**2. Geometrical model of woven structures**

The properties of the fabric depend on the fabric structure. The formal structure of a woven fabric is defined by weave, thread density, crimp and yarn count. The interrelation between fabric parameters can be obtained by considering a geometrical model of the fabric. The model is not merely an exercise in mathematics. It is not only useful in determining the entire structure of a fabric from a few values given in technological terms but it also establishes a base for calculating various changes in fabric geometry when the fabric is subjected to known extensions in a given direction or known compressions or complete swelling in aqueous medium. It has been found useful for weaving of maximum sett structures and also in the analysis and interpretation of structure-property relationship of woven fabrics. Mathematical deductions obtained from simple geometrical form and physical characteristics of yarn combined together help in understanding various phenomena in fabrics.

### **2.1 Basic relationship between geometrical parameters**

The geometrical model is mainly concerned with the shape taken up by the yarn in the warp or weft cross-section of the fabric. It helps to quantitatively describe the geometrical parameters. The basic model (Pierce, 1937) is shown in figure 5. It represents a unit cell interlacement in which the yarns are considered inextensible and flexible. The yarns have circular cross-section and consist of straight and curved segments. The main advantages in considering this simple geometry are:

Fig. 5. Peirce's model of plain weave

Modeling of Woven Fabrics Geometry and Properties 7

When the interlacement pattern is modified by changing the float length, the structure of the fabric changes dramatically. It has a profound effect on the geometry of the yarn interlacement and related properties in the woven fabric. The maximum weavability limit is predicted by extending the Peirce's geometrical model for non-plain weaves by soft computing. This information is helpful to the weavers in avoiding attempts to weave impossible constructions thus saving time and money. It also helps to anticipate difficulty of weaving and take necessary steps in warp preparations. The relationship between the cover factors in warp and weft direction is demonstrated for circular and racetrack cross-section for plain, twill, basket and satin weave in later part of this chapter. Non plain weave fabric affords further flexibility for increasing fabric mass and fabric cover. As such they enlarge scope of the fabric designer and researcher. Figure 6 shows the relationship between warp

Fig. 6. Relation between average thread spacing in warp and weft for different weaves

<sup>2</sup> <sup>1</sup> 1 1

or <sup>1</sup> <sup>2</sup> 1 1 *D*(sec 1) tan 0 *θ θ p h*

<sup>2</sup> 1 1 we get, 0 x x <sup>1</sup> <sup>2</sup> <sup>1</sup> 2 2 *h h <sup>D</sup> <sup>p</sup>* 

1 1 sin (1 cos )

*θ θ*

cos sin *<sup>p</sup> <sup>D</sup> <sup>θ</sup> <sup>h</sup> <sup>D</sup> <sup>θ</sup> <sup>l</sup> <sup>D</sup><sup>θ</sup>*

**2.1.1 Relation between weave composition and structural parameters** 

and weft thread spacing for different weaves for a given yarn.

(Circular cross-section, yarn tex=30, Ø=0.6, ρ=1.52)

1 1

**2.2 Some derivatives** 

Substituting <sup>1</sup> 1

**2.2.1 Relation between** *p, h, θ* **and** *D*

tan <sup>x</sup> <sup>2</sup> *<sup>θ</sup>*

From equations 4 and 5 we get:


From the two-dimensional unit cell of a plain woven fabric, geometrical parameters such as thread-spacing, weave angle, crimp and fabric thickness are related by deriving a set of equations. The symbols used to denote these parameters are listed below.

*d* - diameter of thread

*p* - thread spacing

*h* - maximum displacement of thread axis normal to the plane of cloth ( crimp height)

*θ* - angle of thread axis to the plane of cloth (weave angle in radians)

*l* - length of thread axis between the planes through the axes of consecutive cross- threads (modular length)

*c* - crimp (fractional)

*D = d*1 *+ d*<sup>2</sup>

Suffix 1 and 2 to the above parameters represent warp and weft threads respectively.

In the above figure projection of yarn axis parallel and normal to the cloth plane gives the following equations:

$$c\_1 = \frac{l\_1}{p\_2} - 1\tag{3}$$

$$p\_2 = (l\_1 - D\theta\_1)\cos\theta\_1 + D\sin\theta\_1\tag{4}$$

$$\eta\_1 = (\mathfrak{l}\_1 - D\theta\_1)\sin\theta\_1 + D(1 - \cos\theta\_1) \tag{5}$$

Three similar equations are obtained for the weft direction by interchanging suffix from 1 to 2 or vice-versa as under:

$$\mathbf{r}\_{C2} = \frac{l\_2}{p\_1} - \mathbf{1} \tag{6}$$

$$p\_1 = (\mathfrak{l}\_2 - D\theta\_2)\cos\theta\_2 + D\sin\theta\_2\tag{7}$$

$$\mathbf{r}\_{12} = (\mathbf{l}\_2 - D\boldsymbol{\theta}\_2)\sin\theta\_2 + D(\mathbf{1} - \cos\theta\_2) \tag{8}$$

$$\text{Also},\tag{9}$$

$$d\_1 + d\_2 \equiv l\_1 + l\_2 = D\tag{9}$$

In all there are seven equations connecting eleven variables. If any four variables are known then the equations can be solved and the remaining variables can be determined. Unfortunately, these equations are difficult to solve. Researchers have tried to solve these equations using various mathematical means to find new relationships and also some simplified useful equations.

