We can understand the concept of crack tip plastic zones and the plasticity correction to crack length quite readily. From the definition of the stress intensity, based on the elastic stress field near a crack tip, i.e.:
we can see that, as r tends towards zero, the crack tip stresses become singular. This implies that a yielded region will exist in the material ahead of the crack for all reasonable stress values. The shape and size of the plastic zone can be determined, to a first order, from the simple models first proposed by Irwin. Consider a material with a simple elastic-perfectly plastic response (i.e. no strain hardening occurs). A first estimate of the plastic zone size ahead of the crack tip (Rp), along the plane of the crack, can be obtained by substituting the yield strength into the above equation (see figure below):
The plastic zone size is obtained as:
An approximate idea of the shape can be obtained by substituting the near-tip stresses into a yield criterion, e.g. the von Mises shear strain energy criterion, and allowing the angle of the stressed element to vary.
Importantly, Irwin observed that the presence of significant crack tip plasticity caused the specimen to behave as though it contained a crack of greater length than was actually the case. That is, the compliance of the specimen became greater as plasticity developed at the crack tip. This observation led him to propose a 'plastic zone correction' to crack length, based on a more accurate model of crack tip plastic zone size.
The first model above has truncated the elastic stress field in the near-tip region, where yielding occurs. Irwin calculated a more accurate estimate of plastic zone size, taking the necessary re-distribution of crack tip stresses (which accompanies yielding) into account. This leads to a larger plastic zone size as indicated in the figure below.
Areas B and C are equal, and the effective crack length is Aeff. A simple analysis indicates that rp = 2Rp with Aeff = a + Rp. Thus Irwin proposed that Rp represents a plasticity correction to crack length which should be applied when crack tip plasticity is relatively extensive, e.g. under plane strain conditions. Under such cases the stress intensity factor is corrected iteratively through taking account of the effective crack length. The procedure first calculates K using the actual crack length, then finds Rp using this value of K. Aeff is then found and the K value re-calculated. This iteration can be continued further if necessary.
Plasticity is important in fracture mechanics, as the extent of plasticity, relative to specimen dimensions and crack size, determines the state of stress (plane strain or plane stress) and whether LEFM is applicable or not. In turn, stress state affects the direction of planes of maximum shear stress and hence the fracture plane. Thus fracture proceeds perpendicularly to the maximum principal stress in plane strain, and at 45º to this direction in plane stress.
As a general rule, the stress state approaches plane strain when the plastic zone is about 1/15 of the crack length and material thickness. Plane stress occurs when the size of plastic zone tends towards the material thickness. If the plastic zone is of the same order of size as the crack length, LEFM would not be valid and yielding fracture mechanics (YFM) parameters must be used.