On the Path Independency of the Point-wise J Integral in Three-dimensions

Abstract

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 $J_{X_1}$-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 $J_{X_1}$-integral may be achieved which indeed show path independency. We also show that computation of the path part of the $J_{X_1}$ 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 $J_{X_1}$-integral.