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أخواني في الله أحضرت لكم كتاب Elasticity in Engineering Mechanics Third Edition ARTHUR P. BORESI Professor Emeritus University of Illinois, Urbana, Illinois and University of Wyoming, Laramie, Wyoming KEN P. CHONG Associate National Institute of Standards and Technology, Gaithersburg, Maryland and Professor Department of Mechanical and Aerospace Engineering George Washington University, Washington, D.C. JAMES D. LEE Professor Department of Mechanical and Aerospace Engineering George Washington University, Washington, D.C.
![كتاب Elasticity in Engineering Mechanics E_i_e_10](https://i.servimg.com/u/f61/15/36/74/14/e_i_e_10.jpg) و المحتوى كما يلي :
CONTENTS Preface xvii CHAPTER 1 INTRODUCTORY CONCEPTS AND MATHEMATICS 1 Part I Introduction 1 1-1 Trends and Scopes 1 1-2 Theory of Elasticity 7 1-3 Numerical Stress Analysis 8 1-4 General Solution of the Elasticity Problem 9 1-5 Experimental Stress Analysis 9 1-6 Boundary Value Problems of Elasticity 10 Part II Preliminary Concepts 11 1-7 Brief Summary of Vector Algebra 12 1-8 Scalar Point Functions 16 1-9 Vector Fields 18 1-10 Differentiation of Vectors 19 1-11 Differentiation of a Scalar Field 21 1-12 Differentiation of a Vector Field 21 1-13 Curl of a Vector Field 22 1-14 Eulerian Continuity Equation for Fluids 22 vvi CONTENTS 1-15 Divergence Theorem 25 1-16 Divergence Theorem in Two Dimensions 27 1-17 Line and Surface Integrals (Application of Scalar Product) 28 1-18 Stokes’s Theorem 29 1-19 Exact Differential 30 1-20 Orthogonal Curvilinear Coordiantes in Three-Dimensional Space 31 1-21 Expression for Differential Length in Orthogonal Curvilinear Coordinates 32 1-22 Gradient and Laplacian in Orthogonal Curvilinear Coordinates 33 Part III Elements of Tensor Algebra 36 1-23 Index Notation: Summation Convention 36 1-24 Transformation of Tensors under Rotation of Rectangular Cartesian Coordinate System 40 1-25 Symmetric and Antisymmetric Parts of a Tensor 46 1-26 Symbols δij and ijk (the Kronecker Delta and the Alternating Tensor) 47 1-27 Homogeneous Quadratic Forms 49 1-28 Elementary Matrix Algebra 52 1-29 Some Topics in the Calculus of Variations 56 References 60 Bibliography 63 CHAPTER 2 THEORY OF DEFORMATION 65 2-1 Deformable, Continuous Media 65 2-2 Rigid-Body Displacements 66 2-3 Deformation of a Continuous Region. Material Variables. Spatial Variables 68 2-4 Restrictions on Continuous Deformation of a Deformable Medium 71 Problem Set 2-4 75 2-5 Gradient of the Displacement Vector. Tensor Quantity 76CONTENTS vii 2-6 Extension of an Infinitesimal Line Element 78 Problem Set 2-6 85 2-7 Physical Significance of ii. Strain Definitions 86 2-8 Final Direction of Line Element. Definition of Shearing Strain. Physical Significance of ij(i = j ) 89 Problem Set 2-8 94 2-9 Tensor Character of αβ. Strain Tensor 94 2-10 Reciprocal Ellipsoid. Principal Strains. Strain Invariants 96 2-11 Determination of Principal Strains. Principal Axes 100 Problem Set 2-11 106 2-12 Determination of Strain Invariants. Volumetric Strain 108 2-13 Rotation of a Volume Element. Relation to Displacement Gradients 113 Problem Set 2-13 116 2-14 Homogeneous Deformation 118 2-15 Theory of Small Strains and Small Angles of Rotation 121 Problem Set 2-15 130 2-16 Compatibility Conditions of the Classical Theory of Small Displacements 132 Problem Set 2-16 137 2-17 Additional Conditions Imposed by Continuity 138 2-18 Kinematics of Deformable Media 140 Problem Set 2-18 146 Appendix 2A Strain–Displacement Relations in Orthogonal Curvilinear Coordinates 146 2A-1 Geometrical Preliminaries 146 2A-2 Strain–Displacement Relations 148 Appendix 2B Derivation of Strain–Displacement Relations for Special Coordinates by Cartesian Methods 151 2B-1 Cylindrical Coordinates 151 2B-2 Oblique Straight-Line Coordinates 153viii CONTENTS Appendix 2C Strain–Displacement Relations in General Coordinates 155 2C-1 Euclidean Metric Tensor 155 2C-2 Strain Tensors 157 References 159 Bibliography 160 CHAPTER 3 THEORY OF STRESS 161 3-1 Definition of Stress 161 3-2 Stress Notation 164 3-3 Summation of Moments. Stress at a Point. Stress on an Oblique Plane 166 Problem Set 3-3 171 3-4 Tensor Character of Stress. Transformation of Stress Components under Rotation of Coordinate Axes 175 Problem Set 3-4 179 3-5 Principal Stresses. Stress Invariants. Extreme Values 179 Problem Set 3-5 183 3-6 Mean and Deviator Stress Tensors. Octahedral Stress 184 Problem Set 3-6 189 3-7 Approximations of Plane Stress. Mohr’s Circles in Two and Three Dimensions 193 Problem Set 3-7 200 3-8 Differential Equations of Motion of a Deformable Body Relative to Spatial Coordinates 201 Problem Set 3-8 205 Appendix 3A Differential Equations of Equilibrium in Curvilinear Spatial Coordinates 207 3A-1 Differential Equations of Equilibrium in Orthogonal Curvilinear Spatial Coordinates 207 3A-2 Specialization of Equations of Equilibrium 208 3A-3 Differential Equations of Equilibrium in General Spatial Coordinates 210CONTENTS ix Appendix 3B Equations of Equilibrium Including Couple Stress and Body Couple 211 Appendix 3C Reduction of Differential Equations of Motion for Small-Displacement Theory 214 3C-1 Material Derivative. Material Derivative of a Volume Integral 214 3C-2 Differential Equations of Equilibrium Relative to Material Coordinates 218 References 224 Bibliography 225 CHAPTER 4 THREE-DIMENSIONAL EQUATIONS OF ELASTICITY 226 4-1 Elastic and Nonelastic Response of a Solid 226 4-2 Intrinsic Energy Density Function (Adiabatic Process) 230 4-3 Relation of Stress Components to Strain Energy Density Function 232 Problem Set 4-3 240 4-4 Generalized Hooke’s Law 241 Problem Set 4-4 255 4-5 Isotropic Media. Homogeneous Media 255 4-6 Strain Energy Density for Elastic Isotropic Medium 256 Problem Set 4-6 262 4-7 Special States of Stress 266 Problem Set 4-7 268 4-8 Equations of Thermoelasticity 269 4-9 Differential Equation of Heat Conduction 270 4-10 Elementary Approach to Thermal-Stress Problem in One and Two Variables 272 Problem 276 4-11 Stress–Strain–Temperature Relations 276 Problem Set 4-11 283 4-12 Thermoelastic Equations in Terms of Displacement 285 4-13 Spherically Symmetrical Stress Distribution (The Sphere) 294 Problem Set 4-13 299x CONTENTS 4-14 Thermoelastic Compatibility Equations in Terms of Components of Stress and Temperature. Beltrami–Michell Relations 299 Problem Set 4-14 304 4-15 Boundary Conditions 305 Problem Set 4-15 310 4-16 Uniqueness Theorem for Equilibrium Problem of Elasticity 311 4-17 Equations of Elasticity in Terms of Displacement Components 314 Problem Set 4-17 316 4-18 Elementary Three-Dimensional Problems of Elasticity. Semi-Inverse Method 317 Problem Set 4-18 323 4-19 Torsion of Shaft with Constant Circular Cross Section 327 Problem Set 4-19 331 4-20 Energy Principles in Elasticity 332 4-21 Principle of Virtual Work 333 Problem Set 4-21 338 4-22 Principle of Virtual Stress (Castigliano’s Theorem) 339 4-23 Mixed Virtual Stress–Virtual Strain Principles (Reissner’s Theorem) 342 Appendix 4A Application of the Principle of Virtual Work to a Deformable Medium (Navier–Stokes Equations) 343 Appendix 4B Nonlinear Constitutive Relationships 345 4B-1 Variable Stress–Strain Coefficients 346 4B-2 Higher-Order Relations 346 4B-3 Hypoelastic Formulations 346 4B-4 