Available

Available online
online at
at www.sciencedirect.com
www.sciencedirect.com

ScienceDirect
Available online at www.sciencedirect.com
Structural
Structural Integrity
Integrity Procedia
Procedia 00
00 (2022)
(2022) 000–000
000–000
www.elsevier.com/locate/procedia
www.elsevier.com/locate/procedia
ScienceDirect
Procedia Structural Integrity 41 (2022) 87–93

2nd
2nd Mediterranean
Mediterranean Conference
Conference on
on Fracture
Fracture and
and Structural
Structural Integrity
Integrity

In-plane fracture analysis of bi-material adhesively bonded joints by

using a simple bend beam specimen
M.R.M. Alihaaa*,
M.R.M. Aliha *, H.G.
H.G. Kucheki
Kuchekia,, N.
a
Razavi
S.M.J.
b
Razavib
aWelding
Welding and
and Joining
Joining Research
Research Center,
Center, School
School of
of Industrial
Industrial Engineering,
Engineering, Iran
Iran University
University of
of Science
Science and
and Technology
Technology (IUST),
(IUST), Tehran
Tehran 16846-
a
16846-
13114, Iran
13114, Iran
bDepartment of Mechanical and Industrial Engineering, Norwegian University of Science and Technology, Richard Birkeland vei 2B, 7491,
b
Department of Mechanical and Industrial Engineering, Norwegian University of Science and Technology, Richard Birkeland vei 2B, 7491,
Trondheim,
Trondheim, Norway
Norway

Abstract
Abstract

In
In the
the present
present work,
work, aa new
new and
and novel
novel test
test specimen
specimen is is presented
presented to to solve
solve the
the difficulties
difficulties and
and problems
problems of of traditional
traditional adhesive
adhesive
fracturing
fracturing test specimens. The specimen is a bi-material adhesive joint in the shape of a rectangular beam and
test specimens. The specimen is a bi-material adhesive joint in the shape of a rectangular beam and containing
containing aa
vertical
vertical crack
crack in
in the
the adhesive
adhesive layer.
layer. If
If the
the specimen
specimen is is loaded
loaded by
by aa three-point
three-point bend
bend setup
setup with
with symmetric
symmetric or or asymmetric
asymmetric bottom
bottom
span
span supports
supports relative
relative to
to the
the crack,
crack, opening
opening deformation
deformation any any desired
desired combination
combination of of opening
opening andand shear
shear deformations
deformations is is achieved,
achieved,
respectively.
respectively. The
The results
results demonstrated
demonstrated that
that the
the proposed
proposed testtest specimen
specimen could
could be
be used
used to
to introduce
introduce thethe full
full range
range ofof mode
mode mixities
mixities
from
from pure
pure opening
opening mode
mode to to pure
pure shearing
shearing modemode for
for adhesively
adhesively bonded
bonded joints
joints with
with similar
similar and
and dissimilar
dissimilar adherents
adherents (such
(such as
as
Alumina
Alumina and
and Aluminum)
Aluminum) by by changing
changing the
the position
position ofof the
the bottom
bottom span
span supports.
supports. The
The variations
variations ofof mode
mode II and
and mode
mode IIII geometry
geometry
factors,
factors, normalized
normalized T-stress,
T-stress, biaxiality
biaxiality ratio, crack tip
ratio, crack tip plastic
plastic zone
zone and
and also
also the
the direction
direction of
of fracture
fracture initiation
initiation angle
angle are
are also
also
presented
presented for the analyzed test specimen by using the ABAQUS commercial code. As a result, the suggested specimen can
for the analyzed test specimen by using the ABAQUS commercial code. As a result, the suggested specimen can be
be
recommended
recommended as as aa potential
potential candidate
candidate specimen
specimen forfor fracture
fracture studies
studies ofof adhesively
adhesively bonded
bonded joints
joints subjected
subjected toto in-plane
in-plane loading
loading
conditions.
conditions.
©
© 2022 TheTheAuthors. Published by Elsevier B.V.
© 2022
2022
This is anThe
Authors.
openAuthors.
Published
Published
access article
by
by ELSEVIER
under the ELSEVIER
CC BY-NC-ND
B.V.
B.V.
license (https://creativecommons.org/licenses/by-nc-nd/4.0)
This
This is
is an
an open
open access
access article
article under
under the
the CC
CC BY-NC-ND
BY-NC-ND license
license (https://creativecommons.org/licenses/by-nc-nd/4.0)
(https://creativecommons.org/licenses/by-nc-nd/4.0)
Peer-review under responsibility of the MedFract2Guest Editors.
Peer-review
Peer-review under responsibility of the MedFract2Guest Editors.
under responsibility of the MedFract2Guest Editors.
Keywords: Bi-material
Keywords: Bi-material adhesively
adhesively bonded
bonded joint;
joint; Novel
Novel test
test spacimen;
spacimen; Asymmetric
Asymmetric beam
beam bend;
bend; Fracture
Fracture parameters;
parameters; Mixed
Mixed mode
mode I/II;
I/II; Finite
Finite
element
element analysis
analysis

