On the Hysteresis in Martensitic Transformations

  • Miroslav Šilhavý


The paper proposes an explanation of the hysteresis in shape memory alloys using energy minimization in nonlinear elasticity and the entropy criterion. The stored energy has two wells describing the two phases. At elongations from some interval, the minimum energy is realized on two-phase mixtures. If the loaded phases are incompatible, that minimum energy is a concave function, and so we have a phase equilibrium curve of negative slope. The quasistatic evolution during loading experiments is realized in the class of mechanically, but not thermodynamically, equilibrated mixtures. This family contains states of elongation and force covering the whole area of the hysteresis loop. The evolution must satisfy the entropy criterion for moving phase interfaces which implies that, in the region above the phase equilibrium line, only processes with nondecreasing amount of the second phase are possible while below the situation is the opposite. This picture provides all the elements necessary for the explanation of the hysteresis, including the internal hysteresis loops.


Hysteresis Loop Martensitic Transformation Shape Memory Alloy Nonlinear Elasticity Equilibrium Mixture 
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  1. [1]
    Pedregal, P. (1997): Parametrized measures and variational principles. Birkhäuser, BaselMATHCrossRefGoogle Scholar
  2. [2]
    Pedregal, P. (2000): Variational methods in nonlinear elasticity. SIAM, PhiladelphiaMATHCrossRefGoogle Scholar
  3. [3]
    Müller, S. (1999): Variational models for microstructure and phase transitions. In: Bethuel, F. et al. (eds.) Calculus of variations and geometric evolution problems. (Lecture Notes in Mathematics, vol. 1713). Springer, Berlin, pp. 85–210CrossRefGoogle Scholar
  4. [4]
    Roubíček, T. (1997): Relaxation in optimization theory and variational calculus. W. de Gruyter, BerlinMATHCrossRefGoogle Scholar
  5. [5]
    Šilhavý, M. (1997): The mechanics and thermodynamics of continuous media. Springer, BerlinMATHGoogle Scholar
  6. [6]
    Kohn, R.V, Strang, G. (1986): Optimal design and relaxation of variational problems., I, II, III. Coram. Pure Appl. Math. 39, 113–137MathSciNetMATHCrossRefGoogle Scholar
  7. [6a]
    Kohn, R.V, Strang, G. (1986): Optimal design and relaxation of variational problems., I, II, III. Coram. Pure Appl. Math. 39, 139–182MathSciNetMATHCrossRefGoogle Scholar
  8. [6b]
    Kohn, R.V, Strang, G. (1986): Optimal design and relaxation of variational problems., I, II, III. Coram. Pure Appl. Math. 39, 353–377MathSciNetMATHCrossRefGoogle Scholar
  9. [7]
    Ball, J.M., Chu, C, James, R.D. (1995): Hysteresis during stress-induced variant rearrangement. J. Physique IV C8, 245–251Google Scholar
  10. [8]
    Müller, I. (1989): On the size of the hysteresis in pseudoelasticity. Contin. Mech. Ther-modyn. 1, 125–142ADSCrossRefGoogle Scholar
  11. [9]
    Müller, L, & Xu, H. (1991): On the pseudoelastic hysteresis. Acta Metall. Mater. 39, 263–271Google Scholar
  12. [10]
    Wilmański, K. (1993): Symmetric model of stress-strain hysteresis loops in shape memory alloys. Internat. J. Engrg. Sci. 31, 1121–1138MATHCrossRefGoogle Scholar
  13. [11]
    Kohn, R. V (1991): The relaxation of a double-well energy. Contin. Mech. Thermodyn. 3, 193–236MathSciNetADSMATHCrossRefGoogle Scholar
  14. [12]
    Šilhavý, M. (2001): On the pseudoelastic hysteresis in a double well material. In preparationGoogle Scholar
  15. [13]
    Šilhavý, M. (2001): Rotationally invariant rank 1 convex functions. Appl. Math. Optim. 44, 1–15MathSciNetMATHCrossRefGoogle Scholar

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© Springer-Verlag Italia, Milano 2003

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  • Miroslav Šilhavý

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