Elastoplasticity Theory

Lecture Notes in Applied and Computational Mechanics Volume 42 Series Editors Prof. Dr.-Ing. Friedrich Pfeiffer Prof. Dr.-Ing. Peter Wriggers

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Elastoplasticity Theory Koichi Hashiguchi

123

Koichi Hashiguchi 3-10-10-201, Ohtemon Fukuoka, 810-0074 Japan E-mail: [email protected]

ISBN: 978-3-642-00272-4

e-ISBN: 978-3-642-00273-1

DOI 10.1007/978-3-642-00273-1 Lecture Notes in Applied and Computational Mechanics

ISSN 1613-7736 e-ISSN 1860-0816

Library of Congress Control Number: 2009920691 © Springer-Verlag Berlin Heidelberg 2009 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, 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.

Typeset & Cover Design: Scientific Publishing Services Pvt. Ltd., Chennai, India. Printed on acid-free paper 9876543210 springer.com

Preface Contents

Recent advancements in the performance of industrial products and structures are quite intense. Consequently, mechanical design of high accuracy is necessary to enhance their mechanical performance, strength and durability. The basis for their mechanical design can be provided through elastoplastic deformation analyses. For that reason, industrial engineers in the fields of mechanical, civil, architectural, aerospace engineering, etc. must learn pertinent knowledge relevant to elastoplasticity. Numerous books about elastoplasticity have been published since “Mathematical Theory of Plasticity”, the notable book of R. Hill (1950), was written in the middle of the last century. That and similar books mainly address conventional plasticity models on the premise that the interior of a yield surface is an elastic domain. However, conventional plasticity models are applicable to the prediction of monotonic loading behavior, but are inapplicable to prediction of deformation behavior of machinery subjected to cyclic loading and civil or architectural structures subjected to earthquakes. Elastoplasticity has developed to predict deformation behavior under cyclic loading and non-proportional loading and to describe nonlocal, finite and rate-dependent deformation behavior. The author has been lecturing on applied mechanics and has been investigating elastoplasticity for nearly a half century, during which time elastoplasticity has made great progress. Various lecture notes, research papers, review articles in English or Japanese, and books in Japanese on these subjects are piled at hand. At present, the author is continuing composition of a monograph on elastoplasticity that has been published serially in a monthly journal from June 2007, to be completed at the end of 2009, although it will unfortunately have been written entirely in Japanese. Based on those teaching and research materials, this book comprehensively addresses fundamental concepts and formulations of phenomenological elastoplasticity from the conventional to latest theories. Especially, the subloading surface model falling within the framework of the unconventional plasticity model is delineated in detail, which enables us to predict rigorously the plastic strain rate induced by the rate of stress inside the yield surface, comparing it with the other unconventional models. The viscoplastic model is also presented; it is applicable to prediction of deformation behavior in the wide range of strain rate from the quasi-static to the impact loads. Explicit constitutive equations of metals and soils are given for practical application of the theories. In addition, constitutive models of friction are described because they are indispensable for analysis of boundary

VI

Preface

value problems. Various theories proposed by the author himself are included among the contents in this book in no small number. Their detailed explanations would be possible but, on the other hand, they would easily fall into subjective explanations. For that reason, particular care was devoted to retaining objectivity in the presentation. The main purpose of this book is to expedite the application of elastoplasticity theory to analyses of engineering problems in practice. Consequently, the salient feature of this book is the exhaustive explanation of elastoplasticity, which is intended to be understood easily and clearly not only by researchers but also by beginners in the field of applied mechanics, without reading any other book. Therefore, mathematics including vector-tensor analysis and derivatives and the fundamentals in continuum mechanics are first explained to the degree necessary to understand the elastoplasticity theory described in subsequent chapters. For that reason, circumstantial explanations of physical concepts and formulations in elastoplasticity are given without a logical jump such that the derivations and transformations of all equations are described without abbreviation. Besides, a general formulation unlimited to a particular material is first addressed in detail since deformations of materials obey fundamental common characteristics, which would provide the universal knowledge for deformation of materials more than describing a formulation for a particular material. Thereafter, explicit constitutive equations of metals, soils, and friction phenomena are presented in detail, specifying material functions involved in the general formulation. Without difficulty, readers will be able to incorporate the equations included in this book into their computer programs. The author expects that a wide audience including students, engineers, and researchers of elastoplasticity will read this book and that this work will thereby contribute to the steady development of the study of elastoplasticity and applied mechanics. As a foundation, the mathematical and the physical ingredients of the continuum mechanics are treated in Chapters 1, 2, 3, and 4. Chapter 1 addresses tensor analysis. The physical quantities used in continuum mechanics are tensors; consequently, their relations are described mathematically using tensor equations. Explanations for mathematical properties and rules of tensors are presented to the extent that is sufficient to understand the subject of this book: elastoplasticity. In chapters 2–4, physical quantities for the description of deformation of solids, containing stress (rate) and strain (rate), are described with their fundamental laws. Chapter 2 is devoted to the description of motion and strain (rate) and their related quantities. Chapter 3 presents conservation laws of mass, momentum, and angular momentum, and equilibrium equations and virtual work principles derived from them. In addition, their rate forms used for constitutive equations of inelastic deformation are explained concisely. Chapter 4 specifically addresses the objectivity of constitutive equations, which is required for the description of material properties. The objectivities of various stress, strain and rotation measures are described by examining their coordinate transformation rules. Then, it is shown that the material-time derivative of state variables has no objectivity. The corotational rate tensors fulfilling the objectivity are described in detail.

Preface

VII

Chapter 5 specifically examines the description of elastic deformation. Elastic constitutive equations are classified as either hyperelasticity, Cauchy elasticity, or hypoelasticity depending on their levels of reversibility. The mathematical and physical characteristics of these equations are explained prior to the description of elastoplastic constitutive equations in the subsequent chapters. Elastoplastic constitutive equations are described comprehensively in Chapters 6–8. Chapter 6 presents the basic concepts and formulations of plastic constitutive equation, e.g. the elastic and the plastic strain rates, the consistency condition, the plastic flow rule, and the loading criterion. Then, the elastoplastic constitutive equation is formulated based on them. A description of anisotropy and the tangential inelastic strain rate are also incorporated. However, they fall within the framework of conventional plasticity on the premise that the interior of the yield surface is an elastic domain. Therefore, they are incapable of predicting a smooth transition from the elastic to plastic state and a cyclic loading behavior of real materials. In Chapter 7, the continuity and the smoothness conditions are described first. They are the fundamental requirements for the constitutive equations for irreversible deformation. Then, various unconventional elastoplastic constitutive equations are proposed, with the intention of describing the plastic strain rate induced by the rate of stress inside the yield surface. Among those equations, only the subloading surface model, assuming the subloading surface passing through the current stress point and similar to the yield surface, fulfils the requirements for elastoplastic constitutive equations: continuity and smoothness conditions. Then, the inelastic strain rate attributable to the stress rate tangential to the subloading surface is pertinently incorporated, which is indispensable for the prediction of non-proportional loading behavior observed often in plastic instability problems. In Chapter 8, the initial subloading surface model explained in Chapter 7 is shown to be incapable of describing cyclic loading behavior appropriately because it predicts only an elastic strain rate in the unloading process. Therefore, excessive strain accumulation with open hysteresis loops is predicted in the cyclic loading process. Various cyclic plasticity models have been proposed to date. Among them, only the extended subloading surface model, making a similarity center of the normal-yield and the subloading surfaces move with a plastic strain rate, can represent cyclic loading behavior appropriately, fulfilling the continuity and smoothness conditions. Notable advantages of the extended subloading surface model are presented in comparison with the other cyclic plasticity models. Chapter 9 presents a viscoplastic constitutive equation for describing ratedependent deformation induced for the stress level over the yield surface. A pertinent viscoplastic constitutive equation is described, in which the concept of the subloading surface is incorporated into the overstress model. It is applicable to the prediction of rate-dependent deformation behavior from quasi-static to impact loads. In Chapters 10 and 11, based on the elastoplastic constitutive equations described in the preceding chapters, specific constitutive equations of metals and soils are formulated. They are typical elastoplastic materials related to engineering practice. Specific yield conditions, evolution rules of hardening and softening,

VIII

Preface

anisotropy, stagnation of isotropic hardening, up/degradation of structure, etc. for these materials are incorporated in their formulations. Special issues related to elastoplastic deformation behavior are discussed in Chapters 12–14. Chapter 12 specifically examines corotational rate tensors, the necessity of which is suggested in Chapter 4. Mechanical features of corotational tensors with various spins are examined comparing their simple shear deformation characteristics. The pertinence of the plastic spin is particularly explained. Chapter 13 opens with a mechanical interpretation for the localization of deformation inducing a shear band. Then, the approaches to the prediction of shear band inception condition, inclination and thickness, e.g. eigenvalue analysis and the gradient theory are explained, and the smeared model, i.e., the shear-band embedded model for the practical finite element analysis, is described. Chapter 14 verifies first the distinguished ability of the subloading surface model for the numerical calculation in the yield state. However, it is limited to the yield state. Then, the basic equations for the return-mapping method are shown for subyield state in the subloading surface model. Chapter 15 describes the prediction of friction phenomena. All bodies except those floating in a vacuum are contacting with other bodies so that the friction phenomena occur on their surfaces. Pertinent analyses, not only of the deformation behavior of bodies but also of friction behavior on the contact surface, are necessary for the analyses of boundary-value problems. A constitutive equation of friction is formulated similarly to the elastoplastic constitutive equation. It is subsequently extended to describe the transition from a static to a kinetic friction attributable to plastic softening and the recovery of the static friction attributable to creep hardening. The author wishes to express his thanks to the colleagues at Kyushu University, who have discussed and collaborated for a long time during work undertaken until retirement: Professor M. Ueno (currently at The University of the Ryukyus), and T. Okayasu, S. Tsutsumi, and S. Ozaki (currently at Yokohama National Univ.). In addition, Professor T. Tanaka of The University of Tokyo, Professor Yatomi, C. of Kanazawa Univ., Professor F. Yoshida of Hiroshima Univ., Professor M. Kuroda of Yamagata Univ., Y. Yamakawa of Tohoku Univ., Dr. T. Ozaki of Kyushu Electric Engineering Consultants Inc., and Mr. T. Mase of Tokyo Electric Power Services Co., Ltd. are appreciated for their valuable discussions and collaborations. Furthermore, the author would like to express his sincere gratitude to Professor A. Asaoka and his colleagues at Nagoya University: Professor M. Nakano and Professor T. Noda who have appreciated and used the author’s subloading surface model widely in their analyses and who have offered discussion continually on deformation of geomaterials. In addition, the author thanks Professor T. Nakai and Professor T.F. Zhang of the Nagoya Institute of Technology, for their valuable comments. The author is deeply indebted to Professor Bogdan Raniecki and H. Petryk of the Inst. Fund. Tech. Research, Poland, who have visited Kyushu University several times to deliver lectures on applied mechanics. Bogdan gave me valuable comments and suggestions by the critical reading of the manuscript, which highly

Preface

IX

contributed to the correction. Further, the author thanks Professor I.F. Collins of the University of Auckland, Professor O.T. Bruhns of Ruhr Univ., Bochum, Professor A.C. Aifantis of Michigan Tech. Univ. and Professor I. Vardoulakis of Natl. Univ. Tech. Athens, who have also stayed at Kyushu Univ., delivering lectures and engaging in valuable discussions related to continuum mechanics. The author wishes to acknowledge deeply Professor P. Wrrigers of Hanover Univ. who recommend me to publish this book in the series of Lecture Notes in Applied and Computational Mechanics, Springer. Finally, the author would like to state that the enthusiastic support of Mrs. Heather King from the Springer Publishing Company and SPS data processing team for their help in editing this book. Fukuoka December 2008

Koichi Hashiguchi

Contents Contents

1

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 for Vectors……………………………………. 1.2.3 Component Description 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………………………………………... 1.3.5 Tensor Product and Component…………………………... 1.4 Operations of Second-Order Tensor……………………………... 1.4.1 Trace……………………………………………………… 1.4.2 Various Tensors…………………………………………... 1.5 Eigenvalues and Eigenvectors………………………………….... 1.6 Calculations of Eigenvalues and Eigenvectors…………………... 1.6.1 Eigenvalues……………………………………………….. 1.6.2 Eigenvectors………………………………………………. 1.7 Eigenvalue and Eigenvectors of Skew-Symmetric Tensor………. 1.8 Cayley-Hamilton’s Theorem…………………………………….. 1.9 Positive Definite Tensor…………………………………………. 1.10 Polar Decomposition…………………………………………....... 1.11 Isotropic Tensor-Valued Tensor Function……………………….. 1.12 Representation of Tensor in Principal Space…………………….. 1.13 Two-Dimensional State………………….……………………….. 1.14 Partial Differential Calculi……………..……………………….... 1.15 Time Derivatives……………………..………………………..…. 1.16 Differentiation and Integration in Field……………………..……

1 1 1 1 2 8 8 8 9 14 14 15 17 18 19 20 20 21 26 31 31 32 33 35 35 36 37 40 44 47 50 51

2

Motion and Strain (Rate)……………………………………………. 2.1 Motion and Deformation…………………………………………. 2.1.1 Material, Spatial and Relative Descriptions………………. 2.1.2 Deformation Gradient and Deformation Tensors…………

57 57 57 59

XII

Contents

2.2 Strain Tensor……………………………………………………... 2.3 Strain Rate and Spin Tensors…………………………………….. 2.4 Various Simple Deformations……………………………………. 2.4.1 Uniaxial Loading………………………………………….. 2.4.2 Simple Shear……………………………………………… 2.4.3 Combination of Tension and Distortion…………………... 2.5 Surface Element, Volume Element and Their Rates……………...

64 70 81 81 84 94 97

3

Conservation Laws and Stress Tensors………………………..…… 3.1 Conservation Law of Mass………………………………………. 3.2 Conservation Law of Momentum………………………………... 3.3 Conservation Law of Angular Momentum………………………. 3.4 Stress Tensor……………………………………………………... 3.5 Equilibrium Equation…………………………………………….. 3.6 Equilibrium Equation of Moment………………………………... 3.7 Virtual Work Principle……………………………………………

101 101 101 102 102 105 107 108

4

Objectivity and Corotational Rate Tensor…………………………. 4.1 Objectivity………………………………………………………... 4.2 Influence of Rigid-Body Rotation on Various Mechanical Quantities………………………………………………………… 4.3 Rate of State Variable and Corotational Rate Tensor……………. 4.4 Transformation of Material-Time Derivative of Scalar Function to Its Corotational Derivative……………………………………. 4.5 Various Objective Stress Rate Tensors…………………………... 4.6 Work Conjugacy………………………………………………….

111 111 112 114 119 122 124

5

Elastic Constitutive Equations…….………………………………… 5.1 Hyperelasticity…………………………………………………… 5.2 Cauchy Elasticity………………………………………………… 5.3 Hypoelasticity…………………………………………………….

127 127 130 131

6

Basic Formulations for Elastoplastic Constitutive Equations…….. 6.1 Multiplicative Decomposition of Deformation Gradient and Additive Decomposition of Strain Rate………………………….. 6.2 Conventional Elastoplastic Constitutive Equations……………… 6.3 Loading Criterion……………….………………………………... 6.4 Associated Flow Rule……………………………………………. 6.4.1 Positivity of Second-Order Plastic Work Rate: Prager’s Interpretation ……………………………………………... 6.4.2 Positivity of Work Done During Stress Cycle: Drucker’s Hypothesis………………………………………………… 6.4.3 Positivity of Second-Order Plastic Relaxation Work Rate..……………………………………………………… 6.4.4 Comparison of Interpretations for Associated Flow Rule...

135 135 142 148 151 151 151 152 153

Contents

7

8

9

10

XIII

6.5 Anisotropy………………………………………………………... 6.5.1 Definition of Isotropy.…………………………………….. 6.5.2 Anisotropic Plastic Constitutive Equation………………... 6.6 Incorporation of Tangential-Inelastic Strain Rate………………... 6.7 Hyperelastic-Plastic Constitutive Equation: Finite Strain Theory……………………………………………………….……

156 156 157 159 165

Unconventional Elastoplasticity Model: Subloading Surface Model……………………………………….......................................... 7.1 Mechanical Requirements………………………………………... 7.1.1 Continuity Condition……………………………………… 7.1.2 Smoothness Condition……………………………………. 7.2 Subloading Surface Model……………………………………….. 7.3 Salient Features of Subloading Surface Model…………………... 7.4 On Bounding Surface and Bounding Surface Model…………….. 7.5 Incorporation of Anisotropy……………………………………... 7.6 Incorporation of Tangential Inelastic Strain Rate………………...

171 171 171 172 174 181 184 186 187

Cyclic Plasticity Model: Extended Subloading Surface Model…… 8.1 Classification of Cyclic Plasticity Models……………………….. 8.2 Translation of Subyield Surface(s): Extension of Kinematic Hardening………………………..………...…………. 8.2.1 Multi-surface Model……………………………………… 8.2.2 Two-Surface Model………………………………………. 8.2.3 Infinite-Surface Model……………………………………. 8.2.4 Nonlinear Kinematic Hardening Model…………………... 8.3 Extended Subloading Surface Model…………………………….. 8.4 Modification of Reloading Curve………………………………... 8.5 Incorporation of Tangential-Inelastic Strain Rate………………...

191 191

Viscoplastic Constitutive Equations………………………………… 9.1 History of Viscoplastic Constitutive Equations………………….. 9.2 Mechanical Response of Ordinary Overstress Model……………. 9.3 Modification of Overstress Model: Extension to General Rate of Deformation…………………………………………………… 9.4 Incorporation of Subloading Surface Concept: Subloading Overstress Model…………………………………………………

211 211 214

Constitutive Equations of Metals………………………………….... 10.1 Isotropic and Kinematic Hardening…………………………….. 10.2 Cyclic Stagnation of Isotropic Hardening……………………… 10.3 On Calculation of the Normal-Yield Ratio……………………... 10.4 Comparisons of Test Results…………………………………… 10.5 Orthotropic Anisotropy………………………………………….

191 191 194 195 195 196 205 208

215 217 221 221 225 232 232 238

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Contents

10.6 Representation of Isotropic Mises Yield Condition……………. 10.6.1 Plane Stress State……………………………………….. 10.6.2 Plane Strain State………………………………………..

244 245 248

Constitutive Equations of Soils……………………………………… 11.1 Isotropic Consolidation Characteristics………………………… 11.2 Yield Conditions………………………………………………... 11.3 Isotropic Hardening Function…………………………………... 11.4 Rotational Hardening…………………………………………… 11.5 Extended Subloading Surface Model…………………………... 11.6 Partial Derivatives of Subloading Surface Function……………. 11.7 Calculation of Normal-Yield Ratio……………………………... 11.8 Simulations of Test Results…………………………………….. 11.9 Simple Subloading Surface Model……………………………... 11.10 Super-Yield Surface for Structured Soils in Natural Deposits……………………………………………………….. 11.11 Numerical Analysis of Footing Settlement Problem…………..

249 249 253 259 261 265 267 271 275 281 291 301

12

Corotational Rate Tensor…………….…………….…………….….. 12.1 Hypoelasticity…………….…………….…………….………… 12.1.1 Jaumann Rate…………….…………….………………. 12.1.2 Green-Naghdi Rate…………….…………….………… 12.2 Kinematic Hardening Material…………….…………………… 12.2.1 Jaumann Rate…………….…………….………………. 12.2.2 Green-Naghdi Rate…………….…………….………… 12.3 Plastic Spin…………….…………….…………….…………....

309 309 309 311 313 315 316 317

13

Localization of Deformation………………………………………… 13.1 Element Test…………….…………….…………….………….. 13.2 Gradient Theory…………….…………….…………….………. 13.3 Shear-Band Embedded Model: Smeared Crack Model………… 13.4 Necessary Condition for Shear Band Inception…………………

327 327 328 331 333

14

Numerical Calculation………………………………………………. 14.1 Numerical Ability of Subloading Surface Model………………. 14.2 Return-Mapping Algorithm Formulation for Subloading Surface Model…………………………………………………..

337 337

11

15

Constitutive Equation for Friction………………………………….. 15.1 History of Constitutive Equation for Friction…………………... 15.2 Decomposition of Sliding Velocity………………….................. 15.3 Normal Sliding-Yield and Sliding-Subloading Surfaces……….. 15.4 Evolution Rules of Sliding-Hardening Function and Normal Sliding-Yield Ratio………..…………………............................. 15.4.1 Evolution Rule of Sliding-Hardening Function……….. 15.4.2 Evolution Rule of Normal Sliding-Yield Ratio………...

340 349 349 350 354 355 355 356

Contents

15.5 15.6 15.7 15.8

Relations of Contact Traction Rate and Sliding Velocity……… Loading Criterion………………….............................................. Sliding-Yield Surfaces…………………....................................... Basic Mechanical Behavior of Subloading-Friction Model…….. 15.8.1 Relation of Tangential Contact Traction Rate and Sliding Velocity…………………...…………………..... 15.8.2 Numerical Experiments and Comparisons with Test Data…………………....................................................... 15.9 Extension to Orthotropic Anisotropy…………………................

XV

357 359 360 365 366 367 375

Appendixes……………………………………………………………. Appendix 1: Projection of Area………………………………………... Appendix 2: Proof of ∂ ( F jA / J ) / ∂ x j = 0 ................................................ Appendix 3: Euler’s Theorem for Homogeneous Function…………… Appendix 4: Normal Vector of Surface………………………………... Appendix 5: Convexity of Two-Dimensional Curve………………….. Appendix 6: Derivation of Eq. (11.19) ……………………………….. Appendix 7: Numerical Experiments for Deformation Behavior Near Yield State………………………………………………...

387 387 388 388 389 390 391

References……………………………………………………………... Index……………………………………………………………………

395 407

392

Chapter 1

Tensor Analysis 1 Tensor Analysis Physical quantities appearing in deformation mechanics of solids belong mathematically to tensors. Therefore, their relations are described using tensor equations. Before studying the main theme of this book, elastoplaticity theory, the mathematical properties of tensors and the mathematical rules on tensor operations are explained to the extent necessary to understand elasoplasticity theory. The orthogonal Cartesian coordinate system is adopted throughout this book.

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. A repeated suffix in any term is summed over numbers that the suffix can take. For instance, n n ⎫ ur vr = ∑ ur vr , Tir vr = ∑ Tir vr ⎪ ⎪ r =1 r =1 ⎬ n ⎪ Trr = ∑ Trr ⎪⎭ r =1

where the range of suffixes is

1, 2,

• ••,

(1.1)

n . Because of ur vr = us vs ,

Tir vr = Tis vs , Trr = 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, we assume that a repeated suffix obeys this convention, except for the particular case in which it is stated that this convention is not applied.

1.1.2 Kronecker’s Delta and Permutation Symbol The symbol

δ ιj (i, j = 1, 2,…, n)

defined in the following equation is called the

Kronecker’s delta. K. Hashiguchi: Elastoplasticity Theory, LNACM 42, pp. 1–56. springerlink.com © Springer-Verlag Berlin Heidelberg 2009

2

1 Tensor Analysis

　

(1.2)

δir δrj = δij , δii = n

(1.3)

⎧1: i = j δ ij = ⎨ ⎩0 : i ≠ j for which it holds that

Furthermore, the symbol ε p1 p2 …pn defined by the following equation is called the alternating (or permutation) symbol or Eddington’s epsilon or Levi-Citiva “e” tensor. l p p ... pn 　are different from each other ⎧ ε p1 p2…pn = ⎨(−1) : 1 2

⎩ 0 : others

(1.4)

where l is the number of replacement required to obtain the permutation

p1 , p2 , ..., pn from the regular permutation 1, 2, ..., n . Equation (1.4) is written for the third degree ( n = 3) as ⎧ 1 : for ijk is a cyclic permutation of 123 ε ijk = ⎪⎨−1 : for ijk is an anticyclic permutation of 123 ⎪ 0 : others ⎩

(1.5)

The number of permutations that the suffixes p1 , p2 , , pn in ε p1 p2… pn take different values from each other is n! and, needless to say, the square of

...

ε p1 p2…pn (=1 or − 1)

is

+1 . Therefore, it holds that

ε p1 p2… pn ε p1 p2…pn = n!

(1.6)

1.1.3 Matrix and Determinant When the quantity arrangement

T possessing n × n components Tij is expressed in the

　

⎡T11 T12 …T 1n ⎤ ⎢T T …T ⎥ 2n ⎥ ⎢ 21 22 T = [Tij ] = ⎢ ⎥ ⎢ ⎥ ⎢⎣Tn1 Tn 2 …Tnn ⎥⎦

(1.7)

the expression of T in this form is called a matrix. For the two matrices T and S , their product TS is defined by the matrix having the following components.

1.1 Conventions and Symbols

3

(1.8)

(TS)ij = Tir Srj

Further, the quantity defined by the following equation is called the determinant of T and is shown by the symbol det T , i.e.

det T = ε p1 p2… pn T1 p1T2 p2 …Tn pn

(1.9)

n = 3 becomes

which for

det T = ε pqrT1 pT2 qT3r

(1.10)

Therefore, the determinant is the number obtained using the process by which i) one makes product by selecting elements in different rows from all lines (number of products made in this process is n ! ) and ii) sums them up with signs which are plus for even permutation and minus for odd permutation, i.e. multiplying the permutation symbol. Now, consider the rearrangement of the elements

T1 p1T2 p2 …Tn pn → Tr11Tr2 2 …Trn n

changing the order such that the second suffices becomes the regular permutation, while the first suffices was the regular order before the rearrangement. Here, the , pn numbers of times in the replacement to obtain the permutations p1, p2 ,

…

…

, rn from the regular permutation 1, and r1, r2 , Eq. (1.9) can be rewritten as follows:

2, ..., n are the same. Then,

det T = ε r1r2…rn Tr11Tr2 2 …Trn n

(1.11)

Eventually, “the value of determinant does not change even if the lines and the rows are mutually substituted”, i.e.

det TT = det T where

(

)T

designates the transpose, i.e. the mutual replacement of lines and rows.

Here, the number of permutations that the suffixes can take is

(1.12)

p1 , p2 , ..., pn in ε p1 p2… pn

n! . Therefore, Eqs. (1.9) and (1.11) can be written collectively as

det T = 1 ε r1r2…rn ε p1 p2… pn Tr1 p1Tr2 p2 …Trn pn n!

(1.13)

which is written for the third degree matrix as

det T = 1 ε abcε pqrTapTbqTcr 3!

(1.14)

4

1 Tensor Analysis

Eq. (1.13) is rewritten as

det T = 1n Trs Δrs , det T = 1n TΔT

(1.15)

Δij ≡ 1 ε i r1r2…rn−1ε j p1 p2… p n−1 Tr1 p1Tr2 p2…Trn −1 pn −1 (n − 1) !

(1.16)

where

which is called the cofactor for the i -line and the (1.16) are written for the third degree matrix as

j -row. Equations. (1.13) and

det T = 1 Trs Δ rs, Δij ≡ 1 ε rabε spq Tab Tpq 3 2 Because of

ε p1 p2… pi… p j … pn = − ε p1 p2… p j … pi… pn

of sign and T1 p1T2 p 2

…Tn p

n

…Tipi …Tj p …Tn p j

n

(1.17)

meaning the change

= T1 p1T2 p2 … Tj p j …Ti pi

it follows that

ε p1 p2… pi… p j … pn T1 p1T2 p2 …Ti pi …Tj p j …Tn pn

= −ε p1 p2… p j … pi… pn T1 p1T2 p2 …Tj p j …Ti pi …Tn pn

(1.18)

Then, “the sign of the determinant changes if two lines or two rows ( i -line and

j - row in the above equation) are mutually substituted”. This property engenders the fact “the determinant having two lines or rows that are mutually same is zero”. Multiplying

ε r1r2…rn

to both sides in Eq. (1.9), one has

ε r1r2…rn detT = ε r1r2…rn ε p1 p2… pn T1 p1T2 p2 …Tn pn = ε p1 p2… pn Tr1 p1Tr2 p2 …Trn pn

(1.19)

The transformation from the second side to the third side in Eq. (1.19) results from rn , as the fact that the determinant is zero if the same number exists in r1r2

…

…

rn respectively described in Lemma (1.18), and is detT and − detT if r1r2 signify the even permutation and odd permutation. Here, note that the expression of the determinant in Eq. (1.13) is obtained by multiplying ε r1r2…rn to both sides in Eq. (1.19) and noting Eq. (1.6).

T p T p …Ti p …Tn p

Ti p j in the determinant ε p1 p2… p j… pn = Ai p j + Bi p j leads to

The additive decomposition of the elements

1 1 2 2

j

n

into Ti p j

ε p1 p2… p j… pn T1 p1T2 p2 …( Ai p j + Bi p j )…Tn pn

= ε p1 p2 … p j … pn T1 p1T2 p2 … Ai p j …Tn pn

1.1 Conventions and Symbols

5

+ ε p1 p2… p j … pn T1 p1T2 p2 … Bi p j …Tn pn

(1.20)

Therefore, the value of determinant in which elements in a line (or row) are decomposed additively is the sum of the two determinants made by exchanging the line (or row) of the original determinants into the decomposed elements. Since it is obtained from Eqs. (1.8), (1.9), (1.11) and (1.19) that

ε p1 p2…pn (T1r1 Sr1 p1 )(T2 r2 Sr2 p2 )…(Tnrn Srn pn ) = ε p1 p2… pn T1r1T2 r2 …Tnrn Sr1 p1 Sr2 p2 … Srn pn = ε r1r2…rn T1r1T2 r2 …Tnrn det S

(1.21)

the following product law of determinant holds.

det(TS) = detTdetS

(1.22)

The partial derivative of a determinant is given from Eq. (1.13) as

∂det T = ∂ Tij

1 ∂ n! ε r1r2…rn ε p1 p2… pn Tr1 p1Tr2 p2 …Trn pn ∂ Tij

= 1 (ε r1r2…rn ε p p … pn δ i r1δ j p1Tr p …Trn pn 1 2 2 2 n! + ε r1r2…rn ε p p … pn Tr p δ i r2δ j p2 …Trn pn 1 2

+•••

1 1

+ ε r1r2…rn ε p p … pn Tr p δ i r2δ j p2 …δ i rnδ j pn 1 2

1 1

= 1 (ε ir2…rn ε j p … pn Tr p …Trn pn 2 2 2 n! + ε r1i…rn ε p j… pn Tr p Tr3 p …Trn pn +•••

1

1 1

3

+ ε r1r2…iε p p … jTr p Tr p 1 2

1 1

2 2

…Tr − p − Tr p Tr p …Trn

n = (n − 1) ! ε i r1r2…rn−1ε j p1 p2… p n−1

n 1 n 1

1 1

2 2

−1 pn −1

which leads to

∂ det T = Δ , ∂ det T = Δ ij ∂ Tij ∂T

(1.23)

6

1 Tensor Analysis

Equation (1.23) is derived for the third degree matrix as follows:

1ε ε T T T ∂det T = ∂ 3! abc pqr ap bq cr ∂ Tij ∂ Tij 1 = 3! ε abcε pqr (δ iaδ jpTbqTcr + Tapδ ibδ jqTcr + TapTbqδ icδ jr ) 1 (ε ε = 3! ibc j qr TbqTcr + ε aicε pj r TapTcr + ε abiε pqjTapTbq ) 1 = 3! (ε ibcε j qrTbqTcr + ε bicε qj rTbqTcr + ε cbiε rqjTcrTbq ) 1ε ε = 2! ibc j qr TbqTcr = Δ ij The permutation symbol in the third order, i.e.

ε ijk

appears often hereinafter. It

is related to Kronecker’s delta by the determinants.

δ1i δ1j δ1k ε ijk = δ 2i δ 2 j δ 2 k

δ1i δ 2i δ 3i = δ1 j δ 2 j δ 3 j

δ 3i δ 3 j δ 3 k

δ1k δ 2 k δ 3k

(1.24)

Here, the second side in Eq. (1.24) is expanded as

δ1i δ1j δ1k ε ijk = δ 2i δ 2 j δ 2 k = δ1iδ 2 j δ 3 k + δ1kδ 2iδ 3 j + δ1j δ 2 kδ 3i δ 3i δ 3 j δ 3 k − δ1kδ 2 j δ 3i − δ1iδ 2 kδ 3 j − δ1j δ 2iδ 3 k We can confirm this relation by

ε123 = δ11δ 22δ 33 + δ13δ 21δ 32 + δ12δ 23δ 31 − δ13δ 22δ 31 − δ11δ 23δ 32 − δ12δ 21δ 33 = 1 ε 213 = δ12δ 21δ 33 + δ13δ 22δ 31 + δ11δ 23δ 32 − δ13δ 21δ 32 − δ12δ 23δ 31 − δ11δ 22δ 33 = −1 for instance. The third side in Eq. (1.24) could be confirmed as well.

1.1 Conventions and Symbols

7

The following relation holds from Eqs. (1.24) and (1.22).

ε ijk ε pqr

⎡ δ1i δ 2i δ 3i ⎤ ⎡ δ1 p δ1 q δ1r ⎤ δ1i δ 2i δ 3i δ1 p δ1 q δ1r ⎥ ⎢ ⎥⎢ = δ 1 j δ 2 j δ 3 j δ 2 p δ 2 q δ 2 r = ⎢ δ1 j δ 2 j δ 3 j ⎥ ⎢ δ 2 p δ 2 q δ 2 r ⎥ ⎢ δ δ δ ⎥⎢ δ p δ q δ ⎥ δ1k δ 2 k δ 3k δ 3 p δ 3q δ 3 r ⎣ 1k 2 k 3k ⎦ ⎢⎣ 3 3 3 r ⎥⎦

δ siδ sp δ siδ sq δ s iδ sr

δ ip δ iq δ ir

= δ sjδ sp δ sjδ sq δ s jδ sr = δ jp δ jq δ jr

δ sk δ sp δ sk δ sq δ s k δ s r

(1.25)

δ kp δ kq δ kr

from which further one has

δ ip δ iq δ ik ε ijk ε pqk = δ jp δ jq δ jk δ kp δ kq δ kk = δ ipδ jqδ kk + δ iqδ jk δ kp + δ ik δ jpδ kq − δ ik δ jqδ kp − δ ipδ jk δ kq − δ iqδ jpδ kk = 3δ ipδ jq + δ iqδ j p + δ iqδ j p − δ ipδ jq − δ ipδ jq − 3δ iqδ jp = δ ipδ jq − δ iqδ jp

δ ii δ ij δ iq ε ijpε ijq = δ ji δ jj δ j q δ pi δ pj δ pq = δ iiδ jj δ pq + δ ijδ j qδ pi + δ iqδ jiδ pj − δ iiδ j q δ pj − δ ijδ jiδ pq − δ iqδ jj δ pi

= 9δ pq + δ iqδ pi + δ iqδ ip − 3δ pq − 3δ pq − 3δ pq = 2δ pq

ε ijk ε ijk = 2δ kk = 6 Consequently, the following relation holds.

ε ijk ε pqk = ε kijε kpq = δ ipδ jq − δiq δ jp ⎫⎪ ε ijpε ijq = 2δ pq , ε ijk ε ijk = 6 The last equation can also be obtained directly from Eq. (1.6).

⎬ ⎪⎭

(1.26)

8

1 Tensor Analysis

1.2 Vector 1.2.1 Definition of Vector A quantity having only magnitude is defined as a scalar. On the other hand, a quantity having direction and sense in addition to magnitude and fulfilling the following three properties is defined as a vector. A vector is expressed using lowercase letters in boldface to distinguish it from a scalar. Equivalence: The vectors having same magnitude, direction and sense are equivalent. Here, equivalence of two vectors u and v is expressed by u = v . Addition: The addition of vectors is given by the parallelogram law. Multiplication with scalar: The multiplication of vector and scalar induces a vector whose magnitude is given by the multiplication of the original vector and the scalar, direction is identical to that of the original vector, and sense is same and opposite to that of the original vector if the scalar is positive and negative, respectively. By virtue of the properties presented above, the commutative, distributive, and the associative laws hold as follows:

　　

u + v = v + u, (u + v) + w = u + ( v + w ) ⎫ ⎬ a(bv) = (ab) v = b(av), (a + b)v = (b + a)v, a(u + v) = au + av ⎭ (1.27) where a, b are arbitrary scalars. The magnitude of vector is denoted by || v || . In particular, the vector whose magnitude zero is called the zero vector and is shown as 0 . The vector whose magnitude is unity is called the unit vector.

1.2.2 Operations for Vectors 1) Scalar product Denoting the angle between the two vectors u, v by θ when they are translated to the common initial point, the scalar (or inner) product is defined as ||u |||| v ||cos θ and it is denoted by the symbol u • v , i.e.

u • v ≡ ||u |||| v || cosθ

(1.28)

The magnitude of vector is expressed using the scalar product in Eq. (1.28) as follows:

|| v ||= v • v

(1.29)

The quantity obtained by the scalar product is a scalar and the following commutative, distributive and associative laws hold.

　

u • v = v • u, v • (u + w ) = u • v + u • w, a (u • v) = (au) • v (1.30)

1.2 Vector

9

2) Vector product The operation obtaining a vector having 1) magnitude identical to the area of the parallelogram formed by the two vectors u, v , provided that they are translated to the common initial point, and 2) direction of the unit vector n which forms the right-hand bases u, v, n in this order is defined as the vector (or cross) product and is noted by the symbol u × v . Therefore, denoting the angle between the two vectors u, v by θ when they are translated to the common initial point, it holds that

u × v ≡ || u |||| v || sinșn ( || n|| = 1)

(1.31)

The vector product is not commutative, i.e.

u × v= − v × u

(1.32)

On the other hand, the distributive and the associative laws hold as follows:

u × ( v + w ) = u × v + u × w,

(au) × (bv ) = ab(u × v )

(1.33)

The operation defined by the following equation for the vector and the scalar products of three vectors is called scalar triple product.

[uvw ] ≡ (u × v ) • w

(1.34)

The commutative law for the scalar triple product will be shown in the subsequent section. In addition to the scalar and the vector products, the tensor product is defined as will be described in 1.3.5.

1.2.3 Component Description of Vector The component description of vector is explained here prior to the description of component description of general tensor. 1) Component description Consider the set of the normalized orthonormal vectors

{ei } (i = 1, 2,…, n) .

Here, the “normalized” means the unit vector and “orthonormal” means that they

are mutually orthonormal. Then, adopt the coordinate system {O − x i} , in which the directions of axes are chosen to the directions of normalized orthonormal vectors {ei } . Hereinafter, the set {ei } of vectors is called the normalized

orthonormal base. The scalar and the vector products between the base vectors for n = 3 are given from Eq. (1.2), (1.28) and (1.31) as follows:

ei • e j = δ ij

(1.35)

10

1 Tensor Analysis

ei × ej = ε ijr er Vector

(1.36)

v is described in the linear associative form as follows: v = vr e r (=v1e1 + v2e2 + … + vnen )

(1.37)

v1 , v2 , …, vn are the components of v . Denoting the angle of the direction of vector v from the direction of the base vector e i by θ i , cosθ i = n • ei is called the direction cosine by which the component of v is

where

given as

vi = v • ei = || v || n • ei = || v ||cosθ i The magnitude of vector

(1.38)

v and its unit direction vector n are given from Eqs.

(1.29), (1.35), (1.37) as follows:

|| v || ≡ vr vr , Because of u • v = u r e r • vs es using the components as

v n ≡ v = r er || v || || v ||

(1.39)

= ur vsδ rs the scalar product is expressed

u • v = ur vr

(1.40)

The vector product is expressed from Eqs. (1.36), (1.37) as follows:

u × v =ui ei × v j e j = ε ijk u j vk e i (= (u2 v3 − u3v2 )e1 + (u3v1 − u1v3 )e2 + (u1v2 − u2 v1 )e3 ) (1.41) For sake of Eq. (1.41) the scalar triple product defined in Eq. (1.34) is expressed in a component form as

[uvw ] = (u × v) • w = ε ijr ui v j e r • wk ek = ε ijr ui v j wk δ rk = ε ijk ui v j wk (1.42) or in a matrix form as follows:

u1 v1 w1 [uvw ] = u2 v2 w2

(1.43)

u3 v3 w3 Furthermore, from these equations the following equation holds.

