Grain Interaction and Elastic Strain Distribution in Polycrystalline Materials

Statistical distributions of the elastic strain and stress tensor components in the grains of polycrystalline materials are necessary to calculate the probabilities of various local critical events, such as damage and others, which are of random origin due to the stochastic grain structure. Many experimental and computational studies suggest that these distributions can be approximated by a normal distribution. The normal distribution parameters are determined from histogram-like plots obtained experimentally or by computer simulation. Most published histogram distributions are highly skewed, in contrast to the normal distribution. Here we present a new direct calculation method for the probability densities of the elastic strain tensor components. The method uses an integral equation for strains in heterogeneous solids, which reduces the solution of the boundary value problem of polycrystal deformation to the sum of solutions of some problems for neighboring grains. The focus is on the influence of random grain interactions on the strain distribution. Calculations are carried out for polycrystals with different elastic symmetries and degrees of grain anisotropy. All probability densities are finite, asymmetric, and noticeably different from Gaussian ones. It is shown that very few particularly located neighboring grains (of dozens) have a much greater effect on the distribution pattern and limiting values of the strain tensor components than all the others.