INDEX

Ab initio, 255Abraham, F., 4, 60Acceleration:

convective terms of, 143kinematics of deformable media, 141–144

Acceleration field, 22Acceleration vector, 21, 22, 142, 203, 218Active materials, 2Active stress:

and muscle mechanics, 239–240significance of, 478–483

Actuators, in smart structures/materials, 5Adams, P. H., 10, 61Adiabatic deformation process, 230–232Adjoint matrix, 55Adkins, J. E., 8, 61, 226n.1, 361Admissible functions, 58Advanced material processing, 3Aeolotropic material, 256Agrawal, A., 362Airy, G. B., 9Airy stress function, 9, 383–399, 426–427

complex variables, 428–429displacement components, 401–404, 429–430harmonic functions, 399–400plane theory, 383–399

body forces and temperature effects, 385,389–392, 467–475, 487–489

boundary conditions, 385–388compatibility, 379

harmonic functions, 399–400multiply connected regions, 388–392polar coordinates, 463simply connected regions, 383–385

in polar coordinates, 456–457, 463, 498solutions:

in polar coordinates, 463in rectangular coordinates, 385, 408–415

Almansi strain, 238Almansi strain tensor, 83, 159American Society for Testing and Materials

(ASTM), 10n.2Aneurysm, intracranial saccular, 295–298Angle of twist, 331, 529Anisotropic material, 231, 241–255, 259–261,

494–498elastic coefficients, 242strain energy density, 242, 368–369strain–temperature relation, 288stress-strain relations, 261, 368–369

Anticlastic surface, 322Antisymmetric square arrays, 39Approximate methods, 3, 10Arbitrary square arrays, 39–40Argument function, 58Ariman, T., 421, 453, 519Arrays:

antisymmetric, 39antisymmetric square, 39arbitrary, 39–40

621Elasticity in Engineering Mechanics, Third Edition Arthur P. Boresi, Ken P. Chong and James D. LeeCopyright © 2011 John Wiley & Sons, Inc.

622 INDEX

Arrays: (continued )characteristic equation, 50determinants, 49rectangular, 38–40skew-symmetric, 39square, 39–40stress, 166–167symmetric, 39symmetric square, 39typical element, 39

Arroyo, M., 4, 60Artery wall, atheromatous plaque on, 475–478Associative law of vector addition, 12ASTM (American Society for Testing and

Materials), 10n.2Atheromatous plaque on artery wall, 475–478Atomistic field theory, 352–359

atomistic quantities in physical space,355–357

conservation equations, 357–359phase-space and physical-space descriptions,

353–355Atrek, E., 1, 60Averbach, B. L., 277, 362Axis of twist:

generally, 327, 536transfer of, 549–550

Baker, M. J., 8, 63Balance laws, 214, 218

angular momentum, 205, 358in atomistic field theory, 358, 359linear momentum, 205, 289of micromorphic theory, 350–351

Bar, prismatic, 318–322, 527–591bending, 318–322, 569–577

Bernoulli-Euler equation, 322curved bar, 505–506elliptic cross section, 584–586pure, 318–322, 581–582rectangular cross section, 586–590transverse end force, 569–577

Prandtl torsion theory, 534–538Saint-Venant’s torsion theory, 529–534shear-center, 581–584torsion, 529–568

axis of twist, 327, 536, 549–550boundary conditions, 528elliptic cross section, 538–542, 584narrow rectangular cross section, 560–561Prandtl function of, 535Prandtl membrane analogy, 554–562Prandtl theory, 534–538rectangular section, 562–568

Saint-Venant’s solution, 529–534shear-stress components, 543–544with tubular cavities, 547–549warping, circular cross section, 544

Bathe, K.-J., 106, 159Beams:

cantilever, 506–507tapered, 591–595thermal stress, 274–276

Beltrami–Mitchell compatibility equations, 529Beltrami–Mitchell compatibility relation,

299–305Belytshko, T., 4, 60Bending:

of prismatic bar, 318–322, 569–577Bernoulli-Euler equation, 322curved bar, 505–506elliptic cross section, 584–586pure, 318–322, 581–582rectangular cross section, 586–590transverse end force, 569–577

pure, 318–322bar subjected to transverse end force,

527–529, 569–577Bessel functions, 519cantilever beam, 506–507curved bars, 505–506function (flexural), 572general equations, 569–577plane wedges, 509–510prismatic bars, 318–322

Berendsen, H. J. C., 293, 359Berendsen thermostat, 293, 294Bernoulli-Euler equation, 322Beus, M. J., 504, 525Biharmonic equation, 384. See also Airy stress

functionfunctions, 384solutions of, 385, 465

Bilinear form, 50Bio-inspired sensors, 4Biological tissues:

constitutive equation for, 237–238lifeless material vs. living, 239–240structure of, 5

Biomechanics, 5Bioscience, 5Biot, M. A., 270, 359Biotechnology, 1, 5, 7Birkhoff, G., 55, 60, 96, 97n.5, 159Body couples, 7, 161, 211–214

force, 233, 467–475, 487moments, 166–167

Body force (atomistic field theory), 357

INDEX 623

Boley, B. A., 270, 359Boltzmann constant, 290Bone structure, 5, 6Boresi, A. P., 3, 9, 11, 60, 67, 68n.1, 142, 159,

187, 214, 215n.8, 224, 333, 346, 359Borgman, E. S., 525Born, J. S., 310, 361Boundary conditions, 169, 287, 528

for bars, 528equilibrium, 305–310intracranial saccular aneurysm, 298mixed boundary value problem, 306–307for multiply connected regions, 388for plane polar coordinates, 471–473,

496–497Saint-Venant’s principle, 307–310stress, 169–171, 287, 306in terms of Airy stress function, 385–388in terms of displacement, 313, 315–316for torsion of bars, 547–548

Boundary element method, 3Boundary-value problems, 10–11, 438–440

Dirichlet, 10–11, 534mixed, 11Neumann, 11, 533, 550

Boussinesq, J., 307, 308, 359, 611, 614–615,618

Boussinesq problem, 614–616Brazilian test, 522Brebbia, C. A., 3, 60Brown, G. H., 161, 224Brown, J. W., 384, 399, 436, 437, 441, 451, 453,

533, 542, 564, 577, 595Brown, O. E., 19, 62Buehler, M. J., 62Bulatov, V. V., 359Bulk modulus, 258

CAD (computer-aided design), 2Cai, W., 353, 359Calculus of variations, 56–60

admissible functions, 58argument function, 58conditions of admissibility, 58Euler differential equation, 59first variation of an integral, 60functionals, 58stationary value of an integral, 59variation of a function, 58–60

CAM (computer-aided manufacturing), 2Car, R., 255, 359Carlson, D. E., 212n.5, 426, 453Carrier, G. F., 436, 448, 453Carslaw, H. S., 270, 359

