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The influence of local geometry on the strength of adhesive joints
The influence of local geometry on the strength of adhesive joints R.D. Adams* and J.A. Harris ~ (*University of Bristol/tMaterials Engineering Research Laboratories Ltd, UK)
This paper describes a two-part investigation into the effect of local geometry changes at the edges of the overlap in single lap joints. In the first part, finite element analysis has been used to model the effects on the stress distribution of geometry changes which are small in relation to the dimensions of the local geometry, in order to provide an improved model for failure prediction. The model used is that of an adhesive around a rigid corner, and the effect of rounding the corner has been considered. The second part deals with local geometry changes of the same order of magnitude as the dimensions of the geometry and their effect on joint strength. Three types of joint - - one with a square-edged adhesive layer, one with a fillet of adhesive and one with an adhesive fillet plus a radiused adherend - - were manufactured, tested and analysed. Improvements to the finite element models were also made following the results of part 1. It is found that finite element analysis is capable of predicting the significant strength increases that may be achieved in single lap joints by filleting the adhesive at the edges of the overlap and rounding the ends of the adherends.
Key words: adhesive-bonded joints; stress distribution; strength prediction; overlap geometry; rounding of corners; singularities; finite element analysis
In order to design an adhesive joint for a critical loadbearing application, it is necessary to have a knowledge of the ultimate strength of the joint and an understanding of its failure properties. The application of finite element methods (WM) to the analysis of adhesive joints has been particularly successful: for example, the strength and locus of failure o f single lap joints with a wide range of adhesives have been predicted with reasonable accuracy I. The main reason for this is that when cohesive failure of the adhesive is responsible for joint failure, as is often the case in wellprepared joints, initiation of failure is confined to a ve-'2,' localized region in the adhesive. For the single lap joint, it is found that the highest stresses and strains in the adhesive occur in regions at the edge of the overlap. Closed-form analytical solutions to problems of this ty,pe may give good overall indications of the conditions in the adhesive layer when the joint is loaded, but they are unable to predict the critical conditions in these edge regions in sufficient detail for joint failure to be predicted correctly. In contrast, the use of finite elements enables the distributions in the critical regions to be predicted with much greater definition. A second requirement met by FEM is that the nonlinear properties of the adhesive and adherends may
be modelled, a process which is necessary for all practical problems. W h e n the stress and strain fields in adhesive joints are examined in detail, the critical regions are found to be singular in nature. Fig. I indicates the singular points in a single lap joint, in one case with a squareedged adhesive layer and in another with a fillet of adhesive at the edge. Each point is on the boundary, between two dissimilar materials where there is also an abrupt change in slope giving a perfectly sharp corner. In order to predict joint strength using the results of a finite element analysis of such joints, a failure criterion is required. One possible a p p r o a c h is to treat the singular points in the same way as crack tips are treated in fracture mechanics. Failure may be predicted at a critical stress intensity or fracture energy, assuming that these parameters are purely properties of the adhesive. However, it has been shown for adhesive joints that the fracture energy is not independent of the joint g e o m e t ~ 2. so that no straightforward failure criterion or method of prediction is found. An alternative approach I is to base joint failure on the m a x i m u m conditions of stress or strain predicted by the analysis using an appropriate bulk property of the adhesive as a failure criterion. The difficulty then arises that, because of their nature, the ability of finite
0 1 4 3 - 7 4 9 6 / 8 7 / 0 2 0 0 6 9 - 1 2 $03.00 © 1987 Butterworth 5t Co (Publishers) Ltd INT.J.ADHESION AND ADHESIVES VOL.7 NO.2 APRIL 1987
69
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including singularities, were also considered following the results from the first part of the work. Thus, the aim was to demonstrate that these larger local geometry, changes could have a measurable effect on improving joint strength, and that these improvements may be predicted from finite element analysis of the various joints.
Investigation of small-scale local geometry changes Simplified model
a
B
~iiJi;i;~;~i~i;i !~!;!ii~i~!ii~i i;i~i~i i i ~i~i~i ~i~i~i;i ~;~ii ~i~i i i_~i~i~i~
b ~ - - - - - - - Locus of failure Fig. 1 Initiation of failure from singular points in the single lap joint: (a) square-edged adhesive layer; and (b) with an adhesive fillet
elements to model the singular behaviour in the joint depends on the local degree of refinement of the finite element grid, so the m a x i m u m conditions predicted would depend on the grid used in the analysis. However, in reality, the geometries of the corners will not be absolutely perfect: there will always be some degree of rounding and the singularities will not necessarily exist. Indeed, the loci of failure indicated in Fig. 1 often run adjacent to and not necessarily exactly through the singular points. In the first part of this paper, a study of the effects of local changes in geometry, small in relation to the dimensions of the local geometry (ie, the adhesive layer thickness), is reported. This is a means of overcoming the problems associated with singularities, the aim being to provide an improved model on which to base failure prediction. The region of point "B" in the joint with an adhesive fillet in Fig. 1 was considered. However, because a high degree of refinement of the finite element grid was required, the analysis of the complete joint was avoided and a simplified model of the critical region was considered instead. In the second part, changes of local geometry of the same order of magnitude as the dimensions of the geometry are considered in terms of their effect on joint strength. Three joint types were manufactured, tested and analysed: one had a square-edged adhesive layer, one had a fillet as in Fig. 1 and the third was a joint with a fillet plus a large radius on the adherend. Possible improvements to the models of the joints,
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INT.J.ADHESION AND ADHESIVES APRIL 1987
The simplified model of the critical region of the joint with an adhesive fillet is illustrated in Fig. 2. The model consists of an adhesive surrounding a rigid corner. A uniformly distributed load was applied across the width of the model via a strip of material, relatively stiff compared with the adhesive, to which was assigned the elastic properties of aluminium. By laterally constraining the sides to minimize bending, the action of the applied load was to displace the strip approximately uniformly across its width. Thus a concentration of stress in the adhesive around the corner was ensured, similar to the conditions around the critical corner of the lap joint. Because of symmetry, only half of the model needed to be analysed. Modification of the corner geometry in the model was achieved by transforming the coordinates of a section of the finite element grid so that the boundary became of the form v = x2/2k + C as shown in Fig. 3. The value of k is a measure of the degree of rounding, and "C is the thickness of the adhesive layer at the corner, which is equal to 0.125 m m for no rounding (k = 0) and with rounding is increased by an amount equal to k/2. The effect of rounding on the local stress distributions was investigated by analysing
::::::::::::::::::::::::::::::
a
i
L_ I-
1.46
Uniform loading
b Fig. 2 Critical region of a lap joint with an adhesive fillet: (a) load transfer; and (b) simplified model. All dimensions in mm
nature of the distribution, holds to a distance pan-way
into the element closest to the comer; the numerical results are unreliable closer to the comer. Thus, in the linear region, the distribution can be expressed in the form:
Oprin. max= constant X r -n X f(0)
1
c
EI
Adhesive
x2 .y=~-~ + c
k
C= 0.125 + ~- mm
Fig. 3 Modification of geometry to produce a rounded corner in the simplified model
the model for four cases corresponding to k values of 0.0, 0.00125, 0.0146 and 0.161 mm. Except for the latter case, these modifications are small compared with the local thickness of the adhesive layer. Refinement of the finite element grids was arranged so that the dimensions of the elements increased in equal logarithmic steps away from the comer. For the adhesive, eight rows and twelve columns of eightnoded isoparametric elements were used, the smallest element adjacent to the c o m e r being 0.00125 by 0.00125 mm. The analysis was performed for both a linearly elastic adhesive and an elastic-plastic adhesive with a small degree of strain hardening. For the computation, the adhesive properties were based on the uniaxial tensile stress/strain characteristics shown in Fig. 4. The linear material corresponds to an unmodified epoxy adhesive with a Young's modulus of 2.8 GPa and a Poisson's ratio of 0.4. The elastic-plastic material corresponds to a toughened epoxy with a Young's modulus of 2.5 GPa and a Poisson's ratio of 0.37. The yield and plastic flow of the adhesive was represented in the finite element computation by a curve-fit model of the uniaxial behaviour in Fig. 4. Details of the formulation of the toughened material are given in the section on large-scale local geometry changes, as this material was used in the various lap joints that were tested. A state of plane strain was assumed for the model and a uniform load equivalent to a direct stress of 100 MPa was applied throughout. Results from analysis of simplified model
(1)
where for this case n = 0.185. In theory, for increasingly small values of r, the principal stress would increase to infinity. However, this is not a sensible practical consideration, since the surface roughness can be expected to be of the order of 0.001 to 0.01 mm. Also, in the manufacture of adhesive joints chemical etching and/or mechanical abrasion will remove any sharp 90 ° corner, even if it could be produced to this degree of perfection in the first place. The relationship given in Equation (!) is similar to that for the singularity around a sharp crack in an elastic homogeneous material, where the constant multiplier corresponds to the stress intensity factor and for a sharp crack "n" is equal to 0.5. The principal stress distributions in Fig. 6 for the various cases analysed show that introducing a rounded c o m e r into the model results in a departure from the singular distribution of the stresses as the c o m e r is approached. Rounding the corner effectively removes the singularity, a plateau is reached at a distance from the corner and thereafter the maximum principal stress remains constant. With the exception of the case o f k = 0.161 mm, the change in geometry, of the corner is small relative to the thickness of the adhesive layer in that region. Thus, as might be expected from St Venant's principle, the local geometry change results in only a local modification of the stress distribution; further away from the corner the stresses are unaffected. However, even with k = 0.161 mm, which represents a 64% increase in the thickness of the adhesive layer at the comer, the stresses are only reduced slightly beyond 0.1 mm from the corner. To a first approximation it would appear that:
(O'prin. max)p oC l/k and rpOC k
80
tic
~. 60
\ Elastic-plastic
o~ 2
40
k-
20
Elastic adhesive
The distribution of the maximum principal adhesive stresses for the square-corner model (k = 0) is shown in Fig. 5. The stresses on the centre-line of the model have been linearly interpolated from the Gauss point values. The linear relationship, indicating the singular
0 Fig. 4
[
I
I
0.02
0.04
0.06
I
I
0.08 0.10 Strain
I
I
0.12
0.14
0.16
Adhesive uniaxial tensile stress/strain properties
INT.J.ADHESION A N D ADHESIVES APRIL 1 9 8 7
71
be seen that with the rounded corners a peak in the distribution occurs away from the centre-line of the model. Locally this will be governed by the nature of the curvature around the corner. Again. at a sufficient distance from the centre-line, the stress distribution is unaffected.
