The overall elastic energy of polycrystalline martensitic solids

Rise of Correlations of Transformation Strains in Random Polycrystals

“…As an applied displacement is imposed, our polycrystal may acquire microscopic strains ε which contain combined contributions of elastic and stress-free transformations (see [8]):…”

“…As a displacement (2.13) is imposed on the polycrystal, each grain may undergo a stress-free transformation into one of the three possible stress-free states. The overall energy W n (U, θ) in the polycrystal is determined by global minimization of the misfit energy W (U, ε T ) of the polycrystal over all admissible configurations [8] W n (U, θ) = min…”

“…Our study assumes that the final state of a polycrystal is determined by minimizers of a time-independent, dissipation-free misfit energy (see e.g. [6][7][8] and references therein). Importantly, however, we do not assumec that the grains in the polycrystal transform without elastic stresses (self-accommodation); see e.g.…”

“…The approximation (3.14) provides a justification, in a one dimensional context, of numerical schemes which are used in practical evaluation of the misfit energy [6][7][8]. The approximation (3.14) takes into account only interaction with the nearest neighbors.…”

“…Therefore the computational complexity of finding the misfit energy can be significantly reduced if (3.11) is replaced by (3.14). In [6][7][8] this truncation was implemented for general two-and three-dimensional polycrystals and the convergence was verified numerically. The following Proposition justifies this convergence analytically in the case of chains of rectangular grains, and provides an explicit exponential error estimate.…”