Non-linear Finite Element Analysis of Solids and Structures Volume 2

Non-linear Finite Element Analysis of Solids and Structures Volume 2
اسم المؤلف
M.A. Crisfield
التاريخ
2 فبراير 2019
المشاهدات
التقييم
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Non-linear Finite Element Analysis of Solids and Structures
Volume 2: Advanced Topics
M.A. Crisfield
Imperial College Of Science,
Technology and Medicine, London, UK
Contents
Preface xiii
10 More continuum mechanics 1
10.1 Relationships betweensome strain measures and the structures
10.2 Large strains and the Jaumann rate
10.3 Hyperelasticity
10.4 The Truesdell rate
10.5 Conjugate stress and strain measures with emphasison isotropic
conditions 10
10.6 Furtherwork on conjugate stress and strain measures 13
10.6.1 Relationship betweeni: and U 14
10.6.2 Relationship between the Bio! stress, B and the Kirchhoff stress, T 15
10.6.3 Relationship betweenU, the i’s and the spin of the Lagrangian
triad, W, 15
10.6.4 Relationship between€, the A’s and the spin, W, 16
10.6.5 Relationship between6,the 2’s andthe spin, W, 17
10.6.6 Relationship between€and E 17
10.6.6.1 Specific strain measures 17
10.6.7 Conjugate stress measures 18
10.7 Using log,V with isotropy 19
10.8 Other stress rates and objectivity 20
10.9 Special notation 22
10.10 References 24
11 Non-orthogonal coordinates and CO- and contravariant tensor
components 26
1 1.1 Non-orthogonal coordinates 26
11.2 Transforming the componentsof a vector (first-ordertensor) to a new set of
base vectors 28
11.3 Second-order tensors in non-orthogonal coordinates 30
11.4 Transformingthe componentsof a second-order tensor to a new set of
base vectors 30
11.5 The metrictensor 31
11.6 Work terms and the trace operation 32vi CONTENTS
11.7 Covariant components, natural coordinates and the Jacobian 33
11.8 Green’s strain and the deformation gradient 35
11.8.1 Recoveringthe standard cartesian expressions 35
11.9 The second Piola-Kirchhoff stresses and the variation of the Green’s
strain 36
11.10 Transforming the components of the constitutive tensor 37
11.11 A simple two-dimensional example involving skew coordinates 38
11.12 Special notation 42
11.13 References 44
12 More finite element analysisof continua 45
12.1 A summary of the key equations for the total Lagrangian formulation 46
12.1.1 The internalforce vector 46
12.1.2 The tangentstiffnessmatrix 47
12.2 The internal force vector for the ‘Eulerian formulation’ 47
12.3 The tangent stiffness matrix in relation to the Truesdell rate of Kirchhoff
stress 49
12.3.1 Continuum derivationof the tangentstiffnessmatrix 49
12.3.2 Discretised derivationof the tangent stiffnessmatrix 51
12.4 The tangent stiffness matrix using the Jaumann rate of Kirchhoff stress 53
12.4.1 Alternativederivationof the tangent stiffnessmatrix 54
12.5 The tangent stiffness matrix using the Jaumann rate of Cauchy stress 55
12.5.1 Alternative derivationof the tangentstiffnessmatrix 56
12.6 Convected coordinates and the total Lagrangian formulation 57
12.6.1 Elementformulation 57
12.6.2 The tangentstiffnessmatrix 59
12.6.3 Extensionsto three dimensions 59
12.7 Special notation 60
12.8 References 61
13 Large strains, hyperelasticityand rubber 62
13.1 Introduction to hyperelasticity 62
13.2 Using the principal stretch ratios 63
13.3 Splitting the volumetric and deviatoric terms 65
1 3.4 Development using second Piola-Kirchhoff stresses and Green’s
strains 66
13.4.1 Plane strain 69
13.4.2 Plane stress with incompressibility 69
13.5 Total Lagrangian finite element formulation 71
13.5.1 A mixed formulation 72
12.5.2 A hybridformulation 74
13.6 Developments using the Kirchhoff stress 76
13.7 A ‘Eulerian’ finite element formulation 78
13.8 Working directly with the principal stretch ratios 79
13.8.1 The compressible ‘neo-Hookean model’ 80
13.8.2 Usingthe Green strain relationships in the principal directions 81
13.8.3 Transforming the tangentconstitutive relationshipsfor a ‘Eulerian formulation’ 84
13.9 Examples 86
13.9.1 A simpleexample 86
13.9.2 The compressible neo-Hookeanmodel 89
13.10 Further work with principal stretch ratios 89
13.10.1 An enerav function usina the DrinciPalloa strains fthe Henckvmodel) 90CONTENTS vii
