Analysis of a quasi-three-point bending test for fabrics with friction and extensibility effect

Abstract

This paper presents a theoretical bending model to investigate the quasi-three-point bending step of the comprehensive handle evaluation system for fabrics and yarns (CHES-FY). The effect of the friction and extensibility of fabrics on the bending step of the CHES-FY is discussed based on the theoretical bending model in three cases, and bending parameters, including maximum bending force, linear fitting slope and bending work, are featured from the bending force–displacement curves of each case. Comparisons of the theoretical and experimental results were also conducted to validate the model. The results revealed that the friction effect tended to enhance the bending parameters of the bending force–displacement curve of the bending step, and the effect of the friction could be remarkable at low bending rigidity of samples. However, the effect of the extensibility of samples was almost negligible for the bending test of wear fabrics, as the maximal relative error was less than 11.7% for fabrics where the tensile elastic constant was higher than 0.1 N/cm·%. This work can also provide a theoretical guidance for improving the measurement accuracy of the three-point bending test.

Keywords

quasi-three-point bending friction extensibility comprehensive handle evaluation system for fabrics and yarns theoretical model

The measurement of the bending property of fabrics can be used to predict the handle, comfort and aesthetic characteristics, as well as the application of a fabric, and it has received considerable attention from researchers. A classic characterization of the bending property is the bending rigidity, from a cantilever method first proposed by Peirce.¹ Subsequently, a series of development of the measurement methods for the bending property of fabrics was reported.^2–6 Kocil et al.⁷ employed an Instron tensile apparatus to measure the critical maximum buckling force of a fabric to characterize bending rigidity. Sun⁸ developed a new tester measuring the bending length of a cross-shaped specimen to assess the fabric bending stiffness in both warp and weft directions in one test. Ghane and Hedayati⁹ used a fixed-support beam method to calculate the bending rigidity of fabrics based on the slope of a fitted line. In addition, two popular bending testers, the Kawabata Evaluation System (KES) bending tester and the Fabric Assurance by Simple Testing (FAST) bending meter, have also been reported.^10,11 They were widely used to test the bending property of fabrics under the low stress state. However, the instruments mentioned above were developed to test only one property in one test. It is difficult for these apparatuses to meet the actual application where the fabrics deformed under multiple mechanical actions simultaneously covering bending, friction, stretching and compression properties, as the comprehensive behavior of the fabric was not the simple linear summary of those mechanical properties.

Other attempts have shown effective methods using simultaneous measurement to evaluate the mechanical behaviors of the fabric under complex mechanical action.^12–16 The best known of these methods includes the ring test, and a commercial instrument, that is the PhabrOmeter Fabric Evaluation System, has been developed by Pan.¹⁷ This system featured the handle index by pulling a circle fabric sample through a circle orifice to measure the comprehensive property of fabrics. It successfully measured the comprehensive handle index by the membrane theory and effectively featured multiple mechanical properties by the statistical algorithm.¹⁸

Recently, the comprehensive handle evaluation system for fabrics and yarns (CHES-FY) was developed. The CHES-FY was designed based on a pulling-out test whereby the weight, bending, friction and tensile/shear properties for both fabrics and yarns can be measured simultaneously,¹⁹ and the fabric handle can also be assessed using the pulling-out force–displacement curve.²⁰ The testing process and apparatus parameters of the CHES-FY were discussed based upon experimental and theoretical analysis,^21,22 and the factors in the testing of the weighting and stretching steps have been investigated in previous work.^23,24 For the bending step, it approximates to a three-point bending test that has become an attractive approach to evaluating the bending property of various materials.^25–27 Some corresponding bending models were developed by analyzing the three-point bending process to calculate the bending rigidity in terms of the deflection of the measured materials.^28–30 However, less work has been found focusing on the investigation into the effect of friction and extensibility of measured samples on the three-point bending test. For flexible elastic materials, such as fabrics and yarns, and the bending test under the low stress state, the effect of the friction between samples and the three restraint points, as well as the extension of samples during the test, become relatively large with respect to the bending force. In a previous bending model of the CHES-FY, the friction and extension of samples were ignored, which in some cases may disturb the test results for characterizing the bending property.

Therefore, this paper aimed to develop a theoretical bending model based on the quasi-three-point bending process of the CHES-FY by including the friction and extension factors so that the influencing degree of the friction and extensibility on the results of the bending process could be analyzed mathematically.

Modeling analysis

In order to analyze the bending process, it is necessary to give a brief description of the CHES-FY. The basic structure of the CHES-FY is schematically illustrated in Figure 1. It is composed of a pulling pin, bi-U-shaped pins, a pair of jaws, a sensor recording the pulling-out force and a digital camera capturing the sample profile.

Figure 1.

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

The pulling pin can move up and down with the control of a micro-step motor to pull a sample through the interval of the bi-U-shaped pins. When a sample undergoes complex morphological deformation, the pulling-out force is simultaneously recorded by the sensor connected with the pulling pin. By analyzing the geometry morphology of the deformed fabric, multiple physical and mechanical properties of the sample can be calculated in a pulling-out test. The testing process and the corresponding pulling-out force–displacement curve are shown in Figure 2(a) and (b), where it can be seen that the whole testing process can be divided into four steps, that is, the weighting, bending, friction and stretching steps. They can be used to characterize the weight, bending, friction and tensile properties of the fabric, respectively. In this work, the bending step, which starts at G₀ when the sample is lifted up by the pulling pin to just touch the bi-U-shaped pins and ends at the maximum bending force (MBF; see Figure 2(c)), is investigated and discussed.

