Important Fabric parameters and Formulae

Written by: Manian  

Cover Factor
Cover factor denotes the density of fabric i.e the area occupied by the threads in relation the air space between the threads.

Ratio of threads per inch to square root of count is defined as "cover factor" K. Cover factor determines the appearance,handle, feel, permiability, transparancy, limits of pick insertion and hardness of fabric. If p is spacing between threads in mils(1/1000 inch)
K = 1000/(p√C)
where d1is diameter of thread in mils and specific volume of yarn is v. If v is assumed as 1.1
K = = 28 d1/p.
If count is in tex units, then cover factor is equal to threads per cm multiplied by square root of tex of yarn. When cover dactor is 28,  d = p. The threads will contact at the point where they cross from one face to other phase of cloth. Higher cover factors can be obtained by compression of yarns or by distortion of structure.In practise, cover factor has to be kept lower than 28 to allow space for threads to pass over one another. . So very high values are possible only in one direction in which threads have high crimp. 28 is the limit for canvas. In poplins warp will have a higher cover factor than weft. Normal fabrics will have a cover factor of 12..

Crimp is defined as  the proportion by which straightened length of yarn is higher than the cloth length which contains the yarn. For determining crimp a length of fabric,l is marked. Yarn is removed from marked length of fabric, straightened to remove the waves by application of tension and measuring its length(l1). Fractional Crimp, c = (l1 - l)/l. Tension applied to straighten the yarn is standardised at 16/C oz.

Weight per square yard
Weight per square yard of fabric is equal to weight of warp and weft in a square yard of fabric.
Weight of warp = (K1(1+c1)0.6857)/√C1
Weight of weft = (K2(1+c2)0.6857)/√C2
Weight of one square yd of fabric = (K1(1+c1)0.6857)/√C1+ K2(1+c2)0.6857/√C2Weight per square yd of fabric = (1/√C1)0.6857{K1(1+c1) + K2(1 + c2) ß}
where suffices 1 and 2 refer to warp and weft and ß = √(C1/C2)

Crimp - spacing Relationship
By neglecting bending resistance of yarn ,and assuming yarn cross section in fabric to be cicular Pierce(JTI 1937, T45) developed a geometrical model to determine crimp, thread spacing relationships. The lie of threads in a plain fabric under such conditions is shown in Fig 1.

Let d1 in mils..... denote diameter of warp
p1in mils.....denote spacing between warp threads
Θ1.....denote maximum angle of warp to plane of cloth
l1.....denote length of warp thread axis between axis of consecutive weft threads
h1.......denote maximum displacement of warp thread axis, normal to plane of cloth
c1........denote fractional crimp of warp
 The above terms with subscript 2 denote the corresponding values for weft. Then
D = d1 + d2
c1 = (l1/p2)
 p2 = (l1 - DΘ1)cos Θ1 +
sin Θ1
h1 = (l1 - DΘ1)sin Θ1 +
D(1 - cos Θ1)
h1 +  h2 = D
Upon expanding sinΘ and cosΘ in ascending powers of Θ
c1 = (l1Θ2 - DΘ3 ...)/(l1 -l1Θ2 +DΘ3..)
This approximates to
h1 ≈ p2√ 2c1
In practise however, modification of the factor √ 2  by 4/3 gives a more accurate estimate
h1 = (4/3)p2√c1
As shown earlier, d1 = (1000/29.3)√(v/C) = 34.14√(v/C)
where d1 = diameter of yarn in mils, v = specific volume and C = Count of yarn.
Since D = h1 + h2,
 D = 4/3(p2√c1 + p1√c2)
= 34.14(√(v1/C1) +√(v2/C2)
Jammed Structures
When warp is jammed, the weft starts touching the adjoining warp the momment it leaves previous warp. Length of weft is therefore made of curved wrappings made around warp with no straight portion.In this case l1/D = Θ1, p2/D = sin(l1/D) = sin((1+ c1)p2/D), h2 = 1 - cos(l1/D)

Square cloth - jammed structure
In the case of square cloth, warp and weft have the same diameter, spacing and crimp. p1 = p2, c1 = c2, l1 = l2 = l, h1 = h2 = 0.5
cos(l/D) = 1- (h/D) = 0.5 Θ = 600 and l/D = 1.0472 radians p/D = sin 600 = 0.866
c = (l/p) - 1
l/p = (l/D)× (D/p) = 1.0472/.866 = 1.2092
c = 0.2092 For square cloth, warp threads leave an uncovered portion (p - d)/p = 1 - D/2p = 0.4227. Proportion of space not covered by warp and weft is projection of both sets of threads and is (o.4227)2 = 0.1787

Jammed structure with race course cross section
When warp or weft is jammed the threads get compressed and assume a cross section similar to that of an ellipse or race track. Race track cross section is more easily amenable to mathematical analysis.The lie of threads in jammed structure with race track cross section is shown in Fig 2. 
Fig 2

From Fig it is seen that E =√(F2 -h2)where F = D1 + D2, the sum of warp and weft race track radii and A is width of race track and
p = E + (A - D)
√(F2 - h2) = p - (A - D) = q
h = √(F2 - q2)
√(1 - (q1/F)2) + √(1 - (q2/F)2) = 1. From this the maximum number of picks that can be inserted into a cloth for a given ends per inch and diameter of yarn can be determined and likewise for ends per inch.

Cover Factor for close constructions
If weave is close in both directions p2/D = sinΘ1 = sin l1/D
p1/D = sinΘ2
cosΘ1 = 1 - (h1/D)
cosΘ2 = 1 - (h2/D)
cosΘ2 = √(1 - (p1/D)2)
√(1 - (p1/D)2) + √(1 - (p2/D)2) = 1 This leads to
K1 = 28D/(p1(1 + ß))
K2 = 28D/(p2(1 + ß))
where ß = C1/C2
For a square cloth
p1 = p2 and ß = 1 and the closest construction is given by
√(1 - (p1/D)2) + √(1 - (p2/D)2) = 1
√(1 - (14/K1)2) + √(1 - (14/K2)2) = 1 Table below shows the cover factor for weft for various values of warp cover factor when both are close.

Warp Cover Factor Weft Cover factor
14 20.8
15 18.23
16 16.34
17 15.52
18 15.08
19 14.79
20 14.6
21 14.47
22 14.38
23 14.3
24 14.25
25 14.21
26 14.17

Normal Square Cloth
For square cloth of more open structure
h = (4/3)p√C and D = 2h
8/3(p√c) = 68.28(√(v/C)
√c = 25.59/p(√v/C) and K = 1000/p√C So √c = 25.59K√v/1000 = .02559K√v. If v is assumed as 1.1 then c ≈ (K(0.1)/4)2 ≈(K/4)2%

Crimp alteration
Consider the case when one of the threads is stretched by tension. If the weft threads are pulled straight, then h2becomes zero and h1becomes D.
D = (l1 - DΘ1 )sinΘ1 + D(1 - cosΘ1)
l1/D = Θ1 + cotΘ1 and
p2/D = l1/D cosΘ1 - Θ1cosΘ1 +sinΘ1. This reduces to p2/D = cosecΘ1. If weft threads are too close, warp threads will jam when the straight intermediate portion(l1 -DΘ1) is reduced to zero.
Then l1 - DΘ1 = 0
l1/D = Θ1. These equations help to determine the crimp alteration that takes place when warp or weft is stretched or shrunk.