[uvw ] = [ vwu] = [ wuv ] = −[ vuw ] = −[ wvu] = −[uwv ]

(1.44)

1.2 Vector

11

Here, note that [uvw ] designates the volume of a parallelepiped formed by choosing u, v, w as the three sides to produce the right-handed coordinates in this order. 2) Coordinate transformation

{O - xi∗} with the bases {e*i } in addition to the normalized orthogonal coordinate system {O − x i} with the bases {ei } (Fig. 1.1). Noting v = v j e j = ( v • e j )e j in general, the

Adopt the other normalized orthogonal coordinate system

following relations hold in these bases.

ei = (ei • e*j )e*j ⎫⎪ ⎬ e*i = (e*i • e j )e j ⎪ ⎭

(1.45)

ei = Qri e*r ⎫⎪ ⎬ e*i = Qir er ⎪⎭

(1.46)

where the coordinate transformation operator

Qij is defined by

Qij ≡ cos(angle between e*i and e j ) = e*i • e j 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 in a two-dimensional state

(1.47)

12

1 Tensor Analysis

Moreover, because of

Qir Q jr = (e*i • e r )(e*j • e r ) = e*i • (e*j • e r )e r = e*i • e*j ⎫⎪ ⎬ Qri Qrj = (e*r • ei )(e*r • e j ) = (ei • e*r )e*r • e j = ei • e j ⎪ ⎭ it holds that

Qir Q jr = Qri Qrj = δ ij

(1.48)

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

e*i by ( )* , the coordinate transformation rule, i.e. the transformation rule of the components of v mutually coincide. Then, denoting the component on the base

on these coordinate systems is given by

vi∗ = Qij v j

(1.49)

noting

v*r e*r • e*i = vj e j • e*i and based on

v = v j e j = v*r e*r Furthermore, noting

(1.50)

Qri v∗r = Qri Qrs vs = δ is vs, the inverse relation of Eq. (1.49)

is given as

vi = Qji v∗j

(1.51)

Equations (1.49) and (1.51) are expressed in matrix form as

⎧v* ⎫ ⎡Q Q Q ⎤ ⎪ 1 ⎪ ⎢ 11 12 13 ⎥ ⎧v1 ⎫ ⎪ ⎪ ⎢ ⎪ ⎪ ⎨v*2 ⎬ = Q21 Q22 Q23 ⎥ ⎨v2 ⎬ , ⎥⎪ ⎪ ⎪ ⎪ ⎢ v ⎪⎩v3* ⎪⎭ ⎢⎣Q31 Q32 Q33 ⎥⎦ ⎩ 3 ⎭

⎡ ⎤ ⎧ *⎫ ⎧v1 ⎫ ⎢Q11 Q21 Q31 ⎥ ⎪v1 ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ ⎨v2 ⎬ = Q12 Q22 Q32 ⎥ ⎨v*2 ⎬ ⎥⎪ ⎪ ⎪ ⎪ ⎢ ⎩v3 ⎭ ⎢Q Q Q ⎥ ⎪v3* ⎪ ⎣ 13 23 33 ⎦ ⎩ ⎭

(1.52)

which are often expressed simply as

{v*} = [Q]{v}, {v} = [Q]T {v*}

(1.53)

1.2 Vector

13

Needless to say, an equation involving ( )* or Qi j does not describe the relation between different physical quantities, but describes the relations between components when a certain physical quantity is described by two different coordinate systems. As known from the following equation, the vector of magnitude is not influenced by the coordinate transformation, whilst it is the basic property of the scalar quantity, as described later.

||v *|| = Qir vr Qis vs = Qir Qis vr vs = δ rs vr vs = vr vr = ||v|| The relations expressed above are shown below.

⎡e* • e e* • e ⎤ ⎡(e* ) (e* ) ⎤ ⎡ cos θ sinθ ⎤ 1 1 1 2 1 2 ⎥=⎢ 1 1 ⎥ = ⎢_ [Q ] = ⎢ ⎥ ⎢e* • e e* • e ⎥ ⎢(e* ) (e* ) ⎥ ⎣ sinθ cosθ ⎦ ⎣ 2 1 2 2⎦ ⎣ 2 1 2 2⎦

cos θ sinθ ⎤ ⎡cos θ _sinθ ⎤ ⎡1 Qir Q jr = ⎡⎢ ⎥⎢ ⎥=⎢ _ ⎣ sinθ cos θ ⎦ ⎣ sinθ cos θ ⎦ ⎣ 0

0⎤ = ⎡δ ⎤ 1⎥⎦ ⎣ ij ⎦

e∗i = (e∗i • e1 )e1 + (e∗i • e2 )e 2 = Qi1e1 + Qi 2e2 , e1∗ = cos θ e1 + sin θ e2 ⎫⎪ ⎬ e∗2 = − sin θ e1 + cos θ e2 ⎪⎭ ei = (ei • e1∗ )e1∗ + (ei • e∗2 )e∗2 = Q1i e1∗ + Q2i e∗2 v = v1e1 + v2e2 = v1∗e1∗ + v2∗e∗2

_ ⎧ ⎫ v∗ ⎪⎧ 1 ⎪⎫ ⎡ cos θ sinθ ⎤ ⎧v1 ⎫ ⎨⎧v1 ⎫⎬ = ⎡ cos θ sinθ ⎤ ⎨⎪v1* ⎬⎪ ⎢sinθ cos θ ⎥ v∗ ⎨ ⎬ = ⎢_ ⎥⎨ ⎬, v ⎦ ⎪⎩ 2 ⎭⎪ ⎪⎩ v*2 ⎪⎭ ⎣ sinθ cos θ ⎦ ⎩v2 ⎭ ⎩ 2 ⎭ ⎣ Choosing the position vector x as the vector (1.51)that

x∗i = Qir xr ⎪⎫ ⎬ xi = Q ri x∗r ⎪⎭

v , it holds from Eqs. (1.49) and

(1.54)

14

1 Tensor Analysis

from which one has

⎫ ∂xi* ∂Qir xr Q ∂xr = = ir = Qirδ jr = Qij ⎪ ∂x j ∂x j ∂x j ⎪ ⎬ ∂x j ∂Qrj x∗r Q ∂ x∗r ⎪ = = rj ∗ = Qrjδ ir = Qij ⎪ ∂ xi ∂xi* ∂xi* ⎭ Consequently,

(1.55)

Qij can be also described as Qij =

∂ xi∗ ∂x j = ∂x j ∂ x∗i

(1.56)

1.3 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.

1.3.1 Definition of Tensor n m functions T ( p1 , p2 , ..., pm ) , where each of the suffixes p1 , p2 , ..., pm takes the number 1, 2, , n , is described in the coordinate system {O - xi } , this set of functions is defined as the mth-order tensor in the nth - dimension if the set of functions is observed as When the set of

T * ( p1 , p2 ,

pm ) = Q p1q1 Q p2q2

Q pm qm T (q1 , q2 ,

qm )

(1.57)

or

T * ( p1 , p2 ,

pm ) =

∂x∗p1 ∂xq1

∂x∗p2 ∂xq2

∂x∗pm T (q1 , q2 , ∂xqm

qm )

(1.58)

in the other coordinate system, provided that only the directions of axes are different but the origin is common and the relative motion does not exist. It is denoted by the symbol T p1 p2⋅⋅⋅ pm . Then, Eq. (1.58) is expressed as

Tp∗1 p2 ⋅⋅⋅ pm = Q p1q1 Q p q … Q pm qm Tq1q ⋅⋅⋅qm 2 2 2

(1.59)

1.3 Tensor

15

or

∂x∗p ∂x∗p Tp∗1 p2⋅⋅⋅pm = x 1 x 2 ∂ q1 ∂ q2

∂x∗pm T ∂xqm q1q2⋅⋅⋅qm

Q pm rm Tp∗1 p2 ⋅⋅⋅ pm = Q p1r1 Q p2r2

Q pm rm Q p1q1 Q p2q2

(1.60)

Noting that

Q p1r1 Q p2r2

= (Qp1r1 Qp1q1 )(Q p2r2 Q p2q2 ) = δ r1q1 δ r2 q2

• • •

• • •

Q pm qm Tq1q2 ⋅⋅⋅qm

(Qp r Q p q )Tq1q2 ⋅⋅⋅qm m m m m

δ r q Tq q ⋅⋅⋅q m m

1 2

m

the inverse relation of Eq. (1.59) is given by

Tr1r2 ⋅⋅⋅rm = Q p1r1 Q p2r2

Q pm rm Tp∗1 p2 ⋅⋅⋅ pm

(1.61)

While the transformation rule of the first-order tensor, i.e. vector is given by Eqs. (1.49) and (1.51), the transformation rule of the second-order tensor is given by

Tij∗ = Qir Q jsTrs , Tij = Qri Q jsTrs∗

(1.62)

The transformation between the coordinate systems without relative motion is considered above in the definition of the tensor, whereas the transformation in the form of (1.59) or (1.61) 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. It will be explained below.

T ( p1 , p2 , • • •, pm ) becomes B pl +1 pl +2⋅⋅⋅ pm ( m − l - th order tensor lacking the suffices p1 ∼ pl ) by multiplying it by A p1 p2⋅⋅⋅ pl ( l - th order tensor (l ≤ m) ), the set is a m - th order tensor”. Quotient law: “If a set of functions

(Proof) The proof is achieved by showing that the quantity

T ( p1 , p2 , • • •, pm ) is

the m - th order tensor when it holds that

T ( p1 , p2 , • • •, pm ) Ap1 p2⋅⋅⋅pl = B pl +1 pl +2⋅⋅⋅pm

(1.63)

16

1 Tensor Analysis

which is described in the coordinate system

{O - xi∗} as follows:

T ∗ ( p1 , p2 , • • •, pm ) A∗p1 p2 ⋅⋅⋅ pm = B*pl +1 pl +2⋅⋅⋅pm

⋅⋅⋅

Here, A p1 p2 pl is the l - th order tensor and B pl +1 pl +2 order tensor,. Therefore, the following relation holds.

B*pl +1 pl +2⋅⋅⋅ pm = Q pl +1rl +1 Q pl +2 rl +2

= Q pl +1rl +1 Q pl +2 rl +2

•••

•••

⋅⋅⋅ p

m

(1.64)

is the m − l - th

Q pm rm Brl +1rl +2⋅⋅⋅rm

Q pm rm T (r1 , r2 , • • •, rm ) Ar1r2 ⋅⋅⋅rl

= Q pl+1rl +1 Q pl+2rl+2 • • • Q pm rm T (r1 , r2 , • • •, rm )Q p1r1 Q p2r2 • • • Q pl rl A∗p1 p2⋅⋅⋅ pl (1.65) (l + 1 ∼ m) (1 ∼ l ) Substituting Eq. (1.64) into Eq. (1.65) yields

{T ∗ ( p1 , p2 , • • •, pm ) − Q p1r1 Q p2 r2

•••

Q pm rm T (r1 , r2 , • • •, rm )} Ap1 p2⋅⋅⋅ pl = 0 (1.66)

from which it holds that

T ∗ ( p1 , p2 , • • •, pm ) = Q p1r1 Q p2 r2

•••

Q pmrm T (r1 , r2 , • • •, rm ) (1.67)

Therefore, noting the definition of tensor in Eq. (1.67), the quantity

T ( p1 , p2 , • • •, pm ) is the m - th order tensor. (End of proof) According to the proof presented above, Eq. (1.63) can be written as Tp1 p2⋅⋅⋅pm Ap1 p2⋅⋅⋅pl = B pl +1 pl +2⋅⋅⋅pm

(1.68)

For instance, if the quantity T (i, j ) transforms the first-order tensor, i.e. vector v i to the vector ui by the operation T (i, j )v j =u i , one can regard T (i, j ) 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 (1.59) 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⋅⋅⋅pm (G p1 p2⋅⋅⋅pl + H p1 p2⋅⋅⋅ pl ) = Tp1 p2⋅⋅⋅ pm G p1 p2⋅⋅⋅pl + Tp1 p2⋅⋅⋅pm H p1 p2⋅⋅⋅pl ⎫⎪ ⎬ Tp1 p2⋅⋅⋅pm (aAp1 p2⋅⋅⋅pl ) = aTp1 p2⋅⋅⋅pm Ap1 p2⋅⋅⋅pl ⎪⎭ where a is an arbitrary scalar. Therefore, the tensor has the function 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 Tensor

17

1.3.3 Notations of Tensors When we express the tensor T as T = Tp1 p2⋅⋅⋅ pm e p1 ⊗e p2 • • • ⊗ e pm

(1.69)

in a similar form to the case of vector in Eq. (1.37), Eq. (1.69) is called the component notation with bases, defining e p1 ⊗e p2 • • • ⊗ e pm as the base of m - th order tensor. The transformation of T between the bases in Eq. (1.46) leads Eq. (1.69) to

T = Tp1 p2 ••• pm Qr1 p1 e∗r1 ⊗ Qr2 p2 er∗2 ⊗ • • • ⊗ Qrm p m er∗m = Qr p Qr2 p2 • • • Qrm pm Tp1 p2⋅⋅⋅pm e∗r1 ⊗ e∗r2 ⊗ • • • ⊗ e∗rm 1 1

= Tr1∗r2 ••• rm e∗r1 ⊗ e∗r2 ⊗ • • • ⊗ e∗rm

(1.70)

The following various notations are used for tensors.

Component notation: T p1 p2⋅⋅⋅ pm

Component notation with base: T p1 p2⋅⋅⋅ pm e p1 e p2 ⊗

⋅⋅⋅ ⊗ e p

m

Symbolic (or direct) notation: T Matrix notation: Eq. (1.7) 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 Tj l , Sijk Tkl , Sijk Til ? In other words, symbolic notation has its limit. On the other hand, component notation with bases holds always without setting any special rule. Introducing the notation

(Q T ) p p p ≡ Q p q Q p q • • • Q p q Tq q q ⎫⎪ m m 1 2 ⋅⋅⋅ m 1 1 1 2 ⋅⋅⋅ m 2 2 ⎬ T (Q T ) p p p ≡ Q p q Q p q • • • Q p q Tq1q2⋅⋅⋅qm ⎪ m m 1 2 ⋅⋅⋅ m 1 1 2 2 ⎭

(1.71)

18

1 Tensor Analysis

Eqs. (1.59) and (1.61) can be expressed by the symbolic notation as follows:

⎫⎪ ⎬ T∗ ⎪⎭

T* = Q T T=

QT

(1.72)

In particular, transformations of the vector and the second-order tensor are expressed by

v∗ = Qv, v = QT v ∗

(1.73)

T* = QTQ T , T = Q T T*Q

(1.74)

1.3.4 Orthogonal Tensor The coordinate transformation operator Qi j described in 1.2.3 plays an important role in the coordinate transformation and is called the orthogonal tensor. The component notation with bases is obtained from

Qij ei ⊗ e j = ei ⊗ (e∗i • e j )e j ( = ei ⊗ e∗i ) = (ei • e∗r )e∗r ⊗ (e∗i • e j )e j

= Qri e∗r ⊗ e∗i

(1.75)

as follows:

Q = Qij ei ⊗ e j = Qije∗i ⊗ e∗j Furthermore, considering Eq. (1.46), the direct notation of

(1.76)

Q is given by

Q = ei ⊗ e∗i

(1.77)

Because of

ei = δ ir er = Qis Qrs e r = Qrs er Qis = Qrs er (e s • e∗i ) = (Qrs e r ⊗ e s )e∗i ⎫⎪ ⎬ e∗i = δ ir e∗r = Qsi Qsr e∗r = Qsr e∗r Qsi = Qsr e∗r (e∗s • ei ) = (Qsr e∗r ⊗ e∗s )ei ⎪⎭ it holds that

ei = Qe∗i , e∗i = QT ei Furthermore, changing Eq. (1.48) to the direct notation or noting the relation

QQT = ei ⊗ e∗i e∗j ⊗ e j = ei (e∗i • e∗j ) ⊗ e j = eiδ ij ⊗ e j ⎫⎪ ⎬ QT Q = e∗i ⊗ ei e j ⊗ e∗j = e∗i (ei • e j ) ⊗ e∗j = e∗i δ ij ⊗ e∗j ⎪ ⎭

(1.78)

1.3 Tensor

19

obtained from Eq. (1.77), it holds that

QQT = QT Q = I

(1.79)

from which one has

QT = Q −1

(1.80)

Moreover, from Eqs. (1.12), (1.22) and (1.79), it is obtained that

det Q = det Q T = ±1

(1.81)

Further from Eq. (1.79) one obtains

(Q − I)QT = −(Q − I )T Making the determinant of this equation and noting Eqs. (1.12), (1.22) and (1.81), it holds that

det(Q − I )= − det(Q − I ) → det(Q − I ) = 0

(1.82)

Then, it is known that one of the principal values of the orthogonal tensor is unity as known from the fact described in 1.5.1.

1.3.5 Tensor Product and Component Based on the three vectors v (1) , v (2) , ⋅⋅⋅, v ( m ) , one can make the m - th order tensor as follows: (2) ( m) v (1) ⊗ v (2) ⋅⋅⋅ ⊗ v ( m ) = v (1) p1 v p 2 ⋅⋅⋅ v p m e p1 ⊗e p2 ⋅⋅⋅⊗ e pm

(1.83)

For the two vectors, one has the second-order tensor

u ⊗ v = ui ei ⊗v j e j = ui v j ei ⊗e j

(1.84)

which is expressed in matrix form

⎡u1v1 u1v2 u1v3 ⎤ ⎢u v u v u v ⎥ ⎢ 2 1 2 2 2 3⎥ ⎢⎣u3v1 u3v2 u3v3 ⎥⎦

(1.85)

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, v , one calls u ⊗ v as the tensor (cross) product or dyad which means one set by two. Particularly, it holds for three arbitrary vectors that

20

1 Tensor Analysis

(u ⊗ v)ir (w )r = ui vr wr which can be expressed in the symbolic notation as

(u ⊗ v )w = u( v • w ) and thus the following expression holds.

u ⊗ v = u( v •

(1.86)

The component of vector is expressed by the direct notation in Eq. (1.38). Here, consider the component of second-order tensor in the direct notation. The second-order tensor T is expressed from Eq. (1.69) as

T = Tij ei ⊗ e j from which, noting Eq. (1.86) it holds that

(1.87)

ei • Te j = ei • Trs e r ⊗ e s e j

= Trsδ irδ sj and thus, the component of T in the direct notation is given as Tij = ei • Te j

(1.88)

As known from Eq. (1.88), the orthogonal projection of the vector Te j to the base

e i is the component of the Tij (i ≠ j ) are called the normal

vector

tensor T . Especially, Tij (i = j ) and component and the shear component,

respectively.

1.4 Operations of Second-Order Tensor As described in 1.3.3, tensor operations must be expressed by component notation in general. While the operations Tv , TS ,

ΞT

appear often in the

elastoplasticity, let them be meant the following operations.

(Tv)i = Tir vr , (TS)ij = Tir vrj , (ΞT)ij = Ξijkl Tkl

(1.89)

Various operations of the second-order tensor are described below.

1.4.1 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

trT = Trsδ rs = Trr = T11 + T22 + T33

(1.90)

1.4 Operations of Second-Order Tensor

21

tr(TS) = Tir Sri = T11S11 + T12 S21 + T13 S31

+T21T12 + T22T22 + T23T32 +T31T13 + T32T23 + T33T33

(1.91)

The following relations hold for the trace.

t r (T + S) = t r T + t r S, t r (aT) = a t rT, t r (TS) = t r (ST), t r (u ⊗ v)=u • v (1.92)

1.4.2 Various Tensors 1) Zero tensor A tensor, the components of which are all zero, is called the zero tensor and is denoted by the symbol 0 , i.e. (0)ij = 0 (1.93)

The zero tensor transforms any tensor to a zero tensor. 2) Identity tensor The tensor which transforms a tensor to itself is called the identity tensor. The second-order identity tensor has the components expressed by the Kronecker’s delta and is denoted by the symbol I , i.e.

I ij = δ ij

(1.94)

The fourth-order identity tensor has the components given by the following equation and is denoted by the symbol I .

Iijkl ≡ 1 (δ ik δ jl + δ ilδ jk ) 2

(1.95)

(TT )ij = T ji

(1.96)

3) Transposed tensor

The tensor

TT fulfilling

is called the transposed tensor of T. It holds from

{(TS)T }ij = T jr Sri that (TS)T = ST TT

(1.97)

22

1 Tensor Analysis

Further, the following relation holds for the trace.

t r TT = t r T

(1.98)

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. (1.96) as

T = TijTij = tr (TTT )

(1.99)

The tensor whose magnitude is unity is called the unit tensor. It holds for arbitrary vectors u, v from Eq. (1.96) that

Tu • v = u • TT v,

Tij u j vi = u j (TT ) ji vi

(1.100)

Further, it holds from Eq. (1.100) that

Tu • Sv = u • TT Sv

(1.101)

4) Inverse tensor −1 The tensor T fulfilling the following relation is defined as the inverse tensor of the tensor T . −

TT−1 = I , Tir (T 1 ) rj = δ ij

(1.102)

It holds from Eq. (1.15) with Eq. (1.3) for n = 3 that

δ ij det T = 1n δ ijδ ijTi s Δj s = Ti s Δ j s from which one has T T Δ = I , Tis Δ js = δ ij det T det T

Consequently,

(1.103)

−

T 1 is given by −

T1=

ΔT , det T

−

(T 1 )ij =

Δ ji det T

(1.104)

Then, det T ≠ 0 is required in order that T−1 exists, while the tensor fulfilling this condition is called the non-singular tensor. The partial derivative of Eq. (1.23) is rewritten by Eq. (1.104) as

∂ det T = (det T)T−T ∂T

(1.105)

The derivation of Eq. (1.105) starting from the definition of the total differential equation has been often described in some literatures (cf. Leigh, 1964) but it needs cumbersome manipulations. Compared with it, the derivation shown above would be concise and straightforward.

1.4 Operations of Second-Order Tensor

23

The following relation holds for the inverse tensor. −

−

−

−

(TT ) 1 = (T −1)T ( ≡ T T ), (TS) 1 = S 1T

−1

(1.106)

because of

((TT−1 )T =) (T−1 )T TT = I = (TT ) −1 TT −

−

TS(TS) 1 = I → S(TS) 1 = T−1 Now, when we regard the transformation of the vector tensor T , i.e.

v to the vector u by the

Tv = u, Tij v j = ui

(1.107)

v are the unknown numbers, solution exists for u ≠ 0 if det T ≠ 0 and is given by v = T −T u as as the simultaneous equation in which the components of T Δ ji v = Δ u, vi = uj detT detT

(1.108)

Here, T must be the non-singular tensor fulfilling det T ≠ 0 in order that the

non-trivial solution v ≠ 0 exists for u ≠ 0 . On the other hand, T must be the

singular tensor fulfilling det T = 0 in order that the solution v ≠ 0 exists for

u = 0. 5) Symmetric and skew-symmetric tensors

Tensors

TS and T A fulfilling the following relations are defined as the symmetric

and the skew (or anti)-symmetric tensor, respectively.

TST = TS , T jiS = TijS

(1.109)

T AT = −T A , T jiA = −TijA

(1.110)

and

An arbitrary tensor T is uniquely decomposed into the symmetric and the skew-symmetric tensors.

T = TS + T A T S = 1 (T + TT ), T A = 1 (T − TT ) 2 2

(1.111) (1.112)

24

1 Tensor Analysis

while the components of

TS and T A are often denoted by T(ij ) and T[ij ] ,

respectively. Eq. (1.111) is called the Cartesian decomposition, following the decomposition of a complex number to a real and an imaginary parts. It holds that

(T S )T = T S , (T A )T = −T A ((T A )ii = 0, no sum)

(1.113)

and

t r TS = t r T,

t r TA = 0

(1.114)

The diagonal components of the skew-symmetric tensor are zero, and thus its determinant is zero, i.e.

det T A = 0

(1.115)

6) Mean and deviatoric components

When the tensor T is decomposed as follows:

T = Tm + T' ⎫ Tm ≡ TmI, Tm ≡ 1n (tr T) ⎪ ⎬ T' ≡ T − TmI (trT' = 0) ⎭⎪

(1.116)

(1.117)

Tm and T' are called the mean (or spherical) part and the deviatoric part of the A tensor T . Noting Eq. (1.113), the skew-symmetric tensor T' of the deviatoric tensor T' is given by T' A = T A

(1.118)

Then, the symmetric part of the deviatoric tensor is given by

T' S = T' − T A = T − Tm I − T A

(1.119)

from which one has

T = Tm I + T' S + T A

(1.120)

The decomposition of T into the mean component TmI , the deviatoric symmetric S A component T' and the skew-symmetric component T is called triple decomposition.

1.4 Operations of Second-Order Tensor

25

7) Axial vector A The skew-symmetric tensor T has three independent components in the three-dimensional state. Therefore, vector t A having the following components is called the axial vector.

t Ai = − 1 ε rsiT rsA 2

(1.121)

Inversely from Eq. (1.121) it is obtained that

T ijA = −ε ijr t Ar ,

⎡0 − t A3 t A2 ⎤ ⎢ ⎥ T ijA = ⎢ 0 − t 1A ⎥ ⎢ ant. 0 ⎥ ⎢⎣ ⎥⎦

(1.122)

Furthermore, noting Eq. (1.41) and the relation

T irAvr = −ε irs t sA vr = ε irs t rA vs

(1.123)

the following relation holds.

TAv = t A × v A

tA

(1.124)

tA

The relation of T and is shown in Fig. 1.2 in the case that is the angular velocity vector and v is the position vector of particle. The quantity in Eq. (1.124)

tA

|| v || sinθ n

θ

TAv = t A× v = ( || t A |||| v ||sinθ ) n

v

0 Fig. 1.2 Meaning of axial vector in case of rotation

26

1 Tensor Analysis

designates the peripheral velocity vector, while

T A is called the spin tensor which

induces the peripheral velocity by undergoing the multiplication of the position vector.

1.5 Eigenvalues and Eigenvectors The tensor T is expressed in the component notation having only normal components by choosing the coordinates with special bases. In what follows, consider the special direction for the second-order tensor in the nth dimensional space. The unit vector e fulfilling (1.125)

Tij e j = Tei , Te = Te i.e.

(Tij − Τ δ ij )e j = 0,

( T − Τ Ι )e = 0

(1.126)

for the second-order tensor 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. The necessary and sufficient condition that e has a non-zero solution in the simultaneous equation (1.126) is given by

Tij − T δ ij = 0, det(T − TI ) = 0

(1.127)

noting the description in 1.6. Eq. (1.127) is called the characteristic equation of the tensor, which is regarded as the equation of nth degree of T . Unit vectors eⅠ, eⅡ, , e N are derived for each of solutions T , T , , TN from Eq. (1.126). Eigen values are real numbers and eigenvectors are mutually orthogonal for eigenvalues that differ from each other in the second-order real symmetric tensor, the components of which are real and which fulfill the symmetry Tij = T ji . The simple proof for this fact is presented below. Regarding T and ei as complex numbers in general and taking the conjugate complex number of both sides in Eq. (1.125), one has the following equation, denoting the conjugate complex number of T and ei by T and ei (note ab = a b for arbitrary complex numbers a, b ).

…

Ⅰ Ⅱ…

Tij e j = T ei

(1.128)

Multiplying both sides of Eqs. (1.125) and (1.128) by ei and ei , respectively, and taking the difference in both sides, one has

1.5 Eigenvalues and Eigenvectors

27

Tij e j ei − Tij e j ei = (T − T )ei ei

(1.129)

The left-hand side in Eq. (1.129) becomes zero because of Tij = Tji . Thereby, one has T = T which means that T must be a real number because ei ei Furthermore, from the two equations for the two solutions

TeⅠ = TⅠⅠ e , TeⅡ = TⅡeⅡ

≠ 0. (1.130)

one has

eⅡ • TeⅠ − eⅠ • TeⅡ = (TⅠ − ΤⅡ)eⅠ • eⅡ

(1.131)

The left-hand side of Eq. (1.131) is zero because of the symmetry of the tensor T . Therefore, one has

(TⅠ − Τ Ⅱ)eⅠ• eⅡ = 0

(1.132)

Ⅱ

Consequently, it can be concluded that eⅠ and e are mutually orthogonal for T ≠T . Based on the result described above, denoting the eigenvectors by

Ⅰ Ⅱ

ⅠⅡ …, N ) and the corresponding eigenvalues as T , one can write

eJ ( J = , ,

J

TeJ = TJ eJ (no sum)

(1.133)

In addition, noting that the shear component on the coordinate system with the base vector {eJ } is zero, i.e.

TJK (J ≠ K ) = 0

(1.134)

T = TJ e J ⊗ e J

(1.135)

the tensor T is expressed by

If tensor that

T having eigenvector eJ has the same eigenvalues as tensor T , it holds TeJ = TJ eJ (no sum)

where the orthogonal tensor

Q between the eigenvectors of these tensors is given by

QIJ = e I • eJ , Q = eJ ⊗ eJ , eJ = QeJ , eJ = QT eJ , Applying

(1.136)

Q to Eq. (1.133), one has (QT TeJ =) QT TQQT eJ = TJ QT eJ (no sum)

(1.137)

28

1 Tensor Analysis

from which, considering Eqs. (1.136) and (1.137), it holds that

QT TQ eJ = TJ eJ = TeJ (no sum)

(1.138)

Then, one obtains the relation

T = QT TQ

(1.139)

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 (1.74) of a certain tensor and the relation (1.139) of two tensors having identical eigenvalues but different eigenvectors, are of mutually opposite forms. If the function f of tensor A , B , • • • is observed to be identical independent of observers, i.e. if it fulfills the relation

f ( A, B, • • • ) = f ( Q A , Q B , • • • )

(1.140)

using the symbol in Eq. (1.71), f is called the isotropic scalar-valued tensor function, which is none other than the invariant. In particular, The isotropic scalar-valued tensor function f (T) of single second-order tensor T fulfills

f (T) = f (QTQT )

(1.141)

f (T) is expressed by three principal values in the three-dimensional case, involving 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 equation (1.127) of T for n = 3 leads to

T11 − Τ

T12

T21 T22 − Τ T31

T32

T13 T23 T33 − Τ

= (T11 − Τ )(T22 − Τ )(T33 − Τ ) + T12T23T31 + T21T32T13 − (T11 − Τ )T23T32 − (T22 − Τ )T31T13 − (T33 − Τ )T12T21 = −Τ 3 + (T11 + T22 + T33 )Τ 2 − (T11T22 + T22T33 + T33T11 )Τ + T11T22T33 + 2T12T23T31 + (T12T21 + T23T32 + T31T13 )T − T11T23T32 − T22T31T13 − T33T12T21 = −Τ 3 + (T11 + T22 + T33 )Τ 2 − (T11T22 + T22T33 + T33T11 − T12T21 − T23T32 − T31T13 )Τ + T11T22T33 − T11T23T32 − T22T31T13 − T33T12T21 + 2T12T23T31

1.5 Eigenvalues and Eigenvectors

29

= −Τ 3 + (T11 + T22 + T33 )Τ 2

1 − [{Τ112 + Τ222 + Τ332 + 2(T11T22 + T22T33 + T33T11 )} 2 − {Τ112 + Τ222 + Τ332 + 2(T12T21 + T23T32 + T31T13 )}] Τ

+T11T22T33 +T12T23T31 − T11T23T32 − T22T31T13 − T33T12T21 + 2T12T23T31 = −Τ 3 + (T11 + T22 + T33 )Τ 2 1 − [(T11 + T22 + T33 )2 − {Τ112 + Τ222 + Τ332 + 2(T12T21 + T23T32 + T31T13 )}] Τ 2 + T11T22T33 − T11T232 − T22T312 − T33T122 + 2T12T23T31 = 0 from which the characteristic equation is given as

Ⅱ Ⅲ= 0

(1.142)

+ T22 + T33 = Tii = trT

(1.143)

T 3 −I T 2 + Τ − where

Ⅰ≡ T

11

Ⅱ ≡ TT

11

T12

21

T22

+

T22 T23 T32 T33

+

T11 T13 T31 T33

= Dii = Δii = 12 (TrrTss − TrsTsr )= 12 ( tr 2 T − trT 2 ) (1.144)

T11 T12 T13

Ⅲ≡ T

1 1 3 1 3 T22 T23 = detT = ε rstTr1Ts 2Tt 3 = 6 tr T − 2 tr T trT2 + trT 3 T31 T32 T33 21

(1.145) The direct notation of III is derived by taking the trace of the expression of the Cayley-Hamilton theorem described in the next section and substituting the direct notations of I and II into it. On the other hand, the characteristic equation (1.142) is expressed using the principal values as follows:

(T − TⅠ)(T − TⅡ)(T − TⅢ ) = 0

(1.146)

Comparing Eqs. (1.151) and (1.155), coefficients I, II, and III are described as

Ⅰ= TⅠ+TⅡ+TⅢ ⎫ Ⅱ = TⅠTⅡ + TⅠTⅡ + ΤⅢTⅠ⎪⎬ ⎪ Ⅲ=TⅠTⅡTⅢ ⎭

(1.147)

30

1 Tensor Analysis

Eq. (1.147) can also be derived by substituting T11 = T , T22 = T ,

T33 = TⅢ , T12 = T23 = T31 = 0 in the determinants in Eqs. (1.143)-(1.145). Since I, II, and III are the symmetric functions of principal values, they are the invariants and are called the principal invariants. Next, consider the deviatoric tensor T' . The characteristic equation of given by replacing T to T' in Eq. (1.142) as follows:

T ' 3 − 'ȉ ' − ' = 0

T' is (1.148)

noting

Ⅰ' ≡ trT' = 0

(1.149)

where

Ⅱ' ≡ D'ii = Δii' = 12 trT'

2

1 = 2 Trs' Tsr'

1 = 2 (T11' 2 + T22' 2 + T33' 2 ) + T12' 2 + T23' 2 + T31' 2 2 2 2 1 = {(T11 − T22 ) 2 + (T22 − T33 )2 + (T33 − T11 )2 } + T12' + T23' + T31' 6

1 1 = 2 (TⅠ' 2 + TⅡ' 2 + TⅢ' 2 ) = {(TⅠ − TⅡ) 2 + (TⅡ − TⅢ ) 2 + (TⅢ − TⅠ) 2} (1.150) 6 T11' T12' T13'

Ⅲ' ≡ T '

21

1 1 3 T22' T23' = det T' = 3 trT' = 3 Trs' Tsr' Trt'

T31' T32' T33'

= T11' T22' T33' − T11' T23' − T22' T31' 2 − T33' T12' 2 + 2T12' T23' T31' 2

1 3 = TⅠ'TⅡ'TⅢ' = −TⅠ' 2TⅡ' − TⅠ'TⅡ' 2 = 3 (TⅠ' 3 + TⅡ' + TⅢ' 3)

(1.151)

The direct notation in (1.151) is derived by taking the deviator and the trace of the expression of Cayley-Hamilton’s theorem described in the next section and substituting the direct notations of ' (=0) and ' in Eqs. (1.149) and (1.150) into that.

Ⅰ

Ⅱ

1.6 Calculations of Eigenvalues and Eigenvectors

31

1.6 Calculations of Eigenvalues and Eigenvectors The symmetric tensor T can be represented in the eigendirections as N

T = ∑ TJ eJ ⊗ eJ

(1.152)

J =1

which is called the spectral representation. 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 and Carlson and Hoger, 1986) are shown in this section.

1.6.1 Eigenvalues In order to obtain eigenvalues, one must solve the characteristic equation which is the cubic equation having the coefficients as the functions of invariants. Now, infer the form

Ⅱ

4 ' cosψ (1.153) 3 for the eigenvalues of deviatoric part of tensor T . The substituting Eq. (1.153) into T' =

Eq. (1.148), we have

') ( 4Ⅱ 3

3/ 2

Ⅱ Ⅱ

cos3 ψ − ' ( 4 ' ) 3

1/2

Ⅲ

cosψ − ' = 0

(1.154)

which reduces to

Ⅲ

Ⅱ

4 3 / 2 cos 3ψ − ' =0 ' 3 3

(1.155)

using the trigonometric formula

cos3 ψ = 1 (cos 3ψ + 3 cosψ ) 4 It is obtained from (1.155) that

Ⅲ

cos 3ψ = 3 3 3/ 2' 2 '

Ⅱ

(1.156)

(1.157)

Noting that the cosine is the periodic function with the period 2π , the angle ψ is expressed by the following equation in general.

Ⅲ

ψ J = 1 {cos −1 (3 3 3/ 2' ) − 2π J } 3

Ⅱ

2 '

(1.158)

32

1 Tensor Analysis

Substituting Eq. (1.158) into Eq. (1.153) and adding the isotropic component the eigenvalues of T are given as follows:

TJ = 1 3

(Ⅰ+ 4Ⅱ3 ' cos[13{cos (32Ⅱ3'Ⅲ' ) − 2π J }]) −1

3/ 2

Ⅰ/ 3 ,

(1.159)

1.6.2 Eigenvectors Eq. (1.152) can be expressed as follows: N

T = ∑ TJ EJ

(1.160)

J =1

EJ is called the eigenprojection of T , which is defined by

while the tensor

EJ ≡ eJ ⊗ eJ (no sum)

(1.161)

fulfilling N

E J (= eⅠ⊗ eⅠ+ eⅡ ⊗ eⅡ + …+ e m ⊗ em ) = I ∑ J

(1.162)

­EJ for J =K EJ E K = ® , t r(EJ E K ) = δ IJ ¯ 0 for J ≠ K

(1.163)

=1

It holds that N N ⎫ TEJ = ( ∑ TK EK )e J ⊗ e J = ( ∑ TK e K ⊗ e K) e J ⊗ e J = TJ e J ⊗ e J ⎪ ⎪ K =1 K =1 ⎬ (no sum for J ) N N ⎪ ⊗ e e E J T = e J ⊗ e J ( ∑ TK E K ) = e J ⊗ e J ( ∑ TK K K ) = e J ⊗ e J TJ ⎪⎭ K =1 K =1

and thus one has

TEJ = EJ T = TJ EJ (no sum for J ) On the other hand, it holds from Eq. (1.162) that N

T − TK I = ∑ TJ EJ − ΤK J =1

N

EJ ∑ J= 1

(1.164)

1.7 Eigenvalues and Eigenvectors of Skew-Symmetric Tensor

33

and thus it is obtained that N

T − TK I = ∑ (TJ − ΤK ) EJ J =1

(1.165)

from which one has N

∏ K ≠θ

N

(T − TK I ) = ∏

N

∑ (TJ − ΤK ) E J

(1.166)

K ≠θ J =1 K =1

K =1

The right-hand side in Eq. (1.166) is rewritten as N

N

N

(TJ − ΤK ) EJ = {∏ (Tθ − ΤK )}Eθ ∑ ∏ = J K ≠θ K ≠θ K =1

1

(1.167)

K =1

noting Eq. (1.163). Then, considering E θ = I for the case of N = 1 , i.e. a single root, the following Sylvester’s formula is obtained.

⎧ N T − TK I for N >1 ⎪∏ − T T ≠ θ K K θ Eθ = ⎨ K =1 ⎪ I for N =1 ⎩

(1.168)

For instance, E2 (θ =2) in the popular case of N = 3 is obtained by the above-mentioned method as follows:

∏ (T − TK I ) = (T − T1I )(T − T3I )

K =1, 3

∏

3

(TJ − ΤK ) E J ∑ J=

K =1, 3

1

={( T1 − Τ1) E1 + ( T2 − Τ1) E2 + ( T3 − Τ1) E 3} {( T1 − Τ3 ) E1 + ( T2 − Τ3 ) E2 + ( T3 − Τ3 ) E 3} =( T2 − Τ1)( T2 − Τ3 )E2 = { ∏ (T2 − ΤK )}E 2 = ( T2 − Τ1)( T2 − Τ3 ) E 2 K =1, 3

∏ (T − TK I )

E2 =

K =1, 3

∏ (T2 − TK)

K =1, 3

=

(T − T1I )(T − T3I ) ( T2 − Τ1)( T2 − Τ3 )

1.7 Eigenvalues and Eigenvectors of Skew-Symmetric Tensor The characteristic equation of skew-symmetric tensor is given by substituting T A into Eq. (1.142) as follows:

34

1 Tensor Analysis

1 2 2 T 3 − trT AT 2 + 2 ( tr T A − trT A )Τ − detT A = 0

(1.169)

A A Noting trT = detT = 0 in Eqs. (1.114) and (1.115), Eq. (1.169) leads to 2 T (2T 2 − trT A ) 2 = 0

(1.170)

from which we have 2

T = ±i | trT A | / 2 = ± i|| t A || and 0

(1.171)

noting

trT A 2 = −2( t 1A 2 + t 2A 2 + t 3A 2 ) < 0 obtained from (1.122). It is known from Eq. (1.171) that the real eigenvalue of skew-symmetric tensor is zero. Here, if select one of the principal direction e Ⅲ to the one with the zero principal value, it holds that

T A * = [Q][T A ][Q]T ⎡ cosθ sinθ 0 ⎤ ⎡ 0 ω 0 ⎤ ⎡cosθ _sinθ 0 ⎤ = ⎢⎢ _sinθ cosθ 0 ⎥⎥ ⎢⎢ −ω 0 0 ⎥⎥ ⎢⎢sinθ cosθ 0 ⎥⎥ ⎢⎣ 0 0 1 ⎥⎦ ⎢⎣ 0 0 1⎥⎦ ⎢⎣ 0 0 1 ⎥⎦ ⎡ −ωsinθ cosθ + ωsinθ cos θ ω sin 2 θ + ω cos 2 θ 0⎤ ⎢ _ ⎥ ω cos 2 θ − ωsin 2θ ωsinθ cosθ − ωsinθ cosθ 0 ⎥ =⎢ ⎢ 0 0 1 ⎥ ⎣ ⎦ ⎡ 0 ω 0⎤ = ⎢⎢ −ω 0 0 ⎥⎥ = T A ⎢⎣ 0 0 1⎥⎦

(1.172)

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 base in the coordinate system is chosen to the principal direction of skew-symmetric tensor.

