**4. Crimp in the fabric**

The crimp in fabric is the most important parameter which influences several fabric properties such as extensibility, thickness, compressibility and handle. It also decides quantity of yarn required to weave a fabric during manufacturing. Therefore control of crimp is vital for geometrical analysis of fabric structure.

#### **4.1 Crimp interchange equation**

Normally crimp interchange equation is used to predict the change in crimp in the fabric when it is extended in any direction by keeping the ratio of modular length to the sum of thread diameter ( l1/D and l2/D)constant. An attempt is made by soft computing to exploit the crimp interchange equation in a different way instead of keeping the usual three invariants l1, l2 and D and the relationship between warp crimp (C1 ) and weft crimp (C2 ) is determined by varying l1/D and l2/D. Such a strategy enables bias of crimp in a preferred direction. This is a new concept and entirely a different use of crimp interchange equation.

Following equation gives a useful relationship between the two directions of the fabric.

$$D = h\_1 + h\_2 = |h\_1'| + h\_2'$$

Superscript represents changes in the fabric parameter after modification

$$D = \vec{h\_1} + \vec{h\_2} = \frac{4}{3} \left[ r\_2^{\prime} \sqrt{\vec{c\_1}} + r\_1^{\prime} \sqrt{\vec{c\_2}}^{\prime} \right]$$

$$D = \frac{4}{3} \left[ \frac{l\_1 \sqrt{\vec{c\_1}}}{1 + c\_1} + \frac{l\_2 \sqrt{\vec{c\_2}}}{1 + c\_2} \right]$$

$$\frac{l\_1}{D} \frac{\sqrt{\vec{c\_1}^{\prime}}}{(1 + \vec{c\_1})} + \frac{l\_2}{D} \frac{\sqrt{\vec{c\_2}^{\prime}}}{(1 + \vec{c\_2})} = \frac{3}{4} \tag{52}$$

3 1.732 *<sup>p</sup> <sup>d</sup>*

0.5773 *<sup>p</sup>*

<sup>1</sup> crimp 1 0.2092 20.9% *<sup>d</sup> p d*

The crimp in fabric is the most important parameter which influences several fabric properties such as extensibility, thickness, compressibility and handle. It also decides quantity of yarn required to weave a fabric during manufacturing. Therefore control of

Normally crimp interchange equation is used to predict the change in crimp in the fabric when it is extended in any direction by keeping the ratio of modular length to the sum of thread diameter ( l1/D and l2/D)constant. An attempt is made by soft computing to exploit the crimp interchange equation in a different way instead of keeping the usual three invariants l1, l2 and D and the relationship between warp crimp (C1 ) and weft crimp (C2 ) is determined by varying l1/D and l2/D. Such a strategy enables bias of crimp in a preferred direction. This is a new concept and entirely a different use of crimp

Following equation gives a useful relationship between the two directions of the fabric.

Superscript represents changes in the fabric parameter after modification

4

*D*

 *D* = *h1+h2*= / /

' ' '' '' 1 2 21 12 ' ' 11 22 1 2

4 3

*Dh h p c pc*

31 ' 1 '

*l l c c D D c c* 

*lc lc*

 

*c c*

/ / 1 2 1 2 ' ' 1 2

(1 ) (1 ) 4

1 2 *h h*

3

(52)

3 *θ*

*d* 

2 sin , 1 2 <sup>3</sup> *<sup>θ</sup> p d <sup>θ</sup> <sup>D</sup><sup>θ</sup> <sup>d</sup>* <sup>1</sup> <sup>2</sup>

> *d*

Therefore complete cover is not possible with square cloth.

crimp is vital for geometrical analysis of fabric structure.

**4. Crimp in the fabric** 

interchange equation.

**4.1 Crimp interchange equation** 

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 fabric.

In the crimp interchange equation one of the parameter / 1*c* or / <sup>2</sup>*c* is determined based on the requirement of modification and the other parameter is calculated.

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

$$\frac{\text{\AA} \text{\textquotedblleft C1}}{\text{1} + \text{C1}} \text{ and } \frac{\text{\AA} \text{\textquotedblleft C2}}{\text{1} + \text{C2}} \text{ for constant } \text{l}\_1/\text{D} \text{ and } \text{l}\_2/\text{D}.$$

#### **4.2 Crimp balance equation**

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 ratio of crimp in both warp and weft directions.

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

$$\mathbf{V} = 16 \text{ M } \sin \theta / \mathbf{p}^2$$

The balance of inter yarn force in two direction gives

$$\mathbf{V}\_1 = \mathbf{V}\_2$$

$$\mathbf{M}\_1 \sin \theta\_1 / \ \mathbf{p}\_2^2 = \mathbf{M}\_2 \sin \theta\_2 \ \mathbf{p}\_1^2$$

Since sinθ C

$$\frac{\sqrt{\mathbf{C}\_1}}{\sqrt{\mathbf{C}\_2}} = \frac{\mathbf{M}\_2}{\mathbf{M}\_1} \left(\frac{\mathbf{p}\_2}{\mathbf{p}\_1}\right)^2 \tag{53}$$

$$\frac{\mathbf{C}\_1}{\mathbf{C}\_2} = \left(\frac{\mathbf{l} + \mathbf{C}\_2}{\mathbf{l} + \mathbf{C}\_1}\right)^4 \left(\frac{\mathbf{M}\_2}{\mathbf{M}\_1}\right)^2 \left(\frac{\mathbf{l}\_1 / \mathbf{D}}{\mathbf{l}\_2 / \mathbf{D}}\right)^4 \tag{54}$$

The solution for C1 and C2 for this equation is obtained in terms of M2/M1 and l1/D and L2/D using special algorithm in MATLAB.

