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

@article{citeulike:6101965, abstract = {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. Analytic solutions are known only for up to 3 variants, whereas cases of practical interests involve 7–13 variants. In this study we examine the n -variant case utilizing relaxation theory and produce a seemingly obvious but very powerful observation regarding a lower bound to the quasi-convex relaxation that makes practical evolutionary computations possible. We also examine in detail the 4-variant case where we explicitly show the relation between three different forms of the free energy of mixing: upper bound by lamination, the Reu{\ss} lower bound, and a lower estimate of the -measure bound. A discussion of the bounds and their utility is provided; sample computations are presented for illustrative purposes.}, author = {Govindjee, S. and Mielke, A. and Hall, G.}, citeulike-article-id = {6101965}, citeulike-linkout-0 = {http://dx.doi.org/10.1016/S0022-5096(03)00015-2}, citeulike-linkout-1 = {http://linkinghub.elsevier.com/retrieve/pii/S0022509603000152}, doi = {10.1016/S0022-5096(03)00015-2}, issn = {00225096}, journal = {Journal of the Mechanics and Physics of Solids}, keywords = {non-local-damage-deterministic}, month = {April}, number = {4}, pages = {I--XXVI}, posted-at = {2009-11-12 13:34:13}, priority = {2}, title = {The free energy of mixing for -variant martensitic phase transformations using quasi-convex analysis}, url = {http://dx.doi.org/10.1016/S0022-5096(03)00015-2}, volume = {51}, year = {2003} }

TY - JOUR ID - citeulike:6101965 L3 - citeulike-article-id:6101965 N2 - 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. Analytic solutions are known only for up to 3 variants, whereas cases of practical interests involve 7–13 variants. In this study we examine the n -variant case utilizing relaxation theory and produce a seemingly obvious but very powerful observation regarding a lower bound to the quasi-convex relaxation that makes practical evolutionary computations possible. We also examine in detail the 4-variant case where we explicitly show the relation between three different forms of the free energy of mixing: upper bound by lamination, the Reuß lower bound, and a lower estimate of the -measure bound. A discussion of the bounds and their utility is provided; sample computations are presented for illustrative purposes. IS - 4 JF - Journal of the Mechanics and Physics of Solids SN - 00225096 EP - XXVI TI - The free energy of mixing for -variant martensitic phase transformations using quasi-convex analysis VL - 51 SP - I KW - non-local-damage-deterministic AU - Govindjee, S AU - Mielke, A AU - Hall, G PY - 2003/04// UR - http://dx.doi.org/10.1016/S0022-5096(03)00015-2 ER -