Theoretical and experimental investigations on the effects of friction,bending rigidity,extensibility,and Poisson's ratio on fabric tensile properties

Abstract

A three-point tensile model that consists of a noncontact model and a modified capstan model of contact sections, including coupling effects of factors, is established in this study. The tension ratio calculated using the Runge–Kutta method increases along with the extensibility, surface friction coefficient, and radius ratio and is inhibited by the power-law friction ( $n = 0.67$ ). Moreover, the theoretical model in a case with all the factors and frictional modification shows high accuracy with the actual test of the quick-intelligent handling evaluation system, and the Poisson's effect can be negligible, especially with the power-law friction. It has been confirmed that greater surface roughness and thickness (lower radius ratio) with worse extensibility result in tighter fabric tensile properties. Therefore, this work can provide theoretical guidance for the measurement of fabric tensile properties and the evaluation of practical application of fabrics.

Keywords

three-point tensile model extensibility power-law friction quick-intelligent handle evaluation system

Tensile properties are among the basic mechanical properties of materials (especially fabrics), and directly determine applications of fabrics.¹ Generally, the stability of the fabric structure must be coordinated with comfort and tensile properties, where the tensile properties, along with other mechanical properties, such as bending and friction, work together to give the final properties of fabrics. Nevertheless, the tensile behavior being studied is complex and difficult to detect and predict, as the tensile properties of fabrics are predominantly nonlinear and viscoelastic, exhibiting coupling effects with other mechanical behaviors.² Many theoretical studies have been carried out on the tensile properties of fabrics.^3–6 However, in recent years, research on tensile properties has mainly focused on improving or enhancing the tensile properties through the composition or modification of materials,^7–10 while there is less research on the tensile theory. Various models have been proposed to analyze the tensile properties using geometric, numerical, and mechanistic methods in the past. Olofsson^11, ¹² used a rheological model, consisting of springs and Kelvin elements in series, and constructed a theoretical stress–strain curve for the stretching behavior based on the elastic constants of the fabric obtained through deformation–recovery cycles under constant stress. Grosberg¹³ analyzed the initial load–extension modulus based on Castigliano's theorem and bending behavior, and indicated that the bending modulus and geometry of the yarns would affect the tensile properties. Also, Leaf¹⁴ presented an analysis of the initial load–extension behavior of fabrics by the principle method of Castigliano's theorem, proposed a generalized model based on the bending behavior, and developed a method for calculating the ratio of crimp height to thread spacing and weave angle.¹⁵ Hearle¹⁶ proposed an energy method for properties based on the bearing mechanism of fabric structure by excluding the change of internal energy during deformation, and considering the strain energy of yarns and fibers. Huang¹⁷ studied finite deformation under biaxial stresses and the effect of the contact deformation of yarns, where uniaxial and biaxial extensions with equal stresses were solved by numerical solutions. Meanwhile, Huang¹⁸ calculated finite biaxial extensions, including bending linearly elastic extension, by using a rigorous force method, and solved a nonlinear boundary-value problem by a procedure of trial and error. Kawabata^19,20 proposed a simplified stereo model to analyze biaxial and uniaxial deformation. Jung²¹ developed an inelastic stretch model based on Kelvin's equation that can reproduce the stretch deformation. However, most of these studies lack the application and experimental verification of theoretical models in practice; in this study, models will be established on detection principles and be verified by subjective and experimental methods.

In this modeling, a winding contact between the fabric and a circular object is involved, so the capstan model is introduced. Wei²² presented a capstan equation of the tension, including the influence of the bending behavior. Jung²³ established a model consisting of three compatible ordinary differential equations of force and bending moments, based on the beam–body contact, and improved the model by taking account of power-law friction and bending rigidity and analyzing the tension transmission efficiency.²⁴ Moreover, Jung²⁵ then conducted an in-depth analysis of the capstan model, supplementing the effects of extensibility and Poisson's ratio. The capstan model of fibers and yarns was also investigated by Gao²⁶ and Tu.^27,28 But some of these theoretical studies have not been applied to tensile studies, and some of them only involve the tension ratio, which cannot directly characterize the tensile properties of materials and has not been combined with experiments. This study can compensate for these shortcomings. As explained by cohesion theory, Amonton's classical friction law can satisfactorily explain most metallic friction behaviors. However, it has been indicated that the law fails for nonmetallic materials, especially elastic solids, such as polymer monofilaments and natural fibers.^29–33 In these cases, the friction coefficient is proved not to remain constant, but gradually decreases as the load increases, indicating that the local deformation occurs at the interface between the polymer and the friction object.³⁴ After that, Howell^35,36 proposed that the friction force must be related to the load followed by $F = a R^{n}$ , where R is the applied normal load and a and n are constants (a depends on the properties of the surface materials, while n is determined by the geometry of the contact surface). Thus, the power law is used in this study to further analyze the effect of friction.

