Approximation in multiscale modelling of microstructure evolution in shape-memory alloys

  • Tomáš RoubíčekEmail author
Original Article


Various models of microstructure in deformation gradient and its evolution arising in martensitic mechanically-induced isothermal phase transformation are surveyed and scrutinized, focusing on over-bridging of various scales of the problem and its numerical approximation. In particular, numerically efficient model of a relaxed problem is shown to be approximated by conventional but computationally less efficient model based on standard partial differential inequalities.


Smart materials Rate-independent processes Young measures 

Mathematics Subject Classification (2000)

35K85 49S05 65Z05 74N15 


  1. 1.
    Aranda E., Pedregal P.: On the computation of the rank-one convex hull of a function. SIAM J. Sci. Comput. 22, 1772–1790 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Arndt, M. : Modelling and numerical simulation of martensitic transformation. In: Wendland, W., Efendiev M. (eds.) Anal Simul of Multifield Problems. L.N. Appl. Comput. Mech. vol. 12. Springer, pp. 59–65 (2003)Google Scholar
  3. 3.
    Arndt M., Griebel M., Novák V., Roubíček T., Šittner P.: Martensitic transformation in NiMnGa single crystals: numerical simulations and experiments. Int. J. Plast. 22, 1943–1961 (2006)zbMATHCrossRefGoogle Scholar
  4. 4.
    Aubry S., Fago M., Ortiz M.: A constrained sequential-lamination algorithm for the simulation of sub-grid microstructure in martensitic materials. Comp. Meth. Appl. Mech. Engr. 192, 2823–2843 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Ball J.M., James R.D.: Fine phase mixtures as minimizers of energy. Arch. Rat. Mech. Anal. 100, 13–52 (1988)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bartel T., Hackl K.: A micromechanical model for martensitic transformations in shape-memory alloys based on energy-relaxation. Zeitsch. angewandte Math. Mechanik 89, 792–809 (2009)zbMATHCrossRefGoogle Scholar
  7. 7.
    Bartels S., Carstensen C., Hackl K., Hoppe U.: Effective relaxation for microstructure simulations: algorithms and applications. Comput. Methods Appl. Mech. Engrg. 193, 5143–5175 (2004)MathSciNetADSzbMATHCrossRefGoogle Scholar
  8. 8.
    Bedford A.: Hamilton’s Principle in Continuum Mechanics. Pitman, Boston (1985)zbMATHGoogle Scholar
  9. 9.
    Benešová, B.: Modeling of shape-memory alloys on the mesoscopic level. In: Šittner, P. et al. (eds.), Proceedings of ESOMAT 2009, EDP Sciences, vol. 03003, pp. 1–7 (2009)Google Scholar
  10. 10.
    Benešová, B.: Global optimization numerical strategies for rate-independent processes. J. Global Optim. doi: 10.1007/s10898-010-9560-6 (In print)
  11. 11.
    Bhattacharya K.: Microstructure of Martensite. Why it Forms and How it Gives Rise to the Shape-Memory Effect. Oxford Univ. Press, New York (2003)Google Scholar
  12. 12.
    Carstensen C., Plecháč P.: Numerical analysis of a relaxed variational model of hysteresis in two-phase solids. Math. Model. Numer. Anal. 35, 865–878 (2001)zbMATHCrossRefGoogle Scholar
  13. 13.
    Dolzmann, G.: Variational Methods for Crystalline Microstructure. Analysis and Computations. L. N. Math., vol. 1803, Springer, Berlin (2003)Google Scholar
  14. 14.
    Efendiev M.: On the compactness of the stable set for rate-independent processes. Comm. Pure Appl. Anal. 2, 495–509 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Fonseca I., Müller S., Pedregal P.: Analysis of concentration and oscillation effects generated by gradients. SIAM J. Math. Anal. 29, 736–756 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Francfort G., Mielke A.: An existence result for a rate-independent material model in the case of nonconvex energies. J. reine u. angew. Math. 595, 55–91 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Fried E., Gurtin M.E.: Tractions, balances, and boundary conditions for nonsimple materials with application to liquid flow at small-lenght scales. Arch. Ration. Mech. Anal. 182, 513–554 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Govindjee S., Mielke A., Hall G.J.: Free-energy of mixing for n-variant martensitic phase transformations using quasi-convex analysis. J. Mech. Phys. Solids 50, 1897–1922 (2002)MathSciNetADSzbMATHCrossRefGoogle Scholar
  19. 19.
    Halphen B., Nguyen Q.S.: Sur les matériaux standards généralisés. J. Mécanique 14, 39–63 (1975)zbMATHGoogle Scholar
  20. 20.
