*S e c t i o n 1 . *

### MODELLING OF TRIAXIAL WOVEN FABRIC

In this section it will be introduced the starting finite element model of TWF which will be considered as reference. This starting finite element model was created by Giuseppe Palermo who interpreted the geometry of the yarn and developed a TWF model made up of Solid elements. Thanks to his work, which is a really good starting base, improvements will be introduced and explained in the details. Starting from these improvements several models will be created with different degree of complexity; mechanical simulation tests will be then applied on each one in order to evaluate the E1 modulus and comparisons and comments will be added in order to identify the configuration which better simulates the real behavior of the TWF.

**1 **

**TRIAXIAL WOVEN FABRIC **

As discussed in the introduction, TWF is attractive for light weight structures but one of the disadvantages is its complicated RUC geometry and behaviour under mechanical and thermal loads too. Now it will introduced the starting finite element model that will be considered as reference for all the improvements that will be suggested and applied in the new finite element model which will be shown in the next pages. The concept used is to model the single yarn in a very detail way using Solid elements HEX8. Starting from this the single RUC is created copying and rotating the main yarn of +- 60 degrees angle. The aim was to reproduce a faithful model according to the yarn geometry introduced in the Hoa and Zhao [2] study, to apply mechanical simulation tests in order to evaluate the mechanical properties of the TWF and comparing the results with the ones showed in the literature. In the next page it is possible to check the geometrical parameters used to identify the single RUC cell; they are summarized in Table 1.

*Figure 1: Geometric schematization of the RUC [ 1 ] *

* Geometrical Parameters * _{ Values }

Yarn thickness ty (mm.) 0.07
Resin thickness t_{r} (mm.) 0.001

Yarn width W1 (mm.) 0.85

Larger bridging distance W2 (mm.) 1.10
Smaller bridging distance W3 (mm) 0.1185
Unit cell’s height W4 (mm.) 2.755
Unit cell’s width W5 (mm.) 3.182
Fabric area density (g/cm2_{) } _{ 0.0158 }

The elastic properties of the impregnated yarn of the TWF composites are shown in the table below; they are obtained using the rule of mixtures on the separated properties of the single carbon fiber and resin.

*Table 2: Elastic Properties of the Materials used [ 1 ] *

The geometry of the single yarn is made up of only straight parts with rectangular cross section; this is an approximation of the real geometry, where the cross sections are characterized by lenticular shape and the free parts of the yarn are curve. In Figure 2 the single yarn geometry chosen as reference in the starting finite element model is shown.

*Figure 2: Single yarn geometry *

*Component * *E1 *
*(Gpa.) *
*E2*
*(Gpa.) *
*G12 *
*(Gpa.)* Vf *(g/cm3 _{) }*
Yarn
(carbon/epoxy) 338.57 12.4 5.61 0.287 0.437 0.695 1.816
Matrix 3.5 3.5 1.30 0.35 ~ ~ 1.17

The kinds of elements used to model the single RUC cell are HEX8 elements for each yarn with properties shown in Table 2; in total there are 120 HEX8 elements. Once defined the single RUC cell, it was extended along the X-axis and Y-axis to represent the TWF specimen of given size LxW. It is important to underline that, in the starting model, the material properties are referred to Coordinate Systems which are in the plane of the fabric but rotated. For each of the three different yarn directions (0, +60, -60) in order to represent the orientation of the single yarn, properties were defined and referred to the corresponding Coordinate System. Because of this the material properties didn’t follow the shape of the yarn through the wavy parts of the RUC because the orientation of the Coordinate System was fixed (See Appendix A). This approximation was considered reasonable because the undulations of the yarn are small. However, as will be shown later, the present study showed that this approximation can in certain cases introduce significant errors. Mechanical test were simulated using the TWF model and mechanical properties were evaluated as it is shown in the next pages. The pictures below show the single RUC cell and the TWF specimen obtained with the method introduced.

* *

*Figure 3: Solid Model of the RUC [ 2 ] *

*Figure 4: Triaxial Fabric specimen [ 2 ] *

It was decided to remove the real resin thickness of 0.001mm that is located between the two yarns in the cross region. This assumption will be also respected in the other new models that will be developed

resin layer in between. It was also decided to choose another configuration of TWF specimen in order to work with symmetric structure in all directions. In fact either for the shape of the fabric and either for the particular geometry of the repeated RUC, a TWF specimen is characterized by a not homogeneous and regular boundary because of the presence of a lot of free edges; according to this it was decided to consider as reference a configuration made up of no free edges. In this way it will be easy to apply the boundary conditions of the mechanical simulation test because the boundary frame is well identified. The following pictures show the new configuration derived starting from the original real one.

