ISBN: 3540669604
TITLE: Mechanics of Deformable Solids
AUTHOR: Doghri, Issam
TOC:

1. Basic mechanics 1
1.1 On tensors 1
1.2 Stress 3
1.3 Strain 5
1.4 Principal invariants and eigenvalues of stress and strain 6
1.5 Mohr's stress circles 7
1.6 Equilibrium 8
1.7 Local formulation of static problems 9
1.8 Continuity equations 11
1.9 Compatibility equations 12
1.10 Strength criteria 12
1.11 Linear elasticity 16
1.12 Strain energy 21
1.13 Navier equations 23
1.14 Beltrami-Mitchell compatibility equations 24
1.15 Saint-Venant's principle, Uniqueness, Superposition, Special theories 25
1.16 Solved problem: composite cylinder under axial load 26
2. Variational formulations, work and energy theorems 29
2.1 Local formulation of static problems 29
2.2 Virtual work theorem (VWT) 30
2.3 Displacement-based variational formulation 33
2.4 Potential energy theorem 35
2.5 Stress-based variational formulation 37
2.6 Complementary energy theorem 38
2.7 Energy bounds 39
2.8 Stored energy 40
2.9 Maxwell-Betti reciprocity theorem 41
2.10 Castigliano's theorem 42
2.11 Introduction to numerical methods 45
2.11.1 Method of Ritz 45
2.11.2 Finite element method (FEM) 45
2.12 Solved problem: volume change of an axially compressed body 47
3. Theory of beams (strength of materials) 49
3.1 Definitions and geometric properties 49
3.2 Pure bending of a straight beam: 3D elasticity solution 51
3.3 Basic assumptions of beam theory 55
3.3.1 External loads 56
3.3.2 Internal loads (stress resultants) 57
3.3.3 Equilibrium equations 58
3.3.4 Navier-Bernoulli assumption 59
3.3.5 Constitutive equations 59
3.4 Displacement boundary and continuity conditions 60
3.5 Computation of internal loads (stress resultants) 62
3.6 Computation of normal and shear stresses 64
3.7 Statically determinate or indeterminate problems 66
3.8 Strain energy 67
3.9 Work and energy theorems 68
3.10 Influencelines 69
3.11 Solved problems 70
3.11.1 Shear reduced area 70
3.11.2 Statically determinate problems 71
3.11.3 Statically indeterminate problems 77
3.11.4 Maxwell-Betti reciprocity theorem 80
3.11.5 Castigliano's theorem 85
3.11.6 Virtual work theorem (VWT) 90
4. Torsion of beams 95
4.1 Formulation with a warping function 95
4.2 Formulation with a conjugate function 98
4.3 Formulation with Prandtl's stress function 98
4.4 Contour lines of the stress function 100
4.5 Maximum tangential stress 101
4.6 Membraneanalogy 102
4.7 Multi-connected sections - Particular case 104
4.8 Multi-connected sections - General case 104
4.9 Thin tubes with variable thickness 107
4.10 Potential energy 108
4.11 Cylindrical coordinates 109
4.12 Solved problems 110
4.12.1 Elliptic section 110
4.12.2 Triangular section 113
4.12.3 Notched circular section 114
4.12.4 Thin rectangular section 116
4.12.5 Ritz's method - Square section 117
4.12.6 Hollow elliptic section - Special method 118
4.12.7 Hollow elliptic section - General method 119
4.12.8 Thin circular tube 121
4.12.9 Thin-walled section with multiple voids 121
5. Theory of thin plates 123
5.1 Definitions and notation 123
5.2 Internal loads (stress resultants) 123
5.3 Equilibrium equations 126
5.4 Displacements 127
5.5 Strains 129
5.6 Constitutive equations 129
5.7 Summary: two un-coupled problems 130
5.8 Fundamental P.D.E. for bending problem 131
5.9 Boundary conditions 133
5.10 Contradictions in Kirchhoff-Love theory 135
5.11 Plates with two simply supported opposite edges - Lvy's method 136
5.12 Potential energy 139
5.13 Influence function 140
5.14 Solved problems 140
5.14.1 Uniformly loaded rectangular plate with two simply supported opposite edges and two built-in edges 140
5.14.2 Uniformly loaded rectangular plate with two simply supported opposite edges and two free edges 142
6. Bending of thin plates in polar coordinates 143
6.1 Change of coordinates 143
