The Finite ElementMethod for Three-dimensionalThermomechanical Applications
Guido Dhondt
Munich, Germany
Innodata0470857625.jpg
The Finite ElementMethod for Three-dimensionalThermomechanical Applications
The Finite ElementMethod for Three-dimensionalThermomechanical Applications
Guido Dhondt
Munich, Germany
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To my wife Barbara and my children Jakob and Lea
Contents
Preface xiii
Nomenclature xv
1 Displacements, Strain, Stress and Energy 11.1 The Reference State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The Spatial State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Strain Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4 Principal Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.5 Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.6 Objective Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.7 Balance Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.7.1 Conservation of mass . . . . . . . . . . . . . . . . . . . . . . . . . 251.7.2 Conservation of momentum . . . . . . . . . . . . . . . . . . . . . . 251.7.3 Conservation of angular momentum . . . . . . . . . . . . . . . . . . 261.7.4 Conservation of energy . . . . . . . . . . . . . . . . . . . . . . . . 261.7.5 Entropy inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.7.6 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.8 Localization of the Balance Laws . . . . . . . . . . . . . . . . . . . . . . . 281.8.1 Conservation of mass . . . . . . . . . . . . . . . . . . . . . . . . . 281.8.2 Conservation of momentum . . . . . . . . . . . . . . . . . . . . . . 291.8.3 Conservation of angular momentum . . . . . . . . . . . . . . . . . . 311.8.4 Conservation of energy . . . . . . . . . . . . . . . . . . . . . . . . 311.8.5 Entropy inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.9 The Stress Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.10 The Balance Laws in Material Coordinates . . . . . . . . . . . . . . . . . . 34
1.10.1 Conservation of mass . . . . . . . . . . . . . . . . . . . . . . . . . 351.10.2 Conservation of momentum . . . . . . . . . . . . . . . . . . . . . . 351.10.3 Conservation of angular momentum . . . . . . . . . . . . . . . . . . 371.10.4 Conservation of energy . . . . . . . . . . . . . . . . . . . . . . . . 371.10.5 Entropy inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.11 The Weak Form of the Balance of Momentum . . . . . . . . . . . . . . . . 381.11.1 Formulation of the boundary conditions (material coordinates) . . . 381.11.2 Deriving the weak form from the strong form (material coordinates) 391.11.3 Deriving the strong form from the weak form (material coordinates) 411.11.4 The weak form in spatial coordinates . . . . . . . . . . . . . . . . . 41
1.12 The Weak Form of the Energy Balance . . . . . . . . . . . . . . . . . . . . 421.13 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
viii CONTENTS
1.13.1 Summary of the balance equations . . . . . . . . . . . . . . . . . . 431.13.2 Development of the constitutive theory . . . . . . . . . . . . . . . . 44
1.14 Elastic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471.14.1 General form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471.14.2 Linear elastic materials . . . . . . . . . . . . . . . . . . . . . . . . . 491.14.3 Isotropic linear elastic materials . . . . . . . . . . . . . . . . . . . . 521.14.4 Linearizing the strains . . . . . . . . . . . . . . . . . . . . . . . . . 541.14.5 Isotropic elastic materials . . . . . . . . . . . . . . . . . . . . . . . 58
1.15 Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2 Linear Mechanical Applications 632.1 General Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632.2 The Shape Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.2.1 The 8-node brick element . . . . . . . . . . . . . . . . . . . . . . . 682.2.2 The 20-node brick element . . . . . . . . . . . . . . . . . . . . . . 692.2.3 The 4-node tetrahedral element . . . . . . . . . . . . . . . . . . . . 712.2.4 The 10-node tetrahedral element . . . . . . . . . . . . . . . . . . . 722.2.5 The 6-node wedge element . . . . . . . . . . . . . . . . . . . . . . 732.2.6 The 15-node wedge element . . . . . . . . . . . . . . . . . . . . . . 73
2.3 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752.3.1 Hexahedral elements . . . . . . . . . . . . . . . . . . . . . . . . . . 762.3.2 Tetrahedral elements . . . . . . . . . . . . . . . . . . . . . . . . . . 782.3.3 Wedge elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 782.3.4 Integration over a surface in three-dimensional space . . . . . . . . 81
2.4 Extrapolation of Integration Point Values to the Nodes . . . . . . . . . . . . 822.4.1 The 8-node hexahedral element . . . . . . . . . . . . . . . . . . . . 832.4.2 The 20-node hexahedral element . . . . . . . . . . . . . . . . . . . 842.4.3 The tetrahedral elements . . . . . . . . . . . . . . . . . . . . . . . . 862.4.4 The wedge elements . . . . . . . . . . . . . . . . . . . . . . . . . . 86
2.5 Problematic Element Behavior . . . . . . . . . . . . . . . . . . . . . . . . . 862.5.1 Shear locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 872.5.2 Volumetric locking . . . . . . . . . . . . . . . . . . . . . . . . . . . 872.5.3 Hourglassing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
2.6 Linear Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 912.6.1 Inclusion in the global system of equations . . . . . . . . . . . . . . 912.6.2 Forces induced by linear constraints . . . . . . . . . . . . . . . . . 96
2.7 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 972.8 Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
2.8.1 Centrifugal loading . . . . . . . . . . . . . . . . . . . . . . . . . . . 1032.8.2 Temperature loading . . . . . . . . . . . . . . . . . . . . . . . . . . 104
2.9 Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1062.9.1 Frequency calculation . . . . . . . . . . . . . . . . . . . . . . . . . 1062.9.2 Linear dynamic analysis . . . . . . . . . . . . . . . . . . . . . . . . 1082.9.3 Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
2.10 Cyclic Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1142.11 Dynamics: The -Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
CONTENTS ix
2.11.1 Implicit formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 1202.11.2 Extension to nonlinear applications . . . . . . . . . . . . . . . . . . 1232.11.3 Consistency and accuracy of the implicit formulation . . . . . . . . 1262.11.4 Stability of the implicit scheme . . . . . . .
