Some problems in the antiplane shear deformation of bi-material wedges

Abstract

The antiplane shear deformation of a bi-material wedge with finite radius is studied in this paper. Depending upon the boundary condition prescribed on the circular segment of the wedge, traction or displacement, two problems are analyzed. In each problem two different cases of boundary conditions on the radial edges of the composite wedge are considered. The radial boundary data are: traction–displacement and traction–traction. The solution of governing differential equations is accomplished by means of finite Mellin transforms. The closed form solutions are obtained for displacement and stress fields in the entire domain. The geometric singularities of stress fields are observed to be dependent on material property, in general. However, in the special case of equal apex angles in the traction–traction problem, this dependency ceases to exist and the geometric singularity shows dependency only upon the apex angle. A result which is in agreement with that cited in the literature for bi-material wedges with infinite radii. In part II of the paper, Antiplane shear deformation of bi-material circular media containing an interfacial edge crack is considered. As a special case of bi-material wedges studied in part I of the paper, explicit expressions are derived for the stress intensity factor at the tip of an edge crack lying at the interface of the bi-material media. It is seen that in general, the stress intensity factor is a function of material property. However, in special cases of traction–traction problem, i.e., similar materials and also equal apex angles, the stress intensity factor becomes independent of material property and the result coincides with the results in the literature.

Introduction

Bonded wedges of different materials have been under consideration in the recent decades. Bogy (1971) and Dempsey and Sinclair, 1979, Dempsey and Sinclair, 1981 studied the in-plane problem of two edge-bonded elastic wedges of different materials. Ma and Hour (1989) analyzed the antiplane shear deformation problem of simple isotropic, bi-material and anisotropic wedges with infinite radii. All of the above-mentioned papers were devoted to the analysis of the order of stress singularity at the wedge apex. In fact, the stress fields may have a form of the order O($r^{-{\lambda }_s}$) near the apex (r → 0), in which λ_s is called the order of stress singularity. Ting (1986) studied the order of stress singularity at the tip of an interface crack in a single isotropic and anisotropic material under in-plane loading. Indeed, the base of his studies was the in-plane problem of a simple wedge in special apex angles π and 2π. The antiplane shear deformation problem of isotropic as well as anisotropic finite wedges was solved under different boundary conditions by Kargarnovin et al. (1997) and Shahani (1999), respectively. Analytical solutions for the displacement and stress fields were derived and explicit expressions for the order of stress singularity at the wedge apex were extracted. Meanwhile, the conditions for which the stress singularity does occur at the apex were derived as functions of geometry and material property. In a recent paper (Shahani, submitted), the author have derived closed form relations for the stresses directly from the series form equations of Kargarnovin et al. (1997), using mathematical techniques. It has been shown that the τ_θz-component is unbounded at the point of application of the concentrated tractions, however, the τ_rz-component is convergent at that point. Also, continuity of the τ_rz-component over the entire wedge is studied. The facts which were not apparent from the series form solutions of Kargarnovin et al. (1997). On the other hand, Fariborz (2004) has shown the continuity of the τ_rz-component of the paper Kargarnovin et al. (1997) along the arc r = h, numerically. Shahani and Adibnazari (2000) considered the antiplane shear deformation problem of perfectly bonded dissimilar wedges with infinite radii as well as bonded wedges with an interfacial crack. They extracted explicit expressions for the stress distribution in the bi-material wedge under traction–traction boundary conditions on the radial edges of the composite wedge and showed that in the special case of equal apex angles (but still with different materials), the stress distribution releases its dependency to the material property. In a recent paper, the author (Shahani (2001)) derived a closed form solution for an edge crack lying at the interface of two edge-bonded wedges and terminating to the apex.

The only paper which deals with the antiplane shear deformation of bi-material wedges with finite radius is that of Kargarnovin and Fariborz (2000). This work has been restricted to the derivation of the displacement field, near the wedge apex only. Meanwhile, only “the dominant solution for displacement field near the wedge apex has been obtained (Kargarnovin and Fariborz, 2000)”.

The finiteness of the radius of the wedge causes that the effect of different possible boundary conditions on the circular segment of the wedge become important.

On the other hand, analytical expressions for the stress intensity factors of different geometries and various loadings are important in fracture mechanics. In the area of mode III problems, a number of contributions are related to the problem of finite or semi-infinite cracks in an infinite medium (Suo, 1989, Shiue et al., 1989, Choi et al., 1994, Lee and Earmme, 2000, Shahani and Adibnazari, 2000, Shahani, 2001). However, the interaction of finite boundaries on the cracks affects the severity of the induced stresses near the crack tip. Also, edge cracks vastly occur in composite laminates and bonded structures. Hence, edge delamination or edge debonding between the laminas or dissimilar components has appeared to become the main failure mode of these materials.

Most interfacial edge crack problems analyzed in the literature to date considered cracks between two bonded quarter planes. In a recent paper, the author (Shahani, 2003) has analyzed the antiplane shear deformation of several edge-cracked geometries and derived analytical expressions for the stress intensity factor of single-material circular shafts with edge cracks, bonded half planes containing an interfacial edge crack, bonded wedges with an interfacial edge crack terminating to the apex and also DCB’s. All of the above-mentioned problems have been analyzed under traction–traction boundary conditions.

In the present paper, antiplane shear deformation of two dissimilar edge-bonded wedges with finite radii is studied. The paper is organized in two parts. In part I, stress and singularity analysis in bi-material finite wedges is considered. Two problems related to the type of boundary condition prescribed on the circular portion of the boundary are studied. The traction free and fixed displacement conditions are imposed on the arc for problems I and II, respectively. The boundary conditions on the radial edges of the composite wedge in these problems are: traction–displacement and traction–traction. The tractions are assumed to act concentrically which allows the solutions to be used as the Green’s function for the analysis of a bi-material wedge under general distribution of traction. The solution is accomplished by employing the finite Mellin transforms. The full field solution is obtained for displacement and stresses. Also, analytical expressions are derived for the orders of stress singularity. In general, the order of stress singularity depends on the material property of the bi-material wedge. However, in special cases, the order of stress singularity releases its dependency to the material property, a fact which is in agreement with the published results in the literature. It is shown, as was expected, that in the special case of a bi-material wedge with infinite radius, the results of the two problems become identical.

In part II of the present paper, analytical expressions are derived for the stress intensity factors in different geometries of bi-material circular media containing an interfacial edge crack. Parallel to that followed in part I, the stress intensity factor expressions are extracted for problems I and II, i.e., traction-free and fixed displacement boundary conditions on the circular portion of the boundary, respectively. In fact, the finite boundary (finite radius of the bi-material circular media) affects the analytical expressions for the stress intensity factors. Furthermore, in each problem, two cases of boundary conditions corresponding to the boundary data prescribed on the crack faces are considered, which are: traction–displacement and traction–traction. As in part I of the paper, in all of the problems concentrated antiplane tractions are assumed to act which allows the solutions to be used as the Green’s function for obtaining the stress intensity factor of any General distribution of tractions. Various combinations of the apex angles are considered for which closed form solutions are obtained for the related characteristic equations and explicit relations are derived for the stress intensity factor. Generally, the stress intensity factors are dependent on the material property (mismatch ratio of the composite wedge), however, in special cases of traction–traction problem, i.e., similar materials and also equal apex angles, the stress intensity factor becomes independent of material property and the obtained results coincide with the results in the literature. In addition, in the special case of infinite bi-material media with an interfacial edge crack, the stress intensity factors can easily be obtained by letting the radius of the circular media to approach infinity.