where (aprm. max)p and rp are the peak value of the maximum principal stress and the distance from the corner at which the peak value is attained, respectively. Thus. the predicted stress distribution is extremely sensitive to small changes in the local geometry of the corner. Again. taking into account practical considerations of joint preparation, a typical radius of the order of 0.05 mm would have a significant effect on the local stress distribution. Considering the stresses along a line close to the rigid interface in the adhesive shown in Fig. 7, it can
Elastic-plastic adhesive In order to assess the post-yield behaviour of the adhesive, the plastic energy, density. Wp (that is the plastic work done per unit volume at a point), has been
e (°)
Boundary of fi rstel ement
+
45
• X
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--
I E
i
~
~
1000 A
E
100
I
I
I
J
II
L
t
t
I
I
J
I Lt
0.001
I
~
L
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L L I]
L
0.01
0.1
•(mm) Fig. 5
Maximum
principal ,adhesive
stresses
around
a square
corner with
elastic adhesive
prope~ies
f o r an a v e r a g e
applied
stress of 1 O0 MPa
Boundary of first element 1000 a.
k (ram) E c"
0.0!~....~ 0.00125, ~ ' ~ - ~
- ~ l m ~
I I
~li~ll,.,,.,.lt~t
I ~]ni-,..nnl~
0.161 ~
1:3
0
~
13
D
0
E,
~-,
I [ I 100
I
I
I
I[
I
I
I I
L
I
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L
I
0.001
L
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I
I
0.01
J
J
J
L
L
I
I
L I
0.1
• (mm) Fig. 6 Variation of the adhesive maximum principal stress distribution with corner rounding along the centre-line with elastic adhesive properties (average applied stress = 100 MPa)
72
INT.J.ADHESION A N D A D H E S I V E S APRIL 1 9 8 7
considered based on the elastic-plastic adhesive properties in Fig. 4. The distributions of Wp and O'prin.max for the square-corner case are shown in Fig. 8, from which the relationships:
The Wp distribution departs from the singular form far away from the c o m e r (r > 0.i mm) and close to the c o m e r (r < 0.003 mm). For the latter, the proximity, of the rigid c o m e r laterally restrains the adhesive. This results in an increased hydrostatic component of stress and a suppression of yield and plastic flow, so that Wp appears to attain a m a x i m u m close to the comer. It is worth coml~aring these results with the analytical solution ~'4 for the stress and strain fields
Gprin. max oC r -0"03 WpOC r -°-92
hold over most of the range.
~
~
Location o f values plotted
"" J :-"
Rigid ./
1000G.
k (mm) E
0.0~
,-.
0.161 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
100
=
t
I
,
~ , , ,I 0.001
0.0001
=
A
=
L
L
= L =[
0.01
A
A
~
I
L t
I t I 0.1
x (mm)
Fig. 7 Variationof the adhesive maximum principal stress distribution with corner rounding along the interface with the rigid material, for elastic adhesive properties (averageapplied stress = 100 MPa) 10.0
1000
Wp
"
to
1.0
100 Oprin, roll
0.1
10
I
0.0001
L
I
0.001
0.01
0.1
r (mrn) Fig. 8 Adhesivemaxrmum principal stress and plastic energy density distributions along the joint centre-line with a square corner and elastic-plastic adhesive properties (averageapplied stress = 100 MPa)
INT.J.ADHESION AND ADHESIVES APRIL 1987
73
around an elastic-plastic crack tip, which are of the form:
Further, for k greater than 0.00125 mm, there is a maximum some distance from the corner. This is because, even though the normal stress (cr,,) increases slightly as the corner is approached, the rigidity, of the rounded corner constrains the adhesive in the x (transverse) direction, thus inducing higher transverse tensile stresses (Crx). The net effect is that the hydrostatic component of stress is increased and vield suppressed, so that close to the corner there is a reduction in Wp as the corner is approached. Thus there is departure from singular behaviour in Wp, partly as a result of the restraints imposed by the rigid corner and partly because the load is more evenly distributed through the adhesive as the corner is progressively rounded. Again, the effects are local: the distributions remain relatively unaffected sufficiently far away from the corner. The Wp distributions close to the rigid interface are shown in Fig. 10. The redistribution in load as a result of rounding reduces Wp around the corner but increases it away from the corner, such that there is a peak at a distance of the same order of magnitude as "k" from the centre-line in the x-direction. Still further away, Wp again remains unchanged, so that rounding only has significant effects in the local region two or three times the size of k. The overall distribution of Wp around the corner is illustrated by the contour plots in Fig. 11 for k = 0.0 and 0.0146 mm and for a 100 MPa uniform loading. By
O.ij = r - N / ( I + N) Si j = r - I / ( I
+ N)
where N. the power-law hardening parameter, lies between 0 and 1. For the elastic-plastic adhesive properties in Fig. 4 there is a small degree of strain hardening, so that N is slightly greater than zero. Thus, if it is assumed that: N = 0.05 and Wp c£ o'ij .sij
then the analytical solution would give: O'prin. max o£ r -005
and WpO£? - I
which is of similar form to the numerical results for the rigid corner. The Wp distributions with rounding are plotted in Fig. 9. With increasing k the concentration of loading along part of the centre-line is reduced, leading to a reduction in Wv over part of the region, and in particular to a reduction in the maximum value.
10.0
k (mrn)
O0 0.00125 ~
,
~
~-" 1.0 E
im
0.161-"
"" ""
" "" "''''
" "''
"'"
"'"
g 11OIOOO
O . . . . .
O I O O I U D
QO
O
" "'''"
0.1
0.0001
0.001
0.01
0.1
r(mm) Fig. 9 Variation of the adhesive plastic energy density distribution with corner rounding along the joint centre-line for elastic-plastic adhesive properties (average applied stress = 100 MPa)
74
INT.J.ADHESION AND ADHESIVES APRIL 1987
10
_ _ Location of
k (mm)
00
values plotted
l ¸
0.00125 m
----- jt'
x 0.0146 E
0.161 . . . . . . .
....
