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Department of
Aeronautics
Modelling the mechanical response
of multiphase recycled CFRP
Maciej Lembke
Supervisors:
Soraia Pimenta
Silvestre T. Pinho
London, February 2010

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Table of Contents
Table of Figures ..............................................................  3 
Table of Tables ................................................................  4 
Abstract .........................................................................  5 
1. Introduction .............................................................  6 
2. Literature review .......................................................  9 
2.1. Influence of fibre bundles on the response of thematerials .........................................................................  9 
2.2. Crack propagation techniques in Abaqus ...............  12 
3. Modelling and Numerical Analysis ..............................  13 
3.1. Choice of the modelling technique ........................  13 
3.2. Modelling properties ...........................................  14 
3.2.1 Geometrical properties .....................................  14 
3.2.2 Physical properties ...........................................  15 
3.2.3 Numerical properties ........................................  17 
3.2.4 Element type ..................................................  18 
3.3. Different approaches to modelling the idealised rCFRPmodel............................................................................  18 
3.3.1 2D singleinstance model .................................  18 
3.3.2 2D model with cohesive elements modellinginterface..... ................................................................  23 
3.3.3 2D cohesive contact model ...............................  27 
3.3.4 3D model .......................................................  31 
3.4. Mesh sensitiveness .............................................  33 
4. 2D cohesive contact model approach for differentgeometries.. .....................................................................  35 
4.1. Geometry of the fibre bundle ...............................  35 
4.2. Inclination of the fibre bundle ..............................  38 
5. Conclusions and further work ....................................  41 
Appendix A: Fractals as a tool for analysis of materialsproperties .........................................................................  43 
References ...................................................................  46 

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Table of Figures
Figure 1 rCFRP microstructure (bundle highlighted) ..................  6 
Figure 2 Bundle fracture .......................................................  7 Figure 3 Inplane geometry and dimension of the 2D model ....  14 
Figure 4 Inplane geometry and dimensions of the 3D mode.....................................................................................................  15 
Figure 5 Boundary conditions ..............................................  17 
Figure 6 Crack propagation arrest before the bundle (von Mises avg.
stresses) ..............................................................................  19 
Figure 7 Convergence problems after cracking most of the bundle(von Mises avg. stresses) .......................................................  20 
Figure 8 Damage of crackpath elements (STATUSXFEM status(damaged, failed) of the XFEM elements) .................................  20 
Figure 9 ETOTAL (total energy) and ALLSD (dissipated energy) for2D single instance model........................................................  21 
Figure 10 Magnification of a vertical drop of energy for 2D singleinstance model .....................................................................  21 
Figure 11 Crack propagation in a thinbundle model. a) crackpropagates towards the bundle; b) Failed matrix elements before and
after the bundle; c) completely failed bundle; (STATUSXFEM)... ..  22 
Figure 12 Energy plot for thinbundle model .........................  22 
Figure 13 Layer of cohesive elements around the bundle (SDEG degradation (damaged, failed) of cohesive elements) .................  23 
Figure 14 Crack propagating towards the bundle (von Mises avg.stresses) ..............................................................................  24 
Figure 15 Crack propagation through the bundle (von Mises avg.stresses) ..............................................................................  25 
Figure 16 Damaged cohesive elements (SDEG) .....................  26 
Figure 17 Model behaviour with additional precracks ..............  26 
Figure 19 Crack propagating towards bundle .........................  27 
Figure 18 Partially failed interface ........................................  27 
Figure 20 Fractured bundle (STATUSXFEM) ...........................  27 
Figure 21 Pulledout bundle (von Mises avg. stresses) ............  27 
Figure 22 Delamination of the matrixbundle interface case I weakened interface (CSDEG degradation (damaged, failed) ofcohesive contact) ..................................................................  28 
Figure 23 Delamination of the matrixbundle interface case II weakened interface (CSDEG) ..................................................  29 

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Figure 24 Delamination of the matrixbundle interface case III weakened interface and bundle (CSDEG) ..................................  30 
Figure 25 Pullout of the bundle embedded into the matrix .....  31 
Figure 26 Pullout of the bundle positioned at the surface of the
specimen .............................................................................  31 Figure 27 Crosssection of a 3D plate ...................................  32 
Figure 28 3D fibre bundle ...................................................  32 
Figure 29 Cracked 3D model ...............................................  32 
Figure 30 Nonexpected crackpath (von Mises avg. stresses)...................................................................................................  33 
Figure 31 Too coarse mesh causing stress concentration (von Misesavg. stresses) .......................................................................  33 
Figure 32 2D cohesive contact model for different mesh sizes(CSDEG) ..............................................................................  34 
Figure 33 Crack propagation for baseline geometry (CSDEG, vonMises avg. stresses) ..............................................................  35 
Figure 34 Crack propagation for a thick bundle ......................  36 
Figure 35 Crack propagation for a thin bundle (CSDEG, von Misesavg. stresses) .......................................................................  36 
Figure 36 Crack propagation for a long bundle (CSDEG, von Misesavg. stresses) .......................................................................  37 
Figure 37 Crack propagation for a short bundle (CSDEG, von Misesavg. stresses) .......................................................................  37 
Figure 38 Crack propagation for bundle inclined at 10 (von Misesavg. stresses, CSDEG) ...........................................................  39 
Figure 39 Crack propagation for bundle inclined at 30 (von Misesavg. stresses, CSDEG) ...........................................................  39 
Figure 40 Crack propagation for bundle inclined at 45 (von Misesavg. stresses, CSDEG) ...........................................................  40 
Table of Tables
Table 1 Material properties of the bundle ..........................  16 
Table 2 Material properties of the matrix with applied XFEM  16 
Table 3 Material properties of matrixbundle interface ........  16 
Table 4 Step properties ..................................................  17 
Table 5 Interface properties ............................................  18 
http://c/Users/Maciek/Desktop/Final_Project__Maciej_Lembke__eng__commented_by_soraia_again.docx%23_Toc253129830http://c/Users/Maciek/Desktop/Final_Project__Maciej_Lembke__eng__commented_by_soraia_again.docx%23_Toc253129831http://c/Users/Maciek/Desktop/Final_Project__Maciej_Lembke__eng__commented_by_soraia_again.docx%23_Toc253129831http://c/Users/Maciek/Desktop/Final_Project__Maciej_Lembke__eng__commented_by_soraia_again.docx%23_Toc253129831http://c/Users/Maciek/Desktop/Final_Project__Maciej_Lembke__eng__commented_by_soraia_again.docx%23_Toc253129831http://c/Users/Maciek/Desktop/Final_Project__Maciej_Lembke__eng__commented_by_soraia_again.docx%23_Toc253129830 
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Abstract
Technologies for recycling carbonfibre reinforcedpolymers (CFRP)
have been developed. The mechanical properties of the recyclatesshow they can be used in noncritical structural applications, e.g. in
secondary structures of aircraft and automobiles. The presence of
fibre bundles, groups of aligned fibres held together by residual
virginmatrix, is very common. It was already verified
experimentally that these bundles have a significant influence on
the mechanical properties of the rCFRP and that they add a
considerable degree of complexity to the failure mechanisms.
The objective of this project was to develop numerical models of
idealised rCFRP, focusing on the failure mechanisms in the presence
of fibre bundles bundle pullout and bundle fracture.
Different models were created in order to meet the objectives, both
2D and 3D. In all models, Extended Finite Element Method (XFEM)
was used to model crack propagation. Several different tools were
tried to model the interface, namely cohesive elements and surface
tosurface cohesive contact.
A 2D cohesive contact model that meets the objectives was
developed; using this model, a study of the influence of bundle
geometry and position on the failure mode was run. A 3D model
based on the previous 2D model was also developed. There are
strong premises that, after further work, these models can be a
very good representation of rCFRP.

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1.IntroductionTechnologies for recycling carbonfibre reinforcedpolymers (CFRP)
have been developed in the past years and are now mature.Generically, the process involves two main stages. The first one is
reclaiming the carbonfibres (CF) from an existent CFRP. It is done
by matrix removal, either from an endoflife part or manufacture
scrap. The second step is the reimpregnation of the recycled
carbonfibres (rCF) with new resin, producing a recycled (r) CFRP.
The mechanical properties of the recyclates show they can be used
in noncritical structural applications, e.g. in secondary structures of
aircraft and automobiles.
For rCFRP to be used in real structural applications, it is necessary
to understand its mechanical behaviour and to develop design
methods. However, recycled composites are considerably different
from their virgin precursors: the properties of the CF undergo (little)
degradation during the reclamation, and more important the
typical architecture of the rCFRP is very peculiar [Figure 1].
Figure 1 rCFRP microstructure (bundle highlighted)
The presence of fibre bundles groups of aligned fibres held
together by residual virginmatrix which was not entirely removed

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during the recycling process in addition to individual fibre
filaments dispersed within the resin this dispersed composite
phase will be hereby referred as matrix confers a multiphase /
hierarchical / fractal characteristic to the microstructure of therecyclates. For information about fractals as a tool for analysis of
materials properties see Appendix A.
It was already verified experimentally that these bundles have a
significant influence on the mechanical properties of the rCFRP [1].
They can increase the local toughness of the material up to 3 times.
They also add a considerable degree of complexity to the failure
mechanisms [Figure 2].
Figure 2 Bundle fracture
The overall aim of this project is to develop numerical models of
idealised microstructures of rCFRP, focusing on the failure
mechanism in presence of fibre bundles. Models are expected to
reproduce the two basic behaviour templates: fibre bundle fracture
and fibre bundle pullout. Fibre bundle fracture can be observed
when, after crack propagation through the bundle, it is broken into
two parts, with its crackpath generally inline with the matrix
crackpath. Fibre bundle pullout can be observed when, after crack

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propagation through the bundle, it is not fractured but part or the
entire bundle slides out of its original position; such behaviour is
caused by the failure of matrixbundle interface.
This report opens with chapter 2 which covers a literature review:to select a suitable modelling technique for the rCFRP, it was
necessary to understand the mechanical response of materials in
presence of fibre bundles. Comparison of different crack
propagation methods available in Abaqus 6.9 is also presented. The
development of the numerical model turned out to be much more
challenging than initially considered. Chapter 3 explains the
development and discusses over the results presented by four
different modelling strategies. The following chapter 4 presents the
differences in crack propagation for different geometries; both
geometry and inclination of the bundle are changed. The model
used in this chapter is a best performing 2D model from chapter 3.
The project is summarised by chapter 5 covering conclusions and
recommended future work.

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2.Literature review2.1.Influence of fibre bundles on the response of
the materials
In fibre composites every component of the material and its
production is very important. Highperformance fibres, matrix which
not only binds the material, but also transfers stresses, also the
way the two components are connected highly influences the way
the material behaves. By choosing suitable fibres we can obtain
very tough composites.
With recycled CFRP everything is more complex than with the virgin
material. Length and orientation of the fibres is not uniform. In
rCFRP we find many features specific of the recyclates: variable
fibre length, broken fibre segments, fibre waviness, fibres
aggregated in bundles and high void content [1]. All of these
features influence the mechanical behaviour of the material in
different ways. In this paper, the influence of fibre bundles will beanalysed.
Presence of bundles results in reduced distance between fibres
within the composite, local fibre alignment, end synchronisation,
increase in the variation of local volume fraction over the
composite, and presence of resinrich areas [2]. Mulligan et al. [2]
reviews models for toughness of a material with bundles using
different characteristics as a main factor e.g. local change in volume
fraction or aspect ratio of the bundle. According to Piggott [3], in
cases when the elastic work is similar to the work of fracture, a low
aspect ratio helps reducing the amount of elastic stress transfer and
increases toughness. He also points out that for a low aspect ratio
even when a fibre bundle is two times longer than its critical length
(xo), still more than half of the fibres will pull out, by what they still
contribute to fracture toughness.

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Piggott [3] presented this equation for critical fibre length:
0x  critical length
d  fibre diameter
u  tension strength
s  strength transfer
y  shear strength
This will make 2x0 the best fibre length for maximum fracture
toughness. If the crack propagates further the stresses betweenfibres and matrix will break the fibres that do not pull out [4].
Piggott [3] noticed also that during fibrebridging, the maximum
stress appears on the plane of the crack, making it the place where
the strongest fibres, which are not pulled out, will finally fail.
Trapeznikov [5] considered the influence of the number of fibres
within the fibre bundle and noticed that the fewer fibres in the
bundle, the smaller stress is required to pull them out, which leads
to failure. From that he concluded that the number of fibres per
bundle is counter proportional to the strength of the material.
Piggott [3] developed an expression for fracture toughness, in
which elastic stress transfer from matrix to the fibre is not taken
into account. It is derived based on the examination of the process,
where fracture of the fibre causes plastic flow in the matrix while
relaxing.
The same article presents an equation for the value of fracture
surface energy and toughness of composites however, there are
limitations that apply to these equations. Elastic stress transfer near
cracks for fibres bridging the cracks should be small compared with
stresses transferred by plastic flow at the interface between matrix
and fibres. It is also worth noting that fracture energy is much more
y
sudx
2
)(0

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dependent on the properties of the fibre than on those of the matrix
[6].
Kim and Mai [7] suggested that, depending on the fibre volume
fraction and impregnated material, the fracture toughness of abundled material may increase up to 100% while maintaining its
flexural strength, compared to composites without bundle
impregnation. Kim and Mai [7] based their suggestions on studies of
a composite, in which fibre bundles were impregnated with polymer
before they were connected with the matrix in a range of small
volume fraction. The process of impregnating fibres into the matrix
is very important: it is best to use the matrix with shear flow
properties. When the process is strictly controlled and the bonding
is effective, a much better toughness can be achieved. According to
Fila et al. [4], the fracture toughness is linearly proportional to the
volume fraction of fibre bundle. Fibre bundling also significantly
improves the stability of cracking. It is also worth mentioning that
shear deformation in the fibre bundle is often not taken into account
when calculating fracture toughness [4].
Fila et al. [4] describes how one of the abilities of bundles is to
toughen significantly very brittle materials. The degree of their
influence is not only dependant on their strength. It can also
increase with their diameter or decrease with the modulus, which is
counter proportional to fibre cluster volume fraction [2].
Large diameter fibres (as bundles are often considered [2, 6, 7])
are likely to significantly increase fracture toughness of the
composite. This is caused by increased shear stresses transferred to
the matrix, which are proportional to the diameter of the fibre [6,
7]. Additionally, Piggott [6] concludes that fibre diameter should be
maximized to get the largest value of toughness only for moderately
ductile and brittle fibres, while length of the fibre should be as large
as possible for ductile fibres to achieve the same effect.

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2.2.Crack propagation techniques in AbaqusThere are four main techniques for modelling crack propagation
available in Abaqus 6.9: progressive damage models, cohesive
elements method, contour integral estimates, and Extended Finite
Element Method (XFEM) [8].
Progressive damage models are mainly used for ductile materials or
UD composites. They offer a general capability for modelling
progressive damage and failure of the material. After damage
initiation, the material stiffness is degraded progressively according
to the specified damage evolution response. The progressivedamage models allow for a smooth degradation of the material
stiffness, which makes them suitable for both quasistatic and
dynamic situations.
The cohesive elements method uses cohesive elements to model
crack propagation. It requires the estimation of possible crackpaths
at the stage of modelling. A relatively very thin line of cohesive
elements needs to be placed in all the places that the crack is
expected to propagate throuthg. The crack propagation analysis
consists of analysis of failed or damaged cohesive elements on the
crackpath.
Contour integral estimates are used to study damaged initiation in
quasistatic problems. They cannot be used to predict crack
propagation.
The Extended Finite Element Method (XFEM) is used to study both
damage initiation and crack propagation. The crackpath does not
need to be predicted as crack propagation is found by the method
itself. XFEM uses the partition of unity framework to model strong
and weak discontinuities independent of finite element mesh.

