Nonlinear finite elements for continua and structures:
Gespeichert in:
| Format: | Buch |
|---|---|
| Sprache: | English |
| Veröffentlicht: |
Chichester, West Sussex
Wiley
2014
|
| Ausgabe: | 2. ed. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XXVIII, 804 S. graph. Darst. |
| ISBN: | 9781118632703 |
Internformat
MARC
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| 245 | 1 | 0 | |a Nonlinear finite elements for continua and structures |c Ted Belytschko ... |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Chichester, West Sussex |b Wiley |c 2014 | |
| 300 | |a XXVIII, 804 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Finite element method | |
| 650 | 4 | |a Continuum mechanics | |
| 650 | 4 | |a Structural analysis (Engineering) | |
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| 650 | 0 | 7 | |a Festkörpermechanik |0 (DE-588)4129367-8 |2 gnd |9 rswk-swf |
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Datensatz im Suchindex
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| adam_text |
Titel: Nonlinear finite elements for continua and structures
Autor: Belytschko, Ted
Jahr: 2014
Contents
Foreword xxj
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
1.3.1 Indicial Notation 7
1.3.2 Tensor Notation 8
1.3.3 Functions 8
1.3.4 Matrix Notation 8
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
2.2.1 Nomenclature 21
2.2.2 Motion and Strain Measure 22
2.2.3 Stress Measure 22
2.2.4 Governing Equations 23
2.2.5 Momentum Equation in Terms of Displacements 26
2.2.6 Continuity of Functions 27
2.2.7 Fundamental Theorem of Calculus 28
viii
Contents
2.3 Weak Form for Total Lagrangian Formulation
2.3.1 Strong Form to Weak Form
2.3.2 Weak Form to Strong Form
2.3.3 Physical Names of Virtual Work Terms
2.3.4 Principle of Virtual Work
2.4 Finite Element Discretization in Total Lagrangian Formulation
2.4.1 Finite Element Approximations
2.4.2 Nodal Forces
2.4.3 Semidiscrete Equations
2.4.4 Initial Conditions
2.4.5 Least-Square Fit to Initial Conditions
2.4.6 Diagonal Mass Matrix
2.5 Element and Global Matrices
2.6 Governing Equations for Updated Lagrangian Formulation
2.6.1 Boundary and Interior Continuity Conditions
2.6.2 Initial Conditions
2.7 Weak Form for Updated Lagrangian Formulation
2.8 Element Equations for Updated Lagrangian Formulation
2.8.1 Finite Element Approximation
2.8.2 Element Coordinates
2.8.3 Internal and External Nodal Forces
2.8.4 Mass Matrix
2.8.5 Equivalence of Updated and Total Lagrangian Formulations il !¦
2.8.6 Assembly, Boundary Conditions and Initial Conditions
2.8.7 Mesh Distortion n
2.9 Governing Equations for Eulerian Formulation v,'i
2.10 Weak Forms for Eulerian Mesh Equations « «
2.11 Finite Element Equations
2.11.1 Momentum Equation %\\
2.12 Solution Methods i,t
2.13 Summary
2.14 Exercises ,1,
oil
3 Continuum Mechanics , B )i}
3.1 Introduction
3.2 Deformation and Motion
3.2.1 Definitions
3.2.2 Eulerian and Lagrangian Coordinates
3.2.3 Motion
3.2.4 Eulerian and Lagrangian Descriptions .
3.2.5 Displacement, Velocity and Acceleration
3.2.6 Deformation Gradient
3.2.7 Conditions on Motion ,
3.2.8 Rigid Body Rotation and Coordinate Transformations
3.3 Strain Measures v
3.3.1 Green Strain Tensor
3.3.2 Rate-of-Deformation
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Contents
3.3.3 Rate-of-Deformation in Terms of Rate of Green Strain 98
3.4 Stress Measures 104
3.4.1 Definitions of Stresses 104
