Computational methods for plasticity: theory and applications
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| Main Authors: | , , |
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| Format: | Book |
| Language: | English |
| Published: |
Hoboken, NJ
Wiley
2008
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| Edition: | 1. publ. |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XXII, 791 S. Ill., graph. Darst. |
| ISBN: | 9780470694527 0470694521 |
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| 084 | |a MTA 070f |2 stub | ||
| 084 | |a MAT 674f |2 stub | ||
| 100 | 1 | |a Neto, Eduardo de Souza |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Computational methods for plasticity |b theory and applications |c E. A. de Souza Neto ; D. Perić ; D. R. J. Owen |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Hoboken, NJ |b Wiley |c 2008 | |
| 300 | |a XXII, 791 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Includes bibliographical references and index | ||
| 650 | 4 | |a Mathematisches Modell | |
| 650 | 4 | |a Plasticity |x Mathematical models | |
| 650 | 0 | 7 | |a Computerunterstütztes Verfahren |0 (DE-588)4139030-1 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Plastizität |0 (DE-588)4046283-3 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Mathematisches Modell |0 (DE-588)4114528-8 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Plastizität |0 (DE-588)4046283-3 |D s |
| 689 | 0 | 1 | |a Mathematisches Modell |0 (DE-588)4114528-8 |D s |
| 689 | 0 | 2 | |a Computerunterstütztes Verfahren |0 (DE-588)4139030-1 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Perić, Djordje |e Verfasser |4 aut | |
| 700 | 1 | |a Owens, David |e Verfasser |4 aut | |
| 856 | 4 | 2 | |q text/html |u http://deposit.dnb.de/cgi-bin/dokserv?id=3151997&prov=M&dok_var=1&dok_ext=htm |3 Inhaltstext |
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CONTENTS
Preface
xx
Part One Basic concepts
1
1
Introduction
3
1.1
Aims and scope
. 3
1.1.1
Readership
. 4
1.2
Layout
. 4
1.2.1
The use of boxes
. 7
1.3
General scheme of notation
. 7
1.3.1
Character fonts. General convention
. 8
1.3.2
Some important characters
. 9
1.3.3
Indiciai
notation, subscripts and superscripts
. 13
1.3.4
Other important symbols and operations
. 14
2
Elements of tensor analysis
17
2.1
Vectors
. 17
2.1.1
Inner product, norm and orthogonality
. 17
2.1.2
Orthogonal bases and Cartesian coordinate frames
. 18
2.2
Second-order tensors
. 19
2.2.1
The transpose. Symmetric and skew tensors
. 19
2.2.2
Tensor products
. 20
2.2.3
Cartesian components and matrix representation
. 21
2.2.4
Trace, inner product and Euclidean norm
. 22
2.2.5
Inverse tensor. Determinant
. 23
2.2.6
Orthogonal tensors. Rotations
. 23
2.2.7
Cross product
. 24
2.2.8
Spectral decomposition
. 25
2.2.9
Polar decomposition
. 28
2.3
Higher-order tensors
. 28
2.3.1
Fourth-order tensors
. 29
2.3.2
Generic-order tensors
. 30
2.4 Isotropie
tensors
. 30
2.4.1 Isotropie
second-order tensors
. 30
2.4.2 Isotropie
fourth-order tensors
. 30
viii CONTENTS
2.5 Differentiation. 32
2.5.1 The derivative map.
Directional
derivative. 32
2.5.2
Linearisation of a nonlinear function.
32
2.5.3
The gradient
. 32
2.5.4
Derivatives of functions of vector and tensor arguments
. 33
2.5.5
The chain rule
. 36
2.5.6
The product rule
. 37
2.5.7
The divergence
. 37
2.5.8
Useful relations involving the gradient and the divergence
. 38
2.6
Linearisation of nonlinear problems
. 38
2.6.1
The nonlinear problem and its linearised form
. 38
2.6.2
Linearisation in infinite-dimensional functional spaces
. 39
3
Elements of continuum mechanics and thermodynamics
41
3.1
Kinematics of deformation
. 41
3.1.1
Material and spatial fields
. 44
3.1.2
Material and spatial gradients, divergences and time derivatives
. 46
3.1.3
The deformation gradient
. 46
3.1.4
Volume changes. The determinant of the deformation gradient
. . 47
3.1.5
Isochoric/volumetric split of the deformation gradient
. 49
3.1.6
Polar decomposition. Stretches and rotation
. 49
3.1.7
Strain measures
. 52
3.1.8
The velocity gradient. Rate of deformation and spin
. 55
3.1.9
Rate of volume change
. 56
3.2
Infinitesimal deformations
. 57
3.2.1
The infinitesimal strain tensor
. 57
3.2.2
Infinitesimal rigid deformations
. 58
3.2.3
Infinitesimal isochoric and volumetric deformations
. 58
3.3
Forces. Stress Measures
. 60
3.3.1
Cauchy's axiom. The Cauchy stress vector
. 61
3.3.2
The axiom of momentum balance
. 61
3.3.3
The Cauchy stress tensor
. 62
3.3.4
The First Piola-Kirchhoff stress
. 64
3.3.5
The Second Piola-Kirchhoff stress
. 66
3.3.6
The
Kirchhoff
stress
. 67
3.4
Fundamental laws of thermodynamics
. 67
3.4.1
Conservation of mass
. 67
3.4.2
Momentum balance
. 67
3.4.3
The first principle
. 68
3.4.4
The second principle
. 68
3.4.5
The Clausius-Duhem inequality
. 69
3.5
Constitutive theory
. 69
3.5.1
Constitutive axioms
. 69
3.5.2
Thermodynamics with internal variables
. 71
3.5.3
Phenomenological and micromechanical approaches
. 74
CONTENTS ix
3.5.4
The purely mechanical theory
. 75
3.5.5
The constitutive initial value problem
. 76
3.6
Weak equilibrium. The principle of virtual work
. 77
3.6.1
The spatial version
. 77
3.6.2
The material version
. 78
3.6.3
The infinitesimal case
. 78
3.7
The quasi-static initial boundary value problem
. 79
3.7.1
Finite deformations
. 79
3.7.2
The infinitesimal problem
. 81
4
The finite element method in quasi-static nonlinear solid mechanics
83
4.1
Displacement-based finite elements
. 84
4.1.1
Finite element interpolation
. 85
4.1.2
The discretised virtual work
. 86
4.1.3
Some typical isoparametric elements
. 90
4.1.4
Example. Linear elasticity
. 93
4.2
Path-dependent materials. The incremental finite element procedure
. 94
4.2.1
The incremental constitutive function
. 95
4.2.2
The incremental boundary value problem
. 95
4.2.3
The nonlinear incremental finite element equation
. 96
4.2.4
Nonlinear solution. The Newton-Raphson scheme
. 96
4.2.5
The consistent tangent modulus
. 98
4.2.6
Alternative nonlinear solution schemes
. 99
4.2.7
Non-incremental procedures for path-dependent materials
. 101
4.3
Large strain formulation
. 102
4.3.1
The incremental constitutive function
. 102
4.3.2
The incremental boundary value problem
. 103
4.3.3
The finite element equilibrium equation
. 103
4.3.4
Linearisation. The consistent spatial tangent modulus
. 103
4.3.5
Material and geometric stiffnesses
. 106
4.3.6
Configuration-dependent loads. The load-stiffness matrix
. 106
4.4
Unstable equilibrium. The arc-length method
. 107
4.4.1
The arc-length method
. 107
4.4.2
The combined Newton-Raphson/arc-length procedure
. 108
4.4.3
The predictor solution
.
Ill
5
Overview of the program structure
115
5.1
Introduction
. 115
5.1.1
Objectives
. 115
5.1.2
Remarks on program structure
. 116
5.1.3
Portability
. 116
5.2
The main program
. 117
5.3
Data input and initialisation
. 117
5.3.1
The global database
. 117
5.3.2
Main problem-defining data. Subroutine
INDATA. 119
CONTENTS
5.3.3
External loading.
