Modelling of the Deformation Diffusion Areas on a Para-Aramid Fabric with B-Spline Curves

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.


Introduction
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 [1].Gordon and Riesenfeld proposed curves and surfaces which use basis splines as blending functions.These are called B-spline curves and surfaces.B-spline curves, like Bezier curves, are defined by polygon vertices and have properties similar to those of Bezier curves.That is, B-spline curves are expressed as a convex combination of polygon vertex position vectors and also have the variation diminishing property.Curve shapes are smoothed versions of the polygon shapes and can be roughly predicted from the polygon shapes [2].In plane geometry Holditch Theorem states if a chord of fixed length is allowed to rotate inside a convex closed curve then the locus of a point on the chord a distance  from one end and a distance  from the other is a closed curve whose area is less than that of the original curve by .The theorem was published in 1858 by Rev. Amnet Holditch.While not mentioned by Holditch, the proof of the theorem requires an assumption that the chord be short enough that the traced locus is a simple closed curve [3].
In this research the Classic Holditch Theorem is given in [4].Also in the paper [5] the authors used Holditch Theorem to calculate the areas between two quadratic uniform Bspline curves.In this study, the mathematical modelling of the deformation areas on para-aramid fabric has been investigated using the Holditch Theorem which calculated the areas between two uniform B-spline curves.
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 [6][7][8][9][10][11][12][13][14][15].In our paper, 2 International Journal of Engineering Mathematics we have focused on a type of para-aramid fabric and we have investigated geometrical modelling of the planar energy diffusion behaviors of the deformations on the ballistic fabric.

Material
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 1.
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 1.Fabrics were plied 1, 2, 4, 6, and 8 ply numbers and sewn 2,5 cm from edges with plain stich with high twist para-aramid sewing yarn.Fabric dimensions are 31 × 31 cm and parameters of these fabrics are given in Table 1.
In our experiment step which was performed in the textile laboratory, the drop test was used under the   standards.The first material consists of three different bullets 45 mm in diameter, 1 kg in weight, consisting of a round tip shown with A, a middle butt-end tip shown with B, and a sharp tip shown with C in Figure 2.
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 3.
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.
Note.Type Roma Plastilina (clay) which is given in   standards was used as clay material in this research.Clay was filled in a box as pointed out in the standard.

Classical Holditch Theorem.
Let an AB chord with a constant length of  +  on a circle (C) with a radius  in Euclidean plane  2 be divided by a point D into two segments with lengths of  and , respectively.When the end points A and B of the chord draw the circle in full, then geometric location of D forms an inner circle.The difference between the areas of two closed curves can be calculated from the formula by  ⋅  ⋅  as following Figure 4 in [4].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 Bspline 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 Bspline 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.

De Casteljau Algorithm
Given:  0 ,  1 , . . .,   ∈  3 and  ∈  Set: The polygon  formed by  0 ,  1 , . . .,   is called the Bezier polygon or control polygon of the curve   .Similarly the polygon vertices   are called control points or Bezier points.Sometimes we also write   () = [ 0 , . . .,   ; ] = [, ] or, shorter,   = [ 0 , . . .,   ] = .This notation defines  to be the linear operator that associates the Bezier curve with its control polygon.We say that the curve [ 0 , . . .,   ] is the Bernstein-Bezier approximation to the control polygon, a terminology borrowed from approximation theory.In the Bezier curves, the De Casteljau algorithm is used, because it ensures that the control points are taken in a certain order.

De Boor Algorithm.
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.

Uniform Closed B-Spline.
A class  −2 closed curve (of degree  − 1) consisting of  + 1 curve segments   (),    = 0, 1, . . ., , with a curve defining polygon determined by the  + 1 position vectors  0 ,  1 , . . .,   in Figure 5 can be expressed by the following function: where the initial and finish points   (0) =   ,  = 0, 1, . . ., , are equal to each other, in [1,2].We planned to use the uniform closed B-spline curves in our study because the photos of the deformation traces were taken in 2D during the experiment.

Location of Joints Uniform Quadratic B-Spline Segments.
In [5], specifically each part is expressed as parabolas consisting of control points with three points because local control is ensured in B-spline curves in Figure 6.The matrix formats of these quadratic B-spline curves are as shown in the following equation: In this case, only 3 control vertices are used in Figure 7.By joining these curves which correspond to these matrices, the closed curve is formed.In this situation, the midpoints of the two control points from the starting point are considered as being like tangents to the curves.Since the joining places Figure 7: Location of joints uniform quadratic B-spline segments. of these curves are the tangents, the curves are connected to each other continuously with the C1 Continuity in Figure 8.
When we expand this method by taking the  point, the matrices in the following formulas are formed for each part: () : ( 1 ) ∪ ( 2 ) ∪ ⋅ ⋅ ⋅ ∪ (  ) =  () . ( Uniform B-splines are convenient to represent closed curves.The only thing needed is a change in the number of the curve segments.Figure 9 shows a closed quadratic B-spline curve produced by 6 control points.Also we can find location of joints between uniform quadratic B-spline segments.If we take the vertices of this polygon as control points, quadratic uniform B-spline curve will pass from midpoints of sides.Equations of the curve segments will give us a closed curve for  control points.

The Holditch Theorem by Using B-Splines.
The  and  chords taken in the Classic Holditch Theorem are taken especially in equal lengths in [3,4].We can find the area between the B-spline closed curves as by the help of Holditch Classical Theorem in Figure 10.

Main Results
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 11 and 12.

Modelling of Deformation
Splines for Ball Tip .The coordinates of big polygon for the ball tip in Figure 13 are given in the following: can be represented in the MATLAB program as shown in Figure 15.Also, the area between the B-spline closed curves can be found by using (6); thus Table 2 shows the areas for all tips.

Conclusion
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.

Figure 8 :
Figure 8: The curves with the C1 Continuity.

Table 2 :
Calculating the expansion areas of tips.