3.4.2 Transformation between Stresses 105
3.4.3 Corotational Stress and Rate-of-Deformation 107
3.5 Conservation Equations 111
3.5.1 Conservation Laws 111
3.5.2 Gauss's Theorem 112
3.5.3 Material Time Derivative of an Integral and Reynolds'
Transport Theorem 113
3.5.4 Mass Conservation 115
3.5.5 Conservation of Linear Momentum 116
3.5.6 Equilibrium Equation 119
3.5.7 Reynolds'Theorem for a Density-Weighted Integrand 119
3.5.8 Conservation of Angular Momentum 120
3.5.9 Conservation of Energy 120
3.6 Lagrangian Conservation Equations 123
3.6.1 Introduction and Definitions 123
3.6.2 Conservation of Linear Momentum 124
3.6.3 Conservation of Angular Momentum 126
3.6.4 Conservation of Energy in Lagrangian Description 127
3.6.5 Power of PK2 Stress 129
3.7 Polar Decomposition and Frame-Invariance 130
3.7.1 Polar Decomposition Theorem 130
3.7.2 Objective Rates in Constitutive Equations 135
3.7.3 Jaumann Rate 136
3.7.4 Truesdell Rate and Green-Naghdi Rate 137
3.7.5 Explanation of Objective Rates 142
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
4.3.1 Strong Form to Weak Form 153
4.3.2 Weak Form to Strong Form 154
4.3.3 Physical Names of Virtual Power Terms 156
4.4 Updated Lagrangian Finite Element Discretization 158
4.4.1 Finite Element Approximation 158
4.4.2 Internal and External Nodal Forces 160
4.4.3 Mass Matrix and Inertial Forces 161
4.4.4 Discrete Equations 161
4.4.5 Element Coordinates 163
4.4.6 Derivatives of Functions 165
4.4.7 Integration and Nodal Forces 166
4.4.8 Conditions on Parent to Current Map 166
4.4.9 Simplifications of Mass Matrix 167
X
Contents
4.5 Implementation 1
4.5.1 Indicial to Matrix Notation Translation ?. ' 169
4.5.2 Voigt Notation 1!" ? 171
4.5.3 Numerical Quadrature f- 173
4.5.4 Selective-Reduced Integration t- l7^
4.5.5 Element Force and Matrix Transformations ' 175
4.6 Corotational Formulations - '
4.7 Total Lagrangian Formulation ' • 203
4.7.1 Governing Equations 203
4.7.2 Total Lagrangian Finite Element Equations by Transformation 205
4.8 Total Lagrangian Weak Form • 206
4.8.1 Strong Form to Weak Form • 206
4.8.2 Weak Form to Strong Form 208
4.9 Finite Element Semidiscretization \-;7 209
4.9.1 Discrete Equations . 209
4.9.2 Implementation 211
4.9.3 Variational Principle for Large Deformation Statics 221
4.10 Exercises 225
5 Constitutive Models 227
5.1 Introduction 227
5.2 The Stress-Strain Curve 228
5.2.1 The Tensile Test 229
5.3 One-Dimensional Elasticity 233
5.3.1 Small Strains 233
5.3.2 Large Strains 235
5.4 Nonlinear Elasticity 237
5.4.1 Kirchhoff Material 237
5.4.2 Incompressibility 241
5.4.3 Kirchhoff Stress 242
5.4.4 Hypo elasticity 242
5.4.5 Relations between Tangent Moduli 243
5.4.6 Cauchy Elastic Material 247
5.4.7 Hyperelastic Materials 248
5.4.8 Elasticity Tensors 249
5.4.9 Isotropic Hyperelastic Materials 251
5.4.10 Neo-Hookean Material 252
5.4.11 Modified Mooney-Rivlin Material 253
5.5 One-Dimensional Plasticity 254
5.5.1 Rate-Independent Plasticity in One Dimension 254
5.5.2 Extension to Kinematic Hardening 257
5.5.5 Rate-Dependent Plasticity in One Dimension 260
5.6 Multiaxial Plasticity 262
5.6.1 Hypoelastic-Plastic Materials 263
5.6.2 J2 Flow Theory Plasticity ' 267
5.6.3 Extension to Kinematic Hardening 269
5.6.4 Mohr-Coulomb Constitutive Model 271
Contents
xi
5.6.5 Drucker-Prager Constitutive Model 273
5.6.6 Porous Elastic-Plastic Solids: Gurson Model 274
5.6.7 Corotational Stress Formulation 277
