Abstract
The article presents a variational theory of sharp phase interfaces bearing a deformation dependent energy. The theory involves both the standard and Eshelby stresses. The constitutive theory is outlined including the symmetry considerations and some particular cases. The existence of phase equilibria is proved based on appropriate convexity properties of the interfacial energy. Some generalization of the convexity properties is given and a relationship established to the semiellipticity condition from the theory of parametric integrals over rectifiable currents.
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Bibliography
F. J. Almgren, Jr. Existence and Regularity Almost Everywhere of Solutions to Elliptic Variational Problems among Surfaces of Varying Topological Type and Singularity Structure. Annals of Mathematics, 87:321–391, 1968.
L. Ambrosio, N. Fusco, and D. Pallara. Functions of bounded variation and free discontinuity problems. Oxford, Clarendon Press 2000.
J. M. Ball. Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal., 63:337–403, 1977.
J. M. Ball, and R. D. James. Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal., 100:13–52, 1987.
J. M. Ball, and F. Murat. W 1,p-quasiconvexity and variational problems for multiple integrals. J. Fund. Anal., 58:225–253, 1984.
B. Dacorogna. Direct methods in the calculus of variations. Second Edition. Berlin, Springer, 2008.
J. D. Eshelby. The force on an elastic singularity. Phil. Trans. Royal Soc. London, A244:87–112, 1951.
J. D. Eshelby. Continuum theory of defects. In D. Turnbull editor, Solid State Physics, vol. 3 pp. 79–144: Academic Press, New York, 1956.
H. Fédérer. Geometric measure theory. New York, Springer, 1969.
I. Fonseca. Interfacial energy and the Maxwell rule. Arch. Rational Mech. Anal., 106:63–95,1989.
M. Giaquinta, G. Modica, and J. Souček. Cartesian currents, weak diffeomorphisms and existence theorems in non-linear elasticity. Arch. Rational Mech. Anal., 106:97–160,1989.
M. Giaquinta, G. Modica, and J. Souček. Cartesian Currents in the Calculus of Variations I, II. Berlin, Springer, 1998.
M. E. Gurtin. The nature of configurational forces. Arch. Rational Mech. Anal., 131:67–100, 1995.
M. E. Gurtin. Configurational forces as basic concepts of continuum physics. New York, Springer, 2000.
M. E. Gurtin, and I. Murdoch. A continuum theory of elastic material surfaces. Arch. Rational Mech. Anal., 57:291–323, 1975.
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, 1990.
P. H. Leo, and R. F. Sekerka. The effect of surface stress on crystal-melt and crystal-crystal equilibrium. Acta Metall., 37:3119–3138, 1989.
J. E. Marsden, and T. J. R. Hughes. Mathematical foundations of elasticity. Englewood Cliffs, Prentice-Hall, 1983.
C. B. Morrey, Jr. Quasi-convexity and the lower semicontinuity of multiple integrals. Pacific J. Math., 2:25–53, 1952.
C. B. Morrey, Jr. Multiple integrals in the calculus of variations. New York, Springer, 1966.
S. Müller. Variational Models for Microstructure and Phase Transitions. In S. Hildebrandt, M. Struwe editor, Calculus of variations and geometric evolution problems (Cetraro, 1996) Lecture Notes in Math. 1713 pp. 85–210, Springer, Berlin, 1999.
S. Müller, Q. Tang, and B. S. Yan. On a new class of elastic deformations not allowing for cavitation. Ann. Inst. H. Poincaré, Analyse non linéaire, 11:217–243, 1994.
G. P. Parry. On shear bands in unloaded crystals. J. Mech. Phys. Solids, 35:367–382, 1987.
M. Pitteri, and G. Zanzotto. Continuum models for phase transitions and twinning in crystals. Boca Raton, Chapman & Hall/CRC, 2003.
P. Podio-Guidugli. Configurational balances via variational arguments. Interfaces and Free Boundaries, 3:223–232, 2001.
P. Podio-Guidugli. Configurational forces: are they needed? Mechanics Research Communications, 29:513–519, 2002.
J. Schröder, P. Neff, and V. Ebbing. Anisotropic polyconvex energies on the basis of crystallographic motivated structural tensors. Journal of the Mechanics and Physics of Solids, 56:3486–3506, 2008
M. Ŝilhavý. Phase transitions with inter facial energy: a variational approach. Preprint, Institute of Mathematics, AS CR, Prague. 2008-10-22
P. Steinmann. On boundary potential energies in deformational and configurational mechanics. Journal of the Mechanics and Physics of Solids, 56:772–800, 2008.
H. Whitney. Geometric integration theory. Princeton, Princeton University Press, 1957.
G. Wulff. Zur Frage der Geschwindigkeit des Wachstums and der Auflösung der Kristallflächen. Zeitschrift für Kristallographie, 34:449–530, 1901.
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Šilhavý, M. (2010). Phase transitions with interfacial energy: convexity conditions and the existence of minimizers. In: Schröder, J., Neff, P. (eds) Poly-, Quasi- and Rank-One Convexity in Applied Mechanics. CISM International Centre for Mechanical Sciences, vol 516. Springer, Vienna. https://doi.org/10.1007/978-3-7091-0174-2_6
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DOI: https://doi.org/10.1007/978-3-7091-0174-2_6
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