Statistical Mechanics of Elasticity

This advanced, self-contained treatment illustrates the general principles of statistical mechanics, focusing on their application to the elastic behavior of solids. Part 1 is based solely on classical mechanics, with a summary of thermoelasticity from the continuum viewpoint, followed by a study of the thermoelastic behavior of both crystalline and polymeric solids. Part 2 is based on quantum mechanics, discussing their role in interatomic force laws and their effects on the low-temperature mechanical behavior of solids and on the rates of thermally activated processes. A practical alternative to the usual course in statistical mechanics, this volume will prove valuable to students of physics and chemistry as well as to polymer physicists. 1983 edition. 96 figures.

Jerome H. Weiner is Professor Emeritus of Engineering and Physics at Brown University. His other Dover book, co-written with Bruno A. Boley, is Theory of Thermal Stresses.

PART ONE CLASSICAL THEORY

Thermoelasticity from the Continuum Viewpoint

3

(46)

Introduction

3

(1)

Kinematics of Continua

4

(5)

Mechanics

9

(6)

Thermodynamics

15

(9)

Various Thermodynamic Potentials

24

(3)

Thermoelastic Stress-Strain Relations

27

(3)

Thermoelastic Relations for Small Changes from Reference State

30

(4)

Related Thermodynamic Functions

34

(4)

Elastic Constants in Terms of Displacement Gradients

38

(4)

Isotropic Solids

42

(7)

Appendix: Notation of Thurston (1964)

47

(2)

Concepts of Classical Statistical Mechanics

49

(36)

Introduction

49

(1)

Hamiltonian Mechanics

50

(4)

Use of Statistics in Statistical Mechanics

54

(2)

Phase Functions and Time Averages

56

(5)

Phase Space Dynamics of Isolated Systems

61

(11)

Systems in Weak Interaction

72

(4)

Canonical Distribution

76

(4)

Time Averages versus Ensemble Averages

80

(5)

Corresponding Concepts in Thermodynamics and Statistical Mechanics

85

(42)

Introduction

85

(2)

Empirical Temperature

87

(1)

Quasi-Static Process

88

(1)

Phase Functions for Generalized Forces

89

(1)

First Law of Thermodynamics

90

(3)

Second Law of Thermodynamics

93

(3)

Use of Mechanical Variables as Controllable Parameters

96

(8)

Fluctuations

104

(11)

Partition Function Relations

115

(1)

Continuum Formulations of Nonuniform Processes

116

(2)

Equipartition Theorem

118

(2)

Entropy from the Information Theory Viewpoint

120

(7)

Crystal Elasticity

127

(58)

Introduction

127

(2)

Bravais Lattices

129

(2)

The Atomistic Concept of Stress in a Perfect Crystal

131

(9)

Harmonic and Quasi-Harmonic Approximations

140

(3)

Thermoelastic Stress--Strain Relations Based on the Harmonic Approximation

143

(7)

Cauchy Relations

150

(6)

Stress Ensemble

156

(4)

Linear Chain with Nearest Neighbor Interactions

160

(14)

Lattice Dynamics and Crystal Elasticity

174

(11)

Rubber Elasticity, I

185

(26)

Introduction

185

(1)

Relative Roles of Internal Energy and Entropy

186

(2)

Atomic Structure of Long-Chain Molecules and Networks

188

(5)

One-Dimensional Polymer Model

193

(5)

Three-Dimensional Polymer Models

198

(7)

Network Theory of Rubber Elasticity

205

(6)

Rubber Elasticity, II

211

(46)

Introduction

211

(1)

Curvilinear Coordinates

212

(5)

Geometric Constraints

217

(4)

An Example

221

(2)

Curvilinear Coordinates for Stressed Polymer Chains

223

(5)

Rigid and Flexible Polymer Models

228

(6)

Use of S = k log p for Stretched Polymers

234

(3)

Strain Ensemble for Short Freely Jointed Chains

237

(7)

Stress Ensemble for Chain Molecules

244

(3)

Statistical Mechanics of Phantom Networks

247

(10)

Addendum Atomic View of Stress in Polymer Systems

1

Rate Theory in Solids

257

(40)

Introduction

257

(1)

Impurity Atom Diffusion

258

(3)

A Simple One-Dimensional Rate Theory

261

(4)

Exact Normalization

265

(2)

Many Degrees of Freedom

267

(8)

Transition-State Assumption

275

(1)

Brownian Motion

276

(11)

Kramers Rate Formula

287

(10)

PART TWO QUANTUM THEORY

Basic Concepts of Quantum Mechanics

297

(50)

Introduction

297

(1)

Structure of Classical Mechanics

298

(1)

Structure of Quantum Mechanics

299

(4)

Consequences of the Fourier Transform Relation between ψx(x) and ψp(p)

303

(9)

Extension to Three-Dimensional Motion and to N Particles

312

(3)

Hydrodynamic Analogy to Quantum Mechanics

315

(3)

Initial-Value Problems

318

(2)

Aspects of the Measurement Process

320

(2)

Some Examples of Stationary States

322

(12)

Tensor Product of Two Spaces

334

(2)

Electron Spin

336

(4)

Identical Particles and the Pauli Principle

340

(7)

Interatomic Interactions

347

(25)

Introduction

347

(1)

Hydrogen Molecule

348

(5)

Adiabatic Approximation

353

(4)

Hellmann--Feynman Theorem

357

(3)

Harmonic Elastic Moduli Based on Hellmann-Feynman Theorem

360

(5)

Categories of Interatomic Force Laws

365

(7)

Quantum Statistical Effects

372

(49)

Introduction

372

(1)

Quantum Statistical Mechanics

372

(14)

Quantum Statistical Effects in Crystals

386

(17)

Quantum Statistics for Polymer Models

403

(3)

Gaussian Wave Packet Dynamics

406

(6)

Canonical Ensemble in Terms of Coherent States

412

(4)

Quantum Effects on Rate Processes

416

(5)

References

421

(10)

Index

431

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Statistical Mechanics of Elasticity

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