Two bonded half planes with a crack going through the interface

Abstract

Type of Report and Period CoveredContractor Report 14. Sponsoring Agency Code Supplementary Notes AbstractThe plane problem of two bonded elastic half planes containing a finite crack perpendicular to and going through the interface is considered. The problem is formulated as a system of singular integral equations with generalized Cauchy kernels. Even though the system has three irregular points, it is shown that the unknown functions are algebraically related at the irregular point on the interface and the integral equations can be solved by a method developed previously. The system of integral equations is shown to yield the same characteristic equation as that for two bonded quarter planes in the general case of the through crack, and the characteristic equation for a crack tip terminating at the interface in the special case.The numerical results given in the paper include the stress intensity factors at the crack tips, the normal and shear components of the stress intensity factors at the singular point on the interface, and the crack surface displacements. The plane problem of two bonded elastic half planes containing a finite crack perpendicular to and going through the interface is considered. The problem is formulated as a system of singular integral equations with generalized Cauchy kernels. Even though the system has three irregular points, it is shown that the unknown functions are algebraically related at the irregular point on the interface and the integral equations can be solved by a method developed previously. The system of integral equations is shown to yield the same characteristic equation as that for two bonded quarter planes in the general case of the through crack, and the characteristic equation for a crack tip terminating at the interface in the special case. The numerical results given in the paper include the stress intensity factors at the crack tips, the normal and shear components of the stress intensity factors at the singular point on the interface, and the crack surface displacements.

The inclusion problem with a crack crossing the boundary

“…-7-Singular behavior of the solution of integral equations having similar generalized Cauchy kernels was studied in detail in [1] and [2]. Following the complex function technique outlined in [6] and using the procedure described in [1] and [2], if we define the unknown functions f-j and f« in terms of unknown bounded functions g-j and g 2 and unknown powers a-| , 3-j .…”

“…In studying the fracture of composite materials which consist of more than one perfectly bonded homogeneous elastic phase with different mechanical properties, it was shown that the singular behavior of the stress state in the close neighborhood of a crack tip does not remain "self-simi1ar" as it enters and crosses an interface separating two phases of the composite [1,2]. If the crack tip remains in the same homogeneous medium as it propagates, during the crack propagation the characteristic *This work was supported by the National Aeronautics and Space Administration under the Grant NGR-39-007-011 and by The National Science Foundation under the Grant GK 11977.…”

“…A detailed treatment of this problem was given in [1] and [2] for the case in which the interface is a plane and the crack length and its distance to the interface are sufficiently small so that the perturbed stress field can be approximated by that of a crack in two bonded elastic half planes. However, in materials such as ceramics and fiber reinforced composites, the crack length is usually of the order of inclusion or fiber diameter and the stress state in the uncracked medium is quite different than that of two bonded half planes.…”

A crack perpendicular to the bimaterial interface in finite solid

“…Cook and Erdogan (1972) used the Mellin transform method to derive the governing equation of finite cracks perpendicular to an interface and obtained the stress intensity factors. Erdogan and Biricikoglu (1973) solved the problem of two International Journal of Solids and Structures 40 (2003) www.elsevier.com/locate/ijsolstr bounded half planes with a crack going through the interface. Bogy (1971) investigated the stress singularity of an infinite crack terminated at the interface with an arbitrary angle.…”

Mixed Boundary-value Problems in Mechanics

“…Description of such techniques also falls outside the scope of this article. Among thesê a one may mention the methods usedto evaluate such physical quantities as the stress intensity factors, the strain energy release rate, and the crack 91 opening displacement directly in terms of the (calculated) density functions of the integral equations (see, for example, [28] , [29] , [311- [34]). …”