Intermediate Solid Mechanics

Contents......Page 5
Preface......Page 12
Part I Fundamentals of Solid Mechanics......Page 16
1.1 Traction Vector......Page 18
1.2 Cauchy Relation for Traction Vectors......Page 21
1.3 Normal and Shear Stresses over an Inclined Plane......Page 22
1.4 Tensorial Nature of Stress......Page 24
1.5 Principal Stresses: 2D State of Stress......Page 25
1.6 Maximum Shear Stress: 2D Case......Page 27
1.7 Mohr’s Circle for 2D State of Stress......Page 29
1.8 Principal Stresses: 3D State of Stress......Page 31
1.9 Maximum Shear Stress: 3D Case......Page 33
1.10 Mohr’s Circles for 3D State of Stress......Page 35
1.11 Deviatoric and Spherical Parts of Stress......Page 37
1.12 Octahedral Shear Stress......Page 38
1.13 Differential Equations of Equilibrium......Page 39
Problems......Page 42
2.1 Longitudinal and Shear Strains......Page 46
2.2 Tensorial Nature of Strain......Page 48
2.3 Dilatation and Shear Strain for Arbitrary Directions......Page 49
2.5 Maximum Shear Strain......Page 51
2.6 Areal and Volumetric Strains......Page 52
2.7 Deviatoric and Spherical Parts of Strain......Page 53
2.8 Strain–Displacement Relations......Page 54
2.9 Saint-Venant Compatibility Conditions......Page 57
2.10 Rotation Tensor......Page 59
2.11 Determination of Displacements from the Strain Field......Page 60
Problems......Page 62
3.1 Linear Elasticity and Hooke’s Law......Page 66
3.2 Generalized Hooke’s Law......Page 68
3.3 Shear Stress–Strain Relations......Page 70
3.4 Pressure–Volume Relation......Page 72
3.5 Inverted Form of the Generalized Hooke’s Law......Page 76
3.6 Deviatoric Stress – Deviatoric Strain Relations......Page 80
3.7 Beltrami–Michell Compatibility Equations......Page 82
3.8 Hooke’s Law with Temperature Effects: Duhamel–Neumann Law......Page 83
3.10 Plane Strain with Temperature Effects......Page 88
Problems......Page 92
4 Boundary-Value Problems of Elasticity......Page 96
4.1 Boundary-Value Problem in Terms of Stresses......Page 97
4.2 Boundary-Value Problem in Terms of Displacements: Navier Equations......Page 98
4.3 Principle of Superposition......Page 101
4.5 Saint-Venant’s Principle......Page 102
4.6 Stretching of a Prismatic Bar by Its Own Weight......Page 103
4.7 Thermal Expansion of a Compressed Prismatic Bar in a Rigid Container......Page 106
4.8 Pure Bending of a Prismatic Beam......Page 107
4.9 Torsion of a Prismatic Rod of Circular Cross Section......Page 111
Problems......Page 114
5.1 Equilibrium Equations in Cylindrical Coordinates......Page 118
5.2 Strain–Displacement Relations......Page 121
5.3 Geometric Derivation of Strain–Displacement Relations......Page 123
5.4 Compatibility Equations......Page 126
5.6 Navier Equations in Cylindrical Coordinates......Page 128
5.7 Beltrami–Michell Compatibility Equations in Cylindrical Coordinates......Page 129
5.8 Axisymmetric Plane Strain Deformation......Page 130
5.9 Pressurized Hollow Cylinder......Page 132
5.10 Pressurized Thin-Walled Cylinder......Page 139
5.11 Pressurized Solid Cylinder......Page 140
5.12 Pressurized Circular Hole in an Infinite Medium......Page 142
5.13 Shrink-Fit Problem......Page 143
5.14 Axial Loading of a Hollow Cylinder......Page 144
5.15 Axially Loaded Pressurized Hollow Cylinder......Page 145
5.16 Spherical Symmetry......Page 147
Problems......Page 150
Part II Applications......Page 156
6.1 Plane Stress Problems......Page 158
6.2 Beltrami–Michell Compatibility Equation......Page 160
6.3 Airy Stress Function......Page 161
6.4 Pure Bending of a Thin Beam......Page 162
6.5 Bending of a Cantilever Beam......Page 165
6.6 Bending of a Simply Supported Beam by a Distributed Load......Page 167
6.7 Approximate Character of the Plane Stress Solution......Page 168
6.8 Plane Strain Problems......Page 171
6.10 Transition from Plane Stress to Plane Strain......Page 175
Problems......Page 178
7.1 Introduction......Page 183
7.2 Axisymmetric Problems......Page 187
7.3 Non-axisymmetric Problems......Page 189
7.4 Flamant Problem: Vertical Force on a Half-Plane......Page 193
7.5 Distributed Loading over the Boundary of a Half-Space......Page 197
7.6 Michell Problem: Diametral Compression of a Circular Disk......Page 201
7.7 Kirsch Problem: Stretching of a Perforated Plate......Page 204
7.8 Stretching of an Infinite Plate Weakened by an Elliptical Hole......Page 215
7.9 Stretching of a Plate Strengthened by a Circular Inhomogeneity......Page 218
7.10 Rotating Disk......Page 222
7.11 Stress Field near a Crack Tip......Page 224
7.12 Edge Dislocation......Page 229
7.13 Force Acting at a Point of an Infinite Plate......Page 233
Problems......Page 236
8.1 Governing Equations for Antiplane Shear......Page 243
8.2 Antiplane Shear of a Circular Annulus......Page 246
8.3 Concentrated Line Force on the Surface of a Half-Space......Page 247