5.6.8 Small-Strain Formulation 279
5.6.9 Large-Strain Viscoplasticity 280
5.7 Hyperelastic-Plastic Models 281
5.7.1 Multiplicative Decomposition of Deformation Gradient 282
5.7.2 Hyperelastic Potential and Stress 283
5.7.3 Decomposition of Rates of Deformation 283
5.7.4 Flow Rule 285
5.7.5 Tangent Moduli 286
5.7.6 J2 Flow Theory 288
5.7.7 Implications for Numerical Treatment of Large Rotations 291
5.7.8 Single-Crystal Plasticity 291
5.8 Viscoelasticity 292
5.8.1 Small Strains 292
5.8.2 Finite Strain Viscoelasticity 293
5.9 Stress Update Algorithms 294
5.9.1 Return Mapping Algorithms for Rate-Independent Plasticity 295
5.9.2 Fully Implicit Backward Euler Scheme 296
5.9.3 Application to J2 Flow Theory - Radial Return Algorithm 300
5.9.4 Algorithmic Moduli 302
5.9.5 Algorithmic Moduli: J2 Flow and Radial Return 305
5.9.6 Semi-Implicit Backward Euler Scheme 306
5.9.7 AIgorithmic Moduli - Semi-Implicit Scheme 307
5.9.8 Return Mapping Algorithms for Rate-Dependent Plasticity 308
5.9.9 Rate Tangent Modulus Method 310
5.9.10 Incrementally Objective Integration Schemes for Large Deformations 311
5.9.11 Semi-Implicit Scheme for Hyperelastic-Plastic Constitutive Models 312
5.10 Continuum Mechanics and Constitutive Models 314
5.10.1 Eulerian, Lagrangian and Two-Point Tensors 314
5.10.2 Pull-Back, Push-Forward and the Lie Derivative 314
5.10.3 Material Frame Indifference 319
5.10.4 Implications for Constitutive Relations 321
5.10.5 Objective Scalar Functions 322
5.10.6 Restrictions on Elastic Moduli 323
5.10.7 Material Symmetry 324
5.10.8 Frame Invariance in Hyperelastic-Plastic Models 325
5.10.9 Clausius-Duhem Inequality and Stability Postulates 326
5.11 Exercises 328
6 Solution Methods and Stability 329
6.1 Introduction 329
6.2 Explicit Methods 330
6.2.1 Central Difference Method 330
6.2.2 Implementation 332
6.2.3 Energy Balance 335
xii
Contents
6.2.4 Accuracy 1,5
6.2.5 Mass Scaling, Subcycling and Dynamic Relaxation
6.3 Equilibrium Solutions and Implicit Time Integration
6.3.1 Equilibrium and Transient Problems
6.3.2 Equilibrium Solutions and Equilibrium Points
6.3.3 Newmark P-Equations
6.3.4 Newton's Method '
6.3.5 Newton's Method for n Unknowns
6.3.6 Conservative Problems "
6.3.7 Implementation of Newton's Method »•
6.3.8 Constraints '
6.3.9 Convergence Criteria .• ¦$
6.3.10 Line Search
6.3.11 The (X-Method *' •
6.3.12 Accuracy and Stability of Implicit Methods
6.3.13 Convergence and Robustness of Newton Iteration
6.3.14 Selection of Integration Method
6.4 Linearization
6.4.1 Linearization of the Internal Nodal Forces
6.4.2 Material Tangent Stiffness
6.4.3 Geometric Stiffness
6.4.4 AIternative Derivations of Tangent Stiffness
6.4.5 External Load Stiffness
6.4.6 Directional Derivatives
6.4.7 AIgorithmically Consistent Tangent Stiffness
6.5 Stability and Continuation Methods
6.5.1 Stability
6.5.2 Branches of Equilibrium Solutions
6.5.3 Methods of Continuation and Arc Length Methods
6.5.4 Linear Stability
6.5.5 Symmetric Systems
6.5.6 Conservative Systems
6.5.7 Remarks on Linear Stability Analysis
6.5.8 Estimates of Critical Points
6.5.9 Initial Estimates of Critical Points
6.6 Numerical Stability
6.6.1 Definition and Discussion
6.6.2 Stability of a Model Linear System: Heat Conduction
6.6.3 Amplification Matrices
6.6.4 Amplification Matrix for Generalized Trapezoidal Rul
6.6.5 The z-Transform
6.6.6 Stability of Damped Central Difference Method
