An upper bound to the free energy of mixing by twin-compatible lamination for n-variant martensitic phase transformations
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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.
KeywordsShape memory alloys Martensitic phase transformations Relaxation Quasiconvexification Lamination
PACS62.20.−x 62.40. + i 64.70.Kb
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- 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
- 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.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.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
- 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