Introduction Woven fabric technology is deeply rooted in geometry. A fabric consists of millions of fibers assembled together in a particular geometry. Mutual interlacing of two sets of threads creates woven fabric. The manner of the mutual interlacing of threads defines the final fabric structure. The property of a fabric greatly greatl y affected by its geometry.
Fig: Fabric geometry.
Importance of fabric geometry: Knowing the fabric geometry, various problems can be solved and explained. Such as:
design the fabric with a determined crimp
know warp threads or weft threads will be broken first
Prediction of the maximum sett (density) of fabric and fabric dimensions
Prediction of mechanical properties by combining fabric and yarn properties
Find out relationship between geometrical parameters (picks and ends)
fabric thickness
the characteristics of the fabric surface
the length of warp and weft needed for a unit length fabric
Understanding fabric performance (handle and surface effect)
Parameters affecting fabric geometry Fabric constructional parameters relate to the geometrical structure of the fabric and are classified into primary and secondary parameters of fabric geometry.
Primary parameters of fabric geometry are: 1. Yarn thickness, 2. Weave factor and 3. Thread density. Via the defined selection of primary fabric constructional parameters, all other fabric structure parameters may be seen as constant and dependent on primary parameters. For this reason they are logically classified into the separate category, called secondary woven fabric constructional parameters. 1. 2. 3. 4. 5. 6. 7.
Yarn crimp Fabric cover factor Fabric porosity Fabric mass Fabric thickness Fabric mass density Warp and weft tension
Some fiber and yarn parameters have also effect on fabric structure and geometry.
Primary woven fabric constructional parameters Yarn thickness: By dealing with woven fabric geometry we assume that yarn is cylinder with
circular cross-section. The yarn diameter or thickness is then calculated on the basis of yarn linear density according to following equation:
Where, d = yarn thickness in mm, T = yarn linear density in tex, 3 ρ = density in g per cm ,
Weave factor: It is the pattern of interlacing of warp and weft in a woven fabric, which can
be numerically expressed with weave factor in warp and weft direction (Kienbaum, 1990a):
√ √ √ √
Where, V = weave factor, v = yarn volume coefficient, R = number of threads in weave repeat, a = number of passages of yarn in weave repeat from face to back and vice versa, z = smallest weave shift and f = yarn flexibility.
Subscripts 1 and 2 denote warp and weft yarn, respectively. Thread density: It is primary constructional parameter which is altered by weave and yarn
thickness. It is usually defined as the number of threads per centimeter and expressed for warp (ends) and weft (picks) threads. The term limit thread density was introduced to calculate thread density by limit geometry. Limit geometry refers to the situation where the threads are not deformed and lie in one plane close to each other or there is a minimal space for thread passage. On the basis of limit thread density, woven fabric constructor can decide which actual thread density will be set for finished fabric. Relative thread density (or thread tightness) is the ratio between the actual and limit thread densities, expressed as per cent (Kienbaum, 1990a):
Where, t = thread tightness in percent, G = actual thread density in thread per cm, and Glim = limit thread density calculated according to the Kienbaum’s sett ing theory.
Subscripts 1 and 2 denote warp and weft yarn, respectively. Woven fabric tightness – t is then defined as the geometrical average of warp and weft tightness according to following equation.
√ Secondary woven fabric constructional parameters Yarn crimp: It is the consequence of yarn interlacing in woven fabrics. It is numerically
expressed as percentage crimp, which is 100 divided by the fabric length and multiply by the difference between the yarn length and the fabric length (Denton & Daniels, 2002). Theoretically it can be calculated on the basis of fabric geometry with following equations. (Kienbaum, 1990a):
{ } [ ] { } [ ] Where, ε = yarn crimp in percentage, p = distance between neighbourhood yarns in mm, R = number of threads in weave repeat, m = number of thread passages in weave repeat, d = yarn thickness in mm, and G = actual thread density in raw fabric in threads per cm.