" . ='e
le
a
o
.
ol
=
i
•
o
•
o J o o
0.1
I 0.0001
~
I
K I [ III 0.001
1
I
I
I I I III 0.01 X (mm)
f
I
t
I
I III 0.1
I
L
A
Fig. 10 Variation of the adhesive p l a s t i c e n e r g y density distribution with corner rounding along the interface with the rigid material for e l a s t i c - p l a s t i c adhesive properties (average applied stress = 100 MPa)
rounding, the high concentration in strain is removed from the corner. The redistribution results in one peak in HIp on the centre-line and a second, larger ~eak along the interface. Note that the 1 X 106 J m - " contour which is a significant distance from the c o m e r has not been greatly affected by the rounding of the corner. Conclusions from simplified model The following conclusions may be drawn from the results for the simplified model of the critical region of a lap joint with adhesive fillets. 1) For a perfectly square corner, the numerical solution indicates a singular distribution of stress around the corner when the adhesive is elastic. With an elasticplastic adhesive, the Wp distribution follows the singular form (r - I ) over part of the region but, close to the corner, departs from this and attains a m a x i m u m value because of the restraining effects of the interface. 2) Rounding removes the singularity and can substantially reduce the peak values of the m a x i m u m stress for an elastic adhesive and the m a x i m u m plastic energy density, and thus the m a x i m u m strain, for an elastic-plastic adhesive. 3) The model which includes a degree of rounding is more satisfactory because perfectly square geometries will not occur in practice. 4) With a rounded corner there is a region close to the corner where the stress and plastic energy density vary little. If failure is based on the predicted conditions in this region, it will not be dependent on the degree of refinement of the finite element grid. The problem then becomes that of deciding what is the most relevant degree of rounding.
Investigation of large-scale local g e o m e t r y changes Experimental Three types of aluminium/epoxy single lap joints were manufactured to the dimensions shown in Fig. 12. These were chosen so that the effects of altering the local geometry of the joint (on a larger scale than hereto considered) in the critical regions could be investigated. The basic joint (Fig. 12(a) ) had square edges at the ends of the overlap and two other types (Figs 12(b) and (c) respectively) had adhesive fillets and adhesive fillets with radiused adherends. All the joints were 25.4 m m in width. The adherends were of aluminium alloy, type 2014TB, which has a 0.02% proof stress of 304 MPa. The adhesive used was a rubber-modified epoxy consisting of a diglycidyl ether of bisphenoi A, with 15 parts (by weight) per hundred of resin of carboxyl-terminated butadiene-acrylonitrile rubber and 5 parts per hundred of piperidine as a curing agent. The joints were cured at 120°C for 16 h. Three joints of each type were loaded to failure in a screw-driven tensile testing machine at a constant cross-head speed of 1 m m min -I. The results from the tests are given in Table i. Table 1.
Median Lowest Highest
Experimental joint strengths (kN) Squareedged
Filleted
Filleted and radiused
1 5.9 1 5.4 16.4
20.4 1 9.2 21.0
24.5 23.8 24.7
INT.J.ADHESION A N D ADHESIVES APRIL 1 9 8 7
75
strength and that. by suitable modification on a relatively small scale, significant increases in joint strength may be achieved.
a
Finite element 0.005 mm
1
106 J m -3
b 0.005 mm k {mm) ....
analysis
At first the analysis was performed with linearly elastic material properties for each of the three joint types and a small displacement solution obtained. Subsequently, the elastic-plastic properties of both the adhesive and the adherend were included (the properties of the rubber-toughened adhesive corresponding to those of Fig. 4 ) a n d a large displacement solution obtained to account for the rotation of the overlap under load which was observed during testing. In cases where singular behaviour was thought to be influencing the numerical results in the critical regions, local small-scale rounding of the geometry,, as discussed above, was investigated as a means of improving the theoretical model. The square-edged joint was modified by introducing curvature to the edge of the adhesive layer, as shown in Fig. 12(a), in order to transform the singular corner point A to a nonsingular point A'. For the filleted joint, the corner of the adherend, point B, was modified by introducing the curvature to the corner, as indicated in Fig. 12(b). The changes in geometry that these modifications represent are small in relation to the variations which will occur between nominally similar joints in practice.
0.0 0.0146
• ~/
k1 1.5 /
/ ~2.0
/
/
/
ii
,/7l I / 9 '/6/4
/
! !
f,~// /
,/
...................
! ! I I
t /
I
jl l 0.25
L.
1/1.0 > 106 J m -3
,"
~tlll
3.2
100
=l
M o d i f i e d f i n i t e element models
Basic dimensions, m m
a for = 0 . 2 5 ~
Fig. 11 Plastic energy density distributions around the corner: (a) square corner; and (b) k = 0.0 and k = 0 . 0 1 4 6 mm (average applied stress = 1 O0 MPa)
Although the sample number of each type is small. there are significant differences in joint strength. Based on the median values, the inclusion of an adhesive fillet results in a more than 25% increase in strength over the square-edged joint, and the additional modification of radiusing the adherends gives an increase of more than 50%. For the radiused joints, a significant reduction in stiffness was noted in the loading curves just prior to failure, this being a result of plastic deformation in the adherends, which were permanently deformed after joint fracture. From these results, it is seen that the local edge geometry, of the joint has an important bearing on joint
76
INT.J.ADHESION A N D ADHESIVES APRIL 1 9 8 7
0
b
y = V~-X 2 for [xl ~< 0 . 2 5 V ~
q3.2 mm
C
. . . .
Predicted locus o f failure
Fig. 12 Aluminium/epoxy single lap joints: (a) square-edged adhesive bayer; (b) filleted; and (c) filleted and radiused
Square-edged joint The results presented for the 'square-edged' joint are for a 16 kN applied load corresponding to the median value of the experimentally determined joint strength. The principal stress distributions at the edge of the adhesive layer from the small displacement elastic analysis are shown in Fig. 13, for both the square and the modified curved edge. In the unmodified model, the singular nature of the corner point A is apparent and a m a x i m u m principal stress of 1206 M P a is predicted at the Gauss point (point of numerical integration at which the solution is given) 0.001 m m from the corner. With the modification to the model of the curvature at the edge, the singularity at the corner is clearly removed and the largest value of the m a x i m u m principal stress of 258 MPa occurs 0.055 m m from the interface into the bulk of the adhesive layer. Even with this modified model, the m a x i m u m stress predicted is far greater than the tensile strength of the adhesive. Thus, an analysis which only represents the adhesive as an elastic material is not adequate for predicting joint strength. From the large displacement
x
k
x
x
!
~ A i
a
?'_2
1206 MPa (max)
\
\
\
\
\
\
\
\ \
\ \
\
\
\
\\\
\
\ \\\
\
258 MPa (max)
\
\
\ \
\ \
\ \
\
\ \
\
\\~X/ \
-\
\\X% \
\N\ ~.
b Fig. 13 Principal stress distributions for the square-edged joint with elastic adhesive properties and a 16 kN applied load: (a) unmodified model; and (b) modified model
elastic-plastic analysis, the plastic energy density (Wp) distributions are shown in the critical regions of the two joint models in Fig. 14. Again, by introducing the curved edge, the singular behaviour in the plastic strain is removed. As for the stress in the elastic analysis, the m a x i m u m in Wp occurs in a region of relatively gentle variation away from the interface and is of very much lower magnitude than predicted by the square-edged model, thus providing a new basis for predicting joint strength. Clearly the curved-edge model of this joint provides a far more satisfactory means of predicting joint strength based on a cohesive failure of the adhesive. It is significant that such small changes in the local geometry result in such large changes in the local distributions of stress and strain, especially when in practice much larger variations in the edge geometry between joints are likely to exist. It is proposed, therefore, that the modified-edge model is far more representative of the types of local geometry that will be expected in practice.