In previous research, we mainly focused on fabric performance testing using the specially developed quick-intelligent handling evaluation system (QIHES),^37,38 as well as characterization and algorithm models based on objective testing. In this study, a mechanical three-point tensile model structure was established based on the QIHES for comparison with the experimental results and subjective evaluation, which was not involved in our past work. In particular, this study proposes a broader theory and delves into several important factors that affect tensile properties, such as friction law, surface friction coefficient, bending rigidity, extensibility, and Poisson's ratio, as well as the interaction between them. According to the force–displacement relationships in different cases derived by the model, quantitative measurement and characterization of the tensile properties can be achieved to better meet the requirements of fabrics. For apparel applications, a small stress needs to be considered for mechanical properties, which means that looser performance is required, while the large-scale tensile behavior needs to be considered in industrial applications, which means tighter performance.

Theoretical modeling

Nomenclature

The definition of parameters involved in the theoretical model is given in Table 1.

Table 1.

Definition of parameters

Symbol	Unit	Meaning	Symbol	Unit	Meaning
T	cN	Tension force	ρ		Radius ratio, r₂/r₁
Q	cN	Shear force	ε		Axial strain of infinitesimal element
N	cN	Normal force	θ	rad	Total contact angle of press plate
F _μ	cN	Frictional force	ν		Poisson's ratio of the sample
M	cN·mm	Bending moment	μ		Frictional coefficient between contacting solids
Φ	rad	Contact angle before stretching	B	cN·mm	Bending rigidity
Φ′	rad	Contact angler after stretching	E	MPa	Tensile modulus of the sample
r ₁	mm	Radius of the sample before stretching	A	mm²	Cross-sectional area of the sample
r₁′	mm	Radius of the sample after stretching	l	mm	Half of the gap between both working plates
r ₂	mm	Radius of the press plate	h	mm	Displacement variable of the press plate

Structure of QIHES related to theoretical modeling

The fundamental structure and primary parts of the QIHES, a custom-developed instrument, are shown in Figure 1(a), and the three-point tensile principle is shown in Figure 1(b). Throughout the test, the QIHES is connected to a computer and operated automatically through the control software, and stops when the working conditions reach the setting limits. The parameters used in the model are determined according to the mechanical construction and parameter settings of the QIHES. To analyze the tensile properties, the relevant parameters need to be obtained through the friction and stretching stages. As shown in Figure 1(b), a fabric sample is placed on the working platform, and its two ends are pressed by press roller 1 and press roller 2 with a constant force, which is 50 cN at the friction stage and 100 cN at the stretching stage. Both humanoid finger plates are used to prevent the samples from cocking. The procedure of the friction stage is that the press plate, connected to the displacement sensor and the force sensor, moves down with the fabric sample until the displacement value reaches a maximum. At this stage, the friction coefficient can be determined as the ratio of the obtained friction force to the constant pressure of press rollers. Following the friction stage is the stretching stage, during which the press plate continues to move downward until the force value reaches the preset value.

Figure 1.

Quick-intelligent handling evaluation system: (a) fundamental structure; (b) three-point tensile principle and testing procedure.

Assumptions

For theoretical analysis of the contact and noncontact sections of the stretching stage, the following assumptions are presented, according to the characteristics of the stretching process of the QIHES:

It is assumed that the fabric sample is nearly a linear elastic material. During the stretching stage, the sample is subjected to a large stress and small deformation, so the force–displacement curve is approximately linear.

The sample is assumed to be symmetrical to the centerline $O' D$ (Figure 2(a)). In the theoretical analysis, only the right part, which can be divided into two areas, the contact section $CD$ and the noncontact section $AC$ , is considered for the calculation. The point C is the point of demarcation between the contact and noncontact sections.

The noncontact section is assumed to be approximately symmetric about the midpoint S (Figure 2(b)) which is between the press plate and the working plate.

The Poisson's ratio of the contact section is assumed to be constant, so that the strain used throughout the entire contact section is assumed to be the average value instead of varying.

Figure 2.

(a) Three-point structure; (b) test set-up of the essential structure of QIHES.

Analysis of contact section

Figure 3 shows the typical situation of the contact segment. According to Assumption 2, the right side is the analysis object of the contact segment. The resulting force consists of shear force Q in the normal direction and tensile force T in the tangential direction. Based on Assumption 2, the relationship controlling this mechanism is the capstan equation

F_{1} = F_{0} e^{\frac{μ θ}{2}}

(1)

where

θ / θ 22

is the warping angle of point C and D and μ is the frictional coefficient between contacting solids.

Figure 3.

Force and moment analysis of sample in contact with circular body of press plate.

Regarding the friction performance, textile materials generally do not follow Amonton's law, so the relationship between normal force and frictional force is modified to a power law, as

\begin{matrix} F_{μ} = μ N^{n}, & n \leq 1 \end{matrix}

(2)

where n is the material constant. This becomes the classical Amonton's law,

F_{μ} = μ N

, for the special case of

n = 1

. According to experimental measurements,³⁹ the parameter n is expected to be 0.67, owing to the ideal elastic body, which is chosen in this case. However, values of n from 0.8 to 0.96 indicate that neither pure plastic flow nor ideally elastic deformation is obtained.^40,41

Figure 4 shows the contact section of the sample wound over the press plate. According to Figure 4, the infinitesimal length of the sample increases from $d l$ to $d l'$ , so that the elongation of the microelement is

d l' = (1 + ɛ) d l

(3)

d l' = ({r'}_{1} + r_{2}) d ϕ'

(4)

Figure 4.

