Numerical analysis of the stress distribution in single-lap shear tests under [603095]

Numerical analysis of the stress distribution in single-lap shear tests under
elastic assumption —Application to the optimisation of the
mechanical behaviour
J.Y. Cognardn, R. Cre ´ac’hcadec, J. Maurice
Laboratoire Brestois de Me ´canique et des Syst /C18emes, ENSTA Bretagne/UBO/ENIB/Universite ´Europe ´enne de Bretagne, 2 rue F. Verny, 29806 Brest Cedex 9, France
article info
Article history:
Accepted 25 June 2011
Available online 12 July 2011
Keywords:
Lap shearJoint designFinite element stress analysisStress distributionabstract
The single lap joint is the most used test in order to analyse the behaviour of an adhesive in an
assembly as on one hand, the manufacturing of such specimens is quite easy, and on the other handthey require only a classic tensile testing machine. However, such specimens are associated withcomplex loading of the adhesive, i.e. non-uniform shear stress along the overlap length, quite large peel
stress at the two ends of the overlap and significant edge effects associated with geometrical and
material parameters. In addition, the stress concentrations can contribute to fracture initiation in theadhesive joints and thus can lead to an incorrect analysis of the adhesive behaviour. Therefore,
understanding the stress distribution in an adhesive joint can lead to improvements in adhesively
bonded assemblies. The first part of this paper presents the influence of edge effects on the stressconcentrations in single lap joints under elastic assumption of the material and using a pressure-
dependent elastic limit of the adhesive. In the second part, some usual geometries, proposed in the
literature about stress limitation, are compared with respect to the maximum load transmitted bysingle lap joint. The last part presents some geometries, which significantly limit the influence of edgeeffects and are more appropriate for analysing the behaviour of the adhesive.
&2011 Elsevier Ltd. All rights reserved.
1. Introduction
Adhesively bonded assemblies offer many advantages, espe-
cially when assembling dissimilar materials or composite materi-
als[1,2]. However, a lack of confidence limits the current use of
this technology; in fact, bonded assemblies are often charac-
terised by large stress concentrations, which, in particular, make
the analysis of experimental tests difficult [3,4]. Therefore, the
development of numerical tools in order to accurately predict the
mechanical behaviour of industrial bonded assemblies is also
difficult. Moreover one has to take into account the influence of
various parameters such as the possible complex non-linear
behaviour of the adhesive in an assembly, the geometry of the
different parts, the surface preparations, the influence of possible
defects, etc.
The single-lap joint is the most used test in order to analyse
the behaviour of an adhesive in an assembly as on one hand the
manufacturing of such specimens is quite easy, and on the other
hand they require only a classic tensile testing machine. However,
such specimens are associated with complex loading of theadhesive, i.e. non-uniform shear stress along the overlap length,
quite large peel stress at the two ends of the overlap and
significant edge effects associated with geometrical and material
parameters. Thus, peel and cleavage forces in particular make the
analysis of experimental results quite difficult, and can limit the
transmitted load of the assembly despite various techniques
proposed to limit the influence of edge effects. The stress
concentrations can contribute to fracture initiation in adhesivejoints and thus can lead to an incorrect analysis of the behaviour
of the adhesive [5–7].
In the literature various geometries have been proposed to
reduce the stress concentrations, such as influence of spew and
chamfer [8–11], influence of slots [12]; moreover, some works on
the optimisation of such geometries have also been proposed
[13–15]. However, it has been shown that a coupling between the
non-linear behaviour of the adhesive and the stress concentration
can exist [16]. Thus, in a first step, it is important to design
adhesively bonded joints, which significantly limit the influence
of edge effects under elastic assumption of the different parts.
Various simplified models exist [17], but such models give only
the average stress state in the adhesive and are not able to
analyse the stress concentrations close to the free edges of the
adhesive [16,18]. Therefore, understanding the stress distribution
in an adhesive joint can lead to improvements in adhesivelyContents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/ijadhadhInternational Journal of Adhesion & Adhesives
0143-7496/$ – see front matter &2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijadhadh.2011.07.001nCorresponding author. Tel.: ț33 2 98 34 88 16; fax: ț33 2 98 34 87 30.
E-mail address: jean-yves.cognard@ensta-bretagne.fr (J.Y. Cognard).International Journal of Adhesion & Adhesives 31 (2011) 715–724

bonded assemblies. Under elastic assumption, results of asymp-
totic analysis underline some conditions required in order to
obtain experimental devices, which significantly limit stress
concentrations and the strong influence of the local geometry at
the free ends of the adhesive [3,19]. The use of finite element
calculations is useful in order to take into account the influence of
various parameters, such as the global geometry, the boundary
conditions, the possible coupling between the two adhesive–
substrate interfaces, etc. [4,16]. Numerical determination of how
stresses evolve through the thickness of the adhesive joint
requires refined meshes, especially for large material heteroge-
neity of the structure [16]. Moreover, for some specimens, it has
been shown that a strong limitation of the stress concentrations,
computed under elastic assumption, can increase the load trans-
mitted by the bonded assembly even for ductile adhesive [16,20].
Thus, in order to accurately analyse the maximum transmitted
load of single lap joints, numerical simulations are proposed using
a pressure-dependent elastic limit of the adhesive defined from
the two stress invariants: hydrostatic stress and von Mises
equivalent stress; such models are often used to describe the
polymer behaviour [21,22].
