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EFFECT OF CURVED FIBERS IN COMPOSITE STRUCTURES
Thomas NORHADIAN
Master Thesis LIU-IEI-TEK-A—14/01890-SE
Department of Management and Engineering
Division of Solid Mechanics
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EFFECT OF CURVED FIBERS IN COMPOSITE STRUCTURES
Thomas NORHADIAN
Supervisor: Kjell SIMONSSON
Examiner: Daniel LEIDERMARK
Master Thesis LIU-IEI-TEK-A—14/01890-SE
Department of Management and Engineering
Division of Solid Mechanics
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Acknowledgements
As all of them have been bringing me help in this project in their own ways, I would like to warmly
thank:
- The University of Bordeaux 1 which offered me to broaden my horizons by sending me
abroad with ERASMUS,
- Linköping University for giving me the chance to live the amazing experience of discovering
the Swedish life, and for giving a so great studying environment,
- Kjell for his kindness, his simplicity, his support and his great and joyful pedagogy, which
made me like Finite Elements!
- All of my teachers so far, who gave me the best of their knowledge and made me able to
reach this point in my studies,
- More especially Alexandre Lasserre, Michel Mesnard and Philippe Larcher for making me like
Mechanics and for making me believe in me,
- Airbus Group for offering me an internship which will surely be determinant in my future
carrier,
- Thomas for believing in me in every occasion,
- Frédéric for his everyday help, his incredible kindness and patience,
- Antoine, Thomas and Tristan for always being available when unsolvable problems appeared
in my modellings,
- Brice, Floriane, Adrien, Najwa, Hanane, Clotilde and Pierre for their everyday happy mood,
- Joakim and Emelie for talking patiently to me in Swedish!
- Marie, Max, Johan, Seppi, Nico, Anouk, Dav’, Bella, Gustav, Sophie, Adèle, Guillaume,
Stefano, Sergio(s), Tim, Carol, Steph’, Oscar, Ju’, Violette, Lennart, Em, and many more for
their priceless friendship and affection, which has always been a great source of motivation
and inspiration along my studies!
I also wish to greatly thank Christer and Lillemor for welcoming me so warmly in their family and for
making me feel like home during these 2,5 years in Sweden.
A very special thanks to Lina for her sweet love, her support in the difficult times I have been
through, and for her everyday-will to make me discover plenty of amazing things.
Finally, I want to especially thank my great parents for their unconditional love, their everyday
support in everything I undertake, and just for being as they are.
Thomas
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Abstract
This thesis has been conducted in order to analyze the effect of curved carbon fibers in composite
structures. Two main topics have been treated: the advantages that such fibers can give when
mapped around a hole, and the properties that they can bring when it comes to vibration damping.
This has been developed through different Finite Element Models, used only in static analyses.
All along these works, it has been shown that the fact of curving the fibers can bring real benefits
when they are included in composite structures. Indeed, mapping them around a hole tends to
decrease the stress concentration phenomenon by around 44% in the studied cases, while simply
including them in a structure increases the damping abilities up to 50%, still in the studied
structures.
These benefits are extremely promising in order to apply them to space structures, and improve the
overall properties of the future composite equipments.
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Table of contents
Acknowledgements ................................................................................................................................. 3
Abstract ................................................................................................................................................... 4
1. Introduction .................................................................................................................................... 8
1.1 Airbus Defense & Space .......................................................................................................... 8
1.2 Background ............................................................................................................................. 9
1.3 Global outline .......................................................................................................................... 9
1.4 Aim of the work .................................................................................................................... 10
1.5 State of the art, other studies ............................................................................................... 10
1.6 Other aspects of the work .................................................................................................... 11
2. Preliminary studies / Basics .......................................................................................................... 12
2.1 Used material ........................................................................................................................ 12
2.2 Steering technique ................................................................................................................ 12
2.3 Mapping singularities ............................................................................................................ 13
2.4 Vocabulary: Pre-preg stripe/fiber ......................................................................................... 14
2.5 Effect of curved fibers ........................................................................................................... 14
2.6 “Spring principle”: curved fibers associated with straight ones ........................................... 14
2.7 FEM Procedure ...................................................................................................................... 17
2.7.1 Modelling Procedure plan ............................................................................................. 17
2.7.2 FORTRAN Program ........................................................................................................ 20
2.8 FE Boundary conditions ........................................................................................................ 23
2.9 FE Mesh model ...................................................................................................................... 25
2.10 Failure criterion ..................................................................................................................... 26
3. Structural strength of structures with holes ................................................................................. 27
3.1 Mechanical behavior of a plate with a hole .......................................................................... 27
3.1.1 Isotropic and homogeneous materials ......................................................................... 27
3.1.2 Anisotropic and inhomogeneous materials .................................................................. 28
3.2 Curved fibers around a hole.................................................................................................. 28
3.3 Mapping strategies/designs .................................................................................................. 29
3.3.1 Design choice ................................................................................................................ 29
3.4 Main analysis: “Intuitive design” .......................................................................................... 31
3.4.1 Mathematical modelling ............................................................................................... 31
3.4.2 FEM analysis/Code ........................................................................................................ 33
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3.4.3 Analysis frame ............................................................................................................... 35
3.4.4 Results and interpretations: ......................................................................................... 36
3.4.5 Conclusion: .................................................................................................................... 40
3.5 Main analysis: “Simple crossing design” ............................................................................... 41
3.5.1 Mathematical Modelling ............................................................................................... 41
3.5.2 FEM analysis .................................................................................................................. 42
3.5.3 Comparability ................................................................................................................ 44
3.5.4 Analysis frame ............................................................................................................... 45
3.5.5 Results and interpretations ........................................................................................... 46
3.5.6 Conclusion: strength of structures with hole ................................................................ 48
4. Structural damping ....................................................................................................................... 49
4.1 Structure damping with curved fibers .................................................................................. 49
4.2 Static analysis for dynamic phenomenon: the energy method ............................................ 49
4.3 Energy method: global procedure ........................................................................................ 50
4.4 Mapping strategies/designs .................................................................................................. 52
4.5 Main analysis: “Intuitive design” .......................................................................................... 53
4.5.1 Mathematical modelling (same as in part 3.4.1) .......................................................... 53
4.5.2 Comparability and “iso-stiffness” ................................................................................. 54
4.5.3 FEM analysis/Codes ...................................................................................................... 55
4.5.4 Analysis frame ............................................................................................................... 56
4.5.5 Results and interpretations ........................................................................................... 57
4.5.6 Conclusion: Structural damping .................................................................................... 57
5. Overall conclusions ....................................................................................................................... 58
6. Further work ................................................................................................................................. 58
7. References .................................................................................................................................... 59
Appendixes ............................................................................................................................................ 60
Appendix A: Nomenclature ............................................................................................................... 60
Appendix B: Geometry demonstrations for part 3.4 & 3.5 .............................................................. 61
Coordinate and ............................................................................................................. 61
Coordinate ............................................................................................................................ 61
Coordinate ............................................................................................................................... 62
Effect of the overlapping between the fibers ............................................................................... 62
Appendix C: Comparability justification ........................................................................................... 64
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1. Introduction
1.1 Airbus Defense & Space
Airbus Defense & Space is the new appellation for the ex-Astrium, space division of EADS (European
Aeronautic Defense and Space).
With more than 15 000 employees around the world, its main activities are divided in 3 groups:
Space & Transportation, Satellites and Services:
- The Space & Transportation division develops and manufactures the Ariane launchers and
deals with the logistic vehicles for the International Space Station.
The division is also a main actor in the production of propulsion systems and space
equipment.
- The Satellites division sees its activities cover the telecommunication and observation
systems of the Earth – in the civilian field as well as in the military field – scientific missions,
Universe exploration, and equipment and navigation systems.
- The Services division offers a broad panel of satellite services, but also geo-information
products and services. The division is also a large provide in military secured communication
services.
With design and production sites spread in many countries in Europe and even in the world, every
location has its own specialization. In this context, the division the author works in at Airbus Defense
& Space Paris-Les Mureaux is dedicated to the design, research and development of a crucial part of
the Ariane 5 space shuttle. Therefore, all works presented here have been oriented towards the
design of this part.
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1.2 Background
Along the years, space shuttles and equipment have been redesigned many times in order to
achieve the best possible performances. In this context, due to their great properties when it
comes to light weight and stiffness, composite materials have acquired a special place in the
space and aeronautical field. In this way, composites have been widely used in many sectors
and in many ways: short fibers, long fibers; interlaced fibers, pre impregnated… Though, all
these kinds of designs with composites were dealing with unidirectional fibers, combining
different orientations in order to handle the applied stresses. However, these past few
years, new needs have come up in order to keep improving the structures’ performances.
Also, some months ago, new designs became possible thanks to the creation/improvement
of some new manufacturing tools, such as the automation of the pre-preg mapping process.
In this context, and made possible by the process automation, the perspective of curving the
composite fibers in one or several planes has recently emerged. This opens of course an
extremely broad range of applications and creates high hopes when it comes to new
achievable properties and mechanical behaviors.
1.3 Global outline
For the reasons stated above, this thesis’ topic has been oriented towards the research and
development of curve-fibered pre-impregnated composites. This should as the final goal, be
dedicated to the improvement of 2 main points:
- The strength of holed-structures
- The damping of structural vibrations.
This thesis will thus be divided into 3 parts. The first one is dedicated to some preliminary
studies, common to both of the analyses. The second will concern the reinforcement of
structural features (such as holes) with curved fibers, while the third one will focus on
improving structural damping, also by curving fibers. At the end of the paper, the references,
as well as the appendixes can be found.
Note1: since this work is related to the defense department, parts of it are strictly confidential.
