Advertisement

Introduction: Fifty Years of Research on Evolving Phase Interfaces

  • Eliot Fried

Abstract

A material body most often consists of one or more regions, termed phases, within which the chemical composition and the crystal or molecular structure do not vary with position. A phase transition is a process in which a region occupied by one phase grows at the expense of a region occupied by another phase, with the essential physical activity occuring at the interfaces separating phases. Processes of this type can be used to control the scale and distribution of microstructural features that are associated with different phases and, thus, granted an understanding of the various influences that such features may exert on macroscopic properties, to create materials with optimally determined properties. To realize such a program requires a physically faithful mathematical framework with the capacity to describe and predict the evolution of phase interfaces.1 During the last fifty years, materials scientists, continuum physicists, and mathematicians have made substantial progress toward establishing a broadly applicable continuum-level theory of this nature.2 This theory is based on the approach of Gibbs (1878:1), who treated phase interfaces as surfaces across which bulk material properties may suffer discontinuities and, as a means to account for localized interactions between phases, endowed these surfaces with excess fields. Here, some of the relevant developments are summarized, with emphasis being placed on the contributions made by those researchers whose works are reprinted in this volume.3

Keywords

Phase Interface Diffusion Potential Rational Mechanics Atomic Diffusion Stefan Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. 1.
    J. L. Lagrange, Mécanique Analytique, Desaint, Paris, 1788. Translated as Analytical Mechanics, Kluwer, Dordrecht, 1997. 1788Google Scholar
  2. 1.
    T. Young, An essay on the cohesion of fluids, Philosophical Transactions of the Royal Society of London 95, 65–87, 1805ADSCrossRefGoogle Scholar
  3. 1.
    P. S. Laplace Méchanique Céleste, Supplement au X e Livre, Impresse Imperiale, Paris. Translated as Celestial Mechanics, Volume IV, Chelsea, New York, 1966. 1806Google Scholar
  4. 1.
    G. Lamé & B. D. Clapeyron, Mémoire sur la solidification par refroidissement d’un globe liquide, Annales de Chimie et de Physique 47, 250–256, 1831Google Scholar
  5. 1.
    W. J. M. Rankine, On the thermodynamic theory of waves of finite longitudinal disturbance, Philosophical Transactions of the Royal Society of London 160, 277–288, 1870ADSCrossRefGoogle Scholar
  6. 1.
    G. Erdmann, Über unstetige Lösungen in der Variationsrechnung, Journal für die reine und angewandte Mathematik 82, 21–30, 1877CrossRefGoogle Scholar
  7. 1.
    J. W. Gibbs, On the equilibrium of heterogeneous substances, Transactions of the Connecticut Academy of Arts and Sciences 3 108–248. Reprinted in The Scientific Papers of J. Willard Gibbs, vol. 1, Dover, New York, 1961, 1878Google Scholar
  8. 1.
    M. P. Curie, Sur la formation des critaux et sur les constantes capillaires de leurs différentes faces, Bulletin de la Société Mineralogique de Prance 8, 145–150, 1885Google Scholar
  9. 1.
    I. H. Hugoniot, Mémoire sur la propagation du movement dans un fluid indéfini, Journal de Mathématique Pures et Appliquées 3, 477–492 and 4, 153–167, 1887Google Scholar
  10. 1.
    J. Stefan, Über einige Probleme der Theorie der Wärmeleitung, Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften in Wien, Mathematisch-Naturwissenschaftliche Classe 98, 473–484, 1889Google Scholar
  11. 1.
    C. V. Burton, A theory concerning the constitution of matter, Philosophical Magazine 33, 191–204, 1892Google Scholar
  12. 1.
    J. Larmor, A dynamical theory of the electric and luminiferous medium—III. Relations with material media, Philosophical Transactions of the Royal Society of London A 190 205–300, 1897ADSzbMATHGoogle Scholar
  13. 1.
    G. Wulff, Zur FYage der Geschwindigkeit des Wachsthums und der Auflösimg der Krystallfläehen, Zeitschrift für Krystallographie und Mineralogie 34, 499–530,.1901Google Scholar
  14. 1.
    E. Noether, Invariante Variationsprobleme, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematische-Physikalische Klasse 2, 235–257. Translated in Transport Theory and Statistical Physics 1 (1971), 186–207, 1918Google Scholar
  15. 1.
    K. Weierstrass, Mathematische Werke, VJJ, Mayer and Müller, Berlin, 1927zbMATHGoogle Scholar
  16. 1.
    M. Volmer, Kinetik der Phasenbildung, T. Steinkopff, Dresden, 1939Google Scholar
  17. 1.
    C. Eckart, The thermodynamics of irreversible process, I. The simplefluid, Physical Review 58, 267–269, 1940ADSzbMATHCrossRefGoogle Scholar
  18. 2.
    C. Eckart, The thermodynamics of irreversible process, II. Fluid mixtures, Physical Review 58, 269–275, 1940ADSzbMATHCrossRefGoogle Scholar
  19. 1.
