Continuum Mechanics and Thermodynamics

, Volume 29, Issue 1, pp 225–249 | Cite as

On consistent micromechanical estimation of macroscopic elastic energy, coherence energy and phase transformation strains for SMA materials

Open Access
Original Article

Abstract

An apparatus of micromechanics is used to isolate the key ingredients entering macroscopic Gibbs free energy function of a shape memory alloy (SMA) material. A new self-equilibrated eigenstrains influence moduli (SEIM) method is developed for consistent estimation of effective (macroscopic) thermostatic properties of solid materials, which in microscale can be regarded as amalgams of n-phase linear thermoelastic component materials with eigenstrains. The SEIM satisfy the self-consistency conditions, following from elastic reciprocity (Betti) theorem. The method allowed expressing macroscopic coherency energy and elastic complementary energy terms present in the general form of macroscopic Gibbs free energy of SMA materials in the form of semilinear and semiquadratic functions of the phase composition. Consistent SEIM estimates of elastic complementary energy, coherency energy and phase transformation strains corresponding to classical Reuss and Voigt conjectures are explicitly specified. The Voigt explicit relations served as inspiration for working out an original engineering practice-oriented semiexperimental SEIM estimates. They are especially conveniently applicable for an isotropic aggregate (composite) composed of a mixture of n isotropic phases. Using experimental data for NiTi alloy and adopting conjecture that it can be treated as an isotropic aggregate of two isotropic phases, it is shown that the NiTi coherency energy and macroscopic phase strain are practically not influenced by the difference in values of austenite and martensite elastic constants. It is shown that existence of nonzero fluctuating part of phase microeigenstrains field is responsible for building up of so-called stored energy of coherency, which is accumulated in pure martensitic phase after full completion of phase transition. Experimental data for NiTi alloy show that the stored coherency energy cannot be neglected as it considerably influences the characteristic phase transition temperatures of SMA material.

Keywords

SMA NiTi alloys Adaptive composite Macroscopic free energy functions Gibbs energy Micromechanics Coherence energy Stored coherency energy Ultimate phase transformation eigenstrains Self-equilibrated eigenstrains influence moduli SEIM Effective property estimates Martensitic phase transformation 

