Continuum Mechanics and Theory of Materials

Transcription

1 Peter Haupt Continuum Mechanics and Theory of Materials Translated from German by Joan A. Kurth Second Edition With 91 Figures, Springer

2 Contents Introduction 1 1 Kinematics Material Bodies / Material and Spatial Representation Deformation Gradient Strain Tensors Convective Coordinates Velocity Gradient Strain Rate Tensors Strain Rates in Convective Coordinates Geometric Linearisation Incompatible Configurations Euclidean Space Non-Euclidean Spaces Conditions of Compatibility 64 2 Balance Relations of Mechanics Preliminary Remarks Mass Balance of Mass: Global Form Balance of Mass: Local Form Linear Momentum and Rotational Momentum Balance of Linear Momentum and Rotational Momentum: Global Formulation Stress Tensors Stress Tensors in Convective Coordinates Local Formulation of the Balance of Linear Momentum and Rotational Momentum Initial and Boundary Conditions 101

3 XII Contents 2. 4 Conclusions from the Balance Equations of Mechanics Balance of Mechanical Energy The Principle of d'alembert Principle of Virtual Work Incremental Form of the Principle of d'alembert Balance Relations of Thermodynamics Preliminary Remarks Energy Temperature and Entropy Initial and Boundary Conditions Balance Relations for Open Systems Transport Theorem Balance of Linear Momentum for Systems with Time-Dependent Mass Balance Relations: Conservation Laws Discontinuity Surfaces and Jump Conditions Multi-Component Systems (Mixtures) Summary: Basic Relations of Thermomechanics Objectivity Frames of Reference Affine Spaces f Change of Frame: Passive Interpretation Change of Frame: Active Interpretation Objective Quantities Observer-Invariant Relations Classical Theories of Continuum Mechanics Introduction Elastic Fluid Linear-Viscous Fluid Linear-Elastic Solid Linear-Viscoelastic Solid Perfectly Plastic Solid Plasticity with Hardening Viscoplasticity with Elastic Range Remarks on the Classical Theories Experimental Observation and Mathematical Modelling General Aspects Information from Experiments Material Properties of Steel XCrNi Material Properties of Carbon-Black-Filled Elastomers Four Categories of Material Behaviour Four Theories of Material Behaviour Contribution of the Classical Theories 273

4 Contents XIII 7 General Theory of Mechanical Material Behaviour General Principles Constitutive Equations Simple Materials Reduced Forms of the General Constitutive Equation Simple Examples of Material Objectivity Frame-Indifference and Observer-Invariance Properties of Material Symmetry The Concept of the Symmetry Group Classification of Simple Materials into Fluids and Solids Kinematic Conditions of Internal Constraint General Theory Special Conditions of Internal Constraint Formulation of Material Models General Aspects Representation by Means of Functional Representation by Means of Internal Variables Comparison Dual Variables Tensor-Valued Evolution Equations Introduction / Objective Time Derivatives of Objective Tensors Example: Maxwell Fluid Example: Rigid-Plastic Solid with Hardening The Concept of Dual Variables Motivation Strain and Stress Tensors (Summary) Dual Variables and Derivatives Elasticity Elasticity and Hyperelasticity Isotropic Elastic Bodies General Constitutive Equation for Elastic Fluids and Solids Isotropic Hyperelastic Bodies Incompressible Isotropic Elastic Materials Constitutive Equations of Isotropic Elasticity (Examples) Anisotropic Hyperelastic Solids Approximation of the General Constitutive Equation General Representation of the Strain Energy Function Physical Linearisation 388

5 XIV Contents 10 Viscoelasticity Representation by Means of Functionals Rate-Dependent Functionals with Fading Memory Properties Continuity Properties and Approximations Representation by Means of Internal Variables General Concept Internal Variables of the Strain Type A General Model of Finite Viscoelasticity Plasticity Rate-Independent Functionals Representation by Means of Internal Variables Elastoplasticity Preliminary Remarks Stress-Free Intermediate Configuration Isotropic Elasticity Yield Function and Evolution Equations Consistency Condition Viscoplasticity Preliminary Remarks Viscoplasticity with Elastic Domain A General Constitutive Model Application of the Intermediate Configuration Plasticity as a Limit Case of Viscoplasticity The Differential Equation of the Yield Function Relaxation Property Slow Deformation Processes Elastoplasticity and Arclength Representation A Concept for General Viscoplasticity Motivation Equilibrium Stress and Overstress An Example of General Viscoplasticity Conclusions Regarding the Modelling of Mechanical Material Behaviour Constitutive Models in Thermomechanics Thermomechanical Consistency Thermoelasticity General Theory Thermoelastic Fluid Linear-Thermoelastic Solids Thermoviscoelasticity General Concept Thermoelasticity as a Limit Case of Thermoviscoelasticity Internal Variables of Strain Type 541

6 Contents XV Incorporation of Anisotropic Elasticity Properties Incompressible Materials: An Extension of the Mooney-Rivlin Model to Thermoviscoelasticity Thermoviscoplasticity with Elastic Domain Uniaxial Viscoplasticity General Concept Application of the Intermediate Configuration Thermoplasticity as a Limit Case of Thermoviscoplasticity General Thermoviscoplasticity Small Deformations Finite Deformations Conclusion Anisotropic Material Properties Motivation Axes of Elastic Anisotropy Anisotropic Viscoplasticity General Considerations Free Energy Function Evolution Equations Lattice Spin, Summary: A Constitutive Model of Anisotropic Viscoplasticity Numerical Simulations Closing Remark 618 References 619 Index 635