Introduction to Nonlinear Thermomechanics: Theory and Finite-Element Solutions

I. Basic Considerations and Notions.- 1 A State of Stress and Strain.- 1.1 Stress.- 1.2 Strain.- 2 Finite Strains.- 2.1 Finite Strain Tensor in Material and Spatial Descriptions.- 2.2 Deformation Rate Tensor.- 2.3 Stress Measures.- 2.4 Final Remarks.- 3 Temperature.- 3.1 Heat Conduction.- 3.2 Heat Convection.- 3.3 Heat Radiation.- 3.4 Temperature Field in a Heat-Conducting Body.- 3.5 Navier-Stokes Equation.- 4 Thermodynamical Considerations.- 4.1 Thermomechanical Process.- 4.2 Formulation of the Constitutive Law.- II. Fundamentals of Elasticity and Plasticity Theory.- 5 Stress-Strain Curve.- 6 Elasticity.- 7 Plasticity.- 7.1 Idealization of Tension Test.- 7.2 Ideal Plasticity Theories.- 7.2.1 Yield Criteria.- 7.2.2 Hencky-Iljuszyn Deformation Theory.- 7.2.3 Plastic Flow Theory.- 7.2.4 Comparison of Flow Theory and Deformation Theory.- 7.2.5 Ideal Plasticity Theory for Finite Deformations.- 8 Work-Hardening Equation.- 8.1 Drucker Postulate.- 8.1.1 Stability of Plastic Material in the Drucker Sense..- 8.1.2 Associated Plastic Flow.- 8.2 Yield Surfaces for Work-Hardening Materials.- 8.2.1 Experimental Results.- 8.2.2 Isotropic Hardening.- 8.2.3 Kinematic Hardening.- III. Small Strain Thermo-Elasto-Plasticity.- 9 Equations for Thermo-Elasto-Plasticity.- 9.1 Isotropic Hardening.- 9.2 Kinematic Hardening.- 9.3 Elasto-Visco-Plasticity.- 10 Finite-Element Solution.- 10.1 Finite-Element Solution of Heat Flow Equations.- 10.1.1 Weighted Residual Method.- 10.1.2 Variational Formulation.- 10.1.3 Time Integration Schemes for Nonlinear Heat Conduction.- 10.1.4 Stability Analysis.- 10.2 Finite-Element Solution of Navier-Stokes Equations.- 10.3 Modelling of the Phase Change Process.- 10.4 Examples of Thermal Problems.- 10.4.1 Heat Flow with Phase Change..- 10.4.2 Navier-Stokes Equations.- 10.5 Finite-Element Solution of Thermo-Elasto-Plastic Problems.- 10.5.1 Variational Formulation.- 10.5.2 Integration.- 10.5.3 Methods of Iterative Accumulation.- 10.5.4 Tangent Stiffness Matrices.- 10.6 Examples of Thermo-Elasto-Plastic Analyses.- IV. Creep.- 11 Theoretical Background to Creep.- 11.1 Creep and Relaxation Tests.- 11.2 Creep at Constant Uniaxial Stress.- 11.2.1 Time Functions.- 11.2.2 Stress Functions.- 11.2.3 Temperature Functions.- 11.2.4 Stress and Time Functions.- 11.3 Creep Theories with Time-Dependent Uniaxial Stress.- 11.3.1 Total Strain Theory.- 11.3.2 Time Hardening Theory.- 11.3.3 Strain Hardening Theory.- 11.3.4 Heredity Theory.- 11.4 Creep Theories in Complex Stress State.- 11.4.1 Creep Theory of Deformational Type.- 11.4.2 Flow Theories and Creep Potential.- 11.4.3 Generalization of Strain Hardening Theory.- 12 Creep Rupture.- 12.1 Experimental Studies.- 12.2 Ductile Rupture Theories.- 12.3 Brittle Rupture Theories.- 12.4 Rupture of Mixed Type.- 13 Constitutive Equations for Thermo-Elasto-Plastic and Creep Analysis.- 14 Finite-Element Formulation.- 14.1 Matrix Equation for Thermo-Elasto-Plastic and Creep Problems.- 14.2 Remarks on Solution Procedures.- 14.3 Examples.- V. Finite Strains.- 15 Finite Strain Models.- 16 Constitutive Equations.- 16.1 Non-Isothermal Plastic Flow.- 16.2 Multiplicative Decomposition of the Deformation Gradient.- 17 Finite-Element Formulation for Non-Isothermal Plastic Flow.- 17.1 Total Lagrange Formulation.- 17.2 Updated Lagrange and Updated Lagrange-Jaumann Formulations.- 17.3 Updated Lagrange-Hughes Formulation.- VI. Coupled Thermo-Plasticity.- 18 Equations of Coupled Thermo-Plasticity.- 18.1 Heat Transfer Equations.- 18.2 Finite-Element Formulation for the Heat Flow Equation.- 18.3 Internal Dissipation function.- 18.4 Stress-Strain Relations in Coupled Thermo-Plasticity.- 18.4.1 Thermo-Elasto-Plastic Model Based on Additive Decomposition of Strain.- 18.4.2 Thermo-Rigid Plastic and Thermo-Rigid Visco-Plastic Models.- 18.4.3 Remarks on Other Models.- 18.5 Coupled Thermomechanical Algorithm.- 18.6 Examples.- References and Further Reading.