Crack-tip fields with contained plasticity

At larger stresses, beyond SSY, for which the plastic zone extends deeper into the undeformed ligament t = D - a as shown in Fig. 12.6(b), the stress and strain distributions at the crack tip inside the yield zone are now very significantly altered, such that the strongly attenuated stresses are accompanied by more concentrated crack-tip plastic strains. To deal with a family of such problems a form of non-linear crack-tip analysis was introduced independently by Hutchinson (1968) and by Rice and Rosengren (1968). This approach is referred to as the HRR crack-tip analysis. 

In the analysis a non-linear incompressible (v = 0.5) elastic solid having a power-law constitutive relation of the form 

Alternatively, the total stored elastic energy WA over the face A of the part under a load P can be considered as a load potential, according to 

similar to the LEFM case, the model permits a generalized energy-release rate J that is defined as 

In analogy with the LEFM developments of Section 12.2.2 specific crack-tip-field expressions inside the plastic zone for the stress and strain distributions aij and Ejj = ., where i and j stand more specifically for r and 6, are 

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