Skip to main content
SpringerLink
Log in

Existence of Solutions for a Mathematical Model Related to Solid–Solid Phase Transitions in Shape Memory Alloys

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

Abstract

We consider a strongly nonlinear PDE system describing solid–solid phase transitions in shape memory alloys. The system accounts for the evolution of an order parameter χ (related to different symmetries of the crystal lattice in the phase configurations), of the stress (and the displacement u), and of the absolute temperature ϑ. The resulting equations present several technical difficulties to be tackled; in particular, we emphasize the presence of nonlinear coupling terms, higher order dissipative contributions, possibly multivalued operators. As for the evolution of temperature, a highly nonlinear parabolic equation has to be solved for a right hand side that is controlled only in L 1. We prove the existence of a solution for a regularized version by use of a time discretization technique. Then, we perform suitable a priori estimates which allow us pass to the limit and find a weak global-in-time solution to the system.

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. Auricchio F., Petrini L.: A three-dimensional model describing stress-temperature induced solid phase transformations: solution algorithm and boundary value problems. Int. J. Numer. Methods Eng. 61, 807–836 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  2. Auricchio F., Bonetti E.: A new “flexible” 3D macroscopic model for shape memory alloys. Discret. Cont. Dyn. Syst. Ser. S 6, 277–291 (2013)

    MATH  MathSciNet  Google Scholar 

  3. Auricchio F., Mielke A., Stefanelli U.: A rate-independent model for the isothermal quasi-static evolution of shape-memory materials. Math. Models Methods Appl. Sci. 18, 125–164 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  4. Auricchio F., Reali A., Stefanelli U.: A macroscopic 1D model for shape memory alloys including asymmetric behaviors and transformation-dependent elastic properties. Comput. Methods Appl. Mech. Eng. 198, 1631–1637 (2009)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  5. Baiocchi C.: Soluzioni ordinarie e generalizzate del problema di Cauchy per equazioni differenziali astratte lineari del secondo ordine in spazi di Hilbert. Ric. Mat. 16, 27–95 (1967)

    MATH  MathSciNet  Google Scholar 

  6. Barbu, V.: Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff, Leyden, 1976

  7. Barbu V., Colli P., Gilardi G., Grasselli M.: Existence, uniqueness, and longtime behavior for a nonlinear Volterra integrodifferential equation. Differ. Integral Equ. 13, 1233–1262 (2000)

    MATH  MathSciNet  Google Scholar 

  8. Bhattacharya, K., Schlömerkemper, A.: Stress-induced phase transformations in shape-memory polycrystals. Arch. Ration. Mech. Anal. 196, 715–751 (2010)

  9. Berti, V., Fabrizio, M., Grandi, D.: Phase transitions in shape memory alloys: a non isothermal Ginzburg–Landau model. Phys. D 239, 95–102 (2010)

  10. Berti, V., Fabrizio, M., Grandi, D.: Hysteresis and phase transitions for one-dimensional and three-dimensional models in shape memory alloys. J. Math. Phys. 51 (2010), 13 pp

  11. Boccardo, L., Gallouët, T.: Non-linear elliptic and parabolic equations involving measure data. J. Funct. Anal. 87, 149–169 (1989)

  12. Bonetti, E.: Global solution to a nonlinear phase transition model with dissipation. Adv. Math. Sci. Appl. 12, 355–376 (2002)

  13. Bonetti, E.: Global solvability of a dissipative Frémond model for shape memory alloys. I. Mathematical formulation and uniqueness. Q. Appl. Math. 61, 759–781 (2003)

  14. Bonetti, E.: Global solvability of a dissipative Frémond model for shape memory alloys. II. Existence. Q. Appl. Math. 62, 53–76 (2004)

  15. Bonetti, E., Colli, P., Laurençot, Ph.: Global existence for a hydrogen storage model with full energy balance. Nonlinear Anal. 75, 3558–3573 (2012)

  16. Bonetti, E., Frémond, M., Lexcellent, Ch.: Global existence and uniqueness for a thermomechanical model for shape memory alloys with partition of the strain. Math. Mech. Solids 11, 251–275 (2006)

  17. Bonfanti, G., Frémond, M., Luterotti, F.: Global solution to a nonlinear system for irreversible phase changes. Adv. Math. Sci. Appl. 10, 1–24 (2000)

  18. Brezis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Mathematics Studies, vol. 5. North-Holland, Amsterdam, 1973

  19. Colli, P.: Global existence for the three-dimensional Frémond model of shape memory alloys. Nonlinear Anal. 24, 1565–1579 (1995)

  20. Colli, P., Frémond, M., Rocca, E., Shirakawa, K.: Attractors for a three-dimensional thermo-mechanical model of shape memory alloys. Chin. Ann. Math. Ser. B 27, 683–700 (2006)

  21. Colli, P., Gilardi, G., Grasselli, M.: Well-posedness of the weak formulation for the phase-field model with memory. Adv. Differ. Equ. 2, 487–508 (1997)

  22. Colli, P., Laurençot, Ph., Stefanelli, U.: Long-time behavior for the full one-dimensional Frémond model for shape memory alloys. Contin. Mech. Thermodyn. 12, 423–433 (2000)

