438

14 Viscoplastic Constitutive Equations with Subloading Surface Concept 



by replacing the plastic strain rate e p to the viscoplastic strain rate e vp in the evolution rule of the isotropic hardening variable in Eq. (8.23) for the plastic con stitutive equation. Therefore, H_ is the homogeneous function of e vp in degree-one. In what follows, the isotropic yield condition f ðrÞ ¼ FðHÞ in Eq. (8.16) is used below for the sake of simplicity in explanation up to Sect. 14.4. It should be noted that the response of the overstress model is reduced to that of the elastoplastic constitutive equation because f ðrÞ=FðHÞ  1 ffi 0, i.e. f ðrÞ ffi FðHÞ holds in the quasi-static deformation: dr=dt ffi O and de=dt ffi O. In other words, the overstress model is capable of describing the elastoplastic deformation behavior so that the elastoplastic constitutive equation can be disused if the overstress model is used.

14.3

Irrationality of Creep Model

Based on a concept different from the overstress model, the creep model, which also aims at describing the viscoplastic deformation, has been studied widely. The typical one is the Norton law (Norton 1929) in which the creep strain rate is given as follows: 

e c ¼ d0c krkm n

ð14:14Þ

where d0c and mð 1Þ are the material constants. Further, it has been modified by Rice (1970, 1971), Bonder and Partom (1975), Lemaitre and Chaboche (1990), Chaboche (1997, 2008), Chaboche et al. (1979), Ohno et al. (2018, 2021), etc. (see also de Sauza Neto et al. 2008) for the viscoplastic deformation and Nakada and Keh (1966), Hutchinson (1976), Pan et al. (1983), Peirce et al. (1983), etc. for the crystal plasticity as follows:

m    c c f ðrÞ n e ¼ e n ¼ d0 FðHÞ 

c

ð14:15Þ

where the rate of the isotropic hardening variable is given by      H ¼ H r; H; e c

ð14:16Þ

instead of Eq. (14.13). The strain rate is given by 







e ¼ e e þ e c ¼ E1 : r þ d0c krkm n for Eq. (14.14) and

ð14:17Þ

14.3

Irrationality of Creep Model

439



P

l0

f (σ )/ F ( H ) = 1 yield state

Over - run

Elastically decrease

0

Overstress model



0



Creep model

(a) Unloading behavior by reduction of stress



f (σ )/ F ( H ) = 1 yield state

P



f (σ )/ F ( H ) = 1 yield state

P

Stop

Unlimitedly proceed

l

Overstress model



0



Creep model

∞

0

Bar is stretched endlessly only by dead weight in creep model.

f (σ )/ F ( H ) = 1 P yield state



(b) Creep deformation behavior under constant stress

Fig. 14.3 Comparison of overstress model and creep model: The latter exhibits unrealistic behavior predicting always creep strain rate









e ¼ e e þ e c ¼ E1 : r þ d0c



f ðrÞ FðHÞ

m n

ð14:18Þ

for Eq. (14.15). The creep model is regarded to be merely the nonlinearization of the Maxwell model whose rheology model does not possess the slider which is the basic element for the description of irreversible deformation. The creep model described above has different structures from the overstress model because Eqs. (14.14) and (14.15) possess no threshold value for the generation of the creep strain rate so that the creep strain rate is induced for any low stress level. Then, its deformation behavior is not reduced to that of the elastoplastic constitutive equation at quasi-static deformation. Therefore, this model cannot describe appropriately the deformation behavior at a low strain rate. In fact, the creep strain rate does not diminish even if the stress decreases into the inside of yield surface, exhibiting the overrunning stress–strain curve and the creep deformation proceeds unlimitedly under a constant stress state in the creep model as shown in Fig. 14.3. As a concrete example, it is quite unnatural that a bar subjected to any low tension continues to be elongated endlessly as a time elapses. In addition, it is incapable of describing appropriately the impact loading behavior as it describes an elastic deformation behavior with an infinite strength. On the other hand, the viscoplastic strain rate diminishes immediately after the stress decreases into the inside of yield surface in the overstress model as is observed in real materials.

440

14 Viscoplastic Constitutive Equations with Subloading Surface Concept

Consequently, the creep model is not physically accepted but the overstress model is appropriate for the description of the rate-dependent plastic behavior. As described above, the creep model is physically irrational possessing the mechanical structure different basically from the elastoplasticity because the inelastic strain rate is induced even inside the yield surface. Nevertheless, Ohno et al. (2018), Wu et al. (2018), etc. incorporated the creep model into their cylindrical yield surface model in order to describe the rate-dependent deformation behavior, resulting in the ad hoc model by which the rate-independent and the rate-dependent deformation behaviors cannot be described in a unified equation. Needless to say, the deformation behavior in a general rate varying from the quasi-static to the non-static rate cannot be analyzed by such ad hoc method. Furthermore, various rate-dependent constitutive models including the time itself have been proposed to date (cf. e.g. Conway 1967; Penny and Marriott 1971; Boyce et al. 1988). Here, however, note that the determination of time when the creep deformation starts and stops is accompanied with an ambiguity depending on the subjectivity of observers, especially in a fluctuating rate of deformation. Therefore, these models are impertinent, lacking the objectivity.

