COHESIVE LAWS OF DUCTILE ADHESIVES:
AN EXPERIMENAL STUDY
A. Nøkkentved * , P. Brøndsted * and O.T. Thomsen **
Risø National Laboratory, Materials Research Department, Denmark
Department of Mechanical Engineering, Aalborg University, Denmark
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Abstract
Cohesive laws for two different ductile epoxy polymers were investigated. Epoxy polymer joints with different mechanical
properties and geometries are considered in order to study the effect of plasticity within the bond of ductile adhesives,
since this can enhance their fracture toughness. The J-integral can be established by a general approach and
experimental measurements. Cohesive laws are obtained for pure mode I fracture of double cantilever beam (DCB)
specimens. A pre-crack made by a thin teflon film is imbedded in all the specimens during manufacturing. During the
experiments the load and the end-opening is measured. Retrieving the cohesive law is done by differentiating J in respect
to the opening of the pre-crack tip. Results for 15 specimens are presented and discussed.
Introduction
The evolution of adhesive bonding technology and its current knowledge base was made possible by the rapid growth of
adhesive applications in a great variety of industries in the past few decades. Adhesive joints are found today in the
electronic, automobile, aerospace, wind turbine and shipbuilding industries. While the description of their elastic properties
required for the prediction of deformation is well established, predicting failure evolution is more difficult, thus creating a
great need for development of tools that predict failure evolution. Such failure analysis tools must excel in predicting
initiation, size and propagation of the crack within the adhesive layer, while taking into consideration the bonds
asymmetry, anisotropy and thickness variations. Such analysis can be achieved by the use of the cohesive law approach.
The fracture process zone is characterised by a traction-separation law, a cohesive law, Sørensen [1]. The behaviour of
“ductile” fracture and strength prediction requires modelling of the failure process zone. This can be done with fracture
mechanics and a study of the local crack opening and propagation. The cohesive law establish a relationship between, σ,
the stress across the failure process zone and its stretch, δ, during a monotonic increase in δ, Sørensen and Jacobsen [2].
The cohesive law can be taken as a basic fracture property and this offers the advantage that its shape can be predicted
by micromechanical models, [2] and Bao and Suo [3]. The cohesive laws are determined experimentally. This method is
used in this work in order to derive the cohesive law from measurements taken on the J-integral and the end-opening. In
order to determine the J-integral directly from the specimen’s geometry and load level, a stable crack growth is needed so
that the entire σ(δ)−δ curve can be determined. The double cantilever beam (DSB) loaded with pure bending moments
fulfils these demands, Suo, Bao, Fan and Wang [4]. The DCB specimen used is shown in Fig. 1, and the test set-up used
is shown in Fig. 2.
Figure 1. Drawing of the DCB specimen geometry.
The objective of the project is to establish and demonstrate methods for modelling the mechanical behaviour of adhesively
bonded joints in polymer matrix composite materials. Fracture mechanics will assist to establish the relevant framework for
their mechanical performance. The main results from this study are expected to be part in a set of recommendations
regarding production parameters, design, inspection and control of adhesively bonded joints.
Figure 2. The DCB setup for mode I testing.
Analysis
The entire adhesive layer is modelled by the cohesive law. The J-integral just outside of the failure zone gives, Rice [5]:
δ*
J = ∫ σ (δ )dδ
(1)
0
where δ* is the end-opening of the cohesive zone. The J-integral reaches a steady-state level, J ss , when δ* is equal to δ c ,
which is the opening at which the cohesive stress vanishes. In Eq. (1) the whole entire failure zone is described by one
cohesive law. The J-integral is path-independent, meaning that it will attain the same value as Eq. (1) when calculated
along the external boundaries.
B
B
B
B
The cohesive law is found by differentiation of Eq. (1) with respect to δ* [4]:
σ (δ * ) =
∂J
∂δ *
(2)
By measuring the moment and the end-opening simultaneously we can obtain the cohesive law, Fig.2. Calculation of the
J-integral along the boundary of the DCB specimen when loaded by pure bending moments, as shown in Fig. 2, gives [4]:
M2
J = 12(1 − v ) 2 3
B H E
2
(3)
where M is the applied moment, Fig. 2, E and v are the Young’s modulus and Poisson’s ratio of the adherents; B and H
are the dimensions of the adherents and t the adhesive layer thickness. Notice that the adhesive layer is not represented
in the J-integral formulation above. The contribution from the adhesive layer is neglected, since the layer stiffness and
thickness t are small in comparison to those of the adherents.