2. Able to calculate the resistance of the cloth to mechanical deformation such as initial extension, bending and shear in terms of the resistance to deformation of individual fibers. 3. Provide information on the relative resistance of the cloth to the passage of air, water or

From the two-dimensional unit cell of a plain woven fabric, geometrical parameters such as thread-spacing, weave angle, crimp and fabric thickness are related by deriving a set of

*l* - length of thread axis between the planes through the axes of consecutive cross- threads

In the above figure projection of yarn axis parallel and normal to the cloth plane gives the

1 1

2 <sup>1</sup> *<sup>l</sup>*

Three similar equations are obtained for the weft direction by interchanging suffix from 1 to

2 2

*c*

1 <sup>1</sup> *<sup>l</sup>*

Also, *d*1*+d*2*=h*1*+h*2*=D* (9) In all there are seven equations connecting eleven variables. If any four variables are known then the equations can be solved and the remaining variables can be determined. Unfortunately, these equations are difficult to solve. Researchers have tried to solve these equations using various mathematical means to find new relationships and also some

*<sup>p</sup>* (3)

*<sup>p</sup>* (6)

<sup>2</sup> 11 1 1 *p* ( )cos sin *l D D θθ θ* (4)

11 1 1 <sup>1</sup> *h l* ( )sin (1 cos ) *D D θθ θ* (5)

<sup>1</sup> 22 2 2 *p* ( )cos sin *l D D θθ θ* (7)

22 2 2 <sup>2</sup> *h l* ( )sin (1 cos ) *D D θθ θ* (8)

*h* - maximum displacement of thread axis normal to the plane of cloth ( crimp height)

Suffix 1 and 2 to the above parameters represent warp and weft threads respectively.

*c*

1. Helps to establish relationship between various geometrical parameters

4. Guide to the maximum density of yarn packing possible in the cloth.

equations. The symbols used to denote these parameters are listed below.

*θ* - angle of thread axis to the plane of cloth (weave angle in radians)

light.

*d* - diameter of thread *p* - thread spacing

(modular length) *c* - crimp (fractional)

following equations:

2 or vice-versa as under:

simplified useful equations.

*D = d*1 *+ d*<sup>2</sup>

#### **2.1.1 Relation between weave composition and structural parameters**

When the interlacement pattern is modified by changing the float length, the structure of the fabric changes dramatically. It has a profound effect on the geometry of the yarn interlacement and related properties in the woven fabric. The maximum weavability limit is predicted by extending the Peirce's geometrical model for non-plain weaves by soft computing. This information is helpful to the weavers in avoiding attempts to weave impossible constructions thus saving time and money. It also helps to anticipate difficulty of weaving and take necessary steps in warp preparations. The relationship between the cover factors in warp and weft direction is demonstrated for circular and racetrack cross-section for plain, twill, basket and satin weave in later part of this chapter. Non plain weave fabric affords further flexibility for increasing fabric mass and fabric cover. As such they enlarge scope of the fabric designer and researcher. Figure 6 shows the relationship between warp and weft thread spacing for different weaves for a given yarn.

Fig. 6. Relation between average thread spacing in warp and weft for different weaves (Circular cross-section, yarn tex=30, Ø=0.6, ρ=1.52)

#### **2.2 Some derivatives**

#### **2.2.1 Relation between** *p, h, θ* **and** *D*

From equations 4 and 5 we get:

$$\begin{aligned} \left(l\_1 - D\vartheta\_1\right) &= \frac{p\_2 - D\sin\theta\_1}{\cos\theta\_1} = \frac{\mu\_1 - D(1 - \cos\theta\_1)}{\sin\theta\_1} \\\\ &\text{or} \quad D(\sec\theta\_1 - 1) - p\_2\tan\theta\_1 + \mu\_{I1} = 0 \end{aligned}$$

Substituting <sup>1</sup> 1 tan <sup>x</sup> <sup>2</sup> *<sup>θ</sup>*

$$\text{we get, } \ge\_1^2 \left( D - \frac{h\_1}{2} \right) - p\_2 \ge\_1 + \frac{h\_1}{2} = 0.$$

Modeling of Woven Fabrics Geometry and Properties 9

2 2 1 4

These four equations are not new equations in this exercise. They are derived from the previous seven original equations. However they give simple and direct relationships

A woven fabric in which warp and weft yarns do not have mobility within the structure as they are in intimate contact with each other are called jammed structures. In such a structure the warp and weft yarns will have minimum thread spacing. These are closely woven fabrics and find applications in wind-proof, water-proof and bullet-proof

During jamming the straight portion of the intersecting yarn in figure 5 will vanish so that in

1 *l θ D*

1

1 1 *h D*(1 cos )

<sup>2</sup> <sup>1</sup> *p D*sin*θ*

Similarly, for jamming in the weft direction *l*2 – *Dθ*2 = 0, equations 7 and 8 will reduce to the

1 2 <sup>1</sup> <sup>2</sup> *DD D h h* (1 cos ) (1 cos ) *θ θ*

This is an equation relating warp and weft spacing of a most closely woven fabric.

*θ θ*

1 1 11 cos sin *<sup>l</sup> θ θθ <sup>D</sup>* 

2 2 1 2 1 11 *p p D D* 

above equations with suffix interchanged from 1 to 2 and vice-versa.

For a fabric being jammed in both directions we have:

between four fabric parameters *h, p, c* and *θ*.