14.4

Mechanical Response of Past Overstress Model

The development of rate-dependent elastoplastic constitutive equation is reviewed above and it is described that the overstress model would have a pertinent basic structure. Here, let the mechanical responses at the infinitesimal and the infinite rates of deformation be examined in order to clarify the basic property of this model. The past overstress model advocated by Bingham (1922) and extended by Hohenemser and Prager (1932), Prager (1961) and Perzyna (1963, 1966), Guo et al. (2013) (see also de Souza Neto et al., 2008) describes the elastoplastic deformation in the quasi-static loading since the viscous resistance of the dashpot diminishes but the elastic deformation in the impact loading since the viscous resistance of the dashpot becomes infinite. Therefore, it cannot be applied to the deformation behavior in a high rate as an impact loading. In what follows, we will show this fact on the past overstress model. Equations (14.11) and (14.12) are expressed in the following equations for the incremental forms.  n 1 f ðrÞ  1 n dt l FðHÞ

ð14:19Þ

 n 1 f ðrÞ  1 E : n dt l FðHÞ

ð14:20Þ

de ¼ E1 : dr þ dr ¼ E : de 

14.4

Mechanical Response of Past Overstress Model

441

Equation (14.20) is reduced to the following relation for the infinitesimal rate of deformation (quasi-static deformation) fulfilling dt ! 1: dr=dt ! O and de=dt ! O.  n 1 f ðrÞ  1 E: n OffiO l FðHÞ

ð14:21Þ

f ðrÞ 1!0 FðHÞ

ð14:22Þ

leading to

Then, the stress changes fulfilling the yield condition f ðrÞ ¼ FðHÞ, i.e. obeying the plastic constitutive relation, while the elastic deformation is given by the change of stress, so that Eqs. (14.11) and (14.12) exhibit the response of the elastoplastic constitutive relation in the quasi-static deformation as shown in Fig. 14.4. Then, the overstress model is the extension of the elastoplastic constitutive equation to the non-zero rate of deformation. In fact, the elastoplastic constitutive relation is reproduced by the quasi-static deformation in the overstress model. Here, it is reproduced easily for the non-viscous material, i.e. the material with a small viscoplastic coefficient l ffi 0 causing f ðrÞ ¼ FðHÞ, noting that the Bingham model is reduced to the Prandtl model for l ffi 0 in Fig. 14.2. On the other hand, in the infinite rate of deformation fulfilling dt ffi 0: dr=dt ! 1 and de=dt ! 1, Eq. (14.12) is reduced to 



dr ¼ E : de  O; i.e:r ¼ E : e O

ð14:23Þ

Elastic

 Impact loading  ||ε || 

|| ε || increase

Overstress

0

R =1

Quasi-static loading  || ε || 0

ε

Fig. 14.4 Past overstress model which is inapplicable to prediction of deformation behavior at high rate

442

14 Viscoplastic Constitutive Equations with Subloading Surface Concept

approaching the elastic response as shown in Fig. 14.4. Therefore, it predicts the unrealistic response that the material can bear an infinite load. The past overstress model is inapplicable to the prediction of deformation at high rate in general. In order to modify this defect, Lemaitre and Chaboche (1990) and Chaboche (2008) proposed to add the creep strain rate in addition to the elastic and the viscoplastic strain rate. This modification is irrational spoiling the rational advantage of the overstress model. The rigorous modification of the overstress model so as to describe the elasto-viscoplastic behavior in the general rate ranging from the quasi-static to the impact loading can be attained by the subloading-overstress model described in the next section. Eventually, the existing formulation of overstress model in Eq. (14.11), i.e. (14.12) is inapplicable to the prediction of deformation at a high rate. The material constant n included as the power form in Eq. (14.11), i.e. (14.12) is usually selected to be larger than five, but the fitting to the test data for impact load is impossible even if n is selected as one hundred which, needless to say, results in the inappropriate prediction of deformation in a slow loading process. In addition, the inclusion of a high power in the equation induces difficulty in numerical calculations. The so-called flow stress model was proposed by Johnson and Cook (1983) in which the yield stress, i.e. flow stress depends not only on the viscoplastic strain rate but also on the viscoplastic strain rate. It is the empirical model without a generality, which is concerned with the fast loading behavior but inapplicable to the slow loading behavior. However, unfortunately it is used widely for the deformation analyses in the fast loading behavior by adopting the commercial FEM software, e.g. Abaqus and LD-DYNA. Hereinafter, it is desirable to adopt the subloading-overstress model for the accurate analyses of the general loading behavior ranging from the quasi-static to the impact loadings, which will be described in the next section.

14.5

Subloading Overstress Model: Extension to Description of General Rate of Deformation

The subloading surface model will be extended to the subloading-overstress model in this section, which is capable of describing the deformation in general rate from quasi-static (elastoplastic) to the impact loading. All the aforementioned defects in the traditional overstress model are excluded in the subloading-overstress model.

14.5.1

Static and Limit Subloading Surfaces

The static subloading and the limit subloading surfaces (see Fig. 14.5) are incorporated in addition to the subloading, the normal-yield, the elastic-core and the limit

14.5

Subloading Overstress Model: Extension to Description …

443

Fig. 14.5 Limit subloading, subloading, normal-yield, static-subloading surface, limit elastic-core, elastic-core and static-subloading surfaces in subloading-overstress model

elastic-core surfaces in the subloading surface model for the rate-independent elastoplastic deformation described in Chap. 11. Here, the subloading surface on which the current stress lies may become larger than the normal-yield surface in general. The normal-yield ratio R which may become larger than unity in general is calculated by the identical equation for the rate-independent elastoplastic deformation, i.e. Eq. (11.42) in general and by Eq. (12.32) for metals and by Eq. (13.164) for soils. The viscoplastic strain rate is induced by the overstress from the static subloading surface expressed in the following equation which is given by replacing the current stress r, the center a and the normal-yield ratio R to their conjugate points rs , as and the static normal-yield ratio Rs ð  1Þ, respectively, in the subloading surface in Eq. (11.2) for the elastoplastic deformation. f ðrs Þ ¼ Rs FðHÞ