Subroutine INLOAD
. 119
5.3.4
Initialisation of variable data. Subroutine
INITIA
. 120
5.4
The load incrementation loop. Overview
. 120
5.4.1
Fixed increments option
. 120
5.4.2
Arc-length control option
. 120
5.4.3
Automatic increment cutting
. 123
5.4.4
The linear solver. Subroutine FRONT
. 124
5.4.5
Internal force calculation. Subroutine INTFOR
. 124
5.4.6
Switching data. Subroutine SWITCH
. 124
5.4.7
Output of converged results. Subroutines OUTPUT and RSTART
. . 125
5.5
Material and element modularity
. 125
5.5.1
Example. Modularisation of internal force computation
. 125
5.6
Elements. Implementation and management
. 128
5.6.1
Element properties. Element routines for data input
. 128
5.6.2
Element interfaces. Internal force and stiffness computation
. 129
5.6.3
Implementing a new finite element
. 129
5.7
Material models: implementation and management
. 131
5.7.1
Material properties. Material-specific data input
. 131
5.7.2
State variables and other Gauss point quantities.
Material-specific state updating routines
. 132
5.7.3
Material-specific switching/initialising routines
. 133
5.7.4
Material-specific tangent computation routines
. 134
5.7.5
Material-specific results output routines
. 134
5.7.6
Implementing a new material model
. 135
Part Two Small strains
137
6
The mathematical theory of plasticity
139
6.1
Phenomenological aspects
. 140
6.2
One-dimensional constitutive model
. 141
6.2.1
Elastoplastic decomposition of the axial strain
. 142
6.2.2
The elastic
uniaxial
constitutive law
. 143
6.2.3
The yield function and the yield criterion
. 143
6.2.4
The plastic flow rale. Loading/unloading conditions
. 144
6.2.5
The hardening law
. 145
6.2.6
Summary of the model
. 145
6.2.7
Determination of the plastic multiplier
. 146
6.2.8
The elastoplastic tangent modulus
. 147
6.3
General elastoplastic constitutive model
. 148
6.3.1
Additive decomposition of the strain tensor
. 148
6.3.2
The free energy potential and the elastic law
. 148
6.3.3
The yield criterion and the yield surface
. 150
6.3.4
Plastic flow rule and hardening law
. 150
6.3.5
Flow rales derived from a flow potential
. 151
6.3.6
The plastic multiplier
. 152
CONTENTS xi
6.3.7
Relation to the general continuum constitutive theory
. 153
6.3.8
Rate form and the elastoplastic tangent operator
. 153
6.3.9
Non-smooth potentials and the subdifferential
. 153
6.4
Classical yield criteria
. 157
6.4.1
The
Tresca
yield criterion
. 157
6.4.2
The
von
Mises
yield criterion
. 162
6.4.3
The Mohr-Coulomb yield criterion
. 163
6.4.4
The Drucker-Prager yield criterion
. 166
6.5
Plastic flow rules
. 168
6.5.1
Associative and non-associative plasticity
. 168
6.5.2
Associative laws and the principle of maximum plastic dissipation
170
6.5.3
Classical flow rules
. 171
6.6
Hardening laws
. 177
6.6.1
Perfect plasticity
. 177
6.6.2 Isotropie
hardening
. 178
6.6.3
Thermodynamical aspects. Associative
isotropie
hardening
. . 182
6.6.4
Kinematic hardening. The Bauschinger effect
. 185
6.6.5
Mixed isotropic/kinematic hardening
. 189
7
Finite elements in small-strain plasticity problems
191
7.1
Preliminary implementation aspects
. 192
7.2
General numerical integration algorithm for elastoplastic constitutive
equations
. 193
7.2.1
The elastoplastic constitutive initial value problem
. 193
7.2.2
Euler
discretisation: the incremental constitutive problem
. 194
7.2.3
The elastic predictor/plastic corrector algorithm
. 196
7.2.4
Solution of the return-mapping equations
. 198
7.2.5
Closest point projection interpretation
. 200
7.2.6
Alternative justification:
operatorsplit
method
. 201
7.2.7
Other elastic predictor/return-mapping schemes
. 201
7.2.8
Plasticity and differential-algebraic equations
. 209
7.2.9
Alternative mathematical programming-based algorithms
. 210
7.2.10
Accuracy and stability considerations
. 210
7.3
Application: integration algorithm for the isotropically hardening
von
Mises
model
. 215
7.3.1
The implemented model
. 216
7.3.2
The implicit elastic predictor/return-mapping scheme
. 217
7.3.3
The incremental constitutive function for the stress
. 220
7.3.4
Linear
isotropie
hardening and perfect plasticity: the
closed-form return mapping
. 223
7.3.5
Subroutine SUVM
. 224
7.4
The consistent tangent modulus
. 228
7.4.1
Consistent tangent operators in elastoplasticity
. 229
7.4.2
The elastoplastic consistent tangent for the
von
Mises
model
with
isotropie
hardening
. 232
7.4.3
Subroutine CTVM
. 235
XU CONTENTS
7.4.4
The general elastoplastic consistent tangent operator for implicit
return mappings
. 238
7.4.5
Illustration: the
von
Mises
model with
isotropie
hardening
. 240
7.4.6
Tangent operator symmetry: incremental potentials
. 243
7.5
Numerical examples with the
von
Mises
model
.244
7.5.1
Internally pressurised cylinder
.244
7.5.2
Internally pressurised spherical shell
.247
7.5.3
Uniformly loaded circular plate
.250
7.5.4
Strip-footing collapse
.252
7.5.5
Double-notched tensile specimen
.255
7.6
Further application: the
von
Mises
model with nonlinear mixed hardening
. 257
7.6.1
The mixed hardening model: summary
. 257
7.6.2
The implicit return-mapping scheme
. 258
7.6.3
The incremental constitutive function
. 260
7.6.4
Linear hardening: closed-form return mapping
. 261
7.6.5
Computational implementation aspects
. 261
7.6.6
The elastoplastic consistent tangent
. 262
8
Computations with other basic plasticity models
265
8.1
The
Tresca
model
. 266
8.1.1
The implicit integration algorithm in principal stresses
. 268
8.1.2
Subroutine SUTR
. 279
8.1.3
Finite step accuracy: iso-error maps
. 283
8.1.4
The consistent tangent operator for the
Tresca
model
. 286
8.1.5
Subroutine CTTR
. 291
8.2
The Mohr-Coulomb model
. 295
8.2.1
Integration algorithm for the Mohr-Coulomb model
. 297
8.2.2
Subroutine SUMC
. 310
8.2.3
Accuracy: iso-error maps
. 315
8.2.4
Consistent tangent operator for the Mohr-Coulomb model
. 316
8.2.5
Subroutine CTMC
. 319
8.3
The Drucker-Prager model
. 324
8.3.1
Integration algorithm for the Drucker-Prager model
. 325
8.3.2
Subroutine SUDP
. 334
8.3.3
Iso-error map
. 337
8.3.4
Consistent tangent operator for the Drucker-Prager model
. 337
8.3.5
Subroutine CTDP
. 340
8.4
Examples
. 343
8.4.1
Bending of a V-notched
Tresca bar
. 343
8.4.2
End-loaded tapered cantilever
. 344
8.4.3
Strip-footing collapse
. 346
8.4.4
Circular-footing collapse
. 350
8.4.5
Slope stability
. 351
CONTENTS xiii
9 Plane
stress
plasticity
357
9.1
The basic
plane
stress plasticity problem
. 357
9.1.1
Plane stress linear elasticity
. 358
9.1.2
The constrained elastoplastic initial value problem
. 359
9.1.3
Procedures for plane stress plasticity
. 360
9.2
Plane stress constraint at the Gauss point level
. 361
9.2.1
Implementation aspects
. 362
9.2.2
Plane stress enforcement with nested iterations
. 362
9.2.3
Plane stress
von
Mises
with nested iterations
. 364
9.2.4
The consistent tangent for the nested iteration procedure
. 366
9.2.5
Consistent tangent computation for the
von
Mises
model
. 366
9.3
Plane stress constraint at the structural level
. 367
9.3.1
The method
. 367
9.3.2
The implementation
. 368
9.4
Plane stress-projected plasticity models
. 370
9.4.1
The plane stress-projected
von
Mises
model
. 371
9.4.2
The plane stress-projected integration algorithm
. 373
9.4.3
Subroutine SUVMPS
. 378
9.4.4
The elastoplastic consistent tangent operator
. 382
9.4.5
Subroutine CTVMPS
. 383
9.5
Numerical examples
. 386
9.5.1
Collapse of an end-loaded cantilever
. 387
9.5.2
Infinite plate with a circular hole
. 387
9.5.3
Stretching of a perforated rectangular plate
. 390
9.5.4
Uniform loading of a concrete shear wall
. 391
9.6
Other stress-constrained states
. 396
9.6.1
A three-dimensional
von
Mises
Timoshenko beam
. 396
9.6.2
The beam state-projected integration algorithm
. 400
10
Advanced plasticity models
403
10.1
A modified Cam-Clay model for soils
.403
10.1.1
The model
.404
10.1.2
Computational implementation
.406
10.2
A capped Drucker-Prager model for geomaterials
.409
10.2.1
Capped Drucker-Prager model
.410
10.2.2
The implicit integration algorithm
.412
10.2.3
The elastoplastic consistent tangent operator
.413
10.3 Anisotropie
plasticity: the Hill, Hoffman and Barlat-Lian models
.414
10.3.1
The Hill orthotropic model
.414
10.3.2
Tension-compression distinction: the Hoffman model
.420
10.3.3
Implementation of the Hoffman model
.423
10.3.4
The Barlat-Lian model for sheet metals
.427
10.3.5
Implementation of the Barlat-Lian model
.431
xiv CONTENTS
11 Viscoplasticity 435
11.1 Viscoplasticity: phenomenological
aspects
. 436
11.2
One-dimensional
viscoplasticity
model.
