Table Of ContentNon-linear Finite Element Analysis
of Solids and Structures
~~ ~
Volume 2: Advanced Topics
To
Kiki, Lou, Max, ArabeIIa
Gideon, Gavin, Rosie and Lucy
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
JOHN WILEY & SONS
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Contents
Preface xiii
10 More continuum mechanics 1
10.1 Relationships between some 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 emphasis on isotropic
conditions 10
10.6 Further work on conjugate stress and strain measures 13
10.6.1 Relationshipb etween i: and U 14
10.6.2 Relationship between the Bio! stress, B and the Kirchhoff stress, T 15
10.6.3 Relationship between U, 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 between 6,the 2’s and the 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-a nd contravariant tensor
components 26
11 .1 Non-orthogonalc oordinates 26
11 .2 Transforming the components of a vector (first-ordert ensor) to a new set of
base vectors 28
11.3 Second-ordert ensors in non-orthogonalc oordinates 30
11 .4 Transforming the components of a second-order tensor to a new set of
base vectors 30
11.5 The metric tensor 31
11.6 Work terms and the trace operation 32
vi CONTENTS
11.7 Covariant components, natural coordinates and the Jacobian 33
1 1.8 Green’s strain and the deformation gradient 35
11.8.1 Recovering the standard cartesian expressions 35
1 1.9 The second Piola-Kirchhoff stresses and the variation of the Green’s
strain 36
1 1.10 Transforming the components of the constitutive tensor 37
11.11 A simple two-dimensional example involving skew coordinates 38
1 1.12 Special notation 42
1 1.13 References 44
12 More finite element analysis of continua 45
12.1 A summary of the key equations for the total Lagrangian formulation 46
12.1.1 The internal force vector 46
12.1.2 The tangent stiffness matrix 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 derivation of the tangent stiffness matrix 49
12.3.2 Discretisedd erivation of the tangent stiffness matrix 51
12.4 The tangent stiffness matrix using the Jaumann rate of Kirchhoff stress 53
12.4.1 Alternative derivation of the tangent stiffness matrix 54
12.5 The tangent stiffness matrix using the Jaumann rate of Cauchy stress 55
12.5.1 Alternative derivation of the tangent stiffness matrix 56
12.6 Convected coordinates and the total Lagrangian formulation 57
12.6.1 Element formulation 57
12.6.2 The tangent stiffness matrix 59
12.6.3 Extensions to three dimensions 59
12.7 Special notation 60
12.8 References 61
13 Large strains, hyperelasticity and 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
13 .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 hybrid formulation 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 Using the Green strain relationships in the principal directions 81
13.8.3 Transforming the tangent constitutive relationshipsf or a ‘Eulerian formulation’ 84
13.9 Examples 86
13.9.1 A simple example 86
13.9.2 The compressible neo-Hookean model 89
13.10 Further work with principal stretch ratios 89
13.10.1 An enerav function usina the DrinciPal loa strains fthe Henckv model) 90
CONTENTS 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 plasticity and other material non-linearity-I 99
14.1 Introduction 99
14.2 Other isotropic yield 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 active yield surfaces 107
14.3.2 A consistent tangent modular matrix with two active yield surfaces 108
14.4 Yield functions for shells that use stress resultants 109
14.4.1 The one-dimensionalc ase 109
14.4.2 The two-dimensionalc ase 112
14.4.3 A backward-Eulerr eturn with the lllyushin yield function 113
14.4.4 A backward-Eulerr eturn and consistent tangent matrix for
the llyushin yield criterion when two yield surfaces are active 114
14.5 Implementing a form of backward-Euler procedure for the
Mohr-Coulomb yield criterion 115
14.5.1 Implementinga 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 criteria for anisotropic plasticity 122
14.6.1 Hill’s yield criterion 122
14.6.2 Hardening with Hill’s yield criterion 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 modular matrix 130
14.8 Hoffman’s yield criterion 131
14.8.1 The consistent tangent modular matrix 133
14.9 The Drucker-Prager yield criterion 133
14.10 Using an eigenvector expansion for the stresses 134