ð14:24Þ

with rs ¼ rs  as ¼



Rs rs  as r  a rc ; rs  c ¼ Rs ðry  cÞ ¼ Rs r ¼ R Rs R R ð14:25Þ

444

14 Viscoplastic Constitutive Equations with Subloading Surface Concept

Fig. 14.6 Function

in the evolution rule of static-normal-yield ratio: (a) Re = 0, (b) Re > 0

where Rs evolves by the following equation which is identical to Eq. (9.9) with Eq. (11.53) for the extended subloading surface model with the replacement of the 



plastic strain rate e p to the viscoplastic strain rate e vp (see Fig. 14.6). O

ð14:26Þ

with ð14:27Þ Rs ð  1Þ is called the static normal-yield ratio because the quasi-static deformation proceeds in the state R ¼ Rs . It can be analytically calculated by Eq. (9.15) with the replacement of the plastic strain rate to the viscoplastic strain rate for = const. as follows:

under the initial condition Rs ¼ Rs0 for e ¼ vp



evp 0 ,

where e ¼ vp

R

ð14:28Þ jje jjdt. The  vp

inequality R s \0 holds for Rs [ 1 in Eq. (14.26)1 with Eq. (14.27), bringing about the high numerical efficiency.

14.5

Subloading Overstress Model: Extension to Description …

14.5.2

445

Viscoplastic Strain Rate

Now, let the flow rule of the viscoplastic strain rate be given by incorporating the crucially important variable in the subloading surface model, i.e. the normal-yield ratio R into the past overstress model (Prager 1961; Perzyna 1963, 1966) as follows:  vp

e ¼ C n

ð14:29Þ

where C is the positive viscoplastic multiplier given by C

1 hR  Rs in lv

ð14:30Þ

or exp[

]

,

ð14:31Þ

sinh

lv (viscoplastic coefficient) and n are the material constants (Hashiguchi 2022a). The smooth elastic-viscoplastic transition is described by adopting the overstress due to R  RS instead of R  1, because RS increases gradually up to 1 with the viscoplastic deformation, while RS was fixed as RS ¼ 1 in the past overstress model. The function U for the evolution rule of the normal-yield ratio R and the positive _ are modified to Eqs. (11.53) and (11.54) by incorporating proportionality factor k ^c : n Þ in the extended subloading surface and Cn ð n the variables model for the rate-independent elastoplastic deformation. Concurrently, the function Us for the evolution rule of the static normal-yield ratio Rs is given by Eq. (14.27). Analogously, let Eqs. (14.30) and (14.31) for viscoplastic multiplier C be modified to the following equation (see Fig. 14.7). ð14:32Þ or ð14:33Þ where  uc is the material constant. Thus, the rising and the lowering of the stress are and , respectively. It leads to the description enforced in the state of the global Masing effect in the viscoplastic deformation process as shown in Fig. 14.7. 

On uniaxial deformation at constant strain rate ðea ¼ constantÞ 

The axial stress rate ra increases rapidly (almost elastically) at the beginning of the 

deformation because the variable R  Rs is small and thus e vp a is small as 





e e vp a e a ffi e a . Thereafter, as the elastic-core increases toward

, the

446

14 Viscoplastic Constitutive Equations with Subloading Surface Concept

normalyield ratio R increases rapidly for a certain increase of stress and thus R  Rs increases rapidly so that the viscoplastic strain rate develops rapidly leading to 







e e vp a ¼ e a ðe a ¼ 0Þ resulting in r a ¼ 0 (peak stress state). The further increase of induces the large viscoplastic strain rate R  Rs caused by the increase of 











e vp [ e leading to e e ð¼ e  e p Þ\0, i.e. ra \ 0 if we use Eq. (14.30) or (14.31). Then, the unrealistic stress-strain curve exhibiting the peak point and the decrease of stress is predicted by these equations. This defect can be remedied by sup

. It can be realized by pressing the viscoplastic strain rate e vp a with the increase of leading to dividing the positive viscoplastic multiplier C by the variable the suppression of the viscoplastic strain rate with the increase of R c as shown in Eq. (14.32) or (14.33) (see Fig. 14.7). Further, let Eqs. (14.32) and (14.33) be extended such that the infinite viscoplastic strain rate is induced for R ! cm Rs ðcm : material constantÞ leading to C ! 1 in the impact loading process as follows: ð14:34Þ

Fig. 14.7 Modification of stress vs. viscoplastic strain curve by incorporation of variables Cn leading to expression of generalized Masing effect

and

14.5

Subloading Overstress Model: Extension to Description …

447

or ð14:35Þ The variable cm Rs ð cm Þ is called the limit normal-yield ratio. Here, the surface to which the stress can reach at most is given by setting R ¼ cm Rs in the subloading surface in Eq. (11.2) is called the limit subloading surface. On the other hand, the elastic response is described unrealistically for the impact loading in the past overstress models. Further, the quite irrational model was proposed by Lemaitre and Chaboche (1990) incorporating the extra creep strain rate in addition to the elastic and the overstress strain rates. The rational description of deformation in the general rate ranging from the quasi-static to the impact loading is attained by the above-mentioned subloading-overstress model in which the crucially-important variables, i.e. the and the limit normal-yield ratio R, the static normal-yield ratio normal-yield ratio cm Rs are incorporated. The rates of the internal variables H; a and c are given from Eqs. (11.12), (11.13) 



nÞ as follows: and (11.25) with the replacement of e p to e vp ð¼ C 

H ¼ CfHn 

a ¼ Cf kn 

c ¼ Cf cn :

ð14:36Þ ð14:37Þ ð14:38Þ

The isotropic hardening stagnation can be incorporated by the identical equa

tions formulated in Sect. 12.2 with the replacement of the plastic strain rate e p to 

the viscoplastic strain rate e vp .

