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
solutions.
The stress distribution in a body loaded by prescribed surface traction in plane strain or plane stress depends on only two combinations of the elastic constants. The two elastic mismatch parameters are defined for plane
strain by Dundurs 1969
α = µ
1
κ
2
+ 1 − κ
1
+ 1µ
2
µ
1
κ
2
+ 1 + κ
1
+ 1µ
2
; β =
µ
1
κ
2
− 1 − κ
1
− 1µ
2
µ
1
κ
2
+ 1 + κ
1
+ 1µ
2
, 1
where µ
m
= E
m
21 + ν
m
, E
m
, and ν
m
denote shear modulus, Young’s modulus and Poisson’s ratio for material m m = 1, 2, respectively, and κ
m
= 3 − 4ν
m
for plane strain. The material parameter α is positive when material 2 is more compliant than material 1, and is negative when material 2 is stiffer than material 1.
α = β = 0 when the elastic properties of both materials are identical, and switching materials 1 and 2 reverses the signs of α and β. It is well known that the values of α and β for all physically admissible combinations
of materials are contained within a parallelogram in α-β space. The corners of the parallelogram are located at the point α, β = 1, 0, −1, 0, 1, 0.5 and −1, −0.5. Further, the α, β values for typical material
combinations are concentrated along β = 0 and β = α4 lines in α − β Suga et al., 1988. Hence, we focus our discussion in the current paper on material combinations with β = 0 and β = α4.
3. The asymptotic singular solution
3.1. Intensity H of the free-edge singularity By using a complex variable formulation, it is shown in the Appendix that the singular stress and
displacement fields near the interface corner A see figure 2 are of the form σ
m ij
= H r
λ−1
f
m ij
α, β, γ , θ, λ, 2
u
m i
= 1
2µ
m
H r
λ
g
m i
α, β, γ , θ, λ,
An investigation of the stress singularity near the free edge of scarf joints 447
Figure 3. The non-dimensional function f
θ θ
as a function of θ , for various values of material elastic mismatch parameters α and β = α4 and of scarf angle γ . a γ = 0, b γ = 15
◦
and c γ = 30
◦
.
where i, j ≡ r, θ , m = 1, 2 is the material index, µ
m
denotes shear modulus, λ − 1 is the order of the stress singularity and, f
m ij
and g
m i
are non-dimensional functions of material parameters α, β, the scarf angle γ , the polar co-ordinate θ and of λ. The angular functions f
ij
and g
i
are independent of the remote loading conditions and are given explicitly in the Appendix. The functions f
θ θ
and f
rθ
are plotted in figures 3 and 4 for some values of material parameters α, β = α4, and for a range of scarf angles, 0 6 γ 6 30
◦
. The values of both f
θ θ
and f
rθ
satisfy the traction-free conditions at the free surfaces i.e. θ = ±π2 and the stress continuity conditions along the interface i.e. along θ = γ . The relative amount of shear to normal stress at the interface,
448 Z.Q. Qian, A.R. Akisanya
Figure 4. The non-dimensional function f
r θ
as a function of θ , for various values of material elastic mismatch parameters α and β = α4 and of scarf angle γ . a γ = 0, b γ = 15
◦
and c γ = 30
◦
.
which is measured by the ratio f
rθ
f
θ θ
, is shown in figure 5 as a function of the material parameter α and for a range of scarf angles. A general discussion on these functions is given in Section 4.
One of the aims of the work described in this paper is to evaluate the magnitude of the intensity H of the free-edge singularity that develops in scarf joints. However, this requires a knowledge of the order of the stress
singularity, λ − 1. The determination of λ using various methods has been extensively discussed in the literature, for example, Bogy 1971, Hein and Erdogan 1971 and Kelly et al. 1992. For completeness, a summary of
the evaluation of λ for various material combinations and a range of scarf angles is given in the Appendix.
An investigation of the stress singularity near the free edge of scarf joints 449
Figure 5. The ratio of shear to normal stress f
r θ
f
θ θ
at the interface as a function of material elastic mismatch parameters α and β = 0 and α4, for scarf angles γ = 0, 15
◦
and 30
◦
.
