Calculation method of contact index between loop yarn and abrasive

Abstract

Loop yarn is a typical fancy yarn composed of three strands and differs significantly from conventional yarn. The loop yarn also suffers from friction and wear during use and processing. To create a unique fuzzing effect, loop yarn is usually sanded by rubbing against an abrasive. The frictional force and coefficient can only roughly reflect the frictional effect between yarn and abrasive, and it is challenging to represent the amount of contact between the two at the microscopic level. A Hertz contact model is first established to evaluate the contact between the loop yarn and the abrasive. This paper derives the calculation formula for the contact number and area between the loop yarn and the abrasive. The calculation results indicate that as the mesh of the abrasive increases, the frictional force and coefficient decrease, the contact number between the loop yarn and the abrasive gradually increases, the contact area first increases and then tends to stabilize, and the contact force and stress decrease and stabilize. The ratio of the fiber distribution force to its contact area can be used as an evaluation index for fiber wear resistance. The product of the square root of contact number, abrasive particle radius, and friction index can provide feedback on friction performance.

Keywords

Loop yarn friction contact mode contact number contact area

Fancy yarn is a particular strand composed of three or more single yarns. The differences in raw materials, color coordination, linear density, and twists can give fancy yarn some style. Loop yarn is a typical fancy yarn widely used in clothing and home textile products. The loop yarn is also accompanied by friction or abrasion during wearing and processing. Due to its structure being different from ordinary yarn, the abrasion appearance of the loop yarn is also distinct. After wearing, the fibers of the loop yarn fracture, and the hairiness significantly increases.¹ To produce the fuzzing or sanding yarn,² some technical people drew the loop yarn sliding over the corundum abrasive surface³ in the twisting machine.

Many fibers are wrapped and twisted into yarn, and their apparent morphology is related to the fiber shape and wrapping structure. The fiber arrangement, structure gaps, and distribution characteristics determine that the yarn appearance is not uniform and smooth but instead a deformable rough body. In production and daily use, friction between yarns, between yarn and device, and between yarn and fluid can all cause a change in appearance through several fiber rearrangements, loss, or thinning.

El Mogahzy and Gupta⁴ proposed that fibers contact only at the irregular tips of their surface, and the more irregular tips per unit area, the smoother or lower roughness of the fiber. According to the size of the fiber inside the yarn, the geometry between fibers or yarns belongs to the spatial shape line, while the contact between fibers inside the yarn is point or line segment contact. Ning⁵ discovered three friction forms on the surface and inside of a rope: point, line, and surface. The fibers would undergo cyclic friction and fracture, inducing many short fibers and nonperiodic surface hairiness. The contact between yarn and part varies depending on physical conditions and can be multipoint or point-line mixed contact. The part’s surface is more or less uneven, and the contact between the fibers and the part is mostly point contact.

The contact theory can determine the contact number and area between the fiber and the part to explain the contact mechanism. Hosseinali and Thomasson⁶ designed a Matlab function to calculate the number of fibers in a cotton fiber assembly and its contact area with a smooth surface in the maximum elastic deformation. Cornelissen et al.⁷ used a geometric contact model of cylinder and plane for simulating fiber tow–metal contact, and proposed that the surface topography of the counterface was the main parameter affecting the friction behavior. The contact of fiber tows and metal almost remained fully elastic deformation behaviour. Wu et al.⁸ calculated the fiber contact number under normal load by the cross-sectional parameters of carbon fiber and the fiber contact area by the Hertz contact model of the cylinder and plane, then established a mathematical relationship between the normal load and the contact area. Cornelissen et al.⁹ used the Maugis–Dugdale adhesive contact and general Hertzian elliptic contact model to calculate the fiber–fiber contact area, and proposed that the model is investigated both with and without adhesive. Che et al.¹⁰ used the Hertz theory and Johnson–Kendall–Roberts (JKR) theory to calculate the contact area of the fiber tows and the spools, also proposed a method for combining the friction and adhesion effect. Mulvihill et al.¹¹ invented a smooth glass plate to record fiber indentation for verifying the Hertzian contact model of a classic cylinder-on-flat. The indentation was not only an accurate contact but also an approximate result.

The contact number and area are related to the two’s geometric shape, contact load, and material deformation ability. Choi and Kim¹² proposed that there was an excellent correlation between the fiber length distribution on the surface of yarn and its abrasion resistance. Subramaniam et al.¹³ found an excellent correlation between the dynamic modulus of cotton fibers and their friction, abrasion, and fatigue. Wang et al.¹⁴ found that the abrasion performance of cotton yarn was significantly correlated with linear density, hairiness index, and twist. Palaniswamy and Mohamed¹⁵ studied the relationship between the twist of ply yarn and their uniformity, hairiness, and abrasion resistance. They found that as the twist increased, the hairiness of ply yarn decreased, and its abrasion resistance increased.

Chen et al.¹⁶ observed the abrasion morphology of carbon fiber composite core wires and found that the abrasion of the ply yarn varied with the friction coefficient when the ply yarn was slipped. The larger the friction coefficient, the more severe the abrasion. Chen¹⁷ used a white light interferometer to observe the abrasion morphology of the rope surface and calculated the abrasion area, volume, and abrasion coefficient (the abrasion coefficient refers to the abrasion amount under a unit distance and a unit load). The abrasion coefficient was found to be related to the contact force and to increase with the increase of the contact force.

Friction between loop yarn and part or abrasive is a complex physical phenomenon. The contact form and amount between loop yarn and abrasive are often difficult to observe and calculate effectively, and the estimation of the frictional mechanical indexes is even more challenging. To compare the friction performance differences of different types of loop yarns, a contact mechanic model between loop yarn and abrasive is established from the structural characteristics of loop yarn, and the calculation method for the contact number and contact area between loop yarn and abrasive is derived. The contact index between the loop yarn and the abrasive is calculated using geometric dimensions and mechanical parameters. The relationship between the contact number and contact area and the friction index of the loop yarn is analyzed to provide the calculation and prediction method for the friction index of the loop yarn.

Structural parameters of loop yarn

Structural model and dimensional parameters of loop yarn

Loop yarn consists of core, binding, and effect yarn. The “loop” of the loop yarn is formed using the untwisting method on a fancy twisting machine with a hollow spindle wrapping apparatus. The loop size and arrangement on the loop yarn depend on the length of the core yarn, the overfeed ratio of the effect yarn, and the twist ratio at each stage of yarn formation. The fancy twisting machine applied a false twist to the core yarn and effect yarn and an authentic twist to the binding yarn. The turn number of the effect yarn between adjacent loops is determined by the spinning process of the loop yarn, and it can follow a specific pattern. We assume that the cross-sections of the core yarn, effect yarn, and binding yarn are all circular, and also ignore the stretching or compression deformation of each component yarn in the spinning process. The core and binding yarn are evenly distributed along the yarn axis in a spiral form, winding the effect yarn with occasional protruding loops.

Its basic structural model is shown in Figure 1. The turn number of the effect yarn between adjacent loops can be designed in the spinning process according to a specific distributed or random pattern. The binding yarn is uniformly wrapped around the core-effect ply yarn; its helix number per unit length is the twist of the loop yarn.¹⁸

Figure 1.

Structural diagram of loop yarn.

A schematic diagram of the loop yarn structure is shown in Figure 1. In Figure 1, the thick red solid line is the binding yarn, the thick black dashed line is the core yarn, and the thin blue solid line is the effect yarn. We use P_b, P_x, and P_e to denote the pitch of the binding yarn, core yarn, and effect yarn, r_b, r_x, and r_e to denote the radius of the binding yarn, core yarn, and effect yarn, and D_q to denote the loop diameter of the effect yarn.

Calculation of component yarn length

According to Pythagoras’ theorem,¹⁹ the opened-out axes of the component yarns are shown in Figure 2. In Figure 2, AB is the effect yarn axis of one twist l_e (excluding the loop length), AB′ is the effect yarn pitch of one twist P_e, AD is the binding yarn length of one twist l_b, AD′ is the binding yarn pitch of one twist P_b, AC is the core yarn length of one twist l_x, AC′ is the effect yarn pitch of one twist P_x, β is the helix angle of the effect yarn, γ is the helix angle of the binding yarn, and α is the helix angle of the core yarn.

Figure 2.

Opened-out diagram of component yarn axes.

The component yarn length of one twist l_i is calculated as follows:

l_{i} = \frac{P_{i}}{\cos ψ_{i}}

(1)

where i = e, b, and x, respectively representing effect yarn, binding yarn, and core yarn; Ψ_i is the helix angle of the component yarn, where Ψ_e is β, Ψ_b is γ, and Ψ_x is α.

The effect yarn consists of several loops and spiral segments.²⁰ The loop of the effect yarn can be approximated as a circle, and a loop circumference of the effect yarn is πD_q. The component yarn length per unit loop yarn is given as follows:

L_{e} = n_{e} l_{e} + π k n_{q} D_{q}

(2)

L_{b} = n_{b} l_{b}

(3)

L_{x} = n_{x} l_{x}

(4)

where L_b, L_x, and L_e are the binding yarn, core yarn, and effect yarn length per unit loop yarn, respectively, n_q is the loop number of effect yarn per unit length of loop yarn, and n_b, n_x, and n_e are the twist number of binding yarn, core yarn, and effect yarn per unit length of loop yarn, respectively.

The binding yarn wraps around the core yarn and effect yarn, and the twist number of the loop yarn depends on the binding yarn twist. Based on the ideal structure of a loop yarn, the twist number of the binding yarn, core yarn, and effect yarn per unit loop yarn should be equal. That is, n_e = n_b = n_x.

Calculation of the contact area between the component yarn and abrasive

Calculation of contact force between the component yarn and abrasive

According to the loop yarn model in Figure 1, when wound around an arc-shaped abrasive, the effect yarn, binding yarn, and core yarn of the loop yarn, all come into contact with the abrasive, as shown in Figure 3.

Figure 3.

Contact between loop yarn and abrasive: (a) forces acting on the loop yarn and (b) forces acting on the loop yarn element.

In Figure 3(a), the loop yarn slides over the fixed sleeve and rubs against the abrasive. The contact length between the loop yarn and the abrasive is l_q, the unit number of loop yarn in contact area is n_y = l_q/P_i. The effect yarn and binding yarn intersects on the same side, and the core yarn comes into contact with the abrasive; they intersect on opposite sides, and the effect yarn or binding yarn comes into contact with the abrasive. A unit loop yarn is wrapped around the abrasive, the binding yarn, effect yarn, core yarn, and the abrasive have three contact tips, as shown in Figure 3(b).

Two tensions, T₁ and T₂, act on the two ends of the loop yarn in contact with the abrasive, with T₁ at the relaxed end, T₂ at the tensioned end, and T₂ > T₁. The arc-shaped contact area has n_y units of loop yarn. The tension, T_j and T_j + ΔT, act on the end of each unit. Assuming the tension of the loop yarn gradually increases uniformly from T₁ to T₂, so ΔT = (T₂ – T₁)/n_y. The T_j and T_j + ΔT exerted by the unit component yarn on the abrasive resulted in the three normal forces: N_ej acting on the tip2 of effect yarn, N_rj acting on the tip1 of the binding yarn, and N_xj acting on the tip3 of the core yarn, in turn. The tips 1, 2, and 3 are relatively close in the same unit. Thus, N_ej ≈ N_bj ≈ N_xj. The contact force of the unit component yarn N_ij is

N_{ij} = (2 T_{j} + Δ T) \sin \frac{Δ φ}{2}

(5)

where j denotes the jth unit’s loop yarn in contact with the abrasive.

Deformation of the component yarn

Regarding spatial geometry, the contact between the component yarn and the curved abrasive is similar to the contact between the rope’s strand and the guide pulley or between two strands. The component yarn of the bent loop yarn will form a spatial spiral curve. The contact between component yarn and abrasive occurs between spatial spiral yarn and a large cylinder, similar to the helical interlocking of two-ply yarn.

The loop yarn is placed on an arc-shaped abrasive with a curvature radius ρ. With no tension, the component yarn cross-section can be approximately circular, with a diameter of d_0ij, as shown in Figure 4(a). When there is tension, the component yarn cross-section of the non-contact segment between the component yarn and the abrasive is approximately circular, with a diameter of d_1ij. Assuming the compression deformation of the component yarn in contact with the abrasive is δ_ij, the component yarn cross-section of the contact segment between the component yarn and the abrasive is approximately elliptical, with a semimajor axis of a_1ij and a semiminor axis of d_0ij – 0.5δ_ij, and 2b_0ij = d_0ij – δ_ij, as shown in Figure 4(b) and (c). The contact area between the component yarn and the abrasive is also elliptical, with a semimajor axis of a_1ij and a semiminor axis of b_1ij, as shown in Figure 4(d).

Figure 4.

Cross-sectional morphology of component yarn in contact with abrasive: (a) when the yarn has no tension, the cross-section of the yarn is circular; (b) when the yarn has tension, the cross-section of the yarn is approximately elliptical as shown by the dashed line; (c) the cross-section of component yarn in the deformed state and (d) the contact area between component yarn and abrasive.

According to the calculation method of yarn compression,²¹ the deformation under compression of the unit component yarn in contact with the abrasive is δ_ij, which is related to the yarn pressure. The deformation δ_ij varies versus the contact force exerted by the unit yarn,²² relying on measurements:

δ_{ij} = C (1 - e^{- \frac{K_{y}}{L_{0}} N_{ij}})

(6)

where C is the dimensionless coefficient of component yarn under compression, L₀ is the length of the component yarn with a value of 1000 mm, and K_y is the stiffness of the component yarn, approximated as the tensile stiffness. As the yarn pressure increases, the deformation of the component yarn first increases and then tends to be constant, with a nonlinear curve. Here C and K_y are determined by fitting experimental data.

For simplicity, we assume that the geometric dimension deformation of fiber assembly belongs to elastic deformation; the deformation is caused by the tension acting on the i-component yarn. The i-component yarn strain is ε_i, and Poisson’s ratio of the i-component yarn is v_i, which is approximately the Poisson’s ratio of the component fiber. The following calculation can be obtained considering the longitudinal elongation and radial shrinkage of i-component yarn based on the constant equivalent volume of component yarn:

\frac{π d_{0 ij}^{2}}{4} = π a_{0}_{ij} b_{0}_{ij} (1 - ν_{i} ε_{i}) (1 + ε_{i})

(7)

Here, the i-component yarn strain can be approximately calculated by the elongation of i-component yarn, and the elongation of i-component yarn is linearly related to its tensile force. Neglecting the gaps between fibers inside the i-component yarn, we assume that the deformation of the i-component yarn is synchronized with the loop yarn. According to the static theory, the strain of i-component yarn to extend the yarn can be linked to strain of loop yarn by the relationship

ε_{i} = \frac{(1 + ε_{p}) \cos ψ_{i0}}{\cos ψ_{i}} - 1

(8)

where ε_i is the strain of i-component yarn, ε_p is the strain of loop yarn, Ψ_i0 is the initial helix angle of the i-component yarn when the loop yarn is no tension, Ψ_i is the helix angle of the i-component yarn when the loop yarn has tension. The strain of loop yarn is ε_p = (T₁ + jΔT)/(K_l l₀), where K_l is the tensile stiffness coefficient of loop yarn, l₀ is the loop yarn length, with a value of 250 mm.

The helix angle of the ply yarn is related to its degree of abrasion, and the larger the helix angle, the heavier the abrasion.¹⁵ Stretching can change the structure and size of the loop yarn and reduce the helix angle of the component yarn. After measurement, the variation range of the helix angle caused by stretching is 0–20°, and its cosine value has no significant difference. Therefore, we can ignore the influence of the helix angle. That is, Ψ_i0 ≈ Ψ_i. From Equation (8), we can obtain that ε_i ≈ ε_p.

In Figure 1, the half width of loop is r_q = 0.707d_q. Due to the loop distribution on the yarn’s outer side, it may not have sufficient or occasional contact with the abrasive. Estimating the contact area between the loop and the abrasive is difficult.

Calculation of contact area between the component yarn and abrasive

According to Hertz’s theory, the contact area between the component yarn and the abrasive can be approximately elliptical,²³ and the contact area is related to the force, fineness, twist, etc. From the three-component yarn wrapping around each other, it can be seen that the contact between the binding yarn, effect yarn, core yarn, and the abrasive are similar, and they have contact with the abrasive in sequence. Therefore, the contact between the three-component yarn and abrasive can be used in same calculation method.

Based on the mechanism of contact load between strands, Grishanov et al.²⁴ proposed the geometric shape of the contact area between two strands, and they derived the calculation formulas for the contact area’s long semiaxis a and short semiaxis b. According to Grishanov et al.’s calculation formula, the contact area’s long semiaxis a_ij and short semiaxis b_ij between the i-component yarn and the abrasive are as follows:

a_{1 ij} = \frac{c_{1 ij}}{2} [\frac{3 π {(d_{0 ij} + 2 b_{0 ij} - 2 δ_{ij})}^{2}}{32} (\frac{d_{0 ij} - 2 b_{0 ij} + 2 δ_{ij}}{d_{0 ij}}) (\frac{2}{2 b_{0 ij} - 2 δ_{ij}} + \frac{1}{ρ})]^{1 / 3}

(9)

b_{1 ij} = \frac{c_{2 ij}}{2} {[\frac{3 π {(d_{0 ij} + 2 b_{0 ij} - 2 δ_{ij})}^{2}}{32} (\frac{d_{0 ij} - 2 b_{0 ij} + 2 δ_{ij}}{d_{0 ij}}}^{)}

{(\frac{2}{2 b_{0 ij} - 2 δ_{ij}} + \frac{1}{ρ})]}^{1 / 3}

(10)

\frac{c_{1 ij}}{c_{2 ij}} = \frac{\sqrt{\frac{1}{{(b_{0 ij} - δ_{ij})}^{2}} + \frac{1}{ρ^{2}} + \frac{2 \cos (2 π - ψ_{i})}{(b_{0 ij} - δ_{ij}) ρ}}}{\frac{1}{b_{0 ij}} + \frac{1}{2 ρ}}

(11)

where c_1ij and c_2ij are both dimensionless proportional coefficients.

Contact index between fibers of the component yarn and abrasive

We assume that many abrasive particles with uniform size are densely distributed on the abrasive surface. The contact between fibers and abrasives conforms to the Hertz contact condition between elastic and rigid bodies. Based on the calculation equations of contact mechanics, an ideal contact mechanics model of fibers and abrasives is constructed.

Contact points

The abrasive particles have similar sizes but irregular shapes. To simplify the calculation, we assume that the abrasive particles are a regular spherical and ordered arrangement on the abrasive substrate.

The contact model of the fiber and the abrasive is shown in Figure 5. In Figure 5(a), a fiber is placed on a regular abrasive. The abrasive particle diameter is d_m, and the fiber diameter is d_fi. Neglecting the gaps between abrasive particles and the gaps between fibers, assuming that fibers are arranged on the abrasive in the optimal idealized state, the relationship between fibers and abrasive particles is shown in Figure 5(b) and (c).

Figure 5.

(a) Contact between a fiber of component yarn and the spherical abrasive particles. (b, c) Contact between the fiber and the abrasive particles: (b) d_m < d_fi and (c) d_m > d_fi.

When d_m < d_fi, a fiber is located between two abrasive particles, and their contact number g = 2. When d_m > d_fi, a fiber can only come into contact with one abrasive particle, g = 1. The fiber number between the unit component yarn and abrasive is as follows:

χ_{ij} = [\frac{2 b_{ij}}{d_{fi}}]

(12)

where χ_ij is the fiber number in the elliptical contact area between the i-component yarn of the jth unit and the abrasive particles, b_ij is the short axis radius in the elliptical contact area between the i-component yarn of the jth unit and the abrasive, d_fi is the fiber diameter of the i-component yarn.

The contact number between the unit component yarn and abrasive is as follows:

I_{ij} = \frac{2 g \sum_{i = 1, j = 1, k = 1}^{χ_{ij}} \sqrt{(a_{ij}^{2} b_{ij}^{2} - k^{2} d_{f i}^{2} a_{i j}^{2})}}{d_{m} b_{ij}}

(13)

where I_ij is the contact number between the i-component fibers of the jth unit and the abrasive.

In the contact area, the contact number of the component yarn and the abrasive is I_i = ƩI_ij, the contact number of loop yarn and abrasive is I = ƩI_i. We assume that the normal force of any component fiber in contact with the abrasive particles is all equal. That is, N_fij = N_ij/I_ij.

Contact area

The contact area between the component fiber and the abrasive particle can be approximately elliptica.²⁵ The curvature radius of the contact area between fiber and abrasive particle is R_m, and the elliptical contact area’s long and short axis radius is a_fij and b_fij, respectively. The calculation formula²⁶ is as follows:

R_{m} = {(\frac{2}{d_{fi}} + \frac{2}{d_{m}})}^{- 1}

(14)

a_{fij} = 0.984 {(\frac{Φ_{i} Y (Φ_{i}) N_{fij} R_{m}}{E_{i}^{*}})}^{\frac{1}{3}}

(15)

b_{fij} = 0.984 {(\frac{Y (Φ_{i}) N_{fij} R_{m}}{Φ_{i}^{2} E_{i}^{*}})}^{\frac{1}{3}}

(16)

where

Φ_{i} = 1 + \sqrt{\frac{d_{m} \ln \frac{16 d_{m}}{d_{fi}}}{2 d_{fi}}} - \sqrt{\ln 4} + 0.16 \ln \frac{d_{fi}}{d_{m}}

(17)

Y (Φ_{i}) = 0.634 Φ_{i}^{2} [1 + \frac{0.634 (1 - Φ_{i}^{2})}{Φ_{i}^{2}} - 0.125 \ln Φ_{i}^{2}]

(18)

where Φ_i denotes the ellipticity of the i-component fibers and the abrasive particle, Y(Φ_i) denotes the dimensionless of the second kind complete elliptic integral, N_fij denotes the contact force between the jth unit’s i-component fibers and abrasive particle, and E_i* denotes the comprehensive elastic modulus of i-component fiber and abrasives.

Substituting (5) into (15) and (16), we obtain

a_{fij} = 0.682 {(\frac{Φ_{i} Y (Φ_{i}) R_{m} (2 T_{j} + Δ T) \sin \frac{Δ φ}{2}}{E_{i}^{*} I_{ij}})}^{\frac{1}{3}}

(19)

b_{fij} = 0.682 {(\frac{Y (Φ_{i}) R_{m} (2 T_{j} + Δ T) \sin \frac{Δ φ}{2}}{Φ_{i}^{2} E_{i}^{*} I_{ij}})}^{\frac{1}{3}}

(20)

The contact area between a fiber and one abrasive particle is as follows:

S_{fij} = π a_{f ij} b_{f ij}

(21)

where S_f_ij is the contact area between the fiber of the jth unit of the i-component yarn and the abrasive particle.

The contact area between the component yarns, the loop yarn, and the abrasives can be obtained as follows:

S_{i} = π \sum_{j = 1}^{n_{j}} I_{ij} a_{f ij} b_{f ij}

(22)

S = \sum_{i=1} S_{i}

(23)

where S_i denotes the contact area between the i-component yarn and the abrasive, and S denotes the contact area between the loop yarn and the abrasive.

The maximum contact stress between fiber and abrasive particle σ_fij is as follows:

σ_{fij} = \frac{3 N_{fij}}{2 π a_{f ij} b_{f ij}}

(24)

The maximum dynamic shear stress of fiber and abrasive particle τ_fij is as follows:

τ_{fij} = \frac{σ_{fij} \sqrt{2 t - 1}}{2 t (t + 1)}

(25)

where

\frac{b_{fij}}{a_{fij}} = \sqrt{(t^{2} - 1) (2 t - 1)}

(26)

where t is a dimensionless coefficient.

Experiments

Materials and instrument

The abrasion yarn investigated in this article is loop yarn. The main properties related to the loop yarn and their component yarns are noted in Table 1. This experiment selected #100, #360, #600, #1000, #1500, and #2000 grade SiC sandpapers from China as the abrasives. According to ISO 6344: 1998, the average particle diameter of abrasive is given in Table 2.²⁷ The comprehensive elastic modulus and Poisson’s ratio of fiber and abrasives are noted in Table 3.

Table 1.

Index parameters of loop yarn

Type	Manufacturer	linear density (tex)	Twist (t·10 cm⁻¹)	Twist factor	Twist direction	Material
Type	Manufacturer	linear density (tex)	Twist (t·10 cm⁻¹)	Twist factor	Twist direction	Effect yarn	Binding yarn	Core yarn
Q₁	Dongguan Jinlong Textile Co., Ltd	72.3	71.4	607.1	S	Acrylic	Polyamide
Q₂	Dongguan Jinlong Textile Co., Ltd	73.8	61.2	525.9	S	Cotton
Q₃	Dongguan Jinlong Textile Co., Ltd	64.5	65.0	521.7	S	Cotton
Q₄	Dongguan Feifan Textile Co., Ltd	64.8	62.7	504.9	S	Cotton
Q₅	Suzhou Huicai Textile Co., Ltd	61.7	75.6	593.8	S	Polyester
Q₆	Jiangyin Yingze Yarn Industry Co., Ltd	82.3	61.9	561.6	S	Polyester

Table 2.

Mesh number of abrasive and its average particle diameter²⁷

Type	Average diameter of particles (µm)	Type	Average diameter of particles (µm)
#100	150 ± 3.0	#1000	18.3 ± 1.0
#360	40.5 ± 1.5	#1500	12.6 ± 1.0
#600	25.8 ± 1.0	#2000	10.3 ± 0.8

Table 3.

Comprehensive elastic modulus and Poisson’s ratio of fiber and abrasive^6,28–32

Type	E (MPa)	ν	Type	E* (MPa)
Cotton	7497	0.85	Cotton–silicon carbide	25,761.5
Polyester	1921	0.40	Polyester–silicon carbide	2256.4
Acrylic	5826	0.28	Acrylic–silicon carbide	6850.1
Polyamide	1499	0.42	Polyamide–silicon carbide	1621.8
Silicon carbide	520,000	0.25

The linear density of the yarn was determined by the hank method on a yarn-measuring length instrument. The twist of the loop yarn was determined on a Y331A yarn twist tester. The fiber diameter was determined on a BC4K-3630 digital microscope. The yarn diameter was determined on a BC4K-3630 digital microscope.

A yarn friction tester was built to obtain the worn yarn, generally used for abrading yarn on sandpaper. An XM3-103 high-speed camera system was set up above the yarn friction apparatus and used to capture the images of the yarn sliding over the abrasive.

Friction test of loop yarn

In the experiment apparatus (Figure 6), two 24 V DC deceleration motors with a speed of 296 revs/min were used as the driving motors for the traction wheel and delivery wheel, and their linear velocity was 0.622 m/s. The diameter of the abrasion sleeve was 105 mm, and the contact length of the yarn and the abrasive was 75 mm.

Figure 6.

Yarn friction apparatus.

Sandpaper grades #100, #360, #600, #1000, #1500, and #2000 were selected as the abrasive for yarn friction testing; the pretension of the yarn was 2N(200g) weight). For friction testing, the loop yarn was installed in the yarn-to-abrasive apparatus, and the tension sensor recorded the tensile force value from the loop yarn applied to the tension wheel during the friction process. The loop yarn was subjected to friction testing on the yarn friction apparatus, and two tension sensors collected the normal force value of the loop yarn at both ends of the friction area, which was the contact area of the loop yarn and abrasive.

According to the positional relationship between the tension wheel and the yarn guide wheel of the tension sensor, the normal force can be converted into tension of the loop yarn using the following formula³³:

T = \frac{F}{2 \sin β}

(27)

where T is the loop yarn tension of the tension sensor, F is the yarn normal force collected by the tension sensor, and β is the angle between F and T, and it involves the positional size between the tension wheel and the yarn guide wheel of the tension sensor. One tension sensor located at the front end of the friction area collects the normal force F₂, which corresponds to the tension of the loop yarn, T₂; the other tension sensor located at the back end of the friction area collects the normal force F₁, which corresponds to the tension of the loop yarn, T₁.

When the loop yarn slides over the abrasive, its frictional force F_f is taken as the difference between the tension T₁ and T₂. That is, F_f = T₂ – T₁. The frictional coefficient μ of the loop yarn and the abrasive based on the non-Euler formula³⁴ can be calculated as follows:

μ = \frac{1}{ϕ} ln (\frac{T_{2} - m v^{2}}{T_{1} - m v^{2}})

(28)

where φ is the wound angle of the loop yarn lay on the fixed sleeve, m is the mass of the loop yarn per unit length, and v is the sliding speed of the loop yarn. Ignoring the tensile deformation of the loop yarn, there is no slippage between the loop yarn and the traction wheel, and v can be set as the surface linear velocity of the traction wheel.

Results and analysis

Contact index between fiber and abrasives

Figure 7 shows one cycle contact index of the Q₁ loop yarn on the friction device (sandpaper grade #360).

Figure 7.

Friction index, contact number, contact area, distribution force, and contact stress of the unit fiber when the Q₁ loop yarn comes into contact with #360 abrasive: (a) T₁ and T₂; (b) F_f and μ; (c) S_fij; (d) I_ij; (e) S_ij; (f) N_fij; (g) σ_fij and (h) τ_fij.

In Figures 7(a) and (b), as friction progresses, T₂, F_f, and μ of the Q₁ loop yarn gradually decrease with some fluctuations, whereas T₁ has almost no change with slight fluctuations. The abrasive particles have a hindering effect the sliding of the loop yarn; they scrape off the apparent fibers, causing them to be damaged and fall off, and can also alter the yarn structure. The friction between the loop yarn and the abrasive causes fiber detachment, with some fibers remaining between the abrasive particles. This improves the microcontact environment on the abrasive surface, leading to a decreasing trend in the abrasive’s hindrance effect.

We calculate the contact area and force between the Q₁ loop yarn unit and the abrasive using the max, min, and mean values of T₂ and T₁ in Figure 7(a). The contact area and force between the Q₁ loop yarn unit and the abrasive particles from contact to separation are shown in Figures 7(c)–(h).

As can be found through Figures 7(c)–(e), by increasing l_q or j, T_j will increase, S_fij, I_ij, and S_ij of each unit in contact with the abrasive will also increase, and the fluctuation range of their values will also enlarge. The greater the tension of the loop yarn, the larger the contact number and area. Among them, S_fij and S_ij are Q_1b ≈ Q_1x > Q_1e; I_ij is Q_1e > Q_1x ≈ Q_1b. The effect yarn with thin fiber has more fibers than other component yarns, and its single-fiber contact area with the abrasive is small. In contrast, the contact number between the unit and the abrasive is greater than that of other component yarns.

In Figures 7(f)–(h), as l_q or j increase, N_fij, σ_fij, and τ_fij of each unit in contact with the abrasive all increase, and the fluctuation range of their values increases. The greater the tension of the loop yarn, the larger the contact force and stress. However, the fluctuation range of N_fij is relatively more significant than the fluctuation ranges of σ_fij and τ_fij. The tension on the contact force of the loop yarn unit fiber has a much more significant influence than its maximum contact stress and dynamic shear stress. Among them, N_fij is Q_1b ≈ Q_1x > Q_1e, σ_fij and τ_fij are Q_1e > Q_1b ≈ Q_1x. At the separation point between the loop yarn and the abrasive (l_q = 75 mm, j = n_y), N_fij, σ_fij, and τ_fij of the last unit reach a maximum.

Contact number and the area between loop yarn and abrasives

The frictional indexes of the Q₁-Q₆ loop yarn contact with different abrasives are shown in Figure 8.

Figure 8.

Friction index between loop yarn and abrasives: (a) F_f and (b) μ.

In Figure 8, the F_f and μ of Q₅ loop yarn are much lower than those of other types of loop yarn, while the F_f and μ of other types of loop yarn have no significant difference. As the particle mesh of the abrasive increases, the F_f and μ between the loop yarn and the abrasive generally decrease approximately. This indicates that the tension carried for the loop yarn unit and the force applied to the abrasive gradually decrease as the mesh.

The contact morphology between the loop yarn and the abrasive is very complicated, and the ideal contact model of the two can approximately express their contact number and area. The calculation formulas for the contact number and area are relatively complex, making it challenging to obtain the variation law of contact indexes between loop yarn and abrasives of different particle meshes from the formulas. Based on the idealized contact morphology between fibers, yarns, and abrasives, we use Equation (13) to calculate the contact number between a unit’s component yarns and abrasives, as shown in Figure 9; according to the calculation methods of Equations (22) and (23), the contact area between the component yarn, the loop yarn and the abrasive are shown in Figure 10.

Figure 9.

Contact number between the loop yarn and the abrasive: (a) I_ej; (b) I_e; (c) I_bj; (d) I_b; (c) I_xj; (c) I_x and (d) I.

Figure 10.

Contact area between the component yarn, loop yarn and the abrasive: (a) S_e, does not contain loops of effect yarn; (b) S_b; (c) S_x and (d) S.

In Figure 9, the contact number between the component and loop yarn with the abrasive gradually increases with the abrasiveness of the mesh. The contact number between the effect yarn and the abrasive particles is greater than between the other component yarn and the abrasive. We compare the contact numbers of different types of loop yarn and find that I_e is Q₅ > Q₆ > Q₁ > Q₂ > Q₃ > Q₄, I_b is Q₅ > Q₆ > Q₄ >Q₁ > Q₃ > Q₂, I_x is Q₅ > Q₆≈Q₄ > Q₃≈Q₁ > Q₂, and I is Q₅ > Q₆ > Q₁ > Q₄ > Q₃ > Q₂. It can be seen that the contact numbers of Q₅ are much larger than those of other types of loop yarn. The fibers of Q₅ are finer than others and consist of multiple fibers. The contact numbers between the binding yarn, core yarn, and abrasive are similar and related to the same fiber material: nylon fiber. Twisted nylon yarn is commonly used for binding yarn, while twistless nylon yarn is widely used for core yarn. The contact number between the component yarn and the abrasive is related to the fiber and yarn diameter. Under the same contact conditions, the finer the fiber diameter, the thicker the yarn diameter, the more fibers there are in the yarn, and the more contact number between the component yarn and the abrasive.

In Figure 10, the contact area between the component yarn, loop yarn and the abrasive increases first and then decreases as the abrasive mesh. When the abrasiveness of the mesh is #600 or higher, the abrasive particle size is close to the fiber diameter, and the contact area of the component yarn slightly decreases. We find that S_b ≈ S_x > S_e by comparing the contact area between the component yarn and the abrasive. The contact area of the effect yarn is Q₅ ≈ Q₆ > Q₁ > Q₂ ≈Q₃ ≈ Q₄. The contact area between the binding yarn and the abrasive is not significantly different between the core yarn and the abrasive. If the structural difference of the component yarn is ignored, the contact area is nylon > polyester > acrylic > cotton by material type.

The effect yarns of Q₂–Q₄ loop yarns are all cotton fibers, and their contact area with abrasives is almost equal. The effect yarns of Q₅ and Q₆ loop yarn are made of polyester, and there is no significant difference in the contact area between them and abrasives. The binding and core yarn of loop yarn are made of nylon, and there is no significant difference in their contact area with the abrasive. The contact areas of the component yarns, which are made of the same material, are not significantly different. The fiber diameter and elastic modulus significantly affect the contact area. The smaller the fiber diameter, the lower the elastic modulus, and the higher the contact area.

Contact force and ratio coefficient

By Equations (24)–(26), the N_ij-max, σ_ij-max, and τ_ij-max (l_q = 75 mm, j = n_y) obtained are shown in Figure 11.

Figure 11.

Contact force and stress between the Q₁–Q₆ fibers and the abrasives: (a) N_ej-max; (b) N_bj-max; (c) N_xj-max; (d) σ_ej-max; (e) σ_bj-max; (f) σ_xj-max; (g) τ_ej-max; (h) τ_bj-max and (i) τ_xj-max.

In Figure 11, as the abrasive mesh increases, N_ij-max, σ_ij-max, and τ_ij-max continuously decrease. Under the same conditions, N_bj-max ≈ N_xj-max > N_ej-max, N_ij-max of Q₅ component fiber is the smallest, σ_ej-max and τ_ej-max of fiber of Q₅ effect yarn are also the smallest, σ_ij-max and τ_ij-max of each component fiber are also close. By comparing the contact force and stress, we find that they can be arranged by fiber material for cotton, acrylic, polyester, and nylon, in descending order.

Cornelissen et al.³⁵ proposed that the contact area S_fij between a fiber and an abrasive particle has the following nonlinear relation with the fiber contact force $N_{fij}^{2 / 3}$ . The calculation formula for the contact ratio coefficient Г_i can be obtained from Equations (23)–(25) as follows:

Γ_{i} = \frac{N_{fij}^{\frac{2}{3}}}{S_{fij}} = 0.685 {(\frac{Φ_{i} E_{i}^{* 2}}{Y^{2} (Φ_{i}) R_{m}^{2}})}^{\frac{2}{3}}

(29)

According to Equation (29), Г_i ≈ $σ_{ij}^{2 / 3}$ , Г_i is related to R_m, Φ_i, Y(Φ_i), and E_i*, indicating that the fiber contact stress is related to the physical dimension and modulus of fibers and abrasive particles. It can be seen that the smaller the fiber diameter, the larger the fiber modulus, and the larger the stress under the same abrasive. Under the same pressure distribution conditions, the smaller the contact area between the fiber and abrasive, the greater their contact stress. This rule is also consistent with the mechanical properties of fiber: the greater the stiffness of fiber, the weaker its deformation ability, and it is more prone to wear. Therefore, σ_ij-max and τ_ij-max can be used as an evaluation index for the wear resistance of fiber.

