unitor,
To add to what rb1957 said:
The J-integral is strictly valid for a non-linear elastic material, so 'W' is the theoretically recoverable elastic strain energy. When you are dealing with an elastic-plastic material (that is not being unloaded) then you replace the strain energy term by the density of stress working, as if it was recoverable. You'll notice that if you unload, J becomes path dependent because the elastic-plastic material does not recover some of the strain elastically.