Cartesian coordinate system, see RectangularCartesian coordinates

Castigliano’s theorem:on deflections, 341–342principle of virtual stress, 341–342

Cauchy elastic formulation, 346, 347Cauchy-Riemann equations, 533Cauchy strain tensor, 73–74, 88–89, 158Cauchy stress, 177, 237–240, 351Cayley–Hamilton theorem, 74Cell biomechanics, 5Center of shear (shear center), 581–584Center of twist, 536Cerruti, V., 611, 614–616, 618Cerruti problem, 614–617Chadwick, P., 270, 359Chan, S. S., 504, 525Characteristic roots (eigenvalues), 50Chasles’s theorem, 67, 127Chen, J. L., 360Chen, P., 62Chen, W.-F., 184, 224, 346, 347, 359Chen, Y., 352, 354, 355, 357, 359, 361, 363Cheung, Y. K., 3, 60Chistoffel symbols, 210Choi, I., 307n.9, 310, 359Chong, 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., 360Churchill, R. V., 384, 399, 436, 437, 441, 451,

453, 533, 542, 564, 577, 595Circle of Willis, 295–297Clausius–Duhem inequality, 350Cleary, M. P., 346, 360Clemson University, 591n.9Coefficients of the principal dilatations, 120Column matrix, 52Commutative law of vector addition, 12Compatibility, 317

Beltrami–Mitchell compatibility equations,529

Beltrami–Mitchell compatibility relation,299–305

with couple stress, 425–426displacement, 132–138equation for plane elasticity:

in polar coordinates, 463in rectangular coordinates, 369, 408–415

plane strain, 369, 377–382, 461plane stress, 378–381, 462small displacement, conditions of, 132–138, 371in terms of Airy stress function, 379thermoelasticity, 299–305

624 INDEX

Complementary function (solution or integral),389–391

Complex variables, 399–400, 428–453Airy stress function, 428–429conformal transformation, 440–445in curvilinear coordinates, 445–448displacement components, 429–430plane elasticity boundary value problems,

438–440for plane region bounded by circle in z plane,

448–452resultant force and resultant moment, 433–434stress components, 430–432

Composites, 259–261Compressions, 163Computers, 1–3

microcomputers, 1minicomputers, 1in smart structures/materials control, 5supercomputers, 1, 6

Computer-aided design (CAD), 2Computer-aided manufacturing (CAM), 2Conditions of admissibility, 58Conservation laws:

angular momentum, 357in atomistic field theory, 357energy, 232, 289, 290, 357linear momentum, 357of linear momentum, 357mass, 205, 218, 357in molecular dynamics, 236

Constitutive equations:of elastic solids, 351–352for soft biological tissue, 237–238

Constitutive relations, 9, 246. See alsoStress-strain relations

in atomistic field theory, 359in molecular dynamics, 235–236nonlinear, 345–347

Constraints, 56Contact mechanics, 2Continuity, 65–66

conditions of, 134–140equations of, 134, 140, 145, 146material (Lagrangian) form, 139–140spatial form, 144–146

Continuous body:defined, 68deformation, 68, 72–73

Continuous (deformable) medium (continuum), 7,68, 72–73, 140

Continuum mechanics, 7, 68, 205and atomistic models, 353interfacing molecular dynamics and, 353

Continuum physics, 289Contour map, 542Contravariant tensors, 210–211Cook, N. G. W., 268, 360Coordinate lines, curvilinear, 31–32, 147,

445–448Coordinate surfaces, 31, 147Coordinate systems:

cylindrical, 150–153, 208–209, 491Eulerian, 21, 67–71, 82, 232

deformation, 67–71micromorphic theory, 348–350

intrinsic, 159Lagrangian, 67–71, 149, 214, 232, 348–350left-handed, 14, 15material, 66–71, 214oblique, 154–155, 416–420

plane, 416–420straight-line, 154–155

orthogonal curvilinear, 31–32, 146–151differential length in, 32–33gradient, 33–34Laplacian, 34–36strain-displacement relations, 146–151

plane polar, 210, 455–456polar coordinates:

Airy stress function in, 456–457, 463, 498equilibrium equations in, 455–456plane compatibility equation in, 463strain-displacement relations, 457–461stress components in, 456–457stress-strain temperature relations, 461–462

rectangular Cartesian, 32, 40–46, 70–71,408–415

strain components in, 83–84strain-displacement relations, 366transformation of tensors under, 40–46

right-handed, 14, 15spatial, 21, 66–71spherical, 151, 209–210, 294–299

Corrosion sensors, 5Cosserat, E., 213n.6, 421, 453Cosserat, F., 213n.6, 421, 453Coulomb–Buckingham potential, 250Couple, body, 7Couple stress, 7, 211–214, 420–428

deformation, 421–424equations of compatibility, 425–426equations of equilibrium, 421stress concentration from circular hole in plate,

519–522stress functions for plane problems with,

426–428Couple stress tensor, 213

INDEX 625

Courant, R., 30n.5, 57, 58, 60, 61, 72, 139, 159,534, 550, 595

Covariant tensors, 210–211Creep, 8Cross section, 538, 544, 547–549

deformed shape of, 322elliptical, 538–539warping, 536–537, 544

Crystalline systems:multicomponent, 352–354single-component, 353

Cubical strain, 366Curl of vector field, 22Current density, 23Curvilinear coordinates, 147Cutoff radius, 251Cylindrical coordinate system, 150–153,

208–209, 491

Dally, J. W., 10, 61Dana, G. F., 360Davis, D. C., 4, 60Davis, G., 592, 595Deformable body (medium), 65–66

differential equations of motion, 288equilibrium, three-dimensional, 598–600spatial coordinates, 201–206

incompressible, 204kinematics of, 140–146

acceleration, 22, 141–144convective, 143

Deformation:admissible, 73, 84–85compatibility conditions, small displacement,

132–138condition for continuously possible, 72–73of a continuous region, 68–71couple-stress, 421–424definition, 66deformable, continuous media, 65–66, 71–76extension of infinitesimal line element, 78–86gradient of displacement vector, 76–78homogenous, 118–121kinematics of deformable media, 140–146line element:

direction cosines of, 78–79, 89extension of, 78–86final direction cosines of a deformed, 89–90relative elongation of, 86–89

material (Lagrangian) form, 67–71, 139–140mean and deviator strain tensor, 110–112octahedral strains, 112plane strain, 112principal axes, 101–107

principal strains, 100–101proper, 73reciprocal ellipsoid, 96–100rigid-body displacements, 66–67rotation of volume element, 113–117shearing strain, 90–92spatial (Eulerian) form, 67–71strain definitions, 87–89strain invariants, 108–109strain tensor, 94–96theory of small strains and small angels of

