It is recognized that plastic deformation will occur at the crack tip as a result of the high stresses that are generated by the sharp stress concentration.  To estimate the extent of this plastic deformation, Irwin equated the yield strength to the y-direction stress along the x-axis and solved for the radius.  The radius value determined was the distance along the x-axis where the stress perpendicular to the crack direction would equal the yield strength; thus, Irwin found that the extent of plastic deformation was

Subsequent investigations have shown that the stresses within the crack tip region are lower than the elastic stresses and that the size of the plastic deformation zone in advance of the crack is between r_y and 2r_y.  Models of an elastic, perfectly plastic material have shown that the material outside the plastic zone is stressed as if the crack were centered in the plastic zone.  Figure 2.2.5 describes a schematic model of the plastic zone and the stresses ahead of the crack tip.  Note that the real crack is blunted as a result of plastic deformation.

Figure 2.2.5.  Small-Scale Yield Model for Restricted Crack Tip Plastic Deformation

If the extent of the plastic zone as estimated by Equation 2.2.5 is small with respect to features of the structural geometry and to the physical length of the crack, linear elastic fracture mechanics analyses apply.  Sometimes, the concept of contained yielding, as illustrated in Figure 2.2.5, is referred to as small scale yielding.  Most structural problems of interest to the aerospace community can be characterized by linear elastic fracture mechanics parameters because the extent of yielding is contained within a small region around the crack tip.