Skip to main content
SpringerLink
Log in

Existence and time-discretization for the finite-strain Souza–Auricchio constitutive model for shape-memory alloys

  • Original Article
  • Published:
Continuum Mechanics and Thermodynamics Aims and scope Submit manuscript

Abstract

We prove the global existence of solutions for a shape-memory alloys constitutive model at finite strains. The model has been presented in Evangelista et al. (Int J Numer Methods Eng 81(6):761–785, 2010) and corresponds to a suitable finite-strain version of the celebrated Souza–Auricchio model for SMAs (Auricchio and Petrini in Int J Numer Methods Eng 55:1255–1284, 2002; Souza et al. in J Mech A Solids 17:789–806, 1998). We reformulate the model in purely variational fashion under the form of a rate-independent process. Existence of suitably weak (energetic) solutions to the model is obtained by passing to the limit within a constructive time-discretization procedure.

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. Arghavani J., Auricchio F., Naghdabadi R., Reali A.: On the robustness and efficiency of integration algorithms for a 3D finite strain phenomenological SMA constitutive model. Int. J. Numer. Methods Eng. 85, 107–134 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  2. Arghavani J., Auricchio F., Naghdabadi R., Reali A., Sohrabpour S.: A 3D finite strain phenomenological constitutive model for shape memory alloys considering martensite reorientation. Cont. Mech. Thermodyn. 22, 345–362 (2010)

    Article  MathSciNet  Google Scholar 

  3. Auricchio, F., Bessoud, A.-L., Reali, A., Stefanelli, U.: A phenomenological model for the magnetomechanical response of magnetic shape memory alloys single crystals (in preparation) (2011)

  4. Auricchio F., Bessoud A.-L., Reali A., Stefanelli U.: A three-dimensional phenomenological models for magnetic shape memory alloys. GAMM-Mitt. 34(1), 90–96 (2011)

    Article  MathSciNet  Google Scholar 

  5. Auricchio F., Mielke A., Stefanelli U.: A rate-independent model for the isothermal quasi-static evolution of shape-memory materials. Math. Models Method Appl. Sci. 18(1), 125–164 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Auricchio F., Petrini L.: Improvements and algorithmical considerations on a recent three-dimensional model describing stress-induced solid phase transformations. Int. J. Numer. Methods Eng. 55, 1255–1284 (2002)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  8. Auricchio F., Petrini L.: A three-dimensional model describing stress-temperature induced solid phase transformations. Part II: thermomechanical coupling and hybrid composite applications. Int. J. Numer. Methods Eng. 61, 716–737 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. Auricchio, F., Reali, A., Stefanelli, U.: A phenomenological 3D model describing stress-induced solid phase transformations with permanent inelasticity, In: Topics on Mathematics for Smart Systems (Rome, 2006), pp. 1–14, World Scientific Publishing (2007)

  10. Auricchio F., Reali A., Stefanelli U.: A three-dimensional model describing stress-induced solid phase transformation with permanent inelasticity. Int. J. Plast. 23, 207–226 (2007)

    Article  MATH  Google Scholar 

  11. 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  ADS  MathSciNet  Google Scholar 

  12. Auricchio F., Sacco E.: A one-dimensional model for superelastic shape-memory alloys with different elastic properties between austenite and martensite. Int. J. Non-Linear Mech. 32, 1101–1114 (1997)

    Article  MATH  ADS  Google Scholar 

  13. Auricchio F., Taylor R.L.: Shape-memory alloys: modelling and numerical simulations of the finite-strain superelastic behavior. Comput. Methods Appl. Mech. Eng. 143(1–2), 175194 (1997)

    Google Scholar 

  14. Bessoud A.-L., Stefanelli U.: Magnetic shape memory alloys: three-dimensional modeling and analysis. Math. Models Meth. Appl. Sci. 21(5), 1043–1069 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  15. Bessoud, A.-L., Kružík, M., Stefanelli, U.: A macroscopic model for magnetic shape memory alloys, Preprint IMATI-CNR, 23PV10/21/0 (2010)

  16. Brezis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, no. 5 in North Holland Math. Studies, North-Holland, Amsterdam (1973)

  17. Carstensen C., Hackl K., Mielke A.: Non-convex potentials and microstructures in finite-strain plasticity. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 458(2018), 299–317 (2002)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  18. Christ D., Reese S.: A finite element model for shape memory alloys considering thermomechanical couplings at large strains. Int. J. Solids Struct. 46(20), 3694–3709 (2009)

