J Fail. Anal. and Preven. (2025) 25:2150–2164
https://doi.org/10.1007/s11668-025-02257-w
TOOLS AND TECHNIQUES
Determining Mode I Fracture Energy for Bonded Joints Having
Different Adhesive Thicknesses
Ahmed Mohamed Jubartalla Ali . Faycal Benyahia .
Zeyad Alsuhaibani . Bel Abbes Bachir Bouiadjra . Aamir Dean
Submitted: 10 May 2025 / in revised form: 27 July 2025 / Accepted: 28 July 2025 / Published online: 23 August 2025
Ó The Author(s) 2025
Abstract The bonded composite repair technique has
been widely used to extend the life of cracked structures,
such as aircraft and wind turbine structures. After repair,
the growth of the crack is significantly slowed until the
patch begins to lose its effectiveness. To maintain a low
rate of crack growth, the patch repair needs to be replaced
once a critical level of adhesion damage (separation) is
reached. The objective of this research is to determine the
cohesive zone model (CZM) parameters for simulating
mode I damage initiation and propagation in different
A. M. J. Ali F. Benyahia Z. Alsuhaibani
Mechanical Engineering Department, College of Engineering,
King Saud University, Riyadh, P.O. Box 800, 11421 Riyadh,
Saudi Arabia
F. Benyahia
e-mail: fbenyahia@ksu.edu.sa
Z. Alsuhaibani
e-mail: zeyads@ksu.edu.sa
A. M. J. Ali (&)
ADMiRE Research Center, Carinthia University of Applied
Sciences, Europastraße 4, 9524 Villach, Austria
e-mail: a.ali@fh-kaernten.at; a.jubartalla@gmail.com
B. A. B. Bouiadjra
LMPM, Department of Mechanical Engineering, University of
Sidi Bel Abbes, BP 89, Cité Ben M’hidi, Sidi Bel Abbes, Algeria
e-mail: belabbes.bachirbouiadjra@univ-sba.dz
A. Dean
School of Civil Engineering, College of Engineering, Sudan
University of Science and Technology, P.O. Box 72, Khartoum,
Sudan
e-mail: a.dean@isd.uni-hannover.de
A. Dean
Institute of Structural Analysis, Leibniz Universität Hannover,
Appelstr. 9A, 30167 Hannover, Germany
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thicknesses of AralditeÒ2015 structural adhesive. First,
mode I load–displacement curves were obtained experimentally using double cantilever beam (DCB) tests. Next,
the corresponding fracture toughnesses (GIc) were calculated using the compliance-based beam method (CBBM),
which does not require measuring the crack length during
testing. Finally, finite element analysis (FEA) was used to
predict the progression of adhesion damage using three
different softening laws: triangular, trapezoidal, and
exponential. The study found that mode I fracture energy is
the same for adhesive thickness between 0.20 and
0.30 mm, but then increases by 27% when the adhesive
thickness becomes 0.35 mm. Numerically, the results
showed good agreement across all softening shapes, with
the trapezoidal law being considered the most computationally efficient.
Keywords Double cantilever beam Compliance-based beam method Fracture toughness Traction–separation law Cohesive zone model
Introduction
Fatigue is one of the most critical design and operational
considerations in the aerospace [1–3] and wind energy
[4–7] sectors. For structures subjected to repeated loading,
fatigue failure is inevitable, making effective rehabilitation
techniques essential for extending the fatigue life of
cracked structures. Engineers have explored various
methods for crack retardation to safely prolong the use of
components, such as stop holes [8], overloading [9–11],
fastened repair [12], and bonded composite repair (BCR)
[13–16]. BCR, in particular, offers several advantages: It is
J Fail. Anal. and Preven. (2025) 25:2150–2164
(a) highly cost-effective, (b) allows reinforcement in the
desired direction only, and (c) enables patches to be
detached and replaced without damaging the surrounding
structure.
In recent years, BCR has gained considerable interest in
the structural maintenance of civil aircraft and wind turbine
blades (Fig. 1). Numerous design parameters influence the
efficiency of such repairs, including, but not limited to,
patch material [17, 18], dimensions [17, 19–24], shape
[20, 25–28], stacking sequence, and curing temperature.
To determine the mode I strain energy release rate
(SERR), researchers have employed several testing methods, including double cantilever beams (DCB) [29–32],
tapered double cantilever beams (TDCB) [33–35], cracked
lap shear (CLS) [36, 37], and single-edge notched in
bending (SENB) [38, 39]. A comparison between the DCB
and TDCB tests [40] revealed that the TDCB test tends to
underestimate the critical SERR for ductile adhesives,
while the DCB test is suitable for all types of adhesives.
Additionally, the DCB test offers a distinct advantage as it
is considered the best choice for determining the critical
fracture energy for mode I due to its additional supporting
point. Notably, the DCB test employs a data reduction
method known as the compliance-based beam method
(CBBM) [41], which eliminates the need for measuring
crack length during testing.
Cohesive zone modeling (CZM) was introduced in the
1960 s by Dugdale [42] and Barenblatt [43]. CZM
describes the relationship between the pulling or sliding
forces acting on cracked or joined parts and the corresponding opening displacements between the cohesive
surfaces of the crack or joint. This relationship is known as
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the traction–separation law (TSL). As the contacting surfaces are pulled or slid apart, they are resisted by a
cohesive traction force. This force increases to a peak value
and then gradually decreases to zero, signifying complete
separation or failure of the material.
Several studies [44–50] have evaluated various TSL
shapes to model the tensile behavior of adhesives with
different levels of ductility using DCB tests. The results
indicate that for brittle adhesives, the choice of CZM shape
has minimal impact on accuracy, with the triangular law
providing a suitable representation. In contrast, for ductile
adhesives, the shape of the TSL significantly affects the
accuracy of the model, with the trapezoidal law offering a
more precise representation of their behavior.
Bonding imperfections and adhesive separations inversely affect the fatigue life of repaired structures [40].
Albedah et al. conducted an in-depth investigation into the
impact of adhesive disbanding, revealing that the fatigue
life is significantly reduced by the width of the disband,
while the length has a negligible effect [41]. This finding is
consistent with research conducted by Bouiadjra et al. [42].