14.5.3

Strain Rate Versus Stress Rate Relation

The strain rate versus stress rate relations are given from Eqs. (14.3) and (14.29) as follows:
  e ¼ E1 : r þ Cn   r ¼ E : e CE : n

ð14:39Þ

which is represented in the incremental form as follows:


de ¼ E1 : dr þ Cndt dr ¼ E : de  CE : ndt

ð14:40Þ

Then, it follows from Eq. (14.40) with Eq. (14.30) or (14.31) in the quasi-static deformation that

448

14 Viscoplastic Constitutive Equations with Subloading Surface Concept

Fig. 14.8 Stress–strain curve predicted by subloading-overstress model

C ¼ 0 leading to R ¼ RS for jjdejj=dt ! 0 and jjdrjj=dt ! 0

ð14:41Þ

so that the stress changes along the static subloading surface given by Eq. (14.24). Consequently, the response of the subloading–overstress mode is reduced to that of the original subloading surface model in the rate–independent elastoplastic deformation behavior described in Chaps. 8–13. The subloading–overstress model is no more than the generalization of the subloading surface model to the description of the elasto-viscoplastic deformation in the general strain rate. Irreversible deformations in any rate from the static to the impact loading can be described by the subloading— overstress model. Eventually, the elastoplastic constitutive equation with the plastic modulus for the rate–independent elastoplastic deformation which is derived by consistency condition of the yield condition (subloading surface for the subloading surface model) can be disused by adopting only the subloading–overstress model. In the overstress model, we do not need to calculate the plastic modulus which possesses often a complex form as seen in Eq. (11.45) but instead we have only to perform the update calculation of the viscoplastic internal variables involved in the positive viscoplastic multiplier C in Eq. (14.29) and n. The stagnation of isotropic hardening is incorporated by replacing the plastic 



strain rate e p to the viscoplastic strain rate e vp in the formulation described in Sect. 12.2. The stress–strain curve described by the subloading-overstress model is illustrated in Fig. 14.8. The smooth elastic-viscoplastic transition and the cyclic loading behavior can be described appropriately. The stress is given from Eqs. (8.4) and (14.29) by ð14:42Þ

14.5

Subloading Overstress Model: Extension to Description …

449

The second term in the right-hand sides is induced by the viscoplastic deformation caused from the mutual viscoplastic slips of material particles, which is called the viscoplastic (stress) relaxations. It should be emphasized that the distinguished advantages in the subloading-overstress model is realized by the incorporation of the basic variable in the subloading surface model, i.e. the normal-yield ratio R in which various important information on the hardening/softening, the cyclic loading behaviors, etc. are incorporated. On the other hand, the overstress models other than the subloading-overstress model can never succeed to describe the plastic and the viscoplastic deformation behaviors rigorously, because the normal-yield ratio R has never been incorporated in them, although various complicated exponential-type, hyperbolic sine-type, etc. functions have been incorporated for the positive viscoplastic multiplier C as reviewed by Chaboche (2008). The computer program for the subloading-overstress model for metals is given in the Appendix L(c).

14.6

Comparison with Test Data

The capability of the subloading-overstress model for describing the deformation behavior of metals in the general rate of deformation including the quasi-static deformation has been verified performing the simulation of various test data by Hashiguchi et al. (2023). Some of the simulation results are shown in this section.

14.6.1

Dynamic Loading Process Inducing Elastic-Viscoplastic Deformation

The monotonic loading behaviors at various constant strain rates at 550° C in the test data for Modified 9Cr-1 Mo steel after Abel-Karim and Ohno (2000) and their simulations are shown in Fig. 14.9, where the material parameters are chosen as follows: Material constants: Elastic moduli: E ¼ 150; 000 MPa; m ¼ 0:3; ( isotropic: sr ¼ 0:23; cH ¼ 200; Hardening kinematic: ck ¼ 10 MPa; bk ¼ 0:1; Evolution of normal  yield ratio: u ¼ 1; 000; uc ¼ 6; Re ¼ 0:2; Translation of elastic  core: ce ¼ 5; v ¼ 0:7; c ¼ 5; n ¼ 4; cm ¼ 1:9 Viscoplastic deformation: lv ¼ 2; 000; u

450

14 Viscoplastic Constitutive Equations with Subloading Surface Concept

Fig. 14.9 Uniaxial loading behavior of Modified 9Cr-1 Mo steel under various axial strain rates at 550°C (test data after Abdel-Karim and Ohno 2000)

Initial values: Isotrophic hardening function: F0 ¼ 230MPa: The quite close simulations are also seen in Fig. 14.9. The accuracy of the present simulations are same levels of the simulations by the experimenters Abdel-Karim and Ohno (2000) based the creep model, who have provided the test data, while the creep model is physically irrational lacking the generality, i.e. incapable of describing the rate-independent elastoplastic deformation behavior as explained in Sect. 14.3. The deformation behavior at the variable strain rate between 0.001/s and 0.00001/s for 304 stainless steel at room temperature 20°C in the test data after Krempl (cf. Lemaitre and Chaboche 1979) is shown in Fig. 14.10, where the material parameters are chosen as follows:

14.6

Comparison with Test Data

451

Fig. 14.10 Uniaxial loading behavior under various axial strain rate between 0.1 and 0.001 (%/s) of 304 stainless steel at room temperature 20°C (test data after Krempl: cf. Lemaitre and Chaboche 1990)

Material constants: Elastic moduli: E ¼ 200; 000 MPa; m ¼ 0:3; ( isotropic: sr ¼ 0:2; cH ¼ 200; Hardening kinematic: ck ¼ 500 MPa; bk ¼ 0:5; Evolution of normal  yield ratio: u ¼ 1; 000; uc ¼ 6; Re ¼ 0:2; Translation of elastic  core: ce ¼ 200; v ¼ 0:7; c ¼ 10; n ¼ 4; cm ¼ 2 Viscoplastic deformation: lv ¼ 5; u Initial values: Isotrophic hardening function: F0 ¼ 160 MPa: The quite close simulation is shown in Fig. 14.10.