437
11.2.1
Elastoplastic decomposition of the
axial
strain
. 437
11.2.2
The elastic law
. 438
11.2.3
The yield function and the elastic domain
. 438
11.2.4
Viscoplastic flow rule
. 438
11.2.5
Hardening law
. 439
11.2.6
Summary of the model
. 439
11.2.7
Some simple analytical solutions
. 439
11.3
A
von
Mises-based multidimensional model
. 445
11.3.1
A
von
Mises-type
viscoplastic model with
isotropie
strain
hardening
. 445
11.3.2
Alternative plastic strain rate definitions
. 447
11.3.3
Other
isotropie
and kinematic hardening laws
. 448
11.3.4
Viscoplastic models without a yield surface
. 448
11.4
General viscoplastic constitutive model
. 450
11.4.1
Relation to the general continuum constitutive theory
. 450
11.4.2
Potential structure and dissipation inequality
. 451
11.4.3
Rate-independent plasticity as a limit case
. 452
11.5
General numerical framework
. 454
11.5.1
A general implicit integration algorithm
. 454
11.5.2
Alternative Euler-based algorithms
. 457
11.5.3
General consistent tangent operator
. 458
11.6
Application: computational implementation of a
von
Mises-based model
. 460
11.6.1
Integration algorithm
. 460
11.6.2
Iso-error maps
. 463
11.6.3
Consistent tangent operator
. 464
11.6.4
Perzyna-type model implementation
. 466
11.7
Examples
. 467
11.7.1
Double-notched tensile specimen
. 467
11.7.2
Plane stress: stretching of a perforated plate
. 469
12
Damage mechanics
471
12.1
Physical aspects of internal damage in solids
. 472
12.1.1
Metals
. 472
12.1.2
Rubbery polymers
. 473
12.2
Continuum damage mechanics
. 473
12.2.1
Original development: creep-damage
. 474
12.2.2
Other theories
. 475
12.2.3
Remarks on the nature of the damage variable
. 476
12.3
Lemaitre's elastoplastic damage theory
. 478
12.3.1
The model
. 478
12.3.2
Integration algorithm
. 482
12.3.3
The tangent operators
. 485
CONTENTS
XV
12.4
A simplified version of Lemaitre's model
. 486
12.4.1
The single-equation integration algorithm
. 486
12.4.2
The tangent operator
. 490
12.4.3
Example. Fracturing of a cylindrical notched specimen
. 493
12.5
Gurson's void growth model
. 496
12.5.1
The model
. 497
12.5.2
Integration algorithm
. 501
12.5.3
The tangent operator
. 502
12.6
Further issues in damage modelling
. 504
12.6.1
Crack closure effects in damaged elastic materials
. 504
12.6.2
Crack closure effects in damage evolution
. 510
12.6.3 Anisotropie
ductile damage
. 512
Part Three Large strains
517
13
Finite strain hyperelasticity
519
13.1
Hyperelasticity: basic concepts
. 520
13.1.1
Material objectivity: reduced form of the free-energy function
. . 520
13.1.2 Isotropie
hyperelasticity
. 521
13.1.3
Incompressible hyperelasticity
. 524
13.1.4
Compressible
régularisation
. 525
13.2
Some particular models
. 525
13.2.1
The Mooney-Rivlin and the neo-Hookean models
. 525
13.2.2
The Ogden material model
. 527
13.2.3
The Hencky material
. 528
13.2.4
The Blatz-Ko material
. 530
13.3 Isotropie
finite hyperelasticity in plane stress
. 530
13.3.1
The plane stress incompressible Ogden model
. 531
13.3.2
The plane stress Hencky model
. 532
13.3.3
Plane stress with nested iterations
. 533
13.4
Tangent moduli: the elasticity tensors
. 534
13.4.1
Regularised neo-Hookean model
. 535
13.4.2
Principal stretches representation: Ogden model
. 535
13.4.3
Hencky model
. 537
13.4.4
Blatz-Ko material
. 537
13.5
Application: Ogden material implementation
. 538
13.5.1
Subroutine
SU0GD . 538
13.5.2
Subroutine CSTOGD
. 542
13.6
Numerical examples
. 546
13.6.1
Axisymmetric extension of an annular plate
. 547
13.6.2
Stretching of a square perforated rubber sheet
. 547
13.6.3
Inflation of a spherical rubber balloon
. 550
13.6.4
Rugby ball
. 551
13.6.5
Inflation of initially flat membranes
. 552
xvi CONTENTS
13.6.6
Rubber cylinder pressed between two plates
.555
13.6.7
Elastomeric bead compression
.556
13.7
Hyperelasticity with damage: the Mullins effect
.557
13.7.1
The Gurtin-Francis
uniaxial
model
. 560
13.7.2
Three-dimensional modelling.
