Research Article
Published: 2024-07-09
https://doi.org/10.20935/AcadMatSci7266
Parametric analysis of composite tubular adhesive joints
bonded by the bi-adhesive technique
André Lima Faria1, Raul Duarte Salgueiral Gomes Campilho1,2,*
Academic Editor: Sohaib Zia Khan
Abstract
Adhesive bonding plays a fundamental role in various industries, including aerospace, aeronautics, and automotive sectors. Unlike
traditional mechanical joints, adhesive joints offer an efficient approach with fewer components, leading to weight reduction in the
final structure. Additionally, these joints facilitate the joining of dissimilar materials, while distributing applied loads more uniformly,
resulting in better stress distributions compared to conventional joining techniques. Within this context, the integration of adhesive
bonds in joggle tubular structures presents a viable alternative to join tubes with identical diameter. The bi-adhesive technique involves
using a brittle adhesive in the inner overlap region, and a ductile adhesive at the overlap edges, aiming to improve load transfer. The
objective of this study is to conduct a numerical analysis using cohesive zone modeling (CZM) to investigate the tensile behavior of
joggle tubular adhesive joints between composite adherends bonded by the bi-adhesive technique. Initially, the proposed CZM
approach is validated against experimental data. Subsequently, the study focuses on numerically assessing the tensile strength of the
joints and testing different bi-adhesive joint options, aiming to improve the maximum load (Pm), displacement at Pm ( at Pm), and
energy absorbed at failure (Ef). Validation of the cohesive models has been successfully achieved. In conclusion, it was found that
depending on the bi-adhesive conditions, improvements are possible to obtain over single-adhesive joints.
Keywords: composite material, structural adhesive, bi-adhesive technique, tubular adhesive joints, cohesive zone models
Citation: Faria AL, Campilho RDSG. Parametric analysis of composite tubular adhesive joints bonded by the bi-adhesive technique.
Academia Materials Science 2024;1. https://doi.org/10.20935/AcadMatSci7266
1. Introduction
Adhesive joints find extensive application across industries, such
as aerospace, aeronautics, and automotive, drawing upon principles from physics, chemistry, and mechanics, due to significant
advancements in recent decades [1]. This joining method offers
numerous advantages over traditional joining techniques, including more uniform stress distribution, reduced stress concentration, enhanced fatigue resistance, vibration damping,
sealing capabilities, acoustic insulation, and structural weight
reduction. However, adhesive joints also exhibit drawbacks, such
as susceptibility to crack propagation, low resistance to peel
stresses, limited resistance to high temperatures, finite lifespan,
and environmental concerns due to adhesive toxicity [2]. Among
different joint configurations available to the designer, tubular
adhesive joints are used in applications, such as vehicle and
structure frames, aircraft structures (including fuselage sections
and wing components), and construction engineering (e.g.,
bridge components), aiming for weight reduction and improved
stress distributions [3]. The industrial acceptance of adhesive
joints in general is made possible by the existence of reliable
predictive methodologies, enabling cost minimization and
accelerated manufacturing processes [4]. Adhesive joint analysis
relies either on closed-form (analytical) or on numerical methods. Analytical approaches, tracing back to the Volkersen model
[5] in the 1930s, become intricate when dealing with plastic
deformation, composite adhesives, or dissimilar material junctions. Numerical methods often rely on the finite element method
(FEM) [6], pioneered by Harris and Adams [7] in adhesive joint
analysis, often integrated with fracture mechanics principles, to
predict strength through stress intensity factors or energetic
techniques like virtual crack closure technique (VCCT). However,
FEM necessitates remeshing if crack propagation occurs, leading
to increased computational complexity [8]. Cohesive zone
modeling (CZM) combines conventional FEM modeling with
cohesive elements to simulate crack propagation. The eXtended
Finite Element Method (XFEM) utilizes enriched shape
functions to represent continuous displacement fields and has
seen application in crack growth modeling. Recent works have
extensively explored these techniques. Examples include works
by Razavi et al. [9] (theoretical models), Le Pavic et al. [10]
(fracture mechanics), Zhang et al. [11] (damage mechanics),
1Center for Research and Development in Mechanical Engineering, School of Engineering, Polytechnic of Porto, Porto 4200-072,
Portugal.
2Institute of Science and Innovation in Mechanical and Industrial Engineering—Pólo Faculty of Engineering of University of Porto,
Porto 4200-465, Portugal.
