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Equilibrium for Multiphase Solids with Eulerian Interfaces

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Abstract

We describe a general phase-field model for hyperelastic multiphase materials. The model features an elastic energy functional that depends on the phase-field variable and a surface energy term that depends in turn on the elastic deformation, as it measures interfaces in the deformed configuration. We prove existence of energy minimizing equilibrium states and \(\Gamma \)-convergence of diffuse-interface approximations to the sharp-interface limit.

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References

  1. Ambrosio, L., Braides, A.: Functionals defined on partitions in sets of finite perimeter II: semicontinuity, relaxation and homogenization. J. Math. Pures Appl. 69, 307–333 (1990)

    MathSciNet  MATH  Google Scholar 

  2. Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. Oxford University Press, Oxford (2000)

    MATH  Google Scholar 

  3. Baldo, S.: Minimal interface criterion for phase transitions in mixtures of Cahn-Hilliard fluids. Ann. Inst. Henri Poincaré C 7, 67–90 (1990)

    MathSciNet  MATH  Google Scholar 

  4. Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63, 337–403 (1977)

    MathSciNet  MATH  Google Scholar 

  5. Ball, J.M., Crooks, E.C.M.: Local minimizers and planar interfaces in a phase-transition model with interfacial energy. Calc. Var. Partial Differ. Equ. 40, 501–538 (2011)

    MathSciNet  MATH  Google Scholar 

  6. Ball, J.M., James, R.D.: Fine phase mixtures as minimizers of energy. Arch. Ration. Mech. Anal. 100, 13–52 (1988)

    MathSciNet  MATH  Google Scholar 

  7. Ball, J.M., Mora-Corral, C.: A variational model allowing both smooth and sharp phase boundaries in solids. Commun. Pure Appl. Anal. 8, 55–81 (2009)

    MathSciNet  MATH  Google Scholar 

  8. Ball, J.M., Currie, J.C., Olver, P.L.: Null Lagrangians, weak continuity, and variational problems of arbitrary order. J. Funct. Anal. 41, 135–174 (1981)

    MathSciNet  MATH  Google Scholar 

  9. Bhattacharya, K.: Microstructure of Martensite. Why It Forms and How It Gives Rise to the Shape-Memory Effect. Oxford University Press, New York (2003)

    MATH  Google Scholar 

  10. Bhattacharya, K., James, R.D.: A theory of thin films of martensitic materials with applications to microactuators. J. Mech. Phys. Solids 47, 531–576 (1999)

    ADS  MathSciNet  MATH  Google Scholar 

  11. Bogachev, V.I.: Gaussian Measures. Mathematical Surveys and Monographs, vol. 62. Am. Math. Soc., Providence (1998)

    MATH  Google Scholar 

  12. Ciarlet, P.G., Nečas, J.: Injectivity and self-contact in nonlinear elasticity. Arch. Ration. Mech. Anal. 97, 171–188 (1987)

    MathSciNet  MATH  Google Scholar 

  13. Dacorogna, B.: Direct Methods in the Calculus of Variations, 2nd. edn. Springer, Berlin (2008)

    MATH  Google Scholar 

  14. Dal Maso, G.: An Introduction to \(\Gamma \)-Convergence. Progress in Nonlinear Differential Equations and Their Applications, vol. 8. Birkhäuser, Boston (1993)

    MATH  Google Scholar 

  15. Fonseca, I.: Interfacial energy and the Maxwell rule. Arch. Ration. Mech. Anal. 106, 63–95 (1989)

    MathSciNet  MATH  Google Scholar 

  16. Fonseca, I., Leoni, G.: Modern Methods in the Calculus of Variations: \(L^{p}\)-Spaces. Springer, New York (2007)

    MATH  Google Scholar 

  17. Freddi, L., Paroni, R.: The energy density of martensitic thin films via dimension reduction. Interfaces Free Bound. 6, 439–459 (2004)

    MathSciNet  MATH  Google Scholar 

  18. Fusco, N., Moscariello, G., Sbordone, C.: The limit of \(W^{ 1,1}\) homeomorphisms with finite distortion. Calc. Var. Partial Differ. Equ. 33, 377–390 (2008)

    MATH  Google Scholar 

  19. Gehring, F., Iwaniec, T.: The limit of mappings with finite distortion. Ann. Acad. Sci. Fenn., Ser. A 1 Math. 24, 253–264 (1999)

    MathSciNet  MATH  Google Scholar 

  20. Giacomini, A., Ponsiglione, M.: Non-interpenetration of matter for SBV deformations of hyperelastic brittle materials. Proc. R. Soc. Edinb. A 138, 1019–1041 (2008)

    MathSciNet  MATH  Google Scholar 

  21. Grandi, G., Kružík, M., Mainini, E., Stefanelli, U.: A phase-field approach to Eulerian interfacial energies. Arch. Ration. Mech. Anal. 234, 351–373 (2019)

    MathSciNet  MATH  Google Scholar 

  22. Gurtin, M.E.: On phase transitions with bulk, interfacial, and boundary energy. Arch. Ration. Mech. Anal. 96, 243–264 (1986)

    MathSciNet  Google Scholar 

  23. Gurtin, M.E., Murdoch, A.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323 (1975)

    MathSciNet  MATH  Google Scholar 

  24. Heinonen, J., Koskela, P.: Sobolev mappings with integrable dilatations. Arch. Ration. Mech. Anal. 125, 81–97 (1993)

    MathSciNet  MATH  Google Scholar 

  25. Hencl, S., Koskela, P.: Lectures on Mappings of Finite Distortion. Lecture Notes in Mathematics, vol. 2096. Springer, Berlin (2014)

    MATH  Google Scholar 

  26. Hencl, S., Koskela, P., Malý, J.: Regularity of the inverse of a Sobolev homeomorphism in space. Proc. R. Soc. Edinb. A 136A, 1267–1285 (2006)

    MathSciNet  MATH  Google Scholar 

  27. Javili, A., McBride, A., Steinmann, P.: Thermomechanics of solids with lower-dimensional energetics: on the importance of surface, interface, and curve structures at the nanoscale. A unifying review. Appl. Mech. Rev. 65, 010802 (2013). 31 pp.