Filleted joint The principal stresses around the critical corner of the unmodified and modified models of the filleted joint are shown in Fig. 15 for the elastic analysis with a 16 kN applied load. Comparing the magnitudes of the principal stresses, as given by the Gauss points for each model, the square-cornered model predicts a m a x i m u m value of 210 MPa at a point 0.025 m m from the corner point B, while with the rounded corner the m a x i m u m value of 147 MPa occurs at a point inside the overlap 0.015 m m from the adherend surface. These results are analogous to those for the simplified model given earlier in Fig. 6. If the rounding here is compared with the k value of 0.161 mm, then from Fig. 6 the m a x i m u m principal stress for the square corner (r = 0.025 ram) would be 225 MPa and 174 MPa for the rounded corner (r = 0.015 mm). The ratio of the two stresses is thus approximately !.3, which is roughly the same as the ratio here. It is also apparent that the difference between the m a x i m u m stresses in the two models is limited by the degree of refinement that could be used in the square-cornered model. The influence of the singularity at the corner point B on the m a x i m u m Gauss point stresses predicted is not as great in the model for the square-edged joint around point A , because for the latter there was greater local refinement of the finite element grid. Clearly, the round-cornered model is more satisfactory as the results are essentially independent of the degree of refinement. However, as with the square-edged joint, even with the improved model joint failure cannot be predicted from an elastic analysis of the adhesive stresses. The results from the elastic-plastic analyses are shown in Fig. 16, where the Wp distributions are shown for the two models around the critical corner for an applied load of 20 kN, corresponding to the failure load of the joint. As with the stresses in the elastic case. rounding of the sharp corner has led to a redistribution of the plastic deformations, such that the m a x i m u m in Wp occurs inside the overlap. As already discussed, because of the limited mesh refinement there is little difference in the m a x i m u m value of Wp predicted by the two models. In fact, the influence of the singular point on the final results of the elastic-plastic analysis
INT.J.ADHESlON AND ADHESIVES APRIL 1 987
77
8 ~ 10 6 J m -3
<_ a
75 100 × 106 J m -3
J ~
0.02 mm I
I
b
Fig. 14 Plastic energy density distributions for the square-edged joint with elastic-plastic adhesive properties and a 16 kN applied load: (a) unmodified model: and (b) modified model
is even less than for the elastic case. Thus it is not surprising that, with a similar degree of mesh refinement as here, reasonable predictions of joint strength have previously been obtained based on a m a x i m u m plastic strain in the adhesive derived from the Gauss point values in square-cornered joint models I. Radiused adherends with fillet
The elastic principal stresses in the adhesive around the radiused corner are shown in Fig. 17, again for an applied load of 16 kN. As with the small degree of rounding, the highest value occurs inside the overlap, although somewhat further inside in this case (1.43 m m compared with 0.12 mm). As previously noted, the introduction of a small degree of rounding reduced the peak value from 210 to 147 MPa. Here the larger change has made the stress distribution at the edge of the overlap even more uniform and reduced the peak value further to only 61 MPa. Thus, the potential is seen for introducing local geometrical changes of this order of magnitude leading to significant improvements in joint strength. However. for this joint, a higher value of the maximum principal stress of 82.4 MPa occurs at the edge of the adhesive fillet (equivalent to point C in Fig. 12(c) ), which becomes the site from which failure initiates. When the elastic-plastic properties of both the adherends and adhesive are included in the analysis+ the results predict the onset of large-scale plastic deformation of the adherends prior to joint failure. This is clearly seen in the experimental load/deflection curve for the joint in Fig. 18 where, as the experimental failure load is approached, the stiffness (load/ deflection) of the joint decreases rapidly. A theoretical load/deflection curve is also given in Fig. 18 based on
78
INT.J.ADHESION AND ADHESIVES APRIL 1987
the finite element results. This curve is dominated by the yield behaviour of the aluminium adherends, and agrees well with the experimental results. It has been shown previously I that the combination of tension and bending in the adherends after yielding results in a concentration of plastic strain in the region adjacent to the edge of the fillet point C (Fig. 1). This in turn results in higher concentrations of strain in the adhesive around the edge, so that joint strength is lower than would be expected if a higher yield strength material had been used in the adherends. For the other joint types the load/deflection characteristics in Fig. 18 indicate that yielding of the adherends has little influence on joint failure. Thus. the increases in joint strength seen here with the radiused joint are somewhat limited by the yield behaviour of the adherends. Again. this is in keeping with the observed permanent deformation of the adherends in these joints after failure. Prediction of joint strength
The peak values of the m a x i m u m principal adhesive stresses predicted for each of the joint types are summarized in Table 2, based on the elastic small Table 2. Peak values of adhesive maximum principal stress from elastic analysis with 16 kN applied load Joint type
Peak maximum principal stress (MPa)
Square-edged Filleted Filleted and radiused
258" 147* 83
*From modified joint models
f
210 MPa (max)
\
\
N \
\ \
\ \
\
\ \
\
\
\
",, \
\ \
\ \
X \
X \
\ \
\ ,,,
\ \
\
~.
\
\
\
\
\
\
\
\
\
\
\
\
\
•
a
the former, the prediction would have been extremely low while, for the latter, it would have been almost unchanged. This reflects the relative influence of the singularity in each of the unmodified models as determined by the local refinement of the finite element grid in each case. From an examination of the failure surfaces of the joints tested, it was concluded that the predictions of the loci of failure indicated in Fig. 12 are in agreement with those found experimentally. An exception to this was with the filleted and radiused joints, where an interface crack initiating from "C" is observed at one end of the joint (as expected), but a fillet crack similar to that at point "B" in the filleted joint is observed at the other. It is assumed, therefore, that failure initiates at "C" at one end of the joint as expected, but the result of this initial failure is to induce the different type of failure at the other end of the joint. Conclusions
147 MPa
/~
(max)
. / ~" "~"
o \~,\
\
c
"-, ,~
"\
\
\ \
\ \
\ \ \ \ \ \\
",,
~,, \
,~, \
\
\
\ \ \
\
\
\
\
b Fig. 1 5 Principal stress distributions for the filleted joint with elastic adhesive properties and a 16 kN applied load: (a) unmodified model; and (b) modified model
displacement analyses. Although the relative magnitudes do indicate the possibility of increasing joint strength b y modifying the basic 'square-edged" geometry, the values themselves do not provide a basis for predicting joint strength accurately because of the plastic deformation of the adhesive and adherends. In order to predict joint strength based on the bulk properties of a toughened adhesive, such as that used here, the plastic deformation must be accounted for. Therefore, in association with the non-linear analysis carried out for the joints, a failure criterion must be applied to the results. A limiting value of the plastic energy density equal to 7.3 x 10 J m - - has been used. based on the bulk uniaxial tensile properties of the adhesive at failure (Fig. 4). For the square-edged and filleted joints, predictions were based on the results from the modified models. In Table 3, the predicted joint strengths are compared with the median of the experimental values. Agreement is very good, the predictions being to within _+ 10% of the median values. Had the modifications to the critical corner geometries not been used for the square-edged and filleted joints then. for
The site of initiation of failure in adhesive lap joints is strongly dependent on the geometry of the edges of the overlap. In order to predict failure, the method of analysis must, therefore, be able to account for the effects of local geometry and be sensitive to local variations of stress and strain through the adhesive layer in critical regions. Only by using a numerical approach, such as the finite element method, can these types of variation be included in the analysis. In using finite element techniques, prediction of joint strength may be hampered or be extremely dependent on local mesh refinement because of the existence of singularities in the stress or strain fields at critical points. In adhesive joints, these singularities arise at corners between the adhesive and adherend materials as a result of the perfect squareness in the geometry of the model. It has been demonstrated that by rounding the corner the singularity may be removed and replaced by a more uniform distribution of stress and strain. However, the final distribution depends strongly on the degree of rounding that is introduced. Significant increases in single lap joint strength may be achieved in practice by including an adhesive fillet at the edges of the overlap and further increases may be achieved by additionally rounding the ends of the adherends. These observed increases have been predicted by finite element modelling, taking into account the local geometry changes involved and. where necessary, introducing a degree of rounding into the critical areas of the joint model of the same order of magnitude as the glue-line thickness in the joints. In this way, it is felt that a more realistic and hence reliable basis for predicting joint strength has been achieved. Table 3. Comparison of joint strength predictions (kN) with experimental values Joint type
Predicted
Median of experimental
Square-edged Filleted Filleted and radiused
14.5 21.2 25.3
15.9 20.4 24.5
INT.J.ADHESION AND ADHESIVES APRIL 1987
79
\
~ ~
J 3
.....__..._9 ~
a -
2
~ ~ - 3 .
2
× 10 s J m -3
1 × 106 J
m -3
0.05 mm
a
,
,b
Fig. 16 Plastic energy densiW distributions for the filleted joint with elastic-plastic adhesive properties and a 20 kN applned toad: (a) unmodified model; and (b) modified model
Authors
Professor Adams, to whom inquiries should be addressed, is with the Department of Mechanical Engineering, University of Bristol, Queen's Building, University Walk, Bristol, BS8 1TR, UK. Dr Harris is with Materials Engineering Research Laboratories Ltd, Tamworth Road, Hertford, SGI3 7DG, UK. 25
Filleted and radi
Filleted
20
Square-edged.
