Consider a nonlinear elastic body containing a crack,
the J integral is defined as
where is the strain energy density, is the traction vector, is an arbitrary contour around the tip of the crack, n is the unit vector normal to ; , , and u are the stress, strain, and displacement field, respectively.
Rice, J. R., 1968, showed that the J integral is a path-independent line integral and it represents the strain energy release rate of nonlinear elastic materials:
where is the potential energy, the strain energy U stored in the body minus the work W done by external forces and A is the crack area.
The dimension of J is