1
Finite element modelling of cohesive crack propagation using
crack-bridging rupture elements
S Hadidi-Moud 1, A D Crocombe 2 and G Richardson 3
Mechanical Engineering Department, Ferdowsi University, Mashhad, Iran
Hadidi@Ferdowsi.um.ac.ir
Abstract
Due to the extensive use of adhesive materials in structural bonding on one hand and the lack of a unified design tool for
failure prediction in such structures on the other hand, development of failure prediction techniques has received much
interest in recent years. A series of rupture elements have been developed and used to representing the process of failure
initiation and propagation in elastic and elasto-plastic continua. The rupture elements form a line of crack bridging elements
along the assumed crack extension path and are integrated into a 2-D finite element model. The user-defined elements are
incorporated into the analysis using subroutine UEL called by ABAQUS finite element commercial code.
Pre-cracked compact tension, C(T), specimens of bulk adhesive E27 and double cantilever beam, DCB, specimens with
Aluminium substrates and a 1mm thick adhesive E27 binding layer have been used in an extensive fracture testing
programme and the results are used to validate the predictive models. The proposed models are then discussed in the
framework of conventional fracture mechanics and compared with the well-established linear elastic fracture mechanics
(LEFM) and elasto-plastic or “non-linear” fracture mechanics (NLFM) procedures.
Keywords: cohesive crack propagation; rupture modelling; Finite element; Adhesive materials
Introduction
Force controlled interface elements [1,2] have been
used to model crack propagation in an elastic
continuum. Direct failure prediction in an elastic
continuum, following crack initiation and its
subsequent propagation, has been achieved by use of
strain tripped elements with energy based unloading
[3 - 5]. Related work on energy based modelling of
crack propagation [6] and the use of crack bridging
displacement controlled elements in modelling delamination in composites [7] can be found in the
literature. Development of a local damage based
approach to modelling crack propagation and
predicting the corresponding failure load has been
discussed in this work. The suggested model has
been assessed through the comparison of its
prediction with the experimental results from this
work and previous tests carried out on pre-cracked
compact tension specimens made from an epoxy
adhesive.
First the experimental programme carried out to
provide data for assessment of the predictive models
is briefly explained. Second, different rupture
elements developed to model progressive crack
extension are introduced together with their
limitation of application. The user elements are then
integrated into the finite element model of fracture
specimens to predict the conditions of failure.
Details of finite element simulations are briefly
described and the results of the analyses are
demonstrated. The model predictions are compared
1
with the experimental data. Finally the strengths and
weaknesses of the proposed models are discussed in
relation to their use in predicting failure initiation
and /or propagation in bulk adhesives and adhesively
bonded structures.
The experimental programme
The experimental work includes manufacturing and
testing of bulk epoxy adhesive flat tensile, F(T)
compact tension, C(T) and double cantilever beam,
DCB joint specimens. The results of flat tensile tests
are used to characterise the non-linear constitutive
response of the adhesive continua. Fracture data,
obtained from C(T) and DCB tests together with
results from previous research, where cleavage
fracture specimens were tested, are used in the
assessment of the proposed rupture models.
Development of rupture elements
The load-displacement behaviour of the rupture
elements is shown in Figure 1. A range of user
elements are developed and used to predict the
conditions of failure initiation and/ or propagation.
The rupture elements are initially rigid-like and
represent connectivity along the crack line. The
system is then loaded in a quasi-static manner. Once
the condition of the continuum is reached a critical
state (i.e. strain reaches a specified value
representing a level of continuum plasticity) the
rupture element is allowed to loose rigidity (tripped)
and extend freely while the load acting on it is
dropping to zero representing the release state. A
Assistant Professor, Ferdowsi University of Mashhd, Iran / Dr.
Head of Solid Mechanics and Design, School of MME., University of Surrey, UK / Prof.
3 Director of Mechanical Design, Surrey Satellite Technology, UK / Dr.
2
2
convergence algorithm is used to prevent analysis
instability at the turning points of the stiffness of the
rupture elements (i.e. tripping and release stage).