** Corresponding
Corresponding author.
author. Tel.: +98-21-73225031; fax:
Tel.: +98-21-73225031; fax: +98-21-73225098.
+98-21-73225098.
E-mail address: mrm_aliha@iust.ac.ir
E-mail address: mrm_aliha@iust.ac.ir

2452-3216
2452-3216 © © 2022
2022 The
The Authors.
Authors. Published
Published by
by ELSEVIER
ELSEVIER B.V.
B.V.
This
This is an open access article under the CC BY-NC-ND license
is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0)
(https://creativecommons.org/licenses/by-nc-nd/4.0)
Peer-review
Peer-review under
under responsibility
responsibility of
of the
the MedFract2Guest
MedFract2Guest Editors.
Editors.

2452-3216 © 2022 The Authors. Published by Elsevier B.V.

This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0)
Peer-review under responsibility of the MedFract2Guest Editors.
10.1016/j.prostr.2022.05.011
88 M.R.M. Aliha et al. / Procedia Structural Integrity 41 (2022) 87–93
2 Aliha et al. / Structural Integrity Procedia 00 (2022) 000–000

Nomenclature

a crack length
a/W crack length ratio
ABJs adhesively bonded joints
BR biaxiality ratio
Bi-ASB bi-material asymmetric short beam
d thickness of the adhesive layer
DCB double cantilever beam
E modulus of elasticity
ENF end-notch flexural
F applied load
KI mode I stress intensity factor
KII mode II stress intensity factor
MMB mixed mode bending
S distances between fixed bottom span support and crack edge
S/L fixed span ratio
S′ distances between movable bottom span support and crack edge
S′/L movable span ratio
t thickness of the adherents
T T-stress
T* non-dimensional form of T-stress
W width of the beam
YI mode I geometry factor
YII mode II geometry factor

Greek symbols
θ direction of fracture initiation
ν Poisson's ratio

1. Introduction

Structural adhesively bonded joints (ABJs) are used in various industries such as aerospace, automotive, civil
construction, electronics, biology and medicine. This is due to their ability to bond similar and dissimilar materials,
high fatigue and corrosion resistance, ability to present a uniform distribution of stress at the joint region, good
damping properties, and efficiently bonding of thin plates. Variables such as insufficient surface cleaning and the
presence of voids in the adhesive layer lead to the formation of cracks at the bonding region under applied loads. As
a result, the fracture characteristics of cracked ABJs subjected to a variety of loading conditions should be assessed
in order to assure the structure's health [1]. Double Cantilever Beam (DCB) [2], End-Notch Flexural (ENF) [3], and
Mixed Mode Bending (MMB) [4] are some examples of well-known specimens that can be utilized to evaluate the
fracture parameters of cracked ABJs subjected to mode I, mode II, and mixed-mode I/II loading conditions,
respectively [5]. The mentioned testing techniques have some drawbacks, including the incapacity to present the full
range of mode mixities, the need for extensive testing jigs and fixtures to conduct the experiments, the significant
experimental costs, substantial deformations that occur during the test, non-linearity caused by the weight and curve
of the lever, and comparatively large size of the test specimen [6].
As a result, proposing a suitable testing technique for analyses of mixed-mode I/II fracturing of adhesively
bonded joints is still a necessary subject. As a result, Aliha et al. [6,7] offered the bi-material inclined notch short
bend beam (BISBB) specimen for fracture analysis of ABJs with similar or dissimilar adherent types. This suggested
specimen have some advantages including testing convenience using a typical three-point bend fixture, generating
complete ranges of in-plane mode mixities, and small size of final test sample. However, when employing the
BISBB test setup for mixed-mode I/II testing of ABJs, the specimen must be built with varied crack inclination
angles [6] and this issue can be mentioned as a serious disadvantage for the BISBB specimen.
 M.R.M. Aliha et al. / Procedia Structural Integrity 41 (2022) 87–93 89
Aliha et al. / Structural Integrity Procedia 00 (2022) 000–000 3