8.4 Infinite Medium Weakened by a Circular Hole......Page 249
8.5 Infinite Medium Weakened by an Elliptical Hole......Page 251
8.6 Infinite Medium Strengthened by a Circular Inhomogeneity......Page 253
8.7 Stress Field near a Crack Tip under Remote Antiplane Shear Loading......Page 255
8.8 Screw Dislocation......Page 258
8.9 Screw Dislocation in a Half-Space......Page 260
8.10 Screw Dislocation near a Circular Hole in an Infinite Medium......Page 262
8.11 Screw Dislocation near a Circular Inhomogeneity......Page 266
Problems......Page 268
9.1 Torsion of a Prismatic Rod of Solid Cross Section......Page 274
9.2 Boundary Conditions on the Lateral Surface of a Rod......Page 276
9.3 Boundary Conditions at the Ends of a Rod......Page 278
9.4 Displacement Field in a Twisted Rod......Page 279
9.5 Torsional Stiffness......Page 283
9.6 Membrane Analogy......Page 284
9.7 Torsion of a Rod of Elliptical Cross Section......Page 285
9.8 Torsion of a Rod of Triangular Cross Section......Page 288
9.9 Torsion of a Rod of Grooved Circular Cross Section......Page 290
9.10 Torsion of a Rod of Semi-circular Cross Section......Page 291
9.11 Torsion of a Rod of Rectangular Cross Section......Page 292
9.12 Torsion of a Rod of Thin-Walled Open Cross Section......Page 295
9.13 Warping of a Thin-Walled Open Cross Section......Page 298
9.14 Torsion of a Rod of Multiply Connected Cross Section......Page 302
9.15 Torsion of a Rod of Thin-Walled Closed Cross Section......Page 304
9.16 Warping of a Thin-Walled Closed Cross Section......Page 309
9.17 Torsion of a Rod of Thin-Walled Open/Closed Cross Section......Page 312
9.18 Torsion of a Rod of Multicell Cross Section......Page 313
Problems......Page 317
10.1 Bending of a Cantilever Beam of Solid Cross Section......Page 322
10.2 Differential Equation for the Stress Field......Page 324
10.3 Displacement Field in a Bent Cantilever Beam......Page 327
10.4 Shear (Flexural) Center......Page 328
10.5 Bending of a Beam of Elliptical Cross Section......Page 329
10.6 Bending of a Beam of Circular Cross Section......Page 331
10.7 Bending of a Beam of Rectangular Cross Section......Page 333
10.8 Elementary Theory for Shear Stresses......Page 336
10.9 Bending of a Beam of Thin-Walled Open Cross Section......Page 338
10.10 Skew Bending of a Thin-Walled Cantilever Beam......Page 343
10.11 Bending of a Hollow Prismatic Beam......Page 344
10.12 Bending of a Beam of Hollow Circular Cross Section......Page 346
10.13 Bending of a Beam of Thin-Walled Closed Cross Section......Page 348
10.14 Bending of a Beam of Multicell Cross Section......Page 353
10.15 Stress Expressions with Respect to Non-principal Centroidal Axes......Page 357
Problems......Page 362
11 Contact Problems......Page 368
11.1 Axisymmetric Problems in Cylindrical Coordinates......Page 369
11.2 Concentrated Force in an Infinite Space: Kelvin Problem......Page 370
11.3 Concentrated Force on the Surface of a Half-Space: Boussinesq Problem......Page 373
11.4 Ellipsoidal Pressure Distribution......Page 375
11.5 Indentation by a Spherical Ball......Page 378
11.6 Uniform Pressure within a Circular Area......Page 381
11.7 Flat Circular Frictionless Punch......Page 383
11.8 Hertz Problem: Two Spherical Bodies in Contact......Page 384
11.9 Two Circular Cylinders in Contact......Page 391
Problems......Page 394
12 Energy Methods......Page 401
12.1 Strain Energy in Uniaxial Tension Test......Page 402
12.2 Strain Energy for Three-Dimensional States of Stress and Strain......Page 403
12.3 Volumetric and Deviatoric Strain Energy......Page 407
12.4 Betti’s Reciprocal Theorem......Page 410
12.5 Castigliano’s Theorems......Page 413
12.6 Principle of Virtual Work......Page 414
12.7 Potential Energy and the Variational Principle......Page 416
12.8 Application to Structural Mechanics......Page 417
12.9 Derivation of the Beam Bending Equation from the Principle of Virtual Work......Page 425
12.10 Finite Element Method for Beam Bending......Page 427
12.11 Rayleigh–Ritz Method......Page 435
12.12 Finite Element Method for Axial Loading......Page 438
Problems......Page 448
13 Failure Criteria......Page 453
13.1 Maximum Principal Stress Criterion......Page 454
13.2 Maximum Principal Strain Criterion......Page 455
13.3 Maximum Shear Stress Criterion: Tresca Yield Criterion......Page 457
13.4 Maximum Deviatoric Strain Energy Criterion: Von Mises Yield Criterion......Page 460
13.5 Mohr Failure Criterion......Page 464
13.6 Coulomb–Mohr Failure Criterion......Page 468
13.7 Drucker–Prager Failure Criterion......Page 471
13.8 Fracture-Mechanics-Based Failure Criteria......Page 474
13.9 Double-Cantilever Specimen......Page 477
13.10 Fracture Criterion in Terms of the Stress Intensity Factor......Page 479
13.11 J Integral......Page 485
Problems......Page 487
Further Reading......Page 493
Index......Page 496