Refined 2D finite element simulations are used to analyse the
influences of various geometrical parameters, such as joint thick-
ness, overlap length and the geometries close to the free edges of
the adhesive, on the stress distribution. Next, using some results
of asymptotic analysis and the experience acquired while improv-
ing a modified Arcan device developed to analyse the behaviour of
adhesives in an assembly under tensile-shear loadings [23,24],
specific geometries, which significantly limit the influence of edge
effects, are proposed.
The finite element simulations were made with the code
CAST3M (CEA, Saclay, France) [25].
2. Parameters of the study
Fig. 1 presents the main geometrical parameters of the single-
lap shear joint. The substrate length and the overlap length are
denoted by l1and l2. The substrate and adhesive thicknesses are
denoted by h1and h2. ‘‘O’’ represents the centre of the adhesive
(Fig. 1 b). In order to analyse the stress distribution in the
adhesive, results are presented along the mid-plane of the
adhesive [A B] ( y¼0,Fig. 1 b), along the adhesive–substrate
interface [C D] ( y¼h2/2,Fig. 1 b), and some over the whole
adhesive. In order to correctly determine the way the stress
evolves throughout the thickness of the adhesive, precise finite
element analyses have to be performed, even assuming a linear
elastic behaviour of the components, especially for large material
heterogeneity of the assemblies [16]. Moreover refined meshes
are also needed near the substrate–adhesive interfaces in order to
obtain good numerical results [16].
Computations were made in 2D (under plane stress or plane
strain assumptions) on half of the specimen by applying adequate
boundary conditions. Results are presented for aluminium sub-
strates (Young’s modulus: Ea¼75 GPa; Poisson’s ratio: na¼0.3)and the material parameters for the adhesive are Ej¼2.2 GPa and
nj¼0.3.
An optimisation of the transmitted load of adhesively bonded
joints requires to take into account an accurate failure criterion in
order to represent the real behaviour of the adhesive in an
assembly. Various studies underline that an accurate representa-
tion of the elastic yield surface of an adhesive requires the use of a
pressure-dependent constitutive model, i.e. a model taking into
account the two stress invariants: hydrostatic stress and von
Mises equivalent stress. For this study, only elastic behaviour of
the adhesive is used.
Starting from the experimental results obtained, for the epoxy
resin HuntsmanTMAraldites420 A/B, with the modified Arcan
device (adhesive thickness of 0.4 mm and displacement rate of
the tensile machine crosshead of 0.5 mm/min) it is possible to
define the initial yield function for the adhesive (neglecting the
viscous effects). This test allows us to determine the behaviour of
an adhesive in an assembly under radial loadings (tensile/com-
pression-shear loadings) [23]. An exponential Drucker–Prager
yield function allows a good representation of the experimental
data for the so-called initial elastic limit [24]:
F0¼aðsvmȚbțph2pt0¼0 ð1Ț
where svmis the von Mises equivalent stress and phis the
hydrostatic stress. a,band pt0are the material parameters. The
results of the identification are presented in Table 1 .
3. Behaviour of single lap shear tests
Numerical and experimental results underline the complex
stress states in the adhesive, which depend on the geometry of
the bonded assembly, especially close to the free edges of theadhesive. A limitation of the influence of edge effects can lead to
an increase in the load, which can be transmitted by the
assembly. In order to briefly present the influence of edge effects
on the stress distribution in the adhesive, three geometries of the
free edges of the adhesive are used in order to emphasise the
influence of the geometry of the adhesive-free edge defined by
the radius
r,Fig. 2 (geometry A: straight edges, r¼N; geometry
B: cleaned edges, r¼0.75 h2; geometry C: outer edges, r¼/C00.75
h2;h2is the joint thickness). Moreover those examples are used to
underline the influence of the overlap length and the substrate
rigidity on the stress distribution throughout the adhesive. Fig. 2
also shows the mesh of the adhesive close to the free edges and
the mesh of the substrate close to the substrate–adhesive inter-
face in order to respect the properties previously presented. Fig. 2
presents a part of the mesh associated with an adhesive thickness
ofh2¼0.2 mm (only half of the adhesive is modelled). Various
Fig. 1. Presentation of the single-lap shear joint: (a) geometry and (b) central part.Table 1
Material parameters for the initial yield surface.
a(SI units) bp t0(MPa)
1. E/C06 4.87 31.7J.Y. Cognard et al. / International Journal of Adhesion & Adhesives 31 (2011) 715–724 716

simulations have shown that good numerical results are obtained
using meshes with 20 linear rectangular elements for 0.1 mm
thickness of adhesive, especially when stress concentrations are
not too large. For instance, in the case of geometry B, an increase
in the number of elements in the joint thickness is associated onlywith a little change in the stress distribution. It is well known that
for large stress concentrations, more refined meshes are required,
but it is important to notice that the aim of this study is to analyse
specific geometries, which significantly limit the influence of edge
effects. Such meshes are used for the different computations
presented in the following. To facilitate the analysis of the
numerical results, adhesive is meshed with rectangular elements;
thus the distribution of the stresses throughout the thickness of
the adhesive can easily be analysed. In order to limit the number
of elements, in the model in the middle of the overlap length, the
element size can be increased in the xdirection without influen-
cing the numerical results. Moreover it can be seen that the
substrate is meshed using triangular elements, which allow the
element size in the substrate to be increased quickly in order to
limit the finite element model size.
