An Analytical Solution for an Arbitrary Cavity in an Elastic Half-Plane

The complex variable method is used for an analytical solution for an arbitrary cavity in a homogeneous elastic half-plane. Proposing a method for solving the coefficients of the function by conformally mapping the region that contains an arbitrary cavity onto a circular ring utilizing a mapping function, which can be used for arbitrary shape cavity. Considering the stress-free condition of the upper surface, the relationship between two analytic functions is derived; consequently, the problem of solving two analytic functions is attributed to solving one analytic function, and the solve process for the analytic function in general case is given in detail, finally, achieving the solution of an arbitrary shape cavity in an elastic half-plane. The effects of cavity shape and depth were analyzed to the required terms of analytic function under a given accuracy. Examples for a vertical-wall semicircle cavity and a horseshoe cavity are given through the present method, and verification example is given by ANSYS software.