mechanics of structures and continuamechanics of structures and continua for those who want to learn...

Mechanics of Structures and Continua

for those who want to learn the basics of mechanics of deformable bodies

Tetsuo IWAKUMA and Shigeru KOYAMA

This is a translated version of some parts of the lecture note‘I’ve Got a Bad Headache in Learning Mechanics of Structures and Continua!’ in Japanese.

https://hc2.seikyou.ne.jp/home/Tetsuo.Iwakuma/

Copyleft c⃝1998-2002 by T. Iwakuma, 2004-2019 by T. Iwakuma & S. Koyama (Japanese Edition)2019 by T. Iwakuma & S. Koyama (English Edition)

First Edition in Japanese January 13, 1998 (stardate [-30]0546.6)Latest 27th Edition August 29, 2019 ([-26]0036.2)

Draft Version-48 in English November 30, 2019 ([-26]0500.1)

Email:

https://hc2.seikyou.ne.jp/home/Tetsuo.Iwakuma/

Preface

This is a translated version of some parts of our lecture note “Basics of Mechanics of Structures and Continua (構造と連続体の力学基礎)” written in Japanese. The Japanese edition is basically a collection of lecture notes used inthe classes we have offered for more than thirty years at Tohoku University, and thus has more than 1,000 pagesincluding references and indices. Although the word ‘Basics’ is put in the title, the contents and explanationsare more or less different from those of many standard Japanese textbooks of the structural mechanics and thecontinuum mechanics in the civil engineering field. So, many friends of ours claim that it is not appropriate to usein classes especially for undergraduate students, possibly because too mathematical expressions are often employedin order to avoid some kinds of ambiguous or intuitive (so we think) statements sometimes found in old textbooks.However, honestly speaking, please note that readers must be very careful and may find many mistakes, becausewe are not so good at mathematics and English, Ha Ha Ha. This is one of the reasons why this note has neverbeen published. Also, this note is completely useless for those who are going to take an entrance examination toa graduate school, and who are going to take an examination for public service employment, because very fewexamples of solutions of actual boundary-value problems are shown. In that sense, this note is not suitable forundergraduate students who are going to learn the mechanics of structures and continua for the first time, butcan be referred to by graduate students who have studied such mechanics and have had somewhat odd feelingsor questions about what they have learned. Namely, the main purpose of this note is to learn the straightforwardformulations and the physical meanings of the theorems and the solution procedures.

When the first author was a student a long long time ago, one of the hot topics of research subjects in thefield of applied mechanics was the Finite Element Method. The virtual work principle plays the most importantrole in formulation of this approximation scheme, and is an example of the inner products of functions appliedto mechanics. Also, the distributions such as the Dirac delta function and the Heaviside function quite useful inmechanics are defined through some types of inner products with test functions. In that context, we realize that thereciprocal theorems, the influence lines of responses of structures, the inverse problems, and so on are all basedon application of the inner products of functions. Therefore, we think that the virtual work expression is one ofthe most important keys in the mechanics of deformable bodies, and we employ these kinds of inner products offunctions to express and explain mechanical concepts when necessary. It should be noted that the most importantand useful tool is not the energy principle but the virtual work principle.

This note is prepared by a famous typesetting software LATEX, but we keep the source closed. However, as isdeclared at the colophon, any parts of this note can be copied, printed and used without any further notice to theauthors as long as the usage is non-commercial and limited for educational purposes. When such copies are to beincluded in publications like a research paper and a textbook for college students, please put information of thetitle, the authors and the URL of this note in the section of references.

Special thanks are due to Professors F. Nishino and S. Nemat-Nasser for supervisions of researches and educa-tions at the University of Tokyo and Northwestern University. Finally, we wish to express our thanks to so manypeople including our students at Tohoku University. Their names can be found in the Japanese edition.

TI & SKKagoshima & Nagano Japan, Sector 001

September 2019

iii

Contents

1 Statically Determinate Structures 11.1 Equilibrium Equation . . . . . . . . . . . . . 1

1.1.1 What is Equilibrium? . . . . . . . . . 11.1.2 Applied Force and Reaction Force . . 21.1.3 Resistance of Deformable Body . . . 3

1.2 Statically Determinate Trusses . . . . . . . . 41.2.1 Characteristics of Trusses . . . . . . 41.2.2 Deduction of Axial Force . . . . . . . 5

Homework 1-1 . . . . . . . . 81.2.3 Influence Line and Design . . . . . . 8

Homework 1-2 . . . . . . . . 121.3 Statically Determinate Beams . . . . . . . . . 12

1.3.1 Characteristics of Beams and Sec-tional Forces . . . . . . . . . . . . . 12

1.3.2 Typical Examples . . . . . . . . . . . 14Homework 1-3 . . . . . . . . 20

1.3.3 Stress Distribution . . . . . . . . . . 201.3.4 Influence Line and Design . . . . . . 21

Homework 1-4 . . . . . . . . 241.3.5 Differential Equations of Equilibrium 24

Homework 1-5 . . . . . . . . 271.4 What is Statical Indeterminacy? . . . . . . . 27

1.4.1 Stability of Trusses and Indeterminacy 271.4.2 Hint to solve Statically Indeterminate

Structures . . . . . . . . . . . . . . . 29Homework 1-6 . . . . . . . . 31

2 Mechanics of Deformable Body 332.1 What are Continua? . . . . . . . . . . . . . . 332.2 Definition of Deformation . . . . . . . . . . 34

2.2.1 Displacement Vector . . . . . . . . . 342.2.2 Strain Tensor . . . . . . . . . . . . . 342.2.3 Rigid-Body Motion . . . . . . . . . . 362.2.4 Principal Direction of Strain . . . . . 372.2.5 Coordinate Transformation of Strain . 372.2.6 Volumetric Strain and Shear Strain . . 392.2.7 Compatibility Condition of Strain . . 40

Homework 2-1 . . . . . . . . 412.3 Local Equilibrium of Internal Forces . . . . . 41

2.3.1 What is Resistance of DeformableBody? . . . . . . . . . . . . . . . . . 41

2.3.2 Traction Vector and Stress Tensor . . 412.3.3 Equilibrium in term of Stress . . . . . 432.3.4 Boundary Conditions . . . . . . . . . 44

Homework 2-2 . . . . . . . . 452.3.5 Equilibrium of External and Internal

Forces . . . . . . . . . . . . . . . . . 452.3.6 Principal Direction of Stress and In-

variants . . . . . . . . . . . . . . . . 46Homework 2-3 . . . . . . . . 49

2.3.7 Coordinate Transformation of Stress . 49Homework 2-4 . . . . . . . . 50

2.3.8 Hydrostatic Component and ShearComponent . . . . . . . . . . . . . . 50

2.4 Relation of Deformation and Internal Force . 502.4.1 Constitutive Equations . . . . . . . . 502.4.2 Hooke’s Isotropic Elastic Body . . . 512.4.3 Characteristics of Elastic Moduli . . . 54

Homework 2-5 . . . . . . . . 592.4.4 Inelastic Strain and Incompatible Strain 592.4.5 Basics of Plasticity . . . . . . . . . . 61

Homework 2-6 . . . . . . . . 662.4.6 Brief Note on Viscosity . . . . . . . . 662.4.7 Strain and Stress used in Constitutive

Laws . . . . . . . . . . . . . . . . . 662.5 Principle of Virtual Work . . . . . . . . . . . 66

2.5.1 Two Admissible Fields and Principleof Virtual Work . . . . . . . . . . . . 66

2.5.2 Potential Energy . . . . . . . . . . . 682.5.3 Reciprocal Theorem . . . . . . . . . 722.5.4 Reciprocity about Inelastic Strain . . 742.5.5 Unit Load Method and Green’s Func-

tion . . . . . . . . . . . . . . . . . . 75Homework 2-7 . . . . . . . . 77

2.6 Examples in Elastic Problems . . . . . . . . 782.6.1 Wave propagating in Solid . . . . . . 782.6.2 Basics of Fluid Mechanics . . . . . . 78

Homework 2-8 . . . . . . . . 842.6.3 Plane Problems . . . . . . . . . . . . 84

Homework 2-9 . . . . . . . . 882.6.4 Airy’s Stress Functions . . . . . . . . 88

Homework 2-10 . . . . . . . 90Homework 2-11 . . . . . . . 93Homework 2-12 . . . . . . . 101

2.6.5 3-Dimensional Problems and Potentials101Homework 2-13 . . . . . . . 103

3 Mechanics of Bars 1053.1 Elementary Beam Theory . . . . . . . . . . . 105

3.1.1 What are Beams? . . . . . . . . . . . 1053.1.2 Field Equations . . . . . . . . . . . . 105

Homework 3-1 . . . . . . . . 1113.1.3 Boundary Conditions . . . . . . . . . 1113.1.4 Examples of Boundary-Value Problems113

Homework 3-2 . . . . . . . . 1183.2 Concentrated Forces and Moments . . . . . . 118

3.2.1 Continuity Conditions . . . . . . . . 1183.2.2 Concentrated Shear Forces . . . . . . 119

Homework 3-3 . . . . . . . . 1233.2.3 Concentrated Moment . . . . . . . . 123

v

vi CONTENTS

Homework 3-4 . . . . . . . . 1253.3 Principle of Superposition . . . . . . . . . . 125

Homework 3-5 . . . . . . . . 1283.4 Unit Load Method . . . . . . . . . . . . . . . 128

3.4.1 Expression of Concentrated Forces . 128Homework 3-6 . . . . . . . . 130

3.4.2 Unit Load Method . . . . . . . . . . 1303.5 Influence Line . . . . . . . . . . . . . . . . . 136

3.5.1 Influence Line of Displacement . . . 136Homework 3-7 . . . . . . . . 138

3.5.2 Influence Lines of Bending Momentand Shear Force . . . . . . . . . . . . 138

3.5.3 Design of Beams . . . . . . . . . . . 139Homework 3-8 . . . . . . . . 140

3.5.4 Influence Line of Statically Indeter-minate Beams . . . . . . . . . . . . . 140

Homework 3-9 . . . . . . . . 1433.6 Shear Stress and Deformation of Beams . . . 145

3.6.1 Shear Stress of Elementary Beam . . 145Homework 3-10 . . . . . . . 148Homework 3-11 . . . . . . . 152

3.6.2 Shear Lag . . . . . . . . . . . . . . . 1523.6.3 Beam Theory including Shear Defor-

mation . . . . . . . . . . . . . . . . 152Homework 3-12 . . . . . . . 153

3.7 Various Problems of Beams . . . . . . . . . . 1533.7.1 Beams on Elastic Foundation . . . . . 153

Homework 3-13 . . . . . . . 1543.7.2 Vibration of Bars . . . . . . . . . . . 154

3.8 Trusses and Frames . . . . . . . . . . . . . . 1553.8.1 Generalized Unit Load Method . . . 1553.8.2 Displacement and Axial Force of

Trusses . . . . . . . . . . . . . . . . 1573.8.3 Displacement and Sectional Forces

of Frames . . . . . . . . . . . . . . . 1593.8.4 Mechanics of Arches . . . . . . . . . 163

Homework 3-14 . . . . . . . 1633.9 Virtual Work Approach for Beam Theory . . 163

3.9.1 Virtual Work Equation . . . . . . . . 1633.9.2 Beam Theory in Small Displacement 163

4 Basics of Finite Element Method 1674.1 Weak Forms and Approximate Solutions . . . 167

4.1.1 How to Approximate? . . . . . . . . 1674.1.2 Weak Forms . . . . . . . . . . . . . 167

Homework 4-1 . . . . . . . . 170Homework 4-2 . . . . . . . . 171Homework 4-3 . . . . . . . . 173

4.2 Concept of FEM . . . . . . . . . . . . . . . 1734.2.1 Improvement of Galerkin Method . . 173

Homework 4-4 . . . . . . . . 1744.2.2 Approximation by Piecewise Polyno-

mials . . . . . . . . . . . . . . . . . 174Homework 4-5 . . . . . . . . 176

4.3 FEM of Structural Members . . . . . . . . . 1764.3.1 Element Stiffness Equations . . . . . 176

Homework 4-6 . . . . . . . . 1794.3.2 Direct Stiffness Method . . . . . . . 179

Homework 4-7 . . . . . . . . 1834.3.3 Plane Trusses . . . . . . . . . . . . . 183

Homework 4-8 . . . . . . . . 1864.3.4 Plane Frames . . . . . . . . . . . . . 186

4.3.5 Solution of Global Stiffness Equation 188Homework 4-9 . . . . . . . . 189

4.3.6 Variety of Elements . . . . . . . . . . 190Homework 4-10 . . . . . . . 192

4.4 Characteristics of FEM . . . . . . . . . . . . 1924.4.1 Column and Beam Elements are Exact!1924.4.2 Element Stiffness Matrices are Sin-

gular! . . . . . . . . . . . . . . . . . 1934.4.3 Variational Principles and Approxi-

mation . . . . . . . . . . . . . . . . 195Homework 4-11 . . . . . . . 198

4.5 Classical Methods . . . . . . . . . . . . . . . 1984.5.1 Using some Parts of Stiffness Equations1984.5.2 Usage of Complementary Virtual Work 199

Homework 4-12 . . . . . . . 2024.6 FEM for Plane Strain Problems . . . . . . . . 204

4.6.1 Virtual Work and Constitutive Relations2044.6.2 Constant Strain Triangle Element . . 2044.6.3 Higher Order Elements and Non-

Mechanical Problems . . . . . . . . . 209

5 Elastic Stability and Beam-Column Theory 2115.1 What are Stability Problems? . . . . . . . . . 2115.2 Examples of Spring Model . . . . . . . . . . 212

5.2.1 Finite Displacement Problems andStability . . . . . . . . . . . . . . . . 212

5.2.2 Criterion of Stability using Energy . . 215Homework 5-1 . . . . . . . . 217

5.2.3 Tangent Stiffness and Stability . . . . 217Homework 5-2 . . . . . . . . 217

5.2.4 Stability in Relatively Small Dis-placement . . . . . . . . . . . . . . . 217

5.3 Characteristics and Examples of Buckling . . 2195.3.1 Two Kind of Buckling Phenomena . . 2195.3.2 Buckling and Post-Buckling Behaviors 220

5.4 Beam-Column Theory . . . . . . . . . . . . 2215.4.1 Boundary-Value Problems . . . . . . 221

Homework 5-3 . . . . . . . . 2245.4.2 Bending Buckling of Columns . . . . 226

Homework 5-4 . . . . . . . . 2315.4.3 Application of Compression and

Bending . . . . . . . . . . . . . . . . 2315.4.4 Bars with Initial Imperfections . . . . 2335.4.5 Inelastic Buckling of Bars . . . . . . 2355.4.6 Compressive Strength of Bars . . . . 2375.4.7 Stiffness Equation and Buckling . . . 237

Homework 5-5 . . . . . . . . 2395.5 Variety of Beam-Columns . . . . . . . . . . 239

5.5.1 Beam-Columns supported by Spring . 239Homework 5-6 . . . . . . . . 241

5.5.2 Beam-Columns on Elastic Foundation 241Homework 5-7 . . . . . . . . 242

5.5.3 Timoshenko Beam-Column . . . . . 2425.5.4 Timoshenko Beam-Column on Elas-

tic Foundation . . . . . . . . . . . . 2425.6 Mechanics of Bars in Large Displacement . . 243

5.6.1 Finite Displacement Theory of Bars . 2435.6.2 Post Buckling Behavior . . . . . . . 2445.6.3 Elastica . . . . . . . . . . . . . . . . 2455.6.4 Cables . . . . . . . . . . . . . . . . . 2455.6.5 Strings . . . . . . . . . . . . . . . . 247

CONTENTS vii

6 Torsion 2496.1 Torsion of Circular Column and Pipe . . . . . 249

6.1.1 Torsion of Circular Column . . . . . 2496.1.2 Torsion of Circular Pipe . . . . . . . 251

Homework 6-1 . . . . . . . . 2526.2 Saint-Venant Torsion of Thin-Walled Bars . . 252

6.2.1 Shear Flow and Torsional Moment . . 2526.2.2 Torsional Constants of Closed-Section 254

Homework 6-2 . . . . . . . . 2556.2.3 Section with Multiple Cells . . . . . 255

Homework 6-3 . . . . . . . . 2566.2.4 Equilibrium Equation and Boundary

Conditions . . . . . . . . . . . . . . 2566.3 Torsion of Open-Section . . . . . . . . . . . 256

6.3.1 Thin-Walled Rectangular Section . . 2566.3.2 Torsional Constants . . . . . . . . . . 259

Homework 6-4 . . . . . . . . 2606.4 Flexural-Torsional Behavior . . . . . . . . . 260

6.4.1 Flexural Torsion of I-Section . . . . . 2606.4.2 Boundary-Value Problems . . . . . . 265

Homework 6-5 . . . . . . . . 2686.4.3 Generalized Theory . . . . . . . . . . 268

6.5 System of Structural Mechanics . . . . . . . 2706.5.1 Expressions of Mechanics of Bars . . 2706.5.2 Torsion of Prismatic Bars . . . . . . . 270

6.6 Stiffness Equation for Torsion . . . . . . . . 2726.6.1 Flexural-Torsional Problems . . . . . 272

Homework 6-6 . . . . . . . . 2736.6.2 Saint-Venant Torsion . . . . . . . . . 273

Homework 6-7 . . . . . . . . 2736.6.3 Space Structures . . . . . . . . . . . 274

Homework 6-8 . . . . . . . . 2766.7 Flexural Torsion and Stability . . . . . . . . . 279

6.7.1 Torsional Buckling . . . . . . . . . . 2796.7.2 Lateral-Torsional Buckling . . . . . . 2796.7.3 Bending-Compressive Strength . . . 280

7 Basics of Mechanics of Plate 2817.1 Elementary Parts of Structures . . . . . . . . 2817.2 Field Equations and Boundary Conditions . . 281

7.2.1 Assumption for Strain and Kinematics 2817.2.2 Stress and Stress Resultant . . . . . . 2827.2.3 Field Equations . . . . . . . . . . . . 284

Homework 7-1 . . . . . . . . 2857.2.4 Boundary Conditions . . . . . . . . . 2857.2.5 Formulation of Plate Theory . . . . . 286

7.3 Example of Boundary-Value Problems . . . . 2897.3.1 Governing Equations in terms of Dis-

placement . . . . . . . . . . . . . . . 2897.3.2 Navier’s Solution . . . . . . . . . . . 289

Homework 7-2 . . . . . . . . 2907.4 Stiffness Equation . . . . . . . . . . . . . . . 290

7.4.1 Virtual Work Equation . . . . . . . . 2907.4.2 Displacement Functions . . . . . . . 291

7.5 Stability Problems . . . . . . . . . . . . . . . 2917.5.1 Non-Linear Theory . . . . . . . . . . 2917.5.2 Uniformly Compressed Plates . . . . 2937.5.3 Application of Uniaxial Compression 294

Homework 7-3 . . . . . . . . 2957.5.4 Post Buckling Behavior . . . . . . . 2957.5.5 What are Membranes? . . . . . . . . 296

8 Basics of Vibration of Structures 2978.1 One-Degree of Freedom System . . . . . . . 297

8.1.1 Undamped Free Vibration . . . . . . 297Homework 8-1 . . . . . . . . 301

8.1.2 Damped Free Vibration . . . . . . . . 3018.1.3 Forced Vibration . . . . . . . . . . . 305

Homework 8-2 . . . . . . . . 307Homework 8-3 . . . . . . . . 314

8.1.4 Random Responses . . . . . . . . . . 3168.2 Multi-Degree of Freedom System . . . . . . 321

8.2.1 Two-Degree of Freedom . . . . . . . 321Homework 8-4 . . . . . . . . 327

8.2.2 Multi-Degree of Freedom . . . . . . 327Homework 8-5 . . . . . . . . 333

8.2.3 Numerical Approaches . . . . . . . . 3368.3 Vibration of Continua . . . . . . . . . . . . . 336

8.3.1 Vibration of Straight String . . . . . . 3368.3.2 Vibration of Circular Membrane . . . 341

Homework 8-6 . . . . . . . . 3478.4 Vibration of Beams . . . . . . . . . . . . . . 347

8.4.1 Equation of Motion of Beams . . . . 3478.4.2 Straightforward Approach . . . . . . 3488.4.3 Undamped Free Vibration . . . . . . 3498.4.4 Modal Analysis . . . . . . . . . . . . 351

Homework 8-7 . . . . . . . . 3538.4.5 Free Vibration with Viscous Damping 3538.4.6 Forced Vibration . . . . . . . . . . . 354

Homework 8-8 . . . . . . . . 3628.4.7 Finite Element Approach . . . . . . . 362

Homework 8-9 . . . . . . . . 3668.4.8 Rotational Inertia and Shear Defor-

mation . . . . . . . . . . . . . . . . 3698.5 Non-Linear Vibration . . . . . . . . . . . . . 372

8.5.1 Pendulum . . . . . . . . . . . . . . . 3728.5.2 Limit Cycle . . . . . . . . . . . . . . 3788.5.3 Parametric Oscillation . . . . . . . . 3798.5.4 Duffing Vibration . . . . . . . . . . . 383

9 Introduction to Plasticity 3879.1 Irreversible Deformation . . . . . . . . . . . 387

9.1.1 Irreversible Deformation and Fracture 3879.1.2 One-Dimensional Plastic Models . . 3909.1.3 Examples of Structural Members . . . 393

Homework 9-1 . . . . . . . . 3959.2 Generalization to 3-D Plasticity . . . . . . . . 396

9.2.1 Yield Condition . . . . . . . . . . . . 3969.2.2 Flow Rule . . . . . . . . . . . . . . . 3999.2.3 Incremental Constitutive Equations . 400

Homework 9-2 . . . . . . . . 4049.3 General Elastic-Plastic Models . . . . . . . . 404

9.3.1 Yield Surface and Normality Rule . . 4049.3.2 Introduction of Plastic Potential . . . 407

Homework 9-3 . . . . . . . . 4119.3.3 Examples and History-Dependence . 4119.3.4 Dislocations and Prandtl-Reuss Model 421