*email: rds@isep.ipp.pt; raulcampilho@gmail.com
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Huang et al. [12] (CZM approach), and Xará and Campilho [13]
(XFEM), providing comprehensive insights into each method’s
efficacy and applicability.
Different techniques are addressed in the literature to increase
the joining efficiency of adhesive joints. Bi-adhesive joints are an
example of these techniques, which consists of a joint comprising
two adhesives with dissimilar characteristics, one brittle and the
other ductile [14–16]. The brittle adhesive should be applied in
the inner overlap region, while the ductile adhesive fills the
overlap edges, to reduce the geometry-induced stress concentrations [17]. It should be mentioned that the mechanical behavior,
failure modes, and strength prediction of bi-adhesive joints are
very complex, and the onset of damage generally occurs in a
mixed mode [18, 19]. Bi-adhesive joints are preferably subjected
to tensile and flexural stress, although studies have also been
carried out to predict impact resistance [20–22]. In these studies,
the strength of the analyzed bi-adhesive joints proved to be
higher than the homologous single-adhesive joint geometries
[20–22]. However, the proportion between adhesives must be
rigorously defined since, above a certain amount of ductile
adhesive, there are substantial losses in the joint strength. It
should be emphasized that the correct selection of the adhesive
combination, as well as the definition of geometric parameters,
has a significant influence on the results as well [23]. To date,
research has proposed analyzing parameters and conditions,
such as the length ratio of the adhesives used, Young’s modulus
(E) ratio defined by the quotient between the stiffness of the
adhesives, and the adhesive thickness (tA), among other factors.
da Silva et al. [24] studied bi-adhesive joints at low and high
temperatures, and used similar and dissimilar adhesives in
different analyses. It was necessary to consider a stepwise evolution of the adhesive stiffness along the adhesive layer, to reduce
the concentration of shear and peel stresses at the overlap edges.
da Silva and Adams [25] concluded that in a bi-adhesive joint,
most of the applied load will be borne by the low- instead of hightemperature adhesive. The results proved to be valid for tA < 1
mm. The authors also concluded that a bi-adhesive joint can have
higher mechanical strength at high temperatures (200°C) than a
joint with a high-temperature adhesive subjected to low temperatures (−55°C). Moreover, a low-temperature adhesive would not
degrade after a stage at high temperatures, and high-temperature
adhesives do not break after stages at low temperatures. Ramezani
et al. [26] presented a comprehensive experimental analysis of
single-lap joints with bi-adhesive, by applying the digital image
correlation (DIC) method. Different parameters were considered
for this analysis, such as tA, adherends’ thickness (tAd), and overlap
length (LO) for both a stiff and a compliant adhesive. It was found
that failure begins at the interface between both adhesives. The
DIC results showed that the effect of the LO on the adhesive stresses
increases as tA decreases. On the other hand, it decreases with
increasing the adhesive stiffness. It was found that single-lap biadhesive joints with higher tAd have higher strength.
The objective of this study is to conduct a numerical analysis
using CZM, to investigate the tensile behavior of joggle tubular
adhesive joints between composite adherends bonded by the biadhesive technique. Initially, the proposed CZM approach is
validated against experimental data. Subsequently, the study
focuses on numerically assessing the tensile strength of the joints
and testing different bi-adhesive joint options, aiming to improve
Pm,  at Pm, and Ef.
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https://doi.org/10.20935/AcadMatSci7266
2. Materials and methods
2.1. Materials
The SEAL® (Legnano, Italy) Texipreg HS 160 RM prepreg is the
carbon-fiber-reinforced plastic (CFRP) material chosen for the
present analysis, which is considered a material of excellence in
high-performance applications in the aeronautical industry. The
chosen prepreg is made up of unidirectional prepreg plies of carbon
fibers and epoxy resin, whose modeling considers a unidirectional
arrangement with the fibers oriented longitudinally in the direction
of loading. The CFRP used in this commercial composite has high
mechanical strength and specific stiffness, making this composite
suitable for highly efficient structural applications. Table 1 illustrates the elastic properties of CFRP, according to Campilho et al.
[27], being  Poisson’s coefficient and G the shear modulus. The
interlaminar/intralaminar cohesive properties of the CFRP are illustrated in Table 2, which are used to introduce two fracture planes
inside the composite adherends, so that this failure possibility exists
in the numerical models. The cohesive strengths in axial tension and
shear are denoted by tn0 and ts0, respectively, and the fracture
energies in axial tension and shear by GIC and GIIC, respectively.
