Closed-form solutions for a rectangular layer with central crack under anti-plane shear stress

We consider the problem of determining the stress distribution in a finite rectangular elastic layer containing a Griffith crack which is opened by internal shear stress acting along the length of the crack. The mode III crack is assumed to be located in the middle plane of the rectangular layer. The following two problems are considered: (A) the central crack is perpendicular to the two fixed lateral surfaces and parallel to the other two stress-free surfaces; (B) all the lateral surfaces of the rectangular layer are clamped and the central crack is parallel to the two lateral surfaces. By using Fourier transformations, we reduce the solution of each problem to the solution of dual integral equations with sine kernels and a weight function which are solved exactly. Finally, we derive closed-form expressions for the stress intensity factor at the tip of the crack and the numerical values for the stress intensity factor at the edges of the cracks are presented in the form of tables. © Institute of Mathematics and its Applications 2005; all rights reserved.