S e c t i o n 5 .

SUMMARY AND FUTURE WORK

1.

SUMMARY

The objective of this research was to improve the starting finite element model available from the previous work of Giuseppe Palermo [1] and to predict elastic properties of Triaxial woven fabric (TWF).

It has been seen how important is the role played by the geometric parameters present in the fabric; the correct behaviour of this fabric is completely characterized by the good representation of these parameters on the finite element model. A detailed geometry definition was considered, in particular introducing a curved path (sine function) in the free parts of the yarn; a new Local Element Coordinate System was built and properties were referred to it. A new finite element model was built with higher number of solid elements in the curved parts; homogenized E1-modulus of the TWF was predicted (“E1 modulus simulation test”) and compared to the starting values and literature data. A quite large reduction of the predicted E1 modulus was discovered compared to the starting values. Different Shell element models were built to study this unexpected behaviour. In these models three different ways to interpret the cross region (Shell model, Shell+Hex and Shell+Hex+RBE2 element) were used and subjected to the E1 modulus simulation test. The Shell+Hex+RBE2 is stiffest but it is still far from the target E1 value. Moreover, an additional work was introduced working directly on a fragment of the starting TWF model created in the previous work; in particular it was defined a Local coordinate system, material properties were linked to it instead of a Global coordinate system and E1 simulation was performed. The result shows that the starting TWF model has the same unexpected behaviour revealed by testing the new models introduced in the present work. Starting from this problem, additional tests were done working directly with a single yarn; in particular comparison between beam model and solid model were done. Different configurations of yarn were considered; a non-twisted yarn and a twisted yarn. Different shapes were tested and mechanical tests were simulated on the single yarn. The results underlined that the more the shape is symmetric and regular (like a sine function), the

higher the stiffness of the TWF ; on the other side, the steeper are the slopes, the lower is the stiffness. The results obtained working on a twisted configuration revealed that the twist contribution is dominant when high slopes are involved in the geometry; moreover, micrographs of the TWF show that the high slopes which characterize the starting yarn geometry are not realistic. According to this the first important conclusion is that the

unreal high slopes involved in the geometry defined in the previous work is the cause of the not correct behaviour revealed during the simulations, so it is necessary to define a new yarn geometry set.

Starting from these results and working with a new data base kindly made available by the University of Cambridge, a new Shell+Rbe2 element model was introduced according to a new interpretation of the geometry. In fact the new geometry set was characterized by low slopes in the connection parts in order to better simulate the real shape of the yarn itself. E1 modulus mechanical test was simulated both in this new model and the previous solid model and a comparison was performed with Cambridge beam TWF model. The comparison revealed that the previous model had very low stiffness (according to the not correct definition of the geometry of the yarn) while the new one had a very good response, so it was chosen to consider it the actual best simulation of a TWF

specimen available among all the models introduced in the present work. With respect to

the solid element models it was concluded that elliptical or lenticular cross-section of the yarn has to be considered instead of the rectangular one used. This will reduce the steep slopes and twist to more realistic levels.

Improvements in the reduction of the time required to perform the model analysis were introduced; Superelement theory was applied to both Solid and Shell model and mechanical simulation tests were performed. The results verified that not only this theory is

applicable to a TWF specimen, but also the results are very close to those obtained using the full detailed model. Moreover, gain in the bulk data size was revealed, while no

computing time gain was observed. The results obtained from the mechanical simulation revealed small differences when compared with the values obtained with the original models except for the temperature load for which quite a large error (more or less 20%) is revealed in the out-of-plane deformation. Tests were performed in order to discover the reason of these errors using simple geometries. The results revealed that if a simple geometry is built, the superelement technique gives no difference compared to the starting model under both mechanical and thermal load. According to this, a first conclusion

could be that the too complex geometry of the TWF unit cell is the reason of the errors revealed; anyway other reasonable causes can be investigated. In fact, the

normal application of Superelment technique to a TWF specimen needs a changing in the tolerance of the MSC/NASTRAN job file; the reason is the presence of small difference between the orientation of the boundary points which define the location of the superelements. According to this a not enough detailed definition of the boundary points could be another reasonable reason of the errors discovered. Anyway this problem still remain to be explained.