6.2 Axisymmetric problems 146
6.3 Potential energy 148
6.4 Solved problems 149
6.4.1 Uniformly loaded plate 149
6.4.2 Uniform load along a concentric circle 150
6.4.3 Uniform pressure on a concentric disk 153
6.4.4 Plate simply supported on a number of points 155
6.4.5 Ritz's method 160
7. Two-dimensional problems in Cartesian coordinates 163
7.1 Plane strain 163
7.2 Plane stress 164
7.3 Summary: plane strain versus plane stress 165
7.4 Airy stress function 167
7.5 Polynomial solutions 167
7.6 Solution by Fourier series 169
7.7 Generalized plane stress 170
7.8 Solved problems 173
7.8.1 Concentrated load at the end of a cantilever beam 173
7.8.2 Uniform loads on the upper and lower surfaces of a simply supported beam 177
7.8.3 Uniform load on a cantilever beam 182
7.8.4 Uniform load on a beam with two clamped ends 182
7.8.5 Compression of a beam in the height-direction 182
7.8.6 Body forces - Beam under its own weight 187
8. Two-dimensional problems in polar coordinates 193
8.1 Change of coordinates 193
8.2 Summary: plane strain versus plane stress 195
8.3 Airy stress function 196
8.4 Axisymmetric plane problems 198
8.5 Periodic Airy stress functions 200
8.6 Generalized plane strain 201
8.7 Solved problems 201
8.7.1 Hollow circular cylinder under inner and outer pressures 201
8.7.2 Composite hollow cylinder under inner and outer pressures 205
8.7.3 Coil winding 206
8.7.4 Bending of a curved beam 209
8.7.5 Traction of a circular arch 214
8.7.6 Rotating disk of uniform thickness 215
8.7.7 Rotating disk of variable thickness 218
8.7.8 Stress concentration in a plate with a small circular hole 222
8.7.9 Force on the straight edge of a semi-infinite plate 224
8.7.10 Pressure on the straight edge of a semi-infinite plate 228
8.7.11 Compression of a disk along a diameter 230
8.7.12 Compression of a disk over two opposing arcs 231
9. Thermo-elasticity 233
9.1 Constitutive equations 233
9.2 Heat equation 234
9.3 Thermo-mechanical problem 235
9.3.1 Thermal problem 236
9.3.2 Mechanical problem 237
9.4 Thermal stresses: some remarks 237
9.5 Solved problems 239
9.5.1 Axisymmetric thermal stresses in a hollow cylinder 239
9.5.2 Thermal stresses in a composite cylinder 242
9.5.3 Transient thermal stresses in a thin plate 245
10. Elastic stability 249
10.1 Introduction 249
10.2 Direct and energy methods 250
10.3 Euler's method for axially compressed columns 252
10.3.1 Critical buckling load 252
10.3.2 Critical buckling stress 254
10.3.3 Remarks 256
10.4 Energy-based approximate method 256
10.5 Non-conservative loads 258
10.6 Solved problems 259
10.6.1 Two rigid bars connected with a spring 259
10.6.2 Column clamped at one end 260
10.6.3 Column clamped at one end and simply supported at the other 261
10.6.4 Column clamped at both ends 263
10.6.5 Column elastically built-in at one end and simply supported at the other 264
10.6.6 Column with non-uniform properties 266
10.6.7 Eccentric compressive load 267
10.6.8 Beam-column under compressive and bending forces 268
10.6.9 Energy method 269
11. Theory of thin shells 273
11.1 Geometry of the mid-surface 273
11.2 First fundamental form 274
11.3 Second fundamental form 278
11.4 Compatibility conditions of Codazzi and Gauss 280
11.5 Surface of revolution 280
11.5.1 General case 280
11.5.2 Conic surfaces 283
11.6 Gradient of a vector field in curvilinear coordinates 283
11.7 Kinematics of the mid-surface 285
11.8 Displacements and strains outside the mid-surface 287
11.8.1 General theory 287
11.8.2 Application: plates in rectangular coordinates 289
11.8.3 Application: plates in polar coordinates 290
11.9 Internal loads (stress resultants) 290
11.10 Equilibrium equations 292
11.10.1 General theory 292
11.10.2 Application: plates in rectangular coordinates 295
11.10.3 Application: plates in polar coordinates 295