**2.2.3 Jammed structures** 

equation 4 and 5, *l*1–*Dθ*1 = 0

Equations 4 and 5 will reduce to

**2.2.4 Cross threads pulled straight** 

If the weft yarn is pulled straight *h*2 = 0 and *h*1 = *D*,

Equation 5 will give 111 1 *DD D* ( )sin (1 cos ) *l*

requirements.

<sup>3</sup> *h c <sup>p</sup>* (13)

or cos*θ*1+cos*θ*2=1 (14)

(15)

$$\text{For real fabrics} \quad \chi\_1 = \frac{\tan \theta\_1}{2} \quad = p\_2 - \frac{\sqrt{p\_2^2 - 2h\_1(D - \frac{h\_1}{2})}}{2D - h\_1} = \frac{p\_2 - \sqrt{p\_2^2 - h\_2^2 - D^2}}{D + h\_2}$$

Using value of *x*1, one can calculate *θ*, *l* and *c* and also other parameters.

Similarly, using equation 7 and 8, and by eliminating *l* and substituting x1 as above, we will arrive at a more complex equation as:

$$\frac{c\_1}{2} + \mathbf{x}\_1 \frac{D}{p\_2} - \mathbf{x}\_1^2 (1 + \frac{c\_1}{2}) = \frac{D}{p\_2} (1 - \mathbf{x}\_1^2) \tan^{-1} \mathbf{x}\_1$$

It is difficult to solve this equation algebraically for x1. However one can substitute value of x1 obtained earlier to solve this equation just for an academic interest.

These seven equations have been solved by soft computing in order to establish several useful relationships. However, at this stage, one can generalize the relationship as:

$$h\_1 = f\left(p\_2, c\_1\right)$$

This function *f* can be obtained by plotting *p* and *h* for different values of *c*.

#### **2.2.2 Functional relationship between** *p, h, c*

Trigonometric expansion of equations 4 and 5 gives:

$$\begin{aligned} p\_2 &= l\_1 - \frac{l\_1 \theta\_1^2}{2} + \frac{D \theta\_1^3}{3} + \frac{l\_1 \theta\_1^4}{24} + \dots - \frac{l\_1}{24} \\\\ p\_{l1} &= l\_1 \theta\_1 - \frac{D \theta\_1^2}{2} - \frac{l\_1 \theta\_1^3}{6} + \frac{D \theta\_1^4}{8} + \dots - \frac{l\_1}{2} \end{aligned}$$

When *θ* is small, higher power of *θ* can be neglected which gives:

$$h\_1 = l\_1 \theta\_{1\prime} \ p\_2 = l\_{1\prime} \ c\_2 = \frac{\theta\_1^2}{2} \ , \ \mathbf{h}\_1 = p\_2 \sqrt{2c\_1}$$

and these equations reduce to:

$$
\theta\_1 = (\mathbf{2}\_{\mathcal{C}1})^{\frac{1}{2}} \tag{10}
$$

$$
\theta\_2 = \left(\mathbb{Z}\_{\mathcal{C}\_2}\right)^{\frac{1}{2}} \tag{11}
$$

$$p\_{l1} = \frac{4}{3} p\_2 \sqrt{c\_1} \tag{12}$$

$$M\_2 = \frac{4}{3} p\_1 \sqrt{c\_2} \tag{13}$$

These four equations are not new equations in this exercise. They are derived from the previous seven original equations. However they give simple and direct relationships between four fabric parameters *h, p, c* and *θ*.

### **2.2.3 Jammed structures**

8 Woven Fabrics

<sup>1</sup> <sup>2</sup> <sup>2</sup> <sup>2</sup> <sup>2</sup> <sup>1</sup> 2 2 <sup>2</sup>

*<sup>p</sup> D D h h*

Similarly, using equation 7 and 8, and by eliminating *l* and substituting x1 as above, we will

 1 1 <sup>2</sup> 2 1 1 1 1 1

It is difficult to solve this equation algebraically for x1. However one can substitute value of

These seven equations have been solved by soft computing in order to establish several

 *h*1 = *f* (*p*2,*c*1)

234 1 1 1 11

> 234 111 1

2 1 1 11 1 2 2 2 <sup>1</sup> <sup>1</sup> , , , h <sup>2</sup> <sup>2</sup> *<sup>θ</sup> h l <sup>θ</sup> p p l c <sup>c</sup>*

> <sup>1</sup> 1

 <sup>2</sup> 1

1 1 2 4

*θ*<sup>1</sup> 2*c* 2 (10)

*θ*<sup>2</sup> 2*c* 2 (11)

<sup>3</sup> *h c <sup>p</sup>* (12)

*D D <sup>θ</sup> <sup>l</sup> θ θ h l <sup>θ</sup>*

 

<sup>2</sup> <sup>1</sup> 2 3 24 *l l D*

1 11 268

2 2 x x (1 ) 1 x tan x 2 2

useful relationships. However, at this stage, one can generalize the relationship as:

This function *f* can be obtained by plotting *p* and *h* for different values of *c*.

*c c D D p p*

*h*

1 2

*p p h D*

2 1

*( ) tan<sup>θ</sup>*

2

*p h D*

For real fabrics

1 2

**2.2.2 Functional relationship between** *p, h, c* 

and these equations reduce to:

Trigonometric expansion of equations 4 and 5 gives:

*p l*

When *θ* is small, higher power of *θ* can be neglected which gives:

arrive at a more complex equation as:

2 x 2 2

Using value of *x*1, one can calculate *θ*, *l* and *c* and also other parameters.

x1 obtained earlier to solve this equation just for an academic interest.

