MIXED MODE FRACTURE TOUGHNESS OF AN ADHESIVE BONDED JOINT
M. Alfano, F. Furgiuele, A. Leonardi, C. Maletta
Department of Mechanical Engineering
University of Calabria, Via Ponte P. Bucci, Cubo 44C
87036 Arcavacata di Rende (CS), Italy
ABSTRACT
Adhesive joints are continuously emerging as candidate to replace traditional fastening systems, such as riveting or welding, in
primary structural applications. A wide class of components belonging to electronic, automotive and aerospace industries
contains adhesive joints, however, the reliability during service constitutes a major technical problem as these contain flaws.
Therefore, their use in critical structural applications necessitates the developments of robust integrity assessment
methodologies and testing procedures. In the present paper, the interfacial fracture resistance of Al/Epoxy joints has been
studied using a Cracked Lap Shear (CLS) specimen loaded in four point bending. In particular, experimental results are
rationalized by means of finite element analyses (FEA) of the test specimen and some interesting features concerning
interfacial facture issues are elucidated. Interfacial fracture toughness is pointed out in terms of the critical strain energy
release rate and of the phase angle of the complex Stress Intensity Factor (cSIF). In addition, fracture surfaces are analyzed
and the locus of failure is discussed.
Introduction
Adhesive joints are a subject of growing interest in different fields (e.g. automotive, aerospace, biomedical, microelectronics,
etc) because of the advantages provided with respect to the traditional joining techniques. For example, they allow to reduce
the sources of stress concentration and to obtain a more uniform load distribution and, as a consequence, to extend the
fatigue life of the joint. Moreover, they allow weight reduction and, thus, major flexibility in design and reduced manufacturing
costs; in addition the risk of galvanic corrosion in joining dissimilar metals can be tackled using adhesives. For what about the
strength properties of adhesive bonded joints, these are most commonly evaluated by standard test methods like the butt joint
for rigid adherents or the peel test for non rigid ones; in all the standards, the geometries and the stresses likely to be
encountered in practical joint design are analyzed assuming a defect free bond-line [1]. However, real structures are flawed as
inaccurate joint fabrication or inappropriate curing may cause the presence of bubbles, dust particles or un-bonded areas in
the bondline. When subjected to external loads the thin adhesive layer behaves differently if compared to the adhesive as a
bulk material and the extension of these pre-existing flaws induces failure of the joints either within one of the constituents (i.e.
the sandwiched adhesive layer or the substrates) or along one of the interfaces. The measured macroscopic fracture
toughness is affected by the crack path followed through the joint. The main crack paths observed in literature are
schematically reported in Fig. 1: interfacial fracture (visually) at the adhesive substrate interface (a); cohesive fracture,
whereas the crack propagates in the adhesive layer (b) or in one of the substrates (c); serrated fracture with a crack advancing
from one interface to the other (d) [2].
Figure 1. Common failure path for a sandwiched joint: (a) interfacial, (b) cohesive, (c) substrate and (d) serrated fracture
The appearance of the aforementioned crack paths is dictated by several factors, e.g. specimen geometry, loading conditions,
the presence of residual stresses, substrates and interfacial resistance, etc [3-6]. However, interfacial crack growth is the most
frequently observed among them. Basically, there are two reasons for this: first, following to bonding procedures, defects are
likely to occur at the adhesive/substrate interface and, second, it has been experimentally observed that, even if a cohesive
crack exists in the adhesive layer, a mixed mode opening condition at crack tip directs failure toward the interface [7]. Joint
toughness is lower in value in such case, therefore a knowledge of the interfacial fracture resistance becomes of critical
concern. In the present paper, the interfacial fracture resistance of Al/Epoxy joints has been studied using a Cracked Lap
Shear (CLS) specimen loaded in four point bending. Some preliminary results concerning this test procedure have been
partially reported by the authors elsewhere [8]. However, in this paper those experimental data are rationalized by means of
finite element analyses of the test specimen and some interesting features concerning interfacial fracture issues are
elucidated.
