On the path dependence of the J-integral in notch problems

Abstract

This study attempts to clarify the conditions under which the J-integral is path-independent in U- and V-shaped notch problems. The key is to determine the contribution to the J-integral evaluated in the global coordinate system from the second component of the J k -vector evaluated in the local coordinate system along the traction-free surfaces that form part of the integration path. It is found that the global J-integral is path-independent only if the projected contribution from the J 2-integral to J vanishes. The J-integral for a V-shaped notch is, strictly speaking, path-dependent even under remote symmetrical loading. This is due to the fact that, unlike in the case of a line- or plane-crack, the value of the J-integral calculated along a closed contour surrounding the V-shaped notch is dependent on the selection of the starting and ending points on the notch surface. In other words, the traction-free surface of the V-shaped notch does contribute to the J-integral due to the non-zero projected values induced from the J 2-integral. For a U-shaped notch, the path-independence of the J-integral is established if the integration path completely encloses the notch root. This is because both the upper and lower notch surfaces of the U-shaped notch are parallel to the geometric symmetrical line (the x 1-axis) and hence the projected values from the J 2-integral vanish. Furthermore, it is found that the small arc at the root of the notch (whether U- or V-shaped) also contributes to the J-integral even if the remote loading is symmetrical. These conclusions are derived by detailed analytical manipulations and by numerical examples; the analytical solution obtained by Lazzarin and Tovo [Int. J. Fracture 78 (1996) 3] and Lazzarin et al. [Int. J. Fracture 91 (1998) 269] for stress field in the vicinity of a notch root in an infinite elastic plane is used to calculate the contribution induced from the arc with different radii. Some useful results for studying the fracture and fatigue of notches are discussed.