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Statistical Mechanics of Elasticity

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J. H. Weiner
Dover Publications
eBook Typ:
Adobe Digital Editions
eBook Format:
2 - DRM Adobe

Part 1. Classical Theory Chapter 1. Thermoelasticity from the Continuum Viewpoint 1.1 Introduction 1.2 Kinematics of Continua 1.3 Mechanics 1.4 Thermodynamics 1.5 Various Thermodynamic Potentials 1.6 Thermoelastic Stress--Strain Relations 1.7 Thermoelastic Relations for Small Changes from Reference State 1.8 Related thermodynamic Functions 1.9 Elastic Constants in Terms of Displacement Gradients 1.10 Isotropic Solids Appendix Notation of Thurston (1964) Chapter 2. Concepts of Classical Statistical Mechanics 2.1 Introduction 2.2 Hamiltonian Mechanics 2.3 Use of Statistics in Statistical Mechanics 2.4 Phase Functions and Time Averages 2.5 Phase Space Dynamics of Isolated Systems 2.6 Systems in Weak Interaction 2.7 Canonical Distribution 2.8 Time Averages versus Ensemble Averages Chapter 3. Corresponding Concepts in Thermodynamics and Statistical Mechanics 3.1 Introduction 3.2 Empirical Temperature 3.3 Quasi-Static Process 3.4 Phase Functions for Generalized Forces 3.5 First Law of Thermodynamics 3.6 Second Law of Thermodynamics 3.7 Use of Mechanical Variables as Controllable Parameters 3.8 Fluctuations 3.9 Partition Function Relations 3.10 Continuum Formulations of Nonuniform Processes 3.11 Equipartition Theorem 3.12 Entropy from the Information Theory Viewpoint Chapter 4. Crystal Elasticity 4.1 Introduction 4.2 Bravais Lattices 4.3 The Atomistic Concept of Stress in a Perfect Crystal 4.4 Harmonic and Quasi-Harmonic Approximations 4.5 Thermoelastic Stress-Strain Relations Based on the Harmonic Approximation 4.6 Cauchy Relations 4.7 Stress Ensemble 4.8 Linear Chain with Nearest Neighbor Interactions 4.9 Lattice Dynamics and Crystal Elasticity Chapter 5. Rubber Elasticity, I 5.1 Introduction 5.2 Relative Roles of Internal Energy and Entropy 5.3 Atomic Structure of Long-Chain Molecules and Networks 5.4 One-Dimensional Polymer Model 5.5 Three-Dimensional Polymer Models 5.6 Network Theory of Rubber Elasticity Chapter 6. Rubber Elasticity, II 6.1 Introduction 6.2 Curvilinear Coordinates 6.3 Geometric Constraints 6.4 An Example 6.5 Curvilinear Coordinates for Stressed Polymer Chains 6.6 Rigid and Flexible Polymer Models 6.7 Use of S = k log p for Stretched Polymers 6.8 Strain Ensemble for Short Freely Jointed Chains 6.9 Stress Ensemble for Chain Molecules 6.10 Statistical Mechanics of Phantom NetworksAddendum. Atomic View of Stress in Polymer Systems Chapter 7. Rate Theory in Solids 7.1 Introduction 7.2 Impurity Atom Diffusion 7.3 A Simple One-Dimensional Rate Theory 7.4 Exact Normalization 7.5 Many Degrees of Freedom 7.6 Transition-State Assumption 7.7 Brownian Motion 7.8 Kramers Rate FormulaPart 2. Quantum Theory Chapter 8. Basic Concepts of Quantum Mechanics 8.1 Introduction 8.2 Structure of Classical Mechanics 8.3 Structure of Quantum Mechanics 8.4 Consequences of the Fourier Transform Relation between psi subscript x (x) and psi subscript p (p) 8.5 Extension to Three-Dimensional Motion and to N Particles 8.6 Hydrodynamic Analogy to Quantum Mechanics 8.7 Initial-Value Problems 8.8 Aspects of the Measurement Process 8.9 Some Examples of Stationary States 8.10 Tensor Product of Two Spaces 8.11 Electron Spin 8.12 Identical Particles and the Pauli Principle Chapter 9. Interatomic Interactions 9.1 Introduction 9.2 Hydrogen Molecule 9.3 Adiabatic Approximation 9.4 Hellmann-Feynman Theorem 9.5 Harmonic Elastic Moduli Based on Hellmann--Feynman Theorem 9.6 Categories of Interatomic Force Laws Chapter 10. Quantum Statistical Effects 10.1 Introduction 10.2 Quantum Statistical Mechanics 10.3 Quantum Statistical Effects in Crystals 10.4 Quantum Statistics for Polymer Models 10.5 Gaussian Wave Packet Dynamics 10.6 Canonical Ensemble in Terms of Coherent States 10.7 Quantum Effects on Rate Processes References; Index
An advanced treatment of elasticity from the atomistic viewpoint, this volume offers students and teachers a self-contained text. Its detailed development of the general principles of statistical mechanics leads to a concentration on the principles' application to the elastic behavior of solids. The first part is based solely on classical mechanics, starting with an introductory chapter that summarizes thermoelasticity from the continuum viewpoint. The principles of classical statistical mechanics are then developed and applied to the study of the thermoelastic behavior of both crystalline and polymeric solids. The second part is based on quantum mechanics, discussing their role in interatomic force laws, the manner in which quantum statistical effects modify the low-temperature mechanical behavior of solids, and the nature of quantum effects upon the rates of thermally activated processes. This book provides an alternative to the usual course in statistical mechanics, in which the major emphasis is on applications to gases, liquids, and electronic and magnetic phenomena. Graduate students of physics and chemistry will appreciate the treatment of the basic principles of classical statistical mechanics and quantum statistical mechanics, while polymer physicists will find the discussion of curvilinear coordinates, geometric constraints, and the distinction between rigid and flexible polymer models of particular interest.

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