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The geometrical modelling of the planar energy diffusion behaviors of the deformations on a para-aramid fabric has been performed. In the application process of the study, in the experimental period, drop test with bullets of different weights has been applied. The B-spline curve-generating technique has been used in the study. This is an efficient method for geometrical modelling of the deformation diffusion areas formed after the drop test. Proper control points have been chosen to be able to draw the borders of the diffusion areas on the fabric which is deformed, and then the De Casteljau and De Boor algorithms have been used. The Holditch area calculation according to the beams taken at certain fixed lengths has been performed for the B-spline border curve obtained as a closed form. After the calculations, it has been determined that the diffusion area where the bullet with pointed end was dropped on a para-aramid fabric is bigger and the diffusion area where the bullet with rounded end was dropped is smaller when compared with the areas where other bullets with different ends were dropped.

The exploration of the use of parametric curves and surfaces can be viewed as the origin of Computer Aided Geometric Design (CAGD). The major breakthroughs in CAGD were undoubtedly the theory of Bezier surfaces and Coons patches, later combined with B-spline methods. Bezier curves and surfaces were independently developed by P. De Casteljau at Citröen and by P. Bezier and Renault [

In this research the Classic Holditch Theorem is given in [

On the other hand, para-aramid fabrics produced from high-tenacity fibres are used in ballistic protection due to their high energy absorption ability and low tenacity/weight ratio. Ballistic behavior of textile fibres and fabrics has been investigated experimentally and ballistic behavior of textile fabric systems has been estimated in [

Para-aramid fabrics which are a class of heat-resistant and strong synthetic fabrics are used in aerospace and military applications for ballistic protection. In our research, Twaron CT type 710 fabric which is a type of para-aramid fabrics is used. Twaron fabric is one of ballistic fabrics used in steel vest production and developed in the early 1970s; see Figure

Structure of para-aramid fabric.

With the help of textile engineers, the warp and wept information of the para-aramid fabric, which is in CT710 type, and other parameters of these fabrics as density and weight are shown in Table

Properties of Twaron CT710 type fabrics used in this research, [

Fabric Type | Count Warp/Weft | Meterial Type Warp/Weft | Weave | Density | Fabric Weight | Treatment |
---|---|---|---|---|---|---|

CT 710 | 930/930 | 2040/2000 | Plain | 117/117 | 220 | Scouring/Water repellent treatment |

In our experiment step which was performed in the textile laboratory, the drop test was used under the

Bullets tip types.

The second material is the clay which has the characteristics of the human skin. Then the third material is the panel arm which is 1 meter in size and with which the bullet shots were made. The fourth material is the 50 mm pipe used for guiding the weight. Without using cloth for three bullet types and using 1 fold, 2 folds, 6 folds, and 8 folds, the number of the shots was 72, and 72 photos which were 2-dimensional were taken in Figure

Test apparatus used in drop tests [

The dropping tests are applied onto the fabric layers which are put on a clay base. The clay used is a special one which is in agreement with the characteristics of the human tissue. After the tests, the deformation on the clay is accepted as the possible deformation on the human tissue. The computation proceeded in four steps. Firstly, landmarks are identified from photographs and then landmark coordinates are determined. The coordinates of homologous landmarks are obtained from two shapes. These two shapes may represent either individual specimens or the means of two sets of shapes corresponding target landmarks. A transformation from a source shape to a target shape involves the displacement of the source landmarks to the corresponding target landmarks.

