THEORY OF ELASTICITY
Part I General Considerations 1 Introduction 3 11 Notation for Stress and Displacement 4 111 Stress 4 112 Index and vector notation and the summation convention 6 113 Vector operators in index notation 8 114 Vectors, tensors and transformation rules 9 115 Principal stresses and von Mises stress 13 116 Displacement 14 12 Strains and their Relation to Displacements 15 121 Tensile strain 15 122 Rotation and shear strain 17 123 Transformation of coördinates 18 124 Definition of shear strain 20 13 Stress-strain Relations 21 131 Isotropic constitutive law 22 132 Lamé’s constants 23 133 Dilatation and bulk modulus 24 134 Deviatoric stress 25 Problems 25 2 Equilibrium and Compatibility 27 21 Equilibrium Equations 27 22 Compatibility Equations 28 221 The significance of the compatibility equations 30 23 Equilibrium Equations in terms of Displacements 34 Problems 35 Part II Two-dimensional Problems 3 Plane Strain and Plane Stress 41 31 Plane Strain 41 311 The corrective solution 43 312 Saint-Venant’s principle 44 32 Plane Stress 45 321 Generalized plane stress 46 322 Relationship between plane stress and plane strain 46 Problems 47 4 Stress Function Formulation 51 41 Choice of a Suitable Form 51 42 The Airy Stress Function 52 421 Transformation of coördinates 53 422 Non-zero body forces 54 43 The Governing Equation 54 431 The compatibility condition 55 432 Method of solution 55 433 Reduced dependence on elastic constants 56 Problems 56 5 Problems in Rectangular Coördinates 61 51 Biharmonic Polynomial Functions 61 511 Second and third degree polynomials 63 52 Rectangular Beam Problems 65 521 Bending of a beam by an end load 65 522 Higher order polynomials — a general strategy 67 523 Manual solutions —symmetry considerations 72 53 Fourier Series and Transform Solutions 75 531 Choice of form 75 532 Fourier transforms 79 Problems 80 6 End Effects 83 61 Decaying Solutions 83 62 The Corrective Solution 84 621 Separated-variable solutions 85 622 The eigenvalue problem 86 63 Other Saint-Venant Problems 90 64 Mathieu’s Solution 91 Problems 95 7 Body Forces 97 71 Stress Function Formulation 97 711 Conservative vector fields 98 712 The compatibility condition 99 72 Particular Cases 99 721 Gravitational loading 100 722 Inertia forces 100 723 Quasi-static problems 101 724 Rigid-body kinematics 101 73 Solution for the Stress Function 103 731 The rotating rectangular bar 103 732 Solution of the governing equation 105 74 Rotational Acceleration 106 741 The circular disk 107 742 The rectangular bar 109 743 Weak boundary conditions and the equation of motion 112 Problems 112 8 Problems in Polar Coördinates 117 81 Expressions for Stress Components 117 82 Strain Components 119 83 Fourier Series Expansion 120 831 Satisfaction of boundary conditions 121 832 Circular hole in a shear field 121 833 Degenerate cases 124 84 The Michell Solution 126 841 Hole in a tensile field 128 Problems 129 9 Calculation of Displacements 133 91 The Cantilever with an End Load 133 911 Rigid-body displacements and end conditions 135 912 Deflection of the free end 137 92 The Circular Hole 137 93 Displacements for the Michell Solution 140 931 Equilibrium considerations 141 932 The cylindrical pressure vessel 142 Problems 143 10 Curved Beam Problems 147 101 Loading at the Ends 147 1011 Pure bending 147 1012 Force transmission 150 102 Eigenvalues and Eigenfunctions 152 103 The Inhomogeneous Problem 152 1031 Beam with sinusoidal loading 153 1032 The near-singular problem 156 104 Some General Considerations 159 1041 Conclusions 159 Problems 160 11 Wedge Problems 163 111 Power-law Tractions 163 1111 Uniform tractions 164 1112 The rectangular body revisited 166 1113 More general uniform loading 167 1114 Eigenvalues for the wedge angle 168 112 