9.4 Other Useful Physical Models . . . . . . . . 4229.4.1 Kinematic Hardening Model . . . . . 422

Homework 9-4 . . . . . . . . 4269.4.2 Pressure Sensitive Model . . . . . . . 426

Homework 9-5 . . . . . . . . 4309.4.3 Non-Coaxial Model . . . . . . . . . 4309.4.4 Property of Material Parameters . . . 432

viii CONTENTS

9.4.5 More Realistic Models . . . . . . . . 4349.5 Plastic Analysis . . . . . . . . . . . . . . . . 434

9.5.1 Mechanism and Safety Factor . . . . 4349.5.2 Slip Line Theory . . . . . . . . . . . 4359.5.3 Localization of Deformation . . . . . 4409.5.4 Limit Analysis . . . . . . . . . . . . 441

Homework 9-6 . . . . . . . . 4489.5.5 Progressive Fracture . . . . . . . . . 449

Homework 9-7 . . . . . . . . 452

10 Finite Deformation Theory 45510.1 What is Finite Deformation? . . . . . . . . . 45510.2 Strains and Strain Rates . . . . . . . . . . . . 456

10.2.1 Deformation and Strain . . . . . . . . 456Homework 10-1 . . . . . . . 466

10.2.2 Rate of Change of Deformation . . . 467Homework 10-2 . . . . . . . 473

10.2.3 Elastic and Plastic Strain Rates . . . . 47310.3 Stresses, Equilibrium and Stress Rates . . . . 476

10.3.1 Basic Stresses and Equilibrium . . . . 47610.3.2 Other Stresses and Equilibrium . . . 47910.3.3 Physical Meaning of Stresses . . . . . 482

Homework 10-3 . . . . . . . 48510.3.4 Objective Stress Rates . . . . . . . . 485

10.4 Setting Current Configuration as Reference . 48710.4.1 updated Lagrangian Approach . . . . 48710.4.2 Deformation Rate . . . . . . . . . . . 48710.4.3 Stress Rates . . . . . . . . . . . . . . 48710.4.4 Incremental Equilibrium Equation . . 48910.4.5 Update of Stress . . . . . . . . . . . 490

10.5 Constitutive Laws in Finite Deformation . . . 49110.5.1 Choices of Stress and Strain . . . . . 49110.5.2 Hyperelasticity and Hypoelasticity . . 49310.5.3 Several Definitions of Elasticity . . . 494

Homework 10-4 . . . . . . . 50410.5.4 Elastic-Plastic Models . . . . . . . . 504

10.6 Examples of Analytical Estimates . . . . . . 50910.6.1 Localization Condition . . . . . . . . 50910.6.2 Comparison of Constitutive Models . 510

10.7 Examples of Numerical Estimates . . . . . . 51210.7.1 updated Lagrangian Method . . . . . 51210.7.2 Uniaxial Loading of Elastic Body . . 51810.7.3 Comparison of Constitutive Models . 519

A Timoshenko Beam Theory 531A.1 Governing Equations . . . . . . . . . . . . . 531

A.1.1 Kinematics and Stress Resultants . . 531A.1.2 Equilibrium Equations and Boundary

Conditions . . . . . . . . . . . . . . 532A.1.3 Governing Equations in terms of Dis-

placement . . . . . . . . . . . . . . . 533A.2 Virtual Work Equation and Stiffness Equations 534

A.2.1 Virtual Work Equation . . . . . . . . 534A.2.2 Stiffness Equations . . . . . . . . . . 535

B Finite Displacement Theory of Bars 537B.1 Finite Displacement and Deformation . . . . 537

B.1.1 Definition of Strain . . . . . . . . . . 537B.1.2 Virtual Work Principle and Stress . . 538B.1.3 Assumption of Kinematics . . . . . . 538

B.2 Bernoulli-Euler Beam Theory . . . . . . . . 539B.2.1 Kinematics . . . . . . . . . . . . . . 539B.2.2 Equilibrium and Boundary Conditions 540B.2.3 Constitutive Equation . . . . . . . . . 541B.2.4 Buckling Load . . . . . . . . . . . . 541B.2.5 Variational Principle and Elastica . . 542

B.3 Timoshenko Beam Theory . . . . . . . . . . 543B.3.1 Equilibrium and Boundary Conditions 543B.3.2 Constitutive Equation . . . . . . . . . 544B.3.3 Approximate Field Equation . . . . . 545B.3.4 Buckling Load . . . . . . . . . . . . 545

B.4 Beam-Column Theory in Small Displacement 546B.4.1 Linearization of Theory . . . . . . . 546B.4.2 Stiffness Equation . . . . . . . . . . 547

B.5 Thin-Walled Pipe with Sectional Deformation 549B.5.1 Kinematics . . . . . . . . . . . . . . 549B.5.2 Governing Equation . . . . . . . . . 550B.5.3 Stress Resultants . . . . . . . . . . . 551B.5.4 Stability Problem . . . . . . . . . . . 552

B.6 Numerical Analysis . . . . . . . . . . . . . . 553B.6.1 Two-Point Boundary-Value Problems 553B.6.2 Finite Element Approach . . . . . . . 554

C Basics of Tensor Analysis 559C.1 Coordinate and Base Vector . . . . . . . . . . 559C.2 Metric Tensor and Permutation Tensor . . . . 560C.3 Covariant Derivative . . . . . . . . . . . . . 561C.4 Physical Components in Polar Coordinate

System . . . . . . . . . . . . . . . . . . . . . 562C.5 Coordinate Transformation and Tensors . . . 564

D Average Characteristics of Composites 565D.1 Materials with Micro-Structures . . . . . . . 565D.2 Inhomogeneity and Inclusion . . . . . . . . . 566

D.2.1 Inhomogeneity and Eshelby Solution 566D.2.2 Governing Equations . . . . . . . . . 567D.2.3 Fourier Analysis . . . . . . . . . . . 568D.2.4 Equivalent Inclusion Method . . . . . 571

Homework D-1 . . . . . . . . 572D.3 Average Properties of Composites . . . . . . 573

D.3.1 Mori-Tanaka Average of Elastic Body 573D.3.2 Comparison with Experiments . . . . 575

Homework D-2 . . . . . . . . 576D.3.3 Application to Elastic-Plastic Com-

posites and Improvement . . . . . . . 576

Bibliography 581

Index 587

Mechanics of Structures and Continua

Appendix A

Timoshenko Beam Theory

A.1 Governing Equations

A.1.1 Kinematics and Stress Resultants

xz

z

w(x)

u(x)

ux(x, z)

uz(x, z)

B

A

−w′(x)

nγ(x)

ϑ(x)

Fig. A.1 Kinematics of Timoshenko beam

In Chap. 3, the so-called Bernoulli-Euler beam the-ory has been formulated, in which a cross-section iskept normal to the beam axis; i.e. the shear deforma-tion has been neglected as an assumption. Within thekinematics assumed there, the shear stress is derivedfrom equilibrium condition with the normal bendingstress, and shows a parabolic distribution as has beenshown in Sec. 3.6.1. Also, the result is consistent withthe solution in Eq.(2.199b) in the framework of plane-problems of elastic bodies. We here introduce anotherclassical beam theory which takes the effect of sheardeformation of the beam into account from the begin-ning of the formulation. Namely, the normality con-dition of the cross-section to the beam axis will be re-laxed. Incidentally, a matrix is denoted by a boldfacecharacter in this chapter.

It is called the Timoshenko beam theory which as-sumes a uniform shear deformation. Therefore, instead of using Eq.(3.2) for the kinematics of beams, we assume

2 ϵxz(x, z) = γ(x). (A.1)

However, the shear deformation at the top and bottom of a cross-section must be zero because no shearing forceis applied on the top and bottom surface of the beam. Therefore, this assumption cannot be physically acceptable.Namely, in reality, ϵxz must be a function of z as well just like the parabolic distribution obtained in the Ber-noulli-Euler beam theory and Eq.(2.199b). But, we know that the shear deformation is a secondary quantity incomparison with the bending deformation as long as the beam is long and slender enough as has been shown inFig. 3.4. Moreover, we can compensate unrealistic distribution of shear deformation due to the assumption ofkinematics by introducing another parameter kt explained later on.

Fig. A.1 depicts the assumption of the kinematics above. As is clear from this figure, the displacement compo-nents in the x- and z-directions can be given by

ux(x, z) = u(x) + zϑ(x), uz(x, z) = w(x), (A.2a, b)

in infinitesimal displacements instead of Eq.(3.3), where ϑ(x) expresses not the slope but the rotation of the cross-section. Using the definition of the strain in Eq.(2.6), substitution of Eq.(A.2) into Eq.(A.1) leads to

2 ϵxz(x, z) = ϑ(x) + w′(x) = γ(x) → ϑ = −w′ + γ, (A.3)

where a prime denotes the differentiation with respect to x. Namely, the rotation ϑ corresponding to θ in Eq.(3.4)is not equal to the slope −w′ but has a difference by γ due to the shear deformation. Hence, the extensive strain can

531

532 APPENDIX A. TIMOSHENKO BEAM THEORY

be expressed asϵxx(x, y, z) = u′ + zϑ′ = u′ + z (γ′ − w′′). (A.4)

We set that the normal stress obeys the one-dimensional Hooke law in Eq.(2.183a), and that the shear stresssatisfies Hooke’s law in Eq.(2.46). Substituting the strain components assumed above into these Hooke’s laws, wehave relations as

σxx = E ϵxx = Eu′ + z

(γ′ − w′′) , σxz = 2G ϵxz = G γ, (A.5a, b)

where E and G are the Young modulus and the shear modulus respectively. When the x axis is set to lie throughthe centroid of the cross-section, the corresponding stress resultants can be defined by

N(x) ≡∫

Aσxx dA = EA u′, M(x) ≡

∫A

zσxx dA = EI (γ′ − w′′) = EI ϑ′, V(x) ≡∫

Aσxz dA = GktA γ,

(A.6a, b, c)where the material is assumed to be homogeneous. Also, for simplicity, the cross-section is assumed to be uniformalong the axis of the beam. Note that the coefficient kt must be unity if we use the definition of the shear forcein Eq.(A.6c). However, this coefficient kt is introduced in order to compensate an unexpected situation due to theassumed uniform shear deformation γ(x) on a cross-section, and is an important parameter defined by the shape ofthe cross-section and Poisson’s ratio [11]. For example, we have

kt(circular section) =6(1 + ν)7 + 6ν

, kt(rectangular section) =10(1 + ν)12 + 11ν

. (A.7a, b)

Note that this parameter has nothing to do with the coefficient representing ratio of the maximum and average shearstresses defined in Sec. 3.6.1 (1); e.g. it is 3/2 for a rectangular section.

A.1.2 Equilibrium Equations and Boundary ConditionsWe can formulate the equilibrium equation and the boundary condition through the virtual work principle in three-dimensions by putting the kinematics above just like the process explained in Sec. 3.9. From Eqs.(A.2), (A.3) and(A.4), the corresponding variations of kinematics are expressed as

δux = δu + z δϑ, δuz = δw, δϵxx = δu′ + z δϑ′, 2δϵxz = δϑ + δw′.

Since the shearing component must be included in the Timoshenko beam, we can write the internal virtual work as

(IVW) ≡∫

V(σxx δϵxx + 2σxz δϵxz) dV =

∫ ℓ

0

∫Aσxx

(δu′ + z δϑ′

)+ σxz

(δϑ + δw′

)dA dx,

after substituting the variations above. Replacing the integrals of stress components on a cross-section by the stressresultants defined in Eq.(A.6), we can rewrite this work as

(IVW) =∫ ℓ

0

N δu′ + M δϑ′ + V

(δϑ + δw′

)dx.

Then, its integration by parts leads to the following expression;

(IVW) = [N δu + M δϑ + V δw]∣∣∣∣ℓ0−

∫ ℓ

0

N′ δu +

(M′ − V

)δϑ + V ′ δw

dx.

Eventually, using the symbol defined in Eq.(3.26), we can express the internal virtual work as

(IVW) = ni [N δu + M δϑ + V δw]∣∣∣∣x=0,ℓ−

∫ ℓ

0

N′ δu +

(M′ − V

)δϑ + V ′ δw

dx. (A.8)

As for the virtual works by the body forces and the surface tractions, we can directly use the same expressionsfor the Bernoulli-Euler beam after replacing the slope −w′ by the rotation ϑ, and they are∫ ℓ

0(p δu + m δϑ + q δw) dx, [Fi δu +Ci δϑ + Si δw]

∣∣∣∣x=0,ℓ

. (A.9a, b)

A.1. GOVERNING EQUATIONS 533

Combining Eqs.(A.8) and (A.9), we obtain the virtual work equation of the Timoshenko beam as follows:

[(ni N − Fi) δu + (ni V − Si) δw + (ni M −Ci) δϑ]∣∣∣∣x=0,ℓ

−∫ ℓ

0

[(N′ + p

)δu +

(M′ − V + m

)δϑ +

(V ′ + q

)δw

]dx = 0. (A.10)

From the second line of the virtual work Eq.(A.10), the equilibrium equations in terms of the stress resultants are

N′ + p = 0, V ′ + q = 0, M′ − V + m = 0. (A.11a, b, c)

Hereafter, the distributed moment force m will be neglected for simplicity. Also, the boundary conditions areobtained from the first line of the virtual work Eq.(A.10) as

u = ui or ni N = Fi , w = wi or ni V = Si , ϑ = ϑi or ni M = Ci , (A.12a, b, c)

where ui, wi and ϑi are the displacement components specified at the boundary. Except the rotational boundarycondition in Eq.(A.12c), these are identical with those of the Bernoulli-Euler beam given by Eq.(3.25). But,because of the shear deformation, the slope −w′ is not generally continuous at a loading point of midspan, but therotation of the cross-section ϑ must be continuous. So that the rotation instead of the slope must be specified in therotational boundary condition Eq.(A.12c).

A.1.3 Governing Equations in terms of DisplacementAs has been shown in the previous section, the governing equations in the axial direction are the same as thoseof the elementary beam theory and independent of the bending parts. Therefore, only the governing equationsfor bending are examined hereafter. In order to solve statically indeterminate systems, we need to express thegoverning equations in terms of the deflection w(x). To that end, substituting Eq.(A.6) into Eqs.(A.11) and (A.12),and eliminating γ and ϑ using Eq.(A.3), we can write the equilibrium equation as

−EI w′′′′ + q − αt q′′ = 0, (A.13)

and the boundary condition Eq.(A.12) becomes

w = wi or ni−EI w′′′ − αt q′

= Si, (A.14a)

−w′ − αt(w′′′ + αt

q′

EI

)= ϑi or ni

−EI w′′ − αt q= Ci, (A.14b)

where the folloing parameters are introduced;

αt ≡EI

GktA= ℓ2 αt, αt ≡

EG(λt)2 , λt ≡

ℓ√I/(ktA)

. (A.15a, b, c)

The parameter λt represents slenderness of the Timoshenko beam; i.e. one kind of the slenderness ratio. WhenG → ∞; i.e. if the shear deformation can be neglected, we have αt → 0 and αt → 0, and all the governingequations coincide with those of the elementary beam theory.

A solution of the deflection at the center of the span of a simply supported beam is given in Eq.(3.88). Thecorresponding deflection of the left span can be obtained as

w(x) = − P12EI

x3 +Pℓ2

16EIx +

P2GktA

x. (A.16)

In this case, since the shear deformation γ(x) is

γ(x) =P

2GktA, (A.17)

the third term of the equation above apparently represents the shearing component, while the first and secondterms are the solutions of the Bernoulli-Euler beam. In another case of the simply supported beam subjected to auniformly distributed force q, the deflection is obtained as

w(x) =q

24EIx (x − ℓ)

(x2 − ℓx − ℓ2

)+

q2GktA

x (ℓ − x) . (A.18)


Since the corresponding shear deformation is also obtained as

γ(x) = − qGktA

(x − ℓ

2

), (A.19)

the deflection component due to the shear deformation wshear(x) can be estimated by

wshear(x) =∫ x

0γ(ξ) dξ, (A.20)

and substitution of Eq.(A.19) into Eq.(A.20) of course results in the second term of Eq.(A.18).Let us examine a fixed beam subjected to a concentrated load P at the center as an example of the statically

indeterminate beam. The deflection in the left span is obtained as

w(x) = − P12EI

x3 +Pℓ

16EIx2 +

P2GktA

x. (A.21)

In this case, since the shear deformation γ(x) becomes the same as that of the simple beam, the third term is thecomponent wshear(x) due to shear deformation. Also, application of an uniformly distributed load q yields thedeflection as

w(x) =q

24EIx2 (ℓ − x)2 +

q2GktA

x (ℓ − x) , (A.22)

where the first term is the solution of the Bernoulli-Euler beam due to bending, and the second term is wshear(x).On the other hand, when the right end of the fixed beam is forced to deflect by the amount of ∆, the deflection

is obtained asw(x) = ∆

1ℓ3 x2 (3ℓ − 2x) + ∆

12αtℓ3 (1 + 12αt)

x (ℓ − 2x) (ℓ − x) . (A.23)

Note that the deflection due to shearing becomes zero at the both ends and center, because the shear deformationand the shear force become constant as is expressed by

γ =12αt ∆

ℓ (1 + 12αt), V =

12EI ∆ℓ3 (1 + 12αt)

=12EI ∆ℓ3

(1 − 12αt

1 + 12αt

). (A.24a, b)

Again, the total deflection can be decomposed into the bending and shear parts as before. However, the effect ofthe shear deformation is represented not by a simple parameter αt but by a complicated coefficient as 12αt/(1+12αt).This coefficient will appear in the finite element later on.

A.2 Virtual Work Equation and Stiffness Equations

A.2.1 Virtual Work EquationA basic weak form corresponding to the equilibrium Eq.(A.11) can be expressed by

0 = −∫ ℓ

0

δw (V ′ + q) + δϑ (M′ − V)

dx.

Integration by parts can take the boundary condition Eq.(A.12) into account to obtain

0 =∫ ℓ

0

V δ(ϑ + w′) + M δϑ′

dx −

∫ ℓ

0q δw dx −

[S1 δw1 +C1 δϑ1 + S2 δw2 +C2 δϑ2

].

Considering Eq.(A.3), we substitute Eq.(A.5) into the equation above, and the final form of the virtual workequation is obtained as∫ ℓ

0

GktAγ δγ + EIϑ′ δϑ′

dx −

∫ ℓ

0q δw dx −

[S1 δw1 +C1 δϑ1 + S2 δw2 +C2 δϑ2

]= 0. (A.25)

This derivation is a reversed process of the formulation in Sec. A.1.2. Or, substitution of Eq.(A.3) leads to anotherform of the internal virtual work as∫ ℓ

0

GktAγ δγ + EI

(−w′′ + γ′) δ (−w′′ + γ′) dx. (A.26)

A.2. VIRTUAL WORK EQUATION AND STIFFNESS EQUATIONS 535

A.2.2 Stiffness Equations(1) Lowest Order Elements

On the basis of Eq.(A.25), displacement functions can be simply assumed as follows: γ is set at constant, andϑ is given by a first order polynomial. Therefore, the corresponding deflection w is expressed by a second orderpolynomial from Eq.(A.3). Hence, four unknown coefficients are needed to express γ and w and are related to thefour kinematical quantities wi and ϑi, so that the continuity conditions between adjacent elements can be secured.However, this choice of displacement functions leads to a situation that the second term γ′ of the integrand vanishesin another expression of the internal virtual work Eq.(A.26). This contradiction can be avoided by assuming γ bya first order polynomial, but it needs another unknown coefficient and results in cumbersome deduction of a finiteelement as will be formulated in the next section. On the other hand, since the second term of the integrand ofEq.(A.26) represents the bending property in which the contribution of γ may be considered to have secondaryeffect, we may employ this displacement functions as the lowest order approximation and set

γ(x) =w2 − w1

ℓ+ϑ1 + ϑ2

2, w(x) =

(1 − x

ℓ

)w1 +

x(x − ℓ)2ℓ

ϑ1 +xℓw2 +

x(ℓ − x)2ℓ

ϑ2.

Substitution of these functions into Eq.(A.26) yields the corresponding stiffness matrix as

GktAℓ

−GktA2

−GktAℓ

−GktA2(

EIℓ+

GktAℓ4

)GktA

2

(−EIℓ+

GktAℓ4

)GktAℓ

GktA2

Symm.(

EIℓ+

GktAℓ4

)

. (A.27)

Although the Timoshenko beam theory is one kind of improvement of the Bernoulli-Euler beam theory, thismatrix in Eq.(A.27) does not converge to the stiffness matrix in Eq.(4.25) when G → ∞. Also, too many elementsare needed in finite element calculation using Eq.(A.27); i.e. the rate of convergence is very low, but we can showthat a finite element solution converges to the corresponding exact solution [34].

(2) Higher Order Element — the Appropriate Element

It is obvious that the strong form Eq.(A.13) in terms of the deflection has the fourth-order derivative of w, and thatthe solution may be a polynomial of at least the third order depending on q. So that, we here assume that γ is setat constant for the time being, and we employ that

γ(x) = γ0, (A.28a)w(x) = w1 ψ1(x) − w′ ψ2(x) + w2 ψ3(x) − w′2 ψ4(x) = w1 ψ1 + (ϑ1 − γ0)ψ2 + w2 ψ3 + (ϑ2 − γ0)ψ4

= w1 ψ1(x) + ϑ1 ψ2(x) + w2 ψ3(x) + ϑ2 ψ4(x) + γ0 ψ5(x), (A.28b)

where ψn’s are the displacement functions defined in Eq.(4.22), and an additional function

ψ5(x) ≡ −ψ2(x) − ψ4(x) = x − 3x2

ℓ+ 2

x3

ℓ2

is introduced through Eq.(A.3).It is true that γ′ again vanishes in Eq.(A.26), but γ0 is included in the displacement function of w in Eq.(A.28b).

Hence, we can expect that the shearing effect can be taken into account to some extent. And, it is quite interestingthat the resulting stiffness equation becomes the same even when a first order polynomial is employed for γ.