Subscripts 1 and 2 denote warp and weft yarn, respectively. Fabric cover factor: It indicates the extent to which the area of a woven fabric is covered by
one set of threads according to equation below (Fig. 1). It is calculated on the basis of warp/weft cover factor which indicates the ratio between the yarn thickness and the distance between neighborhood yarns or the ratio between the actual thread density and maximal thread density. However, the faultiness of fabric cover factor is the absence of weave influence.
Fig. 1: Woven fabric geometry by fabric cover factor definition
In this equation, K = fabric cover factor, K 1 = warp cover factor = K 2 = weft cover factor =
d = yarn thickness in mm, p = distance between neighborhood yarns in mm, and Gmax = maximal thread density in threads per cm.
Subscripts 1 and 2 denote warp and weft yarn, respectively. Fabric porosity: It indicates the portion of pores in woven fabrics. While the woven fabric
could be treated as two or three dimensional form, the terms open porosity and volume porosity are distinguished. Open porosity indicates the percentage of macropore’s area in the fabric area unit. It is calculated on the basis of fabric cover factor or on the basis of the number of macropores and the area of macropore’s cross section . (Dubrovski & Brezocnik, 2005):
Where, Po = open porosity in percentage, K = fabric cover factor in percentage, N p = number of pores in pores per cm 2, A p = area of macropore’s cross section in cm2, and
Fabric thickness: It is the distance between the fabric face and back. Theoretically, it
represents the sum of height of warp and weft arc (Sokolovič, 1981):
Here, h = height of thread arc in mm, d1 = warp yarn diameter in mm,
d2 = warp yarn diameter in mm F = number of Novik’s fabric construction phase. While by a new fabric development there is no information in which Novik’s fabric construction phase the woven fabric will appear, only minimal and maximal value of fabric thickness can be predicted. Minimal value of fabric thickness refers to V. Novik’s phase, where warp and weft threads have equal yarn crimp. In this case the fabric thickness is the sum of warp and weft diameter (Fig. 2). Maximal value of fabric thickness refers to I. and IX.
Novik’s phases. In I. Novik’s phase, where warp threads don’t have any yarn crimp but weft threads maximal, fabric thickness is the sum of warp diameter and weft diameter multiple by two. In IX. Novik’s phase, where warp threads have maximal yarn crimp and weft none, fabric thickness is the sum of warp diameter multiply by two and weft diameter.
Fig. 2: I., V. and IX. Novik’s fabric const ruction phases
Fabric mass: It can be expressed as mass per one meter or as mass per square meter. Later is
more useful by comparing different types of woven fabrics. Theoretically it can be calculated according to Eq. (1) and Eq. (2) which refers to the unfinished and finished fabric state, respectively:
Here, Mg = raw fabric mass in g per m2, G = actual thread density in threads per cm, T = yarn fineness in tex, ε = yarn crimp in percentage, Mf = fabric mass in finished state in g per m 2, εfin = yarn shrinkage by finishing in percentage, and ΔM = change of raw fabric by finishing in pecentage.
Fabric mass density: It expresses the mass of volume unit of woven fabric in grams per cm3.
It can be calculated from the following equation.
Here, ρfab = fabric mass density in g per cm 3, M = fabric mass in g per m 2 and D = fabric thickness in mm.
By fabric engineering it is possible only to predict minimal and maximal value of fabric mass density according to the minimal and maximal value of fabric thickness.
Geometry Theories Approach
1. In conventional approaches, the general character of fabrics was idealized into simple geometrical forms (circle, ellipse, rectangle) 2. They treated the micro-mechanics of fabrics on the basis of the unit-cell approach, ie fabrics are considered as a repeating network of identical unit cells in the form of crimp weaves and constant yarn cross-section in the woven structure. 3. By combining this kind of geometry with or without physical parameters (material), mathematical deductions could be obtained.
Fabric Geometry Models By using circle, ellipse, rack-track approaches, four fabric geometrical models are formed. 1. 2. 3. 4.
