Mechanics of solids and materials

MECHANICS OF SOLIDS AND MATERIALS Mechanics of Solids and Materials intends to provide a modern and integrated treatment of the foundations of solid mechanics as applied to the mathematical description of material behavior The book blends both innovative topics (e.g., large strain, strain rate, temperature, time-dependent deformation and localized plastic deformation in crystalline solids, and deformation of biological networks) and traditional topics (e.g., elastic theory of torsion, elastic beam and plate theories, and contact mechanics) in a coherent theoretical framework This, and the extensive use of transform methods to generate solutions, makes the book of interest to structural, mechanical, materials, and aerospace engineers Plasticity theories, micromechanics, crystal plasticity, thin films, energetics of elastic systems, and an overall review of continuum mechanics and thermodynamics are also covered in the book Robert J Asaro was awarded his PhD in materials science with distinction from Stanford University in 1972 He was a professor of engineering at Brown University from 1975 to 1989, and has been a professor of engineering at the University of California, San Diego since 1989 Dr Asaro has led programs involved with the design, fabrication, and full-scale structural testing of large composite structures, including high-performance ships and marine civil structures His list of publications includes more than 170 research papers in the leading professional journals and conference proceedings He received the NSF Special Creativity Award for his research in 1983 and 1987 Dr Asaro also received the TMS Champion H Mathewson Gold Medal in 1991 He has made fundamental contributions to the theory of crystal plasticity, the analysis of surface instabilities, and dislocation theory He served as a founding member of the Advisory Committee for NSF’s Office of Advanced Computing that founded the Supercomputer Program in the United States He has also served on the NSF Materials Advisory Committee He has been an affiliate with Los Alamos National Laboratory for more than 20 years and has served as consultant to Sandia National Laboratory Dr Asaro has been recognized by ISI as a highly cited author in materials science Vlado A Lubarda received his PhD in mechanical engineering from Stanford University in 1980 He was a professor at the University of Montenegro from 1980 to 1989, Fulbright fellow and a visiting associate professor at Brown University from 1989 to 1991, and a visiting professor at Arizona State University from 1992 to 1997 Since 1998 he has been an adjunct professor of applied mechanics at the University of California, San Diego Dr Lubarda has made significant contributions to phenomenological theories of large deformation elastoplasticity, dislocation theory, damage mechanics, and micromechanics He is the author of more than 100 journal and conference publications and two books: Strength of Materials (1985) and Elastoplasticity Theory (2002) He has served as a research panelist for NSF and as a reviewer to numerous international journals of mechanics, materials science, and applied mathematics In 2000 Dr Lubarda was elected to the Montenegrin Academy of Sciences and Arts He is also recipient of the 2004 Distinguished Teaching Award from the University of California Mechanics of Solids and Materials ROBERT J ASARO University of California, San Diego VLADO A LUBARDA University of California, San Diego cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge cb2 2ru, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521859790 © Robert Asaro and Vlado Lubarda 2006 This publication is in copyright Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press First published in print format isbn-13 isbn-10 978-0-511-14707-4 eBook (NetLibrary) 0-511-14707-4 eBook (NetLibrary) isbn-13 isbn-10 978-0-521-85979-0 hardback 0-521-85979-4 hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate Contents Preface page xix PART 1: MATHEMATICAL PRELIMINARIES Vectors and Tensors 