Thermomechanics of Evolving Phase Boundaries in the Plane

Hardcover | April 30, 1999

byMorton E. Gurtin

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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 thedynamics 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 andphysical scientists.

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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 thedynamics of two-phase systems within the fram...

Morton E. Gurtin is at Carnegie Mellon University, Pittsburgh.

other books by Morton E. Gurtin

The Mechanics and Thermodynamics of Continua
The Mechanics and Thermodynamics of Continua

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Format:HardcoverDimensions:160 pages, 9.21 × 6.14 × 0.55 inPublished:April 30, 1999Publisher:Oxford University Press

The following ISBNs are associated with this title:

ISBN - 10:0198536941

ISBN - 13:9780198536949

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Table of Contents

IntroductionPart I: Kinematics1. Curves1.1. Preliminary definitions1.2. Convex curves1.3. Integrals1.4. Piecewise-smooth curves1.5. Infinitesimally wrinkled curves2. Evolving curves2.1. Definitions2.2. Transport identities2.3. Integral identities2.4. Steadily evolving interfaces2.5. Piecewise-smooth evolving curves2.6. Variational lemmas3. Phase regions, control volumes, and inflows3.1. Phase regions and control volumes3.2. Inflows, the pillbox lemma, and infinitesimally thin evolving control volumesPart II: Mechanical theory of interfacial evolution4. Balance of forces4.1. Balances of forces4.2. The power identity5. Energetics and the dissipation inequality6. Constitutive theory6.1. Constitutive equations and the compatibility theorem6.2. Balance of capillary forces revisited; corners7. Digression: Statistical theory of interfacial stability; convexity, the Frank diagram, and corners; Wulff regions7.1. Preliminaries; Polar diagrams7.2. Convexity; the extended and convexified energies, and the Frank diagram7.3. Stability7.4. Instability of the total energy7.5. Equilibria of the total energy; Wulff regions7.6. Wulff's theorem8. Evolution equations for the interface: basic assumptions8.1. Isotropic interface8.2. Anisotropic interface8.2.1. Basic equations8.2.2. Equations when the interface is the graph of a function8.2.3. Equations when the interface is a level set8.3. Plan of the next few chapters9. Stationary interfaces and steadily evolving interfaces9.1. Stationary interfaces9.2. Steadily evolving facets9.3. Steadily evolving interfaces that are not flat10. Global behaviour for an interface with stable energy10.1. Existence of evolving interfaces from a prescribed initial curve10.2. Growth and decay of the interface10.3. Evolution of curvature; fingers11. Unstable interfacial energies and interfaces with corners11.1. Admissibility; corner conditions11.2. The initial-value problem11.3. Facets and wrinklings that connect evolving curves11.4. Equations near a corner when the curve is a graph11.5. Interfaces with arbitrary angle-set; infinitesimal wrinklings11.6. Stationary interfaces and steadily evolving interfaces with corners12. Non smooth interfacial energies: crystalline energies12.1. Crystalline energies12.2. The Wulff region12.3. The capillary force at preferred orientations12.4. Corners between preferred facets12.5. Crystalline motions12.6. Interfaces of arbitrary orientation, infinitesimal wrinklings, and generalized motions12.7. Evolution of a rectangular crystal13. Regularized theory for smooth unstable energies; dependence of interfacial energy on curvature13.1. Balance of forces and moments; power13.2. Energetics and the dissipation inequality13.3. Constitutive equations13.4. Evolution equations for the interface13.5. Linearized equations; spinodal decomposition on the interfacePart III: Thermodynamical theory of interfacial evolution in the presence of bulk heat conduction14. Review of single-phase thermodynamics14.1. Basic equations and the first two laws14.2. Constitutive equations and thermodynamic restrictions14.3. The heat equation15. Thermodynamics of two-phase systems15.1. Basic quantities and the first two laws15.2. Local forms of the interfacial laws16. Constitutive theory16.1. Constituive equations for the bulk material16.2. The transition temperature16.3. Constitutive equations for the interface17. Free-boundary problems17.1. Bulk equations and interface conditions17.2. Initial conditions and boundary conditions17.3. Free-boundary problems near the transition temperature for weak surfaces17.3.1. Approximate interface conditions17.3.2. Approximate free-boundary problems17.3.3. The first two laws for the approximate theories17.3.4. Growth theorems17.3.5. Perfect conductors18. Instabilities induced by supercooling the liquid phase18.1. The one-dimensional problem: growth of the solid phase18.2. Instability of a flat interfaceReferencesIndex

Editorial Reviews

`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 areincluded. This monograph is a very useful work.'Zbl. Math. 787