    J. S. Koehlerw, On the dislocation theory of plastic deformation, Physical Review 60, 397–410, 1941ADSCrossRefGoogle Scholar
  20. 1.
    J. Meixner, Zur Thermodynamik der irreversiblen Prozesse in Gasen mit chemische reagierenden, dissoziierenden und anregbaren Komponenten, Annalen der Physik 43, 244–270, 1943MathSciNetADSCrossRefGoogle Scholar
  21. 2.
    J. Meixner, Zur Thermodynamik der irreversiblen Prozesse, Zeitschrift für Physikalische Chemie B 53, 235–263, 1943Google Scholar
  22. 1.
    D. Harker & E. R. Parker, Grain shape and grain growth, Transactions of the American Society for Metals 34, 156–195, 1945Google Scholar
  23. 1.
    L. Bragg & J. F. Nye, A dynamical model of a crystal structure, Proceedings of the Royal Society of London A 190, 474–481, 1947ADSCrossRefGoogle Scholar
  24. 1.
    C. S. Smith, Grain, phases, and interfaces: An interpretation of microstructure, Transactions of the American Institute of Mining and Metallurgical Engineers 175, 15–51, 1948Google Scholar
  25. 1.
    G. Leibfried, Über die auf eine Versetzung wirkenden Kräfte, Zeitschriftfür Physik 126, 781–789, 1949MathSciNetADSzbMATHCrossRefGoogle Scholar
  26. 2.
    I. Prigogine, Le domaine de la validité de la thermodynamique des phénomènes irreversibles Physica 15, 272–284, 1949MathSciNetADSzbMATHCrossRefGoogle Scholar
  27. 1.
    B. A. Bilby, On the interactions of dislocations and solute atoms, Proceedings of the Physical Society A 63, 191–200, 1950CrossRefGoogle Scholar
  28. 1.
    J. D. Eshelby, The force on an elastic singularity, Philosophical Transactions of the Royal Society of London A 244, 87–112, 1951MathSciNetADSzbMATHGoogle Scholar
  29. 2.
    C. Herring, Surface tension as a motivation for sintering, in The Physics of Powder Metallurgy (W. E. Kingston, Ed.), McGraw-Hill, New York, 1951Google Scholar
  30. 1.
    P. A. Beck, Interface migration in recrystallization, in Metal Interfaces, American Society for Metals, Cleveland, 1952Google Scholar
  31. 2.
    J. E. Burke & D. Turnbull, Structure of crystal boundaries, in Progress in Metal Physics 3 (B. Chalmers, Ed.), Pergamon, New York, 1952Google Scholar
  32. 3.
    C. S. Smith, Grain shapes and other metallurgical applications of topology, in Metal Interfaces, American Society for Metals, Cleveland, 1952Google Scholar
  33. 4.
    C. Truesdell, The mechanical foundations of elasticity and fluid dynamics, Journal of Rational Mechanics and Analysis 1, 125–300, 1952MathSciNetzbMATHGoogle Scholar
  34. 1.
    J. D. Eshelby, The equation of motion of a dislocation, Physical Review 90, 248–255, 1953MathSciNetADSzbMATHCrossRefGoogle Scholar
  35. 2.
    C. Herring, The use of classical macroscopic concepts in surface-energy problems, in Structure and Properties of Solid Surfaces (R. Gomer & C. S. Smith, Eds.), University of Chicago, Chicago, 1953Google Scholar
  36. 3.
    W. T. Read, Dislocations in Crystals, Mc-Graw Hill, New York, 1953.zbMATHGoogle Scholar
  37. 4.
    C. Truesdell, Corrections and additions to “The mechanical foundations of elasticity and fluid dynamics,” Journal of Rational Mechanics and Analysis 2, 593–616, 1953MathSciNetzbMATHGoogle Scholar
  38. 1.
    J. D. Eshelby, The continuum theory of lattice defects, in Progress in Solid State Physics 3 (F. Seitz & D. Turnbull, Eds.), Academic Press, 79–144, 1956Google Scholar
  39. 2.
    W. W. Mullins, Two-dimensional motion of idealized grain boundaries Journal of Applied Physics 27, 900–904, 1956MathSciNetADSCrossRefGoogle Scholar
  40. 1.
    A. E. Green & J. E. Adkins, Large Elastic Deformations, Clarendon Press, Oxford, 1960zbMATHGoogle Scholar
  41. 2.
    W. W. Mullins, Grain boundary grooving by surface diffusion, Transactions of the American Institute of Mining Metallurgical and Petroleum Engineers, 218, 354–361, 1960Google Scholar
  42. 3.
    C. Truesdell, Principles of Continuum Mechanics, Socony Mobil Oil Company, Dallas, 1960Google Scholar
  43. 4.
    C. Truesdell & R. A. Toupin, The Classical Field Theories, in ( Handbuch der Physik III/l(S. Flügge, Ed.), Springer-Verlag, Berlin, 1960Google Scholar
  44. 1.