    Hackl K., Fischer F.D.: On the relation between the principle of maximum dissipation and inelastic evolution given by dissipation potentials. Proc. R. Soc. Lond. A 464, 117–132 (2007)MathSciNetADSGoogle Scholar
  21. 21.
    Heinen R., Hoppe U., Hackl K.: Prediction of microstructural patterns in monocrystalline shape memory alloys using global energy minimization. Mater. Sci. Engrg. A 481(2), 362–365 (2008)CrossRefGoogle Scholar
  22. 22.
    Hill R.: A variational principle of maximum plastic work in classical plasticity. Q. J. Mech. Appl. Math. 1, 18–28 (1948)zbMATHCrossRefGoogle Scholar
  23. 23.
    Hormann K., Zimmer J.: On Landau theory and symmetric energy landscapes for phase transitions. J. Mech. Phys. Solids 55, 1385–1409 (2007)MathSciNetADSzbMATHCrossRefGoogle Scholar
  24. 24.
    Jirásek M., Bažant Z.P.: Inelastic Analysis of Structures. Wiley, London (2002)Google Scholar
  25. 25.
    Junker, P., Hackl, K.: On the numerical simulation of material inhomogeneities due to martensitic phase transformation in polycrystals. In: Šittner, P. et al. (eds.), Proceedigns of ESOMAT 2009, EDP Sciences, vol. 03007, pp. 1–9 (2009)Google Scholar
  26. 26.
    Klouček P., Luskin M.: The computation of the dynamics of the martensitic transformation. Continuum Mech. Thermodyn. 6, 209–240 (1994)ADSzbMATHCrossRefGoogle Scholar
  27. 27.
    Kochmann, D.M., Hackl K.: Influence of hardening on the cyclic behaviour of laminate microstructures in finite crystal plasticity. Technische Mechanik (2010, In print)Google Scholar
  28. 28.
    Kristensen, J.: Lower Semicontinuity of Variational Integrals. PhD. Thesis, Math. Inst., Tech. Univ. of Denmark, Lungby (1994)Google Scholar
  29. 29.
    Kristensen J.: On the non-locality of quasiconvexity. Ann. Inst. H. Poincaré Anal. Non Linéaire 16, 1–13 (1999)MathSciNetADSzbMATHCrossRefGoogle Scholar
  30. 30.
    Kruží k M.: Numerical approach to double-well problem. SIAM J. Numer. Anal. 35, 1833–1849 (1998)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Kružík M., Luskin M.: The computation of martensitic microstructure with piecewise laminates. J. Sci. Comp. 19, 293–308 (2003)zbMATHCrossRefGoogle Scholar
  32. 32.
    Kružík M., Mielke A., Roubíček T.: Modelling of microstructure and its evolution in shape-memory-alloy single-crystals, in particular in CuAlNi. Meccanica 40, 389–418 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Luskin M.: On the computation of crystalline microstructure. Acta Numer. 5, 191–257 (1996)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Mainik A., Mielke A.: Existence results for energetic models for rate-independent systems. Calc. Var. PDEs 22, 73–99 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Mainik A., Mielke A.: Global existence for rate-independent gradient plasticity at finite strain. J. Nonlinear Sci. 19, 221–248 (2009)MathSciNetADSzbMATHCrossRefGoogle Scholar
  36. 36.
    Mataché A.-M., Roubíček T., Schwab Ch.: Higher-order convex approximations of Young measures in optimal control. Adv. Comput. Math. 19, 73–97 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Mielke A.: Evolution of rate-independent systems. In: Dafermos C., Feireisl E. (eds). Handbook of Differential Equations: Evolut. Diff. Eqs., pp. 461–559. Elsevier, Amsterdam (2005)Google Scholar
  38. 38.
    Mielke A., Theil F.: A mathematical model for rate-independent phase transformations with hysteresis. In: Alber, H.-D., Balean, R., Farwig, R. (eds) Models of Continuum Mechanics in Analysis and Engineering, pp. 117–129. Shaker Ver., Aachen (1999)Google Scholar
  39. 39.
    Mielke A., Theil F.: On rate-independent hysteresis models. Nonlin. Diff. Eq. Appl. 11, 151–189 (2004)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Mielke A., Theil F., Levitas V.I.: A variational formulation of rate-independent phase transformations using an extremum principle. Archive. Rat. Mech. Anal. 162, 137–177 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Mielke A. et al.: A mathematical framework for generalized standard materials in rate-independent case. In: Helmig, R. (eds) Multifield Problems in Fluid and Solid Mech, pp. 491–529. Springer, Berlin (2006)Google Scholar
  42. 42.
    Mielke A., Roubíček T., Stefanelli U.: Γ -limits and relaxations for rate-independent evolutionary problems. Calc. Var. Part. Diff. Equ. 31, 387–416 (2008)zbMATHCrossRefGoogle Scholar
  43. 43.
    Müller, S.: Variational models for microstructure and phase transitions. In: Hildebrandt, S. et al. (eds.), Calculus of Variations and Geometric Evolution Problems. Lecture Notes in Math., vol. 1713 (Lecture Notes 2, Max-Planck-Institut für Math., Leipzig). Springer, Berlin, pp. 85–210 (1999)Google Scholar
