The crack growth resistance curve (K_R) has shown good promise for materials where limited (small-scale) yielding occurs in front of the crack tip.  Difficulties in estimating crack tip plasticity under large-scale yielding conditions, led Wilhem [1974] to an alternate failure criterion based on the J-integral [Rice, 1968].  For a basic introduction to the J-integral see Section 11.

Wilhem’s J-integral criterion is similar to the K_R -curve criterion; it states that failure will occur when the following conditions are met:

where J is the value of the applied J-integral and J_R is the value of the J-integral representing the material resistance to fracture.  The applied stress (s_f) corresponding to Equation 4.2.5 is defined as the fracture stress.  Since the effect of large-scale yielding can be appropriately incorporated through a suitable elastic-plastic model in the estimation of J-integral, it becomes an effective parameter for predicting failure under plane stress conditions where the plastic zone size is significantly large.

The crack resistance curve for the tearing failure is now represented by ÖJ_R vs. Da curve instead of K_R vs. Da curve.  The use of ÖJ_R rather than J_R is justified by the fact that ÖJ is directly related to the stress-intensity factor for elastic behavior through the equation

where E¢ is the elastic modulus (E) for plane stress conditions and E/(1-v²) for plane strain conditions.

For different levels of applied load, the J-integral can be computed using finite element techniques for the structure of interest for a series of different crack sizes; the ÖJ versus crack length curve is illustrated in Figure 4.2.12a for a constant level of applied stress.  It is noted that this curve will incorporate the influence of material properties (yield strength and strain hardening exponent) through the finite element analysis.  In a manner similar to the stress-intensity factor type resistance curve, i.e. the K_R curve. The resistance curve based on ÖJ_R can be experimentally obtained [Griffis & Yoder, 1974; Verette & Wilhem, 1973].  A ÖJ_R versus crack movement (Da) curve, i.e. the J-integral resistance curve, is schematically illustrated in Figure 4.2.12b.  The failure criterion is also based on the tangency conditions between the ÖJ versus crack length curve and the ÖJ_R versus crack movement curve.  To obtain this condition, the ÖJ_R vs. Da curve can be superimposed on the plot of ÖJ vs. a curve such that at some crack length these two curves are tangent to each other as shown in Figure 4.2.12c.  The corresponding crack length then defines the critical crack size of the structure for the fracture stress, s_f.

Figure 4.2.12.  Schematic Illustration of the Individual and Collective Parts of a J_R Fracture Analysis