Estimates on the Mixture Function for Multiphase Problems in Elasticity

Abstract

In the theory of shape memory alloys and other crystallographic problems it is the basic assumption, that each mesoscopic part of the crystal can choose to be in one of the n allowed phases. We distinguish these phases by their stored—energy densities

$$W(E,{e_i}) = \frac{1}{2}\langle \langle {C_i}[E - {A_i}],E - {A_i}\rangle \rangle + {\alpha _i},\quad i = 1, \ldots ,n,$$

where e _i ∊ ℝ is the ith unit vector and $E = {\nabla _{sym}}u = \frac{1}{2}(\nabla u + \nabla {u^T}) \in \mathbb{R}_{sym}^{d \times d}$ is the linearized tensor.