1.8 Cayley-Hamilton’s Theorem

35

1.8 Cayley-Hamilton’s Theorem Denoting the eigenvector of the tensor T by unit vector e , it holds from Eq. −1 −1 −1 (1.125) and T e = T Te / T = T Te / T = eT −1 that

Tr e = T r e

(1.173)

for r = ±1 . Now, assuming that Eq. (1.173) holds for r = k , one has

Tk ±1e = Tk T±1e = T k T ±1e = T ±1Tk e = T ±1T k e = T k ±1e Consequently, Eq. (1.173) holds also for r = k ± 1 . Then, it is verified that Eq. (1.173) holds for all integers r by mathematical induction. Eq. (1.173) means that the principal value of Tr is T r and the principal directions of the tensors T and Tr are identical mutually, while the tensors having an identical set of principal directions are called to be coaxial or said to fulfill the coaxiality. Then, the linear associative function f (T) of T is coaxial with T and the principal values are given by f (T ) . For n = 3 , the multiplication of the eigenvector e to the characteristic equation (1.142) leads to

Ⅰ Ⅱ ⅢI )e = 0

( T3 − T 2 + T −

noting Eq. (1.173). Because of e ≠ 0 , the following Cayley-Hamilton theorem holds.

Ⅰ Ⅱ ⅢI = 0

T3 − T 2 + T −

(1.174)

It follows from the Cayley-Hamilton theorem that

Ⅰ Ⅱ ⅢI)T =ⅠT −ⅡT + ⅢT =Ⅰ(ⅠT −ⅡT + ⅢI) −ⅡT + ⅢT

T4 = ( T2 − T +

2

3

2

Ⅰ Ⅱ ⅠⅡ− Ⅲ)T +ⅠⅢI Ⅲ T− = T −ⅠT + ⅡI

= ( 2 − )T2 − ( 1

2

2

(1.175) (1.176)

It is concluded that the power of the tensor T is expressed by the linear associative 2

of T , T, I with coefficients consisting of the principal values.

1.9 Positive Definite Tensor When the second-order tensor P is symmetric and fulfills

Pv • v > 0 for an arbitrary vector v ( ≠ 0) , P is called the positive definite tensor.

(1.177)

36

1 Tensor Analysis

Denoting the principal value and direction of P as

PJ and e J , respectively, it

holds that

Pe J • e J = PJ e J • e J = PJ || e J || 2 > 0 (no sum)

(1.178)

noting Eq. (1.177). Then, it is known that the principal value of positive definite tensor is positive. Taking this fact into account for Eq. (1.147)3, it holds that

det P =

Ⅲ> 0

(1.179)

The positive definite tensor U having the same principal directions and principal 1/ 2

values PJ

is defined as the square root of P , i.e. U = P or U = P 2

1/ 2

.

1.10 Polar Decomposition Assuming that the second-order tensor T is not singular ( det T ≠ 0 ), it holds that

Tv ≠ 0 for an arbitrary vector v ( ≠ 0) as described in 4) of 1.4.2 and thus using Eq. (1.101), one obtains

TT Tv • v = Tv • Tv > 0

(1.180)

T

where T T is the symmetric tensor and thus it is the positive-definite tensor as T described in 1.9. Denoting the square root of T T by U , one can write

U = (TT T)1/2 (U 2 = TT T), UT = U

(1.181)

Then, U is the positive definite tensor. Furthermore, defining the tensor R as

R = TU −1

(1.182)

noting Eq. (1.106), one has

RRT = (TU−1 )(TU −1 )T = TU−1U−1TT = T(U 2 )−1 TT = TT−1T−T TT = I (1.183) Therefore, R is the orthogonal tensor. Furthermore, similarly to Eqs. (1.181) and (1.182), consider

V = (TTT )1/ 2 (V 2 =TT T)

(1.184)

from which we have

V 2 = TTT = (RU) (RU)T = RUURT = RURT RURT = (RURT ) 2 (1.185)

1.11 Isotropic Tensor-Valued Tensor Function

37

noting Eq. (1.182), and thus it holds that

V = RURT ,

U=RT VR

(1.186)

Then, one can write

T = RU = VR

(1.187)

Consequently, an arbitrary non-singular tensor T can be decomposed into two forms in terms of the positive definite tensors U or V and the orthogonal tensor R . Here, based on Eqs. (1.181), (1.182), (1.184), and (1.185), R is expressed by the original tensor T as follows:

R = (TTT )−1/ 2 T = T(TT T)−1/ 2

(1.188)

Based on (1.186), U, V are the mutually similar tensors, as described in 1.5.1. For that reason, they have same positive principal values, denoted as U , U , U , and their unit principal vectors uJ , vJ ( J = , , ) are mutually related by

ⅠⅡⅢ

vJ = Ru J , R =vJ ⊗ uJ

Ⅰ Ⅱ Ⅲ

(1.189)

according to Eqs. (1.137), (1.139) and (1.186). Equation (1.187) is called the polar decomposition in similarity to the polar form

Z = Z eiθ (θ : phase angle) which expresses the complex number by the

decomposition into the magnitude and the direction in the polar coordinate system. Actually, RU and VR are respectively called the right and the left polar decompositions.

1.11 Isotropic Tensor-Valued Tensor Function If the function f of tensors T, S, • • • fulfills the following equation, it is called the isotropic function.

Q f (T, S, • • •) = f ( Q T , Q S , • • • )

(1.190)

where the symbol in Eq. (1.72) is used. If f is a scalar, it is to be the invariant defined in Eq. (1.140) 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 , i.e.

38

1 Tensor Analysis

B = f ( A)

(1.191)

where f fulfills

f (QAQT ) = Qf ( A)QT

(1.192)

First introducing the coordinate system with the bases eI , eⅡ, eⅢ , which are the normalized eigenvector of the tensor A and further adopting the another coordinate system rotated 180°around the base eⅢ , the orthogonal tensor between the bases of these coordinate systems is given by

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

(1.193)

where Q 0 fulfills Q 0 = QT0 resulting in the symmetric tensor and it holds that

⎡ −1 0 0 ⎤ ⎧0⎫ ⎧0 ⎫ ⎢ 0 − 1 0 ⎥ ⎪0⎪ = ⎪0 ⎪ , i.e. Q e (1.194) 0 Ⅲ = eⅢ ⎢ ⎥⎨ ⎬ ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ ⎣⎢ 0 0 1⎥⎦ ⎩1⎭ ⎩1 ⎭ Then, it is known that eⅢ is one eigenvector not only of A but also of Q 0 . Furthermore, denoting the principal values of A by αⅠ, αⅡ, αⅢ , it holds that ⎡ −1 0 0 ⎤ ⎡αⅠ 0 0 ⎤ ⎡ −1 0 0 ⎤ ⎡αⅠ 0 0 ⎤ ⎢ 0 − 1 0 ⎥ ⎢ 0 α 0 ⎥ ⎢ 0 − 1 0 ⎥ = ⎢ 0 α 0 ⎥ , i.e. Q AQ T = A Ⅱ ⎥⎢ Ⅱ ⎥ 0 0 ⎢ ⎥⎢ ⎥ ⎢ ⎢⎣ 0 0 1⎥⎦ ⎢⎣ 0 0 αⅢ⎥⎦ ⎢⎣ 0 0 1⎥⎦ ⎢⎣ 0 0 αⅢ⎥⎦ (1.195) and thus it holds that

f (Q 0 AQ T0 ) = f ( A ) = B

(1.196)

On the other hand, from Eq. (1.192) one has

f (Q 0 AQ T0 ) = Q 0 BQ 0T

(1.197)

Then, the commutative law

Q 0 B = BQ 0

(1.198)

1.11 Isotropic Tensor-Valued Tensor Function

39

holds from Eqs. (1.196) and (1.197), and further, noting Eq. (1.194), the following relation is obtained.

Q 0 BeⅢ = BQ 0eⅢ = BeⅢ

(1.199)

which means that BeⅢ is the eigenvector of Q 0 and thus it has the same direction as eⅢ . Then, denoting the principal value B for the eigenvector eⅢ by βⅢ , one can writes

BeⅢ = βⅢ eⅢ

(1.200)

Performing the similar manipulations also for eⅠ and eⅡ , it can be concluded that the tensor B has the same eigenvectors as the tensor A , leading to the coaxility. Therefore, the principal values βⅠ, β Ⅱ, β Ⅲ of the tensor B can be represented in unique relation to the principal values αⅠ, αⅡ, αⅢ of the tensor A . Now, noting the results obtained above, i.e. the coaxiality and the representation of βⅠ, β Ⅱ, β Ⅲ by αⅠ, αⅡ, αⅢ in view of the coaxiality, consider the following equation by way of trial.

βⅠ(αⅠ, αⅡ, αⅢ ) = φ0 + φ1αⅠ + φ2αⅠ2 ⎫

⎪⎪ βⅡ(αⅠ, αⅡ, αⅢ ) = φ0 + φ1αⅡ + φ2αⅡ2 ⎬ ⎪

(1.201)

βⅢ (αⅠ, αⅡ, αⅢ ) = φ0 + φ1α Ⅲ + φ2α Ⅲ2 ⎪⎭ where φ0 , φ1 , φ2 are symmetric functions of αⅠ, αⅡ, αⅢ . While Eq. (1.201) is regarded as the 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

B = φ0 I + φ1A + φ2 A 2

(1.202)

in which φ0 , φ1 , φ2 are functions of invariants of the principal tensor A . When we regard Eq. (1.201) to be the simultaneous equation for the unknown values φ0 , φ1 , φ2 , we know that the Vandermonde’s determinant is not zero for mutually different values of αⅠ, αⅡ, αⅢ as follows:

1 αⅠ αⅠ2

1 αⅡ αⅡ2 = (αⅠ− αⅡ)(αⅡ − αⅢ )(αⅢ − αⅠ) ≠ 0

1 αⅢ α Ⅲ2

(1.203)

40

1 Tensor Analysis

φ0 , φ1 , φ2 are scalar functions which are uniquely determined by αⅠ, αⅡ, αⅢ and then βⅠ, βⅡ, β Ⅲ are uniquely determined by αⅠ, αⅡ, αⅢ , while φ0 , φ1 , φ2 are invariants of A since they are scalar functions of αⅠ, αⅡ, αⅢ , i.e. A . Thus, we can conclude that Eq. (1.202) is the correct

Therefore,

explicit form of Eq. (1.191). Although Eq. (1.202) would not be the unique explicit form of Eq. (1.191), one can say that the isotropic tensor-valued tensor function of a single tensor reduces to this simplest form. Eventually, Eq. (1.202) is the most concise from of the isotropic second-order tensor-valued function of a single second-order tensor. This fact can also be verified using Cayley-Hamilton’s theorem for the special case that f is the linear associative form of the power of A . However, for the case in which f is the general function of A , one must depend on the proof given in this section. In the particular case in which f is the linear function of the tensor A , Eq. (1.202)reduces to

B = a (trA )I + bA where

(1.204)

a and b are the material constants. Equation (1.204) is rewritten as B = CA

(1.205)

where

C ≡ aI ⊗ I + b I (Cijkl ≡ aδ ijδ kl + 1 b(δ ik δ il + δ ilδ jk ) 2

(1.206)

While the second-order isotropic tensor-valued tensor function of single tensor is considered above, the representation theorem of the second-order isotropic tensor-valued tensor function f of two tensors A and B is sown below (cf. e.g., Spencer, 1971).

f(A, B) = ϕ0 I + ϕ1A + ϕ 2B 2 + ϕ3 A 2 + ϕ4B 2 + ϕ5 ( AB + BA)

+ ϕ6 ( A 2B + BA 2 ) + ϕ7 ( AB 2 + B 2 A ) + ϕ8 ( A 2B 2 + B 2 A 2 ) where

ϕ0 , ϕ1 , • ••, ϕ 8

(1.207)

are scalar functions of invariants

⎪⎫ ⎬ tr( AB), tr( AB 2 ), tr( A 2 B), tr( A 2 B 2 ) ⎪⎭ trA, trA 2 , trA 3 , trB, trB 2 , trB3

(1.208)

1.12 Representation of Tensor in Principal Space Let the second-order tensor T be expressed by the vector from in terms of the eigenvectors

eI , eⅡ, eⅢ and the principal values TI , TⅡ, TⅢ as follows: T = TI eI + TⅡeⅡ + TⅢeⅢ (1.209)

1.12 Representation of Tensor in Principal Space

41

Equation (1.209) is 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 (1.209) is rewritten by decomposing T into the mean and the deviatoric components as follows:

T = (Tm + TI' )eI + (Tm + TⅡ' )eⅡ + (Tm + TⅢ' )eⅢ = Tm + T'

(1.210)

where

Tm ≡ Tm I m = 3Tme m (Tm ≡ (TI + TⅡ + TⅢ )/3)

(1.211)

I m ≡ e I + eⅡ + eⅢ , em ≡ 1 I m (||em || = 1) 3

(1.212)

T' ≡ TI' eI + TⅡ' eⅡ + TⅢ' eⅢ = || T' || t'

TI' ≡ TI − Tm , TⅡ' ≡ TⅡ − Tm , TⅢ' ≡ TⅢ − Tm

(1.213) (1.214)

⎫ ⎪ ⎬ 2 1 ( − ) 2 + ( − ) 2 + ( − )2 ⎪ TI TⅡ TⅡ TⅢ TⅢ TI || T' || = TI' 2 + TⅡ' 2 + TⅢ' = 3 ⎭

t' ≡ T' / || T' || (||t' || = 1)

(1.215)

TⅢ

Space diagonal

T T' TⅢ ' t' TⅠ'

eⅢ 0

Tm

eI

Deviatoric plane

TⅡ'

em TⅡ eⅡ

TI

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

42

1 Tensor Analysis

TⅢ

_

0 _

TⅠ'

TI

TⅡ'

θ T'

TⅡ _

( −) TⅢ'

Fig. 1.4 Coordinate system in deviatoric (Ǹ) plane

Whereas the deviatoric tensor T' lies on the π - plane (Fig. 1.4), the orthogonal projection of T' to the three oblique axes, which are the orthogonal projections of the orthogonal axes TI , TⅡ, TⅢ to the π - plane, are given as

⎫ ⎪ ⎪ ⎪ TⅡ' = || T' || cos (θ − 2 π ), ⎬ 3 ⎪ ⎪ 2 TⅢ' = || T' || cos (θ + π ) ⎪ 3 ⎭

TI' = || T' ||cos θ ,

(1.216)

On the other hand, the deviatoric components TI' , TⅡ' , TⅢ ' are the components on the orthogonal coordinates (TI , TⅡ, TⅢ ) of the diviatoric tensor T' (see Fig. 1.3). Denoting the angle contained between the coordinate axis, e.g. eI and its projected line onto the π - plane by the symbol α (see Fig. 1.5), it holds that

1.12 Representation of Tensor in Principal Space

43

0ABC: Rectangular triangular pyramid

em

A

T' C

TI'

TⅠ

eI θ α

0

_

_

TⅠ' B cos α =

Fig. 1.5 Relation of

T1'

_

TⅠ' = || T' || cos θ T' ≡ TⅠ' eⅠ + TⅡ' eⅡ + TⅢ' e Ⅲ

π-plane

TⅠ' = TⅠ' cos α

2/3

and

TI' =

e m • eI = 1 , e m • eI = cos( π − α ) 2 3

(1.217)

resulting in

cosα = 2/3

(1.218)

Substituting Eq. (1.218) into Eq. (1.216) it is obtained that

⎫ ⎪ ⎪ ⎪ TⅡ' = 2 TⅡ' = 2 || T' || cos (θ − 2 π ), ⎪⎬ 3 3 3 ⎪ ⎪ TⅢ' = 2 TⅢ' = 2 || T' || cos (θ + 2 π ) ⎪ 3 3 3 ⎪⎭ TI' =

2T = ' 3 I

2 || T || cos θ , 3 '

(1.219)

The product of the three components is given by

TI' TⅡ' TⅢ ' = 2 2 cos θ cos (θ − 2 π ) cos (θ + 2 π ) = 1 cos 3θ 3 3 || T' || || T' || || T' || 3 3 3 6 (1.220) Considering Eq. (1.151) to Eq. (1.220), it follows that

cos 3θ = 6 tr t' 3

(1.221)

44

1 Tensor Analysis

It holds from Eq. (1.216) that

μ≡

2TⅡ − TI − TⅢ TI − TⅢ

=

2TⅡ' − TI' − TⅢ' = 3 tan (θ − 1 π ) 6 TI' − TⅢ'

(1.222)

which is called the Lode’s variable. Assuming the principal value

T1' =

2 || T || cos θ 3 '

(1.223)

Ⅰ

and substituting it into in the characteristic equation of deviatoric tensor (1.148) while, taking account of ' = 0 , one has the following equation

(

2 || T || cos θ )3 − 1 || T || 2 ( 2 || T || cos θ ) − 1 t r T 3 = 0 ' 3 3 ' 2 ' 3 '

resulting in

4cos 3θ − 3cos θ − 6 t r t' 3 = 0 from which one can derive Eq. (1.221) and further reach Eq. (1.219) by tracing the above-described process inversely.

1.13 Two-Dimensional State Consider the two-dimensional state in which the components related to the direction in the coordinate system

e3

( x1 , x2 , x 3 ) with the bases (e1 , e 2 , e3 ) are

T33 = T 31 = T23 = 0 . Furthermore, introduce the coordinate system ( x1* , x*2 , x3* ) with the bases (e*1 , e*2 , e*3 (= e3 )) which is rotated by the angle α in the counterclockwise direction around the axis x3 as shown in Fig. 1.6. The zero, i.e.

orthogonal tensor between these bases is given from Eq. (1.47) as follows:

　

　 　

⎡ cos α sinα 0 ⎤ [Q] = ⎢⎢ −sinα cosα 0 ⎥⎥ ⎢⎣ 0 0 1 ⎥⎦

(1.224)

1.13 Two-Dimensional State

x2

45

Tt

Material

x2

x1* (T11* , T12* )

x*2 x1*

RT

(T22 , T12 ) x*2

α

0

x1

TⅠ P

2α 2α p TⅡ

α T11 + T22 2

(T22* , T12* )

Tn

(T11 , T12 ) x1

　　　

(a) Physical plane

(b) (Tn , Tt ) plane

Fig. 1.6 Mohr’s circle

Substituting Eq. (1.224) into Eq. (1.62), one has

⎫ ⎪ ⎪ 2 2 T22* = T11 sin α + T22 cos α − 2T12 sin α cos α ⎬ ⎪ T12* = (T22 − T11 ) cos α sin α + T12 (cos 2α − sin 2α ) ⎪ ⎭

T11* = T11 cos 2α + T22 sin 2α + 2T12 cos α sin α

(1.225)

which is rewritten as

T11* = Tm + T cos 2α + T12 sin 2α ⎫ ⎪ ⎪ T22* = Tm − T cos 2α − T12 sin 2α ⎬ ⎪ T12* = −T sin 2α + T12 cos 2α ⎪ ⎭

(1.226)

T11 + T22 T −T , T ≡ 11 22 2 2

(1.227)

where

Tm ≡

Furthermore, it holds from Eq. (1.226) that

T11* + T22* = T11 + T22

∂T11* ∂T22* = 2T12*, = −2T12* ∂α ∂α

(1.228)

(1.229)

46

1 Tensor Analysis

While the axis x3 (= x* 3 ) is one of the principal directions, the other principal directions exist on the plane

(x1, x2 ) . Denoting the principal direction from the

x1 - axis by α , it is obtained by putting ∂T11* / ∂α = T12* = 0 or ∂T22* / ∂α = 0 in Eq. (1.229) with Eq. (1.226) that

tan 2α p =

T12 T

(1.230)

from which one obtains

T = ± RT cos 2α p , RT = T 2 + Τ122

T12 = ± RT sin 2α p ⎫ ⎪ ⎬ ⎪⎭

(1.231)

Substituting Eq. (1.231) into the upper two of Eq. (1.226) with specifying α as α p , the maximum and the minimum principal values TⅠ and TⅡ are described by

TⅠ ⎫ ⎬ = Tm ± R TⅡ ⎭

(1.232)

Equation. (1.232) can also be derived directly from the quadratic equation

T 2 − (T11 + T22 )T + T11T22 − T122 = 0 which is obtained by inserting = T11 + T22 , (T33 = T31 = T23 = 0) in Eq. (1.142). Furthermore, denoting

α

for the extremum of

= T11T22 − T122 ,

Ⅲ= 0

T12* as α s , it holds by taking

∂T12* / ∂α = 0 in Eq. (1.226) that tan 2α s = − T 2T12 Equations

(1.230)

and

(1.233)

yield

(1.233)

the

relation

αs = α p ± π/4

( tan 2α p tan 2α s = −1 ）and thus there exist the two directions for the extremum

T12* and they divide the two principal directions into two equal angles, i.e. π /4 . The extremum of T12* denoted by TM is given from Eq. (1.231)2 as follows:

of

TM = ± RT

(1.234)

1.14 Partial Differential Calculi

47

which is also expressed by Eq. (1.232) as

T −T TM = ± Ⅰ Ⅱ (1.235) 2 Designating the normal stress Tij∗ (i, j = 1, 2; i = j ) by Tn and the shear stress Tij∗ (i ≠ j ) by Tt , the following equation is derived from Eqs. (1.226) and (1.231)3.

(Tn − Tm ) 2 + Tt = RT2

(1.236)

Consequently, the stress on an arbitrary plane is expressed by the point on the circle with the radius RT centering at (Tm , 0) in the two-dimensional plane (Tn , Tt ) as shown in Fig. 1.5. This circle is called the Mohr’s circle. Substituting Eq. (1.231) into Eq. (1.226), we have the expressions

T11* = Tm + RT cos(2α − 2α p ), ⎫ ⎪ ⎪ * T22 = Tm − RT cos(2α − 2α p ), ⎬ ⎪ T12* = − RT cos(2α − 2α p ) ⎪ ⎭

(1.237)

T11* and T12* are shown by the values in the ordinate and abscissa axes, respectively, of the point rotated 2α counterclockwise from point T11 , T12 on the Therefore,

Mohr’s circle as shown in Fig. 1.5, provided that the definition for the sign of shear stress 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 parallelly to the physical plane

x1 and x1* stemming from the points (T11 , T12 )

and (T11* , T12* ) , respectively, on the Mohr’s circle is α 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 stress Tn and the shear stress

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. 1.14 Partial Differential Calculi Partial derivatives of symmetric tensors appearing often in elastoplasticity are shown below.

48

1 Tensor Analysis

∂Tij 1 = (δ ik δ jl + δ ilδ jk ) , ∂ T = I ∂T ∂Tkl 2

(1.238)

1 ∂ ( 1 tr T) ∂Tm ∂ ( 3 Trsδ rs ) 1 3 1 = = 3 δ irδ jsδ rs = δ ij , = 1 I (1.239) 3 3 ∂Tij ∂Tij ∂T ∂Tij' ∂ (Tij − Tmδ ij ) 1 1 = = (δ ik δ jl + δ ilδ jk ) − 1 δ ijδ kl , ∂ T' = I − 3 I ⊗ I 2 3 ∂T ∂Tkl ∂Tkl (1.240)

∂ Trs' Trs' ∂ (Trs' Trs' ) 1 = 1 (Trs' Trs' )−1/ 2 = (Trs' Trs' ) −1/ 2 2δ irδ jsTrs' = t'ij , 2 2 ∂Tij' ∂Tij'

∂ ||T'|| = T' ≡ t' ||T'|| ∂ T' Tij'

∂ ∂tij' = ∂Tkl'

T'pqT'pq ∂Tkl'

1

=

T'pqT'pq

δ ik δ jl Tpq ' Tpq ' − Tij' =

(1.241)

Tkl' T'pqT'pq

Tpq ' T'pq

{12 (δ ikδ jl + δ ilδ jk ) − tij' tkl' },

∂ t' = 1 ( I − t' ⊗ t' ) T ∂ ' ||T'|| ∂tij' ∂tij' ∂Trs' = = ∂Tkl ∂Trs' ∂Tkl'

=

1 T pq ' T pq '

1 Tpq ' Tpq '

(1.242)

{12 (δ irδ js + δ isδ jr ) − t'ij trs' }(δ rk δ sl − 13 δ rsδ kl )

{12 (δikδ jl + δilδ jk ) − 13 δijδ kl − tij' tkl' },

∂ t' 1 = ∂ T ||T' ||

( I − 13 I ⊗ I − t' ⊗ t' )

(1.243)

1.14 Partial Differential Calculi

49

cos 3θ ≡ 6t r t' 3

(1.244)

∂ cos3θ = ∂ 6 t'pqt'qr t'rp = 3 δ δ t t = 3 t t , 6 ip jq qr 6 'ir 'rj ' 'rp ∂t'ij ∂t'ij

∂ cos3θ = 3 6 t' 2 ∂t'

(1.245)

∂ cos3θ ∂ cos3θ ∂t'rs = ∂t'rs ∂ Tij ∂ Tij = 3 6 t'rt tts'

=3 6

1 Tpq ' Tpq '

{12 (δirδ js + δisδ jr ) − 13 δijδ rs − t'ijt'rs}

1 Tpq ' Tpq '

∂ cos 3θ 1 =− ∂T ||T' || df (T) =

(

(tir' t'rj − 1 t'rt ttr' δ ij − trs' t'st ttr' t'ij ) , 3

2 2 6 ||t' || I + 3cos 3θ t' − 3 6 t' )

∂f ( T ) ∂f (T) d TT) d Trs = tr ( ∂T ∂ Trs

(1.246)

(1.247)

Differentiating Tir−1Trs = δ is , one has 1 ∂ (Tir−1Trs ) ∂Tir−1 ∂Tir− −1 = T T T −1 + Tir−1δ rk δ slTsj−1 = 0 rs + Tir δ rk δ sl = 0 → ∂Tkl ∂Tkl rs sj ∂Tkl

which leads to

∂Ti j−1 = − 1 (Tik−1T jl−1 + Til−1T jk−1) 2 ∂Tkl Differentiating Eq. (1.145), it holds that

Ⅲ

Ⅰ Ⅱ

∂ = T2 − T+ I ∂T

50

1 Tensor Analysis

Then, noting Eq. (1.176), one has

Ⅲ

Ⅰ Ⅱ ⅢT

∂detT = ∂ = T2 − T+ I = ∂T ∂T

−1

= (detT) T−1

which was derived also in Eq. (1.105). For asymmetric tensors, the tensor δ ijδ kl has to be used instead of the symmetric identity tensor (δ ik δ jl + δ il δ jk ) /2 in the above equations.

1.15 Time Derivatives Let the tensor T , describing a physical quantity, be denoted by T( x, t ) in the spatial description fixed in a space and by T( X, t ) in the material description moving with a material, where t is a time. The time derivative in the former, i.e.

∂T(x, t ) ∂t

(1.248)

describes the rate of the tensor T in the fixed point in the space and then it is called the spatial-time (or local) derivative. In many cases of fluid mechanics, a mechanical variation of individual particle of fluid is not important since deformation of fluid from the initial state does not influence the movement of fluid and thus the spatial-time derivative is often adopted. On the other hand, the time derivative in the material description, i.e.

∂T( X, t ) ∂t

(1.249)

describes the rate of the tensor T in the particular particle of material and thus is called the material-time derivative. It is denoted by the symbol •

T≡

∂T( X, t ) ∂T( X, t ) or DT ≡ Dt ∂t ∂t

(1.250)

In solid mechanics, the rate of deformation in individual particles of a solid is important and thus the material-time derivative is used usually. Because of T( X( x, t ), t ) = T( x, t ) the material-time derivative in Eq. (1.250) and the spatial-time derivative in Eq. (1.248) is related as follows: •

T≡

∂T(x, t ) ∂T(x, t ) v + ∂t ∂x

(1.251)

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

1.16 Differentiation and Integration in Field

51

the change with the elapse of time and the second term signifies the steady (or convective) term describing the change attributable to the movement of material, which results from the existence of a gradient in the mechanical state. Rate-type constitutive equations describing the irreversible deformation, e.g. the viscoelastic, the elastoplastic and the viscoplastic deformation, must be described by the material-time derivative pursuing a material particle because it describes the relation of physical quantities in a material particle. Further, as described in 4.3, the corotational derivative based on the rate of mechanical state observed by the coordinate system rotating with a material must be used in general.

1.16 Differentiation and Integration in 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 = ∂ ∂xr ∂x

(1.252)

1) Gradient

Scalar field: grad s = ∇ s =

∂s e r ∂x r

(1.253)

Vector field:

　⎪⎪⎧ v ⊗ ∇ = v e ⊗ ∂∂x e i i

j

j

v = ∂xi ei ⊗ e j : rear (right) form ∂ j

gradv = ⎨ (1.254) v j ∂ ∂ ⎪∇ ⊗ v = e ⊗ v j e j = x ei ⊗ e j : front ( left) form ∂xi i ∂ i ⎪⎩

　

Second-order tensor field:

　⎧⎪⎪ T ⊗ ∇ =T e ⊗ e ⊗ ∂∂x e ij i

j

k

k

∂Tij = x ei ⊗ e j ⊗ e k : rear (right) form ∂ k

grad T = ⎨ ⎪∇ ⊗ T = ∂ ei ⊗ T e ⊗ e = ∂ Tjk ei ⊗ e ⊗ e : front ( left) form j jk j k k ⎪ ∂xi ∂xi ⎩

　 　

(1.255) 2) Divergence

　

Vector field: div v = ∇ • v ( = v • ∇ ) = vi ei • ∂

∂x j

e j = ∂vxi ∂ i

(1.256)

52

1 Tensor Analysis

Second-order tensor field:

　⎧⎪⎪ T

• ∇ = Tij e i

　

∂ ⊗ e j • x e k = ∂Txir ei : rear ( right) form ∂ k ∂ r

(1.257) div T = ⎨ ⎪∇ • T = ∂ ei • Tjk e j ⊗ e k = ∂ Tri e i : front ( left) form ∂ xi ∂x r ⎪⎩

　 　

3) Rotation (or curl)

Vector field:

　⎧⎪⎪ v × ∇ = v e × ∂∂x e = ε i i

j

j

ijk

　

∂vi ek : rear right ( ) form ∂x j

(1.258) rotv = ⎨ ⎪∇ × v = ∂ ei × v e = ε ijk ∂v j ε e : front left j j ( ) form ∂xi ∂xi ijk k ⎪⎩

　

noting Eq. (1.36). Second-order tensor field:

　⎪⎧ T × ∇ = T e ⊗ e × ∂∂x e ij i

j

k

k ⎪ ⎪ ∂Tij ∂Tij ⎪ = x ei ⊗ (e j × e k ) = ε jkr x e i ⊗ e r : rear ( right) form ∂ k ∂ k ⎪ rot T = ⎨ ⎪∇ × T = ∂ ei × Tjk ej ⊗ e k ∂xi ⎪ ⎪ ∂ Tjk ∂Tjk ⎪ = x (ei × e j ) ⊗ ek = ε ijr x e r ⊗ e k : front ( left) form ⎪ ∂ i ∂ i ⎩

　

　

　

(1.259) The symbol ∇ is regarded as a vector, and the scalar product of itself, i.e. 2 Δ ≡ ∇ 2 ≡ ∇ • ∇ = ∂x e r • ∂ e s = x∂ x ∂ r ∂ r∂ r ∂xs

(1.260)

has the meaning of ∇ 2 ( ) ≡ div(grad( ) ) . The symbol Δ is called the Laplacian or Laplace operator, which is often used for scalar or vector fields as 2s Δ s = x∂ x ∂ r∂ r

(1.261)

1.16 Differentiation and Integration in Field

53

2 vs Δv = ∂x x e s ∂ r∂ r

(1.262)

The following relations hold between the above-mentioned operators.

grad( sv ) = v ⊗ grads + sgradv,

½ ° div( sv ) = sdivv + v • grads, ° ° div(u × v ) = v • rotu − u • rotv, ° rot(u × v ) = (gradu) v − (gradv )u + (divv )u − (divu) v, ¾ (1.263) grad(u • v ) = (gradv )u + (gradu) v + u × rotv + v × rotu, ° ° ° div( sT) = TT grads + sdivT, ° div(Tv ) = v • divT + tr(Tgradv ) °¿ 4) Gauss’ divergence theorem Consider the physical quantity T ( x ) 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.

∫v ∂∂xT dv = ∫v ∂∂xT dx dx dx 1

1

1

2

3

+

x = [T ]x1− dx2 dx3 = (T + − T − ) dx2 dx3 (1.264) 1

where ( ) + and ( ) − 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. 1.7. 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 ∗ = dxQ , SS∗ = dx S , the vectors → → PQ, PS are given by → → PQ = dx2e 2 + dxQe1 , PS = dx3e3 + dxSe1 and thus it holds that

→ → n + da + = PQ× PS = (dx2e2 + dxQe1 ) × (dx3e3 + dxSe1 )

(1.265)

54

1 Tensor Analysis

R∗ S∗ dx s S

e3

R

n+

dx3

Q∗

e2

dx Q

0

Q

P

e1

Fig. 1.7 Infinitesimal square pillar cut from a zone in material

= dx2 dx3e 2 × e3 + dxQ dx3e1 × e3 + dxSdx2e 2 × e1 = dx2 dx3e1 − dxQ dx3e 2 − dxS dx2e3 Comparing the components in the base

(1.266)

e1 on the both sides in Eq. (1.266), one has

n1+ da + = dx2 dx3

(1.267)

In a similar manner for the surface of the prism exposed on surface in the minimum x1 -coordinate, one has

n1− da − = − dx2 dx3

(1.268)

The general expression of projected area is given in Appendix 1. Adopting Eqs. (1.267) and (1.268) in Eq. (1.264), it holds for the prism that

∫v ∂∂Tx dv = T +n+ da + − T − (−n− da − ) = T +n+ da + + T − n− da − 1

1

1

1

(1.269)

1

Then, the following equation holds for the whole zone.

∫v ∂∂Tx dv = ∫a T n da 1

(1.270)

1

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

∫v ∂∂Txi dv = ∫aT ni da

(1.271)

1.16 Differentiation and Integration in Field

The following equations for the scalar from Eq. (1.271).

55

s , the vector v and the tensor T

hold

∫v ∂∂xi dv = ∫a s ni da, ∫v ∇s dv = ∫a s n da

(1.272)

∫v ∂∂ xii dv = ∫a vi ni da, ∫v ∇ • vdv = ∫a v • n da v

(1.273)

∂ Tij

(1.274)

s

∫v ∂ xj dv = ∫aTij ni da, ∫vT • ∇ dv = ∫a Tn da

5) Material-time derivative of volume integration Supposing that the zone of material occupying the volume v at the current moment (t = t ) changes to occupy the volume v + δ v after the infinitesimal time

(t = t + δ t ) , the material-time derivative of the volume integration

∫v T (x, t )d v of the physical quantity T (x, t ) involved in the volume is defined by the following equation.

1{ (∫v T dv)• = δlim ∫v+δ v T (x, t + δ t)dv − ∫v T (x, t)dv} t →0

δt

= lim 1 [ ∫ {T (x, t + δ t ) − T (x , t )}dv + ∫ T (x, t + δ t ) dv] δv δ t →0 δ t v (1.275) The integration of the first term in the right-hand side in Eq. (1.275) is transformed as

T (x, t ) lim 1 ∫ {T (x, t + δ t) − T (x, t )}dv = ∫ ∂ dv v δ t →0 δ t v ∂t

(1.276)

On the other hand, the second term in Eq. (1.275) describes the influence caused by the change of volume during the infinite time. Here, the increment of volume, δ v , is given by subtracting the volume going out the boundary of the zone from the volume going into the boundary, which is the sum of dv (=v • ndaδ t ) over the whole boundary surface (Fig. 1.8). Therefore, substituting the Gauss’ divergence theorem (1.271) and ignoring the second-order infinitesimal quantity, the integration of the second term in the right-hand side of Eq. (1.275) is given by

lim 1

∫

δ t →0 δ t δ v

T (x, t + δ t ) dv ≅ lim 1 ∫ T (x, t ) dv δ t →0 δ t δ v

∂T (x, t )vr = lim 1 ∫ T (x, t )vr nr daδ t = ∫ T (x, t )vr nr da = ∫ dv a v δ t →0 δ t a ∂ xr

56

1 Tensor Analysis

vδ t n

a

a

da

da

vδ t

v

n

v +δv

Fig. 1.8 Translation of a zone in material

The sum of the first term in the right-hand side in this equation and the Eq. (1.276) is equal to the material-time derivative of T (x, t ) because of Eq. (1.251) and thus Eq. (1.275) is given by

(∫v T (x, t ) dv)• = ∫v{T (x, t ) + T (x, t ) ∂∂vxrr}dv = ∫v{T (x, t ) + T (x, t )divv}dv •

•

(1.277) which is called the Reynolds’ transportation theorem. Equation (1.277) can be obtained also by the following manner.

(∫v T (x, t ) dv)• = (∫V T ( X, t ) JdV )• = ∫v (T• (X, t ) J + T (X, t ) J• ) dV v = ∫ {T ( x, t ) + T ( x, t ) ∂ r }dv v ∂ xr •

where V is the initial volume of zone and it is set that J ≡ dv/dV . Here, it holds •

that J = J (∂vr / ∂ x r ) as will be described in 2.5. For the physical quantity T kept constant in a volume element, Eq. (1.277) leads to •

∫v (T (x, t ) + T (x, t ) divv)dv = 0

(1.278)

The local (weak) form of Eq. (1.278) is given as •

T (x, t ) + T (x, t ) divv = 0

(1.279)

Chapter 2 Motion and Strain (Rate)

2

Motion and Strain (Rate)

The tensor analysis providing the mathematical foundation for the continuum mechanics is described in Chapter 1. Basic concepts and quantities for continuum mechanics will be studied in the three chapters up to Chapter 4. The description of motion is required in the continuum mechanics. Various measures for the motion and strain (rate) of solids are adopted for the description of reversible and irreversible deformation of materials. Main measures among them will be explained in this chapter.

2.1 Motion and Deformation General, basic quantities and manners for describing the motion and deformation will be shown in this section.

