Thermomechanics and the formulation of the Stefan problem for fully faceted interfaces

Authors:
Morton E. Gurtin and José Matias

Journal:
Quart. Appl. Math. **53** (1995), 761-782

MSC:
Primary 35R35; Secondary 35Q72, 73B30, 80A22

DOI:
https://doi.org/10.1090/qam/1359510

MathSciNet review:
MR1359510

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper develops a thermomechanics of two-phase heat conductors in which the interface between phases is fully faceted. The theory is based on balance of forces, balance of energy, and growth of entropy in conjunction with constitutive equations for the interface; and the chief result is a free-boundary problem of Stefan type in which the classical interface condition is replaced by a condition relating the integral of over each facet to the normal velocity of that facet.

**[1951]**C. Herring,*Surface tension as a motivation for sintering*, The Physics of Powder Metallurgy (W. E. Kingston, ed.), McGraw-Hill, New York**[1963]**B. D. Coleman and W. Noll,*The thermodynamics of elastic materials with heat conduction and viscosity*, Arch. Rational Mech. Anal.**13**, 167-178**[1963]**F. C. Frank,*The geometrical thermodynamics of surfaces*, Metal Surfaces: Structure, Energetics, and Kinetics, Amer. Soc. Metals, Metals Park, Ohio**[1972]**D. W. Hoffman and J. W. Cahn,*A vector thermodynamics for anisotropic surfaces*- 1.*Fundamentals and applications to plane surface junctions*, Surface Sci.**31**, 368-388**[1974]**J. W. Cahn and D. W. Hoffman,*A vector thermodynamics for anisotropic surfaces*- 2.*Curved and faceted surfaces*, Act. Metall.**22**, 1205-1214**[1986]**M. E. Gurtin,*On the two-phase Stefan problem with interfacial energy and entropy*, Arch. Rational Mech. Anal.**96**, 199-241**[1988]**Ben Amar and Y. Pomeau,*Growth of faceted needle crystals: Theory*, Europhys. Lett.**6**, 609-614**[1988]**M. E. Gurtin,*Multiphase thermomechanics with interfacial structure*. 1 .*Heat conduction and the capillarity balance law*, Arch. Rational. Mech. Anal.**104**, 195-221**[1988]**J. E. Taylor,*Constructions and conjectures in crystalline nondifferential geometry*(Proc. Conf. Diff. Geom.), Rio de Janeiro, Pittman, London, 1988**[1989]**S. Angenent and M. E. Gurtin,*Multiphase thermomechanics with interfacial structure*. 2 .*Evolution of an isothermal interface*, Arch. Rational Mech. Anal.**108**, 323-391**[1990]**M. E. Gurtin and J. Matias,*Notes of lectures given by Gurtin at the IMA*, University of Minnesota, September 1990**[1990]**M. E. Gurtin and A. Struthers,*Multiphase thermomechanics with interfacial structure*. 3.*Evolving phase boundaries in the presence of bulk deformation*, Arch. Rational Mech. Anal.**112**, 97-160**[1991]**M. E. Gurtin,*On thermomechanical laws for the motion of a phase interface*, Zeit. Angew. Math. Phys.**42**, 370-388**[1992]**F. Almgren and J. Taylor,*Flat flow is motion by crystalline curvature for curves with crystalline energies*, Rept. GCG43, Geometry Center, U. Minnesota**[1992]**J. Taylor,*Mean curvature and weighted mean curvature*, Act. Metall.**40**, 1475-1485**[1992]**P. Rybka,*A quasi-steady approximation to an integro-differential model of interface motion*, preprint**[1993]**T. Fukui and Y. Giga,*Motion of a graph by nonsmooth weighted curvature*, Proceedings of the First World Congress of Nonlinear Analysts (V. Lakshmikantham, ed.), Walter de Gruyter, Hawthorne, NY (to appear)**[1993]**Y. Giga, M. E. Gurtin, and J. Matias,*On the dynamics of crystalline motions*, forthcoming**[1993]**P. M. Girão,*Convergence of a crystalline algorithm for the motion of a simple closed curve by weighted curvature*, Siam J. Numer. Anal. (to appear)**[1993a]**M. E. Gurtin,*Thermomechanics of Evolving Phase Boundaries in the Plane*, Oxford University Press**[1993b]**M. E. Gurtin,*Thermodynamics and the supercritical Stefan equations with nucleations*, Quart. Appl. Math.**52**, 133-155 (1994)**[1993c]**M. E. Gurtin and P. W. Voorhees,*The continuum mechanics of coherent two-phase elastic solids with mass transport*, Proc. Roy. Soc. Lond. A**440**, 323-343**[1994]**P. M. Girão and R. V. Kohn,*Convergence of a crystalline algorithm for the heat equation in one dimension and for the motion of a graph by weighted curvature*, Numer. Math.**67**, 41-70

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
35R35,
35Q72,
73B30,
80A22

Retrieve articles in all journals with MSC: 35R35, 35Q72, 73B30, 80A22

Additional Information

DOI:
https://doi.org/10.1090/qam/1359510

Article copyright:
© Copyright 1995
American Mathematical Society