Stress-strain state of an anisotropic plate with an elliptic hole and thin rigid inclusions

Abstract

The stress-strain state of an anisotropic plate containing an elliptic hole and thin, absolutely rigid, curvilinear inclusions is studied. General integral representations of the solution of the problem are constructed that satisfy automatically the boundary conditions on the elliptic-hole contour and at infinity. The unknown density functions appearing in the potential representations of the solution are determined from the boundary conditions at the rigid inclusion contours. The problem is reduced to a system of singular integral equations which is solved by a numerical method. The effects of the material anisotropy, the degree of ellipticity of the elliptic hole, and the geometry of the rigid inclusions on the stress concentration in the plate are studied. The numerical results obtained are compared with existing analytical solutions.Let an infinite rectilinearly anisotropic plate of thickness h be weakened by an elliptic hole with the contour L 0 = {(x/a) 2 + (y/b) 2 = 1} and a set of thin, absolutely rigid inclusions shaped like smooth open curves L j (j = 1, k ). The curves do not intersect each other and the contour L 0 . For each contour L j , we determine the normal vector n(t) (t ∈ L j ) directed to the right when passing from the points a j to the points b j (Fig. 1). The plate is loaded by external forces X n + iY n at the hole contour and the forces σ ∞ x , σ ∞ y , and τ ∞ xy at infinity. Each inclusion can perform rigid-body translations and rotations:Here c j is a complex constant and ε j is the unknown or specified rotation of the rigid inclusion L j . The plus and minus signs refer to the left and right faces of the inclusion, respectively. It is assumed that a generalized plane stress state occurs in the plate and the rotation vanishes at infinity. The stresses in the plate can be expressed in terms of two analytical functions Φ ν (z ν ) (ν = 1, 2):(z ν = x + μ ν y and μ ν are the roots of the corresponding characteristic equation with positive imaginary parts [1]). We write the functions Φ ν (z ν ) (ν = 1, 2) as