The Crack Problem for a Nonhomogeneous Plane

TLDR

The paper studies the plane elasticity problem of a nonhomogeneous medium with a crack. Assuming a constant Poisson ratio and an exponentially varying Young’s modulus, the authors formulate the half‑plane problem, derive an integral equation for the crack, and provide solutions for arbitrary boundary tractions and various loading conditions. The derived integral equation has a Cauchy‑type kernel, yielding a conventional square‑root stress singularity at crack tips; stress intensity factors are largely insensitive to Poisson ratio but strongly depend on the nonhomogeneity parameter β.

Abstract

In this paper the plane elasticity problem for a nonhomogeneous medium containing a crack is considered. It is assumed that the Poisson’s ratio of the medium is constant and the Young’s modulus E varies exponentially with the coordinate parallel to the crack. First the half plane problem is formulated and the solution is given for arbitrary tractions along the boundary. Then the integral equation for the crack problem is derived. It is shown that the integral equation having the derivative of the crack surface displacement as the density function has a simple Cauchy-type kernel. Hence, its solution and the stresses around the crack tips have the conventional square-root singularity. The solution is given for various loading conditions. The results show that the effect of the Poisson’s ratio and consequently that of the thickness constraint on the stress intensity factors are rather negligible. On the other hand, the results are highly affected by the parameter β describing the material nonhomogeneity in E (x) = E0exp(βx).