Nonlinear Finite Element Analysis of Solids and Structures

by de Borst, René; Crisfield, Mike A.; Remmers, Joris J. C.; Verhoosel, Clemens V.

• ISBN13: 9780470666449
• ISBN10: 0470666447
• Edition: 2nd
• Format: Hardcover
• Copyright: 2012-09-04
• Publisher: Wiley

Summary

Built upon the two original books by Mike Crisfield and their own lecture notes, renowned scientist René de Borst and his team offer a thoroughly updated yet condensed edition that retains and builds upon the excellent reputation and appeal amongst students and engineers alike for which Crisfield's first edition is acclaimed. Together with numerous additions and updates, the new authors have retained the core content of the original publication, while bringing an improved focus on new developments and ideas. This edition offers the latest insights in non-linear finite element technology, including non-linear solution strategies, computational plasticity, damage mechanics, time-dependent effects, hyperelasticity and large-strain elasto-plasticity. The authors' integrated and consistent style and unrivalled engineering approach assures this book's unique position within the computational mechanics literature. Key features: Combines the two previous volumes into one heavily revised text with obsolete material removed, an improved layout and updated references and notations Extensive new material on more recent developments in computational mechanics Easily readable, engineering oriented, with no more details in the main text than necessary to understand the concepts. Pseudo-code throughout makes the link between theory and algorithms, and the actual implementation. Accompanied by a website (www.wiley.com/go/deborst) with a Python code, based on the pseudo-code within the book and suitable for solving small-size problems. Non-linear Finite Element Analysis of Solids and Structures, 2nd Edition is an essential reference for practising engineers and researchers that can also be used as a text for undergraduate and graduate students within computational mechanics.

Author Biography

Mike Crisfield (deceased), Imperial College, London; René de Borst, Joris Remmers & Clemens Verhoosel, TU Eindhoven, Netherlands

Professor Mike Crisfield (deceased) joined the Transport & Road Research Laboratory (TRRL) in 1971, where he rose to the rank of Deputy Chief Scientific Officer. In 1989 he was appointed as first holder of the FEA Chair in Computational Mechanics in the aeronautics department at Imperial College, London, the department that had pioneered FEA in the 1950s and 1960s. Shortly before he died, a list of the most cited engineering researchers in the UK was published included Mike in the top 20, and he received an IACM Research Achievement Award in recognition of his extraordinary achievements in the field of non-linear computational mechanics. An eminent researcher and a scholar, he was reputed as an innovative thinker who adopted a 'hands-on' approach.

René de Borst was appointed Dean and Distinguished University Professor of the Faculty of Mechanical Engineering of TU Eindhoven in May 2007 after a long tenure as Professor and deputy Dean at TU Delft. He is Editor for the International Journal for Numerical Methods in Engineering and International Journal for Numerical and Analytical Methods in Geomechanics and Editor for the Encyclopedia of Computational Mechanics. His many awards and the outstanding assessment of his work by the scientific community attest to his reputation as a world leading scientist and researcher within the field of computational mechanics.

Joris Remmers is an assistant professor within René de Borst's group at TU Eindhoven.

Clemens Verhoosel is an assistant professor within René de Borst's group at TU Eindhoven.