2.1.1 Material, Spatial and Relative Descriptions A material body is assembly of material particles (or material elements). The positions of all material particles in a laboratory (Euclidean space) is referred to as the configuration of the material body. Here, the configurations in the initial time t = 0 and the current time t are called the initial configuration and the current configuration, respectively, and the position vectors of material particle in the initial and the current configurations are designated by X and x(t ) , respectively. Here, X is fixed and thus it can be regarded as a label of each material particle. At fixed time the unique relation exists between X and x provided that the material does not overlap or separate. The mapping may be symbolically described as

x = χ ( X, t ), X = χ −1 (x, t )

(2.1)

As discussed in 1.15, a physical quantity, say T , representing the state of the body changes with the position of material particle, i.e., with the change of the body-configuration and the time. The special reference configuration is usually selected to identify the material particles of the body and it is called the reference configuration. Here, we adopt the initial configuration to be the reference configuration. When X is regarded as an independent variable, the field of physical quantity is described by T( X, t ) . This type of description of material state and properties is called the Lagrangian (or material) description (cf. Section 1.5). On K. Hashiguchi: Elastoplasticity Theory, LNACM 42, pp. 57–99. springerlink.com © Springer-Verlag Berlin Heidelberg 2009

58

2 Motion and Strain (Rate)

the other hand, when we adopt current position of a material particle as independent variable, then the field of physical quantity in the domain of current configuration is described by T( x, t ) . This way of description is called the Eulerian description or spatial description, as already mentioned in 1.5. The former follows the variation of the state of a material particle, whereas the latter observes the variation of physical quantity at a fixed point of a laboratory. A configuration at a particular time t = τ , i.e. x(τ ) can be described from Eq. (2.1) as follows:

x(τ ) = χ ( X,τ ), X = χ −1 (x(τ ),τ ) The state of physical quantity in a body may be described in terms of −1 T(χ (x(τ ),τ ), t ) = τ T{x(τ ), t}

(2.2)

x(τ ) as (2.3)

Here, the set of x(τ ) is called the relative configuration and the description presented in Eq. (2.3) is called the relative description. Specifically, tT{x(t ), t} (τ =t ) leading to T( x, t ) is called the updated Lagrangian de-

scription. In this case the current configuration is regarded as the reference configuration and, for the sake of clarity the type of description T( X, t ) will be called the total Lagrangian description. The fact that a material does not overlap or separate by the deformation requires the existence of the one-to-one correspondence between X and x ( x is uniquely determined for X and vice versa) so that x1 ( X 1 X 2 X 3 ), x2 ( X 1 , X 2 , X 3 ) and

x3 ( X 1 X 2 X 3 ) must be mutually independent. Now, assume that x1 , x2 , x3 are not mutually independent. Then, there exists a function f such that

f ( x1 ( X 1 X 2 X 3 ), x2 ( X 1 , X 2 , X 3 ), x3 ( X 1 X 2 X 3 )) = 0

(2.4)

from which one has

⎫ ∂f ∂x1 ∂f ∂x2 ∂f ∂x3 + + =0 ⎪ ∂x1 ∂X 1 ∂x2 ∂X 1 ∂x3 ∂X 1 ⎪ ⎪ ∂f ∂x1 ∂f ∂x2 ∂f ∂x3 = 0⎬ + + ∂x1 ∂X 2 ∂x2 ∂X 2 ∂x3 ∂X 2 ⎪ ⎪ ∂f ∂x1 ∂f ∂x2 ∂f ∂x3 = 0⎪ + + ∂x1 ∂X 3 ∂x2 ∂X 3 ∂x3 ∂X 3 ⎪⎭

(2.5)

2.1 Motion and Deformation

Eliminating

59

∂f / ∂x1 , ∂f / ∂x2 , ∂f / ∂x3 from Eq. (2.5), one gets

∂xi ∂x ∂x2 ∂x3 (2.6) = ε IJK 1 =0 ) ∂X J ∂X I ∂X J ∂X K on account of Eq. (1.10), where J is called the functional determinant or Jacobian. This result shows that functions x1 , x2 , x3 are not independent. In order that they

J ≡ det (

are independent, it must hold that

det (

∂xi ∂X J

)≠0

(2.7)

The transformation between x and X is called the admissible transformation, if f1 , f 2 , f 3 in x1 = f1 ( X 1 X 2 X 3 ), x2 = f 2 ( X 1 , X 2 , X 3 ) , x3 = f 3 ( X 1 X 2 X 3 ) are single-valued and continuous functions and the Jacobian is not zero at all points. Further, if the Jacobian is positive at all points, 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 at all points, a right-hand coordinate system is transformed to a left-hand one, and it is called the negative transformation. Admissible and positive transformation is assumed throughout this book.

2.1.2 Deformation Gradient and Deformation Tensors At the initial state of deformation ( t = 0 ), consider the material particle, the position vector of which is X , and let the position vector of the adjacent material point be X + dX . Furthermore, consider the current state ( t = t ) in which these material particles moved to the space points having position vectors x and x + dx . The line elements before and after the deformation are described as

dX = dX A e A , dx(t ) = dxi (t ) ei (t )

(2.8)

where the different bases are assumed for the initial and current configurations so that the observer can move with a deformation. Here, define the deformation gradient

(t ) F(t ) ≡ ∂x , ∂X Then,

FiA (t )ei ⊗ eA ≡

∂xi (t ) e ⊗ eA ≡ xi , A (t )ei ⊗ eA (2.9) ∂X A i

dx(t ) is described by dX as follows:

dx(t ) = F(t )dX, dxi (t ) = FiA (t ) dX A , dxi (t )ei (t ) = FiAei (t ) ⊗ e A dXB e B = FiA dX Aei (t )

(2.10)

60

2 Motion and Strain (Rate)

Equation (2.7) is written in terms of the deformation gradient as

J = det F ≠ 0 (2.11) −1 Therefore, the inverse tensor F exists, which is derived from (∂x/∂X)(∂X/∂x) = I as F −1 = ∂X , ∂x

(F −1 ) Ai eA ⊗ ei =

∂X A e ⊗ ei ≡ X A, i (t )eA ⊗ ei ∂ xi A

(2.12)

Here, consider the unit cubic cell (a parallelepiped) whose sides at initial configuration are given by the triad {eA } . It deforms to the cell whose sides are formed by the triad

{eA } while eA are not unit vectors in general but are given from Eq.

(2.10) as follows:

eA = FeA (FiA = ei • FeA = ei • eA )

(2.13)

Applying the polar decomposition in 1.10 to the deformation gradient F , we have

F = RU = VR ( FiA = RiRU RA = Vir RrA )

(2.14)

U = (FT F)1/2 (U 2 = FT F) (UT = U)

(2.15)

V = (FFT )1 / 2 (V 2 =FFT ) (VT =V )

(2.16)

where

R = FU −1 = F(FT F) −1/ 2 , R = V −1F = (FFT ) −1/ 2 F (det R = 1) (2.17)

V = RURT ,

U=RT VR

(2.18)

Since Eq. (2.18) holds and R is the orthogonal tensor, U and V are the similar tensors as discussed in 1.5.2. Therefore, they possesses the principal values

λ α (α = 1, 2, 3) . Denoting the bases for the principal directions of U by

and V

{Nα (t )} and {n (α ) (t )} , respectively, it can be written as 3

3

α =1

α =1

U = ∑ λα N (α ) (t ) ⊗ N (α ) (t ) , V = ∑ λα n (α ) (t ) ⊗ n (α ) (t ) where the relation of

(2.19)

N (α ) (t ) and n (α ) (t ) is given from Eq. (1.189) as follows:

n(α ) (t ) = R (t )N(α ) (t ), N(α ) (t ) = RT (t )n(α ) (t )

(2.20)

2.1 Motion and Deformation

61

with

R (t ) = n(α ) (t ) ⊗ N(α ) (t )

(2.21)

N (α ) (t ) and n (α ) (t ) are called the Lagrangian triad and the Eulerian triad, respectively. Substituting Eqs. (2.19) and (2.21) into Eq. (2.14),

F is described by

3

F(t ) = ∑ λ α (t )n (α ) (t ) ⊗ N (α ) (t )

(2.22)

α =1

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 = RU noting UN(α ) = λα N(α ) (no sum) as follows: 3

3

α =1

α =1

dx = FdX = RUdX = RU ∑ dXα N (α ) = R ∑ λα dXα N (α ) Equation (2.23) means that the infinitesimal line-elements

(2.23)

dXα N(α ) (no sum) in times to λα dXα N (α )

N (α ) are first stretched λα (no sum) and then undergoes the rotation R as shown in Fig. 2.1.

the principal directions

On the other hand, the change of the infinitesimal line-element by the polar decomposition VR is described as 3

3

3

α =1

α =1

α =1

dx = VRdX = V ∑ R dXα N (α ) = V ∑ dXα n (α ) = λ α ∑ dXα n (α ) (2.24)

dXα N(α ) (no sum) in the principal directions N (α ) first becomes dXα n (α ) (no sum) by rotation R and then are stretched λα times to λα dXα n (α ) (no sum) , noting Vn (α ) = λα n (α ) (see Fig. 2.1). As described above, U, V designates the deformation and R the rotation. λα is called the principal stretch and U and V are called the right and left Equation (2.24) means that the infinitesimal line-elements

stretch tensor, respectively.

RL and RE designate the rotations of the Lagrangian triad {N (α ) } and the Eulerian triad {n (α ) } , respectively, from the fixed base {eα } Letting

(α = 1, 2,3) , they are given by 3

3

α =1

α =1

RL ≡ ∑ N (α ) ⊗ eα , RE ≡ ∑ n (α ) ⊗ eα where the following relations hold.

(2.25)

62

2 Motion and Strain (Rate)

R (2 )

n(

2)

N

dX

2)

e2

(1

n( (1

R

)

n

R

0

X2

RL

X

)

n

RE

1)

N(

dX =F x d

X Ud R =

Rd =V

e1 dX ) 2 ( X 2 d N X d 2 λ 2

0

(2 )

) (2

n

N

UdX X1

(1 )

N dX1 dX1 λ1

e2 (2

1)

1 1

( n ) R RE (1 N e1

)

n U

λ

2

R

N( (1 )

1

λ1

n

R

) N(2 e 2

1

) (1

N e1

2)

λ2

RL

U

Fig. 2.1 Polar decomposition of the deformation gradient

1)

N(

λ1

2.1 Motion and Deformation

63

N (α ) = RL eα , n (α ) = RE eα R E = R RL

(2.26) (2.27)

Considering the particle P and the adjacent particles P' and P'' , we designate their position vectors before and after the deformation by X, X + dX, X + δ X and

x, x + dx, x + δ x , respectively. Then, noting (1.100), one has dx • δ x = FdX • Fδ X = FT FdX • δ X = CdX • δ X −T

(2.28)

dX • δ X = F dx • F δ x = F F dx • δ x −1

−1

−1

= (FFT ) −1 dx • δ x = b −1dx • δ x

(2.29)

where

C ≡ FT F = U 2 (C = CT ), C AB = FkA FkB

(2.30)

b ≡ FFT = V 2 (= RCRT ) ( b = bT ) , bij = FiA F jA

(2.31)

and

are the metric tensors describing how the scalar product of two line-element vectors passing through a material point is influenced by a deformation. They are called the right and left Cauchy-Green deformation tensor. In accordance with Eq. (2.19) they are described by 3

3

α =1

α =1

C = ¦ λα2 N (α ) ⊗ N (α ), b = ¦ λα2 n (α ) ⊗ n (α ) , The principal values equation

λa

(2.32)

are obtained by the solutions of the characteristic

Ⅰ Ⅱ λ +Ⅲ

λ 3 − cλ 2 −

c

c

=0

(2.33)

where

Ⅰ ≡ trC, Ⅱ ≡ 12 (trC − trC ), Ⅲ c

c

2

c

1 1 3 3 2 1 ≡ 6 tr C − 2 tr C tr C + 3 trC (2.34)

The principal values and directions are calculated by the method described in 1.5.2. Using the relative description (2.3),the relative deformation gradient on the reference configuration x(τ ) is defined as

τ F{x(τ ), t} =

∂ τ χ{x(τ ), t} ∂x(τ )

(2.35)

64

2 Motion and Strain (Rate)

which is hereinafter shown in the abbreviated notation as follows:

τ F(t ) ≡ τ F{x(τ ), t}

(2.36)

τ F(t ) is related to the deformation gradient F(t )(≡ 0 F(t )) as

F(t ) =

∂x(t ) ∂x(t ) ∂x(τ ) = = τ F(t )F(τ ) ∂X ∂x(τ ) ∂X

(2.37)

and is further expressed in the polar decomposition as

τ F(t ) = τ R (t ) τ U(t ) = τV (t )τ R (t )

(2.38)

where τ C(t ), τ b(t ) defined by 2 τ C(t ) = ( τ F (t )) T τ F (t ) = τ U (t ) ½ ° 2

¾

τ b(t ) = τ F (t ) (τ F (t ))T = τV (t ) ° ¿

(2.39)

are called the relative right, left Cauchy-Green tensors.

2.2 Strain Tensor Taking the subtraction of Eqs. (2.28) and (2.29), one has

dx • δ x − dX • δ X = 2EdX • δ X ( = 2 E AB dX Aδ X B )

= 2 edx • δ x ( = 2eij dxi δ x j )

(2.40)

where

½ 1 1 T 1 ∂x T ∂x E ≡ 2 (C − I) = 2 (F F − I) = 2 {( ) ( ) − I} ° ∂X ∂X ° ∂x k ∂x k 1 1 ° ) − δ AB E AB ≡ ( FAk FkB − δ AB ) = ( °° 2 2 ∂X A ∂X B ¾ T ∂X −1 X ∂ −T −1 1 1 1 e ≡ (I − b ) = (I − F F ) = {I − ( ) }° 2 2 2 ∂x ∂x ° ° ∂X ∂XK ) ° eij ≡ 1 {δ ij − ( F −1 ) Ki (F −1 ) K j} = 1 (δ ij − K 2 2 ∂xi ∂xj ¿°

(2.41)

Applying the quotient law described in 1.3.2 to Eq. (2.40), it is confirmed that

E and e are the second-order tensors.

2.2 Strain Tensor

65

If a deformation is not induced, the triangle PP' P'' has the same shape as in the initial state and thus the scalar quantities in the left-hand side in Eq. (2.40) are zero so that E and e are independent of rigid-body rotation. Conversely, if E ≠ 0, e ≠ 0 , the scalar quantities in the left-hand side in Eq. (2.40) are not zero so that the shape of the triangle is not same as in the initial state. Therefore, E and e are the quantities describing the deformation independent of rigid-body rotation and called the Green (or Lagrangian) strain tensor and the Almansi (or Eulerian) strain tensor, respectively. Using the displacement vector

u = x − X = uie i

(2.42)

they are expressed by

∂u ∂u ∂uK ⎫ ∂u T ∂u T ∂u 1 ∂u 1 ∂u ) E = 2 { + ( ) + ( ) ( )}, E AB = ( A + B + K 2 ∂XB ∂X A ∂X A ∂XB ⎪ ∂X ∂X ∂X ∂X ⎪ ⎬ 1 ∂u ∂u j ∂uk ∂uk u u T u T u ⎪ ) e = 1 {∂ + ( ∂ ) − ( ∂ ) ( ∂ )}, eij = ( i + − 2 ∂x ∂x ∂x ∂x ⎪⎭ 2 ∂x j ∂xi ∂xi ∂x j (2.43) The following relation exists between them.

E = FT eF,

E AB = FiA Fj B eij

(2.44)

Now, limiting to the infinitesimal deformation and rotation,

dx ≅ d X holds

and thus the third term becomes the second-order infinitesimal quantity in Eq. (2.43) so that the difference between or

E and e can be ignored. Then, denoting E

e by ε, it can be written that

⎧ T⎫ ε ≡ ( ∂u ) S = 1 ⎨ ∂u + ( ∂u ) ⎬ , 2 ⎩ ∂X ∂X X ∂ ⎭

∂ u A ∂ uB + ) (2.45) 2 ∂XB ∂XA

ε AB ≡ 1 (

or T⎫ 1⎧ S + ( ∂u = ⎨ ∂u ε ≡ ( ∂u ) ) ⎬, x x x ∂ ∂ 2 ⎩∂ ⎭

∂ui ∂u j + ) (2.46) 2 ∂xj ∂xi

ε ij ≡ 1 (

ε is called the infinitesimal strain tensor. ε is not an exact measure to describe a deformation for finite deformation and rotation since it does not describe the relation of the infinitesimal line-elements

dX and dx directly, and thus it possesses

various impertinence as will be described in 2.3.

66

2 Motion and Strain (Rate)

Consider the same infinitesimal line-element for PP' and PP'' , i.e. dX = δ X, dx = δ x and denoting its direction vector by N (|| N ||= 1) , it holds from Eq. (2.40) that (2.47) || dx||2 − || dX||2 = 2EN • N ||dX||2 Selecting the X 1 -axis for this line-element, ( N1 , N 2 , N 3 ) = (1, 0, 0) holds and thus we have

|| dx||2 − || dX||2 E11 = 1 2 ||dX||2

(= 12 ||dx||||d−X||||dX|| ( ||||ddXx|||| + 1))

(2.48)

from which the ratio of the line-elements before and after the deformation is given by

||dx|| = 1 + 2 E11 ||dX||

(2.49)

In the case that the variation of the length of the line-element is infinitesimal (||dx|| / ||dX|| ≅ 1) , Eq. (2.48) becomes

E11 ≅ ε11 = Then,

||dx|| − ||dX|| ||dX||

(2.50)

E11 becomes to describe the rate of elongation coinciding with the normal

strain in the infinitesimal strain ε. On the other hand, denoting the direction vectors of the two distinct infinitesimal line-element PP' and PP'' as N' and N'' , respectively, and the angles contained by them as θ , it holds from Eq. (2.40) that

|| dx||||δ x||cosθ − ||dX|||| δX|| cosθ 0 = 2EN' • N'' || dX|||| δX|| (2.51) i.e.

||dx|| ||δ x|| cosθ − cosθ 0 = 2EN' • N'' = 2 E ijN'i N''j ||dX|| || δX|| where

θ0

is the initial value of

θ

(2.52)

. Here, assuming that the infinitesimal

line-elements PP' and PP'' were mutually perpendicular before a deformation ( θ 0 = π /2 ), and taking the X 1 - and X 2 -axes to their directions, i.e.

( N1' , N'2 , N'3 ) = (1, 0, 0), ( N1'' , N''2 , N''3 ) = (0, 1, 0) , it holds that ||dx|| ||δ x|| || dx|| ||δ x|| E12 (= 1 cos θ ) = 1 sin(π /2 − θ ) 2 ||dX|| || δX|| 2 || dX|| || δX||

(2.53)

2.2 Strain Tensor

67

which describes the half of decrease in the sine of angle contained by the two line-elements which were mutually perpendicular before deformation when the

changes

in

lengths

of

these

line-elements

are

infinitesimal

≅1, ||δ x|| / || δX|| ≅ 1 ). Furthermore, when the change in the angle formed by these line-elements is infinitesimal ( θ ≅ π /2 ), one has ( || dx|| / || dX||

E12 ≅ ε12 = (1/2) tan(π /2 − θ ) ≅ (π /2 − θ ) / 2

(2.54)

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 = V = I as follows (Seth, 1964; Hill, 1968):

⎫ 1 m 1 m m (U − I ), m (V − I ) for m ≠ 0 ⎪ ⎬ lnV m ln U, for = 0 ⎪⎭

(2.55)

m is the integer (positive or negative). The Green strain tensor is obtained m = 2 in the equation of U and the Almansi strain tensor is obtained by choosing m = −2 in the equation of V in Eq. (2.55). The Biot strain tensor U − I (Biot, 1965) is given by m = 1 . The strain tensors in Eq. (2.55) are coaxial with U or V and their principal values are given by where

by choosing

⎧1 m − ⎪ (λα 1) for m ≠ 0 f (λα ) = ⎨ m ⎪⎩ ln λα for m = 0 The function

(2.56)

f (λα ) fulfills f (1) = 0, f ' (1) = 1

(2.57)

f ' (s) > 0

(2.58)

and

where s is an arbitrary positive scalar quantity. The function Fig. 2.2. (Note) it holds for m = 0 Eq. (2.56) that

f (λα ) is shown in

1 (λm − 1) = lim exp( m ln λα ) − 1 = lim exp(m ln λα ) ln λα = lim m ln λα α m m →0 m→ 0 1

m →0

68

2 Motion and Strain (Rate)

m=2

f (λα ) 1.5

m =1

1 m=0 m = −1

0.5 0.5

0 −0.5

2

1

m=2

λα

−1

m = 0 m = −1 Fig. 2.2 Function of general principal strain measures

Further, adopt the second-order tensor function f ( U ) which is coaxial with the right stretch tensor

U and has the principal values f (λ α ) . Therefore, we can

define the general strain tensor in the spectral representation as follows: 3

f (U) = ∑ f (λα )N (α ) ⊗ N (α ) α =1

In addition, for the left stretch tensor tensor. 3

3

α =1

α =1

(2.59)

V , we can define the following strain

f (V ) = ∑ f (λα )n (α ) ⊗ n(α ) = ∑ f (λα )RN (α ) ⊗ RN (α ) = Rf (U)RT (2.60) The relation (Tu ⊗ Tv )ij = Tir ur Tjs us = Tir ur usTjs = (T(u ⊗ v )TT )ij leading to Tu ⊗ Tv = T(u ⊗ v )TT is used in the derivation of the last side in Eq. (2.60).

2.2 Strain Tensor

69

In the particular case of m = 0 , noting Uα ≥ 0, Vα ≥ 0 , f ( U ) and f ( V ) defined by the following equation are called the Hencky strain tensor. 3 ½ Right - Hencky strain tensor : Ȝr = ¦ λα N (α ) ⊗ N (α ) ≡ lnU = 1 lnC ° ° 2 α =1 ¾ 3 ° Left - Hencky strain tensor : Ȝl = ¦ λα n (α ) ⊗ n (α ) ≡ lnV = 1 lnb °¿ 2 α =1

(2.61) where

Ȝα ≡ lnUα = 1 lnCα = lnVα = 1 lnbα (Uα = Vα , Cα = bα ) 2 2

(2.62)

which are mutually related as follows.

λ r = RT λ l R

(2.63)

When the principal directions of C and b are fixed, choosing the coordinate axes to these directions FiA = 0 (i ≠ A) ), the following equations hold.

λα = ln ∂xα (no sum) ∂Xα

λ α = λ α = ( ∂xα •

•

∂Xα

•

)

∂x /( ∂Xαα

(2.64)

•

) = ∂∂xxαα = Dαα (no sum)

(2.65)

where λα in Eq. (2.64) is the logarithmic strain and Dαα (no sum) is the normal component in the strain rate tensor defined in the next section. It holds from Eq. (2.64) that 3 3 3 trȜ r = ¦ λα = 1 ¦ ln Cα = ¦ lnUα = ln(U IU IIU III ) ½ ° 2 α= 1 ° α =1 α =1 ¾ 3 3 3 ° trȜ l = ¦ λα = 1 ¦ ln bα = ¦ lnVα = ln(VIVIIVIII ) 2 °¿ α =1 α =1 α =1

(2.66)

which is identical to the logarithmic volumetric strain, i.e. 3 ∂xα v trλ r = trλl = ∑ ln ∂X = ln J = ln V = ε v α

α =1

(2.67)

70

2 Motion and Strain (Rate)

2.3 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 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 know the current states of stress and deformation. Here, introduce the velocity gradient tensor defined as

∂v L ≡ ∂v , Lij ≡ i ≡ ∂ j vi ∂x ∂x j Noting

•

(2.68)

•

•

F = ∂ x / ∂X = ∂v / ∂X (dv = FdX) and the chain rule of derivative,

Eq. (2.68) can be rewritten as • • ∂v ∂X A L = ∂v ∂X = FF -1 (F = LF), Lij = i ∂X ∂x ∂XA ∂x j

(2.69)

Now, differentiating Eq. (2.36) and choosing the current state as the reference state, i.e. t F(t ) = I ( τ

= t ), L can be expressed in the updated Lagrangian

description as follows: •

(2.70)

L = t F(t ) Further,

taking

the

time-derivative

of

Eq.

•

•

(2.38)

and

noting

t R (t ) = t U (t ) = tV (t ) = I , it follows that •

t F (t )

•

•

= t U (t ) + t R (t ) = t V + t R (t )

(2.71)

Decomposing L into the symmetric and the skew-symmetric parts and noting Eqs. (2.69)-(2.71), it is obtained that

L = D+ W

(2.72)

2.3 Strain Rate and Spin Tensors

71

where • • T ⎫ D ≡ Ls = 1 (L + LT ) = 1 {∂v + ( ∂v ) } = t U(t ) = t V (t ) ⎪ 2 ∂x 2 ∂x ⎪ ⎬ (2.73) vi ∂v j ∂ 1 s ⎪ v + Dij ≡ Lij ≡ ∂ ( j i ) = ( 2 ∂ x j ∂ xi ) ⎪⎭ • T ⎫ W ≡ LA = 1 (L − LT ) = 1 {∂v − ( ∂v ) } = t R (t ) ⎪ 2 2 ∂x ∂x ⎪ (2.74) ⎬ v ∂ j v ∂ 1 A i ⎪ − Wij ≡ Lij ≡ ∂ [ j vi ] = ( 2 ∂ x j ∂ xi ) ⎪⎭ where D is called the strain rate tensor or the 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 in-

dependently as the rate variable although it is called the strain rate tensor. Substituting Eqs. (2.14) and (2.69) into Eqs. (2.73) and (2.74), D and W are described by U , R as follows: • • ⎫ D = 1 {F F −1 + (F F −1 )T } = 1 [(RU)• (RU) −1 + (RU) −T {(RU)• }T ] ⎪ 2 2 ⎪ • 1 T −1 −1 • ⎪ = R (U U + U U)R ⎪ 2 ⎬ • −1 • −1 1 1 • • T ⎪ −1 −T T W = {F F − (F F ) } = [(RU) (RU) − (RU) {(RU) } ] 2 2 ⎪ ⎪ • • • ⎪ = R R T + 1 R{U U −1 − U −1 U}R T 2 ⎪⎭

(2.75) Consequently, we obtain

⎫ ⎪ ⎪ ⎬ • • T 1  −U  )R T ⎪ W = Ω R + R (U ⎪⎭ 2

• • +U  T )R T D = 1 R (U 2

where

(2.76)

•

•  ≡U U U −1

•

Ω R ≡ R RT

(2.77) (2.78)

72

2 Motion and Strain (Rate)

Ω R is called the relative (or polar) spin tensor. Further, D and W are described by V , R as follows: ⎫ ⎪ ⎪ • •T • T −1 • ⎪ 1 1 −1 −1 −T T = (V V + V V ) + (V R R V − V R R V ) ⎪ 2 2 ⎬ (2.79) 1 −1 • • T −T ⎪ W = [(VR ) (VR ) − (VR ) {(VR ) } ] ⎪ 2 ⎪ • • • • = 1 (V R R T V −1 + V −1 R R T V ) + 1 (V V −1 − V −1 V ) ⎪ 2 2 ⎪⎭

D = 1 [(VR )• (VR ) −1 + (VR )−T {(VR )• }T ] 2

and thus • •T ⎫  R +Ω  RT ) ⎪  +V  ) + 1 (Ω D = 1 (V 2 2 ⎪ ⎬ • •T ⎪ 1 1 T R R    −V  ) W = (Ω − Ω ) + ( V ⎪⎭ 2 2

(2.80)

where •

•  ≡V V V −1  R ≡ VΩ R V −1 Ω

It holds that

•L

(2.81) (2.82)

Ω L ≡ R RL = (N(α ) ⊗ eα )• (N( β ) ⊗ e β )T T

•

• (α )

= (N (α ) ⊗ e(α ) + N (α ) ⊗ e

)e( β ) ⊗ N ( β )

•

= N (α ) ⊗ N (α )

(2.83)

and thus one has • (α )

N

= Ω L N (α )

(2.84)

noting e• (α )

= 0 since {e(α ) } is the fixed base. Therefore, Ω L describes the spin of the Lagrangian triad {N (α )} of the right stretch tensor U and is called the Lagrangian spin tensor. On the other hand, it holds that •

Ω E ≡ RE RE = (n(α ) ⊗ e(α ) )• (n( β ) ⊗ e( β ) )T T

= n• (α ) ⊗ n (α )

(2.85)

2.3 Strain Rate and Spin Tensors

73

and thus one has

n• (α ) = ΩE n(α ) Therefore, stretch tensor

(2.86)

Ω E describes the spin of the Eulerian triad {n (α )} of the right V and is called the Eulerian spin tensor. Here, it holds that

•

R RT = (n(α ) ⊗ N(α ) )• (n( β ) ⊗ N( β ) )T • (α )

= (n• (α ) ⊗ N (α ) + n (α ) ⊗ N

)N ( β ) ⊗ n( β ) • (γ )

= n• (α ) ⊗ n (α ) + n (α ) ⊗ (N (α ) • N (γ ) ) N

• (γ )

= n• (α ) ⊗ n (α ) + n (α ) ⊗ N (α ) N (γ ) ⊗ N • (γ )

= n• (α ) ⊗ n (α ) − n (α ) ⊗ N (α ) N

N( β ) ⊗ n( β )

N( β ) ⊗ n( β )

⊗ N (γ ) N ( β ) ⊗ n( β )

and thus the following relations hold. T T T Ω R = Ω E − R Ω L R , Ω E = Ω R + R Ω L R , Ω L = R (Ω E − Ω R ) R

(2.87) In the rigid-body rotation ( F = (2.76), (2.82) and (2.87) that

R, U = V = I ), it holds from Eqs. (2.73),

L = W = Ω R = Ω E , ΩL = 0

(2.88)

In what follows consider the physical meanings of D and W . The relative velocity of the particle points P and P' , the position vectors of which are x and x + dx , respectively, is given by

d v = Ld x

(2.89)

from Eq. (2.68) and it is additively decomposed as

dv = dv r + dv d

(2.90)

where

dv d ≡ Ddx

(2.91)

d v r ≡ Wd x

(2.92)

74

2 Motion and Strain (Rate)

Here, denoting dv d in the infinitesimal line-element dxi (= dxi ei ; no sum) in the direction of the orthogonal bases ei as dv id and substituting Eq. (2.91) into Dij = Dei • e j = Ddxi /dxi • e j (no sum) based on Eq. (1.88), one has

Dij =

dvid •e (no sum) dxi j

(2.93)

d while dv i is not parallel to the line-element

dxi in general. Therefore, Dij is the orthogonal projection of dv i /dxi (no sum) onto the e j-direction. Eventually, Dii dxi (= dvid • e i ) (no sum) describes the relative velocity in the direction of d the line-element dx i and Dij dxi (= dvi • e j ) (i ≠ j , no sum) describes the peripheral velocity around e k ( k ≠ i, j ) , as shown in Fig. 2.3. Adopting the bases (eⅠ, eⅡ, eⅢ ) which are the principal directions of D , the direction of dvId coincides with eI , i.e. d

dvId ∝ e I

(2.94)

On the other hand, denoting the axial vector described in Eq. (1.121) for the skew-symmetric tensor W by w , it holds that

wi = − 1 ε rsiWrs , Wij = −ε ijr wr 2

(2.95)

and thus Eq. (2.92) is rewritten from Eq. (1.124) as

dv r = w × dx, dvir (Wis dxs = −ε isr wr dxs ) = ε irs wr dxs

dv1r

x2 dv1

wⅢ + D12 e2

P

e1

dv1d

dx1 P'

D12dx1

D11dx1

Fig. 2.3 Extension and rotation of the line-element

x1

(2.96)

2.3 Strain Rate and Spin Tensors

75

w

DⅢdxⅢ xⅢ

DⅡdxⅡ xⅡ

DIdxI

xI Fig. 2.4 Deformation and rotation for principal directions of strain rate

Therefore, the arbitrary line-element dx rotates in the peripheral velocity dv r and angular velocity w , called often the spin vector, whereas 2w is called the vorticity. The total angular velocity of arbitrary line-element due to D and W is given by the equation

ω i = wi + D jk (i ≠ j ≠ k ≠ i )

(2.97)

w describes the angular velocity of line-element in the principal directions of D , i.e. the mean angular velocity since the second term in Eq. (2.97) is zero in these directions. The movement of the infinitesimal line-element dx 1 around one of the principal directions,

e Ⅲ , of D

is shown in Fig. 2.3. The state of deformation and

rotation is shown in Fig. 2.4 for the the principal directions of The rate of the scalar product of the vectors dx and

δx

D. of the infinitesimal

elements connecting the three particle points P, P' , P'' with the position vectors

x, x + dx, x + δ x , respectively, is given noting Eq. (1.100) as follows:

(dx • δ x)• = d v • δ x + dx • δ v = ∂ v dx • δ x + dx • ∂ v δ x = [{∂ v + ( ∂ v)T }dx] • δ x ∂x ∂x ∂x ∂x

= 2Ddx • δ x

(2.98)

76

2 Motion and Strain (Rate)

If the vicinity of the particle P undergoes the rigid-body rotation, the quantity in Eq. (2.98) for an arbitrary scalar quantity dx • δ x is zero and thus D = 0 has to hold. Inversely, if D = 0 , the quantity in Eq. (2.98) for the scalar quantity dx • δ x 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 = 0 is the necessary and the sufficient conditions for the situation that a deformation is not induced, allowing only a rigid-body rotation. Denoting the lengths of the line-elements PP' and PP'' as dS and δ S and the angle contained by them as θ , it holds that

(dx • δ x )• = (dSδ S cos θ )• (dS )• (δ S )• = [{ + }cos θ − θ• sin θ ]dSδ S dS δS

(2.99)

Further, denoting the unit vectors in the directions of the line-elements PP' and

PP'' as μ and ν , respectively, and noting from Eqs. (2.98) and (2.99) that

) {(dS dS

•

+

dx = μdS , δ x = νδ S , it holds

• (δ S )• cos θ − θ sin θ = 2Dμ • ν ( = 2 Drs μr νs ) (2.100) } δS

If the particles

P' and P'' mutually coincide (θ = 0) , it holds from Eq.

(2.100) that

(dS )• = Dμ • μ dS

(2.101)

The left-hand side of Eq. (2.101) designates the rate of extension of the line-element. Therefore, the rate of extension is given by the normal component of D in the relevant direction, noting Eq. (1.88). On the other hand, choosing the line-element PP'' to be perpendicular to the line-element PP' (θ = π /2) , it holds that •

− θ = 2Dμ • ν

(μ • ν = 0)

(2.102)

The left-hand side of Eq. (2.102) 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. (2.40) is given by •

( dx • δ x ) • = 2 E dx • δ X

(2.103)

2.3 Strain Rate and Spin Tensors

77

It is obtained from Eqs. (2.98) and (2.103) that •

Ddx • δ x = E dX • δ X

(2.104)

Furthermore, substituting Eq. (2.10) into Eq. (2.104) and noting Eq. (1.101), we have the relation of Green strain tensor to the strain rate tensor as follows: •

•

E = FT DF , D = F −T E F −1

(2.105)

which is obtained also from • • • E = 1 (FT F − I ) • = 1 ( FT F + FT F ) 2 2

• • = 1 {FT (F −T FT ) F + FT (F F −1 )F} 2

= 1 (FT LT F + FT LF) = 1 FT (LT + L)F 2 2

= FT DF

(2.106)

Next, the time-differentiation of Eq. (2.41)2 leads to •

e = − 1 {(F −T )• F −1 + F −T (F −1 )• } 2

(2.107)

where

∂ ( ∂X ) ∂ ( ∂X ) ∂ ( ∂X ) ∂ ( ∂X v) ∂F −1 ∂F −1 x ∂ x ∂ ∂x ∂t + v= v= + + − ∂X ∂v (F ) = ∂x ∂x ∂x ∂x ∂t ∂x ∂t ∂x −1 •

v = ∂ ( ∂X + ∂X v) − ∂X ∂ ∂x ∂x ∂x ∂t ∂x

(2.108)

The inside of the bracket ( ) in the last side of Eq. (2.108) is the material-time derivative of the initial configuration X and thus it is zero. Then, it holds that

(F −1 )• = −F −1L

(2.109)

Substituting Eq. (2.109) into Eq. (2.107), one has • e = 1 {(F −1L)T F −1 + F −T F −1L} 2 = LT {1 I − ( 1 I − F −T F −1 )} +{1 I − ( 1 I − F −T F −1 )}L 2 2 2 2 T 1 1 (2.110) = L ( I − e ) + ( I − e )L 2 2

78

2 Motion and Strain (Rate)

from which one has the relation of the rate of the Almansi strain tensor to the strain rate tensor: •

e = D − LT e − eL

(2.111)

Equation (2.111) is rewritten as • 1 1 e = D − {(L + LT ) − (L − LT )}e − e{(L + LT ) + (L − LT )} 2 2

Consequently, we obtain •

e − We + eW = D − De − eD

(2.112)

where ∇

•

e ≡ e − We + eW

(2.113)

is called the Jaumann rate of Almansi strain tensor, while the Jaumann rate will be explained in 4.3. • In the initial state

•

∇

(F = I, E = e = 0) , it holds that E = e = e = D and

thus all the strain rates mutually coincide.

•

In what follows, let D and Ddt be designated as ε and dε , respectively. If the direction of the material line-element always coincides with the xⅠ -axis, the principal strain rate in this direction is given by •

∂ uⅠ • • ∂ uⅠ ∂XⅠ ε•Ⅰ = ∂ uⅠ = = ∂xⅠ ∂ ( XⅠ + uⅠ) ∂u 1+ Ⅰ ∂XⅠ

(2.114)

The time-integration of Eq. (2.114) leads to

εⅠ = 1n(1+

∂uⅠ ∂x ) = 1n Ⅰ = 1n(1+εⅠ) ∂XⅠ ∂XⅠ

Therefore, the integration of principal strain rate, tesimal (nominal) strain

εⅠ ,

(2.115)

differs from the infini-

εⅠ . Putting ∂XⅠ → l , ∂xⅠ → l , ∂uⅠ → l − l 0 , where 0

l 0 and l are the lengths of the line-element in in the initial and the current states, respectively, it holds that

2.3 Strain Rate and Spin Tensors

εⅠ = 1n

79

l l − l0 l − l0 = 1n ( 1+ ) , ε = Ⅰ l0 l0 l0

As described above, the material-time integration of the strain rate

(2.116)

ε•Ⅰ

leads

to the logarithmic (or natural) strain. On the other hand, the infinitesimal strain coincides with the nominal strain. It results that l = 0 : εⅠ = −1

and l =2l 0 : εⅠ = 1 in the nominal strain, whereas l = 0 : εⅠ = −∞ and

l =2l 0 : εⅠ = 1n e 2 (≅ 0.693) in the logarithmic strain. Then, the nominal strain

would be impertinent to use for constitutive equation describing the large deformation. In the logarithmic strain one obtains ln

∫l

0

l1 dl l 2 dl l n dl dl = ∫ 0 +∫1 + " + ∫ n−1 l l l l l l l

i.e.

ln l1 l2 ln 1n( 0 ) = 1n( 0 ) + 1n( 1 ) + " + 1n( n−1 ) l l l l

(2.117)

Consequently, the superposition rule

εⅠ0～n = εⅠ0～1 + εⅠ0～2 + " + εⅠn 1～n −

holds, while

εⅠa～b

designates the normal strain in the a

(2.118)

xⅠ -direction when the length

b

of the line-element changed from l to l , provided that the principal direction of strain rate is fixed. When the direction of principal strain rates are fixed, it holds that

l l lⅢ 1n ( ⅠⅡ ) = 1n( lⅠ0 ) + 1n( lⅡ0 ) + 1n( lⅢ0 ) 0 0 0 lⅠlⅡlⅢ lⅠ lⅡ lⅢ where

(2.119)

lⅠ, lⅡ, lⅢ are the lengths of line-elements in the directions of three prin-

cipal strain rates and thus the logarithmic volumetric strain εν is given by the sum of the principal strains:

Ⅲ

εν = ln v = ∑ ε J V

Ⅰ

(2.120)

J=

It will be shown in 2.5 that the time-integration of the volumetric strain rate Dv ≡ trD coincides with the logarithmic volumetric strain εν in general.

80

2 Motion and Strain (Rate)

On the other hand, in the nominal strain, the above-mentioned convenient properties are not possessed and thus the error increases as the deformation becomes large. In what follows, consider the rotation of the material line-element in the plane motion, where the velocity of particles is given by

v1 = v1 ( x1 , x2 ), v2 = v2 ( x1 , x2 ), v3 = 0

(2.121)

Substituting Eq. (2.121) into Eqs. (2.73) and (2.95), one obtains

⎫⎪ ⎬ w1 = w2 = 0, w3 = −W12 ⎪⎭

D 3 j = 0 (j =1, 2, 3)

(2.122)

e3 , w3 by eⅢ , wⅢ while e 3 is the principal direction of D , the total angular velocity ω around eⅢ is given from (2.97) as Denoting

ω = wⅢ + D12 By choosing

(2.123)

D and Dn , Ds for T and Tn , Ts described in 1.13, the relation

of the rate of extension and the rate of rotation is shown in Fig. 2.5. It is depicted by

[{( D11 + D22 ) / 2}2 + D122 ] centering in (( D11 + D22 ) / 2, wⅢ ) in the two-dimensional plane ( Dn , ω ). the circle of relative velocity with the radius

P''

ω (= wⅢ + D12 )

P' (D11 , D12 + wⅢ ) wⅢ

(D22 , D12 + wⅢ )

0

P

DⅡ

Fig. 2.5 Circle of relative velocity

D11 + D22 2

DⅠ

Dn

2.4 Various Simple Deformations

81

The angle contained by the line-elements PP' and PP" changes as the difference of angular velocity depicted by the difference of the ordinates of the intersecting point of the relative velocity circle and the two straight parallel lines to PP' and PP" stemming from the point P as the pole. In particular, the angle contained by the line-elements in the directions of the maximum and the minimum shear strain rates, which are mutually perpendicular momentarily, changes most quickly by DⅠ − DⅢ . On the other hand, the angle contained by the line-elements for the maximum and the minimum strain rate does not change. In Fig. 2.5 it is confirmed that the mean angular velocity is given by wⅢ (= − W12 ) which is identical to the angular velocity of the line-elements in the direction of the principal strain rates.