Other geometries that limit the influence of edge effects are
presented in the following, but the three geometries examined
here are associated with constant thickness of the adhesive joint
(a modification of the adhesive thickness can change the mechan-
ical behaviour of the joint).
Fig. 3 presents the distribution of the shear ( xycomponent
stress) and peel ( yycomponent stress) stresses along the mid-
plane of the adhesive ( y¼0) and along the adhesive–substrate
interface ( y¼h
2/2) for aluminium substrates with a thickness of
h1¼6 mm, for an adhesive joint thickness of h2¼0.4 mm and for
four overlap lengths l2¼10, 30, 50 and 70 mm. For geometries A
and B it can be seen in Fig. 3 , as it is well known, that the shear
stress is not constant in the mid-plane of the adhesive. Moreover,
with such single lap shear specimens, an increase in the overlap
length leads to an increase in edge effects (results are presented for
an average shear stress of 1 MPa). For straight edges (geometry A)
close to the free edges, significant stress variation throughout the
adhesive thickness can be observed; these edge effects are mainlyassociated with significant change in the peel stress. Geometry B
(cleaned edges) allows extensive limitation of the influence of edge
effects with respect to straight edges. Thus it seems difficult to
analyse results of single lap shear tests only taking into account
simplified methods, which mainly take into account only theaverage shear stress in the adhesive and not the complete stress
distribution throughout the adhesive thickness [17].
As the elastic limit of the adhesive is defined using the
hydrostatic stress and the von Mises equivalent stress diagram
(Eq. (1)), the analysis of the stress stage in the adhesive has to be
done in the hydrostatic stress–von Mises equivalent stress dia-
gram. Fig. 4 a presents, in this diagram, the stress in the adhesive
for a joint thickness of 0.4 mm, for cleaned edges of the adhesive
and for the four overlap lengths. For an average shear stress equal
to 1 MPa, the hydrostatic stress and the von Mises equivalent
stress are computed for each element of the finite element mesh,
and only the envelope of those points is drawn in order to
simplify the presentation (all the points, in the hydrostatic
stress–von Mises equivalent stress diagram, are inside the so-
called envelope). Using the linear property of elastic problems,
one can determine the maximal load transmitted by the bonded
assembly using the elastic limit: for an increase in the load and
under elastic assumption, the maximum load is obtained when a
point of the so-called envelope reaches the elastic limit. Thus one
can define the maximal average shear stress in the adhesive
denoted by
taverage maxi (this stress represents the ratio of the
maximal load, under elastic assumption, to the bonded area).
Results are presented in Table 2 ; it can be noted that the
maximum average shear stress decreases with the overlap length,
but the maximum load transmitted by the assembly, Pl, increases
slowly with the overlap length; these results, obtained with the
specific elastic limit (Eq. (1)), are similar to usual ones. Fig. 4 b
presents, in the hydrostatic stress–von Mises equivalent stress
diagram, the stress state in the adhesive for the maximum load
transmitted and for the four overlap lengths. It can be observed
that the envelopes are nearly similar for the different overlap
lengths. Moreover, those results underline the negative influence
of the positive hydrostatic stress, associated with traction peel
Fig. 2. Presentation of geometries A, B and C and mesh for h2¼0.2 mm (close-up view) (‘‘ o’’ zone of maximal equivalent stress in the adhesive under elastic assumption).
(a) Geometry A: straight edges, (b) geometry B: cleaned edges and (c) geometry C: outer edges.J.Y. Cognard et al. / International Journal of Adhesion & Adhesives 31 (2011) 715–724 717

stress in the adhesive, on the maximum transmitted load by the
assembly. The results presented in Fig. 4 b also allow us to
determine the adhesive points, which first reach the elastic limit.
For the proposed geometries, the position of those points is
represented in Fig. 2 by the mark ‘‘ o’’, and is almost independent
of the adhesive thickness and the overlap length. It can be seen
that the free edges of the adhesive can have an influence on the
position of the more stressed parts of the adhesive: close to the
adhesive–substrate interface for geometries A and C, and in the
thickness of the adhesive for geometry B. As the adhesive–
substrate interface near the free edges is often the weakest part
of the assembly, geometries that are associated with a maximum
stress state inside the joint can be interesting.Fig. 5 presents the influence of the adhesive thickness geometry,
of the free edges of the adhesive and of the substrate thickness
(geometries A, B and C) on the maximal average shear stress, i.e. on
the maximum transmitted load of the bonded assembly. Results,von Mises stress (MPa)
von Mises stress (MPa)
Hydrostatic stress (MPa) Hydrostatic stress (MPa)051015
-512=70mm
12=10mm12=50mm
12=30mm
010203040
-1012=10mm
12=30mm
12=50mm
12=70mm
elastic limit
01 02 03 0 05 1 0 15
Fig. 4. Stress in the adhesive with cleaned edges for geometry B and for a joint thickness of h2¼0.4 mm in the hydrostatic stress–von Mises equivalent stress
diagram: (a) average shear stress of 1 MPa and (b) maximum transmitted load.
Table 2Maximum transmitted load by the assembly, for geometry B, a substrate thicknessofh
1¼6 mm and a joint thickness of h2¼0.4 mm.