Summary 347 Appendix 4C Micromorphic Theory 347 4C-1 Introduction 347 4C-2 Balance Laws of Micromorphic Theory 350 4C-3 Constitutive Equations of Micromorphic Elastic Solid 351CONTENTS xi Appendix 4D Atomistic Field Theory 352 4D-1 Introduction 353 4D-2 Phase-Space and Physical-Space Descriptions 353 4D-3 Definitions of Atomistic Quantities in Physical Space 355 4D-4 Conservation Equations 357 References 359 Bibliography 364 CHAPTER 5 PLANE THEORY OF ELASTICITY IN RECTANGULAR CARTESIAN COORDINATES 365 5-1 Plane Strain 365 Problem Set 5-1 370 5-2 Generalized Plane Stress 371 Problem Set 5-2 376 5-3 Compatibility Equation in Terms of Stress Components 377 Problem Set 5-3 382 5-4 Airy Stress Function 383 Problem Set 5-4 392 5-5 Airy Stress Function in Terms of Harmonic Functions 399 5-6 Displacement Components for Plane Elasticity 401 Problem Set 5-6 404 5-7 Polynomial Solutions of Two-Dimensional Problems in Rectangular Cartesian Coordinates 408 Problem Set 5-7 411 5-8 Plane Elasticity in Terms of Displacement Components 415 Problem Set 5-8 416 5-9 Plane Elasticity Relative to Oblique Coordinate Axes 416 Appendix 5A Plane Elasticity with Couple Stresses 420 5A-1 Introduction 420 5A-2 Equations of Equilibrium 421xii CONTENTS 5A-3 Deformation in Couple Stress Theory 421 5A-4 Equations of Compatibility 425 5A-5 Stress Functions for Plane Problems with Couple Stresses 426 Appendix 5B Plane Theory of Elasticity in Terms of Complex Variables 428 5B-1 Airy Stress Function in Terms of Analytic Functions ψ(z) and χ(z) 428 5B-2 Displacement Components in Terms of Analytic Functions ψ(z) and χ(z) 429 5B-3 Stress Components in Terms of ψ(z) and χ(z) 430 5B-4 Expressions for Resultant Force and Resultant Moment 433 5B-5 Mathematical Form of Functions ψ(z) and χ(z) 434 5B-6 Plane Elasticity Boundary Value Problems in Complex Form 438 5B-7 Note on Conformal Transformation 440 Problem Set 5B-7 445 5B-8 Plane Elasticity Formulas in Terms of Curvilinear Coordinates 445 5B-9 Complex Variable Solution for Plane Region Bounded by Circle in the z Plane 448 Problem Set 5B 452 References 453 Bibliography 454 CHAPTER 6 PLANE ELASTICITY IN POLAR COORDINATES 455 6-1 Equilibrium Equations in Polar Coordinates 455 6-2 Stress Components in Terms of Airy Stress Function F = F (r, θ) 456 6-3 Strain–Displacement Relations in Polar Coordinates 457 Problem Set 6-3 460 6-4 Stress–Strain–Temperature Relations 461 Problem Set 6-4 462CONTENTS xiii 6-5 Compatibility Equation for Plane Elasticity in Terms of Polar Coordinates 463 Problem Set 6-5 464 6-6 Axially Symmetric Problems 467 Problem Set 6-6 483 6-7 Plane Elasticity Equations in Terms of Displacement Components 485 6-8 Plane Theory of Thermoelasticity 489 Problem Set 6-8 492 6-9 Disk of Variable Thickness and Nonhomogeneous Anisotropic Material 494 Problem Set 6-9 497 6-10 Stress Concentration Problem of Circular Hole in Plate 498 Problem Set 6-10 504 6-11 Examples 505 Problem Set 6-11 510 Appendix 6A Stress–Couple Theory of Stress Concentration Resulting from Circular Hole in Plate 519 Appendix 6B Stress Distribution of a Diametrically Compressed Plane Disk 522 References 525 CHAPTER 7 PRISMATIC BAR SUBJECTED TO END LOAD 527 7-1 General Problem of Three-Dimensional Elastic Bars Subjected to Transverse End Loads 527 7-2 Torsion of Prismatic Bars. Saint-Venant’s Solution. Warping Function 529 Problem Set 7-2 534 7-3 Prandtl Torsion Function 534 Problem Set 7-3 538 7-4 A Method of Solution of the Torsion Problem: Elliptic Cross Section 538 Problem Set 7-4 542 7-5 Remarks on Solutions of the Laplace Equation, ∇2F = 0 542 Problem Set 7-5 544xiv CONTENTS 7-6 Torsion of Bars with Tubular Cavities 547 Problem Set 7-6 549 7-7 Transfer of Axis of Twist 549 7-8 Shearing–Stress Component in Any Direction 550 Problem Set 7-8 554 7-9 Solution of Torsion Problem by the Prandtl Membrane Analogy 554 Problem Set 7-9 561 7-10 Solution by Method of Series. Rectangular Section 562 Problem Set 7-10 566 7-11 Bending of a Bar Subjected to Transverse End Force 569 Problem Set 7-11 577 7-12 Displacement of a Cantilever Beam Subjected to Transverse End Force 577 Problem Set 7-12 581 7-13 Center of Shear 581 Problem Set 7-13 582 7-14 Bending of a Bar with Elliptic Cross Section 584 7-15 Bending of a Bar with Rectangular Cross Section 586 Problem Set 7-15 590 Review Problems 590 Appendix 7A Analysis of Tapered Beams 591 References 595 CHAPTER 8 GENERAL SOLUTIONS OF ELASTICITY 597 8-1 Introduction 597 Problem Set 8-1 598 8-2 Equilibrium Equations 598 Problem Set 8-2 600 8-3 The Helmholtz Transformation 600 Problem Set 8-3 601 8-4 The Galerkin (Papkovich) Vector 602 Problem Set 8-4 603CONTENTS xv 8-5 Stress in Terms of the Galerkin Vector F 603 Problem Set 8-5 604 8-6 The Galerkin Vector: A Solution of the Equilibrium Equations of Elasticity 604 Problem Set 8-6 606 8-7 The Galerkin Vector kZ and Love’s Strain Function for Solids of Revolution 606 Problem Set 8-7 608 8-8 Kelvin’s Problem: Single Force Applied in the Interior of an Infinitely Extended Solid 609 Problem Set 8-8 610 8-9 The Twinned Gradient and Its Application to Determine the Effects of a Change of Poisson’s Ratio 611 8-10 Solutions of the Boussinesq and Cerruti Problems by the Twinned Gradient Method 614 Problem Set 8-10 617 8-11 Additional Remarks on Three-Dimensional Stress Functions 617 References 618 Bibliography 619 INDEX INDEX Ab initio, 255 Abraham, F., 4, 60 Acceleration: convective terms of, 143 kinematics of deformable media, 141–144 Acceleration field, 22 Acceleration vector, 21, 22, 142, 203, 218 Active materials, 2 Active stress: and muscle mechanics, 239–240 significance of, 478–483 Actuators, in smart structures/materials, 5 Adams, P. H., 10, 61 Adiabatic deformation process, 230–232 Adjoint matrix, 55 Adkins, J. E., 8, 61, 226n.1, 361 Admissible functions, 58 Advanced material processing, 3 Aeolotropic material, 256 Agrawal, A., 362 Airy, G. B., 9 Airy stress function, 9, 383–399, 426–427 complex variables, 428–429 displacement components, 401–404, 429–430 harmonic functions, 399–400 plane theory, 383–399 body forces and temperature effects, 385, 389–392, 467–475, 487–489 boundary conditions, 385–388 compatibility, 379 harmonic functions, 399–400 multiply connected regions, 388–392 polar coordinates, 463 simply connected regions, 383–385 in polar coordinates, 456–457, 463, 498 solutions: in polar coordinates, 463 in rectangular coordinates, 385, 408–415 Almansi strain, 238 Almansi strain tensor, 83, 159 American Society for Testing and Materials (ASTM), 10n.2 Aneurysm, intracranial saccular, 295–298 Angle of twist, 331, 529 Anisotropic material, 231, 241–255, 259–261, 494–498 elastic coefficients, 242 strain energy density, 242, 368–369 strain–temperature relation, 288 stress-strain relations, 261, 368–369 Anticlastic surface, 322 Antisymmetric square arrays, 39 Approximate methods, 3, 10 Arbitrary square arrays, 39–40 Argument function, 58 Ariman, T., 421, 453, 519 Arrays: antisymmetric, 39 antisymmetric square, 39 arbitrary, 39–40622 INDEX Arrays: (continued) characteristic equation, 50 determinants, 49 rectangular, 38–40 skew-symmetric, 39 square, 39–40 stress, 166–167 symmetric, 39 symmetric square, 39 typical element, 39 Arroyo, M., 4, 60 Artery wall, atheromatous plaque on, 475–478 Associative