**3.2.1 Yarn diameter**

Two important geometrical parameters are needed for calculating weavability for a general case. These are yarn diameter and weave factor.

Yarn diameter in terms of linear density in tex for a general case is given as:

$$d = \frac{\sqrt{T}}{280.2\sqrt{\rho\_{\text{pf}}}} \tag{37}$$

Where *d* = yarn diameter (cm), *T* = yarn linear density (tex, i.e. g/km),

*ρ*f = fiber density(g/cm3), *ρ*y = yarn density(g/cm3), Φ is yarn packing factor.

This equation for the yarn diameter is applicable for any yarn type and fiber type. The packing factor depends on fiber variables such as fiber crimp, length, tex and cross-section shape.

Table 2 and 3 give the fiber density and yarn packing factor for different fiber and yarn type respectively.


Table 2. Fiber density, g/cm3


Table 3. Yarn packing factor

For blended yarns, average fiber density is given by the following

$$\frac{1}{\rho} = \sum\_{1}^{n} \frac{p\_i}{p\_{\text{fit}}} \tag{38}$$

where = avera ge fiber density ,

20 Woven Fabrics

The maximum number of ends and picks per unit length that can be woven with a given yarn and weave defines weavability limit (Hearle et al., 1969). 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 preparations. (Dickson, 1954) demonstrated the usefulness of theoretical weavability limit and found agreement with the loom performance. Most of the work in this area was done using empirical relationships. The geometrical model is very useful in predicting this limit for a given warp, weft diameter (tex) and any weave. Maximum weavability limit is calculated in the model by using jamming

Two important geometrical parameters are needed for calculating weavability for a general

<sup>f</sup> 280.2 *<sup>ρ</sup> <sup>T</sup> <sup>d</sup>*

This equation for the yarn diameter is applicable for any yarn type and fiber type. The packing factor depends on fiber variables such as fiber crimp, length, tex and cross-section shape.

Table 2 and 3 give the fiber density and yarn packing factor for different fiber and yarn type

Ring-spun 0.60 Open-end-spun 0.55 Worsted 0.60 Woolen 0.55 Continuous-filament 0.65

Acetate 1.32 Cotton 1.52 Lycra 1.20 Nylon 6 1.14 Nylon 66 1.13-1.14 Polyester 1.38 Polypropylene 0.91 Rayon 1.52 Wool 1.32

(37)

conditions for plain and non-plain weaves for circular and race track cross-sections.

Yarn diameter in terms of linear density in tex for a general case is given as:

Where *d* = yarn diameter (cm), *T* = yarn linear density (tex, i.e. g/km),

*ρ*f = fiber density(g/cm3), *ρ*y = yarn density(g/cm3), Φ is yarn packing factor.

**3.2 Weavability limit** 

**3.2.1 Yarn diameter** 

respectively.

Table 2. Fiber density, g/cm3

Table 3. Yarn packing factor

case. These are yarn diameter and weave factor.

*p*i = weight fraction of the ith component,

*p*ft = fiber density of the ith component and

*n* = number of components of the blend

#### **3.2.2 Effect of variation in beta (***d***2***/d***1) on the relation between warp and weft cover factor for jammed fabrics**

An increase in the value of beta from 0.5-2 increases the range of warp cover factors but raises the level for the weft cover factor. This means with an increase in beta higher weft cover factors are achievable and vice-versa. However it may be noted that for cotton fibers having higher fiber density the sensitivity range between the warp and weft cover factor is relatively large compared to polypropylene fiber as shown in figure 13a and 13b. This shows a very important role played by fiber density in deciding warp and weft cover factors for the jammed fabrics.

Fig. 13a. Effect of *β* on the relation between warp and weft cover factor

Modeling of Woven Fabrics Geometry and Properties 23

Fig. 14. Jammed structure for 1/3 weave (circular cross-section along warp)

 <sup>2</sup> <sup>2</sup> <sup>2</sup> 2 2 <sup>1</sup> *<sup>p</sup> <sup>P</sup> <sup>d</sup> M M DD D*

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

1 2 1 2

This equation can easily be transformed in terms of warp and weft cover factor (*K*1 and *K*2)

f f 1 2

**3.2.4 Relation between fabric parameters for circular cross-section for different** 

these weaves are 1, 1.5, 2 and 2.5 respectively for all the discussion which follows.

