#### Vol. 1, No. 8, 2006

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A closed-form solution for a crack approaching an interface

### Boris Nuller, Michael Ryvkin and Alexander Chudnovsky

Vol. 1 (2006), No. 8, 1405–1423
##### Abstract

A closed-form solution is presented for the stress distribution in two perfectly bonded isotropic elastic half-planes, one of which includes a fully imbedded semi-infinite crack perpendicular to the interface. The solution is obtained in quadratures by means of the Wiener–Hopf–Jones method. It is based on the residue expansion of the contour integrals using the roots of the Zak–Williams characteristic equation. The closed-form solution offers a way to derive the Green’s function expressions for the stresses and the SIF (stress intensity factor) in a form convenient for computation. A quantitative characterization of the SIF for various combinations of elastic properties is presented in the form of function the $c\left(\alpha ,\beta \right)$, where $\alpha$ and $\beta$ represent the Dundurs parameters. Together with tabulated $c\left(\alpha ,\beta \right)$ the Green’s function provides a practical tool for the solution of crack-interface interaction problems with arbitrarily distributed Mode I loading. Furthermore, in order to characterize the stability of a crack approaching the interface, a new interface parameter $\chi$, is introduced, which is a simple combination of the shear moduli ${\mu }_{s}$ and Poisson’s ratios ${\nu }_{s}\phantom{\rule{2.77626pt}{0ex}}\left(s=1,2\right)$ of materials on both sides of the interface. It is shown that $\chi$ uniquely determines the asymptotic behavior of the SIF and, consequently, the crack stability. An estimation of the interface parameter prior to detailed computations is proposed for a qualitative evaluation of the crack-interface interaction. The propagation of a stable crack towards the interface with a vanishing SIF is considered separately. Because in this case the fracture toughness approach to the material failure is unsuitable an analysis of the complete stress distribution is required.

##### Keywords
stress intensity factor, crack stability, bimaterial plane, analytic function