Lembke Bsc

7/27/2019 Lembke Bsc

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Department of

Aeronautics

Modelling the mechanical response

of multiphase recycled CFRP

Maciej Lembke

[email protected]

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 single-instance 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 In-plane geometry and dimension of the 2D model .... - 14 -

Figure 4 In-plane 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 crack-path 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 thin-bundle 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 thin-bundle 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 Pulled-out bundle (von Mises avg. stresses) ............ - 27 -

Figure 22 Delamination of the matrix-bundle interface case I weakened interface (CSDEG degradation (damaged, failed) ofcohesive contact) .................................................................. - 28 -

Figure 23 Delamination of the matrix-bundle interface case II weakened interface (CSDEG) .................................................. - 29 -


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Figure 24 Delamination of the matrix-bundle interface case III weakened interface and bundle (CSDEG) .................................. - 30 -

Figure 25 Pull-out of the bundle embedded into the matrix ..... - 31 -

Figure 26 Pull-out of the bundle positioned at the surface of the

specimen ............................................................................. - 31 -Figure 27 Cross-section of a 3D plate ................................... - 32 -

Figure 28 3D fibre bundle ................................................... - 32 -

Figure 29 Cracked 3D model ............................................... - 32 -

Figure 30 Non-expected crack-path (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 -



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 matrix-bundle 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 carbon-fibre reinforced-polymers (CFRP)

have been developed. The mechanical properties of the recyclatesshow they can be used in non-critical structural applications, e.g. in

secondary structures of aircraft and automobiles. The presence of

fibre bundles, groups of aligned fibres held together by residual

virgin-matrix, 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 pull-out 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-

to-surface 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 carbon-fibre reinforced-polymers (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 carbon-fibres (CF) from an existent CFRP. It is done

by matrix removal, either from an end-of-life part or manufacture

scrap. The second step is the re-impregnation of the recycled

carbon-fibres (rCF) with new resin, producing a recycled (r-) CFRP.

The mechanical properties of the recyclates show they can be used

in non-critical 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 virgin-matrix 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 pull-out. Fibre bundle fracture can be observed

when, after crack propagation through the bundle, it is broken into

two parts, with its crack-path generally in-line with the matrix

crack-path. Fibre bundle pull-out 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 matrix-bundle 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. High-performance 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 resin-rich 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 fibre-bridging, 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 quasi-static and

dynamic situations.

The cohesive elements method uses cohesive elements to model

crack propagation. It requires the estimation of possible crack-paths

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

crack-path.

Contour integral estimates are used to study damaged initiation in

quasi-static 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 crack-path 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;

matrix-bundle 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 pull-out; 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 matrix-bundle 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 crack-path 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 Two-dimensional (2D) modelAll the 2D models are planar and deformable with plane stress,

thickness of 1mm. Figure 3 represents the in-plane geometry and

dimensions.

Figure 3 In-plane geometry and dimension of the 2D model


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3.2.1.2 Three-dimensional (3D) model

3.2.2 Physical propertiesTable 1 to Table 3 present the material properties of bundle, matrix

and bundle-matrix interface. Table 4 and Table 5 present numerical

properties of the step and the matrix-bundle 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 In-plane 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 matrix-bundle 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 8-node 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 single-instance model

The 2D single-instance 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 bundle-matrix 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 crack-path below the bundle are

damaged, but the job fails to converge before the bundle fails

completely.

Figure 8 Damage of crack-path 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 thin-bundle 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 thin-bundle model


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allow models to capture the dynamic effect which is the cause of the

problem.

The second problem with the single-instance model is that it is not

able to model the pull-out 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 pull-out possible, the next model was developed.

3.3.2 2D model with cohesive elements modellinginterface

This model was designed to enable fibre bundle pull-out. 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 pull-out 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 pull-out 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 crack-pathThe next approach was to substitute the two cohesive elements on

the predicted crack-path 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 pull-out 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 crack-pathThe last approach taken to this model was to delete the two

cohesive elements on the predicted crack-path. Here again,

similarly to the model with cohesive elements on the crack-path,

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 bundlepull-out. 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 pull-out, 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 ofsurface-to-surface 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 pull-out [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 Pulled-out bundle (von

Mises avg. stresses)


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Figure 22 to Figure 24 present different sequences of events of

matrix-bundle 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 matrix-bundle interface case I

weakened interface (CSDEG degradation (damaged, failed) of

cohesive contact)


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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 matrix-bundle interface case II

weakened interface (CSDEG)


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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 two-instance model enables visualisation of both pull-out

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 pulling-out 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 matrix-bundle interface case III

weakened interface and bundle (CSDEG)


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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 two-instance model. There are

two different elements: a fibre bundle [Figure 27] and a plate with a

bundle-shaped hollow space [Figure 28]. The two are also

connected with surface-to-surface cohesive contact. 27 28

Figure 25 Pull-out of the bundle

positioned at the surface of thespecimen

Figure 26 Pull-out of thebundle embedded into the

matrix


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As predicted, the 3D model enabled crack propagation around thebundle and bundle pull-out [Figure 29].

a) Crack propagating around the

bundle, which is being

pulled-out (von Mises avg.

stresses)

b) Cross-section of a half-cracked

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 time-consuming and power-

demanding.

Figure 28 Cross-section of a 3D

plate

Figure 27 3D fibre bundle

Figure 29 Cracked 3D model


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Basing on the 2D two-instance 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 Non-expected

crack-path (von Mises avg.

stresses)

Figure 31 Too coarse mesh

causing stress concentration (von

Mises avg. stresses)


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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 best-performing 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 matrix-bundle

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

matrix-bundle interface, which was expected. If the bundle was

pulled-out for the baseline-geometry model, then it should do thesame when the thickness of the bundle is larger.


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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

pull-out 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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a prediction can be made based on this figure; as the matrix-bundle

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 pull-out 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 matrix-bundle

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 pulled-out and

the crack propagated further into the matrix, going around the

bundle upwards.


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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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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 fibre-reinforced

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 pull-out.

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 pull-out. The next model had cohesive

elements modelling the matrix-bundle 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 pull-out. The third model created used

cohesive contact to model the matrix-bundle 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 pull-out 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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with the thickness (within the geometries tested) was noticed.

When the bundle was made longer, the failure mode changed from

the fibre bundle pull-out 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 (pull-out 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 pull-out.

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

man-made 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 multi-scale

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 scale-invariant 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

self-similarities 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 non-Euclidean 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 stress-strain relationship, valid for

undamaged zones, and by a stress-crack 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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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 Sierpiski-like

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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3. Piggott, M., Theoretical estimation of fracture toughness offibrous composites. Journal of Materials Science, 1970. 5(8):p. 669-675.

4. Fila, M., C. Bredin, and M. Piggott, Work of fracture of fibre-reinforced polymers. Journal of Materials Science, 1972. 7(9):p. 983-988.

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6. Piggott, M., The effect of aspect ratio on toughness incomposites. Journal of Materials Science, 1974. 9(3): p. 494-502.

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heterogeneous materials based on fractal geometry.Theoretical and Applied Fracture Mechanics, 2006. 46(1): p.46-56.

11. Laizet, S. and J.C. Vassilicos, Multiscale generation ofturbulence. Journal of Multiscale Modelling, 2009. 1(1): p.177-196.

12. Carpinteri, A. and N. Pugno, Mechanics of hierarchicalmaterials. International journal of fracture, 2008. 150(1): p.221-226.

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14. Carpinteri, A., B. Chiaia, and P. Cornetti, On the mechanics ofquasi-brittle materials with a fractal microstructure.Engineering Fracture Mechanics, 2003. 70(16): p. 2321-2349.

15. Soare, M. and R. Picu, An approach to solving mechanics

problems for materials with multiscale self-similarmicrostructure. International Journal of Solids and Structures,2007. 44(24): p. 7877-7890.

16. Newman, W. and A. Gabrielov, Failure of hierarchicaldistributions of fibre bundles. I. International journal offracture, 1991. 50(1): p. 1-14.

17. Piggott, M., Mesostructures and their mechanics in fibrecomposites. Advanced Composite Materials, 1996. 6(1): p.75-81.

18. Wang, M. and N. Pan, Elastic property of multiphasecomposites with random microstructures. Journal ofComputational Physics, 2009. 228(16): p. 5978-5988.

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