2.4 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.

2.4.1 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 XⅠ-axis to the axial direction of cylinder, it holds that

xⅠ = λⅠXⅠ, xⅡ = λⅡXⅡ, xⅢ = λⅡX Ⅲ

(2.124)

where

λⅠ = l , λⅡ = λ Ⅲ = r L R

(2.125)

from which one has

⎡λⅠ 0 0 ⎤ F =U =V = ⎢⎢0 λⅡ 0 ⎥⎥ , ⎢⎣0 0 λⅡ⎥⎦ R=I

⎫ ⎡λⅠ−1 0 0 ⎤ ⎪ ⎢ ⎥ F −1 = ⎢ 0 λⅡ−1 0 ⎥ , J = detF = λⅠλⅡ2 ⎪ ⎬ ⎢ 0 0 λ −1 ⎥ Ⅱ⎦ ⎪ ⎣ ⎪ ⎭ (2.126)

82

2 Motion and Strain (Rate)

The aforementioned measures of deformation (rate) are given as

ªλ 2 0 0 º « » C = U 2 , FT F = « 0 λ 2 0 » , «0 0 λ 2 » ¬ ¼

ªλ −2 0 0 º « » b −1 =V −2 =F −T F −1= «0 λ −2 0 » «0 0 λ −2 » ¬ ¼ (2.127)

ªλ 2 − 1 0 0 º « » E = 1 (b − I ) = 1 « 0 λ 2 − 1 0 » , 2 2 « 0 0 λ 2 − 1»¼ ¬ ª1 − λ −2 0 º 0 « » e = 1 (I − b −1 ) = 1 « 0 1 − λ −2 0 » 2 2 −2 » « 0 0 1− λ ¼ ¬

r

λ =

λl

(2.128)

　

0 ⎤ ⎡ln(l/L) 0 0 ⎡lnλⅠ 0 ⎤ ⎢ ⎥ ⎢ ⎥ (2.129) = ⎢ 0 lnλⅡ 0 ⎥ = ⎢ 　 0 ln(r / R) 0 ⎥ ⎢⎣ 0 0 lnλⅡ⎥⎦ ⎢⎣ 0 0 ln(r / R) ⎥⎦

⎡ • −1 ⎤ ⎡• ⎤ 0 ⎥ ⎢λⅠ 0 0 ⎥ ⎡λⅠ−1 0 0 ⎤ ⎢λⅠλⅠ 0 • • −1 ⎢ ⎥ ⎢ ⎥ ⎥⎢ • L = F F = ⎢ 0 λⅡ 0 ⎥ ⎢ 0 λⅡ−1 0 ⎥ = ⎢ 　 0 λⅡλⅡ−1 0 ⎥ ⎥ ⎢ • ⎥ ⎢ 0 0 λ −1 ⎥ ⎢ • Ⅱ⎦ ⎢ 0 0 λⅡλⅡ−1 ⎥ ⎢0 0 λⅡ ⎥ ⎣ ⎣ ⎦ ⎣ ⎦

　

　

⎡ • −1 ⎤ 0 0 ⎥ ⎢l l • •l ⎢ ⎥ • = ⎢ 0 r r −1 0 ⎥ = D = λ r = λ ⎢ ⎥ • ⎢ 0 0 r r −1 ⎥ ⎣ ⎦ W=0

(2.130)

(2.131)

2.4 Various Simple Deformations

83

ˆ for the description of longitudinal deformation one has E 0 ⎤ ⎡ λ1 − 1 0 ⎢ ⎥ ⎥ 0 0 ⎡(l − L) / L ⎤ ⎢ ⎢ ⎥ − 0 λ 1 0 2 ⎥= ˆ =⎢ r − R) / R E 0 ( 0 ⎢ ⎥ = V−I ⎢ ⎥ ⎥ ⎢⎣ 0 0 (r − R) / R ⎥⎦ ⎢ ⎢ 0 0 λ 2 − 1⎥ ⎥ ⎢ ⎢⎣ ⎦⎥ For the nominal strain

(2.132)

⎡• ⎤ ⎢ l /L 0 0 ⎥ • ˆ = ⎢ 0 r• /R 0 ⎥ E ⎢ ⎥ ⎢ ⎥ • ⎢ 0 0 r /R ⎥ ⎣ ⎦

(2.133)

by which the strain rate tensor is described as •

ˆ (E ˆ + Ι ) −1 D=E

(2.134)

Denoting the axial load as F , various stresses (cf. Chapter 3) are shown as follows:

⎡ F ⎤ ⎡ F 0 ⎢ π r 2 0 0⎥ ⎢ λⅡ2π R 2 ⎢ ⎥ ⎢ σ = ⎢ 0 0 0⎥ = ⎢ 0 0 ⎢ 0 0 0⎥ ⎢ 0 0 ⎢ ⎥ ⎢ ⎣ ⎦ ⎢⎣ ⎡ F ⎢λ2A 0 ⎢ Ⅱ 0 2 τ = Jσ = λⅠλⅡ ⎢ 0 0 ⎢ 0 ⎢ 0 ⎢⎣

⎤ ⎡ 0 ⎥ ⎢ 2F 0 λ A ⎥ ⎢ Ⅱ 0 0⎥ = ⎢ 0 0 ⎥ ⎢ 0⎥ ⎢ 0 0 ⎥⎦ ⎢⎣

⎤ ⎡ 0 ⎥ ⎢ F λⅠ 0 A0 ⎥ ⎢ 0⎥ = ⎢ 0 0 ⎥ ⎢ 0⎥ ⎢ 0 0 ⎥⎦ ⎢⎣

⎡λⅠ−1 0 0 ⎤ ⎡ F /(λⅡ2 A0 ) 0 ⎢ ⎥⎢ 0 0 Π = JF −1σ = λⅠλⅡ2 ⎢ 0 λⅡ−1 0 ⎥ ⎢ ⎢ 0 0 λ −1 ⎥ ⎢ 0 0 Ⅱ ⎦⎣ ⎣

　 　

⎤ 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥⎦

⎤ 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥⎦

(2.135)

(2.136)

0 ⎤ ⎡ F / A0 0 0 ⎤ ⎥ 0⎥ = ⎢ 0 0 0⎥ ⎢ ⎥ 0 ⎥⎦ ⎢⎣ 0 0 0 ⎥⎦

　 　

(2.137)

84

2 Motion and Strain (Rate)

S = JF −1σF −T

⎡ ⎢ F 0 ⎢ λⅠA0 =⎢ ⎢ 0 0 ⎢ 0 0 ⎢⎣

　

⎤ 0⎥ ⎥ ⎥ 0⎥ 0⎥ ⎥⎦

　　 　 　　 　

(2.138)

2.4.2 Simple Shear As shown in Fig. 2.6, consider the simple shear in which the shear deformation is induced parallelly to the x1 -axis.

x1 = X 1 + γ X2 , x2 = X 2 , x3 = X3 where

γ (=2D12 )

(2.139)

is the engineering shear strain. Denoting the shear angle by

θ

,

it holds that •

•

γ = tanθ , γ = θ sec2 θ

(2.140)

It holds in this situation that

⎡1 ⎡ ∂xi ⎤ ⎢ = 0 F=⎢ ∂X ⎥ ⎢ ⎣⎢ j ⎦⎥ ⎢0 ⎣

γ 1 0

0⎤ 0 ⎥⎥ , 1 ⎦⎥

⎡1 F −1 = ⎢⎢0 ⎣⎢0

−γ 1 0

0⎤ 0 ⎥⎥ , 1⎦⎥

J =detF =1 (2.141)

−

θ

γX2

X 2 = x2

X

x

e2 e1 Fig. 2.6 Simple shear

X1 x1

2.4 Various Simple Deformations

85

−1

where the inverse tensor F is derived using (1.104). 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.

⎡1 γ ⎤ F=⎢ ⎥, ⎣ 0 1⎦

⎡1 − γ ⎤ F −1 = ⎢ ⎥ ⎣0 1 ⎦

(2.142)

from which it is obtained that

⎡ • ⎤ ⎡0 − γ ⎤ ⎡ •⎤ • L = F F −1 = ⎢ 0 γ ⎥ ⎢ = ⎢0 γ ⎥ ⎥ ⎢⎣0 0 ⎥⎦ ⎣0 1 ⎦ ⎢⎣0 0 ⎥⎦ •

⎡ ⎤ γ D = ⎢0 1 ⎥ , 2 ⎣1 0 ⎦

(2.143)

•

⎡ ⎤ γ W=⎢ 0 1⎥ 2 ⎣ −1 0 ⎦

(2.144)

Further, it holds that

γ ⎤ ⎡1 C ( = U 2 = FT F ) = ⎢ 2⎥ ⎣γ 1 + γ ⎦

，

⎡1 + γ 2 − γ ⎤ C−1 ( = U −2 = F −1FT ) = ⎢ ⎥ 1⎦ ⎣ −γ

(2.145)

ª 1 −γ º ª1+ γ 2 γ º −1 b ( = V 2 = FF T ) = « b ( = V −2 = F − T F −1 ) = « » 2» 1¼ ¬ γ ¬ −γ 1+ γ ¼ (2.146)

1 γ E = 1 (C − I ) = 1 , 2 2 γ γ2

0 −γ = 1 ( − b−1 ) = 1 2 2 −γ γ 2

,

(2.147)

86

2 Motion and Strain (Rate)

ⅠⅡ

Next, derive the principal stretches λα and the eigenvectors n (α ) ( α = , in the present two dimensional state) of V in Eq. (2.19). The following relation holds from Eq. (1.125).

Vn (α ) = λα n (α ) (|| n(α ) || =|| n ( β ) || = 1, n (α ) • n ( β ) = δαβ )

（

where the principal stretches λα of V and U acteristic equation based on Eq. (1.151).

(2.148)

）must fulfill the following char-

V11 − λα V12 = (V11 − λα )(V22 − λα ) − V12V21 V21 V22 − λα = λα2 − (V11 + V22 )λα + V11V22 − V12V21 = λα2 − (trV )λα + det V = 0

(2.149)

where, it holds from Eqs. (2.146) and (2.140) that

det V = det V 2 = 1+ γ 2 −γ 2 = 1

(2.150)

and

trV 2 = 2 + γ 2 = 2 + tan 2 θ

(2.151)

Here, denoting the principal values of the second-order tensor T in the two-dimensional state as TⅠ, TⅡ with TⅢ = 0 , it holds that

trT2 = TⅠ2 + TⅡ2 = (TⅠ + TⅡ)2 − 2TⅠTⅡ = (trT) 2 − 2 det T

(2.152)

in general and thus we have

Γ ≡ trV = trV 2 + 2 det V = 4 + γ 2 = 4 + tan 2θ

(2.153)

where it holds that •

•

Γ = 1 Γ −1 2 γ γ = 2

γ • γ Γ

(2.154)

Substituting Eqs. (2.150) and (2.151) into Eq. (2.149), the principal stretches λ + and λ − are given by

λ ± = 1 (Γ ± γ ) (λ + = 1 → +∞, λ − = 1 → 0 for γ = 0 → ∞) 2

(2.155)

2.4 Various Simple Deformations

87

Ⅲ

(α ) ⊗ n (α ) ) Furthermore, multiplying the identity tensor I (= ∑ n to both

Ⅰ

α=

sides of the last equation in Eq. (2.149), it holds that

V 2 − (trV )V + ( det V )I = 0

，

(2.156)

，

Substituting Eqs. (2.146) (2.150) (2.153) into Eq. (2.156), one has

V=

V 2 + ( det V )I 1 = Γ trV

(

⎡1 + γ 2 γ ⎤ ⎡1 0 ⎤ ⎢ ⎥+⎢ ⎥ 1⎦ ⎣ 0 1 ⎦ ⎣ γ

)

(2.157)

resulting in

⎡ 2 + γ 2 γ ⎤ 1 ⎡ 2 + tan 2θ tanθ ⎤ 1 ⎡ ⎤ V = Vij ei ⊗ e j , ⎣Vij ⎦ = ⎢ ⎥ (2.158) ⎥= ⎢ Γ ⎣ γ 2 ⎦ Γ ⎣ tanθ 2 ⎦ for which the inverse tensor of V is given noting Eq. (1.104) as

−γ ⎤ 1 ⎡ 2 ⎡2 V −1 = 1 ⎢ = Γ ⎣ −γ 2 + γ 2 ⎥⎦ Γ ⎢⎣ − tanθ

− tanθ

⎤ ⎥ 2 + tan 2θ ⎦

(2.159)

Substituting Eqs. (2.142), (2.159) into (2.17), R is described as follows:

− γ ⎤ ⎡1 γ ⎤ 1 ⎡ 2 γ ⎤ ⎡2 = R = V −1F = 1 ⎢ Γ ⎣ −γ 2 + γ 2 ⎥⎦ ⎢⎣0 1⎥⎦ Γ ⎢⎣ −γ 2 ⎥⎦

(2.160)

Here, setting

tan θ R ( = R12/R11) =

γ 2

= tan θ (γ = 2 tan θ R ) 2

(2.161)

R is also expressed from Eqs. (1.88), (2.21), (2.160) and (2.161) as

　

⎡ cos θ R sinθ R ⎤ α α ( ) ( ) ⎡ ⎤ R = ⎣(ei • n )(e j • N ) ⎦ = ⎢ ⎥ R R ⎣ −sinθ cosθ ⎦

(2.162)

It holds from Eq. (2.161) that •

• • • γ • γ = 2 R θ R = 2(1 + tan 2 θ R ) θ R , θ R = = 22 γ 2 2 cos θ 2{1 + (γ / 2) } Γ •

(2.163)

88

2 Motion and Strain (Rate)

The relative spin in Eq. (2.78) is given from Eqs. (2.162) and (2.163) as follows:

⎡ −sinθ R cosθ R ⎤ • R ⎡cos θ R − sinθ R ⎤ ΩR = ⎢ ⎥θ ⎢ ⎥ R R R cosθ R ⎦ ⎣ −cosθ − sinθ ⎦ ⎣sinθ

⎡0 =⎢ ⎣ −1

1⎤ • R ⎡ 0 1⎤ • θ = 22 ⎢ γ (2.164) ⎥ Γ ⎣ −1 0 ⎥⎦ 0⎦ p Next, denoting the expression of V in the principal direction by V , it holds from Eq. (1.73) that p

V = Q E VQ E T

，

p

Vij = (n(i ) • er )Vrs (n ( s ) • e j ) (2.165)

where p

⎡λ + 0 ⎤ V=⎢ ⎥ ⎣0 λ − ⎦

(2.166)

⎡n(1) • e1 n (1) • e 2 ⎤ ⎡cosθ E sinθ E ⎤ ( ) i e • ⎡ ⎤ Q = ⎣n j =⎢ ⎦ n( 2) • e n ( 2) • e 2 ⎥ = ⎢− sinθ E cosθ E ⎥ (2.167) 1 ⎦ ⎣⎢ ⎦⎥ ⎣ E

where

θE

is the rotation angle of the eigenvector n (1) , n ( 2 ) of V from the bases

e1 , e2 measured in a counterclockwise direction. On the other hand, the components of

RE are given from Eq. (2.25) as follows:

3

RijE = ei • RE e j = ei • ∑ n (α ) ⊗ eα • e j = ei • n ( j ) = Q Eji , R E = Q ET α =1

(2.168) The following expressions are obtained by substituting Eq. (2.165) into Eq. (2.168). p

V = R E T VR E ,

p

T

p

Vij = RriEVrs RsjE p

V = R E VR E , Vij = RirE Vrs R jsE

(2.169) (2.170)

V1 ≥ V2 always, choosing the maximum principal value λ + in the direction of n (1) , it holds from Eqs. (2.158), (2.166)，(2.168), (2.170) that Since

2 E E E E 1 ⎡ 2 + γ γ ⎤ = ⎡ cosθ − sinθ ⎤ ⎡ λ + 0 ⎤ ⎡ cosθ sinθ ⎤ ⎥⎢ ⎥ ⎥ ⎢ ⎥⎢ Γ ⎢⎣ γ 2 ⎦ ⎣sinθ E cosθ E ⎦ ⎣ 0 λ − ⎦ ⎣− sinθ E cosθ E ⎦

2.4 Various Simple Deformations

89

⎡λ + cosθ E − λ − sinθ E ⎤ ⎡ cosθ E sinθ E ⎤ =⎢ ⎥⎢ ⎥ E λ − cosθ E ⎦⎥ ⎣− sinθ E cosθ E ⎦ ⎣⎢λ + sinθ ⎡λ + cos 2θ E +λ − sin 2θ E (λ + − λ − )sinθ E cosθ E ⎤ =⎢ ⎥ (2.171) 2 E 2 E E E ⎣⎢(λ + − λ −)sinθ cosθ λ + sin θ + λ − cos θ ⎦⎥ Substituting Eq. (2.155) into Eq. (2.171), one obtains 2 1 ⎡2 + γ γ ⎤ ⎥ Γ ⎢⎣ γ 2⎦

⎡1 ⎤ 1 2 E 2 E γ sinθ E cosθ E ⎢ 2 (Γ + γ )cos θ + 2 (Γ − γ )sin θ ⎥ =⎢ ⎥ 1 ( + γ )sin 2θ E + 1 ( − γ )cos 2θ E ⎥ ⎢ γ sinθ E cosθ E 2 Γ 2 Γ ⎢⎣ ⎥⎦

⎡Γ + γ cos2θ E γ sin2θ E ⎤ = 1⎢ 2 γ sin2θ E Γ − γ cos2θ E ⎥ ⎣ ⎦

(2.172)

from which it holds that

γ 1 ⎫ = γ sin2θ E → sin2θ E = 2 ⎪⎪ Γ Γ 2 ⎬ γ2 γ ⎪ E E (V11 − V22 =) = γ cos2θ → cos 2θ = Γ Γ ⎪⎭ and thus it can be obtained that

γ sinθ E = 1 (1 − 2 Γ

)=

(Γ − γ ) /2

Γ

2 cosθ E = 1 − ( Z − ) = Z + = 1 Z− Γ Γ

−γ tan θ E = Z − = Γ Z+ 2

=

⎫ Γ (Γ − γ ) /2 Z − = = 1 ⎪ Z+ Γ Γ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪⎭

(2.173)

90

2 Motion and Strain (Rate)

with

Ζ ± ≡ Γ (Γ ± γ ) / 2 where the double signs Ζ±.

(2.174)

± take in the same order. The following relations hold for

Ζ +Ζ − = Γ Ζ + Γ +γ Ζ− Γ −γ = 2 , = 2 Ζ+ Ζ−

⎫ ⎪ ⎪ ⎪⎪ ⎬ Ζ +2 + Ζ −2 = Γ 2 , Ζ +2 − Ζ −2 = γΓ ⎪ ⎪ 1 + 1 = 1, 1 − 1 = γ ⎪ Ζ +2 Ζ −2 Ζ −2 Ζ +2 Γ ⎪⎭

(2.175)

The substitution of Eq. (2.167), (2.168), (2.173) into Eq. (2.168) yields.

⎡ 1

⎡Ζ + − Ζ − ⎤ ⎢⎢ Ζ − E 1 R = ⎢ = Γ ⎣Ζ − Ζ + ⎥⎦ ⎢ 1 ⎢Ζ ⎣ +

⎤ − 1 ⎥ Ζ+ ⎥ 1 ⎥ Ζ − ⎥⎦

(2.176)

From Eqs. (2.154) and (2.173) we have •

θ E = 1 ( γ − γ• , • E = Γ (Γ + γ ) /2 1 ( γ − γ• = − 1 γ• 1 θ 2 Γ 1) cos 2θ E 2 Γ ) Γ2 Γ2 (2.177) Substituting Eqs. (2.167), (2.168), (2.177) into Eq. (2.85), it is obtained that

⎡ −sinθ E − cosθ E ⎤ • E ⎡ cosθ E sinθ E ⎤ ΩE = ⎢ ⎥θ ⎢ ⎥ E E E E ⎣ cosθ − sinθ ⎦ ⎣− sinθ cosθ ⎦

⎡0 −1⎤ • E 2 ⎡ 0 1 ⎤ γ• 1 Ω R =⎢ θ = Γ 2 ⎢− ⎥ ⎥ =2 ⎣1 0 ⎦ ⎣ 1 0⎦

(2.178)

The following expression of U is obtained in a similar manner to that used in Eq. (2.157)

U 2 + ( det U)I 1 U= = Γ trU

( ⎡⎢⎣1γ

γ

⎤ ⎡1 0 ⎤ ⎥ +⎢ ⎥ 1 + γ 2 ⎦ ⎣0 1 ⎦

)

2.4 Various Simple Deformations

91

from which it is obtained that

⎤ sinθ ⎡ 2 γ ⎤ ⎡cos θ =⎢ U= 1⎢ ⎥ 2 Γ ⎣γ 2+ γ ⎦ ⎣sinθ (1 + sin 2θ ) / cosθ ⎥⎦

(2.179)

RL of the eigenvector N (α ) of U , denoting the (1) ( 2) angle measured in counterclockwise direction from e1, e 2 to N , N by θ L , In order to obtain the rotation

one has

U = RLV p RL

T

(2.180)

where, setting

⎡e1 • N (1) e1 • N (2) ⎤ ⎡cosθ L − sinθ L ⎤ RL = ⎡⎣ei • N ( j ) ⎤⎦ = ⎢ ⎥=⎢ ⎥ (2.181) ⎢⎣e 2 • N (1) e 2 • N (2) ⎦⎥ ⎣sinθ L cosθ L ⎦ and substituting Eqs. (2.166), (2.179), (2.181) into Eq. (2.180), we have L sinθ L ⎤ ⎡λ + 0 ⎤ ⎡cosθ L − sinθ L ⎤ 1 ⎡ 2 γ ⎤ = ⎡ cosθ (2.182) ⎢ ⎢ ⎥ Γ ⎣γ 2 + γ 2 ⎦ ⎣− sinθ L cosθ L ⎥⎦ ⎢⎣ 0 λ − ⎥⎦ ⎢⎣sinθ L cosθ L ⎥⎦

The substitution of Eq. (2.155) into Eq. (2.182) leads to

1 ⎡2 γ ⎤ = Γ ⎣⎢γ 2 + γ 2 ⎦⎥

L − γ sin2θ L ⎤ 1 ⎡Γ + γ cos2θ ⎢ 2 − γ sin2θ L Γ − γ cos2θ L ⎥ ⎣ ⎦

(2.183)

⎫ ⎪⎪ Γ ⎬ γ2 γ ⎪ L L (U11 − U22 =) − = γ cos2θ → cos 2θ = − Γ Γ ⎪⎭

(2.184)

from which one has

sin2θ L (= sin2θ E ) = − 2

and

⎫ γ 2 Z sin θ L = 1 (1 + ) = + = 1 ⎪ 2 Γ Γ Z− ⎪ ⎪⎪ Z 2 cos θ L = 1 − ( + ) = Z − = 1 ⎬ Γ Γ Z+ ⎪ ⎪ tan θ L = Z + = 2 γ = 1 E ⎪ Z− Γ − tan θ ⎪⎭

(2.185)

92

2 Motion and Strain (Rate)

The substitution of Eq. (2.185) into Eq. (2.181) reads:

⎡ 1 ⎤ − 1 ⎥ ⎢ ⎡Z − − Z+ ⎤ ⎢ Z+ Z − ⎥ = RL = 1 ⎢ Γ ⎣ Z + Z − ⎥⎦ ⎢ 1 1 ⎥ ⎢ Z − Z+ ⎥ ⎣ ⎦ Substituting Eqs. (2.176) and (2.186) into holds that

(2.186)

R = RE RL T based on Eq. (2.27), it

⎡Ζ + − Ζ − ⎤ 1 ⎡ Z − Z + ⎤ 1 ⎡ 2Z + Z − Z +2 − Z −2 ⎤ R= 1 ⎢ = Γ ⎣Ζ − Ζ + ⎥⎦ Γ ⎢⎣−Z + Z − ⎥⎦ Γ 2 ⎢⎢⎣− Z 2 + Z 2 2Z + Z − ⎥⎦⎥ + −

⎡ 2 γ⎤ = 1⎢ Γ ⎣ −γ 2 ⎥⎦

(2.187)

which coincides with Eq. (2.160) obtained by the different approach. Substituting Eqs. (2.162), (2.167), (2.168), (2.181) into

　

R = RE RL T , one obtains

⎡ cos θ R sinθ R ⎤ ⎡ cosθ E − sinθ E ⎤ ⎡ cosθ L sinθ L ⎤ ⎢ ⎥=⎢ ⎥⎢ ⎥ R R E cosθ E ⎦ ⎣− sinθ L cosθ L ⎦ ⎣ −sinθ cosθ ⎦ ⎣sinθ ⎡cosθ E cosθ L + sinθ E sinθ L cosθ E sinθ L − sinθ E cosθ L ⎤ =⎢ ⎥ L L L L E E E E ⎣sinθ cosθ − cosθ sinθ sinθ sinθ + cosθ cosθ ⎦ ⎡cos(θ E − θ L ) − sin(θ E − θ L ) ⎤ =⎢ ⎥ L L E E ⎣sin(θ − θ ) cos(θ − θ ) ⎦

(2.188)

from which the following relation is obtained.

θ E = θ L −θ R

(2.189)

The rotations of n (α ) and N (α ) are shown in Fig. 2.7. Furthermore, it is derived from Eqs (2.154) and (2.185) that •

θ L = 1 ( γ + γ• • L = Γ (Γ − γ ) / 2 1 ( γ + γ• 1 • 1) , θ 2 Γ 1) = Γ 2 γ Γ2 cos 2θ L 2 Γ (2.190)

2.4 Various Simple Deformations

93

e2 n (Ⅱ) (∞ ) N (Ⅰ) (∞ )

n (Ⅱ) (t)

N (Ⅰ) (t) N (Ⅰ) (0) n (Ⅰ) (0)

n (Ⅱ) (0) ( Ⅱ N ) ( 0) N (Ⅱ) (t)

θ L (t)

n (Ⅰ) (t)

θ R (t) θ E (t )

π /4

N (Ⅱ) (∞)

n (Ⅰ) (∞)

Fig. 2.7 Rotation of Lagrangian triad

N (α )

e1

n (α )

and Eulerian triad

Substituting Eqs. (2.181), (2.190) into Eq. (2.83), we obtains

⎡ −sinθ L − cosθ L ⎤ • ⎡ cosθ L sinθ L ⎤ ΩL = ⎢ ⎥θ L ⎢ ⎥ L L L − sinθ L cos θ ⎣ ⎦ ⎣− sinθ cosθ ⎦

⎡0 −1⎤ • L 2 ⎡ 0 −1⎤ γ• − 1 Ω R =⎢ θ =Γ2⎢ ⎥ ⎥ = 2 ⎣1 0 ⎦ ⎣1 0 ⎦ It can be confirmed easily that the three kinds of spins

(2.191)

Ω R , Ω E , Ω L fulfill

the relation (2.87) by Eqs. (2.160), (2.164), (2.178) and (2.191). •r

•

Note here that a simple relation between D and λ = λ l does not hold since their principal directions does not coincide mutually. Denoting τ = σ 12 , various stress tensors are described as follows:

⎡σ 11 τ ⎤ ⎥ ⎣ τ σ 22 ⎦

σ=⎢

(2.192)

94

2 Motion and Strain (Rate)

⎡1 − γ ⎤ ⎡σ 11 τ ⎤ ⎡σ 11 − γτ τ − γσ 22 ⎤ Π = JF −1σ = ⎢ ⎥=⎢ ⎥⎢ τ σ 22 ⎥⎦ ⎣0 1 ⎦ ⎣ τ σ 22 ⎦ ⎣

(2.193)

⎡σ 11 − γτ τ − γσ22 ⎤ ⎡ 1 0 ⎤ S = JF −1σF −T = ⎢ σ22 ⎥⎦ ⎢⎣ −γ 1 ⎥⎦ ⎣ τ ⎡σ 11 − γ 2σ22 − 2γτ τ − γσ22 ⎤ =⎢ ⎥ τ − γσ22 σ22 ⎦ ⎣

(2.194)

2.4.3 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 = α R, θ = Θ + ω Z , z = λ Z

(2.195)

where ( R, Θ , Z) signifies the initial configuration, and α , ω and λ denote the proportionality factors depending on the deformation, while ω is described by the relative distortion angle φ between both ends as follows:

ω ≡φ / L

(2.196)

L being the length of the specimen. Variables depending on the deformation are given as follows:

⎡ FrR FrΘ ⎢ F = ⎢ Fθ R FθΘ ⎢⎣ FzR FzΘ

⎡ ∂r ∂r ⎤ ∂r ⎢ ∂R R ∂Θ ∂ Z ⎥ FrZ ⎤ ⎢ ⎥ ⎥ ⎢ r ∂θ θ θ ∂ ∂ r r ⎥ Fθ Z ⎥ = ⎢ ∂R R ∂Θ ∂Z ⎥ FzZ ⎥⎦ ⎢ ⎥ ∂z ∂z ⎥ ⎢ ∂z ∂Z ⎥⎦ ⎣⎢ ∂R R ∂Θ

⎡ 0 ⎤ ∂α R ∂α R ∂α R ⎤ ⎡α 0 ⎢ ⎥ ⎢ R∂Θ ⎥ ∂R ∂Z ⎢ ⎥ ⎢ ⎥ r (Θ + ω Z ) r ∂ (Θ + ω Z ) r ∂ (Θ + ω Z ) ⎥ ⎢ =⎢ ∂ = 0 α ωα R ⎥ ⎢ ⎥ ⎢ R∂Θ ∂R ∂Z ⎥ ⎢ ⎥ ⎢ ⎥ ∂λ Z ∂λ Z ∂λ Z ⎥ ⎢ ⎢ λ ⎥⎦ R∂Θ ∂Z ⎢⎣ ⎥⎦ ⎣0 0 ∂R (2.197)

2.4 Various Simple Deformations

95

⎡1 ⎢α ⎢ −1 F = ⎢⎢0 ⎢ ⎢0 ⎢⎣

⎤ 0 ⎥ ⎥ 1 − ωR ⎥ α λ ⎥ ⎥ 1 ⎥ 0 λ ⎥⎦ 0

(2.198)

0 0 ⎤ ⎡α ⎢ ⎥ V = ⎢0 α cosφ + αω R sin φ λ sinsφ ⎥ ⎢0 λ sinφ λ cosφ ⎥⎦ ⎣

(2.199)

0 0 ⎡α ⎤ ⎢ ⎥ U = ⎢0 α cosφ α sinφ ⎥ ⎢⎣0 α sinφ αω R sin φ + λ cosφ ⎥⎦

(2.200)

where

cos φ =

λ +α (α + λ ) + (αω R ) 2

2

,

sinφ =

αω R (α + λ ) 2 + (αω R ) 2

(2.201)

R being the mean radius ( R ≅ R ) of the thin cylindrical specimen in the initial state. Here, R is given by Eq. (2.162).

⎡α 2 0 ⎤ 0 ⎢ ⎥ T 2 ωα R ⎥ C = F F = ⎢ 0 α2 ⎢ 0 ωα 2 R λ 2 + ω 2α 2 R 2 ⎥ ⎣ ⎦ ª 1 0 «α 2 « −1 −T −1 1 b = F F = «« 0 α2 « « 0 − ωR «¬ αλ

º » » ω R » − » αλ » ω2R2 + 1 » λ2 λ 2 »¼

(2.202)

0

(2.203)

96

2 Motion and Strain (Rate)

⎡ • ⎤ ⎢ α − ω• Z 0 ⎥ ⎢α ⎥ ⎢ ⎥ • • • −1 ⎢ • ⎥ ω α R α L = F F = ⎢ω Z α λ ⎥ ⎢ ⎥ • ⎥ ⎢ λ ⎥ ⎢0 0 λ ⎥ ⎢ ⎣ ⎦ ⎡• ⎢α ⎢α ⎢ ⎢ D = ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎣

0

α• α ω• α R 2λ

⎤ 0 ⎥ ⎥ ⎥ • R⎥, ωα 2λ ⎥ ⎥ • λ ⎥⎥ λ ⎥ ⎦

(2.204)

⎡ ⎢ 0 − ω• Z ⎢ ⎢ W = ⎢ω• Z 0 ⎢ ⎢ • ⎢ 0 − ωαR ⎢⎣ 2λ

⎤ 0 ⎥ ⎥ • ωαR ⎥ ⎥ 2λ ⎥ ⎥ 0 ⎥ ⎥⎦ (2.205)

Stresses in various definitions are described by the following equations, desig-

nating the normal stress σ zz and the shear stress tion of the cylinder by σ and τ , respectively.

σ rθ

applied to the traverse sec-

⎡0 0 0 ⎤ σ = ⎢⎢0 0 τ ⎥⎥ ⎢⎣0 τ σ ⎥⎦ ⎡1 ⎢α ⎢ ⎢ −1 Π = JF σ = α 2λ ⎢0 ⎢ ⎢0 ⎢⎣

⎤ 0 ⎥ ⎡0 0 0 ⎤ ⎥ ⎥ ⎢⎢ ⎥ 1 − ω R ⎥ ⎢0 0 τ ⎥ α λ ⎥⎢ ⎥ ⎥⎢ ⎥ 1 ⎥ 0 ⎢ ⎥⎦ τ σ 0 λ ⎥⎦ ⎣ 0

(2.206)

2.5 Surface Element, Volume Element and Their Rates

97

0 0 ⎡0 ⎤ ⎢ ⎥ = ⎢0 − α 2ωτ R αλτ − α 2ωσ R ⎥ ⎢0 ⎥ α 2τ α 2σ ⎣ ⎦

(2.207)

　

S = ΠF −T

⎡ ⎢0 ⎢ = ⎢0 ⎢ ⎢ ⎢0 ⎢⎣

⎡1 0 ⎢α 0 0 ⎡0 ⎤⎢ ⎢ ⎥ 1 = ⎢0 − α 2ωτ R αλτ − α 2ωσ R ⎥ ⎢0 α ⎢ ⎢0 ⎥⎢ α 2τ α 2σ ⎣ ⎦ ⎢0 − ω R λ ⎢⎣ 0 2 2 2 − 2αωτ R + α ω σ R

λ

2 ατ − α ωσ R λ

⎤ 0 ⎥ ⎥ 2 R α ωσ ⎥ ατ − λ ⎥ ⎥ α2 ⎥ λ ⎥⎦

⎤ 0⎥ ⎥ 0⎥ ⎥ ⎥ 1 ⎥ λ ⎥⎦

(2.208)

2.5 Surface Element, Volume Element and Their Rates a

c

a

c

Presuming that the line-elements dX , dXb , dX change to dx , dxb , dx by the deformation, the following relation holds for the volume element before and after the deformation from Eqs. (1.22), (1.42), (1.43), (2.6), (2.9) and (2.11).

dv = (dxa × dxb ) • dxc = ε ijk dxia dxjb dxkc = detdx dx1a dx1b dx1c

F1R dX Ra F1R dX Rb F1R dX Rc

= dx2a dxb2 dx2c = F1R dX Ra F2 R dX Rb F2 R dX Rc dx3a dx3b dx3c

F1R dX Rc F3 R dX Rb F3 R dX Rc

a b c F11 F12 F13 dX 1 dX 1 dX 1

= F21 F22 F23 dX 2a dX 2b dX 2c = JdV F31 F32 F33 dX 3a dX 3b dX 3c

98

2 Motion and Strain (Rate)

from which one has

ρ J = det F = dv = ρ0 dV

(2.209)

where ρ 0 and ρ are the initial and the current densities. On the other hand, denoting the areas and the unit normal vectors of the surface elements formed by the line-elements we have

dX a , dXb and dx a , dxb as dA

， N and da , n , respectively,

dV = [dXa dXb dXc ] = (dXa × dXb ) • dXc = dXc • NdA dv

= [ dx a

dx b

dx c ] =

( dx a

× dxb ) • dxc

=

dx c

• nda =

F dX c

• nda =

dXc

⎪⎫ ⎬ • F nda ⎪⎭ T

(2.210) noting Eq. (1.100). The following Nanson’s formula is derived from Eqs. (2.209) and (2.210).

nda = JF −T NdA, NdA = 1 FT nda J

(2.211)

or

da = JF −T dA, dA = 1 FT da J

(2.212)

da ≡ nda, dA ≡ NdA

(2.213)

where Further, noting

∂ det d x = Δ = 1 ε ε dx b dx c pq 2 pjk qbc j k ∂ x pq derived from Eqs. (1.17) and (1.23), one has

( dv )• = (det d x)• = ∂ det dq x (dx pq )• = ∂ det dq x dv pq = 1 ε pjk ε qbc dxbj dxkc 2 ∂d x p ∂d x p

∂v p q dx ∂xr r

1 1 = ε ijk ε abc dxjb dxkc dx ar ∂xvi = ε ijk ε abc dx ra dxjb dxkc Lir 2 ∂ r 2

1 1 = δ irε ijk ε abc dx ra dxjb dxkc Lir = ε ijk ε abc dxia dxjb dxkc Lvv 6 6 1 = ε ijk ε abc dxia dxjb dxkc Dvv = dvDvv 6 or •

J = tr( ∂ det F FT ) = tr{(det F )F −T FT } = tr{J (F F −1 )T } = JtrLT ∂F •

•

•

2.5 Surface Element, Volume Element and Their Rates

99

from which the following relation holds for the rate of volume element, noting

trTT = trT and t rW = 0 . (dv)• J• or (dv)• = dvt rD = • dV = ε v = t rD = J dv J •

(2.214)

Then, the time-integration of the volumetric strain rate leads to the logarithmic volumetric strain.

ε v ≡ ∫ t rDδ t = ln dv

(2.215)

dV

Moreover, it is obtained from Eq. (2.69) and the Nanson’s formula (2.211) that •

• −T

(nda )• = ( JF −T NdA)• = ( J F −T dA + J F )NdA • −T

= {(t rD)I + F

FT }F −T JNdA •

= {(trD)I − F −T FT }nda = {(trD)I − LT}nda

(2.216)

•

On the other hand, noting n • n = 0 from n • n = 1 for the unit vector n , it holds that

(da)• = n • n(da)• = n • {(nda)• − n• da} = n • (nda)•

(2.217)

Substituting Eq. (2.216) into Eq. (2.217), one obtains the rate of the current infinitesimal area as follows:

(da )• = n{(trD) I − LT }nda

(2.218)

(da)• = {(trD) I − n • Dn}da

(2.219)

which reduce to

Further, it holds from Eqs. (2.216) and (2.219) that

n• da = (nda)• − n(da)• = {(trD)I − LT }nda − n{(trD) − n • Dn}da

(2.220)

Then, the rate of the unit normal of the current surface element is given by

n• = {(n • Dn) I − LT }n

(2.221)

Chapter 3

Conservation Laws and Stress Tensors 3 Conservation Laws and Stress Tensors

Conservation laws must be fulfilled for mass, momentum, angular momentum, etc. during a deformation. These laws are described first in detail. Then, the Cauchy stress tensor is defined and further, based on it, various stresses are derived. Introducing the stress tensor, the equilibrium equations of force and moment are formulated from the conservation rules. The virtual work principle required for the analyses of boundary value problems are also described in this chapter.

3.1 Conservation Law of Mass Denoting the field of density in a material as given as

ρ (x, t ) , the mass in a region v

is

m = ∫v ρ (x, t )dv . Therefore, the following conservation law of mass

must hold. • (= (∫v ρ (x, t )dv)• ) = 0 m

(3.1)

from which, noting the Reynolds’ transportation theorem of Eq. (1.277), one has the

continuity equation.

ρ• + ρdivv = 0 , ρ• + ρ ∂v r = 0 ∂ xr

Further, setting

(3.2)

T (x, t ) ≡ ρφ , where φ is a physical quantity per unit mass, it

holds from Eqs. (1.277) and (3.2) that

(∫v ρφ dv)• = ∫v (ρ φ + ρ• φ + ρφ ∂∂vxrr ) dv = ∫v ρ φ• dv •

(3.3)

3.2 Conservation Law of Momentum The momentum possessed by a region

v in a current state is given by ∫v ρ vdv . On

the other hand, denoting the traction (or stress vector) applied to the unit surface area of the region as t, the traction applied to the surface of the region is given as

K. Hashiguchi: Elastoplasticity Theory, LNACM 42, pp. 101–109. springerlink.com © Springer-Verlag Berlin Heidelberg 2009

102

3 Conservation Laws and Stress Tensors

∫a t da and, denoting the body force per unit mass as b, the body force applied to the region is given by ∫ ρ bdv . The rate of momentum has to be equivalent to the v sum of the traction and the body force applied to the region. Therefore, Euler’s first law of motion (or conservation law of momentum) is given as

(∫v ρ vdv)• = ∫a t da + ∫v ρ bdv or

∫v ρ v• dv = ∫a t da + ∫v ρ bdv

(3.4)

by virtue of Eq. (3.3).

