Stress and Strain Field around Crack
To show how the proposal works, assume that the stress field in the elastic domain has been obtained from LEFM analysis. Irwin uses the stress field on the crack line у = 0 in the form
where K] (= S-s/жа) is the stress intensity factor and r is the distance from the crack tip and S is the far-field stress. Consider plane stress, Mises yield criterion and use the effective stress
a11 + a22 a11a12*
Suppose the material yields at r = ry where the effective stress equals the yield stress ay. К/ = S^/lnry. The load transmitted across the surface у = 0 in the plastic zone is P = 2ayry. If the condition a11 = a22 holds along the crack line even in the plastic zone, the equilibrium condition predicts the plastic zone size of
Note that the method of evaluating crack opening displacement involves the use of elastic solution even though a part of the material undergoes plastic deformation. Moreover, the crack opening displacement of elastic solution indicates that the material particles initially on the crack surface (a = 0 or —a0 < x < a0, y = 0) has opened into (a > 0), otherwise there will be no crack opening. Obviously, if the boundary conditions are expressed in terms of true stress, the traction boundary conditions must be satisfied not on the un-deformed (a = 0) surface but on the deformed (a > 0) surface it opens into. This shows blunting and requires ац, vanishing at the crack tip. Further, in the case of plane stress, а22 = aY. This information can be used to select an analytic function f for the plastic zone as follows.