Singular solutions for an anisotropic plate with an elliptical hole

Abstract

A solution of the bending problem for a plate with an elliptical hole subjected to a point force (a singular solution) is obtained using the engineering theory of thin anisotropic plates and Lekhnitskii's complex potentials. The solution is constructed by conformal mapping of the exterior of the elliptical hole onto the exterior of a unit circle with evaluation of the Cauchy-type integrals over closed contours. Different versions of the boundary conditions on the holw contour are considered. In the limiting case where the ellipse becomes a slot, the solution describes the bending of a plate with a rectilinear crack or a rigid inclusion.Let a point bending moment m * = m x +im y be applied at the point with coordinates τ = ξ +iη in an infinite anisotropic plate with an elliptical hole Λ (Fig. 1). Boundary conditions for the bending moments and transverse shear force (static conditions) or deflections and slopes (kinematic conditions) are specified on the hole contour. The solution of this problem reduces to constructing two analytic functions F ν (z ν ) governing the stress-strain state of the plate, where z ν = x + µ ν y are generalized complex coordinates (ν = 1, 2) and µ ν are roots of the characteristicThe static boundary conditions are written in complex form as follows [1]:Here m(s) and p(s) are the normal bending moments and transverse shear forces distributed along the contour and C, C 1 , and C 2 are unknown real constants. Integration is performed along the arc of the contour from the starting point to the current point. Below, the contour is assumed to be traction-free [m(s) = 0 and p(s) = 0]. For a plate subjected to a point bending moment applied at an internal point, the complex potentials are given by ϕ ν (z ν , τ ν ) = B ν ln (z ν − τ ν ) + ϕ ν0 (z ν , τ ν ).