In the studies referenced above, it is evident that the
damage and separation of the adhesive layer significantly
impact the performance of patches in terms of their ability
to retard crack propagation. This study aims to mechanically characterize the patch adhesion strength, degradation,
and separation in mode I fracture for various adhesive
thicknesses through experimental testing. We will determine the parameters required for the CZM and conduct
numerical simulations of DCB experiments using different
TSL shapes. These numerical results will be compared with
experimental data to identify the most effective modeling
Fig. 1 Examples of using BCR in structural maintenance of a civil aircraft wing and wind turbine blade
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approach for cohesion degradation. The findings from this
study will contribute to future research by developing criteria to assess the interaction between progressive fatigue
damage of patch adhesion and its effectiveness in halting
repaired fatigue crack growth.
approximately 0.1 mm thick, was placed between the
mating surfaces to initiate a crack with a length of 35 mm.
The specimens were pressed together, with a dummy
weight of 30 kg, and allowed to cure for 24 h at room
temperature using a specially designed mold, as shown in
Fig. 3a.
Material and Methods
Testing Procedure
In this section, the materials and methods used for sample
preparation, testing, and analysis of results are described.
An Instron E10,000 machine equipped with a 10 kN load
cell was used for all tests. The tests were conducted in
displacement control mode at a speed of 0.5 mm/min, as
specified by the ISO standard [51], and performed at room
temperature. Prior to each test, a real crack was initiated by
applying a slight load beyond the peak load and then
releasing it. Subsequently, an opening load was applied
until the specimen completely failed. This process was
Samples Preparation
The DCB testing procedure was selected due to its simplicity in preparation, testing, and the extraction of fracture
toughness using beam theory.
To comply with the plane stress condition for our
available adherent thickness of 2 mm and to follow the
International Organization for Standardization (ISO)
guidelines [51], a beam width of 25 mm was used. Four
groups (A, B, C, and D) of different adhesive thicknesses
were fabricated, investigated, and then interpolated to
capture the variability encountered in real-world repairs for
aircraft and wind turbine structures. A schematic drawing
of the DCB specimen is shown in Fig. 2, and measured
thicknesses are reported in Table 1.
The specimens were prepared by first cutting a flat plate
of aluminum alloy Al 2024-T3 to the required dimensions,
as illustrated in Fig. 2. The mating surfaces were then
roughened and cleaned using sandpaper and acetone. Equal
quantities of the two components of AralditeÒ2015 were
mixed with a small stick, and thin layers of the mixture
were applied to the mating surfaces. A thin Teflon film,
Fig. 2 Schematic
representation of the specimens
with dimensions in millimeters
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Table 1 Measured thicknesses (mm) of the fabricated specimens
Specimen
Thickness t
Average
Standard deviation
A1
0.23
0.20
0.03
A2
0.20
A3
B1
0.18
0.29
0.31
0.02
B2
0.31
0.36
0.01
0.44
0.04
B3
0.32
C1
0.37
C2
0.35
C3
0.36
D1
0.44
D2
0.48
D3
0.41
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repeated for all specimens, with load–displacement (P d) data recorded at a frequency of 0.3 Hz, corresponding to
18 readings per minute. Figure 4 illustrates the test setup
and captures images taken at different stages of the test.
Determining Fracture Energy
The fracture energy in mode I can be calculated using the
Irwin–Kies equation [52]:
GIc ¼
P2 dC
2B da
ðEq 1Þ
where P represents the applied load, B is the joint width,
C ¼ Pd denotes the compliance of the specimen, and a is the
crack length. However, the Irwin–Kies equation requires
precise monitoring of crack length during propagation,
which can be challenging. Additionally, when ductile
adhesives are employed, the dissipated energy in the
fracture process zone ahead of the crack tip is substantial
and must be considered when determining the SERR.
To address these issues, the CBBM is used. This method
relies on the instantaneous compliance of the specimen to
determine an equivalent crack length, eliminating the need
for direct crack length measurement during the test.
Moreover, the equivalent crack length provided by this
method accounts for the effects of crack tip plasticity.
Using the CBBM, the fracture energy in mode I can be
determined as in Ref. [41]:
6P2 2a2e
1
GIc ¼ 2
þ
ðEq 2Þ
B h Ef h2 5G
where h is the thickness of each arm (see Fig. 2), G is the
shear modulus, ae is the equivalent crack length calculated
by solving Eq. (3) iteratively and taking only the real root,
and Ef is the corrected flexural modulus obtained directly
from Eq. (4).
8
12
3
ðEq 3Þ
a þ
ae C o ¼ 0
5G13 Bh
Bh3 Ef e
12ðae DÞ 1 8ðae DÞ3
Ef ¼ Co ðEq 4Þ
5GBh
Bh3
Fig. 3 (a) Specially designed
curing mold; and (b) ready-totest specimens
Fig. 4 (a) Tensile test setup, and (b) selected snapshot for illustrating the test’s progress
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Fig. 5 (a–c) Traction–separation law, and (d–f) damage progression for: triangular (a & d), trapezoidal (b & e), and exponential (c & f) shapes.
The colors green, red, and blue indicate the intervals in Eqs. (8), (12), and (16)
with Co is the initial compliance, and D is the root rotation
correction factor, which will be determined numerically by
taking the x-axis intercept of the line fitting the relationship
between the cube root of the initial compliance and the
1
3
initial crack length, ao, i.e., ðC o Þ ¼ f ðao Þ:
Numerical Investigation
In the field of damage or fracture mechanics, several
numerical methods can be used to investigate fracture
energy. These methods include but not limited to cohesive
zone modeling (CZM) [53–55], crack tip open displacement (CTOD) [56–58], extended finite element method
(XFEM) [59, 60], phase-field fracture models [61–63],
discrete element method (DEM) [64, 65].
In this work, the approach of the CZM has been chosen
due to the key feature that the presence of an initial crack is
not needed. Detailed descriptions of the models created for
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Table 2 Mechanical properties of the materials used in the
simulation [70, 71]
Parameter
Al 2024-T3
AralditeÒ2015
E
GPa
72.4
1.85
G
GPa
–
0.56
V
1
0.33
–
ru
MPa
–
23.0
su
MPa
–
18.0
GIc
N/mm
–
0.385
the CZM and FEA are provided in the following
subsections.