law of vector addition, 12 ASTM (American Society for Testing and Materials), 10n.2 Atheromatous plaque on artery wall, 475–478 Atomistic field theory, 352–359 atomistic quantities in physical space, 355–357 conservation equations, 357–359 phase-space and physical-space descriptions, 353–355 Atrek, E., 1, 60 Averbach, B. L., 277, 362 Axis of twist: generally, 327, 536 transfer of, 549–550 Baker, M. J., 8, 63 Balance laws, 214, 218 angular momentum, 205, 358 in atomistic field theory, 358, 359 linear momentum, 205, 289 of micromorphic theory, 350–351 Bar, prismatic, 318–322, 527–591 bending, 318–322, 569–577 Bernoulli-Euler equation, 322 curved bar, 505–506 elliptic cross section, 584–586 pure, 318–322, 581–582 rectangular cross section, 586–590 transverse end force, 569–577 Prandtl torsion theory, 534–538 Saint-Venant’s torsion theory, 529–534 shear-center, 581–584 torsion, 529–568 axis of twist, 327, 536, 549–550 boundary conditions, 528 elliptic cross section, 538–542, 584 narrow rectangular cross section, 560–561 Prandtl function of, 535 Prandtl membrane analogy, 554–562 Prandtl theory, 534–538 rectangular section, 562–568 Saint-Venant’s solution, 529–534 shear-stress components, 543–544 with tubular cavities, 547–549 warping, circular cross section, 544 Bathe, K.-J., 106, 159 Beams: cantilever, 506–507 tapered, 591–595 thermal stress, 274–276 Beltrami–Mitchell compatibility equations, 529 Beltrami–Mitchell compatibility relation, 299–305 Belytshko, T., 4, 60 Bending: of prismatic bar, 318–322, 569–577 Bernoulli-Euler equation, 322 curved bar, 505–506 elliptic cross section, 584–586 pure, 318–322, 581–582 rectangular cross section, 586–590 transverse end force, 569–577 pure, 318–322 bar subjected to transverse end force, 527–529, 569–577 Bessel functions, 519 cantilever beam, 506–507 curved bars, 505–506 function (flexural), 572 general equations, 569–577 plane wedges, 509–510 prismatic bars, 318–322 Berendsen, H. J. C., 293, 359 Berendsen thermostat, 293, 294 Bernoulli-Euler equation, 322 Beus, M. J., 504, 525 Biharmonic equation, 384. See also Airy stress function functions, 384 solutions of, 385, 465 Bilinear form, 50 Bio-inspired sensors, 4 Biological tissues: constitutive equation for, 237–238 lifeless material vs. living, 239–240 structure of, 5 Biomechanics, 5 Bioscience, 5 Biot, M. A., 270, 359 Biotechnology, 1, 5, 7 Birkhoff, G., 55, 60, 96, 97n.5, 159 Body couples, 7, 161, 211–214 force, 233, 467–475, 487 moments, 166–167 Body force (atomistic field theory), 357INDEX 623 Boley, B. A., 270, 359 Boltzmann constant, 290 Bone structure, 5, 6 Boresi, A. P., 3, 9, 11, 60, 67, 68n.1, 142, 159, 187, 214, 215n.8, 224, 333, 346, 359 Borgman, E. S., 525 Born, J. S., 310, 361 Boundary conditions, 169, 287, 528 for bars, 528 equilibrium, 305–310 intracranial saccular aneurysm, 298 mixed boundary value problem, 306–307 for multiply connected regions, 388 for plane polar coordinates, 471–473, 496–497 Saint-Venant’s principle, 307–310 stress, 169–171, 287, 306 in terms of Airy stress function, 385–388 in terms of displacement, 313, 315–316 for torsion of bars, 547–548 Boundary element method, 3 Boundary-value problems, 10–11, 438–440 Dirichlet, 10–11, 534 mixed, 11 Neumann, 11, 533, 550 Boussinesq, J., 307, 308, 359, 611, 614–615, 618 Boussinesq problem, 614–616 Brazilian test, 522 Brebbia, C. A., 3, 60 Brown, G. H., 161, 224 Brown, J. W., 384, 399, 436, 437, 441, 451, 453, 533, 542, 564, 577, 595 Brown, O. E., 19, 62 Buehler, M. J., 62 Bulatov, V. V., 359 Bulk modulus, 258 CAD (computer-aided design), 2 Cai, W., 353, 359 Calculus of variations, 56–60 admissible functions, 58 argument function, 58 conditions of admissibility, 58 Euler differential equation, 59 first variation of an integral, 60 functionals, 58 stationary value of an integral, 59 variation of a function, 58–60 CAM (computer-aided manufacturing), 2 Car, R., 255, 359 Carlson, D. E., 212n.5, 426, 453 Carrier, G. F., 436, 448, 453 Carslaw, H. S., 270, 359 Cartesian coordinate system, see Rectangular Cartesian coordinates Castigliano’s theorem: on deflections, 341–342 principle of virtual stress, 341–342 Cauchy elastic formulation, 346, 347 Cauchy-Riemann equations, 533 Cauchy strain tensor, 73–74, 88–89, 158 Cauchy stress, 177, 237–240, 351 Cayley–Hamilton theorem, 74 Cell biomechanics, 5 Center of shear (shear center), 581–584 Center of twist, 536 Cerruti, V., 611, 614–616, 618 Cerruti problem, 614–617 Chadwick, P., 270, 359 Chan, S. S., 504, 525 Characteristic roots (eigenvalues), 50 Chasles’s theorem, 67, 127 Chen, J. L., 360 Chen, P., 62 Chen, W.-F., 184, 224, 346, 347, 359 Chen, Y., 352, 354, 355, 357, 359, 361, 363 Cheung, Y. K., 3, 60 Chistoffel symbols, 210 Choi, I., 307n.9, 310, 359 Chong, K. P., 1–4, 6, 9, 60, 125n.14, 187, 224, 245, 246, 346, 360, 403, 453, 504, 522, 524, 525, 592, 595 Christian, J. T., 360 Churchill, R. V., 384, 399, 436, 437, 441, 451, 453, 533, 542, 564, 577, 595 Circle of Willis, 295–297 Clausius–Duhem inequality, 350 Cleary, M. P., 346, 360 Clemson University, 591n.9 Coefficients of the principal dilatations, 120 Column matrix, 52 Commutative law of vector addition, 12 Compatibility, 317 Beltrami–Mitchell compatibility equations, 529 Beltrami–Mitchell compatibility relation, 299–305 with couple stress, 425–426 displacement, 132–138 equation for plane elasticity: in polar coordinates, 463 in rectangular coordinates, 369, 408–415 plane strain, 369, 377–382, 461 plane stress, 378–381, 462 small displacement, conditions of, 132–138, 371 in terms of Airy stress function, 379 thermoelasticity, 299–305624 INDEX Complementary function (solution or integral), 389–391 Complex variables, 399–400, 428–453 Airy stress function, 428–429 conformal transformation, 440–445 in curvilinear coordinates, 445–448 displacement components, 429–430 plane elasticity boundary value problems, 438–440 for plane region bounded by circle in z plane, 448–452 resultant force and resultant moment, 433–434 stress components, 430–432 Composites, 259–261 Compressions, 163 Computers, 1–3 microcomputers, 1 minicomputers, 1 in smart structures/materials control, 5 supercomputers, 1, 6 Computer-aided design (CAD), 2 Computer-aided manufacturing (CAM), 2 Conditions of admissibility, 58 Conservation laws: angular momentum, 357 in atomistic field theory, 357 energy, 232, 289, 290, 357 linear momentum, 357 of linear momentum, 357 mass, 205, 218, 357 in molecular dynamics, 236 Constitutive equations: of elastic solids, 351–352 for soft biological tissue, 237–238 Constitutive relations, 9, 246. See also Stress-strain relations in atomistic field theory, 359 in molecular dynamics, 235–236 nonlinear, 345–347 Constraints, 56 Contact mechanics, 2 Continuity, 65–66 conditions of, 134–140 equations of, 134, 140, 145, 146 material (Lagrangian) form, 139–140 spatial form, 144–146 Continuous body: defined, 68 deformation, 68, 72–73 Continuous (deformable) medium (continuum), 7, 68, 72–73, 140 Continuum mechanics, 7, 68, 205 and atomistic models, 353 interfacing molecular dynamics and, 353 Continuum physics, 289 Contour map, 542 Contravariant tensors, 210–211 Cook, N. G. W., 268, 360 Coordinate lines, curvilinear, 31–32, 147, 445–448 Coordinate surfaces, 31, 147 Coordinate systems: cylindrical, 150–153, 208–209, 491 Eulerian, 21, 