Look at sect.2.5 J-integral https://en.wikipedia.org/wiki/Energy_release_rate_(fracture_mechanics)#J-integral

to check that the conditions are met for G=J

Hi Vahid,

Would you mind if I clarify what you are interested in?

It might be helpful to set out some of the background based upon a crack in a body of linear elastic material. The crack may be placed in a region of the body where the nominal stresses would be composed of both direct and shear stresses, so the crack tip stress field would be contain components of all three modes of crack tip loading (I, II and III). The principle of elastic superposition can be used to define the crack tip field as the summation of the crack tip stresses that arise from each of the modes individually, and the magnitude of the stresses from each mode are determined by KI, KII and KIII. The previous answer from Victor Adlucky shows that the energy release rate from each mode can be added to make the total energy release rate.

G = J = KI2/E' + KII2/E' + KIII2/2mu

I assume that this could be rewritten as follows:

J = GI + GII + GIII

With

GI = JI = KI2/E'

GII = JII = KII2/E'

GIII = JIII = KIII2/2mu

Does this answer your question, or are you interested in some other conditions such as plasticity?

Regards,

Simon