Cohesive Zone Model
The origin of the cohesive zone model (CZM) is traced
back to the sixties by Dugdale [42] and Barenblatt [43].
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The CZM describes material separation with a traction–
separation law (TSL). It assumes a relationship between the
normal/shear traction and the opening/sliding displacement
and can capture the debonding process of particle/matrix
interfaces. In general, the TSL can be expressed in the form
of:
d
r ¼ rmax f ðEq 5Þ
df
where rmax is the material strength, f is a dimensionless
function that depends on the shape of the TSL, and df is the
separation at failure. The critical energy release rate, GC,
defined as the area under the TSL curve, is:
Z df
Gc ¼
r d dd
ðEq 6Þ
0
The stress components of the TSL are affected by
cohesion damage, or stiffness reduction, and can be
expressed as follows:
r ¼ ð1 DÞ K d
ðEq 7Þ
where K is defined as the ratio between the elastic modulus
in tension or shear (E or G, respectively) and the thickness.
In this work, the cohesive behavior of AralditeÒ2015 in
mode I is reproduced. Accordingly, several shapes of the
TSL are evaluated, and the most appropriate is selected.
Specifically, the triangular, trapezoidal, and exponential
shapes are considered, as shown in Fig. 5. The justification
for this selection is that the triangular and trapezoidal TSLs
are more accurate representations for brittle and ductile
adhesives, respectively [45, 46, 48, 66, 67], where the
exponential shape is already built-in in ABAQUS and not
widely studied in the literature.
For the triangular shape shown in Fig. 5a, the traction, r,
is obtained by:
8
< rmax d ; d\dd
dd r¼
: rmax df d ; d\dd
df dd
ðEq 8Þ
where rmax is the maximum traction, dd is the displacement
at which damage initiates (see Eq. (9)), and df is the displacement at which complete failure occurs (see Eq. (10)).
rmax
ðEq 9Þ
dd ¼
K
2Gc
df ¼
ðEq 10Þ
rmax
The damage parameter is obtained by substituting
Eq. (8) in Eq. (7):
D¼1
dd ð df dÞ
d ð df dd Þ
ðEq 11Þ
For the symmetric trapezoidal shape shown in Fig. 4b,
the traction, r, is calculated by:
8
d
>
r
>
max
<
dd ; d\dd1
rmax;
d\dd2
ðEq 12Þ
r¼
dd1 >
>
: rmax df d ; dd2 \d df
df dd2
Table 3 Converged values of the studied parameters
TSL
Initial
step
(mm)
Max
step
(mm)
Viscosity
coefficient
Element
length
(mm)
CPU
time
(sec)
Triangular
0.12500
0.500
10–3
0.25
765.7
Trapezoidal 0.12500
0.250
10–3
0.25
339.6
Exponential 0.03125
0.125
10–3
0.25
575.5
Fig. 6 Finite element model
including the mesh and
boundary conditions used for
simulating the tensile test
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Fig. 7 Selected sample (two
sides) of tested specimens
showing (a) a cohesive failure,
and (b) an adhesive failure
where dd1 is the first inflection point at which damage
initiates (see Eq. (9)), and dd2 is the second inflection point,
calculated as:
dd2 ¼
Gc
rmax
ðEq 13Þ
The maximum displacement at which complete failure
occurs is then obtained by:
df ¼ dd1 þ dd2
ðEq 14Þ
Similarly, the damage parameter is obtained by
substituting Eq. (12) in Eq. (7):
(
1 ddd1 ; dd1 d\dd2
D¼
2 ddf ; dd2 \d df
For the exponential shape shown in Fig. 4c, the critical
strain energy release rate, Gc, and the effective traction for
mode I are given by Eqs. (16) [68] and (17) [69],
respectively:
GIc ¼ e rmax dd
dI d
dI
r ¼ rmax e d
dd
ðEq 15Þ
ðEq 16Þ
After exceeding dd, the value of r decreases but never
reaches zero, and df cannot be determined mathematically.
Here, it is assumed that complete separation occurs when
df = 6dd. By substituting Eq. (16) into Eq. (7), the damage
parameter for mode I is given by:
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"
D¼1
dI
1¼
dd
e
d
d I
d
#
ðEq 17Þ
Finite Element Analysis
ABAQUS [68] is used to simulate the mechanical behavior
of the adhesive under mode I loading. A plane-strain model
is created in a 2-D space. The model consists of one
deformable part and two materials: (1) Aluminum alloy Al
2024-T3 defined as an isotropic material with elastic
behavior, and (2) AralditeÒ2015 defined as a traction
material with elastic behavior followed by damage. The
mechanical properties required for the analysis are listed in
Table 2. To characterize the cohesive behavior of the
adhesive, a damage model with a quadratic nominal stress
(QUADS) criterion and element deletion is implemented.
The deformable part is partitioned into three subparts: an
adhesive layer (t = 0.36 mm) and two adherents (t = 2 mm
each), c.f. Fig. 2. The adherents are meshed using structured, quad-shaped, linear order, 4-node bilinear planestrain quadrilateral elements (CPE4), and the adhesive is
meshed using structured, quad-shaped, linear order, 4-node
two-dimensional cohesive elements (COH2D4). The seed
size is set to 0.2 mm, with eight elements through the
thickness of the adherents and one element through the
thickness of the adhesive. To replicate the gripping system
of the testing machine, the upper and lower adherents are
pinned (UX = UY = 0) at their leftmost upper and lower
points, respectively. A vertical displacement (UY = 30
mm) is applied at the upper left point of the upper adherent
to simulate the actuator movement, as shown in Fig. 6.
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Fig. 8 Load–displacement curves for the tested specimens. (a–d): Groups A–D. Specimen D2 exhibited an adhesive failure, and specimen D3
was not successful
Due to the high deflections in the adherents, a geometrically nonlinear analysis is considered. Additionally,
damage stabilization is applied with a viscosity coefficient,
g, of 0.001 to overcome the convergence difficulties
associated with the adhesive’s softening behavior [72].
Then, the model is analyzed using three shapes of
material softening: triangular, trapezoidal, and exponential,
and a sensitivity study is conducted to ensure that the
obtained results are independent of the viscosity coefficient, step size, and mesh size (Table 3).