3.3 Conservation Law of Angular Momentum The angular momentum possessed by a region

v in a current state is given as

∫v ρ (x × v)dv. On the other hand, denoting the moment of traction and the moment of body force applied to the region as ∫a ( x × t ) da and ∫ ρ ( x × b) dv , v respectively, Euler’s second law of motion, i.e. conservation law of angular momentum is described as

(∫v ρ x × vdv)• = ∫a x × tda + ∫v ρ x × bdv (∫v ρε ijk xj vk dv)• = ∫a ε ijk xj t k da + ∫v ρε ijk xj bk dv which reduces to •

∫v ρ x × v dv = ∫a x × tda + ∫v ρ x × bdv •

∫v ρε ijk xj v k dv = ∫a ε ijk xj t k da + ∫v ρε ijk xj bk dv noting

•

(3.5)

•

(x × v )• = v × v + x × v = x × v and Eq. (3.3).

3.4 Stress Tensor When the infinitesimal force vector df applies to the surface with infinitesimal area da and the unit normal vector n, the stress vector is given as shown below.

t ≡ df da

(3.6)

3.4 Stress Tensor

103

Now, introduce the following second-order tensor

σ

fulfilling the relation

t = σT n, ti = σ ji n j by the quotient law described in 1.3.2. The components of the tensor by Eq. (1.88) as

(3.7)

σ

σij = ei • σe j = σ e i • e j = σ T e i • ej

are given (3.8)

if σ is the symmetric tensor. 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. (3.7) with n = ei into Eq. (3.8).

σij = t • e j

(3.9)

Therefore, σij is the component in the direction of e j for the stress vector t applying on the surface element having the outward-normal ei . The tensor σ is called the Cauchy stress tensor. Equation (3.7) is called the Cauchy’s fundamental theorem or Cauchy’s stress principle. It holds from the conservation law of angular momentum described in 3.6 that

σ = σT (σij = σji )

(3.10)

which means that σ is the symmetric tensor. Various stress tensors are defined in addition to the Cauchy stress tensor described above. Some of them, which are often used in continuum mechanics, are presented below. The tensor τ defined by the following equation is called the Kirchhoff stress tensor. τ = Jσ (3.11) The vector t defined by the following equation is called the nominal stress vector (see Fig. 3.1).

df t ≡ dA

(3.12)

Tensor Π , which is related to t by the following equation, is called the nominal stress tensor or the first Piola-Kirchhoff stress tensor.

t ≡ ΠN (ti ≡ ΠiA NA )

(3.13)

Here, substituting Eqs. (2.211) and (3.13) into Eq. (3.12), we have

df ≡ ΠNdA = 1 ΠFΤ nda (dfi = Π iΑ NA dA = 1 Π iΑ FrA nr da ) J J

(3.14)

104

3 Conservation Laws and Stress Tensors

On the other hand, the substitution of Eq. (3.7) into Eq. (3.6) yields

df = σT nda

(3.15)

Comparing Eqs. (3.14) and (3.15), it holds that

σT nda = ΠNdA and the relation of

σ

and

(3.16)

Π is obtained noting Eq. (2.211) as shown below.

⎫ σ = 1 FΠT (= 1 ΠFT ) (σ ij = 1 FiAΠ j Α = 1 Π iΑ F jA ) ⎪ J

Π = JσF

J

−T

J

J

(ΠiΑ = J σ ir (F )rA ) (Π ≠ Π ) −1

T

⎬ ⎪ ⎭

(3.17)

df

df

n

N

F

da

dA dX

dv

dV

dx = FdX

Fig. 3.1 Variation of configuration

Further, the tensor S defined by the following equation is called the second Piola-Kirchhoff stress tensor.

t = SN

(3.18)

where −1 t ≡ F df = F −1t dA

(3.19)

Using Eq. (2.211) into these equations, one has the following expression.

df ≡ FSNdA = 1 FSFT nda J

(3.20)

3.5 Equilibrium Equation

105

Comparing Eqs. (3.15) and (3.20), it holds that

⎫ ⎪ ⎬ = J (F −1 ) Ai σ ij (F −1 )jB ) (S = S T )⎪ ⎭

σ = 1 FSF T (σ ij = 1 FiA S AB FBj ) J

J

S = JF σF −1

−T

( S AB

(3.21)

The following stress is called the convected stress.

cσ ≡ F T σF ( cσ = cσT )

(3.22)

The relations of various stress tensors defined above are summarized in Table 3.1. Table 3.1 Relations of various stress tensors Names, Notations Cauchy

ı (= ıT )

IJ (= IJT )

IJ Nominal (1st Piola-Kirchhoff) Ȇ 2nd Piola-Kirchhoff S Convected cı

(Note)

S (= ST )

c ı ( = cı T )

1 FȆT J

1 FSFT J

F −T cı F −1

FȆT

FSFT

JF −T cıF −1

FS

JF −T cıF −1F −T

1 IJ J

ı Kirchhoff

Ȇ ( ≠ ȆT )

Jı

JıF −T

IJF −T

JF −1ıF −T

F −1IJF −T

Ȇ T F −T

F T ıF

1 F T IJF J

1 F T FȆT F J

−1 t ≡ df , t ≡ df , t ≡ F df da dA dA t = σn , Jt = τn , t = ΠN ,

JF −1F −T cıF −1F −T 1 F T FSFT F J

= F −1t t = SN

3.5 Equilibrium Equation Substituting Eq. (3.7) into Eq. (3.4) for the conservation law of momentum and noting Eq. (3.10), the following equation is obtained.

(∫v ρ vdv)• = ∫a σT nda + ∫v ρ bdv, (∫v ρ vi dv)• = ∫a σi r n r da + ∫v ρbidv (3.23) The right had side is given from Eq. (3.3) as

(∫v ρ vdv)• = ∫v ρ v• dv

106

3 Conservation Laws and Stress Tensors

and the first term in the right-hand side of Eq. (3.23) is given by Eq. (1.267) of Gauss’ divergence theorem as

σ

∫a σi r nr da = ∫v ∂∂xirr dv By this equations and Eq. (3.3) the local form of Eq. (3.23) is given as

σ • ∇ + ρ b = ρ v• ,

∂σ ij + ρ bi = ρ v• i ∂xj

(3.24)

This equation is called the Cauchy’s first law of motion, i.e. the equilibrium equation. On the other hand, substituting Eqs. (2.209) and (3.16) into Eq. (3.23), one has

(∫V ρ0 vdV )• = ∫A ΠNd A + ∫V ρ0bdV

(3.25)

which is rewritten by Gauss’ divergence theorem as follows:

∫V ρ where

0

v• d V = ∫V Π • ∇X dV + ∫V ρ0bdV

(3.26)

∇ X ≡ (∂ / ∂X A )e A = ∂/∂ X . The local form of this equation is given as •

Ȇ • ∇X+ ρ0b = ρ 0 v,

∂Ȇ iA + ρ0b = ρ 0 v• i i ∂XA

(3.27)

The equilibrium equation in a rate form is required in constitutive equations for irreversible deformation including elastoplastic deformation. The time-differentiation of Eq. (3.27) engenders the following rate-type (or incremental-type) equilibrium equation, provided that the acceleration does not •• change, i.e. v = 0 . •

•

∂ ȆiA + ρ • = 0 0 bi ∂XA

•

Ȇ • ∇X + ρ0 b = 0, Denoting

∗ 1 • • 1 • σ ≡ J Π F T (= σ T = J F Π T )

(3.28)

(3.29)

and noting Eq. (3.17), we have

∗ 1 • F −T + • −T + J σ • −T) FT = J• σ + σ• + σ • −T FT F F σ = J (Jσ J σF J •

• −T

Substituting (F −TF T ) = F

•

•

F T + F −T F T = F −T F T + LT = 0 due to Eq.

(2.69) and Eqs. (2.73) and (2.214) to this equation, the following holds.

3.6 Equilibrium Equation of Moment

107

: ∗ • (3.30) σ = σ + σ t r L − σ LT = σ + σ t r D − σ D + W σ ∗ : • Therein, σ is designated as the nominal stress rate, whereas σ ≡ σ − w σ + σ w

is the Jaumann stress rate which will be described in detail in the next chapter. The partial derivative of Eq. (3.29) by x j noting ∂ ( F jA / J ) / ∂ x j = 0 (see Appendix 2) leads to the following.

* • • • ∂ σ ij = 1 F ∂Π iΑ = 1 ∂Π iΑ ∂x j = 1 ∂Π iΑ J jA ∂x j J ∂x j ∂X A J ∂X A ∂x j Substitution of this relation into Eq. (3.28) yields the rate-type equilibrium defined in the current configuration:

* • • ∗ ∂ σ ij ı • ∇ + ρ b = 0, ∂x j + ρ bi = 0

(3.31)

This equation is derived also by the following manner. From Eq. (3.23) one has

(∫v ρ v• dv)• = ( ∫a σT nda)• + (∫v ρ bdv)• (∫V ρ 0 v• dV )• = (∫a σT nda)• + ( ∫V ρ0 bdV )• •

•

0 = ³a ı T nda + ³a ıT (nda )• + ³ ρ0 b dV V

•

•

0 = ³a ı T nda + ³a ıT {(trD)I − L }nda + ³v ρ b d v T

•

• T 0 = ³v (ı T + ıT trD − ıT L ) • ∇dv + ³v ρ b d v

which results in Eq. (3.31), noting Eqs. (1.274), (2.216) and

σ = σT .

3.6 Equilibrium Equation of Moment Substituting Eq. (3.7) into Eq. (3.5) of the conservation law of angular momentum and noting (3.10), one has •

∫v ρε ijk xj v k dv = ∫a ε ijk xj σ kr nr da + ∫v ρε ijk xj bk dv

(3.32)

Because the first term in the right-hand side of this equation is rewritten as

∫a ε ijk xj σ kr nr da = ∫a ε ijk

∂ xj σ kr σ dv = ∫v (ε ijkσ k j + ε ijk xj ∂ xkr ) dv x ∂ r ∂ r

108

3 Conservation Laws and Stress Tensors

Eq. (3.32) leads to

∫v{ε ijkσ k j + ε ijk xj (∂∂σxkrr + ρbk − ρ v• k )}dv = 0 Noting the equilibrium equation (3.24) to this equation, it holds that from which we have the symmetry of Cauchy stress tensor, i.e.

ε ijkσ k j = 0

σ ij = σ ji

(3.33)

3.7 Virtual Work Principle The stress (rate) field fulfilling the equilibrium equation and the boundary condition of stress is called the statically admissible filed. On the other hand, the displacement (velocity) field fulfilling the geometrical requirement

FiA (= ∂xi / ∂X A ) = δ iA + ∂ui / ∂X A or Dij = (∂vi / ∂x j +∂v j / ∂xi ) / 2 and the boundary condition of displacement (velocity) is called the kinematically-admissible field. Denoting arbitrary statically admissible stress field and kinematically-admissible velocity field by the following equation from Eq. (3.24).

³v (ı Δ• ∇

+ ρ b − ρ v• ) • u∇ dv = 0,

( ) Δ and ( )∇ respectively, one has Δ

∂σ ij ( ³v ∂xj

+ ρ bi − ρ v• i ) u∇i dv = 0 (3.34)

Using the Eq. (1.264) of Gauss’ divergence theorem, we have ∇ Δ ∂ u i dv σ v ij ∂x j

∫

Δ

Δ

∂σ ij ∇ ∂ (σ ij u ∇i ) u dv =∫ dv − ∫ v v ∂xj ∂xj i ∂σ ijΔ ∇ u i dv v ∂xj

= ∫a σ ij u∇i nj da + ∫ σ ijΔ ui nj da − ∫ av

t

∂σ ijΔ ∇ u i dv av v ∂xj t − where ( ) designates the given boundary condition, and at and a v specify the = ∫a tij u∇i da + ∫ σ ijΔ nju i da − ∫

surfaces of the body on which the traction (rate) and the displacement (velocity) are given, respectively. Substituting Eq. (3.34) into this equation, the following virtual work principle described by the quantities in the current state is obtained.

∂ u∇

∫v σ ijΔ ∂xji dv = ∫at ti u∇i da + ∫av σ ij nj ui da + ∫v ρ biu∇i dv − ∫v ρ vi ui dv Δ

•

∇

(3.35)

3.7 Virtual Work Principle

109

Similarly, the following virtual work principle can be described by the quantities in the initial state from Eq. (3.27).

∂ u∇

∫V Π iAΔ ∂XiA dV = ∫At ti u∇i dA + ∫Av Π iA NJ ui dA + ∫V ρ biu∇i dV − ∫V ρ Δ

0

0

v• i u∇i dV (3.36)

Furthermore, one has the rate-type virtual work principle from Eqs. (3.28) and (3.31) as follows:

• Δ

•

∫vσ ij Dij∇ dv = ∫at t i v∇i da + ∫av σ ij nj vi da + ∫v ρ bi v∇i dv •Δ •∇

•

•

•Δ

•

•Δ

∫V ΠiA F iAdV = ∫At t i vi∇ d A+ ∫Av Π iA NA vi dA + ∫V ρ

0

b i vi∇ dV

(3.37)

(3.38)

Chapter 4

Objectivity and Corotational Rate Tensor 4 Objectivity and Corotatio nal Rate Tensor

The mechanical properties of a material are independent of the reference frame of the observer. Therefore, a constitutive equation describing the properties must be invariant under the change of coordinate system mutually rotating with respect to each other. The physical meaning of this type of the objectivity of constitutive equations and the general form of corotational rate of a tensor that has to be used in stead of the usual material-time derivative (under the fixed coordinate system) in order to fulfill the invariance are described in this chapter.

4.1 Objectivity When a material is subjected to a constant stress, i.e. a stress the components of which are observed to be constant from the material itself, the deformation has not to be induced even if a rigid-body rotation is added. However, the components of stress in a coordinate system fixed in the laboratory change so that the stress rate is observed to be induced in the fixed coordinate system if the material is subjected to a rotation. On the other hand, the strain rate described in 2.3 is given by the symmetric part of the velocity gradient excluded its skew-symmetric, i.e. rotational part and thus its components are not influenced by the rotation. The one-to-one correspondence between the stress and the strain does not exist, and thus the stress rate and the strain rate have to be related mutually in an irreversible constitutive equation leading to the rate-type equation, e.g. the viscoplastic, the elastoplastic and the viscoplastic constitutive equations. Therefore, if the material-time derivative of stress observed in the fixed coordinate system is adopted in these constitutive equations, the incorrect prediction is given such that the strain rate is induced and thus the material deforms even if the stress observed by material itself is constant. Let the general interpretation for this fact and how to remedy this defect be considered below. As already mentioned, the nechanical properties of materials are independent of the mutual rigid body motion of observers. Therefore, the deformation characteristics must be described uniquely in an identical equation independent of the relative position and motion to the coordinate systems by which it is described. This fact is called the principle of objectivity or principle of material-frame indifference or simply objectivity (Oldroyd, 1950). The explicit problem occurs in the deformation analysis of materials subjected to the rotation, while the coordinate system describing a constitutive equation is fixed usually. Therein, the check is K. Hashiguchi: Elastoplasticity Theory, LNACM 42, pp. 111–125. springerlink.com © Springer-Verlag Berlin Heidelberg 2009

112

4 Objectivity and Corotational Rate Tensor

required as to whether or not the deformation behavior is formulated such that it is not influenced by the rotation. As the preliminary examination, how the physical quantities relevant to the stress and deformation are influenced by the rotation will be first considered in the next section. Consequently, how to remedy this defect for the quantity, which is observed to be constant by the material itself but is observed to change in the fixed coordinate system, will be considered. Further, the explicit mathematical process that the material-time derivative of arbitrary scalar-valued tensor function is transformed directly to its corotational derivative is shown, while this transformation is required in the consistency condition which is derived from the yield function on the formulation of plastic constitutive equation described in the later chapter. Mechanical elements are often subjected to a rotation independent of the occurrence of deformation, as seen in gears, wheels, etc. in the case of machinery. Soils near the side edges of footings, at the pointed ends of piles, etc. undergo a large rigid-body rotation in soil structures. Therefore, a formulation of constitutive equations independent of rigid-body rotation is of great importance in practical engineering problems.

4.2

Influence of Rigid-Body Rotation on Various Mechanical Quantities

In order to check whether or not a constitutive equation is formulated in accordance with the objectivity principle, it is expedient first to examine the influence of rigid-body rotation on variables used in constitutive equations. To this end, it is sufficient to derive the transformation rules for the components of constitutive variables under the change of coordinate system from one with the fixed base {ei } T into the other with the rotating base {e∗ i (t ) = Q (t )ei } , which coincide with base {ei } in initial instant and then rotates with the elapse of time. Designating the infinitesimal line-element vectors observed in the above-mentioned coordinate systems before and after the deformation as dX, dx and dX∗ , dx∗ , the relations

dX∗ = dX (Q = I for t = 0), dx∗ = Qdx hold where Q = ei ⊗ e∗ i as shown in Eq. (1.77) and they are related by

dx = FdX, dx∗ ( = Qdx = QFdX) = F∗dX

(4.1)

(4.2)

and consequently, we have

F* = QF

(4.3)

It is known from Eq. (4.3) that the deformation gradient F transforms as the second-order tensor between two fixed coordinate systems, but it would be observed as if it were a vector when a rigid-body rotation is superposed. In other

4.2 Influence of Rigid-Body Rotation on Various Mechanical Quantities

113

words, it obeys the objective transformation as the first-order tensor for the superposition of rigid-body rotation. It is called the two-point tensor because it is described using the two bases in the initial and the current states. Substituting Eq. (4.3) into Eqs. (2.15)-(2.16) and (2.41), the following relations hold for various quantities describing a deformation.

R* = QR

(4.4)

U∗ = U, C∗ = C (F∗T F∗ = (QF)T QF = FT QT QF = FT F)

(4.5)

T T T T V∗ = QVQT , b∗ = Q bQT (F∗F∗ = QF(QF) = QFF Q ) (4.6)

E∗ = E , e ∗ = QeQT

(4.7)

Noting the relation •

•

•

•

(QF)• (QF) −1 = (QF + QF)F −1Q −1 = Q(FF −1 − QT Q)QT •

−

1 and Eq. (4.3), it holds for the velocity gradient L ( ≡ FF ) that

L∗ = Q(L − Ω)QT = QLQT + Ω where

(4.8)

Ω and Ω are given by • • ⎫ Ω ≡ QT Q, Ω ≡ Q ri Qrj ei ⊗ e j ⎪ ⎪ ⎬ • •T Ω ≡ Q Q , Ω ≡ Qir Q jr e i ⊗ e j ⎪ ⎪⎭

(4.9)

and they are related by •

Ω ≡ Q QT = − QΩQT where

(4.10)

•

Q is given by • Q = er ⊗ e•∗r

(4.11)

Substituting Eq. (4.11) into Eq. (4.9) 1 , we have

Ω ≡ e•*r ⊗ e*r

(4.12)

from which it is obtained that

e• *i = Ω e*i It is seen from Eq. (4.13) that

Ω is the spin of the base {e∗i } .

(4.13)

114

4 Objectivity and Corotational Rate Tensor

The substitution of Eq. (4.8) into Eqs. (2.73) and (2.74) yields the following relations.

D∗ = QDQT W∗ = Q(W − Ω)Q T = QWQ T + Ω

(4.14) (4.15)

The conclusions concerning the influence of rigid-body rotation and following from Eqs. (4.5)-(4.15) are: 1) The right Cauchy-Green deformation tensor C and the Green strain tensor E are observed to be unchangeable, 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 obey the transformation rule of second-order tensors. 2) The strain rate tensor D obeys the transformation rule of the second-order tensor. On the other hand, the velocity gradient tensor L and the continuum spin tensor W are directly subjected to the influence of rate of rigid-body rotation. The following relations hold for stress tensors described in Chapter 3.

σ ∗ = QσQT (t ∗ = Qt = Qσn = QσQT n ∗ = σ ∗n ∗)

τ ∗ = Q τ QT

(4.16) (4.17)

Π ∗ = QΠ (Π ∗ = Jσ∗F∗−T = JQσQT (QF) −T = JQσQT Q −T F −T = QJσF −T = QΠ) (4.18)

S∗ = S (S∗ = F ∗ 1Π ∗ = (QF )−1 (QΠ ) = F −1QT QΠ = F −1Π = S) −

(4.19) Then, the Cauchy stress tensor σ and the Kirchhoff stress tensor τ are observed as the second-order tensor. On the other hand, the nominal stress tensor Π is observed as the first-order tensor which is the two-point tensor and the second Piola-Kirchhoff stress tensor S is observed as the scalar under the superposition of rigid-body rotation.

4.3 Rate of State Variable and Corotational Rate Tensor It was stated in 1.15 that the material-time derivatives of state variables pursuing the material particle must be used in rate-type constitutive equations. In addition, as described in 4.1 for the stress rate tensor, if the material-time derivative of physical

4.3 Rate of State Variable and Corotational Rate Tensor

115

quantity observed in the fixed coordinate system is adopted in a constitutive equation, the irrational prediction is resulted. Instead of the material-derivative, considering the rate of the quantity which is observed not only moving but also rotating with the material, we would have to adopt its translation to the fixed coordinate system describing the constitutive equation. The explicit formulation for this quantity will be shown below (Hashiguchi, 2007). Consider the transformation of the material-time derivative of a state variable obeying the objective transformation (1.59) or (1.61). Their material-time derivative reads: •

• T ∗p1 p2 ⋅⋅⋅ pm = Q p1q1 Q p2 q2

+ Q p1q1 Q p2q2 •

• • •

•

• • •

Q pm qm Tq1q2 ⋅⋅⋅qm + Q p1q1 Q p2q2 • • • Q pm qm Tq1q2 ⋅⋅⋅qm + • • •

•

Q pm qm Tq1q2 ⋅⋅⋅qm + Q p1q1 Q p2q2

• • •

•

Q pm qm T q1q2 ⋅⋅⋅qm (4.20)

•

•

T p1 p2 ••• pm = Q q p Q q p • • • Q qm pm Tq q ••• q + Qq p Q q p • • • Qq m pm T ∗q1q ••• qm + • • • m 2 1 1 2 2 1 2 1 1 2 2

+ Qq1 p1 Qq

•

T ∗q1q2 ••• qm + Qq1 p1 Qq2 p2

p • • • Q qm pm

2 2

•

• • • Qq p T ∗ m m q1q2 ••• qm

(4.21) Noting

the

•

•

•

Q pi qi = δ pi s Q s qi = Q pit Qst Q s qi

relation

• = − Q pit Q st Qsqi = − Q pit Ωt qi and replacing t → qi , qi → ri , then Eqs. (4.20)

and (4.21) can be rewritten as follows: •

T ∗p1 p2 ⋅⋅⋅ pm = Q p q Q p q 1 1 2 2 •

T p1 p2 ••• pm = Qq p Qq p 1 1 2 2

•

Q pm qm (T q1q2 ⋅⋅⋅qm − Ωq r Tr q ⋅⋅⋅q − Ωq r Tq r ⋅⋅⋅q 2 2 11 1 2 1 2 m m − • • • − Ωq rmTq q ⋅⋅⋅r ) (4.22) 1 1 2 m • • •

• • • • Qq p (Tq∗q ••• q 1 2 m m m

− Ω q1 r1Tr∗q2 ••• q − Ωq r Tq∗r ••• qm 1 2 2 1 2 m

− • • • − Ω qm r m Tq∗1q2⋅⋅⋅rm )

(4.23)

It is known from Eqs. (4.20)-(4.23) that the material-time derivative cannot be •

adopted in constitutive equations, since the components T p1 p2 ••• pm in the fixed •∗

coordinate system changes even when the components T p1 p2 ⋅⋅⋅ pm observed in the coordinate system rotating with the material does not change. Eqs. (4.20)-(4.23) are expressed for scalar S , vector v and second-order tensor T in symbolic notation as follows: •

•

S∗ = S v• ∗ = Q ( v• − Ω v) , v• = QT ( v• ∗ − Ω v ∗)

(4.24) (4.25)

116

4 Objectivity and Corotational Rate Tensor

•

•

•

T ∗ = Q ( T − Ω T + TΩ) QT , T = Q T (T ∗ − ΩT∗ − T ∗ Ω )Q (4.26) •

,

Now, consider the tensor T having the components obtained using the inverse transformation from the components observed by the coordinate system rotating with the material. Then, using Eq. (4.22), it is represented by ,

T p1 p2 ••• pm = Qq1 p1 Qq2 p2 = Qq1 p1 Qq2 p2

• • • Qq p m m

•

T ∗q1q2 ••• q m

• • • Qq p Q Q q1t1 q2 t2 m m

• • •

Qqmt m

•

(T t1t2 ⋅⋅⋅tm − Ωt r Tr t ⋅⋅⋅t − Ωt2 r Tt r ⋅⋅⋅q − • • • − Ω t rmTt t ⋅⋅⋅r ) m m m 11 1 2 1 1 2 2 1 2

= δ p1t1δ p2t 2 • • • δ pmt m •

(T t1t2 ⋅⋅⋅tm − Ωt r Tr t ⋅⋅⋅t − Ω t2 r Tt r ⋅⋅⋅t m − • • • − Ω t rmTt t ⋅⋅⋅r ) 11 1 2 m m 1 12 2 1 2 •

= T p1 p2 ••• pm − Ω p1r1 Tr1 p2 ••• pm − Ω p2r2T p1r2 ••• pm − • • • − Ω pm rmTp p2 •••rm 1 (4.27) While various spins describing explicit rotational rates of material would be assumed, denoting them by the symbol ω , let the following corotational rate

D

,

tensor T be introduced, following the form of

T in Eq. (4.27), i.e.

D • T p1 p2⋅⋅⋅ pm = T p1 p2⋅⋅⋅pm − ω p1r1 Tr1 p2⋅⋅⋅ pm − ω p2 r2T p1r2⋅⋅⋅ pm − • • •

− ω pm rm Tp1 p2⋅⋅⋅rm

(4.28)

which must obey the objective transformation rule

D T *p1 p2⋅⋅⋅ pm = Q p1q1 Q p2 q2

• • •

D Q pm qm T q1q2⋅⋅⋅qm

(4.29)

between an arbitrary and the reference coordinate systems with the bases {e∗ i } and {ei } . Thus, substituting Eq. (4.28) into Eq. (4.29) and noting Eq. (4.22), it must hold that •

T *p1 p2⋅⋅⋅pm − ω *p1r1 Tr1*p2⋅⋅⋅pm − ω *p2 r2T p*1r2⋅⋅⋅pm − • • • − ω*pmrm Tp*1 p2⋅⋅⋅rm = Q p1q1 Q p2 q2

• • •

Q pm qm

•

(T q1q2 ⋅⋅⋅qm − Ωq r Tr q ⋅⋅⋅q − Ωq2 r Tq r ⋅⋅⋅q − • • • − Ωq rmTq q ⋅⋅⋅r ) m m m 11 1 2 1 2 1 1 2 2 −ω *p1r1 Qr1q1Q p2 q2

• • •

Q pm qm Tq1q2⋅⋅⋅qm − ω *p2 r2 Q p1q1Qr2 q2

• • •

Q pm qm Tq1q2⋅⋅⋅qm

4.3 Rate of State Variable and Corotational Rate Tensor

− • • • − ω*pmrmQr1q1Q p2q2 = Q p1q1 Q p2q2

• • •

117

Q pm qm Tq1*q2⋅⋅⋅qm

Q pm qm

• • •

•

(T p1 p2⋅⋅⋅ pm − ωq1r1 Tr1q2⋅⋅⋅qm − ω q2 r2Tq1r2⋅⋅⋅qm − • • • − ωqm rm Tq1q2⋅⋅⋅rm) from which it obtained that

Q p1q1 Q p2q2

• • •

Q pm qm (Ωq r Tr q ⋅⋅⋅q + Ωq2r Tq r ⋅⋅⋅q + • • • + Ωqm rmTq q ⋅⋅⋅r ) 11 1 2 1 2 1 2 m m m 2

+ ω *p1r1 Qr1q1Q p2 q2

• • •

Q pm qm Tq1q2⋅⋅⋅qm + ω *p2 r2 Qp 1q1Q p2r2

+ • • • + ω*pmrmQ p1q1Q p2q2 = Q p1q1 Q p2q2

• • •

• • •

• • •

Q pm qm Tq1r2⋅⋅⋅qm

Q pm rm Tq1*q2⋅⋅⋅qm

Q pm qm

(ωq1r1 Tr1q2⋅⋅⋅qm + ω q2 r2Tq1r2⋅⋅⋅qm + • • • + ωqmrm Tq1q2⋅⋅⋅rm) which may be arranged into the form

ω *p1r1 Qr1q1Q p2q2

• • •

Q pm qm Tq1q2⋅⋅⋅qm

= Q p1q1 Q p2 q2

• • •

= Q p1α Q p2 q2

• • •

Q pm qm (ωq1r1 − Ωq r )Tr1q2⋅⋅⋅qm 11

Q pm qm (ωα q1 − Ωα q1)T q1q2⋅⋅⋅qm

= Q p1α δq1β Q p2 q2

• • •

Q pm qm (ωαβ − Ωαβ )T q1 q2⋅⋅⋅qm

= Q p1α Qr1 β Qr1q1Q p2q2

• • •

Q pm qm (ωαβ − Ωαβ )T q1q2⋅⋅⋅qm

= Q p1α Qr1 β (ωαβ − Ωαβ )Qr1q1Q p2 q2 Eventually,

• • •

Q pm qm T q1q2⋅⋅⋅qm

ω must obey the following transformation rule. ω∗ = Q(ω − Ω)QT , ωij∗ = Qir Q js (ωrs − Ωrs )

The following equations for scalar hold from Eqs. (4.28).

S , vector v and the second-order tensor T

D • D S = S = S∗

(4.31)

vD = v• − ω v ; vD ∗ = Q vD ; vD = QT vD ∗ D

•

D

(4.30)

D

D

(4.32)

D

T = T − ωT + Tω; T∗ = Q T QT , T = QT T ∗ Q

(4.33)

118

4 Objectivity and Corotational Rate Tensor

Now, introduce the symbols for the coordinate transformation.

(Q aT b) p1 p ⋅⋅⋅ pm ≡ Q p q Q p q • • • Q p q Tq q q ⎫ 2 m m 1 1 1 2 ⋅⋅⋅ m ⎪ 2 2 ⎬ T (Q a Tb) p1 p2⋅⋅⋅ pm ≡ Qq p Qq p • • • Qq p Tq1q qm ⎪ 2 ⋅⋅⋅ m m 1 1 2 2 ⎭ Then, the following expressions for an arbitrary tensor and (4.29).

T∗ = Q aT b , and

D

D

T∗ = Q a T b ,

D

T hold from Eqs. (1.59)

T = Q T a T ∗b D

D

(4.34)

•

(4.35)

D

•

( T∗ = T ∗, T = QT a T∗ b )

T = QT a T ∗b

(4.36) The tensor is calculated by the following time-integration by Eq. (4.28), (4.32) and (4.33).

Tp1 p2⋅⋅⋅pm = (TD p p p + ω p r T ∫ 1 2⋅⋅⋅ m 1 1 r1 p2⋅⋅⋅pm + ω p2 r2T p1r2⋅⋅⋅pm + • • • + ω pmrm Tp1 p ⋅⋅⋅rm ) dt 2

(4.37)

i.e.

v = ∫ ( vD + ω v ) dt

(4.38)

D T = ∫( T + ωT − Tω ) dt

(4.39)

D D T p1 p2⋅⋅⋅pm or vD or T by the constitutive relation. , D While the corotaional rate T in Eq. (4.28) following the tensor T in Eq. (4.27)

after calculating

is introduced above, note that various objective corotational rate have been

D

proposed, which do not belong to T , as will be described in 4.5 for stress rate. When the continuum spin W in Eq. (2.74) is adopted for the material spin

ω,

D the corotational rate tensor T is called the Jaumann rate (Jaumann, 1911) and then : let it be designated by the symbol T , i.e. :

•

T p1 p2⋅⋅⋅pm = T p1 p2⋅⋅⋅pm − Wp1r1 Tr1 p2 ••• pm − Wp2 r2 T p1r ⋅⋅⋅pm − • • • 2

− W pmrmTp1 p ⋅⋅⋅r 2

m

which is represented for the second-order tensor as

(4.40)

4.4 Transformation of Material-Time Derivative :

119

•

T = T − WT + TW

(4.41)

The spin W obeys the transformation rule (4.30) as shown in Eq. (4.15). Furthermore, the corotational rate tensor adopting the relative spin •

Ω R ( = R RT ) in Eq. (2.78) is called the Green-Naghdi rate tensor (Green-Naghdi, 1965) or Dienes rate tensor (Dienes, 1979) and is represented by ⊕

T , i.e. •

⊕

T p1 p2⋅⋅⋅ pm = T p1 p2⋅⋅⋅ pm − ΩpR1r1 Tr1 p2 ••• pm − Ω pR2r2 T p1r ⋅⋅⋅pm − • • • 2 −Ω pRm rmTp1 p ⋅⋅⋅r 2

(4.42)

m

which is represented for the second-order tensor as ⊕

•

T = T − ΩR T + TΩR Note that

(4.43)

Ω R obeys Eq. (4.30) as known from •

•

•

Ω R∗ = R∗ R∗T = (QR )• (QR )T = (Q R + Q R )RT QT •

•

•

= (Q R + Q R )RT QT = Q QT + QΩ R QT = Q(Ω R − Ω)QT

(4.44)

noting Eqs. (4.4) and (4.9). When the deformation can be ignored, all the material spins in the corotational rate described above mutually coincide as described in Eq. (2.88). Here, they are determined solely by a geometrical change of outward appearance, which is independent of individual properties of given material. On the other hand, the spin which influences on the mechanical response is the spin of substructure (microstructure) in the material. For that reason, we must adopt the corotational rate with the material spin which depends on the material property reflecting a microstructure and the deformation process. Generally speaking, the spin of the microstructure is not so large as that given by the continuum spin. An explicit form of the spin of substructure in the elastoplastic deformation will be described in Chapter 12. 4.4 Tra nsfor matio n of Material-Time Derivative

4.4

Transformation of Material-Time Derivative of Scalar Function to Its Corotational Derivative

4.4 Tra nsfor matio n of Material-Time Derivative

As in the plastic constitutive equation described in the later chapters, one takes first the material-time derivative of yield condition as a scalar function and must then transform it to the corotational derivative to formulate the plastic strain rate. Here, needless to say, any scalar function and its material-derivative are observed to be identical to each other independent of coordinate systems and thus it would be

120

4 Objectivity and Corotational Rate Tensor

expected that the material-time derivative is directly transformed to the corotational derivative as shown in Eq. (4.31). On the other hand, the explicit mathematical process by which the material-time derivative of scalar function is transformed to the form that all the involved material-time derivatives are replaced to the corotational derivatives obeying Eq. (4.36) is shown below (Hashiguchi, 2007). Here, adopting the symbol

Tr(ST) ≡ S p1 p 2⋅⋅⋅pmT p1 p 2⋅⋅⋅pm for arbitrary tensors S and T , the scalar-valued tensor function of arbitrary tensors

(4.45)

f ( A , B, • • • )

A, B, • • • is described as shown below.

•

f ( A, B, • • • ) = Tr(

• ∂f ( A, B, • • • ) • + Tr( ∂f ( A, B, • • • ) B )+• • • A) ∂B ∂A

(4.46) Here, because f ( A, B, • • • ) is the isotropic function of the arguments A, B, • • • , it holds that

f ( A, B, • • • ) = f ( A∗ , B∗ , • • • ) ⎫⎪ ⎬ • • f ( A, B, • • • ) = f ( A∗ , B∗ , • • • ) ⎪⎭

(4.47)

where the following pertains. •

f ( A∗ , B∗ , • • • ) = Tr(

∂ f ( A ∗ , B∗ , • • • ) • ∗ ∂ f ( A ∗ , B∗ , • • • ) • ∗ A + B )+• • • ) ( Tr ∂B ∗ ∂A ∗

c ∂ f ( A , B, • • • ) f • c f• g A∗ ) + Tr(Q d ∂ f ( A, B, • • • ) g B = Tr(Q dd gh de gh ∗ ) + • • • ∂A ∂B e = Tr (

∂f ( A, B, • • • ) T • ∗ ∂f ( A, B, • • • ) T • ∗ Q a A b) + tr( Q a B b) + • • • ∂A ∂B

=Tr(

∂f ( A, B, • • • ) D ∂f ( A, B, • • • ) D A) + T r( B) + • • • ∂A ∂B

Equations (4.35), (4.36) and the following equation for arbitrary tensors in the same order are considered in the derivation of Eq. (4.48).

(4.48)

T and S

4.4 Transformation of Material-Time Derivative

121

T r(Q a Tb S) = (Q p1q1 Q p2 q2 • • • Q pm qmTq1q2⋅⋅⋅qm ) S p1 p 2⋅⋅⋅pm = Tq1q2⋅⋅⋅qm (Q p1q1 Q p2 q2 • • • Q pm qm S p1 p 2⋅⋅⋅pm ) = Tr(T QT aS b )

(4.49)

u, v , it holds that u∗ • v∗ = Q u • v∗ = u • QT v∗ , and if they are the second-order tensors U, V , it holds that t r( U∗V∗ ) = t r(QU QT V∗) = t r( U QT V∗Q) . Here, if T, S are vectors

The following expression is obtained from Eqs. (4.47)2 and (4.48). •

f ( A, B, • • • ) = Tr(

∂f ( A, B, • • • ) D ∂f ( A, B, • • • ) D A) + Tr( B) + • • • ∂A ∂B

(4.50) Therefore, the material-derivative of scalar function is transformed directly to the corotational derivative. Here, if function f involves only a single second-order tensor

A , the

transformation to the corotational derivative can be verified using the following equation based on Eq. (1.192), noting that

tr( An ω) = tr(ωAn ) .

∂f ( A) ( Aω − ωA)} = tr{(φ0I − φ1A −φ 2A 2 )( Aω − ωA)} = 0 tr{ ∂A (4.51) Furthermore, for the case in which two tensor variables are involved, the transformation to the corotational derivative is verified using Eq. (4.51) if they are replaced to a single variable, as in the kinematic hardening of metals and the rotational hardening of soils, which will be described in later chapters. However, in the general case for which function f involves plural tensor variables, it consists of many invariants made by them. The scalar function involving two independent tensor variables as the simplest example consists of a lot of scalar variables in Eq. (1.198) in general. Therefore, one has the inequality

Tr(

∂f ( A, B, • • • ) • ∂f ( A, B, • • • ) D A ) ≠ Tr( A) ∂A ∂A

for individual derivative terms. Then, the simple proof by individual transformation of each variable is inapplicable in this case but proofs for the transformation to the corotational derivative are necessary for all invariants including tensor operations between tensors. They usually require quite complicated calculations, while these would be the limit of itself. On the other hand, the mathematical process described

122

4 Objectivity and Corotational Rate Tensor

above gives rise to the general proof for the transformation of the material-time derivative of scalar function to the corotational derivative without such complicated processes.

