A new boundary integral formulation for plane elastic bodies containing cracks and holes

This paper presents a new boundary integral formulation for a plane elastic body containing an arbitrary number of cracks and holes. The body is assumed to be linear elastic and isotropic, but can be of either finite or infinite extend. The cracks inside the body can be either internal or edge crack, and either straight or curvilinear; and the holes can be of arbitrary number and shape. Starting from Somigliana formula, we obtain a system of boundary integral equations by applying integration by parts. In complex variables notation, the stress and displacement components can be expressed in terms of Muskhelishvilis analytic functions, which are in turn written as functions of boundary traction and displacement data in the form of Cauchy integral. The complex boundary integral equations for traction involve only singularity of order 1/r, where r is the distance measured from the singular boundary points, and no hypersingular terms appear. This new boundary integral formulation provides an effective basis in solving problems both analytically and numerically. To illustrate the validity of our new integral formulation, a number of classical problems are re-examined analytically using the present formulation: (i) an infinite body containing a circular hole subject to far field biaxial stress, internal pressure, and a point force on the holes boundary respectively; and (ii) an infinite body containing a circular-arc crack under remote uniaxial tension. To illustrate the applicability of the present formulation for boundary element method analysis, two numerical examples for the interactions between two collinear cracks are considered and the results agree well with the existing solutions by for the case of finite rectangular plates and with Isida (cited in p. 195 of ) for the case of infinite plates.