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Bending Analysis of Nonlinear Material Fibers with a Generalized Elliptical Cross-section Kyung Woo Lee1 Division of Fashion and Textiles, Dong-A University, Busan, South Korea ABSTRACT The bending moment–curvature relation was derived for generalized elliptical crosssection fibers made of nonlinear material and particular values were found to agree with those already obtained for elliptical cross-section and rectangular cross-section. The bending shape factors, the relation between bending rigidity and linear density were explicitly determined. The bending stress–strain relation of the generalized elliptical cross-section fibers was also derived.

Woven and knit fabrics are made using yarns, which are aggregates of oriented fibers. Nonwoven fabrics are made directly using fibers. To bend a woven or knit fabric requires bending of the yarns. To bend a nonwoven fabric requires bending of the fibers. The bending properties of fibers influence the behavior such as the drape, comport and handle of yarn and fabrics. In order to anticipate the bending behavior of yarn and fabrics, we must first understand the bending properties of fibers [8]. Finlayson investigated the bending of fiber made of linear elastic material [4]. Recently the bending formulas of fiber made of nonlinear material were obtained for fibers whose cross-section is an ellipse or a rectangle [5]. In this paper, we consider the boundary of the fiber cross-section is a generalized ellipse given by the equation:
x p
y q + =1 (1) a b where p, q are the shape parameters and a, b are semi-axes. Sections bounded by the ellipse or rectangle are included as particular cases. A fiber with a generalized elliptical cross-section is of practical interest because it can represent various fiber cross-section by putting in different values for p, q, a, b. A few examples are shown in Figures 1 and 2. Note that for the limiting case p, q → ∞, we obtain a rectangular cross-section. The purpose of this paper is to derive the bending formulas for generalized elliptical cross-section fibers made of nonlinear material.

Theoretical Considerations The Bending Moment–Curvature Relation In this study, the stress–strain law of the material of a fiber is assumed to be of the power law form: σ = Eε n

,

(2)

where σ represents the stress, ε is the strain, E is the tensile modulus and n is a constant. The typical values of n for polymers are around 0.9 [9]. In a bending problem, it is well known [6] that the bending moment is given by M = EIn+1 (κ)n

,

(3)

where M is the bending moment, κ is the curvature and In+1 is the n + 1-th area moment defined by  In+1 = yn+1 dA . (4) A

For n = 1, corresponding to linear elastic material, In+1 = I2 = A y2 dA which is the familiar second area moment. Determination of n + 1-th Area Moment for a Generalized Elliptical Cross-Section We may compute In+1 for a generalized elliptical crosssection. By Green’s theorem   1 In+1 = yn+1 dA = − yn+2 dx . (5) n + 2 A C The generalized ellipse may be represented parametrically by

1 Tel.: 8251 880 7334, fax: 8251 880 7335; e-mail: kwlee@mail. donga.ac.kr

Textile Res. J. 75(10), 710–714 (2005) DOI: 10.1177/0040517505059713

2

x = a(cos θ) p ,

2

y = b(sin θ) q ,

© 2005 SAGE Publications

0 ≤ θ ≤ 2π

(6)

www.sagepublications.com

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Figure 2. Various fiber cross-sections of ( ax )p + ( by )q = 1 for a = b = 1: (a) p = q = 0.5; (b) p = q = 1; (c) p = 2, q = 1; (d) p = q = 2; (e) p = 5, q = 1; (f) p = q = 10.

where (r) is the Gamma function defined by 

∞

(r) =

Substituting equation (6) into equation (5) yields  π 2 2n+4 2 8abn+2 In+1 = (sin θ ) q +1 (cos θ ) p −1 dθ . p(n + 2) 0

.

0

( by )q

Figure 1. Various fiber cross-sections of + = 1 for a = b = 1: (a) p = q = 0.5; (b) p = q = 1; (c) p = 2, q = 1; (d) p = q = 2; (e) p = 5, q = 1; (f) p = q = 10. ( ax )p

e−x x r−1 dx

Then we obtain In+1 =

8abn+2 p(n + 2)



π 2

(sin θ)

2n+4 q +1

(cos θ) p −1 dθ 2

0

  4ab n+2 1 = B + 1, p(n + 2) q p n+2

(7)

The integral on the right can be evaluated by using the Beta function B(r, s) defined by the integral [1]  π 2 (r)(s) B(r, s) = 2 , (sin θ )2r−1 (cos θ )2s−1 dθ = (r + s) 0 (8)

Using Beta function identities B(r, s + 1) =

s B(r, s) r+s

B(r, s) = B(s, r)

.

