Nonlinear Finite Elements for Continua and Structures
[BOOK DESCRIPTION]
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.
[TABLE OF CONTENTS]
Foreword xxi
Preface xxiii
List of Boxes xxvii
1 Introduction 1 (18)
1.1 Nonlinear Finite Elements in Design 1 (3)
1.2 Related Books and a Brief History of 4 (3)
Nonlinear Finite Elements
1.3 Notation 7 (2)
1.3.1 Indicial Notation 7 (1)
1.3.2 Tensor Notation 8 (1)
1.3.3 Functions 8 (1)
1.3.4 Matrix Notation 8 (1)
1.4 Mesh Descriptions 9 (4)
1.5 Classification of Partial 13 (4)
Differential Equations
1.6 Exercises 17 (2)
2 Lagrangian and Eulerian Finite Elements 19 (58)
in One Dimension
2.1 Introduction 19 (2)
2.2 Governing Equations for Total 21 (7)
Lagrangian Formulation
2.2.1 Nomenclature 21 (1)
2.2.2 Motion and Strain Measure 22 (1)
2.2.3 Stress Measure 22 (1)
2.2.4 Governing Equations 23 (3)
2.2.5 Momentum Equation in Terms of 26 (1)
Displacements
2.2.6 Continuity of Functions 27 (1)
2.2.7 Fundamental Theorem of Calculus 28 (1)
2.3 Weak Form for Total Lagrangian 28 (6)
Formulation
2.3.1 Strong Form to Weak Form 28 (2)
2.3.2 Weak Form to Strong Form 30 (2)
2.3.3 Physical Names of Virtual Work 32 (1)
Terms
2.3.4 Principle of Virtual Work 33 (1)
2.4 Finite Element Discretization in 34 (6)
Total Lagrangian Formulation
2.4.1 Finite Element Approximations 34 (1)
2.4.2 Nodal Forces 35 (2)
2.4.3 Semidiscrete Equations 37 (1)
2.4.4 Initial Conditions 38 (1)
2.4.5 Least-Square Fit to Initial 39 (1)
Conditions
2.4.6 Diagonal Mass Matrix 39 (1)
2.5 Element and Global Matrices 40 (11)
2.6 Governing Equations for Updated 51 (2)
Lagrangian Formulation
2.6.1 Boundary and Interior Continuity 52 (1)
Conditions
2.6.2 Initial Conditions 53 (1)
2.7 Weak Form for Updated Lagrangian 53 (2)
Formulation
2.8 Element Equations for Updated 55 (12)
Lagrangian Formulation
2.8.1 Finite Element Approximation 55 (1)
2.8.2 Element Coordinates 56 (2)
2.8.3 Internal and External Nodal Forces 58 (1)
2.8.4 Mass Matrix 59 (1)
2.8.5 Equivalence of Updated and Total 60 (1)
Lagrangian Formulations
2.8.6 Assembly, Boundary Conditions and 61 (3)
Initial Conditions
2.8.7 Mesh Distortion 64 (3)
2.9 Governing Equations for Eulerian 67 (1)
Formulation
2.10 Weak Forms for Eulerian Mesh 68 (1)
Equations
2.11 Finite Element Equations 69 (3)
2.11.1 Momentum Equation 71 (1)
2.12 Solution Methods 72 (2)
2.13 Summary 74 (1)
2.14 Exercises 75 (2)
3 Continuum Mechanics 77 (70)
3.1 Introduction 77 (1)
3.2 Deformation and Motion 78 (17)
3.2.1 Definitions 78 (1)
3.2.2 Eulerian and Lagrangian 79 (1)
Coordinates
3.2.3 Motion 80 (1)
3.2.4 Eulerian and Lagrangian 80 (1)
Descriptions
3.2.5 Displacement, Velocity and 81 (2)
Acceleration
3.2.6 Deformation Gradient 83 (1)
3.2.7 Conditions on Motion 84 (1)
3.2.8 Rigid Body Rotation and 85 (10)
Coordinate Transformations
3.3 Strain Measures 95 (9)
3.3.1 Green Strain Tensor 95 (2)
3.3.2 Rate-of-Deformation 97 (1)
3.3.3 Rate-of-Deformation in Terms of 98 (6)
Rate of Green Strain
