On the Elastic Interaction between Self‐stress Fields

Abstract

The physical properties of a solid material is often decisively influenced by its structure faults. Particularly the elastic interaction between the various defects plays an important role. The essential aspects of the interactions between defects can be properly described in the frame of the mechanics of continua (ESHELBY 1955; SCHAEFER, KRONM~LLER). Although the mathematical means to treat such problems are principially to our disposal, only few analytical expressions easily to handle for important interaction energies between structure defects are known in the literature (see e.g. BACON; WOLFER, MANSUR; and above all MI-CHEL 1979a).As according to MICHEL (1979a) the elastic interaction between N precipitates in a matrix can be attributed to the interaction between pairs of precipitates it is of particular interest to calculate usable analytical expressions for the elastic interaction of two defects in various position to each other and with respect to the matrix. For example the case of two spherical volume defects in the vicinity of a flat boundary (elastic semi-space) has been treated in several papers (see e.g. MICHEL 1979b). Considering the problem of the diffusion controlled growth of precipitates the case of two spherical precipitates A and B in a spherical matrix is of particular interest. Thus we will treat this problem.Assuming the precipitate A situated concentrically, but the precipitate B excentrically in the spherical matrix, the generally derived expression for the interaction energy in an elastic linear medium yields an analytically closed expression, namelyIn an elastic isotropic homogeneous medium at each arbitrary point holds and thus (1) can be written in the form which can be simplified tothat means only for the source A of the self-stresses the solution of b A of the self-stress problem has to satisfy the exact boundary condition, whereas for the source of the selfstresscs B the solution for the infinite medium (full space) is sufficient (see ESHELBY 1956; AUGUST).