4.5 Various Objective Stress Rate Tensors Various stress rate tensor with objectivity are shown below. The following Jaumann rate and Green-Naghdi (Dienes) rate of Cauchy stress tensor based on Eqs. (4.40) and (4.42) are often adopted in constitutive equations. :

•

σ ≡ σ− W σ +σ W

(4.52)

σ ≡ R (R T σ R )• RT = σ• − Ω Rσ + σΩ R ⊕

(4.53)

from which various objective stress rates are obtained as follows: •

: • ( J σ ): J σ: + J σ − Wτ + τW τ τ ≡ (= )= J = J τJ J J :

:

:

= σ + σ trL = σ + σ trD Δ σ ≡ Lv (σ ) = F( S )• F T = F(F −1σF −T )• FT J • −1

• −T

•

= F (F σF −T + F −1 σ F −T + F −1σ F •

−1

T •T

•

•

(4.54)

•

• −T

)F T = F F −1 σ + σ• + σ F •

−1

•

FT

−1

= − F F σ + σ − σ F − F = − F F σ + σ − σ (F F ) T

= σ• −Lσ −σLT Δ

σJ ≡

Lv (J σ) J

(4.55)

•

F S F T F(J F −1σF −T )• F T = = J J • −1

•

•

•

− F (J F −1σF −T + J F σF −T + J F −1 σ F −T + J F −1σ F T )F T = J

•

• −1 • • J = J σ + (F F )σ + σ + σ (F F −1)T

•

•

−1 −1 T • = σ − (F F )σ − σ − (F F ) + σ tr D = σ• − Lσ − σ LT + σ tr D :

= σ − Dσ − σD + σ tr D

(4.56)

4.5 Various Objective Stress Rate Tensors

123

•

τ ≡ Lv (τ ) = FSFT = F(F −1τF −T )• FT • −1 • −T • −1 • −1 • • = F F τ + τ + σ F FT = ( − F F ) τ + τ + σ ( − F F ) T • T = − Lτ + τ − τ L Δ

•

•

= −(D + W ) τ + τ − τ (D + W)T = τ − Wτ + τW − Dτ − τD :

= τ − Dτ − τ D

(4.57) •

∇

σ ≡ Lv (σ ) = F −T c σ F −1 = F −T (F T σF )• F −1 •

•

•

= F −T (F T σF + F T σ F + F T σ F )F −1 •

= σ +LTσ + σ L

(4.58)

−1

−T

)• F T (cf. Marsden

In the above equations, the symbol Lv (T) ≡ F (F TF and Hughes, 1983 for general definition) designates the Lie derivative, i.e. the T push-forward operation FTF of the material-time derivative of the pull-back operation of F −1TF −T of the contravariant tensor. In addition, the symbol Lv (T) ≡ F −T (FT TF)• F − 1 designates the push-forward operation F −T TF − 1 of the material-time derivative of the pull-back operation of FT TF of the covariant tensor. The objective stress rate tensors described above are listed in Table 4.1. Table 4.1. Various stress rate tensors :

•

Jaumann rate of Cauchy stress : σ ≡ σ − w σ + σ w Green - Naghdi or Dienes stress rate : σ ≡ R (R T σR )• RT = σ• − Ω R σ + σΩ R ⊕

:

: : Jaumann rate of Kirchhoff stress : τ J ≡ τ = σ + σ trD J Δ S • T Oldroyd stress rate : σ J ≡ Lv (σ) = F( ) F = F (F −1 σF −T )• F T = σ• − L σ − σLT J

Δ

Truesdell stress rate : σ ≡

Lv (J σ) J

•

⊗ F S FT −1 − • = = F (F τF T ) F T = S + σ tr D J

:

= σ − Dσ − σD + σ tr D •

:

Truesdell rate of Kirchhoff stress : τΔ ≡ Lv (τ ) = F S F T = τ − Dτ − τD •

∇

−1 Convected or Cotter - Rivlin stress rate : σ ≡ Lv (σ ) = F −T c σ F = F −T ( F T σF )• F −1

•

= σ + LTσ + σ L

(4.59)

124

4 Objectivity and Corotational Rate Tensor

The corotational stress rate tensors obtained directly from the basic equation (4.28) of corotational tensor fulfill the following relation, noting tr(TTω) = tr(TωT) . : ⊕ t r(σ 2 )• = 2t r(σ σD ) = 2t r(σ σ ) = 2t r(σ σ )

(4.60)

In addition, the corotational rate of deviatoric tensor becomes deviatoric leading to

t r{(σ* ) : } = t r{(σ* ) ⊕ } = 0

(4.61)

Other objective stress tensors do not possess these properties. Although the objective rates are shown above for the stress rate, the objective rate must be adopted also for rates of internal variables.

4.6 Work Conjugacy Designating the volume in a specific region of material in the initial and the current • configurations as V and v , respectively, the work rate W done in this region is given from Eqs. (2.30) (2.41) (2.69) (2.73) (2.105) (3.11) (3.17) and (3.21) as follows:

，

，

，

，

，

，

•

W = ∫ tr(σ D) dv = ∫ tr (σ D ) JdV = ∫ tr( τD) dV v

v

(4.62)

v

• W = ∫ tr(σ D) dv = ∫ 1 {tr(σL) + t r(σLT )}dv v v2

= ∫ 1 {tr(σT LT ) + t r(σLT )}dv = ∫ tr(σLT ) dv v2 v •

•

•

= ∫ tr{σ (F F−1 )T }JdV = ∫ tr( JσF −T F T ) dV = ∫ tr(ΠF T ) dV V

V

V

•

•

(4.63)

•

W = ∫ tr(σD)dv = ∫ tr(σF−T EF−1)dv = ∫ tr( F−1σF−T E)dv v

v

•

v

•

•

= ∫ tr( F −1σ F −T E)JdV = ∫ tr( J F −1σ F −T E)dV = ∫ tr(SE)dV V V V (4.64) •

• • W = ∫ tr(S E)dV = 1 ∫ tr(SC)dV = 1 ∫ tr{S(U 2 )•}dV V 2 V 2 V

• • • • = 1 ∫ tr{S(U U + U U)}dV = 1 ∫ tr{(SU + US) U}dV = ∫ tr(Ζ U)dV V 2 V 2 V

(4.65)

4.6 Work Conjugacy

125

where

Ζ ≡ 1 (SU + US) 2

(4.66)

which is called the Biot stress tensor. The rate of work done per unit volume element in the reference configuration is given as •

•

•

•

• = t r( τD) = t r(Π FT ) = t r(S E) = t r(SC /2) = t r(Ζ U) w

(4.67)

The observation follows directly from the following equation (see Fig. 4.1). •

•

•

• = dfr • e r = Πe r • (Fe r )• = Πer • F e r = er • FT Πe r w •

•

•

= (FT Π ) rr = t r(FT Π ) = t r(Π FT )

(4.68)

dA = 1 . The unit cubic cell having the unit orthonormal outward-normal vectors e r deforms to the current cell having the outward-normal vectors er which are changed from e r , while the tractions df r noting Eqs. (2.10), (3.14) and (4.67) with

are applied to the cell in Fig. 4.1. The work rate influencing the constitutive property is not the one done for a current unit volume but the one done for a reference volume element, i.e. a unit volume element at the initial state of deformation. The pairs of stresses and strain rates (or rates of deformation gradient) shown in Eq. (4.67) are called the work-conjugate pair. Formulation of a constitutive equation relating them pertinently is necessary for the description of finite deformation with finite rotation.

df 2

F

e2

df 3

e−2

− e3

e3

e2

e1

e1

− df1

df1

e−1

−df 3

−df2 Fig. 4.1 Variation of the initial unit cell due to the deformation

Chapter 5

Elastic Constitutive Equations 5 Ela stic Constitutive Equatio ns

Elastic deformation is induced by the reversible change of distances between material particles without a mutual slip between them. They therefore exhibit high stiffness. Elastic constitutive equations are classifiable into several types depending on their levels of reversibility. As preparation for the study of elastoplasticity after the next chapters, they are explained in this chapter.

5.1 Hyperelasticity The elastic material which has the strain energy function so that the one-to-one correspondence between stress and strain exists and the work done by the stress is independent of the loading path is called the hyperelastic material or Green elastic material. The constitutive equation of hyperelastic material is given as the following relation between the first Piola-Kirchhoff Π and the deformation gradient F .

Π = ρ0

∂ψ (F) ∂F

(5.1)

under the condition [∂ψ / ∂F ]F=R = 0, ψ ( R ) = 0 , where ψ (F) is the specific strain energy (per unit mass). Substituting Eq. (5.1) into Eq. (4.63) or (4.68), the mechanical work done by a surface loading on an enclosure of certain region of the initial volume V is described as •T ρ0 t r( ∂ Π F t r( ) dt ] dV = [ [ ∫ V ∫0 V ∫0

W =∫

t

t

=

t•

ψ (F)

•

T ∂ F F )dt ]dV

∫V [∫0 ψ (F)dt ]dV = ∫V ρ0ψ (F) dV

(5.2)

Consequently, the work done in a specific region is independent of the loading path and therefore ψ has the status of the potential energy. The energy dissipation is not induced during the stress or strain cycle. Substituting Eq. Eq. (5.1) into Eqs. (3.11), (3.17), (3.21), we obtain various expressions of the hyperelasticity by the deformation gradient as follows: K. Hashiguchi: Elastoplasticity Theory, LNACM 42, pp. 127–133. springerlink.com © Springer-Verlag Berlin Heidelberg 2009

128

5 Elastic Constitutive Equations

ψ( ) ψ( ) σ = 1 ρ0 ∂ F FT = ρ ∂ F FT J

∂F

∂F

τ = ρ0 ∂ψ (F) FT ∂F

∂ψ (F) Π = ρ0

(5.3)

∂F ∂ψ (F) S = ρ0 F −1 ∂F Furthermore, noting

∂ = ∂ ∂CPQ = ∂ ∂FrP Fr Q = ∂ ( δ riδ PA FrQ + FrPδ riδ QA ) ∂FiA ∂CPQ ∂FiA ∂CPQ ∂FiA ∂CPQ

=

∂ ∂ F = 2F ∂ F + iP ∂C AQ iQ ∂CPA iP ∂CPA

(5.4)

it holds that

∂ψ = 2 F ∂ψ = F ∂ ψ ∂F ∂C ∂E

⎫ ⎪ ⎪ ∂ψ = 1 − 1 ∂ψ = 1 ∂ ψ ⎪ F ⎬ ∂C 2 ∂F 2 ∂E ⎪ ∂ψ = − 1 ∂ψ = 2 ∂ψ ⎪ F ∂F ∂C ⎭⎪ ∂E

(5.5)

Then, substituting Eq. (5.5) into Eq. (5.3), the hyperelasticity is expressed in terms of the Green strain E as follows:

σ = 2 ρ F ∂ ψ (C) FT = ρ F ∂ ψ (E) FT

∂C ∂E ψ( ψ( τ = 2 ρ0F ∂ C) FT = ρ0F ∂ E) FT ∂C ∂E

ψ (C) ψ (E) Π = 2 ρ0F ∂ = ρ0F ∂ ∂C ∂E S = 2 ρ0

∂ ψ (C) = ρ ∂ ψ (E ) 0 ∂C ∂E

(5.6)

5.1 Hyperelasticity

129

It holds from Eq. (1.192) for the isotropic material that

∂ ψ (E) = φ0E I + φ1E E + φ2E E 2 ∂E where φ0E , φ1E , φ2E are the functions of invariants of following equation for the linear elastic material.

(5.7)

E . Eq. (5.7) reduces to the

∂ ψ ( E) = aE ( t rE )I + 2 b E E ∂E

(5.8)

where aE , b E are the material constants. The time-differentiation of Eq. (5.6) 4 leads to •

S = ρ0

∂ 2ψ (E) • E ∂E∂E

(5.9)

Substituting Eq. (2.106) and using the Truesdell rate of Kirchhoff stress Eq. (4.57), Eq. (5.9) is rewritten as

Δ

τ

in

2 Δ ∂ 2ψ (E ) Δ Dkl ) τ = ρ0 F ∂ ψ (E ) FΤ D FFΤ (τ ij = ρ0 FiA FjBFkC FlD ∂E ∂E

∂E∂E

Here,

Δ

τ

AB

CD

(5.10) is related to the Jaumann rate of Cauchy stress as Δ

:

τ = (det F ) (σ −D σ − σD + σ trD)

(5.11)

The relation between the Jaumann rate of Cauchy stress and the strain rate is given by

ρ

σ = det0 F F ∂ :

2

ψ (E)

∂E∂E

FΤ D FFΤ + D σ + σD − σ trD

(5.12)

which is expressed as :

 σ = ED where the elastic modulus tensor

(5.13)

 is given by E

ρ0 ∂ 2ψ (E) E ijkl ≡ det F FiA ∂E ∂E FkC FlD FjB + Σ ijkl − σijδ kl AB CD

(5.14)

130

5 Elastic Constitutive Equations

with

Σ ijkl ≡ 12 (σ ik δ jl + σ ilδ jk + σ jk δ il + σ jlδ ik ) (Σ ijkl =Σ klij =Σ jikl =Σ ijlk ) (5.15)

5.2 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 involves six strain components. The equation of six strain components fulfills the condition of complete integration leading to the strain energy function, i.e. the hyperelasticity only in special cases. Then, the work done by the stress is generally dependent on the deformation path. For that reason, an energy dissipation is induced during the stress or strain cycle. In the above-mentioned definition, the Cauchy elastic material is described as

σ = f (e)

(5.16)

Equation (5.16) reduces to the following equation by virtue of Eq. (1.202) for the isotropic material.

σ = φ0e I + φ1e e + φ2e e 2 where φ0e, φ1e, φ2e are functions of invariants of linear elastic material, Eq. (5.17) reduces to

(5.17)

e . Furthermore, for an isotropic

σ = a ( t r e )I + 2b e

(5.18)

a, b are the material constants. Limiting the infinitesimal strain leading to ≅ e ε , Eq. (5.18) results in σ = a(t r ε )I + 2b ε (5.19)

where

Here, substituting Eq. (5.19) into Eq. (3.24), the Navier’s equation is obtained as follows:

∂ 2u j ∂ 2ui • • (a + b)∇ (∇ • u) + bΔu + ρ b = ρ v , (a + b) ∂xj ∂x + b ∂x ∂xj + ρbi = ρ v i j i (5.20) noting

∂u ∂u j ∂u ∂{a xk δ ij + 2b 12 ( ∂x i + )} ∂ 2u j ∂ k j ∂ 2u j ∂xi ∂ 2ui +b +b =a x ∂ j ∂xi ∂xj ∂xj ∂xj ∂xi ∂xj

5.3 Hypoelasticity

131

5.3 Hypoelasticity The following material, for which the Jaumann rate of Cauchy stress is related linearly to the strain rate, is referred to as the hyperelastic material by Truesdell (1955). D D (5.21)

σ = HD, σ ij = H ijkl Dkl

where H is the function of stress and internal variables. If Eq. (5.21) exhibits the isotropic rate-linearity, it is described as follows:

σD = L(t rD)I + 2GD

(5.22)

Therein, L, G are the material parameters corresponding to the Lame coefficients in Hooke’s law for the infinitesimal deformation of isotropic linear elastic material. The following relation is obtained from Eq. (5.22), and thus G is called the shear modulus. σD (5.23)

'ij = 2GDij'

where ( )' designates the deviatoric part described in 1.4.2. Equation (5.22) is expressed as follows:

σD = ED, σD ij = Eijkl Dkl

(5.24)

E ≡ LI ⊗ I + 2G I , Eijkl ≡ Lδ ijδ kl + G (δ ik δ jl + δ il δ jk )

(5.25)

where

is the tangent elastic modulus tensor, which hereinafter will be simply called the elastic modulus tensor. Equation (5.25) is further rewritten as shown below.

σD = K (t rD)I + 2GD'

(5.26)

K ≡ L+ 2G 3

(5.27)

where

Using these elastic moduli, the elastic modulus tensor E is given as

⎫ ⎪⎪ ⎬ = 1 ( 1 − 1 )δ ij δ kl + 1 (δ ikδ jl + δ ilδ jk ) ⎪ 4G 3 3K 2G ⎪⎭

Eijkl = (K − 2 G)δ ijδ kl + G (δ ik δ jl + δ il δjk ), 3

(E−1)ijkl

in the form of Hooke’s law.

(5.28)

132

5 Elastic Constitutive Equations

The following equation is derived from Eq. (5.26) and thus K is called the bulk modulus. • (5.29)

σ m = KDv

where

( ) m designates the mean part described 1.4.2.

Furthermore, the following is obtained from Eqs. (5.23) and (5.29).

D = 1 σ• m I + 1 σD ' , Dij = 1 σ• m δ ij + 1 σD 'ij 3K 2G 3K 2G

(5.30)

from which we have

D D D = −3 ν σ• m I + 1 +ν σ, Dij = −3 ν σ• m δ ij + 1 +ν σ ij E E E E

(5.31)

where

− E ≡ 9 KG , ν ≡ 3K 2G 2(3K + G ) 3K + G

(5.32)

The expressions below hold from Eq. (5.31) in the uniaxial loading process D ( σ ij = 0 for i = j ≠ 1 ).

D11 = 1 σD 11 , D22 = − ν σD 11 → D22 = −ν E E D11

(5.33)

E is the ratio of the axial stress rate to the axial strain rate and is called the Young’s modulus, and ν is the ratio of the magnitude of lateral strain rate to that of axial strain rate and is called the Poisson’s ratio. The isotropic linear hypoelastic material has two independent material parameters, as described above. They are listed in Table 5.1. Table 5.1 Relationships between two independent material constants E ,ν

G,ν

E, G

E, K

G, K

L, G

μ (3L + 2G ) L+G

E

E

2(1 + ν )G

E

E

9 KG 3K + G

G

E 2(1 +ν )

G

G

3EK 9K − E

G

G

K

E 3(1 − 2ν )

2(1 +ν )G 3(1 − 2ν )

EG 3(3G − E )

K

K

L+ 2G 3

ν

ν

ν

E − 2G 2G

3K − E 6K

3K − 2G 2(3K + G )

L 2( L + G )

L

νE (1 +ν )(1 − 2ν )

2Gν 1 −ν

G ( E − 2G ) 3G − E

3K (3K − E ) 9K − E

2 K − 3G

L

5.3 Hypoelasticity

133

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)

．

σD = f (D, σ ), σD ij = fij ( Dkl , σ kl )

(5.34)

fij is the nonlinear function of Dkl and the stress, and for rate-independent deformation it is the homogeneous function of Dkl in degree-one fulfilling fij ( sDkl ) = s fij ( Dkl ) which implies (∂f ij / ∂Dkl ) Dkl = f ij on account of where

Euler’s theorem for homogeneous function (see Appendix 3). While the tree popular elastic materials are described above, the special elastic material, called the Cosserat elastic material, is 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 6

Basic Formulations for Elastoplastic Constitutive Equations 6 Basic Formulations Constitutive Equationsfor Elastoplastic

As described in Chapter 5, the elastic deformation is induced microscopically by the deformations of the material particles themselves, which returns to the initial state if the applied stress is removed. Therefore, it has a one-to-one correspondence to the stress. On the other hand, when the stress reaches the yield stress, slippage between material particles is induced and they do not return to the initial state even if the stress is removed, which leads macroscopically to the plastic deformation. For that reason, one-to-one correspondence between the stress and the strain, i.e. the stress-strain relation, observed in the elastic deformation does not hold in the elastoplastic deformation process. Therefore, one must formulate the constitutive equation as a relation between the stress rate and the strain rate in that process. This chapter presents a description of the basic concept and formulation for elastoplastic constitutive equations within the framework of conventional plasticity (Drucker, 1988) premised on the assumption that the inside of the yield surface is a purely elastic domain for the introductory to elastoplasticity. The unconventional plasticity describing the plastic strain rate induced by the rate of stress inside the yield surface will be described in subsequent chapters.

6.1 Multiplicative Decomposition of Deformation Gradient and Additive Decomposition of Strain Rate 6.1 Mult iplicative Decompos itio n of Deformatio n Gradie nt

Variation in characteristics of the mechanical responses of materials arises from irreversible changes of their internal structures. The changes of internal structures are induced macroscopically by plastic deformation but they must be free from elastic deformation. Therefore, the rigorous decomposition of deformation into the elastic and the plastic deformations is necessary for description of elastoplastic deformation. Strictly speaking, the elastic deformation must be described by the purely elastic, i.e., hyperelastic constitutive equation, in which the elastic deformation returns to an initial state during a stress cycle and the work done during that cycle is zero, as described in Chapter 5. The decomposition of deformation into elastic and plastic deformations is the basic concept of elastoplasticity. On the premise of the definite decomposition of the deformation into the elastic and the plastic deformations as described above, the individual constitutive relations are formulated for these deformations in the elastoplasticity. Then, one must first decompose the current deformation into the elastic and the plastic deformations K. Hashiguchi: Elastoplasticity Theory, LNACM 42, pp. 135–170. springerlink.com © Springer-Verlag Berlin Heidelberg 2009

136

6 Basic Formulations for Elastoplastic Constitutive Equations

explicitly. The elastic deformation is known from the current stress since it is related uniquely to the stress, then the plastic deformation is known as that remainder. This is realized through an explicit process: incorporating the intermediate configuration unloaded obeying the elastic constitutive relation from the current configuration, one regards deformation from the initial to the intermediate configuration as plastic deformation and the deformation from the intermediate to the current configuration as the elastic deformation. Here, note that a constitutive relation concerns with a homogeneous state of deformation. However, residual stresses exist even if a load is removed from the current configuration of a body undergoing an inhomogeneous deformation because a plastic deformation which is not removed by an unloading is induced non-homogeneously in general. Therefore, in order that all material points reach the stress-free state, individual material points must undergo different amount of de-stressing. Then, intermediate configuration can be defined only locally (point-wise). Thus, we suppose conceptually virtual intermediate configurations for each infinitesimal material elements by cutting a material into them. Based on this idea, the multiplicative (or Lee) decomposition of e the deformation gradient F into the elastic deformation gradient F and the p plastic deformation gradient F was proposed by Eckart (1948), advanced by Kroner (1960) and further developed by Lee and Liu (1967), Lee (1969), Mandel (1972), Lubarda and Lee (1981), etc. (see Fig. 6.1) as follows:

F = FeF p

( ∂∂Xx = ∂∂Xx ∂∂XX ,

∂xi ∂xi ∂Xα = ∂ X A ∂X α ∂X A

)

(6.1)

where e ∂xi ∂x ∂X p F e = ∂X (Fiα = ∂X ), F = ∂X α

( F αp A = ∂∂XXα )

(6.2)

A

X represents the position vector of material particle in the intermediate configuration and the mapping is restricted to the class of homogeneous deformation. F is termed the Eulerian–total Lagrangian two-point tensor,

Fe the Eulerian-mobile

p

Lagrangian two-point tensor, F the mobile Lagrangian-total Lagrangian tensor. When the stress reached the current state, different plastic deformations are induced depending on the loading path along which the current stress is realized and the elastic and plastic deformations are induced in mixed order in real unloading process. Further, a purely elastic unloading path does not necessarily exist in a real loading process. However, it is noteworthy that the real existence of purely elastic unloading process is not required since the intermediate configuration is known by the elastic constitutive relation. It holds from Eq. (6.1) with Eq. (1.22) that

detF = det(F e F p ) = detF e detF p

(6.3)

ΔX a , ΔXb , ΔXc in the initial a c state change to Δx a , Δxb , Δx c in the current state and to ΔX , ΔXb , ΔX in On the other hand, presuming that line-elements

6.1 Multiplicative Decomposition of Deformation Gradient

137

X 3 , X 3 , x3

dx

F

x

Fe

dX

X dX

F

X

p

0

X 2 , X 2 , x2

X1 , X1 , x1 Fig. 6.1 Multiplicative decomposition of deformation gradient

the intermediate state, and denoting the initial, current and intermediate volume elements as

V , v and V , respectively, it holds that Δv = (Δx a × Δxb ) • Δxc = ε ijk Δxia Δxbj Δxkc = detΔx

Δx1a Δx1b Δx1c

F1R ΔX Ra F1R ΔX Rb F1R ΔX Rc

a b c F11 F12 F13 ΔX 1 ΔX 1 ΔX 1

= Δx2a Δxb2 Δx2c = F2 R ΔX Ra F2 R ΔX Rb F2 R ΔX Rc = F21 F22 F23 ΔX 2a ΔX 2b ΔX 2c = J ΔV Δx3a Δx3b Δx3c

F3 R ΔX Rc F3 R ΔX Rb F3 R ΔX Rc

F31 F32 F33 ΔX 3a ΔX 3b ΔX 3c

F1eR ΔX Ra F1Re ΔX Rb F1eR ΔX Rc

F11e F12e F13e ΔX 1a ΔX 1b ΔX 1c

Δx1a Δx1b Δx1c

Δv = Δx2a Δxb2 Δx2c = F2eR ΔX Ra F2eR ΔX Rb F2eR ΔX Rc = F21e F22e F23e ΔX 2a ΔX 2b ΔX 2c = J e ΔV Δx3a Δx3b Δx3c ΔX 1a ΔX 1b ΔX 1c

F3eR ΔX Ra F3eR ΔX Rb F3eR ΔX Rc

F31e F32e F33e ΔX 3a ΔX 3b ΔX 3c

F1R ΔX Ra F1R ΔX Rb F1R ΔX Rc

F11 F12 F13 ΔX 1a ΔX 1b ΔX 1c

p

p

p

p

p

p

ΔV = ΔX 2a ΔX 2b ΔX 2c = F2pR ΔX Ra F2Rp ΔX Rb F2pR ΔX Rc = F21p F22p F23p ΔX 2a ΔX 2b ΔX 2c = J p ΔV ΔX 2a ΔX 3b ΔX 3c

F3pR ΔX Ra F3Rp ΔX Rb F3pR ΔX Rc

F31 F32 F33 ΔX 3a ΔX 3b ΔX 3c p

p

p

It obtained from these equations that

J = J eJ p

(6.4)

138

6 Basic Formulations for Elastoplastic Constitutive Equations

where

J ≡ detF = Δv , J e ≡ detF e = Δv , J p ≡ detF p = ΔV ΔV ΔV ΔV

(6.5)

Equation (6.4) engenders the additive decomposition of the logarithmic volumetric strain

εv

into the elastic logarithmic volumetric strain

logarithmic volumetric strain

ε ve

and the plastic

ε vp , i.e. ε v = ε ve + ε vp

(6.6)

where

ε v ≡ ln J = ln Vv , ε ve ≡ lnJ e = ln v , ε vp ≡ lnJ p = ln V

V

V

(6.7)

for a homogeneous deformation. It holds from Eqs. (6.7) that •

•

•

•p

•

e • • p ε• v = JJ = vv , ε• ve = J e = vv − V , ε• v = J p = V J J V V

(6.8)

Substitutions of Eqs. (6.1) into the velocity gradient L in Eq. (2.69) leads to • •p −1 −1 L = F e F e + F e F F p F e −1 = Le + L p

(6.9)

where

⎫ ⎪ ⎪ • • ⎪ ⎪ • −1 ∂ X ∂X = ∂ X ⎬ Lp ≡ F p F p = ∂X ∂X ∂X ⎪ ⎪ • p p e −1 x X ∂ ∂ X ∂ ⎪  e L ≡F L F = ∂X ∂X ∂x ⎪⎭ •e −1 ∂x • ∂X Le ≡ F F e = ( ) ∂x ∂X

(6.10)

 can be decomposed additively into the symmetric Furthermore, L , L , L and skew-symmetric parts as follows: e

p

p

e Le = D + W e ⎫ p p⎪ Lp = D + W ⎬ ⎪ p  p +W  p⎭ L = D

(6.11)

6.1 Multiplicative Decomposition of Deformation Gradient

139

where

⎫ • De = (Le ) s = (F e F e −1 ) s , ⎪ ⎬ •e − 1 e e W = (L )a = (F F e ) a ⎪⎭

(6.12)

• ⎫ ∂ X ),⎪ D = (L ) s = ( F F ) s = ( ∂X s ⎪ ⎬ • ⎪ • −1 W p = (Lp )a = ( F p F p )a = ( ∂ X )a ⎪ ∂X ⎭ p

•p

p

p −1

(6.13)

 p = (L p ) = (F e Lp F e −1 ) = (F e D p F e −1 ) s + (F e W p F e −1 ) s , ⎫⎪ D s s ⎬ 6.14)  p = (L p ) a = (F e Lp F e −1 ) a = (F e D p F e −1 ) a + (F e W p F e −1 ) a ⎪ W ⎭ where ( ) s and ( )a denotes the symmetric and the skew(anti)-symmetric parts, respectively. In the similar manner as in Eq. (2.98) it holds that •

•

(dX • δ X)• = d X• δ X + dX • δ X •

•

•

•

T = ∂ X dX • δ X + dX • ∂ X δ X = [{∂ X + ( ∂ X ) }dX] • δ X ∂X ∂X ∂X ∂X

e −T e −1 = 2 D p dX • δ X ( = 2F D p F dx • δ x )

(6.15) The strain rate D and the spin W are additively decomposed by substituting Eq. (6.9) into Eqs. (2.73) and (2.74) as follows:

 p = De + (F e D p F e −1 ) s + (F e W p F e −1 ) s D = (Le ) s + (L p ) s = De + D

½° ¾  p = W e + ( F e D p F e −1 ) a + ( F e W p F e −1 ) a ° W = (Le )a + (L p )a = W e + W ¿ (6.16)

Furthermore, the following equations hold for the volumetric strain rate from Eqs. (2.73), (6.10), (6.12)-(6.14).

140

6 Basic Formulations for Elastoplastic Constitutive Equations

⎫ ⎪ ⎪ ⎪ • • • ⎪ • (∂X 1∂X 2∂X 3 ) •p = V = ε v ,⎪ Dvp ≡ t rD p = t r ∂ X = ∂ X i = ⎪ ∂X ∂X i V ∂X 1∂X 2∂X 3 ⎬ • • • ⎪  p = t r( ∂ x ∂ X ∂X ) = t r ∂ X = V = ε• vp = D p ⎪ D vp ≡ t rD v ∂X ∂X ∂x ∂X V ⎪ ⎪ • • ⎪ e • p v V e  = − =εv Dve ≡ t rD = t rD − t rD ⎪ v V ⎭

• • (∂x1∂x2∂x3 )• v• • = v = ε v, Dv ≡ t rD = t r ∂ x = ∂ xi = ∂x ∂xi ∂x1∂x2∂x3

(6.17)

from which the additive decomposition holds exactly for the volumetric strain rate, i.e.

Dv = Dve + Dvp

(6.18)

Now, let Q(t ) be a time-dependent rotation superimposed on the current configuration, by which deformation gradient is given as

F∗ = QF

(6.19)

Here, let it be postulated that the initial and the intermediate configurations are fixed so that they are not affected by the superposition of rigid-body rotation (Dashner, 1986). Then, it holds that

F∗ = F e∗F p∗; F e∗ = QF e , F p∗ = F p

(6.20)

The elastic part of the right Cauchy-Green tensor in Eq. (2.30) defined in the ine

termediate configuration, denoted C , is given by Eq. (6.20) as

Ce∗ = Fe∗T F e∗ = (QF e )T QF e = F eT F e ≡ Ce

(6.21)

Therefore, the hyperelastic constitutive equation described by Eq. (5.6) 4 , i.e.

S = 2 ρ0 ∂ ψ (Ce ) / ∂ Ce holds independent of the superposition of rigid-body rotation, noting S∗ = S in Eq (4.19), where S is the second Piola-Kirchhoff e stress defined in the intermediate configuration ( F is replaced to F ), the explicit expression of which will be shown in 6.7. Eq. (6.20) imposes the basis describing elastic deformation, i.e. the intermediate configuration to be unaffected by the rigid-body rotation. Thus, the multiplicative decomposition is required to obey Eq. (6.20) for the fulfillment of objectivity of elastic constitutive equation.

6.1 Multiplicative Decomposition of Deformation Gradient

141

Now, consider the influence of the rigid-body rotation on the main rate variables for the multiplicative decomposition fulfilling Eq. (6.20). The substitution of Eq. (6.20) into Eq. (6.10) leads to

Le∗ = (F e∗ ) • F e∗ = (QF e ) • (QF e ) −1 −1

•

•

e = (Q F e + Q F )F e −1Q −1 • • = Q F e Fe −1 QT + Q QT

= Q(Le − Ω)QT where Ω is the spin of the base (4.12). Then, we have

(6.22)

{e∗i } rotating with a material as shown in Eq.

⎫ ⎪⎪ ⎬ (6.23) e∗ = 1 (Le∗ − Le∗T ) = Q ( W e − Ω)QT ⎪ W 2 ⎪⎭

D e∗ = 1 (Le∗ + Le∗T ) = QDeQT 2

Next, substituting Eq. (6.20) into Eq.(6.10), one has

⎫ Lp∗ = Lp ⎪ ⎬ −1 p p p − 1 p e e p e e e T T 1 − ∗ ∗ ∗ ∗ e = (QF )L (QF ) = QF L F Q = QL Q ⎪ L = F L F ⎭ (6.24) Then, It holds from Eq. (6.13) and (6.14) that

D p∗ = D p , W p∗ = W p

(6.25)

 p∗ = QD  p QT , W  p∗ = Q W  p QT D

(6.26)

 indicate the Consequently, the elastic strain rate De and the plastic strain rate D objectivity. Apart from the exact formulation conforming to the hyperelasticity, let it be postulated that the unloading process to the intermediate configuration is a purely elastic deformation (Lee, 1969) and the elastic deformation is infinitesimal (Dafalias, 1985). Then, it holds in the polar decomposition of elastic deformation p

e

e

e

gradient F = V R that

F e = V e ≅ I, R e = I

(6.27)

142

6 Basic Formulations for Elastoplastic Constitutive Equations

In this case, substituting Eq. (6.27) into Eq. (6.16), the following equation holds.

⎫ ⎪ ⎬ • e e −1 • p p −1 p p e  (W = ( V V )a , W = W = ( F F )a ) ⎪⎭ •

• • p  p = ( F p F p −1 ) ) D = De + D p (De = (V e ) s = V e , D = D s

W = We + W p

(6.28) with an error of first order in small elastic strain. Here, note that the rigid-body rotation is attributed to the variation of F p in direct opposition to the above-mentioned Dashner’s (1986) notion in Eq. (6.20) and that W p differs substantially from the so-called plastic spin (Dafalias, 1985) as will be described in Chapter 12. The additive decomposition of strain rate D into the elastic strain rate

De and the plastic strain rate D p in Eq. (6.28) is adopted in the formulation of elastoplastic constitutive equations with the hypoelasticity for De in the subsequent sections.

6.2 Conventional Elastoplastic Constitutive Equations First, let the elastic strain rate be given by the following hypoelastic constitutive equation from Eq. (5.24). D (6.29) De = E −1 σ Next, the plastic strain rate must be formulated. As described in the beginning, let the conventional elastoplastic constitutive equations (Drucker, 1988) be described below. Now, to formulate the plastic strain rate, consider first the following isotropic yield condition exhibiting the isotropic hardening/softening.

f (σ ) = F ( H )

(6.30)

where F is the function of the isotropic hardening variable H and is called the hardening function. Hereinafter, assume that the yield stress function f (σ ) is the function of stress invariants and is the homogeneous function of σ in degree-one. Therefore, it holds that f (s σ) = s f (σ) (6.31) for the arbitrary positive scalar

s tr(

and

∂ f (σ ) σ ) = f (σ) ∂σ

(6.32)

for sake of Euler’s theorem for homogeneous function in degree-one (see Appendix 3). Then, it holds from Eqs. (6.30) and (6.32) that

6.2 Conventional Elastoplastic Constitutive Equations

143

σ tr( ∂f ( ) σ) σ f (σ ) ∂f ( ) ∂σ N= N= F N = ∂σ tr(N σ ) tr(N σ ) tr(N σ )

(6.33)

where N is the normalized outward-normal of the yield surface (see Appendix 4), which is described as

N≡

∂f (σ) ∂σ

∂f (σ) ∂σ

( N = 1)

(6.34)

Here, the yield surface in Eq. (6.30) retains the similar shape and orientation with respect to the origin of stress space by virtue of homogeneity of function f (σ ) . Taking the material-time derivative of Eq. (6.30) and considering the transformation to the corotational derivative (Hashiguchi, 2007) described in Chapter 4, one has the consistency condition:

tr( ∂

f (σ) D σ) = F' H• ∂σ

(6.35)

where

F ' ≡ dF / dH ( ≥ 0)

(6.36)

•

H is assumed to be the function of stress, internal variables and the plastic strain rate, i.e. • (6.37) H = h (σ, Hi ; D p ) H i (i = 1, 2,

• •• ) signifies internal variables collectively. Here, note that

h has to be the homogeneous function of D p in degree-one fulfilling h (σ, H i ; s D p ) = s h (σ, H i ; D p ) for the rate-independent deformation behavior, while, needless to say, h is a nonlinear equation of

Dijp in general as seen in metals described

later. The substitution of Eq. (6.33) into the consistency condition (6.35) leads to • tr (N σD ) = F' tr(N σ ) H F

(6.38)

Further, assume the plastic flow rule

D p = λ M (||M|| ≠ 1)

(6.39)

where λ is the positive proportionality factor and M is the second-order tensor function of stress and internal variables. Substituting the flow rule (6.39) into the consistency condition (6.38) with Eq. (6.37), one has

144

6 Basic Formulations for Elastoplastic Constitutive Equations

tr(N σD ) = F' tr(Nσ)λ h ( σ, Hi ; M) F from which it is derived that

D

D

σ σ λ = tr(Np ) , D p = tr(Np ) M M

where

M

(6.40)

M p is called the plastic modulus and is given by F ′ h( σ, H ; M ) tr(Nσ ) i F

Mp≡

(6.41)

Substituting Eq. (6.29), (6.40) into Eq. (6.23)1, the strain rate is given by

D = E −1 σD +

D

tr( N σ ) M Mp

(6.42)

The proportionality factor described in terms of strain rate, denoted by instead of λ , in the flow rule (6.39) is given from Eq. (6.42) as follows:

Λ=

tr(NED) M + tr( NEM) p

Λ

(6.43)

Actually, Eq. (6.43) can be derived directly by substituting the flow rule (6.39) into the following consistency condition in terms of the strain rate obtained by substituting Eqs. (6.23), (6.29), (6.33) and (6.37) into the consistency condition (6.38) in terms of stress rate.

tr{NE(D − D p )} = F' tr(Nσ)λ h ( σ, Hi ; M ) F

(6.44)

The inverse relation of Eq. (6.42) is given by using Eq. (6.43) as follows:

⊗ EN ) D σD = ED − M p + NEM EM = (E − M pEM + tr( NEM) tr( ) tr( NED)

σD ij = Eijrs Drs −

N pq E pqmn Dmn Eijrs M rs M + N ab Eabcd M cd p

(6.45)

Using the plastic relaxation modulus tensor

K pr ≡

Kijklr ≡ p

EM ⊗ EN , M p + tr( NEM)

Eijrs M rs N pq E pqkl M p + N ab Eabcd M cd

(6.46)

6.2 Conventional Elastoplastic Constitutive Equations

145

σD p is described as σD p ≡ ED p = K pr D

the plastic relaxation stress rate

Furthermore, using the elastoplastic stiffness modulus tensor Kep ≡ E − K pr = E −

EM ⊗ EN M p + tr( NEM) ,

e pr K ijkl ≡ Eijkl − K ijkl = Eijkl − p

Eijrs M rs N pq E pqkl M p + N ab Eabcd M cd

(6.47)

the stress rate can be described as

σD = K ep D As known from Eqs. (6.46) and (6.47), the plastic relaxation modulus tensor and the elastoplastic stiffness modulus tensor are the asymmetric tensors, i.e.

K p ≠ K p T , Kep ≠ Kep T in general. As described in the next section, however, these stiffness modulus tensors become symmetric when the direction of plastic strain rate is chosen to be the outward-normal of yield surface, i.e. M = N . In this case the plastic flow rule is called the associated flow rule or the normality rule because the direction of plastic strain rate is given using the yield function as the plastic potential function. Hereinafter, the explicit equations of the above-mentioned constitutive equations adopting the associated flow rule are shown collectively as a support for the explanation of subsequent formulations.