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

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

Modeling of Woven Fabrics Geometry and Properties 31

Fig. 19. Interaction of crimp interchange and crimp balance equations (l1/D > l2/D)

Fig. 20. Interaction of crimp interchange and crimp balance equations (l1/D < l2/D)

solutions using iterations. It is the aim of this paper to facilitate fabric engineer in determining the fabric parameters for a given value of warp and weft crimp. This approach gives another alternative to engineer fabrics. The important variables of crimp balance equations are M2/M1, l1/D and l2/D. For a given crimp interchange equation in terms of l1/D and l2/D, the crimp balance equation gives intersections. The scales are also calibrated in terms of crimp.

Figures 18, 19 and 20 show the interaction of crimp interchange and crimp balance equation corresponding to l1/D = l2/D, l1/D> l2/D and l1/D < l2/D respectively. It is interesting to note that in all these curves with the increase in M2/M1, warp crimp increases and weft decreases. Another interesting result can be seen from these figures when l1/D not equal to l2/D. l1/D > l2/D or l1/D < l2/D causes a reduction in a range and shift towards lower values for both C1 and C2

Fig. 18. Interaction of crimp interchange and crimp balance equations (l1/D = l2/D)

These three curves show very interesting ways in which the values of crimp in warp and weft can be varied in a wide range. Therefore the three parameters M2/M1, l1/D and l2/D can influence the crimp in warp and weft in a wide range and this is what gives maneuverability to the fabric designer.

solutions using iterations. It is the aim of this paper to facilitate fabric engineer in determining the fabric parameters for a given value of warp and weft crimp. This approach gives another alternative to engineer fabrics. The important variables of crimp balance equations are M2/M1, l1/D and l2/D. For a given crimp interchange equation in terms of l1/D and l2/D, the crimp balance equation gives intersections. The scales are also calibrated

Figures 18, 19 and 20 show the interaction of crimp interchange and crimp balance equation corresponding to l1/D = l2/D, l1/D> l2/D and l1/D < l2/D respectively. It is interesting to note that in all these curves with the increase in M2/M1, warp crimp increases and weft decreases. Another interesting result can be seen from these figures when l1/D not equal to l2/D. l1/D > l2/D or l1/D < l2/D causes a reduction in a range and shift towards lower

Fig. 18. Interaction of crimp interchange and crimp balance equations (l1/D = l2/D)

maneuverability to the fabric designer.

These three curves show very interesting ways in which the values of crimp in warp and weft can be varied in a wide range. Therefore the three parameters M2/M1, l1/D and l2/D can influence the crimp in warp and weft in a wide range and this is what gives

in terms of crimp.

values for both C1 and C2

Fig. 19. Interaction of crimp interchange and crimp balance equations (l1/D > l2/D)

Fig. 20. Interaction of crimp interchange and crimp balance equations (l1/D < l2/D)

**2** 

Jeng-Jong Lin

*Taiwan, R.O.C.* 

**An Integration of Design and Production for** 

Nowadays, the enterprises all over the world are approaching toward globalizing in design and production in order to be more sustainable. Integration of interior divisions in a company or cooperation among different companies worldwide is of great importance to the competence enhancement for entrepreneurs. There have been a variety of developed applications to integrating different divisions (Cao et al., 2011) (Yamamoto et al., 2010). Moreover, the range of R&D cycle for textiles is much narrowed than ever. It is necessary for an enterprise to afford the demand of marketing change in small quantity and large variety for the commodity. Thus, it is crucial for textile manufacturer to integrate the design and

Generally speaking, at the very beginning a piece of fabric appeals to a consumer by its appearance, which is related to the weave structure and the colors of warp and weft yarns. Next, the characteristics, e.g., the permeability, the thickness, the tenacity, the elongation et al. of the fabric are required. Finally, the price of the fabric is used as an evaluation basis, by comparing which to the above-mentioned items (i.e., the outlook and the characteristics), the value of the fabric can thus be defined and determined. If the value is satisfactory, the fabric will be accepted by the consumer. Otherwise, it will become a slow-moving-item commodity. It is essential for the fabric with good quality to be of appropriate weaving density except being equipped with satisfactory pattern. If the weaving density is too less, the fabric will seem obviously too sparse to have good enough strength. The more weight consumption of the material yarns is, the higher cost needed for the manufacturing of a piece of fabric is. Thus, it is a crucial issue for a designer to make a good balance between the cost and the

essential consumption of the material yarns during woven fabric manufacturing.

Woven fabric is manufactured through the interlacing between the warp and weft yarn. The pattern of the woven fabric is illustrated through the layout of the different colors of the warp and weft yarn. Therefore, the application of computer-aided design (CAD) (Dan, 2011) (Wang et al., 2011) (Liu et al., 2011) (Gerdemeli et al., 2011) (Mazzetti et al., 2011) to simulated woven-fabric appearance and to the other aspects has been a major interesting research in recent years and various hardware and software systems are now available on the market for widely commercial applications. Until these systems became available, a considerable amount of time and money had been needed to show designers' ideas of fabric

**1. Introduction** 

production processes.

**Woven Fabrics Using Genetic Algorithm** 

*Department of Information Management, Vanung University* 