(9)

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Finally, we find the n + 1-th area moment of generalized ellipse In+1

  4abn+2 1 n+2 = B , p q p(n + 2) + q

.

(10)

In the special case of the ellipse, setting p = q = 2 in equation (10), we find that In+1



√ π



.

  √ 1 = π 2

lim

p,q→∞

p

, n+2 q



p(n + 2) + q

=

1 n+2

,

.

4abn+2 n+2

(11)

(12)

,

(13)

which is the same as that previously found.

Results Shape Factor for Bending Shape factor, η, is defined as the ratio of the bending rigidity EIn+1 of non-circular cross-section to the bending rigidity of circular cross-section with the same area and may be expressed by η=

In+1 (non-circular cross section) In+1 (circular cross section)

.

.

(16)

Equating the area of the circle to that of the generalized 2 ellipse (a = tb, t is a constant) and solving for Rb 2 ,

.

Finally we have the n + 1-th area moment for a rectangle by In+1 =

dA

b2 π(p + q)   = 2 R 4t B 1p , q1

 n  2 n   abn+2 n + 3  n+3 2

1



  4ab 1 1 Area = B , (p + q) p q

which agrees with the previous result [5]. As a further check on equation (10), we can find the n + 1-th area moment for a rectangle, in the limit p, q → ∞ in equation (10). By using the relation B

The area of a generalized ellipse is defined

By setting n = −1 in equations (5) and (10), we obtain the area of a generalized ellipse by

Finally we have the result In+1 =

(15)

A

By using recurrence formula and

In+1

  2Rn+3 1 n+2 = B , n+3 2 2

Area =

  4abn+2 1 n+2 = B , 2 2n + 6 2     n+2  1  n + 1 4ab 2 2   = 2n + 6  n+3 2

(r + 1) = r(r)

In the case of a circular cross-section with radius R, setting p = q = 2, a = b = R, in equation (10), we find

(14)

.

(17)

Substituting equations (10) and (15) into equation (14) and taking into account equation (17), finally we obtain the shape factor for the generalized elliptical crosssection    √ n+2 n+1 π (n + 3)  n+3 1 2  2  η= 2 t p(n + 2) + q  n+2 2

n+3   2 1 n+2 ( p + q) 1 1 ×B , . p q B p, q

(18)

A particular example is now worked out. The shape factor for the rectangular cross-section is easy to obtain from equation (18) for p, q → ∞. Since   B 1p , n+2 1 q lim = , p,q→∞ p(n + 2) + q n+2 (p + q)  =1 . (19) lim  p,q→∞ B 1 , 1 p q We have, therefore    √ n+2 n+1 π 1 2 (n + 3)  n+3  2  η= 2 t (n + 2)  n+2 2 which agrees with the previous result.

.

(20)

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in cross-section and be made of linear elastic material. He derived a relation such that

The Relation Between Bending Rigidity and Linear Density

  b 4M =E π ab2 ρ

Carlene derived the relation between bending rigidity and denier for linear elastic material such that G=

Eη (tex)2 4π S 2

,

(21)

where G is the bending rigidity defined as the product EI2 , η is the shape factor, and S is the density [3]. It follows from this relation that bending rigidity of a fiber depends on its shape, its modulus, its density and, most of all, its thickness (linear density). Since the fineness comes in as a squared term, and in view of the range of values occurring in practice, namely from 0.1 tex for a fine man-made fiber to 1 tex for a coarse wool and higher for some hair fibers and man-made monofilament, it will be the most important factor in determining the bending rigidity [7]. Experimentally, the bending rigidity has been found to be proportional to (tex)m and for the viscose rayon he obtained m = 1.8, which is slightly less than the theoretical value of 2. For fiber made of nonlinear material of the power law form, from equations (11) and (14), we have     √  n2 n  n + 3  Rn+3 . (22) G = Eη π n+3  2 By using the relations Area = πR2 =

tex S

n+3 2

g(n) = Eη(π )



 n 1 M = EIn+1 (κ) = EIn+1 ρ     n 4abn+2 1 n+2 1 =E B , p(n + 2) + q p q ρ n

,

    n2 n −( n +2 3 )   (S) n+3  n+3 2

.

