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
One of the aims of a joint design engineer is to use joint geometries andor material combinations which remove the stress singularity at the free edge i.e. ensures that the stress field is of O1. In the design of a
scarf joint between two given materials, the scarf angle γ may be selected such that the stress singularity is eliminated i.e. λ 1. Alternatively, if the scarf angle is given, the material combinations characterised by
the material parameters α and β may be chosen such that the free-edge stress singularity vanishes. However, it is impossible to completely eliminate the singularity in many practical joint problems, but the effects of the
singular field can be minimised by having a fundamental understanding of joint failure at the free edge. This requires a knowledge of the magnitude of the intensity H and the order of the stress singularity λ − 1.
The stress components in the region dominated by the stress singularity and at a radial distance r from the interface corner are of the form given by 2. The singular stresses and the corresponding displacements
depend on the magnitude of H, λ and on the non-dimensional functions f
ij
and g
i
; where i, j ≡ r, θ . In the following we discuss the results for the non-dimensional functions and for the intensity H .
4.1. Non-dimensional functions f
ij
The non-dimensional functions f
ij
depend on λ, θ, γ and on the material elastic mismatch parameters α, β see the Appendix. Typical results for f
θ θ
and f
rθ
over the full range of θ −π2 6 θ 6 π2 are shown in figures 3 and 4 respectively, for various values of the material parameters α, β = α4 and for scarf angles
γ = 0, 15
◦
and 30
◦
. Recall that the interface is along θ = γ and the material in the interval −π2 θ γ is termed material 2 while the material occupying the region γ θ π2 is referred to as material 1. The
material parameter α 0 if material 2 is more compliant than material 1 and α 0 if material 2 is stiffer than material 1. The function f
θ θ
only reflects a qualitative distribution of the stress component σ
θ θ
; the actual magnitude of σ
θ θ
depends, in addition, on H . By definition Eq. 3 the function f
θ θ
= 1 along the interface θ = γ , and the values of f
θ θ
and f
rθ
are consistent with the stress continuity conditions at the interface and the traction-free conditions at θ = −π2
and at θ = π2. When the scarf angle γ = 0 see figure 3a and the elastic properties of the materials are identical i.e. α = β = 0, the maximum value of f
θ θ
, and hence of the stress component σ
θ θ
, occurs at the interface. However, when the elastic properties of the materials are different, the maximum value of f
θ θ
occurs slightly away from the interface and in the stiffer material. For example, the maximum value of f
θ θ
equals 1.02 at θ = 6
◦
for α, β = 0.5, 0.125, and equals 1.04 at θ = 9
◦
when α, β = 0.8, 0.2. These results indicate that for tensile butt joints γ = 0 between two dissimilar brittle materials, higher stresses are developed in the
stiffer material and away from the interface. The values of f
θ θ
for scarf joints with γ = 15
◦
and γ = 30
◦
are shown in figures 3b and 3c, respectively. As for γ = 0, the maximum values of f
θ θ
occur away from the interface and in the stiffer material. For example, when α = 0.8, the maximum value of f
θ θ
occurs at θ = 17
◦
for γ = 15
◦
figure 3b, and at θ = 10
◦
for γ = 30
◦
figure 3c. The material parameter α has a negligible effect on the magnitude of the function f
θ θ
when material 1 is much more compliant than material 2 i.e α −0.5 and the scarf angle is large, for example, γ = 30
◦
. The function f
rθ
, shown in figure 4, represents a qualitative distribution of the shear stress σ
rθ
at the singular region of the interface corner. As for function f
θ θ
the values of f
rθ
are consistent with the boundary conditions,
452 Z.Q. Qian, A.R. Akisanya
Table Ia. The non-dimensional constant a for a scarf joint between two long strips of materials as a function of the scarf angle γ , and material elastic parameters α and β = 0.
