2.1. Assumptions for studying the compressive behaviour of woven fabrics using a mathematical model

1. The yarns in the fabric are assumed to be elastic and isotropic, although there are some anisotropy and in-elastic properties.
2. The yarn is assumed to be a solid cylinder.
3. The geometry of the yarn surface is assumed to be smooth and uniform (which is in fact not smooth, because of the yarn twist and the protruded fibres present on yarn surface); the resistance of the fine hairs on the yarn surface is therefore ignored.
4. The difference caused by different fibres and fibre arrangements arising from different spinning technologies is assumed to have been captured in the material constant, and is not considered for modelling.
5. The fabric is assumed to be completely relaxed; this means that the fabric does not have any residual force within itself.
6. The response of the fabric to different machine settings (yarn tension, yarns per dent, beat up-force, etc., and post-weaving treatments (desizing, bleaching, softening, dyeing, reducing weight, etc.,) will not be taken into consideration.
7. Change in crimp crowns is assumed to occur in a transverse direction and is not taken into account when calculating geometric thickness; the longitudinal deformation due to lateral pressure is negligible.
8. Yarn rigidity, slippage between the fibres or filaments, crimp balance and crimp interchange effects in longitudinal directions is negligible under low pressure, and its effect is not considered.
9. The compressive force applied on the fabrics is at low pressure, ranging from 0.5 to 50 gf/cm².
10. On compression threads moving closer to attain equilibrium conditions following different cross section models and its phenomenon is not taken into consideration.

2.2 Peirce model for change in thickness

Figure 1 . Peirce unit cell model

From Pierce’s geometry, the fabric thickness ‘t’ is calculated, which is the sum of the crimp height and the diameter. Let the crimp height be ‘h’ and the yarn diameter be ‘d’.

Fabric thickness ‘t’ is given by

t = h1 +d1 or h2 + d2 (5)

Since the yarn diameters are assumed to be circular, we have

t= max (t1, t2) (6)

where:

N1 and N2 are warp and weft counts,

e is the flattening factor, and

t is the thickness of fabric.

In reality, cross sections are far from circular, and later models modified the cross section; the racetrack (the rectangle with circular arcs at the sides) was widely adopted for mathematical tractability. The compressed thread may be considered to have the form of a flattened elliptical cross section, where the area of the ellipse is given by

abπ

(7)

4

where ‘a’ and ‘b’ are major and minor diameters. But this is not true; when pressure is applied, the threads spread unevenly, and can attain a configuration which can be assumed to lie somewhere between the ellipse and the rectangular with circular arcs. For ?structural reasons, the limit of compression where the crowns are flattened must be considered, and hence a flattening factor ‘e’ must be introduced.

By introducing the flattening factor ‘e’, which is a function of crimp of yarn, cloth setting, and count, the flattening factor ‘e’ is obtained by [12].

C% C%

1 ₊ 2 _⎜^ 1 1 _⎟^

= 0.28 e

+ ⎟

nn ⎜

21 _ ^N1 ^N2 ⁽⁸⁾



2.3. Change in thickness for non-circular yarns

Figure 2. Kemp’s race-track geometry cross-section

In the case of industrial fabrics like canvas and tarpaulins subjected to large deformation by compressive force, would result in jamming of either warp or weft. In large deformations, one has to consider different assumptions for the analysis, introduce change into the structural elements and then infer the thickness. It would be appropriate to fix the limits of compression using Kemp’s racetrack model for flattened yarns [3] with changes in pick spacing, diameter of yarns, crimp height etc.

This new shape, which is termed the race-track section, is rectangular with semi-circular ends, and is used as the basis for an extension of Peirce’s theory to the more general case of non-circular threads.

The horizontal and vertical diameters of the threads are denoted by ‘a’ and ‘b’ respectively. The scale unit ‘D’ is equal to the sum of the vertical diameters:

D =b + b = h + h (9)

12 12

Peirce’s formulae and tables of functions may be used for part of this section, the variables being

' ''' ''

p , p ,l ,l ,c ,c

1 21212

The relationships between these variables and the structural particulars p , p ,l ,l ,c ,c are as

1 21212

follows:

p ^' =p − (a − b )

22 22

(10)

l^' = l − (a − b )

11 22

' ''

and c =(l' − p )/ p = cp /[ p − (a − b )] (11)

1 12 2122 22

'' '

similarly, for p₁,l and c₂ .

2

Thus, if the term (a −b ) is known,

22 '''

p,l ,c can be easily calculated from the structural particulars, p,l₁, and c₁ . Similarly, a knowledge

211 2 '' '

of (a₁ − b₁) is required for the calculation of p₁,l₂ and c₂

''

From the circular thread theory, assuming ‘θ’, the angle of crimp, to be small, and p₂ and l₁ large, the following relation holds as a useful approximation:

a

h1		=		3 4		p ' 2		c ' 1		*								(12)
Hence, from equations (11) &, (12), the following relation is obtained:
b1	+b2				=		3 4		c p [ p1 2 2		−(a2	−b2 )]	+	3 4	c p [ p2 1 1	−(a1	−b1)]	(13)
If the thread flattening coefficient ‘e’ is defined by
e =			b															(14)

and ‘d’ is the diameter of the circular thread of equal area, then it can be easily shown that

1

b =d

(15)

 41−e ⎤

1+ ()

⎢⎥

⎣πe 

[(1−e)/ e]

and (a−b) =d ₄ = df (e), say (16) [e/(1−e) + ]

π

hence

4

b₁ +b₂ =

p − ()] +⁴

cp [ d fe

cp [ p −df ()e ] (17)

122 22

211 11

3

3

4

1

1

(18)

p − ()] +⁴

cp [ dfe

p − ()]1 = _d1

cp [ dfe

+d₂

122 22

211 11

3

3

4 ⎛1−e 

4 ⎛1−e ⎞

[1+ ¹ ⎟]

[1+ ² ⎟]

⎜⎟

⎜⎟

π e

π e

 1 

 2 

This equation contains the unknowns d ,d,e and e₂ . Assuming that d₁ and d₂ can be accurately

1 21

estimated from the count of the warp and weft yarns, then if a further relation is evolved between e₁

and e₂ (for example, in special cases it may be valid to assume e =e =1, the two variables e₁ and

12

e₂ can be reduced to one ‘e’.

In such cases, a graphical method of solving the equation may be conveniently be adopted, that is, by plotting both sides of the final equation against ‘e’, the intersection of two curves giving the required value ‘e’.

2.4. Change in the cover factor of the fabric

Hamilton[13] defines the fabric tightness which could be used for comparison of fabrics at different configurations as the ratio of the sum of the fabric cover factors to the sum of the cover factors of the maximum-set fabric with the same ratio of cover factors. This is simply expressed as:

(K +K )

12 actual

T = (19)

(K +K )

1 2 lim it

^d1

Here, the warp cover factor K = where d is the warp yarn diameter and p₁ the thread spacing,

11

p

1

^d2

and the weft cover factor K = where d and p₂ are the weft yarn diameters and thread

22

p

2

spacing respectively. The term '(K₁ +K ) refers to the actual values of a specific fabric, the

2 actual

value (K +K ) is determined graphically from graphs from Hamilton’s paper.

1 2 lim it

Hamilton and others were concerned that yarns in fabrics are not circular in the cross-section, and they therefore used modified equations to allow for yarn flattening. This procedure is no doubt essential for calculating a wide range of properties and dimensions. The Peirce equations assume a circular cross-section, rather than an elliptical or race-track shape, such as may occur in a more open fabric. Hence we need to use yarn diameter rather than arbitrary or measured major & minor diameters.