Williams’ Asymptotic Method 169 1121 Acceptable singularities 169 1122 Eigenfunction expansion 171 1123 Nature of the eigenvalues 173 1124 The singular stress fields 176 1125 Other geometries 178 113 General Loading of the Faces 180 Problems 181 12 Plane Contact Problems 185 121 Self-Similarity 185 122 The Flamant Solution 186 123 The Half-Plane 187 1231 The normal force Fy 188 1232 The tangential force Fx 189 1233 Summary 189 124 Distributed Normal Tractions 190 125 Frictionless Contact Problems 191 1251 Method of solution 192 1252 The flat punch 194 1253 The cylindrical punch (Hertz problem) 195 126 Problems with Two Deformable Bodies 198 127 Uncoupled Problems 201 1271 Contact of cylinders 201 128 Combined Normal and Tangential Loading 202 1281 Cattaneo and Mindlin’s problem 203 1282 Steady rolling: Carter’s solution 206 Problems 209 13 Forces, Dislocations and Cracks 215 131 The Kelvin Solution 215 1311 Body force problems 217 132 Dislocations 218 1321 Dislocations in Materials Science 220 1322 Similarities and differences 221 1323 Dislocations as Green’s functions 222 1324 Stress concentrations 222 133 Crack Problems 223 1331 Linear Elastic Fracture Mechanics 224 1332 Plane crack in a tensile field 225 1333 Energy release rate 228 134 Disclinations 230 1341 Disclinations in a crystal structure 231 135 Method of Images 232 Problems 235 14 Thermoelasticity 239 141 The Governing Equation 239 142 Heat Conduction 241 143 Steady-state Problems 242 1431 Dundurs’ Theorem 243 Problems 245 15 Antiplane Shear 247 151 Transformation of Coördinates 248 152 Boundary Conditions 248 153 The Rectangular Bar 249 154 The Concentrated Line Force 250 155 The Screw Dislocation 252 Problems 252 16 Moderately Thick Plates 257 161 Boundary Conditions 258 162 Edge Effects 260 163 Body Force Problems 261 164 Normal Loading of the Faces 263 1641 Steady-state thermoelasticity 264 Problems 265 Part III End Loading of the Prismatic Bar 17 Torsion of a Prismatic Bar 269 171 Prandtl’s Stress Function 270 1711 Solution of the governing equation 271 1712 The warping function 273 172 TheMembrane Analogy 273 173 Thin-walled Open Sections 274 174 The Rectangular Bar 276 175 Multiply-connected (Closed) Sections 277 1751 Thin-walled closed sections 279 Problems 281 18 Shear of a Prismatic Bar 289 181 The Semi-inverseMethod 289 182 Stress Function Formulation 290 183 The Boundary Condition 291 1831 Integrability 291 1832 Relation to the torsion problem 292 184 Methods of Solution 293 1841 The circular bar 293 1842 The rectangular bar 294 Problems 297 Part IV Complex-Variable Formulation 19 Prelinary Mathematical Results 301 191 Holomorphic Functions 302 192 Harmonic Functions 303 193 Biharmonic Functions 303 194 Expressing Real Harmonic and Biharmonic Functions in Complex Form 305 1941 Biharmonic functions 306 195 Line Integrals 306 1951 The residue theorem 307 1952 The Cauchy integral theorem 310 196 Solution of Harmonic Boundary-value Problems 312 1961 Direct method for the interior problem for a circle 312 1962 Direct method for the exterior problem for a circle 314 1963 The half-plane 315 197 Conformal Mapping 317 Problems 319 20 Application to Elasticity Problems 323 201 Representation of Vectors 323 2011 Transformation of coördinates 324 202 The Antiplane Problem 325 2021 Solution of antiplane boundary-value problems 326 203 In-plane Deformations 328 2031 Expressions for stresses 330 2032 Rigid-body displacement 331 204 Relation between the Airy Stress Function and the Complex Potentials 332 205 Boundary Tractions 334 2051 Equilibrium considerations 336 206 Boundary-value Problems 337 2061 Solution of the interior problem for the circle 338 2062 Solution of the exterior problem for the circle 340 207 Conformal Mapping for In-plane