Figure 2.

The pulling-out process (a), the corresponding pulling-out force—displacement curve (b) and the bending step (c) of the comprehensive handle evaluation system for fabrics and yarns (CHES-FY).

When a sample bends under the interaction of the pulling pin and bi-U-shaped pins, there exists friction between the sample and the bi-U-shaped pins as well as the pulling pin. As a result, the sample may also generate a possible extension due to the friction and pulling-out force during the bending step. Before conducting the theoretical analysis of the bending step, the following assumptions are made based on the bending process of the CHES-FY, and the main parameters used in the theoretical derivation are listed in Table 1.

The sample is assumed to be symmetric with respect to the midline G₀G₁ (see Figure 2) under the bending action of the pulling pin and bi-U-shaped pins. Therefore, only the right-hand and part, that is, section G₁O of the entire sample is analyzed for modeling, and it can be divided into four segments, that is, two contact segments G₁D and CA and two noncontact segments DC and AO. The points D, C and A are the demarcation points between the contact segment and the noncontact segment.

The sample between the bi-U-shaped pins is assumed as a horizontal line at the beginning of the bending step, and the section DC of the sample is approximately symmetric about the middle point S of the noncontact segment DC during the whole bending process (see Figure 2(c)). Therefore, the bending moment at point S is assumed to be zero. This assumption is based on the observation of a preliminary test, as shown in Figure 3, where the basic parameters of the CHES-FY are optimized for bending modeling.

The samples are assumed to be homogeneous flexible materials with a finite bending stiffness, and thus the effect of the interweaving structure is ignored in this work. The material property of samples is assumed to be linearly elastic, and the extension of the samples in the entire noncontact segment is assumed to be uniform.

The effect of the weight, compressibility and Poisson’s ratio of samples is negligible during the bending process.

Throughout the bending process, the samples are assumed to be in an equilibrium state, as the pulling-out speed of the pulling pin is very small.

The jaws of the CHES-FY consist of a rotational pedestal, and thus the bending moment at point O is small and can be negligible.

Table 1.

Definition of parameters

Symbols	Meaning	Symbols	Meaning
F	Resultant force	M	Bending moment
t	Width of the sample	B	Bending rigidity
γ	Deflection angle along sample	ω	Inclined angle of resultant force
r ₁	Half-thickness of sample	r ₂	Radius of curvature of cylindrical testing pin
η	r₂/r₁	R	r₁+r₂
μ	Friction coefficient	E_c	Tensile elastic constant
T	Axial tension force	Q	Shear force
F*	Bending force in bending step	h	Displacement of pulling pin in bending step
H	Displacement of pulling pin in the entire pulling-out process	γ	Deflection angle of sample in noncontact region
θ	Wrap angle in contact region	ϕ	Endpoint deflection angle

Figure 3.

The sample shape of the quasi-three-point bending process at different pulling-out displacements h.

Based on the above assumptions, the theoretical model for quasi-three-point bending can be derived by combining the equivalence of the sample length from mechanical analysis and its original length hanging between the two jaws.

Mathematical derivation

Analysis of the sample noncontact segment with the pin

Referring to the first and the second assumptions, the sections DS and AO of the noncontact segments are segregated for calculation. The bending moment at points S and O, according to the second and sixth assumptions, is zero. Therefore, the deformed configuration and force and moment equilibrium, as well as an infinitesimal element dl of the sample in the noncontact segment, are shown in Figure 4 with the extension of the sample increasing from the original length $L'$ (the dashed curve) to the extended length L (solid curve).

Figure 4.

The configuration (a) and an infinitesimal element (b) of the extended sample in force and moment equilibrium in the noncontact segment.

As seen in Figure 4, the resultant force, F, acting on the sample has an inclined angle ω to the initial axis of the sample due to the complex actions of the shear force and bending moment,³¹ and it can be decomposed into two components that are along the x-axis and y-axis, respectively. Assuming the coordinate of the endpoint as (X,Y) and using the moment equilibrium at point P, the following equation is obtained

M (l) = F (Y - y) \cos ω - F (X - x) \sin ω

(1)

where l is the arc length of the sample profile from the endpoint (see Figure 4(a)). Referring to the Bernoulli–Euler law and the geometric relations shown in the infinitesimal element, we get

\frac{d M (l)}{dl} = - tB \frac{d 2 γ}{dl 2} \frac{dx}{dl} = \cos γ, \frac{dy}{dl} = \sin γ

(2)

Differentiating equation (1) with respect to dl and substituting equation (2) into equation (1) gives

κ = L \sqrt{\frac{F}{tB}}, ρ = \frac{l}{L}

(4)

From Figure 4 and the second assumption, the following boundary conditions are given

ρ = 0, γ = ϕ, \frac{d γ}{d ρ} = 0

ρ = 1, γ = 0

(5)

Integrating equation (3) with respect to γ and submitting the boundary conditions yields

(\frac{d γ}{d ρ}) 2 = 4 κ 2 \cos 2 \frac{ω - ϕ}{2} (1 - \frac{1}{\cos 2 \frac{ω - ϕ}{2}} \cos 2 \frac{ω - γ}{2})

(6)

Introducing two new variables which meet

k = \cos \frac{ω - ϕ}{2}, k \sin ψ = \cos \frac{ω - γ}{2}

(7)

Equation (6) is simplified as

d ρ = \frac{- d ψ}{κ \sqrt{1 - k 2 \sin 2 ψ}}

(8)

