Three-dimensional asymptotic stress field in the vicinity of the circumferential line of intersection of an inclusion and plate surface^*

Abstract

A heretofore unavailable asymptotic solution pertaining to the stress field in the neighborhood of the circumferential line of intersection of an inclusion and one of the bounding (top or bottom) surfaces of a plate, subjected to far-field extension/bending (mode I), inplane shear-twisting (mode II) and torsional (mode III) loadings, is presented. A local orthogonal curvilinear coordinate system (ρ,φ,θ), is selected to describe the local deformation behavior of the afore-mentioned plate in the vicinity of the afore-mentioned circumferential line of intersection. One of the components of the Euclidean metric tensor, namely g₃₃, is approximated (ρ/a<<1) in the derivation of the kinematic relations and the ensuing governing system of three partial differential equations. Four different combinations of boundary conditions are considered: (i) free bounding surface of the plate and uncapped inclusion (free-free), (ii) bounding surface of the plate clamped by an infinitely rigid washer and inclusion covered with an infinitely rigid cap (clamped-clamped), (iii) free bounding surface of the plate and inclusion covered with an infinitely rigid cap (free-clamped), and (iv) bounding surface of the plate clamped by an infinitely rigid washer and uncapped inclusion (clamped-free). For dissimilar inclusion and plate (matrix) materials, the asymptotic stress fields in the vicinity of the line of intersection of an inclusion and the bottom (or top) surface of a plate, subjected to far-field mode I and mode II loadings, are singular, in all cases of the above described boundary conditions. In the case of the mode III (torsional) loading, the asymptotic stress fields are singular only for the free-clamped (or clamped-free) boundary condition. Numerical results presented include the effect of the ratio of the shear moduli of plate and inclusion materials, and also Poisson's ratios on the computed lowest eigenvalues.