Abrief
review
. 562
13.7.3
A simple rate-independent three-dimensional model
. 562
13.7.4
Example: the model problem
. 565
13.7.5
Computational implementation
. 565
13.7.6
Example: inflation/deflation of a damageable rubber balloon
. . . 569
14
Finite strain elastoplasticity
573
14.1
Finite strain elastoplasticity: a brief review
.574
14.2
One-dimensional finite plasticity model
.575
14.2.1
The multiplicative split of the axial stretch
.575
14.2.2
Logarithmic stretches and the Hencky hyperelastic law
.576
14.2.3
The yield function
.576
14.2.4
The plastic flow rule
.576
14.2.5
The hardening law
.577
14.2.6
The plastic multiplier
.577
14.3
General hyperelastic-based multiplicative plasticity model
.578
14.3.1
Multiplicative elastoplasticity kinematics
.578
14.3.2
The logarithmic elastic strain measure
.582
14.3.3
A general
isotropie
large-strain plasticity model
.583
14.3.4
The dissipation inequality
.586
14.3.5
Finite strain extension to infinitesimal theories
.588
14.4
The general elastic predictor/return-mapping algorithm
.590
14.4.1
The basic constitutive initial value problem
.590
14.4.2
Exponential map backward discretisation
.591
14.4.3
Computational implementation of the general algorithm
.595
14.5
The consistent spatial tangent modulus
.597
14.5.1
Derivation of the spatial tangent modulus
.598
14.5.2
Computational implementation
.599
14.6
Principal stress space-based implementation
.599
14.6.1
Stress-updating algorithm
.600
14.6.2
Tangent modulus computation
.601
14.7
Finite plasticity in plane stress
.601
14.7.1
The plane stress-projected finite
von
Mises
model
.601
14.7.2
Nested iteration for plane stress enforcement
.604
14.8
Finite viscoplasticity
.605
14.8.1
Numerical treatment
.606
14.9
Examples
.606
14.9.1
Finite strain bending of a V-notched
Tresca
bar
.606
14.9.2
Necking of a cylindrical bar
.607
14.9.3
Plane strain localisation
.611
14.9.4
Stretching of a perforated plate
.613
14.9.5
Thin sheet metal-forming application
.614
CONTENTS xvii
14.10 Rate
forms: hypoelastic-based plasticity models
. 615
14.10.1
Objective stress rates
. 619
14.10.2
Hypoelastic-based plasticity models
. 621
14.10.3
The Jaumann rate-based model
. 622
14.10.4
Hyperelastic-based models and equivalent rate forms
. 624
14.10.5
Integration algorithms and incremental objectivity
. 625
14.10.6
Objective algorithm for Jaumann rate-based models
. 628
14.10.7
Integration of Green-Naghdi rate-based models
. 632
14.11
Finite plasticity with kinematic hardening
. 633
14.11.1
A model of finite strain kinematic hardening
. 633
14.11.2
Integration algorithm
. 637
14.11.3
Spatial tangent operator
. 642
14.11.4
Remarks on predictive capability
. 644
14.11.5
Alternative descriptions
. 644
15
Finite elements for large-strain incompressibility
647
15.1
The F-bar methodology
. 648
15.1.1
Stress computation: the F-bar deformation gradient
. 649
15.1.2
The internal force vector
. 651
15.1.3
Consistent linearisation: the tangent stiffness
. 652
15.1.4
Plane strain implementation
. 655
15.1.5
Computational implementation aspects
. 656
15.1.6
Numerical tests
. 656
15.1.7
Other centroid sampling-based F-bar elements
. 663
15.1.8
A more general F-bar methodology
. 663
15.1.9
The F-bar-Patch Method for simplex elements
. 665
15.2
Enhanced assumed strain methods
. 669
15.2.1
Enhanced three-field variational principle
. 669
15.2.2
EAS finite elements
. 671
15.2.3
Finite element equations: static condensation
. 676
15.2.4
Implementation aspects
. 678
15.2.5
The stability of EAS elements
. 678
15.3
Mixed u/p formulations
. 683
15.3.1
The two-field variational principle
. 683
15.3.2
Finite element equations
. 685
15.3.3
Solution: static condensation
. 687
15.3.4
Implementation aspects
. 689
16 Anisotropie
finite plasticity: Single crystals
691
16.1
Physical aspects
. 692
16.1.1
Plastic deformation by slip: slip-systems
. 692
16.2
Plastic slip and the
Schmid
resolved shear stress
. 693
16.3
Single crystal simulation: a brief review
. 694
16.4
A general continuum model of single crystals
. 694
16.4.1
The plastic flow equation
. 695
16.4.2
The resolved
Schmid
shear stress
. 696
xviii CONTENTS
16.4.3 Multisurface
formulation of the flow rule
.696
16.4.4 Isotropie
Taylor hardening
.698
16.4.5
The hyperelastic law
.698
16.5
A general integration algorithm
.699
16.5.1
The search for an active set of slip systems
.703
16.6
An algorithm for a planar double-slip model
.705
16.6.1
A planar double-slip model
. 705
16.6.2
The integration algorithm
. 707
16.6.3
Example: the model problem
. 710
16.7
The consistent spatial tangent modulus
. 713
16.7.1
The elastic modulus: compressible neo-Hookean model
.713
16.7.2
The elastoplastic consistent tangent modulus
.714
16.8
Numerical examples
.716
16.8.1
Symmetric strain localisation on a rectangular strip
.717
16.8.2
Unsymmetric localisation
.720
16.9
Viscoplastic single crystals
.721
16.9.1
Rate-dependent formulation
.723
16.9.2
The exponential map-based integration algorithm
.724
16.9.3
The spatial tangent modulus: neo-Hookean-based model
.725
16.9.4
Rate-dependent crystal: model problem
.726
Appendices
729
A
Isotropie
functions of a symmetric tensor
731
A.I
Isotropie
scalar-valued functions
. 731
A.I.I Representation
. 732
A.
1.2
The derivative of an
isotropie
scalar function
. 732
A.
2
Isotropie
tensor-valued functions
. 733
A.2.1 Representation
. 733
A.2.2 The derivative of an
isotropie
tensor function
. 734
A.3 The two-dimensional case
. 735
A.3.1 Tensor function derivative
. 736
A.
3.2
Plane strain and axisymmetric problems
. 738
A.4 The three-dimensional case
. 739
A.
4.1
Function computation
. 739
A.4.2 Computation of the function derivative
. 740
A.5 A particular class of
isotropie
tensor functions
. 740
A.5.1 Two dimensions
. 742
A.5.
2
Three dimensions
. 743
A.6 Alternative procedures
. 744
CONTENTS xix
В
The tensor exponential
747
B.I The tensor exponential function
. 747
B.I
.1
Some properties of the tensor exponential function
. 748
B.I.
2
Computation of the tensor exponential function
. 749
B.2 The tensor exponential derivative
. 750
B.2.1 Computer implementation
. 751
B.3 Exponential map integrators
. 751
B.3.1 The generalised exponential map midpoint rule
. 752
С
Linearisation of the virtual work
753
C.I Infinitesimal deformations
. 753
C.2 Finite strains and deformations
. 755
C.2.
1
Material description
. 755
C.2.
2
Spatial description
. 756
D
Array notation for computations with tensors
759
D.
1
Second-order tensors
. 759
D.2 Fourth-order tensors
. 761
D.2.1 Operations with non-symmetric tensors
. 763
References
765
Index
783 |
| adam_txt |
CONTENTS
Preface
xx
Part One Basic concepts
1
1
Introduction
3
1.1
Aims and scope
. 3
1.1.1
Readership
. 4
1.2
Layout
. 4
1.2.1
The use of boxes
. 7
1.3
General scheme of notation
. 7
1.3.1
Character fonts. General convention
. 8
1.3.2
Some important characters
. 9
1.3.3
Indiciai
notation, subscripts and superscripts
. 13
1.3.4
Other important symbols and operations
. 14
2
Elements of tensor analysis
17
2.1
Vectors
. 17
2.1.1
Inner product, norm and orthogonality
. 17
2.1.2
Orthogonal bases and Cartesian coordinate frames
. 18
2.2
Second-order tensors
. 19
2.2.1
The transpose. Symmetric and skew tensors
. 19
2.2.2
Tensor products
. 20
2.2.3
Cartesian components and matrix representation
. 21
2.2.4
Trace, inner product and Euclidean norm
. 22
2.2.5
Inverse tensor. Determinant
. 23
2.2.6
Orthogonal tensors. Rotations
. 23
2.2.7
Cross product
. 24
2.2.8
Spectral decomposition
. 25
2.2.9
Polar decomposition
. 28
2.3
Higher-order tensors
. 28
2.3.1
Fourth-order tensors
. 29
2.3.2
Generic-order tensors
. 30
2.4 Isotropie
tensors
. 30
2.4.1 Isotropie
second-order tensors
. 30
2.4.2 Isotropie
fourth-order tensors
. 30
viii CONTENTS
2.5 Differentiation. 32
2.5.1 The derivative map.
Directional
derivative. 32
2.5.2
Linearisation of a nonlinear function.
