Regularity condition and numerical examination for degenerate scale problem of BIE for exterior problem of plane elasticity

This paper investigates the regularity condition at infinity in the exterior boundary value problem (EBVP) of plane elasticity. It is found that the usual suggested kernel, or the displacement from a fundamental field in textbooks on BIE, do not satisfy the regularity condition when the loadings on contour are not in equilibrium. A new kernel from a revised displacement expression in the fundamental field is suggested, which satisfies the regularity condition. In this paper, a numerical technique for evaluating the degenerate scale in boundary integral equation (BIE) is suggested. In the technique, the scale of a notch is changed gradually. Once the determinant for an influence matrix is nearly equal to zero, the degenerate scale is found. In addition, the eigenfunctions, or the non-trivial solutions, can also be evaluated. Three examples are presented to examine the results obtained from numerical solution of the BVP. In those examples, the relevant exact solutions are known beforehand. In the examination, a suggested kernel that satisfies the regularity condition is used. It is found that if the used size is near the critical value, the improper solution exists. However, if the used size is different from the critical value by 10%, a sufficient accurate numerical solution can be achieved.