Finally, the accuracy of the FEA model has been validated by comparing the load–displacement curves to actual
testing results, and the results are presented in the following section.
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Table 4 Summary of the experimental results of tested specimens
Specimens
Standard
deviation
Average
#
D
(mm)
Pmax
(N)
GIc (N/
mm)
Pmax
(N)
GIc (N/
mm)
Pmax
(N)
GIc (N/
mm)
A1
–8
59.0
0.3174
66.47
0.3113
6.69
0.0649
A2
–6
71.9
0.3730
A3
–4
68.5
0.2436
B1
–11
63.4
0.3479
69.83
0.2937
8.01
0.0545
B2
–11
78.7
0.2389
B3
–22
67.3
0.2943
C1
–14
79.5
0.3750
91.07
0.3847
12.37
0.0089
C2
–13
89.6
0.3865
C3
3
104.1
0.3925
D1
–17
81.2
0.4262
81.20
0.4262
–
–
D2
–31
54.0
0.1762
D3
–
–
–
Results and Discussion
As discussed in the previous sections, four groups of DCB
specimens were prepared and tested, and the behavior of
one group was simulated using three shapes of TSLs. In the
following subsections, the experimental and numerical
results are presented.
Experimental Results
The tests were conducted successfully for Groups A, B, and
C, but not for Group D. An adhesive failure was observed
in specimen D2, and the test of specimen D3 was not
successful. No plastic deformation was observed in any
specimen. Selected tested specimens are shown in Fig. 7.
The obtained load–displacement curves for the four groups
are plotted in Fig. 8, and a summary is provided in Table 4.
For Group D, the result of specimen D1 is included for
informational purposes only.
Generally, the load–displacement curves (Fig. 8) show
relatively good agreement and test repeatability. In the
initial part of the curves (linear part), stress builds up at the
crack tip until the energy level reaches the adhesive’s
resistance (load peak). The crack then starts to propagate,
causing the load to drop, as shown in the second part of the
curves. The fluctuations during the load drop (crack propagation) are attributed to adhesion imperfections in the
specimens, such as air bubbles and other internal defects.
It is worth mentioning that the difference in the slope of
the linear part of the load–displacement curve of specimen
C3, compared to the other members of the group (C1 and
C2), is due to the difference in the value of its root rotation
factor (D = 3). This indicates that its crack behaved as if it
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was shorter by 3 mm, resulting in a naturally higher stiffness. Conversely, cracks in specimens C1 and C2 behaved
as if they were longer by 14 mm and 13 mm, respectively.
The same observation applies to specimens A1 and B3, but
inversely, as their cracks behaved as if they were longer.
After obtaining the experimental load–displacement
data, the CBBM is used to derive the respective R-curves,
following the procedure outlined in Section ‘‘Determining
Fracture Energy’’. As seen in Fig. 9, the first stage of the
curves displays a relatively vertical line. This is because
the crack length remains constant (unpropagated), and the
stored energy is accumulating at the crack tip. When this
stored elastic energy reaches its maximum allowable value,
additional energy drives the creation of new cracked surfaces, satisfying the damage evolution criterion and
initiating crack propagation. In the second stage of the Rcurve (post-peak), the energy fluctuates around a horizontal
line. The SERR value of this line represents the mode I
fracture energy for the corresponding adhesive thickness:
0.3113 N/mm, 0.2937 N/mm, and 0.3847 N/mm for
thicknesses of 0.20 mm, 0.31 mm, and 0.36 mm, respectively. Specimen C3 lags behind specimens C1 and C2 by
approximately 16 mm due to the difference in their root
rotation factor as previously explained (D1 D2,3 & 16).
The same observation applies to specimens A1 and B3. It is
worth mentioning that the crack lengths plotted in Fig. 9
are the equivalent ones (ae), not the actual ones (ao) which
are designed to be 35 mm for all specimens. The equivalent crack length is calculated by solving Eq. (3) iteratively
and taking only the real root.
To validate the results, they are compared with other
equivalent investigations in the literature, as illustrated in
Fig. 10. The critical mode I SERR of the same adhesive
with a thickness of 0.2 mm was determined by Özer et al.
[73], de Moura et al. [41], and Gheibi et al. [74] to be 0.500
N/mm, 0.430 N/mm, and 0.293 N/mm, respectively. In
contrast, the critical mode I SERR for a thickness of
1.0 mm was calculated by Fernandes et al. [44] and
Teixeira et al. [40] to be 0.539 N/mm and 1.005 N/mm,
respectively. In all of these studies, the adhesive was
allowed to cure at room temperature, and the CBBM was
used for data reduction. The observed deviation in the
results could be attributed to uncontrolled curing temperatures, as room temperature can vary significantly between
countries and overtime.
Numerical Results
To determine the most appropriate TSL shape, three simulations of the DCB test were performed for each of
Group’s A, B, and C using triangular, trapezoidal, and
exponential TSLs. In total, nine simulations were conducted, and for the sake of pithiness, only the results of
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Fig. 9 R-curves for the tested specimens. (a–d): Groups A–D. Specimen D2 exhibited an adhesive failure, and specimen D3 was not successful
Group C are presented. According to the equivalent stress
analysis shown in Fig. 11, no plastic deformation is
observed in the adherents.
The load–displacement curves and the R-curves are
plotted in Fig. 12. Generally, very good agreement is found
between the three TSL shapes. The maximum loads for the
triangular, trapezoidal, and exponential TSLs are 93.46 N,
94.12 N, and 95.26 N, respectively, which differ slightly
from the reference experimental value of 91.07 N. The
deviation between the numerical and experimental results
for the maximum load is 2.6%, 4.6%, and 3.3% for the
triangular, trapezoidal, and exponential TSLs, respectively.
The deviations in the fracture energy are 1.0%, 1.8%, and
1.4%, respectively. Figure 13 compares the numerical
load–displacement curve for the triangular TSL with the
experimental curve.
All the TSL shapes tested (triangular, trapezoidal, and
exponential) yielded similar and accurate results. The
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triangular law is straightforward to implement and is
commonly available in commercial software. However, the
trapezoidal law demonstrated significantly faster computational performance, with results obtained 44% and 59%
faster than the triangular and exponential laws,
respectively.