1.1 Vector Algebra 1.2 Coordinate Transformation: Rotation 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23 of Axes Second-Rank Tensors Symmetric and Antisymmetric Tensors Prelude to Invariants of Tensors Inverse of a Tensor Additional Proofs Additional Lemmas for Vectors Coordinate Transformation of Tensors Some Identities with Indices Tensor Product Orthonormal Basis Eigenvectors and Eigenvalues Symmetric Tensors Positive Definiteness of a Tensor Antisymmetric Tensors 1.16.1 Eigenvectors of W Orthogonal Tensors Polar Decomposition Theorem Polar Decomposition: Physical Approach 1.19.1 Left and Right Stretch Tensors 1.19.2 Principal Stretches The Cayley–Hamilton Theorem Additional Lemmas for Tensors Identities and Relations Involving ∇ Operator Suggested Reading 1 5 7 10 10 11 12 14 14 15 15 17 19 20 21 21 22 23 23 25 v vi Contents Basic Integral Theorems 2.1 Gauss and Stokes’s Theorems 2.1.1 Applications of Divergence Theorem 2.2 Vector and Tensor Fields: Physical Approach 2.3 Surface Integrals: Gauss Law 2.4 Evaluating Surface Integrals 2.4.1 Application of the Concept of Flux 2.5 The Divergence 2.6 Divergence Theorem: Relation of Surface to Volume 2.7 2.8 Integrals More on Divergence Theorem Suggested Reading Fourier Series and Fourier Integrals 3.1 Fourier Series 3.2 Double Fourier Series 3.2.1 Double Trigonometric Series 3.3 Integral Transforms 3.4 Dirichlet’s Conditions 3.5 Integral Theorems 3.6 Convolution Integrals 3.6.1 Evaluation of Integrals by Use of Convolution 3.7 3.8 3.9 3.10 Theorems Fourier Transforms of Derivatives of f (x) Fourier Integrals as Limiting Cases of Fourier Series Dirac Delta Function Suggested Reading 26 26 27 27 28 29 31 31 33 34 35 36 36 37 38 39 42 46 48 49 49 50 51 52 PART 2: CONTINUUM MECHANICS Kinematics of Continuum 4.1 Preliminaries 4.2 Uniaxial Strain 4.3 Deformation Gradient 4.4 Strain Tensor 4.5 Stretch and Normal Strains 4.6 Angle Change and Shear Strains 4.7 Infinitesimal Strains 4.8 Principal Stretches 4.9 Eigenvectors and Eigenvalues of Deformation Tensors 4.10 Volume Changes 4.11 Area Changes 4.12 Area Changes: Alternative Approach 4.13 Simple Shear of a Thick Plate with a Central Hole 4.14 Finite vs Small Deformations 4.15 Reference vs Current Configuration 4.16 Material Derivatives and Velocity 4.17 Velocity Gradient 55 55 56 57 58 60 60 61 62 63 63 64 65 66 68 69 71 71 Contents 4.18 4.19 4.20 4.21 4.22 4.23 4.24 4.25 4.26 4.27 4.28 4.29 vii Deformation Rate and Spin Rate of Stretching and Shearing ˙ vs D Material Derivatives of Strain Tensors: E Rate of F in Terms of Principal Stretches 4.21.1 Spins of Lagrangian and Eulerian Triads Additional Connections Between Current and Reference State Representations Transport Formulae Material Derivatives of Volume, Area, and Surface Integrals: Transport Formulae Revisited Analysis of Simple Shearing Examples of Particle and Plane Motion Rigid Body Motions Behavior under Superposed Rotation Suggested Reading Kinetics of Continuum 5.1 Traction Vector and Stress Tensor 5.2 Equations of Equilibrium 5.3 Balance of Angular Momentum: Symmetry of σ 5.4 Principal Values of Cauchy Stress 5.5 Maximum Shear Stresses 5.6 Nominal Stress 5.7 Equilibrium in the Reference State 5.8 Work Conjugate Connections 5.9 Stress Deviator 5.10 Frame Indifference 5.11 Continuity Equation and Equations of Motion 5.12 Stress Power 5.13 The Principle of Virtual Work 5.14 Generalized Clapeyron’s Formula 5.15 Suggested Reading Thermodynamics of Continuum 6.1 First Law of Thermodynamics: Energy Equation 6.2 Second Law of Thermodynamics: Clausius–Duhem 6.3 6.4 6.5 6.6 6.7 Inequality Reversible Thermodynamics 6.3.1 Thermodynamic Potentials 6.3.2 Specific and Latent Heats 6.3.3 Coupled Heat Equation Thermodynamic Relationships with p, V, T, and s 6.4.1 Specific and Latent Heats 6.4.2 Coefficients of Thermal Expansion and Compressibility Theoretical Calculations of Heat Capacity Third Law of Thermodynamics Irreversible Thermodynamics 6.7.1 Evolution of Internal Variables 74 75 76 78 81 82 83 84 85 87 88 89 90 92 92 94 95 96 97 98 99 100 102 102 107 