A
15
Z
Fig. 1 7 Principal stress distribution for the rediused and filleted joint with elastic adhesive properties and a 16 kN applied load (see Fig. 12(c) for dimensions)
o 10
Experimental ~ m Theoretical prediction
References 1
Harris, J.A. and Adams, R.D. Int J Adhesion and Adhesives 4 No 2 (1984) pp 6 5 - 7 8
2
Kinloch, A.J. and Shaw, S.J. J Adhesion 12 (1981) pp 5 9 - 7 7
3
Rice, J.R. and Rosengren, G.F. J Mech and Phys of Sohds 16 No 1 (1968) pp 1-12
4
Hutchinson, J.W. J M e c h a n d P h y s o f S o l i d s 1 6 N o 1 pp 13-31
80
INT.J.ADHESION
AND ADHESIVES
(1968)
APRIL 1 987
if 0
~ 0.2
h 0.4
I 0.6
I 0.8
I 1.0
I 1.2
Deflection (mm) Fig. 18 Load/deflection curve for a filleted and radiused joint indicating the range of failure load for various joint types
This paper describes a two-part investigation into the effect of local geometry changes at the edges of the overlap in single lap joints. In the first part, finite element analysis has been used to model the effects on the stress distribution of geometry changes which are small in relation to the dimensions of the local geometry, in order to provide an improved model for failure prediction. The model used is that of an adhesive around a rigid corner, and the effect of rounding the corner has been considered. The second part deals with local geometry changes of the same order of magnitude as the dimensions of the geometry and their effect on joint strength. Three types of joint - - one with a square-edged adhesive layer, one with a fillet of adhesive and one with an adhesive fillet plus a radiused adherend - - were manufactured, tested and analysed. Improvements to the finite element models were also made following the results of part 1. It is found that finite element analysis is capable of predicting the significant strength increases that may be achieved in single lap joints by filleting the adhesive at the edges of the overlap and rounding the ends of the adherends.
Key words: adhesive-bonded joints; stress distribution; strength prediction; overlap geometry; rounding of corners; singularities; finite element analysis
In order to design an adhesive joint for a critical loadbearing application, it is necessary to have a knowledge of the ultimate strength of the joint and an understanding of its failure properties. The application of finite element methods (WM) to the analysis of adhesive joints has been particularly successful: for example, the strength and locus of failure o f single lap joints with a wide range of adhesives have been predicted with reasonable accuracy I. The main reason for this is that when cohesive failure of the adhesive is responsible for joint failure, as is often the case in wellprepared joints, initiation of failure is confined to a ve-'2,' localized region in the adhesive. For the single lap joint, it is found that the highest stresses and strains in the adhesive occur in regions at the edge of the overlap. Closed-form analytical solutions to problems of this ty,pe may give good overall indications of the conditions in the adhesive layer when the joint is loaded, but they are unable to predict the critical conditions in these edge regions in sufficient detail for joint failure to be predicted correctly. In contrast, the use of finite elements enables the distributions in the critical regions to be predicted with much greater definition. A second requirement met by FEM is that the nonlinear properties of the adhesive and adherends may
be modelled, a process which is necessary for all practical problems. W h e n the stress and strain fields in adhesive joints are examined in detail, the critical regions are found to be singular in nature. Fig. I indicates the singular points in a single lap joint, in one case with a squareedged adhesive layer and in another with a fillet of adhesive at the edge. Each point is on the boundary, between two dissimilar materials where there is also an abrupt change in slope giving a perfectly sharp corner. In order to predict joint strength using the results of a finite element analysis of such joints, a failure criterion is required. One possible a p p r o a c h is to treat the singular points in the same way as crack tips are treated in fracture mechanics. Failure may be predicted at a critical stress intensity or fracture energy, assuming that these parameters are purely properties of the adhesive. However, it has been shown for adhesive joints that the fracture energy is not independent of the joint g e o m e t ~ 2. so that no straightforward failure criterion or method of prediction is found. An alternative approach I is to base joint failure on the m a x i m u m conditions of stress or strain predicted by the analysis using an appropriate bulk property of the adhesive as a failure criterion. The difficulty then arises that, because of their nature, the ability of finite
0 1 4 3 - 7 4 9 6 / 8 7 / 0 2 0 0 6 9 - 1 2 $03.00 © 1987 Butterworth 5t Co (Publishers) Ltd INT.J.ADHESION AND ADHESIVES VOL.7 NO.2 APRIL 1987
69
'
f~ /Y
including singularities, were also considered following the results from the first part of the work. Thus, the aim was to demonstrate that these larger local geometry, changes could have a measurable effect on improving joint strength, and that these improvements may be predicted from finite element analysis of the various joints.
Investigation of small-scale local geometry changes Simplified model
a
B
~iiJi;i;~;~i~i;i !~!;!ii~i~!ii~i i;i~i~i i i ~i~i~i ~i~i~i;i ~;~ii ~i~i i i_~i~i~i~
b ~ - - - - - - - Locus of failure Fig. 1 Initiation of failure from singular points in the single lap joint: (a) square-edged adhesive layer; and (b) with an adhesive fillet
elements to model the singular behaviour in the joint depends on the local degree of refinement of the finite element grid, so the m a x i m u m conditions predicted would depend on the grid used in the analysis. However, in reality, the geometries of the corners will not be absolutely perfect: there will always be some degree of rounding and the singularities will not necessarily exist. Indeed, the loci of failure indicated in Fig. 1 often run adjacent to and not necessarily exactly through the singular points. In the first part of this paper, a study of the effects of local changes in geometry, small in relation to the dimensions of the local geometry (ie, the adhesive layer thickness), is reported. This is a means of overcoming the problems associated with singularities, the aim being to provide an improved model on which to base failure prediction. The region of point "B" in the joint with an adhesive fillet in Fig. 1 was considered. However, because a high degree of refinement of the finite element grid was required, the analysis of the complete joint was avoided and a simplified model of the critical region was considered instead. In the second part, changes of local geometry of the same order of magnitude as the dimensions of the geometry are considered in terms of their effect on joint strength. Three joint types were manufactured, tested and analysed: one had a square-edged adhesive layer, one had a fillet as in Fig. 1 and the third was a joint with a fillet plus a large radius on the adherend. Possible improvements to the models of the joints,
70
INT.J.ADHESION AND ADHESIVES APRIL 1987
The simplified model of the critical region of the joint with an adhesive fillet is illustrated in Fig. 2. The model consists of an adhesive surrounding a rigid corner. A uniformly distributed load was applied across the width of the model via a strip of material, relatively stiff compared with the adhesive, to which was assigned the elastic properties of aluminium. By laterally constraining the sides to minimize bending, the action of the applied load was to displace the strip approximately uniformly across its width. Thus a concentration of stress in the adhesive around the corner was ensured, similar to the conditions around the critical corner of the lap joint. Because of symmetry, only half of the model needed to be analysed. Modification of the corner geometry in the model was achieved by transforming the coordinates of a section of the finite element grid so that the boundary became of the form v = x2/2k + C as shown in Fig. 3. The value of k is a measure of the degree of rounding, and "C is the thickness of the adhesive layer at the corner, which is equal to 0.125 m m for no rounding (k = 0) and with rounding is increased by an amount equal to k/2. The effect of rounding on the local stress distributions was investigated by analysing
::::::::::::::::::::::::::::::
a
i
L_ I-
1.46
Uniform loading
b Fig. 2 Critical region of a lap joint with an adhesive fillet: (a) load transfer; and (b) simplified model. All dimensions in mm
nature of the distribution, holds to a distance pan-way
into the element closest to the comer; the numerical results are unreliable closer to the comer. Thus, in the linear region, the distribution can be expressed in the form:
Oprin. max= constant X r -n X f(0)
1
c
EI
Adhesive
x2 .y=~-~ + c
k
C= 0.125 + ~- mm
Fig. 3 Modification of geometry to produce a rounded corner in the simplified model