The line of elements along the crack extension path
are tripped and released subsequently throughout
this process in a sequential order (progressive
propagation). The applied load to the system at this
stage is referred to as the failure load. Various
unloading schemes shown in figure 1 have been
considered and discussed. Potentially the rupture
elements used to predict the failure load of a precracked component are also applicable to a noncracked model and similar analyses can be used to
predict the onset (initiation) of fracture. Three types
of rupture elements are briefly described.

Strain tripped rupture element with no
unloading (type 1)
This rupture element follows the load-displacement
scheme of figure 1(a). The element’s stiffness is set
to zero when the critical continuum strain is reached.
The rupture element will then continue to extend at a
constant load, corresponding to the tripping
condition. This represents crack opening but does
not include unloading and release states. Using a
series of rupture elements of this type along the
crack line of a CT specimen model, an alternative
solution to the conventional LEFM has been
achieved. The area under the curve (ER) represents
the energy absorbed by these elements. The element
definition is illustrated in figure 2.

Strain tripped rupture element with energy
based unloading (type 2)
A direct representation of the process of crack
propagation and failure prediction can be achieved
by including unloading and release in the rupture
element definition. The scheme of figure 1(b)
absorbs a critical value of energy (ER) during
unloading. The element itself is quite similar to that
of figure 2.

Strain tripped rupture element with time based
unloading (type 3)
Use of the previous types of rupture elements in
modelling progressive crack propagation in an
elastic-plastic continuum, generally results in a nonsequential release of the elements thus causing an
irregular crack front opening profile. This
phenomenon, referred to as “locking”, is due to the
dependence of the level of plastic deformation
around the crack tip area on the loads acting on the
rupture element. Studies indicate that the unloading
curve may not be controlled independently in
presence of plasticity. To overcome this problem a
rupture element using the multi-step time based
unloading profile as shown in figure 1(c) has been
developed. This element uses an average strain
based on nodal displacements to detect the tripping
condition. This element simulates the process of
progressive crack propagation and yet it has a much
simpler structure compared with the other element
types as shown in figure 3.
Characteristics of rupture elements
Type 1 local damage based rupture elements are
used to predict the failure load in an elastic
continuum while rupture elements type 2 are also
capable of representing the process of failure
initiation and propagation in the presence of
plasticity. It has been shown that in the case of an
elastic continuum this approach is an alternative
representation to the conventional LEFM virtual
crack closure technique with the additional
advantages of visualising the process of progressive
crack propagation and direct calculation of the
failure load. Using the rupture elements rather than
the conventional fracture mechanics, one can also
predict the onset of fracture. Investigations have
revealed that with elastic-plastic continua, the
unloading
process
cannot
be
controlled
independently but it is also governed by the
continuum conditions. Therefore, the definition of
the rupture element has been modified. The result is
the rupture element type 2, a strain controlled
element whose behaviour is decided by the strain
state ahead of the crack extension path. Validation of
these elements in failure prediction is verified
through their application to a pre-cracked C(T)
specimen model made of epoxy adhesive E27. The
predicted failure loads are found in very good
agreement with the experimental results. The
validation of the elements type 1 and type 2 however
is limited to symmetric models failing under mode-I
fracture only. These limitations have been addressed
by introducing the generalised rupture element type
3. This element is tripped at a specified critical strain
parameter of the continuum to which it is
incorporated and follows a time-based unloading
scheme. The strain parameter is calculated from the
incrementally updated elements’ nodal relative
displacement vectors, projected along the element
direction. Details are shown in Figure 4. This would
result in a generalised version of the element that is
applicable to non-symmetric models and may also
be used for failure prediction under mixed mode
loading. This model could also represent progressive
crack propagation along the assumed pre-specified
path.
The local damage based rupture element has been
used to interfacial failure prediction in adhesively
bonded structures. The modelling technique
accounts for material plasticity and is applicable to
both cracked and non-cracked problems under
mode-I and mixed mode quasi-static loading.
Experimental data obtained from extensive tests of
standard fracture specimens, as described earlier,
have been used to validate the developed technique.
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Description of Finite Element models
The FE model of continuum C(T) specimen
The CT specimen model of an epoxy adhesive under
mode I loading for which experimental data was
obtained in previous research has been used in this
work. The continuum is modelled using 8-node
quadrilateral elements and a refined mesh has been
applied to the crack tip area with the smallest
element size equal to around 1.5 m.
To validate the proposed model a line of rupture
elements have been incorporated along the adhesionsubstrate interface of a cleavage fracture specimen
model consisted of Aluminium substrates bonded by
an epoxy adhesive. A Drucker-Prager material
model using the results obtained from tensile tests
carried out on the bonding adhesive in previous
research and confirmed by others has been used. The
FE analysis has been performed for mode-I and
mixed mode configurations using the cleavage
specimen model.
The FE Model of Cleavage joint specimen
The finite element model of the cleavage test
specimen is shown in figure 5. The model consists
of aluminium substrates bonded with a 2 mm-thick
layer of Permabond E27 epoxy adhesive. Using
PATRAN pre-processor, a localised refinement
scheme has been generated at different crack lengths
along the interface as shown in figure 5. The
continuum element used for both the adhesive layer
and the aluminium substrates was the 4-node
quadrilateral plane strain element as defined in
ABAQUS element library. An exponent DruckerPrager material model was used to model the nonlinear response of the adhesive layer whereas the
response of the aluminium substrates was considered
linear elastic.
The Rupture Element
The 3-node rupture element, type 3, has two nodes
connecting the crack faces and a third node that is
coincident with a node in the continuum. The
element structure and its correlation with the
continuum elements is shown in figure 6. The
algorithm used in the development of the rupture
element is described in figure 7.
The Element Implementation
A series of rupture elements incorporated into the
finite element model along the assumed crack line
(the upper interface) are used to model progressive
crack propagation. A calibrated tripping strain is
used for failure prediction in cracked and noncracked specimens under mode-I and also in mixed
mode loading. To assess and validate the predictive
technique, experimental data is compared with the
model predictions. Figure 8 shows the deformed
model with the crack length of around 3 mm. The
extended crack from the initial tip at 3 mm towards
the end of the refined mesh (where the rupture
elements are incorporated), can be seen in figure 9.
Analysis of results
Failure modelling in elastic continuum
The rupture elements of type 1 and 2 have been used
to model crack propagation in the CT specimen
described above. A series of the rupture elements,
forming a line of crack bridging elements, have been
integrated along the assumed crack line in the
continuum model and non-linear finite element
analysis has been performed. Figure 10 shows the
formulation used to calculate the energy release rate
from the results of the analysis of the model that
uses rupture elements of type 1. The critical value
(fracture energy) is compared with the result of
LEFM crack closure technique in table 1. The good
correlation validates the proposed solution. Using
the rupture energy with the energy based unloadingrelease scheme (type 2), a direct prediction of the
failure load has been achieved and the process of
progressive crack propagation in the elastic
continuum has been successfully presented as
illustrated in figure 11.
Failure modelling in plastic continua
Using rupture elements of type 3 the technique could
provide promising results in failure prediction within
an elasto-plastic continua. An exponent DruckerPrager model is used for this purpose. A physically
justifiable presentation of progressive crack
propagation has been obtained as shown in figure 9.
Finite element analyses have been also carried out
for Non-cracked and pre-cracked (for a range of
crack lengths) cases for each configuration and
results have been compared well with available
results reported elsewhere [2,6].
The user routine “Rupture element” has been
developed based on the algorithm shown in figure 7
using Fortran programming language. In the element
definition two nodes (i.e. nodes 2 and 3) span the
potential crack line whilst nodes 1 and 2 are used to
calculate the average strain in the continuum from
the co-related nodal displacements. A very high
stiffness is introduced into the element to represent
the connectivity of the crack surfaces through nodes
2 and 3. This situation remains until the average
strain in the direction normal to the crack reaches a
specified level representing the level of continuum
plasticity. The assigned stiffness is then removed
and the element is unloaded over a very small time
period. The fully unloaded element represents the
material rupture.
Mode-I Analyses
To calibrate the tripping strain, a number of analyses
using different tripping strains were performed and
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the failure predictions were compared with the
experimental failure load. Using the calibrated
values of tripping strain, finite element analyses
were performed for the cleavage model with rupture
elements incorporated along the crack line. The
analyses were repeated for a number of locally
refined crack tip locations (corresponding to a range
of crack lengths from 0.0 to 3.0 mm). Mode-I
predicted failure loads were obtained. Results,
showing the variation of the normalised (based on
experimental failure load for mode-I non-cracked
specimen) experimental and predicted failure load
with the crack length, are plotted in Figure 12.
Mixed-Mode Analyses
Similar analyses were used to validate the modelling
technique for failure prediction under mixed mode
loading. The cleavage model with the localised
refinement as described above, was used for all
mixed mode analyses. The calibrated tripping strain
used for mode-I failure prediction was applied. The
results appear to be in good agreement with the
experimental results corresponding to the mixed
mode loading as shown in Figure 13. This is a
remarkable achievement suggesting that in a plastic
continuum the average strain, in the direction
perpendicular to the crack, characterises the
condition of propagation of the crack. This statement
is valid regardless of the loading mode, at least for
the material used in this investigation, provided that
the appropriate material model is used in the
analysis.
Discussion
The strain tripped rupture elements can represent
physically realistic progressive crack propagation in
plastic continuum that cannot be achieved by the use
of the stress tripped rupture elements. Further
developments of the strain tripped elements
introduced time controlled unloading. This final
formulation has been successfully used to predict
cohesive and interfacial failure in cracked and noncracked configurations under mode-I and mixed
mode loading using a single failure criterion.
This is a significant advance on previous failure
methodologies that generally require separate failure
parameters for cracked and non-cracked and for
different modes of loading.
Concluding remarks
Using the UEL subroutine facility in ABAQUS
finite element code, a range of so called “local
damage based rupture modelling” elements are
developed to simulate progressive cohesive crack
advance in polymeric materials.
Force (stress) controlled rupture modelling elements
are used as crack face bridging element that trigger
the condition required for the onset (initiation) as
well as the propagation of cohesive cracks in elastic
continua
The local damage based rupture elements monitor
the state of stress/strain in the surrounding media
and behave accordingly.
A failure prediction technique that uses a local
damage based rupture element was extensively
assessed through the integration of these elements
along the crack line of a cleavage model. Continuum
plasticity was included and both cracked and noncracked configurations under mode-I and mixed
mode loading were analysed.
The strain tripped rupture element with no unloading
included can be used as an alternative to the
conventional LEFM when integrated into the elastic
continuum. The method is not also sensitive to the
control parameters for the elastic problem.
Using the strain tripped rupture element with energy
based unloading the failure load of a cracked elastic
structure was directly predicted. This is an
improvement to the application of LEFM as a postprocessing design tool.
The rupture elements with time based unloading are
developed and successfully applied to the fracture
specimens and a physically justifiable representation
of the crack propagation process is achieved.
In presence of continuum plasticity however both
the tripping condition and the unloading scheme
have to be adjusted as expected. Investigations have
revealed that the unloading process may not be
controlled independently and like tripping it is also
governed by the continuum conditions.
The predicted failure loads are in very good
agreement with the experimental results obtained in
the earlier research. Good failure prediction for all
loading modes was achieved using a single failure
criterion. Thus the failure prediction technique was
found to be mode independent. This is a noteworthy
and important achievement
The use of the modelling technique applied to both
mode-I and mixed mode loading in plastic
continuum would suggest this approach as a
promising method in achieving a framework for
predicting failure initiation and propagation in
generalised continuum.
Acknowledgements
The authors would like to thank Ferdowsi University
of Mashhad, Iran and Iran Ministry of Science,
5
Research and Technology for sponsoring this
research. Results of experimental work by Dr G
Richardson have been used in model assessment that
is fully appreciated.
References
[1] Richardson G, PhD thesis, University of Surrey,
UK, 1993
[2] Hadidimoud S, Crocombe AD and Richardson ,
G, WCARP 1, pp 219-222, Germany, 1998
[3] Hadidimoud S, PhD thesis, University of Surrey,
UK, 2000
[4] Hadidimoud S, Crocombe AD and Richardson
G, EuAdh2000, France, Sep. 2000
[5] Hadidimoud S, Crocombe AD and Richardson
G, EuAdh2000, France, Sep. 2000
[6] Crocombe AD, Richardson G and Smith PA, J
Adhesion, 49, pp 211-244, 1995
[7] Wisnom MR, Dept of Aerospace Eng, Univ.
Bristol, UK 1996
scheme
No element, LEFM
Element (type 1)
Element (type 2)
Load, N
Virtual crack closure
Method Fig. 10
instability
64
143.01
144.11
143.64
Table 1- Comparison of GIc values from the developed technique of modelling crack propagation in elastic
continuum with the conventional LEFM
6
F
F
F
ER
ER


(a)
T
i
m
(b)
(c)
Fig.1- Unloading schemes for rupture elements
I
+
II
6
5
7
4
3
9
8
14
2
10
11
=
13 12
1
Fig. 2- Correlation of rupture elements (types 1 and 2) with the continuum elements
3
I
II
=
+
I
II
2
1
Fig. 3- Correlation of rupture element type 3 with the continuum elements
u2
r  r1  r2
r
u r  u1  u 2
Tan  ry rx
r2
Y
r1

Tan  u ry u rx
   

 av  u r Cos r

X
ur
u1
-u2
Fig. 4- Description of the generalisation algorithm used in rupture element development; Updating algorithm
used for the rupture element to calculate the average strain
7
Fig. 5- Finite element model of the cleavage test specimen with aluminium substrates bonded with epoxy
adhesive E27, a) whole model, b) localised refinement
F
1
t
I
II
1
2,3
2
Figure 6- Rupture element behaviour and its correlation with the continuum elements
Tripping strain approximation:
F
ur=u2y- u1y
Tripping point (t)
av= ur/l
Ft
Unloading step
t<av<1.02t
Cut back factor (f):
Release point
ti
t
f=0.1(t-b)/ (e-b)
(b: begin inc., e: end inc.)
tr
Unloading factor, if (ti/tf)<1.0
Release point approach:
fi=fi-1+0.2(ti/tf),
0.98< fi<1.0
tf=0.2 tr
Load during unloading:
Fi=(1- fi)Ft
Load reduction step:
Cut back factor (f):
f=0.1(1.0/ fi), if: fi>1.0
F=0.2(ti/tf)Ft<0.2Ft
Fig. 7- Tripping, unloading and release algorithm for the rupture element
8
Figure 8- Deformed cleavage finite element model showing the adhesive layer and the propagated crack along
the interface, whole model and the crack tip area at 3 mm
Fig. 9– Progressive crack propagation in CT model using rupture elements with time based unloading and
exponent Drucker-Prager continuum material model
9
F
Fav
Fi
i


Fi
i
GI=2{Fav Lt,
Fav=(Fi+Fi+1)/2,
i+1- i
Where L, t are element length and specimen thickness respectively
Fig. 10- Virtual rupture element extension technique for calculation of G I from “no softening” elements
Force, N
Time
Fig. 11- Progressive crack propagation in elastic CT model using softening elements; Top: formation of the
process zone, Bottom: rupture elements’ load history
10
3
fa ilu re lo a d
(n o rm a lis e d )
E P 2 7 ; D ru c k e r-P ra g e r/A l; E la s tic
A ll E la s tic
2 .5
E x p e rim e n ta l (p re v io u s re s e a rc h )
2
1 .5
1
0 .5
c ra c k le n g th , m m
0
-0 .2
0 .2
0 .6
1
1 .4
1 .8
2 .2
2 .6
3
Figure 12– Mode-I results for failure load in cleavage test specimen at different crack lengths based on different
continuum models compared with the experimental data
3 .5
fa ilu re lo a d
(n o rm a lis e d )
3
E x p e rim e n ta l(p re v io u s re s e a rc h )
E P 2 7 ; D ru c k e r-P ra g e r/A L ; E la s tic
a ll E la s tic
2 .5
2
1 .5
1
0 .5
c ra c k le n g th , m m
0
-0 .2
0 .2
0 .6
1
1 .4
1 .8
2 .2
2 .6
3
Figure 13– Mixed mode results for the failure load in cleavage test specimen at different crack lengths compared
with the experimental data