Substitution of Eq.(A.28) into Eq.(A.25) results in the element stiffness equation asS1C1S2C20

+

q1q2q3q4q5

=

kb

h

⌊ ht ⌋ h5

w1ϑ1w2ϑ2γ0

, (∗)


where kb is the stiffness matrix in Eq.(4.23a) and (4.25), and qi’s (i=1～ 4) are the equivalent nodal forces definedin Eq.(4.23b). Explicit expressions of h5 and h are given in reference [34], and we set

h ≡ ⌊h1 h2 h3 h4⌋t, q5 ≡∫ ℓ

0qψ5 dx, (A.29a, b)

hn ≡ EI∫ ℓ

0ψ′′n ψ

′′5 dx (n = 1, 2, 3, 4), h5 ≡ GktAℓ +

∫ ℓ

0EI (ψ′′5 )2 dx, (A.29c, d)

where q5 becomes zero under the uniformly distributed loading. The left-hand side of the fifth line of the stiffnessEq.(∗) is zero, because there is no force component corresponding to the internal shear deformation γ in theboundary condition and in the external virtual work term in Eq.(A.25). Therefore, this fifth equation holds onlywithin each finite element, and is a kind of constraint condition on the freedom γ0. In other words, since theboundary condition Eq.(A.12) does not require continuity of γ across the adjacent elements, this freedom γ0 mustbe eliminated in the element stiffness Eq.(∗) before assemblage of elements. From this fifth equation, we have

γ0 =q5 − ht u

h5, (∗∗)

where, for convenience, we putu ≡ ⌊w1 ϑ1 w2 ϑ2⌋t.

Substitution of Eq.(∗∗) into the right-hand side of the other four equations of Eq.(∗) leads toS1C1S2C2

+

q1q2q3q4

= k u + h γ0 = k u + hq5 − ht u

h5.

Eventually, the final form of the element stiffness equation can be expressed as

f + q(T) = kt u, (A.30)

where the following expressions are defined;

f ≡ ⌊S 1 C1 S 2 C2⌋t, q(T) ≡

q(T)i

, q(T)

i ≡ qi −q5 hi

h5, (A.31a, b, c)

kt ≡ kb −h ht

h5=

EI1 + 12αt

12ℓ3 − 6

ℓ2 −12ℓ3 − 6

ℓ2

4 + 12αtℓ

6ℓ2

2 − 12αtℓ

12ℓ3

6ℓ2

Symm.4 + 12αt

ℓ

. (A.31d)

When the shear deformation is neglected, we can set G → ∞ and thus αt → 0, and Eq.(A.31d) coincides withthe stiffness Eq.(4.25) of the elementary beam theory. It should be noted that the shearing elements of the stiffnessmatrix have the influence coefficient 12αt/(1+12αt) in Eqs.(A.23) and (A.24). Furthermore, this Eq.(A.30) happens tobe identical with the exact stiffness equation derived in the field of the matrix structural mechanics. Namely, thesame stiffness Eq.(A.30) can be obtained by integrating Eq.(A.13) under the general boundary condition Eq.(A.14).Although a word ‘higher order’ has been used in the title of this section, the obtained element is found to be themost appropriate one.

Appendix C

Basics of Tensor Analysis

C.1 Coordinate and Base Vector

O

2, g2

u P

1, g1P1g1

P2g2α

P2g2

P1g1

P1 + P2 cosαsinα

Fig. C.1 Skew rectilinear coordinate sys-tem

Referring to the book1 by Flugge [20], we explain basic character-istics of tensor analysis needed in the field of the mechanics of con-tinua. In a skew rectilinear coordinate system shown in Fig. C.1,let gi (i = 1, 2) using subscripts denote unit base vectors along thecorresponding coordinate axes, and an arbitrary vector P can be de-composed with respect to the base vectors g j as

P = P1 g1 + P2 g2, (C.1)

where the quantities Pi (i = 1, 2) using superscripts are calledthe contravariant components. On the other hand, denoting thehorizontal (x-direction; same as the ‘1’-direction) and vertical (y-direction) unit base vectors by i1 and i2 respectively, we can decom-pose the vector P by using the horizontal and vertical components Px and Py as

Px = P1 + P2 cosα, Py = P2 sinα.

Then, suppose that the vector P is a force vector and that a displacement vector is denoted by a vector u, and thework W done by these force and displacement can be expressed by an inner product (dot product; scalar product)of the two vectors as

W = u · P =(u1 + u2 cosα

) (P1 + P2 cosα

)+ u2 sinα P2 sinα.

Namely, we can writeW = u · P =

(u1 + u2 cosα

)P1 +

(u2 + u1 cosα

)P2. (C.2)

Furthermore, if we define the displacement components using subscripts by

u1 ≡ u1 + u2 cosα, u2 ≡ u2 + u1 cosα, (C.3a, b)

the work expressed by Eq.(C.2) can be rewritten as

W =3∑

i=1

ui Pi = ui Pi, (C.4)

where two-dimensional explanation above is extended to three dimensions. It should be noted that the summationsymbol

∑is omitted in the last expression of Eq.(C.4). This expression is possible only when the condition called

the summation convention is satisfied. The rule is briefly given as

Summation Convention: When the same letter appears twice as one subscript and one superscript in aproduct, the symbol

∑can be omitted.

1 We strongly recommend the readers to read this book if available.

559

560 APPENDIX C. BASICS OF TENSOR ANALYSIS

If another set of base vectors gi (i = 1, 2) using superscripts is defined as is shown in Fig. C.1, geometricalexamination of the figure will lead to a conclusion that the quantities u1 and u2 in Eq.(C.3) using subscripts arecomponents with respect to this new base vectors gi. Such decomposition of the vector P is shown explicitly inFig. C.1. Namely, just like Eq.(C.1), an arbitrary vector can be expressed as

u = u1 g1 + u2 g

2. (C.5)

These components ui using subscripts are called the covariant component. Accordingly, the base vector gi iscalled the covariant base vector, while gi is the contravariant base vector.

The work in Eq.(C.2) can be also rewritten as

W = u1(P1 + P2 cosα

)+ u2

(P2 + P1 cosα

). (C.6)

Therefore, quantities defined by

P1 = P1 + P2 cosα, P2 = P2 + P1 cosα, (C.7a, b)

are the covariant components of the vector P as can be understood from comparison with Eq.(C.3). Hence comes

P = P1 g1 + P2 g

2. (C.8)

As can be estimated from the figure, it should be noted that the base vector gi is not a unit vector but has the

absolute value1

sinα. However, these two kinds of base vectors are orthogonal to each other and satisfy

∣∣∣gi∣∣∣ = 1

sinα, gm · gn = δ

mn , (C.9a, b)

where δmn is the Kronecker’s delta (not a tensor component). Eventually, the work can be expressed as

W =(ui g

i)·(P j g j

)= uiP j gi · g j = uiP j δi

j = ui Pi =(ui gi

)·(P j g

j)= u j P j. (C.10)

Incidentally, in a Cartesian coordinate system (α = π/2), no distinction is needed between the covariant componentsand the contravariant components.

In order to consider geometrical relations, let ξi denote coordinates to the gi-direction, and the coordinatetransformation between the ξ system and the Cartesian x system can be written as

x1 = ξ1 + ξ2 cosα, x2 = ξ2 sinα. (C.11a, b)

Since an alternative definition of the base vector is given by

gi =∂x j

∂ξi i j, (C.12)

substitution of Eq.(C.11) into this equation results in geometrical relations as follows:

g1 = i1, g2 = cosα i1 + sinα i2,

which also show that these are non-dimensional unit vectors. However, as is clear from Eq.(C.9), the covariantbase vectors gi are non-dimensional, but their magnitudes are not unity. Furthermore, it should be noted that thebase vectors are no longer non-dimensional in a polar coordinate system (see Sec. C.4).

C.2 Metric Tensor and Permutation TensorAn inner product of the base vectors of the same kind defines the metric tensor as

gi j ≡ gi · g j, gi j ≡ gi · g j. (C.13a, b)

On the other hand, the inner product of the covariant and contravariant base vectors is zero because these vectorsare orthogonal to each other as is shown in Eq.(C.9b). Using these relations, we can relate the covariant componentto the contravariant component as

ui gi = u j g j → ui g

i · gk = u j g j · gk → ui δik = u j g jk → uk = u j g jk; (C.14)

C.3. COVARIANT DERIVATIVE 561

i.e. the metric tensor exchanges the superscript into the subscript and vice versa.In order to express an outer product (cross product; vector product) of two vectors and a determinant of a

matrix in terms of components, we here introduce the permutation symbol ei jk (not a tensor component) as

ei jk = ei jk =

+1 if (i jk) is even permutation of (123)−1 if (i jk) is odd permutation of (123)0 otherwise

. (C.15)

Then, for an arbitrary 3 × 3 matrix(

C)

as

(C

)≡

C11 C1

2 C13

C21 C2

2 C23

C31 C3

2 C33

,its determinant c can be expressed as

c ≡ det(C

)= Ci

1 C j2 Ck

3 ei jk = C1i C2

j C3k ei jk, elmn c = Ci

l C jm Ck

n ei jk, c =16

Cil C j

m Ckn ei jk elmn. (C.16a, b, c)

Furthermore, defining the determinant of the metric tensor in matrix form by

g ≡ det(gi j

),

1g≡ det

(gi j

), (C.17a, b)

we introduce the permutation tensor ϵi jk as

ϵi jk =√g ei jk, ϵ i jk =

1√g

ei jk. (C.18a, b)

Useful relations of these tensors are, for example,

ϵ i jk ϵimn = δjm δ

kn − δ

jn δ

km, ϵ i jk ϵi jn = 2δk

n, ϵ i jk ϵi jk = 6. (C.19a, b, c)

Using this permutation tensor, the component of a vector w = a × b defined by an outer product of the two vectorsa and b can be given by

wk = ai b j ϵi jk = ϵki j ai b j. (C.20a, b)

In the Cartesian coordinate system, the components of the permutation tensor is equivalent to the permutationsymbol because g = 1.

C.3 Covariant DerivativeIn a polar coordinate system, differential operators such as the nabla ∇ and the Laplacian ∇2 have complicatedexpressions in comparison with those in the Cartesian coordinate system. This is because one of the base vectorsis not constant. Suppose that a vector function u is expressed as u = ui gi using the contravariant components, wecan easily realize that the derivative of u needs differentiation of the base vector gi with respect to the coordinates,because, in general, the base vectors gi are not always constant but functions of the independent variables. Let ξ j

denote the independent variable to the direction of the base vector g j, and the derivative of u with respect to thecoordinate ξ j must be

∂u∂ξ j =

∂(ui gi

)∂ξ j =

∂ui

∂ξ j gi + ui ∂gi

∂ξ j ,

where the second term of the rightmost side is a derivative of the corresponding base vector. Recalling that thebase vector can be defined by Eq.(C.12), we can derive the derivative of the base vectors as

∂gi

∂ξ j = gi, j =∂2xk

∂ξi ∂ξ j ik,

where a comma followed by a subscript j in the middle expression represents the differentiation with respect to theindependent variable ξ j. Let us express the derivative of the base vector as

gi, j = Γi jk gk = Γk

i j gk, (C.21)


where the quantity Γi jk is called the Christoffel symbol (not tensor components). Then, the inner product withanother base vector leads to

gi, j · gk = Γi jl gl · gk = Γi jl δ

lk = Γi jk, gi, j · gk = Γk

i j. (C.22a, b)

Using this Eq.(C.21), we can write the derivative above as

∂u∂ξ j =

∂(ui gi

)∂ξ j = ui

, j gi + ui gi, j =(ui, j + uk Γi

jk

)gi = ui

∣∣∣j gi → ui

∣∣∣j = ui

, j + uk Γijk, (C.23a, b)

where ui∣∣∣j is called the covariant derivative.

C.4 Physical Components in Polar Coordinate SystemIt should be noted that the base vectors are not always unit just like the base vector gi of Eq.(C.9) in Fig. C.1.Moreover, the base vectors may be functions of independent variables as has been explained in Sec. C.3. In suchcases, the tensor components do not necessarily have any physical meanings although the corresponding tensorsare of course physical entities. In order to show one typical example, we here examine quantities and differentialoperators in the polar coordinate system as (ξ1 = r, ξ2 = θ, ξ3 = z). First of all, the base vectors are defined by

gr = i1 cos θ + i2 sin θ, gθ = −i1 r sin θ + i2 r cos θ, gz = i3, (C.24a, b, c)

where gθ has a dimension of length r. From these, we have metric tensors as

grr = 1, gθθ = r2, grr = 1, gθθ =1r2 , (C.25a, b, c, d)

and non-zero terms of the Christoffel symbols are given by

Γrθθ = r, Γθθr = −r, Γrθθ = −r, Γθrθ =

1r. (C.26a, b, c, d)

Using these quantities, we can write explicitly the equilibrium equations σ ji∣∣∣j + f i = 0 as follows:

∂σrr

∂r+σrr

r+∂σrθ

∂θ− rσθθ +

∂σrz

∂z+ f r = 0, (C.27a)

∂σrθ

∂r+

3σrθ

r+∂σθθ

∂θ+∂σθz

∂z+ f θ = 0, (C.27b)

∂σrz

∂r+σrz

r+∂σzθ

∂θ+∂σzz

∂z+ f z = 0 (C.27c)

where f r and f θ are the body force components to the directions of gr and gθ respectively. Since the base vectorgθ has a dimension of length r as has been shown in Eq.(C.24), the corresponding components to the θ-directionof any tensors do not have clear physical meanings. Then, define physical components of the stress tensor and theforce vector by adjusting the dimension of the base vector gθ as

τrr ≡ σrr, τrθ ≡ rσrθ, τθθ ≡ r2σθθ, τzz ≡ σzz, τrz ≡ σrz, τθz ≡ rσθz, qr ≡ f r, qθ ≡ r f θ, qz ≡ f z,(C.28a, b, c, d, e, f, g, h, i)

and the equilibrium Eq.(C.27) can be rewritten as

∂τrr

∂r+

1r∂τrθ

∂θ+τrr − τθθ

r+∂τrz

∂z+ qr = 0, (C.29a)

∂τrθ

∂r+

2τrθ

r+

1r∂τθθ

∂θ+∂τθz

∂z+ qθ = 0, (C.29b)

∂τrz

∂r+τrz

r+

1r∂τzθ

∂θ+∂τzz

∂z+ qz = 0, (C.29c)

which are well-known expressions of the equilibrium equation found in many books.

C.4. PHYSICAL COMPONENTS IN POLAR COORDINATE SYSTEM 563

On the other hand, tensor components of the displacement gradients are expressed as

ur |r =∂ur

∂r, ur |θ =

∂ur

∂θ− r uθ, uθ

∣∣∣r =

∂uθ

∂r+

1r

uθ, uθ∣∣∣θ=∂uθ

∂θ+

1r

ur. (C.30a, b, c, d)

We here again define a physically meaningful ‘vector’ u having the physical components corresponding to thetensor components of u by

vr ≡ ur, vθ ≡ r uθ. (C.31a, b)

Then the tensor components of the displacement gradients can be expressed by the physical components as

ur |r =∂vr

∂r, ur |θ =

∂vr

∂θ− vθ, uθ

∣∣∣r =

∂

∂r

(vθ

r

)+

1r2 v

θ, uθ∣∣∣θ=

1r∂vθ

∂θ+

1rvr. (C.32a, b, c, d)

Since the covariant strain tensor ϵi j is conjugate with the contravariant stress tensor σi j with respect to the elasticwork, we can express the covariant components of the displacement tensor above using the metric tensor as

ur |r = grr ur |r =∂vr

∂r, ur |θ = grr ur |θ =

∂vr

∂θ− vθ, uθ |r = gθθ uθ

∣∣∣r = r

∂vθ

∂r, uθ|θ = gθθ uθ

∣∣∣θ= r2

(1r∂vθ

∂θ+

1rvr).

(C.33a, b, c, d)Therefore, the strain tensor components can be defined by

ϵrr =∂vr

∂r, 2ϵrθ = r

1r∂vr

∂θ+∂vθ

∂r− 1

rvθ, ϵθθ = r2

(1r∂vθ

∂θ+

1rvr). (C.34a, b, c)

Incidentally, the covariant base vectors corresponding to the contravariant base vectors in Eq.(C.24) are given by

gr = i1 cos θ + i2 sin θ, gθ = −i11r

sin θ + i21r

cos θ. (C.35a, b)

Hence, physical components εi j of the strain tensor components ϵi j can be defined by

εrr = ϵrr, εrθ =1rϵrθ, εrr =

1r2 ϵθθ. (C.36a, b, c)

Eventually, the physical components of the strain can be defined by

εrr =∂vr

∂r, εrθ =

12

(1r∂vr

∂θ+∂vθ

∂r− 1

rvθ), εθθ =

1r∂vθ

∂θ+

1rvr. (C.37a, b, c)

Components relating to the z-axis are almost the same as those in the Cartesian coordinate system and are

εzz =∂vz

∂z, εrz =

12

(∂vr

∂z+∂vz

∂r

), εθz =

12

(∂vθ

∂z+

1r∂vz

∂θ

). (C.38a, b, c)

As has been shown above, the tensor components do not always represent the corresponding physical quantities.In particular, when a constitutive model is specified by the following tensor equation

σi j = Ci jkl ϵkl, (C.39)

the coefficient tensor C itself may not have a clear physical meaning. The physical meanings must be taken intoaccount on the physical basis using physical components of the stress and strain tensors; e.g.

τi j = Ci jklεkl (C.40)

must express clear physical characteristics in the relation between physical components, although this is not atensor equation. A shell theory in an arbitrary curvilinear coordinate system can be found, for example, in atextbook [84]. Furthermore, a great care must be needed in defining physical quantities of the tensors especially infinite displacement theory, because the base vectors within the framework of Lagrangian approach depend on thedeformation undertaken, and are no longer unit or orthogonal system, as has been shown through the definition ofthe physical stress in Eq.(B.22) for a finite displacement theory of in-plane beams.


C.5 Coordinate Transformation and TensorsWhat are the tensors? They represent physical quantities independent of coordinate systems chosen for the con-venience of people. Therefore, tensor equations such as Eq.(C.39) also hold no matter which coordinate systemsare selected. On the contrary, since the base vectors have different characteristics depending on each coordinatesystem, the tensor components differ according to the coordinate system. For example, consider two different arbi-trary coordinate systems as x-system and x′-system. Then the base vectors are related to each other like Eq.(C.12)as

gk = βki′ g

i′ , βki′ ≡

∂xk

∂xi′ , gk′ = βk′i g

i, βk′i ≡

∂xk′

∂xi . (C.41a, b, c, d)

The quantities βki′ are the coordinate transformation coefficients (coordinate transformation matrix) and satisfy

βik′ β

k′j = δ

ij, βk

i′ βj′

k = δj′

i′ . (C.42a, b)

Therefore, the components of an arbitrary vector u can be related to each other as

u = ui gi = ui β

ij′ g

j′ = u j′ gj′ → u j′ = β

ij′ ui and ui = β

j′

i u j′ . (C.43a, b, c)

This rule of transformation applies to any tensors; e.g. the tensors of second order are transformed as

σi j = βik′ β

jl′ σ

k′l′ , σi′ j′ = βi′k β

j′

l σkl, (C.44a, b)

and four transformation coefficients are necessary for the tensors of fourth order.

Appendix D

Average Characteristics of Composites

concretes soils

rocks and FRP’s polycrystals

averaged materialE2

E1

E

Fig. D.1 Materials with micro-structures and averaging

D.1 Materials with Micro-StructuresComposites such as Glass/Carbon-Fiber-Reinforced-Polymers (GFRP, CFRP), concretes and even crystalline met-als like steel are not homogeneous, and have micro-structures made of different phases inside materials. Forexample, concretes are composed of, at least, two different materials such as cement paste and aggregates whichhave different Young’s moduli, E1 and E2. Rocks have many cracks while many fibers are distributed in FRP’s;solid lines in the bottom-left figure of Fig. D.1 represent cracks in rocks or fibers in FRP’s. Soils are composedof grains, air and water. Polycrystals are made of many randomly distributed single crystals which have differentorientations and shapes. In most practices of the civil engineering field, however, only the average behavior isusually necessary to design structures, but still it is much better if we can take some effects of such microscopiccharacteristics into account explicitly. If you use the Finite Element Method (FEM), it is possible to model suchmicro-structures directly by using certain finite elements. But, since a huge number of elements are needed for thatpurpose, it is not realistic to solve numerically a Boundary-Value-Problem (BVP) of complicated real structuresmade of composites. On the other hand, if we can use some analytical estimates of average properties of com-posites, the number of finite element needed can be significantly reduced. Such analytical approaches can be alsoused to design the micro-structures such as material, orientation, shape and volume fraction of inhomogeneities incomposites. We here explain one of such analytical averaging1 methods.

1 Recently the terminology ‘homogenization’ is frequently used for such averaging estimates, because the word is employed in a famouscontribution [113] utilizing the singular perturbation method between microscopic and macroscopic variables. However, since the methodis used through implementation into a numerical approach, FEM, the word ‘averaging’ is used for analytical approaches in this textbook.

565

566 APPENDIX D. AVERAGE CHARACTERISTICS OF COMPOSITES

E1

E1E2

E2

ϵσ

σϵ

Fig. D.2 Voigt and Reuss models

One of the simplest classical averaging methods is called the Voigtmodel. Suppose that two kinds of springs with different spring constants,E1 and E2, are connected parallel to each other as is shown in the left figureof Fig. D.2, and the overall extension ϵ is applied. Then the total resistancecan be evaluated by

σ = (1 − f ) σ1 + f σ2, σ1 = E1 ϵ, σ2 = E2 ϵ, (D.1a, b, c)

where f expresses the ratio of the number of the second spring to the totalnumber of two springs. This relation can define an average spring constant(corresponding to the average Young modulus) by

σ = E ϵ → E ≡ (1 − f ) E1 + f E2 (Voigt), (D.2a, b)

which is a simple volumetric average of the two Young’s moduli. On the other hand, when two springs are alignedstraight as is shown in the right figure of Fig. D.2, if the overall force σ is applied, the total extension becomes

ϵ = (1 − f ) ϵ1 + f ϵ2, ϵ1 =σ

E1, ϵ2 =

σ

E2, (D.3a, b, c)

which yields another definition of the average Young modulus as

ϵ =σ

E→ E ≡

(1 − f

E1+

fE2

)−1

(Reuss). (D.4a, b)

This is a volumetric average of the two compliances, and is called the Reuss model. Well, then, how do you averagethe Poisson ratios?

