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Fibers and Polymers 2012, Vol.13, No.2, 237-243

DOI 10.1007/s12221-012-0237-2

A New Approach for Numerical Identification of Bending Behavior of Plain Woven Fabric A. R. Moghassem Department of Textile Engineering, Islamic Azad University, Qaemshahr Branch, Qaemshahr, Iran (Received June 10, 2011; Revised August 17, 2011; Accepted September 18, 2011) Abstract: There is a variety of approaches for investigating bending behavior of woven fabrics. Some of them are based on fabric deformation with one edge fixed; the others are based on measurement of force, moment or energy producing bending deformation. In all methods, bending properties is acquired after testing prepared fabric samples. Therefore, in this work an attempt is made by a mechanical model and a novel calculation technique to determine bending characteristics of the plain woven fabrics before sample production. Theoretical data including bending length, bending rigidity and bending modulus were directly determined for supposed fabric samples with a given yarn count and yarn density using Peirce’s structural model for plain woven fabric and a especial code written in Maple12. Besides, fabric samples with the defined characteristics were woven on a Sulzer-Ruti weaving machine. Then, these fabrics were tested for bending behavior using Shirley bending tester. Comparison showed good agreement between predicted and measured bending characteristics of the fabrics. However, theoretical bending rigidities of the samples were more than experimental values. Keywords: Bending stiffness, Cantilever test method, Woven fabric, Bending rigidity, Peirce’s model

expensive and the measurement performed is time consuming [1]. Therefore, researchers focused on simple and speedy methods for investigating bending rigidity of the woven fabrics. Kocik et al. [7] used an Instron tensile tester and principles of buckling in the case of small curvature to evaluate the bending rigidity of flat textiles. Alamdar-Yazdi et al. [4] investigated effect of warp and weft densities on bending properties of polyester-viscose woven fabrics using this apparatus. A variety of the practical end-use considerations of textile structures are associated with the bending rigidity of yarns comprising a structure [7]. Therefore, in recent years, bending behavior of woven fabrics has been studied in terms of the bending properties of their constituent yarns and their constructional parameters [8]. In addition, information about bending behavior is achieved only after testing some prepared fabric samples. Considering above-mentioned problems, this work aims at the development of a new mechanical model and calculation technique to predict bending properties of the plain woven fabrics before producing process on the basis of the cantilever test method and Peirce’s structural model.

Introduction Bending rigidity as an index for bending behavior of fabrics is determined by bending resistance of threads lying in the direction of bending [1,2]. Flexural rigidity influences fabric’s handle and formability [3]. There are various standard approaches for investigating bending behavior of woven fabrics [1,4]. The common method which is based on fabric deformation under its own weight was proposed by Peirce in 1930. This approach is called cantilever test method. Bending parameters including bending length that is related to the fabrics’ ability to drape [4], bending rigidity and bending modulus are calculated using equations that describe pure bending theory of an elastic beam within the limit of linear strain. FAST (Fabric Assurance by Simple Testing) bending tester is the most popular tester in this group [1]. In cantilever test method, an equation related to small deformation is used for large deflection by considering correction coefficient and is different from fabric behavior in use [5]. Experiments showed that, relationship between bending moment (M) and curvature (1/ρ) is not generally linear. Szablewski [6] obtained a certain measure of bending rigidity on the basis of Peirce’s test using a numerical analysis. This measure determines the non-linear relationship between these two parameters. In second group of the methods, force, moment or energy producing bending deformation are measured. The most widely used pure bending tester belonging to the second group is KES-FB (Kawabata Evaluation System) bending tester. Bending rigidity and bending hysteresis are measured by KES-FB bending tester [1,4]. However, this apparatus is

Theoretical Analysis Principles of Cantilever Test Method Employing principles of the cantilever bending, a specimen with dimensions of 20×2.5 cm is supported on a smooth low-friction horizontal platform (Figure 1). A weighted slide is advanced at a constant rate in a direction parallel to its long dimension. As leading edge of fabric specimen projects from platform, it bends under its own mass. Once the fabric flexes enough to touch the bend angle

*Corresponding author: [email protected] 237

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Figure 2. Internal forces and moments at a section of the beam with a distance of (x) from fixed support. Figure 1. Principle of the cantilever test method and bending of the elastic beam under uniformly distributed weight.