A woven fabric in which warp and weft yarns do not have mobility within the structure as they are in intimate contact with each other are called jammed structures. In such a structure the warp and weft yarns will have minimum thread spacing. These are closely woven fabrics and find applications in wind-proof, water-proof and bullet-proof requirements.

During jamming the straight portion of the intersecting yarn in figure 5 will vanish so that in equation 4 and 5, *l*1–*Dθ*1 = 0

$$\frac{l\_1}{D} = \theta\_1$$

Equations 4 and 5 will reduce to

$$\begin{aligned} h\_1 &= D(1 - \cos \theta\_1) \\\\ p\_2 &= D \sin \theta\_1 \end{aligned}$$

Similarly, for jamming in the weft direction *l*2 – *Dθ*2 = 0, equations 7 and 8 will reduce to the above equations with suffix interchanged from 1 to 2 and vice-versa.

For a fabric being jammed in both directions we have:

$$D = h\_1 + h\_2 = D(1 - \cos\theta\_1) + D(1 - \cos\theta\_2)$$
 
$$\text{or } \cos\theta\_1 + \cos\theta\_2 = 1 \tag{14}$$

$$\sqrt{1 - \left(\frac{p\_1}{D}\right)^2} + \sqrt{1 - \left(\frac{p\_2}{D}\right)^2} = 1\tag{15}$$

This is an equation relating warp and weft spacing of a most closely woven fabric.

#### **2.2.4 Cross threads pulled straight**

If the weft yarn is pulled straight *h*2 = 0 and *h*1 = *D*,

Equation 5 will give 111 1 *DD D* ( )sin (1 cos ) *l θ θ*

$$\cos\theta\_1 = \left(\frac{l\_1}{D} - \theta\_1\right)\sin\theta\_1$$

Modeling of Woven Fabrics Geometry and Properties 11

*d ab*

*hh ddbb* 12 1212

11 12 2 1 1 2 4

This can be used to relate yarn diameter and crimp height by simply substituting in

34.14 *v v hhdd <sup>D</sup>*

These are useful equation to be used subsequently in the crimp interchange derivation.

In race track model (Kemp, 1958; Love, 1954) given in figure 8, *a* and *b* are maximum and minimum diameters of the cross-section. The fabric parameters with superscript refer to the zone AB, which is analogous to the circular thread geometry; the parameters without superscript refer to the race track geometry, a repeat of this is between CD. Then the basic

1 2

1 2 f1 f2

 *ρ ρ* 

1212

1 280.2 *T T hhdd*

1212

= 0.65, *ρ<sup>f</sup>* = 1.52 for cotton fiber

Yarn diameter is given by its specific volume, *v* and yarn count as under:

*<sup>N</sup>* , *N* is the English count.

*<sup>ρ</sup>* , assuming,

f Tex Tex

Fig. 7. Elliptical cross-section

equation 19 to obtain:

**2.2.5.2 Race track cross-section** 

equations will be modified as under:

assuming,

280.2 <sup>280</sup> *<sup>d</sup>* 

mils 34.14 *<sup>v</sup>*

*d*

cm

<sup>3</sup> *bbhh c c p p* (19)

1 2

1 2

1 280

1 2

*T T*

(20)

*N N* 

= 0.65, *ρ<sup>f</sup>* = 1.52 for cotton fiber

$$\text{for } \theta\_1 + \cot \theta\_1 = \frac{l\_1}{D} \tag{16}$$

This equation gives maximum value of *θ*1 for a given value of*l*1/*D*

The above equation will be valid for warp yarn being straight by interchange of suffix from 1 to 2.

*However, the weft thread can be restricted in being pulled straight by the jamming of warp threads*. In such a case,

$$I\_1 - D\theta\_1 = 0$$

$$\text{or} \quad \theta\_1 = \frac{I\_1}{D}$$

Equation 5 will become

$$\eta\_{12} = D - \eta\_{11} = D - D(1 - \cos \theta\_1) = D \cos \frac{l\_1}{D} \tag{17}$$

If the weft thread is pulled straight and warp is just jammed

$$\text{Then } \frac{l\_1}{D} = \theta\_1 = \frac{\pi}{2} \tag{18}$$

These are useful conditions for special fabric structure.

#### **2.2.5 Non circular cross-section**

So far, it is assumed that yarn cross-section is circular and yarn is incompressible. However, the actual cross-section of yarn in fabric is far from circular due to the system of forces acting between the warp and weft yarns after weaving and the yarn can never be incompressible. This inter-yarn pressure results in considerable yarn flattening normal to the plane of the cloth even in a highly twisted yarn. Therefore many researchers have tried to correct Peirce's original relationship by assuming various shapes for the cross-section of yarn. Two important cross-sectional shapes such as elliptical and race-track are discussed below.

#### **2.2.5.1 Elliptical cross-section**

Peirce's elliptical yarn cross-section is shown in figure 7; the flattening factor is defined as

$$e = \sqrt{\frac{b}{a}}$$

Where *b* = minor axis of ellipse, *a* = major axis of ellipse

The area of ellipse is (π/4)*ab*. If *d* is assumed as the diameter of the equivalent circular cross– section yarn, then

<sup>1</sup> or cot 1 1 *<sup>l</sup> θ θ <sup>D</sup>*

The above equation will be valid for warp yarn being straight by interchange of suffix from

*However, the weft thread can be restricted in being pulled straight by the jamming of warp threads*.