rotation, 121–132transformations of lines and surfaces,

138–139volumetric strain, 109–110zero state (configuration), 229

Deformation gradient tensor, 73De Koning, M., 359Del (nabla), 17Delange, S. L., 237, 296, 297, 361Delph, T. J., 353, 360Density, 66

at cell level, 358current, 23mass, 7

Density functional theory, 255Desai, C. S., 346, 360Designer materials, 5–7Determinants:

of arrays, 49vector, 14, 16

Determinant notation, 14, 16, 42Development, biomechanics of, 5Diagonal matrix, 54Differential, total, 79–80, 387Differential equations of motion, 204Differential length, in orthogonal curvilinear

coordinates, 32–33Differentiation:

of scalar field, 21of vector field, 21–22of vectors, 19–21

Diffusivity, 271Dilatation:

cubical, 110intracranial saccular aneurysm, 295–298pure, 120–121, 125

Dillon, O. W., 60DiNola, A., 359Directional derivative, 17–18, 550–554Direction cosines:

determinants of, 42in index form, 43orthogonality relations, 41–42

626 INDEX

Direction cosines: (continued )relations between, 41–42table, 41

between two sets of rectangular Cartesian axes,40–43

Dirichlet boundary-value problem, 10–11, 534Disk, 470–471, 487–489, 494–498, 522–525Displacement:

admissible, 73, 75, 84–85of cantilever beam subjected to transverse end

force, 577–581compatibility (continuity), 132–138components of, 71, 82–83

equations, 314–317in terms of Airy stress function, 401–404,

429–430torsion, 536–538

deformable body, 71fluid particles, 21gradient of, 76–78, 115–117particle, 66, 67plane, 67, 366–368, 415–416proper, 73, 75reflection, 73rigid-body, 66–67, 127–130

plane, 67rotation, 67translation, 66, 67

small strains and angles of rotation, 121–132vector, 76–78virtual, 333–338

Displacement potential, 389–392Displacement potential function, 390Divergence, of vector field, 23Divergence theorem, 25–27, 233

Gauss’s theorem, 25–26Green’s theorem, 27Green’s theorem of the plane, 28in two dimensions, 27–28

Dove, R. C., 10, 61Drucker, D. C., 184, 224Duchaineau, M., 60Duhamel, J. M. C., 269, 270, 360Duhamel-Neumann theory, 269–270Dummy indexes, 37, 38Dvorak, G. J., 1, 61

E, W., 353, 360Education, in mechanics, 3Eigenvalues (characteristic roots), 50Eigenvectors, 50–52, 188Eisenhart, L. P., 41, 50, 61, 98, 159Elastic coefficients (stiffnesses), 241–246,

257–261

for general anisotropic elastic material, 242Lame, 33, 257, 266, 311, 312law of transformation, 246–249

Elasticity:anisotropic, 231, 241–255, 259–261axisymmetric problem, 302–304, 467–485in biomechanical problems, 5boundary-value problems, 10–11, 305–307,

438–440, 533–534bulk modulus, 258concept of, 229–230isotropic, 231linear theory, 8, 227nonlinear theory, 8, 345–346perfect, 227, 229–231plane, 9. See also Plane theoryPoisson’s ratio, 267polynomial solution of two-dimensional

problems, 408–415pseudoelasticity, 237–239shear modulus, 267solutions in, 9–11, 317–323, 384, 408–415,

465, 485–489general, 9, 597–618successive elastic, 8three-dimensional, 9, 317–327, 597–618

strain energy density, 234–235theory of, 8–9, 230uniqueness theorem in, 311–314Young’s modulus, 267

Elastic limit, 227, 228, 230Elastic response, 8Elastic strain, 228Elder, A. S., 114n.11, 159Electronic structure theory, 255Electroreheological (ER) fluids, 4Ellis, E. W., 522, 525Ellis, R. W., 421, 453Ellis, T. M. R., 2, 61Emissivity, 272Energy:

internal, 232–234intrinsic density function, 230–232kinetic, 66, 242stress energy density function, 232–235,

256–262, 368Energy methods, 8Energy principles:

Castigliano’s theorem, 341–342conservation energy, 232elasticity, 332–333minimum elastic energy, 338minimum strain energy, 338mixed virtual stress-virtual strain, 342–343

INDEX 627

Reissner’s theorem, 342–343stationary potential energy, 337virtual displacement, 334–338, 343, 345virtual stress, 339–342virtual work, 333–339, 343–345

for elastic bodies, 335–338for particles, 334–335

Energy-related solid mechanics, 2Engquist, B., 360Environmental sensors, 5Equations of constraint, 56Equilibrium:

astatic, 308boundary conditions, 305–310of cubic element, 204differential equations of, 204, 207–211, 421,

598–600in cylindrical coordinates, 208–209in general spatial coordinates, 210–211including couple stress and body couple,

211–214in material coordinates, 210, 218–224in oblique coordinates, 416–420in orthogonal curvilinear spatial coordinates,

207–208in plane polar coordinates, 210, 455plane strain, 366specialization of, 208–210in spherical coordination, 209–210

of infinitesimal cubic element, 165of moments, 166in three dimensions, 317–322uniqueness theorem of, 311–314

Eringen, A. C., 7, 61, 224, 231, 346, 348, 350–352,360

Eskandarian, A., 361Eubanks, R. A., 314, 363Euclidean metric tensor, 155–157Euler angles, 231Euler differential equation, 59Eulerian continuity equation, 22–24Eulerian (spatial) coordinates, 21, 70, 82, 232

deformation, 67–71micromorphic theory, 348–350

Euler’s theorem, 67Exact differential, 30–31Experimental Mechanics, 10n.2Experimental methods, 2Experimental stress analysis, 9–10Experimental Techniques, 10n.2Extreme (extreme values, extrema), 56, 181–183

Failure criteria (modes), 186–189Fairhurst, C., 524, 525

Feshbach, H., 106, 159Fields, 17–19, 21–22

acceleration, 22divergence, 23, 25–27nonstationary (unsteady), 18scalar, 16–18stationary (steady), 18vector, 18–19, 21–23vector lines of, 18velocity, 18, 22–24

Field lines, 18Finite difference method, 3, 8Finite element method, 1, 8, 52, 359Finite layer method, 3Finite prism method, 3Finite strip method, 3Flexural function, 572Fluids:

circulation, 29divergence, 25–27electroreheological, 4Eulerian (spatial) continuity equation, 22–24flow, 22–24, 145–146, 163frictionless, 163ideal, 163incompressible, 24, 146, 345irrotational flow, 24, 145–146magnetorheological, 4momentum, 215

convective, 215local, 215

steady flow, 24, 142, 216unsteady flow, 22velocity fields of, 18, 22–24viscous, 163, 345vorticity, 30

Forces:body, 201–202, 204, 236, 338, 343–344conservative, 231distributed, 161inertial, 202–203, 338, 343–344nonconservative, 231normal, 162–163point, 161shearing, 162, 163statically equivalent systems, 308–309surface, 164, 203tractive, 203

