Some comments on the geometry dependence of theJ versusc relation for plane strain crack growth

Abstract

Rice, Drugan, and Sham have recently presented a discussion on the elastic-plastic stress and deformation fields at the tip of a crack growing in an ideally plastic solid under plane strain small-scale yielding conditions. Coupling their asymptotic analysis results with a crack tip opening angle criterion which requires that a constant angle 0 (measured at a characteristic distancer m behind the tip) be maintained, leads to a differential equation describing crack growth: $$\frac{{dJ}}{{dc}} = \frac{{\sigma _0 \theta }}{\alpha } - \frac{{\beta \sigma _0^2 }}{{\alpha E}}\ln \lgroup {\frac{{eR}}{{r_m }}} \rgroup,$$ wherec = crack length, σo = yield strength,E = Young's modulus, andJ denotes the far-field value of theJ integral: (1 − ν2)K2/E for small-scale yielding conditions, where ν = Poisson's ratio. The asymptotic analysis thatβ = 5.08 (for ν = 0.3), but does not give the values of the parameters α andR. However, comparisons with finite element results suggest that α has approximately the same value for stationary and growing cracks, whileR scales approximately with the plastic zone size. For large-scale yielding, Rice, Drugan, and Sham argue that a similar growth equation is expected to apply with possible variations in α and β at least in cases which maintain triaxial constraint at the crack tip. They speculate thatR increases linearly withJ at first, but then saturates in the general yield state at some fraction of the dimension of the uncracked ligament. The present paper analyses the highly idealized Dugdale-Bilby-Cottrell-Swinden model of a growing crack tip, and the results provide support for the speculations regarding the magnitude ofR in the large-scale plasticity and general yield states.