Overlap length (mm): l2 10 30 50 70
taverage maxi (MPa) 10.86 4.39 3.22 3.01
Pl/Pl¼10 mm 1.00 1.21 1.48 1.94shear and peel stress (MPa)
shear and peel stress (MPa)
x (mm) x (mm)shear and peel stress (MPa)
shear and peel stress (MPa)
x (mm) x (mm)-30369
yy
xyl2=70 mm l2=50 mm
l2=30 mm
l2=10 mm
-60612182430
yy
xyl2=70 mm
l2=50 mm
l2=30 mm
l2=10 mm
-30369
yy
xyl2=70 mml2=50 mm
l2=30 mm
l2=10 mm
-30369
-40yy
xyl2=70 mm
l2=50 mm
l2=30 mm
l2=10 mm
-20 0 20 40 -40 -20 0 20 40-40 -20 02 0 4 0
-40 -20 02 0 4 0
Fig. 3. Shear and peel stresses in the adhesive for an average shear stress of 1 MPa, for a substrate thickness of h1¼6 mm and a joint thickness of h2¼0.4 mm. (a) Geometry
A: mid-plane, (b) geometry A: adhesive–substrate interface, (c) geometry B: mid-plane and (d) geometry B: adhesive–substrate interface.J.Y. Cognard et al. / International Journal of Adhesion & Adhesives 31 (2011) 715–724 718

starting from the stress state in the adhesive, are presented for two
overlap lengths (10 and 70 mm), using large marks. Moreover,
results obtained using only the stress state in the middle line of the
adhesive are also represented with little marks linked together
with thin lines. It is important to notice that a little modification of
the free edges of the adhesive can have a significant influence on
the transmitted load of the assembly (geometries A, B and C, Fig. 5 ).
Thus, in order to optimise the design of bonded assemblies, it is
necessary to significantly limit the influence of edge effects. For a
given thickness of the substrates ( h1¼6 mm), the geometry of the
free edges of the adhesive (A, B and C) has only a very low
influence on the stress distribution in the middle line of the
adhesive, but it has a significant influence on the stress distribution
in the adhesive thickness ( Fig. 5 ).Fig. 5 also underlines that an
increase in the substrate thickness leads to a little increase in the
transmitted load especially for thick adhesives (presented for
geometry B, for h1¼6 and 12 mm). In fact, an increase in the
rigidity of the substrate slowly reduces the influence of edge effects
[7,20]. Moreover, it is important to notice that for geometries A and
C, an increase in the adhesive thickness leads to a small decrease in
the maximum transmitted load. However for geometry B, an
increase in the adhesive thickness leads to quite a large increase
in the maximum transmitted load under the hypothesis used for
this analysis.
For such computations, plane stress and plane stress assump-
tions give quite similar results. Plane stress conditions are repre-
sentative to points close to the free edges of the single-lap shear
specimen and plane strain conditions are nearly representative topoints inside the specimen (for such points, a 3D computation can
give better results).
4. Analysis of usual geometries
Fig. 6 presents four geometries (D, E, F and G) used in various
studies in order to limit the influence of edge effects [6,9,11,26].
Geometry D is defined by an angle b¼451.G e o m e t r yEi sd e fi n e d
with the two following parameters: d1¼h2/2 and R1¼1.5h2;h2is
the adhesive thickness. For geometry F, the main parameters are
d2¼h2/2,d3¼h2,b¼451and linear chamfers. Geometry G is defined
with d4¼d5¼h2/2,d6¼1.5 h2,R2¼2h2and circular chamfers.
For the four geometries the influence of the adhesive thickness on
the maximum transmitted load of the bonded assembly is presented
for two overlap lengths (10 and 70 mm) in Fig. 7 . It can be noted
that the geometries E and G give similar results to those of geometry
B(Fig. 2 )p r e s e n t e di n Fig. 5 . Geometries D and E give better results
than geometries A and C, but they are associated with quite
significant stress concentrations.
Fig. 8 shows, in the hydrostatic stress–von Mises equivalent
stress diagram, the stress in the adhesive for the four geometries(D, E, F and G), for two adhesive thicknesses (0.4 and 0.8 mm) and
for an average shear stress equal to 1 MPa. Those results under-
line the influence of the bond-line thickness. It can be observed
that the envelopes are similar for the different overlap lengths.
Moreover, in the hydrostatic stress–von Mises equivalent stress
diagram, it can be seen that the stress in the adhesive, for the τaverage maxi (MPa)τaverage maxi (MPa)
adhesive thickness: h2 (mm) adhesive thickness: h2 (mm)04812162024
A h1=6mm
B h1=6mm
C h1=6mm
B h1=12mm
024681012
0.0A h1=6mm
B h1=6mm
C h1=6mm
B h1=12mm
0.4 0.8 1.2 1.6 0.0 0.4 0.8 1.2 1.6
Fig. 5. Influence of the adhesive thickness, the substrate thickness and the overlap length on the maximal average shear stress under plane stress hypothesis for
geometries A, B and C. ( ——— : taking only into account the middle line of the adhesive). Overlap length: (a) l2¼10 mm and (b) l2¼70 mm.
