Effect of bending rigidity,Poisson’s ratio and surface friction of fabrics on the stretching step of the comprehensive handle evaluation system for fabrics and yarns

Abstract

The main topic investigated in the present paper is the analysis of the effect of bending rigidity, Poisson’s ratio and surface friction of fabrics on the stretching step of the comprehensive handle evaluation system for fabrics and yarns (CHES-FY) by using a theoretical tensile model. Simulated pulling-out force–displacement curves of the stretching step for several cases were investigated and compared with the experimental curves. The results showed that bending rigidity and surface friction of fabrics were the important factors that affected the test results of the stretching step. The slope of the simulated pulling-out force–displacement curve becomes larger by including the bending rigidity and surface friction of fabrics. However, the effect of Poisson’s ratio was small and was able to be neglected in the formulation of the model. In addition, the relationship between the bending indices, i.e. ω and k, and the diameter of the testing pins was discussed. It has been confirmed that the effect of the bending rigidity of fabrics on the pulling-out force–displacement curve strongly depended on the diameter of the testing pins. The model also detected that the surface friction effect became more remarkable for fabrics with a high tensile modulus.

Keywords

stretching step mechanical modeling bending rigidity Poisson’s ratio surface friction CHES-FY

The tensile property is one of the fundamental mechanical properties of a fabric, which determines the application and the quality of the fabric under low stress. Many theoretical investigations on the tensile property of fabrics have been carried out.^1–9 These theoretical models have been validated only by limited experimental studies and one may face difficulties in evaluating the tensile property of fabrics with different structure parameters. In order to better meet the requirements of end-use products, quantitatively measuring and characterizing the tensile behavior of fabrics in applications, especially for apparel applications, are necessary. The typical tensile tester was the Instron tensile instrument, which was used to test mechanical properties of fabrics based on the principles of constant rate of loading (CRL) and constant rate of extension (CRE).^10,11 The other two popular tensile apparatuses were the Kawabata’s evaluation system (KES) tensile tester¹² and the fabric assurance through simple testing (FAST) tensile meter.¹³ They were mainly used to test fabric tensile properties in a low stress state. The apparatuses mentioned above were designed to measure one pure property during one test; however, the actual application was that fabrics deformed under complex mechanical properties, including bending, friction, compression and stretching, and the comprehensive response of fabrics was not the simple linear summary of those properties. Therefore, it was difficult to obtain characteristic indices to measure effectively the comprehensive mechanical behavior of fabrics by using the above-mentioned apparatuses.

The in-situ complex measurement technology was presented to test the mechanical response behavior of fabric under complex mechanical properties. The corresponding research studies were conducted by Alley and McHatton,¹⁴ Pan and Yen,¹⁵ Strazdiene et al.,¹⁶ Wang et at.,¹⁷ and Liao et al.,¹⁸ wherein Pan¹⁹ developed a commercial tester, i.e. the PhabrOmeter fabric evaluation (PFE) system, to measure the complex response behavior of fabric, i.e. handle index, by constructing complex handle deformation. The method was effective and convenient to acquire the comprehensive properties of fabrics.¹⁹ For the PFE system, a sample was cut so that it was circular, and it was effective in evaluating the handle of camber samples. However, less research was reported concerning flat fabric that was often processed in a textile factory, such as winding fabric in the weaving process.

Therefore, the comprehensive handle evaluation system for fabrics and yarns (CHES-FY) was developed to study extracting deformation of flat fabric. The CHES-FY can measure weight, bending, friction and tensile/shear properties for both yarns and fabrics in-situ by a pulling-out test and by assessing the fabric’s handle.^20,21 The bending and weighting steps of the CHES-FY were investigated in previous papers.^22,23 The stretching step of the CHES-FY was analyzed by a corresponding tension model using the classical capstan equation by Du and Yu.^24,25 It was effective in estimating the tensile behavior of fabrics to a certain degree. However, the previous tension model is only suitable for an ideal situation where the fabric sample is considered as a perfectly flexible material of stationary thickness. For the CHES-FY, the pulling-out force–displacement curve of the stretching step is obtained when the sample is in a quasi-three-point bending state; thus, the bending rigidity of the sample may disturb the test results. In addition, the previous tension model cannot explain the difference between the theoretical results and the experimental results clearly. Therefore, this paper studied the tension model based on the CHES-FY by taking the bending rigidity and Poisson’s ratio, i.e. the strain in the thickness direction to the strain in the stretching direction, of fabrics into consideration so that the influencing degree of the bending rigidity, Poisson’s ratio and surface friction to the test results of the stretching step could be quantitatively analyzed.

Geometrical analysis

Analysis of the CHES-FY and modeling

The schematic structure of the CHES-FY, which is composed of a pulling pin controlled by a motor, a pair of jaws, bi-U-shaped pins, a sensor recording force and a digital camera capturing the sample profile, is illustrated in Figure 1. The radii of the circular cross section of both the pulling pin and the bi-U-shaped pins are set as the same value (r₂).

Figure 1.

Schematic structure of the comprehensive handle evaluation system for fabrics and yarns (CHES-FY).

