Inhomogeneities in grains of polycrystalline materials and Eshelby problem

The paper presents the method aimed at calculating inhomogeneous strain fields in grains of polycrystalline materials. The calculations are based on the earlier developed method of solving boundary values problem for inhomogeneous polycrystalline bodies by means of the original perturbation theory variant based on analogies with the quantum fields theory. The boundary value problem for inhomogeneous strain fields in a differential form transforms into the integral equation for strains tensor. The solution of the integral equation is formed as a series upon the intensity of strains interaction. This allows interpreting inhomogeneous strain at any point in a grain as a superposition of macrostrain, caused by boundary conditions and two components conditioned by intragrain and intergrain interaction. It is shown that in untextured polycrystals, despite the long range type of elastic interaction, one can take into account the interaction only with the nearest and second neighbor grains to evaluate how the intergrain interaction influences the inhomogeneity in the given grain. The contributions of interactions with farther grains mutually annihilate each other. The strain field inhomogeneous within one grain is approximated by the step-wise constant function. For that, each grain is divided into a great quantity of small subgrains, where subgrain strain fields are supposed to be homogeneous. This approximation reduces the integral equations for local strains into linear algebraic ones, which are solved numerically. The application of this method to a classical problem related to calculating strains in a spherical inclusion embedded into the infinite matrix gives Eshelby solution. The numerical evaluation of strain inhomogeneities is made using model zinc polycrystals. Close to boundaries in spherical grains the extreme strain values, caused by intergrain interaction, surpass mean strain values by 30 percent. The strains concentration is much higher in materials with a lower elastic symmetry of grains.