α γ = 0
γ = 15
◦
γ = 30
◦
γ = 45
◦
γ = 60
◦
γ = 75
◦
− 0.99
0.3526 0.4524
0.5141 0.5
0.03017 0.0008
− 0.8
0.4409 0.4920
0.5198 0.5
0.2878 0.0155
− 0.5
0.6350 0.6025
0.5689 0.5
0.2815 0.0380
− 0.2
0.8964 0.7709
0.6535 0.5
0.2638 0.0574
0.0 1.0
0.9326 0.7483
0.5 0.2508
0.0671 0.2
0.8973 1.0424
0.9273 0.5
0.2355 0.0729
0.5 0.6304
0.7249 1.5574
0.5 0.2128
0.0755 0.8
0.4402 0.3769
0.3076 0.5
0.1905 0.0689
0.99 0.3542
0.2422 0.1189
0.5 0.1825
0.0672
Table Ib. The non-dimensional constant a for a scarf joint between two long strips of materials as a function of the scarf angle γ , and material elastic parameters α and β = α4.
α γ = 0
γ = 15
◦
γ = 30
◦
γ = 45
◦
γ = 60
◦
γ = 75
◦
− 0.99
0.4314 0.5717
0.6607 0.6401
0.03828 0.0008
− 0.8
0.5490 0.6234
0.6672 0.6437
0.2424 0.0155
− 0.5
0.7550 0.7254
0.6903 0.5965
0.2808 0.0380
− 0.2
0.9476 0.8492
0.7224 0.5379
0.2662 0.0574
0.0 1.0
0.9326 0.7483
0.5 0.2509
0.0671 0.2
0.9495 0.9778
0.7925 0.4630
0.2318 0.0737
0.5 0.7559
0.8329 0.8225
0.4134 0.2018
0.0752 0.8
0.5479 0.4589
0.4464 0.1587
0.0767 0.0663
0.99 0.4332
0.2898 0.0291
0.1054 –
–
i.e. continuity of σ
rθ
across the interface and traction-free conditions at surfaces θ = ±π2. For all the material combinations and scarf angles shown in figure 4, the absolute maximum of f
rθ
is located within the stiffer material.
Interfacial toughness data are usually presented in terms of the phase angles, which is a measure of the relative amount of the shear stress to the normal stress at the interface. Figure 5 shows the ratio f
rθ
f
θ θ
at the interface in the region dominated by the free-edge singularity for various material combinations and scarf
angles γ = 0, 15
◦
and 30
◦
. The stress ratio σ
rθ
σ
θ θ
= f
rθ
f
θ θ
along the interface is independent of the radial distance from the interface corner and is significantly influenced by the material mismatch parameters α and
β, and by the scarf angle γ . The magnitude of the ratio σ
rθ
σ
θ θ
is greater for β = α4 than for β = 0 when α is positive and it is smaller for β = α4 than for β = 0 when α is negative. The magnitude of the ratio f
rθ
f
θ θ
increases with increasing scarf angle γ . Since interfacial toughness increases with increasing magnitude of interfacial phase angles Akisanya and Fleck, 1992, the results shown in figure 5 confirms that scarf joints are
‘stronger’ than butt joints γ = 0.