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3.Modelling and Numerical Analysis3.1.Choice of the modelling techniqueThe idealised rCFRP consists of four parts:
matrix, which is an homogenised representation of thecomposite phase of individual fibres dispersed within the
polymeric resin;
bundle, which is an homogenised representation of thecomposite bundle;
matrixbundle interface, which connects the bundle to thematrix;
precrack, which is used to initiate failure.The model should be able to model crack propagation from the
direction of the precrack towards the bundle while loaded in tension.
The crucial part is to capture the behaviour of the crack in the area
of the bundle. There are two possible ways in which the bundle can
behave. The first is bundle fracture; in this case, the crack
propagates through the matrix, then fractures the bundle, and
continues cracking the matrix afterwards. The other option is for the
bundle to pullout; in that case, the crack similarly propagates from
the precrack towards the bundle; the next step, instead of bundle
fracture, is the delamination of the matrixbundle interface; after
that, the bundle is only partly attached to the matrix and the crack
propagates further through the matrix.
The crack propagation modelling technique chosen for all models is
the Extended Final Element Method (XFEM). It has an advantage
over other available techniques. Contour integral cannot be used to
predict crack propagation. Progressive damage model can only be
used for ductile materials or UD composites and cannot represent a
sharp crack. Cohesive element method requires a crackpath to be
predicted a priori. XFEM enables sharp crack propagation, allows

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new cracks to appear in all areas of the model where XFEM is
applied to, and enables crack propagation in all directions.
3.2.Modelling properties3.2.1 Geometrical properties
3.2.1.1 Twodimensional (2D) modelAll the 2D models are planar and deformable with plane stress,
thickness of 1mm. Figure 3 represents the inplane geometry and
dimensions.
Figure 3 Inplane geometry and dimension of the 2D model

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3.2.1.2 Threedimensional (3D) model
3.2.2 Physical propertiesTable 1 to Table 3 present the material properties of bundle, matrix
and bundlematrix interface. Table 4 and Table 5 present numerical
properties of the step and the matrixbundle interface.
3.2.2.1 Material propertiesAll the bundle properties were calculated based on data from [9],
considering the bundle as a UD composite with fibre volume fraction
of 65%.
All matrix and interface properties were taken from [9], and
correspond to experimental testing of a rCFRP.
Figure 4 Inplane geometry and dimensions of the 3D model

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Table 1 Material properties of the bundle
Symbol Value Unit Property
E1 142700 MPa Young modulus in direction x
E2 8900 MPa Young modulus in direction y
Nu12 0.33  Poissons ratio
G12 4000 MPa Shear modulus in direction x
G13 4000 MPa Shear modulus in direction y
G23 1200 MPa Shear modulus in direction z
X 2704 MPa Maximum principal stress
G 100 kJ/m2 Fracture Energy / Toughness
Symbol Value Unit Property
E1 28100 MPa Young modulus in direction x
E2 16000 MPa Young modulus in direction y
Nu12 0.42  Poissons ratio
G12 7000 MPa Shear modulus in direction x
G13 1200 MPa Shear modulus in direction y
G23 1200 MPa Shear modulus in direction z
X 194.5 MPa Maximum principal stress
G 2.79 kJ/m2 Fracture Energy / Toughness
Table 3 Material properties of matrixbundle interface
Symbol Value Unit PropertyE/Knn 2810 GPa Young modulus
G1/Kss 700 GPa Shear modulus in direction x
G2/Ktt 120 GPa Shear modulus in direction y
 194.5 MPa Maximum nominal stress in normal direct.
 86.9 MPa Maximum nominal stress in shear direct. 1
 86.9 MPa Maximum nominal stress in shear direct. 2
G 0.2 kJ/m2 Fracture Energy / Toughness
Table 2 Material properties of the matrix with applied XFEM

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3.2.2.2 Boundary conditionsThe displacement in direction x is applied on the top right part of
the model as shown on the figure. The top left part of the model is
fixed in x direction. Both left and right parts are fixed in direction y
[Figure 5].
The maximum displacement applied was 0.67mm.
Figure 5 Boundary conditions
3.2.3 Numerical propertiesThe analyses were run in implicit, static mode. The most important
step properties are presented in Table 4. The numerical properties
of the interface are presented in Table 5.
Table 4 Step properties
Value Property
0.0002  0.2 Damping dissipated energy fraction
20 Maximum number of iterations per increment0.025 Convergence criterion for the ratio of the largest
residual to the corresponding average flux norm for
convergence (5 times larger than the default value)
0.05 Convergence criterion for the ratio of the largest
solution correction to the largest corresponding
incremental solution value (5 times larger than the
default value)

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Table 5 Interface properties
Symbol Value Property
 0.0027 Tolerance for adjustment zone
Knn 281000 Stiffness in normal direction
Kss 500 Stiffness in shear direction 1
Ktt 1000 Stiffness in shear direction 2
3.2.4 Element typeAll 2D models have a structured mesh, with each element
approximately 0.15mm large. The element type used is CPS4I; it is
a linear Quad element shape with incompatible modes to prevent
hourglassing.
In majority of the 3D models the mesh size was 0.45mm. Parts of
the fibre bundle were meshed using Hex element shape with a
sweep (medial axis) technique of meshing. The element type is
C3D8I: An 8node linear brick. It is a Hex element shape with astructured technique of meshing. In all models Incompatible modes
were used.
3.3.Different approaches to modelling the idealisedrCFRP model
3.3.1
2D singleinstance model
The 2D singleinstance model is a very simple model. Only one
part/instance is used. It has a sketched bundle shaped area that
has different material properties assigned. No specific features are
used to model the bundlematrix interface.
When the crack approaches the bundle, the propagation is arrested
because of rapid change of material properties [Figure 6]. Bothfracture energy and maximal principal stress of the bundle are

