In 1968, Rice [1968b] published a paper describing a path independent integral (J) which was noted to be equal to the negative of the change in potential energy of deformation occurring during the infinitesimal growth of a crack in a nonlinear elastic material, i.e. he showed that

Rice’s path independent integral J was defined by [Rice, 1968a; 1968b]

where G is any contour surrounding the crack tip, traversing in a counter clockwise direction (see Figure 2.2.13), W is the strain energy density,  is the traction on G, and  is the displacement on an element along arc s.

Figure 2.2.13.  J-Integral Parameters Illustrated

Before elaborating on a detailed description of the parameters involved in the calculation of the J-Integral, it is useful to note that Equation 2.2.33 is the nonlinear elastic equivalent of Equation 2.2.12.  Thus, for linear elastic materials, J reduces to the value of the strain energy release rate, G, i.e.

and the J-integral is related to the stress-intensity factor through the expression

where E¢ is given by Equation 2.2.18.

Equations 2.2.35 and 2.2.36 are noted to be valid only when the material is behaving in a linear elastic fashion.  When values of the J-Integral are determined via Equation 2.2.34 using finite element methods applied to linear elastic cracked structures, Equation 2.2.36 provides the engineer with a simple energy-based method for obtaining stress-intensity factors as a function of crack length.

In the first subsection below, the calculations associated with developing the J-Integral for an elastic-plastic material are detailed.  In the second subsection, some engineering approximation methods for calculating the J-Integral are outlined.