Identities for the fundamental solution of thin plate bending problems and the nonuniqueness of the hypersingular BIE solution for multi-connected domains

Abstract

Four integral identities for the fundamental solution of thin plate bending problems are presented in this paper. These identities can be derived by imposing rigid-body translation and rotation solutions to the two direct boundary integral equations (BIEs) for plate bending problems, or by integrating directly the governing equation for the fundamental solution. These integral identities can be used to develop weakly-singular and nonsingular forms of the BIEs for plate bending problems. They can also be employed to show the nonuniqueness of the solution of the hypersingular BIE for plates on multi-connected (or multiply-connected) domains. This nonuniqueness is shown for the first time in this paper. It is shown that the solution of the singular (deflection) BIE is unique, while the hypersingular (rotation) BIE can admit an arbitrary rigid-body translation term in the deflection solution, on the edge of a hole. However, since both the singular and hypersingular BIEs are required in solving a plate bending problem using the boundary element method (BEM), the BEM solution is always unique on edges of holes in plates on multi-connected domains. Numerical examples of plates with holes are presented to show the correctness and effectiveness of the BEM for multi-connected domain problems.