14.10.1 An example involving plane-stress plasticity and the von Mises
yield criterion 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 models in concrete 140
14.11.3 Relationship between the ‘rotating crack model’ and
a ‘deformationt heory’ plasticity approach using the ‘square yield criterion’ 142
14.11.4 A flow theory approach for the ‘square yield criterion’ 144
14.12 Damage mechanics 148
14.13 Special notation 152
14.14 References 154
15 More plasticity and other material non-linearity-ll 158
15.1 Introduction 158
15.2 Mixed hardening 163
15.3 Kinematic hardening for plane stress 164
viii 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 procedure for 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 and other hardening models 17 4
15.10 .1 Sophisticated overlay model 178
15.10.2 Relationshipw ith conventional kinematic hardening 180
15.10 .3 Other models 180
15.11 Computer exercises 181
15.12 Viscoplasticity 182
15.12.1 The consistent tangent matrix 184
15.12 .2 Implementation 185
15.13 Special notation 185
15.14 References 186
16 Large rotations 108
16.1 Non-vectoriall arge rotations 188
16.2 A rotation matrix for small (infinitesimal)r otations 188
16.3 A rotation matrix for large rotations (Rodrigues formula) 191
16.4 The exponential form for the rotation matrix 194
16.5 Alternative forms for the rotation matrix 194
16.6 Approximations for the rotation matrix 195
16.7 Compound rotations 195
16.8 Obtaining the pseudo-vector from the rotation matrix, R 197
16.9 Quaternions and Euler parameters 198
16.10 Obtaining the normalised quarternion from the rotation matrix 199
16.11 Additive and non-additiver otation increments 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 Expressions for curvature that directly use nodal triads 204
16.14.2 Curvature without nodal triads 207
16.15 Special notation 21 1
16.16 Refe rences 212
17 Three-dimensional formulations for beams and rods 213
17 .1 A co-rotationalf ramework for three-dimensional beam elements 21 3
17.1.1 Computing the local ‘displacements’ 21 6
17.1.2 Computation of the matrix connecting the infinitesimal local
and global variables 218
17.1.3 The tangent stiffness matrix 22 1
17.1.4 Numerical implementationo f the rotational updates 223
17.1.5 Overall solution strategy with 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 228
CONTENTS ix
17.2.4 Virtual work and the internal force 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 relationshipw ith the formulation of
Dvorkin et al. 239
17.4 Symmetry and the use of different ‘rotation variables’ 240
17.4.1 A simple model showing symmetry and non-symmetry 24 1
17.4.2 Using additive rotation components 242
17.4.3 Considering symmetry at equilibrium for the element of Section 17.2 243
17.4.4 Using additive (in the limit) rotation components with the element
of Section 17.2 245
17.5 Various forms of applied loading including ‘follower levels’ 248
17.5.1 Point loads applied at a node 248
17.5.2 Concentratedm oments applied at a node 249
17.5.3 Gravity loading with co-rotationale lements 251
17.6 Introducing joints 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-rotationala pproach for two-dimensional continua 262
18.3 A co-rotationala pproach for three-dimensional continua 266
18.4 A co-rotational approach for a curved membrane using facet triangles 269
18.5 A co-rotational approach for a curved membrane using quadrilaterals 271
18 .6 A co-rotational shell formulation with three rotational degrees
of freedom per node 273
18.7 A co-rotationalf acet shell formulation based on 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 framework for the semi-loof shells 283
18.10 An alternative co-rotational framework for three-dimensional beams 285
18.10 .1 Two-dimensionalb eams 286
18.11 Incompatible modes, enhanced strains and substitute strains for
continuum elements 287
8.1 1.1 Incompatiblem odes 287
18.11 .2 Enhanced strains 29 1
18.11 .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 variables into the Eulerian formulation 298
18.14 Introducing large elastic strains into the co-rotationalf ormulation 300
18.15 A simple stability test and alternative enhanced F formulations 301
18.16 Special notation 304
18.17 References 305
19 Large strains and plasticity 308
19.1 Introduction 308
19.2 The multiplicative F,F, approach 309
