Geometric structure model of plain woven fabric based on progressive spring-slide mechanics

Abstract

The geometric structure of woven fabric affects the appearance and physical properties and can be used to predict the weavability of the fabric. An accurate description of the structure is beneficial to predict the characteristics and the appearance of the woven fabric. The purpose of the study is to build an accurate, realistic, and stable three-dimensional (3D) geometric structure model based on the mechanics for plain woven fabrics. Parameters such as the initial Young's modulus of the yarn materials, thicknesses, warp density, pre-loaded tension on the yarns, and the amount of letting-off and taking-up are considered in the model. The yarns are simplified as a series of spring-sliders that are stretched in the fabric and moved along the direction of the resultant forces step by step according to the changing forces. The yarns stop moving when all the forces reach equilibrium. The tests demonstrate that the algorithm conforms to the weaving principles. The 3D images of the geometric structure of the woven fabric at different steps are displayed by B-spline surface modeling technology. A Keyence VH600 micro-measurement system is used to measure the 3D coordinates of the real fabric accurately without destroying the fabric structure. The similarity of the real geometric structure and the calculated geometric model is evaluated by calculating the discrete Fréchet distances. The result validates that the similarity of the calculated geometric structure and the measured value is more than 90% for both the warp and weft direction.

Keywords

plain woven fabric geometric phase of woven structure progressive spring–slider mechanical model discrete Fréchet distance B-spline surface

The geometric structure of the woven fabric describes the shape of the warp yarns and weft yarns in the fabric, which in turn affects the weavability, mechanical properties, thermal properties, electrical properties, aesthetical effects of the fabric, etc. The geometric structure of a woven fabric is determined by the weave interlacing, fabric set, yarn thickness, yarn modulus, the amount of taking-up and letting-off, pre-loaded tension on yarns, the position of the beam, and the timing of the picking. Since plain woven fabric is the most widely used fabric in the world, it is important to understand the geometry of plain woven fabric. In the 1930s, Peirce¹ developed a pure mathematical model according to the position of the yarns to study the geometry of the plain weave based on cotton fabrics. He made the assumptions that (1) the yarns are not extensible; (2) all the yarns are uniform; and (3) the cross-sections of the yarns are circular and later elliptical. In order to get a more accurate calculation, Kemp,² Hamilton,³ and Love, Ellis and Hearle⁴ extended his theory and thought the cross-sections of the yarns were race-track shaped or lens shaped. According to Peirce's¹ paper, the accuracy of the calculation can reach as high as 90%. To use Peirce's¹ geometric model, the weaving crimp retraction and the crimp height of the yarn in both the warp and weft direction should be given in advance, which can only be obtained by experience.⁵ Therefore, the result is dependent on the designers' choice. Before the 1970s, when the theory was proposed, slicing technology was the unique method to measure and validate the geometry, which is not suitable for measuring soft matter such as fabric;⁶ in other words, the model is doubtable. At present, although the commercial fabric computer-aided design (CAD) systems, such as EAT, NedGraphicis, and ScotWeave, which are based on Peirce's¹ model can imitate the two-dimensional (2D) or three-dimensional (3D) image of the woven fabrics, they cannot distinguish the differences in normal plain woven fabrics, such as distinguish between poplin and fine plain or cambric, despite some more complex weaves being imitated.

Worse still, Peirce's¹ theory does not take the yarn material, warp tension into account, which is a distinct drawback for predicting the geometry of the structure. Therefore, Hearle,⁷ Hearle et al.,⁸ Shanahan and Hearle,⁹ Sagar and Potluri,¹⁰ Lomov et al.,¹¹ Verpoest and Lomov,¹² and Cui¹³ proposed the minimum energy method to calculate the geometry of the woven fabric. The shape of the central line is represented by a polynomial or cubic spline curve. However, the method can only apply to non-compact fabric, the assumptions are doubtable, and the calculations are not validated.¹⁴ Behera and Hari¹⁵ proposed a method to calculate the yarn's path in the woven fabric, as well, but did not give the result of the calculation to demonstrate its validity. From the viewpoint of the fabric bending and draping properties, Wang¹⁶ believed that the central lines of the yarn in woven fabric were sinusoidal, while Kawabata et al.¹⁷ thought they were in a zigzag pattern. Kawabata et al.¹⁷ even developed an algorithm to calculate the deformed geometric structure of woven fabric considering the tension force in the research of biaxial tensile properties, uniaxial tensile properties, and shear-deformation properties of woven fabrics. However, the initial geometric structures were still based on the Peirce's¹ model. Ji et al.¹⁸ proposed the spring–mass model to investigate the mechanical properties of the fabric, but not the geometry of the structure. The finite element system seems to be an effective method to solve the problem, but no papers have been reported so far. In order to study the geometric structure of the woven fabrics more accurately, Gong et al.¹⁹ proposed a way to describe the structure of the yarn with the European Synchrotron Radiation Facility (ESRF), which may be used to measure the 3D structure of the woven fabrics. By ESRF technology, Ozgen and Gong confirmed^20,21 the non-circular nature of yarn cross-sections and also the flattening of yarns at the warp–weft yarn intersections. But the problems of the long time to wait and the high cost for the measurement result of ESRF still exist, and the cross-section is too complex to use.

