Eur. J. Mech. ASolids 18 1999 443–463
Elsevier, Paris
An investigation of the stress singularity near the free edge of scarf joints
Z.Q. Qian, A.R. Akisanya
Engineering Department, University of Aberdeen, King’s College, Aberdeen AB24 3UE, U.K. Received 1 October 1997; revised and accepted 10 August 1998
Abstract – The singular stress field which develops at the interface corner of a scarf joint between two bonded elastic solids is investigated. Depending upon the scarf angle and the material elastic properties, the singular stresses at a radial distance r from an interface corner may be expressed in the
form H r
λ−1
, where λ − 1 is the order of the stress singularity and H is the intensity of the singularity. The intensity H of the singularity at the interface corner of scarf joints subjected to a remote uniform tension is evaluated for various material combinations and a range of scarf angles using
a combination of the finite element method and a path independent contour integral. Two types of scarf joints are considered: i a scarf joint between two long bi-material strips and ii a scarf joint consisting of a thin elastic layer sandwiched between two substrates. The role of the intensity H , which
determines the amplitude of the stress field within the singularity zone, is examined and the implications of the results for the initiation of joint failure are discussed.
Elsevier, Paris
stress singularity bonded joints contour integral failure initiation finite element method
1. Introduction
Bonded joints containing two or more layers of dissimilar materials are increasingly used for various engineering and structural applications. For example, they are used in the production of electronic, automobile
and aerospace components. The failure of these joints and of many other multi-layer systems often initiates at the interface corner, where the interface of the joint intersects the traction-free surfaces. It is therefore important
to characterise the stress andor strain field near the interface corner so that this type of failure can be accurately predicted and minimised.
In general, the geometrical configuration at the free edge of two bonded dissimilar materials is characterised by the angles θ
1
and θ
2
which the interface makes with the traction-free surfaces at the interface corner, as shown for example in figure 1; the interface corner is denoted by A in figure 1. The two materials in figure 1
are assumed to be elastic and are perfectly bonded along the interface. A stress singularity may develop at the interface corner under an applied loading. Depending on the material elastic properties and on the edge
geometry i.e. θ
1
and θ
2
the stress singularity may be of the form H r
λ−1
. Here, r is the radial distance from the interface corner A, H is the intensity of the singularity and λ − 1 is the order of the stress singularity. We shall
refer to this as the H -field, in contrast with a crack tip K-field Akisanya and Fleck, 1997; Akisanya, 1997. The H -field dominates only a local region near the interface corner of the joint and as such it is sometimes
referred to as a free-edge effect. The intensity H of the free-edge singularity will hereafter be referred to as the free-edge intensity factor.
The value of λ and the corresponding value of the intensity H may be real or complex, depending upon the relative elastic properties of the materials and upon the edge geometry. The magnitude of the intensity H
To whom all correspondence should be addressed.
444 Z.Q. Qian, A.R. Akisanya
Figure 1. General configuration at the free-edge of two bonded dissimilar materials.
of the singularity, which characterises the amplitude of the stress state near the interface corner, depends upon the edge geometry, the material elastic properties and upon the remote loading. In this paper, we focus on scarf
joints for which the material elastic properties and the scarf angle are such that the values of both λ and H are real.
The evaluation of λ for various edge geometries has been discussed extensively in the literature see, for example Williams, 1952; Bogy, 1971; Hein and Erdogan, 1971; Kelly et al., 1992. However, the evaluation
of the intensity of the singularity, H , has received little attention. The significance of H in the design of joints is two-fold. Firstly, a detailed characterisation of the singular field at an interface corner or a free edge requires
a knowledge of the magnitude of both H and λ for various industrially used joint geometries. Secondly, the magnitude of H can be used to predict the initiation of failure at the interface corner in a manner similar to
the use of conventional crack tip stress intensity factor for predicting the onset of crack growth Gradin, 1982; Groth and Brottare, 1988; Hattori et al., 1989. Failure at the interface corner occurs when the magnitude of H
reaches a critical value, say H
c
, which will have to be determined by carefully designed experiments. Therefore, a detailed calibration of the intensity of the free-edge singularity H for various specimen geometries, material
combinations and loading conditions is required for the effective application of a H -based failure initiation criterion.
Reedy 1990, 1993 has recently determined the magnitude of H for i a thin elastic layer bonded to a rigid adherend, and ii for a thin elastic layer sandwiched between two rigid substrates. The calibration H provided
by Reedy is applicable only to joints with substrates or adherends that are much stiffer than the thin elastic layer for example, aluminiumepoxy and ceramicmetal joints. However, many high temperature components
contain ceramic–ceramic joints where the elastic properties of both the thin layer and the adherends may be of the same order of magnitude. In addition, the magnitude of H was determined by Reedy 1990, 1993
via an extrapolation method which involved the matching of the asymptotic free-edge displacement field to the corresponding finite element results. The accuracy of this method depends on the mesh density near the
interface corner. Munz and Yang 1992, 1993 have used a similar extrapolation method to obtain the magnitude of the intensity of the singularity near the interface corner of a bi-material wedge for which both λ and H are
complex constants.
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