Forester, T. R., 293, 363Formula, 350, 564Fosdick, L. D., 1, 2, 61Foster, R. M., 564, 566, 595Fracture gages, 10Frames, 68–71

628 INDEX

Free indexes, 37–38Frequency–wave vector relations, 353Friction coefficient, 293Functionals, 58Functional determinant, 72, 220Fung, Y. C., 5, 61, 237, 239, 360

Galerkin, B., 597, 598, 604–606, 609, 617, 618Galerkin–Papkovich vector, 597–598,

602–608Gallagher, R. H., 60Gao, H., 60Gaussian constraints, 293–294Gauss’s theorem, 25–26. See also Divergence

theoremGeneral tensor notation, 3Geotechnical Testing Journal, 10n.2Gibbs vector notation, 36Gilbert, D., 61Gilbert, L., 55, 61Golsten, S., 361Goodier, J. N., 307n.9, 363, 389, 403, 453, 454,

463n.1, 522, 524, 526, 566, 596, 610, 619Goree, James G., 591n.9Goursat, E., 25, 61Gradient (grad), 33–36, 552

of displacement vector, 76–78in orthogonal curvilinear coordinates, 33–34of scalar function, 17twinned, 611–614

Gradshteyn, I. S., 595Green, A. E., 8, 46, 61, 210, 211n.4, 224, 226n.1,

361, 410, 453Green, R. E., Jr., 10, 62Green–Saint-Venant strain tensor, 83, 159, 237,

238Green’s deformation tensor, 238Greenspan, D., 11, 61Green’s strain tensor, 83, 158, 159Green’s theorem, 27Green’s theorem of the plane, 28Green-type materials, 346Griffith, B. A., 231, 363Griffiths, D. V., 106, 159Grossmann, G., 554n.5, 595Growth, biomechanics of, 5Gunther, W., 543, 545, 560, 596

Haak, J. R., 359Haile, J. M., 291, 361Half-plane, 507–508Hamed, E., 5Hansma, P., 62Hardy, R. J., 353, 358, 361

Hartsock, J. A., 403, 453Hayashi, K., 239, 362Hayes, D. J., 525Health-care delivery, 5Heat conduction equation, 270–272, 289Heat transfer (exchange), 272Helmholtz’s free-energy density, 352Helmholtz transformation, 600–601Higher-order relations, 346Hilbert, D., 58, 534, 550, 595Hildebrand, F. B., 50, 61, 74, 80, 102, 159, 261,

312, 361Hill, R., 231n.3, 361Hodge, P. G., Jr., 186, 225, 229, 362Homeland Security problems, 3Homogenous deformation/state of strain, 118–121Homogenous media, 256Hondros, G., 524, 525Hooke’s law, 241–255, 257, 346Hoover, W. G., 292, 361Horgan, C. O., 307n.9, 310, 359, 361Horvay, G., 310, 361Hsu, C. S., 9, 62Huang, Y., 5, 61Huang, Z., 353, 360Hughes, T. J. R., 231, 362, 618Humphrey, J. D., 5, 61, 239, 296, 297, 361, 478,

525Hutter, J., 255, 362Hydraulic systems, 5Hydrostatic pressure, 238Hydrostatic stress, 287–288, 317–318Hyperelastic materials, 346Hypoelastic materials, 346–347Hysteresis, 230

Ince, E. L., 61Incompressible fluids, 24Incompressible soft biological tissue, 238Indexes:

dummy, 37, 38free, 37–38Latin letter, 38repeated Greek index, 36–38, 43, 117repeated nonsummed, 38rule of substitution, 47summation convention, 36–40, 43–44

Index notation, 3determinant, 42orthogonality relations, 42summation, 36–40, 43–44

Inelastic response, 8Infinitesimal strain, 238Information technology, 1, 7

INDEX 629

Integral:line, 28–30, 136particular, 577stationary value of, 56–60surface, 29volume, 214–218

Integration, constant of, 574–576Intelligent structures, 3–4. See also Smart

structures/materialsInteratomic force, 235, 249–255Intracranial saccular aneurysm, 295–298Intrinsic energy density function, 230–232Invariance (invariants), 43, 100, 108

strain, 108–112strain ellipsoid, 100stress, 180, 182–183

Inverse matrix, 55, 56Irrotational flow, 24Irvine, J. H., 361Irving, J., 353, 358, 521, 525Isotropic material/media (body), 231, 255–256,

280, 312–313higher-order relations, 346strain energy density for, 256–266strain–temperature relation, 289thermoelasticity equations, 269–270

Jacobian, 72–73, 220, 348Jaeger, J. C., 270, 359Jasiuk, I., 5Jeffery, A., 595Jeffreys, H., 257, 361Jiang, H., 61Jones, J. E., 250, 361Jones, R. E., 63Journal of Testing and Evaluation, 10n.2

Kaloni, P. N., 421, 453, 519Kannan, R., 4, 62Kaplan, W., 80, 159Karpov, E. G., 62Keller, H. B., 310, 361Kellogg, O. D., 533, 534, 595Kelvin’s problem, 609–611, 614Ketter, R. L., 595Khang, D.-Y., 5, 61Khattab, M. A., 525Kinetic energy, law of, 333, 334Kirchhoff, G. R., 311n.10, 361Kirchhoff uniqueness theorem, 311–314Kirk, W. P., 5, 62, 65, 159Kirkwood, J. G., 353, 358, 361Kirsch, G., 1, 498, 525Kirsch, U., 61

Kitipornchai, S., 592, 595Kittel, C., 250, 361Knops, R. J., 10, 61Knowles, J. K., 310, 361Koiter, W. T., 421, 453, 510, 526Kronecker delta, 47–48, 73, 211Krook, M., 453Kuruppu, M. D., 522, 525

Lagaros, N. D., 63Lagrange multiplier, 57, 238Lagrange multiplier method, 57–58, 101–105,

181, 238Lagrangian (material) coordinates, 70, 149, 214,

232, 348–350Lamb, R. S., 595Lame elastic coefficients, 33, 257, 266, 311, 312Lamit, L., 2, 61Lancaster, P., 55, 62Langhaar, H. L., 58, 59, 62, 148n.17, 311, 333,

342, 343, 345, 361, 420, 453, 618Laplace equation, 10, 18, 24, 34, 316, 542–546Laplacian:

defined, 18in orthogonal curvilinear coordinates, 34–36

Large-deformation theory, 87Large strain theory, 73La Rubia, T. D., 60Latent roots, 50Latin letter indexes, 38Lattice dynamics, 353, 354Lee, G. C., 592, 595Lee, G. G., 592, 595Lee, J. D., 351, 352, 357, 359, 361–363Leeman, E. R., 525Lei, Y., 359, 362Lekhnitskii, S. G., 245, 362, 617, 618Lennard-Jones potential, 250Level surfaces, 17Li, J. C., 60Li, S., 353, 362, 363Lin, A. Y., 62Linearly elastic materials, 8Linear momentum density, at cell level, 358Linear theory of elasticity, 8Line element:

direction cosines of, 78–79, 89extension of, 78–86final direction of, 89–90relative elongation of, 86–89