As a consequence, an improved design for Short Beam was presented in this work to cover the drawbacks of the
BISBB specimen. In the improved specimen that is called "Bi-material asymmetric short beam (Bi-ASB)", both
adherent parts are rectangular and a vertical crack is considered in the adhesive layer. For altering the contribution of
opening-shearing deformations the specimen is loaded asymmetrically using three-point bend fixture. The
manufacturing procedure for this new specimen is straightforward and there is no need to adjust the crack inclination
angle throughout the manufacturing process for conducting mixed-mode I/II tests on adhesively bonded joint. As a
result, the suggested Bi-ASB specimen keeps all of the benefits of the BISBB specimen while simultaneously
covering its drawbacks. As a result, the cost of producing and fracture testing via the Bi-ASB specimens is
significantly less than that of traditional and previous specimens such as DCB, ENF, MMB, BISBB and etc.
In the following sections of this study, after introducing the proposed test sample, its fracture parameters
including modes I and II stress intensity factors (K I and KII) and T-stress, are determined and presented for different
geometrical and loading conditions and also adherent types.

2. Suggested Test Specimen

The Bi-ASB test specimen suggested for fracture analysis of the ABJs under mixed-mode I/II loading conditions
is shown in Fig. 1. The Bi-ASB specimen's overall shape is a rectangle formed by two adherents (of the same size of
width=W and length = L) bonded together by an epoxy adhesive. The width of adhesive is d and a straight edge
crack with length a is assumed to be introduced in the middle of the adhesive layer. The specimen is loaded
asymmetrically with a three-point bend fixture.
The state of mode mixity in the Bi-ASB specimen can easily be changed by changing the locations of two bottom
span supports (i.e. S and S′). Pure mode I occurs when the bottom span supports are located symmetrically relative
to the crack (i.e. when S′ = S). Mode II, footprint appears when the Bi-ASB specimen is subjected to asymmetric
loading (i.e. S′ ≠ S). Controlling the contribution of mode I and mode II in this specimen is done by selecting
appropriate S′ and S values. In this research Alumina ceramic and Aluminum metal alloy were investigated for
joining using an epoxy adhesive to construct the suggested Bi-ASB specimen. Four distinct permutations of the Bi-
ASB specimen based on the adherent type are seen in Fig. 2. Table 1 shows also the mechanical properties of the
chosen materials.

Fig. 1. Overall schematics of the suggested Bi-ASB specimen for fracture study of adhesively boned joints subjected to mixed mode I/II
loading
90 M.R.M. Aliha et al. / Procedia Structural Integrity 41 (2022) 87–93
4 Aliha et al. / Structural Integrity Procedia 00 (2022) 000–000

Fig. 2. Four distinct permutations of the Bi-ASB specimen based on the adherent type considered for the Finite Element analyses.

Table. 1. Mechanical properties of the adhesive and adherents used to create the Bi-ASB specimen [6]
Material Properties Elastic modulus, E Poisson’s ratio, ν Fracture toughness,
(GPa) KIc (MPa √m)
Alumina 300 0.21 3.5
Aluminum alloy 70 0.33 29
Epoxy adhesive 2.84 0.35 1.5

The stress intensity factors KI and KII for the Bi-ASB specimen are functions of the crack length and the positions
of the bottom span supports described by S′ and S. These two fracture parameters can be expressed as below
equation for the Bi-ASB specimen:

3𝐹𝐹𝐹𝐹𝑆𝑆 ′ (1)
𝐾𝐾𝑖𝑖 = √𝜋𝜋𝜋𝜋 𝑌𝑌𝑖𝑖 ; 𝑖𝑖 = 𝐼𝐼. 𝐼𝐼𝐼𝐼
𝑡𝑡𝑊𝑊 2 (𝑆𝑆 + 𝑆𝑆 ′ )

where F is the applied load, t is thickness of the adherents, and Yi is the geometry factors of modes I and II.
Several finite element (FE) models of the Bi-ASB specimen were created to calculate the mode I and mode II
geometry factors (i.e. YI and YII) by using the ABAQUS commercial code.

