Thermomechanics of Evolving Phase Boundaries in the Plane: (Oxford Mathematical Monographs)

This book is one of the very first on the subject of mathematical materials science and presents a view different from that prevalent in the physical literature. Issues foundational in nature are stressed with the emphasis on the interplay between mathematics and physics. It discusses the dynamics of two-phase systems within the framework of modern continuum thermodynamics. Two general theories are considered and the resulting equations exhibit unstable growth patterns. The free boundary problems that form the basis of the subject should be of great interest to mathematicians and physical scientists.

Table of Contents:
Introduction Part 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 Part 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 assumptions 8.1: Isotropic interface 8.2: Anisotropic interface 8.2.1: Basic equations 8.2.2: Equations when the interface is the graph of a function 8.2.3: Equations when the interface is a level set 8.3: Plan of the next few chapters 9: Stationary interfaces and steadily evolving interfaces 9.1: Stationary interfaces 9.2: Steadily evolving facets 9.3: Steadily evolving interfaces that are not flat 10: Global behaviour for an interface with stable energy 10.1: Existence of evolving interfaces from a prescribed initial curve 10.2: Growth and decay of the interface 10.3: Evolution of curvature; fingers 11: Unstable interfacial energies and interfaces with corners 11.1: Admissibility; corner conditions 11.2: The initial-value problem 11.3: Facets and wrinklings that connect evolving curves 11.4: Equations near a corner when the curve is a graph 11.5: Interfaces with arbitrary angle-set; infinitesimal wrinklings 11.6: Stationary interfaces and steadily evolving interfaces with corners 12: Non smooth interfacial energies: crystalline energies 12.1: Crystalline energies 12.2: The Wulff region 12.3: The capillary force at preferred orientations 12.4: Corners between preferred facets 12.5: Crystalline motions 12.6: Interfaces of arbitrary orientation, infinitesimal wrinklings, and generalized motions 12.7: Evolution of a rectangular crystal 13: Regularized theory for smooth unstable energies; dependence of interfacial energy on curvature 13.1: Balance of forces and moments; power 13.2: Energetics and the dissipation inequality 13.3: Constitutive equations 13.4: Evolution equations for the interface 13.5: Linearized equations; spinodal decomposition on the interface Part III: Thermodynamical theory of interfacial evolution in the presence of bulk heat conduction 14: Review of single-phase thermodynamics 14.1: Basic equations and the first two laws 14.2: Constitutive equations and thermodynamic restrictions 14.3: The heat equation 15: Thermodynamics of two-phase systems 15.1: Basic quantities and the first two laws 15.2: Local forms of the interfacial laws 16: Constitutive theory 16.1: Constituive equations for the bulk material 16.2: The transition temperature 16.3: Constitutive equations for the interface 17: Free-boundary problems 17.1: Bulk equations and interface conditions 17.2: Initial conditions and boundary conditions 17.3: Free-boundary problems near the transition temperature for weak surfaces 17.3.1: Approximate interface conditions 17.3.2: Approximate free-boundary problems 17.3.3: The first two laws for the approximate theories 17.3.4: Growth theorems 17.3.5: Perfect conductors 18: Instabilities induced by supercooling the liquid phase 18.1: The one-dimensional problem: growth of the solid phase 18.2: Instability of a flat interface References Index

Review :
`The book is unique in the topics it discusses and in the approach and the nature of the presentation pertaining to these topics ... it can be used as a reference.' J.A. Gatowski, Applied Mechanics Review, vol. 47, no 7, July 1994 `This very good book, presents a study of the dynamics of two-phase systems within the framework of modern continuum thermodynamics ... a detailed account of a number of important results in a field to which the author has made substantial contributions .., Special problems and exercises are included. This monograph is a very useful work.' Zbl. Math. 787