Fracture toughness is pointed out in terms of the critical strain energy release rate, Gc, and of the phase angle of the complex
stress intensity factor (cSIF), ψc [9]. These last have been extracted from an extensive series of finite element analyses carried
out using a commercial finite element code.
Basic Concepts of Interfacial Fracture Mechanics
In this section the main concepts related to interfacial fracture mechanics are briefly summarized. Fig. 2 gives the generic
configuration of an interface crack: solid 1 is above the interface and solid 2 is below. Both the solids are assumed to be
homogeneous with linear elastic, isotropic behavior.
Figure 2: Conventions of an interface crack
It is well known that, for a crack lying at the interface separating two isotropic elastic materials, an oscillatory singularity exists
at crack tip. Let (r,φ) be the polar coordinate system centered at the crack tip. The near tip William asymptotic field for an
interface crack lying between two infinite, homogeneous and isotropic materials can be described, as r approaches the tip, by
the following equation
σ hk (r ,φ ) =
1
[Re(Kr iε )fhkI (φ ) + Im( Kr iε )fhkII (φ )]
2πr
(1)
where (h,k)=(r,φ), K is a complex quantity, namely the complex Stress Intensity Factor (cSIF), while ε is the bimaterial constant
(oscillation index) defined as:
ε=
1 ⎡ κ 1 µ 2 + µ1 ⎤
ln⎢
⎥
2π ⎣ κ 2 µ1 + µ 2 ⎦
(2)
and
: plane strain
⎧⎪ 3 − 4ν j
⎪⎩ (3 − ν j ) /(1 + ν j ) : plane stress
κj =⎨
( j = 1, 2)
I
II
µj is the shear modulus and νj is the Poisson ratio. The dimensionless angular functions fhk (φ) and fhk (φ) are given in [9] and
these can be scaled so that the tractions ahead of the crack are given by
( σ yy + iσ xy ) φ =0 =
Kr iε
2πr
while the crack faces displacement jump is given by
(3)
(δ y + iδ x ) φ = 0 =
where i =
8Kr iε
r
E (1 + 2iε ) cosh(πε ) 2π
(4)
*
− 1 while 2 / E * = 1 / E1 + 1 / E 2 and E j = 2µ j /(1 − ν j ) ( j = 1, 2) .
It should be noted that the term r iε = e iε ln r = cos(ε ln r ) + i sin(ε ln r ) implies stress oscillation and crack faces
interpenetration. Nevertheless, it has been shown [9] that the size of the interpenetration zone is very small, so this solution is
able to model adequately the stress field. Moreover, the elastic mismatch between the materials forming the interface leads to
a mode mixity at crack tip whatever the applied load is and, therefore, the modulus of K as well as its phase angle should be
given. However, as a consequence of the peculiar singularity the proportion of shear to normal stress varies with the distance
from the crack tip: is not possible to talk about fracture in the same sense as in fracture mechanics for homogeneous solids.
An unambiguous presentation of the phase angle could be obtained, as suggested by Rice [9], scaling the complex SIF to an
arbitrary length parameter l, i.e. K should be multiplied by l raised to the power iε so that
~
~
K = Kl iε = K eiψ = K I + iKII
note that
l iε = 1
and
Kl iε = K
(5)
1/2
. KI and KII have the usual dimensions (MPa m ) as in conventional Stress Intensity
Factor expressions for cracks in homogeneous materials. ψ is the phase angle at distance r and gives the ratio of normal to
shear stress ahead of the crack
⎡ K II ⎤
⎡ σ 12 ⎤
⎥ = arctan ⎢
⎥
K
⎣ I⎦
⎣ σ 22 ⎦ θ =0,r =l
(6)
ψ = arctan ⎢
It does not matter the value chosen for l so long one chooses a fixed length in reporting data. In addition, the modulus of the
cSIF is related to the energy release rate by means of the following IRWIN type equation
G=
K2
E cosh 2 (πε )
(7)
*
and thus for any bimaterial system an assessment of its fracture behavior requires either K or the energy release rate to be
specified together with the mode mix ψ.