67–71, 82, 232 deformation, 67–71 micromorphic theory, 348–350 intrinsic, 159 Lagrangian, 67–71, 149, 214, 232, 348–350 left-handed, 14, 15 material, 66–71, 214 oblique, 154–155, 416–420 plane, 416–420 straight-line, 154–155 orthogonal curvilinear, 31–32, 146–151 differential length in, 32–33 gradient, 33–34 Laplacian, 34–36 strain-displacement relations, 146–151 plane polar, 210, 455–456 polar coordinates: Airy stress function in, 456–457, 463, 498 equilibrium equations in, 455–456 plane compatibility equation in, 463 strain-displacement relations, 457–461 stress components in, 456–457 stress-strain temperature relations, 461–462 rectangular Cartesian, 32, 40–46, 70–71, 408–415 strain components in, 83–84 strain-displacement relations, 366 transformation of tensors under, 40–46 right-handed, 14, 15 spatial, 21, 66–71 spherical, 151, 209–210, 294–299 Corrosion sensors, 5 Cosserat, E., 213n.6, 421, 453 Cosserat, F., 213n.6, 421, 453 Coulomb–Buckingham potential, 250 Couple, body, 7 Couple stress, 7, 211–214, 420–428 deformation, 421–424 equations of compatibility, 425–426 equations of equilibrium, 421 stress concentration from circular hole in plate, 519–522 stress functions for plane problems with, 426–428 Couple stress tensor, 213INDEX 625 Courant, R., 30n.5, 57, 58, 60, 61, 72, 139, 159, 534, 550, 595 Covariant tensors, 210–211 Creep, 8 Cross section, 538, 544, 547–549 deformed shape of, 322 elliptical, 538–539 warping, 536–537, 544 Crystalline systems: multicomponent, 352–354 single-component, 353 Cubical strain, 366 Curl of vector field, 22 Current density, 23 Curvilinear coordinates, 147 Cutoff radius, 251 Cylindrical coordinate system, 150–153, 208–209, 491 Dally, J. W., 10, 61 Dana, G. F., 360 Davis, D. C., 4, 60 Davis, G., 592, 595 Deformable body (medium), 65–66 differential equations of motion, 288 equilibrium, three-dimensional, 598–600 spatial coordinates, 201–206 incompressible, 204 kinematics of, 140–146 acceleration, 22, 141–144 convective, 143 Deformation: admissible, 73, 84–85 compatibility conditions, small displacement, 132–138 condition for continuously possible, 72–73 of a continuous region, 68–71 couple-stress, 421–424 definition, 66 deformable, continuous media, 65–66, 71–76 extension of infinitesimal line element, 78–86 gradient of displacement vector, 76–78 homogenous, 118–121 kinematics of deformable media, 140–146 line element: direction cosines of, 78–79, 89 extension of, 78–86 final direction cosines of a deformed, 89–90 relative elongation of, 86–89 material (Lagrangian) form, 67–71, 139–140 mean and deviator strain tensor, 110–112 octahedral strains, 112 plane strain, 112 principal axes, 101–107 principal strains, 100–101 proper, 73 reciprocal ellipsoid, 96–100 rigid-body displacements, 66–67 rotation of volume element, 113–117 shearing strain, 90–92 spatial (Eulerian) form, 67–71 strain definitions, 87–89 strain invariants, 108–109 strain tensor, 94–96 theory of small strains and small angels of rotation, 121–132 transformations of lines and surfaces, 138–139 volumetric strain, 109–110 zero state (configuration), 229 Deformation gradient tensor, 73 De Koning, M., 359 Del (nabla), 17 Delange, S. L., 237, 296, 297, 361 Delph, T. J., 353, 360 Density, 66 at cell level, 358 current, 23 mass, 7 Density functional theory, 255 Desai, C. S., 346, 360 Designer materials, 5–7 Determinants: of arrays, 49 vector, 14, 16 Determinant notation, 14, 16, 42 Development, biomechanics of, 5 Diagonal matrix, 54 Differential, total, 79–80, 387 Differential equations of motion, 204 Differential length, in orthogonal curvilinear coordinates, 32–33 Differentiation: of scalar field, 21 of vector field, 21–22 of vectors, 19–21 Diffusivity, 271 Dilatation: cubical, 110 intracranial saccular aneurysm, 295–298 pure, 120–121, 125 Dillon, O. W., 60 DiNola, A., 359 Directional derivative, 17–18, 550–554 Direction cosines: determinants of, 42 in index form, 43 orthogonality relations, 41–42626 INDEX Direction cosines: (continued) relations between, 41–42 table, 41 between two sets of rectangular Cartesian axes, 40–43 Dirichlet boundary-value problem, 10–11, 534 Disk, 470–471, 487–489, 494–498, 522–525 Displacement: admissible, 73, 75, 84–85 of cantilever beam subjected to transverse end force, 577–581 compatibility (continuity), 132–138 components of, 71, 82–83 equations, 314–317 in terms of Airy stress function, 401–404, 429–430 torsion, 536–538 deformable body, 71 fluid particles, 21 gradient of, 76–78, 115–117 particle, 66, 67 plane, 67, 366–368, 415–416 proper, 73, 75 reflection, 73 rigid-body, 66–67, 127–130 plane, 67 rotation, 67 translation, 66, 67 small strains and angles of rotation, 121–132 vector, 76–78 virtual, 333–338 Displacement potential, 389–392 Displacement potential function, 390 Divergence, of vector field, 23 Divergence theorem, 25–27, 233 Gauss’s theorem, 25–26 Green’s theorem, 27 Green’s theorem of the plane, 28 in two dimensions, 27–28 Dove, R. C., 10, 61 Drucker, D. C., 184, 224 Duchaineau, M., 60 Duhamel, J. M. C., 269, 270, 360 Duhamel-Neumann theory, 269–270 Dummy indexes, 37, 38 Dvorak, G. J., 1, 61 E, W., 353, 360 Education, in mechanics, 3 Eigenvalues (characteristic roots), 50 Eigenvectors, 50–52, 188 Eisenhart, L. P., 41, 50, 61, 98, 159 Elastic coefficients (stiffnesses), 241–246, 257–261 for general anisotropic elastic material, 242 Lame, 33, 257, 266, 311, 312 ´ law of transformation, 246–249 Elasticity: anisotropic, 231, 241–255, 259–261 axisymmetric problem, 302–304, 467–485 in biomechanical problems, 5 boundary-value problems, 10–11, 305–307, 438–440, 533–534 bulk modulus, 258 concept of, 229–230 isotropic, 231 linear theory, 8, 227 nonlinear theory, 8, 345–346 perfect, 227, 229–231 plane, 9. See also Plane theory Poisson’s ratio, 267 polynomial solution of two-dimensional problems, 408–415 pseudoelasticity, 237–239 shear modulus, 267 solutions in, 9–11, 317–323, 384, 408–415, 465, 485–489 general, 9, 597–618 successive elastic, 8 three-dimensional, 9, 317–327, 597–618 strain energy density, 234–235 theory of, 8–9, 230 uniqueness theorem in, 311–314 Young’s modulus, 267 Elastic limit, 227, 228, 230 Elastic response, 8 Elastic strain, 228 Elder, A. S., 114n.11, 159 Electronic structure theory, 255 Electroreheological (ER) fluids, 4 Ellis, E. W., 522, 525 Ellis, R. W., 421, 453 Ellis, T. M. R., 2, 61 Emissivity, 272 Energy: internal, 232–234 intrinsic density function, 230–232 kinetic, 66, 242 stress energy density function, 232–235, 256–262, 368 Energy methods, 8 Energy principles: Castigliano’s theorem, 341–342 conservation energy, 232 elasticity, 332–333 minimum elastic energy, 338 minimum strain energy, 338 mixed virtual stress-virtual strain, 342–343INDEX 627 Reissner’s theorem, 342–343 stationary potential energy, 337 virtual displacement, 334–338, 343, 345 virtual stress, 339–342 virtual work, 333–339, 343–345 for elastic bodies, 335–338 for particles, 334–335 Energy-related solid mechanics, 2 Engquist, B., 360 Environmental sensors, 5 Equations of constraint, 56 Equilibrium: astatic, 308 boundary conditions, 305–310 of cubic element, 204 differential equations of, 204, 207–211, 421, 598–600 in cylindrical coordinates, 208–209 in general spatial coordinates, 210–211 including couple stress and body couple, 211–214 in material