S e c t i o n 3 .
A MORE DETAILED ANALYSIS
In this section a new analysis will be introduced, with the aim to discover the causes of the inconsistent values of the stiffness obtained with the new models introduced in the previous section. In particular it will be considered as reference the data base available from Cambridge University thanks to the collaboration of Prof. Eng. Pellegrino who worked on TWF developing a model made of beam elements. Not only, but mechanical tests were done on a real TWF specimen, micrographs of the yarn are available, the RUC geometry is clearly defined and all the results obtained are shown in graphs.
Therefore, it will be possible to make true comparisons between the Cambridge beam model and the Solid model introduced in the present work.
1 SOLID ELEMENT MODEL
According to the new data base to be considered as reference, it will be necessary to adapt the geometry of the Solid element model to the new one; in particular this new configuration is characterized by w3
= 0; it means that in the new RUC there is no gap between the three yarns in the middle of it.
According to this particular geometry it was decided to adapt the new RUC geometry to the PCL function which creates the Solid model; for this it was decided to interpret the geometry in order to maintain a gap different from zero but still maintaining the equivalent area of 0.078x0.947mm^2 indicated in the documentation. But a gap different from zero implies a width which is smaller than the indicated value. The new geometrical parameters are summarized below.
Reference model (Cambridge beam model).
Beam elements with the following characteristics:
• Thickness = 0.078 mm (average thickness)
• Cross section = rectangular with equivalent area = 0.078 x 0.947 mm^2
• Yarn width = 0.947 mm
• W3 = 0 mm
• W2 = 1.04 mm
• W2 + W6 = 2.08 mm
Solid+rbe2 model:
• Yarn width = 0.875 mm
• Yarn thickness = 0.0844 mm
• Same equivalent area
• W3 + W6 + W2 = 2.08mm
• (W6 = W1 / sin( pi / 3))
• Cross section = rectangular with the same equivalent area
According to the size of the specimen that was used for the mechanical tests, a 25x110mm solid specimen was used and both linear and non linear analysis was performed by applying progressive steps in strain until 0.4%.
The boundary conditions that were adopted on the two opposite sides are shown below:
U = V = W = 0 U = 110 * STRAIN (MM); V = W = 0
Figure 1: Boundary conditions
In the table below all the results obtained from a linear and non linear analysis are summarized.
• Simulation with a model made up of only Solid HEX8_25x110mm:
HEX8 model Fx reaction (N) Tension (N/mm)
Strain (% of 110mm) .11 -28.6 1.144
Strain (% of 110mm) .22 -57.2 2.288
Strain (% of 110mm) .33 -85.8 3.432
Strain (% of 110mm) .44 -114.4 4.57
Table 1: 25x110 mm linear analysis results
• 25x110 HEX8+RBE2 non linear analysis.
HEX8 model Fx reaction (N) Tension (N/mm)
Strain (% of 110mm) .11 -30.61 1.2754
Strain (% of 110mm) .22 -62.50 2.604
Strain (% of 110mm) .33 -95.36 3.958
Strain (% of 110mm) .44 -129.25 5.40
Table 2: 25x110 mm non linear analysis results
The graph below shows the comparison of the results with the Cambridge beam model made up of a sine function.
Figure 2: comparison of the results by graph
According to the results shown above, the first conclusion is that actually the solid model does not represent faithfully the real behavior of the TWF specimen; in fact the results underline that the solid model has lower stiffness than the beam model. This unexpected behavior suggests a more detailed analysis with the aim to understand the reason of low stiffness in the solid model. The new analysis starts from the definition of new shape beam element characterized by improving non-symmetry in the shape itself. This is to understand what is the contribution of the shape to the stiffness; in particular the analysis will be performed directly testing one single yarn and then, starting from the beam model, the correspondent solid element model will be created. A mechanical test will be simulated and boundary conditions will be applied by fixing the first node on one side of the specimen and applying a displacement of 0.001 mm to the opposite side node. The results will be compared and comments will be added.
• SINE SHAPE BEAM YARN.
Figure 3: sine shape yarn
The following table shows the results in terms of reactions calculated on the fixed node.
Table 3: sine shape results
• MIXED SINE AND STRAIGHT SHAPE.
Figure 4: mixed sine and straight shape
Table 4: mixed sine and straight shape results
Model Fx Fy Fz
Sine shape beam -1.636 0.037 0
Hex8 global System -1.793 0.040 -4.5E-08
Hex8 Local System -1.730 0.039 9.7E-07
Model Fx Fy Fz
Beam model -1.137 0.0161 0
Hex8 global System -1.1885 0.0160 -1.3E-08
Hex8 Local System -1.197 0.0168 2.6E-07
• TWO SINE FUNCTIONS AND TWO STRAIGHT PARTS.
