The elastic equilibrium of an inhomogeneous wedge with varying Poisson's ratio

Abstract

Abstract A system of two elastic equilibrium differential equations is studied in polar coordinates when Poisson's ratio is an arbitrary sufficiently smooth function of an angular coordinate and the shear modulus is constant. The elasticity modulus is also found to depend on the angular coordinate in this case. A general representation of the solution is proposed which leads to a vector Laplace equation and a scalar Poisson equation, the right-hand side of which depends on Poisson's ratio. When projected, the vector Laplace equation reduces to an elliptic system of differential equations with constant coefficients. Exact general solutions of the Laplace and Poisson equations are constructed in quadratures using a Mellin integral transformation and the method of variation of arbitrary constants. A contact problem for an inhomogeneous wedge is considered and the stress concentration at the vertex of the wedge is studied.