32
2.5.3
The gradient
. 32
2.5.4
Derivatives of functions of vector and tensor arguments
. 33
2.5.5
The chain rule
. 36
2.5.6
The product rule
. 37
2.5.7
The divergence
. 37
2.5.8
Useful relations involving the gradient and the divergence
. 38
2.6
Linearisation of nonlinear problems
. 38
2.6.1
The nonlinear problem and its linearised form
. 38
2.6.2
Linearisation in infinite-dimensional functional spaces
. 39
3
Elements of continuum mechanics and thermodynamics
41
3.1
Kinematics of deformation
. 41
3.1.1
Material and spatial fields
. 44
3.1.2
Material and spatial gradients, divergences and time derivatives
. 46
3.1.3
The deformation gradient
. 46
3.1.4
Volume changes. The determinant of the deformation gradient
. . 47
3.1.5
Isochoric/volumetric split of the deformation gradient
. 49
3.1.6
Polar decomposition. Stretches and rotation
. 49
3.1.7
Strain measures
. 52
3.1.8
The velocity gradient. Rate of deformation and spin
. 55
3.1.9
Rate of volume change
. 56
3.2
Infinitesimal deformations
. 57
3.2.1
The infinitesimal strain tensor
. 57
3.2.2
Infinitesimal rigid deformations
. 58
3.2.3
Infinitesimal isochoric and volumetric deformations
. 58
3.3
Forces. Stress Measures
. 60
3.3.1
Cauchy's axiom. The Cauchy stress vector
. 61
3.3.2
The axiom of momentum balance
. 61
3.3.3
The Cauchy stress tensor
. 62
3.3.4
The First Piola-Kirchhoff stress
. 64
3.3.5
The Second Piola-Kirchhoff stress
. 66
3.3.6
The
Kirchhoff
stress
. 67
3.4
Fundamental laws of thermodynamics
. 67
3.4.1
Conservation of mass
. 67
3.4.2
Momentum balance
. 67
3.4.3
The first principle
. 68
3.4.4
The second principle
. 68
3.4.5
The Clausius-Duhem inequality
. 69
3.5
Constitutive theory
. 69
3.5.1
Constitutive axioms
. 69
3.5.2
Thermodynamics with internal variables
. 71
3.5.3
Phenomenological and micromechanical approaches
. 74
CONTENTS ix
3.5.4
The purely mechanical theory
. 75
3.5.5
The constitutive initial value problem
. 76
3.6
Weak equilibrium. The principle of virtual work
. 77
3.6.1
The spatial version
. 77
3.6.2
The material version
. 78
3.6.3
The infinitesimal case
. 78
3.7
The quasi-static initial boundary value problem
. 79
3.7.1
Finite deformations
. 79
3.7.2
The infinitesimal problem
. 81
4
The finite element method in quasi-static nonlinear solid mechanics
83
4.1
Displacement-based finite elements
. 84
4.1.1
Finite element interpolation
. 85
4.1.2
The discretised virtual work
. 86
4.1.3
Some typical isoparametric elements
. 90
4.1.4
Example. Linear elasticity
. 93
4.2
Path-dependent materials. The incremental finite element procedure
. 94
4.2.1
The incremental constitutive function
. 95
4.2.2
The incremental boundary value problem
. 95
4.2.3
The nonlinear incremental finite element equation
. 96
4.2.4
Nonlinear solution. The Newton-Raphson scheme
. 96
4.2.5
The consistent tangent modulus
. 98
4.2.6
Alternative nonlinear solution schemes
. 99
4.2.7
Non-incremental procedures for path-dependent materials
. 101
4.3
Large strain formulation
. 102
4.3.1
The incremental constitutive function
. 102
4.3.2
The incremental boundary value problem
. 103
4.3.3
The finite element equilibrium equation
. 103
4.3.4
Linearisation. The consistent spatial tangent modulus
. 103
4.3.5
Material and geometric stiffnesses
. 106
4.3.6
Configuration-dependent loads. The load-stiffness matrix
. 106
4.4
Unstable equilibrium. The arc-length method
. 107
4.4.1
The arc-length method
. 107
4.4.2
The combined Newton-Raphson/arc-length procedure
. 108
4.4.3
The predictor solution
.
Ill
5
Overview of the program structure
115
5.1
Introduction
. 115
5.1.1
Objectives
. 115
5.1.2
Remarks on program structure
. 116
5.1.3
Portability
. 116
5.2
The main program
. 117
5.3
Data input and initialisation
. 117
5.3.1
The global database
. 117
5.3.2
Main problem-defining data. Subroutine
INDATA. 119
CONTENTS
5.3.3
External loading.
Subroutine INLOAD
. 119
5.3.4
Initialisation of variable data. Subroutine
INITIA
. 120
5.4
The load incrementation loop. Overview
. 120
5.4.1
Fixed increments option
. 120
5.4.2
Arc-length control option
. 120
5.4.3
Automatic increment cutting
. 123
5.4.4
The linear solver. Subroutine FRONT
. 124
5.4.5
Internal force calculation. Subroutine INTFOR
. 124
5.4.6
Switching data. Subroutine SWITCH
. 124
5.4.7
Output of converged results. Subroutines OUTPUT and RSTART
. . 125
5.5
Material and element modularity
. 125
5.5.1
Example. Modularisation of internal force computation
. 125
5.6
Elements. Implementation and management
. 128
5.6.1
Element properties. Element routines for data input
. 128
5.6.2
Element interfaces. Internal force and stiffness computation
. 129
5.6.3
Implementing a new finite element
. 129
5.7
Material models: implementation and management
. 131
5.7.1
Material properties. Material-specific data input
. 131
5.7.2
State variables and other Gauss point quantities.