In this new shape the cross region is modeled as rectangular while the connection between the two cross regions are obtained with sine functions
Figure 5: two sine functions and straight parts shape
Model Fx Fy Fz
Beam model -0.969 0.023 0
Hex8 global System -1.045 0.0215 0
Hex8 Local System -1.058 0.0248 0
Table 5: two sine functions and straight part shape results
Checking the results shown in the tables it is possible to see that the way with which the yarn is modeled has a large influence on its stiffness. It is important to underline that all the three shapes are reasonable interpretation of the real shape of the yarn. Not only, but looking at the results obtained
with the equivalent solid model, it is possible to see that there is a really small difference going from the global system to the local element system; it means that for simple geometries characterized by untwisted configuration, the behavior of the specimen under different material coordinate system definition is more or less the same. Now it is important to discover what is the effect if the real two different shape functions are applied in the two opposites side of the yarn; in other words, it is interesting to discover if the twist involved by the real yarn shape has a relevant role on the stiffness of the yarn itself. Starting from the solid model that was introduced at the beginning of this Section, the same analysis applied until now will be performed. The following pictures show the twisted real shape of the yarn that will be subjected to the mechanical test.
Figure 6: twisted real yarn configuration
According to the same boundary condition used until now, the results obtained in terms of Fx, Fy, Fz reactions are summarized below.
Table 6: tesited real yarn configuration results
Model Fx Fy Fz
Hex8 global System -1.116 0.025 0.012
Hex8 Local System -0.544 0.015 0.006
Looking at the results it is possible to underline that in the stiffness evaluation of the yarn, not only the shape has an important role, but in particular also the twist involved by using two different shapes on the two opposite sides of the yarn (See figure above), has a really significant role; in fact the value of the Fx reaction is almost halved starting from a Global Coordinate System and going to a Local Element System. (see Appendix A). According to these results it will be introduced a new Shell model in which the geometry of the yarn will be interpreted in a different way.
Figure 7: micrographs of the cross section [7]
Looking to the available micrographs of the yarn it is possible to make interesting comments; first of all, thanks to a quasi elliptical section, the yarn has a softer slope than the one introduced in the starting solid model. This is because all the cross section is modeled as rectangular, so with this assumption the climb of the yarn starts just after the end of the cross region.
According to this observation, a new interpretation of the geometry of the yarn will be suggested; it will be chosen as assumption to let the yarn start the climb from the middle of the cross region. With this new assumption the new model will show softer slopes on the curve parts and this new geometry set should better simulate the real way of interlacing of the yarn because of its quasi-elliptical section. It is important to underline that starting from this new geometry it is quite impossible to model the yarn according with the use of solid elements because of the difficulty of the shapes involved (it is suggested the use of other software). According to this problem it was decided to interpret the modeling of the new yarn using shell elements. Both the geometry and the new meshed yarn are shown in the following pictures.
Figure 8: new yarn geometry set
The new geometry is referred to the middle plane of the yarn; this new geometry configuration should better simulate the real behavior of the TWF because the yarn still has different curve functions on each side but they are defined according to the real slope that the yarn should have through the fabric.
For meshing the six regions displayed in the picture above two different shell element were used;
CQUAD and TRIA elements. The disposition of them is shown in the following picture; in particular CQUAD elements are used to mesh the surfaces which define the cross region of the weave and in this way the alignment of the nodes will be guaranteed when Rbe2 elements will be applied on them.
In the following picture it is possible to see the main RUC cell made up of three yarns connected in the cross region by Rbe2 points in order to impose the relative displacements of the nodes being zero.
Figure 10: new RUC cell
According to this new configuration, the same mechanical tests performed at the beginning of this Section will be applied to a Shell specimen model of 25x110mm with progressive steps in strain up to 0.4%. In the following table the results obtained with the new linear mechanical test are summarized.
Table 7: new Shell model results
In the following graph it is represented the comparison between the sine shape beam element model chosen as reference, the solid model (both linear and non linear analysis) and the new Shell+Rbe2 model (linear analysis).
Strain (% of 110mm) Fx reaction (N/mm) 0.1 - 2.33
0.2 - 4.73 0.3 - 7.103 0.4 - 9.47
Figure 11: comparison of the models by graph
The results obtained from the analysis introduced in this Section demonstrate three important aspects;
first of all the details used in the shape approximation of the yarn, speaking about symmetry and slopes, shows that a regular and soft shape improves the stiffness of the yarn if compared to the other non-regular shapes used in the previous analysis. Secondly, if a non-twisted geometry is used, working with solid elements the material orientation is not significant; in fact the results obtained show that the difference of the values of the reaction for the same model but with material properties defined according to different coordinate systems is really low. Third the contribution to the stiffness decrease given by the twist involved by the real shape becomes the main responsible of the reduction of stiffness when shapes with high slopes (exactly like in the starting geometry that was chosen as reference) are used to model the yarn; this statement also justifies the original strange values obtained with the new Solid element model introduced in Section two. The reason of this large effect on the stiffness is difficult to explain; a reason could be related to the twisted geometry of the yarn which facilitates the rotation of the yarn itself around the longitudinal axis under the applied load. This tendency induces in the cross region a load field which facilitates the deformation out of the plane and reduces the real potentialities of the single RUC cell speaking about the stiffness; this assumption is justified looking at the results obtained with the beam model which shows a high stiffness; in fact a beam model implies a
yarn whose geometry is always symmetric and not twisted, so the out-of-plane deformation of all the cross region is not facilitated, so the RUC can maintain all its potentiality.