11.11 Constitutive equations 296
11.12 Membranetheory 298
11.13 Further reading 298
12. Elasto-plasticity 301
12.1 One-dimensional model 301
12.2 Three-dimensional model 304
12.3 Linear elasticity 305
12.4 Equivalent stress 306
12.5 Hardening 306
12.6 Flow rules 307
12.7 Tangent operator, loading/unloading, hardening/softening 308
12.8 Elementary examples 312
12.8.1 Uniaxial tension-compression 312
12.8.2 Simple shear 313
12.9 Boundary-value problem 313
12.10 Numerical algorithms 313
12.10.1 Finite element method (F.E.M.) 313
12.10.2 Return mapping algorithm 315
12.10.3 Consistent tangent operator 318
12.ll A general framework for material models 320
12.11.1 State variables 320
12.11.2 Equations of state 321
12.11.3 Flow rules 322
12.11.4 Rate-independent plasticity 323
12.11.5 Heat equation 324
12.12 A class of non-associative plasticity models 325
12.13 Furtherreading 327
13. Elasto-viscoplasticity 329
13.1 One-dimensional model 329
13.2 Three-dimensional model 331
13.3 Numerical algorithms 332
13.3.1 Return mapping algorithm 333
13.3.2 Consistent tangent operator 333
13.4 Further reading 335
14. Nonlinear continuum mechanics 337
14.1 Kinematics 337
14.1.1 Description of motion 338
14.1.2 Material time derivative 338
14.2 Deformation 339
14.2.1 Deformation gradient 339
14.2.2 Polar decomposition 340
14.2.3 Spectral decompositions 341
14.2.4 Length variation 343
14.3 Strainmeasures 344
14.3.1 One-dimensional case 344
14.3.2 Three-dimensional case 344
14.4 Strain rates  346
14.5 Balance laws 348
14.5.1 Transport formula 348
14.5.2 Conservation of mass 349
14.5.3 Conservation of linear momentum 350
14.5.4 Conservation of rotational momentum 350
14.5.5 Cauchy stress tensor 351
14.5.6 Eulerian strong formulation 351
14.5.7 Eulerian weak formulation 352
14.5.8 Balance of work and energy rates 353
14.5.9 Nominal stress 354
14.5.10 Lagrangian weak formulation 354
14.5.11 Lagrangian strong formulation 355
14.6 Conjugate stress and strain measures 356
14.6.1 Definition and examples 356
14.6.2 Interpretation of the second Piola-Kirchhoff stress 358
14.6.3 Uniaxial tension/compression 358
14.7 Objectivity 359
14.7.1 Definition 360
14.7.2 Examples 361
14.8 Objective stress rates 363
14.8.1 Examples 363
14.8.2 A family of objective rates 365
14.9 Laws of thermodynamics 366
14.9.1 First law 366
14.9.2 Second law 366
14.9.3 Clausius-Duhem inequality 367
14.10 Further reading 367
15. Nonlinear elasticity 369
15.1 Hyperelasticity and hypoelasticity 369
15.1.1 Definitions 369
15.1.2 Hyperelasticity and material objectivity 369
15.1.3 Elasticity tensors 371
15.1.4 Incompressibility constraint 373
15.2 Principal invariants and principal stretches 374
15.3 Isotropic hyperelasticity in principal invariants 375
15.3.1 Formulation 375
15.3.2 Modified neo-Hookean model 377
15.3.3 Modified Mooney-Rivlin model 378
15.4 Isotropic hyperelasticity in principal stretches 379
15.4.1 Formulation 379
15.4.2 Modified Ogden's model 380
15.5 Examples of homogeneous deformations 381
15.5.1 Homogeneous simple shear 381
15.5.2 Uniform extension 383
15.5.3 Pure dilatation 384
15.6 Linearization 385
15.6.1 Linearization of the deformation 386
15.6.2 Linearization of constitutive equations 387
15.6.3 Linearization of the equations of elasto-statics 387
15.6.4 Variational formulations 388
15.6.5 Linearization of the weak formulation 389
15.7 Mixed variational formulation 390
15.7.1 Formulation 390
15.7.2 Incompressibility constraint 392
15.8 Appendices 393
15.8.1 The Piola identity 393
15.8.2 Linearization of a pressure B.C. 393
15.8.3 Differentiation of an isotropic function of a second-order symmetric tensor 394
15.8.4 Elasticity tensors for principal stretch formulation 395
16. Finite-strain elasto-plasticity 397
16.1 First theory 397
16.1.1 Multiplicative decomposition of the deformation gradient 397
