Applications of the two-state conservation integrals for analyzing wedge-type singular boundary layers

Abstract

The determination of the stress intensity as well as the stress singularities of wedge vertices has been a major subject in fracture mechanics community. In two-dimensional wedges, the stress singularities and their stress intensities can be computed accurately with the complex potentials. On the other hand in three-dimensional wedges, most works did not look into the near-tip stress intensities of the singular stress field, but concentrated on calculating only the stress singularities at the intersection of a crack front with a free surface and at various three-dimensional wedge vertices. Only a few studies have tried to compute the stress intensities as well as the stress singularities. This is partly because three-dimensional problems are in itself very complicated and partly because any reliable methodology to compute stress intensities or a fracture parameter like the J-integral for three-dimensional cracks was not available. In the present work we examine the stress state around the vertices of two- and three-dimensional wedges. Moreover, for the first time a general and systematic computational methodology is proposed for computing the singular stress states near two- and three-dimensional wedge vertices with the aid of the two-state M-integral and the eigenfunction expansion. To compute the stress intensity near a wedge vertex, we employ the path or surface independence of the two-state M-integral, and the auxiliary solution. The auxiliary solution is obtained from the complementary eigenvalue, which is also the eigenvalue of the problem under consideration and satisfies the complementarity relationship in the M-integral sense. The existence of the complementary eigenvalue in the M-integral sense, which comprises the key to success of the present computational scheme together with the path or surface independence of the two-state M-integral, is verified numerically for three-dimensional generic wedges. The complementarity relationship of the eigenvalue...