1 1 1 1 1

The effect of weave in the jammed structures is examined using the above equations for plain, twill, basket and satin weave. *M*, the weave factor value (average float length) for

1 1 11 1 1 1 *P P M M <sup>β</sup> M M <sup>D</sup> <sup>β</sup> <sup>D</sup> <sup>β</sup>* 

1 2

1 2

28.02 1 28.02

2 2

*ρ ρ M M <sup>β</sup> M M K K β β*

1 2

1 1

 

1

*<sup>p</sup> <sup>P</sup> <sup>β</sup> M M D D <sup>β</sup>* (41)

2 2

(42)

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

Similarly, interchanging suffix 1, 2 we get

**weaves** 

For a jammed fabric the following equation is valid:

Fig. 13b. Effect of *β* on the relation between warp and weft cover factor

#### **3.2.3 Equation for jammed structure for circular cross-section in terms of weave factor**

Weave factor is useful in translating the effect of weave on the fabric properties. For circular cross-section the general equation for jammed cloth is desired.

Thread spacing *P*t1 for a non-plain weave per repeat is shown in figure 14 and is given as:

$$p\_{t1} = I\_2 p\_1 + (E\_1 - I\_2) d\_1 \tag{39}$$

Average thread spacing 2 12 <sup>1</sup> <sup>1</sup> 1 1 *I EI p* ( )*d P E* 

That means, 11 1 1 1 2 2 <sup>1</sup> *EP E p d I I* 1 11 <sup>1</sup> <sup>1</sup> *M P p* ( 1) *M d* <sup>1</sup> <sup>1</sup> <sup>1</sup> 1 1 <sup>1</sup> *<sup>p</sup> <sup>P</sup> <sup>d</sup> M M DD D* <sup>1</sup> <sup>1</sup> <sup>1</sup> 1 1 1 *p P M <sup>M</sup> D D* (40)

where *β* = *d*2/*d*<sup>1</sup>

Fig. 13b. Effect of *β* on the relation between warp and weft cover factor

cross-section the general equation for jammed cloth is desired.

Average thread spacing 2 12 <sup>1</sup> <sup>1</sup> 1

> 2 2 <sup>1</sup> *EP E p d*

*I I*

That means, 11 1

where *β* = *d*2/*d*<sup>1</sup>

*P*

1 1

 

**factor** 

**3.2.3 Equation for jammed structure for circular cross-section in terms of weave** 

Weave factor is useful in translating the effect of weave on the fabric properties. For circular

12 12 <sup>1</sup> <sup>1</sup> ( ) *P I EI <sup>t</sup> p d* (39)

(40)

Thread spacing *P*t1 for a non-plain weave per repeat is shown in figure 14 and is given as:

1 11 <sup>1</sup> <sup>1</sup> *M P p* ( 1) *M d*

 <sup>1</sup> <sup>1</sup> <sup>1</sup> 1 1 <sup>1</sup> *<sup>p</sup> <sup>P</sup> <sup>d</sup> M M DD D*

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

*p P M <sup>M</sup> D D*

1

1

1 *I EI p* ( )*d*

*E* 

Fig. 14. Jammed structure for 1/3 weave (circular cross-section along warp) Similarly, interchanging suffix 1, 2 we get

$$
\frac{p\_2}{D} = M\_2 \frac{\overline{P\_2}}{D} - (M\_2 - 1) \frac{d\_2}{D}
$$

$$
\frac{p\_2}{D} = M\_2 \frac{\overline{P\_2}}{D} - (M\_2 - 1) \frac{\beta}{1 + \beta} \tag{41}
$$

For a jammed fabric the following equation is valid:

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

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

This equation can easily be transformed in terms of warp and weft cover factor (*K*1 and *K*2)

$$\sqrt{1-\left[\left(\frac{28.02\sqrt{\rho\rho\_f}M\_1}{K\_1}-(M\_1-1)\right)\frac{1}{1+\beta}\right]^2}+\sqrt{1-\left[\left(\frac{28.02\sqrt{\rho\rho\_f}M\_2}{K\_2}-(M\_2-1)\right)\frac{\beta}{1+\beta}\right]^2}=1\tag{42}$$

#### **3.2.4 Relation between fabric parameters for circular cross-section for different weaves**

The effect of weave in the jammed structures is examined using the above equations for plain, twill, basket and satin weave. *M*, the weave factor value (average float length) for these weaves are 1, 1.5, 2 and 2.5 respectively for all the discussion which follows.

