The Thermodynamics of Simple Materials with Fading Memory

This Tract gives an account of certain recent attempts to construct a satisfactory theory of thermodynamics for materials which have a memory for the past. Naturally it draws heavily on the writings of those who have made significant contributions to the field. I am particularly grateful to Professor C. A. Truesdell of The lohns Hopkins University for his invitation to write the Tract and to Professor A. E. Green of Oxford for his comments on various parts of the manuscript. Hertford College, Oxford December 1971 W. A. Day Contents Introduction 1 Chapter 1 Preliminaries 5 1. 1 Vector and Tensor Analysis. 5 1. 2 Paths and Line Integrals . 7 1. 3 Kinematics and the Balance Laws 11 1. 4 Simple Materials with Memory 15 21 Chapter 2 A Theory of Thermodynamics . 2. 1 Processes. 21 2. 2 The Thermodynamic Inequality . 23 2. 3 Heat Conduction Inequalities . 24 2. 4 The Conversion of Heat into Mechanical Work 27 31 The Construction of the Entropy Chapter 3 The Clausius Inequality 31 3. 1 3. 2 Fading Memory . 34 3. 3 The Entropy in Equilibrium. Thermostatics. 38 3. 4 The Entropy away from Equilibrium. The Clausius- Planck Inequality 45 Chapter 4 Applications . . 55 4. 1 Thermoelasticity and Materials of Differential Type 55 4. 2 A Class of Viscoelastic Materials . . . . . . 60 Chapter 5 Thermodynamics based on the Clausius-Duhem Inequality . . . . . . . . . . 77 5. 1 The Clausius-Duhem Inequality. 78 5.

1 Preliminaries.- 1.1 Vector and Tensor Analysis.- 1.2 Paths and Line Integrals.- 1.3 Kinematics and the Balance Laws.- 1.4 Simple Materials with Memory.- 2 A Theory of Thermodynamics.- 2.1 Processes.- 2.2 The Thermodynamic Inequality.- 2.3 Heat Conduction Inequalities.- 2.4 The Conversion of Heat into Mechanical Work.- 3 The Construction of the Entropy.- 3.1 The Clausius Inequality.- 3.2 Fading Memory.- 3.3 The Entropy in Equilibrium. Thermostatics.- 3.4 The Entropy away from Equilibrium. The Clausius-Planck Inequality.- 4 Applications.- 4.1 Thermoelasticity and Materials of Differential Type.- 4.2 A Class of Viscoelastic Materials.- 5 Thermodynamics based on the Clausius-Duhem Inequality.- 5.1 The Clausius-Duhem Inequality.- 5.2 Materials of the Differential Type.- 5.3 Materials with Fading Memory and Instantaneous Elastic Response.- 5.4 Behaviour near Equilibrium.- 5.5 The Internal Energy as an Independent Variable.- 6 Thermodynamic Restrictions on Isothermal Linear Viscoelasticity.- 6.1 Compatibility with Thermodynamics. Dissipative Relaxation Functions.- 6.2 The Symmetry of the Relaxation Function.- 6.3 A Remark on the Monotonicity of Relaxation Functions.- References.