FFT-ASSISTED SOLUTION FOR THE EIGENSTRESS PROBLEM IN AN ELASTIC HALF-SPACE

Abstract

A numerical solution circumventing the mathematical complexity of the fully analytical solution of the eigenstress problem is advanced in this paper, whose strong point is the computational efficiency implied by the convolution and correlation theorems, assisted by the fast Fourier transform. The half-space solution is constructed by superposition of two solutions for the infinitely extended medium and an appropriate boundary value problem. The needed eigenstresses result by superimposing three stress states that can be obtained more easily. The first elastic state places the input eigenstrains in an infinite medium. A second infinite space state has a special distribution of eigenstrains, which geometrically mirrors the input eigenstrains with respect to the half-space boundary. Superposition of the two infinite medium states replicates the problem of the input eigenstrains in the half-space, except for the boundary, where a spurious distribution of normal tractions remains. The latter distribution generates a stress state that needs to be superimposed to the superposition of the first two states, yielding the half-space elastic state. The third state has a well-known solution developed in Contact mechanics, based on the Boussinesq fundamental solutions for the elastic half-space. Scenarios reported in the literature concerning uniform dilatational eigenstrains inside an ellipsoidal inclusion are replicated with the proposed method, validating the newly advanced computer program. New results are then obtained by considering dilatational eigenstrains that decrease proportionally with the ellipsoid radii. The latter scenario is of practical interest because it is met in elastic-plastic contacts under moderate load.