By analyzing the deformation mechanism of the fabric, the testing process is divided into five steps, i.e. non-touching, weighting, bending, friction and stretching steps, as seen in Figure 2(a). In the first step, i.e. the non-touching step, the pulling pin moves up to reach the sample. The other four steps can be utilized to evaluate the weight, bending, surface friction and tensile properties of the fabric, respectively. In the present paper, only the stretching step of the CHES-FY is evaluated and discussed. It starts at point H when the sample is about to be stretched and then ends with the specified length of the sample or the stated maximum pulling-out force at point G under the stretching of the pulling pin, as shown in Figure 2(b).

Figure 2.

(a) The pulling-out process of the comprehensive handle evaluation system for fabrics and yarns (CHES-FY) with (b) the stretching step shown below.

Basic assumptions

Based on the tensile process of the CHES-FY, the following assumptions are made for the theoretical analysis of the stretching step:

The tensile property of the samples is assumed to be linear and elastic in the stretching test process, which is based on the fact that the sample is limited to small deformation; therefore, the stress–strain curve and the moment–curvature curve of the sample are also approximately linear.

The sample is assumed to be symmetric with respect to the midline GH (Figure 2). Therefore, only the right part is taken into consideration for the calculation, and it can be divided into five regions, i.e. two noncontact regions AO and CE and two contact regions AC and GE, as well as the held region in the jaw. Points E, C and A are the demarcation points between the noncontact region and the contact region. Point G (the middle point of the section EE’) is invariable relative to the pulling pin.

The sample between the pulling pin and the bi-U-shaped pins, i.e. section EC, is assumed to be approximately symmetric about the middle point S of the noncontact region EC. (See the Determination of the elongation of the fabric noncontact region with the pin section for more information.) The bending moment at the middle point S is zero.

Poisson’s ratio throughout the entire range of the contact region is assumed to be constant, so the average strain is used to replace the varying strain along the entire contact region.

The sample throughout the stretching process is assumed to be in an equilibrium state as the moving speed of the pulling pin is very small.

The effect of sample weight is negligible.

Based on the above assumptions, the sample elongation by theoretical formulas and the image method is calculated and combined with the theoretical tensile model of the stretching step of the CHES-FY.

Nomenclature

The parameters used in the derivation process are listed in Table 1.

Table 1.

Definition of parameters

Symbols	Meaning	Symbols	Meaning
T	Axial tension force	Q	Shear force
θ ₁	Right-part contact angle of pulling pin	θ ₂	Total contact angle of bi-U-shaped pins
M	Bending moment	ω	Inclined angle of resultant force
N	Normal force	r ₁	Half-thickness of sample
F_μ	Surface frictional force	r ₂	Radius of testing pin
ϕ	Contact angle	R	r₁ + r₂
ɛ	Axial strain of sample	k	r₂/r₁
A	Width of sample	μ	Surface frictional coefficient
E	Tensile modulus of sample	ν	Poisson’s ratio

Theoretical modeling

Determination of the elongation of the fabric contact region with the pin

Theoretical analysis of the contact region

For the complex actions of the shear force and bending moment, the resultant force acting on the sample has an inclined angle ω to the axis of the sample, which can be decomposed into the tension force component along the axial direction and the shear force component perpendicular to axial direction,²⁶ as seen in Figure 3(a). The force and moment equilibrium of any infinitesimal element dl of the sample neutral line in the contact region are illustrated in Figure 3(b) with the extension of the infinitesimal element from dl to dl'.

Figure 3.

(a) Force and moment analysis of two endpoints of the contact regions and (b) an infinitesimal element of the sample.

Based on the force equilibrium along the axial and tangent directions (n, t) of the sample and the moment equilibrium at point O in Figure 3(b), the following equations, according to the fifth assumption, are obtained

dT + Q d ϕ' - {dF}_{μ} = 0

(1)

dQ - T d ϕ' + dN = 0

(2)

dM - Q (r_{1}' + r_{2}) d ϕ' + r_{1}' {dF}_{μ} = 0

(3)

Considering the fourth assumption, the sample deformation in the contact region can be expressed as the following

dl' = (1 + ɛ) dl

(4)

dl = (r_{1} + r_{2}) d ϕ, dl' = (r_{1}' + r_{2}) d ϕ'

(5)

r_{1}' = r_{1} (1 - ν \bar{ɛ})

(6)

where ɛ is the axial strain of infinitesimal element, ν is the Poisson’s ratio of the sample and

\bar{ɛ}

is the average strain throughout the range of the contact region. By setting 1/EA = c,

\bar{ɛ}

is expressed in equations (7) and (8)

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

(7)

λ = \frac{1}{θ} \int_{0}^{θ} T d ϕ

(8)

According to the first assumption, $dM / d ϕ = 0$ . Equation (9) is obtained by substituting equations (2) and (3) into equation (1)

\frac{d 2 T}{d ϕ' 2} + \frac{1 - v λ c + k}{μ (1 - v λ c) d ϕ'} - \frac{k}{1 - v λ c} T = 0

(9)

By solving equations (4)–(6), we yield equation (10)

d ϕ' = \frac{(1 + cT) (r_{1} + r_{2})}{r_{1} (1 - v λ c) + r_{2}} d ϕ = κ (1 + cT) d ϕ