These two estimates are at present known as the upper and lower bounds of the average Young modulus, andmost experimental data are found to lie between these bounds. However, since these bounds are not so close toeach other even when f remains small as can be seen in Fig. D.8, they cannot be used in practice and must beimproved. The most important factor missing in these averages is the mechanical interactions between the twophases (springs) of composites. Furthermore, the geometric properties such as shapes and orientations of phases(e.g. shape of aggregates in concretes) are not taken into account, so that some more sophisticated mechanicalconsideration must be necessary in a microscopic sense. We here explain the formulation of averaging based onthe ‘Micromechanics2’ in the book by Mura [57]. For example, curves indicated by ‘Mori-Tanaka’ in Fig. D.8are such estimates called the ‘Mori-Tanaka average’. In the same figure, the results (‘SC’) of the self-consistentmethod by Hill [28] are also shown for comparison.

D.2 Inhomogeneity and Inclusion

D.2.1 Inhomogeneity and Eshelby Solution

Em

Eiϵ = const.

σ

σ

Ω Ω

Fig. D.3 the Eshelby solution

Consider a two-phase material containing an infinite numberof inhomogeneities randomly distributed in the matrix mate-rial. All the inhomogeneities have the same shape and orien-tation, and have material properties different from those of thematrix. In order to estimate the mechanical field disturbed byexistence of the inhomogeneity, we examine a part surround-ing one inhomogeneity occupying the region Ω illustrated inthe left figure of Fig. D.3. Both materials are assumed to beisotropically elastic; the Young modulus of the matrix materialis Em, and its Poisson’s ratio is νm, while the Young modulusof the inhomogeneity is Ei, and its Poisson’s ratio is νi. Theinhomogeneity has an ellipsoidal shape with three principalradii, a1, a2 and a3.

As is shown in the right figure of Fig. D.3, when only one inhomogeneity is embedded in an infinite bodysubjected to some applied force σ at infinity, Eshelby [19] found that the strain field inside inhomogeneity

2 The terminology ‘micro’ does not directly mean the size of 10−6 m, but implies microscopic mechanics between two regions with dis-tributed ‘eigenstrains’ in a material. Thus ‘Micromechanics’ is a unique name originally designated by Prof. Mura.

D.2. INHOMOGENEITY AND INCLUSION 567

is uniform; i.e. ϵ = const. in Ω. This conclusion is valid only when the materials are isotropic elastic andwhen the inhomogeneity has ellipsoidal shape. This famous finding suggests that the disturbed field due toone inhomogeneity of two-phase composites becomes equivalent to the field of a homogeneous infinite bodywith an ‘appropriate’ constant residual strain distributed only within the region Ω. Such a residual strainis an inelastic strain just like thermal expansion, and is called ‘eigenstrain’ ϵ∗ in the book [57]. Mura calls theregion Ω with eigenstrain ‘inclustion’, while the region of material different from the matrix material is called‘inhomogeneity’. However, in this textbook, we sometimes call the region of ‘inhomogeneity’ by ‘inclusion’.The equivalent replacement of the inhomogeneity by an inclusion is called the ‘equivalent inclusion method’ [19]explained later on.

D.2.2 Governing Equations(1) Statement of Problem with One Inhomogeneity

Suppose that only one inhomogeneity occupies the region Ω in an infinite body D as is shown in the left figure ofFig. D.4. The displacement u is related to the total strain ϵ as

ϵi j =12

(ui, j + u j,i

). (D.5)

In elastic problems, the constitutive relations of the two materials are expressed by Hooke’s law as

σi j = Cmi jkl ϵkl = Cmi jkl uk,l in D −Ω, σi j = Cii jkl ϵkl = Cii jkl uk,l in Ω, (D.6a, b)

where Cm and Ci are the elasticity tensors of the two materials. Incidentally, manipulation about the secondequality of both equations above is based on the relation given in Eq.(D.5) and the symmetry of elastic moduli asCki jkl = Cki jlk (k = m, i). The tensor can be explicitly expressed using the Lame constans µk and λk as

Cki jkl = µk(δik δ jl + δil δ jk

)+ λk δi j δkl, (k = m, i), (D.7)

where δi j is the Kronecker delta. Also, the Lame constants are related to Young’s modulus and Poisson’s ratio as

µk =Ek

2 (1 + νk), λk =

νk Ek(1 + νk) (1 − 2νk)

, (k = m, i). (D.8a, b)

Without the body force, the equilibrium equations of force and moment are given by

σ ji, j = 0, σi j = σ ji → or σi j, j = 0. (D.9a, b, c)

Eq.(D.9c) is the result from substitution of Eq.(D.9b) into Eq.(D.9a). The corresponding boundary condition atinfinity is specified as

n j σ ji = fi at |x| → ∞, (D.10)

where n is a unit outer normal vector of the surface at infinity where the force f is applied. Also, the continuityconditions of displacement u and surface traction (ν · σ) must hold on the interface ∂Ω, where ν is a unit outernormal of the surface.

Therefore, as can be easily imagined, consideration of the continuity condition on the interface ∂Ω makes thesolution procedure of such a two-phase body very complicated. Let the fields of stress and strain be decomposedinto a homogeneous one without any inhomogeneities and a disturbed one by the inhomogeneity as

σ(x) = σ0 + σd(x), ϵ(x) = ϵ0 + ϵd(x), σd(x)→ 0, ϵd(x)→ 0 as |x| → ∞, (D.11a, b, c, d)

where σd and ϵd represent the stress and strain disturbances due to the existence of the inhomogeneity, and σ0 is auniform stress field of the homogeneous body subjected to the applied force f at infinity.

(2) Inclusion Problem with Eigenstrain

We decompose the original problem into a part of homogeneous state of the body subjected to the applied forceand a part of disturbed state due to the inhomogeneity as is shown in the middle two figures of Fig. D.4. Thehomogeneous state can be easily solved to obtain uniform fields of stress and strain as σ0 = Cm : ϵ0. As for thedisturbed state of the stress σd and strain ϵd, instead of solving the problem of a two-phase body, we solve an


CmCi

Ω

Cm

σ0, ϵ0 σ0, ϵ0

Original problem Homogeneous state

Ω

Cm

Disturbed state

Ω

Cm

Auxiliary problem

disturbed field

ϵd

− = =

ϵd

ϵd

Ci Cmϵ∗

Fig. D.4 Decomposition of the problem; inhomogeneity (Ci in Cm) is replaced by an inclusion (ϵ∗ in Cm)

auxiliary problem of a homogeneous body with one inclusion having some residual strain ϵ∗ as is shown in therightmost figure, on the basis of Eshelby’s finding mentioned above. Material all over the domain is homogeneous,and its elastic modulus is Cm in this auxiliary problem. But, in the region Ω, there distributes a certain residualstrain, ϵ∗. This particular residual strain is called the ‘eigenstrain’. Evaluation process of the appropriate eigenstrainwill be explained later in Sec. D.2.4, and, for the time being, the disturbed state is solved with a given eigenstrain.For simplicity, the superscript ‘d’ is omitted below.

Since the eigenstrain is an inelastic strain, the total strain is made of the elastic strain e and the eigenstrain as

ϵi j(x) = ei j(x) + ϵ∗i j(x), ϵ∗i j

, 0 in Ω= 0 in D −Ω . (D.12a, b)

Then, the Hooke law of Eq.(D.6) must be replaced by the relation between the elastic strain and the stress as

σi j = Ci jkl ekl = Ci jkl

(ϵkl − ϵ∗kl

)= Ci jkl

(uk,l − ϵ∗kl

)in D, (D.13)

where Eqs.(D.5) and (D.12) and the symmetry of the elastic moduli are used. For simplicity, the superscript ‘m’ isomitted. Substitution of Eq.(D.13) into Eq.(D.9c) results in the equilibrium equation in terms of the displacementu as

Ci jkl uk,l j = Ci jkl ϵ∗kl, j. (D.14)

Since no force is applied at infinity, the boundary condition is given by

n j σ ji = 0 at |x| → ∞. (D.15)

Putting Eq.(D.13) into Eq.(D.15), we express this boundary condition as

n j Ci jkl uk,l = n j Ci jkl ϵ∗kl = 0 at |x| → ∞ [

∵ ϵ∗(|x| → ∞) = 0], (D.16)

in terms of the displacement u. We call this problem the ‘inclusion problem’.

D.2.3 Fourier Analysis(1) Fourier Integral and Fourier Transformation

The inclustion problem can be solve by the Fourier analysis. First the eigenstrain ϵ∗ is expressed by the Fourierintegral as

ϵ∗i j(x) =$ ∞

−∞ϵ∗i j(ξ) exp (i ξ · x) dξ. (D.17)

The corresponding Fourier transform ϵ∗ can be obtained as

ϵ∗i j(ξ) =1

(2π)3

$ ∞

−∞ϵ∗i j(x) exp (−i ξ · x) dx. (D.18)

Similarly, the Fourier integral of the displacement is

ui(x) =$ ∞

−∞ui(ξ) exp (i ξ · x) dξ. (D.19)


Substitution of Eqs.(D.17) and (D.19) into Eq.(D.14) yields

−∫ ∞

−∞Ci jkluk(ξ)ξlξ j exp (i ξ · x) dξ =

∫ ∞

−∞Ci jklϵ

∗kl(ξ) i ξ j exp (i ξ · x) dξ,

where a triple-integral is abbreviated by a single integral. Equality of the integrands in both sides results in

(Ci jkl ξl ξ j) uk = −i Ci jkl ϵ∗kl ξ j, (D.20)

which is a Fourier transform of the equilibrium Eq.(D.14). Hence comes an algebraic equation for u as

Kik uk = Xi, (D.21)

whereKik ≡ Ci jkl ξl ξ j, Xi ≡ −i Ci jkl ϵ

∗kl ξ j.

Then, the Fourier transform of the displacement is obtained from this equation as

uk = (Kik)−1 Xi =Nki(ξ)D(ξ)

Xi, (D.22)

where Ni j is a cofactor matrix of the matrix Ki j, and D is its determinant calculated by

Ni j =12

eikl e jmn Kmk Knl, D =16

ei jk elmn Kil K jm Kkn, (D.23a, b)

where ei jk is the permutation symbol. Finally, the displacement can be obtained by the Fourier inverse transforma-tion as

ui(x) = −i∫ ∞

−∞C jlmn ϵ

∗mn(ξ) ξl Ni j(ξ) D−1(ξ) exp (i ξ · x) dξ. (D.24)

The corresponding strain and stress are also expressed by the Fourier integrals as

ϵi j(x) =12

∫ ∞

−∞Cklmn ϵ

∗mn(ξ) ξl

ξ j Nik(ξ) + ξi N jk(ξ)

D−1(ξ) exp(i ξ · x) dξ, (D.25)

σi j(x) = Ci jkl

[∫ ∞

−∞Cpqmn ϵ

∗mn(ξ) ξq ξl Nkp(ξ) D−1(ξ) exp(i ξ · x) dξ − ϵ∗kl(x)

]. (D.26)

(2) Green’s Function

Substituting Eq.(D.18) into Eq.(D.24), we have

ui(x) = −i∫ ∞

−∞dξC jlmn

1(2π)3

∫ ∞

−∞dx′ ϵ∗mn(x′) exp

(−i ξ · x′) ξl Ni j(ξ) D−1(ξ) exp (i ξ · x) . (D.27)

If we define a function G by

Gi j(x − x′) ≡ 1(2π)3

∫ ∞

−∞dξ Ni j(ξ) D−1(ξ) exp

i ξ · (x − x′)

. (D.28)

then the displacement in the above equation can be rewritten as

ui(x) = −∫ ∞

−∞C jlmn ϵ

∗mn(x′)

(∂Gi j(x − x′)

∂xl

)dx′. (D.29)

Judging from the form of the integrand, we can call this function G Green’s function, because the derivative of Gin the integrand expresses ‘an influence function of x subjected to a unit eigenstrain at x′.’ Actually, the functionG satisfies the following equation;

Ci jkl Gkm,l j(x − x′) + δim δ(x − x′) = 0, (D.30)

where δim is the Kronecker delta while δ(x − x′) is the Dirac delta function. In three dimensions, δ(x − c) ≡δ(x1 − c1) δ(x2 − c2) δ(x3 − c3). Try to prove it in a problem No.1 of Homework D-1. On the other hand, since theequilibrium equation of an infinite body subjected to an arbitrary body force X is expressed by

Ci jkl uk,l j + Xi = 0, (D.31)

comparison of Eq.(D.30) with this equilibrium Eq.(D.31) indicates a physical meaning of the Green function G asfollows:


Gkm(x− x′) is the displacement component at x in the direction of xk of a body subjected to a unitconcentrated force at x′ in the direction of xm.

Namely, it is the same as the influence line in the structural mechanics.Using derivatives of Eq.(D.29), we can express the strain and stress as

ϵi j(x) = −12

∫ ∞

−∞Cmklmn ϵ

∗mn(x′)

∂2Gik(x − x′)

∂xl ∂xi+∂2G jk(x − x′)

∂xl ∂x j

dx′, (D.32)

σi j(x) = −Cmi jkl

[∫ ∞

−∞Cmpqmn ϵ

∗mn(x′)

∂2Gkp(x − x′)∂xq ∂xl

dx′ + ϵ∗kl(x)]. (D.33)

The explicit form of Green’s function of an isotropic elastic body has been obtained as

D(ξ) = µ2m (λm + 2µm) ξ6, Ni j(ξ) = µm ξ2

(λm + 2µm) δi j ξ

2 − (λm + µm) ξi ξ j

, (D.34a, b)

Gi j(x − x′) =1

4πµm

δi j

|x − x′| −1

16πµm (1 − νm)∂2

∂xi ∂x j

∣∣∣x − x′∣∣∣ , ξ2 ≡ ξk ξk. (D.34c, d)

b

b

x1x2

x3

O

O

x1x3x2

Fig. D.5 Screw dislocation

Example: Infinite number of dislocations exist inside crys-talline metals, and are kinds of gaps in the layers of atom grid.They are sources of plastic deformation explained in Chap. 9,and can be modeled by some distribution of inelastic strains; i.e.eigenstrains. For example, one screw dislocation is the gap b inthe x3-direction lying straight to the x1-direction as is illustratedin Fig. D.5. The gap b is called the Burgers vector. This gap canbe modeled by the distribution of eigenstrain as

ϵ∗23 =12

b H(−x1) δ(x2), (D.35)

where H(x) is the Heaviside function. Since its Fourier transform is

ϵ∗23 = −b δ(ξ2)8π2 i ξ1

, (D.36)

substitution of this expression into Eq.(D.24) results in zero displacement components in the x1- and x2-directions;i.e. u1 = 0 and u2 = 0, while non-zero component in the x3-direction is obtained as

u3 = −i" ∞

−∞

2ξ2

1 + ξ22

−b8π2 i ξ1

ξ2 exp i (ξ1x1 + ξ2x2) dξ1 dξ2 =b

2πtan−1 x2

x1, (D.37)

which is eventually a multivalued function representing the gap of b in the x3-direction along the negative part ofthe x1 axis.

(3) Eshelby Tensor — Isotropic Case

Assume that the material is isotropically elastic, and that the inclusion has an ellipsoidal shape. As a result fromEshelby’s finding, when the eigenstrain inside Ω is uniform, the strain field is also uniform inside Ω but decays as|x| goes to infinity. In such a case, the eigenstrain ϵ∗ in Eq.(D.32) can be put out of the integrand, and the equationcan be integrated to find a symbolic relation as

ϵi j(x) = Si jkl(x) ϵ∗kl. (D.38)

Then, inside the region of inclusion Ω, the fourth-order tensor S(x) becomes uniform; i.e. S(x) = const. (x ∈ Ω),and depends only on the Poisson ratio of the matrix material and the ratios of the principal radii ai (i = 1, 2, 3) ofthe inclusion. This tensor S is called the Eshelby tensor [57]. Especially when the inclusion has a spherical shape,its components inside Ω are expressed by

Si jkl = α13δi j δkl + β

12

(δik δ jl + δil δ jk

)− 1

3δi j δkl

= α Ai jkl + β Bi jkl, (D.39)


Cm

CiΩ

Ω

Cm

Cm

ϵ∗

σ0, ϵ0 σ0, ϵ0

Original problem Auxiliary problemFig. D.6 Equivalent inclusion method

whereα ≡ 1 + νm

3 (1 − νm), β ≡ 2 (4 − 5νm)

15 (1 − νm). (D.40a, b)

And, these two tensors A and B are the basic isotropic tensors defined by

Ai jkl ≡13δi j δkl, Bi jkl ≡

12


)− 1

3δi j δkl, (D.41a, b)

and have a special property of ‘orthogonality’ as

Ai jmn Bmnkl = 0. (D.42)

Using these isotropic tensors, we can express the isotropic elasticity tensor as

Cmi jkl = 3κm13δi j δkl + 2µm

12


)− 1

3δi j δkl

= 3κm Ai jkl + 2µm Bi jkl, (D.43)

where κm is the bulk modulus related to the Lame constants as

κm ≡ λm +23µm. (D.44)

Namely, the component relating to A represents the volumetric part of deformation, while the component of B isthe shearing part. Similarly, the identity tensor I can be written as

Ii jkl = 113δi j δkl + 1

12


)− 1

3δi j δkl

= 1 Ai jkl + 1 Bi jkl. (D.45)

Nemat-Nasser and Hori [61] define a short-hand notation of fourth-order isotropic tensors specifying only thecoefficients of A and B as

S = (α, β), Cm = (3κm, 2µm), I = (1, 1). (D.46a, b, c)

Using the orthogonality of Eq.(D.42), we can carry out manipulations of fourth-order tensors easily as follow:

S − I = (α − 1, β − 1) ,(Cm

)−1 S =(α

3κm,

β

2µm

). (D.47a, b)

This kind of manipulations is quite useful in solving inclusion problems with a spherical inhomogeneity withoutusing any numerical tools of computations; e.g. you can easily solve a problem No.2 of Homework D-1.

D.2.4 Equivalent Inclusion MethodLet’s return to the problem to calculate the fields near one inhomogeneity in an infinite body subjected to theapplied forces at infinity as is depicted in the left figure of Fig. D.6. Eshelby’s finding [19] ensures us that thestrain field inside Ω is uniform, as long as the ellipsoidal elastic inhomogeneity is embedded in the isotropicallyelastic infinite body. This conclusion suggests that it is enough to solve a certain corresponding auxiliary problem


with a certain eigenstrain ϵ∗ given in Ω as is illustrated in the right figure of Fig. D.6. In the previous section, theeigenstrain is a given function. But, in this section, we show one method to evaluate the value of the eigenstrainappropriate for the original inclusion problem.

After the decomposition explained in Fig. D.4, the uniform stress and strain fields σ0 and ϵ0 can be easilysolved to obtain

σ0 = Cm : ϵ0. (D.48)

Let σi j(x) and ϵi j(x) denote the disturbed fields of stress and strain, and we can express the Hooke law as follow:

σ0i j + σi j(x) = Cmi jkl

(ϵ0

kl + ϵkl(x))

in D −Ω, σ0i j + σi j(x) = Cii jkl

(ϵ0

kl + ϵkl(x))

in Ω. (D.49a, b)

And, Eshelby [19] showedϵi j(x) = const. in Ω. (D.50)

However, this original problem is not so easy to solve directly with the continuity condition on the interfacesurrounding the inhomogeneity. Instead, the auxiliary problem as is shown in the rightmost figure of Fig. D.4 canbe used by introduction of a proper eigenstrain. Then the corresponding auxiliary problem subjected to the appliedforce is depicted in the right figure of Fig. D.6. If we can find the eigenstrain ϵ∗ so that the fields in both the figuresof Fig. D.6 become equivalent to each other, the original two-phase problem is considered to be solved indirectly.This approach is called the ‘equivalent inclusion method’, and the expression of the stress field in Eq.(D.49b)can be replaced by the following relation:

σ0i j + σi j(x) = Cmi jkl

(ϵ0

kl + ϵkl(x) − ϵ∗kl

), (D.51)

where the eigenstrain is specified as

ϵ∗i j

= 0 in D −Ω, 0 in Ω (unknown but uniform) . (D.52)

Consequently, the value of the eigenstrain ϵ∗ must be evaluated by equating the right hand sides of Eqs.(D.49b)and (D.51) as

Cii jkl

(ϵ0

kl + ϵkl

)= Cmi jkl

(ϵ0

kl + ϵkl − ϵ∗kl

)in Ω. (D.53)

Since Eq.(D.38) relates the disturbed strain field to the eigenstrain as

ϵkl = Sklmn ϵ∗mn, (D.54)

substitution of this equation into Eq.(D.53) results in

Cii jkl

(ϵ0

kl + Sklmn ϵ∗mn

)= Cmi jkl

(ϵ0

kl + Sklmn ϵ∗mn − ϵ∗kl

)→

Cii jkl Sklmn −Cmi jkl (Sklmn − Iklmn)

ϵ∗mn =

(Cmi jkl −Cii jkl

)ϵ0

kl.

Hence, the solution of the eigenstrain can be obtained as

ϵ∗i j =Cii jmn Smnkl −Cmi jmn (Smnkl − Imnkl)

−1 (Cmklpq −Ciklpq

)ϵ0

pq. (D.55)

Once the eigenstrain is obtained, the disturbed strain field inside the inhomogeneity can be calculated by substitut-ing Eq.(D.55) into Eq.(D.54), and the corresponding stress field is evaluated from Eq.(D.51).

