Structures Under Crash and Impact

Cover -- Contents -- 1 Introduction -- 2 Thermo-Mechanical Basics -- 2.1 Kinematic Equations -- 2.1.1 Coordinates and Displacements in Reference Systems -- 2.1.2 Deformation Gradients and Displacement Gradients -- 2.1.3 Strain Measures -- 2.1.4 Material and Spatial Time Derivatives of Deformations -- 2.1.5 Strain Rate Tensors -- 2.1.6 Compatibility Conditions -- 2.2 Stress Measures -- 2.2.1 Cauchy Stresses -- 2.2.2 Alternative Stress Measures -- 2.2.3 Rate Dependent Stress Measures -- 2.3 Descriptions of Static Equilibrium -- 2.3.1 Direct Formulation of Equilibrium -- 2.3.2 Calculus of Variations -- 2.3.3 Equilibrium Formulated as Variational Problem -- 2.4 Conservation Equations -- 2.4.1 Four Ways of Describing Conservation -- 2.4.2 Conservation of Mass -- 2.4.3 Conservation of Momentum -- 2.4.4 Conservation of Energy -- 2.4.5 Compressed Formulation of the Conservation Equations -- 2.5 Variational Solutions of the Balance Equations -- 2.5.1 What are Weak Forms? -- 2.5.2 Weak Forms of the Equation of Motion -- 2.5.3 Hamilton's Principle of Least Action -- 2.6 Thermodynamic Basics -- 2.6.1 Energy Is Conserved -- The First Law -- 2.6.2 Entropy Increases -- The Second Law -- 2.6.3 Thermodynamic Potentials -- 2.6.4 Formulations of the Clausius-Duhem Inequality -- 2.6.5 Consequences for Constitutive Equations -- 3 Constitutive Equations -- 3.1 Equations of State -- 3.1.1 Axiomatic Equations of State -- 3.1.2 Empirical Equations of State -- 3.2 Constitutive Equations for Total Stresses -- 3.2.1 Cauchy Elasticity -- 3.2.2 General Elastic Anisotropy -- 3.2.3 Elasticity with Symmetry Planes -- 3.2.4 Green Elasticity -- Hyperelastic Behavior -- 3.2.5 Some Examples of Hyperelastic Formulations -- 3.3 Constitutive Equations for Inelastic Deformations -- 3.3.1 Basic Terminology in Plasticity Theory -- 3.3.2 Selected Yield Criteria -- 3.3.3 Flow Rules -- 3.3.4 Strain Rate Dependent Yield Criteria -- 3.3.5 Plasticity Effects at Shock Compression States -- 3.3.6 Meso-Mechanical Calculation of Yield Loci -- 3.3.7 Polymers -- Nonlinear Elasticity, Initial Plastic Softening, Visco-Plastic Hardening -- 4 Shock Waves and Related Equations of State -- 4.1 Elastic Wave Propagation in Solids -- 4.1.1 Wave Equation and Sound Speeds -- 4.1.2 Solution to the One-Dimensional Wave Equation -- 4.2 Shock Wave Formation -- 4.3 Shock Wave Propagation in Solids -- 4.3.1 Conditions for Shock Waves -- Phenomenological Aspects -- 4.3.2 Shock Front Dimensions -- 4.4 Thermo-Mechanics of Shock Waves -- 4.4.1 Dispersion -- Precondition for Shock Wave Evolution and Stability -- 4.4.2 Thermodynamic Conditions upon Shock Wave Transit -- 4.4.3 Riemann Problem and Rankine-Hugoniot Equations -- 4.4.4 Hugoniot Curves and vS-v1 Relations -- 4.4.5 Energy Dissipation upon Shock Wave Transition -- 4.5 Nonlinear Equations of State for Shock Waves.