κ = \frac{1 + k}{1 - v λ c + k}

(10)

By assuming λ is known and by substituting equation (10) into equation (9), the differential equations can be acquired

\frac{d 2 T}{d ϕ 2} + \frac{κ (1 - v λ c + k)}{μ (1 - v λ c)} (1 + cT) \frac{dT}{d ϕ} - \frac{κ c}{1 + cT} (\frac{dT}{d ϕ}) 2 - \frac{k κ 2 (1 + cT) 2}{1 - v λ c} T = 0

(11)

Q = \frac{(1 - v λ c + k) (1 - v λ c)}{k (1 + k) (1 + cT)} \frac{dT}{d ϕ}

{dF}_{μ} = \frac{1 - v λ c + k}{k} dT, dN = \frac{1}{μ} {dF}_{μ}

(12)

Equations (11) and (12) are the governing equations for calculating the tension ratio when considering the effect of bending, Poisson’s ratio and surface friction in the contact region. In addition, the axial tension force ratio K(θ), the normal shear force ratio $Γ (θ)$ and the total tension force ratio $Φ (θ)$ are introduced

K (θ) = \frac{T (θ)}{T (0)}, Γ (θ) = \frac{Q (θ)}{Q (0)}, Φ (θ) = \sqrt{\frac{T 2 (θ) + Q 2 (θ)}{T 2 (0) + Q 2 (0)}}

(13)

Employing the fourth-order Runge–Kutta method and setting the value of parameter λ, equation (11) can be solved numerically based on the following boundary conditions (see Figure 3(a))

T (θ) = F_{1} \cos ω_{1}, U (θ) = \frac{F_{1} k (1 + k) (1 + {cF}_{1} \cos ω_{1})}{(1 - v λ c + k) (1 - v λ c)} \sin ω_{1}

(14)

where

U (ϕ) = dT (ϕ) / d ϕ

. Then, the solution of T should be used to iterate the value of λ until equation (8) is satisfied. When the solution is obtained,

K (θ),

Γ (θ)

and

Φ (θ)

are calculated from equations (12) and (13).

In case of zero Poisson’s ratio, i.e. ν is zero, the axial tension force ratio equation in equation (13) is described in equation (15)

K (θ) = \frac{T (θ)}{T (0)} = \frac{(α - β) \cos ω_{1}}{{(k \sin ω_{1} - β \cos ω_{1}) / e α θ + (α \cos ω_{1} - k \sin ω_{1}) / e β θ}}

(15)

The derivation process is described in the appendix. By letting $ω_{1} = 0$ and setting k → +∞, equation (15) is simplified as the classic capstan equation as follows

K (θ) = \frac{T (θ)}{T (0)} = e μ θ

(16)

Comparing equations (15) and (16), it is confirmed that the classic capstan equation applying to an entirely flexible material is a special case of equation (15) when ω = 0 and k → +∞. In other words, the effect of the bending rigidity of fabrics is governed by the values of ω and k in the equation. Therefore, we define the two parameter, ω and k, as bending indices to analyze the effect of the bending rigidity of fabrics.

Sample axial strain in the contact regions

The axial strain of the sample, ɛ, can be expressed from Figure 3(b) and the first assumption as

ɛ = \frac{T \cos (d ϕ / 2) - Q \sin (d ϕ / 2) + {dF}_{μ}}{EA} \approx \frac{T - Q (d ϕ / 2) + {dF}_{μ}}{EA}

(17)

Taking the limits of $d ϕ \to 0$ and ignoring the infinitesimal values, the simplified expression of the axial strain of the sample yields

ɛ = \frac{T}{EA}

(18)

Therefore, the sample axial strain is mainly determined by the axial tension force.

Elongation of the contact regions: CA and GE of the fabric

The force at point C is equal to the force at points E and E’ based on the second and third assumptions and Figure 4.

Figure 4.

Analysis of the elongation of the contact regions on (a) the bi-U-shaped pins and (b) the pulling pin.

To let the resultant force acting on point E be F₁ and its incline angle be $ω_{1}$ , the axial forces at points C and E, i.e. T_C and T_E, are expressed in equation (19)

T_{E} = T_{C} = F_{1} \cos ω_{1}

(19)

To calculate the elongation of the contact region AC, the sample with length s from points C to F is taken into consideration, and the corresponding subtending angle is ϕ, as seen in Figure 4(a).

The extension l_ds of the differential arc ds can be expressed as²⁴

l_{ds} = \frac{T}{AE + T} ds

(20)

It can be proved that equation (20) is also applicable to region CA and the noncontact regions. T is the axial tension force of the differential arc. For region CA, we obtain

T = F_{1} \cos ω_{1} / K (ϕ)

(21)

By substituting equation (21) into equation (20), the extension l_CA of arc AC by integration is given as follows

l_{CA} = \int_{0}^{θ_{2}} \frac{F_{1} \cos ω_{1}}{AE K (ϕ) + F_{1} \cos ω_{1}} R' d ϕ

(22)

where

R' = r'_{1} + r_{2}

. According to the first assumption,

\frac{F_{1} \cos ω_{1}}{AE K (ϕ)}

is small, and equation (22) is simplified into equation (23)

l_{CA} \approx \int_{0}^{θ_{2}} \frac{F_{1} \cos ω_{1}}{AE K (ϕ)} R' d ϕ

(23)

According to the same principle in calculating the elongation of contact region CA, the elongation of contact region GE is acquired in equation (24)

l_{GE} = \int_{0}^{θ_{1}} \frac{F_{1} \cos ω_{1}}{AE K (ϕ)} R' d ϕ

(24)

The derivation for the contact regions cannot directly calculate the values of F₁ and $ω_{1}$ , which should be determined from the noncontact region in the next section.

Determination of the elongation of the fabric noncontact region with the pin

Theoretical analysis of the noncontact region

Based on the force and moment analysis in Figure 3(a), the reactive forces F₁ and F₂ and the moments M₁ and M₂ are illustrated in Figure 5.

Figure 5.

Force and moment analysis of the noncontact regions.

The resultant force, F₁, in the noncontact region EC, is also divided into two forces (F₁₁ and F₁₂), wherein F₁₁ is along the Y vertical direction, and F₁₂ is along the X horizontal direction. From Figure 5, we obtain

ω_{1} = ϑ_{2} - ϑ_{1}

(25)

Where

ϑ_{1}

is the included angle of the resultant force F₁ and the vertical component force F₁₁, and

ϑ_{2}

is the included angle of the axial tension force at point E and the vertical component force F₁₁. It is necessary to point out that the direction of the vector of the resultant force F₁ at point E is related to the bending stiffness of the fabric. Based on Figure 5 and the third assumption, the moment and force equilibrium equations at point S are

F_{11} X_{ES} - F_{12} Y_{ES} - M_{1} = 0

(26)

F_{11} = \frac{1}{2} F *

(27)

F_{11} = F_{1} \cos ϑ_{1}, F_{12} = F_{1} \sin ϑ_{1}, M_{1} = \frac{B}{ρ}

(28)

where F^* is the extracting force sensed by the sensor connected with the pulling pin, M₁ is the bending moment at point E, X_ES is the horizontal distance from point E to point S, Y_ES is the vertical distance from point E to point S, and ρ is the radius of curvature of the sample. Since ρ = R′ at point E, we obtain equation (29) from equations (26)–(28)

ϑ_{1} = \arctan \frac{F * R' X_{ES} - 2 B}{F * R' Y_{ES}}

(29)

Then, angle $ω_{1}$ and the resultant force F₁ at the demarcation points E and C are

ω_{1} = ϑ_{2} - \arctan \frac{F * R' X_{ES} - 2 B}{F * R' Y_{ES}}

(30)

F_{1} = F * / 2 \cos (\arctan \frac{F * R' X_{ES} - 2 B}{F * R' Y_{ES}})

(31)

When the values of

ω_{1}

and F₁ are obtained,

K (θ)

and

Γ (θ)

can be used to calculate the axial force T_A and the shear force Q_A, respectively; moreover, angle

ω_{2}

and force F₂ can be expressed as follows

ω_{2} = \arctan (\frac{Q_{A}}{T_{A}}), F_{2} = \sqrt{T_{A}^{2} + Q_{A}^{2}}

T_{A} = F_{1} \cos ω_{1} / K (θ_{2}), Q_{A} = F_{1} \sin ω_{1} / Γ (θ_{2})

(32)

Elongation of the noncontact region EC

The noncontact region EC is shown in Figure 6(a).

Figure 6.

Theoretical analysis (a) for section EC and (b) for section AO in the noncontact region.

The axial tension force, T, for EC can be expressed as follows

T = F_{1} \cos (γ - ϑ_{1})

(33)

where γ is the included angle of the axial tension force and the vertical direction along the whole range of EC, and γ = ϑ₂ at point E. By assuming the locus of section EC by the function f(x), γ and ϑ₂ can be approximated as

γ = \arctan [f' (x)], ϑ_{2} = \arctan [f' (0)]

(34(a), (b))

By substituting equations (34(a), (b)) into equation (33), the tension force T is acquired as

T = F_{1} \cos {\arctan [f' (x)] - ϑ_{1}}

(35)

When substituting equation (35) into equation (20), the elongation of the sample in the noncontact region EC is integrated

l_{EC} = \int_{0}^{X} \frac{F_{1} \cos {\arctan [f' (x)] - ϑ_{1}}}{AE + F_{1} \cos {\arctan [f' (x)] - ϑ_{1}}} \times \sqrt{1 + [f' (x)] 2} dx

(36)

Equation (36) can be simplified as equation (37) by the first assumption

l_{EC} = \int_{0}^{X} \frac{F_{1} \cos {\arctan [f' (x)] - ϑ_{1}}}{AE} \sqrt{1 + [f' (x)] 2} dx

(37)

where X is the abscissa of point C, and it has the following relationship

X = h + 2 R' \cos θ_{1}

(38)

Elongation of the noncontact region AO

Figure 6(b) represents the axial force of section AO, and the function g(x) is utilized to fit the locus of section AO. The axial tension force, T, for AO can be expressed as

T = F_{2} \cos {arctan [g' (x)] + ϑ_{3}}

(39)

where

ϑ_{3} = ω_{2} - (π / 2 - θ_{2} + θ_{1})

. Using the same principle for calculating section EC to set the abscissa of point A as

X' = h_{0} + R' \cos (θ_{2} - θ_{1})

, the elongation of section AO is obtained as

l_{AO} = \int_{0}^{X'} \frac{F_{2} \cos {arctan [g' (x)] + ϑ_{3}}}{AE} \sqrt{1 + [g' (x)] 2} dx

(40)

Determination of the elongation of the fabric by the image method

Elongation of the noncontact regions EC and AO

The basic process to determine the elongation of the fabric using the image by image method is shown in Figure 7.

Figure 7.

The flow chart of the image process system for the noncontact region.

The cubic spline function s(x) is used to fit a sample shape by n data points from the fabric image of the noncontact region.²⁷ The noncontact part of the fabric is then divided into n−1 sections, and the function s_i(x) of each section can be generally expressed as equation (41)

s_{i} (x) = M_{j} \frac{(x_{j + 1} - x) 3}{6 h_{j}} + M_{j + 1} \frac{(x - x_{j}) 3}{6 h_{j}} + (y_{j} - \frac{M_{j} h_{j}^{2}}{6}) \frac{x_{j + 1} - x}{h_{j}} + (y_{j + 1} - \frac{M_{j + 1} h_{j}^{2}}{6}) \frac{x - x_{j}}{h_{j}}, i = 1, 2, \dots, n - 1; j = 0, 1, \dots, n - 1

(41)

where M_j (j = 0, 1, … , n) and h_j (j = 0, 1, … , n−1) are constants calculated by the value of the coordinates of the n data points. Splines are n − 1 piecewise polynomials, s_i(x), that are smoothly connected together. The length of the spline curve, L_s₍ _x ₎ is

L_{s (x)} = \sum_{i = 1}^{n - 1} \int_{x_{i - 1}}^{x_{i}} \sqrt{1 + [s'_{i} (x)] 2} dx, i = 1, 2, \dots, n - 1

(42)

When M_j (j = 0, 1, … , n) and h_j (j = 0, 1, … , n−1) are obtained, the function of spline s(x) and its first-order derivative function s’(x) can be solved by substituting M_j and h_j into equation (41). The lengths of sections EC and AO are calculated according to equation (42) as follows

L_{EC} = \sum_{i = 1}^{n - 1} \int_{X_{i - 1}}^{X_{i}} \sqrt{1 + [f'_{i} (x)] 2} dx

(43)

L_{AO} = \sum_{i = 1}^{n - 1} \int_{X'_{i - 1}}^{X'_{i}} \sqrt{1 + [g'_{i} (x)] 2} dx

(44)

Elongation of the noncontact regions GE and AC

The demarcation point between the contact region and the noncontact region is obvious after extracting the noncontact regions, and the wrapping angles θ₁ and θ₂ are measured from the sample image. According to Figure 4, the lengths of the contact regions under tension are acquired as follows

L_{GE} = R' θ_{1}

(45)

L_{AC} = R' θ_{2}

(46)

Determination of the total elongation of the fabric

According to the above analysis and the second assumption, the total elongation l_oo′ of the sample with original length L₀ by the image method is expressed as follows

l_{oo'} = 2 (L_{EC} + L_{AO}) + 2 R' θ_{1} + 2 R' θ_{2} - L_{0}

(47)

where L_EC and L_AO are the lengths of EC and AO after tensile deformation and can be calculated from equations (43) and (44). Then, by combining the total elongation equations derived by the mechanical and image method, we obtain

L_{EC} + L_{AO} + R' θ_{1} + R' θ_{2} - \frac{L_{0}}{2} = l_{CA} + l_{GE} + l_{EC} + l_{AO}

(48)

By submitting equations (23), (24), (37) and (40) into equation (48), the theoretical tensile model is acquired as follows

L_{EC} (h) + L_{AO} (h) + R' θ_{1} + R' θ_{2} - \frac{L_{0}}{2} = \int_{0}^{θ_{2}} \frac{F_{1} \cos ω_{1}}{AE K (ϕ)} R' d ϕ + \int_{0}^{θ_{1}} \frac{F_{1} \cos ω_{1}}{AE K (ϕ)} R' d ϕ + \int_{0}^{h + 2 R' \cos θ_{1}} \frac{F_{1} \cos {arctan [f' (x)] - ϑ_{1}}}{AE} \times \sqrt{1 + [f' (x)] 2} dx + \int_{0}^{h_{0} + R' \cos (θ_{2} - θ_{1})} \frac{F_{2} \cos {arctan [g' (x)] + ϑ_{3}}}{AE} \times \sqrt{1 + [g' (x)] 2} dx

(49)

where

F_{1} = F * / 2 \cos (\frac{π}{2} - θ_{1} - ω_{1}), F_{2} = \sqrt{(\frac{F_{1} \cos ω_{1}}{K (θ_{2})}) 2 + (\frac{F_{1} \sin ω_{1}}{Γ (θ_{2})}) 2}

Based on the theoretical tensile model, the pulling-out force–displacement curve of the tensile process can be simulated and the effect of bending, Poisson’s ratio and surface friction can be analyzed.