Problems 343 2071 The elliptical hole 345 Problems 348 Part V Three-Dimensional Problems 21 Displacement Function Solutions 353 211 The Strain Potential 353 212 The Galerkin Vector 354 213 The Papkovich-Neuber Solution 356 2131 Change of coördinate system 357 214 Completeness and Uniqueness 358 2141 Methods of partial integration 359 215 Body Forces 362 2151 Conservative body force fields 362 2152 Non-conservative body force fields 363 Problems 363 22 The Boussinesq Potentials 365 221 Solution A: The Strain Potential 366 222 Solution B 366 223 Solution E: Rotational Deformation 367 224 Other Coördinate Systems 368 2241 Cylindrical polar coördinates 368 2242 Spherical polar coördinates 369 225 Solutions Obtained by Superposition 371 2251 Solution F: Frictionless isothermal contact problems 371 2252 Solution G: The surface free of normal traction 374 2253 A plane strain solution 374 226 A Three-dimensional Complex-Variable Solution 375 Problems 376 23 Thermoelastic Displacement Potentials 379 231 The Method of Strain Suppression 381 232 Boundary-value Problems 382 2321 Spherically-symmetric Stresses 382 2322 More general geometries 383 233 Plane Problems 384 2331 Axisymmetric problems for the cylinder 384 2332 Steady-state plane problems 385 2333 Heat flow perturbed by a circular hole 387 2334 Plane stress 390 234 Steady-state Temperature: Solution T 390 2341 Thermoelastic plane stress 393 Problems 394 24 Singular Solutions 397 241 The Source Solution 397 2411 The centre of dilatation 398 2412 The Kelvin solution 399 242 Dimensional Considerations 401 2421 The Boussinesq solution 401 243 Other Singular Solutions 404 244 Image Methods 406 2441 The traction-free half-space 407 Problems 409 25 Spherical Harmonics 411 251 Fourier Series Solution 412 252 Reduction to Legendre’s Equation 412 253 Axisymmetric Potentials and Legendre Polynomials 413 2531 Singular spherical harmonics 414 2532 Special cases 415 254 Non-axisymmetric Harmonics 415 255 Cartesian and Cylindrical Polar Coördinates 416 256 Harmonic Potentials with Logarithmic Terms 417 2561 Logarithmic functions for cylinder problems 419 257 Non-axisymmetric Cylindrical Potentials 420 258 Spherical Harmonics in Complex-variable Notation 422 2581 Bounded cylindrical harmonics 423 2582 Singular cylindrical harmonics 425 Problems 425 26 Cylinders and Circular Plates 427 261 Axisymmetric Problems for Cylinders 427 2611 The solid cylinder 428 2612 The hollow cylinder 432 262 Axisymmetric Circular Plates 434 2621 Uniformly loaded plate on a simple support 434 263 Non-axisymmetric Problems 437 2631 Cylindrical cantilever with an end load 437 Problems 439 27 Problems in Spherical Coördinates 443 271 Solid and Hollow Spheres 443 2711 The solid sphere in torsion 444 2712 Spherical hole in a tensile field 445 272 Conical Bars 447 2721 Conical bar transmitting an axial force 447 2722 Inhomogeneous problems 450 2723 Non-axisymmetric problems 451 Problems 453 28 Eigenstrains and Inclusions 457 281 Governing Equations 458 282 Galerkin Vector Formulation 459 2821 Non-differentiable eigenstrains 460 2822 The stress field 461 283 Uniform Eigenstrains in a Spherical Inclusion 462 2831 Stresses outside the inclusion 463 284 Green’s Function Solutions 463 2841 Nuclei of strain 464 285 The Ellipsoidal Inclusion 465 2851 The stress field 468 2852 Anisotropic material 468 286 The Ellipsoidal Inhomogeneity 469 2861 Equal Poisson’s ratios 470 2862 The ellipsoidal hole 470 287 Energetic Considerations 471 2871 Evaluating the integral 472 2872 Strain energy in the inclusion 472 Problems 473 29 Axisymmetric Torsion 475 291 The Transmitted Torque 476 292 The Governing Equation 476 293 Solution of the Governing Equation 477 294 The Displacement Field 479 295 Cylindrical and Conical Bars