Microelement of contact section in equilibrium between force and moment.

As the Poisson's effect is considered, the radius of the sample after stretching is

r'_{1} = r_{1} (1 - ν \bar{ɛ})

(5)

where

\bar{ɛ}

is the average strain. Based on Assumption 4,

\bar{ɛ}

is expressed as

\bar{ɛ} = \frac{\int_{0}^{\frac{θ}{2}} ɛ d ϕ}{\int_{0}^{\frac{θ}{2}} d ϕ} = \frac{1}{θ} \int_{0}^{θ} \frac{T}{EA} d ϕ = c λ

(6)

λ = \frac{1}{θ} \int_{0}^{θ} T d ϕ, c = \frac{1}{EA}

(7)

Substituting equations (6) and (7) into equation (3) and eliminating the higher-order terms gives

d ϕ' = \frac{(1 + c λ) ({r'}_{1} + r_{2})}{{r'}_{1} (1 - ν c λ) + r_{2}} = (1 + c λ) η d ϕ

(8)

η = \frac{1 + ρ}{1 - ν c λ + ρ}, ρ = \frac{r_{2}}{{r'}_{1}}

(9)

These relational equations are used to derive the governing equation of the tensile ratio in the next section.

Based on the force equilibrium along the n and t directions and the moment equilibrium at point O, the following classical capstan equations are derived

d T + Q d ϕ' - d F_{μ} = 0

(10)

d Q - T d ϕ' + d N = 0

(11)

d M - Q (r'_{1} + r_{2}) d ϕ' + r'_{1} d F_{μ} = 0

(12)

Based on Assumption 1, $d M = 0$ and the following equations can be obtained by incorporating equations (10) to (12)

\begin{matrix} \frac{d^{2} T}{d ϕ^{2}} + \frac{1}{n} \frac{ρ (1 + c λ) η}{1 - ν c λ} (\frac{1 - ν c λ + ρ}{μ ρ})^{\frac{1}{n}} T^{\frac{1 - n}{n}} \frac{d T}{d ϕ} \\ - \frac{ρ (1 + c λ) 2 η^{2}}{1 - ν c λ} T = 0 \end{matrix}

(13)

\begin{matrix} Q = \frac{(1 - ν c λ) (1 - ν c λ + ρ)}{ρ (1 + ρ) (1 + c λ)} \frac{d T}{d ϕ}, \\ d F_{μ} = \frac{1 - ν c λ + ρ}{ρ} d T, N = (\frac{F_{μ}}{μ})^{\frac{1}{n}}, \\ M = \frac{EI}{{r'}_{1} + r_{2}} \end{matrix}

(14)

The fourth-order Runge–Kutta method is selected as the numerical procedure, as equation (13) cannot be solved analytically. Thus, equation (13) can be divided into the following two differential equations

\frac{d T (ϕ)}{d ϕ} = U (ϕ)

(15)

\begin{matrix} \frac{d U (ϕ)}{d ϕ} = - \frac{ρ (1 + ρ) (1 + c λ) (1 - ν c λ + ρ) \frac{1}{n}}{n (1 - ν c λ) (1 - ν c λ + ρ) μ^{\frac{1}{n}} ρ^{\frac{1}{n}}} T^{\frac{1 - n}{n}} U (ϕ) \\ + \frac{ρ (1 + c λ) 2 (1 + ρ) 2}{(1 - ν c λ) (1 - ν c λ + ρ) 2} T = 0 \end{matrix}

(16)

And the two equivalent boundary conditions are

\begin{matrix} T (\frac{θ}{2}) = F_{1} cos α_{1}, \\ U (\frac{θ}{2}) = \frac{F_{1} sin α_{1} ρ (1 + ρ) (1 + c λ)}{(1 - ν λ c + ρ) (1 - ν λ c)} \end{matrix}

(17)

As the parameter λ contains the functional solution and independent variable of t equation (13), the value of λ needs to be iterated until equation (7) is satisfied.

Equation (13) is the governing equation used to calculate the tension ratio affected by the surface friction coefficient, bending rigidity, and Poisson's ratio at the contact portion. Moreover, the axial tension ratio $Φ (ϕ)$ is introduced to calculate the deformation of the contact section $CD$

Φ (\frac{θ}{2}) = \frac{T (\frac{θ}{2})}{T (0)}

(18)

The corresponding contact angle is $θ / θ 22$ , as seen in Figure 3. Furthermore, the elongation $Δ l_{CD}$ of $d l'$ with Poisson's ratio is expressed as

\begin{matrix} Δ l_{CD} = \int_{0}^{\frac{θ}{2}} \frac{ɛ}{1 + ɛ} ({r'}_{1} + r_{2}) d ϕ \\ = \int_{0}^{\frac{θ}{2}} \frac{F_{1} cos α_{1}}{EA Φ (ϕ) + F_{1} cos α_{1}} ({r'}_{1} + r_{2}) d ϕ \end{matrix}

(19)

Based on Assumption 1, ɛ is small, and equation (19) can be simplified into

Δ l_{CD} \approx \int_{0}^{\frac{θ}{2}} \frac{F_{1} cos α_{1}}{EA Φ (ϕ)} ({r'}_{1} + r_{2}) d ϕ

(20)

In the calculation of the extension in the contact section, values of $F_{1}$ and $α_{1}$ are required, which are determined by modeling the noncontact section in the next section.