3.4 Stress Measures 104 (7)
3.4.1 Definitions of Stresses 104 (1)
3.4.2 Transformation between Stresses 105 (2)
3.4.3 Corotational Stress and 107 (4)
Rate-of-Deformation
3.5 Conservation Equations 111 (12)
3.5.1 Conservation Laws 111 (1)
3.5.2 Gauss's Theorem 112 (1)
3.5.3 Material Time Derivative of an 113 (2)
Integral and Reynolds' Transport Theorem
3.5.4 Mass Conservation 115 (1)
3.5.5 Conservation of Linear Momentum 116 (3)
3.5.6 Equilibrium Equation 119 (1)
3.5.7 Reynolds' Theorem for a 119 (1)
Density-Weighted Integrand
3.5.8 Conservation of Angular Momentum 120 (1)
3.5.9 Conservation of Energy 120 (3)
3.6 Lagrangian Conservation Equations 123 (7)
3.6.1 Introduction and Definitions 123 (1)
3.6.2 Conservation of Linear Momentum 124 (2)
3.6.3 Conservation of Angular Momentum 126 (1)
3.6.4 Conservation of Energy in 127 (2)
Lagrangian Description
3.6.5 Power of PK2 Stress 129 (1)
3.7 Polar Decomposition and 130 (13)
Frame-Invariance
3.7.1 Polar Decomposition Theorem 130 (5)
3.7.2 Objective Rates in Constitutive 135 (1)
Equations
3.7.3 Jaumann Rate 136 (1)
3.7.4 Truesdell Rate and Green--Naghdi 137 (5)
Rate
3.7.5 Explanation of Objective Rates 142 (1)
3.8 Exercises 143 (4)
4 Lagrangian Meshes 147 (80)
4.1 Introduction 147 (1)
4.2 Governing Equations 148 (4)
4.3 Weak Form: Principle of Virtual Power 152 (6)
4.3.1 Strong Form to Weak Form 153 (1)
4.3.2 Weak Form to Strong Form 154 (2)
4.3.3 Physical Names of Virtual Power 156 (2)
Terms
4.4 Updated Lagrangian Finite Element 158 (10)
Discretization
4.4.1 Finite Element Approximation 158 (2)
4.4.2 Internal and External Nodal Forces 160 (1)
4.4.3 Mass Matrix and Inertial Forces 161 (1)
4.4.4 Discrete Equations 161 (2)
4.4.5 Element Coordinates 163 (2)
4.4.6 Derivatives of Functions 165 (1)
4.4.7 Integration and Nodal Forces 166 (1)
4.4.8 Conditions on Parent to Current 166 (1)
Map
4.4.9 Simplifications of Mass Matrix 167 (1)
4.5 Implementation 168 (26)
4.5.1 Indicial to Matrix Notation 169 (2)
Translation
4.5.2 Voigt Notation 171 (2)
4.5.3 Numerical Quadrature 173 (1)
4.5.4 Selective-Reduced Integration 174 (1)
4.5.5 Element Force and Matrix 175 (19)
Transformations
4.6 Corotational Formulations 194 (9)
4.7 Total Lagrangian Formulation 203 (3)
4.7.1 Governing Equations 203 (2)
4.7.2 Total Lagrangian Finite Element 205 (1)
Equations by Transformation
4.8 Total Lagrangian Weak Form 206 (3)
4.8.1 Strong Form to Weak Form 206 (2)
4.8.2 Weak Form to Strong Form 208 (1)
4.9 Finite Element Semidiscretization 209 (16)
4.9.1 Discrete Equations 209 (2)
4.9.2 Implementation 211 (10)
4.9.3 Variational Principle for Large 221 (4)
Deformation Statics
4.10 Exercises 225 (2)
5 Constitutive Models 227 (102)
5.1 Introduction 227 (1)
5.2 The Stress--Strain Curve 228 (5)
5.2.1 The Tensile Test 229 (4)
5.3 One-Dimensional Elasticity 233 (4)
5.3.1 Small Strains 233 (2)
5.3.2 Large Strains 235 (2)
5.4 Nonlinear Elasticity 237 (17)
5.4.1 Kirchhoff Material 237 (4)
5.4.2 Incompressibility 241 (1)