Experimental details

Sample preparation

Four kinds of cotton plain-woven fabrics with different specifications were selected from a cotton mill for model validation. Each fabric was cut as five pieces of sample with the warp × weft size as 20 cm × 2 cm for the CHES-FY testing and five pieces of sample with the warp × weft size as 20 cm × 20 cm for the Kawabata evaluation system for fabrics (KES-F) testing, respectively. All samples were conditioned at 20 ± 2℃ and 65 ± 3% relative humidity for longer than 24 h. The specifications for the four fabrics are listed in Table 2, where B, μ, E and ν are the bending rigidity, surface friction coefficient, tensile modulus and Poisson’s ratio, respectively. The bending rigidity, surface friction coefficient and tensile modulus are measured by the KES-F system. Poisson’s ratio is set as 0.35, the typical Poisson’s ratio for isotropic solid materials.²⁸ This choice seems a little arbitrary, but it can reflect the primary influencing degree of Poisson’s ratio to the stretching step of the CHES-FY system in the absence of an appropriate testing method.

Table 2.

Basic specifications for the four types of samples

Sample	Weight (g/m²)	Warp yarn (tex)	Weft yarn (tex)	Number of warp threads (/10 cm)	Number of weft threads (/10 cm)	Thickness (mm)	B (cN·cm)	μ	E (N/cm·%)	ν
1	261	78	60	286	152	0.252	0.044	0.210	0.45	0.35
2	312	148	74	197	130	0.708	0.077	0.283	0.41	0.35
3	487	260	132	130	102	1.021	0.129	0.267	0.90	0.35
4	395	160	98	175	116	0.614	0.078	0.260	0.53	0.35

Apparatus settings

The basic parameters of the CHES-FY are shown in Figure 8. The horizontal and vertical distances from the curvature centers of the bi-U-shaped pins to the jaws (S₀ and h₀) are set as 30 mm and 60 mm, respectively; the horizontal distance of the curvature centers of the bi-U-shaped pins and the pulling pin (S₁) is 8 mm; the diameters of the pulling pin and the bi-U-shaped pins are 6 mm; the vertical distance from the hanging point of the jaws to the pulling pin is 70 mm at the beginning of the test, which can be obtained by the data acquisition system connected with the computer during the testing. The vertical speed of the pulling pin is 0.2 mm/s; the sampling frequencies of the analog to digital convertor and the digital camera are set as 100 Hz and 10 Hz, respectively. In the testing, the sample gage length between the two jaws is 18 cm.

Figure 8.

Basic parameters of the comprehensive handle evaluation system for fabrics and yarns (CHES-FY).

Parameters for modeling

Tensile modulus E

The tensile property is characterized in the “V” region, i.e. the stretching step throughout the pulling-out force–displacement curves of the samples (see Figure 9). When the sample is about to be stretched, an extracting force is acting on the pulling pin, and it becomes the preload in the stretching step. Therefore, only a fraction (not from 0 N to the maximum force) of the stress–strain curves from the Kawabata evaluation system for fabrics tensile tester (KES-F1) was used. The tensile modulus is obtained by fitting the linear slope of the effective fraction (from 1 N, i.e. the stated pre-tension of the CHES-FY, to the maximum force, i.e. the stated maximum pulling-out force, 350 cN).

Figure 9.

The entire pulling-out force–displacement curves of the samples (regions I, II, III, IV and V represent non-touching, weighting, bending, surface friction and stretching steps, respectively).

Stretched sample length

By selecting points E and O as the origin points for section EC and AO, respectively, we can obtain the spline curve of the five feature data points from the interpolation algorithm. The length of the spline curve is computed based on equation (42). The typical sample images captured by the digital camera, together with the spline curve and the connecting line between the tangency points E and C (i.e. dash line EC) and points O and A (i.e. dash line AO), are shown in Figure 10.

Figure 10.

Typical sample images captured by digital camera for sections (a) EC and (b) AO.

We can find that the buckling of the sample is small in the noncontact region. Based on the geometry analysis in Figure 8, the estimated lengths of lines EC and AO are calculated by the Pythagorean theorem in RtΔPO₁O₂ and RtΔAO₁O, respectively, and written as

{\overset{\land}{l}}_{EC} = \sqrt{h 2 + s_{1}^{2} - (2 R) 2}

(50)

{\overset{\land}{l}}_{AO} = \sqrt{h_{0}^{2} + s_{0}^{2} - R 2}

(51)

The estimated length from equations (50) and (51) is consistent with that of the spline curve; the relative errors are less than 2%. Therefore, the stretched sample length in the noncontact region can be estimated by the dash line. Then, g(x) and f(x) are also simplified into a linear equation.

The first-order derivative of the spline curve is computed by a MATLAB program, and f′(x) and g′(x) can be acquired from four quadratic polynomials. The typical curves of f′(x) and g′(x) are illustrated in Figure 11.