Linearized Stability Analysis of Newmark P-Method
Eigenvalue Inequality and Time Step Estimates
6.6.9 Element Eigenvalues
6.6.10 Stability in Energy
6.6.7
6.6.8
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Contents
xiii
6.7 Material Stability 407
6.7.1 Description and Early Work 407
6.7.2 Material Stability Analysis 408
6.7.3 Material Instability and Change of Type ofPDEs in ID 411
6.7.4 Regularization 412
6.8 Exercises 415
7 Arbitrary Lagrangian Eulerian Formulations 417
7.1 Introduction 417
7.2 ALE Continuum Mechanics 419
7.2.1 Material Motion, Mesh Displacement, Mesh Velocity,
and Mesh Acceleration 419
7.2.2 Material Time Derivative and Convective Velocity 421
7.2.3 Relationship of ALE Description to Eulerian
and Lagrangian Descriptions 422
7.3 Conservation Laws in ALE Description 426
7.3.1 Conservation of Mass (Equation of Continuity) 426
7.3.2 Conservation of Linear and Angular Momenta 427
7.3.3 Conservation of Energy 428
7.4 ALE Governing Equations 428
7.5 Weak Forms 429
7.5.1 Continuity Equation-Weak Form 430
7.5.2 Momentum Equation - Weak Form 430
7.5.3 Finite Element Approximations 430
7.5.4 The Finite Element Matrix Equations 432
7.6 Introduction to the Petrov-Galerkin Method 433
7.6.1 Galerkin Discretization of the Advection-Diffusion Equation 434
7.6.2 Petrov-Galerkin Stabilization 436
7.6.3 Alternative Derivation of the SUPG 437
7.6.4 Parameter Determination 438
7.6.5 SUPG Multiple Dimensions 441
7.7 Petrov-Galerkin Formulation of Momentum Equation 442
7.7.1 Alternative Stabilization Formulation 443
7.7.2 The Sv['G Test Function 443
7.7.3 Finite Element Equation 444
7.8 Path-Dependent Materials 445
7.8.1 Strong Form of Stress Update 446
7.8.2 Weak Form of Stress Update 446
7.8.3 Finite Element Discretization 446
7.8.4 Stress Update Procedures 447
7.8.5 Finite Element Implementation of Stress Update
Procedures in ID 453
7.8.6 Explicit Time Integration Algorithm 456
7.9 Linearization of the Discrete Equations 457
7.9.1 Internal Nodal Forces 457
7.9.2 External Nodal Forces 459
xiv
Contents
7.10 Mesh Update Equations
7.10.1 Introduction '
7.10.2 Mesh Motion Prescribed A Priori u
7.10.3 Lagrange-Euler Matrix Method .aw 461
7.10.4 Deformation Gradient Formulations w ri .i 463
7.10.5 Automatic Mesh Generation 465
7.10.6 Mesh Update Using a Modified Elasticity Equation 466
7.10.7 Mesh Update Example 467
7.11 Numerical Example: An Elastic-Plastic Wave Propagation Problem 468
7.12 Total ALE Formulations ;-w 471
7.12.1 Total ALE Conservation Laws 'v 471
7.12.2 Reduction to Updated ALE Conservation Laws 473
7.13 Exercises « 475
ww
8 Element Technology 477
8.1 Introduction 477
8.2 Element Performance p 479
8.2.1 Overview " 479
8.2.2 Completeness, Consistency, and Reproducing Conditions 483
8.2.3 Convergence Results for Linear Problems 484
8.2.4 Convergence in Nonlinear Problems 486
8.3 Element Properties and Patch Tests 487
8.3.1 Patch Tests 487
8.3.2 Standard Patch Test 487
8.3.3 Patch Test in Nonlinear Programs 489
8.3.4 Patch Test in Explicit Programs 489
8.3.5 Patch Tests for Stability 490
8.3.6 Linear Reproducing Conditions of Isoparametric Elements 490
8.3.7 Completeness of Subparametric and Superparametric Elements 492
8.3.8 Element Rank and Rank Deficiency 493
8.3.9 Rank of Numerically Integrated Elements 494
8.4 Q4 and Volumetric Locking 496
8.4.1 Element Description 496
8.4.2 Basis Form of Q4 Approximation 497
8.4.3 Locking in Q4 499