  23. Colli, P., Luterotti, F., Schimperna, G., Stefanelli, U.: Global existence for a class of generalized systems for irreversible phase changes. NoDEA Nonlinear Differ. Equ. Appl. 9, 255–276 (2002)

  24. Colli, P., Sprekels, J.: Positivity of temperature in the general Frémond model for shape memory alloys. Contin. Mech. Thermodyn. 5, 255–264 (1993)

  25. Daghia, F., Fabrizio, M., Grandi, D.: A non isothermal Ginzburg–Landau model for phase transitions in shape memory alloys. Meccanica 45, 797–807 (2010)

  26. Dautray, R., Lions, J.L.: Analyse mathématique et calcul numérique pour les sciences et les techniques. Tome 3, Masson, Paris, 1985

  27. Dhote, R.P., Fabrizio, M., Melnik, R.N.V., Zu, J.: Hysteresis phenomena in shape memory alloys by non-isothermal Ginzburg–Landau models. Commun. Nonlinear Sci. Numer. Simul. 18, 2549–2561 (2013)

  28. Fabrizio, M., Pecoraro, M.: Phase transitions and thermodynamics for the shape memory alloy AuZn. Meccanica 48, 1695–1700 (2013)

  29. Falk, F.: Elastic phase transitions and nonconvex energy functions. Free Boundary Problems—Theory and Applications V–VI. (Eds. K.H. Hoffmann, J. Sprekels) Longman, London, 1989

  30. Falk, F., Konopka, P.: Three-dimensional Landau theory describing the martensitic phase transformations of shape-memory alloys. J. Phys. Condens. Matter 2, 61–77 (1990)

  31. Frémond, M.: Non-Smooth Thermomechanics. Springer, Berlin, 2002

  32. Frémond, M.: Phase change in mechanics. Lecture Notes of the Unione Matematica Italiana, xiii + 303. Springer, Berlin, 2012

  33. Frémond, M., Rocca, E.: A model for shape memory alloys with the possibility of voids. Discrete Contin. Dyn. Syst. 27, 1633–1659 (2010)

  34. Gilardi, G., Rocca, E.: Well-posedness and long-time behaviour for a singular phase field system of conserved type. IMA J. Appl. Math. 72, 498–530 (2007)

  35. Grandi, D., Stefanelli, U.: A phenomenological model for microstructure-dependent inelasticity in shape-memory alloys. Meccanica 49, 2265–2283 (2014)

  36. Jerome, J.W.: Approximation of nonlinear evolution systems. Mathematics in Science and Engineering, vol. 164, xx + 280, Academic Press Inc, Orlando, 1983

  37. Laurençot, Ph., Schimperna, G., Stefanelli, U.: Global existence of a strong solution to the one-dimensional full model for irreversible phase transitions. J. Math. Anal. Appl. 271, 426–442 (2002)

  38. Luterotti, F., Stefanelli, U.: Existence result for the one-dimensional full model of phase transitions. Z. Anal. Anwendungen 21, 335–350 (2002)

  39. Roubíček, T.: Modelling of thermodynamics of martensitic transformation in shape-memory alloys. Discret. Contin. Dyn. Syst. 892–902, 2007

  40. Roubíček, T., Stefanelli, U.: Magnetic shape-memory alloys: thermomechanical modeling and analysis. Contin. Mech. Thermodyn. 26, 783–810 (2014)

  41. Roubíček, T., Tomassetti, G.: Phase transformations in electrically conductive ferromagnetic shape-memory alloys, their thermodynamics and analysis. Arch. Ration. Mech. Anal. 210, 1–43 (2013)

  42. Simon, J.: Compact sets in the space L p(0, T; B). Ann. Mat. Pura Appl. (4) 146, 65–96 (1987)

  43. Souza, A.C., Mamiya, E.N., Zouain, N.: Three-dimensional model for solids undergoing stress-induced phase transformations. Eur. J. Mech. A Solids 17, 789–806 (1998)

  44. Sprekels, J., Zheng, S.: Global solutions of a Ginzburg–Landau theory for structural phase transitions in shape memory alloys. Phys. D 39, 59–76 (1989)

  45. Zwicknagl, B.: Microstructures in low-hysteresis shape memory alloys: scaling regimes and optimal needle shapes. Arch. Ration. Mech. Anal. 213, 355–421 (2014)

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica “F. Casorati”, Università di Pavia, via Ferrata 1, 27100, Pavia, Italy

    Elena Bonetti, Pierluigi Colli & Gianni Gilardi

  2. Dipartimento di Matematica, Università di Bologna, piazza di Porta San Donato, 5, 40126, Bologna, Italy

    Mauro Fabrizio

Authors
  1. Elena Bonetti
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pierluigi Colli
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mauro Fabrizio
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Gianni Gilardi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pierluigi Colli.

Additional information

Communicated by I. Fonseca

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bonetti, E., Colli, P., Fabrizio, M. et al. Existence of Solutions for a Mathematical Model Related to Solid–Solid Phase Transitions in Shape Memory Alloys. Arch Rational Mech Anal 219, 203–254 (2016). https://doi.org/10.1007/s00205-015-0896-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-015-0896-4

Keywords

Search

Navigation