Ax = 0

(4)

Ay = γ × L

(5)

indicator 41.5 o or a line with equation of Y = −tanθ × x = −tan(41.5)×x = −0.8847x below the plane of the platform surface the test is stopped. Length of the overhang is measured. Two other bending parameters namely, bending rigidity and bending modulus can be calculated based on Peirce’s equations in bending, using bending length and others fabric constructional characteristics (equations (1)(3)) [9].

γ×L MO = -----------2

L C = --2

(1) 3

G = WC 10

3

(2)

–6

12G10 ϕ = ------------------(3) 3 T Where, C is bending length, L is length of the overhang, G is bending rigidity, W is weight per unit area, ϕ is bending modulus and T is fabric thickness. A Mechanical Model to Identify Relationship between Length of the Overhang and Constructional Parameters of an Elastic Beam Pure bending of an elastic beam with one edge fixed (cantilever beam) was simulated to calculate bending length of the fabric specimen. Dimensions (20×2.5 cm) and weight (w) of the elastic beam are the only given data for the modeling. In the proposed model, beam is loaded under its own uniformly distributed weight (Figure 1). To simplify analysis of bending behavior in large deflection condition, the beam is assumed as a linear elastic and homogeneous structure with constant rectangular section. As can be seen in Figure 1, length of the beam is divided into two different segments when it bends enough and touches the bend angle indicator 41.5 o. Elastic beam with total length of P is consists of a straight segment (K) and a bent segment (L). The length of the overhang is the length of the second segment of the beam (L). To determine deflection of the beam, forces and moments at O (fixed support of the system) equations of static equilibrium at this point was solved. Results of the analysis are as follow.

2

(6)

Where, Ax is force in horizontal direction, Ay is force in vertical direction, MO is bending moment and γ = w/p is weight for unit length of the beam. To analyze the modeled system and determine the length of the overhang (L), equations of equilibrium at a section with a distance of (x) from fixed support of the system were considered. Figure 2 shows existing forces and moments at this section of the beam. The main goal of the analysis was determination of the applied bending moment (M) at O'. 2

2

γL γx – γ Lx + -------- + ------- + M = 0 2 2 2

(7)

2

L x M = γ Lx – γ ---- – γ ----2 2

(8)

In preliminary mathematics it can be shown that the curvature (ρ) of a fixed point is [10] 2

2

d υ/ dx υ″ --1- = -------------------------------------- = ---------------------------ρ [ 1 + [ dυ/ dx ] 2 ]3/2 [ 1 + ( υ′ )2 ] 3/2

(9)

Besides, the curvature of neutral line is as follow, where; E is initial Young’s modulus and I is moment of inertia of section of the beam [10]. M--1- = --------ρ E×I

(10)

Substituting equation (10) into equation (9) and letting y = υ it was concluded that; y″ M - ------------------------------= (11) E × I ( 1 + y′2 )3/2 Letting y'= u and 1/EI = α, equation (11) can be expressed as: u′ M - -----------------------------= E × I ( 1 + u2 )3/2

(12)

u′ M = -------------------------2 3/2 α(1 + u )

(13)

Mechanical Model for Bending Behavior of Woven Fabric

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Since u' = du/dx therefore; du -------------------------= M dx 2 3/2 α(1 + u )

(14)

du

- = ∫ M dx ∫ -------------------------2 3/2

α(1 + u ) Substituting equation (8) into equation (15) 2

(15)

2

L⎞ du - = ⎛ γLx γ x---- ---∫ -------------------------∫ ⎝ – 2 – γ 2-⎠ dx 2 3/2 α(1 + u )

(16)

3

u x γ --------------------- = --- ⎛ Lx2 – ---- – L2 x⎞ + C ⎝ ⎠ 2 2 3 α 1+u

Figure 3. The length of the overhang split into several segments.