Table 1 • SEAL® Texipreg HS 160 RM CFRP lamina elastic
orthotropic properties (with fibers unidirectionally aligned in
the x direction, while y and z represent the transverse
directions) [27]
Ex = 1.09E+05 MPa
νxy = 0.342
Gxy = 4,315 MPa
Ey = 8,819 MPa
νxz = 0.342
Gxz = 4,315 MPa
Ez = 8,819 MPa
νyz = 0.380
Gyz = 3,200 MPa
Table 2 • CFRP interlaminar cohesive properties
E
G
tn0
ts0
[MPa]
108,000
4,315
GIC
GIIC
[N/mm]
40
35
0.39
0.82
Three different types of adhesives were employed to predict the
tensile behavior of tubular adhesive joints, including both ductile
and brittle properties. This broad selection enhances the scope of
the study and enables a more detailed optimization analysis.
Specifically, Araldite® AV138 (an epoxy-based adhesive with
brittle characteristics), Araldite® 2015 (an epoxy-based adhesive
with some degree of ductility), and Sikaforce® 7752 (a polyurethane-based adhesive with ductile properties, albeit less robust)
were chosen for evaluation (Araldite®, Huntsman, The Woodlands, TX, USA, and Sikaforce®, Sika, Baar, Switzerland). Experimental testing of these adhesives was conducted using various
setups, generating the data summarized in Table 3 [28, 29],
which encompasses an extensive array of mechanical and
fracture properties serving as input parameters for CZM simulations. The tensile mechanical properties (E, tensile yield stress or
σe, tensile strength or σf, and tensile failure strain or εf) were
derived from bulk testing of dogbone-shaped specimens, while
the shear mechanical properties (G, shear yield stress or τe, shear
strength or τf, and shear failure strain or γf) were obtained from
thick adherend shear tests (TAST) conducted on specimens with
steel adherends. The values of σe and τe were estimated by
identifying the intersection points between the corresponding
stress-strain curves and a parallel line offset by 0.2%. Furthermore, dedicated fracture tests were performed to determine GIC
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(using the Double-Cantilever Beam or DCB test) and GIIC (via
the End-Notched Flexure or ENF test).
Tensile strength, f [MPa]
2.2. Joint geometries
Tensile failure strain, f [%]
The proposed base geometry for the joggle tubular bi-adhesive
joints is depicted in Figure 1. This configuration comprises an
overlap tubular adhesive joint with an external joggle. The main
parameters defining the base joint geometry are overlap length
(LO), outer diameter of the non-hydroformed adherend (dEANH),
joggle angle (θ), and tAd.
Table 3 • Mechanical and fracture properties of the selected
adhesives [28, 29]
Property
Young’s modulus, E [GPa]
Poisson’s ratio, 
Tensile yield stress, e [MPa]
AV138
4.89 ± 0.81
2015
1.85 ± 0.21
2.47
0.61
0.48
19.18 ±
1.40
25.1 ± 0.33
14.6 ± 1.3
5.16 ± 1.14
Shear strength, f [MPa]
Shear failure strain, f [%]
Toughness in tension, GIC
[N/mm]
3.24 ±
4.77 ± 0.15
Shear yield stress, e [MPa]
0.09
12.63 ±
1.21 ± 0.10
0.19b
Toughness in shear, GIIC
36.49 ±
0.25
Shear modulus, G [GPa]
0.49 ±
0.30a
11.48 ±
1.61
0.70b
7752
0.33a
21.63 ±
3.18
1.81b
[N/mm]
0.35a
39.45 ±
30.2 ± 0.40
7.8 ± 0.7
17.9 ± 1.8
43.9 ± 3.4
0.20c
0.43 ±
10.17 ±
0.64
54.82 ±
6.38
2.36 ± 0.17
0.02
0.38c
4.70 ±
5.41 ± 0.47
0.34
aManufacturer’s data;
bEstimated from Hooke’s law using E and 
cEstimated in reference [28].
Figure 1 • Joggle tubular joint geometry and dimensions.
To account for the possibility of CFRP failures, an intralaminar
layer and an interlaminar layer were introduced into the adherends. The dimensions and position of the interlaminar layer,
in red, and the intralaminar layer, in green, are illustrated in
Figure 2 for the straight adherend. This layout was selected
based on previous evidence for similar joints [30], aiming to
replicate real failure conditions.
inner joggle radius (RIR) = 1, outer joggle radius (RER) = 3, and
inner diameter of non-hydroformed adherend (dIANH), outer
diameter of hydroformed adherend (dEAH), and inner diameter of
hydroformed adherend (dIAH), all dependent on dEANH and tA.