Let an AB chord with a constant length of

Classical Holditch Theorem [

In the early 1960s, J. C. Ferguson from the Boeing Airplane Company in the USA developed a method which defined the curves as vectors. A Ferguson Curve is a cubic vector function obtained with the parameters after determining the starting and ending points of a curve. Later in 1964, S. A. Coons introduced a surface definition method which gave the mathematical definition that provided the border conditions. In this method, the border curves and the position vectors in the 4 corners of the surface patch were taken into consideration. In 1974, P. Bezier from the Renault Company in France defined a curve representation by giving a polygon and called it as the Bezier curve. The Bezier curves were applied in practice to Renault automobile body designs. Gordon and Riesenfeld introduced the curves that used the base splines as combined functions and these curves were called B-spline curves. Like the Bezier curves, the B-spline curves are also defined by polygon corners and have similar properties. However, the B-spline curves differ from the Bezier curves in these points: The control points of the Bezier curves ensure only global control; however, the control points of the B-spline curves ensure local control. In other words, if a point is moved, the whole of the curve changes in the Bezier curve, but in the B-spline curve, only some parts are affected, and the other parts remain unchanged. For this reason, because of their ease in practice, the B-spline curves will be used in our study because they ensure local changes and the other parts of the curve do not change.

Given:

Set:

The polygon

The De Boor algorithm is used in the design of the B-spline curves, because it ensures a certain method in taking the control points. Let

A class

Uniform closed B-spline.

In [

Quadratic B-spline segments.

Location of joints uniform quadratic B-spline segments.

The curves with the C1 Continuity.

When we expand this method by taking the

Different way of modelling uniform closed B-spline [

The

Holditch Theorem with B-spline curves [

We applied these methods to the deformation photos at the end of the experiment. Our purpose is to calculate the area between the projection of the bullet and its effective area border curve with the help of the Holditch Theorem after drawing the B-spline curves. By doing so and by observing the deformation area in the cloth, we aim to determine which weight-end was used; see Figures

Deformation area with ball type.

Deformation areas with sharp type.

The coordinates of big polygon for the ball tip in Figure

Modelling deformation splines with ball type.

−12,7
12,15
15,21
5,04
−12,09
−15,62
12,11
23,35
−4,55
−15,19
−11,56
2,79
−0,275
13,68
10,125
−3,525
−13,855
−14,16
17,73
9,4
−9,87
−13,375
−4,385
7,45

−3,62
−0,09
3,54
3,54
0,01
−3,66
2,22
4,25
2,08
−2,29
−4,32
−2,15
−1,855
1,725
3,54
1,775
−1,825
−3,64
3,235
3,165
−0,105
−3,305
−3,235
0,035

Modelling deformation splines with sharp type.

−9,79
−1,09
10,19
10,55
1,97
−11,71
3,91
12,47
4,13
−3,62
−11,12
−3,86
−5,44
4,55
10,37
6,26
−4,87
−10,75
8,19
8,3
0,255
−7,37
−7,49
0,025

−3,59
0
3,66
3,66
0,06
−3,59
2,19
4,37
2,17
−2,17
−4,36
−2,18
−1,795
1,83
3,66
1,86
−1,765
−3,59
3,28
3,27
0
−3,265
−3,27
0,005

An example of geometric modelling by MATLAB.

Also, the area between the B-spline closed curves can be found by using (

Calculating the expansion areas of tips.

8 Ply of fabric | |
---|---|

Type-A, ball type | 971.6465 |

Type-B, blunt type | 598.7112 |

Type-C, sharp type | 529.1348 |

In our paper, the diffusion areas were modelled by a geometrical spline method. Some special points around the expansion area were taken on the deformation photos. Then the geometric spline modelling was done. As was mentioned above, the data of the Holditch areas that were found by using spline method after the geometric modelling study for each tip are given. The geometric spline method is very useful for the estimation of the expansion areas of tips. When we consider the tables, in estimating the bullet tip by observing the spread area, we can say that the spread area that is left by the bullet with round tip is more than the area which is left by the bullet with sharp tip. In our research we can see that using the Holditch area theorem with geometric spline method is very useful method in determining the tips from deformation photographs.

The authors declare that they have no competing interests.

This study has received support from the Uludağ University Scientific Research Unit with the Project Reference no. UAP(F)-2012/21. The authors would like to thank Professor Ali Çalışkan for his valuable comments and suggestions in improving the quality of the paper.