"FFT-ASSISTED SOLUTION FOR THE EIGENSTRESS PROBLEM IN AN INFINITE ELASTIC MEDIUM "

Abstract

Finding the distributions of eigenstresses induced by eigenstrains regardless of their type is a fundamental problem in mechanical engineering, described by complex mathematical models. Analytical solutions exist only for a small number of particular distributions of eigenstrains. This paper advances a numerical solution for the eigenstresses due to arbitrary distributions of eigenstrains in an infinite space. The imposed discretization transforms the continuous problem space into a set of adjacent cuboids, each characterized by a single value calculated analytically in a chosen point, usually the cuboid centre. In this manner, continuous functions are replaced in the mathematical model by sets of values calculated in discrete points, which, if the discretization is fine enough, replicate well the continuous distributions. The contribution of the uniform eigenstrains from a specific cuboid, to the eigenstresses in the calculation point, expressed analytically in the literature, is used as a starting point. To reduce the high computational requirements for superposition, state-of-the-art spectral methods for the acceleration of convolution products are applied. A Matlab computer program was developed to implement the newly advanced method. The case of a cuboid containing uniform dilatational eigenstrains was first simulated for validation purposes. Small deviations from the analytical solution can be observed near the inclusion boundary, but their magnitude decreases with finer meshes, suggesting it’s a discretization related error. The results were then extended by considering radially decreasing eigenstrains inside an ellipsoid.