An eigenfunction expansion solution for three-dimensional stress field in the vicinity of the circumferential line of intersection of a bimaterial interface and a hole

Abstract

An asymptotic solution pertaining to the stress field in the neighborhood of the circumferential line of intersection of an interface of a two-layer plate made of dissimilar isotropic materials and a through-hole, 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 g ₃₃, 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) open hole (free-free), (ii) hole fully filled with an infinitely rigid plug (clamped-clamped), (iii) hole partially (i.e., in the layer 2) filled with an infinitely rigid plug (free-clamped), and (iv) hole partially (i.e., in the layer 1) filled with an infinitely rigid plug (clamped-free). The computed eigenvalues for the clamped-free boundary condition can be obtained from their free-clamped counterparts by replacing G ₂ by G ₁ and ν₂ by ν₁, and vice versa. These two boundary conditions are then equivalent in the complementary sense. Numerical results presented include the effect of the ratio of the shear moduli of the layer materials, and also Poisson's ratios on the computed lowest eigenvalues.