Peirce’s Model Modified Pierce’s Model (ellipse) Kemp’s racetrack Model (rectangle & circle) Hearle’s Lenticular Model
Now we will discuss in brief about these models. Mathematical notations for each model are given below: d = free circular-thread diameter D = sum of circular diameters (d1 + d2) a = major diameter of flattened thread b = minor diameter of flattened thread e = thread flattening coefficient (a/b) h = height of crimp wave T = fabric thickness (h 1 + b1 or h2 + b2, whichever is greater) p = average thread spacing for the fabric as a whole n = average number of threads per unit length (n = 1/p) c = thread crimp K = cover factor θ = maximum angle of the thread axis to plane of cloth in radius l = length of thread axis between planes containing the axes of consecutive cross threads lc = contact length of yarn N = cotton count of yarn Peirce’s Model
In this model, a two-dimensional unit cell of fabric was built by superimposing linear and circular yarn segments to produce the desired shaped. The yarns were assumed to be circular in cross-section and highly incompressible, but perfectly flexible so that each set of yarns had a uniform curvature imposed by the circular cross-sectional shape of interlacing yarns.
Fig. 3: Pierce’s model for plain-weave fabric geometry
Geometrical parameters such as thread spacing (p), weave crimp, weave angle and fabric thickness (h) can be found. Relations of Peirce’s model:
Pick spacing (p1) and end spacing (p 2), warp thickness (h1), weft thickness (h2) can be found from this model Peirce’s Elliptic Model
In more tightly woven fabrics, however, the inter-thread pressures setup during weaving cause considerable thread flattening normal to the plane of cloth. Pierce recognized this and proposed an elliptic section theory as shown in Fig. 4(a). Because such model would be too complex and laborious in operation, he adopted an approximate treatment, which involved merely replacing the circular thread diameter in his circular-thread geometry with minor diameter as shown in Fig. 4(b). This modified model is good for reasonable open fabric but cannot be applied for very closely jammed fabric.
(a)
(b)
Fig. 4: (a) Pei rce’s elliptic cross-section geometry of plain weave fabric , (b) Pierce’s approximate treatment of flattened yarn geometry of plain-weave fabrics.
Kemp Model (Race-track section)
To overcome the jammed structure, Kemp proposed a racetrack section to modified crosssection shape. The model consisted of a rectangle enclosed by two semi-circular ends and had the advantage that it allowed the relatively simple relations of circular-thread geometry, already worked out by Pierce, to be applied to a flatted threads.
Fig 5: Kemp’s racetrack section geometry of plain-weave fabrics.
Relations of Kemp’s model:
( ) Hearle’s Model Using energy method for calculations in fabric mechanics, a l enticular geometry was proposed by Hearle as shown in Fig. 6.
Fig. 6: Hearle’s lenticular section geometry of plain-weave fabrics.
Relations of Hearle’s model:
( ) Limitations of fabric geometry models 1. Firstly, fabrics are complicated materials that do not conform even approximately to any of the ideal features suggested by these four f abric models. 2. Secondly, the measurement of geometrical parameters is not easy in practice. 3. Thirdly, the relationship between fabric mechanic (tensile, elongation, bending) to fabric geometry is not fully explored.
Kn itted F abri c Geometr y Knitted fabrics consist of only one set of weft yarns. Here the yarns are interlaced with adjacent yarns to construct a self-supporting structure. The different interlacing patterns can be categorised as jersey, rib, interlock, lacoste, pique, etc.
Fig: Knitted fabric geometry. Knitted fabric geometry and parameter
–
air permeability;
– shrinkage of a fabric
– pilling and snagging effect – flexural rigidity and extension – tightly or slackly of fabric – tightness or cover factor The Presented loop Model Assume that we have a rectangle with the length of 8 units and width of 13 units (for drawing suchrect angle we must initially find the exact amount of 13 using Pythagoras). The length of the rectangle must be divided in to 4 equal segments. As shown in figure (1) we connect a line from M to C and then another line from N to D.
Fig 1: The primary frame for drawing the geometrical configuration of the Knitted loop.