While it is known that ductile adhesives are highly
influenced by the shape of the TSL [46], the results
obtained from the three shapes were very similar in this
case. This similarity is attributed to the fact that the
adhesive curing was performed at room temperature, which
likely made the adhesive more brittle, thereby reducing the
Fig. 10 Critical mode I SERR for the tested specimens and other
published works
Fig. 11 Simulation of crack
propagation with element
deletion
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impact of the TSL shape on the results. Another justification could be the data scattering seen in Tables 1 and 4,
which is probably caused by the non-uniformity of the
thickness along the specimen length and width. Besides,
the time spent by each specimen before the testing might
also be a source for this, as polymers continue curing after
they get extracted from the mold, and hence, different
performance at different times might be seen.
Conclusion
The main objective of this study was to characterize the
bonding strength of AralditeÒ2015 in opening mode at
different thicknesses and to develop a CZM that would best
represent adhesion degradation.
Firstly, a group of DCB tests have been conducted to
evaluate the bonding strength of the adhesive at different
thicknesses. Secondly, the CBBM, which does not require
measuring the crack length during the test, was utilized to
determine mode I critical SERR. Thirdly, several shapes of
the TSLs, namely the triangular, trapezoidal, and exponential, were compared in simulating the DCB tests.
The study found that the fracture energy is almost
constant at 0.3025 N/mm for adhesive thickness between
0.20 and 0.30 mm, and then increases linearly up to 0.3847
N/mm at the thickness of 0.35 mm. For thicknesses greater
than 0.35 mm, further investigations were deemed
necessary.
Furthermore, all the tested TSLs produced similar and
accurate load–displacement curves for all tested DCB
groups, with the trapezoidal shape being the fastest one
without any observed limitations.
J Fail. Anal. and Preven. (2025) 25:2150–2164
2161
Fig. 12 Simulated (a) load–displacement curves and (b) R-curves
Data availability All the data required for reproducing our results
are included within this manuscript.
Code availability
Not applicable.
Conflict of interest The authors declare that they do not have a
known conflict of interest.
Ethical approval
Fig. 13 Experimental versus numerical load–displacement curves
The outcomes of this study are expected to assist in the
development of a model for simulating the progressive
fatigue damage of similar bonded joints in future research.
Funding Open access funding provided by Carinthia University of
Applied Sciences (CUAS). This work is financially supported by the
Researchers Supporting Project number (RSPD2024R921), King
Saud University, Riyadh, Saudi Arabia.
Not applicable.
Consent to participate
Not applicable.
Consent for publication
Not applicable.
Open Access This article is licensed under a Creative Commons
Attribution 4.0 International License, which permits use, sharing,
adaptation, distribution and reproduction in any medium or format, as
long as you give appropriate credit to the original author(s) and the
source, provide a link to the Creative Commons licence, and indicate
if changes were made. The images or other third party material in this
article are included in the article’s Creative Commons licence, unless
indicated otherwise in a credit line to the material. If material is not
included in the article’s Creative Commons licence and your intended
use is not permitted by statutory regulation or exceeds the permitted
use, you will need to obtain permission directly from the copyright
holder. To view a copy of this licence, visit http://creativecommons.
org/licenses/by/4.0/.
References
1. Al-Mukhtar, A.M., Rahman, T., Doos, Q.M.: Spot welding joint’s
fracture behavior and fundamental. Proceedings of the 7th
international conference on fracture fatigue and wear (2018)
123
2162
2. A.M. Al-Mukhtar, Case studies of aircraft fuselage cracking.
Adv. Eng. Forum. 33, 11–18 (2019). https://doi.org/10.4028/
www.scientific.net/aef.33.11
3. A.M. Al-Mukhtar, Aircraft fuselage cracking and simulation.
Procedia Struct. Integrity. 28, 124–131 (2020). https://doi.org/10.
1016/j.prostr.2020.10.016
4. L. Li, J. Qin, Y. Pan, J. Xu, M.H. Faber, A trustworthy intelligent
offshore wind turbine fatigue crack propagation prediction
framework from the probabilistic perspective. Ocean Eng. 314,
119739 (2024). https://doi.org/10.1016/j.oceaneng.2024.119739
5. K. Ravikumar, R. Subbiah, N. Ranganathan, J. Bensingh, A.
Kader, S.K. Nayak, A review on fatigue damages in the wind
turbines. Wind Eng. 44(4), 434–451 (2020). https://doi.org/10.
2307/26956683
6. X.-Y. Miao, C.-J. Chen, S. Fæster, S.S. Samareh-Mousavi, X.
Chen, Fatigue and post-fatigue static crack characterization of a
wrinkled thick glass fibre laminate in a composite wind turbine
blade. Int. J. Fatigue. 176, 107855 (2023). https://doi.org/10.
1016/j.ijfatigue.2023.107855
7. J. Tian, X. Li, F. Li, Research on fatigue crack growth life of
offshore wind turbine towers with double cracks in extreme
environments. J. Offshore Mech. Arctic Eng.. 147(5), 052001
(2025)
8. M.R. Ayatollahi, S.M.J. Razavi, H.R. Chamani, Fatigue life
extension by crack repair using stop-hole technique under pure
mode-I and pure mode-II loading conditions. Proce. Eng. 74, 18–
21 (2014). https://doi.org/10.1016/j.proeng.2014.06.216
9. S.M. Mohammed, B.B. Bouiadjra, F. Benyahia, A. Albedah,
Analysis of the single overload effect on fatigue crack growth in
AA 2024–T3 plates repaired with composite patch. Eng. Fracture
Mech. 15(202), 147–61 (2018)
10 S.M. Mohammed, A. Albedah, F. Benyahia, B.B. Bouiadjra,
Effect of single tensile peak overload on the performance of
bonded composite repair of cracked Al 2024–T3 and Al 7075–T6
plates. Compos. Struct. 193, 260–267 (2018)
11. G.A. Harmain, A model for predicting the retardation effect
following a single overload. Theor. Appl. Fract. Mech. 53(1), 80–
88 (2010). https://doi.org/10.1016/j.tafmec.2009.12.008
12. N.K. Salgado, M.H. Aliabadi, The analysis of mechanically fastened repairs and lap joints. Fatigue Fract. Eng. Mater. Struct.
