Nonlinear Finite Elements for Continua and Structures
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Persons
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
Content
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 Elements 501
8.6 Multi-Field Quadrilaterals 514
8.7 One-Point Quadrature Elements 518
8.8 Examples 527
8.9 Stability 531
8.10 Exercises 533
9 Beams and Shells 535
9.1 Introduction 535
9.2 Beam Theories 537
9.3 Continuum-Based Beam 540
9.4 Analysis of the CB Beam 551
9.5 Continuum-Based Shell Implementation 563
9.6 CB Shell Theory 578
9.7 Shear and Membrane Locking 584
9.8 Assumed Strain Elements 589
9.9 One-Point Quadrature Elements 592
9.10 Exercises 595
10 Contact-Impact 597
10.1 Introduction 597
10.2 Contact Interface Equations 598
10.3 Friction Models 609
10.4 Weak Forms 614
10.5 Finite Element Discretization 624
10.6 On Explicit Methods 638
11 EXtended Finite Element Method (XFEM) 643
11.1 Introduction 643
11.2 Partition of Unity and Enrichments 647
11.3 One-Dimensional XFEM 648
11.4 Multi-Dimension XFEM 656
11.5 Weak and Strong Forms 660
11.6 Discrete Equations 662
11.7 Level Set Method 668
11.8 The Phantom Node Method 670
11.9 Integration 673
11.10 An Example of XFEM Simulation 675
11.11 Exercise 678
12 Introduction to Multiresolution Theory 681
12.1 Motivation: Materials are Structured Continua 681
12.2 Bulk Deformation of Microstructured Continua 685
12.3 Generalizing Mechanics to Bulk Microstructured Continua 686
12.4 Multiscale Microstructures and the Multiresolution Continuum Theory 696
12.5 Governing Equations for MCT 699
12.6 Constructing MCT Constitutive Relationships 701
12.7 Basic Guidelines for RVE Modeling 705
12.8 Finite Element Implementation of MCT 710
12.9 Numerical Example 712
12.10 Future Research Directions of MCT Modeling 718
12.11 Exercises 719
13 Single-Crystal Plasticity 721
13.1 Introduction 721
13.2 Crystallographic Description of Cubic and Non-Cubic Crystals 723
13.3 Atomic Origins of Plasticity and the Burgers Vector in Single Crystals 726
13.4 Defining Slip Planes and Directions in General Single Crystals 729
13.5 Kinematics of Single Crystal Plasticity 735
13.6 Dislocation Density Evolution 740
13.7 Stress Required for Dislocation Motion 742
13.8 Stress Update in Rate-Dependent Single-Crystal Plasticity 743
13.9 Algorithm for Rate-Dependent Dislocation-Density Based Crystal Plasticity 745
13.10 Numerical Example: Localized Shear and Inhomogeneous Deformation 747
13.11 Exercises 750
Appendix 1 Voigt Notation 751
Appendix 2 Norms 757
Appendix 3 Element Shape Functions 761
Appendix 4 Euler Angles From Pole Figures 767
Appendix 5 Example of Dislocation-Density Evolutionary Equations 771
Glossary 777
References 781
Index 795
Preface
The objective of this book is to provide a comprehensive introduction to the methods and theory of nonlinear finite element analysis. We have focused on the formulation and solution of the discrete equations for various classes of problems that are of principal interest in applications of the finite element method to solid mechanics, the mechanics of materials, and structural mechanics. The core topics are presented first, which 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; and procedures for the solution of the discrete equations, including considerations of both numerical and physical instabilities. More specialized applications are then presented. These include: the treatment of structural and contact-impact problems; representation of weak and strong discontinuities that evolve in failing solids; and mechanism-based modeling of material nonlinearities, illustrating advanced treatments of their multiscale aspects and microstructural origins. These are the topics which are of relevance to industrial and research applications and which are essential to those in the practice, research, and teaching of nonlinear finite elements.
The book has a mechanics style rather than a mathematical style. Although it includes analyses of the stability of numerical methods and the relevant partial differential equations, the objective is to teach methods of finite element analysis and the properties of the solutions and the methods. Topics such as proofs of convergence and the mathematical properties of solutions are not considered.
In the formulation of the discrete equations, we start with the governing equations based on the mechanics of the system, develop a weak form, and use this to derive the discrete equations. Weak forms and the discrete equations are developed for Lagrangian, arbitrary Lagrangian, and Eulerian meshes, for in the simulation of industrial processes and research, problems with large deformations that cannot be treated by Lagrangian meshes are becoming more common. Both the updated Lagrangian and the total Lagrangian approaches are thoroughly described.
Since a fundamental understanding of the equations requires substantial familiarity with continuum mechanics, Chapter 3 summarizes the continuum mechanics which is pertinent to the topics in this book. The chapter begins with a basic description of motion with an emphasis on rotation. Strain and stress measures are described along with transformations between them, which are later generalized as push-forward and pullback operations. The basic conservation laws are described in both so-called Eulerian and Lagrangian descriptions. Objectivity, often known as frame invariance, is introduced.
