Symmetrically Distributed Eigenstrains In a Thin Sheet of Material

Abstract

A solution for the eigenstress problem in a half-space can be derived based on the following method: stresses in the finite body with traction-free boundary conditions can be derived from those corresponding to the infinitely extended body, by imposing the boundary relief: the stress field induced by the false normal stresses remaining on the boundary from the full-space solution, is subtracted from the latter solution. The method can be further extended to the case of a thin sheet of material, by applying the boundary relief twice, for each of the two limiting parallel planes bounding the sheet. Compared to the half-space case, an additional difficulty arises, as the Boussinesq fundamental solutions needed for the surface relief cannot be directly applied, because the thin sheet is not infinitely extended in one direction as the half-space. A suitable Green’s function for the thin elastic sheet compressed between two equal and collinear concentrated forces is employed in this paper. Four elastic states, three employing the infinite medium solution and one the calculation of stresses in a thin sheet, are superimposed to replicate the needed eigenstresses. Convolution and correlation products are calculated with state-of-the-art numerical methods based on the fast Fourier transform. Spherical and cuboidal inclusions, symmetrically distributed about the sheet mid plane, are simulated with the newly advanced Matlab code, and the results are compared with the ones obtained under the half-space assumption. In the first case, deviations of up to 15% are found, whereas the cuboidal inclusion exhibits more significant differences, due to the eigenstrains concentrated near the sheet limiting planes.