(25)

Finally, we obtain the bending stress and strain for fiber with nonlinear material characteristics of power law form by  n M b , (26) = E 2 ab f (n, p, q) ρ   where f (n, p, q) = p(n +42) + q B 1p , n +q 2 Equation (26) reduces to equation (24) for the special case f (1, 2, 2).

Conclusions

(23)

where − n +2 2

(24)

where ρ is the radius of curvature. He defines the quantity π4M as the bending stress ab2 because it has the dimensions of stress and the quantity ρb as the bending strain. For a generalized elliptical cross-section fiber with nonlinear material characteristics of power law form, equation (3) may be written as

.

Finally we find the relation between the bending rigidity and linear density G = g(n)(tex)

,

The bending moment–curvature relation was obtained for nonlinear material fibers whose cross-section is a general ellipse. The relation between bending rigidity and linear density was also derived. Results obtained in this study can be used to optimize cross-section shape of fibers for bending properties of textile materials.

Appendix

,

which reduces to the linear elastic material case of equation 21 when n = 1.

In this section we will prove the two equations (12) and (19).   B 1p , n+2 q 1 = n+2 (1) Equation (12): lim p(n+2)+q p,q→∞

Bending Stress–Strain Relation for Nonlinear Material Characteristics of Power Law Form Chapman investigated the bending properties of single fiber [2]. In his work, fiber was assumed to be elliptical

To prove equation (12), using identities B(r, s) =

(r)(s) , (r + s)

(r + 1) = r(r)

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Textile Research Journal first we rewrite it in the more useful form for developing asymptotic relation   B 1p , n +q 2 p(n + 2) + q

1 n+2 1 n+2 + q  q 1 p p 1 n+2 = = p(n + 2) + q  p + q p(n + 2) + q  1  1 n+2 1 1  p  q  p  q pq  = 1 n+2  × 1 n+2  p + q +1  p + q +1 1  n+2  1  p +1  q +1   = n + 2  1p + n +q 2 + 1 

Using the known value (1) = 1, we finally arrive at the asymptotic relation   B 1p , n+2 q lim p,q→∞ p(n + 2) + q 1   n+2  1  p +1  q +1   = lim p,q→∞ n + 2  1p + n+2 +1 q =

1 1 (1) = n + 2 (1)(1) n+2

(2) Equation 19: lim

(p+q) 
1 1 p,q

p,q→∞ B

= 1.

In a similar manner, we find that    1p + q1 (p + q)   = (p + q)  1   1  B 1p , q1  p  q   pq  1p + q1 + 1 p+q     = (p + q) p 1p q q1

=



 1 p

 + q1 + 1    + 1  q1 + 1

1 p

.

The final results is thus    1p + q1 + 1 (p + q)     = lim  1 lim  p,q→∞ B 1 , 1 p,q→∞  + 1  q1 + 1 p q p =

(1) =1 (1)(1)

.

Literature Cited 1. Arfken, G. B., and Weber, H. J., “Mathematical Methods for Physicists,” pp. 331–337, 613–619. Academic Press, San Diego, 1995. 2. Chapman, B. M., The Bending Stress–Strain Properties of Single Fibres and the Effect of Temperature and Relative Humidity, J. Textile Inst. 64, 312–327 (1973). 3. Carlene, P. W., The Relation Between Fibre and Yarn Flexural Rigidity in Continuous Viscose Yarns, J. Textile Inst. 41, 159–172 (1950). 4. Finlayson, D., Yarn for Special Purpose – Effect of Filament Size, J. Textile Inst. 37, 168–179 (1946). 5. Lee, K., Bending of Fibres with Non-linear Material Characteristic of Power Law Form, J. Textile Inst. 93, 132–136 (2002). 6. Lewis, G., and Monaso, F., Large Deflections of Cantilever Beams of Non-Linear Materials, Comput. Struct. 14, 357– 360 (1981). 7. Morton, W. E., and Hearle, J. W. S., “Physical Properties of Textiles Fibres,” pp. 399–411, Textile Institute, Manchester, 1993. 8. Warner, S. B., “Fiber Science,” pp. 158–160, Prentice Hall, New Jersey, 1995. 9. Williams, J. G., “Stress Analysis of Polymers,” pp. 142–143, Ellis Horwood, Chichester, 1980.