Homework D-1

1. Prove that the function G satisfies Eq.(D.30). Note that the Dirac delta function is a Fourier transform of ‘1’expressed as

δ(x − x′) =1

(2π)3

∫ ∞

−∞exp

i ξ · (x − x′)

dξ. (D.56)

2. Obtain the ratios of the stresses inside an inhomogeneity; e.g. the ratios of(σ0

12 + σ12

)to σ0

12 and of(σ0

kk + σkk

)to σ0

kk, using the equivalent inclusion method, when the ratio of shear moduli of the two phasesis either µi/µm = 10 or 1/10 with the same Poisson ratios of νi = νm = 0.3.

D.3. AVERAGE PROPERTIES OF COMPOSITES 573

D.3 Average Properties of Composites

D.3.1 Mori-Tanaka Average of Elastic BodyThe disturbed mechanical fields are created by the existence of the inhomogeneities in the matrix material. Thesedisturbances are introduced by the mechanical interactions between the phases, one of which is the interactionbetween one inhomogeneity and the matrix. Another source of disturbances is the interaction between manyinhomogeneities. When the inhomogeneities are distributed far away from each other, the disturbed strain fieldcan be approximated to become uniform on the basis of Eshelby’s discovery, and can be evaluated by the methodexplained in Sec. D.2. However, if the distance between nearby inhomogeneitis is in the same order of the size ofa typical inhomogeneity, the strain in each inhomogeneity is no longer uniform. Recently, the latter interaction hasbeen examined by many researchers (e.g. [41, 77]) under some kinds of assumptions on the geometrical distributionpattern of inhomogeneities. We here neglect this latter interaction, and thus, as an approximate approach, weexplain one of the averaging methods of characteristics of composites called the ‘Mori-Tanaka average3’, using thesolution obtained in Sec. D.2.

Cm

Ci

Cm

ϵ∗

Fig. D.7 An equivalent model of composites

A composite in the left figure of Fig. D.7 is re-placed by a uniform infinite body of the isotropicallyelastic material with ellispoidal inclusions with someeigenstrains inside as is shown in the right figure ofFig. D.7. Let ⟨ϵ⟩m denote the average strain in the ma-trix part, and we can write the Hooke law as

⟨σ⟩m = Cm : ⟨ϵ⟩m , (D.57)

where ⟨·⟩m represents the average over the matrix do-main, and ⟨σ⟩m is the average stress in the matrix. Animportant magical statement of this method is that the average strain ⟨ϵ⟩m is not defined by an actual average ofthe evaluated mechanical field but remains unknown hoping that the effect of the neglected interaction betweeninhomogeneities may be taken into account.

Suppose that one inhomogeneity is added into the portion of the matrix where the strain field is approximately⟨ϵ⟩m. Since there are already a huge number of inhomogeneities, such addition of only one inhomogeneity does notaffect the overall mechanical characteristics at all. Therefore, instead of solving this two-phase problem with manyinhomogeneities directly, Mori and Tanaka [55] consider the problem of an infinite body having the average strain⟨ϵ⟩m with only one inclusion. Then, the strain field in this inhomogeneity can be approximated by the field of oneinclusion in the infinite body with the strain field ⟨ϵ⟩m explained in Sec. D.2. Let Hooke’s law of the inhomogeneitybe given by

⟨σ⟩i = Ci : ⟨ϵ⟩i , (D.58)

and let ⟨γ⟩i denote the disturbed part of the strain, and the total strain inside one inhomogeneity is expressed as

⟨ϵ⟩i = ⟨ϵ⟩m + ⟨γ⟩i , (D.59)

where ⟨·⟩i is the average over the inhomogeneities. On the basis of the equivalent inclusion method, the stress fieldin the inclusion must satisfy the equivalency condition Eq.(D.53) as follows:

⟨σ⟩i = Ci :⟨ϵ⟩m + ⟨γ⟩i = Cm :

⟨ϵ⟩m + ⟨γ⟩i − ⟨ϵ∗⟩i . (D.60)

Since the Eshelby tensor relates the disturbed strain to the eigenstrain as

⟨γ⟩i = S : ⟨ϵ∗⟩i , (D.61)

substitution of Eq.(D.61) into Eq.(D.60) yields

Ci : ⟨ϵ⟩m + S : ⟨ϵ∗⟩i = Cm : ⟨ϵ⟩m + (S − I) : ⟨ϵ∗⟩i .

Several steps of manipulations lead to an expression of the eigenstrain as

⟨ϵ∗⟩i =Cm − (

Cm − Ci)

S−1 (

Cm − Ci)

: ⟨ϵ⟩m =Cm − (

Cm − Ci)

S−1 (

Cm − Ci) (

Cm)−1 : ⟨σ⟩m , (D.62)

3 The idea of this method is the same as the basis of a scheme in physics by which average electric current is estimated from flow of aninfinite number of free electrons with negative electric charge.


where Eq.(D.57) is used, and this Eq.(D.62) is essentially the same as Eq.(D.55). After substitution of Eq.(D.61)into Eq.(D.60), considering the relation of Eq.(D.57), we can express the average stress in the inclusion as

⟨σ⟩i = Cm : ⟨ϵ⟩m + (S − I) : ⟨ϵ∗⟩i = ⟨σ⟩m + Cm (S − I) : ⟨ϵ∗⟩i . (D.63)

Defining the volume fraction of the inhomogeneity by

f ≡∑

VΩV

, (D.64)

we can define an average stress or a macroscopic stress of the composite σ by a volumetric average as

σ ≡ 1V

∫Ω

σ dV +1V

∫D−Ω

σ dV =∑

VΩV⟨σ⟩i +

(V −∑

VΩV

)⟨σ⟩m = f ⟨σ⟩i + (1 − f ) ⟨σ⟩m , (D.65)

which corresponds to the Voigt average. Furthermore, we here define the corresponding average strain ϵ by

ϵ ≡ f ⟨ϵ⟩i + (1 − f ) ⟨ϵ⟩m , (D.66)

which is the same as the Reuss average. It is quite interesting that we employ both the Voigt average Eq.(D.1) andthe Reuss average Eq.(D.3) at the same time. Substituting Eqs.(D.57), (D.61) and (D.62) into Eq.(D.66), we have

ϵ =(Cm

)−1 : ⟨σ⟩m + f S : ⟨ϵ∗⟩i . (D.67)

Consequently, eliminating ⟨σ⟩m, ⟨σ⟩i, ⟨ϵ∗⟩i and ⟨ϵ⟩m from Eqs.(D.62), (D.63), (D.65) and (D.67), we can expressthe overall strain in terms of the macroscopic stress as

ϵ =[Cm − (

Cm − Ci) S − f (S − I)]−1 [

Cm − (1 − f )(Cm − Ci

)S] (

Cm)−1 :σ. (D.68)

As has been mentioned before, no explicit definition of the average strain of the matrix part ⟨ϵ⟩m has been intro-duced4 at all. Let C

−1denote the average compliance of the composite, the right-hand side of the equation above

can be formally put as the right-hand side of Eq.(D.68)

≡ C−1

:σ. (D.69)

Eventually, the average elastic modulus of the composite are obtained as

C ≡[[

Cm − (Cm − Ci

) S − f (S − I)]−1 [Cm − (1 − f )

(Cm − Ci

)S] (

Cm)−1]−1

. (D.70)

In general, it is difficult to calculate the inverse of fourth-order tensors, but the property given in Eq.(D.47)makes it easy as long as the inhomogeneity is spherical in shape, because all the tensors in equations above becomeisotropic. Otherwise, you can use a manipulation explained by Nemat-Nasser and Hori [61]. As an example, themacroscopic elastic constant of CFRP examined in Sec. 2.4.3 is evaluated by the Mori-Tanaka approach. We setthe Young modulus and the Poisson ratio of vinyl-ester synthetic resin matrix as Em = 2.81 GN/m2 and νm = 0.274,while these of the carbon fiber are specified by Ei = 223 GN/m2 and νi = 0.352. The volume fraction of the fiberis set at f = 0.5, and the fiber is modeled by an infinitely long circular cylinder to the x3-direction. Then values ofthe corresponding Eshelby tensor can be found in the book [57], and Eq.(D.70) results in a symmetric matrix bythe Voigt-constant notation as

(C

)≡

8.62 3.17 3.77 0 0 03.17 8.62 3.77 0 0 03.77 3.77 115. 0 0 0

0 0 0 3.20 0 00 0 0 0 3.20 00 0 0 0 0 2.72

GN/m2, (D.71)

showing transverse isotropy.

4 The second author of this textbook recognized it the most important magical statement of this Mori-Tanaka approach.


0 0.5 10

20

40

60polyester

glass

E

(GN/m2)

f

Em = 1.72 GN/m2,

Ei = 70.2 GN/m2,

νm = 0.45

νi = 0.21

SC

Reuss

Voigt

Mori-Tanaka

0 0.5 10

20

40

60

80epoxy Em = 3.01 GN/m2, νm = 0.394

glass Ei = 76 GN/m2, νi = 0.23

E

(GN/m2)

f

SCVoigt

Reuss

Mori-Tanaka

(a) Young’s modulus

0 0.5 10.2

0.3

0.4

0.5ν

f

SCVoigt

Reuss

Mori-Tanaka

0 0.5 1

0.3

0.4

0.5

ν

f

SCVoigt

Reuss

Mori-Tanaka

(b) Poisson’s ratioFig. D.8 Comparison with experimental data in the case of spherical inhomogeneity

D.3.2 Comparison with ExperimentsEstimates of the elastic moduli by the Mori-Tanaka approach are compared with the experimental measurements[66, 71] in Fig. D.8. Inhomogeneity in the experiments is approximated by a sphere. Dot-dashed curves representthe Voigt and Reuss average, while solid curves show the estimates by the Mori-Tanaka approach. The latter liesbetween two classical bounds but is closer to the Reuss average, while the experimental data are rather close tothe Mori-Tanaka averages. Dotted curves indicated by ‘SC’ are predictions by Hill’s self-consistent method [28]explained later on, which predicts Poisson’s ratios quite well independently of the volume fraction. Therefore, aslong as the volume fraction f is small, these two analytical estimates are close to each other suggesting that theinteraction effect is taken into account properly to some extent. In other words, the classical bounds by Voigt andReuss models cannot be used in practice at least for composites with particulate inhomogeneities.

Brief explanation of Hill’s self-consistent method: This method is similar to the Mori-Tanaka approach, butthe inhomogeneities are embedded not in the actual matrix material but in the unknown average material of thecomposite. It may sound quite reasonable if you remember the first assumption of the Mori-Tanaka approachwhich is an addition of one inhomogeneity into the matrix portion with the average strain ⟨ϵ⟩m. Therefore, theprediction Eq.(D.70) of elastic moduli becomes implicit. For example, when the inhomogeneity is spherical inshape, they become

µ = µm +f (µi − µm) µ

µ + 2 S1212 (µi − µ), κ = κm +

f (κi − κm) κκ + 1

3 Sii j j (κi − κ), (D.72a, b)

where unknown average moduli appear even in the right-hand sides. Furthermore, the Eshelby tensors depend onthe unknown Poisson ratio of the average composite material as

2 S1212 =2 (4 − 5ν)15 (1 − ν) ,

13

Si ji j =1 + ν

3 (1 − ν) , ν =3κ − 2µ

2 (µ + 3κ). (D.73a, b, c)

The most controversial result is observed in porous materials. By setting the elastic constants of the inhomo-geneities zero in order to make them voids, the above equations result in

µ

µm= 1 − f

1 − 2 S1212,

κ

κm= 1 − f

1 − 13 Sii j j

, (D.74a, b)


and the average Poisson ratio is calculated from

ν =(7 + 5νm) − 6 f (1 + νm) −

√D

5 2 − 3 f (1 − νm) , (D.75)

whereD ≡ (7 − 5νm)2 − 6 f

(19 − 56νm + 45ν2

m

)+ 9 f 2

(9 − 42νm + 49ν2

m

). (D.76)

Strangely enough, the elastic constants are calculated to be zero at f = 0.5 from these equations. Consideringthat a volume fraction of one big spherical void fitted in a cube is f = π/6 ≃ 0.52, many readers may think thisresult appropriate. However, since absolute sizes of the inhomogeneities need not to be specified in the formulationabove, you can insert as many small spherical voids as possible in the matrix portion outside the one big sphericalvoid at the center, so that you can make the void volume fraction larger than 0.52 just like sponges.

0.45 0.5 0.55 0.60

0.1

0.2

0.3κ

κm

νm = 0.3

k = 0.02

k = 0.1

k = 0.001

k = 10−5

fFig. D.9 Bulk modulus of porous material

The elastic constants of void are set at zero in the previ-ous example, but we here leave the bulk modulus arbitraryand set

µi = 0, νi = 0.5, κi = arbitrary, k ≡ κiκm, 0.

(D.77a, b, c, d)Fig. D.9 shows the results. The average Poisson ratio be-comes smaller as f becomes larger but starts increasingnear f = 0.5 to obtain at f = 0.6

ν = 0.5, µ = 0,κ

κm=

5k3 + 2k

. (D.78a, b, c)

However, since the actual bulk modulus of the air is κi ≃ 0.14 MN/m2, the ratio of bulk moduli k is about 10−6 ∼10−5 when the matrix material is steel, aluminum, glass or even polyvinyl chloride resin. Therefore, the averageelastic constants of porous materials are predicted to be almost zero at f = 0.5 by Hill’s self-consistent method.

Homework D-2

3. Derive Eqs.(D.62) and (D.68) first. Calculate the average elastic constants when the inhomogeneity is spher-ical in shape with

µiµm= 10 and νm = νi = 0.3. Plot the results in a figure just like Fig. D.8. Next, exchange

the materials of the matrix and inhomogeneity and calculate the average elastic constants; i.e. for a two-phasecomposite with materials ‘A’ and ‘B’, solve the following two cases:

1. f % of the spherical material ‘A’ is distributed in the material ‘B’.

2. (1 − f )% of the spherical material ‘B’ is distributed in the material ‘A’.

Although the shape of the matrix part is not spherical, both composites are made of the same two materialswith the same volume fraction. Do these Mori-Tanaka estimates coincide with each other? Plot the resultson a figure relating µ to f . It is well known that these two estimates happen to be equal to the predictionscalled ‘Hashin-Shtrikman’s upper and lower bounds’ [27] explained in Sec. D.3.3 (2) as long as the shape ofthe inhomogeneity is sphere.

D.3.3 Application to Elastic-Plastic Composites and Improvement(1) Incremental Plasticity and Yield Surface

We can easily extend the Mori-Tanaka estimate in elasticity to apply to the incremental mechanics of plasticity,because the incremental governing equations are formally the same as those of the elastic body in infinitesimaldisplacements [93, 129, 134]. The local incremental constitutive relations are written as

⟨σ⟩m = Cm :(⟨ϵ⟩m − ⟨

ϵp⟩m

), ⟨σ⟩i = Ci :

(⟨ϵ⟩m + ⟨γ⟩i − ⟨ϵp⟩i

), (D.79a, b)

where a quantity with a superposed dot indicates an increment or a rate, and ϵp is the plastic strain rate. Substitutionof Eq.(D.79a) into Eq.(D.79b) results in

⟨σ⟩i = Ci :(

Cm)−1 : ⟨σ⟩m + ⟨γ⟩i − ∆

⟨ϵp⟩i

, (D.80)


where∆

⟨ϵp⟩i ≡

⟨ϵp⟩i −

⟨ϵp⟩m , (D.81)

which is a misfit between two plastic strain rates of the matrix and inhomogeneity. The equivalency condition ofthe corresponding equivalent inclusion is given by

⟨σ⟩i = Ci :(

Cm)−1 : ⟨σ⟩m + ⟨γ⟩i − ∆

⟨ϵp⟩i

= Cm :

(Cm

)−1 : ⟨σ⟩m + ⟨γ⟩i −(∆

⟨ϵp⟩i + ⟨ϵ∗⟩i

). (D.82)

Then the disturbance due to the misfit of the plastic strain rates can be evaluated by the Eshelby tensor as

⟨γ⟩i = S :(∆

⟨ϵp⟩i + ⟨ϵ∗⟩i

). (D.83)

Just like the elastic case in the previous sections, you can calculate the eigenstrain rate ϵ∗ from these equations.Eventually, the overall constitutive equation can be expressed as

ϵ = C−1

: σ + F :⟨ϵp⟩m + G :

⟨ϵp⟩i , (D.84)

where C is the global elastic tensor, and the explicit expressions of C, F and G can be found in the reference[93, 129]. Note that

F , (1 − f ) I, G , f I; (D.85a, b)

i.e. the overall plastic strain rate is not equal to a volumetric average of the two plastic strain rates of the matrixand inhomogeneity.

σ3 (MN/m2)

σ2σ1

O

600

400

200

p = 0 GN/m2principal axis

p = 7 GN/m2

p = −7 GN/m2

Fig. D.10 Macroscopic yield surface of composite

As an example, a global yield surface of thecomposite material of 2124Al reinforced by SiCexamined in the reference [77] is calculated. Theyield condition of the aluminum is specified bythe Mises model, but SiC remains elastic. SiCis modeled by an aligned prolate-spheroid, thelongest principal axis of which lies on the x1-x3plane and is oriented to the direction clockwisefrom x3-axis around x2-axis by 60 degrees. Also,we set a1 = a2 and a3/a1 = 2, and the volume frac-tion of the inclusion is 13.2%. The yield stress isgiven by σym = 700 MN/m2, while the elastic con-stants are assumed as follows: Em = 60 GN/m2,νm = 0.3, Ei = 450 GN/m2 and νi = 0.2. Circlesin Fig. D.10 [134] represent the contour lines ofthe macroscopic yield surface. Along each con-tour circle, the global hydrostatic pressure p iskept constant at the indicated value. Although thematrix material cannot yield under the hydrostaticpressure, the composite can become plastic in such an isotropic state of stress because of the microstructures.

(2) Improvement using Three-Phase Model

The Mori-Tanaka averages improve the classical Voigt and Reuss estimates quite well indicating that the mechan-ical interactions are properly taken into account to some extent. However, the larger the volume fraction of theinhomogeneity is, the lower the precision of predictions of average characteristics becomes, as is clear from Fig.D.8. On the other hand, Hill’s self-consistent method can predict experimental measurements even when the vol-ume fraction of the inhomogeneity is large. This comparison suggests that the choice of the matrix material playsan important role on quantitative improvement of evaluation of average properties within the framework of theMori-Tanaka approach. We here apply the Mori-Tanaka scheme to a three-phase composite in order to predict theaverage characteristics of two-phase materials [37].

In estimating the average elasticity of a composite of two materials ‘1’ and ‘2’, we temporarily treat these twomaterials as two inhomogeneities and put them into another arbitrary matrix material ‘M’. After averaging thisthree-phase composite by the Mori-Tanaka approach, we take a limit so that the volume fraction of the matrixmaterial ‘M’ becomes zero to make the composite a two-phase composite of the materials ‘1’ and ‘2’ only. Sincethe temporary matrix material ‘M’ eventually vanishes as a limit, we call this matrix portion the ‘virtual matrix’.


As a typical example, an elastic composite with spherical inhomogeneities is examined; the two materials have thebulk moduli κ1 and κ2, and the volume fraction of the two materials are f1 and f2. Although f1 + f2 = 1 whenthis is a two-phase composite, we set them arbitrary and set both the phases spherical in shape for the time being.We first put these two materials into a virtual matrix of the material ‘M’ with its bulk modulus κm and its volumefraction 1 − ( f1 + f2). After averaging this three-phase composite by the Mori-Tanaka scheme, we take a limit as( f1 + f2)→ 1 to eliminate the virtual matrix portion. Then, the average bulk modulus κ can be obtained as

κ =

2∑i=1

fi κi

κm − (κm − κi) α

2∑i=1

fiκm − (κm − κi) α

, (D.86)

where α and β are calculated from Eq.(D.40) using Poisson’s ratio νm of the non-existing virtual matrix material‘M’. Existence of κm and νm of the non-existing material is the most important characteristics of this approach.Depending on the choice of this material ‘M’, different average elastic moduli can be ‘arbitrarily’ calculated.

For example, let the virtual matrix a rigid body; i.e. κm → ∞, and we have

κ = f1 κ1 + f2 κ2, (D.87)

which coincides with the Voigt estimate of Eq.(D.2b). As you may easily expect, if a vacuum is chosen as thevirtual matrix; i.e. κm → 0, the Reuss estimate of Eq.(D.4b) is obtained as

κ =

(f1κ1+

f2κ2

)−1

. (D.88)

Therefore, when the elastic moduli of the virtual matrix are chosen to be positive and finite, the predictions bythis three-phase approach are bounded by the Voigt and Reuss averages. Namely, these classical averages can beconsidered as the upper and lower ‘limits’ rather than the ‘bounds’.

0 0.5 1

1000

2000

Hashin-Shtrikmanself-consistent

κ1 = 2000 MN/m2, ν1 = 0.3κ2 = 300 MN/m2, ν2 = 0.3

κ (MN/m2)

f2Fig. D.11 Improvement of average bulk modulus

A more interesting result is obtained when the virtualmatrix is made of the material ‘1’. Namely, in the case ofκm = κ1 and νm = ν1, Eq.(D.86) results in

κ

κ1= 1 −

f2

(1 − κ2

κ1

)1 − f1

(1 − κ2

κ1

)α

, (D.89)

which is the Mori-Tanaka average of a composite of the ma-terial ‘1’ reinforced by the material ‘2’. Furthermore, thisaverage coincides with one of the Hashin and Shtrikman up-per and lower bounds [27] as long as the shape of the inho-mogeneity is spherical. Also, another bound of the Hashinand Shtrikman bounds can be obtained when the material‘2’ is chosen for the virtual matrix; i.e. κm = κ2 and νm = ν2.The latter average is the solution of the problem No.3 ofHomework D-2.

