Hypersingular boundary integral equation for axisymmetric elasticity

Abstract

SUMMARYAn axisymmetric hypersingular boundary integral formulation for elasticity problems is presented in this paper. The hypersingular and strong-singular fundamental solutions are derived and their singular behaviour is discussed in detail for di erent locations of the source point. Several free terms arise from the limiting process when generating hypersingular boundary integral equations, including an extra one speciÿc to the axisymmetric formulation which does not appear in two and three dimensional cases. The singularity subtraction technique is used to regularize all strong-singular and hypersingular integrals, and their evaluation procedure is explained. Finally, the developed formulation is assessed through simple numerical tests.

Self-regular stress integral equations for axisymmetric elasticity

“…When the source point nAB approaches some boundary point xAqB, that is when JxÀnJ-0, the kernel function D becomes singular of a type O(1/JxÀnJ), and a kernel function S hypersingular of a type O(1/JxÀnJ 2 ) [1]. Thus, when calculating stresses or deformations at a point, which is placed close enough to the boundary, the integrand in Eq.…”

“…Assume that B is axially symmetric and symmetrically loaded with respect to the axis of symmetry Oz. The integral equation for determination of stresses at an internal source point nAB, neqB according to [1] can be written as…”

“…The pioneering work concerning regularization of hypersingular stress integral equations for the axisymmetric elasticity was that by de Lacerda and Wrobel [1]. The further researches and a new more convenient solution strategy of the hypersingular equations with their previous regularization were presented by Mukherjee [2].…”

“…Therefore, the regularization approaches [1,2] are actually unsuitable for calculation of stresses in the whole domain continuously up to the boundary, because of the boundary layer effect, which arise due to the numerical integration of nearly singular integrals [3]. According to [4], the accurate evaluation of stresses at internal points that are close to the boundary requires the usage of selfregular (continuous) stress boundary integral equations, which limits to the boundary exist.…”

Galerkin boundary integral analysis for the axisymmetric Laplace equation

“…Beginning in the mid-1970s [1][2][3], collocation solutions of integral equations for axisymmetric problems have been extensively considered in the literature [4]. Recent work has focused on axisymmetric elasticity [5][6][7][8], in particular for fracture and contact analysis. Although the Galerkin approach has probably been applied to solve axisymmetric boundary integral equations, we have been unable to locate any papers on this subject.…”