The purpose of the study is to build a convenient, accurate, and stable 3D geometric structure model for plain woven fabric based on the mechanical behavior and to evaluate the calculation. The parameters such as the yarn's initial modulus, thicknesses, fabric set, pre-loaded tension on warp, and the amount of letting-off and taking-up will be considered in the calculation. After the calculation, the 3D realistic image of the woven structure will be displayed based on computer graphics. The calculation will be compared with the measured value of the real fabric. The idea may be promoted to more complicated fabric structures, such as twill, satin, sateen, hopsack, or honeycomb. The study may also have the potential application in engineering, where it can be used to design fiber enhanced composites and garment textiles with computers.

Calculation of the key points on the central line of the yarn

The key area of the study is to calculate the key points on the central line of the yarn in the plain woven structure. As the fabric is comprised of the warp and weft yarns, it is obvious that the key points to refer to are the coordinates of the central point of the cross-section of the weave points on the warp and weft yarns. Figure 1 demonstrates the key points J₁, J₂, J₃, and so on for the warp yarns and W₁, W₂, W₃, and so on for the weft yarns in a woven plain structure.

Figure 1.

The key points determining the geometric structure of a plain woven fabric.

Assumptions for the algorithm

In this paper, we have the following assumptions:

A segment of yarn in the woven structure can be considered as the combination of a spring (or two halves of spring) and a slider as shown in Figure 2. The spring is used to reflect Hook's law and gives the yarn stretch tension. Considering the calculation for more complicated woven structures where friction between yarns may happen, a slider is used to connect the springs and deliver the support or pressure from one system of yarn (warp or weft) to another (weft or warp).

The path of the yarns in a woven structure is of a zigzag shape, which means a segment line can be used to describe the yarn between two weave points. Figure 3 shows the weave pattern, yarn interlacing configuration, and the spring–slider model of a plain woven fabric where the yarns in the woven structure can be regarded as a series of spring–sliders.

The directions of the pre-loaded forces are exactly coinciding with the yarn path; furthermore, they do not elongate the yarn. In other words, the length of the yarn under pre-load is regarded as the original length of the yarn.

The warp and weft yarns keep in contact, which means the sum of the warp diameter and the weft diameter is equal to the sum of the warp crimp height and the weft crimp height

d_{j} + d_{w} = h_{j} + h_{w}

(1)

where d_j and d_w are the diameter of the warp yarn and weft yarn respectively, while h_j and h_w are the crimp height of the warp yarn and weft yarn respectively.

Figure 2.

Spring–slider model of yarn.

Figure 3.

Weave pattern, interlacing configuration, and the spring–slider model for plain woven fabric.

Yarn's mechanical behaviors based on the spring–slide model

Yarn is an assembly of fibers. Its mechanical properties are time dependent because of the viscoelastic nature of fiber materials. Usually, the mechanical model of the yarn mechanical properties consists of a series of elements, such as Hook springs, Newton dashpots, unidirectional friction elements, and inertial elements. For simplicity, the relationship between the stress and the strain can be considered to obey Hook's law. There are some formulas describing the strain of the completely elastic material

σ = E • ɛ

(2)

σ = \frac{F}{S} = E • ɛ = E \frac{Δ L}{L_{0}}

(3)

F = E • S • \frac{Δ L}{L_{0}} = K • Δ L

(4)

where S is the area of the material, E is the elastic modulus, σ is the stress, ɛ is the strain, L₀ is the original length of the spring, and ΔL is the elongation of the spring. The relationship between the modulus E and the coefficient of stiffness K of the spring can be written as

K = E \frac{S}{L_{0}}

(5)

A simplified stress–strain diagram for fibers may be better described in Figure 4. Considering the extension of the yarn in weaving may go beyond the yielding point t₁, equation (6) is given in order to accurately express the relationship between the elongated length L₁ and the tension force F

if \frac{L_{1} - L_{0}}{L_{0}} \leq t_{1} F = K_{1} • (L_{1} - L_{0}) else \frac{L_{1} - L_{0}}{L_{0}} = t_{2} > t_{1} F = K_{1} • L_{0} • t_{1} + K_{2} • (t_{2} - t_{1}) • L_{0}

(6)

where K₁ and K₂ are the stiffness of the yarn before and after the yielding point respectively.

Figure 4.

A simplified stress–strain curve for fiber.

Usually, the fiber's initial Young's modulus is available. The modulus of yarns comprised of filaments will decrease dramatically. Yu et al.²² reported that the relationship between the modulus of filament yarn E_y and the modulus of fiber E_f can be described as the following formula

E_{y} = E_{f} • \cos^{2} β

(7)

where β is the twist angle of the yarns.

When compared to a spun yarn, the modulus will decrease further. Hearle et al.²³ proposed that the modulus of the spun yarn can be described as the following formula

E_{y} = E_{f} • \cos^{2} β • (1 - t \csc β)

(8)

where t is a coefficient between 0.01 and 0.1. Thus, the elastic modulus of the spun yarn can be as low as 20% of the fiber's elastic modulus.