108 109 111 111 113 113 114 116 116 118 119 120 121 122 123 125 127 129 viii Contents 6.8 Gibbs Conditions of Thermodynamic Equilibrium 6.9 Linear Thermoelasticity 6.10 Thermodynamic Potentials in Linear Thermoelasticity 6.10.1 Internal Energy 6.10.2 Helmholtz Free Energy 6.10.3 Gibbs Energy 6.10.4 Enthalpy Function 6.11 Uniaxial Loading and Thermoelastic Effect 6.12 Thermodynamics of Open Systems: Chemical 6.13 6.14 6.15 6.16 6.17 6.18 Potentials Gibbs–Duhem Equation Chemical Potentials for Binary Systems Configurational Entropy Ideal Solutions Regular Solutions for Binary Alloys Suggested Reading Nonlinear Elasticity 7.1 Green Elasticity 7.2 Isotropic Green Elasticity 7.3 Constitutive Equations in Terms of B 7.4 Constitutive Equations in Terms of Principal Stretches 7.5 Incompressible Isotropic Elastic Materials 7.6 Elastic Moduli Tensors 7.7 Instantaneous Elastic Moduli 7.8 Elastic Pseudomoduli 7.9 Elastic Moduli of Isotropic Elasticity 7.10 Elastic Moduli in Terms of Principal Stretches 7.11 Suggested Reading 129 130 132 132 133 134 135 136 139 141 142 143 144 145 147 148 148 150 151 152 153 153 155 155 156 157 158 PART 3: LINEAR ELASTICITY Governing Equations of Linear Elasticity 8.1 Elementary Theory of Isotropic Linear Elasticity 8.2 Elastic Energy in Linear Elasticity 8.3 Restrictions on the Elastic Constants 8.3.1 Material Symmetry 8.3.2 Restrictions on the Elastic Constants 8.4 Compatibility Relations 8.5 Compatibility Conditions: Cesaro ` Integrals 8.6 Beltrami–Michell Compatibility Equations 8.7 Navier Equations of Motion 8.8 Uniqueness of Solution to Linear Elastic Boundary Value Problem 8.8.1 8.8.2 8.9 Statement of the Boundary Value Problem Uniqueness of the Solution Potential Energy and Variational Principle 8.9.1 Uniqueness of the Strain Field 161 161 163 164 164 168 169 170 172 172 174 174 174 175 177 Contents 8.10 Betti’s Theorem of Linear Elasticity 8.11 Plane Strain 8.11.1 Plane Stress 8.12 Governing Equations of Plane Elasticity 8.13 Thermal Distortion of a Simple Beam 8.14 Suggested Reading Elastic Beam Problems 9.1 A Simple 2D Beam Problem 9.2 Polynomial Solutions to ∇ φ = 9.3 A Simple Beam Problem Continued 9.3.1 Strains and Displacements for 2D Beams 9.4 Beam Problems with Body Force Potentials 9.5 Beam under Fourier Loading 9.6 Complete Boundary Value Problems for Beams 9.6.1 Displacement Calculations 9.7 Suggested Reading 10 Solutions in Polar Coordinates 10.1 Polar Components of Stress and Strain 10.2 Plate with Circular Hole 10.2.1 Far Field Shear 10.2.2 Far Field Tension 10.3 Degenerate Cases of Solution in Polar Coordinates 10.4 Curved Beams: Plane Stress 10.4.1 Pressurized Cylinder 10.4.2 Bending of a Curved Beam 10.5 Axisymmetric Deformations 10.6 Suggested Reading 11 Torsion and Bending of Prismatic Rods 11.1 Torsion of Prismatic Rods 11.2 Elastic Energy of Torsion 11.3 Torsion of a Rod with Rectangular Cross Section 11.4 Torsion of a Rod with Elliptical Cross Section 11.5 Torsion of a Rod with Multiply Connected Cross Sections 11.5.1 Hollow Elliptical Cross Section 11.6 Bending of a Cantilever 11.7 Elliptical Cross Section 11.8 Suggested Reading 12 Semi-Infinite Media 12.1 Fourier Transform of Biharmonic Equation 12.2 Loading on a Half-Plane 12.3 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Acta Mater., Vol 50, pp 61–73 Yang, W., and Lee, W B (1993), Mesoplasticity and Its Applications, Springer-Verlag, Berlin Youssef, K M., Scattergood, R O., Murty, K L., and Koch, C C (2004), Ultratough Nanocrystalline Copper with a Narrow Grain Size Distribution, Appl Phys Lett., Vol 85, pp 929–931 Ziegler, H (1959), A Modicification of Prager’s Hardening Rule, Q Appl Math., Vol 17, pp 55–65 Ziegler, H (1983), An Introduction to Thermomechanics, 2nd revised ed., North-Holland, Amsterdam Zyczkowski, M (1981), Combined Loadings in the Theory of Plasticity, PWN, Polish Scientific Publishers, Warszawa Index absolute temperature, 115 acoustic tensor, 266 adaptive elasticity, 609 adiabatic loading, 119 affinity, 127, 129, 618 Airy stress function, 179, 410, 415, 419 angle of twist, 