Discussion

Relationship between contact index and friction index

From Figures 8 and 9, the contact number and friction index trends are opposite, and their curves can both be approximated as straight lines. As the abrasive mesh increases, the contact area curve between the loop yarn and the abrasive follows an approximate parabola, entirely different from the contact number curve. For abrasiveness above #360 mesh, the contact area between the loop yarn (except Q₅) and the abrasive continuously decreases, and this trend is consistent with that of the friction index.

The smaller the abrasive particle size, the lower the hinder force of the abrasive on the loop yarn, and the higher the contact number, whereas the contact area also tends to decrease. This indicates that the friction between the loop yarn and the abrasive is not solely determined by load but is also related to other factors such as fiber diameter, abrasive diameter, fiber modulus, and particle shape. The higher the abrasive diameter, the more significant the size effect of irregular abrasive, and the higher the probability of the irregular protrusions scraping fibers. It is not simply a frictional contact but should include other unpredictable scraping effects.

Kaliski³⁶ found that the smaller the particle size and the more particles on the material surface in contact with fibers, the lower the frictional force. Except for the asperity number and the asperity radius, the other conditions were the same. He proposed an empirical function that multiplied the square root of the asperity number by the asperity radius, and the frictional coefficient was equal to a dimensionless coefficient δ.

According to Kaliski’s empirical formula, multiplying the square root of contact number by abrasive particle radius and frictional force is δ_f = 0.5 ×10⁻³I^0.5d_mF_f, and multiplying the square root of contact number by abrasive particle radius and frictional coefficient is δ_μ= 0.5 × 10⁻³I^0.5d_mμ. Their relationship with abrasive mesh is shown in Figure 12.

Figure 12.

(a) Effect of abrasive mesh on δ_f and (b) effect of abrasive mesh on δ_μ.

In Figure 12, both δ_f and δ_μ decrease with the abrasiveness of the mesh, and their curves are approximately concave shape. However, δ_f and δ_μ of the Q₁–Q₆ loop yarn are not significantly different. We calculated the mean values of F_f and μ of the same mesh separately, then by fitting for obtaining δ_f = 115.720c⁻^0.538, δ_μ = 61.982c⁻^0.515. These can be used as an approximate calculation method for the friction index of loop yarn.

El Mogahzy and Gupta^4,37 derived a fiber contact model and found that the contact number between fibers in line contact was higher than in point contact. There are fewer asperity particles on the surface of rough fibers than on smooth fibers, and the frictional coefficient is lower when the fiber surface is highly smooth. This trend is consistent with the principle of friction and our conclusion. El Mogahzy and Gupta analyzed the relationship between the asperity number on the fiber surface and friction at the microscale. The asperity size on the material surface analyzed is smaller than the contact size between the abrasive and the fiber discussed in this article.

Structural and contact model

The geometrical model of multi-ply yarn covers several types of fancy yarns, such as bouclé, gimp, wavy, and loop yarn. The loop yarns we tested are all composed of three component yarns, and their structures are similar. This model can be extended to account for multi-ply fancy yarns made with two- or three-effect yarns. The structural model of the loop yarn mathematically followed a method suitable for multi-ply fancy yarns.

Real contact area could not be measured directly. Some studies mainly focused on the contact analysis of fiber tows under hypothetical or idealized conditions. The Hertz contact model can be applied to accurately explain the normal contact (without adhesion) between elastic bodies. The Maugis–Dugdale adhesive contact model provides insights into the adhesion contact area. The JKR contact model, which is based on Hertz contact model, was proposed for the contact application with adhesive. Cornelissen et al.⁹ proposed that the results of the Hertz contact and Maugis–Dugdale adhesive contact model had little difference under the same pressure. Che et al.¹⁰ also believed the fiber contact tips were elastic with little adhesion in the sliding process: the contact area under JKR model is approximately equal to that under Hertz model. To analyze the frictional behavior of different types of yarn, we use the contact index to evaluate the contact difference.

The contact model mentioned in this paper can be applied to analyze the contact behavior between any multi-ply fancy yarn and abrasive. To simplify the calculation, the contact form between cylindrical fibers and spherical abrasive particles has been discussed at the microscopic level, ignoring the actual particle distribution on the abrasive and the fibers’ surface morphology characteristics. Subsequent research will obtain the apparent morphology of abrasives using 3D scanners, and statistical methods will also be used to analyze the abrasives’ distribution characteristics, further optimizing the simulation calculation data.

Conclusion

Loop yarn sliding over an abrasive is a typical sliding friction phenomenon. Based on the structural characteristics of the loop yarn, we have assumed that the contact between fibers and abrasives is a typical Hertz model and have derived the calculation formulas for the loop yarn’s contact number, area, and force. The calculation results indicate that as the tensile force of loop yarn increases, the contact number, area, and force of fiber also increase. Under the same contact conditions, the thicker the diameter of the component yarn, the finer the fibers, and the more contact pits between the loop yarn and abrasives. The contact area does not continuously increase with the increase of abrasive mesh, while the contact force and stress are opposite; they are related to the physical indexes of the fibers and component yarns. The ratio of the force applied by the loop yarn on the abrasive to their contact area is related to the material and size of the yarn, which can provide feedback on the wear resistance of the yarn. Multiplying the square root of the contact number of the loop yarn by the radius of the abrasive particles and the frictional forces or coefficient are all dimensionless coefficients, and these coefficients are similar for different types of loop yarn in contact with abrasives of the same mesh, which can be used as a feedback index for friction performance.

Footnotes

Declaration of conflicting interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Bintuan Science &Technology Program (grant numbers 2023CB009-06 and 2024DA037).

ORCID iDs

Li Yong

Zhou Yuyang

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