Experimental procedure
Number of
specimens
The cohesive laws are determined by measuring the J-integral and end-opening of the cohesive zone of a double
cantilever beam which is subjected to pure mode I loading. For each adhesive system, nine DCB specimens with three
different bond thicknesses where produced.
Thickness (mm)
1
4
8
M300 Plexus
3
3
3
M420 Plexus
3
3
3
Table 1. DCB specimens prepared.
Preparation of test specimens. Two adhesives were tested. They were an epoxy M420 Plexus having a Youngs modulus
E = 1800MPa and a tensile-elongation = 100% and an epoxy M300 Plexus with modulus E = 2900MPa and a tensileelongation = 25%. The thicknesses of the adhesive layer were: t 1 =1mm, t 2 =4mm and t 3 =8mm, see Tab. 1. The DCB
beams were 500 mm steel beams (UBH 11, UDDEHOLM) with nominal dimensions: H = 8.6 mm, B = 33mm. The beams
were machined by milling and the surfaces treated by sandblasting. Before bonding the adherents, they were degreased
with acetone. Inserts at the ends of the adherents assured the correct adhesive thickness along the joint. The adhesive
paste was poured between the inserts. A central pre-crack or notch was created at the one end of the adhesive joint, by
inserting a 0.05mm thick film at the centre plane of the adhesive. The adherents are held in place with tape and put in a
fixture while the adhesive was cured. The curing process was done in room temperature (23 o ) and for the prescribed time
span of each adhesive system. After the curing process has been complete metal pins were inserted in the neutral axis of
the adherents. These pins were used for mounting extensometers and they are located at the end of the adhesive layer,
Fig. 3. They are used to measure the end-opening of the cohesive zone.
U
U
B
B
B
B
B
B
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Double Cantilever Beam testing. A test rig which can produce the full range of modes from mode I to mode II developed at
Risø National Laboratory was used for testing the DCB specimens. Bending moments are applied to the specimen
through an arrangement consisting of wires and transverse beams, Fig. 2. The loading rate was 0.15mm/s and each
experiment lasted on average for 6 minutes.
U
U
Results
Cohesive law. To obtain the cohesive law the fracture resistance given by the J-integral is calculated. The J-integral is
calculated from the applied moments and the end-opening, which are measured during testing, as shown in Eq (3). The
measured data points are fitted to an exponential associate function estimated by numerical non-linear regression, Fig. 3:
U
U
-
x
t1
-
x
t2
y = y 0 + A1 (1 - e ) + A 2 (1 - e )
Figure 3. J-δ* curve fitted with exponential functions are illustrated for specimen nr.1 of the M300 Plexus adhesive –
thickness t=8mm.
(4)
In the DCB-specimens, separation occurred by cohesion, i.e. failure inside the adhesive layer. Specimens were failure
occurred by decohesion, i.e. along the interface (between steel and adhesive), were discarded since only pure mode I
results are desired. Of the 18 specimens, 3 were discarded based on the argument above. For these experiments the
recorded value δ* represents the entire deformation of the adhesive layer. The J value increases with increasing endopening, until a steady state value J ss is reached. After the J has attained J ss , the experiments where stopped. The overall
J-δ* curves are fairly close but some variations of the J ss values are seen, Fig. 4. This variation might be caused by local
deviation in local bond strength or material non-linearity, but it is uncertain.
B
B
B
B
B
B
B
B
Figure 4. Measured J-δ* curves for M300 Plexus (left) and M420 Plexus (right). The fitted measured data are represented
by dashed lines. The full lines represent an average fitting.
By numerical differentiation, the cohesive laws are calculated from the average J-δ* curves, Fig.4, according to Eq. (2).
The results are shown in Fig. 5. The effect of thickness is clear. The maximum cohesive stress is about 8.6 MPa higher for
thin (t 1 ) M300 Plexus specimens compared to the thick specimens t 3 . For M420 Plexus the difference is 1.5 MPa
respectively. Fig.6 shows the maximum cohesive stresses as function of the adhesive layer thickness, t, for both adhesive
systems, derived from the average cohesive laws, Fig 5.