214 angular change between fibers, 76 of principal directions, 76 anisotropic elastic solid, 264, 345, 604 elasticity, 332 hardening, 475 antiplane shear, 390 strain, 386 aorta, 630 apposition, 609 arc length, 59 area change, 64 for a membrane, 636 Asaro–Tiller field, 459 associative flow rule, 480 asymptotic stress fields, 459 axial vector, 17, 87 axisymmetric problems, 211 back stress, 475 balance of angular momentum, 95 Barnett–Lothe tensors, 333 Bauschinger effect, 475, 595 Beltrami–Michell equations, 172 bending of beams, 225 Bessel functions, 385 Betti’s reciprocal theorem, 177, 318 couple-stress elasticity, 393 Bianchi conditions, 171 bifurcation, 498 biharmonic equation, 179, 184 polynomial solutions, 185 with body forces, 188 bimaterial interface, 407, 410, 419 binary alloy, 145 biomaterials, 609 Blatz–Ko strain energy, 700 blood vessel, 630 body force potential, 178 Boussinesq–Papkovitch solutions, 248 Burgers vector, 294, 407, 410, 419 Cauchy elasticity, 149 Cauchy stress, 92 nonsymmetric, 398 principal values of, 96 symmetry of, 95 Cauchy tetrahedron, 92 Cauchy–Green deformation tensor, 58 cavitation instability, 500, 501 Cayley–Hamilton theorem, 22 Cesaro ` integrals, 172 characteristic equation, 13, 23 chemical potential, 139, 370, 449 Christoffel stiffness tensor, 266 circular plate, 288 Clapeyron’s formula, 111 Clausius–Duhem inequality, 115 coarse slip bands, 558 coefficient of compressibility, 122 of thermal expansion, 122 compatibility equations, 169, 377 Beltrami–Michell, 172 Saint-Venant, 171 compliance tensor, 118 configurational entropy, 144 force, 400 853 854 conservation integrals, 403 laws, 404 of mass, 34 consistency condition, 472, 481 constitutive equations linear elasticity, 161 nonlinear elasticity, 149 plasticity deformation theory, 485 isotropic hardening, 475 kinematic hardening, 477 nonassociative, 480 pressure-dependent, 480 rate-dependent, 482 constrained equilibrium, 127 field, 338 contact problems, 271 continuity equation, 34, 610 convected lattice vectors, 540 stress rate, 105, 680 convolution integrals, 48 coordinate transformation, core energy, 424, 426 corner theory of plasticity, 487 corotational stress rate, 106 correspondence theorem, 378, 389 Cotter–Rivlin convected rate, 680 Coulomb’s law, 27 couple-stress, 375, 376, 398 coupled heat equation, 120 crack extension force, 323 crack opening displacement, 325 creep, 483 critical conditions for localization, 574 film thickness, 432 hardening rate, 567 nucleus size, 374 resolved shear stress, 505 crystal plasticity, 538, 601 laminate model, 601 single slip, 605 curl, 55, 56 curvature tensor, 376, 399 cylindrical coordinates, 668 void, 500 Debye’s temperature, 124 deformation gradient, 56 cylindrical coordinates, 670 multiplicative decomposition, 490, 622 deformation rate, 69, 73, 74 deformation theory of plasticity, 484, 486, 596 degenerate solutions, 204 densification, 611 Index determinant, deviatoric plane, 469 work, 464 dilatancy factor, 481 dilatant materials, 480 Dirac delta function, 51, 233, 264 Dirichlet conditions, 42 disclination, 760 dislocation, 293, 299 core, 424 density, 511 distribution, 296 driving force, 439, 441 edge, 419, 509 energy, 424, 426 forces on, 365 forest, 515 injection, 444 interactions, 514 line, 299 misfit, 432, 433 multiplication, 517 near free surface, 426 partial, 513 perturbed array, 443 recession, 439 spacing, 432 threading, 432, 437 dislocation array energy, 430 formation, 432, 438 stress field, 428 dislocation force array, 431 edge dislocation, 415, 418, 421, 423 screw dislocation, 408, 410 straight, 427 displacement discontinuity, 421, 424, 430 displacements determination of, 196 half space, 238 in beam, 187 nonsingle valued, 293 distributed contact loading, 274 div, 56 divergence theorem, 26, 33, 34 double Fourier series, 37 double slip, 576 Drucker–Prager yield criterion, 468 Duhamel–Neumann expression, 132 Dulong–Petit limit, 124 dyadic notation, 13 eccentricity, 636 edge dislocation, 299, 410 couple-stress elasticity, 381 near free surface, 417, 422 Index eigenvalues, 