Table 4 • Analyzed geometries of bi-adhesive tubular joints
Adhesives
Stiff (interior)
Geometrical parameters
Ductile
dEANH
LO
(edges)
tAd

L1
[mm]
[°]
0
Araldite®
2015
Figure 2 • Dimensions and position of the interlaminar layer, in
red, and the intralaminar layer, in green (example for the straight
adherend).
20
10
2
0.25 × LO
0.50 × LO
Araldite®
0.75 × LO
AV138
0
Sikaforce®
7752
20
10
2
0.25 × LO
0.50 × LO
45
45
0.75 × LO
Table 4 illustrates the different configurations of analyzed
tubular bi-adhesive joints. L1 corresponds to the length of the
edge adhesive, and four configurations were tested, from L1 = 0
(only brittle adhesive Araldite® AV138) to L1 = 0.75 × LO, that is,
a bigger proportion of outer adhesive. Additional parameters are
as follows (in mm): tA = 0.2, total length of the tubular adhesive
joint (LT) = 80, length of joggle adherend (LAH) dependent on LO,
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2.3. Numerical modeling
The static numerical analysis, conducted using FEM, was performed in Abaqus® (Dassault Systèmes, Vélizy-Villacoublay,
France). Among the various design techniques described in the
Introduction section, CZM was chosen for its accuracy,
availability in commercial software, and computational efficiency
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[31]. CZM involves the application of cohesive laws along
predefined fracture paths, utilizing mixed-mode stress criteria
for damage propagation onset and energetic failure criteria for
subsequent damage progression. This method excels in meshindependent strength predictions by triggering crack growth
based on an energy criterion calculated over an area rather than
stress-based concepts [32]. Moreover, parameter calibration
tools and standardized tests facilitate its widespread application
in the industry [33].
Model preparation entails different modules detailing simulation
features. In the part module, the joint geometry is defined as an
axisymmetric deformable solid, constructed from a two-dimensional sketch revolved around an axis. Materials and crosssections are defined and assigned in the property module, with
the composite adherends being modeled as elastic orthotropic
solids with the properties defined in Table 2. Adhesive types and
properties are defined accordingly (Table 3), with the adhesive
layer represented by CZM elements utilizing cohesive laws derived from experimental data. Failure criteria are assigned for
https://doi.org/10.20935/AcadMatSci7266
CZM elements, incorporating quadratic stress and linear energetic failure criteria for damage initiation and growth, respectively. In the step module, computing parameters for the
numerical analysis are defined, including activation of geometric
non-linearity to account for large displacements. The load module defines boundary conditions replicating tensile loading, with
longitudinal displacement imposed at one end and longitudinal
restraint at the opposite end. Meshing, preceding the job module,
discretizes the adhesive joint into finite elements with the mesh
controls defined as structured (adherends), free (zones with
curvature—joggle), and sweep meshing (adhesive layer). Fournode axisymmetric solid elements (CAX4) were used for the
adherends, and four-node axisymmetric cohesive elements
(COHAX4) were considered for the adhesive layer. Mesh bias
effects are utilized to increase computational efficiency, with
higher refinement at stress-critical regions. The resulting mesh,
exemplified for the base geometry with 2082 elements in Figure 3,
demonstrates enhanced refinement in critical regions corresponding to the initiation of damage. This approach ensures
accurate modeling of adhesive joint behavior under tensile loading conditions.
Figure 3 • Mech for the base geometry and details at the overlap edges.
3. Results
3.1. Validation with experiments
The present section describes the validation process for the
numerical technique with experimental data, aiming to attest the
validity of the CZM technique/axisymmetric approach and adhesive properties for the numerical bi-adhesive joint analysis that
follows. Initially, the geometry of the adhesive joint and the
materials used are described. Subsequently, the collected experimental results obtained are analyzed in comparison with the
numerical results, to validate the method used. The tubular joint
is an overlap geometry identical to that depicted in Figure 1,
with  = 0º. The geometrical parameters are those described in
Section 2.2 (Table 4), while LO = 20 and 40 mm were evaluated.
Due to experimental limitations, the validation process was
carried out with aluminum alloy adherends (AW 6082-T651).