20(4), 583–593 (1997). https://doi.org/10.1111/j.1460-2695.1997.
tb00290.x
13. A.A. Baker, R.J. Callinan, M.J. Davis, R. Jones, J.G. Williams,
Repair of mirage III aircraft using the BFRP crack-patching
technique. Theor. Appl. Fract. Mech. 2(1), 1–15 (1984). https://
doi.org/10.1016/0167-8442(84)90035-1
14. B.B. Bouiadjra, F. Benyahia, A. Albedah, B.A. Bouiadjra, S.M.
Khan, Comparison between composite and metallic patches for
repairing aircraft structures of aluminum alloy 7075 T6. Int. J.
Fatigue. 80, 128–135 (2015). https://doi.org/10.1016/j.ijfatigue.
2015.05.018
15. J.-B. Orsatelli, E. Paroissien, F. Lachaud, S. Schwartz, Bonded
flush repairs for aerospace composite structures: a review on
modelling strategies and application to repairs optimization,
reliability and durability. Compos. Struct. 304, 116338 (2022).
https://doi.org/10.1016/j.compstruct.2022.116338
16. Ashforth, C., Momm, G.G., Seneviratne, W., Trace, A., Ilcewicz,
L.: 23 -aerospace structural bonding: Qualification, quality control, substantiation, and risk mitigation. In: Dillard, D.A. (ed.)
Advances in Structural Adhesive Bonding, Second edition edn.
Woodhead Publishing in Materials, pp. 779–806. Woodhead
Publishing, ??? (2023). https://doi.org/10.1016/B978-0-32391214-3.00029-6
.
https://www.sciencedirect.com/science/
article/pii/B9780323912143000296
17. M. Malekan, C.A. Cimini, Finite element analysis of a repaired
thin-walled aluminum tube containing a longitudinal crack with
123
J Fail. Anal. and Preven. (2025) 25:2150–2164
composite patches under internal dynamic loading. Compos.
Struct. 184, 980–1004 (2018). https://doi.org/10.1016/j.
compstruct.2017.10.079
18. J.J. Schubbe, S.H. Bolstad, S. Reyes, Fatigue crack growth
behavior of aerospace and ship-grade aluminum repaired with
composite patches in a corrosive environment. Compos. Struct.
144, 44–56 (2016). https://doi.org/10.1016/j.compstruct.2016.01.
107
19. P.S. Shinde, P. Kumar, V.K. Tripathi, Dependence of repair
strength on the size of FRP patch bonded to a cracked aluminum
alloy panel. Thin-Walled Struct. 124, 303–311 (2018). https://doi.
org/10.1016/j.tws.2017.12.022
20. Y. Tie, Y. Hou, C. Li, X. Zhou, T. Sapanathan, M. Rachik, An
insight into the low-velocity impact behavior of patch-repaired
CFRP laminates using numerical and experimental approaches.
Compos. Struct. 190, 179–188 (2018). https://doi.org/10.1016/j.
compstruct.2018.01.075
21. S.M. Fekih, A. Albedah, F. Benyahia, M. Belhouari, B.B.
Bouiadjra, A. Miloudi, Optimisation of the sizes of bonded
composite repair in aircraft structures. Mater. Des. 41, 171–176
(2012). https://doi.org/10.1016/j.matdes.2012.04.025
22 A. Albedah, S.M. Mohammed, B.B. Bouiadjra, B.A. Bouiadjra, F.
Benyahia, Effect of the patch length on the effectiveness of onesided bonded composite repair for aluminum panels. Int. J. Adhes.
Adhesiv. 1(81), 83–9 (2018)
23. H. Hosseini-Toudeshky, M.A. Ghaffari, B. Mohammadi, Finite
element fatigue propagation of induced cracks by stiffeners in
repaired panels with composite patches. Compos. Struct. 94(5),
1771–1780 (2012). https://doi.org/10.1016/j.compstruct.2012.01.
002
24. S.M.A. Khan, M. Essaheb, Effect of patch thickness on the repair
performance of bonded composite repair in cracked aluminum
plate. Mater. Today Proc. 4(8), 9020–9028 (2017). https://doi.
org/10.1016/j.matpr.2017.07.255
25. B.B. Bouiadjra, M.F. Bouanani, A. Albedah, F. Benyahia, M. EsSaheb, Comparison between rectangular and trapezoidal bonded
composite repairs in aircraft structures: a numerical analysis.
Mater. Des. 32(6), 3161–3166 (2011). https://doi.org/10.1016/j.
matdes.2011.02.053
26. F. Benyahia, A. Albedah, B.B. Bouiadjra, Analysis of the adhesive damage for different patch shapes in bonded composite
repair of aircraft structures. Mater. Des. 54, 18–24 (2014). https://
doi.org/10.1016/j.matdes.2013.08.024
27 M. Rachid, B. Serier, B.B. Bouiadjra, M. Belhouari, Numerical
analysis of the patch shape effects on the performances of bonded
composite repair in aircraft structures. Compos. Part B Eng. 43(2),
391–397 (2012). https://doi.org/10.1016/j.compositesb.2011.08.
047
28. M. Ramji, R. Srilakshmi, M.B. Prakash, Towards optimization of
patch shape on the performance of bonded composite repair using
FEM. Compos. Part B Eng. 45(1), 710–720 (2013). https://doi.
org/10.1016/j.compositesb.2012.07.049
29. S. Duda, M. Smolnicki, T. Osiecki, G. Lesiuk, Determination of
fracture energy (mode I) in the inverse fiber metal laminates
using experimental–numerical approach. Int. J. Fract. 234(1–2),
213–222 (2021). https://doi.org/10.1007/s10704-021-00566-3
30. T. Chen, C.M. Harvey, S. Wang, V.V. Silberschmidt, Analytical
corrections for double-cantilever beam tests. Int. J. Fract. 229(2),
269–276 (2021). https://doi.org/10.1007/s10704-021-00556-5
31. Beck, R., Prado, A., FM, L., Tita, V., Medeiros, R.d.: Assessing
critical fracture energy in mode i for bonded composite joints: a
numerical–experimental approach with uncertainty analysis.
