Journal of Elasticity

, Volume 92, Issue 2, pp 151–186 | Cite as

Stability and Elastic Properties of the Stress-Free B2 (CsCl-type) Crystal for the Morse Pair Potential Model

  • Venkata Suresh Guthikonda
  • Ryan S. Elliott


Solid-to-solid martensitic phase transformations are responsible for the remarkable behavior of shape memory alloys. There is currently a need for shape memory alloys with improved corrosion, fatigue, and other properties. The development of new accurate models of martensitic phase transformations based on the material’s atomic composition and crystal structure would lead to the ability to computationally discover new improved shape memory alloys. This paper explores the Effective Interaction Potential method for modeling the material behavior of shape memory alloys. In particular, an extensive parameter study of the Morse pair potential model of the stress-free B2 cubic crystal is performed. Results for the stability, potential energy, current unit cell volume, instantaneous bulk modulus, and the two instantaneous cubic shear moduli are presented and discussed. It is found that an Effective Interaction Potential model based on the Morse potential is appropriate for modeling transformations between the B2 cubic structure and the B19 orthorhombic structure, but is not likely to be capable of simulating the B2 cubic to B19′ monoclinic transformation found in the popular shape memory alloy NiTi. In fact, this conclusion may be extended to all types of pair interaction potential models.


Shape memory alloys B2 crystal Martensitic precursor behavior Morse potential 

Mathematics Subject Classifications (2000)