5.4.3 Kirchhoff Stress 242 (1)
5.4.4 Hypoelasticity 242 (1)
5.4.5 Relations between Tangent Moduli 243 (4)
5.4.6 Cauchy Elastic Material 247 (1)
5.4.7 Hyperelastic Materials 248 (1)
5.4.8 Elasticity Tensors 249 (2)
5.4.9 Isotropic Hyperelastic Materials 251 (1)
5.4.10 Neo-Hookean Material 252 (1)
5.4.11 Modified Mooney--Rivlin Material 253 (1)
5.5 One-Dimensional Plasticity 254 (8)
5.5.1 Rate-Independent Plasticity in 254 (3)
One Dimension
5.5.2 Extension to Kinematic Hardening 257 (3)
5.5.3 Rate-Dependent Plasticity in One 260 (2)
Dimension
5.6 Multiaxial Plasticity 262 (19)
5.6.1 Hypoelastic-Plastic Materials 263 (4)
5.6.2 J2 Flow Theory Plasticity 267 (2)
5.6.3 Extension to Kinematic Hardening 269 (2)
5.6.4 Mohr--Coulomb Constitutive Model 271 (2)
5.6.5 Drucker--Prager Constitutive Model 273 (1)
5.6.6 Porous Elastic--Plastic Solids: 274 (3)
Gurson Model
5.6.7 Corotational Stress Formulation 277 (2)
5.6.8 Small-Strain Formulation 279 (1)
5.6.9 Large-Strain Viscoplasticity 280 (1)
5.7 Hyperelastic--Plastic Models 281 (11)
5.7.1 Multiplicative Decomposition of 282 (1)
Deformation Gradient
5.7.2 Hyperelastic Potential and Stress 283 (1)
5.7.3 Decomposition of Rates of 283 (2)
Deformation
5.7.4 Flow Rule 285 (1)
5.7.5 Tangent Moduli 286 (2)
5.7.6 J2 Flow Theory 288 (3)
5.7.7 Implications for Numerical 291 (1)
Treatment of Large Rotations
5.7.8 Single-Crystal Plasticity 291 (1)
5.8 Viscoelasticity 292 (2)
5.8.1 Small Strains 292 (1)
5.8.2 Finite Strain Viscoelasticity 293 (1)
5.9 Stress Update Algorithms 294 (20)
5.9.1 Return Mapping Algorithms for 295 (1)
Rate-Independent Plasticity
5.9.2 Fully Implicit Backward Euler 296 (4)
Scheme
5.9.3 Application to J2 Flow Theory -- 300 (2)
Radial Return Algorithm
5.9.4 Algorithmic Moduli 302 (3)
5.9.5 Algorithmic Moduli: J2 Flow and 305 (1)
Radial Return
5.9.6 Semi-Implicit Backward Euler 306 (1)
Scheme
5.9.7 Algorithmic Moduli -- 307 (1)
Semi-Implicit Scheme
5.9.8 Return Mapping Algorithms for 308 (2)
Rate-Dependent Plasticity
5.9.9 Rate Tangent Modulus Method 310 (1)
5.9.10 Incrementally Objective 311 (1)
Integration Schemes for Large
Deformations
5.9.11 Semi-Implicit Scheme for 312 (2)
Hyperelastic--Plastic Constitutive
Models
5.10 Continuum Mechanics and Constitutive 314 (14)
Models
5.10.1 Eulerian, Lagrangian and 314 (1)
Two-Point Tensors
5.10.2 Pull-Back, Push-Forward and the 314 (5)
Lie Derivative
5.10.3 Material Frame Indifference 319 (2)
5.10.4 Implications for Constitutive 321 (1)
Relations
5.10.5 Objective Scalar Functions 322 (1)
5.10.6 Restrictions on Elastic Moduli 323 (1)
5.10.7 Material Symmetry 324 (1)
5.10.8 Frame Invariance in 325 (1)
Hyperelastic--Plastic Models
5.10.9 Clausius--Duhem Inequality and 326 (2)
Stability Postulates
5.11 Exercises 328 (1)
6 Solution Methods and Stability 329 (88)
6.1 Introduction 329 (1)
6.2 Explicit Methods 330 (7)
6.2.1 Central Difference Method 330 (2)
6.2.2 Implementation 332 (3)
6.2.3 Energy Balance 335 (1)