14.6.2

Quasi-static Loading Process Inducing Elastoplastic Deformation Behaviors

It was verified in the previous subsection that the subloading-overstress model is capable of describing the elasto-viscoplastic deformations induced in the dynamic

452

14 Viscoplastic Constitutive Equations with Subloading Surface Concept

Fig. 14.11 Mechanical ratcheting during pulsating loading between 0 and +830MPa of 1070 steel at room temperature (Test data after Jiang and Zhang 2008)

loading process. Further, it will be shown below that the subloading-overstress model is capable of describing the elastoplastic deformations induced in the quasi-static loading process at the room temperature by the simulations of test data. The comparison with the test data for the 1070 steel under the cyclic loading of axial stress between 0 and +830MPa at room temperature after Jiang and Zhang (2008) is depicted in Fig. 14.11, where the material parameters and the strain rate are selected as shown in the following, while the axial strain state is chosen to be 

quite low, i.e. e a ¼ 107 =s, although the strain rate is not written in the paper of Jiang and Zhang (2008). Material constants: Elastic moduli: E ¼ 160; 000 MPa; m ¼ 0:3; ( isotropic: sr ¼ 0:61; cH ¼ 155; Hardening kinematic: ck ¼ 3; 000 MPa; bk ¼ 0:5; Evolution of normal  yield ratio: u ¼ 90; uc ¼ 6; Re ¼ 0:5; Translation of elastic  core: ce ¼ 7; 000; v ¼ 0:7; c ¼ 6; n ¼ 3; cm ¼ 5 Viscoplastic deformation: lv ¼ 5; 000; u

14.6

Comparison with Test Data

453

Fig. 14.12 Cyclic loading behavior under increasing strain amplitude of 316 steel at room temperature (Test data after Chaboche et al. 1979): (a) Isotropic hardening stagnation is considered, (b) Isotropic hardening is ignored

Initial values: Isotrophic hardening function : F0 ¼ 471 MPa: Quite close simulation is shown in Fig. 14.11, while the isotropic hardening stagnation is hardly relevant to the accuracy of simulation. Furthermore, examine the uniaxial cyclic loading behavior under the constant symmetric strain amplitudes to both postive and negative sides. Comparison with the test data of the 316 steel under the cyclic loading with the increasing axial strain amplitudes ±1.0, ±1.5, ±2.0, ±2.5, ±3.0% after Chaboche et al. (1979) is depicted in Fig. 14.12 where the material parameters and the strain rate are selected as shown in 

the following, while the axial strain rate is chosen to be quite low, i.e. e a ¼ 108 =s, although the strain rate is not written in the paper of Chaboche et al. (1979). Material constants: Elastic moduli: E ¼ 170; 000 MPa; m ¼ 0:3; ( isotropic: sr ¼ 1:2; cH ¼ 8; Hardening kinematic: ck ¼ 2; 000 MPa; bk ¼ 0:4; Evolution of normal  yield ratio: u ¼ 20; uc ¼ 3; Re ¼ 0:5; Translation of elastic  core: ce ¼ 283; v ¼ 0:7; c ¼ 3; n ¼ 2; cm ¼ 10 Viscoplastic deformation: lv ¼ 5; 000; u

454

14 Viscoplastic Constitutive Equations with Subloading Surface Concept

Initial values: Isotrophic hardening function: F0 ¼ 290 MPa: The quite close simulation is shown in Fig. 14.12, while the accuracy of the simulation decreases without the isotropic hardening stagnation since the full inverse loading is repeated mant tymes in this test data. As shown in this section, not only the rate-dependent but also the rate-independent deformation behaviors can be described accurately by the subloading-overstress model and thus the elastoplastic constitutive equation can be disused by adopting the subloading-overstress model.

14.7

Temperature Dependence of Elasto-Viscoplastic Deformation Behavior

The elasto-viscoplastic deformation behavior is influenced by the temperature. However, the dependence of material functions and constants in the elasto-viscoplastic constitutive equation would be complex so that they are specified for each temperature resulting in the piecewise-linear relation in the deformation analysis as seen in the research papers (e.g. Ohno et al. 2018, 2021) and the commercial software (e.g. Marc and Abaqus). Such method for incorporation of the piecewise-linear relation is easy-going but requires the large computational load. Therefore, it is desirable to formulate material functions and constants as functions of temperature. The simple equations of the isotropic hardening function and the Young’s modulus as the functions of temperature are given in this section. The isotropic hardening function FðHÞ would decrease with the elevation of temperature. The following equation taken account of the influence of temperature would be able to be postulated referring to the test data (Poh 2001; Vu and Staat 2007; Cao et al. 2014; Li et al. 2019; Lee et al. 2020) (see Fig. 14.9). h i F ðH; hÞ ¼ F0 f1 þ h1 ½1  expðh2 H Þ g exp Ch ðh  hS Þ2