This material is a structural alloy which, despite having intermediate mechanical strength, has the highest strength of the
6,000 series and excellent corrosion resistance. As it is a relatively new alloy with good mechanical properties, it ends up
replacing the use of the AW 6061 alloy in some applications. The
properties of this alloy are obtained through a process of hardening (stress relief through temperature and mechanical stress) and
artificial aging at a temperature of 180°C. The addition of a large
amount of manganese controls the grain of the structure, resulting
in a stronger alloy. It is difficult to produce thin sheets with
complex shapes in extrusion. Thus, the extrusion surface is not as
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smooth as other similar alloys in the 6000 series. Campilho et al.
[28] characterized this aluminum alloy in previous work. The
results obtained showed E = 70.07 ± 0.83 GPa, e = 261.67 ± 7.65
MPa, f = 324.00 ± 0.16 MPa, and f = 21.70 ± 4.24%. The
adhesives used in the validation study are the ones described in
Section 2.1. The numerical analysis was carried out following the
description of Section 2.3, including using a triangular law CZM
approach, for Pm comparison with experimental data. Figure 4
shows the experimental Pm values as a function of LO and the
numerical predictions. Table 5 shows the comparison between
the experimental and numerical values for the different LO of the
three adhesives analyzed.
Figure 4 • Pm as a function of LO for the three adhesives: experimental results and CZM predictions.
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Figure 4 and Table 5 show that the CZM and experimental
values for Pm are very close for the tubular adhesive joints with the
adhesives Araldite® AV138 and Araldite® 2015. The Araldite®
AV138 is the adhesive with the smallest relative differences
between the experimental and numerical results. For LO = 20 mm,
the relative difference was 2.4%, while for LO = 40 mm, it was 4.7%.
The numerical values slightly overshoot the experimental data, but
the difference is negligible, and these results are therefore
considered adequate. In the case of the adhesive Araldite® 2015,
the percentile difference between the experimental and numerical
Pm results is 6.1% for LO = 20 mm, with the values obtained by CZM
higher than the experimental ones. This discrepancy is acceptable
as it is very small, given that the experimental Pm ≈ 27 kN and the
numerical Pm ≈ 29 kN. This difference reduces for LO = 40 mm
(2.9%). It can be concluded that, as with the adhesive Araldite®
AV138, and despite the small differences observed, the CZM
predictions are accurate. In the case of the tubular joints with the
adhesive SikaForce® 7752, the numerical Pm values are lower by a
non-negligible amount. For adhesive joints with ductile adhesives
simulated with triangular CZM laws, predictions may be lower
than expected, given the immediate depreciation of the stress once
tn0 and ts0 are reached. However, some studies on the delamination
of composite materials have shown that cohesive laws that are not
very suitable to model a given material can still provide a rough
approximation of its behavior [34]. The difference between the
experimental and CZM values for the adhesive Sikaforce® 7752
was 18.4% for LO = 20 mm and 14.3% for LO = 40 mm, with the
experimental values being always higher than the numerical ones.
Therefore, considering the factors involved, the values obtained
numerically are accepted for design purposes, despite the respective dispersion of values. After these analyses, the numerical
results obtained were considered valid and suitable to be used as a
source of comparison for the parametric study that follows on
tubular bi-adhesive joints.
Table 5 • Experimental and numerical Pm [kN] values for the
three adhesives as a function of LO, and their relative difference
Adhesive
Araldite®
AV138
Araldite®
2015
SikaForce®
7752
LO [mm]
20
40
20
40
20
40
Experimental
32.80
37.86
27.24
39.07
23.86
35.93
Numerical
33.57
39.63
28.90
40.21
19.46
30.80
Difference [%]
2.4
4.7
6.1
2.9
−18.4
−14.3
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undamaged element stiffness, and it ranges from 0 (undamaged
material) to 1 (failure). This variable is calculated as
 current = (1 − SDEG ) undamaged ,
(1)
where current is the actual stress transmitted by the cohesive
elements and undamaged is the theoretical stress for the same
applied displacement without SDEG. By plotting SGEG along x/LO,
in which x stands for the normalized longitudinal coordinate, such
that 0 ≤ x ≤ LO (Figure 3), it is possible to analyze the failure
process and critical regions in the joint during loading.