6.2.4 Accuracy 336 (1)
6.2.5 Mass Scaling, Subcycling and 337 (1)
Dynamic Relaxation
6.3 Equilibrium Solutions and Implicit 337 (21)
Time Integration
6.3.1 Equilibrium and Transient Problems 337 (1)
6.3.2 Equilibrium Solutions and 338 (1)
Equilibrium Points
6.3.3 Newmark β-Equations 338 (1)
6.3.4 Newton's Method 339 (2)
6.3.5 Newton's Method for n Unknowns 341 (2)
6.3.6 Conservative Problems 343 (1)
6.3.7 Implementation of Newton's Method 344 (2)
6.3.8 Constraints 346 (7)
6.3.9 Convergence Criteria 353 (1)
6.3.10 Line Search 354 (1)
6.3.11 The α-Method 355 (1)
6.3.12 Accuracy and Stability of 356 (1)
Implicit Methods
6.3.13 Convergence and Robustness of 357 (1)
Newton Iteration
6.3.14 Selection of Integration Method 358 (1)
6.4 Linearization 358 (17)
6.4.1 Linearization of the Internal 358 (2)
Nodal Forces
6.4.2 Material Tangent Stiffness 360 (1)
6.4.5 Geometric Stiffness 361 (1)
6.4.4 Alternative Derivations of 362 (2)
Tangent Stiffness
6.4.5 External Load Stiffness 364 (8)
6.4.6 Directional Derivatives 372 (2)
6.4.7 Algorithmically Consistent 374 (1)
Tangent Stiffness
6.5 Stability and Continuation Methods 375 (16)
6.5.1 Stability 375 (3)
6.5.2 Branches of Equilibrium Solutions 378 (2)
6.5.3 Methods of Continuation and Arc 380 (2)
Length Methods
6.5.4 Linear Stability 382 (1)
6.5.5 Symmetric Systems 383 (1)
6.5.6 Conservative Systems 384 (1)
6.5.7 Remarks on Linear Stability 384 (1)
Analysis
6.5.8 Estimates of Critical Points 385 (1)
6.5.9 Initial Estimates of Critical 386 (5)
Points
6.6 Numerical Stability 391 (16)
6.6.1 Definition and Discussion 391 (1)
6.6.2 Stability of a Model Linear 392 (4)
System: Heat Conduction
6.6.3 Amplification Matrices 396 (1)
6.6.4 Amplification Matrix for 397 (1)
Generalized Trapezoidal Rule
6.6.5 The z-Transform 398 (1)
6.6.6 Stability of Damped Central 399 (2)
Difference Method
6.6.7 Linearized Stability Analysis of 401 (2)
Newmark β-Method
6.6.8 Eigenvalue Inequality and Time 403 (1)
Step Estimates
6.6.9 Element Eigenvalues 404 (2)
6.6.10 Stability in Energy 406 (1)
6.7 Material Stability 407 (8)
6.7.1 Description and Early Work 407 (1)
6.7.2 Material Stability Analysis 408 (3)
6.7.3 Material Instability and Change 411 (1)
of Type of PDEs in 1D
6.7.4 Regularization 412 (3)
6.8 Exercises 415 (2)
7 Arbitrary Lagrangian Eulerian Formulations 417 (60)
7.1 Introduction 417 (2)
7.2 ALE Continuum Mechanics 419 (7)
7.2.1 Material Motion, Mesh 419 (2)
Displacement, Mesh Velocity, and Mesh
Acceleration
7.2.2 Material Time Derivative and 421 (1)
Convective Velocity
7.2.3 Relationship of ALE Description 422 (4)
to Eulerian and Lagrangian Descriptions
7.3 Conservation Laws in ALE Description 426 (2)
7.3.1 Conservation of Mass (Equation of 426 (1)
Continuity)
7.3.2 Conservation of Linear and 427 (1)
Angular Momenta
7.3.3 Conservation of Energy 428 (1)
7.4 ALE Governing Equations 428 (1)
7.5 Weak Forms 429 (4)
7.5.1 Continuity Equation -- Weak Form 430 (1)
7.5.2 Momentum Equation -- Weak Form 430 (1)