13.10.2 Ogden’s energy function 91
13.10.3 An example using Hencky’s model 93
13.11 Special notation 95
13.12 References 97
14 More plasticityand other material non-linearity-I 99
14.1 Introduction 99
14.2 Other isotropicyield criteria 99
14.2.1 The flow rules 104
14.2.2 The matrix ?a/(% 105
14.3 Yield functions with corners 107
14.3.1 A backward-Euler return with two activeyield surfaces 107
14.3.2 A consistent tangent modularmatrix with two active yield surfaces 108
14.4 Yield functions for shells that use stress resultants 109
14.4.1 The one-dimensional case 109
14.4.2 The two-dimensional case 112
14.4.3 A backward-Euler return with the lllyushin yieldfunction 113
14.4.4 A backward-Euler return and consistent tangent matrix for
the llyushinyield criterionwhen two yield surfaces are active 114
14.5 Implementing a form of backward-Euler procedurefor the
Mohr-Coulomb yield criterion 115
14.5.1 Implementing a two-vectored return 118
14.5.2 A return from a corner or to the apex 119
14.5.3 A consistent tangent modular matrix following
a single-vector return 120
14.5.4 A consistent tangent matrix following a two-vectored return 121
14.5.5 A consistent tangent modular matrix following a return from a corner or
an apex 121
14.6 Yield criteriafor anisotropic plasticity 122
14.6.1 Hill’s yield criterion 122
14.6.2 Hardening with Hill’syieldcriterion 124
14.6.3 Hill’s yield criterion for plane stress 126
14.7 Possible return algorithms and consistent tangent modular matrices 129
14.7.1 The consistent tangent modularmatrix 130
14.8 Hoffman’s yield criterion 131
14.8.1 The consistent tangent modularmatrix 133
14.9 The Drucker-Prager yieldcriterion 133
14.10Usingan eigenvector expansionfor the stresses 134
14.10.1 An example involving plane-stress plasticity and the von Mises
yieldcriterion 135
14.11 Cracking, fracturing and softening materials 135
14.11.1 Mesh dependency and alternative equilibrium states 135
14.11.2 ‘Fixed’ and ‘rotating’ crack modelsin concrete 140
14.11.3 Relationship between the ‘rotating crack model’ and
a ‘deformation theory’ plasticity approach usingthe ‘squareyield criterion’ 142
14.11.4 A flow theory approachfor the ‘square yield criterion’ 144
14.12 Damage mechanics 148
14.13 Special notation 152
14.14 References 154
15 More plasticityand other materialnon-linearity-ll 158
15.1 Introduction 158
15.2 Mixed hardening 163
15.3 Kinematic hardeningfor plane stress 164viii CONTENTS
15.4 Radial return with mixed linear hardening 166
15.5 Radial return with non-linear hardening 167
15.6 A general backward-Euler return with mixed linear hardening 168
15.7 A backward-Euler procedurefor plane stress with mixed linear hardening 170
15.8 A consistent tangent modular tensor following the radial return of
Section 15.4 172
15.9 General form of the consistent tangent modular tensor 173
15.10 Overlay andother hardening models 174
15.1 0.1 Sophisticated overlay model 178
15.10.2 Relationship with conventional kinematic hardening 180
15.10.3 Other models 180
15.11 Computer exercises 181
15.12 Viscoplasticity 182
15.12.1 The consistenttangent matrix 184
15.12.2 Implementation 185
15.13Special notation 185
15.14 References 186
16 Largerotations 108
16.1 Non-vectorial large rotations 188
16.2 A rotation matrix for small (infinitesimal)rotations 188
16.3 A rotation matrix for large rotations (Rodrigues formula) 191
16.4 The exponential form for the rotationmatrix 194
16.5 Alternative forms for the rotation matrix 194
16.6 Approximations for the rotationmatrix 195
16.7 Compound rotations 195
16.8 Obtaining the pseudo-vectorfrom the rotation matrix, R 197
16.9 Quaternions and Euler parameters 198
16.10Obtaining the normalised quarternion from the rotation matrix 199
16.11 Additive and non-additive rotationincrements 200
16.12 The derivative of the rotation matrix 202
16.13 Rotating a triad so that one unit vector moves to a specified unit vector
via the ‘smallest rotation’ 202
16.14 Curvature 204
16.14.1 Expressionsfor curvaturethat directly use nodal triads 204
16.14.2 Curvature without nodal triads 207
16.15 Special notation 211
16.16 References 212
17 Three-dimensional formulations for beams and rods 213
