Dual boundary integral equation formulation in antiplane elasticity using complex variable

This paper investigates the dual boundary integral equation formulation in antiplane elasticity using complex variable. Four kinds of boundary integral equation (BIE) are studied, and they are the first complex variable BIE for the interior region, the second complex variable BIE for the interior region, the first complex variable BIE for the exterior region, and the second complex variable BIE for the exterior region. The first BIE for the interior region is derived from the Somigliana identity, or the Betti’s reciprocal theorem in elasticity. A displacement versus traction operator is suggested. After using this operator, the second BIE for the interior region is derived. Similar derivations are performed for the first and second BIEs for the exterior region. In the case of the exterior boundary, two degenerate boundary cases are studied. One is the curved crack case, and other is the case of a deformable line. All kernels in the suggested BIEs are expressed in terms of complex variable.