Rise of Correlations of Transformation Strains in Random Polycrystals

Abstract. We investigate the statistics of the transformation-strains that arise in random martensitic polycrystals as boundary conditions cause its component crystallites to undergo martensitic phase transitions. In our one-dimensional polycrystal model the orientation of the n grains is given by an uncorrelated random array of the orientation angles θi, i = 1, . . . , n. Under imposed boundary conditions the polycrystal grains may undergo a martensitic transformation. The associated transformation strains εi, i = 1, . . . , n depend on the array of orientation angles, and they can be obtained as a solution to a nonlinear optimization problem. While the random variables θi, i = 1, . . . , n are uncorrelated, the random variables εi, i = 1, . . . , n may be correlated. This issue is central in our considerations. We investigate it in following three different scaling limits: (i) Infinitely long grains (L = ∞); (ii) Grains of finite but large height (L = L ≫ 1); and (iii) Chain of short grains (L = l0/(2n), l0 ≪ 1). With references to de Finetti’s Theorem, Riesz’ rearrangement inequality and near neighbor approximations, our analyses establish that under the scaling limits (i), (ii) and (iii) the arrays of transformation strains arising from given boundary conditions exhibit no correlations, long-range correlations and exponentially decaying short-range correlations, respectively.