Depending on the material elastic properties and on the scarf angle, the stress field at an interface corner of the joint geometries shown in figure 2 may take one of three forms, as discussed in the Appendix. In this paper
we focus on the evaluation of the magnitude of H for a free-edge singularity of form H r
λ−1
, where both H and λ are real constants. H is defined such that at a distance r from the interface corner and along the interface
θ = γ , the stress component normal to the interface, σ
θ θ
, in the region dominated by the singularity is given by
σ
1 θ θ
= σ
2 θ θ
= H r
λ−1
along θ = γ , 3
where the superscripts 1 and 2 denote the two materials. The magnitude of H depends, in general, upon the free-edge geometry, elastic mismatch parameters and upon the remote loading. Dimensional considerations
dictate that H be related to the joint geometry and material elastic properties by Akisanya and Fleck 1997, Akisanya 1997, Reedy 1990, 1993.
H = σ l
1−λ
aα, β, γ , 4
where σ is the magnitude of the applied uniform remote tension, l is a characteristic length scale and a is a dimensionless constant function of the scarf angle γ , and of the material elastic parameters α, β. In the current
study, we have chosen l as the smallest length scale in a particular joint geometry. Therefore, l ≡ w for the scarf joint consisting of two long elastic strips of materials each of width w figure 2a, and l ≡ h for the scarf joint
consisting of a thin layer of elastic solid of thickness h sandwiched between two substrates figure 2b. We shall evaluate H , and hence the non-dimensional constant a, by a combination of the finite element method and a
path independent contour integral.
3.2. Evaluation of the free-edge intensity factor H The free-edge intensity factor H is determined by the ‘reciprocal work integral contour method’. The method
involves a convolution of the asymptotic field of the corner singularity with a finite element solution. This contour integral method has been used by various authors to obtain the stress intensities for different crack
and notch geometries Stern et al., 1976; Sinclair et al., 1984; Carpenter and Byers, 1987; Akisanya and Fleck, 1997; Akisanya, 1997. The method is based on Betti’s reciprocal law Sokolnikoff, 1956, and is outlined
below.
450 Z.Q. Qian, A.R. Akisanya
Figure 6. A closed integration path around the interface corner A.
Consider a closed contour C = C
1
+ C
2
+ C
3
+ C
4
around the interface corner A, as shown in figure 6. Betti’s reciprocal law can be stated as
I
C
σ
ij
u
∗ i
− σ
∗ ij
u
i
n
j
ds = 0, 5
where i, j ≡ r, θ represent plane polar co-ordinates centred at the interface corner see figure 6, σ
ij
and u
i
are the free-edge singular stress and displacement fields given by 2, σ
∗ ij
, u
∗ i
are auxiliary fields satisfying the same boundary conditions as σ
ij
and u
i
, n
j
is the outward unit normal to C, and ds is an infinitesimal line segment of C. By appropriate choice of the auxiliary field σ
∗ ij
, u
∗ i
the integral 5 can be used to determine the free-edge intensity factor H .
In the evaluation of H the auxiliary fields σ
∗ ij
and u
∗ i
are chosen as the free-edge singular stress and displacement fields given by 2 with intensity H
∗
and λ replaced by λ
∗
= − λ. The starred fields σ
∗ ij
, u
∗ i
with λ
∗
= − λ satisfy the same boundary conditions as those for the unstarred fields σ
ij
, u
i
. These boundary conditions are as stated in Eq. A.3 of the Appendix. The unstarred fields are obtained for both scarf joints
using the finite element method. The value of H
∗
for the auxiliary field is chosen such that the evaluation of 5 by the domain integration method gives the intensity H for the elastic state of interest. A more detailed
description of the procedure for the evaluation of H is given elsewhere Akisanya and Fleck, 1997. Once the value of H has been obtained, we calculate the non-dimensional constant aα, β, γ via 4.
Elastic analysis of the scarf joints shown in figure 2 has been carried out using the finite element code ABAQUS.
1
The finite element mesh consists of 618 eight-node plane strain isoparametric, quadrilateral elements. For the scarf joint between two long isotropic solids shown in figure 2a the width of the strip w
is taken as unity and L = 10w. Roller boundary conditions are applied along the bottom edge of material 2 i.e. along EF in figure 2a while a uniform remote tension σ is applied along the top edge of material 1 i.e. GH .
In addition, point F at the bottom edge is fixed in the finite element model to prevent rigid body movement. For
1
Hbbitt, Karlsson and Sorenson, ABAQUS Users Manual, Version 5.5, HKS Inc. 1995.
An investigation of the stress singularity near the free edge of scarf joints 451
the scarf joint consisting of a thin elastic layer sandwiched between two identical substrates i.e. figure 2b, the layer thickness h is chosen as unity while w = 10h and L = 10w. The boundary conditions are kept the same
as those for the scarf joint between two long strips of materials.
4. Results and discussion