Material-specific state updating routines
. 132
5.7.3
Material-specific switching/initialising routines
. 133
5.7.4
Material-specific tangent computation routines
. 134
5.7.5
Material-specific results output routines
. 134
5.7.6
Implementing a new material model
. 135
Part Two Small strains
137
6
The mathematical theory of plasticity
139
6.1
Phenomenological aspects
. 140
6.2
One-dimensional constitutive model
. 141
6.2.1
Elastoplastic decomposition of the axial strain
. 142
6.2.2
The elastic
uniaxial
constitutive law
. 143
6.2.3
The yield function and the yield criterion
. 143
6.2.4
The plastic flow rale. Loading/unloading conditions
. 144
6.2.5
The hardening law
. 145
6.2.6
Summary of the model
. 145
6.2.7
Determination of the plastic multiplier
. 146
6.2.8
The elastoplastic tangent modulus
. 147
6.3
General elastoplastic constitutive model
. 148
6.3.1
Additive decomposition of the strain tensor
. 148
6.3.2
The free energy potential and the elastic law
. 148
6.3.3
The yield criterion and the yield surface
. 150
6.3.4
Plastic flow rule and hardening law
. 150
6.3.5
Flow rales derived from a flow potential
. 151
6.3.6
The plastic multiplier
. 152
CONTENTS xi
6.3.7
Relation to the general continuum constitutive theory
. 153
6.3.8
Rate form and the elastoplastic tangent operator
. 153
6.3.9
Non-smooth potentials and the subdifferential
. 153
6.4
Classical yield criteria
. 157
6.4.1
The
Tresca
yield criterion
. 157
6.4.2
The
von
Mises
yield criterion
. 162
6.4.3
The Mohr-Coulomb yield criterion
. 163
6.4.4
The Drucker-Prager yield criterion
. 166
6.5
Plastic flow rules
. 168
6.5.1
Associative and non-associative plasticity
. 168
6.5.2
Associative laws and the principle of maximum plastic dissipation
170
6.5.3
Classical flow rules
. 171
6.6
Hardening laws
. 177
6.6.1
Perfect plasticity
. 177
6.6.2 Isotropie
hardening
. 178
6.6.3
Thermodynamical aspects. Associative
isotropie
hardening
. . 182
6.6.4
Kinematic hardening. The Bauschinger effect
. 185
6.6.5
Mixed isotropic/kinematic hardening
. 189
7
Finite elements in small-strain plasticity problems
191
7.1
Preliminary implementation aspects
. 192
7.2
General numerical integration algorithm for elastoplastic constitutive
equations
. 193
7.2.1
The elastoplastic constitutive initial value problem
. 193
7.2.2
Euler
discretisation: the incremental constitutive problem
. 194
7.2.3
The elastic predictor/plastic corrector algorithm
. 196
7.2.4
Solution of the return-mapping equations
. 198
7.2.5
Closest point projection interpretation
. 200
7.2.6
Alternative justification:
operatorsplit
method
. 201
7.2.7
Other elastic predictor/return-mapping schemes
. 201
7.2.8
Plasticity and differential-algebraic equations
. 209
7.2.9
Alternative mathematical programming-based algorithms
. 210
7.2.10
Accuracy and stability considerations
. 210
7.3
Application: integration algorithm for the isotropically hardening
von
Mises
model
. 215
7.3.1
The implemented model
. 216
7.3.2
The implicit elastic predictor/return-mapping scheme
. 217
7.3.3
The incremental constitutive function for the stress
. 220
7.3.4
Linear
isotropie
hardening and perfect plasticity: the
closed-form return mapping
. 223
7.3.5
Subroutine SUVM
. 224
7.4
The consistent tangent modulus
. 228
7.4.1
Consistent tangent operators in elastoplasticity
. 229
7.4.2
The elastoplastic consistent tangent for the
von
Mises
model
with
isotropie
hardening
. 232
7.4.3
Subroutine CTVM
. 235
XU CONTENTS
7.4.4
The general elastoplastic consistent tangent operator for implicit
return mappings
. 238
7.4.5
Illustration: the
von
Mises
model with
isotropie
hardening
. 240
7.4.6
Tangent operator symmetry: incremental potentials
. 243
7.5
Numerical examples with the
von
Mises
model
.244
7.5.1
Internally pressurised cylinder
.244
7.5.2
Internally pressurised spherical shell
.247
7.5.3
Uniformly loaded circular plate
.250
7.5.4
Strip-footing collapse
.252
7.5.5
Double-notched tensile specimen
.255
7.6
Further application: the
von
Mises
model with nonlinear mixed hardening
. 257
7.6.1
The mixed hardening model: summary
. 257
7.6.2
The implicit return-mapping scheme
. 258
7.6.3
The incremental constitutive function
. 260
7.6.4
Linear hardening: closed-form return mapping
. 261
7.6.5
Computational implementation aspects
. 261
7.6.6
The elastoplastic consistent tangent
. 262
8
Computations with other basic plasticity models
265
8.1
The
Tresca
model
. 266
8.1.1
The implicit integration algorithm in principal stresses
. 268
8.1.2
Subroutine SUTR
. 279
8.1.3
Finite step accuracy: iso-error maps
. 283
8.1.4
The consistent tangent operator for the
Tresca
model
. 286
8.1.5
Subroutine CTTR
. 291
8.2
The Mohr-Coulomb model
. 295
8.2.1
Integration algorithm for the Mohr-Coulomb model
. 297
8.2.2
Subroutine SUMC
. 310
8.2.3
Accuracy: iso-error maps
. 315
8.2.4
Consistent tangent operator for the Mohr-Coulomb model
. 316
8.2.5
Subroutine CTMC
. 319
8.3
The Drucker-Prager model
. 324
8.3.1
Integration algorithm for the Drucker-Prager model
. 325
8.3.2
Subroutine SUDP
. 334
8.3.3
Iso-error map
. 337
8.3.4
Consistent tangent operator for the Drucker-Prager model
. 337
8.3.5
Subroutine CTDP
. 340
8.4
Examples
. 343
8.4.1
Bending of a V-notched
Tresca bar
. 343
8.4.2
End-loaded tapered cantilever
. 344
8.4.3
Strip-footing collapse
. 346
8.4.4
Circular-footing collapse
. 350
8.4.5
Slope stability
. 351
CONTENTS xiii
9 Plane
stress
plasticity
357
9.1
The basic
plane
stress plasticity problem
. 357
9.1.1
Plane stress linear elasticity
. 358
9.1.2
The constrained elastoplastic initial value problem
. 359
9.1.3
Procedures for plane stress plasticity
. 360
9.2
Plane stress constraint at the Gauss point level
. 361
9.2.1
Implementation aspects
. 362
9.2.2
Plane stress enforcement with nested iterations
. 362
9.2.3
Plane stress
von
Mises
with nested iterations
. 364
9.2.4
The consistent tangent for the nested iteration procedure
. 366
9.2.5
Consistent tangent computation for the
von
Mises
model
. 366
9.3
Plane stress constraint at the structural level
. 367
9.3.1
The method
. 367
9.3.2
The implementation
. 368
9.4
Plane stress-projected plasticity models
. 370
9.4.1
The plane stress-projected
von
Mises
model
. 371
9.4.2
The plane stress-projected integration algorithm
. 373
9.4.3
Subroutine SUVMPS
. 378
9.4.4
The elastoplastic consistent tangent operator
. 382
9.4.5
Subroutine CTVMPS
. 383
9.5
Numerical examples
. 386
9.5.1
Collapse of an end-loaded cantilever
. 387
9.5.2
Infinite plate with a circular hole
. 387
9.5.3
Stretching of a perforated rectangular plate
. 390
9.5.4
Uniform loading of a concrete shear wall
. 391
9.6
Other stress-constrained states
. 396
9.6.1
A three-dimensional
von
Mises
Timoshenko beam
. 396
9.6.2
The beam state-projected integration algorithm
. 400
10
Advanced plasticity models
403
10.1
A modified Cam-Clay model for soils
.403
10.1.1
The model
.404
10.1.2
Computational implementation
.406
10.2
A capped Drucker-Prager model for geomaterials
.409
10.2.1
Capped Drucker-Prager model
.410
10.2.2
The implicit integration algorithm
.412
10.2.3
The elastoplastic consistent tangent operator
.413
10.3 Anisotropie
plasticity: the Hill, Hoffman and Barlat-Lian models
.414
10.3.1
The Hill orthotropic model
.414
10.3.2
Tension-compression distinction: the Hoffman model
.420
10.3.3
Implementation of the Hoffman model
.423
10.3.4
The Barlat-Lian model for sheet metals
.427
10.3.5
Implementation of the Barlat-Lian model
.431
xiv CONTENTS
11 Viscoplasticity 435
11.1 Viscoplasticity: phenomenological
aspects
. 436
11.2
One-dimensional
viscoplasticity
model.