Geometric phases of the woven structure

According to the theory of geometric thread disposition by Novikov in Chepelyuk et al.,²⁴ the geometric structure of plain weave can be identified by one of nine structure “phases” as shown in Figure 5. The first phase relates to the extreme case where all the warp yarns remain straight within the woven fabric while all the weft yarns bend to the maximum degree. While at the ninth phase, the weft yarns remain straight, and the warp yarns bend to the maximum degree. Between the first and ninth phases, the warp yarns and the weft yarns bend in different degrees. As the phase value increases, the warp crimp height also increases. At the fifth phase, the crimp height of the warp yarns and the weft yarns are equal.

Figure 5.

Nine phases of geometric disposition of threads in woven structures by Novikov.²⁴

Although the warp yarn and the weft yarn are of a diameter in Figure 5 to aid the explanation, they can be different. Therefore, the relationship between the crimp height h_j or h_w and the phase value V can be determined by equation (9)

V = 9 - 8 • h_{w} / (d_{j} + d_{w})

(9)

In the study, the phase value is unnecessarily an integer. It can be any decimal number from 1 to 9 that describes a more accurate disposition of the yarns.

Description of the algorithm (assumption of initial status of the woven geometric structure)

Considering the weft yarn is straight when it is initially inserted into shedding, the initial position of the woven structure is started at geometric phase 9 as shown in Figure 6. The warp yarn is completely crimped, stretched, and has the largest extension while the weft yarn is totally straight and has no extension.

Figure 6.

Initial status and the forces acting on the woven structure.

At phase 9, the warp yarns are subjected to an elastic tension F_j due to elongation as shown in Figure 6. The direction of F_j points to the center of its neighboring weave point on the same warp yarn. They may also be subject to an additional pre-loaded force F_a. The first weave point J₁ or W₁ is taken as an example. The resultant force of F_j and F_a only makes the J₁ go downward and gives W₁ pressure force F_p, which means J₁ will go downward and W₁ will go upward. It is the same situation for J₃ and W₃. Since the warp yarns and the weft yarns are interlaced alternately, the adjacent J₂ and J₄ will go upward while W₂ and W₄ will go downward, and the geometry of the fabric has the tendency to change to a lower phase.

In the process of forming the structure, the forces that the yarn is subjected to changes continuously. When the warp yarns retract, F_j decreases, which means F_p decreases. At the same time, the weft yarns are elongated and are subjected to an elastic tension F_w, which in turn only gives the warp yarns a perpendicular support resultant force F_s to block the movement of the warp yarns as shown in Figure 7. Likewise, the direction of F_w points to the neighboring weave point on the weft yarn. During the movement, the elongation of the warp yarn decreases while the elongation of the weft yarn increases; that is to say, F_p decreases while F_s increases. The processing continues and the positions of the yarns' center and the forces are calculated step by step according to the previous data. Finally, when the two opposite forces reach equilibrium, the movement stops, and a stable woven structure is formed.

Figure 7.

The schematic diagram of the forces acting on the changing woven structure.

As we can see, only z-coordinates of the weave points change in a plain structure since the component forces along the x or y-direction of the elastic tensions are in equilibrium. The weft yarns and the warp yarns only go upward and downward; they will never go forward, backward, left, or right, theoretically.

In calculating, the tendency of the key points on the warp yarns or weft yarns are analyzed according to the resultant force. These key points will move according to a given tiny step. The initial step length S₁ may be set as 1/16 of the sum of the diameter of the warp yarn and the weft yarn. Then, the new positions of the key points are recalculated and the resultant forces that the key points receive are analyzed again. If the direction of the resultant force of F_b, F_p, F_a, and F_s are the same, the movement tendency of the key points does not change, and the key points move at the same step. The iterated process continues until equilibrium is reached (if the resultant force < 0.00001 N) or the direction of the resultant force changes. The number of the steps at step length S₁ is recorded. Then, if the key points need further movement, the step length is halved to S₂, and all the key points go in the opposite direction at step S₂. The new positions of the key points are calculated and the resultant force will be analyzed again. Since step length S₂ is the half of S₁, the number of the progressive calculating processes is not more than two. If force equilibrium cannot be reached in step length S₂, then S₃ (half of S₂) or even S₄ (half of S₃) will be used. If it still cannot reach equilibrium, the calculating stops since the step length S₄ is so tiny (1/128 of the sum of d_j and d_w) and almost does not affect the final configuration. At last, the total path length L passing the key points is counted according to the step length and the number of the movements. L is also the length of the key point (weave point). J₁ goes downward or the key point (weave point) W₁ goes upward; therefore, the final stable structure of the woven fabric is calculated.

Algorithm chart for calculating the key points on the central line of the yarn path

From the unit cell of a plain woven fabric, geometric parameters such as thread spacing, weave angle, and crimp contraction are related by deriving a set of equations. The symbols used to denote these parameters are listed below: d:

diameter of the yarn

Radius of the yarn

thread spacing or the distance between two neighboring threads

displacement of the thread axis normal to the plane of the fabric (crimp height)

θ:

angle of the thread axis to the plane of the cloth

stiffness of the yarn

fabric density, or fabric setting (yarns per 10 cm)

crimp contraction of the fabric (%)

Subscripts j and w to the above parameters represent warp and weft threads, respectively. Superscript 1 indicates the positions of the points that have changed.