2.

FUTURE WORK

According to the results and all the important aspects underlined in the present work, comments can be added for future works. The actual best TWF model available, which gives a good simulation of a TWF real specimen, is made of Shell and Rbe2 elements. Because of the complexity of the new geometry set adopted in this new model, it was impossible to model it using solid elements because Patran has some limits in this. Therefore a future work could start from this point; it could be interesting to model the TWF yarn using CATIA in order to work with solid elements; in this new configuration it has to be considered a lenticular cross section of the yarn itself instead of a rectangular one. It will be interesting to investigate if this new solid model will be able to better simulate the TWF behaviour. Speaking about the improvements introduced in the model reduction studying, it is possible to say that Superelement technique is a good approach to follow and develop in the future; in particular, since actually it is only possible to look at the deformation of the main superelement, it could be interesting to study the way to observe the deformation of all the specimen. In particular it is suggested to find the way to know the behavior of all the nodes of which the model is made up; this is because actually it is not possible to know the response in terms of forces and displacements of each node which belongs to the secondary Superelement. Moreover, it will be also interesting to develop this technique for large surface simulation, like an antenna reflector for example, remembering that if Temperature is involved, the quality of the solution has to be considered a first time solution order because, the complexity of the TWF geometry (or other undiscovered reasons) involves unavoidably some errors in the results, while, on the other hand if only mechanical loads are applied the quality of the results involved is still good.

APPENDIX A: ABOUT HEX, SHELL, BEAM ELEMENTS

AND COORDINATE SYSTEMS

Hex, Shell and Beam elements: in the following pages it will be introduced the

definition of each element used to mesh the yarn geometry. In particular the Hex, Shell and Beam elements will be defined, in order to understand the difference that there are between these elements.

Hex element: it is a solid element made up of eight GRID points; the figure below shows

a Hex element in which is also defined the local element system by default.

Figure 1: Hex8 element [9]

For non hyperelastic elements, in MSC/NASTRAN the element coordinate system for the CHEXA element is defined in terms of the three vectors R, S, and T, which join the centroids of opposite faces.

• R vector joins the centroids of faces G4-G1-G5-G8 and G3-G2-G6-G7. • S vector joins the centroids of faces G1-G2-G6-G5 and G4-G3-G7-G8. • T vector joins the centroids of faces G1-G2-G3-G4 and G5-G6-G7-G8.

The origin of the coordinate system is located at the intersection of these vectors. The X, Y, and Z axes of the element coordinate system are chosen as close as possible to the R, S, and T vectors and point in the same general direction. (Mathematically speaking, the coordinate system is computed insuch a way that if the R, S, and T vectors are described in the element coordinate system a 3 x 3 positive-definite symmetric matrix would be produced) If orthotropic materials are used it is important to assign the correct orientation. For Hex elements the material orientation is either defined by introducing a special Coordinate System or the Element Coordinate System is used. In the first case, a local C.S. should be introduced for every element in the “wavy” part of the yarn. For the other case, regularly shaped Hex elements with “correct” orientation have to be used.

Shell element: this element defines an isoparametric membrane-bending or plane strain

quadrilateral plate element. The following picture shows the way with which it is defined.

Figure 2: Shell element [9]

The x-axis of the material coordinate system is determined by projecting the x-axis of the MCID coordinate system (defined by the CORDij entry or zero for the basic coordinate system) onto the surface of the element. In this way, it is easy to define the material properties and they are able to follow the shape of the geometry as well as the definition of the local Coordinate system introduced for the solid Hex elements. If 2D Orthotropic materials are used for Shell elements, then the material 1-direction (e.g. E1) is aligned with the element X-material-axis (see figure above) and the material 2-direction (e.g. E2) is aligned with the element Y-material-axis.