3.2.1. Araldite® AV138/Araldite® 2015
Figure 5 illustrates the P-δ curves as a function of L1 for the
Araldite® AV138/Araldite® 2015 bi-adhesive tubular joints. In
view of these numerical data, there is a Pm and  at Pm
improvement between L1/LO = 0 and 25%, followed by a decrease
of these variables for L1/LO ≥ 25%. Thus, the joint geometry with
the highest Pm (21.4 kN) is that with L1/LO = 25%. For the
analyzed L1/LO values, there was a percentile decrease in Pm of
approximately 17% for L1/LO = 75%, compared to the reference
geometry. Table 6 illustrates the percentile Pm variation as a
function of L1/LO for the Araldite® AV138/Araldite® 2015 biadhesive tubular joints, compared to the base geometry. The
adhesive joint with L1/LO = 25% has the highest Pm, with a
percentile increase of 5.00% compared to the base geometry.
Following the tendency observed for Pm,  at Pm is highest for
L1/LO = 25%, and decreases for L1/LO > 25%. In all cases, the
joints behaved linearly until reaching failure.  at Pm was highest
for L1/LO = 25% (0.250 mm), with a percentile increase of 4.17%
compared to the base geometry. The bi-adhesive joint with the
highest Ef follows the trend of Pm and  at Pm, that is, L1/LO = 25%
provides the highest Ef, with a percentage increase in adherend
plasticization of approximately 10% compared to the base
geometry.
3.2. Numerical bi-adhesive composite joint analysis
Following the CZM approach validation in the previous section,
the numerical analysis is carried out considering two bi-adhesive
combinations: Araldite® AV138/Araldite® 2015 and Araldite®
AV138/Sikaforce® 7752, always with the brittle adhesive
(Araldite® AV138) in the inner overlap region, and the other
adhesive at the overlap edges. Comparisons are performed with
a joint only with the Araldite® AV138 (L1/LO = 0%), considering
Pm,  at Pm, Ef, and stiffness degradation (SDEG) damage variable
in the adhesive layer at Pm. Ef is estimated by the area beneath
the load-displacement (P-) curves of each joint configuration up
to failure and is a relevant parameter that assesses the energy
uptake during the loading process. SDEG is the stiffness degradation of the CZM elements representative of the adhesive layer and
composite interlaminar/intralaminar plies, compared to the
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Figure 5 • P-δ curves as a function of L1/LO, for the Araldite®
AV138/Araldite® 2015 bi-adhesive tubular joints.
Table 6 • Percentile variation of different variables as a
function of L1/LO for the Araldite® AV138/Araldite® 2015 biadhesive tubular joints, in relation to the base geometry
L1/LO
0% (base)
25%
50%
75%
Pm
-
+5.00
−5.03
−16.69
 at Pm
-
+4.17
−2.92
−12.50
Ef
-
+9.80
−7.31
−27.33
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An SDEG analysis at Pm was carried out on all L1/LO tubular joint
configurations to assess the damage along the adhesive layer.
Figure 6 illustrates the evolution of SDEG as a function of x/LO.
For the base geometry (L1/LO = 0%), the variation of SDEG
occurred abruptly near to x/LO = 0 from the undamaged state to
the maximum observed degradation of the adhesive layer
elements (Figure 6a). This behavior is closely related to the peak
stresses found at this region in the overlap due to the geometry
transition. At Pm, the maximum SDEG was 0.925. For L1/LO > 0%,
the variation of SDEG is more pronounced in the region
corresponding to the stiffer adhesive (interior region) and toward
the overlap edges (region bonded with the adhesive Araldite®
2015). It should be noted that the effect of peel stresses developed
in the adhesive layer is preponderant at the start of the damage,
and these have higher values than shear stresses along the overlap.
For all the analyzed geometries, the gradient of SDEG is more
abrupt in the transition zone between adhesives. For L1/LO = 25%
(Figure 6b), the observed gradients in the SDEG curves are
directly related to the brittleness of the middle adhesive, which is
dominant over the more ductile adhesive. Although this feature is
not a preponderant factor in the joint’s failure mode, damage may
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occur in some cohesive elements of the intralaminar layer. For
L1/LO = 50% (Figure 6c), damage develops more abruptly in the
transition between adhesives. Although the rupture is mostly
cohesive in the adhesive toward the end adjacent to the curvature
(x/LO = 1), the SDEG analysis predicts a mixed rupture, due to the
elimination of elements from the intralaminar layer, coinciding
with the overlap region bonded with the stiffer adhesive. For
L1/LO = 75% (Figure 6d), the variation of SDEG was more subtle,
although the gradient of the damage variable in the transition zone
between the adhesives used was noticeable. There was negligible
damage evolution in the intralaminar layer, and it can be assumed
that the failure mode was cohesive in the adhesive. Analysis of the
numerical models showed that the joint geometries have identical
maximum values of peak peel stresses in the adhesive layer, which
translates into a similar slope of the curves describing the evolution
of the damage variable. The SDEG curves’ analysis suggests that
failure starts at the end adjacent to the curvature (x/LO = 1) for all
analyzed joint geometries, that is, by the most ductile adhesive,
even though cohesive elements were damaged in the transition
zones between adhesives and at both ends of the overlap.