Figure 8 shows the initial structure diagram of a weave repeat as a calculating unit, in which some key points such as J₁, W₁, E, and O are labeled. Figure 9 shows an intermediate structure stage of the plain woven fabric after some steps of progressing and J₁¹, W₁¹, E¹, and O¹ are the corresponding points of J₁, W₁, E, and O, respectively. Since the warp yarns and the weft yarns move symmetrically, the central plane of the fabric does not change and is stable; thus, the plane is used as the XOY plane of the xyz-coordinate system. In progressive calculating processes, points J₁, W₁, and E move to J₁¹, W₁¹, and E¹ respectively while O remains motionless. In other words, O and O¹ are the same point. The following formulas are listed for a better understanding of the algorithm

J_{1} J_{1}^{1} = {EE}^{1} = \frac{1}{2} h_{w}

(10)

{OE}^{1} = \frac{1}{2} h_{j} = OE - {EE}^{1} = \frac{1}{2} (R_{j} + R_{w}) - {EE}^{1}

(11)

Figure 8.

The geometry of the plain woven structure at the initial status (at phase 9).

Figure 9.

The geometry of the plain woven structure at an intermediate stage.

Since the warp yarns and the weft yarns do not move transversally or longitudinally according to the analysis above, all the weave points are evenly distributed in the x and y-direction theoretically. The algorithm flow chart for calculating the z-coordinates of the weave point J₁ and W₁ is shown in Figure 10. Obviously, the z-coordinates of the weave point J₂ and W₂ can be easily calculated.

Figure 10.

Algorithm flow chart for calculating the geometric structure of woven fabric.

Testing examples

Imagine a set of plain woven fabrics for the calculation. The parameters of the fabrics are varied one by one, and the final geometric phases of the fabrics are calculated and displayed in Table 1 accordingly.

Table 1.

The parameters of the invented plain fabrics for testing.

Sample no.	Warp diameter R_j (mm)	Weft diameter R_w (mm)	Warp modulus E_j (GPa)	Weft modulus E_w (GPa)	Pre-loaded warp tension F_a(N)	Pre-loaded weft tension F_b(N)	Dent Size N_d (dents/10 cm)	Ends per dent N (ends/dent)	Amount of letting-off T_j (mm)	Amount of taking-up g_j (mm)	Final geometric phase V
1	0.20	0.20	2	2	0.2	0	95	2	0.526	0.5	5.3125
2	0.20	0.20	3	1	0.2	0	95	2	0.526	0.5	4.625
3	0.20	0.20	2	5	0.2	0	95	2	0.526	0.5	6.125
4	0.20	0.24	2	5	0.2	0	95	2	0.526	0.5	6.3125
5	0.20	0.20	2	2	0	0	95	2	0.526	0.5	5.435
6	0.20	0.20	2	2	0.2	0	125	2	0.526	0.5	5.8125
7	0.20	0.24	2	2	0.2	0	95	2	0.526	0.5	5.5625
8	0.24	0.20	2	2	0.2	0	95	2	0.526	0.5	5.0625
9	0.20	0.20	2	2	0.2	0	95	2	0.421	0.4	4.8125
10	0.20	0.20	2	2	0.2	0	100	2	0.526	0.5	5.5
11	0.20	0.20	2	2	0.2	0	100	2	0.5	0.475	5.375
12	0.20	0.20	2	2	0.2	0	100	2	0.505	0.5	5.0625
13	0.20	0.20	2	2	0.2	0	95	2	0.51	0.5	5.0
14	0.20	0.20	2	2	0.2	0.1	95	2	0.51	0.5	5.0625
15	0.20	0.20	2	2	0.2	0.2	95	2	0.51	0.5	5.125
16	0.20	0.20	2	2	0.2	0.35	95	2	0.51	0.5	5.1875

From Table 1, it can be seen that with the increment of the modulus of the weft yarn, the warp yarn density (dent size), and the letting-off per pick or pre-loaded weft tension that the final geometric phase increases. On the contrary, when the modulus of the warp yarn, the pre-loaded warp tension, or the weft density (the amount of taking-up) increase, the final geometric phase decreases. The calculated results conform to the principle illustrated in Gu²⁵ and Cai,²⁶ which demonstrates that the algorithm based on the mechanical model is reasonable.

Real fabrics are used to test the algorithm. Table 2 displays the specifications of 12 cotton samples listed in Cai.²⁶ Only warp count, weft count, warp density, weft density, warp crimp contraction, and weft crimp contraction are provided; there is a need to calculate the parameters needed for the software according to equations (12)–(14). The parameter “Pre-loaded warp tension” is not available for all the samples. For the samples that this parameter is not available, it is selected according to Chen.²⁷ The pre-loaded weft tensions are unknown as well, and they are all set zero. Table 3 shows the parameters according to Table 2 and the final calculated geometric phase

N_{d} = \frac{P_{j}}{n} (1 - a_{w})

(12)

g_{j} = 100 / P_{w} (mm)

(13)

T_{j} = g_{j} / (1 - a_{j}) (mm)

(14)

where N_d is the dent size (number of dents per 10 cm), n is the number of the ends threaded-in each dent and T_j is the length of the amount of letting-off.

Table 2.

Specifications of the cotton fabric samples²⁶ for testing.

Sample no.	Warp count (Tex)	Weft count (Tex)	Warp density (ends/10 cm)	Weft density (picks/10 cm)	Warp crimp contraction (%)	Weft crimp contraction (%)
1	42	42	188.5	188.5	7.17	8.33
2	29	29	188.5	188.5	5.0	8.34
3	29	29	236	236	8.0	6.66
4	24	24	261.5	237	7.0	6.34
5	19.5	19.5	267.5	267.5	7.0	5.88
6	18	18	244	236	5.4	6.45
7	14	14	248	244	3.5	5.09
8	29	29	326.5	188.5	9.1	3.58
9	14.5	15.5	503.5	220	8.7	2.16
10	14*2	42	393.5	196.5	16.5	1.09
11	7*2	7*2	567	276	12.2	2.09
12	14*2	21	355.5	265.5	10.36	4.6

Table 3.

Parameters corresponding to the samples in Table 2 and the calculated geometric phase.

Sample no.	Warp diameter R_j (mm)	Weft diameter R_w (mm)	Warp modulus Y_j (GPa)	Weft modulus Y_w (GPa)	Pre-loaded warp tension F_a (N)	Dent size N_d (dents/10 cm)	Ends per dent n (ends/dent)	Amount of letting-off T_j (mm)	Amount of taking-up g_j (mm)	Final geometric phase V
1	0.24	0.24	2	2	0.25	86	2	0.5715	0.5305	5.4375
2	0.20	0.20	2	2	0.20	86	2	0.5584	0.5305	5.3125
3	0.20	0.20	2	2	0.20	110	2	0.4606	0.4237	5.5
4	0.18	0.18	2	2	0.15	122	2	0.4537	0.4220	5.6875
5	0.1633	0.1633	2	2	0.15	126	2	0.4020	0.3738	5.3125
6	0.157	0.157	2	2	0.15	114	2	0.4479	0.4237	5.4375
7	0.1384	0.1384	2	2	0.12	118	2	0.4247	0.4098	5.25
8	0.20	0.20	2	2	0.20	158	2	0.5836	0.5305	6.6875
9	0.1409	0.1457	2	2	0.12	124	4	0.4979	0.4545	7.25
10	0.1958	0.24	2	2	0.20	194	2	0.5883	0.5089	7.3125
11	0.1384	0.1384	2	2	0.12	138	4	0.4127	0.3623	7.875
12	0.1958	0.1696	2	2	0.20	170	2	0.4202	0.3766	6.0

There are no apparent differences between the warp yarn densities and the weft yarn densities for samples 1–7; the final calculated geometric phases are from samples 5 and 6. As for samples 8–12, the warp yarn densities are much larger than the weft yarn densities; the final values of the calculated geometric phases are approximately 7.

3D visualization of the progressive processing for the geometric structure

Simulating the 3D image of the geometric structure according to the calculated data provides a realistic impression of the algorithm. The displaying method in this paper is based on Zheng;²⁸ all the yarns are modeled by a B-spline surface. For a better simulation, the coordinates of more key points including the midpoint of the two neighboring weave points are calculated. For the sake of simplicity, the cross-sections of the yarn are regarded as elliptical. According to the image captured by a Keyence VH600,⁵ it is found that there is a linear relationship among the major axes of the ellipses on the yarn,²⁹ which will not be elaborated on in this paper. The code is complied with the help of Microsoft Visual Studio and OpenGL. The 13 simulated images in Figure 11 show the 3D oblique view figure of sample 7 at different progressive stages from the start phase to the final structure. From the images, the weft yarns start at a straight status and then become bent gradually until the movement stops. The step length is (r_j + r_w)/16 for Figure 11(a)–(i), (r_j + r_w)/32 for Figure 11(j), (r_j + r_w)/64 for Figure 11(k), and (r_j + r_w)/128 for Figure 11(l) and (m). Figure 12 shows the front view of the simulated 3D image of sample 7.

Figure 11.

The 3D simulated oblique view figures of sample 7 at 13 different steps.

Figure 12.

The 3D simulated figures of sample 7: (a) without the twist effect and (b) with the twist effect.

Validation for the geometric model based on progressive mechanics

The 12 samples in Table 2 are not available; however, a real cotton plain woven fabric (29 × 29 × 238 × 210) similar to sample 7 in Table 2 is used to validate the theory. The warp crimp contraction a_j is 8.2% while the weft crimp contraction a_w is 6.7%. It is easy to get the theoretic diameter of the warp or weft yarns; the value is 0.2 mm. The final geometric phase is 5.625 based on the algorithm in this study.

There is still a necessity to validate the accuracy of the calculation by comparing the calculated result with the real fabric. The 3D coordinates of the key points of the fabric structure can be measured with a Keyence VH600 digital measuring system without destroying the structure of the fabric.⁵ Discrete Fréchet distance criterion^29–31 can be used to judge the similarity of the theoretic values and the measured values. Figure 13 illustrates how two curves are compared by the method. Supposing both curve L₁ and L₂ are composed of a set of discrete points, S_p is the distances between the two corresponding peak points on the curves, and S_t is the distances between the two corresponding trough points on the curves. Given a threshold δ, if all the absolute differences of the discrete Fréchet distances are within a given threshold, or if $| S_{pi} - S_{ti} | \leq δ$ , (i = 1, 2, 3…), the two curves L₁ and L₂ are regarded as similar. Otherwise, L₁ and L₂ are not similar. When the threshold δ is smaller, the similarity of the two curves is increased.

Figure 13.

Schematic of the similarity of the curves.