    S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover, New York, 1961Google Scholar
  45. 1.
    P. M. Blatz & W. L. Ko, Application of finite elastic theory to the deformation of rubbery materials, Transactions of the Society of Rheology 6, 223–251, 1962CrossRefGoogle Scholar
  46. 2.
    W. Günther, Über einige Randintegrale der Elastomechanik, Abhandlungen der Braunschweigischen wissenschaftliche Gesellschaft 14, 54–72, 1962Google Scholar
  47. 1.
    B. D. Coleman & V. J. Mizel, Thermodynamics and departures from Fourier’s law of heat conduction, Archive for Rational Mechanics and Analysis 13, 245–261, 1963MathSciNetADSzbMATHCrossRefGoogle Scholar
  48. 2.
    B. D. Coleman & W. Noll, The thermodynamics of elastic materials with heat conduction and viscosity, Archive for Rational Mechanics and Analysis 13, 167–178, 1963MathSciNetADSzbMATHCrossRefGoogle Scholar
  49. 3.
    W. W. Mullins & R. F. Sekerka, Morphological stability of a particle growing by diffusion or heat flow, Journal of Applied Physics 34, 323-329, 1963ADSCrossRefGoogle Scholar
  50. 1.
    B. D. Coleman & V. J. Mizel, Existence of caloric equations of state inthermodynamics, Journal of Chemical Physics 40 1116–1125, 1964MathSciNetADSCrossRefGoogle Scholar
  51. 1.
    W. W. Mullins & R. F. Sekerka, Stability of a planar interface during solidification of a binary alloy, Journal of Applied Physics 35, 444–451, 1964ADSCrossRefGoogle Scholar
  52. 1.
    C. Truesdell & W. Noll, The Non-Linear Field Theories of Mechanics, in (Handbuch der Physik III/3 (S. Fliigge, Ed.), Springer-Verlag, Berlin, 1965Google Scholar
  53. 2.
    B. D. Coleman, H. Markovitz & W. Noll, Viscometric Flows of Non-Newtonian Fluids, Springer-Verlag, Berlin, 1966Google Scholar
  54. 2.
    E. Varley & A. Day, Equilibrium phases of elastic materials at uniform temperature and pressure, Archive for Rational Mechanics and Analysis 22, 253–269, 1966MathSciNetADSCrossRefGoogle Scholar
  55. R. M. Bowen, Toward a thermodynamics and mechanics of mixtures, Archive for Rational Mechanics and Analysis 24, 370–403, 1967MathSciNetADSzbMATHCrossRefGoogle Scholar
  56. 2.
    M. E. Gurtin & W. O. Williams, Phases of elastic materials Zeitschrift fur angewandte Maihematik und Physik 13, 132–135, 1967ADSCrossRefGoogle Scholar
  57. 1.
    R. M. Bowen, The thermochemistry of elastic materials with diffusion, Archive for Rational Mechanics and Analysis 34, 97–127, 1969MathSciNetADSzbMATHCrossRefGoogle Scholar
  58. J. D. Eshelby, The elastic field of a crack extending non-uniformly under general anti-plane loading, Journal of the Mechanics and Physics of Solids 17, 177–189, 1969ADSzbMATHCrossRefGoogle Scholar
  59. C. Truesdell, Rational Thermodynamics, McGraw-Hill, New York, 1969Google Scholar
  60. J. D. Eshelby, Energy relations and the energy-momentum tensor in continuum mechanics, in Inelastic Behavior of Solids (M. F. Kanninen, W. F. Alder, A. R. Rosenfield & R. I. Jaffe, Eds.), McGraw-Hill, New York, 1970Google Scholar
  61. A. I. Roitburd, Equilibrium of crystals formed in the solid phase, SovietPhysics Doklady 16, 305–308, 1971ADSGoogle Scholar
  62. 2.
    L. I. Rubenštein, The Stefan Problem, American Mathematical Society, Providence, 1971Google Scholar
  63. 1.
    J. K. Knowles & E. Sternberg, On a class of conservation laws in linearized and finite elastostatics, Archive for Rational Mechanics and Analysis 44, 187–211, 1972MathSciNetADSzbMATHCrossRefGoogle Scholar
  64. 1.
    F. C. Larche & J. W. Cahn, A linear theory of thermochemical equilibrium of solids under stress, Acta Metallurgica 21, 1051–1063, 1973CrossRefGoogle Scholar
  65. 2.
    D. P. Woodruff, The Solid-Liquid Interface, Cambridge University Press, London, 1973Google Scholar
  66. J. L. Ericksen, Equilibrium of bars, Journal of Elasticity 5, 191–202, 1975MathSciNetzbMATHCrossRefGoogle Scholar
  67. J. D. Eshelby, The elastic energy-momentum tensor, Journal of Elasticity 5 321–335, 1975.MathSciNetzbMATHCrossRefGoogle Scholar
  68. J. K. Knowles & E. Sternberg, On the ellipticity of the equations of nonlinear elastostatics for a special material, Journal of Elasticity 5, 341–362, 1975MathSciNetzbMATHCrossRefGoogle Scholar
  69. G. P. Moeckel, Thermodynamics of an interface, Archive for Rational Mechanics and Analysis 57, 255–280, 1975.MathSciNetADSzbMATHCrossRefGoogle Scholar
  70. 1.