12, 63 symmetric tensors, 14 eigenvectors, 12, 63 orthogonality of, 63 stretch of, 63 elastic compliance tensor, 118 constants, 169 deformation gradient, 490 moduli tensor, 158, 627 pseudomoduli, 155 stiffness tensor, 118 unloading, 489 elastic-plastic interface, 497 elasticity 3D, 264 Cauchy, 149 crystal elasticity, 547 Green, 148 isotropic, 150 nonlinear, 148 energy Gibbs, 117, 128 Helmholtz free, 116, 128 internal, 113 kinetic, 113 of a dislocation line, 366 total, 113 energy equation, 114 with mass growth, 616 energy factor matrix, 511 tensor, 315 energy momentum tensor, 358 finite deformations, 361 micropolar elasticity, 404 energy release rate, 315, 323 enthalpy, 117, 128 of mixing, 145 entropic elasticity, 116 entropy, 114 of mixing, 144 epitaxial growth, 432 layer, 432 equilibrium equations, 95 Eshelby inclusion problem, 335 tensor, 341 Euler’s laws of motion, 614 Eulerian strain, 676 rate of, 76 triad, 80 evolution equation, 129, 479 for mass growth, 631 for stretch ratio, 630 evolution of back stress, 476, 477 855 extended dislocation, 513 Taylor model, 587 fiber rotation of, 19 stretch, 60 stretching rate, 73 film thickness, 432 first law of thermodynamics, 113 first Piola–Kirchhoff stress, 99 Flamant solution, 723 flat punch, 278 flexural rigidity, 284 flow potential, 482 flow rule associative, 480 nonassociative, 480 flux, 31, 127, 129, 618 force generalized, 357 on a dislocation, 365 on a precipitate interface, 373 on an interface, 359 forest dislocations, 515 Fourier double series, 37 integral theorem, 46 kernel, 41 law of conduction, 119 loading, 191, 193 series, 36 transform, 39, 48, 265 frame indifference, 102 Frank and van der Merwe criterion, 434 Frank–Read source, 530 free energy, 356 Galerkin vector, 256 Gauss divergence theorem, 26 law, 28 generalized force, 357 plastic strain, 474 geometrical hardening, 607 softening, 508, 580, 607, 816 Gibbs conditions of equilibrium, 129 energy, 117, 128, 141, 356 Gibbs–Duhem equation, 142 grad, 56 Green elasticity, 148 function, 264 isotropic elasticity, 150 lattice strain, 603 strain, 59 stretch tensor, 86 856 Griffith crack, 332, 773 criterion, 320 Gurson yield criterion, 470 half-plane loading, 232 half-space solutions, 229 Hall–Petch relation, 517 hard tissue, 609 hardening, 473 isotropic, 474 kinematic, 475 heat capacity, 124 conduction, 113 Fourier law, 119 equation, 120 flow, 113 Helmholtz equation, 378 Helmholtz free energy, 116, 128 with mass growth, 620 Hertz problem, 259 hollow dislocation, 384 Hooke’s law, 150, 161 hyperelasticity, 149 hypertension, 630 hypertrophy, 609 ideal plasticity, 473 solution, 144 ideally plastic material, 497 identity tensor, 6, 10 image force on a defect, 371 inclusion elastic energy of, 343 field at the interface, 352 field in the matrix, 352 inhomogeneous, 344 isotropic spherical, 353 problem statement, 335 incompressibility constraint, 153 incompressible elasticity, 153 infinite strip, 242 inflation of balloon, 702 inhomogeneous inclusion, 344 instantaneous elastic moduli, 155 integral derivative, 83 transform, 39 interaction energy, 362, 432, 442, 444 between dislocations, 318 interface dislocation edge, 415, 420 screw, 409 interface force, 359 intermediate configuration, 489 internal energy, 113 with mass growth, 619 internal variables, 127 Index invariant functional, 401 invariants, 6, 12, 14 inverse Fourier transform, 48 of a tensor, pole figures, 593 irreversible thermodynamics, 127 isochoric plastic deformation, 474 isotropic Green elasticity, 150 hardening, 474 inclusion, 350 material, 150 J integral, 358, 404 J2 deformation theory of plasticity, 486 Jaumann stress rate, 106, 107, 541 cylindrical coordinates, 685 on crystal spin, 541 Johnson–Cook model, 483 Joule’s effect, 137 kernel, 41 kinematic hardening, 475 linear, 477 nonlinear, 477 Prager, 477 Ziegler, 477 kinetic energy, 113 Kirchhoff stress, 101 Kolosov constant, 180, 411 Kronecker delta, L integral, 404 Lagrangian multiplier, 153 strain, 59 triad, 79 Lame´ constants, 168 problem, 251 laminate plasticity, 601 Laplace’s equation, 172, 378 