    Article  MATH  Google Scholar 

  19. Duerig, T.W., Melton, K.N., Stökel, D., Wayman, C.M. (eds): Engineering aspects of shape memory alloys. Butterworth-Heinemann, Oxford (1990)

    Google Scholar 

  20. Duerig, T.W., Pelton, A.R. (eds.): In: SMST-2003 Proceedings of the International Conference on Shape Memory and Superelastic Technology Conference, ASM International (2003)

  21. Eleuteri M., Lussardi L., Stefanelli U.: A rate-independent model for permanent inelastic effects in shape memory materials. Netw. Heterog. Media 6(1), 145–165 (2011)

    Article  MathSciNet  Google Scholar 

  22. Eleuteri, M., Lussardi, L., Stefanelli, U.: Thermal control of the Souza-Auricchio model for shape memory alloys, Preprint IMATI-CNR 6PV11/4/0 (2011)

  23. Evangelista V., Marfia S., Sacco E.: Phenomenological 3D and 1D consistent models for shape-memory alloy materials. Comput. Mech. 44(3), 405–421 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  24. Evangelista V., Marfia S., Sacco E.: A 3D SMA constitutive model in the framework of finite strain. Int. J. Numer. Methods Eng. 81(6), 761–785 (2010)

    MATH  Google Scholar 

  25. Frémond M.: Matériaux à mémoire de forme. C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre 304, 239–244 (1987)

    Google Scholar 

  26. Frémond M.: Non-smooth Thermomechanics. Springer, Berlin (2002)

    MATH  Google Scholar 

  27. Gurtin M.E.: An Introduction to Continuum Mechanics. Mathematics in Science and Engineering, vol. 158. Academic Press, New York (1981)

    Google Scholar 

  28. Helm D., Haupt P.: Shape memory behaviour: modelling within continuum thermomechanics. Int. J. Solids Struct. 40, 827–849 (2003)

    Article  MATH  Google Scholar 

  29. Krejčí P., Stefanelli U.: Existence and nonexistence for the full thermomechanical Souza–Auricchio model of shape memeory wires. Math. Mech. Solids 16(4), 349–365 (2011)

    Article  MathSciNet  Google Scholar 

  30. Krejčí P., Stefanelli U.: Well-posedness of a thermo-mechanical model for shape memory alloys under tension. M 2AN Math. Model. Numer. Anal. 44(6), 1239–1253 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  31. Lagoudas D.C., Entchev P.B., Popov P., Patoor E., Brinson L.C., Gao X.: Shape memory alloys, Part II: modeling of polycrystals. Mech. Mater. 38, 391–429 (2006)

    Article  Google Scholar 

  32. Panico M., Brinson L.: A three-dimensional phenomenological model for martensite reorientation in shape memory alloys. J. Mech. Phys. Solids 55(11), 2491–2511 (2007)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  33. Popov P., Lagoudas D.C.: A 3-D constitutive model for shape memory alloys incorporating pseudoelasticity and detwinning of self-accommodated martensite. Int. J. Plast. 23, 1679–1720 (2007)

    Article  MATH  Google Scholar 

  34. Lee E.: Elastic-plastic deformation at finite strains. J. Appl. Mech. 36, 1–6 (1969)

    Article  MATH  ADS  Google Scholar 

  35. Levitas V.I.: Thermomechanical theory of martensitic phase transformations in inelastic materials. Int. J. Solids Struct. 35, 889–940 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  36. Mielke A.: Finite elastoplasticity Lie groups and geodesics on SL(d), in geometry, mechanics, and dynamics, pp. 61–90. Springer, New York (2002)

    Google Scholar 

  37. Mielke A.: Energetic formulation of multiplicative elasto-plasticity using dissipation distances. Cont. Mech. Thermodyn. 15(4), 351–382 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  38. Mielke A.: Evolution in rate-independent systems (Ch 6). In: Dafermos, C., Feireisl, E. (eds) Handbook of Differential Equations, Evolutionary Equations, vol. 2, pp. 461–559. Elsevier, Amsterdam (2005)

    Google Scholar 

  39. Mainik A., Mielke A.: Existence results for energetic models for rate-independent systems. Calc. Var. Partial Differ. Equ. 22(1), 73–99 (2005)

    MATH  MathSciNet  Google Scholar 

  40. Mainik A., Mielke A.: Global existence for rate-independent gradient plasticity at finite strain. J. Nonlinear Sci. 19, 221–248 (2009)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  41. Mielke, A.: A new approach to elasto-plasticity using energy and dissipation functionals. In: Applied Mathematics Entering the Twenty-first Century, pp. 315–335. SIAM, Philadelphia, PA (2004)