Figure 6 • Evolution of SDEG as a function of L1/LO for the Araldite® AV138/Araldite® 2015 bi-adhesive tubular joints: 0% (a), 25%
(b), 50% (c), and 75% (d).
3.2.2. Araldite® AV138/Sikaforce® 7752
Figure 7 depicts the P- curves as a function of L1/LO for the
bi-adhesive tubular joints with the adhesives Araldite® AV138/
Sikaforce® 7752. In view of the obtained joint behavior, there is a
gradual decrease of Pm by increasing L1/LO from the L1/LO = 0%
joint configuration. The adhesive Sikaforce® 7752 is the most
ductile between the three adhesives used in this work, but it has
a much lower mechanical strength than the others, which explains the Pm decrease, that is, despite the adhesive’s plasticization the transferred load between adherends is not enough to
promote the improvement of Pm. The adhesive joint with the
highest Pm is thus the reference geometry. Table 7 illustrates the
percentile variation of Pm as a function of L1/LO for the Araldite®
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AV138/Sikaforce® 7752 bi-adhesive tubular joints, compared to
the base geometry. The adhesive joint with the highest Pm is the
reference geometry, while an almost 50% reduction was obtained for
L1/LO = 75%. The evolution of  at Pm with L1/LO followed the trend
of Pm. Thus,  at Pm is gradually reduced by increasing L1/LO. The
value of  at Pm was highest for the joint corresponding to the base
geometry, with  at Pm = 0.240 mm, and reduced up to almost 40%
for L1/LO = 75%. The adhesive joint with the highest Ef value
corresponds to the base geometry, in agreement with the P- curves
shown in Figure 7. The percentile reduction over the base geometry
was highest among the three evaluated variables in Table 7,
reaching almost 70% of reduction in the deformation capacity.
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A similar SDEG analysis of the adhesive layer at Pm is presented
in Figure 8 along x/LO. The base geometry (L1/LO = 0%) was
already described, although it is included in the figure to make
possible the direct comparison with the other geometries. As
previously mentioned, SDEG varied abruptly near to x/LO = 0
from the undamaged state to the maximum observed degradation of the adhesive layer elements (Figure 8a), up to SDEG =
0.925. The variation of SDEG along x/LO is dissimilar for the
different joint geometries analyzed. It should be noted that peel
and shear stresses have critical values at the overlap ends.
According to Figure 8b, for L1/LO = 25%, the SDEG evolution
suggests failure near to x/LO = 1, regardless of CZM elements’
damage at the opposite end. For L1/LO = 50% (Figure 8c),
analysis of the numerical models showed that shear stresses
along the adhesive layer are the main factor to justify the failure
mode. Although failure is cohesive in the adhesive layer, there is
a short-length damaged region in the interlaminar layer,
coinciding with the middle region of the adhesive overlap. For
L1/LO = 75% (Figure 8d), the failure mode is again cohesive in
the transition zones between adhesives, and the dominance of the
more ductile adhesive (Sikaforce® 7752) does not change the
failure behavior. For this combination of adhesives, there is an
expected change in the damage initiation site, which was not
observed in the previous combination of adhesives.
https://doi.org/10.20935/AcadMatSci7266
Figure 7 • P-δ curves as a function of L1/LO, for the Araldite®
AV138/Sikaforce® 7752 bi-adhesive tubular joints.
Table 7 • Percentile variation of different variables as a
function of L1/LO for the Araldite® AV138/Sikaforce® 7752 biadhesive tubular joints, in relation to the base geometry
L1/LO
0% (base)
25%
50%
75%
Pm
-
−6.51
−23.32
−47.48
 at Pm
-
−5.83
−18.75
−38.75
Ef
-
−12.10
−37.91
−68.17
Figure 8 • Evolution of SDEG as a function of L1/LO for the Araldite® AV138/Sikaforce® 7752 bi-adhesive tubular joints: 0% (a), 25%
(b), 50% (c), and 75% (d).