Analytical solution for the energy release rate
The specimen adopted in this study, as shown in Fig. 3a), is a cracked lap shear loaded in four point bending.
Fig. 3. a) Schematic representation of the fracture specimen; b) details about pre-notch fabrication
Similarly to [10], a steady state energy release rate, Gss, independent of crack length, exists for this testing procedure, so the
crack length does not enter the calculations. In particular, for a crack located between the inner loading lines, the cracked
ligament is subjected to a constant bending moment M=Fd/2, with F being the applied load and d the moment arm. Therefore,
Gss can be obtained as the difference between the strain energy evaluated in the cracked and in the crack free portions of the
specimen. When the crack length/beam thickness ratio is large, the strain energy per unit area of crack extension, U, can be
2
expressed as M /2EIB (B is the specimen width) ignoring the shear contribution and thus using the Euler-Bernoulli beam
theory. Therefore, the strain energy release rate, Gss, can be expressed as the difference between the strain energy stored in
a unit section of composite beam from the strain energy stored in the uncracked beam
Gss = U cracked − Uuncracked =
M 2 ⎛ 1 1⎞ P 2 d 2
⎜ − ⎟=
2Es B ⎜⎝ Ic I ⎟⎠ 4B 2Es
⎡ 1 1⎤ P 2 dC P 2
=
ξ
⎢ − ⎥=
⎣ Ic I ⎦ 4B da 4B
(8)
where Ic is the second moment of area per unit width of the cracked section, I is the second moment of area per unit width of
2
the whole specimen, E s is equal to Es in plane stress and to Es/(1-νs ) in plane strain, while C=δ/P is the specimen
compliance. As a consequence, from Eq. 8 it follows
C = ξ a + C0
(9)
with C0 being the fictitious compliance at zero crack length; thus, assuming a constant displacement condition, it is possible to
write
G=
ξ δ2
4B C 2
=
ξ
δ2
(10)
4B (ξ a + C0 )2
In Fig. 4, Eq. 10 is plotted in order to show the behavior of the energy release rate as a function of the crack length for
constant displacement. In particular, a iso-fracture toughness model is assumed for the joint (R=cost.).
Fig. 4. Energy release rate versus the crack length
As it can be seen, a initial crack of length a0 (δ1) will propagate to a1 as the load decreases following to specimen compliance
increase. At this point G=R so that equilibrium is reached and crack growth is arrested. If the displacement is increased quickly
(δ2) the crack will growth until a new equilibrium position is reached again (a2) and so on. If the sample is displaced at a
constant rate it is possible to expect many displacements levels closely located to each other so that the crack extends at
constant G, i.e. G=R.
Numerical modeling
®
2D finite element models of the test specimen were developed using the FEA package ABAQUS [11]; it provides numerical
procedures for evaluating the J-integral and the components of the complex SIF using the interaction integral method and the
virtual crack extension/domain integral methods, respectively. Both interfacial (upper and lower) and cohesive (in the middle of
the layer) cracks were modeled. The initial notch a0 is 50 mm, specimen depth is B=15mm while the total length is 300mm.
Substrates thickness has been varied in the range (6÷8)mm and two adhesive bondline thicknesses have been investigated:
0.2 and 0.5mm, respectively.
The substrate material is aluminum (Es=70GPa and νs=0.33) and the adhesive is epoxy (Ea=1.3GPa and νa=0.35). The
calculated oscillation index in plane strain condition ε is ±0.072, where the sign plus is for the lower interface while minus for
the upper one. About 70000 eight node quadrilateral plane strain elements (CPE8) were used for the whole domain. Plane
strain condition at the crack tip was considered since the adhesive layer presents a small thickness with respect to the
substrate; moreover the great mismatch in the elastic moduli of the materials forming the interface ensures a great constraint
to the transverse deformation. The adhesive layer with thickness 0.2 mm has been modeled using 16 layers of element while
30 layers have been used for 0.5 mm. It is worth noting that at macroscopic level, the toughness of the joint could be
∞
∞
expressed using the remote Stress Intensity Factors KI and KII [12]. These are determined neglecting the presence of the
adhesive layer and they represent a convenient choice provided the layer thickness, h, is small compared with all the other
specimen dimensions and crack length. However, in the present study, h is not negligible compared to the other relevant
geometrical dimensions and as a consequence the adhesive layer has been modeled as previously said.