Determination of forces and angles

The three-point tensile model ( $A' DA$ ) of the sample is shown in Figure 5. Moreover, the moments $M_{1}$ and $M_{2}$ , and the reactive forces $F_{1}$ and $F_{2}$ with its component forces are explained in Figure 5. The reactive force, $F_{1}$ , is divided into $T_{1}$ along the tangential direction, and $Q_{1}$ along the normal direction. Also, $F_{1}$ is divided into $F_{1 x}$ and $F_{1 y}$ , along the horizontal and vertical directions, respectively. The relationships between the forces are established as

F_{1 y} = F_{2 y} = \frac{1}{2} F^{*}

(21)

T_{1} = F_{1} cos α_{1}, Q_{1} = F_{1} sin α_{1}

(22)

F_{1 x} = F_{1} sin ω_{1}, F_{1 y} = F_{1} cos ω_{1}

(23)

where

F^{*}

is the force of the press plates tested by the QIHES,

α_{1}

is the angle between

F_{1}

and

T_{1}

, and

ω_{1}

is the angle between

F_{1}

and the vertical force

F_{1 y}

Figure 5.

Force and moment analysis of sample (including contact and noncontact sections).

According to Assumption 2, the right part is considered in the analysis and the calculation is as Figure 6. Based on the mechanical analysis of Figure 6, the relationships between resultant forces and component forces at point A, C, and D are

T_{0} = F_{0} cos α_{0}, Q_{0} = F_{0} sin α_{0}

(24)

where

F_{0}

is the resultant force and

T_{0}

is the horizontal force at point D, and

α_{0}

is the included angle between them. It is important to analyze and calculate the mechanical behavior of contact and noncontact sections. Thus, the equilibrium equations of moment and force are established as equations (25) to (27), according to Figure 6 and Assumption 3

T_{0} = F_{1 x}, Q_{0} = 0

(25)

F_{1 x} y_{AS} - F_{1 y} x_{AS} - M_{1} = 0

(26)

M_{1} = \frac{B}{{r'}_{1} + r_{2}}

(27)

where

x_{AS}

is the horizontal distance between points A and S,

y_{AS}

is the vertical distance between points A and S, and

M_{1}

is the bending moment at point A. Combining equations (21) to (27), we can obtain expressions about several angle parameters

ω_{1}

α_{0}

α_{1}

, as

ω_{1} = \arctan \frac{F^{*} x_{AS} ({r'}_{1} + r_{2}) + 2 B}{F^{*} y_{AS} ({r'}_{1} + r_{2})}

(28)

α_{1} = \arccos [e^{\frac{μ θ}{2}} sin (\arctan (\frac{F^{*} x_{AS} ({r'}_{1} + r_{2}) + 2 B}{F^{*} y_{AS} ({r'}_{1} + r_{2})}))], α_{0} = 0

(29)

F_{1} = \frac{F^{*}}{2 cos (\arctan (\frac{F^{*} x_{AS} ({r'}_{1} + r_{2}) + 2 B}{F^{*} y_{AS} ({r'}_{1} + r_{2})}))}

(30)

Figure 6.

Mechanical analysis of the separated right part of the three-point tensile model.

According to the structural parameters of the QIHES, the expressions of several length parameters from equations (31) and (32) are

x_{AS} = \frac{l}{2} - \frac{1}{2} (r'_{1} + r_{2}) sin \frac{θ}{2}

(31)

y_{AS} = \frac{h}{2} - \frac{1}{2} (r'_{1} + r_{2}) (1 - cos \frac{θ}{2})

(32)

Analysis of noncontact section

The geometric model of the noncontact section AC of the sample during the stretching stage is shown in Figure 7. The abscissa and ordinate of points A and C are expressed as

x_{A} = 0, y_{A} = 0

(33)

x_{C} = h - ({r'}_{1} + r_{2}) (1 - cos \frac{θ}{2}), y_{C} = l - ({r'}_{1} + r_{2}) sin \frac{θ}{2}

(34)

Figure 7.

Geometrical relationships and basic parameters of the noncontact section.

During the actual stretching test of the QIHES, the shape of the sample is similar to a straight line, owing to the large force on the sample. Therefore, the locus equation of noncontact section AC can be expressed by the two-point form of the straight-line equation, as

f_{AC} (x) = \frac{l - ({r'}_{1} + r_{2}) sin \frac{θ}{2}}{h - ({r'}_{1} + r_{2}) (1 - cos \frac{θ}{2})} x

(35)

The axial tension force, $T_{1}$ , for noncontact section AC can be expressed as

T_{1} = F_{1} cos (δ - ω_{1})

(36)

where δ is the angle of any point between

T_{1}

and the vertical direction in the section AC; δ can be approximately expressed as equation (37) through the function

f_{AC} (x)

, which simulates the trajectory of the section AC

δ = \arctan [f' (x)]

(37)

