Periodic displacement and stress fields near a phase boundary in the isotropic elasticity theory

Abstract

The elastic stress and displacement fields near a periodic boundary separating two different media are derived in the frame of the classical isotropic elasticity theory by using a Fourier series analysis. The coefficients of the trigonometric terms are found from the solution of a linear system of twelve equations with twelve unknowns for any planar boundary structure which has a one-or a two-dimensional periodicity. Analytical solutions of this system are given for a structure having a one-dimensional periodicity only, or for two media having the same Poisson's ratio. Applications to extrinsic and intrinsic (or misfit) dislocation networks emphasize the close connection between the two corresponding displacement fields. For cases where the dislocations are parallel, coplanar and without any related long-range stresses, calculation gives estimates of the elastic energy contribution to the phase-boundary free energy.