4. Conclusions
This work aimed to numerically address the bi-adhesive technique
on the tensile strength of composite tubular adhesive joints,
following validation of the CZM technique against experimental
data. This work was motivated by the growing applications of
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tubular adhesive joint applications in different industries, and
advantages in the applications of composite materials for the tubes
to be joined. Initial validation of the CZM technique against
experimental data on tubular single-adhesive joints with aluminum alloy adherends and  = 0º showed accurate results for the
joints bonded with the adhesives Araldite® AV138 and Araldite®
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2015, while the predictions for the tubular joints bonded with the
adhesive Sikaforce® 7752 showed Pm deviations up to −18.4%.
However, this behavior was expected when modeling ductile
adhesives with a triangular CZM law and was not considered to
invalidate the bi-adhesive joints’ design. Two bi-adhesive joint
configurations were tested: Araldite® AV138/Araldite® 2015 and
Araldite® AV138/Sikaforce® 7752, by comparing results with the
tubular single-adhesive joint bonded with the adhesive Araldite®
AV138, which provides the best Pm for a single adhesive. The
numerical analysis showed that the Araldite® AV138/Araldite®
2015 configuration with L1/LO = 25% was able to improve Pm,  at
Pm, and Ef from the single-adhesive condition by 5.00, 4.17, and
9.80%, respectively, while higher L1/LO ratios would slightly
decrease these three values. On the other hand, the Araldite®
AV138/Sikaforce® 7752 configuration is not recommended over
the single-adhesive configuration, since Pm,  at Pm, and Ef suffer
from severe reductions by using this combination of adhesives,
which can reach almost 70% for Ef. In view of the obtained results,
it can be concluded that CZM is a viable technique for tubular
adhesive joint design and that improvements are possible to obtain
due to the bi-adhesive technique by a careful selection of geometry
and adhesives. Nonetheless, other adhesive combinations can be
further analyzed to maximize the joints’ performance.
A.L.F. and R.D.S.G.C.; writing—original draft preparation,
A.L.F.; writing—review and editing, R.D.S.G.C.; supervision,
R.D.S.G.C. All authors have read and agreed to the published
version of the manuscript.
The main constraints on the development of this work are directly related to limitations and limited information in the current literature, restrictions on the manufacturing of hydroformed
tubular joints, and limitations of the triangular cohesive law
applied to ductile adhesives. Regarding the limitations of applying the triangular cohesive law to ductile adhesives, and
according to the CZM available in the literature, the triangular
law may not be the most suitable to model an adhesive with
significant ductility. The preference for this law is justified by the
possibility of avoiding possible convergence problems with the
analyzed model, allowing the solution to be obtained more
quickly. According to Campilho et al. [35], ductile adhesives are
highly influenced by the shape of the CZM, and the influence of
the law used is inversely proportional to the value of LO. The
biggest limitations in the manufacture of hydroformed joints are
the higher initial investment required [36], the lower production
rate [37], the difficulty in obtaining tight radii and angles without
the use of pressure amplifiers [38], and the loss of ductility of the
adherends due to hardening [39].
Sample availability
Based on the topics covered, the following points emerge as
proposals for future work: experimental and numerical analysis
of bi-adhesive joints bonded with adhesives that cure at different
temperatures; experimental analysis of bi-adhesive joints based
on a hydroformed CFRP exterior adherend; and variation of two
geometric control factors simultaneously, around the geometric
parameter that contributes most to the strength of the adhesive
joints.
Funding
Conflict of interest
The authors declare no conflict of interest.
Data availability statement
Data supporting these findings are available within the article, at
https://doi.org/10.20935/AcadMatSci7266, or upon request.
Institutional review board statement
Not applicable.
Informed consent statement
Not applicable.
The authors declare no physical samples were used in the study.
Additional information
Received: 2024-03-26
Accepted: 2024-06-23
Published: 2024-07-09
Academia Materials Science papers should be cited as Academia
Materials Science 2024, ISSN 2997-2027, https://doi.org/
10.20935/AcadMatSci7266. The journal’s official abbreviation is
Acad. Mat. Sci.
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Copyright
The authors declare no financial support for the research,
authorship, or publication of this article.
© 2024 copyright by the authors. This article is an open access
article distributed under the terms and conditions of the
Creative Commons Attribution (CC BY) license (https://
creativecommons.org/licenses/by/4.0/).
Author contributions
References
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