(17)

Value of the constant (C) is calculated using boundary condition where; u 0 = 0 → C = 0 therefore; 3

u x --------------------- = --γ- ⎛ Lx2 – ---- – L2 x⎞ ⎝ ⎠ 2 2 3 α 1+u 2

2 2

3

u +1–1 α γ ⎛ 2 x -------------------- = ---------- Lx – ---- – L2 x⎞ 2 ⎠ 4 ⎝ 3 1+u

(18) 2

(19)

3

2 x 2 αγ ⎛ Lx – ---- – L x⎞ ⎝ ⎠ 3 u = -----------------------------------------------------------2 3 2 2 2 x 2 4 – α γ ⎛ Lx – ---- – L x⎞ ⎝ ⎠ 3

(20)

2

Since 1 + u = 1 + y′2 and, the equation governing the bent lengthx of the beam in Cartesian coordinate system is L = ∫ 1 + y′2 dx [11], therefore, the length of the overhang 0 can be calculated by: x 2 L = ∫ ----------------------------------------------------------- dx 0 3 2 2 2 x 2 4 + α γ ⎛ Lx – ---- – L x⎞ ⎝ ⎠ 3

(21)

Calculation Techniques A Novel Calculation Approach for Determining Theoretical Length of the Overhang To determine the length of the overhang of the beam based on the new mechanical model equation (21) should be solved. But, it was not easy to solve such kind of equations. The core idea for determining the bending length of the beam was this fact that, the length of the overhang for each specimen is equal to (L) that is between zero and uncertain value when the free edge of sample toughs the bend line indicator with equation of Y = −0.8847x. Because there was not a certain value for bending length prior to test the prepared fabric specimen therefore, different values were assumed for the length of the overhang (L). Then bending behavior of specimen with the assumed value

was simulated using originality approach and relevant codes written in Maple12 software as described below. Since values of the parameters γ and α could be obtained from elastic beam (woven fabric) specifications therefore, protrusion in horizontal direction (x) was calculated for each (L) using equation (21). In this equation all the parameters had a certain value except the protrusion in horizontal direction. Several values less than assumed length of the overhang (x < L) were suggested for x. Based on the numerical calculations, the proper value of x is the number in which left side and right side of equation (21) are equaled. Finally, a table including different values of L and x was presented. In the next step, to simulate bending behavior of the beam, each obtained protrusion in horizontal direction (x) was divided to many small segments namely (xsi). xAfterward, si these data were used to solve equation ysi = ∫0 u dx and protrusion in vertical direction was calculated. Figure 3 shows a scheme of partition of the length of the overhang. For each assumed value of the length of the overhang a curve was obtained by plotting (xsi) against (ysi) using Tecplot7 software. This curve simulates bending deformation of the specimen on the Cantilever testing apparatus in real experimental condition. Successive curves were obtained by plotting the bending deformation of a sample for all assumed values of L in the same Cartesian coordinate system. However, it was not clear which curve shows the bending deformation of the specimen when its free edge toughs the bend angle 41.5 o. Therefore, a line with equation of Y = −tanθ × x = −tan(41.5)×x = −0.8847x was plotted in the same Cartesian coordinate system. If we assume small enough values for L, we will find a curve that its free edge sits on the plotted line (Y = −0.8847x). Specifications of this curve show theoretical length of the overhang (L), protrusion in horizontal direction (x) and protrusion in vertical direction (y) of the beam. Determining Fabric Constructional Parameters Using Its Constituent Fibers and Yarns Characteristics Initial Young’s Modulus To determine bending properties of fabric based on the

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above-mentioned technique, two main parameters that are γ = w/p and 1/EI = α should be calculated using fabric constructional characteristics. P and w were the given data for the fabric specimen. However, initial Young’s modulus of the fabric (E) is determined in weft and warp directions based on its constructional parameters using equations (22), (23) [12]. 3