In the reference [37], we make a proposal that the mate-rial ‘M’ of the virtual matrix is selected so that the interac-tion elastic energy becomes minimum. And, we found that two different predictions were obtained from Eq.(D.86)depending on how the elastic energy is expressed. Namely, the result by minimizing the elastic energy expressedin terms of the global strain rate ϵ is different from the result from the elastic energy in terms of the global stressrate σ. In other words, the former estimate is related to the property of the macroscopic modulus, while the latter isrelated to the macroscopic compliance. Two solid curves in Fig. D.11 are the corresponding results from Eq.(D.86)by the two choices of the material ‘M’ using the two different expressions of the interaction elastic energy. In thesame figure, the prediction by Hill’s self-consistent method is drawn by a dashed curve, and the Hashin and Shtrik-man upper and lower bounds are also shown by two dot-dashed curves. In this case, the Hashin and Shtrikmanupper bound is the same as the Mori-Tanaka average, while the lower bound is the same as the (inverted) Mori-Tanaka average of a composite in which the materials of the matrix and the inhomogeneity are exchanged. Themost important characteristics of three kinds of approaches observed in this figure are as follows:


• The present averages by the three-phase body result in predictions narrower than the Hashin-Shtrikmanbounds.

• The present averages converge to Hill’s self-consistent estimates when one of the volume fractions of thetwo phases becomes small.

• When f2 is small, the present estimate converges to the Hashin-Shtrikman upper bound, while it convergesto the lower bound when f2 is large.

• Although no plot is given in this figure, the classical Voigt and Reuss averages are quantitatively and quali-tatively different from these three kinds of predictions.

(3) Elastic-Plastic Behavior

The three-phase approach in the previous section can be also applied to elastic-plastic materials [44, 106]. Forsimplicity of formulation, the virtual matrix is chosen to be elastic. The average incremental constitutive laws ofthe matrix and inhomogeneity are given by

⟨σ⟩m = Cm : ⟨ϵ⟩m , ⟨σ⟩i = Ci :⟨ϵ⟩i − ⟨

ϵp⟩i. (D.90a, b)

Since the incremental strain in the inhomogeneity can be ⟨ϵ⟩i = ⟨ϵ⟩m + ⟨γ⟩i including the disturbance component,Eq.(D.90b) can be expressed by

⟨σ⟩i = Ci :⟨ϵ⟩m + ⟨γ⟩i − ⟨

ϵp⟩i. (D.91)

Then, the incremental constitutive equation of the corresponding auxiliary problem can be expressed by

⟨σ⟩i = Cm :[⟨ϵ⟩m + ⟨γ⟩i − ⟨

ϵp⟩i + ⟨ϵ∗⟩i

](D.92)

using the eigenstrain rate. Hence, the disturbed component can be evaluated as

⟨γ⟩i = Si :⟨ϵp⟩

i + ⟨ϵ∗⟩i

(D.93)

by using the Eshelby tensor. Finally, from these equations, the equivalent inclusion method determines the incre-mental eigenstrain ⟨ϵ∗⟩i.

0.002 0.004O

0.1

0.2

σ11 (GN/m2)

ϵ11

Mori-Tanaka

Ju[41]

Experiments

3-phase approachσ is specifiedϵ is specified

Fig. D.12 Aluminum reinforced by boron fibers

Several steps of manipulations using the definitions of theoverall stress rate and strain rate corresponding to Eqs.(D.65)and (D.66) lead to the overall constitutive equation as

ϵ = C−1

: σ +2∑

i=1

fi(Pi − C

−1Mi

):⟨ϵp⟩

i , (D.94)

where the volume fraction of the virtual matrix is set at zeroas a limit. Explicit definitions of C and other tensors aregiven in the reference [106]. The second term of the right-hand side of Eq.(D.94) represents the macroscopic plasticstrain rate. Fig. D.12 shows comparisons with the experimen-tal measurements examined in the reference indicated in thefigure. The material is an aluminum composite reinforcedby long boron fibers. The aluminum has Young’s modulusE1 = 55.85 GN/m2 and Poisson’s ratio ν1 = 0.32, while theboron has E2 = 379.23 GN/m2 and ν2 = 0.2. The volumefraction of boron is 34%, and the radii of ellipsoidal inclusionare set at a3/a1 = 1000 and a2 = a1. The boron fibers remain elastic, while the aluminum yields by the von Misescondition as

f ≡√

J2 − F(⟨ϵp⟩

1)= 0, (D.95)

where the tensile yield stress F (⟨ϵp⟩1) is specified by a power law as

F(⟨ϵp⟩

1)=

1√

3

σy1 + h1

ϵeq1√3

n1 (D.96)


0 0.005 0.01

0.08

0.09

0.1

0.11

σ11 (GN/m2)

ϵp11

Experiments

3-phase approach

Mori-Tanaka

Tandon and Weng [78]

Doghri and Ouaar [15]

(a) In the case of 15% volume fraction of silica

0 0.005 0.01

0.1

0.12

0.14

σ11 (GN/m2)

ϵp11

Experiments

3-phase approach

Mori-Tanaka

Tandon and Weng [78]

Doghri and Ouaar [15]

(b) In the case of 35% volume fraction of silica

Fig. D.13 Epoxy reinforced by silica particles

with σy1 = 79.29 MN/m2, h1 = 827.4 MN/m2 and n1 = 0.6. The equivalent plastic strain ϵeq1 is defined by

ϵeq1 ≡

∫history

√2 ⟨ϵp⟩1 : ⟨ϵp⟩1 dt. (D.97)

Two solid curves by the three-phase approach show good precision in relatively small deformation states, and arecloser to the experimental measurements than the predictions by the original Mori-Tanaka scheme. However, inlarge deformation states, the predictions in [41] seem to be much better, because the debonding along the interfaceof the inhomogeneity is taken into account. So that, in the reference [44], we further add one kind of debondingmodels to obtain improved results.

Another comparison is made with experimental results of epoxy composites reinforced by spherical silicaparticles in Fig. D.13. Epoxy has Young’s modulus E1 = 3.16 GN/m2 and Poisson’s ratio ν1 = 0.35, while silicahas E2 = 73.1 GN/m2 and ν2 = 0.18. The silica remains elastic, while the epoxy satisfies the same power law of theMises yield condition as that of the previous example with σy1 = 75.86 MN/m2, h1 = 32.18 MN/m2 and n1 = 0.26.In these cases, the three-phase approach improves the original Mori-Tanaka scheme, but the experimental resultsare not predicted so well when the volume fraction of the inhomogeneity is large.

Bibliography

[1] R. J. Asaro: Geometrical effects in the inhomogeneous deformation of ductile single crystals, Acta Met.,Vol.27, pp.445-453, 1979.

[2] R. J. Asaro: Micromechanics of Crystals and Polycrystals, Advances in Appl. Mech., ed. by J. W. Hutchinsonand T. Y. Wu, Academic Press, Vol.23, pp.1-115, 1983.

[3] R. J. Asaro and J. R. Rice: Strain localization in ductile single crystals, J. Mech. Phys. Solids, Vol.25,pp.309-338, 1977.

[4] K.-J. Bathe and E. L. Wilson: Numerical Methods in Finite Element Analysis, Prentice-Hall Inc., 1976.

[5] Z. P. Bazant and L. Cedolin: Stability of Structures: Elastic, Inelastic, Fracture, and Damage Theories,Dover Publ. Inc., 2003.

[6] K. E. Bisshopp and D. C. Drucker: Large deflection of cantilever beams, Q. Appl. Math., Vol.3, pp.272-275,1945.

[7] J. Bonet and R. D. Wood: Nonlinear Continuum Mechanics for Finite Element Analysis, Cambridge Univ.Press, 1997.

[8] B. Budiansky: Theory of buckling and post-buckling behavior of elastic structures, Advances in Appl.Mech., Vol.14, pp.1-65, 1974.

[9] R. V. Churchill, J. W. Brown and R. F. Verhey: Complex Variables and Applications, Third Edition,McGraw-Hill Co., 1976.

[10] R. W. Clough and J. Penzien: Dynamics of Structures, McGraw-Hill Co., 1975.

[11] G. R. Cowper: The shear coefficient in Timoshenko’s beam theory, J. Appl. Mech., Trans. ASME, Vol.33,pp.335-340, 1966.

[12] D. A. DaDeppo and R. Schmidt: Instability of clamped-hinged circular arches subjected to a point load, J.Appl. Mech., Trans. ASME, Vol.42, pp.894-896, 1975.

[13] C. S. Desai and J. F. Abel: Introduction to the Finite Element Method, Van Nostrand Reinhold, 1972.

[14] E. A. de Souza Neto, D. Peric and D. R. J. Owen: Computational Methods for Plasticity: Theory andApplication, John Wiley & Sons, Inc., 2008.

[15] I. Doghri and A. Ouaar: Homogenization of two-phase elasto-plastic composite materials and structures:Study of tangent operators, cyclic plasticity and numerical algorithm, Int. J. Solids Structures, Vol.40,pp.1681-1712, 2003.

[16] D. C. Drucker: Plasticity, Structural Mech., Proc. 1st Sympo. Naval Struct. Mech., ed. by J. N. Goodier andN. J. Hoff, Pergamon Press, pp.407-455, 1960.

[17] D. C. Drucker and W. Prager: Soil mechanics and plastic analysis or limit design, Q. Appl. Math., Vol.10,pp.157-165, 1952.

[18] A. E. Elwi and D. W. Murray: Skyline algorithms for multilevel substructure analysis, Int. J. Numer. Meth.Eng., Vol.21, pp.465-479, 1985.

581

582 BIBLIOGRAPHY

[19] J. D. Eshelby: The determination of the elastic field of an ellipsoidal inclusion, and related problems, Proc.Roy. Soc. London, Vol.A241, pp.376-396, 1957.

[20] W. Flugge: Tensor Analysis and Continuum Mechanics, Springer Verlag, 1972.

[21] Y. C. Fung: Foundations of Solid Mechanics, Prentice-Hall Inc., 1965.

[22] A. L. Gurson: Continuum theory of ductile rupture by void nucleation and growth: Part I – Yield criteriaand flow rules for porous ductile media, J. Eng. Mater. Tech., ASME, Vol.99, pp.2-15, 1977.

[23] N. D. Hai, H. Mutsuyoshi, S. Asamoto and T. Matsui: Structural behavior of hybrid FRP composite I-beam,Constr. Build. Mater., Vol.24, pp.956-969, 2010.

[24] J. Har: A unified stress update algorithm for explicit transient shell dynamics with combined isotropic-kinematic hardening in Eulerian rate-type phenomenological finite elasto-plasticity models, Compt. Meth.Appl. Mech. Engrg., Vol.196, pp.3248-3275, 2007.

[25] A. Hasegawa, T. Iwakuma and S. Kuranishi: A linearized Timoshenko beam theory in finite displacements,Structural Eng./ Earthquake Eng., Vol.2, pp.321s-326s, (Proc. JSCE, No.362/ I-4) 1985.

[26] A. Hasegawa, T. Iwakuma, K. Liyanage and F. Nishino: A consistent formulation of trusses and non-warping beams in linearized finite displacements, Structural Eng./ Earthquake Eng., Vol.3, pp.477s-480s,(Proc. JSCE, No.374/ I-6) 1986.

[27] Z. Hashin and S. Shtrikman: A variational approach to the theory of the elastic behaviour of multiphasematerials, J. Mech. Phys. Solids, Vol.11, pp.127-140, 1963.

[28] R. Hill: Continuum micro-mechanics of elastoplastic polycrystals, J. Mech. Phys. Solids, Vol.13, pp.89-101,1965.

[29] R. Hill: On constitutive inequalities for simple materials — I & II, J. Mech. Phys. Solids, Vol.16, pp.229-242, pp.315-322, 1968.

[30] R. Hill: The Mathematical Theory of Plasticity, Oxford Classic Texts in the Physical Sciences, ClarendonPress, 1998.

[31] R. Hill and J. W. Hutchinson: Bifurcation phenomena in the plane tension test, J. Mech. Phys. Solids,Vol.23, pp.239-264, 1975.

[32] J. W. Hutchinson: Plastic buckling, Advances in Appl. Mech., Vol.14, pp.67-144, 1974.

[33] T. Iwakuma: Timoshenko beam theory with extension effect and its stiffness equation for finite rotation,Comput. Structures, Vol.34, pp.239-250, 1990.

[34] T. Iwakuma, M. Ai and F. Nishino: On derivation of Timoshenko beam stiffness equation, Proc. JSCE,No.312, pp.119-128, 1981.

[35] T. Iwakuma, K. Ikeda and F. Nishino: Consistency of straight-beam approximation of a thin-walled circularbeam, Comput. Structures, Vol.60, pp.87-93, 1996.

[36] T. Iwakuma, A. Hasegawa, F. Nishino and S. Kuranishi: Principle and numerical check of a stiffnessequation for plane frames, Structural Eng./ Earthquake Eng., Vol.4, pp.73s-83s, (Proc. JSCE, No.380/ I-7)1987.

[37] T. Iwakuma and S. Koyama: An estimate of average elastic moduli of composites and polycrystals, Mech.Mater., Vol.37, pp.459-472, 2005.

[38] T. Iwakuma and S. Kuranishi: How much contribution does the shear deformation have in a beam theory?,Proc. JSCE, No.344/ I-1, pp.141-152, 1984.

[39] T. Iwakuma and S. Nemat-Nasser: An analytical estimate of shear band initiation in a necked bar, Int. J.Solids Structures, Vol.18, pp.69-83, 1982.

[40] T. Iwakuma and S. Nemat-Nasser: Finite elastic-plastic deformation of polycrystalline metals, Proc. R. Soc.London, Vol.A394, pp.87-119, 1984.

BIBLIOGRAPHY 583

[41] J. W. Ju and X. D. Zhang: Effective elastoplastic behavior of ductile matrix composites containing randomlylocated aligned circular fibers, Int. J. Solids Structures, Vol.38, pp.4045-4069, 2001.

[42] T. von Karman: Uber die Formanderung dunnwandiger Rohre, insbesondere federnder Ausgleichrohre,Zeit. des Vereines deutscher Ingenieure, Vol.55, pp.1889-1895, 1911.

[43] T. Kawai: New element models in discrete structural analysis, J. SNAJ, No.141, pp.174-180, 1977.

[44] S. Koyama, S. Katano, I. Saiki and T. Iwakuma: A modification of the Mori-Tanaka estimate of averageelastoplastic behavior of composites and polycrystals with interfacial debonding, Mech. Mater., Vol.43,pp.538-555, 2011.

[45] E. H. Lee, R. L. Mallett and T. B. Wertheimer: Stress analysis for anisotropic hardening in finite-deformationplasticity, J. Appl. Mech., Trans. ASME, Vol.50, pp.554-560, 1983.

[46] R. Lin: Hypoelasticity-based analytical stress solutions in the simple shearing process, ZAMM, Vol.83,pp.163-171, 2003.

[47] R. C. Lin, U. Shomburg and T. Kletschkowski: Analytical stress solutions of a closed deformation path withstretching and shearing using the hypoelastic formulations, Europ. J. Mech. Vol.22, pp.443-461, 2003.

[48] S. C. Lin, C. C. Yang, T. Mura and T. Iwakuma: Average elastic-plastic behavior of composite materials,Int. J. Solids Structures, Vol.29, pp.1859-1872, 1992.

[49] L. E. Malvern: Introduction to the Mechanics of a Continuous Medium, Prentice-Hall Inc., 1969.

[50] M. M. Mehrabadi and S. C. Cowin: Initial planar deformation of dilatant granular materials, J. Mech. Phys.Solids, Vol.26, pp.269-284, 1978.

[51] M. M. Mehrabadi and S. C. Cowin: Prefailure and post-failure soil plasticity models, J. Eng. Mech., Proc.ASCE, Vol.106, pp.991-1003, 1980.

[52] M. M. Mehrabadi and S. C. Cowin: On the double-sliding free-rotating model for the deformation ofgranular materials, J. Mech. Phys. Solids, Vol.29, pp.269-282, 1981.

[53] M. M. Mehrabadi and S. C. Cowin: Eigentensors of linear anisotropic elastic materials, Q. J. Mech. Appl.Math., Vol.43, pp.15-41, 1990.

[54] A. Meyers, H. Xiao and O. T. Bruhns: Choice of objective rate in single parameter hypoelastic deformationcycles, Comput. Struct., Vol.84, pp.1134-1140, 2006.

[55] T. Mori and K. Tanaka: Average stress in matrix and average energy of materials with misfitting inclusions,Acta Metall., Vol.21, pp.571-574, 1973.

[56] P. Morrison, P. Morrison and The Office of C. and R. Eames: Powers of Ten, About the Relative Size ofThings in the Universe, Scientific American Books, 1982.

[57] T. Mura: Micromechanics of Defects in Solids, Martinus Nijhoff Publ., 1982; Micromechanics of Defectsin Solids, Second, Revised Edition, Kluwer Academic Publ., 1998.

[58] G. C. Nayak and O. C. Zienkiewicz: Elaso-plastic stress analysis. A generalization for various constitutiverelations including strain softening, Int. J. Numer. Meth. Eng., Vol.5, pp.113-135, 1972.

[59] S. Nemat-Nasser: On finite deformation elasto-plasticity, Int. J. Solids Structures, Vol.18, pp.857-872,1982.

[60] S. Nemat-Nasser: Plasticity, A Treatise on Finite Deformation of Heterogeneous Inelastic Materials, Cam-bridge Monographs on Mechanics, Cambridge Univ. Press, 2005.

[61] S. Nemat-Nasser and M. Hori: Micromechanics: Overall Properties of Heterogeneous Materials, North-Holland, Elsevier Science Publ., 1993.

[62] S. Nemat-Nasser and A. Shokooh: On finite plastic flows of compressible materials with internal friction,Int. J. Solids Structures, Vol.16, pp.495-514, 1980.

584 BIBLIOGRAPHY

[63] M. Oda and K. Iwashita (eds.): Mechanics of Granular Materials, An Introduction, A. A. Balkema, 1999.

[64] D. L. Powers: Boundary Value Problems and Partial Differential Equations, Fifth Edition, Elsevier Aca-demic Press, 2006.

[65] E. Reissner: On finite bending of pressurized tubes, J. Appl. Mech., Trans. ASME, Vol.26, pp.386-392,1959.

[66] T. G. Richard: The mechanical behavior of a solid microsphere filled composite, J. Comp. Mat., Vol.9,pp.108-113, 1975.

[67] S. M. Roberts and J. S. Shipman: Two-Point Boundary Value Problems: Shooting Methods, Elsevier, 1972.

[68] J. W. Rudnicki and J. R. Rice: Conditions for the localization of deformation in pressure-sensitive dilatantmaterials, J. Mech. Phys. Solids, Vol.23, pp.371-394, 1975.

[69] F. R. Shanley: Inelastic column theory, J. Aeronaut. Sci., Vol.14, pp.261-267, 1947.

[70] J. C. Simo and T. J. R. Hughes: Computational Inelasticity, Springer Verlag, 1998.

[71] J. C. Smith: Experimental values for the elastic constants of a particulate-filled glassy polymer, J. Res. NBS— A. Phys. Chem., Vol.80A, pp.45-49, 1976.

[72] A. J. M. Spencer: Deformation of ideal granular materials, Mechanics of Solids; the Rodney Hill 60thAnniversary Volume, ed. by H. G. Hopkins and M. J. Sewell, Pergamon Press, pp.607-652, 1982.

[73] I. Stakgold: Green’s Functions and Boundary Value Problems, a volume in Pure and Applied Mathemat-ics: A Wiley-Interscience Series of Texts, Monographs, and Tracts, found. by R. Courant, ed. by L. Bers,P. Hilton and H. Hochstadt, John Wiley & Sons, Inc., 1979.

[74] E. Sternberg and R. A. Eubanks: On the singularity at a concentrated load applied to a curved surface, Proc.2nd US Nat’l Congress Appl. Mech., pp.237-246, 1954.

[75] S. Storen and J. R. Rice: Localized necking in thin sheets, J. Mech. Phys. Solids, Vol.23, pp.421-441, 1975.

[76] G. Strang and G. J. Fix: An Analysis of the Finite Element Method, Prentice-Hall Inc., 1973.

[77] L. Z. Sun and J. W. Ju: Effective elastoplastic behavior of metal matrix composites containing randomlylocated aligned spheroidal inhomogeneities, Int. J. Solids Str., Vol.38, pp.203-225, 2001.

[78] G. P. Tandon and G. J. Weng: A theory of particle-reinforced plasticity, J. Appl. Mech., Trans. ASME,Vol.55, pp.126-135, 1988.

[79] S. P. Timoshenko and J. M. Gere: Theory of Elastic Stability, Int. Student Edition, McGraw-Hill Kogakusha,1961.

[80] S. P. Timoshenko and J. N. Goodier: Theory of Elasticity, Int. Student Edition, McGraw-Hill Kogakusha,1970.

[81] C. Truesdell and R. Toupin: The classical field theories, Encyclopedia of Physics, ed. by S. Flugge, Vol.III/1,Springer-Verlag, 1960.

[82] C. Truesdell and W. Noll: The non-linear field theories of mechanics, Encyclopedia of Physics, ed. byS. Flugge, Vol.III/3, Springer-Verlag, 1965.

[83] V. Tvergaard: Infulence of voids on shear band instabilities under plane strain conditions, Int. J. Fracture,Vol.17, pp.389-407, 1981.

[84] G. Wempner: Mechanics of Solids with Applications to thin bodies, Monographs and Textbooks on Me-chanics of Solids and Fluids, Mechanics of Elastic and Inelastic Solids, ed. by G. Æ. Oravas, Sijthoff &Noordhoff, 1981.

[85] G. J. Weng: The overall elastoplastic stress-strain relations of dual-phase metals, J. Mech. Phys. Solids,Vol.38, pp.419-441, 1990.

BIBLIOGRAPHY 585

[86] 青木徹彦: 例題で学ぶ構造力学 I —静定編—, コロナ社, 2015.

[87] 荒川淳平, 岩熊哲夫: 増分弾塑性構成則に用いる応力速度の選択, 土木学会論文集A2（応用力学）,Vol.70, pp.I 365–I 374, 2014.