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B
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Figure 5. Cohesive laws derived for Steel/M300 Plexus and Steel/M420 Plexus.
Likewise, δ*, increases for M300 Plexus adhesive from about 1.25 mm for the thin specimens (t 1 & t 2 ) to almost 2 mm for
the thick specimens. For the M420 Plexus adhesive the same δ* behaviour is observed. δ* increase’s from 4.3 mm for thin
specimens (t 1 & t 2 ) to 5.7mm for the thick (t 3 ) specimens.
The shape of the cohesive laws is seen to be very different for the various adhesive thicknesses. For the M300 Plexus
adhesives thinner specimens (t 1 & t 2 ) the cohesive law increases dramatically reaching high cohesive stresses at rather
small openings (δ* = 0.05215 mm), and decreases almost linearly with increase of the end-opening. For specimens with
thickness t 3 , the cohesive stress rises to a maximum very quickly (δ* = 0.05717 mm) but stays at this stress level until δ* is
about 0.45 mm, and then decreases slowly with increase of the end-opening. Same cohesive stress behaviour is observed
for the M420 Plexus adhesive. The specimens with t 1 thickness, reach maximum cohesive stress with a δ* = 0.15 mm and
decrease slowly with increase of the end-opening. The specimens the t 2 and t 3 reach maximum cohesive stress with a δ*
= 0.6 mm and also decrease very slowly with increase of the end-opening. Note that all M420 Plexus specimens stay in a
high stress level for about 1 mm.
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B
B
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Figure 6. The maximum cohesive stresses, vs. bond thickness for Steel/M300 Plexus and Steel/M420 Plexus, derived
from the average cohesive laws.
Discussion
The cohesive laws for the two different ductile adhesives are evaluated op to steady state energy take-up. The “trend” of
the cohesive laws where found to be quite different for the two adhesive systems and for their different thicknesses. A
dramatic decrease of the maximum cohesive stresses was shown with the increase of specimen thickness for the M300
Plexus adhesive. It is well known within ductile metals, that, the fracture resistance depends on specimen thickness, Broek
[7], due to difference in the crack tip plastic zone size. However, the M420 Plexus adhesive’s maximum cohesive stresses
seem to have little dependency on specimen thickness, suggesting that the M300 Plexus adhesive might be notch
sensitive. As mentioned earlier the measured δ* represents the entire deformation of the adhesive layer. Sørensen [1]
states that as long as since the adhesive possesses reversible stress-strain behaviour the only contribution to the J
integral comes from the cohesive zone itself.
An increase of J ss with larger adhesive thicknesses is also observed for both M300- and M420 Plexus adhesives, see Fig.
4. This suggests that for those specimens an additional toughening happens due to plasticity, Tvergaard and Hutchinson
[6].
B
B
References
1.
2.
3.
4.
5.
6.
7.
Sørensen, B.F.,“Cohesive law and notch sensitivity of adhesive joints”, Acta Materialia, 50 1053-1061 (2002).
Sørensen, B.F., and Jacobsen, T.K., “Large-scale bridging in composites: R-curves and bridging laws”, Composites
Part A, 29A, 1443-1451 (1998).
Bao, G. and Suo, Z., “Remarks on crack-bridging concepts”, Appl. Mech. Rev., 45, 355-361 (1992)..
Suo, Z., Bao, G., Fan, B. and Wang, T.C., “Orthotropy rescaling and implications for fracture in composites”,
International Journal of Solids and Structures, 28(2), 235-248 (1991).
Rice, J.R., Elastic Fracture Mechanics Concepts for Interfacial Cracks”, American Society of Mechanical Engineers,
Paper No. 88-WA/APM-13, 1988.
Tvergaard, V., Hutchinson, J.W., “Toughness of an interface along a thin ductile layer joining elastic solids”,
Philosophical Magazine A, 70, 641-656 (1994).
Broek, D. “Elementary engineering fracture mechanics”, 4 th ed., Dordrecht and Boston: Mertinus Nijhoff Publishers ;
(1986)
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