The Finite Element Method for Three-Dimensional Thermomechanical Applications

Though many 'finite element' books exist, this book provides a
unique focus on developing the method for three-dimensional,
industrial problems. This is significant as many methods which work
well for small applications fail for large scale problems, which
generally:

• are not so well posed

• introduce stringent computer time conditions

• require robust solution techniques.

Starting from sound continuum mechanics principles, derivation in
this book focuses only on proven methods. Coverage of all different
aspects of linear and nonlinear thermal mechanical problems in
solids are described, thereby avoiding distracting the reader with
extraneous solutions paths. Emphasis is put on consistent
representation and includes the examination of topics which are not
frequently found in other texts, such as cyclic symmetry, rigid
body motion and nonlinear multiple point constraints.

Advanced material formulations include anisotropic
hyperelasticity, large strain multiplicative viscoplasticity and
single crystal viscoplasticity. Finally, the methods described in
the book are implemented in the finite element software CalculiX,
which is freely available (www.calculix.de; the GNU General Public
License applies).

Suited to industry practitioners and academic researchers alike,
The Finite Element Method for Three-Dimensional Thermomechanical
Applications expertly bridges the gap between continuum
mechanics and the finite element method.

Autorentext
Guido Dhondt obtained his civil engineering degree at the Catholic University of Leuven, Belgium (1983), going on to undertake a Ph.D. in Civil Engineering at Princeton University, USA (1987). Presently, he works in the field of fracture mechanics and finite element analysis at MTU Aero Engines, Germany. He is one of the authors of the free software finite element program CalculiX.

Klappentext
Though many 'finite element' books exist, this book provides a unique focus on developing the method for three-dimensional, industrial problems. This is significant as many methods which work well for small applications fail for large scale problems, which generally:

• are not so well posed
• introduce stringent computer time conditions
• require robust solution techniques. Starting from sound continuum mechanics principles, derivation in this book focuses only on proven methods. Coverage of all different aspects of linear and nonlinear thermal mechanical problems in solids are described, thereby avoiding distracting the reader with extraneous solutions paths. Emphasis is put on consistent representation and includes the examination of topics which are not frequently found in other texts, such as cyclic symmetry, rigid body motion and nonlinear multiple point constraints. Advanced material formulations include anisotropic hyperelasticity, large strain multiplicative viscoplasticity and single crystal viscoplasticity. Finally, the methods described in the book are implemented in the finite element software CalculiX, which is freely available (www.calculix.de; the GNU General Public License applies).

Suited to industry practitioners and academic researchers alike, The Finite Element Method for Three-Dimensional Thermomechanical Applications expertly bridges the gap between continuum mechanics and the finite element method.

Inhalt

Preface xiii

Nomenclature xv

1 Displacements, Strain, Stress and Energy 1

1.1 The Reference State 1

1.2 The Spatial State 4

1.3 Strain Measures 9

1.4 Principal Strains 13

1.5 Velocity 19

1.6 Objective Tensors 22

1.7 Balance Laws 25

1.7.1 Conservation of mass 25

1.7.2 Conservation of momentum 25

1.7.3 Conservation of angular momentum 26

1.7.4 Conservation of energy 26

1.7.5 Entropy inequality 27

1.7.6 Closure 28

1.8 Localization of the Balance Laws 28

1.8.1 Conservation of mass 28

1.8.2 Conservation of momentum 29

1.8.3 Conservation of angular momentum 31

1.8.4 Conservation of energy 31

1.8.5 Entropy inequality 31

1.9 The Stress Tensor 31

1.10 The Balance Laws in Material Coordinates 34

1.10.1 Conservation of mass 35

1.10.2 Conservation of momentum 35

1.10.3 Conservation of angular momentum 37

1.10.4 Conservation of energy 37

1.10.5 Entropy inequality 37

1.11 The Weak Form of the Balance of Momentum 38

1.11.1 Formulation of the boundary conditions (material coordinates) 38

1.11.2 Deriving the weak form from the strong form (material coordinates) 39

1.11.3 Deriving the strong form from the weak form (material coordinates) 41

1.11.4 The weak form in spatial coordinates 41

1.12 The Weak Form of the Energy Balance 42

1.13 Constitutive Equations 43

1.13.1 Summary of the balance equations 43

1.13.2 Development of the constitutive theory 44

1.14 Elastic Materials 47

1.14.1 General form 47

1.14.2 Linear elastic materials 49

1.14.3 Isotropic linear elastic materials 52

1.14.4 Linearizing the strains 54

1.14.5 Isotropic elastic materials 58

1.15 Fluids 59

2 Linear Mechanical Applications 63

2.1 General Equations 63

2.2 The Shape Functions 67

2.2.1 The 8-node brick element 68

2.2.2 The 20-node brick element 69

2.2.3 The 4-node tetrahedral element 71

2.2.4 The 10-node tetrahedral element 72

2.2.5 The 6-node wedge element 73

2.2.6 The 15-node wedge element 73

2.3 Numerical Integration 75

2.3.1 Hexahedral elements 76

2.3.2 Tetrahedral elements 78

2.3.3 Wedge elements 78

2.3.4 Integration over a surface in three-dimensional space 81

2.4 Extrapolation of Integration Point Values to the Nodes 82

2.4.1 The 8-node hexahedral element 83

2.4.2 The 20-node hexahedral element 84

2.4.3 The tetrahedral elements 86

2.4.4 The wedge elements 86

2.5 Problematic Element Behavior 86

2.5.1 Shear locking 87

2.5.2 Volumetric locking 87

2.5.3 Hourglassing 90

2.6 Linear Constraints 91

2.6.1 Inclusion in the global system of equations 91

2.6.2 Forces induced by linear constraints 96

2.7 Transformations 97

2.8 Loading 103

2.8.1 Centrifugal loading 103

2.8.2 Temperature loading 104

2.9 Modal Analysis 106

2.9.1 Frequency calculation 106

2.9.2 Linear dynamic analysis 108

2.9.3 Buckling 112

2.10 Cyclic Symmetry 114

2.11 Dynamics: The -Method 120

2.11.1 Implicit formulation 120

2.11.2 Extension to nonlinear applications 123

2.11.3 Consistency and accuracy of the implicit formulation 126

2.11.4 Stability of the implicit scheme 130

2.11.5 Explicit formulation 136

2.11.6 The consistent mass matrix 138

2.11.7 Lumped mass matrix 140

2.11.8 Spherical shell subject to a suddenly applied uniform pressure 141

3 Geometric Nonlinear Effects 143

3.1 General Equations 143 3.2 Application to a Snappi...