On the path independency of the point-wise J integral in three-dimensions

The asymptotic solution in the vicinity of a crack front in a three-dimensional (3-D) elastic domain is provided explicitly following the general framework in M. Costabel, M. Dauge and Z. Yosibash, 2004, SIAM Journal of Mathematical Analysis, 35(5), 1177-1202. Using it, we show analytically for several fully 3-D displacement fields (which are neither plane strain nor plane stress) that the pointwise path-area J(x1)-integral in 3-D is path-independent. We then demonstrate by numerical examples, employing p-finite element methods, that good numerical approximations of the path-area J(x1)-integral may be achieved which indeed show path independency. We also show that cornputation of the path part of the J(x1) on a plane perpendicular to the crack front is path dependent. However, one may still use this path integral computed at several radii, followed by the application of Richardson's extrapolation technique (as R -> 0) to obtain a good estimate for J(x1) -integral.