1．General problems in solid mechanics and non-linearity 1
1.1 Introduction 1
1.2 Small deformation solid mechanics problems 4
1.3 Variational forms for non-linear elasticity 12
1.4 Weak forms of governing equations 14
1.5 Concluding remarks 15
References 15
2．Galerkin method of approximation-irreducible and mixed forms 17
2.1 Introduction 17
2.2 Finite element approximation-Galerkin method 17
2.3 Numerical integration-quadrature 22
2.4 Non-linear transient and steady-state problems 24
2.5 Boundary conditions:non-linear problems 28
2.6 Mixed or irreducible forms 33
2.7 Non-linear quasi-harmonic field problems 37
2.8 Typical examples of transient non-linear calculations 38
2.9 Concluding remarks 43
References 44
3．Solution of non-linear algebraic equations 46
3.1 Introduction 46
3.2 Iterative techniques 47
3.3 General remarks-incremental and rate methods 58
References 60
4．Inelastic and non-linear materials 62
4.1 Introduction 62
4.2 Viscoelasticity-history dependence of deformation 63
4.3 Classical time-independent plasticity theory 72
4.4 Computation of stress increments 80
4.5 Isotxopic plasticity models 85
4.6 Generalized plasticity 92
4.7 Some examples of plastic computation 95
4.8 Basic formulation of creep problems 100
4.9 Viscoplasticity-a generalization 102
4.10 Some special problems of brittle materials 107
4.11 Non-uniqueness and localization in elasto-plastic deformations 112
4.12 Non-linear quasi-harmonic field problems 116
4.13 Concluding remarks 118
References 120
5．Geometrically non-linear problems-finite deformation 127
5.1 Introduction 127
5.2 Governing equations 128
5.3 Variational description for finite deformation 135
5.4 Two-dimensional forms 143
5.5 A three-field,mixed finite deformation formulation 145
5.6 A mixed-enhanced finite deformation formulation 150
5.7 Forces dependent on deformation-pressure loads 154
5.8 Concluding remarks 155
References 156
6．Material constitution for finite deformation 158
6.1 Introduction 158
6.2 Isotropic elasticity 158
6.3 Isotropic viscoelasticity 172
6.4 Plasticity models 173
6.5 Incremental formulations 174
6.6 Rate constitutive models 176
6.7 Numerical examples 178
6.8 Concluding remarks 185
References 189
7．Treatment of constraints-contact and tied interfaces 191
7.1 Introduction 191
7.2 Node-node contact:Hertzian contact 193
7.3 Tied interfaces 197
7.4 Node-surface contact 200
7.5 Surface-surface contact 218
7.6 Numerical examples 219
7.7 Concluding remarks 224
References 224
8．Pseudo-rigid and rigid-flexible bodies 228
8.1 Introduction 228
8.2 Pseudo-rigid motions 228
8.3 Rigid motions 230
8.4 Connecting a rigid body to a flexible body 234
8.5 Multibody coupling by joints 237
8.6 Numerical examples 240
References 242
9．Discrete element methods 245
9.1 Introduction 245
9.2 Early DEM formulations 247
9.3 Contact detection 250
9.4 Contact constraints and boundary conditions 256
9.5 Block deformability 260
9.6 Time integration for discrete element methods 267
9.7 Associated discontinuous modelling methodologies 270
9.8 Unifying aspects of discrete element methods 271
9.9 Concluding remarks 272
References 273
10．Structural mechanics problems in one dimension-rods 278
10.1 Introduction 278
10.2 Governing equations 279
10.3 Weak(Galerkin)forms for rods 285
10.4 Finite element solution:Euler-Bernoulli rods 290
10.5 Finite element solution:Timoshenko rods 305
10.6 Forms without rotation parameters 317
10.7 Moment resisting frames 319
10.8 Concluding remarks 320
References 320
11．Plate bending approximation:thin(Kirchhoff)plates and C1 continuity requirements 323
11.1 Introduction 323
11.2 The plate problem:thick and thin formulations 325
11.3 Rectangular element with corner nodes(12 degrees of freedom) 336
11.4 Quadrilateral and parallelogram elements 340
11.5 Triangular element with corner nodes(9 degrees of freedom) 340
11.6 Triangular element of the simplest form(6 degrees of freedom) 345
11.7 The patch test-ananalytical requirement 346
11.8 Numerical examples 348
11.9 General remarks 357
11.10 Singular shape functions for the simple triangular element 357
11.11 An 18 degree-of-freedom triangular element with conforming shape functions 360
11.12 Compatible quadrilateral elements 361
11.13 Quasi-conforming elements 362
