For a two-phase elastic medium with anisotropic residual strain tensors we compute the phase transition temperatures t±. We find an explicit expression for the quasiconvex hull of strain energy densities and obtain all solutions to the relaxed variational problem and limit points of equilibrium states as the surface tension coefficient tends to zero. We show that there are no equilibrium states for the initial energy functional if t ∈ (t−, t+). Bibliography: 18 titles.
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L. D. Landau and E. M. Lifshits, Elasticity Theory [in Russian], Nauka, Moscow (1965).
M. A. Grinfel’d, Methods of Continuum Mechanics in the Theory of Phase Transitions [in Russian], Nauka, Moscow (1990).
V. G. Osmolovskii, “A variational problem of phase transitions for a two-phase elastic medium with zero coefficient of surface tension” [in Russian], Algebra Anal. 22, No. 6, 214–234 (2010); English transl.: St. Petersbg. Math. J. 22, No. 6, 1007–1022 (2011).
V. G. Osmolovskii, “On the phase transition temperature in a variational problem of elasticity theory for two-phase media” [in Russian], Probl. Mat. Anal. 41, 37–47 (2009); English transl.: J. Math. Sci., New York 159, No. 2, 168–179 (2009).
V. G. Osmolovskii, “Independence of temperatures of phase transitions on the domain occupied by a two-phase elastic medium” [in Russian], Probl. Mat. Anal. 66, 147–151 (2012); English transl.: J. Math. Sci., New York 186, No. 2, 302–306 (2012).
B. Dacorogna, Direct Methods in the Calculus of Variations, Springer, Berlin (2008).
E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, Singapore (2003).
L. C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, Am. Math. Soc., Providence RI (1990).
L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, New York etc. (1992).
E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Boston (1984).
V. G. Osmolovskii, “An existence theorem and weak Lagrange equations for a variational problem of the theory of phase transitions” [in Russian], Sib. Mat. Zh. 35, No. 4, 835–846 (1994); English transl.: Sib. Math. J. 35, No. 4, 743–753 (1994).
V. G. Osmolovskii, “The behavior of the area of the boundary of phase interface in the problem about phase transitions as the surface tension coefficient tends to zero” [in Russian], Probl. Mat. Anal. 62, 101–108 (2012); English transl.: J. Math. Sci., New York 181, No. 2, 223–231 (2012).
V. G. Osmolovskii, “Quasiconvex hull of energy densities in a homogeneous isotropic two-phase elastic medium and solutions of the original relaxed problems” [in Russian], Probl. Mat. Anal. 70, 161–170 (2013); English transl.: J. Math. Sci., New York 191, No. 2, 280–290 (2013).
V. G. Osmolovskii, “Description of the set of all solutions to the relaxed problem for a homogeneous isotropic two-phase elastic medium” [in Russian], Probl. Mat. Anal. 72, 147–155 (2013); English transl.: J. Math. Sci., New York 195, No. 5, 730–740 (2013).
V. G. Osmolovskii, “Exact solutions to the variational problem of the phase transition theory in continuum mechanics” [in Russian], Probl. Mat. Anal. 27, 171–205 (2004). English transl.: J. Math. Sci., New York 120, No. 2, 1167–1190 (2004).
V. G. Osmolovskii, “Existence of equilibrium states in the one-dimensional phase transition problem” [in Russian], Vest. S. Peterburg. State Univ. Ser. 1, No. 3, 54–65 (2006).
B. Dacorogna, G. Pisante, and A. M. Ribeiro, “On non quasiconvex problems of the calculus of variations,” Discr. Continuous Syst. 13, No. 4, 961–983 (2005).
V. G. Osmolovskii, Mathematical Questions of the Theory of Phase Transitions in Continuum Mechanics Preprint No 2014-04, http://www.mathsoc.spb.ru/preprint/2014/index.htm1#04.
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Translated from Problemy Matematicheskogo Analiza 77, December 2014, pp. 119-128.
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Osmolovskii, V.G. Temperatures of Phase Transitions and Quasiconvex Hull Of Energy Functionals for a Two-Phase Elastic Medium with Anisotropic Residual Strain Tensor. J Math Sci 205, 255–266 (2015). https://doi.org/10.1007/s10958-015-2246-3
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DOI: https://doi.org/10.1007/s10958-015-2246-3