74N10 74N05 70G99 70C20 70B99 


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  1. 1.
    Bhattacharya, K.: Microstructure of Martensite: Why It Forms and How It Gives Rise to the Shape-Memory Effect. Oxford University Press (2003)Google Scholar
  2. 2.
    Carlsson, A.E.: Solid State Physics. Chapter Beyond Pair Potentials in Elemental Transition Metals and Semiconductors, vol. 43, pp. 1–91. Academic Press (1990)Google Scholar
  3. 3.
    Chadwick, P., Ogden, R.W.: Definition of elastic moduli. Arch. Ration. Mech. Anal. 44(1), 41–53 (1971)MATHMathSciNetGoogle Scholar
  4. 4.
    Duerig, T.W., Melton, K.N., Stockel, D., Wayman, C.M.: Engineering Aspects of Shape Memory Alloys. Butterworth-Heinemann (1990)Google Scholar
  5. 5.
    Elliott, R.S.: Multiscale bifurcation and stability of multilattices. J. Computer-aided Mat. Des. doi:10.1007/s10820-007-9075-8 (2008)
  6. 6.
    Elliott, R.S., Shaw, J.A., Triantafyllidis, N.: Stability of pressure-dependent, thermally-induced displacive transformations in bi-atomic crystals. Int. J. Solids Struct. 39(13–14), 3845–3856 (2002)MATHCrossRefGoogle Scholar
  7. 7.
    Elliott, R.S., Shaw, J.A., Triantafyllidis, N.: Stability of thermally-induced martensitic transformations in bi-atomic crystals. J. Mech. Phys. Solids 50(11), 2463–2493 (2002)MATHCrossRefADSMathSciNetGoogle Scholar
  8. 8.
    Elliott, R.S., Shaw, J.A., Triantafyllidis, N.: Stability of crystalline solids—II: application to temperature-induced martensitic phase transformations in bi-atomic crystals. J. Mech. Phys. Solids 54(1), 161–192 (2006)MATHCrossRefADSMathSciNetGoogle Scholar
  9. 9.
    Elliott, R.S., Triantafyllidis, N., Shaw, J.A.: Stability of crystalline solids—I: continuum and atomic-lattice considerations. J. Mech. Phys. Solids 54(1), 193–232 (2006)MATHCrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Ericksen, J.L.: The Cauchy and Born Hypothesis for Crystals, in Phase Transformations and Material Instabilities in Solids. Academic Press (1984)Google Scholar
  11. 11.
    Friesecke, G., Theil, F.: Validity and failure of the Cauchy-Born hypothesis in a two-dimensional mass-spring lattice. J. Nonlinear Sci. 12(5), 445–478 (2002)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Girifalco, L.A., Weizer, V.G.: Application of the morse potential function to cubic metals. Phys. Rev. 114(3), 687–690 (1959)CrossRefADSGoogle Scholar
  13. 13.
    Huang, K., Born, M.: Dynamical Theory of Crystal Lattices. Oxford University Press (1962)Google Scholar
  14. 14.
    Huang, X., Ackland, G.J., Rabe, K.M.: Crystal structures and shape-memory behavior of NiTi. Nat. Mater. 2(5), 307–311 (2003)CrossRefADSGoogle Scholar
  15. 15.
    Huang, X., Bungaro, C., Godlevsky, V., Rabe, K.M.: Lattice instabilities of cubic NiTi from first principles. Phys. Rev. B-Condensed Matter 65(1), 014108/1–5 (2002)ADSGoogle Scholar
  16. 16.
    Janssen, T., Tjon, J.A.: Microscopic model for incommensurate crystal phases. Phys. Rev. B 25(6), 3767–3785 (1982)CrossRefADSGoogle Scholar
  17. 17.
    Laing, P.G., Ferguson, A.B., Hodges, E.S.: Tissue reaction in rabbit muscle exposed to metallic implants. J. Biomed. Materi. Res. 1(1), 135–149 (1967)CrossRefGoogle Scholar
  18. 18.
    Milstein, F.: Mechanical stability of crystal lattices with 2-body interactions. Phys. Rev. B 2(2), 512–517 (1970)CrossRefADSMathSciNetGoogle Scholar
  19. 19.
    Milstein, F.: Morse function description of anharmonicity in pressure-volume relations of cubic metals. Phys. Status Solidi B-Basic Res. 48(2), 681–688 (1971)CrossRefGoogle Scholar
  20. 20.
    Milstein, F., Hill, R.: Theoretical properties of cubic-crystals at arbitrary pressure—I density and bulk modulus. J. Mech. Phys. Solids 25(6), 457–477 (1977)CrossRefADSGoogle Scholar
  21. 21.
    Milstein, F., Hill, R.: Theoretical properties of cubic-crystals at arbitrary pressure—II shear moduli. J. Mech. Phys. Solids 26(4), 213–239 (1978)CrossRefADSGoogle Scholar
  22. 22.
    Milstein, F., Hill, R.: Divergences among the Born and classical stability-criteria for cubic-crystals under hydrostatic loading. Phys. Rev. Letters 43(19), 1411–1413 (1979)CrossRefADSGoogle Scholar
  23. 23.
    Milstein, F., Hill, R.: Theoretical properties of cubic-crystals at arbitrary pressure–III stability. J. Mech. Phys. Solids 27(3), 255–279 (1979)CrossRefADSGoogle Scholar
  24. 24.
    Otsuka, K., Wayman, C.M.: Shape Memory Materials. Cambridge University Press (1998)Google Scholar
  25. 25.
    Pitteri, M., Zanzotto, G.: Continuum Models for Phase Transitions and Twinning in Crystals. Applied Mathematics, vol. 19. CRC Press (2002)Google Scholar
  26. 26.
    Putters, J.L.M., Sukul, D.M.K., de Zeeuw, G.R., Bijma, A., Besselink, P.A.: Comparative cell culture effects of shape memory metal <Nitinol®>, nickel and titanium: a biocompatibility estimation. Eur. Sur. Res. 24(6), 378–382 (1992)CrossRefGoogle Scholar
  27. 27.
    Ren, X., Miura, N., Zhang, J., Otsuka, K., Tanaka, K., Koiwa, M., Suzuki, T., Chumlyakov, Y.I.: A comparative study of elastic constants of Ti-Ni-based alloys prior to martensitic transformation. Mater. Sci. Eng., A Struct. Mater.: Prop. Microstruct. Process. 312(1–2), 196–206 (2001)Google Scholar
  28. 28.
    Riks, E.: Incremental approach to the solution of snapping and buckling problems. Int. J. Solids Struct. 15(7), 529–551 (1979)MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Soliqo, D., Zanzotto, G., Pitteri, M.: Non-generic concentrations for shape-memory alloys; the case of CuZnAl. Acta Mater. 47(9), 2741–2750 (1999)CrossRefGoogle Scholar
  30. 30.
    Truskinovsky, L., Vainchtein, A.: Quasicontinuum modelling of short-wave instabilities in crystal lattices. Philos. Mag. 85(33–34), 4055–4065 (2005)CrossRefADSGoogle Scholar
  31. 31.
    Watah, J.C., O’Dell, N.L., Singh, B.B., Ghazi, M., Whitford, G.M., Lockwood, P.E.: Relating nickel induced tissue inflammation to nickel release in vivo. J. Biomed. Materi. Res. 58(5), 537–544 (2001)CrossRefGoogle Scholar
  32. 32.
    Zhorovkov, M.F., Kulagina, V.V.: Phonon anomalies and martensitic transitions in BCC materials. Russ. Phys. J. 36(10), 917–923 (1993)CrossRefADSGoogle Scholar

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© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of Aerospace Engineering and MechanicsThe University of MinnesotaMinneapolisUSA

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