An investigation of the stress singularity near the free edge of scarf joints 445
Figure 2. Scarf joint geometries subjected to a uniform remote tension σ . a A scarf joint consisting of two long elastic materials. b A scarf joint
consisting of a thin layer of elastic solid sandwiched between two substrates.
Akisanya and Fleck 1997 and Akisanya 1997 have recently evaluated the magnitude of H via a combination of the finite element method and a path independent contour integral for: i a butt joint between
two long strips of materials and ii a butt joint consisting of a thin elastic layer sandwiched between two elastic and isotropic solids. The advantage of this method over the extrapolation method is that the magnitude of H
can be determined by using the stress and displacement fields away from the interface corner, and hence the accuracy does not depend critically on the mesh density close to the free edge.
In this paper, the singular stress field which develops at the interface corner of a scarf joint between two dissimilar materials is examined. Two joint configurations are considered: i a scarf joint between two long
strips of elastic materials figure 2a, and ii a scarf joint consisting of a thin layer of elastic solid sandwiched between two identical substrates figure 2b. Both joints are subjected to a uniform remote tension σ and the
intensity H of the singularity at one of the interface corners is evaluated for a wide range of material properties and various scarf angles, using a combination of the finite element method and a path independent contour
integral. The role of the intensity H in controlling the stress field within the singularity zone is examined. The implications of the results for the failure of scarf joints are discussed.
2. Statement of problem
The scarf joints considered in this paper are shown in figure 2. Figure 2a shows a joint consisting of two elastic, isotropic and homogeneous long strips bonded together to form a scarf joint of width w and length
2L, where L ≫ w. The interface between the two materials intersects the traction-free surfaces at the interface corners A and D, as shown in figure 2a. Let x, y and r, θ be rectangular Cartesian and plane polar co-
446 Z.Q. Qian, A.R. Akisanya
ordinates centred at the interface corner A. The interface AD is inclined at a scarf angle γ to the x-axis. The material in the region γ θ π2 is termed material 1 while the material in the region −π2 θ γ is
referred to as material 2.
The other scarf joint geometry considered in this paper is shown in figure 2b. It consists of a thin layer of elastic solid material 2 of thickness h sandwiched between two identical substrates material 1. For
consistency, the width of the joint is denoted by w and the total length by 2L; both L and w in figure 2b are assumed to be much greater than the layer thickness h. The two interfaces between the thin elastic layer
and the substrates intersect the traction-free surfaces at the interface corners A, B, C and D, as shown in figure 2b. As for the scarf joint shown in figure 2a, rectangular Cartesian and plane polar coordinates x, y
and r, θ respectively, centered at the interface corner A are used to describe the joint geometry and the stress distribution. The thin layer is inclined at a scarf angle γ to the x-axis as shown in figure 2b. For both scarf
joints i.e. figures 2a and 2b the strips are subjected to a uniform remote tension σ , and are assumed to be sufficiently thick for plane strain conditions to prevail. We further assume that the two materials are perfectly
bonded along the interface.
The stress field within a local region around an interface corner, say A, may be of the form H r
λ−1
, where r is the radial distance from the corner and λ − 1 is the order of the stress singularity. In this paper, the intensity H
of the singularity is evaluated for the scarf joints shown in figures 2a and 2b. The extent of the region dominated by the free edge singularity is estimated by comparing the finite element solutions with the singular asymptotic