the model for four cases corresponding to k values of 0.0, 0.00125, 0.0146 and 0.161 mm. Except for the latter case, these modifications are small compared with the local thickness of the adhesive layer. Refinement of the finite element grids was arranged so that the dimensions of the elements increased in equal logarithmic steps away from the comer. For the adhesive, eight rows and twelve columns of eightnoded isoparametric elements were used, the smallest element adjacent to the c o m e r being 0.00125 by 0.00125 mm. The analysis was performed for both a linearly elastic adhesive and an elastic-plastic adhesive with a small degree of strain hardening. For the computation, the adhesive properties were based on the uniaxial tensile stress/strain characteristics shown in Fig. 4. The linear material corresponds to an unmodified epoxy adhesive with a Young's modulus of 2.8 GPa and a Poisson's ratio of 0.4. The elastic-plastic material corresponds to a toughened epoxy with a Young's modulus of 2.5 GPa and a Poisson's ratio of 0.37. The yield and plastic flow of the adhesive was represented in the finite element computation by a curve-fit model of the uniaxial behaviour in Fig. 4. Details of the formulation of the toughened material are given in the section on large-scale local geometry changes, as this material was used in the various lap joints that were tested. A state of plane strain was assumed for the model and a uniform load equivalent to a direct stress of 100 MPa was applied throughout. Results from analysis of simplified model
(1)
where for this case n = 0.185. In theory, for increasingly small values of r, the principal stress would increase to infinity. However, this is not a sensible practical consideration, since the surface roughness can be expected to be of the order of 0.001 to 0.01 mm. Also, in the manufacture of adhesive joints chemical etching and/or mechanical abrasion will remove any sharp 90 ° corner, even if it could be produced to this degree of perfection in the first place. The relationship given in Equation (!) is similar to that for the singularity around a sharp crack in an elastic homogeneous material, where the constant multiplier corresponds to the stress intensity factor and for a sharp crack "n" is equal to 0.5. The principal stress distributions in Fig. 6 for the various cases analysed show that introducing a rounded c o m e r into the model results in a departure from the singular distribution of the stresses as the c o m e r is approached. Rounding the corner effectively removes the singularity, a plateau is reached at a distance from the corner and thereafter the maximum principal stress remains constant. With the exception of the case o f k = 0.161 mm, the change in geometry, of the corner is small relative to the thickness of the adhesive layer in that region. Thus, as might be expected from St Venant's principle, the local geometry change results in only a local modification of the stress distribution; further away from the corner the stresses are unaffected. However, even with k = 0.161 mm, which represents a 64% increase in the thickness of the adhesive layer at the comer, the stresses are only reduced slightly beyond 0.1 mm from the corner. To a first approximation it would appear that:
(O'prin. max)p oC l/k and rpOC k
80
tic
~. 60
\ Elastic-plastic
o~ 2
40
k-
20
Elastic adhesive
The distribution of the maximum principal adhesive stresses for the square-corner model (k = 0) is shown in Fig. 5. The stresses on the centre-line of the model have been linearly interpolated from the Gauss point values. The linear relationship, indicating the singular
0 Fig. 4
[
I
I
0.02
0.04
0.06
I
I
0.08 0.10 Strain
I
I
0.12
0.14
0.16
Adhesive uniaxial tensile stress/strain properties
INT.J.ADHESION A N D ADHESIVES APRIL 1 9 8 7
71
be seen that with the rounded corners a peak in the distribution occurs away from the centre-line of the model. Locally this will be governed by the nature of the curvature around the corner. Again. at a sufficient distance from the centre-line, the stress distribution is unaffected.
where (aprm. max)p and rp are the peak value of the maximum principal stress and the distance from the corner at which the peak value is attained, respectively. Thus. the predicted stress distribution is extremely sensitive to small changes in the local geometry of the corner. Again. taking into account practical considerations of joint preparation, a typical radius of the order of 0.05 mm would have a significant effect on the local stress distribution. Considering the stresses along a line close to the rigid interface in the adhesive shown in Fig. 7, it can
Elastic-plastic adhesive In order to assess the post-yield behaviour of the adhesive, the plastic energy, density. Wp (that is the plastic work done per unit volume at a point), has been
e (°)
Boundary of fi rstel ement
+
45
• X
135 90
--
I E
i
~
~
1000 A
E
100
I
I
I
J
II
L
t
t
I
I
J
I Lt
0.001
I
~
L
I
L L I]
L
0.01
0.1
•(mm) Fig. 5
Maximum
principal ,adhesive
stresses
around
a square
corner with
elastic adhesive
prope~ies
f o r an a v e r a g e
applied
stress of 1 O0 MPa
Boundary of first element 1000 a.
k (ram) E c"
0.0!~....~ 0.00125, ~ ' ~ - ~
- ~ l m ~
I I
~li~ll,.,,.,.lt~t
I ~]ni-,..nnl~
0.161 ~
1:3
0
~
13
D
0
E,
~-,
I [ I 100
I
I
I
I[
I
I
I I
L
I
I
L
I
0.001
L
I
I
I
0.01
J
J
J
L
L
I
I
L I
0.1
• (mm) Fig. 6 Variation of the adhesive maximum principal stress distribution with corner rounding along the centre-line with elastic adhesive properties (average applied stress = 100 MPa)
72
INT.J.ADHESION A N D A D H E S I V E S APRIL 1 9 8 7
considered based on the elastic-plastic adhesive properties in Fig. 4. The distributions of Wp and O'prin.max for the square-corner case are shown in Fig. 8, from which the relationships:
The Wp distribution departs from the singular form far away from the c o m e r (r > 0.i mm) and close to the c o m e r (r < 0.003 mm). For the latter, the proximity, of the rigid c o m e r laterally restrains the adhesive. This results in an increased hydrostatic component of stress and a suppression of yield and plastic flow, so that Wp appears to attain a m a x i m u m close to the comer. It is worth coml~aring these results with the analytical solution ~'4 for the stress and strain fields
Gprin. max oC r -0"03 WpOC r -°-92
hold over most of the range.
~
~
Location o f values plotted
"" J :-"
Rigid ./
1000G.
k (mm) E
0.0~
,-.
0.161 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
100
=
t
I
,
~ , , ,I 0.001
0.0001
=
A
=
L
L
= L =[
0.01
A
A
~
I
L t
I t I 0.1
x (mm)
Fig. 7 Variationof the adhesive maximum principal stress distribution with corner rounding along the interface with the rigid material, for elastic adhesive properties (averageapplied stress = 100 MPa) 10.0
1000
Wp
"
to
1.0
100 Oprin, roll
0.1
10
I
0.0001
L
I
0.001
0.01
0.1
r (mrn) Fig. 8 Adhesivemaxrmum principal stress and plastic energy density distributions along the joint centre-line with a square corner and elastic-plastic adhesive properties (averageapplied stress = 100 MPa)
INT.J.ADHESION AND ADHESIVES APRIL 1987
73
around an elastic-plastic crack tip, which are of the form:
Further, for k greater than 0.00125 mm, there is a maximum some distance from the corner. This is because, even though the normal stress (cr,,) increases slightly as the corner is approached, the rigidity, of the rounded corner constrains the adhesive in the x (transverse) direction, thus inducing higher transverse tensile stresses (Crx). The net effect is that the hydrostatic component of stress is increased and vield suppressed, so that close to the corner there is a reduction in Wp as the corner is approached. Thus there is departure from singular behaviour in Wp, partly as a result of the restraints imposed by the rigid corner and partly because the load is more evenly distributed through the adhesive as the corner is progressively rounded. Again, the effects are local: the distributions remain relatively unaffected sufficiently far away from the corner. The Wp distributions close to the rigid interface are shown in Fig. 10. The redistribution in load as a result of rounding reduces Wp around the corner but increases it away from the corner, such that there is a peak at a distance of the same order of magnitude as "k" from the centre-line in the x-direction. Still further away, Wp again remains unchanged, so that rounding only has significant effects in the local region two or three times the size of k. The overall distribution of Wp around the corner is illustrated by the contour plots in Fig. 11 for k = 0.0 and 0.0146 mm and for a 100 MPa uniform loading. By
O.ij = r - N / ( I + N) Si j = r - I / ( I
+ N)
where N. the power-law hardening parameter, lies between 0 and 1. For the elastic-plastic adhesive properties in Fig. 4 there is a small degree of strain hardening, so that N is slightly greater than zero. Thus, if it is assumed that: N = 0.05 and Wp c£ o'ij .sij
then the analytical solution would give: O'prin. max o£ r -005
and WpO£? - I
which is of similar form to the numerical results for the rigid corner. The Wp distributions with rounding are plotted in Fig. 9. With increasing k the concentration of loading along part of the centre-line is reduced, leading to a reduction in Wv over part of the region, and in particular to a reduction in the maximum value.
10.0
k (mrn)
O0 0.00125 ~
,
~
~-" 1.0 E
im
0.161-"
"" ""
" "" "''''
" "''
"'"
"'"
g 11OIOOO
O . . . . .