3. Finite Element Analyses

Table 2 shows the overall dimensions of the Bi-ASB specimen utilized in the Finite Element (FE) modeling of
this research. Fig. 3 illustrates the FE model of the Bi-ASB specimen, built-in ABAQUS commercial code based on
the J-contour integral method. This model is composed of 7600 CPE8 (i.e. 8-node biquadratic plane strain
quadrilateral) elements. The produced FE models were subjected to a vertical load of F = 100 N, and the mechanical
properties of each part listed in Table 1 were assigned to the models. Very fine and singular type fine elements were
employed to simulate the crack tip region and construct the crack tip stress/strain singularity [8]. Also finer meshes
were used at the interface between adhesive and adherent due to high-stress concentration at this region, as shown in
Fig. 3.

Table. 2. Overall dimensions of the Bi-ASB specimen utilized in the finite element modeling.
Parameter L (mm) W (mm) t (mm) d (mm) a (mm) S (mm) S′ (mm)
value 54 15 5 0.4 7.5 37.8 In the range of
37.8 to 2.16.
 M.R.M. Aliha et al. / Procedia Structural Integrity 41 (2022) 87–93 91
Aliha et al. / Structural Integrity Procedia 00 (2022) 000–000 5

Fig. 3. FE model and boundary conditions of Bi-ASB specimen.

4. Results

The variations of fracture parameters including the stress intensity factors (K I and KII), T-stress (that is first non-
singular stress term ahead of crack tip), Biaxiality ratio (that is the ratio of singular terms over non-singular term),
and the direction of fracture initiation or kinking angle relative to the crack line) are investigated in this section for
the analyzed Bi-ASB specimens. Parts (a), (b), (c), (d), and (e) shown in Fig. 4 demonstrate how the adhering
material types considered in the left and right sides of the Bi-ASB specimen can affect YI, YII, non-dimensional T-
stress (T*), Biaxiality ratio (BR), and the direction of fracture initiation (θ0). T* and BR for the Bi-ASB specimen are
determined from Eqs. (2) and (3), respectively. The position of adhering materials on the left and right sides of the
adhesive affects the fracture parameters slightly, but Y I, YII, and T* are sensitive noticeably to the type of adherent
materials. Parts (a) and (b) of Fig. 4 show that YII equals to zero for symmetric span supports conditions (i.e. S′ = S),
and therefore the Bi-ASB specimen is subjected to pure mode I loading. When the movable support span distance
(S′ሻ becomes closer to the crack, mode II footprint appears in the Bi-ASB specimen, and YI finally equals to zero
when S′/L reaches a specific value of 0.04. In this condition, pure mode II condition occurs in the Bi-ASB specimen.
Part (c) of Fig. 4 shows the variations of T* for the investigated Bi-ASB specimen. It can be seen that the T-stress
is significantly negative for dominantly mode I conditions (i.e., higher S′/L values), whereas the corresponding
value of T* tends to zero as the value of S′/L decreases (i.e. where mode II becomes dominant). According to the
fracture mechanics literature for bonded components, if the magnitude of T-stress is large enough relative to the
singular terms, the non-singular term can play a substantial role in mixed mode I/II fracture [9-12].

𝑡𝑡𝑊𝑊 2 (𝑆𝑆 + 𝑆𝑆 ′ ) (2)

𝑇𝑇 ∗ = 𝑇𝑇
3𝐹𝐹𝐹𝐹𝑆𝑆 ′

In the process of mixed-mode fracture, the importance of the non-singular term relative to the singular terms (KI
and KII) is commonly characterized by a variable called Biaxiality ratio (BR) defined as:

𝑇𝑇 √𝜋𝜋𝜋𝜋 (3)
𝐵𝐵𝐵𝐵 =
√𝐾𝐾𝐼𝐼 2 + 𝐾𝐾𝐼𝐼𝐼𝐼 2
92 M.R.M. Aliha et al. / Procedia Structural Integrity 41 (2022) 87–93
6 Aliha et al. / Structural Integrity Procedia 00 (2022) 000–000

The variations of BR for the analyzed Bi-ASB specimen are shown in part (d) of Fig. 4. The value of BR is
notably negative under dominant mode I loading conditions (i.e. S′/L>0.2), indicating that the non-singular stress
component plays a considerable role in the process of dominantly pure mode I brittle fracture of the Bi-ASB
specimen. However, once the loading condition is changed towards pure mode II, the negative magnitude of
Biaxiality ratio decreases, indicating that T-stress has minimal influence on mode II fracture of this specimen.
Fig. 4e shows the variations of θ0 (defined schematically in Fig. 4e) for different mode mixities in the analyzed
Bi-ASB specimens. This parameter rises from zero (for the pure mode I condition) to roughly 70o (for the pure mode
II condition). For mode II dominated state, this parameter varies within a small range.
The plastic zone region around the crack tip of analyzed Bi-ASB specimen is also shown in Fig. 5 for a/W = 0.5,
S/L = 0.7, and three distinct mode mixities (i.e. pure mode I, mixed mode I+II loading, and pure mode II). The
plastic zone region is symmetric relative to the crack plane under pure mode I conditions, but its shape becomes
asymmetric with regard to the crack plane under mixed-mode I/II loading conditions.