The boundary conditions are transformed as follows

ρ = 0, γ = ϕ, ψ = \frac{π}{2}

ρ = 1, γ = 0, ψ = ψ_{0}

(9)

ψ_{0} = \arcsin (\frac{\cos (ω / 2)}{\cos [(ω - ϕ) / 2]})

It should be pointed out that the value of ω is generally greater than that of ϕ. Integrating equation (8) based on the transformed boundary conditions and submitting equation (4), we get

κ = L \sqrt{\frac{F}{tB}} = \int_{ψ_{0}}^{π / 2} \frac{d ψ}{\sqrt{1 - k 2 \sin 2 ψ}} = F (k, \frac{π}{2}) - F (k, ψ_{0})

(10)

where

F (k, ψ)

is the Legendre elliptic integrals of the first kind. From the force analysis of the infinitesimal element in Figure 4, the axis strain of the sample in the noncontact segment can be defined as

ɛ = \frac{T \cos (d γ / 2) - Q \sin (d γ / 2)}{{tE}_{c}}

(11)

Taking the limits of $d γ \to 0$ yields

ɛ \approx cT = cF \cos (ω - γ) c = \frac{1}{{tE}_{c}}

(12)

It can be proved that equation (12) is also applicable to the sample contact segment with the pin. Referring to the third assumption and based on equation (12), the elongation, $Δ l$ , of the sample in the noncontact segment can be expressed as follows

Δ l = L - L' = L (\frac{\bar{ɛ}}{1 + \bar{ɛ}}) \bar{ɛ} = τ ɛ_{0}, ɛ_{0} = cF \cos (ω - ϕ)

(13)

Thus, the primary task of analyzing the extension in the noncontact segment is to determine the value of the parameter τ connecting the endpoint strain, $ɛ_{0}$ , and average strain, $\bar{ɛ}$ . In the case where the deflection of the sample is small, we have $τ = 1$ , which means the strain throughout the noncontact segment depending on the axis force acting on the terminal point. This actually expresses the maximal effect of the extensibility of samples.

Bearing in mind that segment DC is symmetrical about point S, the length L_DC of the sample with extension in the noncontact segment DC can be obtained from equation (10) as

L_{DC} = 2 (F (k_{1}, \frac{π}{2}) - F (k_{1}, ψ_{0_{1}})) \cdot \sqrt{\frac{tB}{F_{1}}}

(14)

where F₁ represents the resultant force acting at point S, and it equals the value of the resultant force acting at point D according to the force equilibrium.

The elongation of the noncontact segment DC can also be obtained from equations (10) and (13) as

Δ l_{DC} = 2 τ (F (k_{1}, \frac{π}{2}) - F (k_{1}, ψ_{0_{1}})) \cdot (\frac{F_{1} \cos (ω_{1} - ϕ_{1})}{tB + F_{1} \cos (ω_{1} - ϕ_{1})}) \sqrt{\frac{tB}{F_{1}}}

(15)

Similarly, the length of the sample in the noncontact segment AO can be acquired in equation (16)

L_{AO} = (F (k_{2}, \frac{π}{2}) - F (k_{2}, ψ_{0_{2}})) \cdot \sqrt{\frac{tB}{F_{2}}}

(16)

and the elongation of segment AO is expressed as

Δ l_{AO} = τ (F (k_{2}, \frac{π}{2}) - F (k_{2}, ψ_{0_{2}})) \cdot (\frac{F_{2} \cos (ω_{2} - ϕ_{2})}{tB + F_{2} \cos (ω_{2} - ϕ_{2})}) \sqrt{\frac{tB}{F_{2}}}

(17)

where

k_{i} = \cos \frac{ω_{i} - ϕ_{i}}{2}

and

ψ_{0_{i}} = \arcsin (\frac{\cos (ω_{i} / 2)}{k_{i}})

, (i = 1, 2). It can be found that the length of the sample in the noncontact segment depends upon the resultant force and its inclined angle ω, as well as the bending properties of the sample. Therefore, the main task for figuring out the length of the sample noncontact segment is to determine the resultant force, F, and the angle parameter, ω.

Calculation of the resultant force and angle parameter

For facilitating the analysis, a Cartesian coordinate system whose original point is located at the starting point G₀ is established. Based on the force and moment analysis in Figure 4 and the force and moment equilibrium, the reactive forces F₁ and F₂, as well as the moments M₁ and M₂, are shown in Figure 5.

Figure 5.

Force and moment analysis at the demarcation points of the contact segment and the noncontact segment.

The resultant force, F₁, at point D can be divided into two component forces, that is, F₁ _x along the x-axis direction and F₁ _y along the y-axis direction, which are expressed as

F_{1 y} = F_{1} \cos ϑ_{1}, F_{1 x} = F_{1} \sin ϑ_{1}

(18)

where

ϑ_{1}

is the included angle of the resultant force F₁ and the component force F₁ _y . Referring to Figure 5 and the second assumption, the moment equilibrium equation at point S and force equilibrium equation along the y-axis direction are given as follows

F_{1 y} (\frac{S_{1}}{2} - R \sin θ_{1}) - F_{1 x} (\frac{h}{2} - R (1 - \cos θ_{1})) - M_{1} = 0

(19)

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

(20)

where S₁ is the horizontal distance of the curvature centers of the bi-U-shaped pins and the pulling pin, h is the displacement of the pulling pin during the bending process, F* is the extracting force sensed by the sensor connected with the pulling pin and M₁ is the bending moment and it is equal to B/R. Arranging equations (18)–(20), we obtain