Fig. 6. Presentation of geometries D, E, F and G. (‘‘ o’’ zone of maximal equivalent stress in the adhesive under elastic assumption): (a) geometry D, (b) geometry E,
(c) geometry F and (d) geometry G.J.Y. Cognard et al. / International Journal of Adhesion & Adhesives 31 (2011) 715–724 719

different studied bonded joints ( Figs. 4 a and 8), is in almost the
same direction as that associated with the shear stress (loading of
the specimen) and the peel stress (influence of edge effects). Fig. 6
gives the position of the more stressed parts of the adhesive using
the mark ‘‘ o’’. On the one hand, circular free edges of adhesive give
better results than linear ones (phenomena associated with the
local geometry of the free edges of the adhesive); on the other
hand a chamfered substrate is associated with quite extensive
stress concentration close to the substrate–adhesive interface, not
close to the free edges of the joint, especially for linear chamfers
(geometry F) with respect to circular chamfers (geometry G).
The use of such geometries, associated with an increase in the
thickness of the adhesive close to the free edges, the manufacturing
of which can be quite complex, allows only a little increase in
the transmitted load with respect to the use of straight edges.
An optimisation of the parameters of geometry G could give quite
good results.
5. Analysis of stress singularities
The results presented in the previous sections underline the
strong influence of stress concentrations in the adhesive on thetransmitted load by the bonded joint. Those results are obtained
under elastic assumption, but failure initiation in an adhesive
joint between two metallic structures always starts at stress
concentration points, which are often associated with stress
singularities [6,7]. In order to design bonded specimens, which
significantly limit the influence of edge effects, results ofasymptotic analysis can be useful [3,19]. In the case of 2D
problems under elasticity assumption close to a corner, the
relevant parts of the displacement field uand the stress tensor
r, using the polar coordinates ( r: radius, y: angle) are given by
uðr,yȚ/C25krluðyȚð 2Ț
rðr,yȚ/C25krl/C01rðyȚð 3Ț
The singularity exponent lis the solution of an eigenvalue
problem defined by the boundary close to the singular point;
thus, this parameter depends on the geometry close to the
singular point and on the elastic properties of the materials
(Dundur’s constants) [3,4]. The intensity factor kof the problem
depends on the complete geometry of the structure and on theexternal loads.
uðyȚandrðyȚare angular shape functions; uðyȚis
also a solution of an eigenvalue problem [3,4].
Numerical computations have been made under 2D hypothesis to
analyse the influence of the local geometry close to the substrate–
adhesive interface, on one hand close to the free edges of the adhesive
and on the other hand for a point along the interface ( Fig. 9 ). Close toτaverage maxi (MPa)
τaverage maxi (MPa)
adhesive thickness (mm) adhesive thickness (mm)04812162024
D
E
F
G
024681012
0.0D
E
F
G
0.4 0.8 1.2 1.6 0.0 0.4 0.8 1.2 1.6
Fig. 7. Influence of the adhesive thickness and the overlap length on the maximal average shear stress for geometries D, E, F and G and for a substrate thickness o f
h1¼6 mm. (——— : taking only into account the middle line of the adhesive). Overlap length (a) l2¼10 mm and (b) l2¼70 mm.von Mises equivalent stress (MPa)
von Mises equivalent stress (MPa)
Hydrostatic stress (MPa) Hydrostatic stress (MPa)0246810121416
-2ED
F
G
0246810121416
ED
F
G
0 2 4 6 8 10 12 14 -2 0 2 4 6 8 10 12 14
Fig. 8. Stress in the adhesive for geometries D, E, F and G for an average shear stress of 1 MPa, a substrate thickness of h3¼6 mm and an overlap length of l2¼70 mm in the
hydrostatic stress–von Mises equivalent stress diagram: (a) h2¼0.4 mm and (b) h2¼0.8 mm.
Fig. 9. Definition of the local geometry close to the substrate–adhesive interface:
(a) close to the free edges and (b) along the interface.J.Y. Cognard et al. / International Journal of Adhesion & Adhesives 31 (2011) 715–724 720

ac o r n e r ,t h eg e o m e t r yo ft h es u b s t r a t ea n dt h ea d h e s i v ec a nb e
defined using two angles dsand da(Fig. 9 a).Fig. 10 a presents the
results for aluminium substrates under plane stress assumptions: it
can be seen that in order to obtain a regular solution it is necessary to
decrease the substrate angle dsto obtain a quite large admissible
adhesive angle da. For instance, for ds¼901, one has to verify dao601
in order to prevent stress concentrat ions; therefore, sharp beaks and a
cleaning of the adhesive-free edges is useful in order to limit the
influence of edge effect [16,20].Fig. 10 b presents the evolution of the
singularity exponent lwith respect to the substrate angle dsin the
case of a point along the interface defined in Fig. 8 b(dsțda¼3601).
I tc a nb es e e nt h a t dshas to be equal to 180 1in order to prevent stress
concentrations; therefore, only s mooth variations of the substrate
geometry on the adhesive–substrate interface are required. Results
under plane stress or plane strain assumptions are quite similar
(Fig. 10 b).
These results give some interesting rules in order to design
bonded specimens, which significantly limit the influence of edge
effects.
6. Geometries that significantly limit edge effects
It has been shown on one hand that thin beak machining on
substrates can limit the influence of edge effects [15] and on the
other hand substrates with inverse beaks can also give goodresults [24,27]. Moreover, results presented in the previous
sections underline the properties that have to be respected in
order to reduce the stress concentrations. Fig. 11 presents four
geometries (H, I, J and K) taking into account those properties
(for some of them) and taking into account some machining
constraints. Geometry H is defined using d1¼2 mm, d2¼0.05 mm
and a¼251. The parameters for geometry I are d3¼1 mm and
d4¼0.5 h2. Geometry J is defined with d1¼2 mm, a¼251,
d3¼1 mm, d4¼h2,d5¼0.5 mm and d6¼5 mm. The parameters
for geometry K are a¼251,d2¼0.05 mm, d5¼0.5 mm, d4¼25 mm
and d7¼0.5 mm.