    A. I. Murdoch, A thermodynamical theory of elastic material surfaces, Quarterly Journal of Mechanics and Applied Mathematics 29, 245–275, 1976MathSciNetzbMATHCrossRefGoogle Scholar
  71. 1.
    M. E. Glicksman, Capillary phenomena during solidification, Journal of Crystal Growth 42, 347–356, 1977ADSCrossRefGoogle Scholar
  72. P. C. Hohenberg & B. I. Halperin, Theory of dynamic critical phenomena, Reviews of Modern Physics 49, 435–479, 1977ADSCrossRefGoogle Scholar
  73. J. K. Knowles & E. Sternberg, On the failure of ellipticity of the equations for finite elastostatic plane strain, Archive for Rational Mechanics and Analysis 63, 321–336, 1977MathSciNetzbMATHCrossRefGoogle Scholar
  74. 1.
    K. A. Brakke, The Motion of a Surface by its Mean Curvature, Princeton University Press, Princeton, 1978zbMATHGoogle Scholar
  75. J. L. Ericksen, On the symmetry and stability of thermoelastic solids, Journal of Applied Mechanics 45, 740–744, 1978ADSCrossRefGoogle Scholar
  76. J. K. Knowles & E. Sternberg, On the failure of ellipticity and the emergence of discontinuous deformation gradients in plane finite elastostatics, Journal of Elasticity 8, 329–380, 1978MathSciNetzbMATHCrossRefGoogle Scholar
  77. F. C. Larché & J. W. Cahn, A nonlinear theory of thermochemical equilibrium of solids under stress, Acta Metallurgica 26, 53–60, 1978CrossRefGoogle Scholar
  78. F. C. Larché & J. W. Cahn, Thermochemical equilibrium of multiphase solids under stress, Acta Metallurgica 26, 1579–1589, 1978CrossRefGoogle Scholar
  79. P. Régnier, Surface tension and surface energies, in Handbook of Surfaces and Interfaces, Volume 2 (L. Dobrzynski, Ed.), Garland STMP Press, New York, 1978Google Scholar
  80. 1.
    J. Fernandez-Diaz & W. O. Williams, A generalized Stefan condition Zeitschrift fur angewandte Mathematik und Physik 30, 749–755, 1979MathSciNetADSzbMATHCrossRefGoogle Scholar
  81. J. K. Knowles, On the dissipation associated with equilibrium shocks in finite elasticity, Journal of Elasticity 9, 131–158, 1979MathSciNetzbMATHCrossRefGoogle Scholar
  82. R. D. James, Co-existent phases in the one-dimensional static theory of elastic bars, Archive for Rational Mechanics and Analysis 72, 99–140, 1979MathSciNetADSzbMATHCrossRefGoogle Scholar
  83. 1.
    R. Abeyaratne, Discontinuous deformation gradients in plane finite elastostatics of incompressible materials, Journal of Elasticity 10, 255–293, 1980MathSciNetzbMATHCrossRefGoogle Scholar
  84. J. W. Cahn, Surface stress and the equilibrium of small crystals—I. The case of the isotropic surface, Acta Metallurgica 28, 1333–1338, 1980CrossRefGoogle Scholar
  85. J. D. Eshelby, The energy-momentum tensor of complex continua, in Continuum Models of Discrete Systems (CMDS3), (E. Kröner & K.-H. Anthony, Eds.), University of Waterloo Press, Waterloo, 1980Google Scholar
  86. J. D. Eshelby, The force on a disclination in a liquid crystal, Philosophical Magazine A 42, 359–367, 1980ADSCrossRefGoogle Scholar
  87. C. Gerhardt, Evolutionary surfaces of prescribed mean curvature, Journal of Differential Equations 36, 139–172, 1980MathSciNetzbMATHCrossRefGoogle Scholar
  88. R. D. James, The propagation of phase boundries in elastic bars, Archive for Rational Mechanics and Analysis 73, 125–158, 1980MathSciNetADSzbMATHCrossRefGoogle Scholar
  89. J. S. Langer, Instabilities and pattern formation in crystal growth, Reviews of Modern Physics 52, 1–28, 1980ADSCrossRefGoogle Scholar
  90. G. P. Parry, Twinning in nonlinearly elastic monatomic crystals, International Journal of Solids and Structures 16, 43–80, 1980CrossRefGoogle Scholar
  91. R. Abeyaratne, Discontinuous deformation gradients in finite twisting of an incompressible elastic tube, Journal of Elasticity 11, 43–80, 1981MathSciNetzbMATHCrossRefGoogle Scholar
  92. J. L. Ericksen, Continuous martensitic transitions in thermoelastic solids, Journal of Thermal Stresses 4, 107–119, 1981CrossRefGoogle Scholar
  93. M. A. Grinfeld, On heterogeneous equilibrium of non-linear elastic phases and chemical potential tensors, Letters in Applied Engineering Science 19, 1031–1039, 1981Google Scholar
  94. M. E. Gurtin & R. Temam, On the anti-plane shear problem in finite elasticity, Journal of Elasticity 11, 197–206, 1981MathSciNetzbMATHCrossRefGoogle Scholar
  95. R. D. James, Finite deformation by mechanical twinning, Archive for Rational Mechanics and Analysis 77, 143–176, 1981MathSciNetADSzbMATHCrossRefGoogle Scholar
  96. D. A. Porter & K. E. Easterling, Phase Transformations in Metals and Alloys, Chapman & Hall, London, 1981Google Scholar
  97. 1.