*l*1 1 *Dθ* 0

<sup>1</sup> or 1 *<sup>l</sup> <sup>θ</sup> <sup>D</sup>*

<sup>1</sup> Then <sup>1</sup> <sup>2</sup> *l θ D*

So far, it is assumed that yarn cross-section is circular and yarn is incompressible. However, the actual cross-section of yarn in fabric is far from circular due to the system of forces acting between the warp and weft yarns after weaving and the yarn can never be incompressible. This inter-yarn pressure results in considerable yarn flattening normal to the plane of the cloth even in a highly twisted yarn. Therefore many researchers have tried to correct Peirce's original relationship by assuming various shapes for the cross-section of yarn. Two important cross-sectional shapes such as elliptical and race-track are discussed

Peirce's elliptical yarn cross-section is shown in figure 7; the flattening factor is defined as

*e <sup>a</sup>*

*b*

The area of ellipse is (π/4)*ab*. If *d* is assumed as the diameter of the equivalent circular cross–

This equation gives maximum value of *θ*1 for a given value of*l*1/*D*

If the weft thread is pulled straight and warp is just jammed

These are useful conditions for special fabric structure.

Where *b* = minor axis of ellipse, *a* = major axis of ellipse

**2.2.5 Non circular cross-section** 

**2.2.5.1 Elliptical cross-section** 

section yarn, then

1 to 2.

below.

In such a case,

Equation 5 will become

(16)

1 2 1 <sup>1</sup> (1 cos ) cos *<sup>l</sup> h h D DD D <sup>θ</sup> <sup>D</sup>* (17)

(18)

$$d = \sqrt{ab}$$

$$h\_1 + h\_2 = d\_1 + d\_2 = b\_1 + b\_2$$

$$b\_1 + b\_1 = h\_1 + h\_2 = \frac{4}{3} \left[ p\_1 \sqrt{c\_2} + p\_2 \sqrt{c\_1} \right] \tag{19}$$

Yarn diameter is given by its specific volume, *v* and yarn count as under:

Fig. 7. Elliptical cross-section

This can be used to relate yarn diameter and crimp height by simply substituting in equation 19 to obtain:

$$
\eta\_{11} + \eta\_{12} = d\_1 + d\_2 = D = 34.14 \left( \sqrt{\frac{\upsilon\_1}{N\_1}} + \sqrt{\frac{\upsilon\_2}{N\_2}} \right) \tag{20}
$$

$$
\eta\_{11} + \eta\_{12} = d\_1 + d\_2 = \frac{1}{280.2} \left( \sqrt{\frac{T\_1}{\varphi\_1 \rho\_{f1}}} + \sqrt{\frac{T\_2}{\varphi\_2 \rho\_{f2}}} \right) = \frac{1}{280} \left( \sqrt{T\_1} + \sqrt{T\_2} \right)
$$

assuming, = 0.65, *ρ<sup>f</sup>* = 1.52 for cotton fiber

These are useful equation to be used subsequently in the crimp interchange derivation.

#### **2.2.5.2 Race track cross-section**

In race track model (Kemp, 1958; Love, 1954) given in figure 8, *a* and *b* are maximum and minimum diameters of the cross-section. The fabric parameters with superscript refer to the zone AB, which is analogous to the circular thread geometry; the parameters without superscript refer to the race track geometry, a repeat of this is between CD. Then the basic equations will be modified as under:

Modeling of Woven Fabrics Geometry and Properties 13

*h*1*+d*1 *or h*2*+d*2, whichever is greater.

In a fabric with coarse and fine threads in the two directions and by stretching the fine thread straight, maximum crimp is obtained for the coarse thread. In this case the fabric

Maximum Thickness =*D* + *d*coarse , since *h*coarse = D

In fabric, cover is considered as fraction of the total fabric area covered by the component

*d ET K*

*ρ*

f f 280.2 28.02

*<sup>ρ</sup>*

<sup>1</sup> *K ET K* 10 is cover factor

*T* is yarn tex, *E* is threads per cm = 1/*p* suffix 1 and 2 will give warp and weft cover

max <sup>f</sup> *K* 28.02

1 2

1

*ρ*

1 2

1 2 12 Fabric cover factor K K – K K / 28 (28)

28.02 28.02 *K K K K* 

In this case the fabric gives *minimum thickness* =1/2(*h*1*+d*1*+ h*2*+d*2) *=D*; *h*1*=D* – *d*<sup>1</sup> Such a fabric produces a smooth surface and ensures uniform abrasive wear.