latent hardening, 519, 547 heat, 118 lattice base vectors, 540 parameter, 433 lattice rotation, 506 geometrical softening, 580, 582 shear bands, 580, 582 left stretch tensor, 21 line integrals material derivative, 85 linear dependence, 13 linearly hardening material, 499 loading index kinematic hardening, 476 pressure-dependent plasticity, 479 Index localized plastic deformation, 557 Love’s potential, 257 M integral, 404, 773 macroscopic shear bands, 558 mass conservation, 34 flow, 31 growth, 609, 611, 630 resorption, 631 material derivative, 71 line integrals, 85 surface area, 84 surface integrals, 84 volume integrals, 84 material length, 383 Matthews–Blakeslee criterion, 433 maximum shear stress, 97 Maxwell relations, 128 mechanical power input, 108 Mellin transforms, 40 Michell’s solution, 724 micropolar continuum, 375 elasticity, 398 Mindlin’s stress functions, 380 misfit dislocation, 432, 433 mismatch strain, 433, 434 mixed dislocations, 304 Mohr’s envelope, 467 Mohr–Coulomb yield criterion, 467 molar Gibbs energy, 142 Mooney–Rivlin material, 153, 698 morphogenesis, 610 multiple slip, 576 multiplicative decomposition, 490 in biomechanics, 622 multiply connected cross section, 222 regions, 172 Murnaghan’s constants, 150 nanocrystalline grains, 530 Nanson’s relation, 65 necking, 487 neo-Hookean material, 153 Newtonian fluid, 701 Noether’s theorem, 400 nominal stress, 98, 99, 101 traction, 678 non-Schmid effect, 571 stress, 563 nonassociative flow rule, 480 nonlinear elasticity, 148 norm of a function, 37 857 objective rate, 90, 105 octahedral plane, 464, 468 shear stress, 464 Oldroyd rate, 154, 679, 680 Onsager reciprocity relations, 129 open thermodynamic system, 139 opening angle, 631 ordered crystals localized deformation, 560 orthogonal tensors, 17 geometrical interpretation, 18 specific forms of, 88 orthonormal basis, 2, 11 Papkovich–Neuber potentials, 769 partial dislocations, 513, 531 Peach–Koehler force, 129, 365, 512, 812 Peierls–Nabarro dislocation, 759 permutation tensor, phenomenological plasticity, 461 plane strain, 178 stress, 179 plane stress modulus, 567 plastic potential, 480 deformation gradient, 490 potential surface, 480 rate potential, 488 secant modulus, 486 strain, 484 tangent modulus, 474, 486 void growth, 495 plasticity, 461 associative, 480 corner theory, 487 deformation theory, 484, 486 ideal, 473 nonassociative, 480 strain hardening, 461 plates equilibrium, 282 flexural rigidity, 284 point force, 237, 257, 261 Poisson’s equation, 172 polar decomposition, 20, 63 polynomial solutions, 184 porosity, 470 positive definite tensors, 14 potential energy, 175, 355 bent plate, 284 couple-stress elasticity, 396 Poynting effect, 700 Prager’s hardening, 477 Prandtl stress function, 217 precipitation, 373 858 pressure-dependent plasticity, 478 pressure-sensitive plastic flow, 572 pressurized cylinder, 209 sphere, 250 principal directions, 76 stresses, 96 stretch, 21, 62, 78, 152 principle of virtual work, 109 proportional loading, 484, 486 pseudotraction, 678 pseudomoduli, 155 quadratic forms, 22 rate of deformation, 69, 627 of working, 101 potential, 488 sensitivity, 602 tangent modulus, 548 rate-dependent plasticity, 482 slip, 547, 592 recall term, 478 reciprocal symmetry, 480 regular solution, 145 Reiner–Rivlin fluid, 700 remodeling, 610 residual strain, 631 resolved shear stress, 482 Reuss estimates, 825 reversible thermodynamics, 116 Reynolds transport theorem, 613 right stretch tensor, 21 rigid inclusion, 252 indenters, 271 rigid body motion, 88 Rivlin–Ericksen tensors, 657 rocks, 480 rotation tensor, 376 rubber model, 153 Saint-Venant compatibility equations, 170, 171 principle, 187 Saint-Venant−Kirchhoff assumption, 150, 696 scalar field, 55 gradient, 70 scalar product, Schmid rule, 505, 560 stress, 561 screw dislocation, 299, 302, 407 couple-stress elasticity, 391 near free surface, 409 Index secant modulus, 484 second law of thermodynamics, 114 second Piola–Kirchhoff stress, 102 semi-inverse method, 225 shear center, 227 modulus, 161 strain, 60 shear stress maximum, 97 resolved, 482 