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Nonequilibrium thermodynamics of pseudoelasticity

Continuum Mechanics and Thermodynamics volume 5, pages 163–204 (1993)Cite this article

Abstract

Solid-solid phase transitions often exhibit hystereses, and a hysteresis indicates energy dissipation. Pseudoelasticity refers to a hysteretic loadingunloading characteristic observed in the stress-induced martensitic transformation of shape memory alloys.

This paper describes the thermodynamic model ofideal pseudoelasticity, a largely schematized adaptation of the experimental observations, and it reviews the works of other authors on thermodynamics of pseudoelasticity. Different approaches vary widely and we have chosen to put them into perspective by contrasting their assumptions and predictions against those of ideal pseudoelasticity.

Ideal pseudoelasticity receives support from the experimental results of Fu [1] and its thermodynamic properties have been exploited by Huo [2]. The model makes use of an analytical ansatz proposed by Müller [3] in which the hysteresis is assumed to be due to the presence of a coherency energy in solid phase mixtures. This model permits the study of stability of the equilibrium states and the calculation of the energy dissipation or entropy production during the phase transition: The equilibrium states of a phase mixture are found to be unstable in load-controlled processes and the dissipated energy is related to the coherency coefficient.

We also discuss some open problems concerning the states inside the hysteresis loop and the formation of interfaces.

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References

  1. Fu, S.: Dissertation, in preparation.

  2. Huo, Y.-Z.: On the thermodynamics of pseudoelasticity, Dissertation, TU Berlin, 1992.

  3. Müller, I.: On the size of the hysteresis in pseudoelasticity, Cont. Mech. Thermodyn. 1 (1989), 125–142.

    Google Scholar 

  4. Funakubo, H. (ed.).: Shape memory alloys, Gordon and Breach Sci. Publ., London, 1987.

    Google Scholar 

  5. de Groot, S. E. and P. Mazur: Non-equilibrium thermodynamics, North-Holland Publ., Amsterdam, 1962.

    Google Scholar 

  6. Müller, I.: Thermodynamics, Pitman Publ., London, 1985.

    Google Scholar 

  7. Christian, J. W.: The theory of transformations in metal and alloys, Pergamon Press, 1965.

  8. Nishiyama, Z.: Martensitic transformations, Acad. Press, New York, 1978.

    Google Scholar 

  9. Tolédano, J. C. and P. Tolédano: The Landau theory of phase transition, World Sci. Publ. Singapore, 1987.

  10. Falk, F.: Model free energy, mechanics and thermodynamics of shape memory alloys. Acta Metall. 28 (1980).

  11. Visintin, A.: Mathematical models of hysteresis. In: Topics in non-smooth mechanics, (J. J. Moreau, P. D. Panagiotopoulos, G. Strang, eds.), Birkhäuser, Basel (1987), 295–326.

    Google Scholar 

  12. Krasnosel'skií, M. A. and A. V. Pokrovskií: Systems with hysteresis, Springer, Berlin, 1989.

    Google Scholar 

  13. Mayergoyz, I. D.: Mathematical models of hysteresis, Springer, New York, 1991.

    Google Scholar 

  14. Huo, Y.-Z.: A mathematical model for the hysteresis in shape memory alloys, Cont. Mech. Thermodyn. 1 (1989), 283–303.

    Google Scholar 

  15. Ortín, J.: Preisach modelling of hysteresis for a pseudoelastic Cu−Zn−Al single crystal, J. Appl. Phys. 71 (1992), 1454–1461.

    Google Scholar 

  16. Lü, L.; E. Aernoudt and L. Delaey: Hysteresis effects of martensitic transformation during thermomechanical cycling, Scr. metall. 22 (1988), 1435–1440.

    Google Scholar 

  17. Cesari, E.; J. Pons and C. Segni: Simple model of hysteresis in thermoelastic martensitic transformation. J. de Phys. IV, Colloque C4. 1 (1991), 41–46.

    Google Scholar 

  18. Eshelby, J. D.: The determination of the elastic field of an ellipsoidal inclusion, and related problem, Proc. Roy. Soc., A, 241 (1957), 376–396.