437
11.2.1
Elastoplastic decomposition of the
axial
strain
. 437
11.2.2
The elastic law
. 438
11.2.3
The yield function and the elastic domain
. 438
11.2.4
Viscoplastic flow rule
. 438
11.2.5
Hardening law
. 439
11.2.6
Summary of the model
. 439
11.2.7
Some simple analytical solutions
. 439
11.3
A
von
Mises-based multidimensional model
. 445
11.3.1
A
von
Mises-type
viscoplastic model with
isotropie
strain
hardening
. 445
11.3.2
Alternative plastic strain rate definitions
. 447
11.3.3
Other
isotropie
and kinematic hardening laws
. 448
11.3.4
Viscoplastic models without a yield surface
. 448
11.4
General viscoplastic constitutive model
. 450
11.4.1
Relation to the general continuum constitutive theory
. 450
11.4.2
Potential structure and dissipation inequality
. 451
11.4.3
Rate-independent plasticity as a limit case
. 452
11.5
General numerical framework
. 454
11.5.1
A general implicit integration algorithm
. 454
11.5.2
Alternative Euler-based algorithms
. 457
11.5.3
General consistent tangent operator
. 458
11.6
Application: computational implementation of a
von
Mises-based model
. 460
11.6.1
Integration algorithm
. 460
11.6.2
Iso-error maps
. 463
11.6.3
Consistent tangent operator
. 464
11.6.4
Perzyna-type model implementation
. 466
11.7
Examples
. 467
11.7.1
Double-notched tensile specimen
. 467
11.7.2
Plane stress: stretching of a perforated plate
. 469
12
Damage mechanics
471
12.1
Physical aspects of internal damage in solids
. 472
12.1.1
Metals
. 472
12.1.2
Rubbery polymers
. 473
12.2
Continuum damage mechanics
. 473
12.2.1
Original development: creep-damage
. 474
12.2.2
Other theories
. 475
12.2.3
Remarks on the nature of the damage variable
. 476
12.3
Lemaitre's elastoplastic damage theory
. 478
12.3.1
The model
. 478
12.3.2
Integration algorithm
. 482
12.3.3
The tangent operators
. 485
CONTENTS
XV
12.4
A simplified version of Lemaitre's model
. 486
12.4.1
The single-equation integration algorithm
. 486
12.4.2
The tangent operator
. 490
12.4.3
Example. Fracturing of a cylindrical notched specimen
. 493
12.5
Gurson's void growth model
. 496
12.5.1
The model
. 497
12.5.2
Integration algorithm
. 501
12.5.3
The tangent operator
. 502
12.6
Further issues in damage modelling
. 504
12.6.1
Crack closure effects in damaged elastic materials
. 504
12.6.2
Crack closure effects in damage evolution
. 510
12.6.3 Anisotropie
ductile damage
. 512
Part Three Large strains
517
13
Finite strain hyperelasticity
519
13.1
Hyperelasticity: basic concepts
. 520
13.1.1
Material objectivity: reduced form of the free-energy function
. . 520
13.1.2 Isotropie
hyperelasticity
. 521
13.1.3
Incompressible hyperelasticity
. 524
13.1.4
Compressible
régularisation
. 525
13.2
Some particular models
. 525
13.2.1
The Mooney-Rivlin and the neo-Hookean models
. 525
13.2.2
The Ogden material model
. 527
13.2.3
The Hencky material
. 528
13.2.4
The Blatz-Ko material
. 530
13.3 Isotropie
finite hyperelasticity in plane stress
. 530
13.3.1
The plane stress incompressible Ogden model
. 531
13.3.2
The plane stress Hencky model
. 532
13.3.3
Plane stress with nested iterations
. 533
13.4
Tangent moduli: the elasticity tensors
. 534
13.4.1
Regularised neo-Hookean model
. 535
13.4.2
Principal stretches representation: Ogden model
. 535
13.4.3
Hencky model
. 537
13.4.4
Blatz-Ko material
. 537
13.5
Application: Ogden material implementation
. 538
13.5.1
Subroutine
SU0GD . 538
13.5.2
Subroutine CSTOGD
. 542
13.6
Numerical examples
. 546
13.6.1
Axisymmetric extension of an annular plate
. 547
13.6.2
Stretching of a square perforated rubber sheet
. 547
13.6.3
Inflation of a spherical rubber balloon
. 550
13.6.4
Rugby ball
. 551
13.6.5
Inflation of initially flat membranes
. 552
xvi CONTENTS
13.6.6
Rubber cylinder pressed between two plates
.555
13.6.7
Elastomeric bead compression
.556
13.7
Hyperelasticity with damage: the Mullins effect
.557
13.7.1
The Gurtin-Francis
uniaxial
model
. 560
13.7.2
Three-dimensional modelling.
Abrief
review
. 562
13.7.3
A simple rate-independent three-dimensional model
. 562
13.7.4
Example: the model problem
. 565
13.7.5
Computational implementation
. 565
13.7.6
Example: inflation/deflation of a damageable rubber balloon
. . . 569
14
Finite strain elastoplasticity
573
14.1
Finite strain elastoplasticity: a brief review
.574
14.2
One-dimensional finite plasticity model
.575
14.2.1
The multiplicative split of the axial stretch
.575
14.2.2
Logarithmic stretches and the Hencky hyperelastic law
.576
14.2.3
The yield function
.576
14.2.4
The plastic flow rule
.576
14.2.5
The hardening law
.577
14.2.6
The plastic multiplier
.577
14.3
General hyperelastic-based multiplicative plasticity model
.578
14.3.1
Multiplicative elastoplasticity kinematics
.578
14.3.2
The logarithmic elastic strain measure
.582
14.3.3
A general
isotropie
large-strain plasticity model
.583
14.3.4
The dissipation inequality
.586
14.3.5
Finite strain extension to infinitesimal theories
.588
14.4
The general elastic predictor/return-mapping algorithm
.590
14.4.1
The basic constitutive initial value problem
.590
14.4.2
Exponential map backward discretisation
.591
14.4.3
Computational implementation of the general algorithm
.595
14.5
The consistent spatial tangent modulus
.597
14.5.1
Derivation of the spatial tangent modulus
.598
14.5.2
Computational implementation
.599
14.6
Principal stress space-based implementation
.599
14.6.1
Stress-updating algorithm
.600
14.6.2
Tangent modulus computation
.601
14.7
Finite plasticity in plane stress
.601
14.7.1
The plane stress-projected finite
von
Mises
model
.601
14.7.2
Nested iteration for plane stress enforcement
.604
14.8
Finite viscoplasticity
.605
14.8.1
Numerical treatment
.606
14.9
Examples
.606
14.9.1
Finite strain bending of a V-notched
Tresca
bar
.606
14.9.2
Necking of a cylindrical bar
.607
14.9.3
Plane strain localisation
.611
14.9.4
Stretching of a perforated plate
.613
14.9.5
Thin sheet metal-forming application
.614
CONTENTS xvii
14.10 Rate
forms: hypoelastic-based plasticity models
. 615
14.10.1
Objective stress rates
. 619
14.10.2
Hypoelastic-based plasticity models
. 621
14.10.3
The Jaumann rate-based model
. 622
14.10.4
Hyperelastic-based models and equivalent rate forms
. 624
14.10.5
Integration algorithms and incremental objectivity
. 625
14.10.6
Objective algorithm for Jaumann rate-based models
. 628
14.10.7
Integration of Green-Naghdi rate-based models
. 632
14.11
Finite plasticity with kinematic hardening
. 633
14.11.1
A model of finite strain kinematic hardening
. 633
14.11.2
Integration algorithm
. 637
14.11.3
Spatial tangent operator
. 642
14.11.4
Remarks on predictive capability
. 644
14.11.5
Alternative descriptions
. 644
15
Finite elements for large-strain incompressibility
647
15.1
The F-bar methodology
. 648
15.1.1
Stress computation: the F-bar deformation gradient
. 649
15.1.2
The internal force vector
. 651
15.1.3
Consistent linearisation: the tangent stiffness
. 652
15.1.4
Plane strain implementation
. 655
15.1.5
Computational implementation aspects
. 656
15.1.6
Numerical tests
. 656
15.1.7
Other centroid sampling-based F-bar elements
. 663
15.1.8
A more general F-bar methodology
. 663
15.1.9
The F-bar-Patch Method for simplex elements
. 665
15.2
Enhanced assumed strain methods
. 669
15.2.1
Enhanced three-field variational principle
. 669
15.2.2
EAS finite elements
. 671
15.2.3
Finite element equations: static condensation
. 676
15.2.4
Implementation aspects
. 678
15.2.5
The stability of EAS elements
. 678
15.3
Mixed u/p formulations
. 683
15.3.1
The two-field variational principle
. 683
15.3.2
Finite element equations
. 685
15.3.3
Solution: static condensation
. 687
15.3.4
Implementation aspects
. 689
16 Anisotropie
finite plasticity: Single crystals
691
16.1
Physical aspects
. 692
16.1.1
Plastic deformation by slip: slip-systems
. 692
16.2
Plastic slip and the
Schmid
resolved shear stress
. 693
16.3
Single crystal simulation: a brief review
. 694
16.4
A general continuum model of single crystals
. 694
16.4.1
The plastic flow equation
. 695
16.4.2
The resolved
Schmid
shear stress
. 696
xviii CONTENTS
16.4.3 Multisurface
formulation of the flow rule
.696
16.4.4 Isotropie
Taylor hardening
.698
16.4.5
The hyperelastic law
.698
16.5
A general integration algorithm
.699
16.5.1
The search for an active set of slip systems
.703
16.6
An algorithm for a planar double-slip model
.705
16.6.1
A planar double-slip model
. 705
16.6.2
The integration algorithm
. 707
16.6.3
Example: the model problem
. 710
16.7
The consistent spatial tangent modulus
. 713
16.7.1
The elastic modulus: compressible neo-Hookean model
.713
16.7.2
The elastoplastic consistent tangent modulus
.714
16.8
Numerical examples
.716
16.8.1
Symmetric strain localisation on a rectangular strip
.717
16.8.2
Unsymmetric localisation
.720
16.9
Viscoplastic single crystals
.721
16.9.1
Rate-dependent formulation
.723
16.9.2
The exponential map-based integration algorithm
.724
16.9.3
The spatial tangent modulus: neo-Hookean-based model
.725
16.9.4
Rate-dependent crystal: model problem
.726
Appendices
729
A
Isotropie
functions of a symmetric tensor
731
A.I
Isotropie
scalar-valued functions
. 731
A.I.I Representation
. 732
A.