simple beam Fourier loading, 191 simple shear, 58, 85 single valued displacements, 208 singular integral equation, 296 slip steps, 502 system, 482 traces, 502 slip-plane hardening, 547 small strain, 61 soft tissue, 609, 629 softening, 473 geometric, 816 vertex, 487 soil mechanics, 468 specific heat, 118 sphere subject to temperature gradient, 254 spherical coordinates, 248 indentation, 255 void, 495 spin rate, 74 plastic part, 604 spin tensor, 79 stability array, 441 bounds, 439, 445 dislocation array, 439 stacking faults, 513 emission from grain boundaries, 531 state variables, 116 static equilibrium, 92, 96 stationary discontinuity, 563 stiffness tensor, 118 Stokes theorem, 26 Stoney’s formula, 786 strain definitions, 56 Eulerian, 56 integration of, 187 Lagrangian, 56 logarithmic, 56 natural, 56 nominal, 56 potentials, 256 Index rate, 76 relaxation, 437 shear, 60 small, 61 strain energy, 148 for soft tissues, 629 isotropic elasticity, 151 strain hardening, 461 crystal plasticity, 544 localization, 574 origins of, 505 single crystal, 603 strength-differential effect, 480 stress Cauchy, 92 first Piola–Kirchoff, 99 function, 330 invariants, 462 Kirchhoff, 101 nominal, 98, 101 power, 109 rate, 106 second Piola–Kirchhoff, 102 work conjugate, 99 stretch, 56 principal, 62 tensor, 59 Stroh formalism, 329 structural rearrangements, 127 substrate, 432 surface energy, 449 instability, 455 surface integrals, 30 material derivative, 84 symmetric tensor, 14 tangent modulus, 484 plastic, 486 secant, 486 Taylor lattice, 510 model, 587 temperature, 115, 483 tensile crack, 296 tensors antisymmetric, 14 axial vector, 17 characteristic equation, 13 conductivities, 119 determinant, field, 27 invariants, 6, 14 inverse, orthogonal, 17 positive definite, 14 product, 10 spectral forms, 14 859 symmetric, 14 trace, transpose, texture determination, 593 thermal strains, 178 thermodynamic force, 127 potential, 127, 128 system, 113 thermoelastic effect, 136 thermoelasticity, 127, 131, 178 thin films, 432 plates, 280 thin-walled section, 727 third law of thermodynamics, 126 threading dislocation, 432, 437 time-independent behavior, 483 tissue, 609 torsion, 214 displacements, 214 elliptical cross section, 221 energy of, 216 function, 214 multiply connected cross sections, 222 rectangular cross section, 217 rigidity, 216 total energy, 355 total strain theories, 596 trace, 6, 10 traction vector, 92 transformation strain, 335, 336 polynomial, 345 transmural cut, 631 transport formulae, 83 Tresca yield criterion, 465 triple product, 1, 2, 7, 13 uniform contact pressure, 276 uniqueness of solution, 174 unit cells, 505 vector field, 27 curl of, 55 differentiable, 55 divergence, 55 gradient, 70 vector product, Vegard’s rule, 433 velocity, 71 velocity gradient, 71 antisymmetric part, 74 crystal plasticity, 539 elastoplastic deformation, 490 symmetric part, 74 velocity strain, 73 vertex softening, 487 virtual work, 107 860 void growth, 479, 495 Voigt estimates, 824 notation, 164 Volterra’s integral, 301 volume change, 63 integrals, 84 rate of change, 84 volumetric strain, 161 strain rate, 479 von Mises yield criterion, 463 wedge problem, 271 width of dislocation, 759 work conjugate stress, 99 Index yield cone, 469 surface, 475 vertex, 487 yield criterion Drucker–Prager, 468, 479 Gurson, 470, 479 Mohr–Coulomb, 467 pressure-dependent, 468, 478 Tresca, 465 von Mises, 463 yield surface, 474 Young’s modulus, 161 Ziegler’s hardening, 477 [...]... the introduction to continuum mechanics, linear and nonlinear elasticities, theory of dislocations, fracture mechanics, theory of plasticity, and selected topics from thin films and biomechanics At the end of each chapter we offer a list of recommended references for additional reading, which aid further study and mastering of the particular subject Standard notations and conventions are used throughout... the study of strain relaxation in thin films and stability of planar interfaces Part 6 is devoted to mathematical and physical theories of plasticity and viscoplasticity The phenomenological or continuum