    J. W. Cahn & F. C. Larche, Surface stress and the equilibrium of small crystals—II. Solid particles embedded in a solid matrix, Acta Metallurgica 30, 51–56, 1982CrossRefGoogle Scholar
  98. F. C. Larché & J. W. Cahn, The effect of self-stress on diffusion in solids, Acta Metallurgica 30, 1835–1845, 1982CrossRefGoogle Scholar
  99. W. W. Mullins, The thermodynamics of critical phases with curved interfaces: specific case of interfacial isotropy and hydrostatic pressure, in Proceedings of an International Conference on Solid→Solid Phase Transformations(H. I. Aaronson, D. E. Laughlin, R. F. Sekerka & C. M. Way-man, Eds.), American Institute of Mining, Metallurgical and Petroleum Engineers, New York, 1982Google Scholar
  100. M. Shearer, The Riemann problem for a class of conservation laws of mixed type, Journal of Differential Equations 46, 426–443, 1982MathSciNetzbMATHCrossRefGoogle Scholar
  101. 1.
    J. W. Cahn & F. C. Larché, An invariant formulation of multicomponent diffusion in crystals Scripta Metallurgica 17, 927–932, 1983CrossRefGoogle Scholar
  102. M. G. Crandall & P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Transactions of the American Mathematical Society 277, 1–42, 1983MathSciNetzbMATHCrossRefGoogle Scholar
  103. R. L. Fosdick & G. MacSithigh, Helical shear of an elastic, circular tube with a non-convex stored energy, Archive for Rational Mechanics and Analysis 84, 31–53, 1983MathSciNetADSzbMATHCrossRefGoogle Scholar
  104. M. E. Gage, An isoperimetric inequality with applications to curve shortening, Duke Mathematical Journal 50, 1225–1229, 1983MathSciNetzbMATHCrossRefGoogle Scholar
  105. M. E. Gurtin, Two-phase deformations of elastic solids, Archive for Rational Mechanics and Analysis 84, 1–29, 1983MathSciNetADSzbMATHCrossRefGoogle Scholar
  106. M. Shearer, Admissibility criteria for shock wave solutions of a system of conservation laws of mixed type, Proceedings of the Royal Society of Edinburgh A 93, 233–244, 1983MathSciNetzbMATHGoogle Scholar
  107. M. Slemrod, Admissibility criteria for propagating phase boundaries in a van der Waals fluid, Archive for Rational Mechanics and Analysis 81, 301–315, 1983MathSciNetADSzbMATHCrossRefGoogle Scholar
  108. H. C. Simpson & S. J. Spector, On copositive matrices and strong ellipticity for isotropic elastic materials, Archive for Rational Mechanics and Analysis 84, 55–68, 1983MathSciNetADSzbMATHCrossRefGoogle Scholar
  109. L. M. Truskinovsky, On the chemical potential tensor, Geokhimiya 12, 1730–1744, 1983Google Scholar
  110. L. Zee & E. Sternberg, Ordinary and strong ellipticity in the equilibrium theory of incompressible hyperelastic solids, Archive for Rational Mechanics and Analysis 83, 53–90, 1983MathSciNetADSzbMATHCrossRefGoogle Scholar
  111. 1.
    M. G. Crandall, L. C. Evans & P.-L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Transactions of the American Mathematical Society 282, 487–502, 1984MathSciNetzbMATHCrossRefGoogle Scholar
  112. M. E. Gage, Curve shortening makes convex curves circular, Inventiones Mathematicae 76, 357–364, 1984MathSciNetADSzbMATHCrossRefGoogle Scholar
  113. G. Huisken, Flow by mean curvature of convex surfaces into spheres, Journal of Differential Geometry 20, 237–266, 1984MathSciNetzbMATHGoogle Scholar
  114. W. W. Mullins, Thermodynamic equilibrium of a crystalline sphere in a fluid, Journal of Chemical Physics 81, 1436–1442, 1984MathSciNetADSCrossRefGoogle Scholar
  115. M. Slemrod, Dynamics of first order phase transitions, in Phase Transformations and Material Instabilities in Solids (M. E. Gurtin, Ed.), Academic Press, Orlando, 1984Google Scholar
  116. 1.
