An ill-posed Cauchy type problem for an elastic strip

Abstract

This study deals with an incorrectly posed, plane elasticity, boundary value problem for a strip. The strip is loaded by a concentrated load of known intensity applied to one side and the displacements on this side are also known. The problem is therefore over-determined on one side of the boundary; in contrast no boundary conditions are specified on the other side of the strip. Therefore, the problem is ill-posed with the specified boundary conditions. The problem can be reduced to a system of integral equations derived from basic properties of holomorphic functions, which are used to prove uniqueness of the considered boundary value problem. An analytical solution of the problem is obtained by applying Fourier transforms. The inversion of the Fourier transform is performed with the use of the Stieltjes integral. This is a non-stable operation, which necessitates the application of a regularisation technique in order to build stable solutions. For numerical implementation we discuss the regularisation procedure based on the singular value decomposition truncation method.