Chapter 4 describes the formulation of the discrete equations for Lagrangian meshes. We start with the development of the weak forms of momentum balance and use these to develop the discrete equations. Both the total Lagrangian and the updated Lagrangian formulations are thoroughly described, and methods and approaches for transforming between these formulations are discussed. Examples are given of the development of various elements in two and three dimensions.
Chapter 5 treats constitutive equations, with particular emphasis on the aspects of material models that are relevant to the treatment of material nonlinearities and large deformations.
Solution procedures and analyses of stability are described in Chapter 6. Both explicit and implicit integration procedures are described for transient processes and solutions; continuation procedures for equilibrium problems are considered. Newton methods and the linearization procedures required for the construction of the Newton equations are developed. In the solution of nonlinear problems, the stability of the numerical procedures and of the physical processes is crucial. Therefore, the theory of stability is summarized and applied to the determination of the stability of solutions and numerical procedures. Both geometric and material stability are considered.
Chapter 7 deals with arbitrary Lagrangian Eulerian methods. This chapter also provides the tools for Eulerian analysis. Numerical techniques needed for this class of meshes, such as upwinding and the SUPG formulation, are described.
Chapter 8 deals with element technology, the special techniques which are needed for the successful design of elements in constrained media problems. Emphasis is placed on the problem of incompressible materials but the techniques are described in a general context. One-point quadrature elements and hourglass control are also described.
Chapter 9 is devoted to structural elements, particularly shells and beams; plates are not treated separately because they are special cases of shells. We emphasize continuum-based structural formulations because they are more easily learned and more widely used for nonlinear analysis. The various assumptions are carefully studied and continuum-based formulations for beams and shells are developed. Much of this chapter rests heavily on the preceding chapters, since continuum-based elements can be developed from continuum elements with minor modifications. Therefore, topics such as linearization and material models are treated only briefly.
Contact-impact is described in Chapter 10. Contact-impact is viewed as a variational inequality, so that the appropriate contact inequalities are met in the discrete equations. Both displacement-based and velocity-based formulations are described. Attention is focused on the nonsmooth character of contact-impact and its effect on solution procedures and simulations.
Chapter 11 covers the modeling of strong and weak discontinuities. An overview of methods in classical finite elements is provided as a historical introduction. The chapter focuses on using the extended finite element method (XFEM) to model discontinuities with non-conforming meshes. For strong discontinuities the emphasis is on modeling fracture, with extensions to other problems. For weak discontinuities emphasis is on material interfaces, but the developments presented are easily extendable to other weak discontinuities. The discussion begins with the 1D formulation and then builds to multiple dimensions. Discussions are included for both implementation and integration of XFEM as well as a brief overview of the level set method, which is often coupled with XFEM. The chapter concludes with an example.
The role of material microstructure in defining material nonlinearities is introduced in Chapter 12. Emphasis is made on the multiresolution continuum theory, a multiscale mechanics theory for the large deformation of heterogeneous materials. Its aim is to link the mechanics of solids to materials science. The theory is developed from variational principles and discretized for finite element implementation. Representative volume elements (RVEs) and their role in developing mechanism-based multiscale constitutive formulations are then discussed and integrated in the multiresolution framework.
RVE modeling of single crystals by finite elements is discussed in Chapter 13, as an example of mechanism-based modeling of non-linear materials. From materials science, the crystallographic description of cubic and non-cubic crystals and the theory of dislocation densities are linked to a non-linear constitutive algorithm that governs inhomogeneous deformation in crystalline materials at the continuum level.
This book is intended for beginning graduate students in programs in mechanical engineering, civil engineering, applied mathematics, and engineering mechanics. The book assumes some familiarity with the finite element method, such as a one-semester course or a four- to five-week section in a larger course. The student should be familiar with shape functions, stiffness, and force assembly; it is also helpful to have some background in variational or energy methods. In addition, students should have had some exposure to strength of materials and continuum mechanics. Familiarity with indicial notation and matrix notation is essential.
Most instructors will choose not to cover this entire book. To do so would require a one-year course. Our aim has been to include a wide selection of material to suit the needs and preferences of many instructors. Moreover, the additional material provides the interested student with a source of background reading before embarking into the literature.
Shorter courses, such as a 10-week quarter or a 16-week semester, require a judicious selection of material which reflects the aims and taste of the instructor. The book presents most material in both the total and the updated Lagrangian format. Thus, an introductory course can focus on the updated Lagrangian viewpoint from Chapter 2 to Chapter 4, with selected topics from Chapters 5 and 6 to familiarize the student with material models and solution procedures. Some instructors may opt to skip the one-dimensional treatment in Chapter 2, leaving it as perhaps required reading. The total Lagrangian formulation can then be introduced by simply showing the transformation in Chapter 4. A similar course can be designed with an emphasis on the total Lagrangian formulation.
We have endeavored to use a unified style and notation throughout this book. This is important because, for students, drastic changes in notation and formalism often impede learning. This, at times, causes divergence from notation customary in the literature of a particular area, but we hope that the consistency of presentation will help the student.
For the second edition of this book a solution manual is available, which includes solutions to all exercises in the book,...
