A Crack Induced by a Thin Rigid Inclusion Partly Debonded from the Matrix

The interaction of a thin rigid inclusion with a finite crack is studied. Two plane problems of elasticity are considered. The first one concerns the case when the upper side of the inclusion of length $2b$ is completely debonded from the matrix, and the crack formed symmetrically penetrates into the medium; its length is $2a>2b$. In the second model, the upper side of the inclusion is partly separated from the matrix, and $a<b$. It is shown that both problems are governed by singular integral equations that share the same kernel but have different right-hand sides, and their solution satisfy different additional conditions. Derivation of a closed-form solution to these integral equations is one of the main results of the article. The solution is found by reducing the integral equation to a vector Riemann–Hilbert problem with the Chebotarev–Khrapkov matrix coefficient. A feature of the method proposed is that the vector Riemann–Hilbert problem is set on a finite segment, while the original Khrapkov method of matrix factorization is developed for a closed contour. In the case, when the crack and inclusion lengths are the same, the solution is derived by passing to the limit $b/a\to 1$. It is demonstrated that the limiting case $a=b$ is unstable, and when $a<b$ and the crack tips approach the inclusion ends, the crack tends to accelerate in order to penetrate into the matrix. It is shown that the stresses and the tangential derivative of the displacement have the square root singularity at the crack and the inclusion tips. The stresses and the displacement derivative are monotonic at the external singular points, $\pm a$ and $\pm b$ for Models 1 and 2, and oscillate in small neighborhoods of the internal singular points, $|x/a-b/a|<\varepsilon$ and $|x/b-a/b|<\varepsilon$, for the first and second problems, respectively, and $0<\varepsilon\le 10^{-6}$. The potential energy release rate and the Griffith crack growth criterion are established for both models.