Modeling of Woven Fabrics Geometry and Properties 25

2 *a b <sup>A</sup>*

12 12 <sup>1</sup> <sup>1</sup> ( )4 <sup>2</sup> *<sup>t</sup> a b P I EI <sup>p</sup> <sup>a</sup>*

21 21 <sup>2</sup> <sup>2</sup> ( )4 <sup>2</sup> *<sup>t</sup> a b P I EI <sup>p</sup> <sup>a</sup>*

Where, *p*1 and *p*2 are horizontal spacing between the semi-circular threads in the intersection

1 11 2 1 4 <sup>1</sup>

2 22 1 1 4 <sup>1</sup>

As such analysis of circular thread geometry can be applied for the intersection zone of the

<sup>1</sup> 4 () 21 1 2 1 <sup>2</sup>

1

t2 <sup>1</sup> *<sup>L</sup> <sup>C</sup> P* 

2 2 1 2 1 11 *p p B B* 

*p*1 and *p*2 can be calculated from the jamming considerations of the circular thread geometry

It should be remembered that *p*/*B* corresponds to the semi-circular region of the race track cross-section and is similar to *p*/*D* for circular cross-section. As such the values of *p/D* ratio

1

*<sup>p</sup> a b <sup>P</sup> <sup>a</sup> M MM I*

2 2 2

*<sup>p</sup> a b <sup>P</sup> <sup>a</sup> M MM I*

zone. Here, *a* and *b* are the major and minor diameters of race track cross-section.

1 1

2 2

2 2

1 1

(43)

(44)

2

2

*a b <sup>L</sup> EI I a l* (47)

(46)

(45)

Here, 1 1

Simillarly,

Similarly,

using:

can be used for *p*/*B*

The average thread spacing

race track cross-section.

Total warp crimp in the fabric is given by:

Thread spacing *Pt*1 for a non-plain weave per repeat is given as:

2 2

<sup>1</sup> 1 1

The relation between *p*1avg/*D* and *p*2avg/*D* is established and it is seen that with the increase in float length, the sensitivity of the curve decreases in general. Also the range of *p*1/*D* and *p*2/*D* values gets reduced. This means a weave with longer float length decreases the flexibility for making structures.

Figure 15 shows the relationship between the warp and weft cover factor for circular crosssection. It is interesting to note that the behavior is similar for different weaves. However with the increase in float, the curve shifts towards higher values of weft cover factor. It should be borne in mind that the behavior shown in this figure is for virtual fabrics. In real fabrics jammed structure is unlikely to retain circular cross-section.

Fig. 15. Relation between warp and weft cover factor for jammed fabric (circular crosssection)

#### **3.2.5 Equation for jammed structure for a race track cross-section in terms of weave factor**

In jammed fabrics, the yarn cross-section cannot remain circular. The cross-section will change. It is easy to modify the geometry for circular cross-section by considering race track cross-section. Figure 16 shows the configuration of jammed structure for 1/3 weave for race track cross-section along weft direction of the fabric.

Fig. 16. Jammed structure for 1/3 weave (race track cross-section along warp)

Here, 1 1

24 Woven Fabrics

The relation between *p*1avg/*D* and *p*2avg/*D* is established and it is seen that with the increase in float length, the sensitivity of the curve decreases in general. Also the range of *p*1/*D* and *p*2/*D* values gets reduced. This means a weave with longer float length decreases the

Figure 15 shows the relationship between the warp and weft cover factor for circular crosssection. It is interesting to note that the behavior is similar for different weaves. However with the increase in float, the curve shifts towards higher values of weft cover factor. It should be borne in mind that the behavior shown in this figure is for virtual fabrics. In real

Fig. 15. Relation between warp and weft cover factor for jammed fabric (circular cross-

Fig. 16. Jammed structure for 1/3 weave (race track cross-section along warp)

track cross-section along weft direction of the fabric.