Because of this, no material data will appear in this paper, and no precise mission/structure parts’
names will be mentioned. All analyses will thus focus on the method and the global results
Note2: a nomenclature of all the widely used variables is attached as the Appendix A.
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1.4 Aim of the work
As stated previously, this thesis is dedicated to two main goals:
- The improvement of the strength of a structure with a hole
- The improvement of the damping abilities of a structure
All of this will be made through Finite Element analyses. Therefore, the aim of the work is the
following:
- To create a FE model of a plate with hole (Figure 1) with properties characteristic
from curved fibers, and run a tensile analysis. Then, to see if these properties lead
to a higher strength than for a structure with only straight fibers. This will be
measured through failure analyses, and comparisons between maximum stresses
and Kt coefficients.
- To create a FE model of a plate without hole with properties characteristic from
curved fibers, and run a tensile analysis. From that, to determine if the structural
damping properties are improved compared to a case with only straight fibers.
As mentioned earlier, this work will start by a preliminary studies part, dedicated to the used
methods and necessary knowledge for both parts coming afterwards.
1.5 State of the art, other studies
As the fiber curving is a very new technique brought by the advanced/automatic fiber placement
(AFP) technique, these studies are very innovative in their field. Therefore, it has not been possible
to find similar papers on this subject. However, related to this topic, the paper of Z.Gürdal and
R.Olmedo has been a first and a precursory step in the studies of varying fiber orientations [1].
Moreover, some works have been conducted in the Netherlands in order to determine a design
methodology for variable stiffness composite materials [2]. In the same area, the team of Larry
Lessard at the University of McGill and some researchers in the UK have been working on the
structure singularities/flaws generated by the AFP [3], [4], [5] & [6].
Figure 1: Tensile FE analysis on a plate with hole
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1.6 Other aspects of the work
The implication of Airbus Defense & Space in the military field has always been a sensitive subject
when it comes to ethical questions. However, despite the controverted picture that one can have in
mind when military aspects are evocated, advanced and performant technologies in these kinds of
fields can also be seen as a guaranty for a certain safety of the population, as far as it is used in
dissuasive purposes.
Moreover, with about 200 space shuttles take-offs every year, questions about the environmental
impacts have been raised. Indeed, as the launchers are reaching the high atmosphere lay-ups to
drop their satellites, a part of their body stays in space while another falls down into the oceans.
Since the number of space missions is increasing every year, the Earth’s high atmosphere lay-ups are
becoming full of a mix between satellites and space equipment (space shuttle, satellites…) debris.
In the same way, the parts falling into the oceans are not made biodegradable and become thus an
important source of pollution. To solve this, some international committees (IADC – Inter-Agency
Space Debris Committee) specially created for this purpose have been, and are still writing some
laws about that issue. Moreover, Airbus is now implicated in the development of some missions
aiming to reduce the amount of spatial debris. Nevertheless, the big amount of fuel consumed
during the different missions can also be considered as a major issue. In this way, improving the
structural strength can be a way to reduce the equipment’s’ weigh, implying thus a decrease of the
total amount of fuel. All of these reasons make therefore the environmental impacts one of the
important aspects to take into account in the very near future.
Finally, one can however state that this topic does not raise any gender related issues.
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2. Preliminary studies / Basics
This part should be seen as an important reference for the rest of the work.
It is composed of several parts common to both FEM studies about the application of curved
fibers. Thus the next parts will be treated as if all the coming concepts and methods have
been already introduced.
2.1 Used material
The studied material, all along this paper, is a composite material of the type pre-preg, with
carbon fibers and epoxy matrix, with the stacking sequence: [90 16 -16 0 -16 16 90]
It can be represented through the following “thickness-view” of the composite material’s
stacking sequence:
Moreover, the specified angles are taken from the vertical axis and the positive angles are
taken in the trigonometric way (it will be recalled all along the report through small sketches
with the angle α, being an “example angle”).
2.2 Steering technique
Curving the fibers is a technique which has been allowed by the automation of the mapping
process (Advanced/Automatic Fiber Placement – AFP). Indeed, it now consists in using
unidirectional pre-preg stripes instead of unidirectional pre-preg sheets (see Figure 3 & 4). A
robot will thus map these stripes on a 3D/2D mold, in order to achieve the desired shape.
Thanks to that and to the fact that stripes are more shapeable than sheets, the design
flexibility is widely increased, and many more geometries become possible.
However, this new mapping process implies other kinds of disadvantages. The coming part is
dedicated to this.
90° 16°
-16°
-16° 16° 90°
0°
Figure 2: Illustration of the stacking sequence
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2.3 Mapping singularities
Singularity: mapping defect due to the automatic fiber placement (AFP)
As the mapping process for pre-preg composites is now automatized, the method has also
been changed in order to make it more accurate. For this reason, pre-preg stripes have been
preferred with respect to big pre-preg sheets. In the same way as before, all the fibers are
parallel to the mean line of the stripe. However, some singularities can be created
depending on the mapped geometry. Some of these are listed here:
Overlap:
Gap:
Triangle:
This might imply some special structural mechanical behaviors, depending on the size and
the exact type of the triangles/gaps/overlaps. Therefore, some studies have been and are
currently conducted in order to characterize these singularities [3] [4] [5].
In this thesis, some of these will be mentioned but it will be shown that the studied cases do
not imply big enough singularities to consider them as critical.
Overlap
Stripes
Gap
Triangles
Figure 3: Pre-preg stripes causing some mapping defects (overlaps, gap, triangles…)
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2.4 Vocabulary: Pre-preg stripe/fiber
Here is a very important vocabulary concept for the coming analyses. The pre-preg stripes
are commonly called “fibers”, since they are composed of carbon fibers, parallel to the
stripe’s mean line. Indeed, fiber steering is made possible by the fact that these stripes are
themselves curved. Here is illustrated how fiber steering is achieved:
Therefore, when it comes to mathematical modelling, mapping design and mapping
singularities, the term “fiber” can be a synonym of “pre-preg stripe”.
2.5 Effect of curved fibers
The main mechanical difference between curved and straight (in the loading direction) fibers
is the way they interact with the matrix around them. Indeed, when stresses are applied on
a curved-fibered structure, more shear stresses will be generated between both
components and the matrix will be more solicited than usually. As the matrix has a much
lower Young’s modulus than the fibers, the global stiffness will be decreased as the fiber
curvature increases!
2.6 “Spring principle”: curved fibers associated with straight ones
In a composite plate, the stress map is mostly determined by the fibers. Thus, in a situation
including curved fibers, the stress map should change quite a lot. In one way, this can be
explained by a mechanically trivial example:
k k
F
1
k 3k
F Spring 1
Spring 2
x = x1 = x2
Figure 5: Illustration of the "spring principle"
Fiber Stripe
Matrix
Figure 4: Fiber steering illustration
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In the case where two springs of the same stiffness (left case of Figure 5) are in parallel, one
can easily demonstrate that the stresses are equal in both. Since the displacement is the
same for both, as well as the stiffness coefficient, the internal force – and thus the stresses
since the section area is also set to be equal – is also equal.
But and so .
However, in the right case (Figure 5) and reasoning in the same way, one will notice that the
stiffer spring (3 times stiffer than the other one) will be submitted to 3 times as much
stresses as the other one:
But and
⇔
The same idea can be applied in the case of fibers. The more they are getting curved, the
more the fibers will interact with the matrix around them by creating shear stresses, when
submitted to tension/compression. Since the matrix has a low stiffness and is more solicited
than before, the achieved stiffness will be lower with curved fibers than with straight ones.
Therefore, according to the “spring principle” mentioned before, the stresses will tend to
pass more through straight fibers than through the curved ones (in a case such as Figure 6
for instance), and a new stress map will be generated.
(1.1)
(1.2)
Figure 6: Both straight and curved fibers in parallel
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Seen from a more microscopic level, it can be assumed that since the curved fibers interact
with the matrix around them by creating shear stresses, a bigger stress-transfer will take
place between fibers through the matrix:
Another important thing is that this stress transfer does not happen only in one plan, but in
several. This implies for example that a situation of fiber i crossing a fiber j might create a
stress transfer between both.
σ
Figure 7: Curved fibers generating shear stresses
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2.7 FEM Procedure
In order to model the studied parts (including curved fibers), the Finite Element Method will
be used. However, since the material under study is neither homogeneous, nor isotropic, a
complete modelling of the internal structure would become too complex. Therefore,
another principle is used and the procedure plan is described below.
2.7.1 Modelling Procedure plan
The modelling procedure in this case is split between several softwares: I-deas [8], used as a
pre and post processor, and Nastran [9] used as a FE solver. Indeed, to model fibers inside a
matrix, CAD can become very tricky, especially in the case of curved fibers. Therefore, it has
been chosen to divide the work into several parts:
- Specify the geometry of the model and mesh it in I-deas. One gets then a text file, with
the list of all the nodes and all the elements in the mesh. This file (mesh file) will be the
base for the rest.
- The idea is then to assign a material property to every element and, at the end, plug it
into NASTRAN and make the calculations. One of the most important things here is that
the assigned material properties are depending on the fiber orientations, in the
element. In the case of unidirectional fibers, even combined with several plies, the fiber
orientations (and thus the properties) would be the same for every element and would
be easy to determine:
For curved fibers, the considered principle is the following: the fiber segment going
through an element is associated with the tangent to the fiber and is thus the derivative
of the fiber’s equation. It can be illustrated as follows:
Fibers
Element
Node
Figure 9: Illustration of a curved fiber in several elements
Figure 8: Elements containing straight fibers
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This is justified by the fact that the curvature radii are extremely big compared to the
elements’ size (unlike the situation in Figure 9). However, the tricky point comes here
since the fact of curving the fibers implies that the fiber orientation (and thus the
properties) is now different depending on the element (see Figure 9). Moreover, in
order to find the tangents’ orientations, the equation of the fiber is necessary. The
formulas will be discussed in the next part.
- Once the fibers’ mathematical model is determined, it will be possible to find the fiber
orientation – the tangent – at any point and thus in every element, depending on its
location. However, as the composite parts are composed of several plies of pre-preg,
this implies a combination of different orientations for every element. In theory, every
combination should be unique, due to the curving of the fibers. Because of this, as
stated before, every element must have its own properties. As an example, one can find
that in the element 243, due to 3 different plies, the different fiber orientations are -16°,
3° and 38°.
Finally, as the number of elements should be really high, calculating the fiber
orientations for every element by hand would become extremely unpractical.
Therefore, a program in FORTRAN has been set up in order to shorten the procedure’s
time.
- Next, material properties have to be assigned to every element, knowing which fiber
orientations there are in each of them. However, it is first needed to convert the only
information we have so far (fiber orientations, plies’ material) into material properties.
This is done by creating a “PCOMP”. This gathers information about the combination of
several plies of different (or not) materials with different (or not) fiber orientations.
Here is an example-case, based on the previous example:
PCOMP n°243 Material of the
ply
Orientation of
the ply
Ply 1 Material 1 -16°
Ply 2 Material 1 3°
Ply 3 Material 1 38°
Note: the term “material” is used to designate the combination of fiber + matrix
Table 1: Example of a PCOMP
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NASTRAN will then be able to calculate the material properties corresponding to this
PCOMP, and assign them to the element. The important point here is that each element
is then considered by the solver (in the calculations) as an element with isotropic
material, having its mechanical properties calculated taking into account the properties
of the fiber and the matrix’s materials, and the fiber orientation (through the
mentioned PCOMPs).
By following the principle stated before, there should be as many PCOMP (material
property) as the number of elements. All the PCOMPs are then stored in a single
material data file. Finally, it is needed, in the output mesh file from I-deas, to associate
every element with its “PCOMP”.
- Some boundary conditions also need to be set in order to start the analysis. This is also
done in I-deas, which will create both a load and a restraint data file.
- The last step is to launch the analysis in NASTRAN, based on the mesh, the material and
the load data files.
- To be able to better visualize the results after calculation, the result file will be exported
to I-deas again and displayed in a more convenient manner.
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2.7.2 FORTRAN Program
The final goal is to get two input files for NASTRAN, allowing the FEM calculations to be run.
These two files are:
- One “mesh” file containing the following main information:
For the nodes: their name, their location
For the elements: their name, their associated nodes, their PCOMP,
their coordinate system orientation
- One “material” file containing the PCOMP data of every element (see previous
part), as well as the material properties of the plies’ material used in the
PCOMPs.
Concerning the load data file, it will be generated by I-deas itself and does not need to be
modified by a program.
In this way, the implemented program will go along the following path – assuming that there
is just one ply:
1. Select one element among the element list
2. Read its nodes'numbers
3. Read each element node's
coordinates
4. Calculate the element center's coordinates (x ; y)
5. Insert x and y in the fiber's
mathematical model to...
6. ...get the fiber orientation (tangent) at
the element's center
7. Store the orientation in one PCOMP
8. Store the PCOMP in the input material file
9. Assign the PCOMP to the corresponding
element in the mesh file
Figure 10: Principle of the code generating the NASTRAN files
21
However, in the case of several plies, several orientations must be calculated for every
element. One way to do this is to insert a small loop in the previous big one.
Instead of calculating a new fiber equation for each ply, the general idea is to use a new
coordinate system in which the fiber (for the ply α° for example) would have exactly the
same equation as before. This allows us to use the exact same mathematical model as
previously. This is illustrated in Figure 11:
As one can observe, the orange curve has the same equation in the coordinate system
R1(x1,y1) as the blue one has in the coordinate system R0(x0,y0).
Following that principle, one must “convert” the coordinates of the element (expressed in
R0) in order to get the coordinates expressed in R1. This is done by some simple calculations:
( ) ( )
( ) ( )
The next step is to apply the exact same procedure as before, i.e. inserting the coordinates in
the fiber equation, and get the fiber orientation. However, this orientation is expressed in R1
– since we used the coordinates expressed in R1. Rather trivially, since the value of is
known, the last step is to subtract to the orientation found in R1. This gives us the fiber
orientation in R0. Therefore, this procedure has to be repeated for every ply and will slightly
modify the previous big loop.
y1 x1
x0
y0
α
Figure 11: Illustration of the coordinate change
22
This can be seen on Figure 12:
This whole procedure needs now to be converted into the FORTRAN programming language.
However, as the programs themselves can differ depending (for example) on the mapping
strategy to be achieved, they will be treated later in the paper, in their own chapter.
1. Select one element among the element list
2. Read its nodes'numbers
3. Read each element node's
coordinates
4. Calculate the element center's coordinates (x ; y)
4.b Choose a α and convert (x ; y) to (xi ; yi)
5. Insert xi and yi in the fiber's mathematical
model to... 6. ...get the fiber orientation (tangent) at
the element's center in Ri
6.b Substract α to get the orientation in R0
7. Store the orientation in one
PCOMP
8. Store the total PCOMP in the input
material file
9. Assign the PCOMP to the corresponding
element in the mesh file
Figure 12: Principle of the code generating the NASTRAN files
Until all the plies
are done…
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2.8 FE Boundary conditions
In an FE analysis, the boundary conditions are essential parameters in order to get realistic
results. For this reason, a special attention has been given to this topic.
All the following simulations and analyses will be carried out for tension loaded rectangular
plates:
A small study has been dedicated to the choice of having a fixed force or a fixed
displacement at the top border of the plate, while the bottom border is set to be clamped.
Also, the top border is fixed in the x-direction and thus no x-contraction is allowed there.
The first aspect which came up is the big deformations created on the top border’s corners
by a fixed (tensile) force:
y
x
Figure 13: Illustration of a tensile test
Figure 14: Illustration of the FE defaults
24
This is due to the fact that the fixed force is modelled by applying a constant force F to each node of the top border. Therefore, the forces applied on the corners are not “distributed” in the same way as those applied on the “middle” nodes. It can be explained on the Figure 15: Here it can be seen that the orange forces, applied on the corner nodes are “distributed” on a smaller area (orange lines) than the other forces (blue, green and purple lines). Indeed, since the orange forces are not located in between two elements, the load has to be applied only on one element. Therefore, the total force applied on the corner element is bigger than the total force applied on the “middle” elements. Because of this, the effect of the forces is higher at the plate corners. As a result, these big deformations lead to very high stresses at these places and might alter the studies concerning the maximum stresses. For this reason, a fixed displacement would be preferred. However, if such a plate is considered being a part of a total big loaded structure, a fixed displacement would not correspond to the reality. Therefore, the high stresses at theses points will just be ignored and not included in the analyses. Furthermore, if we consider the plate as a part of a big structure, it would be relevant to lock the displacement of the vertical borders in the x-direction, in order to get the most accurate results. However, due to the fact that these analyses further are to be compared with real samples, this will not be done. Indeed, locking only the x-displacement of a whole border is a hardly realizable task. For this reason, the vertical borders will be set as free (in all directions) in the coming models. Finally, the bottom border is clamped even if it might prevent the plate from contracting at these places, since it will correspond to the future tests. As a conclusion a sketch showing the boundary conditions applied on the subsequent modelled plates (with F>0, with or without hole) can be found in Figure 16 below:
F
Figure 16: Boundary conditions applied to the coming models
y
x
Figure 15: Forces distribution
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2.9 FE Mesh model
The meshing process for the further analyzed plates is rather simple, as the models themselves do
not have a complicated geometry. Indeed, the part is not big and is modelled in 2D. Therefore, a fine
and homogeneous mesh can be allowed everywhere without any risk of having huge calculation
times. However, a special attention has been given to the symmetry of the mesh towards the Y axis
(loading direction). Indeed, this becomes necessary in order to better see if the results are coherent:
the stress maps should be symmetric across the Y-axis, when one looks at the plies 1, 4 and 7 (90°, 0°
and 90°). Indeed, the mapping design is itself symmetric across the Y-axis on these plies.
Moreover, all elements are quadrilaterals of type CQUAD4, and have been “checked” in order to
make them fill the usual quality criteria about their shape (skew, warp, aspect ratio…). This aims to
insure a good accuracy of the results, which would not be guaranteed if the elements are shaped too
far from a regular square. On Figure 17 is represented the used mesh models (for a plate with hole,
and a plate without hole). The amount of elements used is respectively 11807 for the left model, and
11189 for the right one. The dimensions for the plates are 5000x2000 with a hole of Ø300mm in the
left case. These geometries will also be reminded later in the paper.
Figure 17: Meshes used in the coming models
y
x
26
Finally, a convergence study has been carried out in order to determine if a finer mesh would be
relevant or not. During this analysis, it has been noticed that the stress value in the element near the
hole keeps increasing when the mesh gets finer. This can be explained by the fact that the stress
value in the element is an average of the stresses in the whole zone that it covers. On Figure 18, it
can be seen that the stress value in a small element will be higher than the stress value in a big
element:
Due to this it has been concluded that fine meshes as the ones above show good enough precisions,
and do not need to be even more refined.
2.10 Failure criterion
In the coming analyses, failure studies will be conducted in order to determine the strength of the
considered structures. As the materials under study are exclusively composite materials, an adapted
failure criterion needs to be used. Here, the Hill criterion is chosen since it has been widely used at
Airbus for all kind of studies. Moreover, the material data needed for its applications are easily
accessible. Finally, the accumulated experience in this context makes it more appropriate than other
criteria.
It is considered that if the value of H reaches 1, then failure will occur:
(
)
(
)
(
)
With This criterion is a good way to realize how close to failure – and thus how strong – the structure is. Hence, the conclusions will mostly be based on the comparison of “Hill numbers”.
(1.3)
σ
σaverage
σaverage
Distance
Big element
Small element
Figure 18: Consequence of a refined mesh
27
3. Structural strength of structures with holes
In order to develop techniques dedicated to the strength of structures with holes, it is
necessary to divide the work into several distinct parts. First, a study of the mechanical
behavior of a plate with hole will be conducted in order to better understand the induced
phenomenon – for isotropic materials as well as for composite materials. After that, the
question of how curved fibers can be used to improve a structure with hole(s) will be
discussed. Next, the most interesting curving designs will be chosen. Finally, the modelling
procedure as described in the previous part is presented.
3.1 Mechanical behavior of a plate with a hole
3.1.1 Isotropic and homogeneous materials
As mentioned earlier in the “introductory” part, holes and other geometric features will
create stress concentrations. The reason for that can be explained and seen in many ways.
For instance, as a load is applied on a plate, it is easily understandable that due to the hole,
the transmitted stresses will be concentrated within a smaller section (2xa) near the hole:
At a macroscopic level, one can explain the phenomenon by the lack of material (due to the
hole) to transmit the stresses. Therefore, the stresses need to “move” to a place where they
can find a path to be transmitted to the other side of the plate. This way, a stress transfer
will happen towards the outer border of the plate. This will happen until the stress can find
an “easy path” from one side to the other. For this reason, an accumulation of stresses
appears near the hole, which decreases towards the outer border. This can be seen on the
Figure 19b.
b
a
y
x
Figure 19a: Plate with a hole Figure 19b: Stresses repartition in a holed plate
28
Moreover, a 3D study would also consider the effect of a hole on the stress components out
of the plan xy. However, as the studied plates are thin compared to their width and length, a
plane-stress state is assumed. Thus σzz, σyz and σxz are considered to be equal to 0 and an
exclusive attention will be given to the xy-plane stress maps.
3.1.2 Anisotropic and inhomogeneous materials
In the studied case, the material is not homogeneous. However, the principle is rather
similar if one looks at it for a “unidirectional” and “symmetric” case. As the fibers above and
under the hole are not continuous, the stresses cannot be transmitted directly from the top
border to the bottom one.
Therefore, some shear stresses will be generated in the matrix in order to transmit the
loading from the cut fibers to the continuous ones. At the end, as seen on Figure 20, the
situation will end up being similar to the one before: a big stress concentration, near the
hole.
3.2 Curved fibers around a hole
Now if the “spring principle” from the part 2 is applied to a situation with a hole, this can be
an effective way to decrease the stress concentrations around the hole. It may thus improve
the part’s strength properties. Indeed, as curved fibers are placed around the hole, the
stiffness of the material at those places will be changed, and some of the stresses near the
hole will be transferred to the straight fibers further away from the hole.
y
x
Figure 20: Stress concentration in a composite structure
Figure 21: Curved fibers around a hole
29
3.3 Mapping strategies/designs
As one decides to study the effect of curved fibers, it becomes necessary to first investigate
which kind of mapping designs to focus on. Indeed, there are many ways to curve the fibers,
and the properties can vary quite a lot depending on the chosen design. As an example, here
are some mapping strategies that can be utilized:
Since some hypotheses have been made about the mechanical behavior of a curved fibered
structure, it will be possible to choose the most relevant mapping strategies based on these
hypotheses.
3.3.1 Design choice
Thanks to the design principles mentioned earlier, 2 types of mapping strategies have been
chosen. They have been seen as the best compromises in order to:
- Move the stresses further away from the hole by facilitating stress transfers
towards the outside borders.
- Avoid too many/big singularities
In this way, the following strategies have been chosen:
1. « Intuitive design »
- Every fiber symmetric across the horizontal
axis.
- 2 sub-designs possible (to choose)
- Gaps/overlaps between the pre-preg stripes
depending on the chosen sub-design
OR
Figure 23: Intuitive design
Figure 22: Different imaginable designs
30
Now that the global ideas have been set, some parameters/rules are remaining (for a
chosen hole diameter dh = 300mm) in order to decide the exact shape. In the next sketches,
the fibers will be represented as horizontal in order to make the modelling more intuitive.
Circles’ tangency & radii
In order to go around the hole, 4 (2 in Design 2) curvature circles are used. The first
assumption made is that all curvature circles have the same radius R. Moreover, the
curvature circle 1 has its center (O1) placed on the vertical axis, allowing the fiber to be
horizontal (vertical in our previous drawings) at x = 0. Its location must also be such that the
first fiber is tangent to the hole at x = 0. Finally, both curvature circles are tangent to each
other, allowing the fiber equation to be continuous.
Constant radius
Still in order to make the modelling smoother, the radius R is considered constant whatever
the value of n (discrete fiber number) is.
2. « Simple crossing design »
- Crossings overlaps
- Stress transfer in several planes
R1 R2
O2
O1
O
y
x n = 1 n = 2 n = 3
Design 2
Design 1
Figure 24: Simple crossing design
Figure 25: Curved fibers’ definition
31
3.4 Main analysis: “Intuitive design”
3.4.1 Mathematical modelling
This type of fiber (pre-preg stripe) profile has been made taking into account the minimum
curvature radius that the material fibers are able to handle. For the used material, this
minimum radius is 1500mm. Moreover, in order to go around as close as possible to the
hole, the curvature radii have been set to the smallest possible value. Thus, in the following
equations, R = 1500mm. Finally, as there is symmetry across the local X axis, the model is
made only for the stripes located on the upper half of the plate. The stripes on the lower half
will be treated in the code with a simple procedure detailed later on.
| | | |
√
√
√ | | |
|
√
The mathematical/geometrical demonstrations to determine the centers’ coordinates, as
well as some explicative sketches, can be found in Appendix B.
(2.1)
(2.2)
(2.3)
(2.4)
Figure 26: Modelling of the Intuitive design
32
If the derivatives are performed, we get the tangent-to-the-curve’s coefficients:
| | | |
√
√
√ | | |
|
In order to make this model simpler, the vertical distance between the stripes is always the
same, whatever the value of x is. This is done by linking the fiber equations in the following
way:
Because of this, the stripes’ derivatives are always the same, whatever the value of n is.
Furthermore, even if the vertical distance is constant, it can be noticed that the distance
“normal-to-the-stripe”, between every stripe, is not always the same along the x axis.
Therefore, planning a perfect stripe alignment where they are straight (and above the hole)
implies some overlapping where they are curved. However, it is possible to demonstrate
that this singularity is not of first importance (see Appendix B). Indeed, the overlapping is
worth, at most – with a curvature radius of 1500mm and a Φ300mm hole – 6% of a fiber
width. Therefore, this mapping singularity will be neglected for the moment.
(2.9)
(2.5)
(2.6)
(2.7)
(2.8)
33
Moreover, it is necessary, in order to keep the model close to the reality, to have straight
fibers (pre-preg stripes) on the outer borders of the plate. To do that, straight stripes are
introduced from a certain point. Therefore, a distance between the hole’s center and the
first straight stripe has to be set. In this case, yf = 300 mm (Figure 27b). However, combining
straight stripes and curved ones creates a zone empty from any material (Figure 27a).
For this reason, this zone will be filled with cut straight stripes (green fibers on Figure 27b).
3.4.2 FEM analysis/Code
As in all the coming Finite Element analyses, a program will be used in order to assign
material properties to every element, depending on their location. As discussed previously,
these properties are calculated depending on the fiber orientation in each element. Finally,
this orientation is determined from the stripe mathematical model set above, and from the
element’s coordinates. Here are detailed some important ideas, to be added to the main
principle from Part 2.7.
“Empty zone”
As the stripes are curved with a minimum curvature radius and as no cut stripes are
added near the hole, this kind of design creates an area without any material. The
reason for not filling this zone with cut straight fibers is simply that the necessary
extremely short fibers for this process cannot be delivered by the AFP robot.
Yellow
fibers
Figure 28: Illustration of the "empty zone"
Figure 27a: 2 choices for the Intuitive design Figure 27b: 2 choices for the Intuitive design
yf
34
This “empty zone” changes for every ply and will thus be filled by the next plies.
However, this zone needs to be modelled as a zone which cannot handle loads.
This is done by creating an “if-loop” and stating that:
“if the current element is located between both yellow stripes (Figure 28) – in the
curved area – then, assign a material with weak properties to the element”
0° direction
In the aeronautical sector, the 0° direction is the vertical direction, and is not the
same as in Ideas or usual mathematical conventions. Therefore, the lay-up specified
angles need to be adapted to the model which has been developed in a “horizontal
way”. This is simply made by adding 90° to the lay-up rotation angles in the code.
Upper/Lower half-plate
The mathematical model has been developed only for the fibers (and thus the
elements) on the upper half-plate (see Mathematical Modelling from 2.d)
Because of this, the stripes “under” the hole are modelled by applying a “mirror
code” stating that:
“if the element is located on the lower half-plate (y<0), then set the orientation to
the opposite to the one with the same x, but y>0”. This can be justified on Figure 29
where it is seen that the angle – to the horizontal purple dotted line – of the red
tangent at (x0;y>0) is the opposite to the one of the green tangent (x0;y<0).
However, this is valid only in the coordinate system of the ply, and not in the
absolute one. Therefore, the stated calculation has to be done first in the relative
coordinate system, before converting the angle to the absolute coordinate system.
x0
y = 0
Figure 29: Orientation of symmetrical fibers
35
3.4.3 Analysis frame
The analysis is a comparative study, and has been conducted step by step, always in order to
see the effect of curved fibers on the stress map. For this reason, 3 models have been
considered, all of them with the same geometry, the same boundary conditions and the
same stacking sequence [90 16 -16 0 -16 16 90]s:
- One reference model (0) only with straight fibers/stripes
- One “curved model” (1) with only the middle ply (0°) including curved
fibers/stripes around the hole
- One “curved model” (2) with all plies including curved fibers around the hole
Here, the plate’s geometry has been set big enough in order to avoid any border effect in the
zone of interest (near the hole). Moreover, such dimensions correspond quite to the type of
structures the curved fibers are dedicated to. In this case, the geometry, the boundary
conditions and the mesh look as follows:
Finally, the applied force F is a tensile force on the top border, and has an arbitrary value of 71000N – the absolute value is not important since the analysis is linear and since it is a comparative study. As mentioned in Part 2.8, the bottom border is clamped. The results have been displayed in a comparative table (see next page). Furthermore, these results will be complemented by comparative pictures of the stress maps.
5000 mm
2000 mm
Φ 300
1000
2500
α
0° α°
F
Figure 30: Plate’s geometry/mesh
y
x
36
3.4.4 Results and interpretations:
Note: Specifications of the table can be found on the next page, just before the results’ interpretations.
Case n° 0 1 Variation
(n°1/n°0) 2
Variation
(n°2/n°0)
Yf - 300 mm 300 mm 300 mm 300 mm
Curved plies None 0° 0° All All
Max Y Stress 1 8,61E+06 7,93E+06 -8% 7,15E+06 -17%
Max X Stress 1 1,88E+07 1,46E+07 -22% 8,90E+05 -95%
Max XY Stress 1 8,31E+06 7,25E+06 -13% 6,13E+06 -26%
Max Hill 1 5,28E-02 4,48E-02 -15% 4,05E-02 -23%
Max Y Stress 2 2,11E+08 1,92E+08 -9% 1,28E+08 -39%
Max X Stress 2 1,99E+07 1,76E+07 -12% 2,34E+07 18%
Max XY Stress 2 6,30E+07 5,59E+07 -11% 3,74E+07 -41%
Max Hill 2 2,98E-02 2,63E-02 -12% 5,76E-02 93%
Max Y Stress 3 2,11E+08 1,92E+08 -9% 1,28E+08 -39%
Max X Stress 3 1,99E+07 1,76E+07 -12% 2,34E+07 18%
Max XY Stress 3 6,30E+07 5,59E+07 -11% 3,74E+07 -41%
Max Hill 3 2,98E-02 2,63E-02 -12% 5,76E-02 93%
Max Y Stress 4 2,20E+08 2,04E+08 -7% 1,37E+08 -38%
Max X Stress 4 2,42E+06 7,63E+06 215% 9,25E+06 282%
Max XY Stress 4 8,31E+06 2,39E+07 188% 2,69E+07 224%
Max Hill 4 2,42E-02 1,97E-02 -19% 1,32E-02 -45%
Kt 4 2,99 2,78 -7% 1,86 -38%
Y-displacement 1,96E-03 1,96E-03 0% 1,98E-03 +1%
Table 2: Results of the Intuitive design
37
Notes about the table: Since the lay-up is symmetric, the studied plies are 90°, 16°, -16° and 0°. They are numbered
in this order (1 to 4) and a specific color has been assigned to each of them.
The row “Y-displacement” is there to better visualize the influence of curved fibers on the stiffness of the global structure.
The “variation” columns aim to better show the changes created by the inclusion of curved fibers.
An important parameter to be noticed is the coefficient Kt. It gives information on the stress concentration coefficient, and therefore in which extent the stresses are redistributed in the rest of the plate.
As mentioned before, yf is the distance between the hole’s center and the first straight fiber.
All results are displayed in the International Unit System (stresses in Pa, displacements in meters)
Interpretations of the results for Case n°1 (Design1, 0° ply curved):
4th ply:
- Max Y stresses: as curved fibers have been introduced, the stresses should be redistributed and the stress concentration near the hole should be less “intense” (this is shown by Kt). Since the maximum Y stresses are located near the hole, they should be lowered as the fibers are curved (the blue zone in the middle is the “empty zone”):
- Max X stresses: a fiber with such a curved profile, when applying a vertical tension to it, will tend to get closer to a straight vertical fiber.
Thus, its X-displacement will be more important than the displacement of the “already-straight” fibers, which is only due to the Poisson’s ratio effect. This creates a sort of X-tensile-loading between the elements of the curved fibers, and the elements of the straight fibers. Moreover, as the fibers are curved, their orientation is closer to 90°, which is also the X loading direction. The elements’ X-stiffness increases and offers thus a “better” path to the X stresses, which increase. Therefore, the maximum X stresses are located at the place where the fibers are the most curved:
Figure 31: Effect of pulling on a curved fiber
a)
b)
38
However, the point of maximum X stresses is not the same as the one of maximum Y stresses. At that point even, the X stresses are equal to zero.
- Max XY stresses: as stated earlier, curving the fibers makes their orientation deviate from the loading direction, and makes them carry more shear stresses – especially concentrated at the places where the fibers’ angle are the biggest:
- Max Hill number: The point of maximum Hill number is situated on the hole’s border. There, all the stresses tend to decrease. Therefore, the Hill number decreases too.
3rd & 2nd plies:
- Max Y stresses: the maximum Y stresses are located around the hole (rather similar to the 4th ply). Since the global material is softened by adding curved fibers on the 4th ply, the stress paths will tend to go away from the hole and load the outer areas. Hence we end up with a diminution of the Y stresses near the hole.
- Max X stresses: due to the X-stiffness brought by curved fibers in the 4th ply, the X stresses in the other plies tend to decrease.
- Max XY stresses: shear stresses are coming from the fact that fibers are loaded in a direction which is not theirs. Therefore, these shear stresses are directly related to the loading intensity. Because the maximum Y stresses are decreasing, and since the maximum shear stresses are located at the same place as the maximum Y stresses, the maximum shear stresses also tend to decrease.
- Max Hill number: The maximum Hill number is located at the same “place” as all the maximum stresses. Since all of them decrease, the Hill number should decrease as well
1st ply:
- Max Y stresses: the maximum Y stresses are located around the hole. Since the global material is softened by adding curved fibers on the 4th ply, the stress paths will tend to go away from the hole and load the outer areas. Hence we end up with a decrease of the Y stresses near the hole.
- Max X stresses: since the X stresses on the 4th ply are increasing a lot, it tends to unload the other plies, which see their X stresses decrease.
- Max XY stresses: these shear stresses are too small compared to the ones in the other plies to make a significant contribution
- Max Hill number: The maximum Hill number is located at the same “place” as all the maximum stresses. Since all of them decrease, the Hill number should decrease as well
Figure 32: a) Y stress map b) X stress map c) XY stress map c)
39
Interpretations of the results for Case n°2 (Design1, All plies curved):
4th ply:
- Max Y stresses: as curved fibers have been introduced in all the plies, the zone near the hole should be less inclined to handle loads. Thus, the stresses are redistributed and the stress concentration near the hole is much less “intense” (this is shown by Kt). Since the maximum Y stresses are located near the hole, they are also lowered as the fibers are curved:
- Max X stresses: as curved fibers are introduced in the plies 2 and 3, an “empty zone” also appears. This zone is covering a part of the highly X-loaded zone of the 4th ply. Therefore, the 4th ply sees its max X stresses increase compared to when the 2nd and 3rd plies were not curved.
However, the point of maximum X stresses is not the same as the one of maximum Y stresses. At that point, the X stresses are decreasing.
- Max XY stresses: due to the “empty zone” created by the plies 2 and 3, some of the elements where the maximum shear stresses were located (in ply 4) are loaded more than before. This implies therefore a higher maximum shear stress at these points:
- Max Hill number: The point of maximum Hill number is situated on the hole’s border. There, all the stresses tend to decrease. Therefore, the Hill number decreases too.
3rd & 2nd plies:
- Max Y stresses: the Y stresses are totally redistributed and there is no big stress concentration. Therefore, the maximum Y stresses are located a little bit further from the hole and the plies logically see their maximum Y stresses going down.
- Max X stresses: in these plies, the fibers see their orientation getting closer to 90° (the X-loading direction), due to the fiber curving. This implies a higher X-stiffness at some places, which increases the maximum X stresses.
- Max XY stresses: since the shear stresses are directly linked to the loading, a decrease of the Y stresses near the hole will imply a decrease of the max shear stresses (which were also located near the hole). Moreover, curving the fibers on these plies orientate them further from the loading direction, away the hole. Combining these 2 reasons gives a reduction of the maximum shear stresses, and a new location for these.
- Max Hill number: Since curving all the plies creates an “empty zone” for every ply, there will be some elements which will be located where only the 2nd (or the 3rd) ply exhibits some material.
b)
c)
a)
d)
Figure 33: a) Y stress map for ply 4 b) X stress map for ply 4 c) XY stress map for ply 4 d) XY stress map for plies 2 and 3
40
Therefore, these elements will be isolated and will be submitted to much more stresses than in the previous case. At that places, the Hill number increases radically. Consequently, the strength of the whole structure even tends to decrease!
1st ply:
- Max Y stresses: the “empty zone” created by the curved fibers acts like an extremely large hole. This implies that the ply becomes much weaker than previously. Consequently, less stresses will go through this ply, and the max Y stresses tend to decrease.
