Formulation of indirect BIEs in plane elasticity using single or double layer potentials and complex variable

Abstract

This paper provides a formulation for indirect BIEs in plane elasticity using single or double layer potentials and complex variable. There are two ways to obtain two kinds of layer and the relevant indirect BIEs. In the first way, the displacement expression at domain point is directly obtained from the Somigliana identity with necessary modification. In the second way, after placing some density functions, for example, the body force or the dislocation doublet, along the layers, one can obtain the displacement expression at domain point. For both single and double layers, the continuous or discontinuous properties for the displacement and traction are studied in detail when a moving point is passing through the boundaries. Formulations of the Dirichlet and the Neumann problems are proposed. The ranges for solving the boundary value problem by using the single or double layer potentials are clearly indicated. For the case of single layer, the degenerate scale problems for the finite multiply connected region and infinite multiply connected region are studied. For the case of double layer, a hypersingular BIE for crack can be formulated by assuming that the density functions are vanishing along a portion of boundary.