theory of plasticity, single crystal, polycrystalline, and laminate plasticity are presented The micromechanics of crystallographic slip is addressed in detail, with an analysis of the nature of crystalline... deformation Part 7 is an introduction to biomechanics, particularly the formulation of governing equations of the mechanics of solids with a growing mass and constitutive relations for biological membranes Part 8 is a collection of 180 solved problems covering all chapters of the book This is included to provide additional development of the basic theory and to further illustrate its application The... semi-infinite media, and threedimensional isotropic and anisotropic elastic problems Part 4 is concerned with micromechanics, which includes the analyses of dislocations and cracks in isotropic and anisotropic media, the well-known Eshelby elastic inclusion problem, energy analyses of imperfections and configurational forces, and micropolar elasticity In Part 5 we analyze dislocations in bimaterials and thin films,... the orthogonal set of base vectors in question Other more or less standard notations are used, e.g., the left- or right-hand side of an equation is referred to as the lhs, or r hs, respectively The commonly used phrase with respect is abbreviated as wr t, and so on We are grateful to many colleagues and students who have influenced and contributed to our work in solid mechanics and materials science... courses in solid mechanics and materials science, as well as from our own published work We have also consulted and used major contributions by other authors, their research work and written books, as cited in the various sections As such, this book can be used as a textbook for a sequence of solid mechanics courses at the graduate level within mechanical, structural, aerospace, and materials science... kinematics, kinetics, and thermodynamics of a continuum and an application to nonlinear elasticity Part 3 is devoted to linear elasticity The governing equations of the three-dimensional elasticity with appropriate specifications for the two-dimensional plane stress and plane strain problems are given The applications include the analyses of bending of beams and plates, torsion of prismatic rods, contact... graduate students in solid mechanics and materials science and should also be useful to researchers in these fields The book consists of eight parts Part 1 covers the mathematical preliminaries used in later chapters It includes an introduction to vectors and tensors, basic integral theorems, and Fourier series and integrals The second part is an introduction to nonlinear continuum mechanics This incorporates... follows that |A · A−1 | = |I| = 1 = |A||A−1 |, |A−1 | = (1.51) 1 = |A|−1 |A| 1.7 Additional Proofs We deferred formal proofs of several lemmas until now in the interest of presentation We provide the proofs at this time LEMMA 1.1: If a and b are two vectors, a × b = 0 iff a and b are linearly dependent Proof: If a and b are linearly dependent then there is a scalar such that b = αa In this case, if we express... PART 7: BIOMECHANICS 32 Mechanics of a Growing Mass 32.1 Introduction 32.2 Continuity Equation 32.2.1 Material Form of Continuity Equation 32.2.2 Quantities per Unit Initial and Current Mass 32.3 Reynolds Transport Theorem 32.4 Momentum Principles 32.4.1 Rate-Type Equations of Motion 32.5 Energy Equation 32.5.1 Material Form of Energy Equation 32.6 Entropy Equation 32.6.1 Material Form of Entropy Equation .. .MECHANICS OF SOLIDS AND MATERIALS Mechanics of Solids and Materials intends to provide a modern and integrated treatment of the foundations of solid mechanics as applied... dislocation theory, damage mechanics, and micromechanics He is the author of more than 100 journal and conference publications and two books: Strength of Materials (1985) and Elastoplasticity Theory... recipient of the 2004 Distinguished Teaching Award from the University of California Mechanics of Solids and Materials ROBERT J ASARO University of California, San Diego VLADO A LUBARDA University of