    ADS  Google Scholar 

  28. Kohn, R.V., Müller, S.: Surface energy and microstructure in coherent phase transitions. Commun. Pure Appl. Math. 47(4), 405–435 (1994)

    MathSciNet  MATH  Google Scholar 

  29. Levitas, V.I.: Phase field approach to martensitic phase transformations with large strains and interface stresses. J. Mech. Phys. Solids 70, 154–189 (2014)

    ADS  MathSciNet  MATH  Google Scholar 

  30. Levitas, V.I., Javanbakht, M.: Surface tension and energy in multivariant martensitic transformations: phase-field theory, simulations, and model of coherent interface. Phys. Rev. Lett. 105, 165701 (2010)

    ADS  Google Scholar 

  31. Levitas, V.I., Warren, J.A.: Phase field approach with anisotropic interface energy and interface stresses: large strain formulation. J. Mech. Phys. Solids 91, 94–125 (2016)

    ADS  MathSciNet  Google Scholar 

  32. Modica, L.: The gradient theory of phase transitions and the minimal interface criterion. Arch. Ration. Mech. Anal. 98, 123–142 (1987)

    MathSciNet  MATH  Google Scholar 

  33. Modica, L., Mortola, S.: Un esempio di \(\Gamma \)-convergenza. Boll. Unione Mat. Ital., B 14, 285–299 (1977). Italian

    MathSciNet  MATH  Google Scholar 

  34. Müller, S.: Higher integrability of determinants and weak convergence in \(L^{1}\). J. Reine Angew. Math. 412, 20–34 (1990)

    MathSciNet  MATH  Google Scholar 

  35. Müller, S.: Variational models for microstructure and phase transitions. In: Hildebrandt, S., Struwe, M. (eds.) Calculus of Variations and Geometric Evolution Problems, pp. 85–210. Springer, Berlin (1999)

    Google Scholar 

  36. Onninen, J., Tengvall, V.: Mappings of \(L^{p}\)-integrable distortion: regularity of the inverse. Proc. R. Soc. Edinb. A 146, 647–663 (2016)

    MATH  Google Scholar 

  37. Parry, G.P.: On shear bands in unloaded crystals. J. Mech. Phys. Solids 35, 367–382 (1987)

    ADS  MathSciNet  MATH  Google Scholar 

  38. Reshetnyak, Y.G.: Some geometrical properties of functions and mappings with generalized derivatives. Sib. Math. Zh. 7, 886–919 (1966)

    MATH  Google Scholar 

  39. Šilhavý, M.: 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 (2010)

    Google Scholar 

  40. Šilhavý, M.: Phase transitions with interfacial energy: interface null Lagrangians, polyconvexity, and existence. In: Hackl, K. (ed.) IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials, pp. 233–244. Springer, Dordrecht (2010)

    Google Scholar 

  41. Šilhavý, M.: Equilibrium of phases with interfacial energy: a variational approach. J. Elast. 105, 271–303 (2011)

    MathSciNet  MATH  Google Scholar 

  42. Sternberg, P.: The effect of a singular perturbation on nonconvex variational problems. Arch. Ration. Mech. Anal. 101, 209–260 (1988)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

This research of M.K. was supported by the FWF-GAČR project 19-29646L and by the OeAD-MŠMT project 8J19AT013. E.M. acknowledges support from the MIUR-PRIN project No 2017TEXA3H. U.S. is supported by Austrian Science Fund (FWF) projects F 65, W 1245, I 4354, and P 32788 and by the Vienna Science and Technology Fund (WWTF) project MA14-009.

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università degli Studi di Ferrara, Via Machiavelli 30, 44121, Ferrara, Italy

    Diego Grandi

  2. Institute of Information Theory and Automation, Czech Academy of Sciences, Pod vodárenskou veží 4, 182 08, Prague 8, Czech Republic

    Martin Kružík

  3. Faculty of Civil Engineering, Czech Technical University, Thákurova 7, 166 29, Prague 6, Czech Republic

    Martin Kružík

  4. Dipartimento di Ingegneria meccanica, energetica, gestionale e dei trasporti, Università degli studi di Genova, Via all’Opera Pia, 15, 16145, Genova, Italy

    Edoardo Mainini

  5. Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria

    Ulisse Stefanelli

  6. Vienna Research Platform on Accelerating Photoreaction Discovery, University of Vienna, Währingerstraße 17, 1090, Wien, Austria

    Ulisse Stefanelli

  7. Istituto di Matematica Applicata e Tecnologie Informatiche E. Magenes, via Ferrata 1, 27100, Pavia, Italy

    Ulisse Stefanelli

Authors
  1. Diego Grandi
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  2. Martin Kružík
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  3. Edoardo Mainini
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  4. Ulisse Stefanelli
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Correspondence to Ulisse Stefanelli.

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Grandi, D., Kružík, M., Mainini, E. et al. Equilibrium for Multiphase Solids with Eulerian Interfaces. J Elast 142, 409–431 (2020). https://doi.org/10.1007/s10659-020-09800-w

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  • DOI: https://doi.org/10.1007/s10659-020-09800-w

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