8.5 Multi-Field Weak Forms and Elements 501
8.5.1 Nomenclature 501
8.5.2 Hu-Washizu Weak Form 501
8.5.3 Alternative Multi-Field Weak Forms 503
8.5.4 Total Lagrangian Form of the Hu-Washizu 504
8.5.5 Pressure-Velocity (p-v) Implementation 505
8.5.6 Element Specific Pressure 507
8.5.7 Finite Element Implementation of Hu-Washizu 508
8.5.8 Simo-Hughes B-Bar Method 5 jq
8.5.9 Simo-Rifai Formulation 511
8.6 Multi-Field Quadrilaterals 5J4
Contents
xv
8.6,1
Assumed Velocity Strain to Avoid Volumetric Locking
514
8.6.2
Shear Locking and its Elimination
516
8.6.3
Stiffness Matrices for Assumed Strain Elements
517
8.6.4
Other Techniques in Quadrilaterals
517
8.7
One-Point Quadrature Elements
518
8.7.1
Nodal Forces and B-Matrix
518
8.7.2
Spurious Singular Modes (Hourglass)
519
8.7.3
Perturbation Hourglass Stabilization
521
8.7.4
Stabilization Procedure
522
8.7.5
Scaling and Remarks
522
8.7.6
Physical Stabilization
523
8.7.7
Assumed Strain with Multiple Integration Points
525
8.7.8
Three-Dimensional Elements
526
8.8
Examples
527
8.8.1
Static Problems
527
8.8.2
Dynamic Cantilever Beam
528
8.8.3
Cylindrical Stress Wave
530
8.9
Stability
531
8.10 Exercises
533
9 Beams and Shells
535
9.1
Introduction
535
9.2
Beam Theories
537
9.2.1
Assumptions of Beam Theories
537
9.2.2
Timoshenko (Shear Beam) Theory
538
9.2.3
Euler-Bernoulli Theory
539
9.2.4
Discrete Kirchhoffand Mindlin-Reissner Theories
540
9.3
Continuum-Based Beam
540
9.3.1
Definitions and Nomenclature
541
9.3.2
Assumptions
542
9.3.3
Motion
543
9.3.4
Nodal Forces
545
9.3.5
Constitutive Update
545
9.3.6
Continuum Nodal Internal Forces
547
9.3.7
Mass Matrix
549
9.3.8
Equations of Motion
550
9.3.9
Tangent Stiffness
550
9.4
Analysis of the CB Beam
551
9.4.1
Motion
551
9.4.2
Velocity Strains
554
9.4.3
Resultant Stresses and Internal Power
555
9.4.4
Resultant External Forces
556
9.4.5
Boundary Conditions
557
9.4.6
Weak Form
558
9.4.7
Strong Form
558
9.4.8
Finite Element Approximation
559
xvi
Contents
¦0 v'H
9.5 Continuum-Based Shell Implementation
9.5. J Assumptions in Classical Shell Theories
9.5.2 Coordinates and Definitions v
9.5.3 Assumptions «¦
9.5.4 Coordinate Systems 'A:
9.5.5 Finite Element Approximation of Motion
9.5.6 Local Coordinates w
9.5.7 Constitutive Equation
9.5.8 Thickness
9.5.9 Master Nodal Forces
9.5.10 Mass Matrix
9.5.11 Discrete Momentum Equation
9.5.12 Tangent Stiffness
9.5.13 Five Degree-of-Freedom Formulation
9.5.14 Large Rotations
9.5.15 Euler's Theorem
9.5.16 Exponential Map
9.5.17 First-and Second-Order Updates
9.5.18 Hughes-Winget Update
9.5.19 Quaternions
9.5.20 Implementation
9.6 CB Shell Theory
9.6.1 Motion
9.6.2 Velocity Strains
9.6.3 Resultant Stresses
9.6.4 Boundary Conditions
9.6.5 Inconsistencies and Idiosyncrasies of Structural Theories
9.7 Shear and Membrane Locking
9.7. / Description and Definitions
9.7.2 Shear Locking liin
9.7.3 Membrane Locking k ,iv,
9.7.4 Elimination of Locking ,* Cltv
9.8 Assumed Strain Elements w,
9.8.1 Assumed Strain 4-Node Quadrilateral w
9.8.2 Rank of Element - ;.r,
9.8.3 Nine-Node Quadrilateral jS,
9.9 One-Point Quadrature Elements •*
9.10 Exercises
10 Contact-Impact
10.1 Introduction
10.2 Contact Interface Equations
10.2.1 Notation and Preliminaries
10.2.2 Impenetrability Condition
10.2.3 Traction Conditions
10.2.4 Unitary Contact Condition
A.v
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Contents
xvii
10.2.5 Surface Description 603
10.2.6 Interpenetration Measure 604
