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This updated and expanded edition of the bestselling textbook provides a comprehensive introduction to the methods and theory of nonlinear finite element analysis. New material provides a concise introduction to some of the cutting-edge methods that have evolved in recent years in the field of nonlinear finite element modeling, and includes the eXtended finite element method (XFEM), multiresolution continuum theory for multiscale microstructures, and dislocation-density-based crystalline plasticity.
Nonlinear Finite Elements for Continua and Structures, Second Edition focuses on the formulation and solution of discrete equations for various classes of problems that are of principal interest in applications to solid and structural mechanics. Topics covered include the discretization by finite elements of continua in one dimension and in multi-dimensions; the formulation of constitutive equations for nonlinear materials and large deformations; procedures for the solution of the discrete equations, including considerations of both numerical and multiscale physical instabilities; and the treatment of structural and contact-impact problems.
Key features:
* Presents a detailed and rigorous treatment of nonlinear solid mechanics and how it can be implemented in finite element analysis
* Covers many of the material laws used in today's software and research
* Introduces advanced topics in nonlinear finite element modelling of continua
* Introduction of multiresolution continuum theory and XFEM
* Accompanied by a website hosting a solution manual and MATLAB(r) and FORTRAN code
Nonlinear Finite Elements for Continua and Structures, Second Edition is a must have textbook for graduate students in mechanical engineering, civil engineering, applied mathematics, engineering mechanics, and materials science, and is also an excellent source of information for researchers and practitioners in industry.
List of contents
Foreword xxi
Preface xxiii
List of Boxes xxvii
1 Introduction 1
1.1 Nonlinear Finite Elements in Design 1
1.2 Related Books and a Brief History of Nonlinear Finite Elements 4
1.3 Notation 7
1.4 Mesh Descriptions 9
1.5 Classification of Partial Differential Equations 13
1.6 Exercises 17
2 Lagrangian and Eulerian Finite Elements in One Dimension 19
2.1 Introduction 19
2.2 Governing Equations for Total Lagrangian Formulation 21
2.3 Weak Form for Total Lagrangian Formulation 28
2.4 Finite Element Discretization in Total Lagrangian Formulation 34
2.5 Element and Global Matrices 40
2.6 Governing Equations for Updated Lagrangian Formulation 51
2.7 Weak Form for Updated Lagrangian Formulation 53
2.8 Element Equations for Updated Lagrangian Formulation 55
2.10 Weak Forms for Eulerian Mesh Equations 68
2.11 Finite Element Equations 69
2.12 Solution Methods 72
2.13 Summary 74
2.14 Exercises 75
3 Continuum Mechanics 77
3.1 Introduction 77
3.2 Deformation and Motion 78
3.3 Strain Measures 95
3.4 Stress Measures 104
3.5 Conservation Equations 111
3.6 Lagrangian Conservation Equations 123
3.7 Polar Decomposition and Frame-Invariance 130
3.8 Exercises 143
4 Lagrangian Meshes 147
4.1 Introduction 147
4.2 Governing Equations 148
4.3 Weak Form: Principle of Virtual Power 152
4.4 Updated Lagrangian Finite Element Discretization 158
4.5 Implementation 168
4.6 Corotational Formulations 194
4.7 Total Lagrangian Formulation 203
4.8 Total Lagrangian Weak Form 206
4.9 Finite Element Semidiscretization 209
4.10 Exercises 225
5 Constitutive Models 227
5.1 Introduction 227
5.2 The Stress-Strain Curve 228
5.3 One-Dimensional Elasticity 233
5.4 Nonlinear Elasticity 237
5.5 One-Dimensional Plasticity 254
5.6 Multiaxial Plasticity 262
5.7 Hyperelastic-Plastic Models 281
5.8 Viscoelasticity 292
5.9 Stress Update Algorithms 294
5.10 Continuum Mechanics and Constitutive Models 314
5.11 Exercises 328
6 Solution Methods and Stability 329
6.1 Introduction 329
6.2 Explicit Methods 330
6.3 Equilibrium Solutions and Implicit Time Integration 337
6.4 Linearization 358
6.5 Stability and Continuation Methods 375
6.6 Numerical Stability 391
6.7 Material Stability 407
6.8 Exercises 415
7 Arbitrary Lagrangian Eulerian Formulations 417
7.1 Introduction 417
7.2 ALE Continuum Mechanics 419
7.3 Conservation Laws in ALE Description 426
7.4 ALE Governing Equations 428
7.5 Weak Forms 429
7.6 Introduction to the Petrov-Galerkin Method 433
7.7 Petrov-Galerkin Formulation of Momentum Equation 442
7.8 Path-Dependent Materials 445
7.9 Linearization of the Discrete Equations 457
7.10 Mesh Update Equations 460
7.11 Numerical Example: An Elastic-Plastic Wave Propagation Problem 468
7.12 Total ALE Formulations 471
7.13 Exercises 475
8 Element Technology 477
8.1 Introduction 477
8.2 Element Performance 479
8.3 Element Properties and Patch Tests 487
8.4 Q4 and Volumetric Locking 496
8.5 Multi-Field Weak Forms and Ele
About the author
Ted Belytschko, Northwestern University, USA Wing Kam Liu, Northwestern University, USA Brian Moran, King Abdullah University of Science and Technology, The Kingdom of Saudi Arabia Khalil I. Elkhodary, The American University in Cairo, Egypt
Summary
This updated and expanded edition of the bestselling textbook provides a comprehensive introduction to the methods and theory of nonlinear finite element analysis.