11.14 Hermitian rectangle shape function 363
11.15 The 21 and 18 degree-of-freedom triangle 364
11.16 Mixed formulations-general remarks 366
11.17 Hybrid plate elements 368
11.18 Discrete Kirchhoff constraints 369
11.19 Rotation-free elements 371
11.20 Inelastic material behaviour 374
11.21 Concluding remarks-which elements? 376
References 376
12．'Thick'Reissner-Mindlin plates-irreducible and mixed formulations 382
12.1 Introduction 382
12.2 The irreducible formulation-reduced integration 385
12.3 Mixed formulation for thick plates 390
12.4 The patch test for plate bending elements 392
12.5 Elements with discrete collocation constraints 397
12.6 Elements with rotational bubble or enhanced modes 405
12.7 Linked interpolation-an improvement of accuracy 408
12.8 Discrete'exact'thin plate limit 413
12.9 Pefformance of various'thick'plate elements-limitations of thin plate theory 415
12.10 Inelastic material behaviour 419
12.11 Concluding remarks-adaptive refinement 420
References 421
13．Shells as an assembly of flat elements 426
13.1 Introduction 426
13.2 Stiffness of a plane element in local coordinates 428
13.3 Transformation to global coordinates and assembly of elements 429
13.4 Local direction cosines 431
13.5 'Drilling'rotational stiffness-6 degree-of-freedom assembly 435
13.6 Elements with mid-side slope connections only 440
13.7 Choice of element 440
13.8 Practical examples 441
References 450
14．Curved rods and axisymmetric shells 454
14.1 Introduction 454
14.2 Straight element 454
14.3 Curved elements 461
14.4 Independent slope-displacement interpolation with penalty functions(thick or thin shell formulations) 468
References 473
15．Shells as a special case of three-dimensional analysis-Reissner-Mindlin assumptions 475
15.1 Introduction 475
15.2 Shell element with displacement and rotation paranrters 475
15.3 Special case of axisymmetric,curved,thick shells 484
15.4 Special case of thick plates 487
15.5 Convergence 487
15.6 Inelastic behaviour 488
15.7 Some shell examples 488
15.8 Concluding remarks 493
References 495
16．Semi-analytical finite element processes-use of orthogonal functions and'finite strip'methods 498
16.1 Introduction 498
16.2 Prismatic bar 501
16.3 Thin membrane box structures 504
16.4 Plates and boxes with flexure 505
16.5 Axisymmetric solids with non-symmetrical load 507
16.6 Axisymmetric shells with non-symmetrical load 510
16.7 Concluding remarks 514
References 515
17．Non-linear structural problems-large displacement and instability 517
17.1 Introduction 517
17.2 Large displacement theory of beams 517
17.3 Elastic stability-energy interpretation 523
17.4 Large displacement theory of thick plates 526
17.5 Large displacement theory of thin plates 532
17.6 Solution of large deflection problems 534
17.7 Shells 537
17.8 Concluding remarks 542
References 543
18．Multiscale modelling 547
18.1 Introduction 547
18.2 Asymptotic analysis 549
18.3 Statement of the problem and assumptions 550
18.4 Formalism of the homogenization procedure 552
18.5 Global solution 553
18.6 Local approximation of the stress vector 554
18.7 Finite element analysis applied to the local problem 555
18.8 The non-linear case and bridging over several scales 560
18.9 Asymptotic homogenization at three levels:micro,meso and macro 561
18.10 Recovery of the micro description of the variables of the problem 562
18.11 Material characteristics and homogenization results 565
18.12 Multilevel procedures which use homogenization as an ingredient 567
18.13 General first-order and second-order procedures 570
18.14 Discrete-to-continuum linkage 572
18.15 Local analysis of a unit cell 578
18.16 Homogenization procedure-definition of successive yield surfaces 578
18.17 Numerically developed global self-consistent elastic-plastic constitutive law 580
18.18 Global solution and stress-recovery procedure 581
18.19 Concluding remarks 586
References 587
19．Computer procedures for finite element analysis 590
19.1 Introduction 590
19.2 Solution of non-linear problems 591
19.3 Eigensolutions 592
19.4 Restart option 594
19.5 Concluding remarks 595
References 595
Appendix A Isoparametric finite element approximations 597
Appendix B Invariants of second-order tensors 604
Author index 609
Subject index 619