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Definition of J Integral

Consider a nonlinear elastic body containing a crack,

the J integral is defined as

J Integral

where is the strain energy density, is the traction vector, Gamma is an arbitrary contour around the tip of the crack, n is the unit vector normal to Gamma; Stress, Strain, 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 Pi=U-W 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. G and K

For linear elastic materials, the J integral J is in fact the strain energy release rate, G, and both are related to the stress intensity factor K in the following fashion: