Modelling the Toughness of Rubber-Toughened Epoxies:
A Critical Distance Criterion
F.J. Guild, D. Alvarez, A.J. Kinloch and A.C. Taylor
Imperial College London
London SW7 2AZ, UK
F.Guild@imperial.ac.uk
Introduction
The high values of toughness for rubbertoughened epoxies and other filled adhesives arise from
many different failure mechanisms occurring during crack
growth in these materials which all combine to absorb energy and increase toughness. Modelling the toughness of
such materials must attempt to include all the mechanisms
involved. We have now developed a model for toughness
which has been successfully applied to two different rubber-filled epoxy adhesives.
The model includes the
growth of the crack radius which has been observed experimentally, followed by simulations of the crack at the point
of fracture using a modified J-integral approach. The energy arising from cavitation of the particles around the
crack is taken into account.
b)
Finite Element Models
Two sets of finite element models are required
for this work. The first type of model is on the microstructural scale including individual rubber particles or voids.
These are 3-dimensional models, drawn for a unit cell assuming all the particles are equidistant from each other.
The material properties used for the epoxy matrix and the
rubber particles were gained from experimental results.
These models were used to derive the cavitation criterion
[1] and to find the values of plastic dissipation as a function of strain level. Figure 1a shows the models used to
a)
Figure 1. Finite Element Models for Microstructural
Analyses. a) Simulation of Cavitation b) Analysis of Energy Dissipation.
derive the cavitation criterion. The model used to derive
the values of strain energy is shown in Figure 1b; this
model required rigid surfaces attached to the planes loaded
in compression to prevent twisting of the model. It was
carefully checked that these planes did not cause extra
energy contribution.
The second type of model is on the mesostructural scale including a crack. These models use the homogenized material properties of the voided epoxy since
the cavitation criterion demonstrates that material around
the crack would have undergone cavitation of the rubber
particles. These are 2-dimensional models in plane strain.
The finite element package used was ABAQUS
version 6.9. Most of the models were directly run in Implicit code. However, the geometry of small cracks, needed for the simulation of the pure epoxy resins, did not allow direct convergence in Implicit code. These were initially run in Explicit code, and then converted to Implicit
code to allow deduction of J-integral values.
The Toughness Model
The toughness is postulated to arise from two
distinct fracture phases, which have been observed experimentally. They are:
1. Growth of the crack radius
2. Extension of the crack
The energy arising from the growth of the crack
radius is calculated assuming that the crack starts at the
crack radius found for the pure resin and grows to the
crack radius for the voided resin. For such crack radius
growth there are corresponding increases in both the plastic zone size and the cavitation zone size. The energy arising from plastic zone growth is calculated from the difference in total plastic energy for the two crack sizes, which
is found from the two finite element simulations. Note that
this calculation ignores the extra energy arising from the
additional microstructural fracture mechanisms but since
this is a difference calculation the error is expected to be
negligible. The energy arising from the cavitation zone
growth is calculated from the two finite element simulations to find the change in size of the cavitation zone. The
energy associated with this change in size is found from
separate microstructural simulations of cavitation and the
cavitation criterion [1].
The energy arising from the extension of the
crack is calculated using a modified J-integral approach
which calculates the energy required to extend the crack.
Our modified J-integral approach allows the extra energy
arising from the additional microstructural fracture mechanisms to be taken into account. The finite element model is
the stationary crack at the point of extension in the voided
epoxy. The formulation of this model requires a failure
criterion. The deduction of this failure criterion can only
be made using a critical distance approach as described in
the next section.
Figure 2. Contours of Strain around a Crack.
The variation of strain ahead of the crack, along the profile
shown by the red line in Figure 1, has been extracted. The
profiles have been extracted for different values of initial
unloaded crack radius, all loaded to the value of fracture
toughness of the epoxy resin. The profiles are compared in
Figure 3; the profiles are for initial values of crack radius
between 1 and 3 µm. These results in Figure 2 show the
expected variation of maximum strain at the crack tip; the
value of strain is higher for the smaller initial crack sizes.
However, close to the crack tip the values appear to converge to a constant value. This convergence has been studied carefully using curve fitting. Convergence is found at
strain of 0.26 at a distance of 2.73 µm ahead of the crack.
We propose that this strain value should be used as the
failure criterion for the voided epoxy.