O I O O I U D
QO
O
" "'''"
0.1
0.0001
0.001
0.01
0.1
r(mm) Fig. 9 Variation of the adhesive plastic energy density distribution with corner rounding along the joint centre-line for elastic-plastic adhesive properties (average applied stress = 100 MPa)
74
INT.J.ADHESION AND ADHESIVES APRIL 1987
10
_ _ Location of
k (mm)
00
values plotted
l ¸
0.00125 m
----- jt'
x 0.0146 E
0.161 . . . . . . .
....
" . ='e
le
a
o
.
ol
=
i
•
o
•
o J o o
0.1
I 0.0001
~
I
K I [ III 0.001
1
I
I
I I I III 0.01 X (mm)
f
I
t
I
I III 0.1
I
L
A
Fig. 10 Variation of the adhesive p l a s t i c e n e r g y density distribution with corner rounding along the interface with the rigid material for e l a s t i c - p l a s t i c adhesive properties (average applied stress = 100 MPa)
rounding, the high concentration in strain is removed from the corner. The redistribution results in one peak in HIp on the centre-line and a second, larger ~eak along the interface. Note that the 1 X 106 J m - " contour which is a significant distance from the c o m e r has not been greatly affected by the rounding of the corner. Conclusions from simplified model The following conclusions may be drawn from the results for the simplified model of the critical region of a lap joint with adhesive fillets. 1) For a perfectly square corner, the numerical solution indicates a singular distribution of stress around the corner when the adhesive is elastic. With an elasticplastic adhesive, the Wp distribution follows the singular form (r - I ) over part of the region but, close to the corner, departs from this and attains a m a x i m u m value because of the restraining effects of the interface. 2) Rounding removes the singularity and can substantially reduce the peak values of the m a x i m u m stress for an elastic adhesive and the m a x i m u m plastic energy density, and thus the m a x i m u m strain, for an elastic-plastic adhesive. 3) The model which includes a degree of rounding is more satisfactory because perfectly square geometries will not occur in practice. 4) With a rounded corner there is a region close to the corner where the stress and plastic energy density vary little. If failure is based on the predicted conditions in this region, it will not be dependent on the degree of refinement of the finite element grid. The problem then becomes that of deciding what is the most relevant degree of rounding.
Investigation of large-scale local g e o m e t r y changes Experimental Three types of aluminium/epoxy single lap joints were manufactured to the dimensions shown in Fig. 12. These were chosen so that the effects of altering the local geometry of the joint (on a larger scale than hereto considered) in the critical regions could be investigated. The basic joint (Fig. 12(a) ) had square edges at the ends of the overlap and two other types (Figs 12(b) and (c) respectively) had adhesive fillets and adhesive fillets with radiused adherends. All the joints were 25.4 m m in width. The adherends were of aluminium alloy, type 2014TB, which has a 0.02% proof stress of 304 MPa. The adhesive used was a rubber-modified epoxy consisting of a diglycidyl ether of bisphenoi A, with 15 parts (by weight) per hundred of resin of carboxyl-terminated butadiene-acrylonitrile rubber and 5 parts per hundred of piperidine as a curing agent. The joints were cured at 120°C for 16 h. Three joints of each type were loaded to failure in a screw-driven tensile testing machine at a constant cross-head speed of 1 m m min -I. The results from the tests are given in Table i. Table 1.
Median Lowest Highest
Experimental joint strengths (kN) Squareedged
Filleted
Filleted and radiused
1 5.9 1 5.4 16.4
20.4 1 9.2 21.0
24.5 23.8 24.7
INT.J.ADHESION A N D ADHESIVES APRIL 1 9 8 7
75
strength and that. by suitable modification on a relatively small scale, significant increases in joint strength may be achieved.
a
Finite element 0.005 mm
1
106 J m -3
b 0.005 mm k {mm) ....
analysis
At first the analysis was performed with linearly elastic material properties for each of the three joint types and a small displacement solution obtained. Subsequently, the elastic-plastic properties of both the adhesive and the adherend were included (the properties of the rubber-toughened adhesive corresponding to those of Fig. 4 ) a n d a large displacement solution obtained to account for the rotation of the overlap under load which was observed during testing. In cases where singular behaviour was thought to be influencing the numerical results in the critical regions, local small-scale rounding of the geometry,, as discussed above, was investigated as a means of improving the theoretical model. The square-edged joint was modified by introducing curvature to the edge of the adhesive layer, as shown in Fig. 12(a), in order to transform the singular corner point A to a nonsingular point A'. For the filleted joint, the corner of the adherend, point B, was modified by introducing the curvature to the corner, as indicated in Fig. 12(b). The changes in geometry that these modifications represent are small in relation to the variations which will occur between nominally similar joints in practice.
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Although the sample number of each type is small. there are significant differences in joint strength. Based on the median values, the inclusion of an adhesive fillet results in a more than 25% increase in strength over the square-edged joint, and the additional modification of radiusing the adherends gives an increase of more than 50%. For the radiused joints, a significant reduction in stiffness was noted in the loading curves just prior to failure, this being a result of plastic deformation in the adherends, which were permanently deformed after joint fracture. From these results, it is seen that the local edge geometry, of the joint has an important bearing on joint
76
INT.J.ADHESION A N D ADHESIVES APRIL 1 9 8 7
0
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q3.2 mm
C
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Predicted locus o f failure
Fig. 12 Aluminium/epoxy single lap joints: (a) square-edged adhesive bayer; (b) filleted; and (c) filleted and radiused
Square-edged joint The results presented for the 'square-edged' joint are for a 16 kN applied load corresponding to the median value of the experimentally determined joint strength. The principal stress distributions at the edge of the adhesive layer from the small displacement elastic analysis are shown in Fig. 13, for both the square and the modified curved edge. In the unmodified model, the singular nature of the corner point A is apparent and a m a x i m u m principal stress of 1206 M P a is predicted at the Gauss point (point of numerical integration at which the solution is given) 0.001 m m from the corner. With the modification to the model of the curvature at the edge, the singularity at the corner is clearly removed and the largest value of the m a x i m u m principal stress of 258 MPa occurs 0.055 m m from the interface into the bulk of the adhesive layer. Even with this modified model, the m a x i m u m stress predicted is far greater than the tensile strength of the adhesive. Thus, an analysis which only represents the adhesive as an elastic material is not adequate for predicting joint strength. From the large displacement
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b Fig. 13 Principal stress distributions for the square-edged joint with elastic adhesive properties and a 16 kN applied load: (a) unmodified model; and (b) modified model
elastic-plastic analysis, the plastic energy density (Wp) distributions are shown in the critical regions of the two joint models in Fig. 14. Again, by introducing the curved edge, the singular behaviour in the plastic strain is removed. As for the stress in the elastic analysis, the m a x i m u m in Wp occurs in a region of relatively gentle variation away from the interface and is of very much lower magnitude than predicted by the square-edged model, thus providing a new basis for predicting joint strength. Clearly the curved-edge model of this joint provides a far more satisfactory means of predicting joint strength based on a cohesive failure of the adhesive. It is significant that such small changes in the local geometry result in such large changes in the local distributions of stress and strain, especially when in practice much larger variations in the edge geometry between joints are likely to exist. It is proposed, therefore, that the modified-edge model is far more representative of the types of local geometry that will be expected in practice.