7.5.3 Finite Element Approximations 430 (2)
7.5.4 The Finite Element Matrix 432 (1)
Equations
7.6 Introduction to the Petrov--Galerkin 433 (9)
Method
7.6.1 Galerkin Discretization of the 434 (2)
Advection--Diffusion Equation
7.6.2 Petrov--Galerkin Stabilization 436 (1)
7.6.3 Alternative Derivation of the SUPG 437 (1)
7.6.4 Parameter Determination 438 (3)
7.6.5 SUPG Multiple Dimensions 441 (1)
7.7 Petrov--Galerkin Formulation of 442 (3)
Momentum Equation
7.7.1 Alternative Stabilization 443 (1)
Formulation
7.7.2 The δviPG Test Function 443 (1)
7.7.3 Finite Element Equation 444 (1)
7.8 Path-Dependent Materials 445 (12)
7.8.1 Strong Form of Stress Update 444 (2)
7.8.2 Weak Form of Stress Update 446 (1)
7.8.3 Finite Element Discretization 446 (1)
7.8.4 Stress Update Procedures 447 (6)
7.8.5 Finite Element Implementation of 453 (3)
Stress Update Procedures in 1D
7.8.6 Explicit Time Integration 456 (1)
Algorithm
7.9 Linearization of the Discrete 457 (3)
Equations
7.9.1 Internal Nodal Forces 457 (2)
7.9.2 External Nodal Forces 459 (1)
7.10 Mesh Update Equations 460 (8)
7.10.1 Introduction 460 (1)
7.10.2 Mesh Motion Prescribed A Priori 461 (1)
7.10.3 Lagrange--Euler Matrix Method 461 (2)
7.10.4 Deformation Gradient Formulations 463 (2)
7.10.5 Automatic Mesh Generation 465 (1)
7.10.6 Mesh Update Using a Modified 466 (1)
Elasticity Equation
7.10.7 Mesh Update Example 467 (1)
7.11 Numerical Example: An 468 (3)
Elastic--Plastic Wave Propagation Problem
7.12 Total ALE Formulations 471 (4)
7.12.1 Total ALE Conservation Laws 471 (2)
7.12.2 Reduction to Updated ALE 473 (2)
Conservation Laws
7.13 Exercises 475 (2)
8 Element Technology 477 (58)
8.1 Introduction 477 (2)
8.2 Element Performance 479 (8)
8.2.1 Overview 479 (4)
8.2.2 Completeness, Consistency, and 483 (1)
Reproducing Conditions
8.2.3 Convergence Results for Linear 484 (2)
Problems
8.2.4 Convergence in Nonlinear Problems 486 (1)
8.3 Element Properties and Patch Tests 487 (9)
8.3.1 Patch Tests 487 (1)
8.3.2 Standard Patch Test 487 (2)
8.3.3 Patch Test in Nonlinear Programs 489 (1)
8.3.4 Patch Test in Explicit Programs 489 (1)
8.3.5 Patch Tests for Stability 490 (1)
8.3.6 Linear Reproducing Conditions of 490 (2)
Isoparametric Elements
8.3.7 Completeness of Subparametric and 492 (1)
Superparametric Elements
8.3.8 Element Rank and Rank Deficiency 493 (1)
8.3.9 Rank of Numerically Integrated 494 (2)
Elements
8.4 Q4 and Volumetric Locking 496 (5)
8.4.1 Element Description 496 (1)
8.4.2 Basis Form of Q4 Approximation 497 (2)
8.4.3 Locking in Q4 499 (2)
8.5 Multi-Field Weak Forms and Elements 501 (13)
8.5.1 Nomenclature 501 (1)
8.5.2 Hu--Washizu Weak Form 501 (2)
8.5.3 Alternative Multi-Field Weak Forms 503 (1)
8.5.4 Total Lagrangian Form of the 504 (1)
Hu--Washizu
8.5.5 Pressure--Velocity (p--v) 505 (2)
Implementation
8.5.6 Element Specific Pressure 507 (1)
8.5.7 Finite Element Implementation of 508 (2)
Hu--Washizu
8.5.8 Simo--Hughes B-Bar Method 510 (1)
