Eigenvalue and eigenfunction analysis arising from degenerate scale problem of BIE in plane elasticity

A new boundary integral equation (BIE) of plane elasticity is suggested with the use of a novel kernel. The relevant homogenous equation is also suggested. The equation is studied in a discrete form, or it is reduced to an algebraic equation. From the condition that the value of a determinant vanishes, the degenerate scale (or the eigenvalue) and the non-trivial solution (or the eigenfunction) are obtained approximately. Except for the notch with symmetric configuration for two axes, computed results prove that there are two degenerate scales in general. The dependence of the eigenvalue and eigenfunction with respect to the translation or the rotation of notch is investigated. It is found that the eigenvalues are invariant with respect to the translation and the rotation of notch. However, the eigenfunctions are changed when the notch has a rotation. Several numerical examples that include a rectangular notch, a half-ring-shaped notch and a complicated notch configuration are presented with the computed eigenvalues and eigenfunctions.