A rate-independent model for the isothermal quasi-static evolution of shape-memory materials. (English summary) Math. Models Methods Appl. Sci. 18 (2008), no. 1, 125–164. The paper addresses a phenomenological 3-dimensional model of martensitic transformation in polycrystalline shape-memory alloys. The quasistatic evolution of the displacement u and the internal variable z describing phenomenologically a mechanical (tensorial) effect of the detwinning is considered governed by the system ∂εW (ε, z) ∋ σ and ∂D(˙z) + ∂zW (ε, z) ∋ 0 where ε = ε(u) = 1 2 (∇u) ⊤ + 1 2 ∇u is the small-strain tensor, W a stored energy density, D a dissipation energy assumed positively homogeneous, and σ a prescribed external force. Here, “∂ ” denotes subdifferentials or differentials while ˙z is the time derivative of z. Due to positive homogeneity of D, this system is rate independent. The particular form W (ε, z) = 1 2 C(ε − z): (ε − z) + c1|z | + c2|z | 2 + I(z) + 1 2 ν|∇z|2 is considered, where C is the elastic-moduli tensor, c1, c2, ν> 0 are constants, and I is the indicator function of a ball describing the maximal possible transformation strain obtained by the detwinning of the martensite. The theory of rate-independent processes by A. Mielke et al. (see [A. Mielke and F. Theil, NoDEA Nonlinear Differential Equations Appl. 11 (2004), no. 2, 151–189; MR2210284 (2006m:47119)], or a survey by Mielke [in Evolutionary equations. Vol
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