 Solution

Linear elastic fracture mechanics predicts a near-tip tensile stress varying approximately asA ductile material yields near the tip before that mathematical singularity can be realized. Plastic deformation blunts the crack and redistributes stress over a finite process zone, as shown in the right-hand diagram. Constraint can sustain stresses above the uniaxial yield stress, so the schematic plateau should not be interpreted as a universal exactly-${\sigma }_y$ cap.

A useful process-zone scale is $r_p\sim (K_I/{\sigma }_y{)}^2$ with a geometry- and constraint-dependent coefficient. Advancing the crack then requires energy for plastic deformation and other irreversible processes as well as forming new surfaces. Consequently the critical energy release rate is much larger than the reversible surface-energy term $2\gamma$ of a perfectly brittle idealization:In a small-scale-yielding plane-strain description, $G_c=K_{IC}^2/E^{\prime }$ with $E^{\prime }=E/(1-{\nu }^2)$. Large-scale yielding requires an elastic-plastic measure rather than treating a linear-elastic stress-intensity criterion as exact. The increased dissipation and crack-tip blunting explain the larger energy requirement.