It is obvious that if δ is large enough, all the curves can be considered to be similar. For the accuracy of similarity, the thickness of the fabric τ should be considered. The maximum absolute difference of the discrete Fréchet distances of the corresponding points of the theoretical value and the measured value is set at δ, then the curves passing through the discrete points can be considered similar. The similarity of the accuracy η can be obtained by the following formula

δ = | Max (S_{pi} - S_{ti}) | i = 1, 2, 3 \dots η = \frac{τ - δ}{τ} \times 100 %

(15)

For a higher accuracy, more discrete points should be compared.

The calculated coordinates, the measured coordinates of the weave points, and the discrete Fréchet distances of the corresponding points on six warp yarns and six weft yarns are listed on Tables 4 and 5, respectively.

Table 4.

The calculated and measured coordinates of the weave points on the warp yarns.

		Calculated coordinate (mm)			Measured coordinate (mm)			Distance
		x	y	z	x	y	z	(mm)	\|Dp-Dt\|
Warp 1	Weft 1	2.0000	2.2400	0.1141	2.10	1.28	0.066	0.9664	0.0158
	Weft 2	2.0000	2.7200	−0.1141	2.03	1.77	−0.13	0.9506	0.0158
	Weft 3	2.0000	3.2000	0.1141	2.00	2.25	0.064	0.9513	0.0207
	Weft 4	2.0000	3.6800	−0.1141	2.03	2.75	−0.13	0.9306	0.0207
	Weft 5	2.0000	4.1600	0.1141	2.00	3.21	0.068	0.9511	0.0690
	Weft 6	2.0000	4.6400	−0.1141	2.00	3.62	−0.13	1.0201	0.0690
Warp 2	Weft 1	2.4202	2.2400	−0.1141	2.45	1.31	−0.13	0.9306	0.0402
	Weft 2	2.4202	2.7200	0.1141	2.43	1.75	0.075	0.9708	0.0402
	Weft 3	2.4202	3.2000	−0.1141	2.37	2.27	−0.13	0.9315	0.0003
	Weft 4	2.4202	3.6800	0.1141	2.41	2.75	0.068	0.9312	0.0003
	Weft 5	2.4202	4.1600	−0.1141	2.41	3.20	−0.13	0.9602	0.1384
	Weft 6	2.4202	4.6400	0.1141	2.41	3.82	0.061	0.8218	0.1384
Warp 3	Weft 1	2.8403	2.2400	0.1141	2.90	1.30	0.074	0.9427	0.0076
	Weft 2	2.8403	2.7200	−0.1141	2.82	1.77	−0.13	0.9504	0.0076
	Weft 3	2.8403	3.2000	0.1141	2.79	2.28	0.077	0.9221	0.0124
	Weft 4	2.8403	3.6800	−0.1141	2.75	2.75	−0.13	0.9345	0.0124
	Weft 5	2.8403	4.1600	0.1141	2.79	3.20	0.067	0.9625	0.0008
	Weft 6	2.8403	4.6400	−0.1141	2.79	3.68	−0.14	0.9617	0.0008
Warp 4	Weft 1	3.2605	2.2400	−0.1141	3.27	1.33	−0.13	0.9102	0.0105
	Weft 2	3.2605	2.7200	0.1141	3.25	1.80	0.08	0.9207	0.0105
	Weft 3	3.2605	3.2000	−0.1141	3.27	2.27	−0.12	0.9301	0.0007
	Weft 4	3.2605	3.6800	0.1141	3.27	2.75	0.077	0.9308	0.0007
	Weft 5	3.2605	4.1600	−0.1141	3.28	3.25	−0.13	0.9103	0.1392
	Weft 6	3.2605	4.6400	0.1141	3.28	3.87	0.076	0.7712	0.1392
Warp 5	Weft 1	3.6807	2.2400	0.1141	3.82	1.28	0.064	0.9713	0.0187
	Weft 2	3.6807	2.7200	−0.1141	3.75	1.77	−0.13	0.9527	0.0187
	Weft 3	3.6807	3.2000	0.1141	3.75	2.30	0.076	0.9035	0.0292
	Weft 4	3.6807	3.6800	−0.1141	3.75	2.75	−0.13	0.9327	0.0292
	Weft 5	3.6807	4.1600	0.1141	3.75	3.25	0.067	0.9138	0.0287
	Weft 6	3.6807	4.6400	−0.1141	3.75	3.70	−0.12	0.9426	0.0287
Warp 6	Weft 1	4.1008	2.2400	−0.1141	4.18	1.32	−0.13	0.9235	0.0005
	Weft 2	4.1008	2.7200	0.1141	4.18	1.80	0.08	0.9240	0.0005
	Weft 3	4.1008	3.2000	−0.1141	4.13	2.27	−0.12	0.9305	0.0008
	Weft 4	4.1008	3.6800	0.1141	4.13	2.75	0.076	0.9312	0.0008
	Weft 5	4.1008	4.1600	−0.1141	4.13	3.21	−0.12	0.9505	0.0095
	Weft 6	4.1008	4.6400	0.1141	4.13	3.70	0.083	0.9410	0.0095
			h_j =	0.228		$\bar{h_{j}}$ =	0.206	Max =	0.1392

Table 5.

The calculated and measured coordinates of the weave points on the weft yarns.

