Elastic-Plastic Analysis of Growing Cracks

In an extension of earlier studies by Rice and Sorensen, a discussion is presented on the elastic-plastic stress and deformation fields at the tip of a crack which grows in an ideally plastic solid under plane strain, small-scale yielding conditions. The results of an asymptotic analysis suggest the existence of a crack-tip stress state similar to that of the classical Prandtl field, but containing a zone of elastic unloading between the centered fan region and the trailing constant stress plastic region. The near tip expression for the rate of opening displacement δ at distance r from the growing tip is found to have the same form suggested by Rice and Sorensen ˙δ=α˙J∕σo+β(σo∕E)˙aln(R∕r) but now the presence of the elastic wedge causes β to have the revised value of 5.08 (for Poisson ratio ν = 0.3). Here, a = crack length, σo = yield strength, E = elastic modulus, and J denotes the far-field value, namely, (1 − ν2)K2/E for the small scale yielding conditions considered. The parameters α and R cannot be determined from the asymptotic analysis, but comparisons with finite element solutions suggest that, at least for small amounts of growth, α is approximately the same for stationary and growing cracks, and R scales approximately with the size of the plastic zone, being about 15 percent to 30 percent larger. For large scale yielding it is argued that a similar form applies with possible variations in α and β, at least in cases which maintain triaxial constraint at the crack tip, but in the fully yielded case R is expected to be proportional to the dimension of the uncracked ligament. The model crack growth criterion of Rice and Sorensen, requiring a critical δ at some fixed r from the tip, is reexamined in light of the more accurate solution. The results suggest that the J versus Δa relation describing growth will be dependent on the extent of yielding, although it is suggested that this dependency might be small for highly ductile materials, provided that a similar triaxial constraint is maintained in all cases.