**3. Application of geometrical model**

#### **3.1 Computation of fabric parameters**

The basic equations derived from the geometrical model are not easy to handle. Research workers (Nirwan & Sachdev, 2001; Weiner, 1971) obtained solutions in the form of graphs and tables. These are quite difficult to use in practice. It is possible to predict fabric parameters and their effect on the fabric properties by soft computing (Newton, 1995). This information is helpful in taking a decision regarding specific buyers need. A simplified algorithm is used to solve these equations and obtain relationships between useful fabric parameters such as thread spacing and crimp, fabric cover and crimp, warp and weft cover. Such relationships help in guiding the directions for moderating fabric parameters.

Peirce's geometrical relationships can be written as

$$\frac{p\_2}{D} = (\mathbf{K}\_1 - \theta\_\mathbf{l}) \cos \theta\_\mathbf{l} + \sin \theta\_\mathbf{l} \tag{34}$$

$$\frac{h\_1}{D} = (\mathbf{K}\_1 - \theta\_1)\sin\theta\_1 + (1 - \cos\theta\_1) \tag{35}$$

Where K1= *l*1/*D* and two similar equations for the weft direction will be obtained by interchanging the suffix 1 with 2 and vice versa. The solution of *p*2*/D* and *h*1*/D* is obtained for different values of *θ*1 (weave angle) ranging from 0.1– π/2 radians. Such a relationship is shown in figure 9.

It is a very useful relationship between fabric parameters for engineering desired fabric constructions. One can see its utility for the following three cases


#### **3.1.1 Jammed structures**

Figure 9 shows non linear relationship between the two fabric parameters *p* and *h* on the extreme left. In fact, this curve is for jamming in the warp direction. It can be seen that the jamming curve shows different values of *p2*/*D* for increasing *h*1/*D*, that is warp crimp. The theoretical range for *p2*/*D* and *h*1/*D* varies from 0-1. Interestingly this curve is a part of circle and its equation is:

$$\left(\frac{p\_2}{D}\right)^2 + \left(\frac{h\_1}{D} - 1\right)^2 = 1\tag{36}$$

*l*2= *Dθ*2= 0.5236

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

The basic equations derived from the geometrical model are not easy to handle. Research workers (Nirwan & Sachdev, 2001; Weiner, 1971) obtained solutions in the form of graphs and tables. These are quite difficult to use in practice. It is possible to predict fabric parameters and their effect on the fabric properties by soft computing (Newton, 1995). This information is helpful in taking a decision regarding specific buyers need. A simplified algorithm is used to solve these equations and obtain relationships between useful fabric parameters such as thread spacing and crimp, fabric cover and crimp, warp and weft cover.

Such relationships help in guiding the directions for moderating fabric parameters.

2

1

constructions. One can see its utility for the following three cases

3. Special case in which cross-threads are straight

= 0.0472

11 1

11 1

(34)

(35)

(36)

<sup>1</sup> (K )cos sin *<sup>p</sup> θθ θ <sup>D</sup>*

<sup>1</sup> (K )sin (1 cos ) *<sup>h</sup> θθ θ <sup>D</sup>*

Where K1= *l*1/*D* and two similar equations for the weft direction will be obtained by interchanging the suffix 1 with 2 and vice versa. The solution of *p*2*/D* and *h*1*/D* is obtained for different values of *θ*1 (weave angle) ranging from 0.1– π/2 radians. Such a relationship is

It is a very useful relationship between fabric parameters for engineering desired fabric

Figure 9 shows non linear relationship between the two fabric parameters *p* and *h* on the extreme left. In fact, this curve is for jamming in the warp direction. It can be seen that the jamming curve shows different values of *p2*/*D* for increasing *h*1/*D*, that is warp crimp. The theoretical range for *p2*/*D* and *h*1/*D* varies from 0-1. Interestingly this curve is a part of

> 2 2 <sup>2</sup> <sup>1</sup> 1 1 *<sup>p</sup> <sup>h</sup>*

*D D*

= 0.45, *c*2'

*c*1'

twice that of picks per cm.

shown in figure 9.