1.2
The derivative of an
isotropie
scalar function
. 732
A.
2
Isotropie
tensor-valued functions
. 733
A.2.1 Representation
. 733
A.2.2 The derivative of an
isotropie
tensor function
. 734
A.3 The two-dimensional case
. 735
A.3.1 Tensor function derivative
. 736
A.
3.2
Plane strain and axisymmetric problems
. 738
A.4 The three-dimensional case
. 739
A.
4.1
Function computation
. 739
A.4.2 Computation of the function derivative
. 740
A.5 A particular class of
isotropie
tensor functions
. 740
A.5.1 Two dimensions
. 742
A.5.
2
Three dimensions
. 743
A.6 Alternative procedures
. 744
CONTENTS xix
В
The tensor exponential
747
B.I The tensor exponential function
. 747
B.I
.1
Some properties of the tensor exponential function
. 748
B.I.
2
Computation of the tensor exponential function
. 749
B.2 The tensor exponential derivative
. 750
B.2.1 Computer implementation
. 751
B.3 Exponential map integrators
. 751
B.3.1 The generalised exponential map midpoint rule
. 752
С
Linearisation of the virtual work
753
C.I Infinitesimal deformations
. 753
C.2 Finite strains and deformations
. 755
C.2.
1
Material description
. 755
C.2.
2
Spatial description
. 756
D
Array notation for computations with tensors
759
D.
1
Second-order tensors
. 759
D.2 Fourth-order tensors
. 761
D.2.1 Operations with non-symmetric tensors
. 763
References
765
Index
783 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Neto, Eduardo de Souza Perić, Djordje Owens, David |
| author_facet | Neto, Eduardo de Souza Perić, Djordje Owens, David |
| author_role | aut aut aut |
| author_sort | Neto, Eduardo de Souza |
| author_variant | e d s n eds edsn d p dp d o do |
| building | Verbundindex |
| bvnumber | BV035125545 |
| callnumber-first | T - Technology |
| callnumber-label | TA418 |
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| callnumber-search | TA418.14 |
| callnumber-sort | TA 3418.14 |
| callnumber-subject | TA - General and Civil Engineering |
| classification_rvk | UF 3100 |
| classification_tum | PHY 213f MTA 070f MAT 674f |
| ctrlnum | (OCoLC)490108914 (DE-599)BVBBV035125545 |
| dewey-full | 531/.385 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 531 - Classical mechanics |
| dewey-raw | 531/.385 |
| dewey-search | 531/.385 |
| dewey-sort | 3531 3385 |
| dewey-tens | 530 - Physics |
| discipline | Physik Mathematik |
| discipline_str_mv | Physik Mathematik |
| edition | 1. publ. |
| format | Book |
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| id | DE-604.BV035125545 |
| illustrated | Illustrated |
| index_date | 2024-07-02T22:22:45Z |
| indexdate | 2024-07-20T09:53:46Z |
| institution | BVB |
| isbn | 9780470694527 0470694521 |
| language | English |
| lccn | 2008033260 |
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| publisher | Wiley |
| record_format | marc |
| spelling | Neto, Eduardo de Souza Verfasser aut Computational methods for plasticity theory and applications E. A. de Souza Neto ; D. Perić ; D. R. J. Owen 1. publ. Hoboken, NJ Wiley 2008 XXII, 791 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index Mathematisches Modell Plasticity Mathematical models Computerunterstütztes Verfahren (DE-588)4139030-1 gnd rswk-swf Plastizität (DE-588)4046283-3 gnd rswk-swf Mathematisches Modell (DE-588)4114528-8 gnd rswk-swf Plastizität (DE-588)4046283-3 s Mathematisches Modell (DE-588)4114528-8 s Computerunterstütztes Verfahren (DE-588)4139030-1 s DE-604 Perić, Djordje Verfasser aut Owens, David Verfasser aut text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3151997&prov=M&dok_var=1&dok_ext=htm Inhaltstext Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016793135&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Neto, Eduardo de Souza Perić, Djordje Owens, David Computational methods for plasticity theory and applications Mathematisches Modell Plasticity Mathematical models Computerunterstütztes Verfahren (DE-588)4139030-1 gnd Plastizität (DE-588)4046283-3 gnd Mathematisches Modell (DE-588)4114528-8 gnd |
| subject_GND | (DE-588)4139030-1 (DE-588)4046283-3 (DE-588)4114528-8 |
| title | Computational methods for plasticity theory and applications |
| title_auth | Computational methods for plasticity theory and applications |
| title_exact_search | Computational methods for plasticity theory and applications |
| title_exact_search_txtP | Computational methods for plasticity theory and applications |
| title_full | Computational methods for plasticity theory and applications E. A. de Souza Neto ; D. Perić ; D. R. J. Owen |
| title_fullStr | Computational methods for plasticity theory and applications E. A. de Souza Neto ; D. Perić ; D. R. J. Owen |
| title_full_unstemmed | Computational methods for plasticity theory and applications E. A. de Souza Neto ; D. Perić ; D. R. J. Owen |
| title_short | Computational methods for plasticity |
| title_sort | computational methods for plasticity theory and applications |
| title_sub | theory and applications |
| topic | Mathematisches Modell Plasticity Mathematical models Computerunterstütztes Verfahren (DE-588)4139030-1 gnd Plastizität (DE-588)4046283-3 gnd Mathematisches Modell (DE-588)4114528-8 gnd |
| topic_facet | Mathematisches Modell Plasticity Mathematical models Computerunterstütztes Verfahren Plastizität |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=3151997&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016793135&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT netoeduardodesouza computationalmethodsforplasticitytheoryandapplications AT pericdjordje computationalmethodsforplasticitytheoryandapplications AT owensdavid computationalmethodsforplasticitytheoryandapplications |