Figure 11.

First-order derivative graph of the spline curves for (a) EC and (b) AO.

Results and discussion

Comparisons of the pulling-out force–displacement curves with the effects of four different factors, including the bending rigidity, Poisson’s ratio and surface friction of fabrics have been conducted between the experimental curves from the CHES-FY and from the simulated curves from the theoretical model. Four simulated cases are set as follows:

Case 1: with bending rigidity, Poisson’s ratio and surface friction effect;

Case 2: with bending rigidity and surface friction effect (setting $ν = 0$ );

Case 3: with surface friction effect only (setting $ν = 0$ , $ω = 0$ ; k → +∞);

Case 4: no bending rigidity, Poisson’s ratio and surface friction effect (setting $ω, μ, ν \to 0;$ k → +∞).

By substituting the basic property parameters of the samples, i.e. the tensile modulus, bending rigidity, Poisson’s ratio, surface friction coefficient, original length, width and apparatus parameters into equation (49), the simulated curves of the theoretical tensile model are calculated. The experimental curves and the simulated curves are shown in Figure 12.

Figure 12.

Comparison of the simulated and experimental pulling-out force–displacement curves of samples 1(a), 2(b), 3(c) and 4(d).

The linear fitted slope (calculated by the least square method) and the pulling work (the area under the pulling-out force–displacement curve) are calculated for both the experimental curves and the simulated curves, which are shown in Table 3.

Table 3.

Tension parameters from the simulated and experimental curves

			Simulated curve
	Experimental curve		Case 1		Case 2		Case 3		Case 4
Sample	Fitting Slope	Pulling Work (N·mm)	Slope	Pulling work (N·mm)	Slope	Pulling work (N·mm)	Slope	Pulling work (N·mm)	Slope	Pulling work (N·mm)
1	0.405	9.680	0.369	9.536	0.367	9.512	0.299	8.682	0.199	7.413
2	0.360	9.417	0.351	9.309	0.348	9.275	0.272	8.400	0.164	7.030
3	0.943	15.403	0.819	15.161	0.815	15.118	0.581	12.262	0.379	9.476
4	0.476	10.748	0.451	10.561	0.445	10.487	0.344	9.298	0.218	7.664

For comparing the deviation rate (D.R.) of the slope and the pulling work from case_a to case_b, the following expression is given

D . R . = \frac{S_{{case}_{a}} - S_{{case}_{b}}}{S_{{case}_{a}}}

(52)

Comparison of the theoretical and experimental results

It can be seen from Figure 12 that the experimental curve shows a reasonable agreement with that of the simulated results in case 1. It indicates that the pulling-out force–displacement curve from the theoretical tensile model considering the effects of bending rigidity, Poisson’s ratio and surface friction can accurately predict the experimental curve. Referring to the slope and the pulling work of experimental curve and the simulated curve in case 1 (see Table 3), the largest relative error between the experimental curve and the simulated curve is not more than 13.1% (i.e. (0.943 − 0.819)/0.943). It shows that the theoretical tensile model covers the primary influencing factors which affect the pulling-out force–displacement curve of the stretching step. The basic parameters included in the theoretical model, e.g. the apparatus setting parameters and the original sample length and width, are easily controlled. Therefore, the basic property parameters of the samples and their influence on the characterization of the tensile property of the fabrics in the stretching step are worth investigating.

Effect of the surface friction

Comparison results of the curves for case 3 and case 4 show that both the slope and the pulling work of the simulated curves increase when the effect of surface friction is considered (see Figures 12 and 13 as well as Table 3). In other words, this increase occurs because a greater pulling-out force is required to overcome the friction between the fabric and the testing pins for the same extension of the sample with the friction effect in comparison with the sample without the surface friction effect. It is obvious from Figure 12 that the effect of the surface friction is remarkable, and the simulated curve in case 3 is closer to the experimental curve compared with the simulated curve in case 4. The minimum deviation rate of the slope and the pulling work of the simulated curve from case 3 to case 4 are more than 33.4% and 14.6% (calculated by equation (52)), respectively. It also reveals that the surface friction effect is supposed to be noticed in the stretching step of the CHES-FY. The slope of the simulated curve for the different surface friction coefficients and the different tensile moduli are computed by substituting equations (50) and (51) into the theoretical tensile model, and the increased ratio of the slope is also calculated. Both of these items are shown in Figure 13. It shows that when the surface friction coefficient of the fabrics is larger, the effect of the surface friction on the slope of the simulated curve is greater, which is a natural conclusion. From the increased ratio of the slope, the typical results are that the effect of the surface friction on the slope of the simulated curve becomes larger as the tensile modulus increases.

Figure 13.

Effect of the surface friction coefficient on the slope of the simulated curve at different tensile moduli.