D

σ D = λ N , D p = tr(Np ) N p

M

Mp≡

F ′ h ( σ, H i ; N) tr(Nσ) F

(6.48)

(6.49)

D

tr( N σ ) D = E σD + p N −1

(6.50)

M

Λ=

tr(NED) M + tr( NEN) p

⊗ EN ) D σD = ED − M p + NEN EN = (E − M EN p + tr( ) tr( NEN) tr( NED)

(6.51)

(6.52)

146

6 Basic Formulations for Elastoplastic Constitutive Equations

K pr ≡

EN ⊗ EN M p + tr( NEN)

， Ke

p

≡ E−

EN ⊗ EN M p + tr( NEN)

(6.53)

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 with the associated flow rule is assumed for metals.

f (σ ) = σ e , σ e ≡ 3 ||σ' || 2 F ( H ) = F (ε p e ), H = ε p e ≡ ∫ M=N

⎫ ⎪ ⎪ ⎪⎪ 2 || D p || dt ⎬ 3 ⎪ ⎪ ⎪ ⎪⎭

(6.54)

In the monotonic simple tension ( σ ij = 0 except for i = j = 1 , D22p = D33p = − D11p / 2 p (trD p =0) , Dij = 0 (i ≠ j ) ) it holds that

⎫ ⎪ ⎬ p 2 p2 p p 2 D11 + 2(− D11 / 2) dt = ∫ D11 dt = ε11 ⎪ 3 ⎭

σ e = 3/2 (σ 11 − σ 11 / 3) + 2(0 − σ 11 / 3) = σ 11 ε pe ≡ ∫

(6.55)

Then, σ e and ε coincide with the axial stress and the axial plastic strain in that loading and thus are called the equivalent stress and the equivalent (or accumulated) plastic strain, respectively. Substituting Eq. (6.54) into Eqs. (6.50)-(6.52) and using the relations pe

⎫ ⎪ ⎪ ⎪ •e ⎪ D σ = 3 tr ( σ' σD ' ) = 3 tr(N σ ) ⎪ 2 ||σ'|| 2 ⎪ (6.56) ⎬ D p = D p' ⎪ • p ⎪ 2 2 2 p H ( σ, H i ; D ) = ||N|| = || D ||, h( σ, H i ; N) = 3 3 3 ⎪ ⎪ 2 2 ⎪ p F || σ || = F' M = ' ⎪ 3 F 3 ' ⎭ ∂f (σ ) = ∂σ

3 σ' , N = σ' , tr(Nσ ) = || σ || ' 2 || σ'|| ||σ'||

6.2 Conventional Elastoplastic Constitutive Equations

147

the constitutive equation of the isotropic Mises material is given as follows:

2 •e 3σ σ σ' = E−1 σD + 3 1 σ• e σ −1 D + tr( N ) N −1 D + D=E σ =E σ 2 F ||σ' || 2 F' σ e ' 2F ' ' 3 3 D

(6.57) which is called the Prandtl-Reuss equation. The plastic work rate of this material is described as

t r(σ D p ) = t r (σ λ σ' ) = || σ' || λ = || σ' |||| D p || = σ e ε• pe = F ε• pe (6.58) ||σ' || which is the product of the hardening function and the equivalent plastic strain and thus the hardening attributable to the equivalent plastic strain is also called the work hardening. The traction t acting on the plane having the normalized outward-normal vector

em

inclined identically to the principal stress direction, called the octahedral plane, is given by

t = σ em = (σⅠeⅠ ⊗ eⅠ + σⅡeⅡ ⊗ eⅡ + σ ⅢeⅢ ⊗ eⅢ )( 1 eⅠ + 1 eⅡ + 1 eⅢ ) 3 3 3

= 1 (σⅠeⅠ + σⅡeⅡ + σ ⅢeⅢ ) 3 noting Eq. (3.7). Then, the normal component

τ oct

σ oct

(6.59)

and the tangential component

of the traction t is given as

σ oct ≡ t • em = 1 (σⅠeⅠ + σⅡeⅡ + σ ⅢeⅢ ) • ( 1 eⅠ + 1 eⅡ + 1 eⅢ ) 3

3

3

= 1 (σⅠ + σⅡ + σ Ⅲ ) = σ m 3

3

(6.60)

2 τ oct ≡ || t ||2 −σ oct

= 1 (σⅠ2 + σⅡ2 + σ Ⅲ2 ) − 1 (σⅠ + σⅡ + σ Ⅲ ) 2 9 3 =

2 {σ 2 + σ 2 + σ 2 − (σ σ + σ σ + σ σ )} Ⅱ Ⅲ Ⅰ Ⅱ Ⅱ Ⅲ ⅢⅠ 9 Ⅰ

= 1 (σⅠ − σⅡ ) 2 + (σⅡ − σ Ⅲ ) 2 + (σ Ⅲ − σⅠ) 2 = 3

2 J 3 2

(6.61)

148

6 Basic Formulations for Elastoplastic Constitutive Equations

where τ oct is called the octahedral shear stress and J 2 is given by choosing the deviatoric Cauchy tensor σ' as the second invariant of the deviatoric tensor T' in Eq. (1.150), i.e.

J 2 ≡ 1 || σ' || 2 = 1 tr σ' 2 = 1 σ 'rsσ 'sr 2 2 2 1 ' 2 + σ 2 + σ 2 ) + σ 2+ σ 2 + σ 2 = 2 (σ 11 '22 '33 '12 '23 '31

1 = 2 (σⅠ' 2 + σⅡ ' 2 + σⅢ ' 2 ) = 16 {(σⅠ − σⅡ) 2 + (σⅡ − σ Ⅲ ) 2 + (σ Ⅲ − σⅠ) 2} (6.62) It is interpreted from Eqs. (6.54), (6.61) and (6.62) for the Mises yield condition that the yielding is induced when the octahedral shear stress reaches a certain value. Eqs. (6.60) and (6.61) are also derived as the components in the directions I m ( = 3e m ) and 3 t' , i.e. Tm and || T' || / 3 of Tm and T' , respectively., regarding T as σ in Eq. (1.210).

6.3 Loading Criterion The judgment of whether or not the plastic strain rate is induced for a given incremental loading is required for the elastoplastic deformation analysis. The criterion for this judgment is called the loading criterion. In what follows, this criterion is formulated (Hashiguchi, 2000). 1. It is required that

λ =Λ>0

in the loading (plastic deformation) process D 2. It holds that

p

(6.63)

≠ 0.

D

tr(N σ ) ≤ 0

(6.64) p

in the unloading (elastic deformation) process D = 0. Further, substituting

D

D = D e leading to tr(N ED) = tr(N EDe ) = tr(N σ) into Eq. (6.43), Λ is described in this process as

D

tr(N σ ) Λ= p M + t r (NEM ) .

(6.65)

in this process. 3. The plastic modulus M p takes both positive and negative signs in general. On the other hand, noting that the elastic modulus E is the positive definite tensor and thus it holds that t r( NEM )  M p in general and postulating that the plastic relaxation does not proceed infinitely, let the following inequality be assumed.

6.3 Loading Criterion

149

M p + t r(NEM ) > 0 .

(6.66)

The ultimate state fulfilling M + t r(NEM ) → 0 leading to illustrated for the uniaxial loading process in Fig. 6.2. p

p K ijklr → ∞ is

p

Then, in the unloading process D = 0 , the following inequalities hold from Eqs. (6.40) and (6.63)-(6.66), depending on the sign of the plastic modulus M p leading to the hardening, perfectly-plastic and softening state.

λ ≤ 0 and Λ ≤ 0 when M p > 0

⎫ ⎪ λ → −∞ or indeterminate and Λ ≤ 0 when M p = 0 ⎬ (6.67) ⎪ λ ≥ 0 and Λ ≤ 0 when M p < 0 ⎭ Consequently, the sign of

λ

at the moment of unloading from the state m p ≤ 0 is

not necessarily negative. On the other hand, Λ is necessarily negative in the unloading process. Thus, the distinction between a loading and an unloading process cannot be judged by the sign of λ but it can be done by that of Λ . Therefore, the loading criterion is given as either

D p ≠ 0 : f (σ) = F ( H ) and Λ > 0 ⎪⎫ ⎬ D p = 0 : otherwise ⎪⎭

σ

D

D

tr(N σ ) > 0 t r(NED) > 0

}

M p =0

D

D

tr(N σ ) < 0 t r( NED) < 0

}

t r( N σ ) = 0 t r( NED) > 0

Mp<0

Mp >0

}

(6.68)

t r( N σ ) < 0 t r( NED) < 0

}

D

t r( N σ ) < 0 t r( NED) < 0

0 D Fig. 6.2 Signs of tr(N σ) and t r(NED) in uniaxial loading

D

t r( N σ ) < 0 t r( NED) > 0

}

}

M p + t r( NEM ) → 0 p K ijklr → ∞

ε

150

6 Basic Formulations for Elastoplastic Constitutive Equations

or

D p ≠ 0 : f (σ) = F ( H ) and tr(NED) > 0 ⎫⎪ ⎬ D p = 0 : otherwise ⎪⎭

(6.69)

(Hill, 1967, 1983) in lieu of Eq. (6.66). Limiting to the hardening process, Eq. (6.69) leads to

⎫ D p ≠ 0 : f (σ) = F ( H ) and tr(N σD ) > 0 ⎪ ⎬ p ⎪⎭ D = 0 : otherwise Denoting the infinitesimal total strain and plastic strain by tively, the elastic strain

ε

(6.70) and

ε p, respec-

εe = ε − ε p

(6.71)

is related uniquely to the stress. Then, substituting Eq. (6.71) in to Eq. (6.30), the yield surface can be described in the strain space as follows:

g (εe ) = F ( H ) or g (ε − ε p ) = F ( H )

(6.72)

setting g (ε e ) ≡ f (σ (ε e)) . Then, one has

∂f (σ ) ∂g (ε e ) ∂ε e ∂g (ε e ) D e ∂g (ε − ε p ) E−1 ∂g (ε − ε p ) E−1 = = = ( = ∂σ ∂ε ∂ε e ∂ σ ∂ε e σD ∂ ε −εp)

(6.73)

The substitution of Eq. (6.73) into tr( N ED) > 0 in Eq. (6.69) leads to

tr( ∂

g(ε − ε p ) D) > 0 ∂ε

(6.74)

Therefore, it can be interpreted that loading occurs when the strain rate is directed the outward-normal of the yield surface in the strain space. D In elastoplastic deformation analysis, suppose to calculate first σ and D by either of the elastic or the elastoplastic constitutive equation. Then, check the sign of t r(NED) . If the sign conflicts with the loading criterion, it is required to recalculate them using other 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.

6.4 Associated Flow Rule

151

6.4 Associated Flow Rule The associated flow rule holds under some assumptions for a wide range of materials. Some mechanical interpretations for the associated flow rule are delineated in this section.

6.4.1

Positivity 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.

w p ≡ tr(σD D p) / 2 ≥ 0

(6.75)

However, Eq. (6.75) holds only for the hardening process in which the stress rate is directed the outwards of the yield surface.

6.4.2

Positivity of Work Done During Stress Cycle: Drucker’s Hypothesis

Drucker (1951) postulated “the work done during the stress cycle by the external agency is positive”. It is described mathematically as follows:

vσ∫ t r{( σ −σ )Ddt } ≥ 0 0

(6.76)

where σ 0 denotes the initial stress. The following inequality is obtained from Eq. (6.76) under the assumption that the inside of the yield surface is elastic domain (see Fig. 6.3). (6.77) ( σ y − σ 0 )D p ≥ 0 where

σy

designates the stress on the yield surface in which the plastic strain rate

p

D is induced. The followings should hold in order to fulfill Eq. (6.77). 1) The plastic strain rate is directed outward-normal of the yield surface. 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. 6.4). The result 1) is called the associated flow rule or the normality rule and the result 2) is called the convexity of yield surface.

152

6 Basic Formulations for Elastoplastic Constitutive Equations

σ σy

t r{(σ y − σ 0 ) D p dt} ≥ 0

σ0 D p dt

ε

0

Fig. 6.3 Positive work done by external agency in Drucker’s (1951) postulate (Illustration in uniaxial loading process)

Dp

Yield surface

ıy

ı0 ı0 ı0

ı0

ı0

Dp

Elastic region

.

σij , ε ijp

0

ıy

ı0

t r{(ı y − ı 0 ) D p } < 0

Normality rule

Concave yield surface: Violation of Drucker (1951)’s postulate.

Fig. 6.4 Associated flow (normality) rule and convexity of yield surface based on the Drucker’s (1951) postulate

6.4.3 Positivity 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

6.4 Associated Flow Rule

153

second-order work rate w is smaller than the second-order elastic stress work rate wse calculated by presuming that the strain rate is induced elastically” or “the second-order plastic relaxation work rate w pr , i.e. the work rate done during the infinitesimal strain cycle is positive” (Hill, 1968; Hashiguchi, 1993a; Petryk, 1993; see Fig. 6.5). It is described mathematically as follows:

w ≤ wse , w pr ≥ 0

(6.78)

⎫ ⎪ ⎪ D wse ≡ tr(σe D) / 2 = tr(DED) / 2 ⎬ ⎪ Dp p p r w ≡ tr(σ D) / 2 = tr(D ED) / 2 ⎪ ⎭

(6.79)

where

w ≡ tr(σD D) / 2 = tr(DeED) / 2

with

σD e

σD e ≡ ED, σD p ≡ ED p

(6.80)

D p are called the elastic stress rate and the plastic relaxation stress rate, and σ

respectively. It should be noted that the associated flow rule must hold for Eq. (6.78) 2 to meet the loading condition in Eq. (6.68).

6.4.4 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. On the other hand, the interpretation of the positivity of work done by the external agency, i.e. the additional stress during the stress cycle by Drucker (1951) and the positivity of the second-order plastic relaxation work rate are based on postulates of the dissipation energy of materials independent of hardening behavior. Here, Drucker’s (1951) postulate is related to the stress cycle but the 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 the 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.

154

6 Basic Formulations for Elastoplastic Constitutive Equations

σ d dσ p dσ

σ

E

e

1

dσ

a

w pr e

b f

1

E

c

0

ε

d ε p dε e dε

Hardening process ( M p > 0 )

σ d dσ p

E

dσ e

1

a

σ dσ

f e

w pr 1

b

E

c

0 Softening process ( M p < 0 )

dε p dε e dε

ε

Fig. 6.5 Positiveness of second order work rate (Illustration in uniaxial loading)

6.4 Associated Flow Rule

155

(2) Limiting to the infinitesimal cycles, consider the stress and strain cycles. The second-order work increment done during the infinitesimal stress cycle is given D by t r (σ dt D pdt ) / 2 ( Δabe in the hardening process in Fig. 6.5). On the p p other hand, the additional work increment tr(D dt ED dt ) / 2 ( Δaec in the hardening process in Fig. 6.5) must be done to close the strain cycle, whilst tr(D pdt ED pdt ) / 2 > 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) The strain (increment) has one-to-one correspondence to the displacement (increment) induced in the material. Therefore, the configuration of material returns to the initial state only if the strain in any definition returns to the initial value. In other words, if the strain cycle in any definition of strain (logarithmic strain, nominal strain for instance) closes, the strain cycle in the other definition also closes. On the other hand, 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 stress, nominal stress for instance). Eventually, the strain cycle possesses the objectivity, but the stress cycle does not possess it. (4) The assumption that the interior of the yield surface is the purely elastic domain is adopted in Drucker’s postulate. On the other hand, it 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 unloading. Eventually, it can be stated that postulate of the positivity of second-order plastic relaxation work rate is more general than 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 tangential component will be described in 6.6. Regarding the case in which the plastic strain rate is directed the outward-normal of convex yield surface, i.e. the case in which the associated flow rule holds, the plastic work rate done by the actual stress σ y on the convex yield surface is greater than that done by any statically-admissible stress σ * as depicted in Fig. 6.6. It is called the principle of maximum plastic work. It is beneficial for the formulation of variational principles.

156

6 Basic Formulations for Elastoplastic Constitutive Equations

N

σy

σ* σ ij

0

Yield surface Fig. 6.6 Principle of maximum plastic work

6.5 Anisotropy The plastic strain rate described in 6.2 concerns the yield condition with the function f involving only the stress invariants. Therefore, it is limited to the materials exhibiting the isotropy 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.

6.5.1 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 system by which the response is described. Here, the input/output variables are the stress rate and the strain rate in the irreversible deformation. The rate-type constitutive equation in a rate form is described in general as follows: D (6.81) f (σ, σ, Hi , D) = 0 When the following equation holds by giving coordinate transformations only for stress (rate) and strain rate tensors in the function f , it can be stated that Eq. (6.81) describes the constitutive equation of isotropic material.

D D f (Q σ QT , QσQT , Hi , QDQT ) = Qf (σ, σ, Hi , D)QT

(6.82)

In the plastic constitutive equation formulated incorporating the yield and/or plastic potential function, the isotropy holds if the yield and/or plastic potential

6.5 Anisotropy

157

function is given by the function of stress invariants and scalar internal variables. Then, designating these functions by f , it must fulfill the equation.

f (QσQT , Hi) = Qf (σ, Hi)QT

(6.83)

In contrast, the anisotropic plastic constitutive equation is described by incorporating the yield and/or plastic potential function involving tensor-valued parameters in addition to the stress invariants and scalar internal variables. Then, Eqs. (6.82) and (6.83) do not hold in anisotropic constitutive equations.

6.5.2 Anisotropic Plastic Constitutive Equation 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. This phenomenon is induced by the statically-indeterminable 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 developing towards the loading direction and replacing the stress tensor with the 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 between 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 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. Therefore, the kinematic hardening cannot be applied but the rotation of the yield surface, i.e. the rotational hardening, is pertinent to soils. Now, let the yield condition (6.30) be extended to describe the anisotropy by introducing the internal variables of second-order tensors as follows:

f (σˆ ,β ) = F ( H )

(6.84)

σˆ ≡ σ − α

(6.85)

where

α

being the back stress (kinematic hardening variable) (Prager, 1956). Here, it is

assumed that f in Eq. (6.84) is also the function of

σˆ

in the homogeneous de-

f ( sσˆ , β) = sf (σˆ , β) . Then, the yield surface (6.84) maintains the similar shape and orientation with respect to σ = α when β = const. β is the second-order tensor of non-dimension describing the rotation of the yield gree-one fulfilling

158

6 Basic Formulations for Elastoplastic Constitutive Equations

surface by replacing the deviatoric stress ratio tensor

σ' / p

into

σ' / p − β and is

called the rotational hardening variable (Sekiguchi and Ohta, 1977). The isotropic deformation, i.e. expansion/contraction would not influence the anisotropy. Therefore, the anisotropic hardening variables

α

and β are assumed to be the

deviatoric tensors, the evolution rules of which are given by

D

where

αD = a(σ, H i ) || D p|| , β = b (σ, Hi ) || D p' || a and b are functions of σ, H i and are deviatoric,

(6.86) fulfilling

tra = trb = 0 . The symbol for deviatoric tensor is eliminated in (6.86)1 because of trD p = 0 in metals. The material-time derivative of Eq. (6.84) leads to the consistency condition

tr( ∂

ˆ β ˆβ f (σˆ , β) D σ) − tr(∂ f (σ, ) αD ) + tr( ∂ f (σ, ) βD) = F ′ H• β ∂ ∂σˆ ∂σˆ

(6.87)

Substituting Eqs. (6.37) and (6.86) into Eq. (6.87), one has

ˆ β ˆβ f (σˆ , β) D σ) − tr(∂ f (σ, ) a|| D p|| ) + tr( ∂ f (σ, ) b || D p' || ) ∂β ∂σˆ ∂σˆ (6.88) = F ′h (σ, H i ; D p ) Further, substituting the associated flow rule ( λˆ positive proportionality tr( ∂

：

factor)

ˆ ≡ ∂ f (σˆ , β) ˆ, N D p = λˆ N ∂σ into Eq. (6.87) and noting

tr( ∂

/ || ∂

f (σˆ , β) ∂σ ||

(6.89)

∂ f (σˆ , β) / ∂ σ = ∂ f (σˆ , β) / ∂ σˆ , it follows that

ˆ ˆ f (σˆ , β) D σ) − tr(∂ f (σ, β) a λˆ) + tr( ∂ f (σ, β)b || Nˆ ' || λˆ ) ∂β ∂ σˆ ∂ σˆ ˆ) = F ′λˆ h ( σ, H i ; N

(6.90)

Taking account of the Euler’s theorem for homogeneous function in order-one, i.e.

f (σˆ , β) t r( ∂ σ) β ˆ ∂ σˆ ∂ f (σ, ) ˆ = f (σˆ , β) N ˆ = F N ˆ N = ˆ ˆ ˆ σˆ ) ∂ σˆ tr(N σˆ ) tr(N σˆ ) tr(N

(6.91)

6.6 Incorporation of Tangential-Inelastic Strain Rate

159

into Eq. (6.90), it is obtained that

F tr(N ˆ σD ) − F tr(N ˆ a) λˆ + tr( ∂ f (σˆ , β) b || N ˆ ' || ) λˆ ˆ β ∂ ) ˆ tr( ˆ σ ) N tr(N σˆ ˆ) (6.92) = F ′λˆ h (σ, H i ; N from which we obtains

λˆ =

D

ˆ σ) tr(N f (σˆ , β) F ′ (σ, H ; ˆ ) ˆ i N tr(Nσ ˆ ) + tr(Nˆ a) − F1 ( ∂ ∂ β b || Nˆ ' || ) tr(Nˆ σˆ ) Fh D

=

ˆ σ) tr(N ˆ ) − 1 ( ∂ f (σˆ , β) b || N ˆ ' ||)}σˆ + a] ˆ [{( F ′ h (σ, H i ; N tr N F F ∂β

(6.93)

)

(

Consequently, we have

λˆ =

ˆ σD ) tr(N Mˆ p

，D

p

=

ˆ σD ) tr(N ˆ N Mˆ p

(6.94)

where

ˆ ) − 1 ( ∂ f (σˆ , β) b || N' || )} σˆ + a ] ˆ [{( F ′ h (σ, H i ; N Mˆ p ≡ tr N F ∂β F

)

(

(6.95) and

D

ˆ σ) tr(N ˆ D = E σD + ˆ p N M −1

Λˆ = ˆ D) tr( NE

(6.96)

ˆ ED) tr(N ˆ ˆ) Mˆ p + tr( NEN

(6.97)

ˆ N

ˆ

ˆ = (E − E ⊗ EN ) D EN σD = ED − ˆ p ˆ ˆ ˆ ˆ) M + tr( NEN) Mˆ p + tr( NEN

(6.98)

6.6 Incorporation of Tangential-Inelastic Strain Rate As presented in Eqs. (6.40), (6.48) and (6.94), the inelastic strain rate, i.e. the plastic strain rate in the traditional constitutive equation has the following limitations.

160

6 Basic Formulations for Elastoplastic Constitutive Equations

(i) It 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 is determined solely by the current state of stress and internal variables but it is independent of the stress rate. (iii) The principal directions of plastic strain rate tensor coincide with those of stress tensor, exhibiting the so-called coaxiality, in the case of isotropy in which the direction of plastic strain rate depends only on the direction of stress by the fact described in 1.11. On the other hand, it has been verified by experiments that an inelastic strain rate that influences deformation considerably is induced by the deviatoric tangential stress rate, and it is called the tangential inelastic strain rate, in the non-proportional loading process deviating from the proportional loading path normal to the yield surface. Here, the mean part of the tangential stress rate does not induce an inelastic strain rate, as Rudnicki and Rice (1975) verified based on the fissure model. 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 the macroscopic deformation and strength characteristics even in the macroscopically proportional loading process. To remedy these insufficiencies, vairous 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 has been 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, 1997). 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 by the latter models. It was described by Hecker (1972; Ikegami, 1979) that the yield surface projects towards the loading direction generally but the formation of the so-called vertex is doubtful. 3) Hypoplasticity: This term was first used by Dafalias (1986) in the analogy to the term hypoelasticity introduced by Truesdell (1955) described in 5.3. Models in

6.6 Incorporation of Tangential-Inelastic Strain Rate

161

this category are classified into models in which the direction of plastic strain D D rate depends on the direction of the stress rate σ / || σ || (Mroz, 1966; Dafalias and Popov, 1977; Hughes and Shakib, 1986; Wang et al., 1990; Hashiguchi, 1993a) and the models in which the direction of the plastic strain rate depends on the direction of stress rate D/ || D || (Hill, 1959; Simo, 1987; Hashiguchi, 1997; Kuroda and Tvergaard, 2001). The singularity in the filed of direction of plastic strain rate is introduced algebraically 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-strain rate relation becomes nonlinear. Therefore, the inverse expression cannot be derived, which renders deformation analysis as difficult. 4)

J 2 - deformation theory: Budiansky (1959) and later Rudnicki and Rice (1975) incorporated the tangential-inelastic strain rate into the constitutive equation (6.57) with the isotropic Mises yield condition as follows:

σ• e σ' + φ (σ e ){σD − t r ( σ ' σD ) σ ' } D = E −1 σD + 3 1 σ ' 2 F' e ||σ' || ' ||σ' ||

(6.99)

which can be rewritten as •e D = E−1 σD + 3 1 σ e σ' + φ (σ e ){σD ' − 2 / 3 σ• e 2 F' σ

σ'

2 / 3σ e

•e e D = E −1 σD + ( 3 1 − φ (σ e )) σ σ e σ' + φ (σ ) σ ' 2 F'

} (6.100)

where the rate-linearity is retained. On the other hand, Hencky’s deformation theory (Hencky, 1924) is described as

ε = E−1 σ + φ (σ e ) σ'

(6.101)

The material-time derivative of Eq. (6.101) leads to

εD = E−1 σD + φ ' (σ e ) σ• e σ' + φ (σ e )σD '

(6.102)

162

6 Basic Formulations for Elastoplastic Constitutive Equations

Comparing Eq. (6.100) with Eq. (6.102), choosing

F (σ e ) so as to fulfill

1 F ' (ε e p(σ e )) = 3 2 φ (σ e )+ φ ' (σ e )σ e and regarding

εD

as D , it is known that the

(6.103)

J 2 - deformation theory (6.99) coin-

cides with Hencky’s deformation theory (6.101). In what follows, extend the J 2 - deformation theory for the general yield condition unlimited to the Mises yield condition (Hashiguchi, 1998, 2005; Hashiguchi and Tsutsumi, 2001,2003) First, assume that the strain rate is decomposed additively into elastic and plastic strain rates in Eq. (6.28) and further assume the tangential-inelastic strain Dt as

．

D = De + D p + Dt

(6.104)

D which is decomwhere Dt is induced by the deviatoric tangential stress rate σ ' D and the deviatoric tangential stress posed into the deviatoric normal stress rate σ n' D (Fig. 6.7): rate σ 't

σD' = σD n' + σD 't

(6.105)

σD ' = σD − 13 t r σD

(6.106)

where

⎫ ⎪⎪ ⎬ D D D D D ˆ ˆ σ't ≡ I' σ = σ' − σ 'n (tr(N σ't ) = 0) ⎪⎪ ⎭

σD 'n ≡ nˆ' ⊗ nˆ ' σD = tr(nˆ ' σD )nˆ'

f ( ˆ , β) nˆ' ≡ (∂ σ )' ∂σ

Iˆ' ≡ I ' − nˆ ' ⊗ nˆ '

( ∂ f ∂(σσˆ , β) )'

=

ˆ' N (||nˆ' || =1) ˆ ' || || N

' ≡ 12 (δ ik δ jl + δ ilδ jk ) − 13 δ ijδ kl − nˆ'ijnˆ'kl ) ( Iˆijkl

I' ≡ I − 1 I ⊗ I 3

' ≡ 1 (δ ikδ jl + δ ilδ jk ) − 1 δ ijδ kl ) ( Iijkl 2 3

(6.107)

(6.108)

(6.109) (6.110)

In these expressions, the fourth-order tensor I ' plays the role of transforming an arbitrary second-order tensor into its deviatoric second-order tensor. Therefore, it might be called the deviatoric transformation (or projection) tensor. Furthermore, the fourth-order tensor Iˆ' plays the role of transforming an arbitrary

6.6 Incorporation of Tangential-Inelastic Strain Rate

163

second-order tensor into its deviatoric second-order tensor tangential to the yield surface. For that reason, it might be called the deviatoric tangential (or projection) tensor. Hereinafter, the deviatoric tangential tensor is denoted as ( )'t , i.e. Tt' ≡ Iˆ' T for arbitrary second-order tensor T .

σD n'

nˆ'

σD '

σ

σD 't

σ ij'

0

Yield surface Fig. 6.7 Normal and tangential stress rates in the deviatoric stress plane

Now, assume that the tangential inelastic strain rate D t is related linearly to the D . tangential deviatoric stress rate σ 't

Dt = T σD t' 2G

(6.111)

where T is a function of stress and internal variables in general. Substituting Eqs. (6.29), (6.94), (6.111) into Eq. (6.104), the strain rate is given by

D

ˆ σ) tr( N ˆ + T σD ' D = E σ+ N p 2G t Mˆ −1

D

ˆ ˆ = (E −1 + N ⊗ N + 2TG Iˆ' ) σD Mˆ p

(6.112)

For derivation of the inverse expression of Eq. (6.112), let the elastic modulus tensor (5.28) in the Hooke’s type be adopted. Therefore, note that Λˆ is given by D Eq. (6.97) itself since it holds that tr( NEσ 't ) = 2Gtr( NσD 't ) = 0 because of

Eq. (6.107)2.

164

6 Basic Formulations for Elastoplastic Constitutive Equations

Noting (5.28), it holds from Eq. (6.112) that

D

ˆ σ) tr( N D ˆ ' + T σD ' D' = 1 σ' + N 2G t 2G p ˆ M

(6.113)

from which one has the following relation, considering

ˆ=N ˆ 't ≡ Iˆ' N ˆ ' − tr(nˆ' N ˆ ' ) nˆ ' = 0 . N

D't = 1 (1 + T ) σD 't 2G

(6.114)

D't ≡ Iˆ' D = D' − tr(nˆ ' D' )nˆ'

(6.115)

where

Substituting further Eq. (6.114) into Eq. (6.111), the tangential inelastic strain rate is given by

T Dt = 1 + T D't

(6.116)

The stress rate is derived from Eqs. (6.29), (6.89), (6.97) and (6.116) as follows:

σD = ED −

(

ˆ ) EN t r ( NED ˆ − 2GT D't p 1+ T ˆ ˆ ˆ M + t r ( NEN )

ˆ ⊗ = E − EN

(6.117)

ˆ EN − 2+GT Iˆ ' D p ˆ ˆ ˆ M + t r ( NEN ) 1 T

)

(6.118)

Here, since there exists the relation

tr (

∂f (σ ) ∂f (σ) D ∂f (σ ) E(D − D p − Dt )} = dF σ ) = tr( EDe ) = tr{ ∂σ ∂σ ∂σ (6.119)

it holds that

tr{

∂f (σ ) EDt } = 0 ∂σ

(6.120)

Then, it is known that the tangential inelastic strain rate D t does not influence the hardening behavior. Furthermore, the loading criterion is given by Eq. (6.69) as it is, and the mathematical structure is rate-linear identically to the common elastoplastic constitutive equation, whilst it is called sometimes the linear comparison material since it is linear in the continuation of loading. Consequently, no difficulty is brought in the analysis of boundary value problems.

6.7 Hyperelastic-Plastic Constitutive Equation: Finite Strain Theory

165

The tangential inelastic strain rate has no loading criterion and is induced only if the deviatoric tangential stress rate is induced, falling within the framework of hypoelasticity in which the complete integrability condition does not hold and the time-integration depends on the loading path. However, the continuity condition is violated because it falls within the framework of the conventional plasticity assuming the interior of the yield surface to be an elastic domain. The tangential inelastic strain rate is induced suddenly when the stress reaches the yield surface. Therefore, the range of application is limited to the proportional loading process in which the tangential component of stress rate is far smaller than the normal component.

6.7 Hyperelastic-Plastic Constitutive Equation: Finite Strain Theory Strictly speaking, the elastic strain rate has to be formulated based on the hyperelastic constitutive equation in which the elastic deformation is determined uniquely for a given stress and the elastic work done during the stress cycle is zero. On the other and, these basic requirements are not fulfilled exactly in the hypoelasticity as described in Chapter 5. In addition, the stress is calculated directly for a given elastic deformation in the hyperelastic constitutive equation. On the other hand, the hypoelastic constitutive equation must be integrated in time in order to calculate the stress. Therein, it is required to adopt an appropriate corotational stress rate in order to predict the stress pertinently. One of hyperelastic-plastic constitutive equations is shown below, which is based on the postulate of Eq. (6.20) (cf. Mandel, 1973, 1974; Simo, 1998; Belytschko et al., 2001). The hyperelastic constitutive equation is given by replacing F to F e in Eq. (5.9) for the unit initial density as 2ψ ( e • • E) S = ∂ e e Ee ∂E ∂E

(6.121)

where −1

S ≡ Fe τFe

−T

e e E e ≡ 1 (C − I ) = 1 (F eT F e − I ), C ≡ F eT F e 2 2

(6.122) (6.123)

S is interpreted to be the pull-back of the Kirchhoff stress τ to the intermediate configuration. Here, note that S∗ = Fe ∗−1τ ∗Fe ∗−T = (QFe ) −1 Qτ QT (QFe ) −T −1 = Fe τFe −T = S leading to the frame-indifference of the hyperelastic constitutive

relation.

166

6 Basic Formulations for Elastoplastic Constitutive Equations

One can formulate the following relation from Eq. (6.19).

D = De + D p

(6.124)

where •e

 pFe D ≡ F eT DF e, D e ≡ E = F eTD e F e , D p ≡ F eTD Here, note that

(6.125)

D p is expressed as T D p = 1 (Ce Lp + Lp Ce ) 2

(6.126)

because of −1 −1 T D p = F eT 1 {F e Lp F e + (F e Lp F e )T }F e = 1 ( F eT F eLp + Lp F eT F e) 2 2

The following equation is obtained from Eq. (6.121), adopting Eq. (6.124). •

el p S = Cel De = C ( D − D )

(6.127)

where e ∂ 2 ψ (E ) C ≡ e e ∂E ∂E el

(6.128)

Now, we adopt the following yield condition for the sake of simplicity of explanation, while the extension to the anisotropy is not difficult.

f (S ) = F ( H ) while f is the homogeneous function of ferentiation of Eq. (6.129) leads to

tr(

(6.129)

S in degree-one. The material-time dif-

• ∂f (S ) • S) = F' H ∂S

(6.130)

Noting the relation

∂f (S ) = ∂S

∂f (S ) S) f (S ) ∂S NS = NS = F ( H ) NS N S tr( ) tr(NS S) tr(NS S) S

tr(

NS ≡

∂f (S ) ∂S

/ ||

∂f (S ) ∂S

||

(6.131)

(6.132)

the consistency condition is given from Eq. (6.130) as • • tr(N S S ) = F' H tr(NS S) F

(6.133)

6.7 Hyperelastic-Plastic Constitutive Equation: Finite Strain Theory

167

Here, let the plastic flow rule be given by

L p = λr (S )

(6.134)

p

where r (S ) stands for the direction of L , which is function of S . The substitution of Eq. (6.134) into Eq. (6.133), one has • tr( N S S ) = F' λ h (r )tr( N S S ) F

where h (S) is related to (6.135) that

•

•

H as H = λ h (S) . It holds from Eqs. (6.134) and

•

λ=

(6.135)

•

tr(N S S )

tr(N S S )

r (S ) , Lp = F F' h ( ) ' h ( S) tr(N S S) tr(N S S ) S F F

(6.136)

p

D is given from Eqs. (6.126) and (6.136) by •

p

D =

tr(N S S )

1 (Cer + rT Ce ) 2 F' h ( ) F S tr(N S S)

(6.137)

D is given from Eqs. (6.124), (6.127) and (6.137) as D=C

el −1

•

S+

•

tr( N S S )

1 (C e r + r T C e ) 2 F' h ( ) F S tr( N S S )

(6.138)

from which the proportionality factor in the flow rule in Eq. (6.134) is described in terms of D as

Λ=

tr(N S Cel D)

(6.139)

F' h ( S) tr(N S S ) + tr{NS Cel(Cer + rT Ce ) / 2} F

It is obtained from Eqs. (6.138) and (6.139) that •

[

S= C

el −

Cel 1 (Cer + rT Ce ) ⊗ tr(N S Cel ) 2 D F' h ( r) tr(N S S ) + tr{NS Cel (Cer + rT Ce ) / 2 } F

]

On the other hand, Lie derivative of S due to •

(6.140)

F p defined in 4.5, is given by

Lvp S = S −Lp S − SLpT

(6.141)

168

6 Basic Formulations for Elastoplastic Constitutive Equations

noting

L vp S = F p (F p −1SF p −T )• F pT • p −1

= F p (F •p

= (− F F

•

• −T

SF p −T + F p −1 S F p −T + F p −1S F p

p −1

•

)F pT

•

) S + S + S ( − F p F p −1 )T

The substitution of Eqs. (6.134) and (6.140) into Eq. (6.141) leads to

Lvp S = C ep D

where

C

ep ≡

C

el

(6.142)

1 Cel 2 (Cer + rT Ce ) + (r S + Sr )}⊗ tr(N S Cel ) { − el

(6.143)

F' h ( ) tr(N S S) + tr{NS C (C r + rT C ) / 2} F S e

e

It holds from Eqs. (3.21), (4.57) and (6.122) for the Truesdell rate of Kirchhaff stress that Δ

τ = L v τ = F (F −1τF −T )• F T = F eF p ( F p −1F e−1F e S F eT F e −T F p −T )• F p T F eT = F eF p ( F p −1S F p − T )•F p T F eT = F e (Lvp S) F e T

(6.144)

Substituting Eqs. (6.125) and (6.142), it is derived that Δ

τ = F e Cep DF e T = F e Cep ( F eT D F e) F e T = C ep D

(6.145)

⎛ Δ ij ⎞ ep e e ⎜τ = Fim (C mnpq Dpq ) Fjn ⎟ ⎜ = F e (C ep F e D F e ) F e = F e F e F e F eC ep D ⎟ im jn im jn kp lq mn pq kl ⎠ mn pq kp kl lq ⎝ where ep

ep C ijkl = Fime Fjne Fpke Fqel Cmn pq 

(6.146)

Noting Eq. (4.57), Eq. (6.145) is rewritten as :

Δ

τ (= τ + Dτ + τD) = (C ep + Σ)D

(6.147)

by the Jaumann rate of Kirchhoff stress or, noting Eq. (4.54), :

: σ ( = τ − σ trD) = ( 1 C ep + Σ − σ ⊗ I )D J

J

by the Jaumann rate of Cauchy stress, where Σ is defined in Eq. (5.15).

(6.148)

6.7 Hyperelastic-Plastic Constitutive Equation: Finite Strain Theory

169

In the above-mentioned constitutive equation, the flow rule (6.134) for L p has to be formulated rigorously so as to provide the plastic strain rate pertinently noting the associated flow rule. Further, it has to be done also for the spin due to the plastic deformation, i.e. W p = λ (r (S ))a . Further discussions for the finite elastoplastic Experiment Elastic state Elastoplastic state Stress

Stress Unrealistic prediction: Excessively high peak stress

Almost real prediction

0

Prediction by conventional plasticity

Hardening

Strain

0

Softening

Strain

Fig. 6.8 Prediction of monotonic loading behavior by conventional plasticity

Experiment Elastic state Stress

Elastoplastic state

Prediction by conventional plasticity

Cyclic loading with constant stress amplitude

0 Fig. 6.9 Cyclic loading behavior: Inability of conventional plasticity

Strain

170

6 Basic Formulations for Elastoplastic Constitutive Equations

strain theory are referred to the literatures (cf. Kleiber and Raniecki,1985; Raniecki and Mroz, 1990; Raniecki and Nguyen, 2005; Raniecki et al., 2008). 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 vs. 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. 6.8. This results in the serious defect in the prediction of softening behavior. Further, only an elastic deformation is repeated for the cyclic loading of stress below the yield stress. In fact, however, plastic deformation is accumulated for stress cycles less than the yield stress and the strain is amplified leading to the failure as depicted in Fig. 6.9. Therefore, it has various limitations in the application to the mechanical design of machines and structures in engineering practice.

Chapter 7

Unconventional Elastoplasticity Model: Subloading Surface Model 7 Unconventional Model: Subloading SurfaceElastoplasticity 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 designated as the conventional model in 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 elastoplastic constitutive equations. These mechanical requirements are first described and then the subloading surface model is explained in detail.

7.1 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, although their importance for formulation of constitutive equations has not been sufficiently recognized to date. Before formulation of the plastic strain rate, these requirements will be explained below (Hashiguchi, 1993a, b, 1997, 2000).

7.1.1 Continuity Condition It is observed in experiments that “the stress rate changes continuously for a continuous change of the strain rate”. This fact is called the continuity condition and is expressed mathematically as follows (Hashiguchi, 1993a, b, 1997; 2000).

D (σ D (σ lim σ , Hi ; D + δ D) = σ , H i ; D)

δ D→ 0

(7.1)

where H i (i =1, 2, 3, • ••) collectively denotes scalar-valued or tensor-valued internal state variables. In addition, δ ( ) stands for an infinitesimal variation. The response of the stress rate to the input of strain rate in the current state of stress and D internal variables is designated by σ (σ, Hi ; D). Uniqueness of the solution is not guaranteed in constitutive equations violating the continuity condition, predicting K. Hashiguchi: Elastoplasticity Theory, LNACM 42, pp. 171–189. springerlink.com © Springer-Verlag Berlin Heidelberg 2009

172

7 Unconventional Elastoplasticity Model: Subloading Surface Model

jump

jump

Input : D

Output : σD

Fig. 7.1 Violation of continuity condition

different stresses or strains. The violation of this condition is schematically shown in Fig. 7.1. 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, the identical stress rate directed the inward of yield surface can induce different strain rates in loading and unloading states in