17.1 A co-rotationalframework for three-dimensional beam elements 213
17.1.1 Computing the local‘displacements’ 216
17.1.2 Computationof the matrix connecting the infinitesimallocal
and global variables 218
17.1.3 The tangent stiffness matrix 221
17.1.4 Numerical implementationof the rotational updates 223
17.1.5 Overall solution strategywith a non-linear ‘local element’ formulation 223
17.1.6 Possible simplifications 225
17.2 An interpretation of an element due to Simo and Vu-Quoc 226
17.2.1 The finite element variables 227
17.2.2 Axial and shear strains 227
17.2.3 Curvature 228CONTENTS ix
17.2.4 Virtual work and the internalforce vector 229
17.2.5 The tangent stiffness matrix 229
17.2.6 An isoparametric formulation 231
17.3 An isoparametric Timoshenko beam approach using the total
Lagrangian formulation 233
17.3.1 The tangent stiffness matrix 237
17.3.2 An outline of the relationship with the formulationof
Dvorkinet al. 239
17.4 Symmetry and the use of different ‘rotation variables’ 240
17.4.1 A simple model showingsymmetry and non-symmetry 241
17.4.2 Using additive rotationcomponents 242
17.4.3 Considering symmetry at equilibriumfor the element of Section 17.2 243
17.4.4 Using additive (in the limit) rotationcomponents with the element
of Section 17.2 245
17.5 Various forms of applied loading including ‘follower levels’ 248
17.5.1 Point loads appliedat a node 248
17.5.2 Concentratedmoments appliedat a node 249
17.5.3 Gravity loadingwith co-rotationalelements 251
17.6 Introducingjoints 252
17.7 Special notation 256
17.8 References 257
18 More on continuum and shell elements 260
18.1 Introduction 260
18.2 A co-rotationalapproachfor two-dimensional continua 262
18.3 A co-rotationalapproachfor three-dimensional continua 266
18.4 A co-rotational approachfor a curved membrane using facet triangles 269
18.5 A co-rotational approachfor a curved membrane using quadrilaterals 271
1 8.6 A co-rotational shell formulation with three rotational degrees
of freedom per node 273
18.7 A co-rotationalfacet shellformulation basedon Morley’s triangle 276
18.8 A co-rotational shell formulation with two rotational degrees
of freedom per node 280
18.9 A co-rotational frameworkfor the semi-loof shells 283
18.10An alternative co-rotational frameworkfor three-dimensionalbeams 285
18.1 0.1 Two-dimensionalbeams 286
18.1 1 Incompatible modes, enhanced strains and substitute strainsfor
continuum elements 287
8.1 1.1 Incompatiblemodes 287
18.11.2 Enhanced strains 291
18.1 1.3 Substitute functions 293
18.1 1.4 Numerical comparisons 295
18.12 Introducing extra internal variables into the co-rotational formulation 296
18.13 Introducing extra internal variablesinto the Eulerian formulation 298
18.14 Introducing large elastic strains into the co-rotationalformulation 300
18.15 A simple stability test and alternative enhancedF formulations 301
18.16Special notation 304
18.17 References 305
19 Large strains and plasticity 308
19.1 Introduction 308
19.2 The multiplicativeF,F, approach 309X CONTENTS
19.3 Usingthe F,F, approachto arrive at the conventional ‘rate form’ 31 2
19.4 Usingthe rate form with an ‘explicit dynamic code’ 315
19.5 Integrating the rate equations 316
19.6 An F,F, update basedon the intermediate configuration 320
19.7 An F,F, update basedon the final (current) configuration 324
19.7.1 The flow rule 326
19.8 The consistent tangent 326
19.8.1 The limitingcase 327
19.9 Introducing large elasto-plastic strains into the finite element
formulation 328
19.10 A simple example 332
19.11 Special notation 334
19.12 References 335
20 Stability theory 338
20.1 Introduction 338
20.2 General theory without ‘higher-order terms’ 338
20.2.1 Limit point 343
20.2.2 Bifurcation point 343
20.3 The introductionof higher-order terms 344
20.4 Classificationof singular points 346
20.4.1 Limit points 346
20.4.2 Bifurcation points 347
20.4.3 Symmetric bifurcations 347
20.4.4 Asymmetric bifurcations 347
20.5 Computationof higher-order derivativesfor truss elements 349
20.5.1 Amplificationof notation 349
20.5.2 Truss element usingGreen’s strain 350
20.5.3 Truss elements usinga rotated engineering strain 351