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much bigger than those of the matrix. Crack is arrested for the
second time when it is propagating back into the matrix [Figure 7],
the reason this time are problems with convergence.
Figure 6 Crack propagation arrest before the bundle (von Mises avg.
stresses)

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Figure 7 Convergence problems after cracking most of the bundle
(von Mises avg. stresses)
The first problem with the model appears at that point. When thecrack is about to entirely fracture the bundle, the job aborts due to
too low increment time needed for further iterations. Figure 8
presents how elements on the crackpath below the bundle are
damaged, but the job fails to converge before the bundle fails
completely.
Figure 8 Damage of crackpath elements (STATUSXFEM status(damaged, failed) of the XFEM elements)

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After analysis of the Total Energy plot [Figure 9][Figure 10], it is
observed that the energy drops vertically, hence the very low
increment time requested by the job.
The only reason for such behaviour is a physical instability: the next
increment would cause the bundle to fail completely and, with the
amount of accumulated energy, the entire plate would also fail. The
confirmation of that is a different model. It is very similar to the one
described above with only one difference: the width of the bundle is
two times smaller. Figure 11 shows how the bundle fractures
completely and the crack propagates. In spite of similarities, the
model with the thinner bundle behaves differently, because the
elastic energy stored in the plate is not so high. This allows it to be
more stable and converge.
Figure 9 ETOTAL (total energy)
and ALLSD (dissipated energy)
for 2D single instance model
Figure 10 Magnification of a
vertical drop of energy for 2D
single instance model

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a) b) c)
In Figure 12, we can see the Total Energy plot of the model with a
thin bundle. A very similar effect of a sudden energy drop appears,
only that in this case it is not vertical but very steep.
This problem can be minimised with launch of Abaqus 6.10, which
will enable XFEM in implicit dynamics and explicit code. This will
Figure 11 Crack propagation in a thinbundle model. a) crack
propagates towards the bundle; b) Failed matrix elements before
and after the bundle; c) completely failed bundle; (STATUSXFEM)
Figure 12 Energy plot for thinbundle model

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allow models to capture the dynamic effect which is the cause of the
problem.
The second problem with the singleinstance model is that it is not
able to model the pullout of the fibre bundle. The reason for this isXFEM not being able to effectively make a 90 turn at the point of
touching the bundle and therefore following the interface. In order
to make pullout possible, the next model was developed.
3.3.2 2D model with cohesive elements modellinginterface
This model was designed to enable fibre bundle pullout. For this to
be possible, a very thin layer (0.01mm thickness) of cohesive
elements was added around the bundle [Figure 13]. Cohesive
elements are modelling the interface between fibre bundle and
matrix, which enables the pullout without directly damaging any of
them.
The main problem with the model is crack propagation through the
interface. The cohesive elements can only open in one direction. As
in this case they are to model the interface, they are set to open in
the direction normal to bundle outline; for that reason, crack
propagation across a cohesive element is not possible. In order to
deal with this problem, three different approaches were taken.
Figure 13 Layer of cohesive elements around the bundle (SDEG
degradation (damaged, failed) of cohesive elements)

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3.3.2.1 Cohesive element on the crack pathThis model has a continuous layer of cohesive elements around the
fibre bundle. As XFEM cannot be applied to cohesive elements, it is
impossible for the crack to propagate into the bundle [Figure 14].
As result a pullout is also impossible, because for it to happen all
the elements between the precrack and the bundle must be failed.
3.3.2.2 Matrix element on crackpathThe next approach was to substitute the two cohesive elements on
the predicted crackpath with two matrix elements . This enabled
crack propagation across the interface and through the bundle
[Figure 15].
Figure 14 Crack propagating towards the bundle (von Mises avg.stresses)

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The fibre bundle pullout was still not possible. The replaced
elements cracked vertically, so a horizontal crack, which would
correspond to delamination, was not possible. This restriction is
caused by the numerical possibilities of XFEM (only one crack per
element allowed).
3.3.2.3 No element on crackpathThe last approach taken to this model was to delete the two
cohesive elements on the predicted crackpath. Here again,
similarly to the model with cohesive elements on the crackpath,
problems appeared on the stage of crack getting near the bundle.
An obvious problem with this model is present before starting the
analysis. The lack of the cohesive elements is not realistic, even
though the elements are relatively small. This causes cohesive
elements to fail in an unexpected location and in an unexpected
mode. Although some hope for a failure of cohesive elements
appear as the crack draws nearer to the bundle [Figure 16], the
model has problems with further converging.
Figure 15 Crack propagation through the bundle (von Mises avg.
stresses)

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Even with help of additional precracks in the area of the deleted
elements, the model does not produce results it is expected to
[Figure 17].
All the models where cohesive elements were modelling the
interface between fibre bundle and matrix have not brought much
improvement. As with the 2D single instance model, it was possible
to model the bundle fracture but still impossible to model a bundlepullout. The most promising results came from the model with
matrix like elements substituted for cohesive elements. It was
possible to see the model leaning towards the pullout, but
unfortunately, because of the models structure, it was impossible
to completely do so. Based on all that, it is safe to conclude that the
idea to model an interface between bundle and matrix turned out to
be good.
Figure 16 Damaged cohesive elements (SDEG)
Figure 17 Model behaviour with additional precracks