References

  1. 1.
    Abeyaratne, R., Knowles, J.K.: On the driving traction acting on a surface of strain discontinuity in a continuum. J. Mech. Phys. Solids 38, 345–360 (1990)ADSMathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bernardini, D.: On the macroscopic free energy functions for shape memory alloys. J. Mech. Phys. Solids 49, 813–837 (2001)ADSCrossRefMATHGoogle Scholar
  3. 3.
    Bernardini, D., Pence, T.J.: Shape-memory materials, modeling. In: Schwartz, M.J. (ed.) Encyclopedia of Smart Materials, vol. 1, 2nd edn, pp. 964–980. Wiley, London (2002)Google Scholar
  4. 4.
    Bernardini, D., Masiani, R.: New micromechanical estimates of the interaction energy for shape memory alloys modeled by a two-phases microstructure. Solids Math, Mech (2014). doi:10.1177/1081286514562291 Google Scholar
  5. 5.
    Dvorak, G.J., Benveniste, Y.: On transformation strains and uniform fields in multiphase elastic media. Proc. R. Soc. Lond. A 437, 291–310 (1992)ADSMathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Dvorak, G.J.: Micromechanics of Composite Materials. Springer, New York (2013)CrossRefGoogle Scholar
  7. 7.
    Follansbee, P.S.: Fundamentals of Strength—Principles, Experiment, and Application of an Internal State Variable Constitutive Model. The Minerals, Metals, & Materials Society, Wiley, Hoboken (2014)Google Scholar
  8. 8.
    Hill, R.: Elastic properties of reinforced solids: some theoretical principles. J. Mech. Phys. Solids 11, 357–372 (1963)ADSCrossRefMATHGoogle Scholar
  9. 9.
    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, 430–462 (2006)CrossRefGoogle Scholar
  10. 10.
    Lagoudas, D.C. (ed.): Shape Memory Alloys Modeling and Engineering Applications. Springer, New York (2008)MATHGoogle Scholar
  11. 11.
    Levin, V.M.: On the coefficients of thermal expansion of heterogeneous materials. Mech. Solids 2, 58–61 (1967)Google Scholar
  12. 12.
    Milton, G.W.: Theory of Composites, Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2002)Google Scholar
  13. 13.
    Müller, I., Seelecke, S.: Thermodynamic aspects of shape memory alloys. Math. Comput. Model. 34, 1307–1355 (2001)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Nemat-Nasser, S., Hori, M.: Micromechnics: Overall Properties of Heterogeneous Materials. North-Holland, Amsterdam (1993)MATHGoogle Scholar
  15. 15.
    Nishida, M., Kawano, H., Nishiura, T., Inamura, T.: Self-accommodation of B19\(\prime \) martensite in Ti–Ni shape memory alloys—Part I. Morphological and crystallographic studies of the variant selection rule. Philos. Mag. 92(17), 2215–2233 (2012)ADSCrossRefGoogle Scholar
  16. 16.
    Otsuka, K., Wayman, C.M. (eds.): Shape Memory Materials. Cambridge University Press, Cambridge (1998)Google Scholar
  17. 17.
    Patoor, E., Eberhardt, A., Berveiller, M.: Micromechanical modelling of superelasticity in shape memory alloys. J. Phys. IV C16, 277–292 (1996)MATHGoogle Scholar
  18. 18.
    Peultier, B., Ben Zineb, T., Patoor, E.: Macroscopic constitutive law of shape memory alloy thermomechanical behaviour. Application to structure computation by FEM. Mech. Mater. 38, 510–524 (2006)CrossRefGoogle Scholar
  19. 19.
    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)CrossRefMATHGoogle Scholar
  20. 20.
    Raniecki, B., Lexcellent, C., Tanaka, K.: Thermodynamic models of pseudoelastic behaviour of shape memory alloys. Arch. Mech. 44(3), 261–284 (1992)MathSciNetMATHGoogle Scholar
  21. 21.
    Raniecki, B., Tanaka, K.: On the thermodynamic driving force for coherent phase transformations. Int. J. Eng. Sci. 32(12), 1845–1858 (1994)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Raniecki, B., Lexcellent, C.: R\(_{L}\)-models of pseudoelasticity and their specification for some shape memory solids. Eur. J. Mech. A/Solids 13(1), 21–50 (1994)MathSciNetMATHGoogle Scholar
  23. 23.
    Raniecki, B., Lexcellent, C.: Thermodynamics of isotropic pseudoelasticity in shape memory alloys. Eur. J. Mech. A/Solids 13(1), 185–205 (1998)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Raniecki, B., Tanaka, K., Ziółkowski, A.: Testing and modeling of NiTi SMA at complex stress state—selected results of Polish–Japanese Research Cooperation, Material Sciences Research Internal, Special Technical Publication, vol. 2, pp. 327–334 (2001)Google Scholar
  25. 25.
    Rychlewski, J.: Unconventional approach to linear elasticity. Arch. Mech. 47(2), 149–171 (1995)MathSciNetMATHGoogle Scholar
  26. 26.
    Rosen, B.W., Hashin, Z.: Effective thermal expansion coefficients and specific heats of composite materials. Int. J. Eng. Sci. 8, 157–173 (1970)CrossRefGoogle Scholar
  27. 27.
    Sun, Q.P., Hwang, K.C.: Micromechanics constitutive description of thermoelastic martensitic transformation. In: Huchinson, J., Wu, T.W. (eds.) Advances in Applied Mechanics, vol. 31, pp. 249–298. Academic Press, New York (1994)Google Scholar
  28. 28.
    Volokh, K.Y.: Stresses in growing soft tissues. Acta Biomater. 2, 493–504 (2006)CrossRefGoogle Scholar
  29. 29.
    Walpole, L.J.: Elastic behaviour of composite materials: theoretical foundations. In: Yih, C.S. (ed.) Advances in Applied Mechanics, vol. 21, pp. 169–242. Academic Press, New York (1981)Google Scholar
  30. 30.
    Xiao Guo, Z. (ed.): Multiscale Materials Modeling Fundamentals and Applications. Woodhead Publishing Limited, Cambridge (2007)Google Scholar
  31. 31.
    Ziółkowski, A., Raniecki, B.: On the free energy function for Shape Memory Alloys treated as a two-phase continuum. Arch. Mech. 51(6), 785–911 (1999)MATHGoogle Scholar
  32. 32.
    Ziółkowski, A.: Pseudoelasticity of Shape Memory Alloys, Theory and Experimental Studies. Butterworth-Heinemann, Amsterdam (2015)Google Scholar

Copyright information

© The Author(s) 2016

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Institute of Fundamental Technological ResearchPolish Academy of SciencesWarsawPoland

Personalised recommendations