Table of Contents

Basic Concepts and Solution Techniques	p. 21
Preliminaries	p. 23
A simple example of non-linear behaviour	p. 23
A review of concepts from linear algebra	p. 25
Vectors and tensors	p. 32
Stress and strain tensors	p. 36
Elasticity	p. 42
The PYFEM finite element library 44 Bibliography	p. 48
Non-linear finite element analysis	p. 49
Equilibrium and virtual work	p. 49
Spatial discretisation by finite elements	p. 51
PyFEM: Shape function utilities	p. 55
Incremental-iterative analysis	p. 59
Load vs displacement control	p. 68
PyFEM: A linear finite element code with displacement control 71 Bibliography	p. 80
Geometrically non-linear analysis	p. 81
Truss elements	p. 82
Total Lagrange formulation	p. 85
Updated Lagrange formulation	p. 88
Corotational formulation	p. 89
PyFEM: The shallow truss problem	p. 94
Stress and deformation measures in continua	p. 103
Geometrically non-linear formulation of continuum elements	p. 109
Total and Updated Lagrange formulations	p. 109
Corotational formulation	p. 113
Linear buckling analysis	p. 117
PyFEM: A geometrically non-linear continuum element 119 Bibliography	p. 127
Solution techniques in quasi-static analysis	p. 129
Line searches 129 2 Contents
Path-following or arc-length methods	p. 132
PYFEM: Implementation of Riks’ arc-length solver	p. 140
Stability and uniqueness in discretised systems	p. 145
Stability of a discrete system	p. 145
Uniqueness and bifurcation in a discrete system	p. 146
Branch switching	p. 150
Load stepping and convergence criteria	p. 150
Quasi-Newton methods 153 Bibliography	p. 156
Solution techniques for non-linear dynamics	p. 157
The semi-discrete equations	p. 157
Explicit time integration	p. 159
PYFEM: Implementation of an explicit solver	p. 162
Implicit time integration	p. 167
The Newmark family	p. 167
The HHT _-method	p. 168
Alternative implicit methods for time integration	p. 169
Stability and accuracy in the presence of non-linearities	p. 171
Energy-conserving algorithms	p. 175
Time step size control and element technology 178 Bibliography	p. 179
Material Non-linearities	p. 181
Damage mechanics	p. 183
The concept of damage	p. 183
Isotropic elasticity-based damage	p. 185
PYFEM: A plane-strain damage model	p. 188
Stability, ellipticity, and mesh sensitivity	p. 192
Stability, ellipticity, and mesh sensitivity	p. 192
Mesh sensitivity	p. 195
Cohesive-zone models	p. 198
Element technology: Embedded discontinuities	p. 203
Complex damage models	p. 211
Anisotropic damage models	p. 211
Microplane models	p. 212
Crack models for concrete and other quasi-brittle materials	p. 214
Elasticity-based smeared crack models	p. 214
Reinforcement and tension stiffening	p. 219
Regularised damage models	p. 223
Non-local damage models	p. 223
Gradient damage models 224 Bibliography	p. 227
Plasticity 231 Contents	p. 3
A simple slip model	p. 231
Flow theory of plasticity	p. 235
Yield function	p. 235
Flow rule	p. 240
Hardening behaviour	p. 244
Integration of the stress-strain relation	p. 250
Tangent stiffness operators	p. 261
Multi-surface plasticity	p. 264
Koiter’s generalisation	p. 264
Rankine plasticity for concrete	p. 267
Tresca and Mohr-Coulomb plasticity	p. 272
Soil plasticity: Cam-clay model	p. 279
Coupled damage-plasticity models	p. 282
Element technology: volumetric locking 283 Bibliography	p. 289
Time-dependent material models	p. 293
Linear visco-elasticity	p. 293
One-dimensional linear visco-elasticity	p. 293
Three-dimensional visco-elasticity	p. 296
Algorithmic aspects	p. 297
Creep models	p. 299
Visco-plasticity	p. 301
One-dimensional visco-plasticity	p. 302
Integration of the rate equations	p. 303
Perzyna visco-plasticity	p. 304
Duvaut-Lions visco-plasticity	p. 306
Consistency model	p. 308
Propagative or dynamic instabilities	p. 310
Bibliography	p. 315
Structural Elements	p. 317
Beams and arches	p. 319
A shallow arch	p. 319
Kirchhoff formulation	p. 319
Including shear deformation: Timoshenko beam	p. 326
PYFEM: A Kirchhoff beam element	p. 329
Corotational elements	p. 333
Kirchhoff theory	p. 333
Timoshenko beam theory	p. 337
An isoparametric degenerate continuum 2D beam element	p. 339
An isoparametric degenerate continuum 3D beam element	p. 344
Bibliography	p. 352
Plates and shells	p. 355
Contents
Shallow-shell formulations	p. 356
An isoparametric degenerate continuum shell element	p. 363
Solid-like shell elements	p. 368
Shell plasticity: Ilyushin’s criterion	p. 369
Bibliography	p. 373
Large Strains	p. 375
Hyperelasticity	p. 377
More continuum mechanics	p. 377
Momentum balance and stress tensors	p. 377
Objective stress rates	p. 380
Principal stretches and invariants	p. 384
Strain-energy functions	p. 386
Incompressibility and near-incompressibility	p. 388
Strain energy as a function of stretch invariants	p. 390
Strain energy as a function of principal stretches	p. 394
Logarithmic extension of linear elasticity: Hencky model	p. 398
Element technology	p. 400
u/p formulation	p. 401
Enhanced Assumed Strain elements	p. 404
F-bar approach	p. 406
Corotational approach	p. 407
Bibliography	p. 409
Large-strain elastoplasticity	p. 411
Eulerian formulations	p. 412
Multiplicative elastoplasticity	p. 417
Multiplicative elastoplasticity vs rate formulations	p. 421
Integration of the rate equations	p. 424
Exponential return-mapping algorithms	p. 428
Bibliography	p. 432
Advanced Discretisation Concepts	p. 435
Interfaces and discontinuities	p. 437
Interface elements	p. 437
Discontinuous Galerkin methods 446 Bibliography	p. 450
Meshless and partition-of-unity methods	p. 451
Meshless methods	p. 452
The element-free Galerkin method	p. 452
Application to fracture	p. 456
Higher-order damage mechanics 458 Contents	p. 5
Volumetric locking	p. 460
Partition-of-unity approaches	p. 461
Application to fracture	p. 465
Extension to large deformations	p. 470
Dynamic fracture	p. 476
Weak discontinuities	p. 479
Bibliography	p. 480
Isogeometric finite element analysis	p. 483
Basis functions in computer aided geometric design	p. 483
Univariate B-splines	p. 484
Univariate non-uniform rational B-splines	p. 487
Multivariate B-splines and NURBS patches	p. 488
T-splines	p. 490
Isogeometric finite elements	p. 493
B'ezier element representation	p. 493
B'ezier extraction	p. 495
PYFEM: Shape functions for isogeometric analysis	p. 497
Isogeometric analysis in non-linear solid mechanics	p. 500
Design-through-analysis of shell structures	p. 500
Higher-order damage models	p. 505
Cohesive-zone models	p. 510
Bibliography	p. 518
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