With the help of triangles AOC and AOE, and according to the geometric relationship of angles and the Pythagorean theorem, the expression of the contact angle $θ / θ 22$ is

\frac{θ}{2} = \frac{π}{2} - \arccos \frac{{r'}_{1} + r_{2}}{\sqrt{(h - {r'}_{1} - r_{2}) 2 + l^{2}}} + \arctan \frac{h - {r'}_{1} - r_{2}}{l}

(38)

The strain and elongation of the noncontact section AC can be expressed as

ɛ = \frac{L - L'}{L'} = \frac{T_{1}}{AE}

(39)

Δ l_{AC} = L_{AC} - L'_{AC} = (\frac{ɛ}{1 + ɛ}) L_{AC}

(40)

where ɛ is the strain of the sample material, L is the length of the sample after stretching,

L'

is the original length, and

Δ l_{AC}

is the elongation of the sample. By combining equation (39) with equation (40), the extension integral and length of section AC with specific parameters can be expressed as

\begin{matrix} Δ l_{AC} = \int_{0}^{x_{C}} \frac{T_{1}}{T_{1} + AE} d s_{AC} = \int_{0}^{h - ({r'}_{1} + r_{2}) (1 - cos \frac{θ}{2})} \\ \times \frac{F_{1} cos {\arctan [f' (x)] - ω_{1}}}{AE} \sqrt{1 + [f' (x)] 2} d x \end{matrix}

(41)

L_{AC} = \sqrt{L_{AO}^{2} - L_{OC}^{2}} = \sqrt{{(h - {r'}_{1} - r_{2})}^{2} + l^{2} - ({r'}_{1} + r_{2}) 2}

(42)

According to this analysis and Assumption 2, the equilibrium equation of the sample length in the working area during the stretching stage is expressed as

\begin{matrix} L' - L_{R} = 2 \sqrt{(h - {r'}_{1} - r_{2}) 2 + l^{2} - ({r'}_{1} + r_{2}) 2} \\ + 2 ({r'}_{1} + r_{2}) \frac{θ}{2} + 2 \int_{0}^{h - ({r'}_{1} + r_{2}) (1 - cos \frac{θ}{2})} \\ \times \frac{F_{1} cos {\arctan [f' (x)] - ω_{1}}}{AE} \sqrt{1 + [f' (x)] 2} d x \\ + 2 \int_{0}^{\frac{θ}{2}} \frac{F_{1} cos α_{1}}{EA Φ (ϕ)} ({r'}_{1} + r_{2}) d ϕ \end{matrix}

(43)

where

L_{R}

is the total length left on the working platform during the stretching. By submitting equations (28) to (32) and (35) to (38) into the equation (43), the three-point tensile model can be established. After the specific parameters of different samples are brought into the model, the simulated curves of force and displacement can be obtained.

Experimental

Sample preparation

Four plain-woven fabrics were selected from a home textile company for the theoretical validation and experimental test. Each fabric sample was cut into three pieces 500 mm long and 50 mm wide for QIHES testing, and was cut into 10 pieces of 200 mm in both length and width for subjective evaluation. This means that $L'$ = 500 mm in this study. All samples were preprocessed under the conditions of the standard atmosphere ( $20 \pm 2$ ℃ and $65 \pm 3$ % relative humidity (RH)) for over 24 h. The basic specifications of the fabric samples and the modeling parameters are shown in Table 2; W is the mass of the fabric sample per unit area, measured using a Mettler balance with an accuracy of $\pm 0.1$ mg, t is the thickness, determined using a YG141 thickness gauge, B is the bending rigidity, tested using a LLY-01 stiffness instrument, μ is a friction coefficient, measured at the friction stage of QIHES, and E is the tensile modulus tested at the stretching stage of the QIHES. The specific test method is introduced in the previous section. The surface friction coefficient is tested in stage III of the QIHES, and the force–displacement curves during stretching and the tensile properties, including tensile slope and work, are obtained in the stage IV, as shown as Figure 8. The working method of the stretching stage is elongation at fixed load, and the force limit is 400 cN. In general modeling analysis, calculation for elastic or ductile regimes usually assumes that ν is fixed at about 0.3.^42,43 Thus, Poisson's ratio in this study is taken as 0.3.

Figure 8.

(a) Characteristic curves for four samples tested using the QIHES; (b) interface diagram of the QIHES.

Table 2.

Sample specifications and modeling parameters

Sample	Contents (warp/weft)	W (g/m²)	t (mm)	B (cN·mm)	μ	E (MPa)	ν	$r_{1}$ (mm)	$r_{2}$ (mm)	l (mm)	A (mm²)
1	Silk/cotton	69.53	0.126	0.109	0.60	0.105	0.30	0.063	1.5	7.5	6.3
2	Silk/cotton	80.72	0.140	0.243	0.57	0.228	0.30	0.070	1.5	7.5	7.0
3	Lyocell	138.24	0.213	0.640	0.75	0.220	0.30	0.107	1.5	7.5	10.7
4	Cotton	147.89	0.246	1.393	0.96	0.305	0.30	0.123	1.5	7.5	12.3

Subjective evaluation

The panel employed in the subjective evaluation of tightness was composed of 10 experts on research in the textile field. Each judge was trained to familiarize the handling gesture and evaluation rating scale of tightness, which is rated on a five-point scale, that 5 is most loose and 1 is most tight. The specific gesture is shown in Figure 9; holding and pulling both sides of the fabric samples with two hands to evaluate the difficulty of stretching. The judges were required to wash and dry their hands, and to sit in a laboratory at constant temperature and humidity at standard conditions ( $20 \pm 2$ ℃ and $65 \pm 3$ % RH) for 30 min before assessment. Each fabric needs to be evaluated three times, and after measuring each fabric, each judge takes a 10 min rest before the next evaluation.