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3.3.3 2D cohesive contact modelThis model, in contrary to the two models described above, has
two separate instances for the bundle and the matrix plate. The
interface between these two elements is modelled with use ofsurfacetosurface cohesive contact [Figure 18][Figure 19].1819
This type of interface solved the problems with crack propagation
that were present in the previous model. Modelling of both bundle
fracture [Figure 20] and fibre bundle pullout [Figure 21] was
possible, by changing the properties of the bundle and/or interface.
Figure 19 Crack propagating
towards bundle
Figure 18 Partially failed
interface
Figure 20 Fractured bundle
(STATUSXFEM)
Figure 21 Pulledout bundle (von
Mises avg. stresses)

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 28 
Figure 22 to Figure 24 present different sequences of events of
matrixbundle interface failure.
a) The first degradation of the
interface appears near the left
end of the bundle
b) Delamination propagates up
and around in the direction of the
main crack
c) After almost half of the bundle
is delaminated a new crack
appears near the left end of the
bundle
d) Crack propagates downwards
Figure 22 Delamination of the matrixbundle interface case I
weakened interface (CSDEG degradation (damaged, failed) of
cohesive contact)

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 29 
a) The first degradation of the
interface appears near the left
end of the bundle
b) Another delamination begins
near the crack tip
c) Two delamination are
propagating towards each other
d)Almost half of the bundle is
delaminated
Figure 23 Delamination of the matrixbundle interface case II
weakened interface (CSDEG)

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 30 
a) Bundle is damaged, but not
fractured
b) Only after bundle is damaged
the delamination begins
c) Delamination proceeds around
the bundle
d)New cracks appear after the
bundle is pulled out
The 2D twoinstance model enables visualisation of both pullout
and crack propagation through fibre bundle. There is only one
problem with this model apart from difficulties with convergence.
The path of the crack after either pullingout or breaking the bundle
is on the side of the bundle. The cause of such behaviour is the
nature of the 2D planar model. The experimental results behave in
the same way only when the bundle is on the surface of the
specimen [Figure 25]. When the bundle is embedded within the
matrix, the crack should propagate after the bundle in the same
direction it propagated towards the bundle, either fracturing or
Figure 24 Delamination of the matrixbundle interface case III
weakened interface and bundle (CSDEG)

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 31 
pulling it out [Figure 26]. In order to achieve crack continuity a new
model had to be created. 25 26
3.3.4 3D modelThis model was developed to ensure the crack continuity around the
bundle. It is very similar to the 2D twoinstance model. There are
two different elements: a fibre bundle [Figure 27] and a plate with a
bundleshaped hollow space [Figure 28]. The two are also
connected with surfacetosurface cohesive contact. 27 28
Figure 25 Pullout of the bundle
positioned at the surface of thespecimen
Figure 26 Pullout of thebundle embedded into the
matrix

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 32 
As predicted, the 3D model enabled crack propagation around thebundle and bundle pullout [Figure 29].
a) Crack propagating around the
bundle, which is being
pulledout (von Mises avg.
stresses)
b) Crosssection of a halfcracked
plate (STATUSXFEM)
There are some problems with the model. One of them is a difficult
convergence. There are two causes of this problem. The first is the
current limitations of the XFEM tool [Figure 30]. The second reason
is a too coarse mesh [Figure 31]. Unfortunately, when the mesh is
finer, the jobs become much more timeconsuming and power
demanding.
Figure 28 Crosssection of a 3D
plate
Figure 27 3D fibre bundle
Figure 29 Cracked 3D model

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 33 
Basing on the 2D twoinstance model performance, it is safe to
assume that this 3D model is also able to model a fibre bundle pull
out [Figure 29]. When combined with the crack continuity that the
3D approach delivers, it presents an interesting possibility to be a
very accurate failure model of idealised rCFRP.
3.4.Mesh sensitivenessTo validate the 2D cohesive contact model, it was run with different
mesh sizes, and the results of these jobs were compared.
Five different mesh sizes were used: 0.45mm, 0.3mm, 0.15mm,
0.075mm.
Figure 30 Nonexpected
crackpath (von Mises avg.
stresses)
Figure 31 Too coarse mesh
causing stress concentration (von
Mises avg. stresses)

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 34 
a) Mesh size 0.45mm b) Mesh size 0.3mm
c) Mesh size 0.15mm d) Mesh size 0.075mm
The resemblance between different meshes is clearly visible in
Figure 32. The variations between directions of crack propagation
depend on small deviations of the point in which the crack
propagating from the top touches the fibre bundle. Also on the
force/displacement graphs the same trends are present.
Figure 32 2D cohesive contact model for different mesh sizes
(CSDEG)

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4.2D cohesive contact model approach fordifferent geometries
Different modelling techniques of idealised rCFRP were shown in the
previous chapter. The bestperforming one was the 2D cohesive
contact model. In this chapter, it is going to be used to study the
influence of both fibre bundle geometry and inclination on crack
propagation.
4.1.Geometry of the fibre bundleThe baseline geometry of the bundle is presented on Figure 3. It is
2.3 mm long and the bundle is 0.7 mm thick.
Figure 33 presents a model with baseline geometry of the fibre
bundle. After the crack propagated to the bundle, the matrixbundle
interface delaminated at both ends of the bundle and new cracks
started appearing in the matrix below the bundle, near both ends.
Figure 33 Crack propagation for baseline geometry (CSDEG, vonMises avg. stresses)
Model presented on Figure 34 has doubled bundle thickeness now
it is 1.4 mm. In this case, crack propagated by delaminating the
matrixbundle interface, which was expected. If the bundle was
pulledout for the baselinegeometry model, then it should do thesame when the thickness of the bundle is larger.

