Elastic interaction of grains in polycrystalline materials

The method of solution of the problem of elastic interaction of grains in polycrystals based on field theory approach is proposed. The solution of boundary value problem for strains in grains decomposed into two parts - zero order solution, which coresponds to formal absence of intergranular interactions, and the part which responds for this interaction. The zero order solution takes into account the intragrain interaction of strains. Intergranular interaction is considered as perturbation to zero order solution or small parameter of the task. Representation of the exact solution in the form of infinite series of perturbation theory (in field theory terminology) transforms the integral equation of initial boundary values problem into infinite consequence of interconnected systems of integral equations for corrections of different orders to zero order solution. From mathematical point of view it is analogous to situation in the theory of many interacting particles in statistical physics (the chain of Bogolubov equations for multiparticles distribution functions) and quantum theory of interacting fields (chain of Dyson-Shwinger equations for field Green's functions). The neglection of inhomogenuity of strains within individual grain (but taking into account the difference of strains in different grains) reduces the infinite consequence of integral equations to infinite chain of interconnected linear algebraic equations systems which can be solved by contemporary numerical procedures. Coefficients of linear equations depend on shape and positional relationship of the grains, id est they are determined by microstructure of material. Interaction with nearer and more remote grains is taken into account. The numerical evaluations for influence of intergranular interaction on deformation fields in the closest neighbours approximation, adopted from quantum condensed matter theory, are obtained.