When yarn cross- section is flattened, the fabric thickness can be expressed as

**2.3.1 Fabric thickness** 

gives *maximum thickness* as under;

*h*1*+b*<sup>1</sup> or *h*2*+b*2, whichever is greater

Fractional fabric cover is given by:

**2.3.2 Fabric cover** 

factors.

for 1 *<sup>d</sup>*

Fabric thickness for a circular yarn cross-section is given by

When the two threads project equally, then *h*1*+d*1 *= h*2*+d*<sup>2</sup>

yarns. For a circular cross-section cover factor is given as:

*<sup>p</sup>* , cover factor is maximum and given by,

For race track cross-section the equation will be

*p*

1 2 12 1 2 12 *d d dd p p pp*

Multiplying by 28.02 and taking 28.02 ≈ 28 we get fabric cover factor as under:

Fig. 8. Race track cross-section

$$p\_2^"=p\_2 - (a\_2 - b\_2) \tag{21}$$

$$l\_1^\prime = l\_1 - (a\_2 - b\_2) \tag{22}$$

$$\vec{r\_{C1}} = \frac{\vec{l\_1} - \vec{p\_2}}{\vec{p\_2}} = \frac{c\_1 p\_2}{p\_2 - (a\_2 - b\_2)}\tag{23}$$

Similarly,

$$c\_2^{\prime} = \frac{c\_2 p\_1}{p\_1 - (a\_1 - b\_1)}\tag{24}$$

$$
\mu\_{l1} = \frac{4}{3} p\_2^\dagger \sqrt{\mathcal{C}\_1} \tag{25}
$$

$$
\eta\_{12} = \frac{4}{3} p\_1^\dagger \sqrt{c\_2} \tag{26}
$$

*h*1+*h*2 = *B* = *b*1 +*b*<sup>2</sup>

And also if both warp and weft threads are jammed, the relationship becomes

$$\sqrt{B^2 - \left(\vec{p\_1}\right)^2} + \sqrt{B^2 - \left(\vec{p\_2}\right)^2} = B \tag{27}$$

#### **2.3 Prediction of fabric properties**

Using the fabric parameters discussed in the previous section it is possible to calculate the *Fabric thickness, Fabric cover, Fabric mass and Fabric specific volume.*

#### **2.3.1 Fabric thickness**

12 Woven Fabrics

'

'

1 '

'

*c*

2

And also if both warp and weft threads are jammed, the relationship becomes

*Fabric thickness, Fabric cover, Fabric mass and Fabric specific volume.*

*c*

' ' '

1 1 2 2

2 1

*l c p p*

<sup>2</sup> 2 2 <sup>2</sup> ( )

*p p a b*

<sup>1</sup> 1 1 ( ) *c p*

' ' 1 1 2 4

' ' 2 2 1 4

2 2 2 2 ' '

Using the fabric parameters discussed in the previous section it is possible to calculate the

2 2 2 2 *p p* ( ) *a b* (21)

11 2 2 *ll ab* ( ) (22)

(23)

*<sup>p</sup> a b* (24)

<sup>3</sup> *h c <sup>p</sup>* (25)

<sup>3</sup> *h c <sup>p</sup>* (26)

1 2 *B B p pB* (27)

Fig. 8. Race track cross-section

Similarly,

*h*1+*h*2 = *B* = *b*1 +*b*<sup>2</sup>

**2.3 Prediction of fabric properties** 

Fabric thickness for a circular yarn cross-section is given by

*h*1*+d*1 *or h*2*+d*2, whichever is greater.

When the two threads project equally, then *h*1*+d*1 *= h*2*+d*<sup>2</sup> In this case the fabric gives *minimum thickness* =1/2(*h*1*+d*1*+ h*2*+d*2) *=D*; *h*1*=D* – *d*<sup>1</sup>

Such a fabric produces a smooth surface and ensures uniform abrasive wear.

In a fabric with coarse and fine threads in the two directions and by stretching the fine thread straight, maximum crimp is obtained for the coarse thread. In this case the fabric gives *maximum thickness* as under;

$$\text{Maximum Thickness} = \text{D} + d\_{\text{coarse},} \text{ since } h\_{\text{coarse}} = \text{D}$$

When yarn cross- section is flattened, the fabric thickness can be expressed as *h*1*+b*<sup>1</sup> or *h*2*+b*2, whichever is greater

#### **2.3.2 Fabric cover**

In fabric, cover is considered as fraction of the total fabric area covered by the component yarns. For a circular cross-section cover factor is given as:

$$\frac{d}{p} = \frac{E\sqrt{T}}{280.2\sqrt{\rho\rho\_f}} = \frac{K}{28.02\sqrt{\rho\rho\_f}}$$

$$K = E\sqrt{T} \times 10^{-1} \text{ K is cover factor}$$

*T* is yarn tex, *E* is threads per cm = 1/*p* suffix 1 and 2 will give warp and weft cover factors.

for 1 *<sup>d</sup> <sup>p</sup>* , cover factor is maximum and given by,

$$K\_{\text{max}} = 28.02 \sqrt{\rho \rho\_{\text{f}}}$$

Fractional fabric cover is given by:

$$\frac{d\_1}{p\_1} + \frac{d\_2}{p\_2} - \frac{d\_1 d\_2}{p\_1 p\_2} = \frac{1}{28.02} \left[ K\_1 + K\_2 - \frac{K\_1 K\_2}{28.02} \right]$$

Multiplying by 28.02 and taking 28.02 ≈ 28 we get fabric cover factor as under:

$$\text{Fabric cover factor} = \mathbf{K}\_1 + \mathbf{K}\_2 - \left(\mathbf{K}\_1 \mathbf{K}\_2 / 28\right) \tag{28}$$

For race track cross-section the equation will be

Modeling of Woven Fabrics Geometry and Properties 15

Maximum cover in a jammed fabric is only possible by keeping the two consecutive yarns (say warp) in two planes so that their projections are touching each other and the cross thread (weft) interlaces between them. In this case the weft will be almost straight and

d1/p1 = 1 will give K1 = Kmax

*d d <sup>d</sup> <sup>d</sup>*

1. Use fine yarn in the direction where maximum cover is desired and keep them in two planes so that their projections touch each other and use coarse yarn in the cross direction.

