EFFECT OF CURVED FIBERS IN COMPOSITE STRUCTURES

Transcription

1 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

2 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 2

3 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 3

4 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. 4

5 Table of contents Acknowledgements... 3 Abstract Introduction Airbus Defense & Space Background Global outline Aim of the work State of the art, other studies Other aspects of the work Preliminary studies / Basics Used material Steering technique Mapping singularities Vocabulary: Pre-preg stripe/fiber Effect of curved fibers Spring principle : curved fibers associated with straight ones FEM Procedure Modelling Procedure plan FORTRAN Program FE Boundary conditions FE Mesh model Failure criterion Structural strength of structures with holes Mechanical behavior of a plate with a hole Isotropic and homogeneous materials Anisotropic and inhomogeneous materials Curved fibers around a hole Mapping strategies/designs Design choice Main analysis: Intuitive design Mathematical modelling FEM analysis/code

6 3.4.3 Analysis frame Results and interpretations: Conclusion: Main analysis: Simple crossing design Mathematical Modelling FEM analysis Comparability Analysis frame Results and interpretations Conclusion: strength of structures with hole Structural damping Structure damping with curved fibers Static analysis for dynamic phenomenon: the energy method Energy method: global procedure Mapping strategies/designs Main analysis: Intuitive design Mathematical modelling (same as in part 3.4.1) Comparability and iso-stiffness FEM analysis/codes Analysis frame Results and interpretations Conclusion: Structural damping Overall conclusions Further work References Appendixes Appendix A: Nomenclature Appendix B: Geometry demonstrations for part 3.4 & Coordinate and Coordinate Coordinate Effect of the overlapping between the fibers Appendix C: Comparability justification

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8 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 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. 8

9 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. 9

10 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 K t coefficients. Figure 1: Tensile FE analysis on a plate with hole - 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]. 10

11 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. 11

12 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: [ ] It can be represented through the following thickness-view of the composite material s stacking sequence: Figure 2: Illustration of the 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. 12

13 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: Stripes Overlap Gap: Gap Triangle: Triangles Figure 3: Pre-preg stripes causing some mapping defects (overlaps, gap, triangles ) 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. 13

14 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: Fiber Stripe Matrix Figure 4: Fiber steering illustration 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: Spring 2 k k k 3k Spring 1 F x = x 1 = x 2 1 Figure 5: Illustration of the "spring principle" F 14

15 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. (1.1) 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 (1.2) 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. Figure 6: Both straight and curved fibers in parallel 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. 15

16 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: σ Figure 7: Curved fibers generating shear stresses 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. 16

17 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 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: Fibers Node Element Figure 8: Elements containing straight fibers 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: Figure 9: Illustration of a curved fiber in several elements 17

18 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 Table 1: Example of a PCOMP Note: the term material is used to designate the combination of fiber + matrix 18

19 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. 19

20 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: 9. Assign the PCOMP to the corresponding element in the mesh file 1. Select one element among the element list 2. Read its nodes'numbers 8. Store the PCOMP in the input material file 3. Read each element node's coordinates 7. Store the orientation in one PCOMP 4. Calculate the element center's coordinates (x ; y) 6....get the fiber orientation (tangent) at the element's center 5. Insert x and y in the fiber's mathematical model to... Figure 10: Principle of the code generating the NASTRAN files 20

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: y 0 y 1 x 1 α x 0 Figure 11: Illustration of the coordinate change As one can observe, the orange curve has the same equation in the coordinate system R 1 (x 1,y 1 ) as the blue one has in the coordinate system R 0 (x 0,y 0 ). Following that principle, one must convert the coordinates of the element (expressed in R 0 ) in order to get the coordinates expressed in R 1. 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 R 1 since we used the coordinates expressed in R 1. Rather trivially, since the value of is known, the last step is to subtract to the orientation found in R 1. This gives us the fiber orientation in R 0. Therefore, this procedure has to be repeated for every ply and will slightly modify the previous big loop. 21

22 This can be seen on Figure 12: 9. Assign the PCOMP to the corresponding element in the mesh file 1. Select one element among the element list 2. Read its nodes'numbers 8. Store the total PCOMP in the input material file Until all the plies are done 3. Read each element node's coordinates 7. Store the orientation in one PCOMP 4. Calculate the element center's coordinates (x ; y) 6.b Substract α to get the orientation in R 0 4.b Choose a α and convert (x ; y) to (x i ; y i ) 6....get the fiber orientation (tangent) at the element's center in R i 5. Insert x i and y i in the fiber's mathematical model to... Figure 12: Principle of the code generating the NASTRAN files 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. 22

