This paper presents the exact solution of the linear elastic problem of crack interaction with an interface in a laminated media, and application of the solution to the problem of hydraulic fracture confinement. The effect of an interface on hydraulic fracture confinement in a laminated (stratified) medium has been addressed in a number of analytical and experimental investigations since the early 1960s. The problem of crack interaction with a bimaterial interface, that includes an identification of the power of the stress singularity and evaluation of the stress intensity factor (SIF), is conventionally reduced to a solution of singular integral equations. Numerical techniques are commonly employed in solving the problem. However, numerical solutions of a singular integral equation have in general certain limitations that can lead to numerical errors. Particularly, a recently published solution of the crack interaction with an interface presented in graphic form and tables convenient for applications contains significant numerical errors. The exact (analytical) solution of the problem presented in this paper has been obtained by means of the Weiner-Hopf method. An asymptotic representation of the stress field around the crack tip in a vicinity of an interface is extracted from the exact solution. Particularly, the SIF Green's function due to the unit double force applied at the crack faces is derived for arbitrary distance s of the crack tip from the interface. This paper also introduces an interface toughness index (ITI) X to reflect the potential fracture confinement by the interface. ITI characterizes the crack approach to the interface as stable if X > 1 (K1 --> 0 with å--> 0), or unstable if X<1 (K1 --> 8 with å --> 0). The interface toughness index ?is a function of both the shear moduli and Poisson's ratios of the materials on both sides of the interface. The stability conditions are different from those published previously. An experimental investigation of this proposition is outlined and various scenarios of fracture propagation and confinement based on crack stability analysis and stress distribution in the vicinity of the crack tip on both sides of the interface are discussed.
The problem of determining the elastic stress field in the vicinity of a crack tip, approaching a bimaterial interface, has various practical applications in layered materials such as laminated composites and stratified rocks and has been addressed by many authors. Here we refer to only a few publications that are directly relevant to the scope of the present work. The very first formulation and solution of the problem for semi-infinite crack perpendicular to the bimaterial interface with the crack tip positioned on the interface is due to Zak & Williams (1963). The crack tip fields for a finite crack with its tip at the interface were considered by Khrapkov (1968). This problem is the conjugate one to the problem of a semi-infinite crack perpendicular to the interface with its tip located some distance from the interface (one of the problems considered in the present work) in the sense that both lead to the same Weiner-Hopf equation. Erdogan with co-workers derived singular integral equations and solved them numerically for semi-infinite and finite cracks perpendicular to the interface and fully imbedded in one of the half planes, as well as cracks terminating at and crossing the interface (Cook & Erdogan 1972, Erdogan & Biricikoglu 1973).