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 36 
Figure 34 Crack propagation for a thick bundle
(von Mises avg. stresses)
The next model was changed in an opposite way the bundle was
made thinner (0.35 mm bundle thickness). In this case, fibre bundle
pullout was also present [Figure 35]. This was not as likely as in
the case of the thick bundle: the bigger the length to thickness
ratio, the more prone to fracture the bundle should be.
Figure 35 Crack propagation for a thin bundle (CSDEG, von Misesavg. stresses)
The next presented model has doubled value of length to 4.6 mm,
and thickness left at the baseline value of 0.7 mm. Figure 36
presents neither fibre bundle pullout nor fibre bundle fracture,
because the model had problems to converge further. Nevertheless

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 37 
a prediction can be made based on this figure; as the matrixbundle
interface is not damaged and high stress concentrations appear in
the middle of the bundle (and not at its ends), it can be predicted
that bundle fracture is the most likely mechanism for this model. Aswritten before, this pattern of behaviour is expected from models
with high length to thickness ratio of the fibre bundle.
Figure 36 Crack propagation for a long bundle (CSDEG, von Misesavg. stresses)
In the next model, length has been made two times smaller than
the baseline value (1.15 mm). Results from this model are
presented on Figure 37. As expected, bundle pullout is present in
this case.
Figure 37 Crack propagation for a short bundle (CSDEG, von Mises
avg. stresses)

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Drawing conclusion from the presented four cases, it is suggested
that bundle length has more influence on crack propagation
behaviour than bundle thickness. Changes in fibre bundle length
translate directly to change of the amount of interface that needs tofail when loaded in shear in order for the bundle to pull out.
Changes in fibre bundle thickness translate to change of strength of
the bundle which can prevent or allow a bundle fracture. Such
geometrical dependencies are only valid for 2D model: in case of a
3D bundle, a change of length or thickness (diameter) translates to
change of both volume of the bundle and area of interface.
4.2.Inclination of the fibre bundleThe baseline angle of inclination of the longitudinal axis of the
bundle is 0 [Figure 3, page  14 ]. As stated above, the crack
propagation behaviour for this type of geometry is the following:
after the crack propagated to the bundle, the matrixbundle
interface delaminated at both ends of the bundle and new cracks
started appearing in the matrix below the bundle, near both ends.
[Figure 33].
The inclination of the longitudinal axis of fibre bundle in the model
presented in Figure 38 is equal to 10. The bundle pulledout and
the crack propagated further into the matrix, going around the
bundle upwards.

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 39 
Figure 38 Crack propagation for bundle inclined at 10 (von Misesavg. stresses, CSDEG)
The next model presents an inclination of the longitudinal axis of
the bundle of 30 [Figure 39]. Because of convergence problems, it
is not obvious what happens with the crack. Based on failure of
interface on one end of the bundle and little stress concentration in
the middle of the bundle, it is safe to assume that the bundle pulls
out. Unfortunately it is impossible to state on which side of the
bundle the crack propagated further in the matrix.
Figure 39 Crack propagation for bundle inclined at 30 (von Misesavg. stresses, CSDEG)
Figure 40 presents a model with an inclination of the longitudinal
axis of the fibre bundle of 45. Similarly to the previous two
models, also in this model the bundle pulls out. Unfortunately, again
due to convergence problems the model was not able to complete

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 40 
the analysis. Luckily enough, it is possible to state that delamination
is propagating towards the lower end of the bundle and a new crack
in the matrix started appearing also at the lower end of the bundle.
Figure 40 Crack propagation for bundle inclined at 45 (von Mises
avg. stresses, CSDEG)
Drawing conclusions from the three models with different
inclinations of the longitudinal axis of the fibre bundle described,
above two assumptions can be made. Change of the said inclination
does not change the crack propagation behaviour from pulling out
to fracturing the bundle. Direction of crack propagation in the
bundle depends of the value of inclination. For small values
definitely not higher than 45  crack propagates after the bundle
into the matrix from the higher end of the bundle. For inclinations
bigger than 45 the crack propagates after the bundle into the
matrix from the lower end of the bundle; this was valid for the
geometries analysed

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5.Conclusions and further workNumerical models of idealised recycled carbon fibrereinforced
polymer (rCFRP) were developed. Using Extended Finite ElementMethod (XFEM) a primary objective was achieved. The two
dimensional (2D) cohesive contact model is able to model failure
mechanisms existing in the presence of fibre bundle: fibre bundle
fracture and fibre bundle pullout.
Three different 2D models were created to capture these
mechanisms. First one was a single instance model. It was able to
model crack propagation towards and through the bundle, but
aborted when the bundle was about to fracture;it was also not able
to model fibre bundle pullout. The next model had cohesive
elements modelling the matrixbundle interface. It was possible to
propagate the crack into, through and past the bundle.
Unfortunately, similarly to the single instance model, it was unable
to model fibre bundle pullout. The third model created used
cohesive contact to model the matrixbundle interface. It made it
possible to model both mentioned failure mechanisms. It was a
good representation of bundle present at the surface of the
specimen, because of the way crack propagated back into the
matrix after pulling out the bundle.
In order to model a situation when fibre bundle is embedded into
the matrix a three dimensional (3D) model, based on the 2D model
using cohesive contact, was created. It was possible to achieve fibre
bundle pullout with a continuous crack propagation using this
model. With further development this model can be a very accurate
representation of idealised rCFRP.
The 2D cohesive model was used to conduct a study on the
influence of bundle geometry and position on the failure mode. It
was concluded that length of the fibre bundle has influence on the
failure mode of the bundle, whereas no variation of the failure mode