10.2.7 Path-Independent Interpenetration Rate 605
10.2.8 Tangential Relative Velocity for Interpenetrated Bodies 606
10.3 Friction Models 609
10.3.1 Classification 609
10.3.2 Coulomb Friction 609
10.3.3 Interface Constitutive Equations 610
10.4 Weak Forms 614
10.4.1 Notation and Preliminaries 614
10.4.2 Lagrange Multiplier Weak Form 615
10.4.3 Contribution of Virtual Power to Contact Surface 617
10.4.4 Rate-Dependent Penalty 618
10.4.5 Interpenetration-Dependent Penalty 620
10.4.6 Perturbed Lagrangian Weak Form 620
10.4.7 Augmented Lagrangian 621
10.4.8 Tangential Tractions by Lagrange Multipliers 622
10.5 Finite Element Discretization 624
10.5.1 Overview 624
10.5.2 Lagrange Multiplier Method 624
10.5.3 Assembly of Interface Matrix 629
10.5.4 Lagrange Multipliers for Small-Displacement Elastostatics 629
10.5.5 Penalty Method for Nonlinear Frictionless Contact 630
10.5.6 Penalty Method for Small-Displacement Elastostatics 631
10.5.7 Augmented Lagrangian 631
10.5.8 Perturbed Lagrangian 633
10.5.9 Regularization 637
10.6 On Explicit Methods 638
10.6.1 Explicit Methods 638
10.6.2 Contact in One Dimension 639
10.6.3 Penalty Method 641
10.6.4 Explicit Algorithm 642
11 Extended Finite Element Method (XFEM) 643
11.1 Introduction 643
11.1.1 Strong Discontinuity 643
11.1.2 Weak Discontinuity 645
11.1.3 XFEM for Discontinuities 646
11.2 Partition of Unity and Enrichments 647
11.3 One-Dimensional XFEM 648
11.3.1 Strong Discontinuity 648
11.3.2 Weak Discontinuity 652
11.3.3 Mass Matrix 655
11.4 Multi-Dimension XFEM 656
11.4.1 Crack Modeling 656
xviii
Contents
11.5
11.6
11.7
11.8
11.9
11.10
11.11
11.4.2 Tip Enrichment
11.4.3 Enrichment in a Local Coordinate System ' *
Weak and Strong Forms • ¦¦ •' ¥
Discrete Equations v * M '
11.6.1 Strain-Displacement Matrix for Weak Discontinuity
Level Set Method • '
11.7.1 Level Set in ID .» i a*
11.7.2 Level Set in 2D ¥
11.7.3 Dynamic Fracture Growth Using Level Set Updates
The Phantom Node Method .*»«• '•
11.8.1 Element Decomposition in ID t
11.8.2 Element Decomposition in Multi-Dimensions it
Integration *•-1
11.9.1 Integration for Discontinuous Enrichments ^j \
11.9.2 Integration for Singular Enrichments uit
An Example of XFEM Simulation ¦ « : , ¦ \«
Exercise wiv
¦'V.r
12 Introduction to Multiresolution Theory
12.1 Motivation: Materials are Structured Continua
12.2 Bulk Deformation of Microstructured Continua
12.3 Generalizing Mechanics to Bulk Microstructured Continua
12.3.1 The Need for a Generalized Mechanics
12.3.2 Major Ideas for a Generalized Mechanics A?
12.3.3 Higher-Order Approach m s niv
12.3.4 Higher-Grade Approach w
12.3.5 Reinterpretation of Micromorphismfor Bulk
Microstructured Materials
12.4
12.5
12.6
12.7
12.8
12.9
Governing Equations for MCT
12.5.1 Virtual Internal Power
12.5.2 Virtual External Power ss
12.5.3 Virtual Kinetic Power
12.5.4 Strong Form of MCT Equations
Constructing MCT Constitutive Relationships
Basic Guidelines for RVE Modeling
12.7.1 Determining RVE Cell Size
12.7.2 RVE Boundary Conditions
Finite Element Implementation of MCT
Numerical Example
12.9.1
12.9.2
12.9.3
Void-Sheet Mechanism in High-Strength Alloy
MCT Multiscale Constitutive Modeling Outline
Finite Element Problem Setup for a Two-Dimensional
Tensile Specimen
12.9.4 Results
12.10 Future Research Directions of MCT Modeling
12.11 Exercises
"v
658
660
5. " -V
660
V -A "¦¦¦¦
662
4 -t •
665
11'
668
V
668
668
¦if v
669
/ *
670
670
¦; .