For the proposed four geometries, the influence of the adhesive
thickness on the maximum transmitted load of the bonded
assembly is presented for two overlap lengths (10 and 70 mm)
inFig. 12 . Geometries H and I ( Fig. 11 ) give similar results to those
of geometry B ( Fig. 2 ) presented in Fig. 5 . It can be noted that
geometries H and I do not respect the properties presented in the
previous section. Geometries J and K allow a quite large increase
in the maximum transmitted load with respect to other geome-
tries; they allow the stress concentrations to be significantly
limited. It is necessary to use grooves as for geometries J and K
in order to limit the stress concentrations.
Table 3 presents the load transmitted by the single lap shear
specimen using geometry K for an adhesive thickness of
h2¼0.4 mm. It can be noted that the maximum average shear
stress is higher for geometry K than for geometry B ( Table 2 ); the020406080100120140
200.50.60.70.80.91.0
plane stressplane strain
/afii9829s (°) /afii9829s (°)/afii9829a (°)
/afii9838 ≥ 1: regular solution/afii9838 < 1: edge effects
/afii9838
60 100 140 180 30 60 90 150 120 180
Fig. 10. Influence of the local geometry on the singularity exponent lfor bi-material applications for aluminium substrates: (a) close to the free edges (plane stress) and
(b) along the interface (plane stress and plane strain).
Fig. 11. Presentation of geometries H, I, J and K (‘‘ o’’ zone of maximal equivalent stress in the adhesive under elastic assumption). (a) Geometry H: beak on one side,
(b) geometry I: inverse ‘‘linear’’ beak, (c) geometry J: inverse ‘‘parabolic’’ beak and (d) geometry K: beak and grooves.J.Y. Cognard et al. / International Journal of Adhesion & Adhesives 31 (2011) 715–724 721

ratio is of about 2. These results underline the significant
influence of the local geometry close to the free edge of the
adhesive on the stress state in the joint, but the influence of the
overlap length is similar for the different geometries.
As presented in Fig. 4 b, for the maximum load transmitted and
for a given assembly only one of the characteristic points reaches
the elastic limit in the hydrostatic stress–von Mises equivalent
stress diagram; for this considered point, F0¼0; thus the distance
between this point and the elastic limit is equal to 0. Therefore,
for the different points, the distance to the elastic limit, in the
hydrostatic stress–von Mises equivalent stress diagram, can be
computed using the properties of proportional loadings: it is
determined in the direction of the stress path associated to an
increase in the loading. Fig. 13 shows for geometries J and K, onhalf of the joint ( x40,Fig. 11 ), for the maximum load transmitted
of the assemblies, the variation in the distance F0of a character-
istic point to the elastic surface in the thickness of the adhesive
for an adhesive thickness of h2¼0.4 mm, for an overlap length of
l2¼10 mm and for two thicknesses of the substrates ( h1¼6 and
18 mm). In order to simplify the analysis, this figure presents the
results on the complete thickness of the joint but on half of the
overlap length. For geometry J, the elastic limit is first reached at
the root of the inverse beak (grey part in Fig. 13 ), as represented
inFig. 11 (‘‘o’’), and an increase in the substrate thickness ( h1¼6
and 18 mm) leads only to few changes in the stress state in the
adhesive ( Fig. 13 a and b). For geometry K, the stress state is lower
close to the free edges of the adhesive, and an area close to the
end of the overlap length can be observed, where the stress state
is maximum and nearly constant; moreover an increase in thesubstrate thickness leads to an increase in this area (which leads
to an increase in the maximum transmitted load).
Fig. 14 presents, for geometry J, for an adhesive thickness of
h
2¼0.4 mm and for four overlap lengths ( l2¼10, 30, 50 and
70 mm), the maximum and minimum values of the peel stresses
throughout the adhesive thickness (for the upper half
of the adhesive, y40,Fig. 11 ) with respect to the overlap length.
Since the stress developed throughout the thickness of the
adhesive is quite small, only the maximum and the minimum
values are drawn for a given abscissa x. It may be noted that there
is a little change in the peel stress close to free edges of the
adhesive.
Fig. 15 presents, for geometry K and for the same parameters
as used in Fig. 14 , the minimum and maximum values of the
peel stress throughout the adhesive thickness with respect to
the overlap length. For such geometries, it may be noted that
the different stress components are not homogeneous along the
overlap length, but they are nearly constant in the adhesive
thickness for a given abscissa x.
Fig. 16 presents the influence of some parameters for geome-
tries J and K. Fig. 16 a underlines the influence of some parameters
defining the so-called inverse beak: an increase in the adhesive
thickness close to the free edges of the joint from d4¼0.5h2to
d4¼h2(Fig. 11 ) allows an increase in the maximal average shear
stress, and a chamfer length of d1¼2 mm gives better results than
d1¼8 mm. An increase in the length of the inverse beak from
d3¼1m m t o d3¼2 mm also improves a little the performance.