		Calculated coordinate (mm)			Measured coordinate (mm)			Distance
		x	y	z	x	y	z	(mm)	\|Dp-Dt\|
Weft 1	Warp 1	2.0000	2.2400	−0.0859	2.10	1.28	−0.14	0.9667	0.0361
	Warp 2	2.4202	2.2400	0.0859	2.45	1.31	0.072	0.9306	0.0361
	Warp 3	2.8403	2.2400	−0.0859	2.90	1.30	−0.13	0.9429	0.0327
	Warp 4	3.2605	2.2400	0.0859	3.27	1.33	0.07	0.9102	0.0327
	Warp 5	3.6807	2.2400	−0.0859	3.82	1.28	−0.13	0.9711	0.0433
	Warp 6	4.1008	2.2400	0.0859	4.22	1.32	0.078	0.9277	0.0433
Weft 2	Warp 1	2.0000	2.7200	0.0859	2.03	1.77	0.068	0.9506	0.0204
	Warp 2	2.4202	2.7200	−0.0859	2.43	1.75	−0.13	0.9711	0.0204
	Warp 3	2.8403	2.7200	0.0859	2.82	1.77	0.072	0.9503	0.0296
	Warp 4	3.2605	2.7200	−0.0859	3.25	1.80	−0.12	0.9207	0.0296
	Warp 5	3.6807	2.7200	0.0859	3.77	1.77	0.068	0.9544	0.0303
	Warp 6	4.1008	2.7200	−0.0859	4.18	1.80	−0.12	0.9240	0.0303
Weft 3	Warp 1	2.0000	3.2000	−0.0859	2.00	2.25	−0.14	0.9515	0.0201
	Warp 2	2.4202	3.2000	0.0859	2.37	2.27	0.073	0.9314	0.0201
	Warp 3	2.8403	3.2000	−0.0859	2.79	2.28	−0.12	0.9220	0.0081
	Warp 4	3.2605	3.2000	0.0859	3.27	2.27	0.075	0.9301	0.0081
	Warp 5	3.6807	3.2000	−0.0859	3.75	2.30	−0.12	0.9033	0.0356
	Warp 6	4.1008	3.2000	0.0859	4.23	2.27	0.08	0.9389	0.0356
Weft 4	Warp 1	2.0000	3.6800	0.0859	2.03	2.75	0.067	0.9307	0.0004
	Warp 2	2.4202	3.6800	−0.0859	2.41	2.75	−0.13	0.9311	0.0004
	Warp 3	2.8403	3.6800	0.0859	2.75	2.75	0.071	0.9345	0.0034
	Warp 4	3.2605	3.6800	−0.0859	3.25	2.75	−0.13	0.9311	0.0034
	Warp 5	3.6807	3.6800	0.0859	3.75	2.75	0.074	0.9327	0.0016
	Warp 6	4.1008	3.6800	−0.0859	4.13	2.75	−0.12	0.9311	0.0016
Weft 5	Warp 1	2.0000	4.1600	−0.0859	2.00	3.21	−0.13	0.9510	0.0091
	Warp 2	2.4202	4.1600	0.0859	2.41	3.20	0.072	0.9602	0.0091
	Warp 3	2.8403	4.1600	−0.0859	2.79	3.20	−0.13	0.9623	0.0519
	Warp 4	3.2605	4.1600	0.0859	3.28	3.25	0.066	0.9104	0.0519
	Warp 5	3.6807	4.1600	−0.0859	3.75	3.25	−0.13	0.9137	0.0415
	Warp 6	4.1008	4.1600	0.0859	4.20	3.21	0.078	0.9552	0.0415
Weft 6	Warp 1	2.0000	4.6400	0.0859	2.00	3.62	0.070	1.0201	0.0186
	Warp 2	2.4202	4.6400	−0.0859	2.41	3.64	−0.14	1.0015	0.0186
	Warp 3	2.8403	4.6400	0.0859	2.78	3.68	0.07	0.9620	0.0212
	Warp 4	3.2605	4.6400	−0.0859	3.28	3.70	−0.12	0.9408	0.0212
	Warp 5	3.6807	4.6400	0.0859	3.75	3.70	0.077	0.9426	0.0015
	Warp 6	4.1008	4.6400	−0.0859	4.13	3.70	−0.12	0.9411	0.0015
			h_w =	0.178		$\bar{h_{w}}$ =	0.205	δ = Max =	0.0519

Since h_j = 0.228 mm, the thickness of the fabric would be 0.428 mm theoretically. The maximum value of the discrete Fréchet distance |d_pi − d_ti| is 0.1392. Thus, if δ_j is set at 0.1392, the two curves of the calculated warp yarn and the measured warp yarn are considered to be similar. However, the woven plain fabric is usually uniform. Most values of |d_pi − d_ti| (I = 1, 2, 3, ...) are below 0.03 according to Table 4; therefore, the weave point where |d_pi − d_ti| is more than 0.03 can be regarded as an abnormal point due to the measuring accuracy or to random distribution. The accuracy of the calculated warp yarn curve is

η_{j} = \frac{τ - δ_{j}}{τ} \times 100 % = \frac{0.428 - 0.03}{0.428} \times 100 % = 93 %

For the weft yarn, if δ_w is set at 0.0519, the two types of weft yarn curves are similar according to Table 5. For the same reason as δ_j, δ_w is set at 0.03, then the η_w is 93%, as well. That is to say, the algorithm based on the spring–slide mechanics model is accurate enough. Figure 14 shows the photo and the 3D images of the fabric based on the measured value and the calculated value, respectively.

