For a variational problem of the theory of phase transitions in continuum mechanics with the same moduli of elasticity we obtain explicit formulas for the phase transition temperatures and equilibrium energy. The existence of equilibrium states is studied in some particular cases. Bibliography: 15 titles.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
V. G. Osmolovskii, “Independence of temperatures of phase transitions of the domain occupied by a two-phase elastic medium,” J. Math. Sci., New York 186, No. 2, 302–306 (2012).
V. G. Osmolovskii, “A criterion for the coincidence of the phase transition temperatures in the variational problem on the equilibrium of a two-phase elastic medium,” J. Math. Sci., New York 242, No. 2, 299–307 (2019).
V. G. Osmolovskii, “Boundary value problems with free surface in the theory of phase transitions,” Differ. Equ. 53, No. 13, 1734-1763 (2017).
V. G. Osmolovskii, “Exact solutions to the variational problem of the phase transition theory in continuum mechanics,” J. Math. Sci., New York 120, No. 2, 1167–1190 (2004).
V. G. Osmolovskii, “A variational problem of phase transitions for a two-phase elastic medium with zero coefficient of surface tension,” St. Petersbg. Math. J. 22, No. 6, 1007–1022 (2011).
V. G. Osmolovskii, “The volume fraction of one of the phases in equilibrium two-phase elastic medium,” J. Math. Sci., New York 236, No. 4, 419–429 (2019).
G. Allaire and R. V. Kohn, “Optimal bounds on the effective behavior of a mixture of two well-ordered elastic materials,” Q. Appl. Math. 51, No. 4, 643–674 (1993).
I. M. Glazman and Yu. I. Lyubich, Finite-Dimensional Linear Analysis, The M.I.T. Press, London (1974).
V. G. Osmolovskii, “Sufficient conditions for absence of two-phase equilibrium states of elastic media with different phase transition temperatures,” J. Math. Sci., New York 244, No. 3, 497–508 (2020).
B. Dacorogna, Direct Methods in the Calculus of Variations, Springer, Berlin (2008).
L. C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, Am. Math. Soc., Providence RI (1990).
G. Allaire and R. V. Kohn, “Optimal lower bounds on the elastic energy of a composite made from two non-well-ordered isotropic materials,” Q. Appl. Math. 52, No. 2, 311–333 (1994).
G. Allaire and V. Lods, “Minimizers for a double-well problem with affine boundary conditions,” Proc. R. Soc. Edinb., Sect. A, Math. 129, No. 3, 439–446 (1999).
G. Allaire and G. Francfort, “Existence of minimizers for non-quasiconvex functionals arising in optimal design,” Ann. Inst. Henry Poincaré, Anal. Non Linéaire 15, No. 3, 301–339 (1998).
Y. Grabovsky, “Bounds and extremal microstructures for two-component composites: A unified treatment based on the translation method,” Proc. R. Soc. Lond., Ser. A 452, 919–944 (1996).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Problemy Matematicheskogo Analiza 106, 2020, pp. 139-147.
Rights and permissions
About this article
Cite this article
Osmolovskii, V.G. Phase Transitions in Two-Phase Media with the Same Moduli of Elasticity. J Math Sci 251, 713–723 (2020). https://doi.org/10.1007/s10958-020-05124-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-020-05124-1