Line integral, 28–30, 136Lines of force, 18Liu, S. C., 60Liu, W. K., 4, 62

630 INDEX

Liu, X., 362Log, natural (base e), 88Londer, R., 1, 62Loughlan, J., 125n.14Love, 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, 159Lure, A. I., 610, 618Lutsko, J. F., 353, 362

McCulloch, A. D., 238, 363McDowell, D. L., 353, 363McDowell, E. L., 392, 453MacLane, S., 55, 60, 96, 97n.5, 159McLennan, J. A., 355, 362Macroscale, 5Macroscale interactions, simulation of, 4Macroscale technologies, 6–7Magnetorheological (MR) fluids, 4Magnification factor, 83Makeev, M. A., 63Many-body effects, 251Marsden, J. E., 231, 362, 618Marx, D., 255, 362Mass, conservation of, 218Mass density, at cell level, 358Masud, A., 4, 62Materials:

designer, 5–7smart, 1–5

Material coordinates, see Lagrangian (material)coordinates

Material derivative, 214–215Material derivative of a volume integral, 214–218Material equation of continuity, 145Matlock, R. B., 595Matrix:

adjoint, 55column, 52defined, 38diagonal, 54inverse, 55, 56null, 53of order m by n, 52reciprocal, 55, 56row, 52scalar, 54square, 43transpose of, 54unit, 54

Matrix algebra, 52–56Matrix methods, 8Matrix theory, 38

Maxima, 56Maximum principal stress criterion, 187Maximum shearing stress criterion, 187Mazurkiewicz, S. B., 525MD, see Molecular dynamicsMembrane analogy, 10Mendelson, A., 8, 62Menon, M., 63Meshless method, 3Mesoscale technologies, 6–7Method of series:

for bending, 586–590for torsion, 562–568

Metric tensor of space, 33Meyers, M. A., 5, 62Michell, J. H., 463, 525Micro-cantilevers, 5Microcomputers, 1Microcontinuum field theories, 347Microcontinuum of grade N, 347Microelectronics, 1, 7Microgyration tensors, 349Microinertia density, at cell level, 358Micromechanics, 2Micromorphic theory, 347–352

balance laws of, 350–351constitutive equations of elastic solids,

351–352Microscale technologies, 4–7Microscopic space-averaging, 350, 358Milne-Thompson, L. M., 410, 454Mindlin, R. D., 213, 224, 421, 454, 519, 598, 619Minicomputers, 1Minima, 56Minimum strain energy (elastic energy), theorem

of, 338Mixed boundary value problems, 11Mohr, O., 196, 224Mohr–Coulomb failure criterion, 187Mohr’s circles, 195–198Moire method, 10Molecular biomechanics, 5Molecular dynamics (MD), 4, 205

ab initio, 255classical, 254constitutive relation in, 235–236general form of potential energy, 249–250governing equations, 235quantum, 255stiffness matrix in, 253–255temperature in, 289–294

Berendsen thermostat, 293, 294Gaussian constraints, 293–294Nose–Hoover thermostat, 292–294

INDEX 631

random number generation, 292, 294velocity upgrade, 291–292, 294

Moment:body, 166–167equilibrium, 166twisting, 539–540

Moment of momentum density, at cell level, 358Moment stress, 351Momentum:

balance of angular momentum, law of, 205balance of linear momentum, law of, 205, 289time rate, change of, 215–217

Monatomic lattices, 353Moon, F. C., 2, 62Moore’s Law, 7Morrell, M. L., 595Morris, M., 19, 62Morse, P. M., 106, 159Motion, differential equations of, 204

deformable body/medium, 201–206, 288equilibrium, three-dimensional, 598–600spatial coordinates, 201–206stress:

of deformable body relative to spatialcoordinates, 201–206

for small-displacement theory, 214–224MR (magnetorheological) fluids, 4Mullineux, N., 521, 525Multiply connected region, 388–392, 467,

547–549, 557–558Multiscale problems, modeling, 2Munari, A. C., 360Muscle mechanics, 55, 239–240Muskhelishvili, N. I., 9, 62, 307, 362, 365, 389,

410, 437, 438, 440, 454

Nabla (del), 17Naghdi, P. M., 9, 62Nair, S., 74, 159Nanomechanics, 2Nanoscale, 5Nanotechnology, 1, 4–7National Science Foundation (NSF), 2Navier-Stokes equations, 343–345Nearly incompressible soft biological tissue,

238Necessary conditions:

for compatible small-displacement strain,132–138

for exact differential, 30for extreme values, 58–59for rigid-body displacement, 127–130for single-valued Airy stress function, 388

Necking down, 228

Neou, C. Y., 408, 410, 411, 413, 454Neou method, 408–411Neumann, F. E., 269, 270, 362Neumann boundary-value problem, 11, 533, 550Nonclassical materials, 2Nonelastic material response, 228Nonhomogenous material, 256Nonisotropic material, 256Nonlinear constitutive relationships, 345–347

higher-order relations, 346hypoelastic formulations, 346–347variable stress-strain coefficients, 346

Nonlinear theory of elasticity, 8Nonstationary field, 18Nose–Hoover thermostat, 292–294Novozhilov, V. V., 221, 224, 226n.1, 232, 362Nowacki, W., 270, 362NSF (National Science Foundation), 2Nucleation, 2Null matrix, 53Numerical stress analysis, 3, 8–9Nye, J. F., 242, 257, 362

Oblique coordinates, 154–155, 416–420plane, 416–420straight-line, 154–155

Oblique plane, stress on, 169–171Octahedral planes, 186Octahedral shearing strain, 112Octahedral shearing stress, 186–187Octahedral shearing stress criterion, 187Octahedral strain, 112Oden, J. T., 1, 2, 62Optical fibers, 5, 10Optimization methods, 2Orr, C. M., 526Orson, L. A., 62, 363Orthogonal curvilinear coordinates, 31–32

differential length in, 32–33gradient, 33–34Laplacian, 34–36strain-displacement relations, 146–151

Orthogonality relations, 41–42Osman, M., 63

Pan, Y., 2, 63Papadrakakis, M., 63Papklovich, P. F., 597, 598, 619Park, H. S., 62Parks, M. L., 63Parkus, H., 270, 362Parrinello, M., 255, 359Particle(s):

displacement, 72initial location, 69

632 INDEX

Passive materials, 239Payne, L. E., 10, 61Pearson, C. E., 188, 224, 362, 453Pestel, E., 554n.5, 595Peters, T., 526Phase-space, description, 353–355Phase-space coordinates, 354Photoelasticity, 10Physical space:

atomistic quantities in, 355–357description, 353–355

Pierce, B. O., 564, 566, 595Piezoelectric composites, 4, 5Pindera, J. T., 524, 525Pinkerton, C. A., 526Pinter, W. J., 504, 525Piola–Kirchhoff (PK1 and PK2) stress tensors,