4.2. The magnitude of intensity H The free-edge intensity factor H is related to the geometry, material elastic properties and remote loading
as given by Eq. 4. The values of the non-dimensional constant a for the two scarf joint geometries shown in figure 2 are listed in tables I and II, for various material combinations and a range of scarf angles. The results
An investigation of the stress singularity near the free edge of scarf joints 453
Table II. The non-dimensional constant a for a sandwiched scarf as a function of the scarf
angle γ , and material elastic parameters α and β. β = 0
β = α4 α
γ = 0 γ = 15
◦
γ = 30
◦
γ = 0 γ = 15
◦
γ = 30
◦
− 0.99
0.8540 0.9187
0.8408 0.8668
0.9205 0.8130
− 0.8
0.9214 0.9400
0.7966 0.9311
0.9431 0.7794
− 0.5
1.0199 0.9452
0.7551 1.0234
0.9432 0.7547
− 0.2
1.0712 0.9477
0.7451 1.0493
0.9414 0.7442
0.0 1.0
0.9330 0.7500
1.0 0.9330
0.7500 0.2
0.7921 0.8266
0.7788 0.8873
0.8825 0.7558
0.5 0.5045
0.4048 0.9051
0.6972 0.6782
0.7635 0.8
0.3109 0.1632
0.0714 0.5302
0.3796 0.3130
0.99 0.1469
0.0102 0.0179
0.4449 0.2406
0.0587
Figure 7. The non-dimensional constant a as a function of scarf angle γ and material elastic mismatch parameters α and β, for the scarf joint consisting
of two long elastic strips. a β = 0 and b β = α4.
for the non-dimensional constant a are also plotted in figure 7 for the scarf joint between two long strips of material. In general, the magnitude of a at the interface corner A of the scarf joint between two long strips of
materials figure 2a is within the range 0 a 6 1 when λ 1 and a 1 when λ 1; the free-edge stresses for λ 1 are non-singular. However, a 1 for large scarf angles γ 45
◦
irrespective of the value of λ. We
note that a is symmetric with respect to α for a butt joint i.e. γ = 0; a decreases with increasing magnitude of the material parameter α see figure 7. For example, when γ = 0, a decreases from a value of unity in the
homogeneous limit α = β = 0 to a value of about 0.4 when |α| = 0.99 and |β| = 0 and α4. We note also
454 Z.Q. Qian, A.R. Akisanya
Figure 8. The non-dimensional constant a as a function of scarf angle γ and material elastic mismatch parameters α and β, for the sandwiched scarf
joint. a β = 0 and b β = α4.
that when λ 1, a increases with increasing β. The dependence of a on the scarf angle γ do not follow any consistent pattern except when the material parameter α is in the range −0.6 α 0 where a increases with
decreasing magnitude of γ .
The values of the non-dimensional constant a for the sandwiched scarf joint figure 2b are plotted in figure 8
for γ = 0, 15
◦
and 30
◦
. When γ = 0, the values of a are identical to those obtained by Akisanya 1997. In contrast to a butt joint γ = 0 between two long strips of materials, the magnitude of a for a sandwiched butt
joint is not symmetric with respect to α = 0. For a given magnitude of α, the value of a for a sandwiched butt joint γ = 0 is lower for α 0 than for α 0 figure 8. Since λα, β, γ = 0 = λ−α, −β, γ = 0 see
figure A.1, the free-edge stresses in a butt joint are minimised when the more compliant material is used as the sandwiched layer. For all the values of γ considered, the magnitude of a decreases with increasing magnitude
of the material parameter α; the rate of decrease is faster for α 0 than for α 0. The magnitude of a is lower for β = 0 than for β = α4 for all the cases except when γ = 0 and −0.4 α 0, where the magnitude of a is
higher for β = 0 than for β = α4.
In general, the magnitude of H , which is obtained from Eq. 4 and figures 7 and 8 for different combinations of materials, may not be directly comparable since the order of the stress singularity λ − 1 may be different.
Only in butt joint is λ symmetric about the α = 0 line. The units in which the stress intensity H should be expressed are MPa mm
1−λ
, and a comparison of two quantities having different units is not possible. This problem can be avoided if the layer thickness h in the case of the sandwiched joint or the joint width w in the
case of the bi-material strips is made dimensionless by dividing it by a specific length r in a manner similar
to that of Rice 1988 for interfacial cracks. This introduces another problem, namely the determination of
An investigation of the stress singularity near the free edge of scarf joints 455
such a characteristic length r , which is material dependent. It has been suggested by Akisanya 1997 that the
characteristic length r be chosen as the minimum extent of the singular field from the interface corner. The
results for the minimum extent of the singular field from the interface corner for a butt joint as a function of material mismatch parameters α and β are available elsewhere in Akisanya 1997.