1. Jammed structures 2. Non-jammed fabrics

**3.1.1 Jammed structures** 

circle and its equation is:

**3. Application of geometrical model 3.1 Computation of fabric parameters** 

Peirce's geometrical relationships can be written as

Fig. 9. Relation between thread spacing and crimp height

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

For jamming in the warp direction of the fabric the parameters *p*2/*D* and corresponding *h*1/*D* can be obtained either from this figure or from the above equation.

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 the previous section.

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 intersecting threads.

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

Other practical relations are obtained between the warp and weft cover factor and between cloth cover factor and fabric mass (gsm).

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

Modeling of Woven Fabrics Geometry and Properties 19

Fig. 12. Relation between warp and weft cover factor for different *β* in jammed fabric

It can be seen that the relation between *p*2/*D* and corresponding *h*1/*D* is linear for different values of crimp. This relationship is useful for engineering non-jammed structures for a range of values of crimp. The fabric parameters can be calculated from the above non-jammed linear relation between *p*2/*D* and *h*1/*D* for any desired value of warp crimp. Then *h*2/*D* can be obtained from (1–*h*1/*D*) and for this value of *h*2/*D* one can obtain the corresponding value of *p*1/*D* for the desired values of weft crimp. Thus all fabric parameters can be obtained for desired value of *p*2/*D*, picks per cm, warp and weft yarn tex, warp and weft crimp. One can choose any other four parameters to get all fabric

The intersection of horizontal line corresponding to *h*1/*D*=1 gives all possible structures ranging from relatively open to jammed configurations. In this case *h*2= 0*, h*1= D; This gives interesting structures which have stretch in one direction only, enabling maximum fabric thickness and also being able to use brittle yarns. The fabric designer gets the options to choose from the several possible fabric constructions. These options include jamming and other non jammed constructions. Using the above logic it is also possible to get fabric

2. fabric with maximum crimp in one direction and cross-threads being straight. 3. fabric which is neither jammed nor has zero crimp in the cross-threads.

**3.1.2 Non-jammed structure** 

**3.1.3 Straight cross threads** 

1. fabric jammed in both directions.

parameters.

parameters for:

yarn is coarser than the weft, this increases the warp cover factor and decrease the weft cover factor. This is due to the coarse yarn bending less than the fine yarn. Similar effect can be noticed for *β* =2, in which the weft yarn is coarser than the warp yarn. These results are similar to earlier work reported by Newton (Newton, 1991 & 1995; Seyam, 2003).

Fig. 10. Relation between warp and weft crimp for jammed fabric

Fig. 11. Relation between warp and weft crimp in jammed fabric

The relation between fabric mass, (gsm) with the cloth cover (*K*1+*K*2) is positively linear (Singhal & Choudhury, 2008). The trend may appear to be self explanatory. Practically an increase in fabric mass and cloth cover factor for jammed fabrics can be achieved in several ways such as with zero crimp in the warp direction and maximum crimp in the weft direction; zero crimp in the weft direction and maximum crimp in the warp direction; equal or dissimilar crimp in both directions. This explanation can be understood by referring to the non-linear part of the curve in figure 11.

yarn is coarser than the weft, this increases the warp cover factor and decrease the weft cover factor. This is due to the coarse yarn bending less than the fine yarn. Similar effect can be noticed for *β* =2, in which the weft yarn is coarser than the warp yarn. These results are

similar to earlier work reported by Newton (Newton, 1991 & 1995; Seyam, 2003).

Fig. 10. Relation between warp and weft crimp for jammed fabric

Fig. 11. Relation between warp and weft crimp in jammed fabric

the non-linear part of the curve in figure 11.

The relation between fabric mass, (gsm) with the cloth cover (*K*1+*K*2) is positively linear (Singhal & Choudhury, 2008). The trend may appear to be self explanatory. Practically an increase in fabric mass and cloth cover factor for jammed fabrics can be achieved in several ways such as with zero crimp in the warp direction and maximum crimp in the weft direction; zero crimp in the weft direction and maximum crimp in the warp direction; equal or dissimilar crimp in both directions. This explanation can be understood by referring to

Fig. 12. Relation between warp and weft cover factor for different *β* in jammed fabric