Proceedings of the Institution of Mechanical Engineers Part L
Journal of Materials Design and Applications (2024) https://doi.
org/10.1177/14644207241229601
J Fail. Anal. and Preven. (2025) 25:2150–2164
32. M. Sterley, J. Blokland, Mode i fracture energy release rates of
European beech wood-adhesive bonds. Wood Mater. Sci. Eng.
20(1), 232–235 (2024). https://doi.org/10.1080/17480272.2024.
2390587
33. W.L. Tsang, The use of tapered double cantilever beam (TDCB)
in investigating fracture properties of particles modified epoxy.
SN Appl. Sci. (2020). https://doi.org/10.1007/s42452-020-2487-8
34. M.D. Kempe, J. Morse, J. Eafanti, S.E. Julien, K.-T. Wan, L.S.
Bruckman, Y. Wang, R.H. French, A. Fairbrother, X. Gu, S.
Napoli, G.S. O’Brien, A.W. Hauser, L. Ji, K.P. Boyce, Measurement of crack length in width tapered beam experiments. J.
Adhes. Sci. Technol. 35(4), 357–374 (2020). https://doi.org/10.
1080/01694243.2020.1801271
35. M.K. Budzik, S. Heide-Jørgensen, R. Aghababaei, Fracture
mechanics analysis of delamination along width-varying interfaces. Compos. Part B Eng. 215, 108793 (2021). https://doi.org/
10.1016/j.compositesb.2021.108793
36. M. Azimi, S.S. Mirjavadi, S.A. Asli, A.M.S. Hamouda, Fracture
analysis of a special cracked lap shear (CLS) specimen with
utilization of virtual crack closure technique (VCCT) by finite
element methods. J. Fail. Anal. Prev. 17(2), 304–314 (2017).
https://doi.org/10.1007/s11668-017-0243-1
37. R.B. Barros, R.D. Campilho, I.J. Sánchez-Arce, J.M. Dionı́sio, K.
Madani, Evaluation of the cracked lap shear test for mixed-mode
fracture toughness estimation of adhesive joints. Proc. Inst. Mech.
Eng. Part L J. Mater. Des. Appl. 237(12), 2537–2550 (2022).
https://doi.org/10.1177/14644207221140296
38. Y. Huang, X. Wang, On the fracture toughness testing for singleedge notched bend specimen of orthotropic materials. Compos.
Struct. 281, 114970 (2022). https://doi.org/10.1016/j.compstruct.
2021.114970
39. H.-C. Dan, Z. Zhang, J. Chen, W. Cao, Low-temperature fracture
characteristics of asphalt mixtures using the eccentric single-edge
notched bend test: a 3D discrete element study. Constr. Build.
Mater. 344, 128–182 (2022). https://doi.org/10.1016/j.
conbuildmat.2022.128182
40 J.M. Teixeira, R.D. Campilho, F.J. Da Silva, Numerical assessment of the double-cantilever beam and tapered double-cantilever
beam tests for the G IC determination of adhesive layers. J.
Adhes. 94(11), 951–73 (2018)
41. M.F.S.F. de Moura, R.D.S.G. Campilho, J.P.M. Gonçalves, Crack
equivalent concept applied to the fracture characterization of
bonded joints under pure mode I loading. Compos. Sci. Technol.
68(10–11),
2224–2230
(2008).
https://doi.org/10.1016/j.
compscitech.2008.04.003
42. D.S. Dugdale, Yielding of steel sheets containing slits. J. Mech.
Phys. Solids. 8(2), 100–104 (1960). https://doi.org/10.1016/00225096(60)90013-2
43. G.I. Barenblatt, The formation of equilibrium cracks during
brittle fracture. General ideas and hypotheses. Axially-symmetric
cracks. J. Appl. Math. Mech. 23(3), 622–636 (1959). https://doi.
org/10.1016/0021-8928(59)90157-1
44 R.L. Fernandes, R.D.S.G. Campilho, Testing different cohesive
law shapes to predict damage growth in bonded joints loaded in
pure tension. J. Adhes. 93(1–2), 57–76 (2017)
45. R.L. Fernandes, R.D.S.G. Campilho, Numerical evaluation of
dissimilar cohesive models to predict the behavior of doublecantilever beam specimens. Proced. Struct. Integr. 1, 42–49
(2016). https://doi.org/10.1016/j.prostr.2016.02.007
46. R.D.S.G. Campilho, M.D. Banea, J.A.B.P. Neto, L.F.M. Silva,
Modelling adhesive joints with cohesive zone models: effect of
the cohesive law shape of the adhesive layer. Int. J. Adhes.
Adhes. 44, 48–56 (2013). https://doi.org/10.1016/j.ijadhadh.2013.
02.006
47 R.L. Fernandes, R.D.S.G. Campilho, Accuracy of cohesive laws
with differ-ent shape for the shear behaviour prediction of bonded
2163
joints. J. Adhes. 95(4), 325–347 (2019). https://doi.org/10.1080/
00218464.2018.1438895
48. R.D.S.G. Campilho, D.C. Moura, D.J.S. Gonçalves, J.F.M.G.
Silva, M.D. Banea, L.F.M. Silva, Fracture toughness determination of adhesive and co-cured joints in natural fibre composites.
Compos. Part B Eng. 50, 120–126 (2013). https://doi.org/10.
1016/j.compositesb.2013.01.025
49. P.K.A.V. Kumar, A. Dean, S. Sahraee, J. Reinoso, M. Paggi,
Non-linear thermoelastic analysis of thin-walled structures with
cohesive-like interfaces relying on the solid shell concept. Finite
Elements Anal. Design. 202, 103696 (2022). https://doi.org/10.
