|Definition of J Integral|
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
|J vs. and K|
For linear elastic materials, the J integral J is in fact the strain energy release rate, , and both are related to the stress intensity factor K in the following fashion: