Effect of the frictional coefficient and pre-tension on stress distribution of fabric during the weaving process

Abstract

The non-uniform stress distribution of woven fabric has a significant influence not only on its mechanical performance in service, but also on its weaving efficiency in the fabrication process. For investigating the stress distribution in woven fabric, a numerical model at the yarn scale was established to simulate the interlacing process between the weft and warp yarns using an explicit finite element solver. The yarns were assumed to be a homogeneous continuum and the transversal isotropic constitutive equation was used. A modified lenticular initial shape was used as the cross-section of the yarn and trajectories of warp and weft yarns were set to be straight. The classical Amonton–Coulomb law was used for the tangential behavior between the weft and warp yarns. The simulation results reveal that the interaction between weft and warp yarns consists of three phases in terms of contact, adhesion and sliding. The sectional stress distribution in the weft yarn determined by multi-points contact between a single weft yarn and a group of warp yarns was also analyzed. The tension stress of the weft yarn was larger in the middle part than that in both sides. Based on the numerical model, the effects of two key parameters, namely the frictional coefficient and weft pre-tension, on the stress distribution were discussed in detail. The weft crimp angle and warp tension distribution uniformity decreased as the frictional coefficient decreased, whereas the warp tension fluctuation range did not obviously decrease. However, an improved method by exerting pre-tension in two ends of weft yarn was proposed and the warp tension fluctuation range was significantly decreased. The distribution trend of warp tension obtained from the numerical simulation showed an acceptable tendency with experiment measurements.

Keywords

stress distribution finite element analysis frictional coefficient pre-tension

Woven fabric interlaced by high-performance fiber is widely used as a technical textile that has found increasing applications in the aerospace, automobile and ocean engineering industries. The yarn crimp, which is an important structure characteristic of fabric, has a significant influence on its mechanical performance^1–3 and permeability.⁴ In addition, the variation of yarn crimp in fabric will cause another severe result in terms of weaving termination. The warp tensions on the middle part were higher than those on both sides of the loom due to different yarn crimps.^5,6 However, there is a large variation of warp tension along the loom width, which may cause an unclear passage for filling.

Because the crimp of warp and weft yarn is formed in the weaving process, it is necessary to investigate the interaction process between warp and weft yarns. Many studies have already investigated the yarn crimp in woven fabric. Chepelyuk⁷ developed a method of determining the parameters of yarns and that could approximate the crimp height ratio of the warp and weft within the woven fabric structure accurately. Shahabi et al.⁸ used an image processing procedure to observe the crimp interchange behavior, which is dependent on the competitiveness of yarn tension between warp and weft yarns. Realff⁹ used a series of experimental methods including the traditional ravel strip test, videotaping, encapsulation and photography to research yarn crimp and cross-sectional change. However, with the measuring method it has been difficult to discover the entire process of interaction between warp and weft yarns up to now. Therefore, the knowledge of local deformation of yarns is necessary to understand how the stress distribution of yarn influences the performance of woven fabric.

With the development of the finite element method, the contact and deformation of the yarn cross-section has been addressed. Page and Wang¹⁰ used solid brick elements to model yarn, for which the side curvature was approximated using a 45°bevel. Adanur and Vakalapudi^11,12 used an elliptic cross-section and a sinusoidal curve to generate one yarn that had been woven inside the fabric. However, the initial cross-section of the yarn, which is an isolate, is of irregular shape. Since the reorganization of fibers appears to be a significant factor influencing the yarn cross-section, an ellipse and convex that are similar to the cross-section of realistic yarn were considered by Hivet et al.¹³ However, Kawabata and Kawai¹⁴ depicted in detail the cross-section of yarn that is affected by normal loads and other applied force. In the present model, the cross-section and trajectory of yarn changes along the yarn path depending on the position of the contact.

Yarn is a non-isotropic material and its tensile stiffness differs greatly in the longitudinal and transverse directions. In addition, the material constitution model of fabric, which is multi-scale at different levels, is quite different. Hyperelastic and hypoelastic material constitutive laws were used to model the fabric in the macroscopic scale in the literature.¹⁵ At the mesoscopic scale in the literature,¹⁶ the material properties of yarn were assumed to be linear orthotropic. Yarn consists of thousands of individual filaments at the microscopic scale. A digital chain technique was developed by Mahadik and Hallett¹⁷ using a commercial off-the-shelf code that analogs to a bundle of fibers within a yarn to predict the internal architecture of fabric. However, the above finite element methods always focus on the woven fabric geometric modeling, not the weaving process. Vilfayeau et al.¹⁸ proposed a kinematic model to simulate the weaving process of an industrial dobby loom. Although an accurate geometric representation of the woven elementary cell was achieved, the non-uniform stress is not completely covered in the previous work.

In this paper, a model concerning contact–friction behavior at the shedding step of weaving is developed to analyze non-uniform stress distribution. The effect of the frictional coefficient on the weft crimp angle and warp tension distribution are discussed, which is followed by a proposed improved method of increasing pre-tension on the weft ends to decrease the warp tension fluctuation range along the loom width. For comparison, an experiment was also performed to validate the numerical result.

Weaving parameters and experimental methods

Weaving parameters

The woven fabrics were fabricated on a rapier loom (WL-450-1500). The rapier loom and fabric parameters are given in Table 1. The E-glass yarns were twisted at 25 turns/meter, and the frictional coefficient was measured by a DP-Y151 frictional-meter.

Table 1.

Loom, yarn and fabric parameters

Loom parameters	Fabric parameters	Yarn parameters
Weft yarn insertion type: rapier	Weave: plain weave	Type: E-glass
Shed formation: electronic dobby	Warp density: 6 ends/cm	Density: 2.6g/cm³
Machine speed: 350 r/min	Weft density: 3 ends/cm	Tex: 600
Machine width: 1500 mm	Warp width: 1350 mm	Tensile modulus: 52.5 GPa
Let off: servo type electronic	Warp tension: 1 kN	Twist: 25 turns/meter
Take up: servo type electronic		Frictional coefficient: 0.132
Reed density: 6 dents/cm

Warp tension measurement

The objective of the warp tension tests is to obtain the trend of warp tension distribution along the width direction of the weaving machine. A weaving trial was done on a rapier loom, producing a plain weave fabric. Warp yarn tension was measured using a single yarn tension meter. The equidistance of the measure points along the loom width were located on the warp yarns, as shown in Figure 1(a). The warp tension was captured on each position for single yarn when harnessed at the horizontal position. As shown in Figure 1(b), after polynomial curve fitting the warp tensions were larger in the middle part than that in the two sides of the loom.

Figure 1.

Experiment of warp tension measurement: (a) the positions of the measuring points on the rapier loom and warp tension meter (SFY-13); (b) warp tension distribution trends after polynomial curve fitting.

Finite element modeling

In this section, a numerical model was established to simulate the weaving process by commercial finite element analysis (FEA) software (ABAQUS). In the model, the interaction between weft and warp yarns at the step of shedding is presented in detail.

Weaving process

The interaction and deformation between yarns with the mechanism motion, such as shedding and filling, are the most important behaviors in the forming process of woven fabric. As shown in Figure 2, the warp yarns are divided into two groups and move in opposite directions by the traction of the shedding device. When one group of yarns is moved up and the other group is moved down from the horizontal position, namely backward motion, a shed space was formed to fill the straight weft yarn. Before the two groups of yarns move back to the horizontal position, the weft yarn should be drawn into the shed space by the insertion device. With the movement of warp yarns, the interlacing points between the weft yarn and warp yarn occur while the warp yarn moves to the horizontal position and touches with the weft yarn, namely forward motion. Due to the tension in warp yarns, strong squeezing and consequent undulation occur between weft and warp yarns and then one shedding cycle is finished. In the present paper, the model area as shown in Figure 2(c) is used to simulate the interaction of weft and warp yarns.

Figure 2.

Schematic of weaving processes: (a) simplified view of fabric forming; (b) opening of the shed and then filling the weft yarn; (c) closing of the shed.

Geometrical modeling

The aim of this geometrical model is to take into account the deformation of the yarn cross-section and the waviness of the yarn trajectory. The cross-section of yarn is a broadly circle of 1.00 mm mean diameter, which is obtained by the resin-coated method. In order to reflect the yarn cross-section deformation transfer from the initial circle section to a convex shape, a modified lenticular initial shape is used in this model and the initial trajectories of the warp and weft yarns are set to be straight, as shown in Figure 3. The dimensions of the yarn geometrical and weaving pattern are listed in Table 2.

Figure 3.

The geometry of the simulation model: (a) yarn cross-section; (b) simplified initial assembly of the warp and weft yarns.

Table 2.

Dimensions of the geometrical model

Description	Symbol	Value (mm)
Yarn width	w	1.00
Height of curved surface	t	0.25
Height of plane surface	f	0.30
Weft yarn length	L	Undetermined
Warp yarn length	D	4.00
Warp yarn horizontal interval	e	1.50
Warp yarn altitude interval	h	1.80

It is imperative that the model replicates the actual geometry of the structure as realistically as possible to attain reliable results. The warp yarns were numbered from the left- to right-hand side along the weft yarn, divided into up–down groups and constrained with the displacement boundary, which is set to move to the opposite side with specified distance (H/2). However, no boundary condition at the two ends of the weft yarn is made. The weft yarn length is undetermined.

Interaction between yarns

As previously mentioned, contact–friction between the surface of the warp and weft is considered in this model. In the modeling of the weaving process, the sliding distance in each crossing point is unpredictable. The finite-sliding contact method has been considered in this model and the surface-to-surface interaction property with the classical Amonton–Coulomb law is used for the tangential behavior with a constant isotropic frictional coefficient μ between the contact surfaces. The penalty function method assumes that no sliding occurs if the equivalent frictional stress ( $τ^{eq}$ ) is less than the critical stress ( $τ^{crit}$ ), which is proportional to the contact pressure (p). The equivalent frictional stress and critical stress can be calculated as

τ^{eq} = \sqrt{τ_{1}^{2} + τ_{2}^{2}}

(1)

τ^{crit} = μ p

(2)

where

τ_{1}

and

τ_{2}

are two stress components in the contacted tangential plane.

Yarn constitutive model

The mechanical behavior of the yarn is complex because of the possible relative displacements of fibers, which turns the bending stiffness soft, related to the tensile stiffness in the longitudinal direction. In this paper, the weft and warp yarns were assumed to have an homogeneous continuum and the transversal isotropic constitutive equation was used. The axial direction of the yarn was considered as direction 3 and the 1–2 plane to be the plane of isotropy at every point; transverse isotropy requires that $E_{1} = E_{2} = E_{P}$ , $υ_{31} = υ_{32} = υ_{tp}$ , $υ_{13} = υ_{23} = υ_{pt}$ and $G_{13} = G_{23} = G_{t}$ , where E and G represent the elastic modulus and the shear modulus, υ is the Poisson’s ratio and p and t stand for “in-plane” and “transverse”, respectively. Thus, $υ_{tp}$ is the physical interpretation of the Poisson’s ratio that characterizes the strain in the plane of isotropy resulting from stress in the plane of isotropy. In general, the quantities $υ_{tp}$ and $υ_{pt}$ are not equal and are related by $υ_{tp} / E_{t} = υ_{pt} / E_{p}$ . The stress–strain laws reduce to

{ɛ_{11} ɛ_{22} ɛ_{33} γ_{12} γ_{13} γ_{23}} = [A 00 B] {σ_{11} σ_{22} σ_{33} σ_{12} σ_{13} σ_{23}}

(3)

A = [1 / E_{p} - υ_{p} / E_{p} - υ_{tp} / E_{t} - υ_{p} / E_{p} 1 / E_{p} - υ_{tp} / E_{t} - υ_{pt} / E_{p} - υ_{pt} / E_{p} 1 / E_{t}]

(4)

B = [1 / G_{p} 000 1 / G_{t} 000 1 / G_{t}]

(5)

where ɛ and γ represent strain, and σ represents stress. Here,

G_{P} = E_{p} / 2 (1 + υ_{p})

and the total number of independent constants is only five. In the transversely isotropic case the stability relations for orthotropic elasticity simplify to

E_{p}, E_{t}, G_{p}, G_{t} > 0, | υ_{p} | < 1, | υ_{pt} | < (E_{p} / E_{t})^{\frac{1}{2}}, | υ_{tp} | < (E_{t} / E_{p})^{\frac{1}{2}}, 1 - υ_{p}^{2} - 2 υ_{tp} υ_{pt} - 2 υ_{p} υ_{tp} υ_{pt} > 0

The elastic properties of yarn that are chosen according to Vilfayeau et al.¹⁸ are summarized in Figure 4.

Figure 4.

(a) Cartesian coordinate system of yarn. (b) Elastic parameters of the transversal isotropic constitutive.

Results and discussion

For a repetitive structure, simplifying the model, such as by shrinking the number of interlacements, is feasible. This model is aimed at revealing the mechanism during interaction between the weft and warp yarns. Several dimensions of the weft yarn length have been considered and 50 mm is a suitable value that could effectively reflect the interactions. The following results were based on the model at the scale of 50 mm weft yarn length.

Three phases of interaction

One interlacement that is near the middle line of the weft is presented in Figure 5(a). A₁, B₁, B₂ are the initial positions of the weft yarn. After weft yarn contract, A₁, B₁, B₂ were moved to $A_{1}'$ , $B_{1}'$ , $B_{2}'$ , respectively. The crimp angle α of the weft yarn increases until the warp yarns reach the fully horizontal position. As is shown in Figure 5(b), c₁ and c₂ represent the initial and final contact points, respectively. The interaction procedure between the weft and warp yarns can be divided into three phases: contact, adhesion and sliding.

Figure 5.

Crimp formed of weft yarn: (a) three phases of the interactive process: contact, adhesion, sliding; (b) the interlacement near the middle line.

The curve of frictional force on the fifth crossing point with respect to vertical displacement of the warp yarn is shown in Figure 6. The fifth crossing point that had distributed in Figure 3(b) is generally representative for most interactions along the weft wide. While the warp yarns driven by the shedding device move to champ the weft yarn, the unstressed weft yarns are stretched in the axial direction by the frictional force exerted by the warp yarns, which immediately increases the tensile stress in the weft yarns. Meanwhile, the frictional force determined by the normal pressure is enhanced due to strong squeezing between the weft and warp yarn, which is induced by the increasing tensile stress of the weft yarn. However, the equivalent frictional force remained at a small range because cross-section deformations happened first and the stretch force was too small to overcome the frictional force. The warp yarns kept clamping to the weft yarn thereafter, and the critical frictional force that is calculated by equation (2) was enhanced more than the equivalent frictional force that is calculated by equation (1). Therefore, the sliding behavior is captured when the frictional force is weaker than the tensile stress of the weft yarn. The state of interaction retains adhesion until the equivalent frictional force is equal to the critical frictional force and converts to a sliding base in equations (4) and (5).

Figure 6.

The frictional force on the fifth interlacement from simulation results: (a) μ = 0.1; (b) μ = 0.2; (c) μ = 0.3.

The frictional force that is determined by frictional coefficient and induced by the stretch force in μ = 0.1 is less than that in μ = 0.2. However, the frictional force in μ = 0.3 is larger than the previous two conditions in Figure 6(c), but no sliding occurs since the stretch force is not large enough to overcome the critical frictional force process overall. It is deducible that the stress of the weft yarn, which depends on stretch force, would be stronger than that of a small frictional coefficient.

Weft crimp motion

The two ends of weft yarn are set to be free in the present model. The weft yarn will achieve crimp when it encounters pressures from mutual dislocation warp yarns. The friction distribution on the weft yarn is symmetric, like push-and-pull, which results in the middle line of the weft being fixed. The weft yarn is contracted from two opposite sides to the middle line due to the stretch force, as presented in Figure 7(a).

Figure 7.

The right-hand side of the weft yarn: (a) the weft contract direction is from two sides to the middle; (b) horizontal direction frictional forces on the weft yarn; (c) mechanical analysis of interlacement on the weft yarn.

The horizontal frictional forces on the weft are symmetric with respect to the middle line of the weft, and which on the crossing point in the right-hand side of the weft are shown in Figure 7(b). The frictional force on the middle point of the weft is calculated from the following equation

F_{r} = \sum_{i = r}^{R} f_{i}

(6)

where R represents the total number of warp yarns in the right-hand side of the weft, f represents the horizontal frictional force and r represents the sequence number of warp yarns in the right-hand side. The frictional force on the most lateral crossing point of the weft yarn can be calculated as

F_{R} = f_{R}

(7)

The mechanical analysis of interlacement, which is on the right-hand side of middle line, is noted in Figure 7(c). The following relation could be used as

T_{i} cos α_{i} = f_{i} + T_{i + 1} cos α_{i + 1}

(8)

where T represents the stretch force and α represents the weft crimp angle (

α_{i} \neq α_{i + 1}

). The following equation is derived combined with equation (6)

T_{r} cos α_{r} = \sum_{i = r}^{R - 1} f_{i} + T_{R} cos α_{R}

(9)

It is clear that the stretch force in the middle part is larger than that in both sides of the weft yarn, based on equation (9). In other words, the stress would be greater closer to the middle of the weft yarn.

Discussion of influencing factors

Based on the above discussion, the reason for uneven stress distribution along the loom width is the sliding between weft and warp yarns. Therefore, the effect of the frictional coefficient is discussed in this section.