2

12B1 P2 B2 l1 cos θ1 - 1 + -----------------------Ee = ----------------------3 3 2 2 P1 l1 sin θ1 B1 l2 cos θ2 3

(22)

2

12B2 P1 B1 l2 cos θ2 - 1 + -----------------------Ep = ----------------------3 3 2 2 P2 l2 sin θ2 B2 l1 cos θ1

(23)

Where, E is the initial Young’s modulus of the fabric (N/ cm), B is yarn flexural rigidity (mNmm2), P is thread spacing (mm), l is yarn modular length (mm), and θ is weave angle in degree that should be determined. In these equations subscripts 1 and 2 refer to parameters in the warp and weft directions respectively. Yarn Flexural Rigidity On the basis of Zurek et al. [13] and Park et al. [14] researches, we assumed that filament yarns are consisted of several straight mono-filaments. The whole rigidity of yarn bending equals the sum of the bending rigidity of the fibers in a simple case. Therefore, bending rigidity of the filament yarn can be obtained by: Bfy = NEmf Imf

(24)

Where N, Emf and Imf are number of mono-filaments, elastic modulus of mono-filament and moment of inertia of section of mono-filament. Considering lower variation of mono-filament diameter, elastic modulus is obtained using an Instron tensile tester. Assuming a circular cross section for polyester mono-filament, moment of inertia of the 4 section is determined by I = 1--4- πr where, r is the radius of section [10]. Thread Spacing In the case of the plain weave, thread spacing can be calculated by P = 1/n where n is yarn density (1/cm) [15]. Crimp Height Constructional parameters of the fabric were determined based on Peirce’s structural model for plain woven fabric. However, there are not adequate equations for exploring unknown fabric structural parameters due to the lack of yarn crimp ratio. Hence, new relationships (equations (25), (26)) were derived in accordance with Peirce’s rigid-thread model of plain woven fabric. To verify accuracy of proposed equations, all the parameters were calculated after preparing the sample fabrics and the results were compared with theoretical data (Table 3). n1 – n2 h1 = -------------d +d n1 1 2

(25)

n h2 = -----2 × d1 n1

(26)

Where; n is yarn density, h is crimp height and d is yarn diameter. Subscripts 1 and 2 refer to parameters in warp and weft directions respectively. Yarn Diameter A high twist filament yarn was used as warp and weft yarns in structure of sample fabrics. Therefore, diameter of polyester filament yarn was measured directly by a microscope with high degree of accuracy [15]. To verify accuracy of the results packing fraction of the yarns were calculated using equation (27). 2

2

nπ ( d′ ) πd- ----------------------= 4 4φ

(27)

Where d is filament yarn diameter, n is number of fibers, d' is fiber diameter and φ is packing fraction. For filament yarns φ1/2 may be somewhat nearer its maximum possible value of 1.0. Also, packing fractions range from 0.3 for some spun yarns to nearly 0.9 for a highly twisted nylon filament yarn [15]. Based on the warp and weft yarns specifications packing fractions were calculated 0.8948 and 0.8507 for 75 denier/36 filament and 100 den/7 filament yarns respectively. Crimp Ratio, Modular Length and Weave Angle After calculating crimp height, warp and weft yarns crimps (c) can be obtained by [15]: 4 h1 = --- p2 c1 3

(28)

4 h2 = --- p1 c2 3

(29)

Also, yarn modular length is determined by [15]: l 1 = p2 ( 1 + c 1 )

(30)

l 2 = p1 ( 1 + c 2 )

(31)

Then we can calculate weave angle by [15]: θ1 = 106 C1

(32)

θ2 = 106 C2

(33)

Moment of Inertia of the Section of Fabric Samples Supposing a rectangular shape for the section of the fabric specimen, moment of inertia of section can be calculated for a given width (b = 2.5 cm) and thickness (T = MAX(d1 + h2 and d2 + h1) by [10]: 1 3 I = ------ × b × T 12

(34)

Weight of the Unit Area of the Fabric Weight of a unit area of the fabric is sum of the weights of