Both options will give maximum cover in warp and weft but first option will give more

The cover factor indicates the area covered by the projection of the thread. The ooziness of yarn, flattening in finishing and regularity further improves the cover of cloth. It also gives a basis of comparison of hardness, crimp, permeability, transparency. Higher cover factor can be obtained by the lateral compression of the threads. It is possible to get very high values only in one direction where threads have higher crimp. Fabrics differing in yarn counts and average yarn spacing can be compared based on the fabric cover. The degree of flattening for race track and elliptical cross-section can be estimated from fabric thickness measurements to evaluate *b* and *a* from microscopic measurement of the fabric surface. The classical example in this case is that of a poplin cloth in which for warp threads

*p*1 = *D* sin *θ*<sup>2</sup>

*d* = *D/*2 = *D* sin *θ*<sup>2</sup>

*θ*2 = 30º = 0.5236

<sup>1</sup> 82 18 1.4364(using cos cos 1)

*p*2 = *D*sin*θ*1 = 0.991*D* ≈ 2*p*<sup>1</sup>

*l*1 = *Dθ*1 = 1.14364

 1 2

2 1

2 for <sup>2</sup> <sup>3</sup>

and the spacing between the weft yarn, *p*2 =*D* sin*θ*1 = *D* (for *θ* = 900) , *p*2= *d*1+*d*<sup>2</sup>

2 2

*p d d*

2. As in (1) instead of coarse yarn insert two fine yarns in the same shed.

2 1 2

**2.4 Maximum cover and its importance** 

maximum bending will be done by the warp.

If *d*1 = *d*2 then *d*1 = *d*2/*p*2 =0.5 and *K*<sup>2</sup> =0.5 *K*max

This is the logic for getting maximum cover in any fabric.

*p*1 = *d*1 and for *d*1 = *d*2 = *D*/2 and for jamming in both directions

0 '

This will give *K*2 = 2/3 *K*max,

The principles are as under:

thickness than the second case.

$$\frac{a}{p} = \frac{d}{ep\sqrt{1 + \frac{4}{\pi} \left(\frac{1}{c} - 1\right)} \times 28.02 \sqrt{\wp \rho\_f}} \quad \text{here} \quad e = b/a$$

$$= \frac{K}{e\sqrt{1 + \frac{4}{\pi} \left(\frac{1}{c} - 1\right)} \times 28.02 \sqrt{\wp \rho\_f}}$$

For elliptical cross-section the equation will be;

$$\frac{a}{p} = \frac{d}{ep} = \frac{E\sqrt{T}}{280.2 \ e\sqrt{\rho\_{\text{-f}}}} = \frac{K}{28.02 \ e\sqrt{\rho\_{\text{-f}}}} \tag{29}$$
 
$$\text{Here } e = \sqrt{\frac{b}{a}} \text{ and } d = \sqrt{ab}$$

#### **2.3.3 Fabric mass (Areal density)**

$$\mathbf{g}\,\mathbf{g}\,\mathbf{sm} = \left[T\_1E\_1(\mathbf{1}+\mathbf{c}\_1) + T\_2E\_2(\mathbf{1}+\mathbf{c}\_2)\right] \times \mathbf{10}\,\mathbf{^1}\tag{30}$$

$$\mathbf{g}\text{sgn} = \sqrt{T\_1}\begin{bmatrix} \left(1+c\_1\right)\,\,K\_1+\left(1+c\_2\right)\,\,K\_2\boldsymbol{\beta} \end{bmatrix} \tag{31}$$

*E*1*, E*2 are ends and picks per cm.

*T*1*, T*2 are warp and weft yarn tex

Here *K*1 and *K*2 are the warp and weft cover factors, *c* is the fractional crimp and *d*2/*d*1= *β*.

In practice the comparison between different fabrics is usually made in terms of gsm. The fabric engineer tries to optimize the fabric parameters for a given gsm. The relationship between the important fabric parameters such as cloth cover and areal density is warranted.

#### **2.3.4 Fabric specific volume**

The apparent specific volume of fabric, *v*F is calculated by using the following formula:

$$\sigma\_{\upsilon F} = \frac{\text{fabric thickness (cm)}}{\text{fabric mass (g/cm}^2)} \tag{32}$$

Fabric mass (g/cm2) = 10–4 x gsm

$$\text{Fabric packing factor} \\ \text{\(\sigma\)} \tag{33}$$

Here *v*f, *v*F are respectively fiber and fabric specific volume.

A knowledge of fiber specific volume helps in calculating the packing of fibers in the fabric. Such studies are useful in evaluating the fabric properties such as warmth, permeability to air or liquid.