1.8
Max. True
Strain
1.6
1.4
1
1.2
1.25
1.5
1
2
0.8
Failure Criterion
2.5
0.6
3
0.4
We propose that the criterion for the extension of the
crack in the voided epoxy must be the failure strain of the
pure epoxy. This failure strain must be derived from the
known fracture toughness of the epoxy. From finite element simulations, the failure strain at the crack tip can be
found for a given fracture toughness, using the J-integral
results from the simulations. However, the value of the
strain at the crack tip is dependent on the initial radius of
the crack tip of the mesh before loading. Thus a failure
criterion based on the strain at the crack tip cannot be deduced.
Further simulations have been carried for different
values of initial crack tip radius. An example of a simulation, showing only the detailed mesh around the crack, is
shown in Figure 2. The initial crack radius for this simulation was 1.5 µm, and the contours are for maximum logarithmic strain. The range of the contours is between zero
and 1.2 logarithmic strain.
0.2
0
0
0.002
0.004
0.006
0.008
0.01
Distance from crack tip (mm)
0.012
Figure 3. Profiles of Strain ahead of Cracks, of different
initial radius, loaded to the same G value.
Simulations of cracks in the voided epoxy have been
carried out to find the deformed crack size when the strain
ahead of the crack is equal to 0.26. Simulations with different values of initial crack radius have been loaded to a
chosen G value, and the strain at the convergence of the
profiles ahead of the crack has been observed. Iteration
was required to gain the required strain value.
Critical Distance
The theory of critical distances is fully presented by
Taylor [2]. For a linear elastic material, the value of critical distance, L/2, is shown to be a function of the critical
fracture toughness Kc and critical stress, σc via:
𝐿
1
𝐾 2
= 2𝜋 [𝜎𝑐]
2
Table 1. Summary of Contributions to Fracture Energy
(KJm-2).
Plastic zone
Cavitation
J-integral Total
Cure
growth
zone growth
160oC
2.3
0.82
3.1
6.2
120oC
0.45
0.092
0.37
0.91
Conclusions
(1)
𝑐
The results shown in Figure 2 are for critical fracture energy, Gc, of 0.46 KJm-2. The critical stress related to the
critical strain of 0.26 has been found from the material
model. Using the relationship between critical fracture
energy and critical fracture toughness and the Young’s
modulus, E:
𝐾𝑐2 = 𝐸 𝐺𝑐
(2)
The calculated value of critical distance is 34.2 µm, an
order of magnitude higher than the value of 2.73 µm found
from the finite element analyses. We postulate that this
difference may arise from the elastic-plastic material model used for the finite element analyses, which leads to localization of the strain field around the crack tip as shown
in Figures 2 and 3.
Results
This predictive model for toughness has been applied
to two different rubber-toughened epoxies. These are
‘model’ materials, prepared in the laboratory, containing
20% volume fraction of rubber particles. The first material, which is cured at 160oC, has mean rubber particle radius of 1.7 µm. The fracture energy of the pure epoxy is 0.46
KJ m-2, which is increased to 5.9 KJ m-2 by the presence of
the rubber particles. The second material, which is cured
at 120oC, has mean rubber particle radius of 1.4 µm. The
fracture energy of the pure epoxy is 0.21 KJ m-2, which is
increased to 2.23 KJ m-2 by the presence of the rubber particles [3].
The predicted contributions to the fracture energy for
the two materials are summarized in Table 1. Comparing
the values of total fracture energy, the value for the tougher material is close to the measured value of 5.9 KJ m-2.
However, the predicted value for the less tough material
cured at 120oC is only around half the measured value.
The contributions from the cavitation zone growth and the
J-integral value appear small compared to the tougher material. These differences are currently under further investigation.
Toughness is a vital property for adhesives and the
prediction of toughness is an important step to allow further developments of tough adhesives. We have developed
a model which includes the different contributions to the
total fracture energy. The importance of these different
contributions can be assessed; further sources of energy for
more complex materials could be added. We have applied
this model to two different rubber-toughened epoxies and
gained good agreement for the tougher adhesive and an
underestimate of toughness for the less tough adhesive.
This is believed to arise from an under-prediction of the
contribution from the J-integral; this is under current investigation. The model has been able to capture the reduction in toughness arising from the different cure conditions.
Acknowledgements
This work is an extension of a past project which
was jointly funded by the Engineering and Physical Sciences Research Council, UK, and the Defence Science and
Technology Laboratory, UK.
References
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2.
3.
F.J. Guild, A.J. Kinloch and A.C. Taylor, J. Mater.
Sci.. 2010, 45, pp 3882-3894.
D. Taylor, The Theory of Critical Distances, 2007,
Elsevier Ltd.
C.A. Finch, S. Hashemi and A.J. Kinloch Polymer
Communications 28 1987 p. 322.