Thermomechanics of Evolving Phase Boundaries in the Plane [Hardcover]

This is one of the few books on the subject of mathematical materials science. It discusses the dynamics of two-phase systems within the framework of modern continuum thermodynamics, stressing fundamentals. Two general theories are discussed: a mechanical theory that leads to a generalization of the classical curve-shortening equation and a theory of heat conduction that broadly generalizes the classical Stefan theory. This original survey includes simple solutions that demonstrate the instabilities inherent in two-phase problems. The free-boundary problems that form the basis of the subject should be of great interest to mathematicians and physical scientists.

Introduction
I. Kinematics
1. Curves
1.1. Preliminary Definitions
1.2. Convex Curves
1.3. Integrals
1.4. Piecewise-smooth Curves
1.5. Infinitesimally Wrinkled Curves
2. Evolving Curves
2.1. Definitions
2.2. Transport Identities
2.3. Integral Identities
2.4. Steadily Evolving Interfaces
2.5. Piecewise-smooth Evolving Curves
2.6. Variational Lemmas
3. Phase Regions, Control Volumes, and Inflows
3.1. Phase Regions and Control Volumes
3.2. Inflows, the Pillbox Lemma, and Infinitesimally Thin Evolving Control Volumes
II. Mechanical Theory of Interfacial Evolution
4. Balance of Forces
4.1. Balances of Forces
4.2. The Power Identity
5. Energetics and the Dissipation Inequality
6. Constitutive Theory
6.1. Constitutive Equations and the Compatibility Theorem
6.2. Balance of Capillary Forces Revisited; Corners
7. Digression: Statistical Theory of Interfacial Stability; Convexity, the Frank Diagram, and Corners; Wulff Regions
7.1. Preliminaries; Polar Diagrams
7.2. Convexity; the Extended and Convexified Energies, and the Frank Diagram
7.3. Stability
7.4. Instability of the Total Energy
7.5. Equilibria of the Total Energy; Wulff Regions
7.6. Wulff's Theorem
8. Evolution Equations for the Interface: Basic AssumptilÓQ