Figure 9.

Evaluation gesture of tightness.

Results and discussion

The theoretical curves of the three-point tensile model under different situations were computed by substituting the parameters in Table 2 into equation (43). The effects of different factors were explored further through the comparison between the curves of theoretical modeling and the experimental testing. In addition, subjective evaluation was introduced to prove the effectiveness of the experiments and modeling. Therefore, in this section, the different cases designed by the multifactor orthogonal method are used to investigate the effects of factors in the tensile properties. The specific settings of the five simulated cases are:

Case 1.1. With extensibility, bending rigidity, and classic frictional law ( $n = 1$ ); no Poisson's effect ( $ν = 0$ );

Case 1.2. With extensibility, bending rigidity, and frictional modification ( $n = 0.67$ ); no Poisson's effect ( $ν = 0$ );

Case 2.1. With extensibility, bending rigidity, classic frictional law ( $n = 1$ ) and Poisson's effect ( $ν = 0.3$ );

Case 2.2. With extensibility, bending rigidity, frictional modification ( $n = 0.67$ ) and Poisson's effect ( $ν = 0.3$ );

Case 3. With extensibility and classic frictional law only; no bending rigidity or Poisson's effect (n = 1, $ν = 0$ , ω₂ = 0, B = 1).

Comparisons between theoretical and experimental results

The comparisons between the experiment and theoretical modeling are shown in Figure 10, where the abscissa is the vertical displacement of the stretching stage, that is, the parameter h measured in real time minus the displacement value at the end of the previous friction stage. The tensile slope (S) and the tensile work (W) of the experimental and the theoretical curves are shown in Table 3, where the tensile slope is the average linear slope during the stretching and the tensile work is the stretching area under the curve.

Figure 10.

Comparisons of the experimental and simulated force-stretching displacement curves of samples (a) 1, (b) 2, (c) 3, and (d) 4.

Table 3.

Experimental and simulated results of tensile properties

	Experimental curve		Case 1.1		Case 1.2		Case 2.1		Case 2.2		Case 3
Sample	S	W (N·mm)	S	W (N·mm)	S	W (N·mm)	S	W (N·mm)	S	W (N·mm)	S	W (N·mm)
1	0.631	13.612	0.467	10.081	0.540	12.393	0.468	10.114	0.542	12.445	0.360	8.005
2	1.609	5.650	1.023	3.408	1.465	5.138	1.048	3.493	1.466	5.140	0.624	2.125
3	2.149	3.682	1.028	1.851	1.709	3.348	1.088	1.963	1.712	3.356	0.754	1.390
4	3.549	2.552	1.565	1.094	2.949	2.345	1.667	1.168	2.956	2.352	1.437	1.022

To compare the difference of tensile properties between different cases, the expression of the deviation rate is given as

P_{S (W)} = \frac{{S_{case}}_{a} ({W_{case}}_{a}) - {S_{case}}_{b} ({W_{case}}_{b})}{{S_{case}}_{a} ({W_{case}}_{a})}

(44)

By referring to Table 3, it can be seen that the model in Case 2.2 is the most accurate compared with the experimental test, for which the maximal deviation rates of tensile slope and work of the four fabric samples are 20.34% and 9.30% respectively. The reason for the larger error rate of the tensile slope is the mechanical fluctuation of the testing instrument, not the theoretical model, as it is sensitive to the four samples, according to the simulated results of tensile properties.

Effect of bending rigidity

As shown in Figure 10, the slope and work in Case 3 are always smaller than those in Case 1.1, which indicates that the slope and work increase when the bending rigidity is included. This result can be explained by Figure 3, as the axial tension is originally the component force along the axial direction. However, in the absence of bending rigidity, the sample is regarded as a completely flexible material, at which time there is no component force and the resultant force directly becomes the axial tension. So in this case, the tension ratio follows equation (1), which leads to errors in the strain analysis of the contact section and the stress analysis of the noncontact section. By referring to Table 3, between Case 1.1 and Case 3, the maximal deviation rates of the tensile slope and work of four samples are 39.01% and 37.65%, and the minimum deviation rates are 8.18% and 6.58%. This proves that the bending rigidity is a crucial factor affecting the tensile properties. Moreover, it can be seen that the deviation rate of Sample 2 is a maximum and that of Sample 4 is a minimum, that is, the trend is completely contradictory to that of the friction coefficient of the sample. This situation is because Case 3 only includes friction and extensibility in the model, and, regardless of Case 1.1 or Case 3, the tension ratio only depends on the value of friction coefficient when there is no Poisson's effect. Therefore, the influence of the surface friction coefficient is greater, and a larger friction coefficient can reduce the effect of bending rigidity. In addition, the theoretical model considering bending rigidity is more accurate to the experimental curve than the one without bending rigidity.