Frictional coefficient

The weft crimp angle distribution along the loom width is presented in Figure 8(a), which clearly illustrates that the weft crimp angle is smaller in the middle than in the two sides of the weft yarn. A modified standard deviation is used to represent the uniformity of the crimp, and warp tension distribution is considered

δ = \sqrt{\frac{1}{N} \sum_{n = 1}^{N} (x_{n} - x_{c}) 2}

(10)

where δ represents the uniformity coefficient and x_c represents the value of the crimp angle or warp tension at the middle of width. The middle part of the curve where it is flat becomes narrowed as the frictional coefficient is reduced, namely the uniformity is poor as the frictional coefficient is small. In addition, the uniformity of the warp tension is poor with a smaller frictional coefficient, as shown in Figure 8(b). This warp tension distribution is in contrast to the weft crimp distribution along the loom width. It is obvious that widthwise warp tension has its maximum value in the middle part, while the minimum value is in two sides of the loom. The distribution tendency can be explained as follows: it is easy to slide between weft and warp yarns with a small frictional coefficient and that induces the stretch force, which is derived from frictional force, to be too small to maintain the weft yarn straight. Furthermore, the closer to the middle of the weft yarn, the greater the stretch force in the weft yarn, which results in smaller weft crimp. For a large frictional coefficient, the stretch force is large enough to oppose the clamp force from warp yarns and keep the weft yarn crimp small.

Figure 8.

Various frictional coefficients in the simulation model: (a) weft crimp angle distribution; (b) warp tension distribution.

Figure 8 also shows that the uniformity of warp tension distribution is obviously improved by the variation of the frictional coefficient from 0.1 to 0.2, but the magnitude of improvement is slight, from 0.2 to 0.3, compared to the former. These investigations revealed that a significant effect on the stress distribution will happen in the condition of a small frictional coefficient.

The crimp angles displayed large variations in two sides of the weft yarn. The reason for the higher weft crimp is that weft ends that have no constraints induce only a small amount of stretch force in both sides, according to equation (9). At the stage of manufacture, a large warp tension fluctuation range is harmful to the weaving process. Therefore, it is necessary to decrease the warp tension fluctuation range for improving the weaving efficient and fabric performances. The reason for lower warp tension in both sides is that stretch force in the weft yarn is low. Hence, setting pre-tension at two ends of the weft yarn may be a feasible method to increase the stretch force in both sides.

Weft pre-tension

Based on the previous analysis, it can be concluded that the main reason for stretch force variation along the loom width is the unconfined sliding of the weft yarn in both sides. Therefore, the pre-tension in the reverse direction with the same value on the two weft ends along the horizontal direction is made based on the same model. When warp yarns are clamped on the weft yarn and the weft crimp happens over the weft width, but resistance forms not only for frictional force between the weft and warp yarns but also for the force from pre-tension. The distributions of the weft crimp angle and warp tension in the condition of pre-tension are shown in Figure 9. The uniformity of warp tension is obviously improved with pre-tension.

Figure 9.

Various pre-tensions in the simulation model with $μ$ = 0.1: (a) weft crimp angle distribution; (b) warp tension distribution.

Warp tension tests were carried out on the rapier loom and set with different pre-tensions. The experimental warp tension distribution trend is shown in Figure 10. For low pre-tension the warp tension fluctuation range is 38cN, which is much larger than 29cN in the case of high pre-tension. The warp tension fluctuation range deceased 23.7% after increasing pre-tension. The results of the experiment validated that increasing the pre-tension on weft yarn ends is effective to improve the uniformity of warp tension. The fitted curves were not strictly symmetrical for the reason of random error. However, the symmetrical warp tension distribution can be speculated from Figure 10. The numerical results have an acceptable trend compared to the experimental results.

Figure 10.

Experiment of warp tension measure with different pre-tensions.

Conclusions

In this study, a numerical simulation aiming at modeling the interlacing action of yarns during the weaving process was performed. The linear transversal isotropic material model was used to present the strong anisotropic characteristic of yarn. For modeling the contact between yarns, the penalty function method was employed to calculate the contact force in the normal direction between the contact pair, whereas frictional force in the tangential direction was modeled by assigning a frictional coefficient. The simulation results reveal that there are three phases in terms of contact, adhesion and sliding during the interaction between yarns. The stretch force was analyzed, which was larger in the middle than in both sides of the weft yarn.

Two parameters of the frictional coefficient and weft pre-tension were discussed. The uniformity of the weft crimp angle and warp tension along the loom width were poor with a small friction coefficient. Then the warp tension fluctuation range was improved as the pre-tension worked. The trend of warp tension distribution is similar for the simulation and experiment, which shows that high tension occurs in the middle and the low tension in both sides of loom.

Based on the above analysis, in the actual operation, increasing the frictional coefficient and pre-tension appropriately can improve the uniformity of warp tension distribution and decrease the warp tension fluctuation range, respectively. For future work, modeling of the interaction between the weft and warp yarns under the microscope is possible. Then, the influence of yarn twist or other parameters can be studied.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the National Natural Science Foundation of China (grant number 51275482) and the Public Projects of Zhejiang Province (grant number 2016C34G2060012).

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