Mechanical Model for Bending Behavior of Woven Fabric

warp and weft yarns with unit length in that area. Weight of the unit area can be obtained on the basis of warp and weft yarns specifications by:

Fibers and Polymers 2012, Vol.13, No.2

Tex1 Tex2 - × n2 × ---------W = n1 × ---------5 5 10 10

Sample No.1 Plain 75 denier/ 36 monofilament 75 denier/ 36 monofilament 32 29

Description Weave type Warp yarn number Weft yarn number Warp density (1/cm) Weft density (1/cm)

Table 2. Specifications and bending characteristics of sample fabrics in experimental condition Sample No.2 Weft Warp direction direction 100 100 71

71

26

30

0.00615

0.00615

0.2075 3.38 1.69 29.68 39.87

0.2075 3.40 1.70 30.21 40.57

(35)

Experimental

Table 1. Characteristics of investigated fabric samples Sample No.2 Plain 100 denier/ 71 monofilament 100 denier/ 71 monofilament 30 26

241

Sample No.1 Description Weft Warp direction direction 75 75 Yarn count (den) Number of mono36 36 filament 29 32 Yarn density (1/cm) Weight per unit area 0.005 0.005 (g/cm2) 0.187 0.187 Thickness (mm) 3.84 3.96 L (cm) 1.92 1.98 C (cm) 35.38 38.81 G (mg·cm) 64.73 70.99 ϕ (kg/cm2)

Samples and Methods To verify accuracy and precision of proposed model, two different plain woven fabrics were woven on a Sulzer Ruti weaving machine. Specifications of the fabrics have been shown in Table 1. After weaving, five specimens with dimensions of 20× 2.5 cm in both warp and weft directions were taken from each fabric sample. Bending lengths of the specimens were obtained by Shirley bending tester according to ASTMD1388-D96. In addition, thickness and weight per unit area of 10 random specimens of each fabric sample were measured according to ASTM-D3776-96 and ASTM-D573695. Table 2 shows the results of experimental study. Besides, all theoretical constructional parameters of the plain woven fabrics with the same yarn count and yarn density were determined using described calculation processes. Results of the calculations have been shown in Table 3. Bending length of the fabrics was determined on basis of equation (21) through previous-mentioned calculation approach. Two different figures (Figure 4) including a set of curves that simulate the bending behavior of sample fabric were plotted. For each figure, bending length, protrusion in horizontal direction and protrusion in vertical direction of a curve that its free edge sits on the line with equation of Y = −0.8847x was considered as results of theoretical analysis. Obtained results were used to determine other bending

Table 3. Constructional and bending parameters of sample fabrics in theoretical condition Sample No.2 Weft Warp direction direction 0.00622 0.00622 0.01098 0.01098 0.03846 0.033 0.00952 0.01244 0.02196 0.02196 0.04681 0.05893 0.03454 0.04072 22.93 25.73 0.04510 0.04510 1.55 1.55 0.000001287 0.000001287 1.272 1.214 3.42 3.47 1.71 1.735 31.10 32.48 35.24 36.81

Weft direction (experimental) 0.005 0.009324 0.0340 0.00832 0.0187 0.040 0.0324 20.78 1.250 3.84 1.92 35.38 64.73

Sample No.1 Weft direction Warp direction (theoretical) (experimental) 0.00507 0.005 0.009324 0.009324 0.0312 0.0312 0.00844 0.01013 0.01951 0.0187 0.04116 0.050 0.03248 0.0357 21.50 23.07 0.0587 1.267 1.250 0.0000015519 0.000001023 4.04 3.96 2.02 1.98 41.78 38.81 67.51 70.99

Warp direction (theoretical) 0.00507 0.009324 0.0312 0.01019 0.01951 0.05052 0.03571 23.82 0.0587 1.267 0.0000015519 0.00000968 4.00 2.00 40.56 65.54

Description Weight per unit area (g/cm2) d (cm) P (cm) h (cm) T (cm) c l (cm) θ (o) B (mn·mm2) γ (g/m) I (cm4) E (N/m2) L (cm) C (cm) G (mg·cm) ϕ (kg/cm2)