    Google Scholar 

  19. Ling, H. C. and W. S. Owen: A model of the thermoelastic growth of martensite, Acta Metall., 29 (1981), 1721–1736.

    Google Scholar 

  20. Deng, Y. and G. S. Ansell: Investigation of thermoelastic martensitic transformations in a Cu−Zn−Al alloy, Acta Metall. 38 (1990), 69–76.

    Google Scholar 

  21. Liu, I. S.: On interface equilibrium and inclusion problems, Cont. Mech. Thermodyn. 4 (1992), 177–186.

    Google Scholar 

  22. Robin, P. F.: Thermodynamic equilibrium across a coherent interface in a stresses crystal, Am. Miner., 59 (1974); 1286–1298.

    Google Scholar 

  23. Larché, F. and J. W. Cahn: A linear theory of thermodynamical equilibrium of solids under stress, Acta Metall., 21 (1973), 1051–1063.

    Google Scholar 

  24. Larché, F. and J. W. Cahn: A nonlinear theory of thermodynamical equilibrium of solids under stress, Acta Metall. 26 (1978), 53–60.

    Google Scholar 

  25. Larché, F. and J. W. Chan: Thermodynamical equilibrium of multiphase solids under stress, Acta Metall. 26 (1978), 1579–1589.

    Google Scholar 

  26. Heidug, W. and F. K. Lehner: Thermodynamics of coherent phase transitions in non hydrostatically stressed solids, Pure Appl. Geophys. 123 (1985), 91–98.

    Google Scholar 

  27. Johnson, W. C. and J. I. D. Alexander: Interfacial conditions for thermodynamical equilibrium in two-phase crystals, J. Appl. Phys., 59 (1986), 2735–2746.

    Google Scholar 

  28. Abeyaratne, R. and J. K. Knowles: On the driving traction acting on a surface of discontinuity in a continuum, J. Mech. Phys. Solids 38 (1990), 345–360.

    Google Scholar 

  29. Abeyaratne, R. and J. K. Knowles: Kinetic relations and the propagation of phase boundaries in solids. Arch. Rational. Mech. Anal. 114 (1991), 119–154.

    Google Scholar 

  30. Gurtin, M. and A. Struthers: Multiphase thermomechanics with interfacial structures, Arch. Rational Mech. Anal. 112 (1990), 97–160.

    Google Scholar 

  31. Voorhees, P. W. and W. C. Johnson: The thermodynamics of coherent interface, J. Chem. Phys., 90 (1989), 2793–2801.

    Google Scholar 

  32. Salzbrenner, R. J. and M. Cohen: On the thermodynamics of thermoelastic martensitic transformations, Acta Metall. 27 (1979), 739–748.

    Google Scholar 

  33. Grujicic, M.: Kinetic of martensitic interface motion, Ph. D. Thesis at MIT, 1983.

  34. Grujicic, M.; G. B. Olson and W. S. Owen: Mobility of martensitic interfaces, Metall. Trans. 16A (1985), 1713–1722. Mobility of the 203-1 martensitic interfaces: Part I: Experimental measurements. Metal. Trans. 16A (1985), 1723–1734. Part II: Model calculations, Metall. Trans. 16A (1985), 1735–1744.

    Google Scholar 

  35. Olson, G. B. and M. Cohen: Dislocation theory of martensitic transformations. In: Dislocations in solids, N. Nabarro (ed.), North-Holland Publ. Amsterdam, 1986.

    Google Scholar 

  36. Wollants, P.; M. De Bonte and J. R. Roos: A thermodynamic analysis of the stress-induced martensitic transformation in a single crystal, Z. Metallkunde. 70 (1979), 113–117.

    Google Scholar 

  37. Cory, J. S. and J. L. McNichols, Jr. Nonequilibrium thermostatics, J. Appl. Phys. 58 (1985), 3282–3294.

    Google Scholar 

  38. McNicols, Jr.; J. L. and J. S. Cory: Thermodynamics of Nitinol, J. Appl. Phys. 61 (1987), 972–984.

    Google Scholar 

  39. Ortín, J. and A. Planes: Thermodynamics of thermoelastic martensitic transformations, Acta Metall. 37 (1989), 1873–1881.

