Maximum Dissipation Non-Equilibrium Thermodynamics and its Geometric Structure (eBook)

Preface 4
Contents 7
1 Short History of Non-equilibrium Thermodynamics 13
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
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
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
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
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
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 Relaxation Modulus 148
4 Modification of the Long-Term Energy Density Functionto Account for Moisture Content 150
4.1 Water-Induced Swelling of Elastin 151
4.2 Model to Account for Swelling in the Strain EnergyDensity Function 152
4.3 Shear Modulus as a Function of Swelling Ratio 153
4.4 Neo-Hookean Long-Term Strain Energy Densityas a Function of Moisture Content 154
4.5 Zulliger Long-Term Strain Energy Density as a Functionof Moisture Content 155
5 Numerical Determination of the Multi-Scale Thermodynamic Relaxation Modulus 155
5.1 Linear Elastic Long-Term Strain Energy Density 156
5.2 The Moisture Content Function, g(rh) 157
5.3 The Frequency Function, f() 157
5.4 Estimated Values of the Multi-Scale Thermodynamic Relaxation Modulus and Other Parameters 158
6 Recovering the Lillie-Gosline Data for the Frequency Dependence of the Glass Transition in the Elastin-Water System 158
6.1 Application for the Neo-Hookean and the Zulliger Long-Term Quasi-static Strain Energy Densities 159
7 Application to the Response of Arterial Elastin 161
7.1 Uniaxial Creep 161
7.2 Pressure Loaded Elastin Cylinder 164
References 168
7 Contact Geometric Structure for Non-equilibriumThermodynamics 171
1 Introduction 171
2 The Geometry of Continuum Mechanics 172
2.1 Manifolds 172
2.2 The Tangent Space of a Manifold 175
2.3 The Cotangent Space of a Manifold 177
2.4 Tensors 179
2.5 Strain Tensors 182
2.6 Stress Tensors 183
2.7 Thermodynamic Variables 184
3 Contact Structures and Thermostatics 185
3.1 Lift of the Energy Surface in n+1 in the Contact Bundle 186
3.2 Thermostatics in a Contact Manifold 187
3.3 Interpretations of C(n+1,n) 188
3.4 Symplectic Representation of the Thermostatic Manifoldin 2n 188
4 Geometry of Maximum Dissipation Non-equilibrium Thermodynamics for Small Displacements 190
4.1 Morse Family Formulation of the Generalized EnergyFunction 191
4.2 Legendre Transformations 193
5 Compound Systems and Chemical Reactions 195
5.1 Chemical Reactions 196
References 197
8 Bifurcations in the Generalized Energy Function 199
1 Introduction 199
1.1 Lavis and Bell Generalized Thermodynamic Function:Van der Waals Fluid 200
1.2 Rubber Sheet Under Biaxial Loading 202
2 Stability in Energy Density Functions for Which the Equilibria Are Not Critical Points 204
3 Stability, Equivalence and Unfoldings 206
3.1 Equivalence, Unfoldings, and Perturbations of Real Valued Functions 207
3.2 The Simple Catastrophes of One State Variable 209
4 Asymmetric Deformations in Experiments on a Rubber Sheet 210
5 Incompressible Elastic Energy Functions 212
5.1 A Bifurcation Condition 213
6 Bifurcation Types 215
6.1 The Liapunov-Schmidt Reduction for the Equilibria 217
6.2 Bifurcation with Respect to the Load Parameter 220
7 Rubber Constitutive Models Without a Bifurcation 226
7.1 The Neo-Hookean Model 226
7.2 The Arruda-Boyce Model 227
7.3 The Valanis-Landel Hypothesis and Model 228
7.4 The Gent-Thomas Model 229
8 Rubber Constitutive Models that Produce a Bifurcation 230
8.1 The Mooney-Rivlin Model 230
8.2 Alexander Model 235
8.3 The Ogden Models 237
9 Rubber Constitutive Model with a Three Bifurcation PointsStructure 239
10 Influence of Bifurcations on Maximum Dissipation Non-equilibrium Evolution Processes 242
10.1 Dynamic Behavior in ``Snap-Through'' 244
References 246
9 Maximum Dissipation Evolution Constructionfor Non-homogeneous Thermodynamic Systems 248
1 Introduction 248
2 Generalized Entropy Production and Flux Evolution 249
2.1 Relaxation Towards Equilibrium 251
3 Examples of Stationary Manifolds and Evolution of Fluxes 252
3.1 Thermal Gradients and Fluxes 252
3.2 Non-steady Transport in Porous BiologicalMembranes 256
3.3 Electromagnetic Fluxes 258
3.4 Fluids 258
4 Admissible Non-homogeneous Processes 259
4.1 Relation to the Clausius-Duhem Inequality 260
4.2 The Balance Laws as Differential Forms 261
4.3 Non-homogeneous Examples 262
References 265
10 Electromagnetism and Joule Heating 266
1 Introduction 266
2 Constitutive Models 267
2.1 Electromagnetic Relations and the Maxwell Equations 269
2.2 Energy Balance 270
2.3 The Maxwell Equations as Differential Forms 271
2.4 Joule Heating 272
3 Unsteady Thermoelectric and Electromagnetic Evolution 272
3.1 Unsteady Ohm's Law 272
3.2 Classical Joule Heating with the Maxwell-CattaneoHeat Flux 273
3.3 Transient Model of Joule Heating 274
References 276
11 Fracture 278
1 Introduction 278
2 Construction of the Model for the Non-equilibrium Thermodynamics of Fracture 281
3 Linear Elastic Instantaneous Maximum Dissipation CrackPropagation 282
3.1 Freund Equation of Motion as a Maximum Dissipation Evolution Equation 283
3.2 Stability in the Griffith-Irwin Theory Viewed as Maximum Dissipation Fracture 286
3.3 Craze Growth in PMMA Under Creep 289
4 Temperature at the Crack Tip 290
References 293
12 Conclusion 295
1 Some Features of the Maximum Dissipation Construction 295
2 Arrow of Time 296
References 299
Index 300