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.

Table of Contents:
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 α-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 Snapping-through Plate 148 3.3 Solution-dependent Loading 150 3.3.1 Centrifugal forces 150 3.3.2 Traction forces 151 3.3.3 Example: a beam subject to hydrostatic pressure 154 3.4 Nonlinear Multiple Point Constraints 154 3.5 Rigid Body Motion 155 3.5.1 Large rotations 155 3.5.2 Rigid body formulation 159 3.5.3 Beam and shell elements 162 3.6 Mean Rotation 167 3.7 Kinematic Constraints 171 3.7.1 Points on a straight line 171 3.7.2 Points in a plane 173 3.8 Incompressibility Constraint 174 4 Hyperelastic Materials 177 4.1 Polyconvexity of the Stored-energy Function 177 4.1.1 Physical requirements 177 4.1.2 Convexity 180 4.1.3 Polyconvexity 184 4.1.4 Suitable stored-energy functions 189 4.2 Isotropic Hyperelastic Materials 190 4.2.1 Polynomial form 191 4.2.2 Arruda–Boyce form 193 4.2.3 The Ogden form 194 4.2.4 Elastomeric foam behavior 195 4.3 Nonhomogeneous Shear Experiment 196 4.4 Derivatives of Invariants and Principal Stretches 199 4.4.1 Derivatives of the invariants 199 4.4.2 Derivatives of the principal stretches 200 4.4.3 Expressions for the stress and stiffness for three equal eigenvalues 206 4.5 Tangent Stiffness Matrix at Zero Deformation 209 4.5.1 Polynomial form 210 4.5.2 Arruda–Boyce form 211 4.5.3 Ogden form 211 4.5.4 Elastomeric foam behavior 211 4.5.5 Closure 212 4.6 Inflation of a Balloon 212 4.7 Anisotropic Hyperelasticity 216 4.7.1 Transversely isotropic materials 217 4.7.2 Fiber-reinforced material 219 5 Infinitesimal Strain Plasticity 225 5.1 Introduction 225 5.2 The General Framework of Plasticity 225 5.2.1 Theoretical derivation 225 5.2.2 Numerical implementation 232 5.3 Three-dimensional Single Surface Viscoplasticity 235 5.3.1 Theoretical derivation 235 5.3.2 Numerical procedure 239 5.3.3 Determination of the consistent elastoplastic tangent matrix 242 5.4 Three-dimensional Multisurface Viscoplasticity: the Cailletaud Single Crystal Model 244 5.4.1 Theoretical considerations 244 5.4.2 Numerical aspects 248 5.4.3 Stress update algorithm 249 5.4.4 Determination of the consistent elastoplastic tangent matrix 259 5.4.5 Tensile test on an anisotropic material 260 5.5 Anisotropic Elasticity with a von Mises–type Yield Surface 262 5.5.1 Basic equations 262 5.5.2 Numerical procedure 263 5.5.3 Special case: isotropic elasticity 270 6 Finite Strain Elastoplasticity 273 6.1 Multiplicative Decomposition of the Deformation Gradient 273 6.2 Deriving the Flow Rule 275 6.2.1 Arguments of the free-energy function and yield condition 275 6.2.2 Principle of maximum plastic dissipation 276 6.2.3 Uncoupled volumetric/deviatoric response 278 6.3 Isotropic Hyperelasticity with a von Mises–type Yield Surface 279 6.3.1 Uncoupled isotropic hyperelastic model 279 6.3.2 Yield surface and derivation of the flow rule 280 6.4 Extensions 284 6.4.1 Kinematic hardening 284 6.4.2 Viscoplastic behavior 285 6.5 Summary of the Equations 287 6.6 Stress Update Algorithm 287 6.6.1 Derivation 287 6.6.2 Summary 291 6.6.3 Expansion of a thick-walled cylinder 293 6.7 Derivation of Consistent Elastoplastic Moduli 294 6.7.1 The volumetric stress 295 6.7.2 Trial stress 295 6.7.3 Plastic correction 296 6.8 Isochoric Plastic Deformation 300 6.9 Burst Calculation of a Compressor 302 7 Heat Transfer 305 7.1 Introduction 305 7.2 The Governing Equations 305 7.3 Weak Form of the Energy Equation 307 7.4 Finite Element Procedure 309 7.5 Time Discretization and Linearization of the Governing Equation 310 7.6 Forced Fluid Convection 312 7.7 Cavity Radiation 317 7.7.1 Governing equations 317 7.7.2 Numerical aspects 324 References 329 Index 335