    J. I. D. Alexander & W. C. Johnson, Thermomechanical equilibrium in solid-fluid systems with curved interfaces, Journal of Applied Physics 58, 816–824, 1985ADSCrossRefGoogle Scholar
  117. W. Heidug & F. K. Lehner, Thermodynamics of coherent phase transformations in nonhydrostatically stressed solids Pure and Applied Geophysics 123, 91–98, 1985ADSCrossRefGoogle Scholar
  118. F. C. Larché & J. W. Cahn, The interactions of composition and stress in crystalline solids, Acta Metallurgica 33, 331–357, 1985ADSCrossRefGoogle Scholar
  119. W. W. Mullins & R. F. Sekerka, On the thermodynamics of crystalline solids, Journal of Chemical Physics 82, 5192–5202, 1985MathSciNetADSCrossRefGoogle Scholar
  120. M. Pitteri, On the kinematics of mechanical twinning in crystals Archive for Rational Mechanics and Analysis 88, 25–57, 1985MathSciNetADSzbMATHCrossRefGoogle Scholar
  121. J. A. Sethian, Curvature and the evolution of fronts, Communications in Mathematical Physics 101, 487–495, 1985MathSciNetADSzbMATHCrossRefGoogle Scholar
  122. I. J. I. D. Alexander & W. C. Johnson, Interface conditions for thermo mechanical equilibrium in two-phase crystals, Journal of Applied Physics 59, 2735–2746, 1986ADSCrossRefGoogle Scholar
  123. M. Gage & R. S. Hamilton, The heat equations shrinking convex curves, Journal of Differential Geometry 23, 69–96, 1986MathSciNetzbMATHGoogle Scholar
  124. M. E. Gurtin, On the two-phase Stefan problem with interfacial energy and entropy, Archive for Rational Mechanics and Analysis 96, 199–241, 1986MathSciNetADSzbMATHGoogle Scholar
  125. R. D. James, Phase transformations and non-elliptic free energy functions, in New Perspectives in Thermodynamics (J. Serrin, Ed.), Springer-Verlag, Berlin, 1986Google Scholar
  126. R. D. James, Displacive phase transformations in solids, Journal of the Mechanics and Physics of Solids 34, 359–394, 1986ADSzbMATHCrossRefGoogle Scholar
  127. 1.
    R. Abeyaratne & J. K. Knowles, Non-elliptic elastic materials and the modeling of dissipative mechanical behavior: an example, Journal of Elasticity 18, 227–278, 1987MathSciNetzbMATHCrossRefGoogle Scholar
  128. J. M. Ball & R. D. James, Fine phase mixtures as minimizers of energy Archive for Rational Mechanics and Analysis 100, 13–52, 1987MathSciNetADSzbMATHCrossRefGoogle Scholar
  129. C. L. Epstein & M. I. Weinstein, A stable manifold theorem for the curve shortening equation, Communications on Pure and Applied Mathematics 40, 119–139, 1987MathSciNetzbMATHCrossRefGoogle Scholar
  130. M. A. Grayson, The heat equation shrinks embedded plane curves to round points, Journal of Differential Geometry 26, 285–314,1987MathSciNetzbMATHGoogle Scholar
  131. L. M. TVuskinovsky, Dynamics of nonequilibrium phase boundaries in a heat conducting non-linearly elastic medium, Journal of Applied Mathematics and Mechanics (PMM) 51, 777–784, 1987ADSCrossRefGoogle Scholar
  132. 1.
    R. Abeyaratne & J. K. Knowles, On the dissipative response due to dis continuous strains in bars of unstable elastic material, International Journal of Solids and Structures 24, 1021–1044, 1988MathSciNetzbMATHCrossRefGoogle Scholar
  133. M. Chipot & D. Kinderlehrer, Equilibrium configurations of crystals, Archive for Rational Mechanics and Analysis 103, 237–278, 1988MathSciNetADSzbMATHCrossRefGoogle Scholar
  134. M. E. Gurtin, Toward a nonequilibrium thermodynamics of two phase materials, Archive for Rational Mechanics and Analysis 100, 275–312, 1988MathSciNetADSzbMATHGoogle Scholar
  135. M. E. Gurtin, Multiphase thermomechanics with interfacial structure 1. Heat conduction and the capillary balance law, Archive for Rational Mechanics and Analysis 104, 195–221, 1988MathSciNetADSzbMATHCrossRefGoogle Scholar
  136. R. Jensen, The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations, Archive for Rational Mechanics and Analysis 101, 350–362, 1988CrossRefGoogle Scholar
  137. R. Jensen, P.-L. Lions & P. E. Souganidis, A uniqueness result for viscosity solutions of second order fully nonlinear partial differential equations, Proceedings of the American Mathematical Society 102, 975–978, 1988MathSciNetzbMATHCrossRefGoogle Scholar
  138. S. Osher & J. Sethian, Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations, Journal of Computational Physics, 79, 12–49, 1988MathSciNetADSzbMATHCrossRefGoogle Scholar
  139. 1.