4.3. The extent of the singular stress field The asymptotic singular stresses near the interface corner are calculated using Eq. 2 by making use of the
numerically obtained values of H and λ, and of the analytically derived expressions for the angular functions f
ij
and g
i
. The extent of the region dominated by the free-edge singularity is estimated by comparing the finite element solutions with the asymptotic singular solutions. Typical results of the stress component σ
θ θ
near the interface corner along various radial directions are plotted against the distance from the interface corner in
figure 9, for the scarf joint geometry shown in figure 2a. The stress component σ
θ θ
has been normalised by the applied remote tension σ , while the radial distance r, has been normalised by the width of the joint w.
When α = −0.5, β = α4 and γ = 15
◦
figure 9a, the asymptotic and finite element results for σ
θ θ
are in good agreement in the stiffer material i.e. material 2 and away from the interface, up to a radial distance
r = 0.2w; the maximum difference between the two solutions is about 5 of the finite element solution. However, near the interface i.e. along θ = 13
◦
and θ = 17
◦
and also in material 1 i.e. along θ = 30
◦
and θ = 60
◦
the two solutions are only in agreement up to a radial distance r = 0.04w. The comparison between the asymptotic and finite element solutions for σ
θ θ
when α = −0.5, β = 0 and γ = 15
◦
is shown in figure 9b; the effect of the material parameter β on the extent of the singular field is negligible. The actual magnitude of
σ
θ θ
depends, however, on β. Figure 9c shows the comparison between the asymptotic solutions for σ
θ θ
and the finite element solutions when α = 0.2 and β = α4 with γ = 15
◦
. Here λ = 1.01534, and the stresses at the interface corner are non-singular. The two solutions are in very good agreement along all directions and up to
a radial distance r = 0.2w and the stress component σ
θ θ
decreases as the interface corner is approached from within the materials. This is in contrast with other cases with λ 1 where the stresses increase as the interface
corner is approached see figures 9a and 9b. The extent of the singular field for the scarf joint consisting of a thin layer of elastic solid sandwiched
between two identical substrates is qualitatively similar to that shown in figure 9 for a scarf joint between two strips of materials; the singular filed dominates a large region near the interface corner relative to the layer
thickness h.
4.4. Fracture criterion for bonded joints Equation 2 shows that the strength of the stress field near the interface corner is determined by the order of
the stress singularity, λ − 1, and the intensity, H ; λ depends only on the elastic mismatch parameters, α and β, and the scarf angle, γ . If the value of λ is identical for two different scarf joints, the free-edge intensity factor
H will have equal units and may be used for comparing their respective failure loads.
For perfectly bonded materials the critical value of H say H
c
at failure can be measured by experiments using the calibration given by Eq. 4. Once the critical value of H has been determined for a particular value
of joint width figure 2a or layer thickness figure 2b, Eq. 4 can be used to estimate the failure strength of an identical joint with different width or layer thickness. The effective use of the intensity factor H as a
failure criterion requires that the extent of the free-edge singularity must be significantly greater than the size of any inelastic deformation zone and the size of any intrinsic flaw contained within the H -field. The free-edge
456 Z.Q. Qian, A.R. Akisanya
Figure 9. Comparison of the linear elastic finite element and asymptotic singular solutions for stress component σ
θ θ
. The stress component σ
θ θ
is normalised by the applied remote tension σ , while the radial distance r from the interface corner is normalised by the joint width w. a γ = 15
◦
, α = −
0.5, β = α4; b γ = 15
◦
, α = −0.5, β = 0; and c γ = 15
◦
, α = 0.2, β = α4.
singularity magnifies the crack-tip stress intensity factors of an edge crack embedded within the H -field, and thus encourages the initiation and growth of cracks at the free edge Akisanya and Fleck, 1997.
5. Concluding remarks
