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The free-energy of mixing for n-variant martensitic phase transformations using quasi-convex analysis

Govindjee, Sanjay; Mielke, Alexandar; Hall, Garrett J.

UCB/SEMM-2001/04, Dept. of Civil and Environmental Engineering, University of California, Berkeley, 2001, 33 pages

The construction of effective models for materials that undergo martensitic phase transformations requires usable and accurate functional representations for the free energy density. The general representation of this energy is known to be highly non-convex; it even lacks the property of quasi-convexity. A quasi-convex relaxation, however, does permit one to make certain estimates and powerful conclusions regarding phase transformation. The general expression for the relaxed free energy is however not known in the n-variant case. Analytical solutions are known only for up to 3 variants, whereas cases of practical interest involve 7 to 13 variants. The authors examine the n-variant case utilizing relaxation theory. A seemingly obvious but very powerful observation regarding a lower bound to the quasi-convex relaxation that makes practical evolutionary computations possible is produced. The authors also examine in detail the 4-variant case where they explicitly show the relation between three different forms of the free energy of mixing: Kohn's upper bound by lamination, the Reuss lower bound, and a lower estimate of the H-measure bound. The bounds and their utility are discussed. Sample computations are presented for illustrative purposes.

Available online: http://nisee.berkeley.edu/documents/SEMM/SEMM-2001-04.pdf (2 MB)