Maximum Dissipation Non-Equilibrium Thermodynamics and its Geometric Structure

von: Henry W. Haslach Jr.

Springer-Verlag, 2011

ISBN: 9781441977656 , 297 Seiten

2. Auflage

Format: PDF

Kopierschutz: Wasserzeichen

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Preis: 192,59 EUR

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Mehr zum Inhalt

Maximum Dissipation Non-Equilibrium Thermodynamics and its Geometric Structure


 

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