Beam elements: it is an element characterized by an equivalent section that has to be

defined in terms of width and height of it. The beam elements works only with a material whose properties are isotropic. The single element coordinate system definition is particular, as shown in the following picture

Figure 3: beam element [9]

While x-axis is defined by default, the definition of y-z axis are linked to the definition of the orientation vector v. In particular, in order to have a coordinate system with x direction

aligned with the longitudinal direction of the beam, y through the thickness and z through the width as it is shown above, it is necessary to define the v vector so that it belongs to the 1-2 plane of the local beam system.

It is important to explain the meaning of Coordinate System and Local Coordinate System. For this it will be shown an easy example in order to clear every doubt about it.

Coordinate System: this coordinate system is characterized by fixed directions; it is

important to understand what happens when material properties are linked to a Coordinate system. In the following picture an example of model is shown and the Coordinate system that will be used to define its material properties.

Figure 4: Coordinate System

In the following picture, it will be displayed the material properties defined according to the Coord A..

Coord A

Figure 5: Material orientation

As it is possible to see above, the definition of the material properties according to the Coord A, implies that they are not able to follow the shape of the model itself; in fact, being the orientation of Coord A fixed, also the material properties will follow the same behavior.

Local Coordinate System: This new system ids characterized by being defined in each

part of the model; speaking about the modeling of the single yarn that was introduced in the previous sections, it means that it will be defined in each Shell or Hex element used to mesh the yarn geometry. In the following picture it is possible to see the model and the Local Coordinate systems: Coord A, Coord B, Coord C.

Figure 6: Local Coordinate System

Coord A Coord B Coord C

Coord A

As it is possible to see above, the three Coordinate systems defined in the model describe its shape. In the following picture will be shown what happens if the material properties are referred to these three systems.

Figure 7: material orientation in a Local C.S.

The picture above shows that now the material properties are able to follow the shape of the model; this implies a more realistic behaviour.

List of symbols

N number of fibers in the yarn

pd yarn packing density

df fiber diameter

CS Coordinate System

t thickness

a yarn spacing A cross section area Vf fiber volume fraction

WF woven fabric TWF triaxial woven fabric RUC repeated unit cell

R.F. reaction force

L length W width

TEC thermal expansion coefficient MPC multipoint constraint

RBE2 rigid body element two DOF degrees of freedom

REFERENCES

[1] Giuseppe Palermo , Finite Element Modelling of Triaxial Woven Fabric for Aerospace

Structures, 2004

[2] Ahmad Kueh, Omer Soykasap, and Sergio Pellegrino, Thermo-mechanical behavior of

single-ply triaxial weave carbon fiber reinforced plastic

[3] Zhao Q., Hoa SV. Triaxial woven fabrics with open hole, J. of Composite Materials, vol v37 n 9, 2003

[4] Zhao Q., Hoa SV. , Oullette P. Triaxial woven fabrics with open hole, J. of Composite

Materials, vol v37 n 10, 2003

[5] P. Fortescue, J. Stark. G. Swinerd, Spacecraft systems engineering, Wiley2003 [6] Zhao Q., Hoa SV. , Thermal deformation behaviour of triaxial woven fabric, J. of

Composite Materials, vol v37 n 18, 2003

[7] L. Datashvili, M. Lang, N. Nathrath, Ch. Zauner, H. Baier, (all LLB TUM), O.Soykasap, L.T. Tan, A. Kueh, S. Pellegrino (all DSL UC), D. Fasold (FH-M), Technical Assessment of

High Accuracy Large Space Borne Reflector Antenna

[8] R.M. Jones, Mechanics of composite material, Hemispere publishing 1975 [9] Getting started with MSC.Nastran User’s Guide, MSC Nastran 2001 [10] Getting started with MSC.Patran User’s Guide.