ð14:43Þ

F ðH; hÞ ¼ F0 f1 þ h1 ½1  expðh2 H Þ gsech2 ½Ch ðh  hS Þ

ð14:44Þ

or

where h is the temperature (centigrade). hs is the material constant designating the temperature below which FðH; hÞ is highest and constant, while it is lower than the room temperature. Ch is the material constant designating the intensity of the temperature-dependence. Equation (14.43) is the slight modification of the normal (Gaussian) distribution function. Equation (14.44) is invoked from the crystal plasticity (Peierce et al. 1983). It should be noticed that the consideration of the

14.7

Temperature Dependence of Elasto-Viscoplastic Deformation Behavior

455

temperature-dependence at the temperature much higher than hs , i.e. h  hS or h  hS is required in many engineering cases. The viscoplastic multiplier C in Eq. (14.30) or (14.31) would be extended to the temperature dependence by multiplying the Arrhenius type term (cf. e.g. Conrad 1964; Skrzypek and Hetnarski 1993; Rusinek et al. 2007) i.e. C ! C exp

Q RT

ð14:45Þ

where Q is the activation energy, R is the gas constant and C is the absolute temperature. Besides, the following equation for the temperature dependence of the Young’s modulus E was proposed by Watchman et al. (1961) (see also Rusinek et al. 2007) based on the experimental data.





CT dE CT CT ; exp  ¼ B 1 þ E ¼ ET0  BT exp  T T T dT

ð14:46Þ

where ET0 is the Young’s modulus at the absolute temperature zero T ¼ 0, B and CT being the material constants, while expðCT =T Þ is the Boltzmann factor. Equation (14.46) leads to the linear relation between E and T, i.e. dE=dT ¼ B for the above-room temperature, i.e. T [ 273ðKÞ as shown in Fig. 14.10. Therefore, the following linear relation would hold for the above room temperature from Eq. (14.46). E ¼ E0  Bh

ð14:47Þ

where E0 is the Young’s modulus at h ¼ 0. On the other hand, the Poisson’s ratio m would not be much affected by the temperature. The strain eh induced by the temperature elevation would be given by eh ¼ aðh  h0 ÞI

ð14:48Þ

where a is the material constant. The formulations of the temperature-dependence of material constants other than the isotropic hardening function and the Young’s modulus are also required in the elastoplastic deformation analysis. In particular, the temperature-dependences of the material constants involved in the kinematic hardening and the translation of the elastic-core will have to be formulated in the near future.

Chapter 15

Continuum Damage Model with Subloading Surface Concept

In the brittle materials such as rocks, concrete, glass, and ceramics, the growth of small cracks and their coalescence leading to the macroscopic failure are caused when the stress reaches the yield surface and thereby a plastic deformation is induced. On the other hand, in the ductile materials such as metals, the growth of small voids and their coalescence leading to the failure are caused when a large plastic deformation is induced. These phenomena are called the damage. The phenomenological mechanics describing the damage phenomena within the framework of the continuum mechanics without resorting to a microscopic description is called the continuum damage mechanics. The continuum damage mechanics for the brittle materials has been developed by Kachanov (1958), Rabotnov (1969), Lemaitre and Chaboche (1990), Lemaitre (1992), Lemaitre and Desmoral (2005), de Sauza Neto et al. (2008), Murakami (2012), etc. and the one for the ductile materials has been developed by Gurson (1977), Needleman and Rice (1978) and others. The incorporation of the subloading surface model would be inevitable in the constitutive equation of the damage, because the damage progresses remarkably under the cyclic loading even in the low stress level inside the yield surface and the remarkable softening occurs. The elastoplastic-damage and elasto-viscoplasticdamage models for the brittle and the ductile materials and their extensions by the incorporation of the extended subloading surface model to describe the cyclic loading behavior are explained in this chapter.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 K. Hashiguchi, Foundations of Elastoplasticity: Subloading Surface Model, https://doi.org/10.1007/978-3-030-93138-4_15

457

458

15

15.1

Continuum Damage Model with Subloading Surface Concept

Basic Hypothesis of Strain Equivalence in Constitutive Equation with Brittle Damage

The hypothesis of strain equivalence (Lemaitre 1971) insists “The deformation of the damaged actual material subjected to the actual stress is identical to the deformation of the undamaged material subjected to the effective stress”, i.e. “The strain in the current damaged state is given by the constitutive equation for the undamaged state with the replacement of the stress to the effective stress”. Let this hypothesis be applied also to the elastic and the plastic strains, i.e. ep ðr 6¼ r Þ e ¼ e ; ee ¼ ee; ep ¼  

ð15:1Þ

εe = −1: σ = (D)−1: σ (σ, D) % % %

ð15:2Þ

where the variables in the undamaged effective configuration are specified by adding the under tilde ( ). The hypothesis of strain equivalence holds based on the fact that only the microcracks without thickness are induced. Therefore, it holds for the brittle damage but does not hold for the ductile damage in which microvoids possessing the volumes are induced.