- Max X stresses: since the X stresses in all the other plies are increasing quite a lot, the first ply tends to be unloaded (especially in the zones where the other plies are over-loaded). Therefore the maximum X stresses tend to decrease as well. Moreover, the fact of curving the fibers orientates them further from the X-loading direction. Thus, the X-stiffness decreases and the max X stresses as well.
- Max XY stresses: the fact of curving the fibers orientates them in a direction which creates more shear stresses than if they were straight. However, having a very low loading on this ply makes them decrease. The combination gives an overall decrease compared to the reference model.
- Max Hill number: As most of the stresses decrease in this ply, the maximum Hill number does the same.
Comparison conclusions:
- Curving the fibers in one ply can be very beneficial for the same ply, in the way that the stress map is “redrawn”. When studying the case n°1, it can be shown that the Hill number in the weakest ply shows a gain of 15% strength!
- Moreover, curving the fibers also implies a stress redistribution in the other plies, which can be beneficial. This should of course be taken into account.
- Because of this, the consequences of having several curved plies in one lay-up can become hard to predict. In the case studied previously, it even lowers the performances of the whole lay-up due to the superposition of “empty” zones creating a stress concentration.
For all these reasons, the case n°1 (0° ply curved only) will be the one chosen from this analysis.
3.4.5 Conclusion:
Without trying to optimize the curving design, it has been shown that the Design n°1, in the case under study, leads to an overall gain of 15% strength! This confirms the hypotheses made at the beginning, and shows how big the improvements can be if such a mapping design is used in a composite structure.
41
3.5 Main analysis: “Simple crossing design”
3.5.1 Mathematical Modelling
As previously, the curvature radius is equal to 1500mm.
In this case, the fiber/stripe is set to become straight and with a derivative equal to 0
for .
However, for , the equations are the same as for the case “Intuitive design”.
√
√
√
Once more, the vertical distance between the stripes is always the same, whatever the value
of x is. And we get still:
(2.10)
(2.11)
(2.12)
(2.13)
Figure 34: Modelling of the Simple crossing design
42
If the derivatives are performed, we get:
√
√
In the same way as before, it is possible to state that the effect of the overlapping is
negligible.
Finally, as before, a distance yf needs to be set in order to decide where the first straight
stripe is:
3.5.2 FEM analysis
As in all the coming Finite Element analyses, a program will be used in order to assign
material properties to every element, depending on their location. As discussed previously,
these properties are calculated depending on the fiber orientation in every element. Finally,
this orientation is determined from the mathematical stripe model set above.
“Empty zone”
In the same way as before, there is an empty zone (the exact same one as in the
Intuitive Design from Part 3.4) around the hole which needs to be modelled. It is once
more included in the code by assigning a weak material to the elements in that zone.
(2.14)
(2.15)
(2.16)
(2.17)
yf
Figure 35: Determination of the 1st straight fiber's location
43
Superposition principle
In order to compensate for the fact that the fibers are not symmetrical across the y axis,
every mapped ply will be composed of 2 “fiber patterns” which are symmetrical across
the y axis. They are completed by straight fibers on the plate borders. This is illustrated
in the Figure 36:
However, as crossings on the same ply are hard to model, a superposition principle is
introduced in a special way. It is considered that the fibers symmetrical to each other
across the y axis are making a pair. They have therefore been modelled as the
superposition of a straight fiber and of a fiber from the “intuitive design”. This is
illustrated as follows:
Therefore, the code will “split” every pair of fiber on two plies: one ply with the curved
green fiber from above, and one ply with the purple straight one. It should however be
recalled that the way to model this is arbitrary, and that some other modelling method
could beneficially be tested in the future, in order to confirm/infirm the coming results.
1 ply
Figure 36: Definition of 1 ply
Figure 37a: Modelling of 1 ply
44
At the end, every pair of curved fibers generates two plies in the material file:
Finally, the elements for which no fibers (none of the green or blue – for Ply 1 – or purple – for Ply 2)
are going through are filled with the weak material used for the “empty zone”. These are
represented by orange stripped areas. This solution has been chosen since the FE software can only
take into account 1 orientation of fiber per ply. However, every crossing creates 2 orientations in the
element. Therefore, it became necessary to create 2 plies in order to take into account the crossing.
Intuitively, this choice implies that the model does not entirely correspond to the reality. Despite
this, the author and the company believe that this solution is giving a fairly good estimation of the
structure behavior.
3.5.3 Comparability
As it has been seen in the previous parts, this design includes some overlapping between the stripes.
In this case, this implies that the amount of material in the Design 2 will be more important than in
the Design 1.Therefore, depending on the amount of material added, this could become a handicap
due to the difference of weight of the total structure. However, in the plates under study, the
amount of material added compared to the other designs is small enough to be neglected. Indeed,
as it is demonstrated explicitly in the Appendix C, the difference is less than +3%. For this reason, this
aspect will not be taken into account and the different designs can be compared with each other.
2 plies in the code
Figure 30b: Modelling 1 ply as 2 plies in the code
Code’s ply 1 Code’s ply 2
45
3.5.4 Analysis frame
The analysis is once more a comparative study, in order to see the effect of curved fibers on
the stress map. In this part again, 3 models have been considered, all of them with the same
geometry, the same boundary conditions and the same stacking sequence [90 16 -16 0/2]s:
- One reference model only with straight fibers
- One “curved model” with only the middle ply (0°) including curved fibers
around the hole (the same one as the one in Design 1)
- One “curved model” with only the middle ply (0°) including curved fibers
around the hole (with the studied Design 2 this time)
Since the previous design showed that curving all the plies generates too many stress
concentrations, no model with only curved plies is studied here. Here again, the plate’s
geometry has been set big enough in order to avoid any border effect in the zone of interest
(near the hole). Once more, such dimensions correspond quite to the type of structures the
curved fibers are dedicated to. In this case, the geometry, the boundary conditions and the
mesh look as follows:
Finally, the applied force F is a tensile force on the top border, and has once more an arbitrary value of 71000N. As mentioned in Part 2.8, the bottom border is clamped. The results have been displayed in a comparative table (see next page). Furthermore, these results will be complemented by comparative pictures of the stress maps.
5000
mm
2000 mm
Φ 300
1000
2500
α
0°
α°
F
y
x
Figure 38: Plate's geometry/Mesh
46
3.5.5 Results and interpretations
Note: Specifications of the table can be found on the next page, just before the results’ interpretations.
Design n° Reference
case 1
Variation
(n°1/n°0) 2
Variation
(n°2/n°0)
Yf - 300 mm 300 mm 300 mm 300 mm
Curved plies None 0° 0° 0° 0°
Max Y Stress 1 8,61E+06 7,93E+06 -8% 6,42E+06 -25%
Max X Stress 1 1,88E+07 1,46E+07 -22% 1,67E+07 -11%
Max XY Stress 1 8,31E+06 7,25E+06 -13% 6,60E+06 -21%
Max Hill 1 5,28E-02 4,48E-02 -15% 2,94E-02 -44%
Max Y Stress 2 2,11E+08 1,92E+08 -9% 1,72E+08 -18%
Max X Stress 2 1,99E+07 1,76E+07 -12% 1,66E+07 -17%
Max XY Stress 2 6,30E+07 5,59E+07 -11% 5,52E+07 -12%
Max Hill 2 2,98E-02 2,63E-02 -12% 2,22E-02 -26%
Max Y Stress 3 2,11E+08 1,92E+08 -9% 1,72E+08 -18%
Max X Stress 3 1,99E+07 1,76E+07 -12% 1,66E+07 -17%
Max XY Stress 3 6,30E+07 5,59E+07 -11% 5,52E+07 -12%
Max Hill 3 2,98E-02 2,63E-02 -12% 2,22E-02 -26%
Max Y Stress 4 2,20E+08 2,04E+08 -7% 1,79E+08 -19%
Max X Stress 4 2,42E+06 7,63E+06 215% 7,55E+06 212%
Max XY Stress 4 8,31E+06 2,39E+07 188% 2,34E+07 182%
Max Hill 4 2,42E-02 1,97E-02 -19% 1,53E-02 -37%
Kt 4 2,99 2,78 -7% 2,44 -19%
Y-displacement 1,96E-03 1,96E-03 0% 1,95E-03 -0,5%
Table 3: Results for Simple crossing design
47
Notes about the table (same as in Part 3.4.4): Since the lay-up is symmetric, the studied plies are 90°, 16°, -16° and 0°. They are numbered
in this order (1 to 4) and a specific color has been assigned to each of them. However, in this case, since the 4th ply is split into 2 plies, the values in the table correspond to the maximum value of both plies.
The row “Y-displacement” is there to visualize better the influence of curved fibers on the stiffness of the global structure.
The “variation” columns aim to better show the changes created by the inclusion of curved fibers.
An important parameter to be noticed is the coefficient Kt. It gives information on the stress concentration coefficient, and therefore in which extent the stresses are redistributed in the rest of the plate.
As mentioned before, yf is the distance between the hole’s center and the first straight fiber.
All results are displayed in the International Unit System (stresses in Pa, displacements in meters)
Interpretations of the results for Case n°2 (Design2, 0° ply curved):
As the gains from the reference model are similar to the ones of Design 1, the interpretations will be
directed to the comparison of Design 1 with Design 2.
4th ply:
- Max Y stresses: since there is more material which is able to take the loads, the
stresses are distributed on more fibers. Therefore, the Y stresses tend to decrease.
- Max X stresses: in the same way as for the Y stresses, more material implies a
decrease of the X stresses.
- Max XY stresses: since the global Y loading per element tends to decrease, the shear
stresses do the same.