ϑ_{1} = \arctan \frac{F * R (S_{1} - 2 R \sin θ_{1}) - 4 B}{F * R (h - 2 R (1 - \cos θ_{1}))}

(21)

Using equation (20), the angle $ω_{1}$ and the resultant force F₁ at the point D are calculated, respectively

F_{1} = F * / 2 \cos (\arctan \frac{F * R (S_{1} - 2 R \sin θ_{1}) - 4 B}{F * R (h - 2 R (1 - \cos θ_{1}))})

(23)

Then, the forces T_C and Q_C at point C are also obtained from the force equilibrium and symmetry. Once the forces T_A and Q_A at point A are figured out, the angle $ω_{2}$ and force F₂ can be calculated as follows

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

(24)

Therefore, the calculation of F₂ and $ω_{2}$ is transformed as the derivation of the relationship between T_A and T_C, as well as the relationship between Q_A and Q_C, which is analyzed in the next section in terms of the contact segment with the pin.

Analysis of the sample contact segment with the pin

The force and moment equilibrium of an infinitesimal element and how it extends from dl′ to dl are shown in Figure 6.

Figure 6.

Force and moment analysis of an infinitesimal element of the extended sample in the contact segment.

Referring to the fifth assumption, the equilibrium relationship at the n and t directions can be expressed as follows³²

dT + Q d φ - df = 0

(25)

dQ - T d φ + dN = 0

(26)

dM - QR d φ + r_{1} df = 0

(27)

The elongation of the sample in the contact segment, together with equation (12), can be given by

dl' = (1 + ɛ) dl = (1 + cT) dl

(28)

where dl′ and dl, according to the fourth assumption, are expressed as

dl = R d φ, dl' = R d φ'

(29)

Based on the third assumption, $dM / d ϕ = 0$ . The following differential equations are obtained from equations (25)–(29)

\frac{d 2 T}{d φ' 2} + \frac{1 + η}{μ} (1 + cT) \frac{d T}{d φ'} - \frac{c}{1 + cT} (\frac{d T}{d φ'}) 2 - η (1 + cT) 2 T = 0

(30)

Q = \frac{1 + η}{η (1 + η) (1 + cT)} \frac{d T}{d φ'}

df = \frac{1 + η}{η} d T, dN = \frac{1}{μ} df

(31)

Note that the wrap angle is always measured from the active force, that is, the force at points D and C, to the passive force, that is, the force at points G₁ and A. Referring to Figure 5, the boundary conditions of the contact segments can be given by

T (θ) = F_{1} \cos ω_{1}

Employing the fourth-order Runge–Kutta method and using the above boundary conditions, equations (30) and (31) can be solved numerically. When the solutions of $T (φ)$ and $Q (φ)$ are obtained, the tension force ratio, $K (θ)$ , and the shear force ratio, $Γ (θ)$ , can be calculated and expressed as follows

K_{θ} = \frac{T (θ)}{T (0)}, Γ_{θ} = \frac{Q (θ)}{Q (0)}

(33)

Thus, the forces acting at point A can be obtained from equation (33) as

T_{A} = \frac{T_{C}}{K_{θ}} = \frac{F_{1} \cos ω_{1}}{K_{θ}}

(34)

Q_{A} = \frac{Q_{C}}{Γ_{θ}} = \frac{F_{1} \sin ω_{1}}{Γ_{θ}}

(35)

F_{2} = \frac{F_{1}}{K_{θ} Γ_{θ}} \sqrt{Γ_{θ}^{2} \cos 2 ω_{1} + K_{θ}^{2} \sin 2 ω_{1}}

(36)

Based on equation (33) and referring to Figure 5, the elongation $Δ l_{CA}$ of section CA can be acquired by³³

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

(37)

and the elongation

Δ l_{G_{1} D}

of section G₁D is also given as follows

Δ l_{G_{1} D} = \int_{0}^{θ_{1}} \frac{F_{1} \cos ω_{1}}{tE K (φ) + F_{1} \cos ω_{1}} R d φ

(38)

Of course, the lengths of the sample contact segments with extension can be acquired based on the wrapping angles $θ_{1}$ and $θ_{2}$ , which are expressed as

L_{G_{1} D} = R θ_{1}

(39)

L_{CA} = R θ_{2}

(40)

Determination of the total length of the sample

Bearing in mind that the sample original gage length between the two jaws is set as L₀ in the test, it can be calculated by combining the length and extension of the sample in both contact and noncontact segments and expressed in the following equation

\frac{L_{0}}{2} = L_{G_{1} D} + L_{DC} + L_{CA} + L_{AO} - Δ l_{G_{1} D} - Δ l_{DC} - Δ l_{CA} - Δ l_{AO}

(41)

Substituting equations (14)–(17) and (37)–(40) into equation (41), it is modified as

L_{0} = 4 (F (k_{1}, \frac{π}{2}) - F (k_{1}, ψ_{0_{1}})) \times \sqrt{\frac{tB}{F_{1}}} [1 - τ (\frac{F_{1} \cos (ω_{1} - ϕ_{1})}{tB + F_{1} \cos (ω_{1} - ϕ_{1})})] + 2 (F (k_{2}, \frac{π}{2}) - F (k_{2}, ψ_{0_{2}})) \sqrt{\frac{tB}{F_{2}}} \times [1 - τ (\frac{F_{2} \cos (ω_{2} - ϕ_{2})}{tB + F_{2} \cos (ω_{2} - ϕ_{2})})] + 2 R (θ_{1} + θ_{2}) - \int_{0}^{θ_{1}} \frac{2 F_{1} \cos ω_{1}}{{tE}_{c} K (φ, μ) + F_{1} \cos ω_{1}} R d φ - \int_{0}^{θ_{2}} \frac{2 F_{1} \cos ω_{1}}{{tE}_{c} K (φ, μ) + F_{1} \cos ω_{1}} R d φ

