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 |  موضوع: كتاب Elasticity in Engineering Mechanics الأحد 01 أغسطس 2021, 12:50 am | |
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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.
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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