| Preface | 4 |
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| Contents | 7 |
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| 1 Short History of Non-equilibrium Thermodynamics | 13 |
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| 1 Introduction | 13 |
| 2 Gibbs Thermodynamics | 13 |
| 3 Twentieth Century Thermodynamic Theories | 14 |
| 3.1 Carathéodory | 15 |
| 3.2 Linear Irreversible Thermodynamics | 16 |
| 3.3 Extended Irreversible Thermodynamics | 20 |
| 3.4 Continuum Thermodynamics | 21 |
| 3.5 Extended Rational Thermodynamics | 23 |
| 4 Maximum Dissipation Criteria | 23 |
| 5 Nonlinear Dynamical Systems | 25 |
| 5.1 Equilibrium States as Attractors | 26 |
| 6 Goals for a Non-equilibrium Thermodynamic Construction | 28 |
| References | 30 |
| 2 Thermostatics and Energy Methods | 32 |
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| 1 Introduction | 32 |
| 2 The Principle of Virtual Work | 32 |
| 3 The Principle of Stationary Potential Energy | 34 |
| 4 Stability of Equilibria in Conservative Systems | 35 |
| 5 Hyperelastic Thermostatic Energy Density Functions | 36 |
| 5.1 Linear Elastic | 37 |
| 5.2 Nonlinear Elastic | 39 |
| 6 Stability of Classical Thermostatic Energy Functions | 40 |
| References | 41 |
| 3 Evolution Construction for Homogeneous ThermodynamicSystems | 42 |
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| 1 Introduction | 42 |
| 2 Thermostatics | 43 |
| 2.1 Thermodynamic Variables | 45 |
| 2.2 Construction of Thermostatic Energy Density Functions | 45 |
| 3 Generalized Thermodynamic Functions | 46 |
| 3.1 Stability in the Distinguished Manifold | 48 |
| 3.2 Examples of Generalized Thermodynamic Functions | 49 |
| 4 Evolution Equations for Non-equilibrium Processesin a Thermodynamic System Defined by a Generalized Function | 50 |
| 4.1 Affinities | 51 |
| 4.2 Objective Rates | 54 |
| 4.3 Gradient Relaxation Processes | 54 |
| 4.4 Relaxation Convergence to Equilibrium | 59 |
| 4.5 The Gibbs One-Form | 61 |
| 4.6 Maximum Dissipation in Gradient Processes | 62 |
| 4.7 The Gibbs Form and the Clausius-Duhem Inequality | 63 |
| 4.8 Admissible Processes | 64 |
| 5 Forced Non-equilibrium Processes | 65 |
| 5.1 Numerical Methods | 66 |
| 6 Generalized Nonlinear Onsager-Type Relations | 67 |
| References | 70 |
| 4 Viscoelasticity | 71 |
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| 1 Introduction | 71 |
| 2 Brief History of Viscoelastic Models | 71 |
| 2.1 Contemporary Linear Viscoelasticity | 72 |
| 2.2 Ad Hoc Non-integral Creep Models Explicit in Time | 74 |
| 2.3 Viscoelasticity in Classical Continuum Thermodynamics | 75 |
| 2.4 Recent Ad Hoc Nonlinear Viscoelastic Models | 77 |
| 3 Nonlinear, Maximum Dissipation, Viscoelastic Model | 80 |
| 4 Classical Models That May Be Interpreted as a Maximum Dissipation Models | 80 |
| 4.1 Linear Uniaxial Long-Term Behavior | 81 |
| 4.2 Nonlinear Uniaxial Examples Solvable in Closed Form | 84 |
| 5 Nonlinear Maximum Dissipation Viscoelastic Model for Rubber | 85 |
| 5.1 Uniaxial Dynamic Response of Isothermal Rubber | 85 |
| 5.2 A Thermostatic Constitutive Model for Rubber | 87 |
| 5.3 A Nonlinear Thermoviscoelastic Model for Rubber | 89 |
| 5.4 Sudden Stress Perturbations in an Isothermal Rubber Sheet | 90 |
| 5.5 The Sheet Response at Different Constant Temperatures | 91 |
| 5.6 The Nonlinear Thermoviscoelastic Behaviorof a Rubber Rod | 94 |
| 5.7 The Adiabatic Gough-Joule Effect as a Non-equilibrium Relaxation Process | 97 |
| 6 Nonlinear Maximum Dissipation Viscoelastic Models for Soft Biological Tissue | 99 |
| 6.1 Uniaxial Nonlinear Viscoelastic Models for BiologicalTissue | 102 |
| 6.2 Temperature Dependence in Uniaxial Loading | 106 |
| 6.3 Evolution Equations Based on the Holzapfel et al. Long-Term Three-Dimensional Model for Healthy Artery Tissue | 108 |
| 6.4 Viscoelastic Saccular Aneurysm Model | 111 |
| Appendix: Evolution Equation When the Strain EnergyIs a Function of a Tensor | 114 |
| References | 116 |
| 5 Viscoplasticity | 119 |
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| 1 Introduction | 119 |
| 2 Maximum Dissipation Models for Viscoplasticity | 122 |
| 2.1 Thermoviscoplastic Generalized Energy | 123 |
| 2.2 Admissible Thermodynamic Processes and Dissipation | 124 |
| 2.3 Maximum Dissipation and Gradient RelaxationProcesses | 125 |
| 2.4 The Thermodynamic Relaxation Modulus | 126 |
| 2.5 Relaxation Examples | 128 |
| 3 Forced Non-equilibrium Processes | 133 |
| 3.1 Simple Monotonic Loading | 134 |
| 4 A Three-Dimensional Model | 134 |
| References | 139 |
| 6 The Thermodynamic Relaxation Modulusas a Multi-Scale Bridge from the Atomic Level to the Bulk Material | 141 |
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| 1 Introduction | 141 |
| 1.1 Multi-Scale and Dynamic Modeling | 142 |
| 1.2 The Viscoelastic Response of the Elastin-Water System | 142 |
| 2 Background | 144 |
| 2.1 The Structure of Arterial Elastin | 144 |
| 2.2 Experimental Stress-Strain Relations in Elastin | 145 |
| 2.3 The Glass Transition Temperature of the Moisture-Elastin System | 146 |
| 3 The Maximum Dissipation Multi-Scale Viscoelastic Modelfor the Elastin-Water System | 148 |
| 3.1 The Multi-Scale Thermodynamic Relaxat
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