coordinates, 210, 218–224 in oblique coordinates, 416–420 in orthogonal curvilinear spatial coordinates, 207–208 in plane polar coordinates, 210, 455 plane strain, 366 specialization of, 208–210 in spherical coordination, 209–210 of infinitesimal cubic element, 165 of moments, 166 in three dimensions, 317–322 uniqueness theorem of, 311–314 Eringen, A. C., 7, 61, 224, 231, 346, 348, 350–352, 360 Eskandarian, A., 361 Eubanks, R. A., 314, 363 Euclidean metric tensor, 155–157 Euler angles, 231 Euler differential equation, 59 Eulerian continuity equation, 22–24 Eulerian (spatial) coordinates, 21, 70, 82, 232 deformation, 67–71 micromorphic theory, 348–350 Euler’s theorem, 67 Exact differential, 30–31 Experimental Mechanics, 10n.2 Experimental methods, 2 Experimental stress analysis, 9–10 Experimental Techniques, 10n.2 Extreme (extreme values, extrema), 56, 181–183 Failure criteria (modes), 186–189 Fairhurst, C., 524, 525 Feshbach, H., 106, 159 Fields, 17–19, 21–22 acceleration, 22 divergence, 23, 25–27 nonstationary (unsteady), 18 scalar, 16–18 stationary (steady), 18 vector, 18–19, 21–23 vector lines of, 18 velocity, 18, 22–24 Field lines, 18 Finite difference method, 3, 8 Finite element method, 1, 8, 52, 359 Finite layer method, 3 Finite prism method, 3 Finite strip method, 3 Flexural function, 572 Fluids: circulation, 29 divergence, 25–27 electroreheological, 4 Eulerian (spatial) continuity equation, 22–24 flow, 22–24, 145–146, 163 frictionless, 163 ideal, 163 incompressible, 24, 146, 345 irrotational flow, 24, 145–146 magnetorheological, 4 momentum, 215 convective, 215 local, 215 steady flow, 24, 142, 216 unsteady flow, 22 velocity fields of, 18, 22–24 viscous, 163, 345 vorticity, 30 Forces: body, 201–202, 204, 236, 338, 343–344 conservative, 231 distributed, 161 inertial, 202–203, 338, 343–344 nonconservative, 231 normal, 162–163 point, 161 shearing, 162, 163 statically equivalent systems, 308–309 surface, 164, 203 tractive, 203 Forester, T. R., 293, 363 Formula, 350, 564 Fosdick, L. D., 1, 2, 61 Foster, R. M., 564, 566, 595 Fracture gages, 10 Frames, 68–71628 INDEX Free indexes, 37–38 Frequency–wave vector relations, 353 Friction coefficient, 293 Functionals, 58 Functional determinant, 72, 220 Fung, Y. C., 5, 61, 237, 239, 360 Galerkin, B., 597, 598, 604–606, 609, 617, 618 Galerkin–Papkovich vector, 597–598, 602–608 Gallagher, R. H., 60 Gao, H., 60 Gaussian constraints, 293–294 Gauss’s theorem, 25–26. See also Divergence theorem General tensor notation, 3 Geotechnical Testing Journal, 10n.2 Gibbs vector notation, 36 Gilbert, D., 61 Gilbert, L., 55, 61 Golsten, S., 361 Goodier, J. N., 307n.9, 363, 389, 403, 453, 454, 463n.1, 522, 524, 526, 566, 596, 610, 619 Goree, James G., 591n.9 Goursat, E., 25, 61 Gradient (grad), 33–36, 552 of displacement vector, 76–78 in orthogonal curvilinear coordinates, 33–34 of scalar function, 17 twinned, 611–614 Gradshteyn, I. S., 595 Green, A. E., 8, 46, 61, 210, 211n.4, 224, 226n.1, 361, 410, 453 Green, R. E., Jr., 10, 62 Green–Saint-Venant strain tensor, 83, 159, 237, 238 Green’s deformation tensor, 238 Greenspan, D., 11, 61 Green’s strain tensor, 83, 158, 159 Green’s theorem, 27 Green’s theorem of the plane, 28 Green-type materials, 346 Griffith, B. A., 231, 363 Griffiths, D. V., 106, 159 Grossmann, G., 554n.5, 595 Growth, biomechanics of, 5 Gunther, W., 543, 545, 560, 596 ¨ Haak, J. R., 359 Haile, J. M., 291, 361 Half-plane, 507–508 Hamed, E., 5 Hansma, P., 62 Hardy, R. J., 353, 358, 361 Hartsock, J. A., 403, 453 Hayashi, K., 239, 362 Hayes, D. J., 525 Health-care delivery, 5 Heat conduction equation, 270–272, 289 Heat transfer (exchange), 272 Helmholtz’s free-energy density, 352 Helmholtz transformation, 600–601 Higher-order relations, 346 Hilbert, D., 58, 534, 550, 595 Hildebrand, F. B., 50, 61, 74, 80, 102, 159, 261, 312, 361 Hill, R., 231n.3, 361 Hodge, P. G., Jr., 186, 225, 229, 362 Homeland Security problems, 3 Homogenous deformation/state of strain, 118–121 Homogenous media, 256 Hondros, G., 524, 525 Hooke’s law, 241–255, 257, 346 Hoover, W. G., 292, 361 Horgan, C. O., 307n.9, 310, 359, 361 Horvay, G., 310, 361 Hsu, C. S., 9, 62 Huang, Y., 5, 61 Huang, Z., 353, 360 Hughes, T. J. R., 231, 362, 618 Humphrey, J. D., 5, 61, 239, 296, 297, 361, 478, 525 Hutter, J., 255, 362 Hydraulic systems, 5 Hydrostatic pressure, 238 Hydrostatic stress, 287–288, 317–318 Hyperelastic materials, 346 Hypoelastic materials, 346–347 Hysteresis, 230 Ince, E. L., 61 Incompressible fluids, 24 Incompressible soft biological tissue, 238 Indexes: dummy, 37, 38 free, 37–38 Latin letter, 38 repeated Greek index, 36–38, 43, 117 repeated nonsummed, 38 rule of substitution, 47 summation convention, 36–40, 43–44 Index notation, 3 determinant, 42 orthogonality relations, 42 summation, 36–40, 43–44 Inelastic response, 8 Infinitesimal strain, 238 Information technology, 1, 7INDEX 629 Integral: line, 28–30, 136 particular, 577 stationary value of, 56–60 surface, 29 volume, 214–218 Integration, constant of, 574–576 Intelligent structures, 3–4. See also Smart structures/materials Interatomic force, 235, 249–255 Intracranial saccular aneurysm, 295–298 Intrinsic energy density function, 230–232 Invariance (invariants), 43, 100, 108 strain, 108–112 strain ellipsoid, 100 stress, 180, 182–183 Inverse matrix, 55, 56 Irrotational flow, 24 Irvine, J. H., 361 Irving, J., 353, 358, 521, 525 Isotropic material/media (body), 231, 255–256, 280, 312–313 higher-order relations, 346 strain energy density for, 256–266 strain–temperature relation, 289 thermoelasticity equations, 269–270 Jacobian, 72–73, 220, 348 Jaeger, J. C., 270, 359 Jasiuk, I., 5 Jeffery, A., 595 Jeffreys, H., 257, 361 Jiang, H., 61 Jones, J. E., 250, 361 Jones, R. E., 63 Journal of Testing and Evaluation, 10n.2 Kaloni, P. N., 421, 453, 519 Kannan, R., 4, 62 Kaplan, W., 80, 159 Karpov, E. G., 62 Keller, H. B., 310, 361 Kellogg, O. D., 533, 534, 595 Kelvin’s problem, 609–611, 614 Ketter, R. L., 595 Khang, D.-Y., 5, 61 Khattab, M. A., 525 Kinetic energy, law of, 333, 334 Kirchhoff, G. R., 311n.10, 361 Kirchhoff uniqueness theorem, 311–314 Kirk, W. P., 5, 62, 65, 159 Kirkwood, J. G., 353, 358, 361 Kirsch, G., 1, 498, 525 Kirsch, U., 61 Kitiكلمة محذوفةchai, S., 592, 595 Kittel, C., 250, 361 Knops, R. J., 10, 61 Knowles, J. K., 310, 361 Koiter, W. T., 421, 453, 510, 526 Kronecker delta, 47–48, 73, 211 Krook, M., 453 Kuruppu, M. D., 522, 525 Lagaros, N. D., 63 Lagrange multiplier, 57, 238 Lagrange multiplier method, 57–58, 101–105, 181, 238 Lagrangian (material) coordinates, 70, 149, 214, 232, 348–350 Lamb, R. S., 595 Lame elastic coefficients, 33, 257, 266, 311, 312 ´ Lamit, L., 2, 61 Lancaster, P., 55, 62 Langhaar, H. L., 58, 59, 62, 148n.17, 311, 333, 342, 343, 345, 361, 420, 453, 618 Laplace equation, 10, 18, 24, 34, 316, 542–546 Laplacian: defined, 18 in orthogonal curvilinear coordinates, 34–36 Large-deformation theory, 87 Large strain theory, 73 La Rubia, T. D., 60 Latent roots, 50 Latin letter indexes, 38 Lattice dynamics, 353, 354 Lee, G. C., 592, 595 Lee, G. G., 592, 595 Lee, J. D., 351, 352, 357, 359, 361–363 Leeman, E. R., 525 Lei, Y., 359, 362 Lekhnitskii, S. G., 245, 362, 617, 618 Lennard-Jones potential, 250 Level surfaces, 17 Li, J. C., 60 Li, S., 353, 362, 363 Lin, A. Y., 62 Linearly elastic materials, 8 