Physically appropriate characterization of fatigue crack propagation rate in elastic–plastic materials using the $J$-integral concept

Abstract

The current paper discusses the physically correct evaluation of the driving force for fatigue crack propagation in elastic–plastic materials using the $J$-integral concept. This is important for low-cycle fatigue and for short fatigue cracks, where the conventional stress intensity range ($\Delta K$) concept cannot be applied. Using the configurational force concept, Simha et al. (J Mech Phys Solids 56:2876–2895, 2008) , have derived the $J$-integral for elastic–plastic materials with incremental theory of plasticity, $J^{\mathrm{ep}}$, which is applicable for cyclic loading and/or for growing cracks, in contrast to the conventional $J$-integral. The variation of this incremental plasticity $J$-integral $J^{\mathrm{ep}}$ is studied in numerical investigations conducted on two-dimensional C(T)-specimens with long cracks under cyclic Mode I loading. The crack propagates by an increment after each load cycle. The maximum load is varied so that small- and large-scale yielding conditions prevail. Three different load ratios are considered, from pure tension to tension-compression loading. By theoretical considerations and comparisons with the variation of the crack tip opening displacement $\delta _{\mathrm{t}}$, it is demonstrated that the cyclic, incremental plasticity $J$-integral $\Delta J_{\mathrm{actPZ}}^{\mathrm{ep}}$, which is computed for a contour around the active plastic zone of the growing crack, is physically appropriate to characterize the growth rate of fatigue cracks. The validity of the experimental cyclic $J$-integral, $\Delta J^{\mathrm{exp}}$, proposed by Dowling and Begley (ASTM STP 590:82–103, 1976), is also investigated. The results show that $\Delta J^{\mathrm{exp}}$ is correct for the first load cycle, however, not fully appropriate for a growing fatigue crack.