Then, from the middle of length AB (O) will draw a semicircle to arc 2 and to arc 1 with the same center. As per figure 2 will draw semicircles to arc 1 and 2 on CD with the center of Q and P.
Fig 2: Analysis of the loop geometry.
From center M, we draw a curve with the arc of 4 units so that it cuts MC. Then at the same M point, we draw another curve with the arc of 3 units cutting MC. Likewise, we draw curves with D, C, N centers. Finally a shape is resulted which is so similar to the Knitted loop (fig 3). Calculating the total loop length (L): The loop length consists: 1. Needle loop (semicircle) 2. Sinker loop (a quarter of a circle) 3. Arm (circle curves) From the relation (3), we get to the L which is the loop length. L= L1 (sinker and needle loop) + L2 (arm) As we know:
And the ratio of C to W will be calculated as show in figure (3):
Fig 3: The geometry of knitted loop model. A) Calculation of the loop length (L1):
The loop length L1 (needle loop and sinker loop) will be equal to a circle perimet er with regard to two circle square of the sinker and semicircle of the top of the loop. The perimeter
of this circle will be calculated with the diameter
(Calculation will be considered based on the thread diameter). (figure 3)
B) Calculation of the loop length (L2): For finding the numerical value of L2 as per figure (3), we know that L2 is equal to totally 4 identical curves. Having the arc and the in betwee n angle, we can calculate the length of each curve. Then L2 will be computed by multiplying the length of the curve in 4.
L2= 4xα curve length The length of each curve is equal to the curve arc (taking in to consideration the diameter of
the thread, the curve arc is equal to DFP in fig (3),
) x angle α (with regard to the rectangular triangle
Cosα is equal to ).
Application of knitted fabric geometry
1. Fabric weight per unit area (g/m 2 )
– let S is stitches per cm2; – l is loop length in mm; and – T is yarn count in Tex 2 4 No. of stitches in 1 m = S x10 ; 4 Fabric weight = S x 10 X l /1000 X T /1000
2
– As S=K s/l ; – g/m2= K s/l 2 X 104 X l /1000 X T/1000 2 g/m =K ) s T/(100 l 2. Fabric weight per running meter Running meter is one meter length of fabric measured along the direction of production irrespective of width. Number of courses in one meter = cpcm X 100 If n is no. of needles, total stitches = n X cpcm X 100 g/m = n X cpcm X 100 X l /1000 X T /1000
– Since cpcm= K c/l – g/m = n X K c/l X 100 X l /1000 X T /1000 – g/m = nK cT /10000 Fabric Width
Width = number of needles in knitting / W = n/ W where W is wales per unit width (wpcm) – Since W =K w/l – Width = nl /K w
Conclusion: Many features of the cloth are essentially dependent on geometrical relationships. For description of real fabric geometry is necessary to know behavior of individual threads in woven fabric. Mutual relations between warp and weft threads during weaving on weaving loom together with input fabric parameters define position of warp and weft threads in woven fabric as well as mechanical and end-use properties. Important fabric parameters connected with fabric geometry is warp and weft crimp. Warp and weft crimp influence on threads waviness as well as threads set and weave.
Reference A. Class Lecture. B. Website. 1. http://www.texeng.co.uk/papers/Modelling_Fabric_Mechanics.pdf 2. http://www.intechopen.com/download/pdf/36900 3. http://158.132.122.156/knit/woven/.../fabric%20geometry.pp 4. http://www.intechopen.com/download/pdf/12243 5. http://ningpan.net/Publications/1-50/39.pdf 6. http://www.intechopen.com/download/pdf/12245 7. http://www.amazon.com/EFFECTS-FABRIC-GEOMETRY...PERMEABILI.. 8. http://trj.sagepub.com/content/18/11/650.refs 9. http://www.intechopen.com/download/pdf/12251 10. http://www.ellisdev.co.uk/fabric.html 11. http://nopr.niscair.res.in/handle/123456789/406 12. http://www.ft.tul.cz/studenti/seminar.../seminare.../kyosev_2011_cast2.pdf
