The Finite Element Method in Thermomechanics

The rapid advances in the nuclear and aerospace technologies in the past two decades compounded with the increasing demands for high performance, energy-efficient power plant components and engines have made reliable thermal stress analysis a critical factor in the design and operation of such equipment. Recently, and as experienced by the author, the need for sophis­ ticated analyses has been extended to the energy resource industry such as in-situ coal gasification and in-situ oil recovery from oil sands and shales. The analyses in the above applications are of a multidisciplinary nature, and some involve the additional complexity of multiphase and phase change phenomena. These extremely complicated factors preclude the use of classical methods, and numerical techniques such as the finite element method appear to be the most viable alternative solution. The development of this technique so far appears to have concentrated in two extremes; one being overly concerned with the accuracy of results and tending to place all effort in the implementation of special purpose element concepts and computational algorithms, the other being for commercial purposes with the ability of solving a wide range of engineering problems. However, to be versatile, users require substantial training and experience in order to use these codes effectively. Above all, no provision for any modifi­ cation of these codes by users is possible, as all these codes are proprietary and access to the code is limited only to the owners.

1 Fundamentals of the Finite Element Method.- 1.1 Introduction.- 1.2 The concept of discretization.- 1.3 Steps in the finite element method.- References.- 2 Finite Element Analysis in Heat Conduction.- 2.1 Introduction.- 2.2 Review of basic formulations.- 2.3 Finite element formulation of transient heat conduction in solids.- 2.4 Transient heat conduction in axisymmetric solids.- 2.5 Computation of the thermal conductivity matrix.- 2.6 Computation of the heat capacitance matrix.- 2.7 Computation of thermal force matrix.- 2.8 Transient heat conduction in the time domain.- 2.9 Boundary conditions 45 2.10 Solution procedures for axisymmetric structures.- References.- 3 Thermoelastic-Plastic Stress Analysis.- 3.1 Introduction.- 3.2 Mechanical behavior of materials.- 3.3 Review of basic formulations in linear elasticity theory.- 3.4 Basic formulations in nonlinear elasticity.- 3.5 Elements of plasticity theory.- 3.6 Strain hardening.- 3.7 Plastic potential (yield) function.- 3.8 Prandtl-Reuss relation.- 3.9 Derivation of plastic stress-strain relations.- 3.10 Constitutive equations for thermoelastic-plastic stress analysis.- 3.11 Derivation of the [Cep] matrix.- 3.12 Determination of material stiffness (H’).- 3.13 Thermoelastic-plastic stress analysis with kinematic hardening rule.- 3.14 Finite element formulation of thermoelastic-plastic stress analysis.- 3.15 Finite element formulation for the base TEPSAC code.- 3.16 Solution procedure for the base TEPSA code.- References.- 4 Creep Deformation of Solids by Finite Element Analysis.- 4.1 Introduction.- 4.2 Theoretical background.- 4.3 Constitutive equations for thermoelastic-plastic creep stress analysis.- 4.4 Finite element formulation of thermoelastic-plastic creep stress analysis.- 4.5 Integration schemes.- 4.6 Solution algorithm.- 4.7 Code verification.- 4.8 Closing remarks.- References.- 5 Elastic-Plastic stress analysis with Fourier Series.- 5.1 Introduction.- 5.2 Element equation for elastic axisymmetric solids subject to nonaxisymmetric loadings.- 5.3 Stiffness matrix for elastic solids subject to nonaxisymmetric loadings.- 5.4 Elastic-plastic stress analysis of axisymmetric solids subject to nonaxisymmetric loadings.- 5.5 Derivation of element equation.- 5.6 Mode mixing stiffness equations.- 5.7 Circumferential integration scheme.- 5.8 Numerical example.- 5.9 Discussion of the numerical example.- 5.10 Summary.- References.- 6 Elastodynamic stress analysis with Thermal Effects.- 6.1 Introduction.- 6.2 Theoretical background.- 6.3 Hamilton’s variational principle.- 6.4 Finite element formulation.- 6.5 Direct time integration scheme.- 6.6 Solution algorithm.- 6.7 Numerical illustration.- References.- 7 Thermofracture Mechanics.- 1: Review of fracture mechanics concept.- 7.1 Introduction.- 7.2 Linear elastic fracture mechanics.- 7.3 Elastic-plastic fracture mechanics.- 7.4 Application of the finite element method to fracture mechanics.- 2: Thermoelastic-plastic fracture analysis page.- 7.5 Introduction.- 7.6 Fracture criteria.- 7.7 J integral with thermal effect.- 7.8 Numerical illustrations of J integrals with thermal effect.- 7.9 The “breakable element”.- 7.10 Numerical illustrations of stable crack growth.- 3: Thermoelastic-plastic creep fracture analysis.- 7.11 Literature review.- 7.12 Generalized creep fracture model.- 7.13 Path dependence of the Cg* integral.- 7.14 Creep crack growth simulated by “breakable element” algorithm.- References.- 8 Thermoelastic-Plastic Stress Analysis By Finite Strain Theory.- 8.1 Introduction.- 8.2 Lagrangian and Eulerian coordinate systems.- 8.3 Green and Almansi strain tensors.- 8.4 Lagrangian and Kirchhoff stress tensors.- 8.5 Equilibrium in the large.- 8.6 Equilibrium in the small.- 8.7 The boundary conditions.- 8.8 The constitutive equation.- 8.9 Equations of equilibrium by the principle of virtual work.- 8.10 Finite element formulation.- 8.11 Stiffness matrix [K2].- 8.12 Stiffness matrix [K3].- 8.13 Constitutive equations for thermoelastic-plastic stress analysis.- 8.14 The finite element formulation.- 8.15 The computer program.- 8.16 Numerical examples.- References.- 9 Coupled Thermoelastic-Plastic Stress Analysis.- 9.1 Introduction.- 9.2 The energy balance concept.- 9.3 Derivation of the coupled heat conduction equation.- 9.4 Coupled thermoelastic-plastic stress analysis.- 9.5 Finite element formulation.- 9.6 The y matrix.- 9.7 The thermal moduli matrix ?.- 9.8 The internal dissipation factor.- 9.9 Computation algorithm.- 9.10 Numerical illustration.- 9.11 Concluding remarks.- References.- 10 Application of Thermomechanical Analyses in Industry.- 10.1 Introduction.- 10.2 Thermal analysis involving phase change.- 10.3 Thermoelastic-plastic stress analysis.- 10.4 Thermoelastic-plastic stress analysis by TEPSAC code.- 10.5 Simulation of thermomechanical behavior of nuclear reactor fuel elements.- References.- Appendix 1 Area coordinate system for triangular simplex elements.- Appendix 2 Numerical illustration on the implementation of thermal boundary conditions.- Appendix 3 Integrands of the mode-mixing stiffness matrix.- Appendix 4 User’s guide for TEPSAC.- Appendix 5 Listing of TEPSAC code.- Author Index.