Contacts and cracks of complex shapes: Crack-contact dualities and relations between normal and shear compliances

Abstract

We discuss contacts and cracks of complex shapes, and focus on the following issues: (1) Crack-contact duality – the correspondence between compliances of contacts and cracks of the same shape. For a broad class of shapes (all convex and some concave ones) the correspondence involves shape factor M ≡ π〈 a〉〈 a −1〉 −1/ A where a( ϕ) is the distance from the centroid to boundary points and A is the area. It is controlled mostly by the extent of shape elongation. (2) Relations between the normal and shear compliances of cracks and contacts. The two are relatively close, and this has implications for the anisotropy due to multiple cracks or contacting rough surfaces. It also allows extension of the elasticity-conductivity connections to the shear compliances; (3) For the overlapping shapes with known solutions for each of the component shapes, a simple “summation rule” is suggested. (4) Comparison of two approximate methods of finding compliances of non-elliptical domains – of Fabrikant and of Boyer–Greenwood.