16.1.2 Hyperelastic-plastic constitutive equations 399
16.1.3 Stress-strain relations 401
16.1.4 Flow rules 402
16.1.5 Elastic predictor 404
16.1.6 Time discretization of the plastic flow rule 404
16.1.7 Return mapping algorithm in principal stresses and strains 406
16.1.8 Algorithmic tangent moduli 407
16.1.9 Summary of the algorithm 408
16.1.10 Application: Quadratic logarithmic free energy and J_2 flow 410
16.2 Second theory 413
16.2.1 Additive decomposition of the rate of deformation and hypoelasticity 413
16.2.2 Computation of the strain increment 415
16.2.3 Polar decomposition algorithm 417
16.2.4 A time-integration algorithm for the rotation matrix 419
16.2.5 Application: the Jaumann objective stress rate 420
16.2.6 Summary 421
16.2.7 Incremental objectivity 422
17. Cyclic plasticity 423
17.1 One-dimensional model 423
17.2 Three-dimensional model 425
17.3 Dissipation inequality 428
17.4 Plastic multiplier 428
17.5 Tangent operator 429
17.6 Hardening modulus 429
17.7 Return mapping algorithm 430
17.8 Consistent tangent operator 433
17.9 Numerical simulation 435
18. Damage mechanics 439
18.1 Damage variable 439
18.2 Three-dimensional constitutive model 442
18.3 Dissipation inequality 445
18.4 Plastic multiplier 445
18.5 Tangent operator 445
18.6 Hardening modulus 446
18.7 Closed-form solutions for loadings with constant triaxiality 447
18.8 Return mapping algorithm 450
18.8.1 Corrections over the elastic predictor 458
18.8.2 Summary of the algorithm 458
18.8.3 Non-damaged case 458
18.9 Consistent tangent operator 459
18.9.1 Non-damaged case 461
18.10 Numerical simulations 461
18.10.1 Ductile failure under uniaxial tension 462
18.10.2 Ductile failure under simple shear 462
18.10.3 A post-processor for crack initiation 462
18.11 Further reading 465
19. Strain localization 469
19.1 Motivation: a one-dimensional example 469
19.2 Uniqueness and ellipticity 472
19.3 Strain localization 474
19.3.1 Continuous bifurcation 475
19.3.2 Discontinuous bifurcation 475
19.3.3 Summary 477
19.4 Analytical results for initially homogeneous plane problems 477
19.4.1 General strain-softening models 477
19.4.2 A ductile damage model 481
19.5 Numerical results for a ductile damage model 481
19.5.1 Biaxial loadings in plane stress 481
19.5.2 Notched plate with a macro-defect 484
19.6 Nonlocal or internal-length models 496
19.7 A two-scale homogenization procedure 498
19.8 Numerical algorithms 503
19.9 Elasticity with damage- Model without threshold 506
19.9.1 Local constitutive equations 506
19.9.2 Nonlocal macroscopic formulation 506
19.9.3 Numerical simulations 507
19.10 Elasticity with damage- Model with threshold 509
19.10.l Local constitutive equations 509
19.10.2 Non local macroscopic formulation 510
19.10.3 Numerical simulations 510
19.11 Appendices 513
19.11.1 Strain localization criterion in 2D 513
19.11.2 Macroscopic free-energy potential 515
19.11.3 Macroscopic dissipation potential 517
20. Micro-mechanics of materials 519
20.1 Micro/macro approach 519
20.2 Homogenization schemes 521
20.2.1 Average strains and stresses 521
20.2.2 Voigt model 523
20.2.3 Reuss model 524
20.2.4 Self-Consistent model 524
20.2.5 Mori-Tanaka model 526
20.3 Micro/macro constitutive model for semi-crystalline polymers 527
20.3.1 Crystalline phase 528
20.3.2 Amorphous phase 531
20.3.3 Intermediate phase 536
20.3.4 Single inclusion 538
20.3.5 Overall behavior 538
20.3.6 Numerical simulations 538
20.4 Further reading 543
A. Cylindrical coordinates 545
B. Cardan's formulae 551
C. Matrices for the representation of second- and fourth-order tensors 553
C.1 Storage 553
C.2 Change of coordinates 556
D. Zero-stress constraints 559
D.1 Small-strain J_2 elasto-plasticity 559
D.2 General small-strain models 561
D.3 General finite-strain models 562
END