177–178, 237, 239–240Pipes, L., 50, 62Pippard, A. B., 232, 362PK2 and PK2, see Piola–Kirchhoff stress tensorsPlanck, M., 166, 242, 362Plane elasticity, 9Plane polar coordinates, 151Plane strain, 9, 112, 489–490

compatibility, 369, 377–382, 461defined, 366deformation, 112differential equations of equilibrium, 366strain energy density, 368–369, 421

Plane stress, 9, 193–194, 375–376compatibility equation, 378–381, 462generalized, 371–379, 462graphical interpretation, 195–197Mohr’s circles in three dimensions, 197–198orthotropic elastic coefficients for, 248–249

Plane stress tensor, 193Plane theory, 365–518

Airy stress function in, 383–399body forces and temperature effects, 385,

389–392, 467–475, 487–489boundary conditions, 385–388compatibility, 379harmonic functions, 399–400multiply connected regions, 388–392polar coordinates, 463simply connected regions, 383–385

compatibility equation, 369, 377–382, 462couple stress, 420–428, 519–522displacement components, 401–407, 415–416,

447–448, 457–461, 485–489Airy stress function, 401–404polar coordinates, 485–489

generalized plane stress, 371–379, 462

oblique coordinates, 416–420plane strain, 365–371, 377–382, 461plane stress, 371–377, 380–381polar coordinates, 455–518

Plane wedge, 509–510Plaque, atheromatous, 475–478Plastic, perfectly, 229Plasticity, 8, 229, 230Plate, with circular hole, 498–504

stress concentration problem, 498–504stress-couple theory of, 519–522

Poisson equation, 272Poisson’s ratio, 121, 245, 258, 267, 312, 319, 486,

521, 522, 611, 615–617Polar coordinates:

Airy stress function in, 456–457, 463, 498equilibrium equations in, 455–456plane compatibility equation in, 463strain-displacement relations, 457–461stress components in, 456–457stress-strain temperature relations, 461–462

Postma, J. P. M., 359Potential energy, in molecular dynamics,

249–250Potential field, 19Potential function, 19Prager, W., 186, 225, 229, 362Prandtl, L., 66, 159, 530, 534, 537, 538, 543, 554,

556, 572, 595Prandtl membrane analogy, 554–562Prandtl torsion function, 534–538Pressures, 163Principal axes, 259Principal planes of stress, 179Principal strains, 96–100Principal values of the deformation, 101Processors, in smart structures/materials, 5Proportional limit, 227Pseudoelasticity, 237–239

Quadratic forms:characteristic equation of, 50characteristic roots (latent roots; eigenvalues),

50determinant, 49eigenvectors, 50–52homogeneous, 49–52

Quantum MD, 255

Rachev, A., 239, 362Ragsdell, K. M., 60Random number generation, 292, 294Reciprocal matrix, 55, 56Rectangular arrays, 38–40

INDEX 633

Rectangular Cartesian coordinates, 32strain components in, 83–84strain-displacement relations, 366transformation of tensors under, 40–46

Reed, M. A., 5, 62, 65, 159Reissner, E., 342, 343, 362Reissner’s theorem, 342–343Relative emissivity, 272Remodeling, biomechanics of, 5Repeated Greek index, 36–38, 43Rigid body:

definition of, 65displacement, 66–67, 127–130

Riley, W. F., 10, 61Ritchie, R. O., 5, 62Rogers, C. A., 4, 5, 10, 62Rogers, J. A., 61Rogers, R. C., 4, 5, 10, 62Rosenfeld, H. R., 277, 362Rotation, mean, 78

small angles, 115–116vectors, 115volume element, 78, 113–117

Row matrix, 52Ruoff, R. S., 60Ruud, C. O., 10, 62Ryzhik, I. M., 564, 566, 595

Sadd, M. H., 296, 363, 476, 525Saigal, S., 3, 60Saint-Venant, 158, 534n.3Saint-Venant semi-inverse method, 569Saint-Venant’s principle, 307–310, 317–322, 529,

530, 581Saint-Venant’s torsion theory, 529–534Saint-Venant warping function, 538, 544Saleeb, A. F., 184, 224, 346, 359Savin, G. N., 502, 525Scalars, 43Scalar field, 16, 21Scalar matrix, 54Scalar methods, 333Scalar point functions, 16–18Scalar product of vectors, 12–13

applications, 28–29triple product, 14, 16

Scalzi, J. B., 60Schatz, G. C., 60Schijve, J., 421, 454Schild, A., 40, 45n.6, 63, 95, 111, 159Schmidt, R. J., 67, 68n.1, 142, 159, 187, 214,

215n.8, 224, 333, 359Schreiber, E., 9, 62, 246, 363Schrodinger equation, 255

Seager, M., 60Seki, Y., 62Self-diagnosis materials, 2Self-healing materials, 2Semenkov, O. I., 61Semi-inverse method, 317–322Sen, B., 392, 454Sensors:

bio-inspired, 4in smart structures/materials, 5

Set (nonelastic material response), 228Shaft, circular cross section, 327–332Shape memory alloys, 4, 5Sharma, B., 392, 454Shear center, 581–584Shearing components, 164Shearing strain, 90–92Shearing stress, see Stress, shearingShear modulus, 245, 267Shepherd, W. M., 593, 596Sidebottom, O. M., 199, 225, 229, 359, 363Simple connectivity, 30n.5Simulation, atomic-scale-based, 4Simulation-based engineering science, 2Siriwardane, H. J., 346, 360Skew-symmetric square arrays, 39Slack, J. D., 526Smart structures/materials, 1–5Smith, C. W., 421, 453, 522, 525Smith, I. M., 106, 159Smith, J. O., 199, 225, 229, 363Smith, J. W., 9, 60, 187, 224, 360, 525Smith, W., 293, 363Sneddon, I., 410, 454Snell, C., 595Society for Experimental Mechanics (SEM), 10n.2Soft biological tissue:

constitutive equation for, 237–238incompressible and nearly incompressible, 238

Soga, N., 62, 363Sokolnikoff, I. S., 365, 454Sokolovski, V. V., 186, 225Solids, 163

elastic and nonelastic response of, 226–230micromorphic, 351–352semi-infinite, 617

Solid–fluid interactions, 55Solid mechanics research, priorities in, 2–3Space-averaged temperature, 290Spain, B., 62Spatial coordinates, 21, 67–71, see Eulerian

(spatial) coordinatesSpatial equation of continuity, 145Spatial form (continuity equation), 24n.4