    Google Scholar 

  40. Ortín, J. and A. Planes: Thermodynamics of thermoelastic martensitic transformations, Acta Metall. 37 (1989), 1433–1441.

    Google Scholar 

  41. Ortín, J. and A. Planes: Thermodynamics and hysteresis behaviour of thermoelastic transformations, J. de Phys. IV, Colloque C4, 13–23.

  42. Müller, I. and H. Xu: On the pseudo-elastic hysteresis, Acta Metall., 39 (1991), 263–271.

    Google Scholar 

  43. Fu, S.; I. Müller and H. Xu: The interior of the pseudoelastic hysteresis, Mat. Res. Soc. Symp. Proc. Vol. 246 (1992).

  44. Fu, S.; Y.-Z. Huo and I. Müller: Thermodynamics of pseudoelasticity — an analytical approach, to appear in Acta Mechanica.

  45. Bornert, M. and I. Müller: Temperature dependence of hysteresis in pseudo-elasticity, Proc. Conf. on Free Boundary Problems, Oberwolfach (1989).

  46. Huo, Y.-Z. and I. Müller: Thermodynamics of pseudoelasticity—a graphical approach, to appear in Proc. Conf. on Model of Hysteresis, Trento (1991).

  47. Landsberg, P. T.: Thermodynamics and statistical mechanics, Oxfort Uni. Press, Oxford, 1978.

    Google Scholar 

  48. Fédélich, B. and G. Zanzotto: One-dimensional quasistatic non-isothermal evolution of shape-memory material inside the hysteresis loop, Cont. Mech. Thermodyn., 3 (1991), 251–276.

    Google Scholar 

  49. Wollants, P.; de Bonte, M. and J. R. Roos: The Stress-Dependency of the Latent Heat of Transformation in β-Cu−Zn−Al Single Crystals, Z. f. Metallkunde 74 (1983).

  50. Devonshire, A. F.: Theory of Ferroelectrics, Adv. Phys. 3 (1954).

  51. Falk, F.: Ginzburg Landau theory and solitary waves in shape momory alloys, Z. Phys. B—Condensed Matter 54 (1984).

  52. Müller, I.: Pseudoelasticity in shape memory alloys—an extreme case of thermoelasticity. Acc. Naz. dei Lincei Contributi Centro Linceo Interdisc. N 76 Rom (1986).

  53. Achenbach, M. and I. Müller: Shape memory as a thermally activated process., Plasticity Today, Modelling, Methods and Applications, A. Sawczuk, G. Brandir (eds.) Elsevier Appl. Sci. Publ. London, New York (1985).

    Google Scholar 

  54. Achenbach, M.: Simulation des Spannungs-Dehnungs-Temperaturverhaltens von Legierungen mit Form-Gedächtnis-Vermögen, Dissertation, TU Berlin, 1980.

  55. Achenbach, M.: A Model for an Alloy with Shape Memory, Int. J. Plast. 5 (1989).

  56. Wilmanski, K.: On pattern formation in stress-induced martensitic transformation, to appear in Non-linear thermodynamical processes in continua. G. Maugin, W. Muschik (ed.), (1993).

  57. Wilmanski, K.: A model of stress-induced patterns in shape memory alloys, submitted to Acta Metall.

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Authors and Affiliations

  1. Institut für Angewandte Mathematik und Statistik, Technische Universität München, Dachauer Str. 9a, 80335, München 2

    Y. Huo

  2. Institut für Thermodynamik und Reaktionstechnik, HF 2, Technische Universität Berlin, Straße des 17. Juni 135, 10623, Berlin 12

    I. Müller

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  1. Y. Huo
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Huo, Y., Müller, I. Nonequilibrium thermodynamics of pseudoelasticity. Continuum Mech. Thermodyn 5, 163–204 (1993). https://doi.org/10.1007/BF01126524

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  • DOI: https://doi.org/10.1007/BF01126524

Keywords

  • Entropy
  • Phase Transition
  • Equilibrium State
  • Hysteresis Loop
  • Experimental Observation