671
673
673
:¦
675
P, ' ;V
675
A : ' \
678
,3 ' ¦¦
681
681
h
685
\
686
A
686
•% . (
687
/"* \ •
688
i .4
689
1: 1
691
\ Theory
696
}
' 699
\ .)
699
V -
699
700
700
701
705
706
707
710
712
712
713
714
716
718
719
Contents xix
13 Single-Crystal Plasticity 721
13.1 Introduction 721
13.2 Crystallographic Description of Cubic and Non-Cubic Crystals 723
13.2.1 Specifying Directions 724
13.2.2 Specifying Planes 725
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
13.5.1 Relating the Intermediate Configuration to Crystalline Mechanics 735
13.5.2 Constitutive Definitions of the Plastic Parts of Deformation
Rate and Spin 737
13.5.3 Simplification of the Kinematics by Restriction to
Small Elastic Strain 738
13.5.4 Final Remarks 739
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
13.8.1 The Resolved Shear Stress 743
13.8.2 The Resolved Shear Stress Rate 743
13.8.3 Updating Resolved Shear Stress in Rate-Dependent Materials 744
13.8.4 Updating the Cauchy Stress 745
13.8.5 Adiabatic Temperature Update 745
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 |
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| building | Verbundindex |
| bvnumber | BV041556914 |
| classification_rvk | SK 910 UF 2000 |
| classification_tum | MTA 010f |
| ctrlnum | (OCoLC)869872103 (DE-599)HBZHT017622550 |
| dewey-full | 620.001515355 |
| dewey-hundreds | 600 - Technology (Applied sciences) |
| dewey-ones | 620 - Engineering and allied operations |
| dewey-raw | 620.001515355 |
| dewey-search | 620.001515355 |
| dewey-sort | 3620.001515355 |
| dewey-tens | 620 - Engineering and allied operations |
| discipline | Physik Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV041556914 |
| illustrated | Illustrated |
| indexdate | 2024-11-05T09:01:11Z |
| institution | BVB |
| isbn | 9781118632703 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-027002516 |
| oclc_num | 869872103 |
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| physical | XXVIII, 804 S. graph. Darst. |
| publishDate | 2014 |
| publishDateSearch | 2014 |
| publishDateSort | 2014 |
| publisher | Wiley |
| record_format | marc |
| spelling | Nonlinear finite elements for continua and structures Ted Belytschko ... 2. ed. Chichester, West Sussex Wiley 2014 XXVIII, 804 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Finite element method Continuum mechanics Structural analysis (Engineering) Strukturanalyse (DE-588)4183787-3 gnd rswk-swf Festkörpermechanik (DE-588)4129367-8 gnd rswk-swf Nichtlineare Finite-Elemente-Methode (DE-588)4217996-8 gnd rswk-swf Nichtlineare Finite-Elemente-Methode (DE-588)4217996-8 s Festkörpermechanik (DE-588)4129367-8 s Strukturanalyse (DE-588)4183787-3 s DE-604 Belytschko, Ted 1943-2014 Sonstige (DE-588)12264686X oth HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027002516&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Nonlinear finite elements for continua and structures Finite element method Continuum mechanics Structural analysis (Engineering) Strukturanalyse (DE-588)4183787-3 gnd Festkörpermechanik (DE-588)4129367-8 gnd Nichtlineare Finite-Elemente-Methode (DE-588)4217996-8 gnd |
| subject_GND | (DE-588)4183787-3 (DE-588)4129367-8 (DE-588)4217996-8 |
| title | Nonlinear finite elements for continua and structures |
| title_auth | Nonlinear finite elements for continua and structures |
| title_exact_search | Nonlinear finite elements for continua and structures |
| title_full | Nonlinear finite elements for continua and structures Ted Belytschko ... |
| title_fullStr | Nonlinear finite elements for continua and structures Ted Belytschko ... |
| title_full_unstemmed | Nonlinear finite elements for continua and structures Ted Belytschko ... |
| title_short | Nonlinear finite elements for continua and structures |
| title_sort | nonlinear finite elements for continua and structures |
| topic | Finite element method Continuum mechanics Structural analysis (Engineering) Strukturanalyse (DE-588)4183787-3 gnd Festkörpermechanik (DE-588)4129367-8 gnd Nichtlineare Finite-Elemente-Methode (DE-588)4217996-8 gnd |
| topic_facet | Finite element method Continuum mechanics Structural analysis (Engineering) Strukturanalyse Festkörpermechanik Nichtlineare Finite-Elemente-Methode |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027002516&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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