**3.2.5 Equation for jammed structure for a race track cross-section in terms of weave** 

In jammed fabrics, the yarn cross-section cannot remain circular. The cross-section will change. It is easy to modify the geometry for circular cross-section by considering race track cross-section. Figure 16 shows the configuration of jammed structure for 1/3 weave for race

fabrics jammed structure is unlikely to retain circular cross-section.

flexibility for making structures.

section)

**factor** 

Thread spacing *Pt*1 for a non-plain weave per repeat is given as:

$$P\_{I1} = I\_2 p\_1 + (E\_1 - I\_2) a\_1 + 4\left(\frac{a\_1 - b\_1}{2}\right) \tag{43}$$

Simillarly,

$$P\_{t2} = I\_1 p\_2 + (E\_2 - I\_1) a\_2 + 4\left(\frac{a\_2 - b\_2}{2}\right) \tag{44}$$

Where, *p*1 and *p*2 are horizontal spacing between the semi-circular threads in the intersection zone. Here, *a* and *b* are the major and minor diameters of race track cross-section.

2 *a b <sup>A</sup>*

The average thread spacing

$$\overline{P\_1} = \frac{p\_1}{M\_1} + \left(1 - \frac{1}{M\_1}\right)a\_1 + \frac{4}{M\_1 I\_2} \left(\frac{a\_1 - b\_1}{2}\right) \tag{45}$$

Similarly,

$$\overline{P\_2} = \frac{p\_2}{M\_2} + \left(1 - \frac{1}{M\_2}\right)a\_2 + \frac{4}{M\_2 I\_1} \left(\frac{a\_2 - b\_2}{2}\right) \tag{46}$$

As such analysis of circular thread geometry can be applied for the intersection zone of the race track cross-section.

$$\mathbf{r}\_{L1} = \mathbf{4}\left(\frac{a\_2 - b\_2}{2}\right) + (\mathbf{E}\_2 - I\_1)a\_2 + I\_1 \times l\_1 \tag{47}$$

Total warp crimp in the fabric is given by:

$$C\_1 = \frac{L\_1}{P\_{t2}} - 1$$

*p*1 and *p*2 can be calculated from the jamming considerations of the circular thread geometry using:

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

It should be remembered that *p*/*B* corresponds to the semi-circular region of the race track cross-section and is similar to *p*/*D* for circular cross-section. As such the values of *p/D* ratio can be used for *p*/*B*

Modeling of Woven Fabrics Geometry and Properties 27

<sup>2</sup> <sup>2</sup> tan / 2 0.75

Fig. 17. Relation between warp and weft cover factor for jammed fabric (race track cross-

*p/d* ≥ 1.732; *d/p* ≤ 0.5773

Also *D* = 2*d* = *h*1 + *h*2 = 2×(4/3)*p*√*c* 

4 280.2 *d Tex*

0.02677

 *ρ* 

3.57 *K <sup>c</sup>*

cos θ1 + cos θ2 = 1 will give

cos θ = ½ and θ = 600

f *K*

2

*<sup>ρ</sup>*

3 0.75

*p p*

*c*

%

*c*

f

2

This is valid for all values of (*p/D*) 2 ≥ 0.75 or *p/D* ≥ 0.866

Crimp in % can be calculated from,

For jammed square cloth

section)

*D D* 

(50)

(51)

3 *<sup>p</sup> <sup>p</sup> <sup>θ</sup>*

$$\sqrt{1-\left(M\_1\frac{P\_1}{B}-\frac{4}{B\_{I\_2}}\left(\frac{a\_1-b\_1}{2}\right)-(M\_1-1)\frac{a\_1}{B}\right)^2} + \sqrt{1-\left(M\_2\frac{P\_2}{B}-\frac{4}{B\_{I\_1}}\left(\frac{a\_2-b\_2}{2}\right)-(M\_2-1)\frac{a\_2}{B}\right)^2} = 1$$

This equation can be simplified to the following usable forms.