(42)

where

ω_{1} = - \arctan \frac{F * R (S_{1} - 2 R \sin θ_{1}) - 4 B}{F * R (h - 2 R (1 - \cos θ_{1}))} + \frac{π}{2} - θ_{1} ω_{2} = \arctan (\frac{K_{θ}}{Γ_{θ}} \tan ω_{1}), F_{1} = F * / 2 \cos (\arctan \frac{F * R (S_{1} - 2 R \sin θ_{1}) - 4 B}{F * R (h - 2 R (1 - \cos θ_{1}))}) F_{2} = \frac{F_{1}}{K_{θ} Γ_{θ}} \sqrt{Γ_{θ}^{2} \cos 2 ω_{1} + K_{θ}^{2} \sin 2 ω_{1}}

Based on the above equations, the pulling force–displacement curve can be simulated using the given fabric property parameters and the values of the deflection angle ϕ and the wrap angle θ from the image of a digital camera. In addition, the bending rigidity can also be calculated in reverse by using the experimental pulling force and corresponding displacement.

Experimental details

Materials and design

In order to validate the theoretical bending model, four kinds of cotton plain-woven fabrics were selected; the details of the fabrics used in this study have been described previously.²³ The primary specifications and the property parameters of the four kinds of samples are listed in Table 2.

Table 2.

The primary specifications and property parameters of the fabric samples

Sample no.	Thickness (mm)	Weight (g/m²)	Number of warp threads (/10 cm)	Number of weft threads (/10 cm)	Bending rigidity (mN·cm)	Friction coefficient	Tensile elastic constant (N/cm·%)
1	0.25	261	286	152	0.44	0.210	0.45
2	0.71	312	197	130	0.77	0.283	0.41
3	1.02	487	130	102	1.29	0.267	0.90
4	0.61	395	175	116	0.78	0.260	0.53

The thickness of fabrics was determined under pressure of 1 kPa according to the ISO standard 5084-1996. The bending rigidity, friction coefficient and tensile elastic constant of the samples were determined by the KES bending tester, surface tester and tensile/shear tester, respectively, where the tensile elastic constant is the ratio of the tensile force acting on the unit width of the sample to strain. The property parameters were used in the model to simulate the the oretical pulling force–displacement curves. Meanwhile, a quasi-three-point bending test was conducted using the CHES-FY to obtain the experimental pulling force–displacement curves. It should be pointed out that the property parameters were measured in the warp direction of samples by the KES, and the samples were also tested in the warp direction using the CHES-FY.

Before the testing, the basic parameters of the CHES-FY should be set as follows: the diameters of the pulling pin and bi-U-shaped pins were 4 mm; the distances from the curvature centers of the bi-U-shaped pins to the jaws were 40 mm in the horizontal direction and 50 mm in the vertical direction, respectively; the interval distance of the bi-U-shaped pins was set as 14 mm; the moving speed of the pulling pin was 0.2 mm/s; the sampling frequencies of the analog-to-digital convertor and the digital camera were set as 100 and 10 Hz, respectively. The fabrics were cut into strip samples with warp × weft size as 20 cm × 2 cm and stored in a conditioned lab at 20 ± 2℃ and 65 ± 3% relative humidity for over 24 h prior to testing on the CHES-FY. The gage length of the samples between the two jaws was 18 cm at the beginning of the test.

Estimation of the parameters for modeling

Wrap angle θ

To reduce the actual analysis of the image captured by the digital camera of the CHES-FY and facilitate the discussion of the theoretical modeling, it is necessary to investigate the underlying regularities of the dynamic wrap angles based on the experimental observations. Figure 7 shows the evolution of the wrap angles of the four kinds of fabric samples with the displacement of the pulling pin.

Figure 7.

Wrap angles at different displacements of the pulling pin with a fitting trend.

It can be seen that both angles $θ_{1}$ and $θ_{2}$ present a typical nonlinearity with the displacement. Ignoring the small effect of the property parameters of fabrics on the wrap angles, the quantitative relations of any general validity between the wrap angles and displacement can be statistically figured out, respectively, in the quadratic and cubic equations

θ_{1} = - 6 \times 10 - 4 h 2 + 0.022 h

(43)

θ_{2} = 10 - 3 h 3 - 4.1 \times 10 - 3 h 2 + 0.048 h

(44)

Equations (43) and (44) show the primary trend of the wrap angles with the evolution of displacement. At the beginning part, $θ_{2}$ is a little bigger than $θ_{1}$ due to the original bending shape of the sample between the jaw and bi-U-shaped pins, and then $θ_{2}$ drops when the displacement is above 10 mm. In the descending part of $θ_{2}$ with the displacement, the disparity among the fabrics is relatively large. However, in the bending step where the displacement is within 10 mm, the fitting results from equations (43) and (44) show an acceptable accuracy with not more than 10% relative error generally among the fabrics where the bending rigidity, friction coefficient and tensile elastic constant ranges are 0.44–1.29 mN·cm, 0.210–0.283 and 0.41–0.90 N/cm·%, respectively.

