Partial differential equations in mechanics

Cover......Page 1
Title page......Page 3
Copyright page......Page 4
Dedication......Page 5
Preface......Page 7
Acknowledgements......Page 13
Contents......Page 15
8. The biharmonic equation......Page 19
8.1 The concept of a continuum......Page 21
8.2 Displacements and strains in continua......Page 23
8.2.1 Physical interpretations of the strain matrix......Page 30
8.2.2 Physical interpretation of the rotation matrix......Page 34
8.2.3 Indicial notation representations of strain and rotation......Page 36
8.2.4 Transformation of the strain matrix......Page 37
8.2.5 Principal strains and strain invariants......Page 42
8.2.6 Compatibility of strains......Page 44
8.3 Stresses in a continuum......Page 54
8.3.1 The stress dyadic and the stress matrix......Page 55
8.3.2 Tractions on an arbitrary plane......Page 57
8.3.3 Equations of equilibrium......Page 59
8.3.4 Symmetry of the stress matrix......Page 61
8.3.5 Sign convention for stresses......Page 63
8.3.6 Transformation of the stress matrix......Page 65
8.3.7 Principal stresses and stress invariants......Page 67
8.4.1 Generalized Hooke\'s Law......Page 75
8.4.2 The strain energy density......Page 76
8.4.3 Symmetry of the elasticity matrix......Page 78
8.4.4 Isotropic elastic material......Page 80
8.4.5 Thermodynamic constraints on the elastic constants......Page 84
8.4.6 Boundary conditions......Page 86
8.4.7 Time derivatives......Page 89
8.5 Uniqueness theorem in the classical theory of elasticity......Page 90
8.6 Plane problems in classical elasticity......Page 99
8.7 The Airy stress function......Page 110
8.8 Methods of solution of the biharmonic equation......Page 122
8.8.1 Series solution in Cartesian coordinates......Page 123
8.8.2 Variables separable solution - rectangular Cartesian coordinates......Page 136
8.8.3 Integral transform solution of the biharmonic equation governing plane problems......Page 148
8.8.4 Line load problems for an infinite plane......Page 156
8.8.5 Application of complex variable methods......Page 163
8.9 Polar coordinate formulation of the plane problem in elasticity......Page 169
8.9.1 Definition of stresses in plane polar coordinates......Page 170
8.9.2 Definition of strains in plane polar coordinates......Page 172
8.9.3 Elastic stress-strain relations in polar coordinates......Page 176
8.9.4 Transformation from Cartesian equations......Page 177
8.9.5 The Airy stress function in plane polar coordinates......Page 179
8.9.6 Boundary conditions in plane polar coordinates......Page 180
8.9.7 Development of the Airy stress function in polar coordinates......Page 182
8.9.8 An application of complex variable techniques......Page 194
8.10 Biharmonic function formulation of three-dimensional problems in elasticity......Page 199
8.10.1 The strain potential......Page 200
8.10.2 The Galerkin vector......Page 202
8.10.3 General properties of biharmonic functions......Page 204
8.10.4 Love\'s strain function......Page 211
8.10.5 Axisymmetric problem formulation......Page 215
8.10.6 Polynomial solutions of the biharmonic equation; the interior solution......Page 217
8.10.7 Love\'s strain function approach: spherical polar coordinate formulation for unbounded domains......Page 227
8.10.8 Kelvin\'s problem......Page 231
8.10.9 A centre of dilatation......Page 240
8.10.10 Boussinesq\'s problem......Page 243
8.10.11 The spherical cavity problem......Page 251
8.10.12 Application of integral transforms......Page 256
8.10.13 Mixed boundary value problems......Page 276
8.11 Flexure of thin elastic plates......Page 285
8.11.1 Deformation of the plate region......Page 287
8.11.2 Flexural stresses and stress resultants......Page 297
8.11.3 Plate stress-strain relations......Page 300
8.11.4 Equation of equilibrium for the plate......Page 304
8.11.5 Strain energy of a plate......Page 307
8.11.6 The principle of virtual work......Page 309
8.11.7 Boundary conditions for plate problems......Page 311
8.11.8 The classical theory of thin plates......Page 315
8.11.9 Flexure of thin circular plates......Page 323
8.11.10 Complex variable method for circular plates......Page 343
8.11.11 Green\'s function for a thin plate......Page 348
8.11.12 Flexure of rectangular plates......Page 350
8.11.13 Certain general solutions of the biharmonic equation......Page 357
8.11.14 Navier and Levy solutions for rectangular plates......Page 363
8.11.15 Application of integral transform techniques......Page 384
8.11.16 Complex variable methods for the solution of plate problems......Page 392
8.11.17 Uniqueness of solution governing deflections of a plate......Page 404
8.11.18 Uniqueness of solution for plates with clamped boundaries......Page 408
8.12 Slow viscous flow......Page 411
8.12.1 Kinematics of fluid flow......Page 412
8.12.3 Constitutive equations for a viscous fluid......Page 415
8.12.4 Thermodynamic constraints on the viscosity coefficients......Page 417
8.12.5 Navier-Stokes equation......Page 421
8.12.6 Biharmonic formulations of problems in slow viscous flow......Page 422
8.12.7 Planar problems in slow viscous flow......Page 425
8.12.8 Axisymmetric problems in slow viscous flow......Page 427
8.12.9 Boundary conditions......Page 429
8.12.10 Stokes* paradox and related problems......Page 437
8.12.11 Slow viscous flow past a sphere......Page 442
8.12.12 Application of integral transform techniques......Page 448
8.12.13 Diffusive motions in viscous fluids......Page 458
8.13 The compatibility conditions......Page 472
8.14 A proof of Stokes paradox......Page 477
8.15 A uniqueness theorem for viscous flows......Page 480
8.16 PROBLEM SET 8......Page 483
9. Poisson\'s equation......Page 521
9.1 Flow through porous media with internal sources......Page 522
9.2 Heat conduction in the presence of heat sources......Page 525
9.4 Boundary conditions......Page 526
9.4.1 Boundary conditions for flow in porous media......Page 527
9.4.2 Boundary conditions for heat conduction......Page 528
9.4.3 Boundary conditions for loaded membranes......Page 529
9.5 Generalized results......Page 531
9.6 Green\'s function for Poisson\'s equation......Page 544
9.6.1 Integral transform techniques......Page 549
9.6.2 Series solution techniques......Page 553
9.6.3 Method of eigenfunctions......Page 558
9.6.4 Symmetry of the Green\'s function......Page 569
9.6.5 The method of images......Page 571
9.7 A monotonicity result for Poisson\'s equation......Page 575
9.8 Viscous flow in conduits......Page 577
9.9 Torsion of prismatic elastic solids......Page 583
9.9.1 General formulation of the torsion problem......Page 586
9.9.2 Torsion of prismatic elements with multiply connected cross-sections......Page 595
9.9.3 Solutions for torsion problems derived from equations of boundaries......Page 599
9.9.4 Variables separable solutions for the torsion problem......Page 605
9.9.5 A complex variable formulation of the torsion problem......Page 618
9.10 An alternative derivation of the elastic torsion problem......Page 626
9.11 The gravitational potential......Page 632
9.11.1 Spatial distribution of matter......Page 638
9.12 PROBLEM SET 9......Page 652
Bibliography......Page 667
Index......Page 697