20.5.4 Computationof the stability coefficientsS,-S, 352
20.6 Special notation 352
20.7 References 353
21 Branch switching and further advanced solution procedures 354
21.1 Indirect computationof singular points 355
21.2 Simplebranchswitching 359
21.2.1 Corrector basedon a linearised arc-length method 360
21.2.2 Corrector using displacement control at a specified variable 361
21.2.3 Corrector usinga ‘cylindricalarc-lengthmethod’ 361
21.3 Branch switching using higher-order derivatives 36 1
21.4 General predictorsusing higher-order derivatives 362
21.4.1 Loadcontrol 363
21.4.2 Displacement control at a specified variable 363
21.4.3 The ‘cylindricalarc-lengthmethod’ 364
21.5 Correctorsusinghigher-order derivatives 365
21.6 Direct computationof the singular points 366
21.7 Line-searcheswith arc-length and similar methods 368
21.7.1 Line-searches with the RiksMlempnerarc-length method 368
21.7.2 Line-searches with the cylindrical arc-length method 370
21.7.3 Uphillor downhill? 373
21.8 Alternativearc-lengthmethods using relative variables 373
21.9 An alternative methodfor choosing the root for the cylindrical
arc-lengthmethod 374CONTENTS xi
21.10 Statiddynamicsolution procedures 376
21.1 1 Special notation(seealso Section 20.6) 378
21.12 References 379
22 Examplesfrom an updated non-linear finite element computer
program using truss elements
(written in conjunctionwith Dr Jun Shi) 381
22.1 A two-bar truss with an asymmetric bifurcation 382
22.1.1 Bracketing
22.1.2 Branch switching
22.2 The von Misestruss
22.2.1 Bracketing
22.2.2 Branch switching
22.3 A three-dimensional dome
22.3.1 Bracketing
22.3.2 Branch switching
22.3.3 The higher-order predictor
22.3.4 The higher-order correctors
22.3.5 Line searches
22.4 A three-dimensional arch truss
22.5 A two-dimensional circular arch
22.6 References
23 Contactwith friction
23.1 Introduction
23.2 A two-dimensionalfrictionlesscontact formulation using a penalty approach 412
23.2.1 Some modifications 415
23.3 The ‘contact patchtest’ 417
23.4 Introducing ‘sticking friction’ in two dimensions 420
23.5 IntroducingCoulomb ‘sliding friction’ in two dimensions 422
23.6 Using Lagrangian multipliers insteadof the penalty approach 424
23.7 The augmented Lagrangian methods 426
23.8 An augmented Lagrangian technique with Coulomb ’sliding friction’ 429
23.8.1 A symmetrised version 430
23.9 A three-dimensional frictionless contact formulationusinga penalty
approach 431
23.9.1 The consistent tangent matrix 434
23.10 Adding ‘sticking friction’ in three dimensions 435
23.10.1 The consistent tangent matrix 437
23.11 Coulomb ‘sliding friction’ in three dimensions 430
23.12 A penalty/barrier methodfor contact 439
23.1 3 Amendments to the solution procedures 441
24.14 Special notation 442
23.15 References 444
24 Non-linear dynamics 447
24.1 Introduction 447
24.2 Newmark’s method 447
24.3 The ‘average acceleration method’or ‘trapezoidalrule’ 448
24.4 The ‘implicit solution procedure’ 448xii CONTENTS
24.4.1 The ‘predictorstep’ 449
24.2.2 The ‘corrector’ 449
24.5 An explicit solution procedure 450
24.6 A staggered, central difference, explicit solutionprocedure 451
24.7 Stability 452
24.8 The Hilber-Hughes-Taylor s( method 455
24.9 More on the dynamic equilibriumequations 456
24.10 An energy conserving total Lagrangian formulation 458
24.10.1 The ‘predictor step’ 460
24.10.2 The ‘corrector’ 460
24.11 A co-rotational energy-conserving procedurefor two-dimensionalbeams 461
24.11.1 Sophistications 463
24.11.2 Numericalsolution 464
24.12 An alternative energy-conservingprocedure for two-dimensionalbeams 466
24.13 Automatic time-stepping 468
24.14 Dynamic equilibrium with rotations 470
24.15 An ‘explicit co-rotational procedure’for beams 473
24.16 Updating the rotational velocities and accelerations 474
24.17 A simple implicit co-rotational procedure using rotations 476
24.18 An isoparametric formulationfor three-dimensionalbeams 477
24.19 An alternative implicit co-rotational formulation 479
24.20 (Approximately) energy-conserving co-rotational procedures 480
24.21 Energy-conserving isoparametricformulations 483
24.22 Special notation 485
24.23 References 486
Index
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