Endpoint deflection angle ϕ

To obtain the deflection angle ϕ at point S, the sample image was captured by the digital camera of the CHES-FY during the test process and processed by the high-pass filter method to highlight the profile of the sample, as shown in Figure 8.

Figure 8.

The typical sample image of segment DC after the high pass.

Based on the Cartesian coordinate system with the origin point G₀, we can obtain the cubic spline curve s(x) of the sample profile by the interpolation algorithm using the MATLAB program. The angle ϕ can be obtained by equation (45)

ϕ = \arctan [s' (\frac{S_{1}}{2})] - \arctan (s' (R \sin θ_{1}))

(45)

Assuming the deflection segment of the sample approximates to a circle arc and referring to the geometrical relationship in Figure 8, the deflection angle ϕ can also be described by

ϕ = \hat{ϕ} = π - 2 ∠ SDH, ∠ SDH = \arcsin \frac{| SH |}{| SD |}

(46)

Bearing in mind that the coordinates of points D, S and O₂ are ( $R \sin θ_{1}$ , $h - R (1 - \cos θ_{1})$ ), ( $S_{1} / 2$ , $h / 2$ ) and (0, $h - R$ ), the length of the lines SD and SH (see Figures 5 and 8) can be figured out using the analytic geometry method as follows

| SD | = \sqrt{(R \sin θ_{1} - \frac{S_{1}}{2}) 2 + (\frac{h}{2} - R (1 - \cos θ_{1})) 2}

(47)

| SH | = | \frac{S_{1}}{2} \cos θ_{1} + (\frac{h}{2} - R) \sin θ_{1} |

(48)

Substituting equations (43), (44), (47) and (48) into equation (46), the estimated values of the endpoint deflection angle can be calculated, and the comparisons with experimental values obtained from equation (45) are shown in Table 3. It can be seen that the estimated values are generally a little higher than the experimental values. This difference may come from the assumption of the sample profile as a circle shape. However, the maximal relative error of the experimental and estimated values is less than 0.17. Therefore, the estimated value of ϕ can be used to carry out the modeling to analyze the effect level of the friction and extensibility by estimating the angle ϕ of any general validity to avoid the image processing, with an acceptable accuracy.

Table 3.

Comparisons of the endpoint deflection angle between estimated values and experimental values

Displacement (mm)		Experimental value (rad)				Maximal relative error
Displacement (mm)	Estimated value (rad)	Sample 1	Sample 2	Sample 3	Sample 4	Maximal relative error
2	0.24	0.20	0.26	0.29	0.27	0.17
4	0.42	0.41	0.45	0.48	0.45	0.13
6	0.57	0.58	0.60	0.62	0.58	0.08
8	0.68	0.69	0.72	0.76	0.73	0.11

Results and discussion

In order to analyze the effect of the friction and extensibility, the theoretical curves of the four samples were calculated by substituting the corresponding property parameters and the apparatus setting parameters given in the experiment part into equation (42), and three main cases, that is, case 1: with both friction and extensibility effects ( $τ = 1$ ); case 2: with the friction effect only ( $τ, Δ l_{CA}, Δ l_{G_{1} D} = 0$ ); and case 3: no friction or extensibility effect ( $μ = 0$ , $τ, Δ l_{CA}, Δ l_{G_{1} D} = 0$ ), were simulated, and the simulated curves of each case with the experimental curve are shown in Figure 9.

Figure 9.

Comparison between theoretical and experimental curves of samples 1 (a), 2 (b), 3 (c) and 4 (d).

In order to make a better comparison, the MBF, bending work (defined as the area under the curve) and linear fitting slope (obtained by the least square method) of both the experimental and theoretical curves are figured out, and the relative error between the experimental curves and theoretical curves in each case is also calculated. Both of these items are listed in Table 4.

Table 4.

Comparisons of bending parameters of the theoretical and experimental curves

				Theoretical curve
	Experimental curve			Case 1			Case 2			Case 3
Sample	MBF (N)	Slope (N/mm)	Work (N·mm)	MBF (N)	Slope (N/mm)	Work (N·mm)	MBF (N)	Slope (N/mm)	Work (N·mm)	MBF (N)	Slope (N/mm)	Work (N·mm)
#1	1.13	0.157	4.81	1.12 (0.88)	0.137 (12.7)	4.89 (1.66)	1.19 (5.31)	0.145 (7.64)	4.99 (3.74)	0.61 (46.0)	0.075 (52.2)	2.73 (43.2)
#2	1.41	0.210	6.06	1.43 (1.42)	0.177 (15.7)	6.31 (4.13)	1.45 (2.84)	0.179 (14.8)	6.39 (5.45)	1.01 (39.0)	0.123 (41.4)	4.73 (21.9)
#3	1.98	0.284	8.79	2.08 (5.05)	0.254 (10.6)	9.65 (9.78)	2.14 (8.08)	0.260 (8.45)	9.73 (10.69)	1.73 (12.6)	0.210 (26.1)	7.94 (9.67)
#4	1.52	0.221	6.07	1.53 (0.66)	0.188 (14.9)	6.57 (8.24)	1.55 (1.97)	0.191 (13.6)	6.65 (9.56)	1.02 (32.9)	0.125 (43.4)	4.47 (26.4)

The value in the bracket is the absolute relative error (unit: %) between the theoretical value and corresponding experimental value.

MBF: maximum bending force.