Filleted joint The principal stresses around the critical corner of the unmodified and modified models of the filleted joint are shown in Fig. 15 for the elastic analysis with a 16 kN applied load. Comparing the magnitudes of the principal stresses, as given by the Gauss points for each model, the square-cornered model predicts a m a x i m u m value of 210 MPa at a point 0.025 m m from the corner point B, while with the rounded corner the m a x i m u m value of 147 MPa occurs at a point inside the overlap 0.015 m m from the adherend surface. These results are analogous to those for the simplified model given earlier in Fig. 6. If the rounding here is compared with the k value of 0.161 mm, then from Fig. 6 the m a x i m u m principal stress for the square corner (r = 0.025 ram) would be 225 MPa and 174 MPa for the rounded corner (r = 0.015 mm). The ratio of the two stresses is thus approximately !.3, which is roughly the same as the ratio here. It is also apparent that the difference between the m a x i m u m stresses in the two models is limited by the degree of refinement that could be used in the square-cornered model. The influence of the singularity at the corner point B on the m a x i m u m Gauss point stresses predicted is not as great in the model for the square-edged joint around point A , because for the latter there was greater local refinement of the finite element grid. Clearly, the round-cornered model is more satisfactory as the results are essentially independent of the degree of refinement. However, as with the square-edged joint, even with the improved model joint failure cannot be predicted from an elastic analysis of the adhesive stresses. The results from the elastic-plastic analyses are shown in Fig. 16, where the Wp distributions are shown for the two models around the critical corner for an applied load of 20 kN, corresponding to the failure load of the joint. As with the stresses in the elastic case. rounding of the sharp corner has led to a redistribution of the plastic deformations, such that the m a x i m u m in Wp occurs inside the overlap. As already discussed, because of the limited mesh refinement there is little difference in the m a x i m u m value of Wp predicted by the two models. In fact, the influence of the singular point on the final results of the elastic-plastic analysis
INT.J.ADHESlON AND ADHESIVES APRIL 1 987
77
8 ~ 10 6 J m -3
<_ a
75 100 × 106 J m -3
J ~
0.02 mm I
I
b
Fig. 14 Plastic energy density distributions for the square-edged joint with elastic-plastic adhesive properties and a 16 kN applied load: (a) unmodified model: and (b) modified model
is even less than for the elastic case. Thus it is not surprising that, with a similar degree of mesh refinement as here, reasonable predictions of joint strength have previously been obtained based on a m a x i m u m plastic strain in the adhesive derived from the Gauss point values in square-cornered joint models I. Radiused adherends with fillet
The elastic principal stresses in the adhesive around the radiused corner are shown in Fig. 17, again for an applied load of 16 kN. As with the small degree of rounding, the highest value occurs inside the overlap, although somewhat further inside in this case (1.43 m m compared with 0.12 mm). As previously noted, the introduction of a small degree of rounding reduced the peak value from 210 to 147 MPa. Here the larger change has made the stress distribution at the edge of the overlap even more uniform and reduced the peak value further to only 61 MPa. Thus, the potential is seen for introducing local geometrical changes of this order of magnitude leading to significant improvements in joint strength. However. for this joint, a higher value of the maximum principal stress of 82.4 MPa occurs at the edge of the adhesive fillet (equivalent to point C in Fig. 12(c) ), which becomes the site from which failure initiates. When the elastic-plastic properties of both the adherends and adhesive are included in the analysis+ the results predict the onset of large-scale plastic deformation of the adherends prior to joint failure. This is clearly seen in the experimental load/deflection curve for the joint in Fig. 18 where, as the experimental failure load is approached, the stiffness (load/ deflection) of the joint decreases rapidly. A theoretical load/deflection curve is also given in Fig. 18 based on
78
INT.J.ADHESION AND ADHESIVES APRIL 1987
the finite element results. This curve is dominated by the yield behaviour of the aluminium adherends, and agrees well with the experimental results. It has been shown previously I that the combination of tension and bending in the adherends after yielding results in a concentration of plastic strain in the region adjacent to the edge of the fillet point C (Fig. 1). This in turn results in higher concentrations of strain in the adhesive around the edge, so that joint strength is lower than would be expected if a higher yield strength material had been used in the adherends. For the other joint types the load/deflection characteristics in Fig. 18 indicate that yielding of the adherends has little influence on joint failure. Thus. the increases in joint strength seen here with the radiused joint are somewhat limited by the yield behaviour of the adherends. Again. this is in keeping with the observed permanent deformation of the adherends in these joints after failure. Prediction of joint strength
The peak values of the m a x i m u m principal adhesive stresses predicted for each of the joint types are summarized in Table 2, based on the elastic small Table 2. Peak values of adhesive maximum principal stress from elastic analysis with 16 kN applied load Joint type
Peak maximum principal stress (MPa)
Square-edged Filleted Filleted and radiused
258" 147* 83
*From modified joint models
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the former, the prediction would have been extremely low while, for the latter, it would have been almost unchanged. This reflects the relative influence of the singularity in each of the unmodified models as determined by the local refinement of the finite element grid in each case. From an examination of the failure surfaces of the joints tested, it was concluded that the predictions of the loci of failure indicated in Fig. 12 are in agreement with those found experimentally. An exception to this was with the filleted and radiused joints, where an interface crack initiating from "C" is observed at one end of the joint (as expected), but a fillet crack similar to that at point "B" in the filleted joint is observed at the other. It is assumed, therefore, that failure initiates at "C" at one end of the joint as expected, but the result of this initial failure is to induce the different type of failure at the other end of the joint. Conclusions
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displacement analyses. Although the relative magnitudes do indicate the possibility of increasing joint strength b y modifying the basic 'square-edged" geometry, the values themselves do not provide a basis for predicting joint strength accurately because of the plastic deformation of the adhesive and adherends. In order to predict joint strength based on the bulk properties of a toughened adhesive, such as that used here, the plastic deformation must be accounted for. Therefore, in association with the non-linear analysis carried out for the joints, a failure criterion must be applied to the results. A limiting value of the plastic energy density equal to 7.3 x 10 J m - - has been used. based on the bulk uniaxial tensile properties of the adhesive at failure (Fig. 4). For the square-edged and filleted joints, predictions were based on the results from the modified models. In Table 3, the predicted joint strengths are compared with the median of the experimental values. Agreement is very good, the predictions being to within _+ 10% of the median values. Had the modifications to the critical corner geometries not been used for the square-edged and filleted joints then. for
The site of initiation of failure in adhesive lap joints is strongly dependent on the geometry of the edges of the overlap. In order to predict failure, the method of analysis must, therefore, be able to account for the effects of local geometry and be sensitive to local variations of stress and strain through the adhesive layer in critical regions. Only by using a numerical approach, such as the finite element method, can these types of variation be included in the analysis. In using finite element techniques, prediction of joint strength may be hampered or be extremely dependent on local mesh refinement because of the existence of singularities in the stress or strain fields at critical points. In adhesive joints, these singularities arise at corners between the adhesive and adherend materials as a result of the perfect squareness in the geometry of the model. It has been demonstrated that by rounding the corner the singularity may be removed and replaced by a more uniform distribution of stress and strain. However, the final distribution depends strongly on the degree of rounding that is introduced. Significant increases in single lap joint strength may be achieved in practice by including an adhesive fillet at the edges of the overlap and further increases may be achieved by additionally rounding the ends of the adherends. These observed increases have been predicted by finite element modelling, taking into account the local geometry changes involved and. where necessary, introducing a degree of rounding into the critical areas of the joint model of the same order of magnitude as the glue-line thickness in the joints. In this way, it is felt that a more realistic and hence reliable basis for predicting joint strength has been achieved. Table 3. Comparison of joint strength predictions (kN) with experimental values Joint type
Predicted
Median of experimental
Square-edged Filleted Filleted and radiused
14.5 21.2 25.3
15.9 20.4 24.5
INT.J.ADHESION AND ADHESIVES APRIL 1987
79
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a -
2
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2
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a
,
,b
Fig. 16 Plastic energy densiW distributions for the filleted joint with elastic-plastic adhesive properties and a 20 kN applned toad: (a) unmodified model; and (b) modified model
Authors
Professor Adams, to whom inquiries should be addressed, is with the Department of Mechanical Engineering, University of Bristol, Queen's Building, University Walk, Bristol, BS8 1TR, UK. Dr Harris is with Materials Engineering Research Laboratories Ltd, Tamworth Road, Hertford, SGI3 7DG, UK. 25
Filleted and radi
Filleted
20
Square-edged.
A
15
Z
Fig. 1 7 Principal stress distribution for the rediused and filleted joint with elastic adhesive properties and a 16 kN applied load (see Fig. 12(c) for dimensions)
o 10
Experimental ~ m Theoretical prediction
References 1
Harris, J.A. and Adams, R.D. Int J Adhesion and Adhesives 4 No 2 (1984) pp 6 5 - 7 8
2
Kinloch, A.J. and Shaw, S.J. J Adhesion 12 (1981) pp 5 9 - 7 7
3
Rice, J.R. and Rosengren, G.F. J Mech and Phys of Sohds 16 No 1 (1968) pp 1-12
4
Hutchinson, J.W. J M e c h a n d P h y s o f S o l i d s 1 6 N o 1 pp 13-31
80
INT.J.ADHESION
AND ADHESIVES
(1968)
APRIL 1 987
if 0
~ 0.2
h 0.4
I 0.6
I 0.8
I 1.0
I 1.2
Deflection (mm) Fig. 18 Load/deflection curve for a filleted and radiused joint indicating the range of failure load for various joint types