$$\sqrt{1-\left(M\_1\frac{P\_1}{B}-\frac{2(1-\varepsilon)}{e(1+\beta)I\_2}-\frac{(M\_1-1)}{e(1+\beta)}\right)^2} + \sqrt{1-\left(M\_2\frac{P\_2}{B}-\frac{2(1-\varepsilon)\beta}{e(1+\beta)I\_1}-\frac{(M\_2-1)}{e(1+\beta)}\right)^2} = 1$$

It is assumed that *e*1 = *e*2 = *e*

where *e = b/a*

The above equation can easily be transformed in terms of warp and weft cover factor as under:

$$\sqrt{1-\left(\frac{28.02\sqrt{\rho\rho\_{f}}M\_{1}}{(1+\beta)K\_{1}}\sqrt{1+\frac{4}{\pi}\left(\frac{1}{e}-1\right)}-\frac{2(1-e)}{e(1+\beta)I}-\frac{(M\_{1}-1)}{e(1+\beta)}\right)^{2}}$$

$$+\sqrt{1-\left(\frac{28.02\sqrt{\rho\rho\_{f}}M\_{2}\beta}{(1+\beta)K\_{2}}\sqrt{1+\frac{4}{\pi}\left(\frac{1}{e}-1\right)}-\frac{2(1-e)\beta}{e(1+\beta)I}-\frac{(M\_{2}-1)\beta}{e(1+\beta)}\right)^{2}}=1\tag{48}$$

#### **3.2.6 Relationship between fabric parameters in race track cross-section**

The relationship between fabric parameters such as *p*2 and *p*1, *p*1 and *c*2 for the race track cross-section in jammed condition is discussed below.

The parameters are similar to that for the circular cross-section but it shifts towards higher values of thread spacing.

Figure 17 shows the relationship between warp and weft cover factors for different weaves. As discussed above in real fabrics the weaves show distinct differences between them unlike in circular cross-section. Increase in float length decreases the scope of cover factors.

From these equations crimp and fabric cover can be evaluated using the above two equations along with:

$$a^2 = b^2 \left[ \frac{\pi}{4} + \left( \frac{1}{e} - 1 \right) \right] \text{ and } \frac{b\_2}{b\_1} = \beta \tag{49}$$

#### **3.3 Square cloth**

A truly square fabric has equal diameter, spacing and crimp.

$$p\_1 = p\_{2'} \ \mathsf{C}1 = \mathsf{C}2 \ \mathsf{A}1 = \mathsf{d}2 \ \mathsf{A}1 = \mathsf{f}\_{\mathsf{I}2} = \mathsf{D} \ / \ \mathsf{2} \ \mathsf{A} \ \mathsf{e}\_1 = \mathsf{e}\_2 \ \mathsf{A}$$

From the basic equations of the geometrical model from the previous section we have:

1 2 1 1 1 2 2 2 1 12 2 2 1

4 4 <sup>1</sup> ( 1) 1 ( 1) 1 2 2 *P P ab a ab a M MM M B BI I <sup>B</sup> B B <sup>B</sup>* 

1 2 1 2

The above equation can easily be transformed in terms of warp and weft cover factor as

28.02 4 1 2(1 ) ( 1) 1 11 (1 ) (1 ) (1 ) *ρ M e M β K e e β I e β*

**3.2.6 Relationship between fabric parameters in race track cross-section** 

*P P <sup>e</sup> M M <sup>e</sup> <sup>β</sup> M M B e <sup>β</sup> I I <sup>e</sup> <sup>β</sup> B e <sup>β</sup> <sup>e</sup> <sup>β</sup>* 

2 1 2(1 ) ( 1) 2(1 ) ( 1) 1 11 (1 ) (1 ) (1 ) (1 )

f 1 1

f 2 2

28.02 4 1 2(1 ) ( 1) <sup>1</sup> 1 1 <sup>1</sup> (1 ) (1 ) (1 ) *ρ M β e β β M β K e e β I e β*

The relationship between fabric parameters such as *p*2 and *p*1, *p*1 and *c*2 for the race track

The parameters are similar to that for the circular cross-section but it shifts towards higher

Figure 17 shows the relationship between warp and weft cover factors for different weaves. As discussed above in real fabrics the weaves show distinct differences between them unlike

From these equations crimp and fabric cover can be evaluated using the above two

2 2 2

 

*<sup>b</sup> d b <sup>β</sup>*

1 2 1 21 21 2 1 2 *p p* , , , / 2, *c cd dh h D*

From the basic equations of the geometrical model from the previous section we have:

4

A truly square fabric has equal diameter, spacing and crimp.