Comparison of the theoretical and experimental results

As shown in Figure 9, the theoretical curves from the theoretical bending model including friction and extensibility effects show a reasonable agreement with the experimental curves, although the first half of the theoretical curves is visibly lower than the experimental curves. This difference may result from the ideal assumption of samples as homogeneous flexible materials, ignoring the structure of the samples, but the fabric as a fuzzy interwoven material can generate structural adjustment and compression between yarns and between testing pins and the fuzzy surface of samples, especially at the beginning of the bending test. However, both the experimental and theoretical curves present the same variation trend of the pulling-out force with the displacement, and the latter half of the theoretical curves fit well with the experiment. This means that the theoretical bending model can be used to predict the experimental force–displacement curve of the bending step. Referring to Table 4, the maximal relative error of the bending parameters between the theoretical and experimental curve is less than 15.7% (i.e. (0.210–0.177)/0.210), which further reveals that the theoretical bending model is reliable and has included the major factors that act on the bending step of the CHES-FY. Moreover, the model can enable the bending rigidity of fabric samples to be reliably evaluated by eliminating the effects of friction and extensibility of samples using the pulling-out force and corresponding displacement measured by the CHES-FY and friction coefficient and tensile elastic constant of the samples.

Effect of the friction

As can be seen from Figure 9 and Table 4, the pulling-out force–displacement curves show high sensibility to the friction effect. The bending parameters, that is, MBF, linear fitting slope and bending work, from the theoretical curves significantly increase when considering the friction effect. The minimal variation rate of the bending parameters of the theoretical curves from case 2 (with the friction effect) to case 3 (no friction effect) are 19.1%, 19.2% and 18.4%, respectively, and the variation rate of the MBF of sample 1 is even up to 48.7%. This result shows that the friction effect is remarkable in the bending test for fabrics where the friction coefficient range is 0.210–0.283. Moreover, a tendency can be found by comparing the force–displacement curves of the four samples, that is, the friction effect becomes more outstanding when the bending rigidity decreases. This is a natural conclusion, but it indicates the importance of considering the effect of friction on the quasi-three-point bending test when measuring the fabric-similar flexible materials.

In order to analyze further the effect of friction on the force–displacement curve, the theoretical curves at four values of (0, 0.2, 0.3 and 0.5) are computed by combining equation (42) with equations (44), (47) and (48), where B is set as 0.7 mN·cm, and E_c is set as 0.5 N/cm·% in the formula. It is interesting to note from Figure 10 that the bending parameters increase significantly when μ changes from 0 to 0.2, while the increase of the bending parameters is not remarkable, expectedly, when μ shifts from 0.2 to 0.3 and 0.5. Thus, including the friction effect in the model formula is of great importance, but the bending parameters may not be sensitive to the change of the friction coefficient μ.

Figure 10.

Theoretical force–displacement curves with different friction effects.

Effect of the extensibility

The extensibility of the sample always reduces the bending parameters of force–displacement curves, as shown in Figure 9. However, the variation rate of the bending parameters from case 2 (no extensibility effect) to case 1 (with extensibility effect) is not dominant for fabrics whose tensile elastic constant ranges from 0.41 to 0.90 N/cm·%, and the maximal variation rate of bending parameters is not more that 5.88% (i.e. (1.19–1.12)/1.19). Figure 9 also shows that only the theoretical curves with the friction effect can fit the experimental curves well, which indicates that the extensibility of samples is not a major factor affecting the bending test.

Considering these results, we simulate an extra force–displacement curve of a sample with a relatively low tensile elastic constant (E_c = 0.1 N/cm·%) and compare it with the other curves where B is set as 0.7 mN·cm and μ is set as 0.3, as shown in Figure 11. It is shown that the bending parameters increase with the increase of the tensile elastic constant, E_c, from 0.1 to a maximum value (i.e. no extensibility effect). However, the variation rate of the bending parameters is still relatively small; even the tensile elastic constant is at a low value and the variation rate of the bending parameters is not more than 11.7% between the curve with no extension and the curve with a low tensile elastic constant (E_c = 0.1 N/cm·%). Therefore, the effect of the extensibility is almost negligible for the bending test of wear fabrics with maintaining the relative error at less than 11.7% when the tensile elastic constant of the fabric is larger than 0.1 N/cm·%.

Figure 11.

Theoretical force–displacement curves with different extensibility effects.

Conclusions

This paper developed a theoretical bending model to investigate the quasi-three-point bending step of the CHES-FY. Good agreements were obtained between the theoretical and experimental curves, confirming the rationality of the established model, and it also indicates that the theoretical bending model has included the major factors affecting the bending step of the CHES-FY. The model considered, in detail, the effect of the friction and extensibility on the bending test, based on the simulated curves in three cases. The results show that it is of great importance to consider the friction effect, and the friction effect can enhance the bending parameters of the force–displacement curve significantly with the variation rate of the bending parameters, while this effect could be reduced if the bending rigidity of measured samples becomes large. However, the effect of the extensibility of samples on the bending test is negligible, as the variation rate of the bending parameters of the curves is not more than 5.88% for the tested samples when considering the extensibility effect, and the relative error can be limited within 11.7% when the tensile elastic constant is not lower than 0.1 N/cm·%. Therefore, the effect of the extensibility of the wear fabrics on the bending test can be ignored in general. The work also has an application for predicting the bending rigidity using the pulling-out force by the displacement in the measurement of the CHES-FY.