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 42 
with the thickness (within the geometries tested) was noticed.
When the bundle was made longer, the failure mode changed from
the fibre bundle pullout to fracture.
There were also two conclusions from the study of differentinclinations of the longitudinal axis of the bundle. First is that,
within the geometries tested, the change of the inclination does not
influence the failure mode (pullout or fracture of the bundle).
Second was that the magnitude of the inclination has influence on
crack propagation back into the matrix after fibre bundle pullout.
For the geometries tested, when the angle was lower than 45 the
crack propagated from the higher end of the bundle and when it
was higher the crack propagated from the lower end of the bundle.
Further work, as stated earlier, should involve further development
of the 3D model. As the second point, a more extensive study on
the influence of geometry and position of the bundle should be
conducted, involving both 2D and 3D cohesive contact models. This
should help to understand in detail the mechanisms involved in each
failure mode.
After completion of two points listed above work should progress
out of numerical modelling. Based on developed numerical models
and analytical theories as well as multiphase / hierarchical / fractal
theories for materials properties analytical models for strength and
toughness are to be developed. Further, analytical results should be
compared with numerical and experimental data to validate the
analytical models.

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Appendix A: Fractals as a tool for analysis ofmaterials properties
Similarly to many of biological, natural materials, there are
manmade materials that manifest complex, irregular and often at
the first sight chaotic patterns [10]. There are multiple examples of
such materials and objects e.g.: human lungs, veins, trees, fluid
turbulence, polymers, fractures and coastline shapes [11]. All of
these materials have one thing in common they are hard to be
described by means of Euclidean geometry. That is the situation
when fractal geometry applies  it makes use of the multiscale
nature of the materials. Different mathematical descriptions,
formulas and patterns can be easily found in appropriate literature
[10]. One of the amazing fractals characteristics is described as
ability to fit an effectively infinite length within a finite area and an
effectively infinite area within a finite volume. Fractals geometry
can have very broad applications. It is widely uses in mathematics,
physics but less in mechanics and even less in engineering [11, 12].The key to understanding mechanics of materials representing
fractal properties is to understand the correlation between their
structure and mechanical performance. There are two approaches
as to the range of steps in which fractal theory can be applicable
it can be infinite or finite. When the first one is taken into account
new properties such as scaleinvariant quantities or abnormal
physical dimensions must be defined. For many cases only the
second, finite range, approach is applicable [13]. For example, sea
shells have been found to have only 2 3 orders of lamellar
structures [12]. However, despite the fact that some materials show
selfsimilarities only in a small range, it is possible to model their
scaling properties at each level by means of a fractal approach,
using recursive relationships [13].

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Carpinteri et al. [14] described size effect as a common
phenomenon in multiscale materials. With change of structural size
some of the mechanical properties, which would be constant in
classical mechanics, change. A good example of such properties istensile strength, which decreases with an increase of the size of a
structure [14]. Unfortunately the size effect has not been
interpreted yet, due to not knowing its source [10]. Soare [15], in
turn attempted to model the size effect in the deformation and
fracture process in a heterogeneous material by reformulating the
field equations using a nonEuclidean metric. He suggested that for
the processes that take place in the fractal space it should be
assumed that the deformation is localized on a fractal.
Newman and Gabrielov [16] tested failure properties for multiscale
fibre bundles with equal load sharing. What they found is a
universal asymptotic scaling law which links system size to failure
stress threshold, what they believe will bring advantages to
constructing fibre bundles in engineering applications.
Cohesive crack model as a most commonly used in purpose of
damage localization is described by Carpinteri [14]. In this model
the material is characterised by a stressstrain relationship, valid for
undamaged zones, and by a stresscrack opening relationship,
describing how the stress decreases from its maximum value to
zero as the distance between the crack lips increase from zero to
the critical displacement.
Piggott [17] points out that vast majority of available models for
fibre composites have straight, regular, aligned fibres, what is far
from reality and can lead to unexpected results. Presented by he
author concept of mesostructures includes possible mentioned
defects and is applicable to composites with randomly distributed
short fibres. Wang and Pan [18] in their work on numerical
approach for elastic properties prediction of multiphase composites
also emphasised that if not taken into account, the small amounts

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 45 
of voids in a composite may lead to relevant differences between
theoretical and experimental values. In their study, when
calculating the material characteristics with voids taken into
account, all Youngs modulus, shear modulus and Poissons ratiovalues coincided much better with experimental data.
Two main sources of common difficulties with modelling behaviour
of multiscale materials or structures are pinpointed by Soare [15].
First of them is the complexity of geometry in a micro scale and
difficulties selecting the constitutive behaviour. The second problem
concerns the macro scale often the deformation leads to the self
organization of internal structures, which control the constitutive
behaviour. An efficient approach for solving boundary value
problems on domains with complex microstructures based on
multiple scales was proposed. Solving this problem with use of finite
element method seemed too expensive. Soare proposed to use
enriched shape function to capture the complexity of the geometry
and using the outcome as a parametrically defined boundary value
problem. After that all refinement levels can be obtained without
any additional cost.
Oshmyan et al. [19] studied composite materials containing rigid
inclusions/voids with distributions similar to the Sierpiskilike
carpet. They used the finite element method to determine the
effective elastic constants for the first steps of the iteration and
renormalization group techniques to identify the scaling properties.
The scaling exponents turn out to be functions of the fractal
dimension of the microstructure. Similar results were obtained by
others in similar cases.

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