Skip to main content
SpringerLink
Log in

A Phase-Field Approach to Eulerian Interfacial Energies

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

We analyze a phase-field approximation of a sharp-interface model for two-phase materials proposed by Šilhavý (in: Hackl (ed) IUTAM symposium on variational concepts with applications to the mechanics of materials, pp 233–244, Springer, Dordrecht, 2010; J Elast 105:271–303, 2011). The distinguishing trait of the model resides in the fact that the interfacial term is Eulerian in nature, for it is defined on the deformed configuration. We discuss a functional frame allowing for the existence of phase-field minimizers and \(\Gamma \)-convergence to the sharp-interface limit. As a by-product, we provide additional detail on the admissible sharp-interface configurations with respect to the analysis in Šilhavý (2010, 2011).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ambrosio, L., Crippa, G., Maniglia, S.: Traces and fine properties of a BD class of vector fields and applications. Ann. Fac. Sci. Toulouse Math. (6) 14, 527–561, 2005

    Article  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

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Ball, J.M.: Global invertibility of Sobolev functions and the interpenetration of matter. Proc. R. Soc. Edinb. Ser. A 88, 315–328, 1981

    Article  MathSciNet  MATH  Google Scholar 

  5. Barchiesi, M., DeSimone, A.: Frank energy for nematic elastomers: a nonlinear model. ESAIM Control Optim. Calc. Var. 21, 372–377, 2015

    Article  MathSciNet  MATH  Google Scholar 

  6. Barchiesi, M., Henao, D., Mora-Corral, C.: Local invertibility in Sobolev spaces with applications to nematic elastomers and magnetoelasticity. Arch. Ration. Mech. Anal. 224, 743–816, 2017

    Article  MathSciNet  MATH  Google Scholar 

  7. Bielski, W., Gambin, B.: Relationship between existence of energy minimizers of incompressible and nearly incompressible magnetostrictive materials. Rep. Math. Phys. 66, 147–157, 2010

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Bojarski, B., Iwaniec, T.: Analytical foundations of the theory of quasiconformal mappings in \({\mathbb{R}}^n\). Ann. Acad. Sci. Fenn. Ser. A I. Math. 8, 257–324, 1983

    Article  MathSciNet  MATH  Google Scholar 

  9. Braides, A.: \(\Gamma \) -Convergence for Beginners, vol. 22. Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press, Oxford, 2002

  10. Chen, G.Q., Torres, M., Ziemer, W.P.: Gauss–Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws. Commun. Pure Appl. Math. 62, 242–304, 2009

    Article  MathSciNet  MATH  Google Scholar 

  11. Ciarlet, P.G.: Mathematical Elasticity, vol. I. Three-Dimensional Elasticity. North-Holland, Amsterdam, 1988

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Csörnyei, M., Hencl, S., Malý, J.: Homeomorphisms in the Sobolev space \(W^{1, n- 1}\). J. Reine Angew. Math. 644, 221–235, 2010

    MathSciNet  MATH  Google Scholar 

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

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

  16. DeSimone, A., James, R.D.: A constrained theory of magnetoelasticity. J. Mech. Phys. Solids 50, 283–320, 2002

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Fischer, F.D., Waitz, T., Vollath, D., Simha, N.K.: On the role of surface energy and surface stress in phase-transforming nanoparticles. Progress Mater. Sci. 53, 481–527, 2008

    Article  Google Scholar 

  18. Fitzpatrick, R.: An Introduction to Celestial Mechanics. Cambridge University Press, Cambridge, 2012

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

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Hencl, S., Koskela, P.: Regularity of the inverse of a planar Sobolev homeomorphism. Arch. Ration. Mech. Anal. 180(2006), 75–95, 2006

    Article  MathSciNet  MATH  Google Scholar 

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

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

    Article  MathSciNet  MATH  Google Scholar 

  25. Izzo, A.: Existence of continuous functions that are one-to-one almost everywhere. Math. Scand. 118, 269–276, 2016

    Article  MathSciNet  MATH  Google Scholar 

  26. 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

    Article  ADS  Google Scholar 

  27. Kružík, M., Stefanelli, U., Zeman, J.: Existence results for incompressible magnetoelasticity. Discrete Contin. Dyn. Syst. 35, 2615–2623, 2015

    Article  MathSciNet  Google Scholar 

  28. 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

    Article  ADS  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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. 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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  33. Nix, W.D., Gao, H.J.: An atomistic interpretation of interface stress. Scr. Mater. 39, 1653–1661, 1998

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  35. Reshetnyak, Y.G.: Some geometrical properties of functions and mappings with generalized derivatives. Sibirsk. Math. Zh. 7, 886–919, 1966

    MATH  Google Scholar 

  36. Richter, T.: Fluid–Structure Interactions. Models, Analysis and Finite Elements, vol. 118. Lecture Notes in Computational Science and Engineering. Springer, Cham, 2017

  37. Rosato, D., Miehe, C.: Dissipative ferroelectricity at finite strains. Variational principles, constitutive assumptions and algorithms. Int. J. Eng. Sci. 74, 162–189, 2014

    Article  MathSciNet  MATH  Google Scholar 

  38. Rybka, P., Luskin, M.: Existence of energy minimizers for magnetostrictive materials. SIAM J. Math. Anal. 36, 2004–2019, 2005

    Article  MathSciNet  MATH  Google Scholar 

  39. Šilhavý, M.: The Mechanics and Thermodynamics of Continuous Media. Texts and Monographs in Physics. Springer, Berlin, 1997

  40. Šilhavý, M.: Divergence measure fields and Cauchy’s stress theorem. Rend. Sem. Mat. Univ. Padova 113, 15–45, 2005

    MathSciNet  MATH  Google Scholar 

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

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

    Article  MathSciNet  MATH  Google Scholar 

  43. Stefanelli, U.: Existence for dislocation-free finite plasticity. ESAIM Control Optim. Calc. Var., 2018. https://doi.org/10.1051/cocv/2018014

Download references

Acknowledgements

This research is supported by the FWF-GAČR Project I 2375-16-34894L and by the OeAD-MŠMT Project CZ 17/2016-7AMB16AT015. M.K. was further supported by the GAČR Project 18-03834S. U.S. acknowledges the support by the Vienna Science and Technology Fund (WWTF) through Project MA14-009 and by the Austrian Science Fund (FWF) Projects F 65 and P 27052. The authors are indebted to L. Ambrosio and M. Šilhavý for inspiring conversations.

Author information

Authors and Affiliations

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

    Diego Grandi

  2. Czech Academy of Sciences, Institute of Information Theory and Automation, 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, Genoa, Italy

    Edoardo Mainini

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

    Ulisse Stefanelli

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

    Ulisse Stefanelli

Authors
  1. Diego Grandi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martin Kružík
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Edoardo Mainini
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ulisse Stefanelli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ulisse Stefanelli.

Additional information

Communicated by I. Fonseca

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Grandi, D., Kružík, M., Mainini, E. et al. A Phase-Field Approach to Eulerian Interfacial Energies. Arch Rational Mech Anal 234, 351–373 (2019). https://doi.org/10.1007/s00205-019-01391-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-019-01391-8

Search

Navigation