Footnotes

Acknowledgement

The authors would like to thank Dr Del Giudice Michelina (CSIRO Material Science and Engineering, Waurn Ponds, Australia) for her help in the KES test.

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 the National Natural Science Foundations of China project (grant numbers 11272086 and 51203022), the Fundamental Research Funds for the Central Universities (CUSF-DH-D-2016006 and 2232014A3-02) the “DHU Distinguished Young Professor Program (B201307)” and the Fok Ying Tung (huoyingdong) Education Foundation (151071). The authors are grateful to the China Scholarship Council (CSC) for the scholarship (number 201506630046) provided to one of the authors (Sun).

References

Peirce

. The “handle” of cloth as a measurable quantity. J Text Inst 1930; 21: 377–416.

Abbott

. The measurement of stiffness in textile fabrics. Text Res J 1951; 21: 435–441.

Grosberg

. The mechanical properties of woven fabrics part II: the bending of woven fabrics. Text Res J 1966; 36: 205–211.

Livesey

Owen

. Cloth stiffness and hysteresis in bending. J Text Inst 1964; 55: 516–530.

Rudra

Mills

JCT

. Flaw in the Shirley Stiffness Tester. Text Res J 1986; 56: 712–712.

Cassidy

Cassie

et al.

The stiffness of knitted fabrics: a new approach to the measurement of bending-part 1: development. Int J Clothing Sci Technol 1991; 3: 14–19.

Kocil

Zurek

Krucinska

et al.

Evaluating the bending rigidity of flat textiles with the use of an Instron tensile tester. Fibres Text East Eur 2005; 13: 31–34.

Sun

. A new tester and method for measuring fabric stiffness and drape. Text Res J 2008; 78: 761–770.

Ghane

Hedayati

. A fixed-supported beam modelling for measuring fabric rigidity using small deflection equations. J Text Inst 2014; 105: 93–99.

10.

Tester

Buckenham

et al.

Simple instruments for quality control by finishers and tailors. Text Res J 1991; 61: 402–406.

11.

Kawabata

. Characterization method of the physical property of fabrics and the measuring system for hand–feeling evaluation. J Text Mach Soc Jpn 1973; 26: 721–728.

12.

Alley

. Revised theory for the quantitative analysis of fabric hand. J Eng Ind 1980; 102: 25–31.

13.

Pan

Yen

Zhao

et al.

A new approach to the objective evaluation of fabric handle from mechanical properties part I: objective measure for total handle. Text Res J 1988; 58: 438–444.

14.

Wang

Mahar

Hall

. Prediction of the handle characteristics of lightweight next-to-skin knitted fabrics using a fabric extraction technique. J Text Inst 2012; 103: 691–697.

15.

Naebe

Tester

McGregor

. The effect of plasma treatment and loop length on the handle of lightweight jersey fabrics as assessed by the Wool HandleMeter. Text Res J 2015; 85: 1190–1197.

16.

Liao

et al.

A simultaneous measurement method to characterize touch properties of textile materials. Fiber Polym 2014; 15: 1548–1559.

17.

Pan

. Quantification and evaluation of human tactile sense towards fabrics. Int J Des Nat 2007; 1: 48–60.

18.

Pan

Yen

. Physical interpretations of curves obtained through the fabric extraction process for handle measurement. Text Res J 1992; 62: 279–290.

19.

. Physical interpretation of pulling-out curve based on a new apparatus. J Text Inst 2008; 99: 399–406.

20.

. A comprehensive handle evaluation system for fabrics: I. Measurement and characterization of mass and bending properties. Meas Sci Technol 2007; 18: 3547–3554.

21.

Zhou

Yan

et al.

Measurement and characterization of bending stiffness for fabrics. Fiber Polym 2011; 12: 104–110.

22.

. Determination of the bending characteristic parameters of the bending evaluation system of fabric and yarn. Text Res J 2006; 76: 702–711.

23.

Gao

Zhou

et al.

Effects of parameters on mass index of the CHES-FY system. Fiber Polym 2014; 15: 175–180.

24.

Sun

et al.

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. Text Res J. 2016; 86:1947–1961.

25.

Mannocci

Sherriff

Watson

. Three-point bending test of fiber posts. J Endodontic 2001; 27: 758–761.

26.

Fang

Jia

Feng

. Three-point bending test at extremely high temperature enhanced by real-time observation and measurement. Measurement 2015; 59: 171–176.

27.

Yan

Han

et al.

Three-point bending of sandwich beams with aluminum foam-filled corrugated cores. Mater Des 2014; 60: 510–519.

28.

. Analysis of bending properties of worsted wool yarns and fabrics based on quasi-three-point bending. J Text Inst 2005; 96: 389–399.

29.

Steeves

Fleck

. Collapse mechanisms of sandwich beams with composite faces and a foam core, loaded in three-point bending. Part I: analytical models and minimum weight design. Int J Mech Sci 2004; 46: 561–583.

30.

Jamsa

Jalovaara

Peng

et al.

Comparison of three-point bending test and peripheral quantitative computed tomography analysis in the evaluation of the strength of mouse femur and tibia. Bone 1998; 23: 155–161.

31.

Jung

Kang

Youn

. Effect of bending rigidity on the capstan equation. Text Res J 2004; 74: 1085–1096.

32.

Jung

Pan

Kang

. Generalized capstan problem: bending rigidity, nonlinear friction, and extensibility effect. Tribol Int 2008; 41: 524–534.

33.

. Study on strength property of fabric under low-stress condition. J Text Inst 2008; 99: 265–272.