Linear momentum density, at cell level, 358 Linear theory of elasticity, 8 Line element: direction cosines of, 78–79, 89 extension of, 78–86 final direction of, 89–90 relative elongation of, 86–89 Line integral, 28–30, 136 Lines of force, 18 Liu, S. C., 60 Liu, W. K., 4, 62630 INDEX Liu, X., 362 Log, natural (base e), 88 Londer, R., 1, 62 Loughlan, J., 125n.14 Love, A. E. H., 7, 62, 121n.12, 121n.13, 159, 166, 208, 224, 230n.2, 232, 242, 257, 308, 311n.10, 362, 389, 453, 608, 618 Ludwig, P., 87, 159 Lure, A. I., 610, 618 ´ Lutsko, J. F., 353, 362 McCulloch, A. D., 238, 363 McDowell, D. L., 353, 363 McDowell, E. L., 392, 453 MacLane, S., 55, 60, 96, 97n.5, 159 McLennan, J. A., 355, 362 Macroscale, 5 Macroscale interactions, simulation of, 4 Macroscale technologies, 6–7 Magnetorheological (MR) fluids, 4 Magnification factor, 83 Makeev, M. A., 63 Many-body effects, 251 Marsden, J. E., 231, 362, 618 Marx, D., 255, 362 Mass, conservation of, 218 Mass density, at cell level, 358 Masud, A., 4, 62 Materials: designer, 5–7 smart, 1–5 Material coordinates, see Lagrangian (material) coordinates Material derivative, 214–215 Material derivative of a volume integral, 214–218 Material equation of continuity, 145 Matlock, R. B., 595 Matrix: adjoint, 55 column, 52 defined, 38 diagonal, 54 inverse, 55, 56 null, 53 of order m by n, 52 reciprocal, 55, 56 row, 52 scalar, 54 square, 43 transpose of, 54 unit, 54 Matrix algebra, 52–56 Matrix methods, 8 Matrix theory, 38 Maxima, 56 Maximum principal stress criterion, 187 Maximum shearing stress criterion, 187 Mazurkiewicz, S. B., 525 MD, see Molecular dynamics Membrane analogy, 10 Mendelson, A., 8, 62 Menon, M., 63 Meshless method, 3 Mesoscale technologies, 6–7 Method of series: for bending, 586–590 for torsion, 562–568 Metric tensor of space, 33 Meyers, M. A., 5, 62 Michell, J. H., 463, 525 Micro-cantilevers, 5 Microcomputers, 1 Microcontinuum field theories, 347 Microcontinuum of grade N, 347 Microelectronics, 1, 7 Microgyration tensors, 349 Microinertia density, at cell level, 358 Micromechanics, 2 Micromorphic theory, 347–352 balance laws of, 350–351 constitutive equations of elastic solids, 351–352 Microscale technologies, 4–7 Microscopic space-averaging, 350, 358 Milne-Thompson, L. M., 410, 454 Mindlin, R. D., 213, 224, 421, 454, 519, 598, 619 Minicomputers, 1 Minima, 56 Minimum strain energy (elastic energy), theorem of, 338 Mixed boundary value problems, 11 Mohr, O., 196, 224 Mohr–Coulomb failure criterion, 187 Mohr’s circles, 195–198 Moire method, 10 ´ Molecular biomechanics, 5 Molecular dynamics (MD), 4, 205 ab initio, 255 classical, 254 constitutive relation in, 235–236 general form of potential energy, 249–250 governing equations, 235 quantum, 255 stiffness matrix in, 253–255 temperature in, 289–294 Berendsen thermostat, 293, 294 Gaussian constraints, 293–294 Nose–Hoover thermostat, 292–294INDEX 631 random number generation, 292, 294 velocity upgrade, 291–292, 294 Moment: body, 166–167 equilibrium, 166 twisting, 539–540 Moment of momentum density, at cell level, 358 Moment stress, 351 Momentum: balance of angular momentum, law of, 205 balance of linear momentum, law of, 205, 289 time rate, change of, 215–217 Monatomic lattices, 353 Moon, F. C., 2, 62 Moore’s Law, 7 Morrell, M. L., 595 Morris, M., 19, 62 Morse, P. M., 106, 159 Motion, differential equations of, 204 deformable body/medium, 201–206, 288 equilibrium, three-dimensional, 598–600 spatial coordinates, 201–206 stress: of deformable body relative to spatial coordinates, 201–206 for small-displacement theory, 214–224 MR (magnetorheological) fluids, 4 Mullineux, N., 521, 525 Multiply connected region, 388–392, 467, 547–549, 557–558 Multiscale problems, modeling, 2 Munari, A. C., 360 Muscle mechanics, 55, 239–240 Muskhelishvili, N. I., 9, 62, 307, 362, 365, 389, 410, 437, 438, 440, 454 Nabla (del), 17 Naghdi, P. M., 9, 62 Nair, S., 74, 159 Nanomechanics, 2 Nanoscale, 5 Nanotechnology, 1, 4–7 National Science Foundation (NSF), 2 Navier-Stokes equations, 343–345 Nearly incompressible soft biological tissue, 238 Necessary conditions: for compatible small-displacement strain, 132–138 for exact differential, 30 for extreme values, 58–59 for rigid-body displacement, 127–130 for single-valued Airy stress function, 388 Necking down, 228 Neou, C. Y., 408, 410, 411, 413, 454 Neou method, 408–411 Neumann, F. E., 269, 270, 362 Neumann boundary-value problem, 11, 533, 550 Nonclassical materials, 2 Nonelastic material response, 228 Nonhomogenous material, 256 Nonisotropic material, 256 Nonlinear constitutive relationships, 345–347 higher-order relations, 346 hypoelastic formulations, 346–347 variable stress-strain coefficients, 346 Nonlinear theory of elasticity, 8 Nonstationary field, 18 Nose–Hoover thermostat, 292–294 Novozhilov, V. V., 221, 224, 226n.1, 232, 362 Nowacki, W., 270, 362 NSF (National Science Foundation), 2 Nucleation, 2 Null matrix, 53 Numerical stress analysis, 3, 8–9 Nye, J. F., 242, 257, 362 Oblique coordinates, 154–155, 416–420 plane, 416–420 straight-line, 154–155 Oblique plane, stress on, 169–171 Octahedral planes, 186 Octahedral shearing strain, 112 Octahedral shearing stress, 186–187 Octahedral shearing stress criterion, 187 Octahedral strain, 112 Oden, J. T., 1, 2, 62 Optical fibers, 5, 10 Optimization methods, 2 Orr, C. M., 526 Orson, L. A., 62, 363 Orthogonal curvilinear coordinates, 31–32 differential length in, 32–33 gradient, 33–34 Laplacian, 34–36 strain-displacement relations, 146–151 Orthogonality relations, 41–42 Osman, M., 63 Pan, Y., 2, 63 Papadrakakis, M., 63 Papklovich, P. F., 597, 598, 619 Park, H. S., 62 Parks, M. L., 63 Parkus, H., 270, 362 Parrinello, M., 255, 359 Particle(s): displacement, 72 initial location, 69632 INDEX Passive materials, 239 Payne, L. E., 10, 61 Pearson, C. E., 188, 224, 362, 453 Pestel, E., 554n.5, 595 Peters, T., 526 Phase-space, description, 353–355 Phase-space coordinates, 354 Photoelasticity, 10 Physical space: atomistic quantities in, 355–357 description, 353–355 Pierce, B. O., 564, 566, 595 Piezoelectric composites, 4, 5 Pindera, J. T., 524, 525 Pinkerton, C. A., 526 Pinter, W. J., 504, 525 Piola–Kirchhoff (PK1 and PK2) stress tensors, 177–178, 237, 239–240 Pipes, L., 50, 62 Pippard, A. B., 232, 362 PK2 and PK2, see Piola–Kirchhoff stress tensors Planck, M., 166, 242, 362 Plane elasticity, 9 Plane polar coordinates, 151 Plane strain, 9, 112, 489–490 compatibility, 369, 377–382, 461 defined, 366 deformation, 112 differential equations of equilibrium, 366 strain energy density, 368–369, 421 Plane stress, 9, 193–194, 375–376 compatibility equation, 378–381, 462 generalized, 371–379, 462 graphical interpretation, 195–197 Mohr’s circles in three dimensions, 197–198 orthotropic elastic coefficients for, 248–249 Plane stress tensor, 193 Plane theory, 365–518 Airy stress function in, 383–399 body forces and temperature effects, 385, 389–392, 467–475, 487–489 boundary conditions, 385–388 compatibility, 379 harmonic functions, 399–400 multiply connected regions, 388–392 polar coordinates, 463 simply connected regions, 383–385 compatibility equation, 369, 377–382, 462 couple stress, 420–428, 519–522 displacement components, 401–407, 415–416, 447–448, 457–461, 485–489 Airy stress function, 401–404 polar coordinates, 485–489 generalized