Fine phase mixtures as minimizers of energy

Solid-solid phase transformations often lead to certain characteristic microstructural features involving fine mixtures of the phases. In martensitic transformations one such feature is a plane interface which separates one homogeneous phase, austenite, from a very fine mixture of twins of the other phase, martensite. In quartz crystals held in a temperature gradient near the α-β transformation temperature, the α-phase breaks up into triangular domains called Dauphine twins which become finer and finer in the direction of increasing temperature. In this paper we explore a theoretical approach to these fine phase mixtures based on the minimization of free energy.

[1]  J. Gibbs.  On the equilibrium of heterogeneous substances , 1878, American Journal of Science and Arts.

[2]  J. W. Gibbs,et al.  Scientific Papers , 1997, Nature.

[3]  D. A. G. Bruggeman.  Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen , 1935 .

[4]  H. Weyl.  The Classical Groups , 1939 .

[5]  H. Weyl.  The Classical Groups , 1940 .

[6]  J. Mackenzie,et al.  The crystallography of martensite transformations II , 1954 .

[7]  J. Christian,et al.  Crystallography of deformation by twin boundary movements in indium-thallium alloys , 1954 .

[8]  J. Christian,et al.  Experiments on the martensitic transformation in single crystals of indium-thallium alloys , 1954 .

[9]  S. Agmon,et al.  Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I , 1959 .

[10]  C. M. Wayman,et al.  Introduction to the crystallography of martensitic transformations , 1964 .

[11]  J. Christian,et al.  The theory of transformations in metals and alloys , 2003 .

[12]  Yu. G. Reshetnyak.  On the stability of conformal mappings in multidimensional spaces , 1967 .

[13]  Yu. G. Reshetnyak.  Liouville's theorem on conformal mappings for minimal regularity assumptions , 1967 .

[14]  K. Shimizu,et al.  Morphology and Crystallography of Thermoelastic γ' Cu-Al-Ni Martensite , 1969 .

[15]  V. Mizel,et al.  Absolute continuity on tracks and mappings of Sobolev spaces , 1972 .

[16]  J. Ball.  Convexity conditions and existence theorems in nonlinear elasticity , 1976 .

[17]  G. Tendeloo,et al.  The α β phase transition in quartz and AlPO4 as studied by electron microscopy and diffraction , 1976 .

[18]  W. Noll.  A General Framework for Problems in the- Statics of Finite Elasticity , 1978 .

[19]  C. Truesdell.  Some Challenges Offered to Analysis by Rational Thermomechanics: Three Lectures for the International Symposium on Continuum Mechanics and Partial Differential equations , 1978 .

[20]  G. M. D. Penha.  Contemporary developments in continuum mechanics and partial differential equations , 1978 .

[21]  L. Young.  Lectures on the Calculus of Variations and Optimal Control Theory , 1980 .

[22]  John M. Ball,et al.  Strict convexity, strong ellipticity, and regularity in the calculus of variations , 1980, Mathematical Proceedings of the Cambridge Philosophical Society.

[23]  J. Ericksen,et al.  Some phase transitions in crystals , 1980 .

[24]  J. C. Currie,et al.  Weak continuity and variational problems of arbitrary order , 1981 .

[25]  Richard D. James,et al.  Finite deformation by mechanical twinning , 1981 .

[26]  J. Ball,et al.  Discontinuous equilibrium solutions and cavitation in nonlinear elasticity , 1982, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[27]  Andrej Cherkaev,et al.  Regularization of optimal design problems for bars and plates, part 1 , 1982 .

[28]  Andrej Cherkaev,et al.  Regularization of optimal design problems for bars and plates, part 2 , 1982 .

[29]  Luc Tartar,et al.  The Compensated Compactness Method Applied to Systems of Conservation Laws , 1983 .

[30]  Characteristics of stress-strain behaviour associated with thermoelastic martensitic transformation , 1983 .

[31]  Robert V. Kohn,et al.  Explicit relaxation of a variational problem in optimal design , 1983 .

[32]  Morton E. Gurtin,et al.  Two-phase deformations of elastic solids , 1983 .

[33]  Jerrold E. Marsden,et al.  Quasiconvexity at the boundary, positivity of the second variation and elastic stability , 1984 .

[34]  J. Ball,et al.  W1,p-quasiconvexity and variational problems for multiple integrals , 1984 .

[35]  L. Tartar,et al.  Etude des oscilaltions dans les equations aux derivees partielles non lineaires , 1984 .

[36]  Andrej Cherkaev,et al.  On the existence of solutions to some problems of optimal design for bars and plates , 1984 .

[37]  H. Simpson,et al.  On failure of the complementing condition and nonuniqueness in linear elastostatics , 1985 .

[38]  J. Ericksen.  Some surface defects in unstressed thermoelastic solids , 1985 .

[39]  Some Examples of Crinkles , 1986 .

[40]  D. Kinderlehrer,et al.  Homogenization and effective moduli of materials and media , 1986 .

[41]  Graeme W. Milton,et al.  Modelling the Properties of Composites by Laminates , 1986 .

[42]  John M. Ball,et al.  Lyapunov functions for thermomechanics with spatially varying boundary temperatures , 1986 .

[43]  R. James.  Displacive phase transformations in solids , 1986 .

[44]  J. Ericksen,et al.  Constitutive theory for some constrained elastic crystals , 1986 .

[45]  Ingo Müller,et al.  Metastability and incompletely posed problems , 1987 .

[46]  H. Simpson,et al.  On the positivity of the second variation in finite elasticity , 1987 .

[47]  Irene Fonseca,et al.  Variational methods for elastic crystals , 1987 .

[48]  R. James.  The Stability and Metastability of Quartz , 1987 .

[49]  D. Kinderlehrer,et al.  Remarks about Equilibrium Configurations of Crystals , 1987 .

[50]  David Kinderlehrer,et al.  Equilibrium configurations of crystals , 1988 .