  44. 44.
    Nicolaides R.A., Walkington N.J.: Computation of microstructure utilizing Young measure representations. In: Rogers, C.A., Rogers, R.A. (eds) Recent Advances in Adaptive and Sensory Materials and their Appl, pp. 131–141. Technomic Publ., Lancaster (1992)Google Scholar
  45. 45.
    Pedregal P.: Parametrized Measures and Variational Principles. Birkhäuser, Basel (1997)zbMATHCrossRefGoogle Scholar
  46. 46.
    Peerlings R.H.J., de Borst R., Brekelmans W.A.M., de Vree J.H.P.: Gradient enhanced damage for quasi-brittle materials. Intl. J. Numer. Meth. Engr. 39, 3391–3403 (1996)zbMATHCrossRefGoogle Scholar
  47. 47.
    Peerlings R.H.J., Geers M.G.D., de Borst R., Brekelmans W.A.M.: Critical comparison of nonlocal and gradient-enhanced softening continua. Int. J. Solids Struct. 38, 7723–7746 (2001)zbMATHCrossRefGoogle Scholar
  48. 48.
    Plecháč P., Roubíček T.: Visco-elasto-plastic model for martensitic phase transformation in shape-memory alloys. Math. Methods Appl. Sci. 25, 1281–1298 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Pitteri M., Zanzotto G.: Continuum Models for Phase Transitions and Twinning in Crystals. Chapman & Hall, Boca Raton (2003)zbMATHGoogle Scholar
  50. 50.
    Podio-Guidugli P., Vergara Caffarelli G.: Surface interaction potentials in elasticity. Arch. Rat. Mech. Anal. 109, 343–381 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Rajagopal K.R., Roubíček T.: On the effect of dissipation in shape-memory alloys. Nonlinear Anal. Real World Appl. 4, 581–597 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Ren X., Truskinovsky L.: Finite scale microstructures in nonlocal elasticity. J. Elast. 59, 319–355 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Rogers R.C.: Some remarks on nonlocal interactions and hysteresis in phase transitions. Continuum Mech. Thermodyn. 8, 65–73 (1994)ADSCrossRefGoogle Scholar
  54. 54.
    Rogers R., Truskinovsky L.: Discretization and hysteresis. Phys. B 233, 370–375 (1997)ADSCrossRefGoogle Scholar
  55. 55.
    Roubíček T.: Relaxation in Optimization and Variational Calculus. Gruyter, Berlin (1997)zbMATHCrossRefGoogle Scholar
  56. 56.
    Roubíček, T. : Convex locally compact extensions of Lebesgue spaces and their applications. In: Ioffe, A., Reich, S., Shafrir, I. (eds.), Calculus of Variations and Optimal Control Chapman & Hall / CRC Res. Notes in Math., vol. 411, pp. 237–250. CRC Press, Boca Raton, FL (1999)Google Scholar
  57. 57.
    Roubíček, T.: Models of microstructure evolution in shape memory materials. In: Ponte Castaneda, P. et al. (eds.), Nonlinear Homogeneous and its Application to Composites, Polycrystal and Smart Mater. NATO Sci.Ser.II/170, pp. 269–304. Kluwer, Dordrecht (2004)Google Scholar
  58. 58.
    Roubíček, T.: Numerical techniques in relaxed optimization problems. In: Kurdila, A.J. Pardalos, P.M., Zabrankin, M. (eds.), Proceedings of Robust Optimization-Directed Design. Springer, New York, pp. 145–161 (2006)Google Scholar
  59. 59.
    Roubíček T.: Rate independent processes in viscous solids at small strains. Math. Methods Appl. Sci. 32, 825–862 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    Roubíček T., Hoffmann K.-H.: About the concept of measure-valued solutions to distributed parameter systems. Math. Methods Appl. Sci. 18, 671–685 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    Roubíček T., Kruží k M., Koutný J.: A mesoscopical model of shape-memory alloys. Proc. Estonian Acad. Sci. Phys. Math. 56, 146–154 (2007)MathSciNetGoogle Scholar
  62. 62.
    Stupkiewicz S., Petryk H.: Modelling of laminated microstructures in stress-induced martensitic transformations. J. Mech. Phys. Solids 50, 2303–2331 (2002)MathSciNetADSzbMATHCrossRefGoogle Scholar
  63. 63.
    Šverák V.: Rank-one convexity does not imply quasiconvexity. Proc. R. Soc. Edinb. 120, 185–189 (1992)zbMATHCrossRefGoogle Scholar
  64. 64.
    Toupin R.A.: Elastic materials with couple stresses. Arch. Rat. Mech. Anal. 11, 385–414 (1962)MathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    Visintin A.: A Weiss model of ferromagnetism. Phys. B 275, 87–91 (2000)ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Mathematical InstituteCharles UniversityPrague 8Czech Republic
  2. 2.Institute of Thermomechanics of the ASCRPrague 8Czech Republic

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