Figure 14.

The comparison of the structure of the real fabric and the images based on the measured and calculated values: (a) photo of the sample fabric, (b) 3D image based on the measured value, and (c) 3D image based on the calculated value.

The calculation should be compared with Peirce's¹ model. According to Peirce's¹ theory, the crimp height of the yarn can be roughly calculated by equation (16)

h_{j} = k • g_{w} \sqrt{a_{j}} h_{w} = k • g_{j} \sqrt{a_{w}}

(16)

The coefficient k is set between 1.29 ∼ 1.36. If k is given, the crimp height of both the warp yarn and the weft yarn or the Peirce¹ geometric phase of the fabric will be determined. If k is set at 1.34, then

h_{j} = 1.34 \times \frac{100}{210} \times \sqrt{0.082} = 0.183 (mm) h_{w} = 1.34 \times \frac{100}{238} \times \sqrt{0.067} = 0.146 (mm)

If the flatness of the yarn is set at 0.8, then the Peirce¹ geometric phase is 5.35 by using equation (9). Obviously, there is a larger difference between the result based on Peirce's¹ model and the measured value or the theoretical value based on this study. Even if the flatness (which may vary from 0.6–0.85) of the yarns is considered to enhance the accuracy, there is still the problem of estimating the flatness of the yarns, which relates to the shape of the cross-section of the yarn, and in turn, leads to a more complicated hypothesis. This situation is why the paper confirms that the Peirce¹ model is not reliable to calculate the geometric phase of the woven structure.

In the weaving process, the warp crimp contraction is controlled by the amount of letting-off and taking-up, which is listed in Table 2. However, the weft crimp contraction has not been covered in Table 2. The parameters “Dent size” and “Ends per dent” in Table 2 can only reflect the warp yarn density of the fabric, not the weft crimp contraction. Due to the lack of the real length of the weft yarn inserted, a larger elongation and in turn a larger weft yarn tension is estimated in the calculation, which leads to a higher warp crimp contraction or a higher geometric phase value. For example, the calculated geometric phase value of fabric samples 1 and 2 in Table 2 should be below 5. If the real length of the inserted pick is determined, a more accurate geometric structure will be obtained. Therefore, in a later study, the weft crimp contraction should be controlled or estimated in some manner; then, the difference between the amount of letting-off and the taking-up will be a help in calculating the geometric structure of the woven fabric.

Another interesting phenomenon is that the flatness of the yarn does not affect the sum of h_j and h_w according to the measurement. In Tables 4 and 5, the sum of h_j and h_w of the true fabric is 0.41 mm, which is approximated to 0.4 mm, the sum of the theoretic warp diameter and theoretic weft diameter.

It is worth noting that the calculated values of h_j and h_w of the real fabric are 0.228 mm and 0.178 mm respectively while the average measured values of h_j and h_w are 0.206 mm and 0.205 mm respectively. There are three possible reasons that cause this difference. Firstly, the weft crimp retraction is not considered, which leads a higher h_j as explained above. Secondly, the real parameters for the weaving procedure are unknown. For example, if the amount of letting-off is set at 0.48 instead of 0.50 with the other parameters unchanged, the final geometric phase is 5.25, and the calculated h_j and h_w will be 0.2125 and 0.1875 respectively, which are closer to the measured values. Finally, the accuracy of the measurement could be a reason for the difference. The fabric to measure has to be turned over from face to back manually, which may result in a measuring error.

Conclusions

In this paper, an algorithm based on a progressive spring–slide mechanics model for calculating the geometric structure of plain woven fabric is attempted. Some conclusions can be made as follows:

Progressive spring–slide mechanics is demonstrated to be effective in calculating the geometric structure of the woven fabric. The yarns in the woven structure can be regarded as a series of spring–sliders to simulate their behaviors after tension. The parameters of the woven fabrics such as yarn thickness, fabric setting, yarn material, crimp contraction, and pre-loaded tension are considered in the progressive spring–slide mechanics model. The structure of the woven fabric is changed from the initial geometric phase, phase 9. Each weave point is analyzed according to the resultant force received and moved in a given tiny step. The processing continues until all the weaving points reach equilibrium and the final stable structure is formed. The algorithm is simple, stable, and easy to understand and realize. By changing the parameters of the intended fabric, the geometric structure is calculated and the ultimate geometric phase conforms to the weaving principle. The features of the principle make it possible to be used with other basic weaves such as twill, satin, and sateen, or even to be used with more complicated woven structure such as piqué, honeycomb, and mock leno.

The calculated geometric structure of the plain woven fabric is validated by judging the similarity with the measured geometric structure based on the discrete Fréchet distance principle. The accuracy of the algorithm is 93% in both the warp direction and the weft direction. The calculation indicates that the progressive spring–slide model is superior to the Peirce¹ model. Likewise, the principle can be used in judging the accuracy of more complex woven structures.

Footnotes

Funding

This work was supported by the Zhengzhou Municipal Bureau of Science and Technology (grant number 083SGYG25124-2).

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