*Figure 5: New TWF specimen configuration.[ 2] *

**2 **

**E1 MODULUS TEST SIMULATION **

In this paragraph it will be introduced the mechanical test simulated on the starting finite element model that was summarized in the past paragraph; despite several tests were applied in order to evaluate all the complete set of mechanical properties of the TWF, for the aim of this present work it will be only considered the E1 modulus test. This is because in the next Section improvements will be

applied, new models will be defined and only the E1 modulus test will be simulated on them in order to compare the results obtained to the starting ones and the literature to.

* Figure 6: E modulus test machine[ 2 ] *

In the picture above it is shown a typical test set-up to determine the E1 modulus available in a laboratory test. The specimen is located between two grips and a force is applied. The load cell verify the intensity of the force and by an electronic machine the displacement is calculated and with a mathematical formula is found the modulus. A collection of specimens of different aspect ration were tested but having a fix width dimension value of 5 RUCs. The boundary conditions applied are as follow: at one side it was applied a fixed displacement [0,0,0] while at the other side it was applied a translational displacement of 0.001 mm along the X-direction and Z-axis translation was imposed to be zero. In the following picture it is possibile to look at the boundary condition set used. It is important to underline that the same test will be applied in the next Section on the new models that will be introduced and the same boundary conditions will be adopted in order to make a comparison of the results.

The E1 modules is calculated according to the following relation:
1
*F L*
*E*
*w H d*
×
=
× × (4.1)

** where F is the reaction force at the fixed border in the X-direction, L is the length of the specimen **
**tested, w is the width, H is the thickness of the RUC and d is the displacement applied. In the **
following page it is possible to look at the results obtained with the starting Solid element model.

*Figure 8: E1 Modulus[2] *

*Aspect Ratio * 0.923 1.85 3 3.93

*E1 * (Gpa.) **30.1 28.8 28 28 **

*Table 2: E1 Modulus[2] *

How is it visible in the graph, a convergence of the result exists if it is respected an aspect ratio of at least 3. This convergence of the results is determined from the fact that the longer is the specimen the less the boundary constraints influence the stiffness of the specimen. Besides more the boundary

constraints are far from each other, more the tows are free to rotate as they want, and this explain also the why more the length is big more the E1 modulus decreases; and also shorter is the specimen more clamped yarn (a clamped yarn is a yarn that both ends clamped by the grips) there will be in the specimen giving to the structure more stiffness.

*Figure 9: Different specimens of TWF[2] *

The results obtained show that it could be considered an E1 average value of 28 Gpa; this E1 modulus test can only be compared to the documentation that is available from University of Quebec Montreal whose results are shown in the table below and that will be considered as reference for the next E1 modulus tests simulated in Section 2.

Longitudinal Young Modulus

(Gpa)

Transverse Young Modulus (Gpa) B2 33.05 B1 22.29 B4 30.92 B3 20.08 C2 34.84 B5 22.25 C4 28.42 B6 19.60 D3 27.87 Group Ave 23.40 D4 35.02 Group Std 25.40 E1 33.71 E4 23.34 G1 37.87 E5 23.34

University of Quebec tested simple sheets made up of one layer of TWF supplied from YLA
Corporation. A total of thirteen specimens were cut from sheets of the TWFs which were square with
dimension of about 300 by 300 mm. Specimens were nominally 300 by 50 mm, the cut being made
through the centers of the hexagonal openings. Aluminum alloy gripping tabs, two at each end, were
bonded to the specimens. The tests are quasi-static, uniaxial tensile tests with a load speed of 0.2
mm/min. A non contact measurement was chosen using a laser extensometer (MTS LX300). The laser
extensometer scans the tape separation distance 100 times per second; during the tensile tests, the
separation between the reflective tapes and the applied loads were recorded and stored on disks. The
stress is determined by dividing the load by the cross section area which is obtained as the product of
width and thickness; both width and thickness are measured values . Looking at the results shown in
**the table 6 an average E1 modulus value of 32.24 GPa can be chosen as reference; validation of the **
model is obtained by comparing this value with the E1 modulus simulation results. It is possible to
check that the starting finite element model introduced in this Section has an error less than 15% that
for this kind of analysis is really good. According to this result Section 2 will started and improvements
in both detail of the yarn and material property definition will be added, new finite element models will
be introduced in the detail, E1 mechanical test will be simulated on them and comparison of the results
and comments will be added.