#### **2.4 Maximum cover and its importance**

14 Woven Fabrics

*a d eba*

4 1 1 1 28.02 *K*

 

*a d ET K p ep e e* 

*e ρ e*

f f 280.2 28.02 *ρ ρ*

Here and *<sup>b</sup> e d ab <sup>a</sup>*

Here *K*1 and *K*2 are the warp and weft cover factors, *c* is the fractional crimp and

In practice the comparison between different fabrics is usually made in terms of gsm. The fabric engineer tries to optimize the fabric parameters for a given gsm. The relationship between the important fabric parameters such as cloth cover and areal density is

The apparent specific volume of fabric, *v*F is calculated by using the following formula:

fabric thickness (cm) fabric mass (g/ ) cm

Fabric packing factor, Φ = υf/ υF (33)

A knowledge of fiber specific volume helps in calculating the packing of fibers in the fabric. Such studies are useful in evaluating the fabric properties such as warmth, permeability to

4 1 1 1 28.02

 

*e p ρ e*

*p*

For elliptical cross-section the equation will be;

**2.3.3 Fabric mass (Areal density)** 

*E*1*, E*2 are ends and picks per cm. *T*1*, T*2 are warp and weft yarn tex

**2.3.4 Fabric specific volume** 

Fabric mass (g/cm2) = 10–4 x gsm

Here *v*f, *v*F are respectively fiber and fabric specific volume.

*d*2/*d*1= *β*.

warranted.

air or liquid.

f

f

 (29)

gsm = [*T*1*E*1(1+*c*1)+*T*2*E*2(1+*c*2)]×10-1 (30)

gsm =√*T*1 [(1+*c*1) *K*1 + (1+*c*2) *K*2*β*] (31)

2

*vF* (32)

here /

Maximum cover in a jammed fabric is only possible by keeping the two consecutive yarns (say warp) in two planes so that their projections are touching each other and the cross thread (weft) interlaces between them. In this case the weft will be almost straight and maximum bending will be done by the warp.

$$\mathbf{d\_1/p\_1 = 1} \text{ will give } \mathbf{K\_1 = K\_{\text{max}}}$$

and the spacing between the weft yarn, *p*2 =*D* sin*θ*1 = *D* (for *θ* = 900) , *p*2= *d*1+*d*<sup>2</sup>

$$\frac{d\_2}{p\_2} = \frac{d\_2}{d\_1 + d\_2} = \frac{2}{3} \quad \text{for } d\_2 = 2d\_1$$

This will give *K*2 = 2/3 *K*max,

If *d*1 = *d*2 then *d*1 = *d*2/*p*2 =0.5 and *K*<sup>2</sup> =0.5 *K*max

This is the logic for getting maximum cover in any fabric.

The principles are as under:


Both options will give maximum cover in warp and weft but first option will give more thickness than the second case.

The cover factor indicates the area covered by the projection of the thread. The ooziness of yarn, flattening in finishing and regularity further improves the cover of cloth. It also gives a basis of comparison of hardness, crimp, permeability, transparency. Higher cover factor can be obtained by the lateral compression of the threads. It is possible to get very high values only in one direction where threads have higher crimp. Fabrics differing in yarn counts and average yarn spacing can be compared based on the fabric cover. The degree of flattening for race track and elliptical cross-section can be estimated from fabric thickness measurements to evaluate *b* and *a* from microscopic measurement of the fabric surface.

The classical example in this case is that of a poplin cloth in which for warp threads

*p*1 = *d*1 and for *d*1 = *d*2 = *D*/2 and for jamming in both directions

$$p\_1 = D \sin \theta\_2$$

$$d = D/2 = D \sin \theta\_2$$

$$\theta\_2 = 30^\circ = 0.5236$$

$$\theta\_1 = 82^\circ 18^\circ 1.4364 \text{(using } \cos \theta\_1 + \cos \theta\_2 = 1\text{)}$$

$$p\_2 = D \sin \theta\_1 = 0.991D \approx 2p\_1$$

$$l\_1 = D \theta\_1 = 1.14364$$

Modeling of Woven Fabrics Geometry and Properties 17

For jamming in the warp direction of the fabric the parameters *p*2/*D* and corresponding

The relationship between the fabric parameters over the whole domain of structure being jammed in both directions can be obtained by using an algorithm involving equations from

Another useful relationship between the crimps in the two directions is shown in figure 10. It indicates inverse non-linear relationship between *c*1 and *c*2. The intercepts on the X and Y axis gives maximum crimp values with zero crimp in the cross-direction. This is a fabric configuration in which cross-threads are straight and all the bending is being done by the

Figure 11 shows the relation between *h*1*/p*2 and *h*2*/p*1*.* The figure shows inverse linearity between them except at the two extremes. This behavior is in fact a relationship between the

Other practical relations are obtained between the warp and weft cover factor and between

Figure 12 gives the relation between warp and weft cover factor for different ratio of weft to warp yarn diameters (*β*). The relation between the cover factors in the two directions is sensitive only in a narrow range for all values of *β.* The relation between the cover factors in the two directions are inter- dependent for jammed structures. Maximum threads in the warp or weft direction depend on yarn count and weave. Maximum threads in one direction of the fabric will give unique maximum threads in the cross-direction. The change in the value of *β* causes a distinct shift in the curve. A comparatively coarse yarn in one direction with respect to the other direction helps in increasing the cover factor. For *β* = 0.5, the warp

Fig. 9. Relation between thread spacing and crimp height

square root of crimp in the two directions of the fabric.

cloth cover factor and fabric mass (gsm).

*h*1/*D* can be obtained either from this figure or from the above equation.

with centre at (0, 1) and radius equal to 1.

the previous section.

intersecting threads.

*l*2= *Dθ*2= 0.5236

*c*1' = 0.45, *c*2' = 0.0472

This is a specification of good quality poplin which has maximum cover and ends per cm is twice that of picks per cm.