- Max Hill number: as a result of a decrease of all the stresses, the Hill number goes
down and the global ply shows a higher strength.
3rd & 2nd plies:
- Max Y stresses: the intermediate plies are unloaded since the 4th ply is able to take
more loads. Thus, the max Y stresses decrease.
- Max X stresses: in the same ways as for the Y stresses, the 4th ply can take more loads
and unloads the other plies. Thus the X stresses in the 2nd and 3rd plies decrease.
48
- Max XY stresses: following the same principle as before, less Y stresses combined with
fibers oriented diagonally compared to the loading direction implies less shear
stresses.
- Max Hill number: since all the stresses tend to decrease, the max Hill number does
the same.
1st ply:
- Max Y stresses: following the same principle as for the intermediate plies, the reinforced 4th ply tends to unload the others
- Max X stresses: the fact of having a decrease of the X-stresses in the -16° and 16° plies tends to lead to an increase of the ones on this ply
- Max XY stresses: these shear stresses are too small compared to the ones in the other plies to make a significant contribution
- Max Hill number: The maximum Hill number is located at the same “place” as all the
maximum stresses. Since all of them decrease, the Hill number should decrease as
well
3.5.6 Conclusion: strength of structures with hole
Once more, this analysis has been conducted without trying to optimize the curving design.
However, even without that, the weakest ply has increased its strength of 44% from the reference
model! Therefore, the benefits of “Simple crossing design” are much higher than the ones of the
“Intuitive design”. Indeed, from the reference model, the improvements compared to the “Intuitive
design” are +33% of strength. Thus, it can be logically concluded that the “Simple crossing design” is
the one to be chosen in order to improve as much as possible, the performances of a structure with
hole.
However, it should be recalled that a simplification has been made in the model, in order to adapt
the analysis to the FE solver. Therefore, even if the global structural behavior of the structure might
be well represented, the “numbered” results may differ from the reality, and might need to be
confirmed with some concrete tests.
49
4. Structural damping
The Ariane space shuttles are widely used in some freight purposes, such as to carry space
satellites and other important structures. Due to the extremely huge power developed by
the launchers during take-off, a big amount of vibrations are transmitted through the
structures to the satellites. As some studies showed that the main loads applied on the
space shuttles’ payloads are caused by these vibrations, it has been decided to investigate
the effect of curved fibers on the structural damping properties.
4.1 Structure damping with curved fibers
As discussed earlier in Part 2, curved fibered composites show a different mechanical
behavior due to the fact that the matrix is more solicited than with straight fibers. Indeed,
the global stiffness of the material will be decreased as the fiber curvature increases.
However, as really high mechanical performances need to be achieved, an important
stiffness decrease cannot be allowed. Therefore, the damping ability of curved fibers will be
tested, having for constraint an “iso-stiffness”. In this way, the damping abilities will not be
linked to the stiffness decrease, but only to the fact that the fibers are curved.
4.2 Static analysis for dynamic phenomenon: the energy method
Damping properties are characteristics which are usually linked to a notion of time, as it is
related to wave propagation. For this reason, determining these properties for a whole
structure (a plate in this case) is typically a problem of the dynamics field. However, in order
to solve this dynamic problem with the Finite Element Method, damping properties of the
used material need to be inserted into the FE software. In the studied case, these are
properties of a composite material with different fiber orientations. However, even if
damping properties are known for the fibers or the matrix alone, the ones for the
combination of both are not. In order to solve this problem, a simplified solution is
suggested, which has been utilized several times at Airbus: make an approximation through
a static analysis. The approach is then different, as the “time aspect” is not directly taken
into account.
In this case, the analysis will focus on the energy in every element. This energy is stored in
the fiber and in the matrix, depending on the fiber orientation. As the damping properties
for the matrix and the fibers are known, the energy losses for both components will be
calculable, for every element. Extrapolated to the whole plate, the global damping
properties will be determined. By this, the damping properties of a material will be defined
as the ability to dissipate the energy stored. However, since the analysis is made in 2D, it has
been considered more relevant to utilize the energy per unit volume.
The global procedure will be detailed in the next part.
50
4.3 Energy method: global procedure
As the energy losses cannot be calculated directly in the FE software, the chosen approach is divided
in several steps (considering only 1 ply for now):
- Find the elastic strains for every element in the global coordinate system (x0,y0), from the FE
solver
- Find the elastic strain in the fiber (with, for instance, a α° orientation) and in the “matrix”
(normal-to-the-fiber) directions via the mathematical relationship [10]:
[
] [
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ] [
]
With:
Indeed, the coordinate systems are defined as illustrated on the Figure 39:
- In order to calculate the energy per unit volume in the matrix and in the fiber, it will be
considered here that the strain in the fiber is equivalent to the strain in the fiber direction
(1-direction). In the same way, the strain in the matrix is considered to be the “combination”
of and . This could be seen as a big simplification since the matrix is also deformed in
the fiber direction. Thus should be taken into account when considering the matrix
deformation. Despite this, due to the high E-modulus of the fiber, the deformation in that
direction will be extremely small compared to the ones in the “matrix direction”.
Consequently, the simplifications are considered to be small enough to be acceptable.
Finally, by these assumptions, the energy per unit volume “in the fiber” and “in the matrix”
can be calculated.
- Calculate the elastic strain energies per unit volume contained in the fiber and in the matrix
thanks to the formulas (3.2) (no summation on ij) [11]:
(3.2)
(3.1)
y
x
1 2
α Figure 39: Coordinate system attached to the fibers
51
With:
(
)
- Apply the matrix and fiber’s damping coefficients in order to find the energy losses in the
element:
( )
( )
( )
The past experiences of the engineering teams at Airbus allow to state that the typical damping
values for the fibers and the matrix in the composite material under study are around 0,2% and 5%
respectively.
Therefore, it becomes, for 1 element and 1 ply:
The final step to obtain the losses of the whole lay-up is to add all the energy losses (in all the
elements of all the plies) and divide by the total stored energy per unit volume. It can be illustrated
by the following example:
Plies \ Elements Element 1 Element 2 Element3
Ply 1 Uloss11 & Ustored11 Uloss21 & Ustored21 Uloss31 & Ustored31
Ply 2 Uloss12 & Ustored12 Uloss22 & Ustored22 Uloss32 & Ustored32
Table 4: Energy stored/lost for every element in every ply
(3.3)
52
The way to obtain the total losses in this example structure containing 3 elements and 2 plies is
thus the following:
By combining the FE solver and a newly created FORTRAN code, the total damping ability of the
structure can be calculated.
4.4 Mapping strategies/designs
In this part again, a choice is to be made concerning the curved mapping designs to test on
the middle ply. For this study, the tested mapping model is corresponding to the one from
Part 3:
- “Intuitive design” non-symmetrical across the X-axis (no hole to map around)
and only with curved fibers:
This way, this analysis will determine the effect of curved fibers on the structure damping properties. Finally, this type of design will be tested in 2 cases:
- “Intuitive design” with curved fibers only on the 0° ply,
- “Intuitive design” with curved fibers on the 0, 16 and -16° plies.
Figure 40: Intuitive design applied to a plate without hole
y
x
(3.4)
53
4.5 Main analysis: “Intuitive design”
4.5.1 Mathematical modelling (same as in part 3.4.1)
This type of fiber (pre-preg stripe) profile has been made taking into account the minimum
curvature radius that the material fibers are able to handle. For the used material, this
minimum radius is 1500mm. Moreover, since this design corresponds to the Design 1 of the
previous part, the profile still takes into account a “virtual hole” around which the fiber
should pass. Finally, as there is symmetry across the local X axis, the model is made only for
the stripes located on the upper half of the plate. The stripes on the lower half will be
treated in the code with a simple procedure detailed later on.
| | | |
√
√
√ | | |
|
√
The mathematical/geometrical demonstrations to determine the centers’ coordinates, as
well as some explicative sketch can be found in Appendix B.
Figure 41: Modelling of the Intuitive design
(4.1)
(4.2)
(4.3)
(4.4)
54
If the derivatives are performed, we get the tangent-to-the-curve’s coefficients:
| | | |
√
√
√ | | |
|
This way, the fiber orientation can be obtained in every element depending on its location.
4.5.2 Comparability and “iso-stiffness”
As mentioned previously, the analysis is to be made respecting an “iso-stiffness”. Indeed, if the fibers are curved, they tend to interact more with the matrix and thus soften the material. The fact of getting a softer material can be a cause of a higher damping ability. However, curved fiber should be able to increase the damping characteristics even is the stiffness is the same. In order to verify this, the compared models need to have the same stiffness. Therefore, it is first necessary to test the stiffness difference between the models. It appears that curving all fibers in the 0° ply decreases the stiffness by 1%. Consequently, to balance this and achieve the same stiffness for both models, the plies’ thickness of the softest one is also increased of 1%. The result is indeed the expected one: same stiffness in both models. In the same way, this process is made for the 2nd studied case. Thanks to these adjustments, the models can be compared, and the results directly related to the fiber curving.
(4.5)
(4.6)
(4.7)
(4.8)
55
4.5.3 FEM analysis/Codes
In this analysis, 2 codes have to be implemented:
- The code (a) in order to assign static material properties to every element (as it
was done in Part 3)
- The code (b) in order to calculate the energy losses for every element (and thus for
the whole plate), from the results given by the FE software
Here are listed some important ideas of it.
No hole or “mirrored fibers”
In this part, the tested structures are not containing any kind of holes. Therefore, the fibers
are not symmetrical across the vertical axis, and no empty zone is created. All fibers are thus
having the exact same profile. Because of this, the orientation of the fiber in an element is
only depending on its Y-coordinates (Figure 42). Illustrated in the Part 4.4, the 0° ply will look
like this:
y
x
Figure 42: Curved 0° ply for damping analysis
56
4.5.4 Analysis frame
In this comparative analysis, the boundary conditions are the same as for the Part 3 and 4.
