Dual boundary integral equation formulation in plane elasticity using complex variable

This paper investigates the dual boundary integral equation formulation in plane elasticity using a complex variable. Four kinds of BIE are studied, and they are: (1) the first complex variable BIE for the interior region, (2) the second complex variable BIE for the interior region, (3) the first complex variable BIE for the exterior region, and (4) the second complex variable BIE for the exterior region. Using the Somigliana identity and letting the domain point approach a boundary point, the first complex variable BIE is obtained. Displacement versus traction operator is suggested. Using this operator and letting the domain point approach a boundary point, the second complex variable BIE is obtained. When the domain point approaches a boundary point, all limit processes are performed exactly through the generalized Sokhotski–Plemelj’s formulae. For the exterior problems, two degenerate boundary cases, the curved crack and the deformable curved line, are studied. Particularly, for the degenerate boundary case, or the shrinking curved crack case, four kinds of BIE are obtained.