1016/j.finel.2021.103696
50 P.K.A.V. Kumar, A. Dean, J. Reinoso, M. Paggi, A multiphasefield-cohesive zone model for laminated composites: application
to delamination migration. Compos. Struct. 276, 11471 (2021).
https://doi.org/10.1016/j.compstruct.2021.114471
51. ISO 25217:2009, Adhesives—Determination of the mode 1
adhesive fracture energy of structural adhesive joints using
double cantilever beam and tapered double cantilever beam
specimens (2009). https://www.iso.org/standard/42797.html
52. C.-T. Sun, Z. Jin, Fracture mechanics. (Elsevier Science, Amsterdam, The Netherlands, 2016)
53. A. Akhavan-Safar, E.A.S. Marques, R.J.C. Carbas, L.F.M. Silva,
Cohesive zone modelling for fatigue life analysis of adhesive
joints. (Springer, Cham, Switzerland, 2022)
54 I. Katsivalis, O.T. Thomsen, S. Feih, M. Achintha, Development
of cohesive zone models for the prediction of damage and failure
of glass/steel adhesive joints. Int. J. Adhes. Adhesiv. 97, 102479
(2019). https://doi.org/10.1016/j.ijadhadh.2019.102479
55. A.V.M. Rocha, A. Akhavan-Safar, R. Carbas, E.A.S. Marques, R.
Goyal, M. El-zein, L.F.M. Silva, Numerical analysis of mixedmode fatigue crack growth of adhesive joints using CZM. Theor.
Appl. Fract. Mech. 106, 102493 (2020). https://doi.org/10.1016/j.
tafmec.2020.102493
56. Y. Li, Y. Yang, J. Li, B. Wang, Y. Liao, Experimental-numerical
analysis of failure of adhesively bonded lap joints under transverse impact and different temperatures. Int. J. Impact Eng. 140,
103541 (2020). https://doi.org/10.1016/j.ijimpeng.2020.103541
57. Y. Qiao, Z.-X. Zhang, J. Zhou, A modified model for calculating
crack tip opening displacement (CTOD) considering fracture
process zone (FPZ) in rock. Geomech. Geophys. Geo-Energy
Geo-Resour. (2024). https://doi.org/10.1007/s40948-024-00779-5
58. M.K. Budzik, M. Wolfahrt, P. Reis, M. KozLowski, J. Sena-Cruz,
L. Papadakis, M. Nasr Saleh, K.V. Machalicka, S. Freitas, A.P.
Vassilopoulos, Testing mechanical performance of adhesively
bonded composite joints in engineering applications: an overview. J. Adhes. (2021). https://doi.org/10.1080/00218464.2021.
1953479
59. R.V.F. Faria, R.D.S.G. Campilho, P.J.A. Gonçalves, R.J.B.
Rocha, I.J. Sánchez-Arce, F.J.G. Silva, Numerical analysis by
XFEM of hybrid T-peel joints. Proce. Struct. Integr. 33, 673–684
(2021). https://doi.org/10.1016/j.prostr.2021.10.075
60. J.P.J.R. Santos, D.S. Correia, E.A.S. Marques, R.J.C. Carbas, F.
Gilbert, L.F.M. Silva, Extended finite element method (XFEM)
model for the damage mechanisms present in joints bonded using
adhesives doped with inorganic fillers. Materials. 16(23), 74–99
(2023). https://doi.org/10.3390/ma16237499
61. C. Gorkem Dengiz, M. Dorduncu, A unified phase-field approach
for failure prediction in modulus graded adhesively bonded single-lap joints. Theor. Appl. Fract. Mech. 127, 104062 (2023).
https://doi.org/10.1016/j.tafmec.2023.104062
62. C.G. Dengiz, Effect of adhesive material properties on phasefield analysis of T-peel adhesive joints using the Taguchi method.
Int. J. Adhes. Adhes. 139, 103977 (2025). https://doi.org/10.1016/
j.ijadhadh.2025.103977
123
2164
63. Y.G. Motlagh, Rd. Borst, Considerations on a phase-field model
for adhesive fracture. Int. J. Numer. Methods Eng. 121(13),
2946–2963 (2020). https://doi.org/10.1002/nme.6341
64. J.-P. Li, Q.-H. Rao, T.-B. Huang, W.-b Ma, Discrete element
method for adhesion properties evaluation of deep-sea sediment
from macro and micro perspectives. J. Cent. South Univ. 29(5),
1701–1716 (2022). https://doi.org/10.1007/s11771-022-5013-x
65. Z. Fang, X. Qian, Y. Zhang, W. Liu, S. Li, A new discrete element method for small adhesive non-spherical particles. J.
Comput. Phys. 513, 113193 (2024). https://doi.org/10.1016/j.jcp.
2024.113193
66. R.L. Fernandes, R.D.S.G. Campilho, Testing different cohesive
law shapes to predict damage growth in bonded joints loaded in
pure tension. J. Adhes. 93(1–2), 57–76 (2016). https://doi.org/10.
1080/00218464.2016.1169176
67. R.L. Fernandes, R.D.S.G. Campilho, Accuracy of cohesive laws
with different shape for the shear behaviour prediction of bonded
joints. J. Adhes. 95(4), 325–347 (2018). https://doi.org/10.1080/
00218464.2018.1438895
68. X.-P. Xu, A. Needleman, Numerical simulations of fast crack
growth in brittle solids. J. Mech. Phys. Solids. 42(9), 1397–1434
(1994). https://doi.org/10.1016/0022-5096(94)90003-5
123
J Fail. Anal. and Preven. (2025) 25:2150–2164
69. X.-P. Xu, A. Needleman, Void nucleation by inclusion debonding
in a crystal matrix. Model. Simul. Mater. Sci. Eng. 1(2), 111–132
(1993). https://doi.org/10.1088/0965-0393/1/2/001
70 N. Demmouche, A. Albedah, S.M. Mohammed, B.
BachirBouiadjra, F. Benyahia, Interaction between adherend
plasticity and adhesive damage in metal/composite joints: application to bonded composite repair of metallic structures. Welding
World. 63(2), 211–9 (2019)
71. MatWeb.
https://www.matweb.com/search/DataSheet.
aspx?MatGUID=57483b4d782940faaf12964a1821fb61&ckck=1
72. Abaqus 2016 Documentation. Dassault systèmes (2015)
73 H. Özer, E. Erbayrak, The effects of curing temperature on fracture energy and cohesive parameters for the adhesive Araldite
2015. J. Adhes. Sci. Technol. 32(12), 1287–1312 (2018). https://
doi.org/10.1080/01694243.2017.1410080
74 M.R. Gheibi, M.H. Shojaeefard, G.H. Saeidi, Direct determination
of a new mode-dependent cohesive zone model to simulate metalto-metal adhesive joints. J Adhes. 95(10), 943–970 (2019)
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