    S. Angenent & M. E. Gurtin, Multiphase thermomechanics with interfacial structure 2. Evolution of an isothermal interface, Archive for Rational Mechanics and Analysis 108, 323–391, 1989MathSciNetADSzbMATHCrossRefGoogle Scholar
  140. K. Ecker & G. Huisken, Mean curvature evolution of entire graphs, Annals of Mathematics 130, 453–471, 1989MathSciNetzbMATHCrossRefGoogle Scholar
  141. I. Fonseca, Variational methods for elastic crystals, Archive for Rational Mechanics and Analysis 107, 195–224, 1989MathSciNetADSzbMATHCrossRefGoogle Scholar
  142. M. A. Grayson, The shape of a figure-eight under the curve shortening flow, Inventiones Mathematicae 96, 177–180, 1989MathSciNetADSzbMATHCrossRefGoogle Scholar
  143. M. A. Grayson, A short note on the evolution of a surface by its mean curvature, Duke Mathematical Journal 58, 555–558, 1989MathSciNetzbMATHCrossRefGoogle Scholar
  144. H. Ishii, On uniqueness and existence of viscosity solutions of fully nonlinear second order elliptic PDE’s, Communications on Pure and Applied Mathematics 42, 15–45, 1989MathSciNetzbMATHCrossRefGoogle Scholar
  145. P. Leo, & R. F. Sekerka, The effect of surfacestress on crystal-melt and crystal-crystal equilibrium, Acta Metallurgica 37, 3119–3138, 1989CrossRefGoogle Scholar
  146. A. Visintin, Stefan problem with surface tension, in Mathematical Models for Phase Change Problems (J. F. Rodrigues, Ed.), Birkhauser Verlag, Basel, 1989Google Scholar
  147. 1.
    R. Abeyaratne & J. K. Knowles, On the driving traction acting on a surface of strain discontinuity in a continuum, Journal of the Mechanics and Physics of Solids 38, 345–360, 1990MathSciNetADSzbMATHCrossRefGoogle Scholar
  148. P. Bauman & D. Phillips, A nonconvex variational problem related to change of phase, Applied Mathematics and Optimization 21, 113–138, 1990MathSciNetzbMATHCrossRefGoogle Scholar
  149. F. Davi & M. E. Gurtin, On the motion of a phase interface by surface diffusion, Zeitschrift fur angewandte Mathematik und Physik 41, 782–811, 1990MathSciNetADSzbMATHCrossRefGoogle Scholar
  150. M. E. Gurtin & A. Struthers, Multiphase thermomechanics with inter-facial structure 3. Evolving phase boundaries in the presence of bulk deformation, Archive for Rational Mechanics and Analysis 112, 97–160, 1990MathSciNetADSzbMATHCrossRefGoogle Scholar
  151. S. Luckhaus, Solutions for the two-phase Stefan problem with the Gibbs-Thomson law for the melting temperature, European Journal of Applied Mathematics 1, 101–111, 1990MathSciNetzbMATHCrossRefGoogle Scholar
  152. O. Penrose & P. C. Fife, Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Physica D 43, 44–62, 1990MathSciNetADSzbMATHCrossRefGoogle Scholar
  153. P. Rosakis, Ellipticity and deformations with discontinuous gradients in finite elastostatics, Archive for Rational Mechanics and Analysis 109, 1–37, 1990MathSciNetADSzbMATHCrossRefGoogle Scholar
  154. J. A. Sethian, Recent numerical algorithms for hypersurfaces moving with curvature dependent speed: Hamilton-Jacobi equations and conservation laws, Journal of Differential Geometry 31, 131–161, 1990MathSciNetzbMATHGoogle Scholar
  155. R. Abeyaratne & J. K. Knowles, Kinetic relations and the propagation of phase boundaries in solids, Archive for Rational Mechanics and Analysis 114, 119–154, 1991MathSciNetADSzbMATHCrossRefGoogle Scholar
  156. K. Bhattacharya, Wedge-like microstructure in martensite, Acta Metallurgica 39, 2431–2444, 1991CrossRefGoogle Scholar
  157. Y.-G. Chen, Y. Giga & S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, Journal of Differential Geometry 33, 749–786, 1991MathSciNetzbMATHGoogle Scholar
  158. K. Ecker & G. Huisken, Interior estimates for hypersurfaces moving by mean curvature, Inventiones Mathematicae 105, 547–569, 1991MathSciNetADSzbMATHCrossRefGoogle Scholar
  159. L. C. Evans & J. Spruck, Motion of level sets by mean curvature, Journal of Differential Geometry 33, 635–681, 1991MathSciNetzbMATHGoogle Scholar
  160. 1.
