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Temperatures of Phase Transitions and Quasiconvex Hull Of Energy Functionals for a Two-Phase Elastic Medium with Anisotropic Residual Strain Tensor

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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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References

  1. L. D. Landau and E. M. Lifshits, Elasticity Theory [in Russian], Nauka, Moscow (1965).

    Google Scholar 

  2. M. A. Grinfel’d, Methods of Continuum Mechanics in the Theory of Phase Transitions [in Russian], Nauka, Moscow (1990).

    Google Scholar 

  3. 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).

    Article  MathSciNet  Google Scholar 

  4. 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).

    Article  MATH  MathSciNet  Google Scholar 

  5. 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).

    Article  MATH  MathSciNet  Google Scholar 

  6. B. Dacorogna, Direct Methods in the Calculus of Variations, Springer, Berlin (2008).

    MATH  Google Scholar 

  7. E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, Singapore (2003).

    Book  MATH  Google Scholar 

  8. L. C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, Am. Math. Soc., Providence RI (1990).

    MATH  Google Scholar 

  9. L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, New York etc. (1992).

    MATH  Google Scholar 

  10. E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Boston (1984).

    Book  MATH  Google Scholar 

  11. 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).

    Article  MathSciNet  Google Scholar 

  12. 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).

    Article  MATH  MathSciNet  Google Scholar 

  13. 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).

    Article  MATH  MathSciNet  Google Scholar 

  14. 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).

    Article  MATH  MathSciNet  Google Scholar 

  15. 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).

    Article  MathSciNet  Google Scholar 

  16. 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).

  17. 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).

    Article  MATH  MathSciNet  Google Scholar 

  18. 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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  1. St. Petersburg State University, 28, Universitetskii pr., Petrodvorets, St. Petersburg, 198504, Russia

    V. G. Osmolovskii

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  1. V. G. Osmolovskii
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Correspondence to V. G. Osmolovskii.

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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

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