plane stress, 371–379, 462 oblique coordinates, 416–420 plane strain, 365–371, 377–382, 461 plane stress, 371–377, 380–381 polar coordinates, 455–518 Plane wedge, 509–510 Plaque, atheromatous, 475–478 Plastic, perfectly, 229 Plasticity, 8, 229, 230 Plate, with circular hole, 498–504 stress concentration problem, 498–504 stress-couple theory of, 519–522 Poisson equation, 272 Poisson’s ratio, 121, 245, 258, 267, 312, 319, 486, 521, 522, 611, 615–617 Polar coordinates: Airy stress function in, 456–457, 463, 498 equilibrium equations in, 455–456 plane compatibility equation in, 463 strain-displacement relations, 457–461 stress components in, 456–457 stress-strain temperature relations, 461–462 Postma, J. P. M., 359 Potential energy, in molecular dynamics, 249–250 Potential field, 19 Potential function, 19 Prager, W., 186, 225, 229, 362 Prandtl, L., 66, 159, 530, 534, 537, 538, 543, 554, 556, 572, 595 Prandtl membrane analogy, 554–562 Prandtl torsion function, 534–538 Pressures, 163 Principal axes, 259 Principal planes of stress, 179 Principal strains, 96–100 Principal values of the deformation, 101 Processors, in smart structures/materials, 5 Proportional limit, 227 Pseudoelasticity, 237–239 Quadratic forms: characteristic equation of, 50 characteristic roots (latent roots; eigenvalues), 50 determinant, 49 eigenvectors, 50–52 homogeneous, 49–52 Quantum MD, 255 Rachev, A., 239, 362 Ragsdell, K. M., 60 Random number generation, 292, 294 Reciprocal matrix, 55, 56 Rectangular arrays, 38–40INDEX 633 Rectangular Cartesian coordinates, 32 strain components in, 83–84 strain-displacement relations, 366 transformation of tensors under, 40–46 Reed, M. A., 5, 62, 65, 159 Reissner, E., 342, 343, 362 Reissner’s theorem, 342–343 Relative emissivity, 272 Remodeling, biomechanics of, 5 Repeated Greek index, 36–38, 43 Rigid body: definition of, 65 displacement, 66–67, 127–130 Riley, W. F., 10, 61 Ritchie, R. O., 5, 62 Rogers, C. A., 4, 5, 10, 62 Rogers, J. A., 61 Rogers, R. C., 4, 5, 10, 62 Rosenfeld, H. R., 277, 362 Rotation, mean, 78 small angles, 115–116 vectors, 115 volume element, 78, 113–117 Row matrix, 52 Ruoff, R. S., 60 Ruud, C. O., 10, 62 Ryzhik, I. M., 564, 566, 595 Sadd, M. H., 296, 363, 476, 525 Saigal, S., 3, 60 Saint-Venant, 158, 534n.3 Saint-Venant semi-inverse method, 569 Saint-Venant’s principle, 307–310, 317–322, 529, 530, 581 Saint-Venant’s torsion theory, 529–534 Saint-Venant warping function, 538, 544 Saleeb, A. F., 184, 224, 346, 359 Savin, G. N., 502, 525 Scalars, 43 Scalar field, 16, 21 Scalar matrix, 54 Scalar methods, 333 Scalar point functions, 16–18 Scalar product of vectors, 12–13 applications, 28–29 triple product, 14, 16 Scalzi, J. B., 60 Schatz, G. C., 60 Schijve, J., 421, 454 Schild, A., 40, 45n.6, 63, 95, 111, 159 Schmidt, R. J., 67, 68n.1, 142, 159, 187, 214, 215n.8, 224, 333, 359 Schreiber, E., 9, 62, 246, 363 Schrodinger equation, 255 ¨ Seager, M., 60 Seki, Y., 62 Self-diagnosis materials, 2 Self-healing materials, 2 Semenkov, O. I., 61 Semi-inverse method, 317–322 Sen, B., 392, 454 Sensors: bio-inspired, 4 in smart structures/materials, 5 Set (nonelastic material response), 228 Shaft, circular cross section, 327–332 Shape memory alloys, 4, 5 Sharma, B., 392, 454 Shear center, 581–584 Shearing components, 164 Shearing strain, 90–92 Shearing stress, see Stress, shearing Shear modulus, 245, 267 Shepherd, W. M., 593, 596 Sidebottom, O. M., 199, 225, 229, 359, 363 Simple connectivity, 30n.5 Simulation, atomic-scale-based, 4 Simulation-based engineering science, 2 Siriwardane, H. J., 346, 360 Skew-symmetric square arrays, 39 Slack, J. D., 526 Smart structures/materials, 1–5 Smith, C. W., 421, 453, 522, 525 Smith, I. M., 106, 159 Smith, J. O., 199, 225, 229, 363 Smith, J. W., 9, 60, 187, 224, 360, 525 Smith, W., 293, 363 Sneddon, I., 410, 454 Snell, C., 595 Society for Experimental Mechanics (SEM), 10n.2 Soft biological tissue: constitutive equation for, 237–238 incompressible and nearly incompressible, 238 Soga, N., 62, 363 Sokolnikoff, I. S., 365, 454 Sokolovski, V. V., 186, 225 Solids, 163 elastic and nonelastic response of, 226–230 micromorphic, 351–352 semi-infinite, 617 Solid–fluid interactions, 55 Solid mechanics research, priorities in, 2–3 Space-averaged temperature, 290 Spain, B., 62 Spatial coordinates, 21, 67–71, see Eulerian (spatial) coordinates Spatial equation of continuity, 145 Spatial form (continuity equation), 24n.4634 INDEX Special states of stress: hydrostatic, 317–318 irrotational, 126–127 plane, 193–201 pure shear, 267–268 simple tension, 266–267 Specific heat, 271–272 Spherical coordinate system, 151 Spherically symmetrical stress distribution, 294–299 Spitzig, W. A., 60 Split cylinder test, 522–524 Square arrays, 39–40 Square matrix, 43 Srivastava, D., 4, 63 State of plane stress with respect to the (X,Y) plane, 193 Stationary field, 18 Stationary potential energy, principle of, 337 Stationary value of integrals, 56–60 Steady field, 18 Stern, M., 467, 525 Sternberg, E., 9, 63, 213, 225, 307, 314, 363, 392, 453, 454, 510, 526, 618, 619 Stevenson, A. C., 410, 454 Stiffness matrix, in molecular dynamics, 253–255 Stippes, M., 9, 63, 618 Stokes’s theorem, 29–30 Strain: components, 78, 457–461 cylindrical coordinates, 150–153 orthogonal curvilinear coordinates, 146–151, 457 plane polar coordinates, 151, 457–461 rectangular Cartesian coordinates, 83–84 spherical coordinates, 151 definition: cubical, 257 engineering, 87 large-deformation, 85 logarithmic, 87–88 natural or true, 88 deviator, 110–112 elastic, 228 Eulerian (spatial) components, 82 in index notation, 82 invariants, 108–112 Lagrangian (material) components of, 82 of a line element, 86–89 mean, 110–112 notations of, 7 octahedral, 112 octahedral shearing, 112 plane, 112, 365–371 principal, 96–104 principal axes (directions), 96, 101, 104–107, 259 principal values, 96, 100–107 reciprocal ellipsoid, 96–100 invariants of, 97–98, 100 principal axes (directions), 98 principal directions (axes), 98–99 relative elongations, 98–99 set nonelastic, 228 shearing, 90–94 small, theory of, 121–132 special types, 110 dilatation, 110, 120–121, 125 homogeneous, 118–121 irrotational, 126–127 pure, 120–121, 267–268 rigid displacement, 127–130 simple shear, 126, 266 transformation of components, 95–96 volumetric (cubical), 109–110, 257, 366 Strain-displacement relations: cylindrical coordinates, 150–153 Euclidean metric tensor, 155–157 general coordinates, 155–159 geometric preliminaries, 146–148 oblique straight-line plane coordinates, 154–155 orthogonal curvilinear coordinates, 82, 146–151, 367, 457–461 plane polar coordinates, 151, 457–461 rectangular Cartesian coordinates, 366 spherical coordinates, 151 Strain energy density: for anisotropic linearly elastic material, 242, 368–369 for certain symmetry conditions, 242–246, 257, 259–261 for composites, 259, 261 for elastically isotropic medium, 256–266 function, 234, 237 in index notation, 277–279 plane strain, 368–369, 421 relation to stress components, 232–240 for soft biological tissues, 237 in terms of principal strains (invariants), 257 for thermoelasticity, 278 Strain gage methods, 10 Strain tensors, 83, 95–96, 110, 158–159 Almansi, 83, 159 Cauchy, 86, 351 components of, 83, 95–96 generalized Lagrangian, 351 Green–Saint-Venant, 83, 159 in