Elastic fields of surface steps and notches and associated cracks

Abstract

Surface notches or steps may induce local stress concentration. As a consequence, they play a central role in oxidation, corrosion, and fatigue cracking in engineering practice. In this paper, we supply analytic solutions of half-space steps and notches with or without cracks. Full-field displacements and stresses, general stress intensity factors (SIFs) at the root of steps and notches or the crack tips after crack initiation, as well as separation across the crack plane, are readily calculated. For a step of height $h$ on the half-space, it generates a stress field around the root in the form of $\sigma (r,\gamma ,\alpha )={\sigma }_0{K}^{\ast }(\gamma ,\alpha )\left({\left(\frac{r}{h}\right)}^{{\lambda }_1\left(\alpha \right)}+o\left({\left(\frac{r}{h}\right)}^{{\lambda }_1\left(\alpha \right)}\right)\right)$, where ${\lambda }_1$ is the order of singularity given in terms of the slope angle $\alpha$ of the step with ${\lambda }_1\left(\alpha \right)\approx \frac{\pi }{\alpha +\pi }\sqrt{\frac{1+\text{sinc}(\alpha +\pi )}{1-\frac{5}{6}\text{sinc}(\alpha +\pi )}}-1\text{for}\text{sinc}\left(x\right)=\frac{sinx}{x}$, and $K(\gamma ,\alpha )$ is the general SIFs associated with steps and notches. The same conclusion applies to a V-notch of depth $h$ and an opening angle of $\pi -\alpha$. When a crack emanating from the root of the step or notch, the SIFs in terms of the newly formed crack of length $a$ follows $\frac{K}{{\sigma }_0\sqrt{\pi a}}=Q_{{\lambda }_1}(\alpha ,\beta ){\left(\frac{a}{h}\right)}^{{\lambda }_1}$ for short cracks, and $\frac{K}{{\sigma }_0\sqrt{\pi a}}=Q_{-1}\left(\beta \right){\left(\frac{a}{h}\right)}^{-1}+k^{\ast }$ for long cracks, where $\beta$ describes the direction of the crack and $Q_{{\lambda }_1}$ and $Q_{-1}$ are dimensionless factors accounting for the geometry difference. The analysis presented here not only reveals the intriguing interplay between cracks and steps (or notches), but also supplies an analytical tool for reliability analysis of steps and notches beyond the reach of perturbation based analysis: the distinct stress singularity before and after crack initiation excludes an expression of SIFs of the cracked part based on the linear composition of SIFs from its uncracked status.