15.2

Hyperelastic Equation in Undamaged Variables

The hyperelastic constitutive equation is described by the elastic strain energy function wðee Þ and the Gibbs’ free energy function /ðr Þ noting Eq. (7.74) as follows: r ¼ 

@wðee Þ ; @ee

ee ¼

@/ðr Þ  @r 

ð15:3Þ

Now, adopt the quadratic forms of wðee Þ and /ðr Þ as follows: 
 1 e 1 e e wðe Þ ¼ e : E :e ¼ r :e ;  2 2 e


 1 1 e 1 /ðr Þ¼ r :E : r ¼ r :e   2  2

ð15:4Þ

leading to : ee ; r ¼ E 

1 ee ¼ E : r 

is the elastic modulus tensor in the undamaged state. where E 

ð15:5Þ

15.2

Hyperelastic Equation in Undamaged Variables

459

Here, let E be given by the Hooke’s law in Eq. (7.91) under the assumption that the Young’s modulus is influenced but the Poisson’s ratio m is not influenced by the damage, i.e. ; v ¼ v E E 

ð15:6Þ

leading to E ⎧ = % (S + ν T ), ⎪E 1 − 2ν % 1 +ν ⎪ ⎨ −1 1 ⎪E = [(1 +ν )S −νT ], E ⎪ ⎩% %

E ijkl = E% [ 1 (δ ik δ jl + δ ilδ jk ) + ν δ ijδ kl ] 1 +ν 2 1 − 2ν % −1 E ijkl = 1 [ 1 (1 + ν )(δ ik δ jl + δ ilδ jk ) −νδ ijδ kl ]

%

ð15:7Þ

E 2 %

Then, the elastic strain energy function wðee Þ and the elastic complementary strain energy function /ðr Þ in the undamaged state are given for Eq. (15.7) referring to Eq. (7.98) as follows: 8 1 E m > e e e 2 e > > < wðe Þ ¼ 2 1 þ m ½eij eij þ 1  2m ðekk Þ  1 > 2 > > : /ðr Þ ¼ 2 ½ð1 þ mÞr ij r ij  mðr kk Þ  E

ð15:8Þ

The effective stress r and the elastic strain ee are derived from Eq. (15.3) with (15.8) as follows: ⎧ E ∂ψ (ε e ) ν e e e ⎪σ% = ∂ ε e = E: ε = 1 +% ν [ε + 1 − 2ν (tr ε ) I ] % ⎪ ⎪⎛ ⎞ e ⎪⎜ σ ij = ∂ψ (εe ) = E ijkl ε e = E% (ε ij + ν ε kk δ ij ) ⎟ kl 1 +ν 1 − 2ν ∂ε ij ⎟ ⎪⎜ % % ⎪⎝ ⎠ ⎨ σ φ ( ) ∂ 1 ⎪ εe = = E−1 : σ = E [(1+ν ) σ −ν (tr σ)I] ∂σ% ⎪ % % % % % % ⎪ ⎛ ⎞ ⎪ ∂φ (σ ) −1 σ kl = 1 [(1+ν )σ ij −νσ kk δ ij ] ⎟ = E ijkl ⎪⎜ ε ije = % E ∂σ ij ⎟ % % % % ⎪⎩⎜⎝ % ⎠ %

ð15:9Þ

The relation of the elastic strain rate and the undamaged fictitious stress rate is given as follows: 



1 ee ¼ E :r ;  





r ¼E : ee  

ð15:10Þ

460

15

15.3

Continuum Damage Model with Subloading Surface Concept

Hyperelastic Equations with Damage

Let the quadratic free energy functions for the damaged state be given by
 8 1 e 1 D e e e e > < w ðe ; DÞ ¼ e : EðDÞ: e ¼ rðe ; DÞ: e 2 2 1 > : /D ðr; DÞ ¼ rðee ; DÞ: E1 ðDÞ: rðee ; DÞ 2 where Dð0  D  1Þ is the damage variable, from which one has 8 > @wD ðee ; DÞ > < rðee ; DÞ ¼ ¼ EðDÞ: ee ; @ee > @/D ðr; DÞ > : ee ¼ ¼ E1 ðDÞ: rðee ; DÞ @r

ð15:11Þ

ð15:12Þ

The relation between the actual stress rðee ; DÞ and the effective stress r is given  from Eqs. (15.5) and (15.12) as 1 : ee ¼ E : E1 ðDÞ: rðee ; DÞ; rðee ; DÞ ¼ EðDÞ: E : r r ¼ E   

15.3.1

ð15:13Þ

Bilateral Damage

We consider the bilateral damage in which the identical damage is induced in both positive and negative principal stress directions. We adopt the Hooke’s law and it is assumed that the Young’s modulus E decreases by the following equation but the Poisson’s ratio V is not influenced by the damage. ; E ¼ ð1  DÞ E 

v ¼ const:

ð15:14Þ

Then, the actual damaged elastic modulus tensor in the Hooke’s law is given by taking account of Eq. (15.14) into Eq. (15.7) as follows: (1 − D ) E ν ⎫ δ δ + 1 (δ δ + δ ilδ jk )] ⎪ [ 1 + ν % 1 − 2ν ij kl 2 ik jl ⎪ ⎬ 1 1 (1 + ν )(δ δ + δ δ ) −νδ δ ]⎪ [ = ij kl ik jl il jk (1 − D)E 2 ⎪ ⎭ %

E ijkl ( D) = (1 − D)E ijkl = %

−1

E ijkl ( D) =

−1 1 E (1 − D) % ijkl

ð15:15Þ

15.3

Hyperelastic Equations with Damage

461

which leads to ; E ¼ EðDÞ=ð1  DÞ EðDÞ ¼ ð1  DÞ E   EðDÞ1 ¼ E  

1

=ð1  DÞ; 

