Evaluation of the degenerate scale for BIE in plane elasticity and antiplane elasticity by using conformal mapping

For a better understanding for the formulation of the degenerate scale problem by using the complex variable, preliminary knowledge is introduced. Formulation for the degenerate scale problem is based on the direct usage of the complex variable and the conformal mapping. After using the conformal mapping, the vanishing displacement condition is assumed on the boundary of unit circle. The complex potentials on the mapping plane are sought in a form of superposition of the principal part and the complementary part. The principal part of the complex potentials is given beforehand, and the complementary part plays a role for compensating the displacement along the boundary from the principal part. After using the appropriate complex potentials, the boundary displacement becomes one term with the form of g(R)−c (g(R) a function of R), where R denotes a length parameter. By letting the vanishing displacement on the boundary, or g(R)−c=0, the degenerate scale “R” is obtained. For four cases, the elliptic contour, the triangle contour, the square contour and the ellipse-like contour, the degenerate scales are evaluated in a closed form. For the case of antiplane elasticity, similar degenerate scale problems are solved.