Similarly, the geometry, the boundary conditions and the mesh look as follows, for the same
reasons evocated in the parts about structural strength:
Moreover, the applied force F is a tensile force on the top edge, and has a value of 71000N. As mentioned in Part 2.8, the bottom edge is clamped. The results are displayed below in a small comparative table (see next page).
5000 mm
2000 mm
α
0° α°
F
Figure 43: Plate's geometry/ Mesh
57
4.5.5 Results and interpretations
Despite the fact that the global stiffness is almost the same, such a mapping design implies that the
matrix is more solicited and stressed, while the opposite phenomenon happens to the fibers. Since
the matrix dissipates more energy than the fibers, it was expected that the total amount of
dissipated energy tends to increase when curving the fibers.
Indeed, with this mapping design, the results are the following (the considered “energies” are
energies per unit volume [J.m-3]):
Reference case
(no curved fibers)
“Intuitive design”
(only on the 0° ply)
“Intuitive design”
(on plies 0°, 16° -16°)
Total “energy” stored 7,187E+08 7,116E+08 6,437E+08
Total “energy” dissipated 2 295 132 2 365 275 3 131 138
Total losses in the structure 0,317% 0,33% 0,48%
Variation of dissipation - +4,1% +49,8%
4.5.6 Conclusion: Structural damping
As it can be seen, the fact of curving the fibers in this way implies a better vibrational damping. In
the studied case, the benefit is about 4% when curving the 0° ply, while it increases up to 50% when
all 0°, 16° and -16° plies are curved.
However, one can recall that some small simplifications have been made during this analysis.
Therefore, these results should be confirmed by some further studies or tests, in order to be sure
that the simplifications are indeed relevant.
Table 5: Results of the Intuitive design
58
5. Overall conclusions
Throughout this thesis, it has been shown that curved carbon fibers can be a very good path to
explore in order to improve the strength of aerospace structures. The final results show indeed an
improvement of about 44% in the studied structures with the Design 2.
Moreover, the damping properties of curved fibers suggest some new kinds of mapping designs, to
be applied to structures submitted to strong vibrational loadings. In the studied structures, the gains
are close to 50% improvement – for the best cases.
For all these reasons, it can be concluded that such promising properties open a wide range of new
possibilities for the future aeronautical equipment.
6. Further work
Even if the conducted studies show a real potential and a bright future for this new kind of material,
it is necessary to deepen the investigations with concrete tests and more refined analyses.
For instance, an interesting aspect to investigate would be the influence of the modelling on the
results of Design 2. Indeed, as it has been mentioned earlier, the chosen model is arbitrary and it
would be beneficial to try new ones. In that case, a suggestion would be to try the following model:
Moreover, as two designs have been chosen, many more can be imagined and some optimization
studies could be determinant in order to keep improving the performances.
Finally, concrete tests on manufactured samples are mandatory steps in order to confirm/infirm the
theoretical calculations which have been made all along the different studies.
2 plies in the code
Figure 44: New model suggestion
59
7. References
[1] Zafer Gurdal and Reynaldo Olmedo, (1993) "In-plane response of laminates with spatially varying
fiber orientations - Variable stiffness concept", AIAA Journal, Vol. 31, p751-758
[2] Julien van Campen, Christos Kassapoglou, Zafer Gürdal, (2011) “Design of Fiber-steered Variable-
stiffness Laminates Based on a Given Lamination Parameters Distribution”, AIAA Journal, Vol. 52.
[3] Kazem Fayazbakhsh, Mahdi Arian Nick, Damiano Pasini, Larry Lessard, (2014) “Optimization of
variable stiffness composites with embedded defects induced by Automated Fiber Placement »,
Composite structures, Vol.107, p160-166
[4] A.T Rhead, T.J. Dodwell, R. Butler, (2013) “The effect of tow gaps on compression after impact
strength of robotically laminated structures” Computers, Materials and Continua, 35 (1), pp. 1-16.
[5] Croft, K.; Lessard, L.; Pasini, D.; Hoijati, M.; Chen, J. H.; Yousefpour, A. (2011): Experimental study of the effect of automated fibre placement induced defects on performance of composite laminates. Compos Part A: Appl. Sci. Manuf., vol. 42 no. 5, pp. 484-491. [6] Fayazbakhsh, K.; Arian Nik, M.; Pasini, D.; Lessard, L. (2012): “The effect of gaps and overlaps on the inplane stiffness and buckling load of variable stiffness laminates made by automated fiber placement”, Proceedings of 15th European Conference on Composite Materials, Venice, Italy. [7] Thomas Norhadian (2014), “Company Description” translated from French to English from the
Airbus’ intranet company description, by the author.
[8] NX I-deas, software from SIEMENS,
www.plm.automation.siemens.com/en_us/products/nx/ideas/
[9] Quick Reference Guide of Nastran, MSC. NASTRAN 2001, www.mscsoftware.com
[10] M.Vable (2012), “Mechanics of Materials”, Chapter 9, p7 from
http://www.me.mtu.edu/~mavable/MoM2nd.htm
[11] Dr P.Kelly (2013), “Solid Mechanics”, Part I, p245, from
http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/
60
Appendixes
Appendix A: Nomenclature
All along this paper, the mathematical designation of the analyses’ parameters keep being the same.
Therefore, a nomenclature of these designations has been done to facilitate the understanding of
some of the mathematical equations.
Figure 45: Illustration of the nomenclature
R R
y
x
𝒚𝟎𝟐
𝒙𝑰
𝒚𝒇⬚
𝒙𝟎𝟐
𝒚𝟎𝟏
𝑶𝟐
𝑶𝟏
𝑰
𝑶⬚
61
Appendix B: Geometry demonstrations for part 3.4 & 3.5
Here are gathered several important mathematical demonstrations in order to better understand
some statements from the Part 3. This Appendix is starting with a sketch showing the part of
interest.
Coordinate and
For the fiber n°0, the curvature circle’s center 2 has to be placed so that before
getting curved.
For the fiber n°1, the center 2 has to be placed so that before getting curved.
For the fiber n°2, before getting curved.
Therefore, the general expression is
being the fiber number and the fiber width.
With the same principle, the fiber n°0 has to be placed so that ( ) .
The fiber n°1: ( ) and the fiber n°2: ( )
Therefore, the general expression is .
Coordinate
In the triangle ABC, rectangle in B:
‖ ‖
‖ ‖
‖ ‖ | |
According to Pythagoras:
‖ ‖ ‖ ‖ ‖ ‖
⇔ (
) (
)
y
x 𝒙𝟎𝟐
𝒚𝟎𝟏
𝒚𝟎𝟐
𝒙𝑰
𝜽𝟏 𝜽𝟐
R
R B
A
C
(1)
(2)
Figure 46: Sketch of the fiber around the hole
62
⇔ √ (
)
√
Another way to write it would be by inserting (1) and (2) in first part of (3):
⇔ √ ( )
Coordinate
⇔ ( ) ( )
⇔
⇔
Effect of the overlapping between the fibers
In the first two mapping strategies of Part 3, the mathematical model implies that the distance
between the fibers varies when they get curved. Therefore, some overlapping occurs and it becomes
important to know if this can strongly influence the material properties. A way to do that is to
determine the overlapping percentage between the fibers by calculating the distance between 2
fibers where their trajectories get curved. In the Figure 47 below is a sketch representing 2 curved
fibers, only shifted vertically from each other:
y
x x3 x2 x1
d1
d2
d3
𝑓
(3)
(5)
(4)
Figure 47: Illustration of the distance between the fibers
63
On this sketch representing the mathematical model used, it is easy to see that the “vertical
distance” between the fibers always stays equal to whatever the value of x is. However, the
“normal-to-the-fiber distance” varies. The aim here is to determine the “normal-to-the-fiber
distance” ‖ ‖
‖ ‖ is equal to twice ‖ ‖. Moreover, if θ small, the triangle ABC can be considered as right-
angled at A. A usual definition of small is that ( ) .
( ) | |
(| |
) (
√ ( )
)
With R>1500mm and rh=150mm:
and ( )
From the Figure 49 it is now possible to calculate ‖ ‖:
‖ ‖ ( )
⇔ ( )
⇔ ‖ ‖
It is now shown that there is less than 6% of overlapping between the fibers, due to the
mathematical model of fiber curving. It is thus relevant to neglect this phenomenon in the
calculations.
θ A B
C
D
θ
θ
|𝒙𝟎𝟐|
|𝒙𝐈|
R
y
x
Figure 48: Distance to the fiber AD
Figure 49: Sketch of the fiber curveting
(6)
(7)
64
Appendix C: Comparability justification
As stated in the Part 3, the Design 2 is characterized by some fiber crossings implying that the
amount of material on this design is more important than on Design 1. This part aims to show that
this amount of material “added” is relatively small.
Both designs are looking as follows:
Therefore, the difference between both can be schematized by purple fibers (the same thing can be
done on the other side of the hole). The total amount of added material is thus corresponding to
twice the yellow shaded area (Figure 51):
In order to calculate the yellow area, an approximation has been done, since the fiber curvature is
quite small. This yellow area is assimilated to the area of the triangle ABC. In this way:
( )
Thus,
However, it should be recalled that this value is valid only for the plate under study, since it depends on arbitrary parameters such as and the hole’s diameter.
A
B C H
Figure 50a: Simple crossing design Figure 50b: Intuitive design
Figure 51: Difference between both designs
(9)
(8)
65