634 INDEX

Special states of stress:hydrostatic, 317–318irrotational, 126–127plane, 193–201pure shear, 267–268simple tension, 266–267

Specific heat, 271–272Spherical coordinate system, 151Spherically symmetrical stress distribution,

294–299Spitzig, W. A., 60Split cylinder test, 522–524Square arrays, 39–40Square matrix, 43Srivastava, D., 4, 63State of plane stress with respect to the (X,Y)

plane, 193Stationary field, 18Stationary potential energy, principle of, 337Stationary value of integrals, 56–60Steady field, 18Stern, M., 467, 525Sternberg, E., 9, 63, 213, 225, 307, 314, 363, 392,

453, 454, 510, 526, 618, 619Stevenson, A. C., 410, 454Stiffness matrix, in molecular dynamics, 253–255Stippes, M., 9, 63, 618Stokes’s theorem, 29–30Strain:

components, 78, 457–461cylindrical coordinates, 150–153orthogonal curvilinear coordinates, 146–151,

457plane polar coordinates, 151, 457–461rectangular Cartesian coordinates, 83–84spherical coordinates, 151

definition:cubical, 257engineering, 87large-deformation, 85logarithmic, 87–88natural or true, 88

deviator, 110–112elastic, 228Eulerian (spatial) components, 82in index notation, 82invariants, 108–112Lagrangian (material) components of, 82of a line element, 86–89mean, 110–112notations of, 7octahedral, 112octahedral shearing, 112plane, 112, 365–371

principal, 96–104principal axes (directions), 96, 101, 104–107,

259principal values, 96, 100–107reciprocal ellipsoid, 96–100

invariants of, 97–98, 100principal axes (directions), 98principal directions (axes), 98–99relative elongations, 98–99

set nonelastic, 228shearing, 90–94small, theory of, 121–132special types, 110

dilatation, 110, 120–121, 125homogeneous, 118–121irrotational, 126–127pure, 120–121, 267–268rigid displacement, 127–130simple shear, 126, 266

transformation of components, 95–96volumetric (cubical), 109–110, 257, 366

Strain-displacement relations:cylindrical coordinates, 150–153Euclidean metric tensor, 155–157general coordinates, 155–159geometric preliminaries, 146–148oblique straight-line plane coordinates, 154–155orthogonal curvilinear coordinates, 82,

146–151, 367, 457–461plane polar coordinates, 151, 457–461rectangular Cartesian coordinates, 366spherical coordinates, 151

Strain energy density:for anisotropic linearly elastic material, 242,

368–369for certain symmetry conditions, 242–246, 257,

259–261for composites, 259, 261for elastically isotropic medium, 256–266function, 234, 237in index notation, 277–279plane strain, 368–369, 421relation to stress components, 232–240for soft biological tissues, 237in terms of principal strains (invariants), 257for thermoelasticity, 278

Strain gage methods, 10Strain tensors, 83, 95–96, 110, 158–159

Almansi, 83, 159Cauchy, 86, 351components of, 83, 95–96generalized Lagrangian, 351Green–Saint-Venant, 83, 159in terms of rotation vector components, 123–124

INDEX 635

Stream lines, 18Stress:

array, 166–167boundary conditions, 169–171, 306characteristic values, 50, 179components, 430–432

boundary conditions at point 0, 169normal to a plane, 169–171notation, 164–165on oblique plane, 169–171relation to strain energy density, 232–240,

278–279symmetry of, 167tangent to a plane, 169–171thermal-stress problem in terms of, 287–288torsion, 540–541transformation of, 175–179

concentration, 498–504definition, 161–163differential equations of equilibrium:

in curvilinear spatial coordinates, 207–211including couple stress and body couple,

211–214differential equations of motion:

of deformable body relative to spatialcoordinates, 201–206

for small-displacement theory, 214–224direction, 179eigenvalues, 50, 179eigenvectors, 50, 181extreme values, 179–182, 198–200functions, three-dimensional, 317–327index notation, 165–166invariants, 180, 182–183, 257Mohr’s circles, 195–198and muscle mechanics, 239–240normal:

extreme values, 198–200on oblique plane, 169–171

notation, 7, 164–166on oblique plane, 169–171plane, 193–201, 491

components in terms of Airy stress function,384, 456, 457

extreme values of, 198–200Mohr’s circle of, 195–198

at a point, 167–169principal axes, 180–181principal directions, 181principal planes, 179principal values, 179principle stresses, 179–180shearing, 169, 181–183

component, 169, 176

component in any direction, 543–544extreme values, 181–183, 198–200on oblique plane, 169–171octahedral, 186–187

sign convention, 162special states of, 266–269

hydrostatic, 317–318irrotational, 126–127plane, 193–201pure shear, 267–268simple tension, 266–267

summation of moments, 166–167tensor character of, 175–178in terms of Galerkin vector, 603–604theory of, 161thermal, 269–295transformation of components, 175–179vector, 177virtual, 339–342yield, 227

Stress analysis:experimental, 9–10finite element method in, 8–9numerical, 3, 8–9

Stress couples, 161, 167, 211–212Stress notation, 164–166Stress-strain relations, 241–255, 536

anisotropic, 261, 368–369for bars, 528beryllium, 279–281composites, 259–261generalized Hooke’s law, 241–255higher-order, 346including temperature effects, 276–285in index notation, 246, 257for isotropic media, 256–266, 366nonlinear, 346in oblique coordinates, 420relative to axes inclined to crystal axis, 281–283for soft biological tissues, 237special states, 266–269

Stress-strain-temperature relations, 276–285for beryllium, 279–280polar coordinates, 461–462relative to axes inclined to crystal axis, 281–283

Stress tensors, 177, 184–185, 210, 212character of, 175deviator, 185–193invariants of, 180mean, 185–193notation, 165, 166Piola–Kirchhoff, 177–178, 237, 239–240plane, 193

Stretchable electronics/sensors, 5

636 INDEX

Stretch ratio, 240Substitution of indexes, rule of, 47Successive elastic solutions, 8Sufficient conditions:

for compatible small-displacement strain,132–138

for exact differential, 30–31for rigid-body displacement, 127–130

Suhubi, E. S., 348, 350, 360Summation convention, 36–40Summation notation, 43–44Summing index, 37Supercomputers, 1, 6Superposition method, 501, 502Surfaces, level, 17Surface integral, 29Swanson, W. D., 595Symmetric square arrays, 39Synge, J. L., 40, 45n.6, 63, 95, 111, 159, 231n.4,

363Szabo, B. A., 595

Tangential components, 164Taylor, R. L., 1, 9, 63Taylor series, 115–116Temperature distribution:

diffusivity, 271space-averaged, 290specific heat, 271–272stationary, 271steady-state, 271time-averaged, 290

