A thin-film-covered mode III crack with dislocation-free zones

Abstract

We use complex variable methods and the theory of singular integral equations to study a thin-film-covered mode III crack with dislocation-free zones (DFZs) under uniform remote anti-plane shear stress. The equilibrium condition is formulated in terms of a single singular integral equation constructed in the image plane via the solution of the problem of a single screw dislocation interacting with a completely coated crack and the method of continuously distributed dislocations. The singular integral equation is solved numerically using the Gauss–Chebyshev integration formula to arrive at the dislocation distribution function, the DFZ condition, the total number of dislocations in the plastic zone and the local mode III stress intensity factor at the crack tip.

Appendix

The general solution simultaneously satisfying the traction-free condition along the surface of the central crack and the continuity conditions of traction and displacement across the perfect film-substrate elliptical interface L can be derived as follows:

$$ \begin{gathered} f_{1} (\xi ) = h(\xi ) - \overline{h}\left( {\frac{1}{\xi }} \right), \, 1 \le \left| \xi \right| \le \rho^{{ - \tfrac{1}{2}}} ; \hfill \\ f_{2} (\xi ) = \frac{\Gamma + 1}{2}\left[ {h(\xi ) - Kh(\rho \xi ) - \overline{h}\left( {\frac{1}{\xi }} \right) + K\overline{h}\left( {\frac{1}{\rho \xi }} \right)} \right], \, \left| \xi \right| \ge \rho^{{ - \tfrac{1}{2}}} , \hfill \\ \end{gathered} $$

(A1)

in which, for convenience, we write $f_{2} (\xi ) = f_{2} (\omega (\xi )) = f_{2} (z)$. Once the single analytic function $h(\xi )$ is determined for different specific loadings, the two original analytic functions $f_{1} (\xi )$ and $f_{2} (\xi )$ can obtained via Eq. (A1).

For the case of a single screw dislocation with Burgers vector b located at $z = z_{0} = \omega (\xi_{0} )$ inside the cracked thin film, the analytic function $h(\xi )$ is given quite simply by

$$ h(\xi ) = \frac{b}{2\pi }\sum\limits_{j = 0}^{ + \infty } {K^{j} \ln (\rho^{j} \xi - \xi_{0} )} - \frac{b}{2\pi }\sum\limits_{j = 1}^{ + \infty } {K^{j} \ln (\rho^{j} \xi - \overline{\xi }_{0}^{ - 1} )} . $$

(A2)

Substitution of Eq. (A2) into Eq. (A1)₁ yields Eq. (5). We can check that the resulting $f_{2} (\xi )$ exhibits the following remote asymptotic behavior: $f_{2} (\xi ) \cong \frac{b}{2\pi }\ln \xi ,{\text{ as }}\xi \to \infty$.

For the case of remote loading, the analytic function $h(\xi )$ can be derived as

$$ h(\xi ) = \frac{{C\sigma_{32}^{\infty } }}{{\mu_{2} (\Gamma + 1)(1 - K\rho )}}\xi . $$

(A3)

Substitution of Eq. (A3) into Eq. (A1)₁ will yield Eq. (7).