This subsection outlines the calculation of parameters involved in the J-Integral.  Consideration is given to W, , and G as well as the choice of material stress-strain behavior.

The strain energy density W in Equation 2.2.34 is given by

and for generalized plane stress

In Equation 2.2.34, the second integral involves the scalar product of the traction stress vector  and the vector  whose components are the rate of change of displacement with respect to x.  The traction vector is given by

and the displacement rate vector is given by

where u and v are the displacements in the x and y directions, respectively.

Typically, when evaluating the J-Integral value via computer, rectangular paths such as the one illustrated in Figure 2.2.14 are chosen.  Noted on Figure 2.2.14 are the values of the outward unit normal components and the ds path segment for the four straightline segments.  For loading symmetry about the crack axis (x-axis), the results of the integration on paths 0-1, 1-2 and 2-3 are equal to the integrations on paths 6-7, 5-6 and 4-5, respectively.  Thus, for such loading symmetry, one can write

Figure 2.2.14.  Rectangular Path for J Calculation

For paths of the type shown in Figure 2.2.14, the J-Integral can be evaluated by the integrations indicated in Equation 2.2.41.  The strain energy density W, appearing in Equation 2.2.41, is given by Equation 2.2.37, or by Equation 2.2.38 for plane stress conditions.  To integrate according to Equations 2.2.37 or 2.2.38, a relationship between stresses and strains is required.  For material exhibiting plastic deformations, the Prandtl-Reuss equations provide a satisfactory relationship.  For the case of plane stress, when the Prandtl-Reuss relations are introduced into Equation 2.2.38, Equation 2.2.38 becomes

where and  are the equivalent stress and equivalent plastic strain, respectively.  The strain energy density will have a unique value only if unloading is not permitted.  If loading into the plastic range followed by unloading is permitted, then W becomes multi-valued.  It follows that J is also multi-valued for this occurrence.

The statements made in the preceding paragraph would appear to seriously limit the use of J as a fracture criterion since the case of loading into the plastic range followed by unloading (i.e., the case for which J is multi-valued) occurs when crack extension takes place.  On the basis of a number of examples, Hayes [1970] deduced that monotonic loading conditions prevail throughout a cracked body under steadily increasing load applied to the boundaries, provided that crack extension does not occur.  Thus, valid J calculations can be performed for this case.