8.5.9 Simo--Rifai Formulation 511 (3)
8.6 Multi-Field Quadrilaterals 514 (4)
8.6.1 Assumed Velocity Strain to Avoid 514 (2)
Volumetric Locking
8.6.2 Shear Locking and its Elimination 516 (1)
8.6.3 Stiffness Matrices for Assumed 517 (1)
Strain Elements
8.6.4 Other Techniques in Quadrilaterals 517 (1)
8.7 One-Point Quadrature Elements 518 (9)
8.7.1 Nodal Forces and B-Matrix 518 (1)
8.7.2 Spurious Singular Modes 519 (2)
(Hourglass)
8.7.3 Perturbation Hourglass 521 (1)
Stabilization
8.7.4 Stabilization Procedure 522 (1)
8.7.5 Scaling and Remarks 522 (1)
8.7.6 Physical Stabilization 523 (2)
8.7.7 Assumed Strain with Multiple 525 (1)
Integration Points
8.7.8 Three-Dimensional Elements 526 (1)
8.8 Examples 527 (4)
8.8.1 Static Problems 527 (1)
8.8.2 Dynamic Cantilever Beam 528 (2)
8.8.3 Cylindrical Stress Wave 530 (1)
8.9 Stability 531 (2)
8.10 Exercises 533 (2)
9 Beams and Shells 535 (62)
9.1 Introduction 535 (2)
9.2 Beam Theories 537 (3)
9.2.1 Assumptions of Beam Theories 537 (1)
9.2.2 Timoshenko (Shear Beam) Theory 538 (1)
9.2.3 Euler--Bernoulli Theory 539 (1)
9.2.4 Discrete Kirchhoff and 540 (1)
Mindlin--Reissner Theories
9.3 Continuum-Based Beam 540 (11)
9.3.1 Definitions and Nomenclature 541 (1)
9.3.2 Assumptions 542 (1)
9.3.3 Motion 543 (2)
9.3.4 Nodal Forces 545 (1)
9.3.5 Constitutive Update 545 (2)
9.3.6 Continuum Nodal Internal Forces 547 (2)
9.3.7 Mass Matrix 549 (1)
9.3.8 Equations of Motion 550 (1)
9.3.9 Tangent Stiffness 550 (1)
9.4 Analysis of the CB Beam 551 (12)
9.4.1 Motion 551 (3)
9.4.2 Velocity Strains 554 (1)
9.4.3 Resultant Stresses and Internal 555 (1)
Power
9.4.4 Resultant External Forces 556 (1)
9.4.5 Boundary Conditions 557 (1)
9.4.6 Weak Form 558 (1)
9.4.7 Strong Form 558 (1)
9.4.8 Finite Element Approximation 559 (4)
9.5 Continuum-Based Shell Implementation 563 (15)
9.5.1 Assumptions in Classical Shell 564 (1)
Theories
9.5.2 Coordinates and Definitions 564 (1)
9.5.3 Assumptions 565 (1)
9.5.4 Coordinate Systems 565 (1)
9.5.5 Finite Element Approximation of 566 (2)
Motion
9.5.6 Local Coordinates 568 (1)
9.5.7 Constitutive Equation 569 (1)
9.5.8 Thickness 570 (1)
9.5.9 Master Nodal Forces 570 (1)
9.5.10 Mass Matrix 571 (1)
9.5.11 Discrete Momentum Equation 571 (1)
9.5.12 Tangent Stiffness 572 (1)
9.5.13 Five Degree-of-Freedom 572 (1)
Formulation
9.5.14 Large Rotations 573 (1)
9.5.15 Euler's Theorem 573 (2)
9.5.16 Exponential Map 575 (1)
9.5.17 First-and Second-Order Updates 576 (1)
9.5.18 Hughes--Winget Update 577 (1)
9.5.19 Quaternions 577 (1)
9.5.20 Implementation 578 (1)
9.6 CB Shell Theory 578 (6)
9.6.1 Motion 578 (2)
9.6.2 Velocity drains 580 (1)
9.6.3 Resultant Stresses 581 (1)
9.6.4 Boundary Conditions 582 (1)
9.6.5 Inconsistencies and 583 (1)
Idiosyncrasies of Structural Theories
9.7 Shear and Membrane Locking 584 (5)
9.7.1 Description and Definitions 584 (1)
9.7.2 Shear Locking 585 (2)
9.7.3 Membrane Locking 587 (1)
9.7.4 Elimination of Locking 588 (1)
9.8 Assumed Strain Elements 589 (3)