<sup>1</sup> 1 and

*e b*

1

(49)

 in circular cross-section. Increase in float length decreases the scope of cover factors.

This equation can be simplified to the following usable forms.

It is assumed that *e*1 = *e*2 = *e*

values of thread spacing.

equations along with:

**3.3 Square cloth** 

where *e = b/a*

under:

1 2

1

2

cross-section in jammed condition is discussed below.

2 2

2 2

2

2

(48)

Fig. 17. Relation between warp and weft cover factor for jammed fabric (race track crosssection)

This is valid for all values of (*p/D*) 2 ≥ 0.75 or *p/D* ≥ 0.866

$$p/d \ge 1.732; \ d/p \le 0.5773$$

$$\text{Also } D = 2d = h\_1 + h\_2 = 2 \times (4/3)p \cdot \text{c}$$

$$\sqrt{c} = \frac{3}{4} \frac{d}{p} = \frac{0.75}{p} \times \frac{\sqrt{Tex}}{280.2 \sqrt{\rho \rho\_f}}$$

$$c = \left(\frac{0.02677 \,\text{K}}{\sqrt{\rho \rho\_f}}\right)^2 \tag{51}$$

$$\text{from, } \%c = \left(\frac{\text{K}}{3.57}\right)^2$$

Crimp in % can be calculated from,

For jammed square cloth

$$
\cos \theta\_1 + \cos \theta\_2 = 1 \text{ will give}
$$

cos θ = ½ and θ = 600

fabric.

1 1 C 1 C

Since sinθ C

and <sup>2</sup>

2 C 1 C

**4.2 Crimp balance equation** 

Modeling of Woven Fabrics Geometry and Properties 29

The above equation is called crimp interchange equation. It gives the relationship between the warp and weft crimp for the new configuration after the application of stretch in warp/weft direction. It may be noted that the parameters l1, l2 and D are invariant; they have the same value in the original fabric and in the new configuration. This basically means it is assumed that the geometry in deformed fabric is same as in undeformed

The most general manner of solving crimp interchange problems is getting relation between

Textile yarns are not flexible as assumed in Peirce's geometrical model. They offer resistance to bending. The elastica model demonstrates the existence of inter yarn force at the crossover points during fabric formation. The crimp balance equation is an offshoot of this analysis. It shows the importance of bending rigidity of warp and weft yarns in influencing

The analysis using the rigid thread thread model [R] gives the value of inter yarn force

V= 16 M sinθ/p2

V1=V2

M1 sinθ1/ p22 = M2sinθ2 p1

1 2 2 2 1 1 C Mp C M p 

1 2 21 2 1 12 C 1 C M l /D C 1 C M l /D 

**4.3 Interaction of crimp interchange and crimp balance equations** 

The solution for C1 and C2 for this equation is obtained in terms of M2/M1 and l1/D and

The interaction of crimp interchange and crimp balance equations for given values of l1/D , l2/D and M2/M1 (ratio of bending moment of warp and weft) gives desired C1 and C2. It is impossible to solve these equations mathematically however soft computing facilitates

2

(53)

(54)

2

42 4

1*c* or /

<sup>2</sup>*c* is determined based on

In the crimp interchange equation one of the parameter /

for constant l1/D and l2/D.

the ratio of crimp in both warp and weft directions.

The balance of inter yarn force in two direction gives

L2/D using special algorithm in MATLAB.

the requirement of modification and the other parameter is calculated.

$$\begin{aligned} p &= 2d \sin \theta \,\,\, 1 = D\theta = 2d \frac{\theta}{3} \quad \therefore \,\, \frac{1}{d} = 2 \frac{\theta}{3} \\\\ \frac{p}{d} &= \sqrt{3} = 1.732 \\\\ &= > \frac{p}{d} = 0.5773 \\\\ \text{crimp} &= \frac{1/d}{p/d} - 1 = 0.2092 = 20.9\% \end{aligned}$$

Therefore complete cover is not possible with square cloth.