Effect of Poisson's ratio

Under the effect of Poisson's ratio, when the sample is subjected to tensile force, it produces an elongation in the axial direction and a change in the thickness in the direction perpendicular to the tensile loading. Based on equations (3) to (9), the variation in thickness directly affects the radius ratio parameter ρ and the sum of the radius $r'_{1} + r_{2}$ . According to the tensile ratio equation (equation (13)) and the three-point tensile model (equation (43)), the Poisson's ratio determines the elongation of the samples in the contact section. For this reason, the effects of Poisson's ratio ( $ν = 0.3$ ) are analyzed through comparisons between Case 1.1 and Case 2.1, and between Case 1.2 and Case 2.2. It can be seen from Figure 10(a) to (d), that the yellow dashed lines are substantially coincident with the brown solid lines in the case of frictional modification ( $n = 0.67$ ), and the red dashed lines and the green lines are slightly deviated in the case of classic frictional law ( $n = 1$ ). Moreover, according to equation (44) and Table 3, the maximal deviation rate of the tensile slope between Case 1.1 and Case 2.1 is 6.12% and is 0.37% between the Case 1.2 and Case 2.2, and the maximum of the tensile work is 6.34% and 0.42%, respectively. Thus, it can be interpreted that the Poisson's effect is very small, and can even be ignored when the power-law friction is adopted as the frictional modification. To further verify this conclusion, the effects of Poisson's ratio between Case 1.1 and Case 2.1, and between Case 1.2 and Case 2.2 are analyzed under different radius ratios, as shown in Figure 11. In the range of radius ratio from 10 to 50, the deviation rates of the tensile slope and the tensile work with the frictional modification are always less than the values in the situation of the classic friction law. Furthermore, it is interesting that when ρ = 20, all the tensile parameters are the closest, that is, the Poisson's effect has a minimal impact on the tensile properties at this time.

Figure 11.

Relationship between Poisson's ratio and radius ratio with different frictional laws.

Effect of extensibility

As the sample extensibility is directly determined by the tensile modulus of the sample, based on equations (6) and (7), different tensile moduli are selected and substituted into the theoretical model to investigate the effect of the sample extensibility on the tensile properties. Figure 12(a) shows the simulated curves with different tensile moduli based on the basic parameters of Sample 1 in Case 1.1. It is obvious that the higher the tensile modulus, the greater the slope of the force–displacement curves. In addition, as the tensile modulus increases from 0.1 MPa to 0.7 MPa, the rate of increase of the tensile slope gradually decreases. Conversely, since the extensibility is inversely proportional to the tensile modulus, the greater the tensile modulus, the more the extensibility decreases. This indicates that the smaller the extensibility, the greater the tensile slope, and the tighter the sample, as shown in Figure 12(a), which is consistent with the theory of tensile mechanics. Through the analysis of equations (18) and (43), the relationship between extensibility and tension ratio is further analyzed, as can be seen from Figure 12(b). Obviously, at a given radius ratio, whether it is classic friction law or power-law friction, the tension ratio increases with extensibility. Figure 12(c) also shows the same change trend of tension ratio along with the extensibility; the only change is the increment of the radius ratio ρ from 5 to 50. Moreover, the average deviation rate caused by the radius ratio is 4.38% and 10.87% (comparing solid and dashed lines, respectively, of two colors), while that caused by the extensibility is 0.75% and 1.09% (comparing solid and dashed-dotted lines) in the case of the same increase in radius ratio and extensibility. It can be concluded that both extensibility and radius ratio enhance the tension ratio, where the effect of extensibility is far smaller than that of the radius ratio, but this effect becomes more pronounced as the radius ratio increases. By comparing Figure 12(a) and (b), the greater extensibility or the smaller radius ratio lead to the smaller tensile slope and work. This is consistent with the results of the testing and subjective evaluation of fabric tightness, for which the greater extensibility, the thinner and looser the fabric.

Figure 12.

Effect of extensibility under different tensile modulus: (a) comparison of simulated force–displacement curves; (b) comparison of tension ratios; (c) effect of radius ratio.

Effect of friction

It is clear that in the comparisons between Case 1.1 and Case 1.2, and between Case 2.1 and Case 2.2 from Figure 10(a) to (d), the slopes of the simulated curves with the frictional modification of the power-law friction ( $n = 0.67$ ), which is nonlinear, increase significantly compared with the curves with the classic frictional law ( $n = 1$ ), and that each sample conforms to this rule. Moreover, the effect of friction laws is investigated when $ν = 0.3$ and $ν = 0$ , and it is found that the influence caused by the Poisson's ratio effect is far less than that caused by the friction law. Not only that, all the four fabrics in Cases 2.1 and 2.2 have a lower deviation rate from the experimental curve of the QIHES, which indicates that the simulated curves with the frictional modification are more accurate with respect to the experimental results.