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Figure 4. Successive curves that simulate bending behavior of the fabric samples.

characteristics of samples that were bending rigidity and bending modulus. Comparison showed good agreement between predicted and measured bending characteristics of the sample plain woven fabrics. However, the theoretical bending rigidities of the fabric specimens were more than the experimental values that have been reported in other research works [5]. Because in the proposed mechanical model bending behavior of the fabric was studied in real large deflection condition but, Peirce’s Cantilever test method describes pure bending theory of an elastic beam bending within the limit of linear strain. Differences between predicted and measured bending parameters have been shown in Table 4. One of the most outstanding results of this work is representing bending behavior of both prepared and nonproduced fabric samples for each value of advancement on horizontal platform of the Cantilever testing apparatus in the form of the successive curves. It is clear from plotted curves that, a convex region is formed when the fabric specimen is bent under its own weight. However, this special shape of deformation is not considered in Peirce's analysis. Besides, an apparatus could be design for directly determining

Table 4. Differences between predicted and measured bending parameters Sample No.2 Weft Warp direction direction -1.18 % -2.05 % -1.18 % -2.05 % -4.78 % -7.51 % +13.27 % +10.36 %

Sample No.1 Description Weft Warp direction direction -5.20 % -1.01 % L (cm) -5.20 % -1.01 % C (cm) -18.08 % -4.50 % G (mg·cm) -4.29 % +7.77 % ϕ (kg/cm2)

bending rigidity of a fabric specimen based on the new mechanical model.

Conclusion This work aims at the representation of bending behavior of plain woven fabrics before producing process using mechanical analysis of bending deformation of an elastic beam with one fixed edge. A novel relationship was derived between length of the overhang and constructional parameters

Mechanical Model for Bending Behavior of Woven Fabric

of the fabric. Theoretical bending characteristics were determined after plotting several curves that represent bending behavior of the sample fabrics with a given yarn count and yarn density on the basis of the proposed model using special numerical calculation technique and the constructional parameters of the fabrics determined by Peirce's structural model for plain woven fabric. Besides, fabrics with the same specifications were produced on a weaving machine. Then, these samples were tested for bending properties using Shirley bending tester. Although there were good agreement between predicted and measured bending properties, the theoretical bending rigidities of the samples were more than the experimental values that was in consistency with other previous researches.

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5. M. Ghane and M. Javadi-Toghchi, “Proceeding of the 1st International and the 7th National Iranian Textile Engineering Conference”, Rasht, Iran, 2009. 6. P. Szablewski, Autex. Res. J., 4, 206 (2004). 7. M. Kocik, W. Zurek, I. Krucinska, J. Gersak, and J. Jakubczyk, Fibers Text. East. Eur., 13, 31 (2005). 8. T. Matsuo, J. Text. Mach. Soc. Jap., 21, 260 (1968). 9. Standard Test Method for Stiffness of Fabrics, ASTMD1388-96, “Annual Book of ASTM Standards”, International Standards Worldwide, Section Seven, Textile, 2003. 10. S. P. Timoshenko and J. N. Goodier, “Theory of Elasticity”, 2th ed., pp.41-52, McGraw-Hill Book Company, New York, 1951. 11. B. George, Jr. Thomas, “Thomas’s Calculus”, 10th ed., pp.416-424, Massachusetts Institute of Technology, 2000. 12. G. A. V. Leaf and K. H. Kandil, J. Text. Inst., 71, 1 (1980). 13. W. Zurek, “The Structure of Yarn”, p.206, Foreign Scientific Publication Department of the National Center for Scientific, Technical and Economic Information, Warsaw, Poland, 1975. 14. J. W. Park and A. G. Oh, Text. Res. J., 76, 478 (2006). 15. J. W. S. Hearle, “Structural Mechanism of Fiber, Yarn and Fabric”, pp.92-99, 323-333, Cambridge Massachusetts, USA, 1969.