Comments on the eigenvalue formulation of problems with cracks, V-notched cracks, and corners

Abstract

An eigenvector formulation is often used for configurations containing cracks, V-notched cracks, and corners. Coefficients associated with eigenvectors which give a stress singularity at the crack tip or corner are the stress intensities which can be used in a failure criterion. A complex variable formulation is often used to obtain eigenvalues and eigenvectors. In such a formulation, a solution is assumed in terms of a complex parameter, z, raised to the power λ, where λ is the eigenvalue. The quantity zλ is multi-valued. However, previous authors have only considered one of the possible values of the function. In doing so, they may have overlooked important eigenvalues and/or eigenvectors associated with the problem. This paper reformulates, for the general finite opening crack and real eigenvalues, the eigenvector problem so as to consider the multi-valued nature of zλ. For the zero opening crack, V-notched crack and corner problem, it is shown that with the new formulation, no new real eigenvalues exist and for the zero-opening crack problem, it is also shown that no new eigenvector mode shapes exist. The first eigenvalue for a 90° corner problem was considered. No new eigenvector mode shapes were found. These findings suggest that the linear elastic results found in the literature are complete and correct, even if their derivation is somewhat deficient in generality.