    J. M. Ball & R. D. James, Proposed experimental tests of a theory of fine structure and the two well problem, Philosophical Transactions of the Royal Society of London A 338 389–450, 1992ADSzbMATHCrossRefGoogle Scholar
  161. K. Bhattacharya, Self-accommodation in martensite, Archive for Rational Mechanics and Analysis 120, 201–244, 1992MathSciNetADSzbMATHCrossRefGoogle Scholar
  162. X. Chen & F. Reitich, Local existence and uniqueness of solutions of the Stefan problem with surface tension and kinetic undercooling, Journal of Mathematical Analysis and Applications 164, 350–362, 1992MathSciNetzbMATHCrossRefGoogle Scholar
  163. M. G. Crandall, H. Ishii & P.-L. Lions, User’s guide to viscosity solutions of second order partial differential equations, Bulliten of the American Mathematical Society 27, 1–67, 1992MathSciNetzbMATHCrossRefGoogle Scholar
  164. L. C. Evans, H. M. Soner & P. E. Souganidis, Phase transitions and generalized motion by mean curvature, Communications on Pure and Applied Mathematics 45, 1097–1123, 1992MathSciNetzbMATHCrossRefGoogle Scholar
  165. A. M. Meirmanov, The Stefan Problem, de Gruyter, Berlin, 1992zbMATHCrossRefGoogle Scholar
  166. 1.
    G. Barles, H. M. Soner & P. E. Souganidis, Front propagation and phasefield theory, SI AM Journal on Control and Optimization 31, 439–469, 1993MathSciNetzbMATHCrossRefGoogle Scholar
  167. R. L. Fosdick & Y. Zhang, The torsion problem for a non-convex stored energy function, Archive for Rational Mechanics and Analysis 122, 291–322, 1993MathSciNetADSzbMATHCrossRefGoogle Scholar
  168. E. Fried & M. E. Gurtin, Continuum theory of thermally induced phase transitions based on an order parameter, Physica D 68, 326–343, 1993MathSciNetADSzbMATHCrossRefGoogle Scholar
  169. M. E. Gurtin, The dynamics of solid-solid phase transitions 1. Coherent interfaces, Archive for Rational Mechanics and Analysis 123, 305–335, 1993MathSciNetADSzbMATHCrossRefGoogle Scholar
  170. M. E. Gurtin & P. W. Voorhees, The continuum mechanics of coherent two-phase elastic solids with mass transport, Proceedings of the Royal Society of London A 440, 323–343, 1993MathSciNetADSzbMATHCrossRefGoogle Scholar
  171. E. V. Radkevich, On conditions for the existence of a classical solution of the modified Stefan problem (the Gibbs-Thomson law), Russian Academy of Sciences Sbornik Mathematics 75, 221–246, 1993MathSciNetADSCrossRefGoogle Scholar
  172. H. M. Soner, Motion of a set by the curvature of its boundary, Journal of Differential Equations 101, 313–372, 1993MathSciNetzbMATHCrossRefGoogle Scholar
  173. L. M. Truskinovsky, Kinks versus shocks, in Shock Induced Transitions and Phase Structures in General Media (J. E. Dunn, R. L. Fosdick & M. Slemrod, Eds.), Springer-Verlag, Berlin, 1993Google Scholar
  174. S.-L. Wang, R. F. Sekerka, A. A. Wheeler, B. T. Murray, S. R. Coriell, R. J. Braun & G. B. McFadden, Thermodynamically-consistent phase-field models for solidification, Physica D 69, 189–200, 1993MathSciNetADSzbMATHCrossRefGoogle Scholar
  175. P. Cermelli & M. E. Gurtin, The dynamics of solid-solid phase transitions 2. Incoherent interfaces, Archive for Rational Mechanics and Analysis 127, 41–99, 1994MathSciNetADSzbMATHCrossRefGoogle Scholar
  176. M. E. Gurtin, The characterization of configurational forces, Archive for Rational Mechanics and Analysis 126, 387–394, 1994MathSciNetADSzbMATHCrossRefGoogle Scholar
  177. M. E. Gurtin, The nature of configurational forces, Archive for Rational Mechanics and Analysis, 67–00,1995MathSciNetADSzbMATHCrossRefGoogle Scholar
  178. H. M. Soner, Convergence of the phase-field equations to the Mullins-Sekereka problem with kinetic undercooling, Archive for Rational Mechanics and Analysis 131, 139–197. 1995MathSciNetADSzbMATHCrossRefGoogle Scholar
  179. H. W. Alt & I. Pawlow, On the entropy principle of phase transition models with a conserved order parameter, Advances in Mathematical Sciences and Applications 6, 291–376, 1996MathSciNetzbMATHGoogle Scholar
  180. M. E. Gurtin & P. Podio-Guidugli, Configurational forces and the basic laws for crack propagation, Journal of the Mechanics and Physics of Solids 44, 905–927, 1996MathSciNetADSzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Eliot Fried
    • 1
  1. 1.Department of Theoretical and Applied MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA

Personalised recommendations