  42. Mielke A.: Existence of minimizers in incremental elasto-plasticity with finite strains. SIAM J. Math. Anal. 36(2), 384–404 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  43. Mielke A.: Energetic formulation of multiplicative elasto-plasticity using dissipation distances. Cont. Mech. Thermodyn. 15(4), 351–382 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  44. Mielke A., Müller S.: Lower semicontinuity and existence of minimizers in incremental finite-strain elastoplasticity. ZAMM Z. Angew. Math. Mech. 86(3), 233–250 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  45. Mielke A., Paoli L., Petrov A.: On existence and approximation for a 3D model of thermally-induced phase transformations in shape-memory alloys. SIAM J. Math. Anal. 41(4), 1388–1414 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  46. Mielke A., Paoli L., Petrov A., Stefanelli U.: Error estimates for discretizations of a rate-independent variational inequality. SIAM J. Numer. Anal. 48(5), 1625–1646 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  47. Mielke, A., Paoli, L., Petrov, A., Stefanelli, U.: Error bounds for space-time discretizations of a 3D model for shape-memory materials. In: Proceedings of the IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials (Bochum 2008), IUTAM Bookseries, Springer (2009)

  48. Mielke A., Petrov A.: Thermally driven phase transformation in shape-memory alloys. Adv. Math. Sci. Appl. 17, 160–182 (2007)

    MathSciNet  Google Scholar 

  49. Mielke A., Theil F.: On rate-independent hysteresis models. NoDEA Nonlinear Differ. Equ. Appl. 11, 151–189 (2004)

    MATH  MathSciNet  Google Scholar 

  50. Müller C., Bruhn O.: A thermodynamic finite-strain model for pseudoelastic shape memory alloys. Int. J. Plast. 22(9), 1658–1682 (2006)

    Article  MATH  Google Scholar 

  51. Peultier B., Ben Zineb T., Patoor E.: Macroscopic constitutive law for SMA: application to structure analysis by FEM. Mater. Sci. Eng. A 438–440, 454–458 (2006)

    Google Scholar 

  52. Raniecki B., Lexcellent Ch.: R L models of pseudoelasticity and their specification for some shape-memory solids. Eur. J. Mech. A Solids 13, 21–50 (1994)

    MATH  MathSciNet  Google Scholar 

  53. Reese S., Christ D.: Finite deformation pseudo-elasticity of shape memory alloys–Constitutive modelling and finite element implementation. Int. J. Plast. 28, 455–482 (2008)

    Article  Google Scholar 

  54. Roubíček T.: Models of microstructure evolution in shape memory alloys. In: Ponte Castaneda, P., Telega, J.J., Gambin, B. (eds) Nonlinear Homogenization and its Applications to Composites, Polycrystals and Smart Materials, pp. 269–304. NATO Sci. Series II/170, Kluwer, Dordrecht (2004)

    Google Scholar 

  55. 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)

    Article  MATH  ADS  Google Scholar 

  56. Stefanelli, U.: Magnetic control of magnetic shape-memory single crystals. Phys. B (to appear) (2011)

  57. Thiebaud F., Lexcellent Ch., Collet M., Foltete E.: Implementation of a model taking into account the asymmetry between tension and compression, the temperature effects in a finite element code for shape memory alloys structures calculations. Comput. Mater. Sci. 41(2), 208–221 (2007)

    Article  Google Scholar 

  58. Ziolkowski A.: Three-dimensional phenomenological thermodynamic model of pseudoelasticity of shape memory alloys at finite strains. Cont. Mech. Thermodyn. 19, 379–398 (2007)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica ’F. Casorati’, Università di Pavia, v. Ferrata 1, 27100, Pavia, Italy

    Sergio Frigeri

  2. IMATI - CNR, v. Ferrata 1, 27100, Pavia, Italy

    Ulisse Stefanelli

  3. WIAS, Mohrenstr. 39, Berlin, 10115, Germany

    Ulisse Stefanelli

Authors
  1. Sergio Frigeri
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. 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 Andreas Öchsner.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Frigeri, S., Stefanelli, U. Existence and time-discretization for the finite-strain Souza–Auricchio constitutive model for shape-memory alloys. Continuum Mech. Thermodyn. 24, 63–77 (2012). https://doi.org/10.1007/s00161-011-0221-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-011-0221-x

Keywords

Mathematics Subject Classification (1991)

Search

Navigation