Limitations to leading-order asymptotic solutions for ELASTIC–PLASTIC crack growth

Abstract

Previous work has shown that there are significant discrepancies between leading-order asymptotic analytical solutions for the elastic–plastic fields near growing crack tips and detailed numerical finite element solutions of the same problems. The evidence is clearest in the simplest physically realistic case : quasi-static anti-plane shear crack growth in homogeneous, isotropic elastic–ideally plastic material. There, the sole extant asymptotic analytical solution involves a plastic loading sector of radial stress characteristics extending about 20° from ahead of the crack, followed by elastic unloading, whereas detailed numerical finite element solutions show the presence of an additional sector of plastic loading, extending from about 20 to about 50°, that is comprised of non-radial characteristics. To explore how the asymptotic analysis can completely miss this important solution feature, we derive an exact representation for the stress and deformation fields in such a propagating region of non-radial characteristics, as well as in the other allowable solution regions. These exact solutions contain arbitrary functions, which are determined by applying asymptotic analysis to the solutions and assembling a complete near-tip solution, valid through second order, that is in agreement with the numerical finite element results. In so doing, we prove that the angular extent of the sector of non-radial characteristics, while substantial until extremely close to the crack tip, vanishes in the limit as the tip is approached, and that the solution in this sector is not of variable–separable form. Beyond resolving the analytical–numerical discrepancies in this specific anti-plane shear problem, the analysis serves to caution, by explicit example, that purely leading-order asymptotic solutions to nonlinear crack growth problems cannot in general capture all essential physical features of the near-tip fields, and that the often-invoked assumption of variable–separable solutions is not always valid.