9.8.1 Assumed Strain 4-Node 589 (2)
Quadrilateral
9.8.2 Rank of Element 591 (1)
9.8.3 Nine-Node Quadrilateral 591 (1)
9.9 One-Point Quadrature Elements 592 (3)
9.10 Exercises 595 (2)
10 Contact-Impact 597 (46)
10.1 Introduction 597 (1)
10.2 Contact Interface Equations 598 (11)
10.2.1 Notation and Preliminaries 598 (2)
10.2.2 Impenetrability Condition 600 (2)
10.2.3 Traction Conditions 602 (1)
10.2.4 Unitary Contact Condition 603 (1)
10.2.5 Surface Description 603 (1)
10.2.6 Interpenetration Measure 604 (1)
10.2.7 Path-Independent 605 (1)
Interpenetration Rate
10.2.8 Tangential Relative Velocity for 606 (3)
Interpenetrated Bodies
10.3 Friction Models 609 (5)
10.3.1 Classification 609 (1)
10.3.2 Coulomb Friction 609 (1)
10.3.3 Interface Constitutive Equations 610 (4)
10.4 Weak Forms 614 (10)
10.4.1 Notation and Preliminaries 614 (1)
10.4.2 Lag range Multiplier Weak Form 615 (2)
10.4.3 Contribution of Virtual Power to 617 (1)
Contact Surface
10.4.4 Rate-Dependent Penalty 618 (2)
10.4.5 Interpenetration-Dependent 620 (1)
Penalty
10.4.6 Perturbed Lagrangian Weak Form 620 (1)
10.4.7 Augmented Lagrangian 621 (1)
10.4.8 Tangential Tractions by Lagrange 622 (2)
Multipliers
10.5 Finite Element Discretization 624 (14)
10.5.1 Overview 624 (1)
10.5.2 Lagrange Multiplier Method 624 (5)
10.5.3 Assembly of Interface Matrix 629 (1)
10.5.4 Lagrange Multipliers for 629 (1)
Small-Displacement Elastostatics
10.5.5 Penalty Method for Nonlinear 630 (1)
Frictionless Contact
10.5.6 Penalty Method for 631 (1)
Small-Displacement Elastostatics
10.5.7 Augmented Lagrangian 631 (2)
10.5.8 Perturbed Lagrangian 633 (4)
10.5.9 Regularization 637 (1)
10.6 On Explicit Methods 638 (5)
10.6.1 Explicit Methods 638 (1)
10.6.2 Contact in One Dimension 639 (2)
10.6.3 Penalty Method 641 (1)
10.6.4 Explicit Algorithm 642 (1)
11 Extended Finite Element Method (XFEM) 643 (38)
11.1 Introduction 643 (4)
11.1.1 Strong Discontinuity 643 (2)
11.1.2 Weak Discontinuity 645 (1)
11.1.3 XFEM for Discontinuities 646 (1)
11.2 Partition of Unity and Enrichments 647 (1)
11.3 One-Dimensional XFEM 648 (8)
11.3.1 Strong Discontinuity 648 (4)
11.3.2 Weak Discontinuity 652 (3)
11.3.3 Mass Matrix 655 (1)
11.4 Multi-Dimension XFEM 656 (4)
11.4.1 Crack Modeling 656 (2)
11.4.2 Tip Enrichment 658 (2)
11.4.3 Enrichment in a Local Coordinate 660 (1)
System
11.5 Weak and Strong Forms 660 (2)
11.6 Discrete Equations 662 (6)
11.6.1 Strain--Displacement Matrix for 665 (3)
Weak Discontinuity
11.7 Level Set Method 668 (2)
11.7.1 Level Set in 1D 668 (1)
11.7.2 Level Set in 2D 668 (1)
11.7.3 Dynamic Fracture Growth Using 669 (1)
Level Set Updates
11.8 The Phantom Node Method 670 (3)
11.8.1 Element Decomposition in 1D 670 (1)
11.8.2 Element Decomposition in 671 (2)
Multi-Dimensions
11.9 Integration 673 (2)
11.9.1 Integration for Discontinuous 673 (2)
Enrichments
11.9.2 Integration for Singular 675 (1)
Enrichments
11.10 An Example of XFEM Simulation 675 (3)
11.11 Exercise 678 (3)