Fig. 4. Variations of mode I geometry factor (YI), mode II geometry factor (YII), non-dimensional T-stress (T*), Biaxiality ratio (BR), and
fracture initiation angle (θ0) with different loading span ratios (S′/L) and adherent material types (a/W = 0.5, S/L = 0.7)

Fig. 5. Crack tip plastic zone of investigated Bi-ASB specimens with a/W = 0.5, S/L = 0.7, and F = 100 N for pure mode I, mixed-mode I+II,
and pure mode II loading conditions.
 M.R.M. Aliha et al. / Procedia Structural Integrity 41 (2022) 87–93 93
Aliha et al. / Structural Integrity Procedia 00 (2022) 000–000 7

5. Conclusions

In this research the ability of Bi-ASB test specimen to produce any desired deformation of mode I and mode II
was proved using several finite element analyses. Due to some advantages such as, simple geometry, modular
design of sample for easily producing different mode mixities, small size of sample and ease of test setup compared
to conventional test samples, the Bi-ASB sample can be proposed as suitable test candidate for fracture studies of
adhesively bonded joints under in-plane loading conditions. The presented fracture parameters (obtained using
several finite element analyses and for different geometrical, material and loading conditions) allow the researchers
to utilize this specimen for determining the load carrying capacity and analyzing the ABJs made of Bi-ASB test
specimen.

References

[1] da Silva, L. F. M., Öchsner, A., & Adams, R. D. (Eds.). (2011). Handbook of adhesion technology (Vol. 1, p. 1543). Heidelberg: Springer.
[2] Standard, I. S. O. Fibre-reinforced plastic composites–determination of mode I interlaminar fracture toughness. GIC, for uni-directionally
reinforced materials.
[3] ASTM, D. (2014). Standard test method for determination of the mode II interlaminar fracture toughness of unidirectional fiber-reinforced
polymer matrix composites. ASTM Standard.
[4] AC09349168, A. (Ed.). (2006). Standard test method for mixed mode I-mode II interlaminar fracture toughness of unidirectional fiber
reinforced polymer matrix composites. ASTM Internat.
[5] Nunes, F. A. A., & Campilho, R. D. S. G. (2018). Mixed-mode fracture analysis of adhesively-bonded joints using the ATDCB test specimen.
International Journal of Adhesion and Adhesives, 85, 58-68.
[6] Aliha, M. R. M., Kucheki, H. G., & Mirsayar, M. (2021). Mixed Mode I/II Fracture Analysis of Bi-Material Adhesive Bonded Joints Using a
Novel Short Beam Specimen. Applied Sciences, 11(11), 5232.
[7] Aliha, M. R. M., Kucheki, H. G., & Berto, F. (2022). Numerical analysis of crack initiation angles and propagation paths in adhesively
bonded joints under mixed mode I/II loading using a novel test specimen. Procedia Structural Integrity, 39, 393-402.
[8] Mousavi, A., Aliha, M. R. M., & Imani, D. M. (2020). Effects of biocompatible Nanofillers on mixed-mode I and II fracture toughness of
PMMA base dentures. Journal of the mechanical behavior of biomedical materials, 103, 103566.
[9] Mirsayar, M. M. (2019). T-strain effects in kinked interfacial fracture of bonded composites. Theoretical and Applied Fracture Mechanics,
104, 102381.
[10] Mirsayar, M. M., & Park, P. (2016). Modified maximum tangential stress criterion for fracture behavior of zirconia/veneer interfaces.
Journal of the mechanical behavior of biomedical materials, 59, 236-240.
[11] Mirsayar, M. M., & Park, P. (2015). The role of T-stress on kinking angle of interface cracks. Materials & Design, 80, 12-19.
[12] Mirsayar, M. M. (2014). On fracture of kinked interface cracks–The role of T-stress. Materials & Design, 61, 117-123.