; EðDÞ ¼  D E 

E  

1

¼ ð1  DÞEðDÞ1

EðDÞ1 ¼

ð15:16Þ ð15:17Þ



D E 1 2 ð1  DÞ

ð15:18Þ

Incidentally, the strain energy function and the complementary energy function are given by 8
 1 1 1 > D e e e > > r e w ð e ; D Þ ¼ ð ; D Þ: e : ee ¼ ee : EðDÞ : ee ¼ ð1  DÞee : E >  > 2 2 2 > > > > > 1 ð1  DÞ E m >  e e > ðtr ee Þ2  ¼ ½e : e þ > > > 2 1þm 1  2m > >
 > h > > 1 ð1  DÞ E v  e 2 i > e e  e > w e e ð ; D Þ ¼ e þ e > ij ij kk < 2 1þv 1  2v
 1 1 1 > D e e e 1 > rðee ; DÞ: E : rðee ; DÞ / ðr; DÞ ¼ rðe ; DÞ: e ¼ rðe ; DÞ: EðDÞ1 : rðee ; DÞ ¼ > >  > 2 2 2ð1  DÞ > > > h i > 1 > > > ¼ ð1 þ vÞrðee ; DÞ: rðee ; DÞ  vðtrrðee ; DÞÞ2 > > 2ð1  DÞ E >  > ! > > h i > > 1 2 > D e e e > ð1 þ vÞrij ðe ; DÞrij ðe ; DÞ  vðrkk ðe ; DÞÞ / ðr; DÞ ¼ > : 2ð1  DÞ E 

ð15:19Þ Then, the stress tensor in the current damaged state is given from Eq. (15.13) with Eq. (15.16) as follows: 8  ð1  DÞ E  > @wD m  > e e e e > tre e r ð e ; D Þ ¼ ¼ ð1  DÞ r ¼ ð1  DÞ E : e ¼ þ ð ÞI ; > e  >  > 1  2v @e 1þm < r ¼ rðee ; DÞ=ð1  DÞ  > > > > > 1 1 > : ee ¼ EðDÞ1 :rðee ; DÞ ¼ E 1 : rðee ; DÞ ¼ ½ð1 þ mÞr  mðtrrÞI 1D ð1  DÞE

ð15:20Þ from which one has 8    > D < r ¼ ð1  DÞ r þ Dr ¼ 1 r þ r  1D ð1  DÞ2 ð1  DÞ2 >  :  r ¼ ð1  DÞ r  D r

ð15:21Þ

462

15

Continuum Damage Model with Subloading Surface Concept

8 >  > >

> > : r ¼ ð1  DÞ E : e e  

ð15:22Þ



D r 1D

noting  

ee¼

15.3.2

D E ð1  DÞ2 

1

:rþ

1  E 1 : r 1D

ð15:23Þ

Unilateral Damage

The cracks on the planes to which the negative (compressive) principal stresses act would close partly, which is called the crack-closure effect, and this phenomenon is called the unilateral damage. Constitutive equations for the unilateral damage will be given in the following. (1) General tensor formulation with stress transformation tensor The stress tensor is represented as r¼

3 X i¼1

ri nri



nri

¼

3 X 3 X

dij ri nri



nrj

i¼1 j¼1

¼

3 X 3 X

! rij ei  ej

ð15:24Þ

i¼1 j¼1

 where ri ði ¼ 1; 2; 3Þ are the principal stresses and nri are the unit vector base in the principal stress directions of the stress tensor r. It is postulated that the openings of the cracks are induced on the plane to which the positive principal stresses apply. Then, let the stress tensor be decomposed into the stress tensors with the positive and the negative principal values as follows: r ¼ rþ þ r

ð15:25Þ

8 3 2 0 0 hr1 i > > 3 3 > X > 7 X 6 þ r r > > ¼ r ¼ r  n ¼ H ðri Þri nri  nri 0 0 r h i h in 5 4 2 i > i i > > > i¼1 i¼1 < 0 0 hr 3 i 3 2 > r 0 0 h i > 1 > 3 3 > X X >  7 6 > > r ¼ 4 0 hr2 i 0 5¼ hri inri  nri ¼ H ðri Þri nri  nri : > > > : i¼1 j¼1 hr3 i 0 0

ð15:26Þ

15.3

Hyperelastic Equations with Damage

463

where h i is the Macaulay’s bracket and Hð Þ is the Heaviside’s step function, i.e. HðsÞ ¼ 1 for s  0 and HðsÞ ¼ 0 for s\0 (s: arbitrary scalar variable), fulfilling

8 1 for a1 ; a2 ;    ; an [ 0 > >    ð ÞH ð a Þ H ð a Þ ¼ H a < 1 2 n 0

for others 1 for a1 ; a2 ;    ; an \0 > > : H ða1 ÞH ða2 Þ    H ðan Þ ¼ 0 for others

ð15:27Þ

Further, introduce the coordinate transformation tensor from the fixed base fei g to  the positive principal stress bases Hðri Þnri and the negative principal stress base  Hðri Þnri , i.e.
3  8 3 P P > rþ r r > H ð r Þn  e ¼ H ð r Þd n  e Q < i i i i ij i j i¼1 i¼1
 3 3 P P > r > r r : Q  H ðri Þni  ei ¼  H ðri Þdij ni  ej i¼1

ð15:28Þ

i¼1

the components of which are given by 8 3 3   P P > rþ > H ðrr Þnrr  er ej ¼ H ðrr Þei  nrr djr ¼ H rj ei  nrj < Qij ei  r¼1

r¼1

3 3   P P > > : Qr H ðrr Þnrr  er ej ¼  H ðrr Þei  nrr djr ¼ H rj ei  nrj ij ei  r¼1

r¼1

ð15:29Þ The stress tensor r is transformed to r þ and r by applying the novel positive or negative orthogonal projection tensors Pr þ and Pr , respectively, as follows: r þ ¼ Pr þ : r r ¼ Pr : r

ð15:30Þ

where 8 i 3 P 3 P 3 P 3 h   P > rþ > > H ðri ÞH rj H ðrk ÞH ðrl Þdik djl nri  nrj  nrk  nrl