Effect of the bending rigidity

It can be seen from Figure 12 that the slope of the simulated curve in case 2 is larger than the slope in case 3, which shows that the slope of the curve also increases when the bending rigidity of the samples is taken into consideration. By referring to Table 3, we can see that the maximum and minimum deviation rate of the slope and the pulling work of the simulated curve from case 2 to case 3 are about 28.7% and 18.5%, and 18.8% and 8.7% (calculated by equation (52)), respectively, among the fabrics where the bending rigidity range is 0.044–0.129 cN·cm. It shows that the bending rigidity of the fabrics is the important factor affecting the results of the stretching testing of the CHES-FY. This result can be explained from the analysis of the force in Figure 3, because the tension along the axial direction is only a component force of the resultant force that is regarded as the axial direction tension for a completely flexible fabric. The slope of the simulated curve in case 2 is much closer to the experimental curve than the slope in case 3, and the relative error of the slope is less than 13.5% (ranging from 2.3–13.5% for the different samples). It reveals that the theoretical model including the bending effect is more accurate than the one only containing the surface friction effect.

Figure 14 shows the effect of k on the slope of the simulated curve. It is interesting that the slope of the simulated curve increases first and then decreases as k increases, which is not completely consistent with Jung et al.’s²⁹ analysis. This situation is due to that fact that R is also included in our theoretical tensile model, and this parameter, R, generally becomes larger as k increases, resulting in a relatively prominent effect of R when k is small, which makes a slight increase in the slope of the simulated curve at the beginning. From the analysis of equations (15) and (16), we find that when k becomes extremely large, the effect of the bending rigidity is negligible. It is clear that the effect of the bending rigidity of fabrics can be lessened by selecting testing pins with a large diameter as we expected.

Figure 14.

The relationship between the slope of the simulated curve and k.

Figure 15 shows the effect of ω on the simulated curve of the stretching step. If we take $ω_{1}$ as an example to analyze the effect of ω, the tendency is that the slope of the simulated curve increases when the value of ω increases. From equation (30), there is a shallow inverse relationship between ω and R. Therefore, the testing pins with a larger diameter can also lessen the value of ω in order to reduce the effect of the fabric’s rigidity.

Figure 15.

The relationship between the slope of the simulated curve and ω.

Effect of Poisson’s ratio

Tension causes a change in the fabric thickness in addition to the elongation in the axial direction for Poisson’s effect. Based on the analyses of equations (4)–(8), the deformation of the fabric thickness mainly affects the value of k and R. Therefore, the effect of Poisson’s ratio can be analyzed by the two parameters. For the four different samples, the effect of R is relatively prominent as the slope of the curves is greater when taking Poisson’s ratio into consideration. In order to analyze further the effect of Poisson’s ratio on the pulling-out force–displacement curve of the stretching step, three values of ν (0, 0.3 and 1) are chosen.

It can be seen from Figure 16 that for the fabric with a tensile modulus of 0.3 N/cm·% and a surface friction coefficient of 0.3 the slope of the pulling-out force–displacement curve increases with ν; it increases from 0 to 1. However, the deviation of the slope is not as large as expected. This difference is even negligible. It is also shown from Figure 12 that the effect of Poisson’s ratio is much smaller than that of the bending and surface friction. By referring to equation (52) and Table 3, we find that the maximum deviation rate of the slope from case 1 to case 2 is 1.3%, and the maximum deviation rate of the pulling work is 0.7%, which also demonstrates that the effect of Poisson’s ratio is negligible. The slope of the experimental curve is close to the curve of case 1, and the maximum deviation rate is 13.1%. However, the curve of case 2 is also close to the experimental curve. Its deviation compared with the experimental curve is also small (not more than 13.9%), which illustrates that the model with the bending and surface friction is enough to accurately simulate the pulling-out force–displacement curve of the fabric.

Figure 16.

Pulling-out force–displacement curves from the theoretical tensile model with different Poisson effect.

Conclusions

The paper established a theoretical tensile model to investigate the effects of bending rigidity, Poisson’s ratio and surface friction of fabrics on the stretching step of the CHES-FY for four cases. The results show that the bending rigidity and the surface friction of the fabrics are the important factors affecting the test results of the stretching step. Both the bending rigidity and the surface friction effects enhance the slope of the simulated pulling-out force–displacement curve. In addition, the surface friction effect becomes more remarkable for fabrics with a high tensile modulus. However, the effect of Poisson’s ratio is very small and almost negligible because the deviation rate of the slope of the simulated curve generated when considering Poisson’s effect is not more than 1.3% in general. Therefore, good results can be achieved by only considering the effect of bending rigidity and surface friction.

Moreover, comparisons of the simulated and experimental pulling-out force–displacement curves indicate that the theoretical tensile model has covered the primary factors related to the stretching step of the CHES-FY, for the maximum error of the slope between the simulated and experimental pulling-out force–displacement curves is not more than 13.1%. In addition, the theoretical tensile model can be used in reverse to calculate the tensile modulus, E, of the fabric based on its pulling-out force–displacement curve measured from the CHES-FY more accurately by eliminating the effect of bending rigidity, surface friction and Poisson’s ratio of the fabrics.

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 a National Natural Science Foundation of China project (Grant 11272086 and 51203022), supported by the Fundamental Research Funds for the Central Universities (2232014A3-02) and supported by the Donghua University Distinguished Young Professor Program (B201307). The authors are also grateful to the China Scholarship Council (CSC) for the 12-month scholarship provided to one of the authors (Sun).

Appendix

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