Fig. 16 b presents mainly the influence of the substrate thickness
for geometry J. An increase in the adhesive thickness increases the
maximum transmitted load of the bonded assembly especially for
thin substrates. Moreover for such geometries the cleaning of the
free edge of the adhesive has only a little influence on the stress
distribution.τaverage maxi (MPa)
τaverage maxi (MPa)
adhesive thickness (mm) adhesive thickness (mm)04812162024
HIJK
024681012
0.0H
I
J
K
0.4 0.8 1.2 1.6 0.0 0.4 0.8 1.2 1.6
Fig. 12. Influence of the adhesive thickness and the overlap length on the maximal average shear stress for geometries H, I, J and K and for a substrate thickness o f
h1¼6 mm. (——— : taking only into account the middle line of the adhesive). Overlap length (a) l2¼10 mm (b) l2¼70 mm.
Table 3
Maximum transmitted load by the assembly for geometry K and an adhesive
thickness of h2¼0.4 mm.
Overlap length (mm): l2 10 30 50 70
taverage maxi (MPa) 20.38 8.83 6.51 6.15
Pl/Pl¼10 mm 1.00 1.30 1.60 2.11
Fig. 13. Distance to the elastic surface through the thickness of the adhesive on
half the joint ( x40,Fig. 11 ) for the maximum transmitted load, for an adhesive
thickness of h2¼0.4 mm, for an overlap length of l2¼10 mm and for geometries J
and K. (a) Geometry J and a substrate thickness of h1¼6 mm, (b) geometry J and a
substrate thickness of h1¼18 mm, (c) geometry K and a substrate thickness of
h1¼6 mm and (d) geometry K and a substrate thickness of h1¼18 mm.J.Y. Cognard et al. / International Journal of Adhesion & Adhesives 31 (2011) 715–724 722

7. Conclusions
This paper has analysed the influence of some geometrical
parameters on the stress concentrations in the case of the single
lap joints, which are the most used tests to analyse the behaviour
of an adhesive in an assembly. The stress concentrations can
contribute to fracture initiation in the adhesive joints and thus
can lead to an incorrect analysis of the adhesive behaviour.
Therefore, understanding the stress distribution in an adhesive
joint can lead to improvements in adhesively bonded assemblies.
This study has been accomplished using finite element simula-
tions under elastic assumption using refined meshes and using apressure-dependent model, which accurately describe the elastic
limit of the adhesive. It has been shown that geometries, used
in various studies in order to reduce the influence of edge
effects, can generate quite large stress gradients close to the
adhesive-free edges. Using results of asymptotic analysis and
the experience acquired while improving a modified Arcan device
previously developed to analyse the behaviour of adhesives in an
assembly under tensile-shear loadings, specific geometries that
significantly limit the influence of edge effects have been pro-
posed. The use of substrates with beaks and a cleaning of the
free edges of the adhesive allow a large reduction in the stress
state close to the free edges of the adhesive and thus allow anpeel stress (MPa)
peel stress (MPa)
x (mm) x (mm)-2-101234567
12=70 mm
12=50 mm
12=30 mm
12=10 mm
-2-101234567
-4012=70 mm
12=50 mm
12=30 mm
12=10 mm
-20 0 20 40 -40 -20 0 20 40
Fig. 14. Minimum and maximum values of the peel stress in the thickness of the adhesive for an average shear stress of 1 MPa for a joint thickness of h2¼0.4 mm and for
geometry J: (a) minimum values of the peel stress and (b) maximum values of the peel stress.
peel stress (MPa)
peel stress (MPa)
x (mm) x (mm)-2-101234567
12=70 mm
12=50 mm
12=30 mm
12=10 mm
-2-101234567
12=70 mm
12=50 mm
12=30 mm
12=10 mm
-40 -20 0 20 40 -40 -20 0 20 40Fig. 15. Minimum and maximum values of the peel stress in the thickness of the adhesive for an average shear stress of 1 MPa for a joint thickness of h2¼0.4 mm and for
geometry K: (a) minimum values of the peel stress and (b) maximum values of the peel stress.
τaverage maxi (MPa)
τaverage maxi (MPa)
adhesive thickness (mm) adhesive thickness (mm)0510152025
0.0i
j
k
l 12 = 10 mm
12 = 70 mm0510152025
12 = 10 mm
12 = 70 mm
0.4 0.8 1.2 1.6 0.0 0.4 0.8 1.2 1.6i
j
k
l
Fig. 16. Influence of the adhesive thickness, the overlap length and some geometrical parameters on the maximal average shear stress for geometries (a) J and (b )K .J.Y. Cognard et al. / International Journal of Adhesion & Adhesives 31 (2011) 715–724 723

optimisation of the maximum transmitted load of single lap
joints. The numerical results underline that the overlap length
and the adhesive thickness also have an influence on the stress
distribution in the joint. Moreover, it has been shown that
significant evolution of the stress can exist throughout the
thickness of the adhesive; therefore, simplified methods that
roughly analyse the average stress state through the joint can
overestimate the maximal transmitted load of single lap joints.
Thus refined analyses of the stress distribution in the adhesive are
necessary to obtain precise dimensioning of adhesively bonded
assemblies.
In order to complete this numerical study, 3D simulations have
to be developed as 2D stress or strain plane simulations are not
exactly representative of the behaviour of single lap joints, but the
numerical cost of 3D simulations is much higher. Moreover such
computations can give information on the stress state all around
the adhesive, and thus can lead to a geometry optimisation in
order to increase the transmitted load. Experimental tests also
have to be developed in order to analyse the real behaviour of the
proposed geometries using brittle or ductile adhesives.