[88] 石川信隆,大野友則: 入門・塑性解析と設計法, 森北出版, 1988.

[89] 泉満明: ねじりを受けるコンクリート部材の設計法, 技報堂, 1975.

[90] 伊藤学: 構造力学, 森北出版, 1971.

[91] 井上達雄: 弾性力学の基礎, 日刊工業新聞社, 1982.

[92] 岩熊哲夫,斉木功,藤本真明: せん断抵抗則に着目した 2種類の基本的な亜弾性モデルの特性, 土木学会論文集A2（応用力学）, Vol.73, pp.22-33, 2017.

[93] 岩崎智明,岩熊哲夫,小山茂: 複合材料の巨視的降伏および流れ則の予測,応用力学論文集,土木学会, Vol.5,pp.273-282, 2002.

[94] 岡本舜三: 建設技術者のための振動学（第 2版）, オーム社, 1986.

[95] 奥井義昭: 限界状態設計法と維持管理, 片山技報, No.33, pp.2-7, 2014.

[96] 河合貴行,谷和夫: 縦ずれ断層によって未固結被覆層に発達するせん断帯が地表面に出現する位置の予測,土と基礎,地盤工学会, Vol.51, no.11, pp.23-25, 2003.

[97] 菊池文雄: 幾何学的非線型と FEM, 特集有限要素法,数理科学, No.144, pp.18-26, 1975.

[98] 北川浩: 塑性力学の基礎, 日刊工業新聞社, 1979.

[99] 北川浩: 弾・塑性力学—非線形解析のための基礎理論—, 裳華房, 1987.

[100] 京谷孝史: よくわかる連続体力学ノート, 森北出版, 2008.

[101] 国井隆弘,宮田利雄,片山恒雄: 構造物の振動解析, 新体系土木工学 10,技報堂出版, 1979.

[102] 久保慶三郎: 構造力学演習, 学献社, 1974.

[103] 小坪清真: 土木振動学, 森北出版, 1973.

[104] 小林俊一: 地盤の変形解析—土水連成場・構成式—, http://basewall.kuciv.kyoto-u.ac.jp/archive/20020913-seminar.pdf, 2002.

[105] 小林寿彦,斉木功,岩熊哲夫: 構成則に用いる応力速度と変形の局所化, 土木学会論文集A2（応用力学）,Vol.67, pp.I 281–I 291, 2011.

[106] 小山茂, 片野俊一, 大上俊之, 岩熊哲夫: 複合材料や多結晶金属の平均弾塑性挙動予測の一手法, 土木学会論文集A, Vol.64, pp.121-132, 2008.

[107] 近藤恭平: ［工学基礎］振動論, 培風館, 1993.

[108] 斉木功,後藤文彦,岩熊哲夫: 有限回転を許容する棒部材の三次元動的数値解析の一手法, 構造工学論文集,土木学会, Vol.44A, pp.315-322, 1998.

[109] 鈴木基行: ステップアップで実力がつく構造力学徹底演習, 森北出版, 2006.

[110] 田村武: 連続体力学入門, 朝倉書店, 2000.

[111] 田村武: 構造力学—仮想仕事の原理を通して—, 朝倉書店, 2003.

[112] 寺沢直樹, 岩熊哲夫, 後藤文彦, 白戸真大: 地盤の不安定問題における変形局所化の数値予測, 応用力学論文集,土木学会, Vol.3, pp.283-294, 2000.

[113] 寺田賢二郎,菊池昇: 均質化法入門, 計算力学レクチャーシリーズ 1,丸善, 2003.

[114] 土木学会編: 構造力学公式集昭和 61年版,土木学会, 1986.

586 BIBLIOGRAPHY

[115] 土木学会編: 土木工学ハンドブック第 6編「固体力学」, 技報堂出版, pp.215-256, 1989.

[116] 土木学会編: 土木用語大辞典, 技報堂出版, 1999.

[117] 土木学会鋼構造委員会鋼構造物設計指針小委員会編: 鋼構造物設計指針 PART A一般構造物, 土木学会,1997.

[118] 土木学会構造工学委員会計算力学とその応用に関する研究小委員会編: 構造工学における計算力学の基礎と応用, 土木学会, 1996.

[119] 土木学会構造工学委員会構造力学小委員会構成則分科会編: 材料特性の数理モデル入門～構成則主要用語解説集～, 構造工学シリーズ 4,土木学会, 1989.

[120] 中沢正利, 池田清宏, 和知聡, 倉西茂: 等曲げを受ける弾性矩形板に生じる二次座屈現象の解明, 土木学会論文集, No.519/ I-32, pp.67-78, 1995.

[121] 長井正嗣: 橋梁工学［第 2版］, テキストシリーズ土木工学 3,共立出版, 2003.

[122] 西村直志: トラスの力学, http://gspsun1.gee.kyoto-u.ac.jp/nchml/kouriki/kouriki.html,1998.

[123] 西野文雄: 連続体の力学 (II), 土木工学体系 6,彰国社, 1984.

[124] 西野文雄,長谷川彰夫: 構造物の弾性解析, 新体系土木工学 7,技報堂出版, 1983.

[125] 日本材料学会編: 固体力学の基礎, 日刊工業新聞社, 1981.

[126] 日本道路協会: 道路橋示方書・同解説, I共通編, II鋼橋編, 1996.

[127] 野村卓史: 構造力学, 土木・環境系コアテキストシリーズB-1,コロナ社, 2011.

[128] 林毅,村外志夫: 変分法, コロナ社, 1980.

[129] 廣瀬恒太,岩熊哲夫,小山茂: 材料内部の微視構造が初期降伏特性に及ぼす影響と巨視的構成則, 応用力学論文集,土木学会, Vol.6, pp.355-366, 2003.

[130] 福本士: 構造物の座屈・安定解析, 新体系土木工学 9,技報堂出版, 1982.

[131] 細野透: 弧長法による弾性座屈問題の解析（その 2）数値解析方法としての弧長法, 日本建築学会論文報告集,第 243号, pp.21-30, 1976.

[132] 三木千壽: 鋼構造, テキストシリーズ土木工学 10,共立出版, 2000.

[133] 山崎徳也,彦坂煕: 構造解析の基礎, 共立出版, 1978.

[134] 湯本健寛, 岩熊哲夫: 複合材料の平均的初期降伏関数の陽な近似表現, 応用力学論文集, 土木学会, Vol.7,pp.515-525, 2004.

[135] 吉田輝,後藤正司,亀谷泰久,龍岡文夫,木幡行宏,薫軍: 砂礫の平面ひずみ圧縮試験におけるせん断層の応力・変形関係, 地盤の破壊とひずみの局所化に関するシンポジウム発表論文集, 土質工学会, pp.189-196,1994.

Index

Author IndexAbel, J.F., 204Ai, M., 273, 535Asamoto, S., 56Asaro, R.J., 441, 473, 474,

507, 520, 527Bathe, K.-J., 192, 312, 314,

336Bazant, Z.P., 557Bisshopp, K.E., 245Bonet, J., 493, 496Brown, J.W., 83Bruhns O.T., 494Budiansky, B., 219, 244Cedolin, L., 557Churchill, R.V., 83Clough, R.W., 334Cowin, S.C., 59, 440Cowper, G.R., 153, 370, 502,

532DaDeppo, D.A., 555Desai, C.S., 204de Souza Neto E.A., 429, 430Doghri, I., 580Drucker, D.C., 245, 405, 426,

429Eames, C., 33Eames, R., 33Elwi, A.E., 555Eshelby, J.D., 60, 92, 566,

567, 571, 572Eubanks, R.A., 101Fix, G.J., 176, 192, 196Flugge, W., 455, 537, 559Fung, Y.C., 40, 51, 63, 73,

402, 483Gere, J.M., 218, 223, 227,

229, 230, 245, 503, 542,546

Goodier, J.N., 89, 90, 101,270, 285

Gurson, A.L., 453Hai, N.D., 56Har, J., 494Hasegawa, A., 273, 548, 554Hashin, Z., 576, 578Hill, R., 64, 66, 399, 406,

407, 434, 440, 477, 509,

510, 522, 526, 527, 566,575

Hori, M., 54, 571, 574Hughes, T.J.R., 409, 416Hutchinson, J.W., 221, 237,

509, 510, 522Ikeda, K., 163Iwakuma, T., 163, 242, 273,

389, 406, 441, 509, 511,527, 535, 538, 541, 546,548, 554, 556, 577–580

Iwashita, K., 440Ju, J.W., 453, 577, 579, 580von Karman, T., 221Katano, S., 579, 580Kawai, T., 447Kletschkowski, T., 494Koyama, S., 577–580Kuranishi, S., 538, 541, 548,

554Lee, E.H., 486, 500Lin, R.C., 494Lin, S.C., 389Liyanage, K., 273Mallett, R.L., 486, 500Malvern, L.E., 43, 44, 55,

412, 477, 487, 554Matsui, T., 56Mehrabadi, M.M., 59, 440Meyers A., 494Mori, T., 573Morrison, P., 33Mura, T., 60, 389, 421, 566,

567, 570, 574Murray, D.W., 555Mutsuyoshi, H., 56Nayak, G.C., 405, 430Nemat-Nasser, S., 54, 66,

389, 396, 406, 426, 441,455, 468, 474, 477, 481,485, 489, 505, 509, 511,527, 571, 574

Nishino, F., 163, 273, 535,554

Noll, W., 477Oda, M., 440Ouaar, A., 580Owen, D.R.J., 429, 430Penzien, J., 334

Peric, D., 429, 430Powers, D.L., 341, 346Prager, W., 426, 429Reissner, E., 552Rice, J.R., 406, 430–432,

506, 507, 520Richard, T.G., 575Roberts, S.M., 553Rudnicki, J.W., 406, 430–

432, 506Saiki, I., 579, 580Schmidt, R., 555Shanley, F.R., 236Shipman, J.S., 553Shokooh, A., 426, 505Shomburg, U., 494Shtrikman, S., 576, 578Simo, J.C., 409, 416Smith, J.C., 575Spencer, A.J.M., 440Stakgold, I., 129, 194Sternberg, E., 101Storen, S., 432Strang, G., 176, 192, 196Sun, L.Z., 577Tanaka, K., 573Tandon, G.P., 433, 580Timoshenko, S.P., 89, 90,

101, 218, 223, 227, 229,230, 245, 270, 285, 503,542, 546

Toupin, R., 477Truesdell, C., 477Tvergaard, V., 453Verhey, R.F., 83Wempner, G., 163, 563Weng, G.J., 433, 580Wertheimer, T.B., 486, 500Wilson, E.L., 192, 312, 314,

336Wood, R.D., 493, 496Xiao H., 494Yang, C.C., 389Zhang, X.D., 453, 579, 580Zienkiewicz, O.C., 405, 430青木徹彦, 109荒川淳平, 510池田清宏, 295石川信隆, 393, 438, 449

泉満明, 267伊藤学, 109, 202, 450井上達雄, 41岩熊哲夫, 379, 404, 440, 501,

510, 520, 576, 577, 579岩崎智明, 576, 577大上俊之, 579大野友則, 393, 438, 449岡本舜三, 297, 361奥井義昭, 451片野俊一, 579片山恒雄, 385亀谷泰久, 441河合貴行, 440菊池昇, 374, 377, 565菊池文雄, 220, 221北川浩, 66, 396, 405, 406,

434, 438, 445, 473, 489京谷孝史, 204, 455国井隆弘, 385久保慶三郎, 109, 154倉西茂, 295小坪清真, 297後藤文彦, 379, 440後藤正司, 441木幡行宏, 441小林俊一, 430小林寿彦, 501小山茂, 576, 577, 579近藤恭平, 328斉木功, 379, 501, 520白戸真大, 440鈴木基行, 20, 105龍岡文夫, 441谷和夫, 440田村武, 109, 490寺沢直樹, 440寺田賢二郎, 374, 377, 565薫軍, 441長井正嗣, 280中沢正利, 295西野文雄, 125, 153, 163, 192,

216, 249, 258–260, 264,268, 269, 276, 279–281,285

西村直志, 67, 114, 200野村卓史, 125長谷川彰夫, 125, 153, 163,

192, 216, 249, 258, 276,

587

588 INDEX

279–281, 285林毅, 246彦坂煕, 109, 152廣瀬恒太, 576, 577福本士, 237藤井宏, 176, 192, 196

藤本真明, 520細野透, 555本多勝一, 41, 321三木千壽, 149宮田利雄, 385三好哲彦, 176, 192, 196

村上公子, 33村上陽一郎, 33村外志夫, 60, 246, 421, 566,

567, 570, 574森勉, 60, 421, 566, 567, 570,

574

山崎徳也, 109, 152山本善之, 204湯本健寛, 404, 576, 577吉田輝, 441和知聡, 295

SymbolsMathematical⟨ , ⟩, 197( ), j, 457δ, 195δ(x − ξ), 129, 309δi j, 38det, 30( ), 78ei jk, 45, 561ϵi jk, 561F , 317H(t − τ), 310ℑ, 356i, 154L, 91O( ), o( ), 232∥, 46( )′, 114tensor product ·, :,⊗, 455

tr, 39( )t, 38|| · ||, 197

( )| j, 562

MechanicalA, 21, 108α, 60αt, 533αs, 117αt, 533arrow ⊗, ⊙, 31

→→, 249→→→, 276

b, 460β, 301βe, 232C, 53, 457χ, 403cl, 78ct, 78D, 283D, 54d, 467∆, 39( ), 469E, 51

E, 457, 537e, 461EA, 109EE , 463EI, 109EIω, 262EL, 463( )e, 392( )ep, 403ϵ, 36ϵp, 397ϵp, 397Er, 236Et, 235F, 457G, 51γ, 531Gi, 537GJ, 250Gy, 262Gz, 147g0, 243, 538G(p)

z , 283H , 315

I, 108Ii, 49Iω, 262J, 250J3, 396J2, 62, 396Jz, 108K, 51, 227κ, 243, 538kb, 227kp, 295kt, 532l, 467Λ, 459λ, 51, 153, 225,

406λb, 280λ, 235λω, 266λt, 533lim0→t

, 487

M, 108, 540Mω, 262µ, 51

Mx, 283Mxy, 283My, 283N, 108, 540ni, 112n, 489ν, 51ω, 37ω, 269, 299Pcr, 213, 227Pe, 225φ, 249( )p, 392( )′, 39q, 253Qω, 263r, 225Re, 80ρ, 1rω, 266S, 480, 482, 538σ, 42, 476, 482σave, 50σ, 62, 396

σcr, 225σi, 46σ, 63, 397SN , 477, 482∇σ, 486⊔σ, 486⋆σ, 485∨σ, 488T, 481, 482τK , 479τy, 63, 391, 397TR, 480, 482Θ, 268θ, 107ϑ, 531TS , 250U, 459u, 34V , 544u, 460, 467W, 21, 109w, 107w, 468

Aacceleration, 469accumulated plastic strain, 397adaptive FEM, 208adjoint problem, 311, 358adjoint system, 73, 137, 138advection/advective term, 79, 469Airy stress function, 89, 293Almansi strain, 461amplitude, 298angle of internal friction, 430, 439angular velocity, 468anisotropy, 55antisymmetric part, 468approximation error, 191, 197arch, 163area coordinate, 209aseismatic connector, 18aspect ratio, 295associated (associative) flow rule, 410assumption

Bernoulli-Euler —, 105Kirchhoff-Love —, 281— of no cross-sectional deformation, 105— of no thickness change, 281

autocorrelation function, 318average stress, 50axial force, 5, 108axial force diagram, 17

Bband matrix, 328bar, 105, 113Bauschinger effect, 389, 422beach mark, 389beam, 13, 16, 105, 113beam axis, 106beam-column, 221BEM, 136bending moment, 12, 13, 108bending vibration, 155, 347Bernoulli equation, 82Bernoulli-Euler assumption, 105Bessel functions, 230, 343Betti’s reciprocal theorem, 73, 138bifurcation buckling, 219bifurcation condition, 215bifurcation load, 213bifurcation phenomenon, 219bifurcation point, 213, 219biharmonic function, 89, 285, 293Biot’s strain, 463Biot stress, 481, 482body force, 44boundary condition

— of beam-columns, 223— of beams, 112Dirichlet —, 44, 111— of displacements, 44, 111

INDEX 589

essential —, 44, 111, 172— of flexural torsion, 263— of forces, 44, 112geometric —, 44, 111natural —, 44, 112, 172Neumann —, 44, 112— of plates, 286Robin —, 44, 112— of Saint-Venant’s torsion, 250static —, 44, 112— of the third kind, 44, 112— of Timoshenko beam-columns on elastic foundation,

242— of Timoshenko beams, 533

boundary element method, BEM, 136boundary value problem, 26, 113Boussinesq problem, 96, 102Brazilian test, 97brittle fracture, 388brittleness, 388buckling, 219

bifurcation —, 219Euler —, 225extremum —, 220flexural —, 226inelastic —, 236lateral-torsional —, 280torsional —, 279

buckling coefficient— of columns, 227— of plates, 295

buckling load, 213buckling mode, 225, 227buckling point, 220bulk modulus, 51buoyancy, 83Burgers vector, 99, 421

Ccantilever beam, 13carbon-fiber-reinforced polymer, 56, 87Castigliano’s second theorem, 200catenary, 246Cauchy-Green deformation tensor

left —, 460right —, 457

Cauchy integral theorem, 357Cauchy-Riemann equations, 83Cauchy’s theorem, 43Cauchy stress, 476, 482central difference, 312, 336centrifugal force, 300centrifugal moment, 275centripetal force

Eulerian, 81Lagrangian, 300

centroid, 109chain rule, 469channel section, 152, 267characteristic length, 434, 454Christoffel symbol, 562circular motion with constant speed, 300

clamshell mark, 389classical Galerkin form, 169closed section, 252coaxiality, 399, 400, 406coefficient

— of effective length, 227— of kinematic friction, 304linear — of thermal expansion, 60, 93, 474— of static friction, 61, 304, 390

cofactor matrix, 30, 458, 569cohesion, 430, 439collapse, 442collocation method, 168column, 16, 105, 113compact section, 451compatibility condition

strain —, 40, 293compatible strain, 41complementary potential energy, see ‘total complementary po-

tential energy’complementary strain energy, 199, 201complex Fourier series, 317complex velocity potential, 83compliance, 54composite material, 57compressional wave, 78compressive strength, 11, 237condensation, 402condition

Dirichlet —, 44, 111Neumann —, 44, 112Robin —, 44, 112— of the first kind, 44, 111— of the second kind, 44, 112— of the third kind, 44, 112

configurationcurrent —, 456initial —, 456reference —, 456

conformal transformation, 84conjugateness, 481conservation law, [see also ‘law of conservation’], 299, 308,

458conservative, 54conservative force, 229consistency condition, 65, 393, 400, 408consistent mass matrix, 192, 363constant strain triangle element, 205constitutive equation, [see also ‘Hooke’s law’], 51

— of generalized Prandtl-Reuss’s plasticity, 410— of Prandtl-Reuss’s plasticity, 403

constitutive law, 51contact problems, 543continuous beam, 29, 123, 127continuum, 33contradiction, 109, 258, 282contravariant base vector, 560contravariant component, 537, 559convected coordinate, 456convergence, 176convolution [integral], 318corner force, 286

590 INDEX

corotational stress rate, 486couple, 1couple stress, 43, 44covariant base vector, 560covariant component, 537, 560covariant derivative, 562crack, 147creep, 66critical damping, 301cross beam, 5, 18cross-sectional area, 108cubic crystal, 56

face-centered —, 398current configuration, 456curvature, 107, 539curved beam, 105

Ddamage theory, 452damping, 301damping coefficient, 301damping constant, 301damping factor, 301damping matrix, 322damping rate, 302damping ratio, 302deflection, 107deformation, 34deformation gradient, 457deformation rate, 467deformation tensor

left Cauchy-Green —, 460right Cauchy-Green —, 457

deformation theory, 63, 392, 402, 432degree of statical indeterminacy, 27, 28delta function, 72, 129, 309, 569depth-thickness ratio, 295deviatoric strain, 39deviatoric stress, 50deviator stress, 52, 398diffusion, 210dilatancy, 64, 399, 427, 439dilatancy angle, 430Dirac’s delta function, 72, 129, 309, 569direct stiffness method, 180Dirichlet condition, 44, 111dislocation, 63, 398

edge —, 387mathematical —, 99, 421screw —, 387, 570

dislocation line, 387, 421displacement, 34, 456displacement function, 177displacement gradient, 37, 458displacement method, 188distortion, 36distribution, 129divergence theorem, 45double Fourier series, 289drain, 18Drucker-Prager’s model, 426, 429Drucker’s postulate, 405

ductile fracture, 388Duffing oscillator, 383Duhamel’s integral, 310, 355dynamic instability, 230, 328dynamic magnification factor, 306

Eearth pressure, 154edge dislocation, [see also ‘dislocation’], 387effective length of buckling, 227effective stress, 63, 397, 397effective width, 152eigenfunction, 339eigenstrain, 568eigenvalue problem

— for boundary value problems, 227— of matrices, 238

elastica, 245, 542elastic compliance, 54elastic foundation, 153elasticity, 51elastic modulus, 51elastic perfectly-plastic body, 235, 391elastic potential, 54elastic strain, 59, 61elastic tensor, 51elementary beam theory, 106element stiffness equation

— of beam-columns, 238, 547— of beams, 178— of beams in motion, 191— of beams on elastic foundation, 190— of columns, 177— of planar frames, 187— of planar frames in finite displacements, 554— of planar trusses, 184— for plane strain problems, 206— of space frames, 275— of Timoshenko beams, 536— of Timoshenko beams in finite displacements, 554— of torsion, 272

elliptic integral, 245elongation, 458embedded coordinate, 456empirical formula, 454energy norm, 197Engesser’s buckling formula, 242, 546engineering strain, 55, 397engineering stress, 485equation of continuity, 79, 469equation of motion