The energy release rate and the phase angle of the calculated complex SIF have been evaluated. The former has been
compared with the values obtained analytically using equation (8). For an interface crack in a sandwiched layer the
characteristic length can be chosen as the thickness of the adhesive layer. In particular, in the present paper, the adhesive
bond line thickness, ha, was chosen as characteristic length, l=ha.
Finite element analyses have been carried out for different values of the normalized crack length a/d, (crack length/moment
arm). It has been found that G assumes a constant value for a crack length equal to a≈4d, and this values are in excellent
agreement with the values obtained using Eq. 8, i.e. the difference is less than 1%. In a similar manner the phase ψ assumes
a steady state behavior for values of crack length equal to a≈3d. This steady state values of the phase angle are reported in
Tab. 1. In particular, only the data for hs=6 mm have been reported herein as the substrate thickness has a negligible influence
on the values of the phase angle of the cSIF. However, as it can be seen from Tab. 1, a small variation of the phase angle has
been observed in the range of adhesive layer thickness investigated in the paper. In addition, a great mismatch between the
upper and the lower interface phase angle exists, thus proving that the adhesive layer needs to be modeled into the FE
analysis, as previously said. Obviously, the phase angle reported in Tab. 1 in the case of a cohesive crack is referred to the
SIF of an homogeneous material.
Table 1: FE values of the phase angle
ψ (grad)
0.2 mm 0.5 mm
b
-54±
-55±
Upper interface
a
-42±
-41±
Cohesive
b
Lower interface
-27±
-29±
a
b
in the middle of the adhesive layer
l=ha
Fig. 5. Direction of crack propagation as a function of
the adimensional crack depth, c/ha
Moreover, the direction of crack propagation, θ, of a cohesive crack located in the adhesive layer has been evaluated using
the maximum tangential stress criterion [13]. Both the adhesive layer thicknesses, ha=0.2 and 0.5 mm, have been investigated
and the results obtained for different values of crack depth, c/ha, where c is the vertical distance of the crack tip from the lower
interface, are reported in Fig. 5. As it can be seen, if a crack is present into the adhesive layer, it should be directed, under
these loading conditions, toward the upper interface. Furthermore, a negligible influence of the adhesive layer thickness on the
values of θ has been observed.
Materials and test procedures
The test specimen used in this study is a Cracked Lap Shear (CLS) whose dimensions are shown in Fig. 3a) (all the
dimensions were measured with an accuracy of ± 0.02 mm). The substrates are aluminium 6061-T6 alloys while the resin is
Loctite Hysol® 3422 A&B (Henkel-Germany), a two component, medium viscosity and fast curing industrial grade epoxy
adhesive. Each material is assumed isotropic and linear elastic; the material properties of the aluminium substrates are
estimated to be: Young’s modulus Es=70GPa and Poisson’s ratio νs=0.33, while those of the adhesive are Ea=1.3GPa, νa=0.35
respectively. Since adhesion strength is sensitive to the surface conditions of the materials to be joined, substrates were
treated with an alumina grit blast and degreased with acetone in order to remove waste particles and to prevent the formation
of any weak surface boundary layer that could lead to a premature joint failure. In addition, an enhancement of the intimate
molecular contact between the adhesive and the substrate during bonding is generated. Moisture and atmospheric
contamination were avoided by bonding the substrates immediately after surfaces pre-treatments. An initial pre-crack (a0=25
®
mm) was introduced simply by placing a thin Teflon tape at the Al/Epoxy interface as shown in Fig. 3b); in such a way it is
possible to simulate those cracks that commonly occur during service condition. As the epoxy resin do not bond with the
Teflon tape, the tape area can be taken as an interfacial crack. A bond line thickness equal to 0.2 mm was obtained by placing
metallic wires as spacers at each end of the joint. Moreover, bubbles within adhesive are driven out by shearing and
establishing bonding from one edge. All the specimens were cured at room temperature (25°C) for about 24h, we are thus
able to exclude the insurgence of residual stresses. Tests were performed at room temperature using an Instron 8500 plus
universal test machine. Specimens were pin loaded and tested under displacement control at a constant cross-head feed rate
of 0.06 mm/min; the surfaces of the loading pin were lubricated to reduce frictional effects. In addition, the upper loading frame
is allowed to pivot about the centerline in order to compensate the loading asymmetry that appears as the crack advances and
that can be attributed mainly to the asymmetric configuration of the test specimen.