12 Introduction to Multiresolution Theory 681 (40)
12.1 Motivation: Materials are Structured 681 (4)
Continua
12.2 Bulk Deformation of Microstructured 685 (1)
Continua
12.3 Generalizing Mechanics to Bulk 686 (10)
Microstructured Continua
12.3.1 The Need for a Generalized 686 (1)
Mechanics
12.3.2 Major Ideas for a Generalized 687 (1)
Mechanics
12.3.3 Higher-Order Approach 688 (1)
12.3.4 Higher-Grade Approach 689 (2)
12.3.5 Reinterpretation of 691 (5)
Micromorphism for Bulk Microstructured
Materials
12.4 Multiscale Microstructures and the 696 (3)
Multiresolution Continuum Theory
12.5 Governing Equations for MCT 699 (2)
12.5.1 Virtual Internal Power 699 (1)
72.5.2 Virtual External Power 699 (1)
12.5.3 Virtual Kinetic Power 700 (1)
12.5.4 Strong Form of MCT Equations 700 (1)
12.6 Constructing MCT Constitutive 701 (4)
Relationships
12.7 Basic Guidelines for RVE Modeling 705 (5)
12.7.1 Determining RVE Cell Size 706 (1)
12.7.2 RVE Boundary Conditions 707 (3)
12.8 Finite Element Implementation of MCT 710 (2)
12.9 Numerical Example 712 (6)
12.9.1 Void-Sheet Mechanism in 712 (1)
High-Strength Alloy
12.9.2 MCT Multiscale Constitutive 713 (1)
Modeling Outline
12.9.3 Finite Element Problem Setup for 714 (2)
a Two-Dimensional Tensile Specimen
12.9.4 Results 716 (2)
12.10 Future Research Directions of MCT 718 (1)
Modeling
12.11 Exercises 719 (2)
13 Single-Crystal Plasticity 721 (30)
13.1 Introduction 721 (2)
31.2 Crystallographic Description of 723 (3)
Cubic and Non-Cubic Crystals
13.2.1 Specifying Directions 724 (1)
13.2.2 Specifying Planes 725 (1)
13.3 Atomic Origins of Plasticity and the 726 (3)
Burgers Vector in Single Crystals
13.4 Defining Slip Planes and Directions 729 (6)
in General Single Crystals
13.5 Kinematics of Single Crystal 735 (5)
Plasticity
13.5.1 Relating the Intermediate 735 (2)
Configuration to Crystalline Mechanics
13.5.2 Constitutive Definitions of the 737 (1)
Plastic Parts of Deformation Rate and
Spin
13.5.3 Simplification of the Kinematics 738 (1)
by Restriction to Small Elastic Strain
13.5.4 Final Remarks 739 (1)
13.6 Dislocation Density Evolution 740 (2)
13.7 Stress Required for Dislocation 742 (1)
Motion
13.8 Stress Update in Rate-Dependent 743 (2)
Single-Crystal Plasticity
13.8.1 The Resolved Shear Stress 743 (1)
13.8.2 The Resolved Shear Stress Rate 743 (1)
13.8.3 Updating Resolved Shear Stress 744 (1)
in Rate-Dependent Materials
13.8.4 Updating the Cauchy Stress 745 (1)
13.8.5 Adiabatic Temperature Update 745 (1)
13.9 Algorithm for Rate-Dependent 745 (2)
Dislocation-Density Based Crystal
Plasticity
13.10 Numerical Example: Localized Shear 747 (3)
and Inhomogeneous Deformation
13.11 Exercises 750 (1)
Appendix 1 Voigt Notation 751 (6)
Appendix 2 Norms 757 (4)
Appendix 3 Element Shape Functions 761 (6)
Appendix 4 Euler Angles From Pole Figures 767 (4)
Appendix 5 Example of Dislocation-Density 771 (6)
Evolutionary Equations
Glossary 777 (4)
References 781 (14)
Index 795
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