— of beams, 155, 347— of frame elements, 367— of membranes, 296, 342— of multi-degree-of-freedom systems, 327— of single-degree-of-freedom systems, 297— of strings, 247, 337— of two-degree-of-freedom systems, 322

equilibrium equation— of axial forces in beams, 110— of beam-columns, 222, 547— of beam-columns on elastic foundation, 241

INDEX 591

— of beams on elastic foundation, 154— of bending of beams, 110— of bending of plates, 284— of Cauchy stress, 477— of flexural torsion, 263— of forces, 44— of in-plane forces of plates, 284— of membranes, 296— of moments, 44— of nominal stress, 479— of rubber, 543— of Saint-Venant’s torsion, 250— of second Piola-Kirchhoff stress stress, 480— of strings, 247— of Timoshenko beam-columns on elastic foundation,

242— of Timoshenko beams, 533

equivalent inclusion method, 571equivalent nodal force

— of beams, 178— of columns, 177

equivalent plastic strain, 397equivalent stress, 63, 397, 397error, 168, 197Eshelby tensor, 570essential boundary condition, 44, 111, 172Euler buckling, 225Euler curve, 225Euler equation, 70, 165, 195, 247Eulerian formulation, 79, 461Euler load, 225Euler’s formula, 317even permutation, 46evolution rule, 63, 399explicit integration method, 313extension, 35, 107extensional stiffness, 109extensional strain, 35, 463external force, 3extremum buckling, 220

Fface-centered cubic crystal, 398factor of safety, 237, 280fatigue fracture, 388FEM, [see also ‘finite element method’], 176fiber-reinforced material, 87fiber-reinforced polymer

carbon- — (CFRP), 56, 87glass- — (GFRP), 87

finite deformation, 455finite difference, 312, 336finite displacement theory, 212

linearized —, 217finite element, 176

— for diffusion, 210— for seepage flow, 210

finite element method, FEM, 176adaptive —, 208X- —, 209

finite rotation, 459finite strip method, 174

first Piola-Kirchhoff stress, 44, 480, 482first sectional moment, 108first sectional moment function, 147first variation, 69, 195Flamant solution, 96flexural compressive strength, 23, 139, 280flexural rigidity

— of beams, 109— of plates, 283

flexural tensile strength, 23, 139, 280flexural-torsional moment, 262flexural-torsional stiffness, 262Floquet theorem, 382flow rule, 63, 392, 399, 406, 408flow theory, 63, 399follower force, 229formulation

Eulerian —, 79, 461Lagrangian —, 79, 458updated Lagrangian —, 482, 487

Fourier series, 197complex —, 317double —, 289

Fourier spectrum, 318Fourier transform, 317fracture toughness, 101frame, 16, 113, 155free fall, 300free vibration, 297frequency response function, 315FRP, 56, 87functional, 69, 195fundamental solution, 136

GGalerkin form, 170Galerkin method, 171Gauss’s theorem, 45Geiringer’s equations, 438generalized

— coordinate, 331, 352— damping constant, 334, 354— displacement, 331, 352— force, 332, 354— Hooke’s law, 54, 61— mass, 329, 352— Poisson’s ratio, 56— Young’s modulus, 56

geometrical nonlinearity, 221geometric boundary condition, 44, 111geometric stiffness matrix, 238, 548

— of Timoshenko beam, 549, 557Gerber beam, 18glass-fiber-reinforced polymer, 87governing equation, 58Green’s function, 77, 96, 136, 310, 569Green strain, 457, 537Gurson damage model, 453gusset plate, 5

HHagen-Poiseuille flow, 80

592 INDEX

hair clip, 220Hamilton’s principle, 195hardening, 66, 392

isotropic —, 422kinematic —, 422, 509

hardening coefficient, 65, 393, 401, 407–409, 424, 427, 506hardening rule, 65Heaviside function, 310helical spring, 546Helmholtz decomposition theorem, 78Hencky’s theorem, 437Hertz solution, 97hexagonal crystal, 56high seas, 100history-dependency, 402, 415homogenization method, 377Hooke’s law, 51

generalized —, 54, 61incremental —, 401— in one dimension, 85— in plane strain, 85, 204— in plane stress, 85, 282

horizontal stiffener, 22hydrostatic pressure, 50, 58hyperelasticity, 493, 498, 504hypergeometic function, 230hypoelasticity, 494, 497

Iidentity tensor, 457imaginary unit, 154impulse response, 309inclusion, 60, 566incompatible strain, 396incompressibility, 57, 58, 79, 401, 469incremental equilibrium equation

— of Cauchy stress, 490Lagrangian —, 489— of nominal stress rate, 489

incremental Hooke’s law, 401incremental theory, 63, 399inelastic buckling, 236inelasticity, 61, 387inelastic strain, 59, 61infinitesimal displacement theory, 212infinitesimal rotation, 37influence line

— of axial force, 8— of bending moment, 21, 138, 141— of displacements, 136— of shear force, 21, 141

inhomogeneity, 60, 566initial condition, 155initial configuration, 456initial imperfection, 218, 237initial-value/boundary-value problem, 155inner product

— in polar coordinate system, 345weighted —, 197, 329, 340, 352

integral equation, 100internal force, 3, 41interpolation function, 177

invariant, 48first — of strain, 49first — of stress, 50second — of deviatoric strain, 52second — of deviatoric stress, 52, 396— of stress, 49third — of deviatoric stress, 396

inverse Fourier transform, 317inverse matrix, 30, 193irrotational flow, 81isoparametric element, 209isotropic elasticity, 51isotropic hardening, 422isotropic tensor, 53isotropy, 51Iwakuma’s buckling formula, 545

JJacobian, 458Jaumann rate

— of Cauchy stress, 486— of Kirchhoff stress, 487

Jaumann stress rate, 486joint equation, 198joint of Gerber beam, 18Joukowski transformation, 84J2 flow theory, 410

KKarman, see ‘von Karman’Karman’s plate theory, see ‘von Karman’s plate theory’Kelvin-Voigt model, 301kernel, 100kinematically admissible field, 67, 443

— of beams, 199, 447kinematic friction, 304kinematic hardening, 422, 509kinematics, 164, 456, 531, 538kinetic energy, 299Kirchhoff-Love assumption, 281Kirchhoff stress, 479Kotter’s equation, 439Kronecker delta, 38

LLagrange equation, 195Lagrange function, 195Lagrange multiplier, 71, 170, 245, 247, 542Lagrangian formulation, 79, 458Lame constants, 51laminar flow, 80Landau’s symbol, 232large deformation theory, 212, 455large displacement theory, 212large earthquake, 307lateral buckling, 280lateral force, 13lateral-torsional buckling, 280law of conservation

— of energy, 299— of mass, 458— of momentum, 308

INDEX 593

layered plate, 87least square method, 168, 176, 182, 197left Cauchy-Green deformation tensor, 460left stretch tensor, 460Legendre polynomials, 346limit analysis, 441limit cycle, 378linear coefficient of thermal expansion, 60, 93, 474linear distribution of bending strain, 107linear elasticity, 51linearized finite displacement theory, 217L-load, 8loading, 64, 391, 401, 407local buckling, 281, 291localized deformation, 440, 485Lode angle, 405logarithmic decrement, 303logarithmic strain, 463logarithmic strain rate, 472longitudinal vibration, 155longitudinal wave, 78lower bound theorem, 444Luders band, 485, 509, 527lumped mass matrix, 328

Mmain girder, 5mass, 1mass density, 458mass matrix, 191, 322

consistent —, 192, 363lumped —, 328

material derivative, 79, 469material nonlinearity, 221material point, 34, 456material time derivative, 79, 469mathematical dislocation, 99, 421Mathieu equation, 379matrix analysis of structures, 182, 536maximum axial force, 11maximum bending moment, 22, 139maximum friction force, 304maximum shear force, 22, 139Maxwell’s reciprocal theorem, 73, 137member, 3membrane, 296membrane analogy, 272method of strained coordinates, 377metric tensor, 560microtremor, 320Miller index, 398Mindlin-Reissner plate, 281Mises’ yield condition, see ‘von Mises’ yield condition’mixed condition, 44, 112modal analysis

— of beam, 351— of multi-degree-of-freedom system, 331

modebuckling —, 225, 227natural vibration —, 323orthogonal —, 329vibration —, 323, 328, 339, 342, 349

modelDrucker-Prager’s —, 426, 429Gurson damage —, 453Kelvin-Voigt —, 301Mooney-Rivlin —, 496phenomenological —, 434Prager’s —, 423Reuss —, 453, 566Shanley —, 236Voigt —, 452, 566Ziegler’s —, 424

modified Engesser’s formula, 242, 546modulated pendulum, 379Mohr-Coulomb’s failure criterion, 429, 439Mohr’s strain circle, 86Mohr’s stress circle, 436moment diagram, 14moment distribution method, 198moment of inertia, 2Mooney-Rivlin model, 496Muller-Breslau’s theorem, 140multiply connected, 271

Nnatural angular frequency, 299natural boundary condition, 44, 112, 172natural circular frequency, 299natural coordinate, 209natural frequency, 299natural period, 298natural vibration mode, 323Navier’s solution, 289Navier-Stokes equation, 79necking, 440, 522Neumann condition, 44, 112neutral, 219neutral axis, 109neutral loading, 401, 407neutral plane, 109nodal point, 5node, 176nominal stress, 477, 482, 485nominal stress rate, 489nonassociated flow rule, 410, 426, 505noncoaxiality, 430noncompact section, 451nonconforming element, 291nonconservative force, 229nonlinear elasticity, 235non-local theory, 33, 434norm, 197normality rule, 406normal stress, 43

Oobjective stress rate, 485objectivity, 62, 396octahedral stress, 63odd permutation, 46Oldroyd stress rate, 486open section, 252, 256orthogonal function, 197, 290

594 INDEX

orthogonality— of natural vibration mode, 329— of vibration mode of beams, 352— of vibration mode of membranes, 344, 345— of vibration mode of strings, 340

orthogonal mode, 329orthotropy, 55over damping, 301

Ppanel point, 3Papkovich-Neuber potentials, 101parametric oscillation, 383passive damper, 327peak strength, 451penalty method, 543perfect fluid, 81perfect plasticity, 66perfect system, 216permutation

even —, 46odd —, 46

permutation symbol, 45, 561permutation tensor, 561personal computer, 20, 182perturbation, 244, 541, 545perturbation method, 244, 373phase plane, 303, 372phase velocity, 78, 337phenomenological model, 434piecewise polynomial, 176Piola-Kirchhoff stress

first —, 44, 480, 482second —, 480, 482, 538

π-plane, 404plane strain, 84

plastically —, 428, 435, 441, 511plane stress, 85plastically plane strain, 428, 435, 441, 511plastic collapse, 442plastic deformation, 61, 387plastic hinge, 395, 445plasticity, 61, 236plastic moment, 394, 446plastic potential, 63, 408plastic section modulus, 446plastic volumetric strain, 427Poincare frequency shift, 377Poiseuille flow, see ‘Hagen-Poiseuille flow’Poisson’s ratio, 51

generalized —, 56polar decomposition theorem, 459, 554pole, 357polycrystalline metal model, 509positive-definiteness, 57, 239post-buckling, 244potential energy, see ‘total potential energy’, 299potential energy of external forces, 69, 196power law, 432power spectrum density function, 318Prager’s model, 423Prandtl-Reuss’s equation, 64, 399, 404, 407

pressure wave, 78primary wave, 78principal axis of cross sections, 275principal deviatoric stress, 396principal direction of strain, 37, 49principal direction of stress, 46principal strain, 37, 49principal stress, 46principle

— of action and reaction, 3–5— of complementary virtual work, 68, 70, 199Hamilton’s —, 195— of maximum plastic work, 406— of minimum strain energy, 201— of minimum total potential energy, 69, 196Saint-Venant’s —, 90, 148— of stationary total potential energy, 69, 196— of superposition, 125variational —, 195— of virtual work, 67, 67, 172, 195, 196

prismatic bar, 270program, 185, 189, 207progressive failure, 452proportional damping, 334P-wave, 78Pythagorean theorem, 37, 243

Qquadratic form, 196

Rradius of gyration, 225radius thickness ratio, 251, 260Rahmen, 16, 113, 155ramp, 17rank, 194rate of change of twist, 250rate of deformation, 467rate of equivolumetric deformation, 472rate of irrotational deformation, 472rate of plastic work, 400rate of stress-work, 478Rayleigh damping, 334Rayleigh-Ritz method, 196reaction force, 3reaction moment, 12reciprocal theorem

— of beam vibration, 360Betti’s —, 73, 138Maxwell’s —, 73, 137— of one degree-of-freedom vibration, 312

reciprocity, 75reduced modulus theory, 236redundancy, 28reference configuration, 456residual deformation, 61residual strength, 451residual stress, 237residue, 358resonance, 306Reuss model, 453, 566Reynolds number, 80

INDEX 595

right Cauchy-Green deformation tensor, 457right stretch tensor, 459rigid body displacement, 36rigid body rotation, 36rigid body translation, 36rigid perfectly-plastic body, 391Ritz method, 196Robin condition, 44, 112rolling, 237rosette gauge, 86rotation

finite —, 459infinitesimal —, 37

rotational inertia, 369rotation vector, 468

Ssafety factor, 237, 280Saint Venant-Kirchhoff material, 493, 496Saint-Venant’s principle, 90, 148Saint-Venant’s torsional constant, [see also ‘torsional con-

stant’], 250Saint-Venant’s torsional moment, 250Saint-Venant’s torsional stiffness, 250scale effect, 434, 454Schmid law, 508screw dislocation, [see also ‘dislocation’], 387secant modulus, 432secondary wave, 78second invariant of deviatoric strain, 52second invariant of deviatoric stress, 52, 396second Piola-Kirchhoff stress, 480, 482, 538second variation, 196sectional moment of inertia, 108sectional polar moment, 250, 279section modulus, 21, 109, 233seepage flow, 210seismic coefficient method, 307seismograph, 308self-adjoint system, 73self-consistent method, 575self-deployable structure, 221Shanley model, 236shape function, 209shear band, 432, 485, 509shear center, 151, 269shear diagram, 14shear flow, 150, 253shear force, 12, 13shearing strain, 35shearing yield stress, 63, 397shear lag, 152shear modulus, 51shear strength, 23, 397shear stress, 43shear wave, 78shell, 281shoe, 14sifting property, 129simple beam, 15single crystal model, 507singly connected, 271

singular matrix, 193singular perturbation method, 374, 376sink, 80slenderness parameter, 235

— for lateral buckling, 280slenderness ratio, 202, 225

— for flexural torsion, 266— of Timoshenko beams, 533

slide rule, 159slip line theory, 435slope, 107slope-deflection method, 198snap-through, 220softening, 406, 410, 432source, 80Southwell method, 234space structure, 220specific body force, 44spectral representation, 48, 459spin, 81, 468, 468splice plate, 22stability criterion, 215, 218stability problem, 212statically admissible field, 67, 436, 443

— of beams, 199, 447statically determinate, 4statically determinate equivalent system, 125statically indeterminate force, 125statically indeterminate structure, 30static boundary condition, 44, 112static friction, 61, 304, 390step function, 310stiffener

horizontal —, 22vertical —, 18, 22

stiffness equation, [see also ‘element stiffness equation’], 177stiffness matrix, 322

— of beam-columns, 238— of beams, 178— of beams on elastic foundation, 190— of columns, 177— of planar frames, 187— of planar frames in finite displacements, 557— of planar trusses, 184— for plane strain problems, 206— of space frames, 276— of Timoshenko beams, 535, 536— of Timoshenko beams in finite displacements, 557— of torsion, 272

story equation, 198strain, 36

Almansi —, 461Biot’s —, 463engineering —, 55, 397extensional —, 463Green —, 457, 537logarithmic —, 463

strain compatibility condition, 40, 293strain energy, 55, 69, 196strain energy density function, 493strain energy function, 54strain-gauge rosette, 86

596 INDEX

strain gradient, 434stream function, 83streamline, 82, 83strength, 3

compressive —, 237flexural compressive —, 280flexural tensile —, 280shear —, 397tensile —, 237

strength correlation— of axial force and bending, 233

stress, 20, 42Biot —, 481, 482Cauchy —, 476, 482first Piola-Kirchhoff—, 44, 480, 482Kirchhoff—, 479nominal —, 477, 482, 485second Piola-Kirchhoff—, 480, 482, 538true —, 476, 482

stress concentration factor, 91, 98, 102stress function, 89stress intensity factor, 101stress invariant, 49stress rate

corotational —, 486Jaumann —, 486nominal —, 489Oldroyd —, 486Truesdell —, 488

stress relaxation, 66stress resultant, 43stress-strain relation, [see also ‘Hooke’s law’], 51stress vector, 42stretch, 35, 459stretch tensor

left —, 460right —, 459

string, 247stringer, 5strong form, 169structure for falling prevention of bridge, 18Sturm-Liouville problem, 173subgrade reaction, 154submission, 221subparametric element, 209summation convention, 559super-convergence, 176superparametric element, 209surface force, 44surface to the negative direction, 13, 43surface to the positive direction, 13, 43S-wave, 78

Ttangent compliance

— of generalized Prandtl-Reuss’s plasticity, 409— of Prandtl-Reuss’s plasticity, 403

tangent modulus, 432— of generalized Prandtl-Reuss’s plasticity, 410— of Prandtl-Reuss’s plasticity, 403

tangent modulus theory, 235tangent stiffness, 217, 555

tensile strength, 11, 237tensile yield stress, 63, 397tensor, 36tensor product, 48test function, 129, 177theorem

Castigliano’s second —, 200Cauchy integral —, 357Cauchy’s —, 43divergence —, 45Floquet —, 382Gauss’s —, 45Helmholtz decomposition —, 78Hencky’s —, 437— of Kelvin and Tait, 466Muller-Breslau’s —, 140polar decomposition —, 459, 554Pythagorean —, 37, 243reciprocal —, 137, 138, 360three-moment —, 198upper-lower bound —, 444

theorydamage —, 452deformation —, 63, 392, 402, 432elementary beam —, 106finite displacement —, 212flow —, 63, 399Gurson’s —, 453incremental —, 63, 399infinitesimal displacement —, 212J2 flow —, 410large deformation —, 212, 455large displacement —, 212linearized finite displacement —, 217non-local —, 33, 434reduced modulus —, 236slip line —, 435tangent modulus —, 235von Karman’s plate —, 291

thickness parameter, 546thin-walled member, 148, 249, 252third invariant of deviatoric stress, 396three-moment theorem, 198Timoshenko beam, 152, 531torsional buckling, 279torsional constant

— of circular columns, 250— of circular tubes, 251— of thin-walled circular tubes, 251— of thin-walled closed multicell sections, 256— of thin-walled closed sections, 254— of thin-walled open sections, 259— of thin-walled rectangular sections, 259

torsional constants ratio, 266total complementary potential energy, 70, 199total potential energy, 69, 196total-strain theory, [see also ‘deformation theory’], 392toughness, 388trace, 39traction, 42trajectory, 303, 372, 461transient response, 305

INDEX 597

transversely isotropic, 56transverse wave, 78Tresca’s yield condition, 62, 404trial function, 177triaxial compression test, 52, 398trivial solution, 225, 226, 238Truesdell stress rate, 488true stress, 476, 482truss, 4, 155

Uultimate lateral strength, 441unit impulse response, 309unit impulsive force, 309unit load method, 131, 135, 156, 200

— of elastic media, 77— of truss structures, 157

unit tensor, 457unit-warping function, 264, 269unloading, 65, 235, 391, 401, 408updated Lagrangian formulation, 482, 487upper bound theorem, 444upper-lower bound theorem, 444

Vvan der Pol oscillator, 378variational principle, 195variational problem, 195velocity, 467velocity gradient, 467velocity potential, 83vertical stiffener, 18, 22vibration mode, 323, 328, 339, 342, 349vidro, 220vinyl-ester synthetic resin, 56virtual displacement, 69, 172virtual work equation, 172

— of beam-columns, 238, 546— of beams, 131, 165, 178— of beams in motion, 195— of beams on elastic foundation, 190— of columns, 176— in finite displacements, 538— of flexural torsion of bars, 272— for plane strain problems, 204— of plates, 291— of Saint-Venant’s torsion of bars, 273— for three-dimensional problems, 204

— of Timoshenko beams, 533, 534viscoelasticity, 155viscosity, 66, 387viscous damping coefficient, 301void formation, growth and coalescence, 453Voigt constants, 52, 55Voigt model, 452, 566volumetric strain, 39von Karman’s plate theory, 291von Mises’ yield condition, 63, 397, 404vorticity, 468

Wwarping, 260warping constant, 262warping displacement, 261warping function, 264wave equation, 78, 155, 247, 296, 337weak form, 169weight, 168, 197weighted inner product, 197, 329, 340, 352weighted residual method, 168, 197weight function, 170white noise, 318Wiener-Khintchine relation, 319William’s toggle, 220

XX-FEM, 209

Yyield condition, 62, 391, 393, 397

Tresca’s —, 404von Mises’ —, 63, 397, 404

yield function, 64, 393, 397, 404yield moment, 394, 446yield stress, 63, 225, 235, 397

shearing —, 63, 397tensile —, 63, 391, 397

yield surface, 404yield-vertex, 406Young’s modulus, 51

generalized —, 56yo-yo balloon, 308

ZZiegler’s model, 424Zimmermann effect, 361

Acknowledgment on documentation: The optional style files ‘eepic.sty’, ‘eepicsup.sty’, ‘epic.sty’, ‘hhline.sty’,‘hyperref.sty’, ‘multicol.sty’, ‘ovudbrac.sty’, ‘txfonts.sty’, ‘ulem.sty’, ‘wrapfig.sty’ and ‘xbmkanji.sty’are used to enhance the original LATEX definitions of macros. Since this note includes Japanese characters, the NTT JLATEX [rel-atively old version of JTEX, Version 2.2, based on TEX Version 3.14159265 (TEX Live 2016/W32TEX) (preloaded format=jlatex2016.10.19)] is used.

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