Results and discussion
Tests have shown a stick-slip fracture characterized by cycles of fast unstable crack propagation and arrest. Fracture in itself
was interfacial, in particular, the upper interface decohered, leaving an unbroken epoxy layer on the aluminum substrate and
confirming the general finding that an adhesive bonded joint subject to mixed mode load fails at the adhesive-substrate
interface. Crack growth jumps and periodic load fluctuations do affect the resulting load versus crosshead displacement curve
that presents a typical saw-tooth shape (Fig. 6). Besides, the F-δ curve does not deviate from the linearity prior to fracture
indicating that non linear phenomena occur on a scale much smaller than specimen dimensions. Average values of the post
1/2
peak loads were taken as critical loads, Fc, for calculating the fracture toughness of the joint: Kc=(0.267 ± 3%) MPa m
(Gc=26.54 N/mm) and ψ=54º. This low value is usual for epoxy resin tested at room temperature [14].
Fig. 6. Typical load displacement curves
A detailed examination (e.g., SEM analyses) of the fracture surfaces has not been performed in this preliminary work,
however, after failure, visual inspections and optical analyses of the fracture surfaces have been carried out in order to infer
some conclusions about the locus of failure. In Fig. 7 a schematic representation of the fracture surfaces and of the crack path
followed during crack growth has been reported.
Fig. 7: Schematic representation of fracture surfaces and of the crack path into the adhesive layer.
As it can be seen the initial crack, located on the lower interface, abruptly kinks onto the upper interface and it remains there
showing a stick-slick behavior. Marks associated to crack jump/arrest events are clearly visible. It is worth noting that after
each tests substrates have been completely separated using a wedge (i.e. in mode I loading condition): interfacial to cohesive
(in layer) failure transition has been observed (see Fig. 7). This last feature is not surprising as the crack path is dictated by the
mode mixity of the applied loads. Besides, it could be considered as a proof of the good quality bond achieved during
fabrication, indeed, if the interface is poorly bonded the crack will stay on the interface regardless of the mixed mode of the
applied external load. Unless the aforementioned characteristics, the fracture surfaces observed herein are essentially
featureless, in particular their smoothness suggests the absence of toughening mechanisms and that a brittle fracture
occurred. This, in turn, explains the low values of the fracture toughness observed.
Conclusions and perspectives
In the present paper the interfacial fracture toughness of an Al/Epoxy adhesive system with a crack lying at the interface has
been evaluated by means of a cracked lap shear loaded in four point bending. An extensive series of finite element analyses,
carried out using a commercial finite element code, has shown that the energy release rate is independent of crack length for
this testing procedure and has allowed for some interesting features concerning interfacial facture issues to be elucidated.
Interfacial to cohesive (in layer) failure transition has been observed varying the loading condition: the crack is directed toward
the upper interface when loaded in mixed mode while it kinks from the upper interface into the adhesive layer when a wedge is
inserted causing mode I loading conditions. This is in agreement with what is reported in literature, i.e. crack path is dictated by
the mode mixity of the applied loads. Anyhow, the results obtained have shown a reduced scatter, so it is possible to state that
the experimental procedures (i.e., specimens preparation and testing) are quite reliable. Future work will be focused on the
analysis of the adhesion strength for different values of the mode mixity; this can be simply obtained varying the specimen
arms thicknesses.
Acknowledgments
The authors wish to thank Dr. Piero Mauri (Loctite-Henkel, Brugherio, Italy) for providing the epoxy resin.
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