Some problems in the antiplane shear deformation of bi-material wedges
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() near the apex (r → 0), in which λs 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 τθz-component is unbounded at the point of application of the concentrated tractions, however, the τrz-component is convergent at that point. Also, continuity of the τrz-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 τrz-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.
Section snippets
Formulation and problem solution
Consider two dissimilar isotropic wedges with finite radii, a, apex angles α and β, shear moduli μ1, μ2, and infinite lengths in the direction perpendicular to the plane of the wedge, which are bonded together along a common edge (Fig. 1). The common edge is chosen as the reference axis for defining the coordinate θ. The condition of antiplane shear deformation is imposed on the composite wedge. This implies that the only nonzero displacement component be the out of plane component, Wi, which
Part II: mode III stress intensity factor in bi-material media containing an interfacial edge crack
In part II, we are going to use the results of part I and derive the stress intensity factors at the tip of an interfacial edge crack in bi-material circular media. Indeed, the geometry under consideration is the special case of two bonded dissimilar finite wedges, whose apex angles are such that they form a circular bi-material shaft containing an interfacial edge crack, i.e., α + β = 2π, as shown in Fig. 2.
The radial edges of the composite wedge (the crack faces in this special case), can be
Special case of bonded wedges with similar materials
In this case μ1 = μ2 or R = 1 and the twin problems are considered here separately.
Various combinations of the apex angles of the composite wedge
Here, three combinations of the apex angles are considered which are α = β, β = 2α, β = 3α. In these cases, the corresponding characteristic equations can be solved analytically, which causes that the simplified relations for the stress intensity factors to be obtained.
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