Continuum Mechanics and Thermodynamics

, Volume 18, Issue 7–8, pp 443–453 | Cite as

An upper bound to the free energy of mixing by twin-compatible lamination for n-variant martensitic phase transformations

  • S. Govindjee
  • K. HacklEmail author
  • R. Heinen
Original Article


Modeling the energetic behavior of martensitic (phase transforming) materials usually leads to non quasiconvex energy formulations. For this reason, researchers often employ quasiconvex relaxation methods to improve the character of the formulation. Unfortunately, explicit expressions for the relaxed free energy density for multi-variant martensitic materials are typically not available. Thus, some researchers have employed a Reuβ-like convex lower bound, which neglects compatibility constraints, as an estimate on the free energy of mixing. To be confident with such a technique, one needs a measure of the quality of the lower bound. In this paper, we seek such a measure by comparing the Reuβ-like lower bound to an upper bound. The upper bound is constructed upon assumptions on the type of microstructures that form in such alloys. In particular, we consider lamination type microstructures which form by temperature- or stress-induced transformation in monoclinic and orthorhombic Copper-based alloys with cubic austenitic symmetry. Our results display a striking congruence of upper and lower bounds in the most relevant cases.


Shape memory alloys Martensitic phase transformations Relaxation Quasiconvexification Lamination 


62.20.−x 62.40. + i 64.70.Kb 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Avellaneda, M., Milton, G.: Bounds on the effective elasticity tensor of composites based on two-point correlations. In: Hui, D., Koszic, T (eds) Proceedings of the ASME Energy-Technology Conference and Exposition ASME New York (1989)Google Scholar
  2. 2.
    Ball J.M. and James R.D. (1987). Fine phase mixtures as minimizers of energy. Arch. Rat. Mech. Anal. 100: 13–52 CrossRefMathSciNetGoogle Scholar
  3. 3.
    Ball J.M. and James R.D. (1992). Proposed experimental tests of a theory of fine microstructure and the two-well problem. Phil. Trans. R. Soc. London A 338: 389–450 Google Scholar
  4. 4.
    Bhattacharya K. (2003). Microstructure of martensite: why it forms and how it gives rise to the shape-memory effect. Oxford University Press, New York Google Scholar
  5. 5.
    Govindjee, S., Mielke, A., Hall, G. J.: The free energy of mixing for n-variant martensitic phase transformations using quasi-convex analysis. J. Mech. Phys. Solids 51, 763+ (2003)Google Scholar
  6. 6.
    Govindjee S. and Miehe C. (2001). A multi-variant martensitic phase transformation model: formulation and numerical implementation. Comput. Meths. Appl. Mech. Eng. 191: 215–238 CrossRefGoogle Scholar
  7. 7.
    Hall G.J. and Govindjee S. (2001). Application of a partially relaxed shape memory free energy function to estimate the phase diagram and predict global microstructure evolution. J. Mech. Phys. Solids 50: 501–530 CrossRefMathSciNetGoogle Scholar
  8. 8.
    Hall G.J. and Govindjee S. (2002). Application of the relaxed free energy of mixing to problems in shape memory alloy Simulation. J. Intelligent Mat. Sys. Struct. 13: 773–782 CrossRefGoogle Scholar
  9. 9.
    Hall G.J., Govindjee S., Šittner P. and Novák V. (2007). Simulation of cubic to monoclinic-II transformations in a single crystal Cu–Al–Ni tube. Int. J. Plasticity 23: 161–182 CrossRefGoogle Scholar
  10. 10.
    Kohn R. (1991). The relaxation of a double-well problem. Contin. Mech. Thermodyn. 3: 193–236 CrossRefMathSciNetGoogle Scholar
  11. 11.
    Otsuka K., Nakamura T. and Shimizu K. (1974). Electron microscopy study of stress induced acicular β′ − 1 Martensite in Cu–Al–Ni alloy. Trans. Natl Res. Inst. Met. 15: 200–210 Google Scholar
  12. 12.
    Shield T. (1995). Orientation dependence of the pseudoelastic behavior of single crystals of Cu–Al–Ni in tension. J. Mech. Phys. Solids 43: 869–895 CrossRefGoogle Scholar
  13. 13.
    Sedlák P., Seiner H., Landa M., Novák V., Šittner P. and Mañosa L.l. (2005). Elastic constants of bcc austenite and 2H orthorhombic martensite in CuAlNi shape memory alloy. Acta Materialia 53: 3643–3661 CrossRefGoogle Scholar
  14. 14.
    Tartar L. (1990). H-measures, a new approach for studying homogenization, oscillation and concentration effects in partial differential equations. Proc. R. Soc. Edinb A 115: 193–230MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Center of MechanicsETH ZurichZurichSwitzerland
  2. 2.Lehrstuhl für Allgemeine MechanikRuhr-Universität BochumBochumGermany

Personalised recommendations