Templeton, J. A., 63Tensors:

alternating, 49antisymmetric parts of, 47, 77, 78Cauchy strain, 73–74conjugate, 46contravariant, 210–211covariant, 210–211deformation gradient, 73Euclidean metric, 155–157first-order, 44invariants, 108isotropic, 48Kronecker delta (substitution tensor), 47–48mean strain, 110–112metric, 33, 155–157, 210–211microgyration, 349nth-order, 45, 46second-order, 45–48, 77–78, 177special third-order (alternating), 49strain, 95–96, 158–159stress, 177, 210

substitution, 48–49symmetric, 177symmetric parts, 46–47, 77, 78third-order, 45, 48–49transformation under rotation of axes, 40–46zero order, 43–44

Tensor algebra, 36–52homogeneous quadratic forms, 49–52index notation, 36–40notation, 47–49symmetric and antisymmetric tensor parts,

46–47transformation under rotation of axes, 40–46

Tersoff, J., 251, 363Tersoff potential, 251–253Tham, L. G., 3, 60Thermal conductivity, 271Thermal expansion coefficient, 272–273, 277Thermal stress:

in beams, 274–276displacement potential, 303Duhamel-Neumann theory, 269–270elementary approach, 272–276equivalent displacement problem, 269,

285–287physical interpretation, 287–288plane theory, 389–392, 489–494spherically symmetrical, 294–299

Thermal treatment, 55Thermodynamics, first law, 234Thermoelasticity:

axially symmetric case, 302–304equations:

for beryllium, 279–281boundary conditions, 287compatibility (stress), 299–305isotropic media, 269–270, 279–281physical interpretation of thermal-stress

problem, 287–288temperature in molecular dynamics,

289–294in terms of displacement, 285–294thermomechanical coupling, 288–289

plane theory, 389–392, 489–494Thermomechanical coupling, 288–289Thoft-Christensen, P., 8, 63Three-dimensional elasticity, 9Three-dimensional stress functions, 617–618Tiersten, H. F., 213, 224Tietjens, O. G., 66, 159Time-averaged temperature, 290Time evolution law of physical quantities, 355Timoshenko, S., 307n.9, 403, 454, 463n.1, 522,

524, 526, 610, 619

INDEX 637

Timoshenko, S. P., 363, 561, 566, 573n.7, 581, 596Timp, G., 5, 63, 65, 160Timpe, A., 463n.1, 526Ting, T. C. T., 510, 526Tismenetsky, M., 55, 62To, A. C., 353, 362, 363Todorov, I. T., 363Torsion:

of prismatic bars, 529–568axis of twist, 327, 536, 549–550boundary conditions, 528displacement components, 536–538,

560–561elliptic cross section, 538–542, 584moment angle of twist relation, 539–542narrow rectangular cross section, 560–561Prandtl function of, 535Prandtl membrane analogy, 554–562Prandtl theory, 534–538Prandtl torsion function, 534–538rectangular section, 562–568Saint-Venant’s solution, 529–534shear-stress components, 543–544stress components, 540–541with tubular cavities, 547–549warping, circular cross section, 544

of shaft with constant circular cross section,327–332

Torsional rigidity, 540Total energy density, at cell level, 358Toupin, R. A., 421, 454Trahair, N. S., 592, 595Transform methods, 440–445Translation, of a mechanical system, 66, 67Transpose of matrix, 54Transposition, 54Tresca–Saint-Venant–Coulomb–Guest criterion,

187Tribology, 2Trimmer, W., 63Truesdell, C., 347, 363Tsompanakis, Y., 1, 63Turner, J. P., 360Twinned gradient, 611–614Twist:

angle of, 539–540axis of, 327

generally, 327, 536transfer of, 549–550

center of, 536twisting moment, 539–540

Udd, E., 4, 63Uenishi, K., 360

Uniqueness theorem of elasticity (equilibrium),311–314

Unit matrix, 54Unit vectors, 16–17University of Illinois, Theoretical and Applied

Mechanics Dept., 148n.17Unsteady field, 18

Van Gunsteren, W., 309, 359Van Tassel, J., 526Variables, 68–71

complex, 399–400, 428–453material, 70spatial, 70

Variable coefficient approach, 346Vectors:

acceleration, 21, 22, 142, 203, 210, 218body force, 201–202Cartesian, 12components of, 12differentiation, 19–21displacement, 71, 76–78gradient (grad), 17–18, 33–34, 76–78inertial, 202–203interatomic force, 235magnitude, 12notation, 12operator, 18point function, 16–18projections, 12relative position, 249rotation, 78scalar product, 12–14, 16, 28–29stress, 161–171tractions, 203triple product:

scalar, 14, 16vector, 16

unit, 16–17velocity, 141–144vorticity, 30

Vector addition:associative law, 12commutative law, 12in index notation, 43–44

Vector algebra, 12–16Vector fields, 18–19, 146. See also

Vectorscurl of, 22differentiation of, 21–22divergence of, 23irrotational, 145–146potential function of, 19

Vector lines, 18–19

638 INDEX

Vector product of vectors, 14, 15properties, 14vector triple product, 16

Velocity, 21, 345phase-space vs. physical-space, 357vectors, 141–144

Velocity fields, 18, 22–24Velocity upgrades, 291–292, 294Vetter, F. J., 238, 363Virtual stress, 339–342Virtual testing, 2Virtual work:

application to deformable medium, 343–345for particles, 334–335principle of, 333–339

Viscoelasticity, 229, 230Voigt, W., 213n.6Volumetric strain, 109–110Von Mises, R., 307, 308, 363Von Mises–Hencky criterion, 187Vorticity vector, 30

Wagner, G. J., 4, 63Walbert, J. N., 159Walkup, R., 60Waller, B. F., 476, 526Wall stresses, 298Wang, X., 361Wark, K., 232, 363Warping function, 530, 544Weber, C., 543, 545, 560, 596, 618, 619Weiner, J. H., 270, 359

Weitsman, Y., 421, 454, 519Wen, Y.-K., 8, 63Westergaard, H. M., 598, 611–613, 619Whittaker, E. T., 67, 160Wong, Eugene, 7Work:

of body forces, 232–233, 236, 334of external forces, 232–233, 334–335of force acting on particle, 334of internal forces, 334–335of surface force, 232–233, 334virtual, 333–339, 343–345

Xiao, S. P., 4, 60Xiong, L., 359, 363

Yang, H. Y., 3, 63Yang, L. T., 2, 63Yao, J. T. P., 8, 63Yield point, 227Yield stress, 187Yip, S., 359Young’s modulus, 245, 258, 267, 312, 319, 486

Zeng, X., 362Zerna, W., 46, 61, 210, 211n.4, 224Zero state (zero configuration), 229Zhang, S., 62Zhou, M., 353, 363Zienkiewicz, O. C., 1, 9, 60, 63Zimmerman, J. A., 353, 363Zimmerman, K. L., 159