Acknowledgements
The authors wish to acknowledge Dominique Leguillon from
Institut JLRA (CNRS, Universite ´Pierre et Marie Curie, Paris, France)
for his collaboration on the asymptotic analysis.
References
[1] Adams RD. Adhesive bonding: science, technology and applications. Bristol:
Woodhead Publishing Ltd.; 2005.
[2] da Silva LFM, O ¨chsner A. Modeling of adhesive bonded joints. Berlin:
Springer; 2008.
[3] Leguillon D, Sanchez- Palancia E. Computation of singular solutions in elliptic
problems and elasticity. Paris: Editions Masson; 1987.
[4] Wang CH, Rose LRF. Compact solutions for the corner singularity in bonded
lap joints. Int J Adhes Adhes 2000;20:145–54.
[5] Kadioglu F, Vaughn LF, Guild FJ, Adams RD. Use of the thick adherend shear
test for shear stress-strain measurements of stiff and flexible adhesives.J Adhes 2002;78:355–81.
[6] Dean G, Crocker L, Read B, Wright L. Prediction of deformation and failure of
rubber-toughened adhesive joints. Int J Adhes Adhes 2004;24:295–306.[7] Cognard JY, Cre ´ac’hcadec R, Sohier L, Davies P. Analysis of the non linear
behaviour of adhesives in bonded assemblies. Comparison of TAST and
ARCAN tests. Int J Adhes Adhes 2008;28:393–404.
[8] Adams RD, Peppiat NA. Stress analysis of adhesive-bonded lap joints. J Strain
Anal 1974;3:185–96.
[9] Adams RD, Harris JA. The influence of local geometry on the strength of
adhesive joints. Int J Adhes Adhes 1987:69–80.
[10] Hildebrand M. Non-linear analysis and optimization of adhesively bonded
simple lap joints between fibre-reinforced plastic and metals. Int J Adhes
Adhes 1987:261–7.
[11] Belingardi G, Goglio L, Tarditi A. Investigating the effect of spew and chamfer
size on the stresses in metal/plastics adhesive joints. Int J Adhes Adhes2002;22:273–82.
[12] Yan ZM, You M, Yi XS, Zheng XL, Li Z. A numerical study of parallel slot in
adherend on the stress distribution in adhesively bonded aluminium single
lap joint. Int J Adhes Adhes 2007;27:687–95.
[13] Rispler AR, Tong L, Steven GP, Wisnom MR. Shape optimisation of adhesive
fillets. Inter J Adhes Adhes 2000;20:221–31.
[14] Wang J, Rider AN, Heller M, Kaye R. Theoretical and experimental research
into optimal edge taper of bonded repair patches subject to fatigue loadings.
Int J Adhes Adhes 2005;25:410–26.
[15] Marques EAS, da Silva LFM. Joint strength optimization of adhesively bonded
patches. J Adhes 2008;84:917–36.
[16] Cognard JY. Numerical analysis of edge effects in adhesively-bonded assem-
blies. Application to the determination of the adhesive behaviour. Comput
Struct 2008;86:1704–17.
[17] da Silva LFM, das Neves PJC, Adams RD, Spelt JK. Analytical models of
adhesively bonded joints —part I. Literature survey. Int J Adhes Adhes
2009;29:319–30.
[18] Wang J, Zhang C. Three-parameter elastic foundation model for analysis of
adhesively bonded joints. Int J Adhes Adhes 2009;29:495–502.
[19] Reedy ED, Guess TR. Comparison of butt tensile strength data with interface
corner stress intensity factor prediction. Int J Solids Struct 1993;30:2929–36.
[20] Cognard JY, Creac’hcadec R. Analysis of the non linear behaviour of an
adhesive in bonded assemblies under shear loadings. Proposal of an
improved TAST. J Adhes Sci Technol 2009;23:1333–55.
[21] Raghava RS, Cadell RM. The macroscopic yield behaviour of polymers. J Mater
Sci 1973;8:225–32.
[22] Cognard JY, Cre ´ac’hcadec R, Maurice J, Davies P, Peleau M, da Silva LFM.
Analysis of the influence of hydrostatic stress on the behaviour of an
adhesive in an assembly. J Adhes Sci Technol 2010;24:1977–94.
[23] Cognard JY, Davies P, Sohier L, Cre ´ac’hcadec R. A study of the non-linear
behavior of adhesively-bonded composite assemblies. Compos Struct
2006;76:34–46.
[24] Bordes M, Davies P, Cognard JY, Sohier L, Sauvant-Moynot V, Galy J.
Prediction of long term strength of adhesively bonded joints in sea water.
Int J Adhes Adhes 2009;29:595–608.
[25] Cast3m documentation. /www-cast3m.cea.fr/cast3m S.
[26] da Silva LFM, Adams RD, Gibbs M. Manufacture of adhesive joints and bulk
specimens with high-temperature adhesives. Int J Adhes Adhes 2004;24:
69–83.
[27] Cognard JY, Devaux H, Sohier L. Numerical analysis and optimisation of
cylindrical adhesive joints under tensile loads. Int J Adhes Adhes 2010;30:706–19.J.Y. Cognard et al. / International Journal of Adhesion & Adhesives 31 (2011) 715–724 724