The effect of friction on tensile properties is analyzed by assuming different friction coefficients. As shown as Figure 13(a), the tensile slope increases along with the friction coefficient in Case 1.1 with classical friction law. The reason for this increase is that a larger pull-out force is needed to overcome the friction between the sample with a higher surface friction coefficient and the contact plate. By comparing Figure 12(a) and Figure 13(a), the effect of extensibility is more remarkable than that of friction coefficient for the tensile slope and work in the case of the same growth rate in extensibility and friction coefficient, because extensibility is directly related to the tensile properties of the sample. This shows that samples with low friction coefficients have lower tightness and are more easily stretched. For classical friction law, that is linear friction ( $n = 1$ ), and frictional modification ( $n = 0.67$ ), both friction coefficients enhance the tension ratio at $ρ = 10$ . The effect of friction on the tension transmission of the contact section with the capstan is also analyzed. Figure 13(b) is sufficient to show the effect of friction law; the most distinct feature between Case 1.1 and Case 1.2 is the dependence of tension ratio on the friction law. The maximum tension ratio of classical friction is larger than that for the frictional modification, up to 1.72 times, and the minimum is about 1.31 times. As shown in Figure 13(b), the increment of the friction coefficient also causes an increase in the tension ratio. The deviation rates between the different friction coefficients under the two friction laws (compare the solid and dashed lines in the same color) are 25.84% and 18.45%, which verifies that the effect of friction coefficient decreases when selecting power-law friction. Moreover, the effect of the friction coefficient on the increase of the tension ratio is much greater than the effect of extensibility. This is because when both the friction coefficient and the extensibility increase by three times the initial values, the average deviation rates caused by the extensibility are only about 1%. To further determine which factor is more significant, the effects of the friction coefficient and radius ratio are analyzed by orthogonal methods, as shown in Figure 13(c). The average deviation rates caused by the friction coefficient are 26.33% and 18.71%, and those caused by the radius ratio are 1.83% and 0.63%, which shows that the effect of friction coefficient is more significant. In addition, both of the effects are reduced in the case of frictional modification.

Figure 13.

Effect of friction under different friction coefficients: (a) comparison of force–displacement simulated curves; (b) comparison of tension ratios; (c) comparison of the effects of radius ratio and friction coefficient.

Comparing theoretical, experimental, and subjective evaluation

According to the rating scale of subjective evaluation, that is, the tighter the sample, the lower the score, the average scores rated by 10 experts in textiles are shown in Figure 14, where Sample 1 is the loosest and Sample 4 is the tightest. In addition, the trends of experimental and theoretical tensile slope of the four samples are exactly the same, and both the values of tensile slope are inversely proportional to the subjective scores. This is because the greater the tensile slope, the smaller the stretching displacement under a constant load, and the smaller the elongation of the sample. Therefore, this means that samples with high tensile slope have poor extensibility and low subjective scores. Conversely, this also proves the validity and reliability of the theoretical model established, as it is consistent with the results of the actual experimental test of the QIHES and subjective evaluation. In summary, this model provides a theoretical basis for the fabric tensile properties, and guidance for the production of fabrics that need to be loose or tight, according to different purposes.

Figure 14.

Relationship between subjective evaluation, experimental test, and theoretical model.

Conclusions

In this paper, we have established a three-point tensile model consisting of a noncontact model and a capstan model of contact section, aiming to calculate the tension properties of the fabric sample to explore the factors affecting the quality of the fabric sample, including bending rigidity B, Poisson's ratio ν, extensibility, and friction. A comprehensive analysis of these factors has been implemented and the following conclusions are presented:

For the tension ratio Φ of the contact section, the radius ratio ρ, extensibility, and surface friction coefficient μ of the fabric sample increase the tension ratio by compounding the effect of the frictional modification of power-law friction ( $n = 0.67$ ). There is a coupling effect between them, and the friction coefficient has the most significant impact, while the extensibility has minimal impact. In addition, the effects of all three factors are reduced when frictional modification of the power law is adopted.

For the theoretical and experimental studies of the tension properties, the effects of bending rigidity B, surface friction coefficient μ, and friction law all clearly enhance the tensile slope and work, and the model including these three factors has high accuracy with the actual testing results. Furthermore, the Poisson's effect of error is less than 0.5% and can even be neglected under the power-law friction ( $n = 0.67$ ), and the simulated results of power-law friction are much better than those of linear friction ( $n = 1$ ).

In terms of guiding practical applications, this study can be applied to the characterization of fabric tensile properties and the design and production of fabrics. It is necessary to increase the surface roughness and the thickness of the fabric and reduce the extensibility, so as to obtain tight performance. The opposite is applicable for a loose performance.

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 was supported by Jiangxi Provincial Bureau for Quality and Technical Supervision (Grant no. GZJKY201807), Jiangxi Provincial Administration for Market Regulation (Grant no. GSJK201909), Natural Science Foundation Project of Shanghai 2020 “science and technology innovation action plan” (Grant no. 20ZR1400200), Fujian Provincial Key Laboratory of Textiles Inspection Technology (Fujian Fiber Inspection Bureau) of China (Grant no. 2018-MXJ-01) and the China Scholarship Council (Grant no. 201806630060).

ORCID iDs

Yi Sun

Zhaoqun Du

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