terms of rotation vector components, 123–124INDEX 635 Stream lines, 18 Stress: array, 166–167 boundary conditions, 169–171, 306 characteristic values, 50, 179 components, 430–432 boundary conditions at point 0, 169 normal to a plane, 169–171 notation, 164–165 on oblique plane, 169–171 relation to strain energy density, 232–240, 278–279 symmetry of, 167 tangent to a plane, 169–171 thermal-stress problem in terms of, 287–288 torsion, 540–541 transformation of, 175–179 concentration, 498–504 definition, 161–163 differential equations of equilibrium: in curvilinear spatial coordinates, 207–211 including couple stress and body couple, 211–214 differential equations of motion: of deformable body relative to spatial coordinates, 201–206 for small-displacement theory, 214–224 direction, 179 eigenvalues, 50, 179 eigenvectors, 50, 181 extreme values, 179–182, 198–200 functions, three-dimensional, 317–327 index notation, 165–166 invariants, 180, 182–183, 257 Mohr’s circles, 195–198 and muscle mechanics, 239–240 normal: extreme values, 198–200 on oblique plane, 169–171 notation, 7, 164–166 on oblique plane, 169–171 plane, 193–201, 491 components in terms of Airy stress function, 384, 456, 457 extreme values of, 198–200 Mohr’s circle of, 195–198 at a point, 167–169 principal axes, 180–181 principal directions, 181 principal planes, 179 principal values, 179 principle stresses, 179–180 shearing, 169, 181–183 component, 169, 176 component in any direction, 543–544 extreme values, 181–183, 198–200 on oblique plane, 169–171 octahedral, 186–187 sign convention, 162 special states of, 266–269 hydrostatic, 317–318 irrotational, 126–127 plane, 193–201 pure shear, 267–268 simple tension, 266–267 summation of moments, 166–167 tensor character of, 175–178 in terms of Galerkin vector, 603–604 theory of, 161 thermal, 269–295 transformation of components, 175–179 vector, 177 virtual, 339–342 yield, 227 Stress analysis: experimental, 9–10 finite element method in, 8–9 numerical, 3, 8–9 Stress couples, 161, 167, 211–212 Stress notation, 164–166 Stress-strain relations, 241–255, 536 anisotropic, 261, 368–369 for bars, 528 beryllium, 279–281 composites, 259–261 generalized Hooke’s law, 241–255 higher-order, 346 including temperature effects, 276–285 in index notation, 246, 257 for isotropic media, 256–266, 366 nonlinear, 346 in oblique coordinates, 420 relative to axes inclined to crystal axis, 281–283 for soft biological tissues, 237 special states, 266–269 Stress-strain-temperature relations, 276–285 for beryllium, 279–280 polar coordinates, 461–462 relative to axes inclined to crystal axis, 281–283 Stress tensors, 177, 184–185, 210, 212 character of, 175 deviator, 185–193 invariants of, 180 mean, 185–193 notation, 165, 166 Piola–Kirchhoff, 177–178, 237, 239–240 plane, 193 Stretchable electronics/sensors, 5636 INDEX Stretch ratio, 240 Substitution of indexes, rule of, 47 Successive elastic solutions, 8 Sufficient conditions: for compatible small-displacement strain, 132–138 for exact differential, 30–31 for rigid-body displacement, 127–130 Suhubi, E. S., 348, 350, 360 Summation convention, 36–40 Summation notation, 43–44 Summing index, 37 Supercomputers, 1, 6 Superposition method, 501, 502 Surfaces, level, 17 Surface integral, 29 Swanson, W. D., 595 Symmetric square arrays, 39 Synge, J. L., 40, 45n.6, 63, 95, 111, 159, 231n.4, 363 Szabo, B. A., 595 Tangential components, 164 Taylor, R. L., 1, 9, 63 Taylor series, 115–116 Temperature distribution: diffusivity, 271 space-averaged, 290 specific heat, 271–272 stationary, 271 steady-state, 271 time-averaged, 290 Templeton, J. A., 63 Tensors: alternating, 49 antisymmetric parts of, 47, 77, 78 Cauchy strain, 73–74 conjugate, 46 contravariant, 210–211 covariant, 210–211 deformation gradient, 73 Euclidean metric, 155–157 first-order, 44 invariants, 108 isotropic, 48 Kronecker delta (substitution tensor), 47–48 mean strain, 110–112 metric, 33, 155–157, 210–211 microgyration, 349 nth-order, 45, 46 second-order, 45–48, 77–78, 177 special third-order (alternating), 49 strain, 95–96, 158–159 stress, 177, 210 substitution, 48–49 symmetric, 177 symmetric parts, 46–47, 77, 78 third-order, 45, 48–49 transformation under rotation of axes, 40–46 zero order, 43–44 Tensor algebra, 36–52 homogeneous quadratic forms, 49–52 index notation, 36–40 notation, 47–49 symmetric and antisymmetric tensor parts, 46–47 transformation under rotation of axes, 40–46 Tersoff, J., 251, 363 Tersoff potential, 251–253 Tham, L. G., 3, 60 Thermal conductivity, 271 Thermal expansion coefficient, 272–273, 277 Thermal stress: in beams, 274–276 displacement potential, 303 Duhamel-Neumann theory, 269–270 elementary approach, 272–276 equivalent displacement problem, 269, 285–287 physical interpretation, 287–288 plane theory, 389–392, 489–494 spherically symmetrical, 294–299 Thermal treatment, 55 Thermodynamics, first law, 234 Thermoelasticity: axially symmetric case, 302–304 equations: for beryllium, 279–281 boundary conditions, 287 compatibility (stress), 299–305 isotropic media, 269–270, 279–281 physical interpretation of thermal-stress problem, 287–288 temperature in molecular dynamics, 289–294 in terms of displacement, 285–294 thermomechanical coupling, 288–289 plane theory, 389–392, 489–494 Thermomechanical coupling, 288–289 Thoft-Christensen, P., 8, 63 Three-dimensional elasticity, 9 Three-dimensional stress functions, 617–618 Tiersten, H. F., 213, 224 Tietjens, O. G., 66, 159 Time-averaged temperature, 290 Time evolution law of physical quantities, 355 Timoshenko, S., 307n.9, 403, 454, 463n.1, 522, 524, 526, 610, 619INDEX 637 Timoshenko, S. P., 363, 561, 566, 573n.7, 581, 596 Timp, G., 5, 63, 65, 160 Timpe, A., 463n.1, 526 Ting, T. C. T., 510, 526 Tismenetsky, M., 55, 62 To, A. C., 353, 362, 363 Todorov, I. T., 363 Torsion: of prismatic bars, 529–568 axis of twist, 327, 536, 549–550 boundary conditions, 528 displacement components, 536–538, 560–561 elliptic cross section, 538–542, 584 moment angle of twist relation, 539–542 narrow rectangular cross section, 560–561 Prandtl function of, 535 Prandtl membrane analogy, 554–562 Prandtl theory, 534–538 Prandtl torsion function, 534–538 rectangular section, 562–568 Saint-Venant’s solution, 529–534 shear-stress components, 543–544 stress components, 540–541 with tubular cavities, 547–549 warping, circular cross section, 544 of shaft with constant circular cross section, 327–332 Torsional rigidity, 540 Total energy density, at cell level, 358 Toupin, R. A., 421, 454 Trahair, N. S., 592, 595 Transform methods, 440–445 Translation, of a mechanical system, 66, 67 Transpose of matrix, 54 Transposition, 54 Tresca–Saint-Venant–Coulomb–Guest criterion, 187 Tribology, 2 Trimmer, W., 63 Truesdell, C., 347, 363 Tsompanakis, Y., 1, 63 Turner, J. P., 360 Twinned gradient, 611–614 Twist: angle of, 539–540 axis of, 327 generally, 327, 536 transfer of, 549–550 center of, 536 twisting moment, 539–540 Udd, E., 4, 63 Uenishi, K., 360 Uniqueness theorem of elasticity (equilibrium), 311–314 Unit matrix, 54 Unit vectors, 16–17 University of Illinois, Theoretical and Applied Mechanics Dept., 148n.17 Unsteady field, 18 Van Gunsteren, W., 309, 359 Van Tassel, J., 526 Variables, 68–71 complex, 399–400, 428–453 material, 70
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