23 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: y x Figure 13: Illustration of a tensile test 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: Figure 14: Illustration of the FE defaults 23

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: Figure 15: Forces distribution 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 y x Figure 16: Boundary conditions applied to the coming models 24

25 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 for the left model, and 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. y x Figure 17: Meshes used in the coming models 25

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: Big element σ σ average Small element σ average Distance Figure 18: Consequence of a refined mesh 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 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: ( ) ( ) ( ) (1.3) 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. 26

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 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: y x b a Figure 19a: Plate with a hole Figure 19b: Stresses repartition in a holed plate 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. 27

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 σ zz, σ yz and σ xz are considered to be equal to 0 and an exclusive attention will be given to the xy-plane stress maps 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. y x Figure 20: Stress concentration in a composite structure 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. Figure 21: Curved fibers around a hole 28

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: Figure 22: Different imaginable designs 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 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» OR - 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 Figure 23: Intuitive design 29

30 2. «Simple crossing design» - Crossings overlaps - Stress transfer in several planes Figure 24: Simple crossing design Now that the global ideas have been set, some parameters/rules are remaining (for a chosen hole diameter d h = 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 (O 1 ) 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. y O 2 Design 2 n = 3 n = 2 n = 1 R2 R 1 O 1 O Design 1 x Figure 25: Curved fibers definition 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. 30

31 3.4 Main analysis: Intuitive design Mathematical modelling Figure 26: Modelling of the Intuitive design 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. (2.1) (2.2) (2.3) (2.4) The mathematical/geometrical demonstrations to determine the centers coordinates, as well as some explicative sketches, can be found in Appendix B. 31

32 If the derivatives are performed, we get the tangent-to-the-curve s coefficients: (2.5) (2.6) (2.7) (2.8) 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: (2.9) 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. 32

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, y f = 300 mm (Figure 27b). However, combining straight stripes and curved ones creates a zone empty from any material (Figure 27a). y f Figure 27a: 2 choices for the Intuitive design Figure 27b: 2 choices for the Intuitive design For this reason, this zone will be filled with cut straight stripes (green fibers on Figure 27b) 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" 33

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 (x 0 ;y>0) is the opposite to the one of the green tangent (x 0 ;y<0). y = 0 x 0 Figure 29: Orientation of symmetrical fibers 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. 34

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 [ ] 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: F α α y mm x Φ mm Figure 30: Plate s geometry/mesh 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. 35

36 3.4.4 Results and interpretations: Case n 0 1 Variation (n 1/n 0) 2 Variation (n 2/n 0) Y f 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 % 9,25E % Max XY Stress 4 8,31E+06 2,39E % 2,69E % Max Hill 4 2,42E-02 1,97E-02-19% 1,32E-02-45% K t 4 2,99 2,78-7% 1,86-38% Y-displacement 1,96E-03 1,96E-03 0% 1,98E-03 +1% Note: Specifications of the table can be found on the next page, just before the results interpretations. Table 2: Results of the Intuitive design 36

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 K t. 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, y f 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): 4 th 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 K t ). 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. a) Figure 31: Effect of pulling on a curved 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: 37 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. 3 rd & 2 nd plies: Figure 32: a) Y stress map b) X stress map c) XY stress map c) - Max Y stresses: the maximum Y stresses are located around the hole (rather similar to the 4 th ply). Since the global material is softened by adding curved fibers on the 4 th 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 4 th 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 1 st 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 4 th 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 4 th 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 38

39 Interpretations of the results for Case n 2 (Design1, All plies curved): 4 th ply: a) - 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 K t ). 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 4 th ply. Therefore, the 4 th ply sees its max X stresses increase compared to when the 2 nd and 3 rd 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. b) - 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. 3 rd & 2 nd plies: c) - 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 2 nd (or the 3 rd ) ply exhibits some material. 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 39

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! 1 st 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 overloaded). 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 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. 40

41 3.5 Main analysis: Simple crossing design Mathematical Modelling Figure 34: Modelling of the Simple crossing design 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. (2.10) (2.11) (2.12) (2.13) Once more, the vertical distance between the stripes is always the same, whatever the value of x is. And we get still: 41