The Crack Extension Force of a Curved Crack Derived from the Principle of Virtual Work

A general method to determine the crack extension force F related to a finite segment of a curved crack is presented. A path independent integral expression that holds in curvilinear coordinates is derived from the principle of virtual work. An appropriate virtual displacement field allows variation of the position of a crack tip. F is related to the path independent integral expression through variation of a total energy expression. To illustrate the applicability of the method F is derived for the conical crack in axisymmetric loading. For a plane crack with a straight front F is identical to the J-integral, Rice (1968).