Cloth Geometry

8/13/2019 Cloth Geometry

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abric GeometryIntroduction

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 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.


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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. Yarn crimp

2. Fabric cover factor

3. Fabric porosity

4. Fabric mass

5. Fabric thickness6. Fabric mass density

7. 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,

= density in g per cm3,

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):


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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 Kienbaums setting theory.


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):


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[

{ } ]

[ { } ]

Where,

= 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.


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


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In this equation,

K = fabric cover factor,

K1= warp cover factor =

K2= weft cover factor =

d = yarn thickness in mm,

p = distance between neighborhood yarns in mm, and

Gmax= maximal thread density in threads per cm.


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 macropores area in thefabric 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 macropores cross section . (Dubrovski & Brezocnik,

2005):

Where,

Po= open porosity in percentage,K = fabric cover factor in percentage,

Np= number of pores in pores per cm2,

Ap= area of macropores 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 (Sokolovi, 1981):

Here, h = height of thread arc in mm,d1 = warp yarn diameter in mm,

d2= warp yarn diameter in mm

F = number of Noviks fabric construction phase.

While by a new fabric development there is no information in which Noviks 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. Noviks 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.


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Noviks phases. In I. Noviks phase, where warp threads dont have any yarn crimp but weft

threads maximal, fabric thickness is the sum of warp diameter and weft diameter multiple by

two. In IX. Noviks 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. Noviks fabric construction 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 m

2,

G = actual thread density in threads per cm,

T = yarn fineness in tex,

= yarn crimp in percentage,

Mf= fabric mass in finished state in g per m2,

fin= yarn shrinkage by finishing in percentage, and

M = 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, fab= fabric mass density in g per cm

3,

M = fabric mass in g per m2and

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.


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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, iefabrics 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. Peirces Model2. Modified Pierces Model(ellipse)

3. Kemps racetrack Model(rectangle & circle)

4. Hearles 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 threadb = minor diameter of flattened thread

e = thread flattening coefficient (a/b)

h = height of crimp wave

T = fabric thickness (h1+ b1or 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

= maximum angle of the thread axis to plane of cloth in radiusl = length of thread axis between planes containing the axes of consecutive cross threads

lc= contact length of yarn

N = cotton count of yarn

Peirces 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.


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Fig. 3: Pierces 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 Peirces model:

Pick spacing (p1) and end spacing (p2), warp thickness (h1), weft thickness (h2) can be found

from this model

Peirces 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.

Fig. 4: (a) Peirces elliptic cross-section geometry of plain weave fabric, (b) Pierces

approximate treatment of flattened yarn geometry of plain-weave fabrics.

(a) (b)


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Kemp Model (Race-track section)

To overcome the jammed structure, Kemp proposed a racetrack section to modified cross-

section 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: Kemps racetrack section geometry of plain-weave fabrics.

Relations of Kemps model:

( )

Hearles Model Using energy method for calculations in fabric mechanics, a lenticular

geometry was proposed by Hearle as shown in Fig. 6.

Fig. 6: Hearles lenticular section geometry of plain-weave fabrics.


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Relations of Hearlesmodel:

( )

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 fabric 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.

Knitted Fabric Geometry

Knitted fabrics consist of only one set of weft yarns. Here the yarns are interlaced withadjacent yarns to construct a self-supporting structure. The different interlacingpatterns 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


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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.


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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 perimeter 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


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(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 between angle, we can calculate the length of each

curve. Then L2 will be computed by multiplying the length of the curve in 4.

L2= 4x 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 ) x angle (with regard to the rectangular triangleDFP in fig (3),Cos is equal to

).

Application of knitted fabric geometry

1. Fabric weight per unit area (g/m2) let Sis stitches per cm2;

lis loop length in mm; and

Tis yarn count in Tex

No. of stitches in 1 m

2

=Sx10

4

; Fabric weight = Sx 104X l/1000 X T/1000


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As S=Ks/l2;

g/m2= Ks/l2X 104X l/1000 X T/1000

g/m2=KsT/(100l)

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

Ifnis no. of needles, total stitches = nX cpcm X 100

g/m = nX cpcm X 100 X l/1000 X T/1000

Since cpcm= Kc/l

g/m = nX Kc/lX 100 X l/1000 X T/1000

g/m = nKcT/10000

Fabric Width

Width = number of needles in knitting / W= n/ W

whereWis wales per unit width (wpcm)

Since W=Kw/l

Width = nl/Kw

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.


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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


5. http://ningpan.net/Publications/1-50/39.pdf


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/1225110.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

The nd
http://www.texeng.co.uk/papers/Modelling_Fabric_Mechanics.pdfhttp://www.intechopen.com/download/pdf/36900http://www.intechopen.com/download/pdf/12243http://www.intechopen.com/download/pdf/12245http://www.amazon.com/EFFECTS-FABRIC-GEOMETRY...PERMEABILIhttp://www.intechopen.com/download/pdf/12251http://www.ellisdev.co.uk/fabric.htmlhttp://nopr.niscair.res.in/handle/123456789/406http://nopr.niscair.res.in/handle/123456789/406http://www.ft.tul.cz/studenti/seminar.../seminare.../kyosev_2011_cast2.pdfhttp://www.ft.tul.cz/studenti/seminar.../seminare.../kyosev_2011_cast2.pdfhttp://nopr.niscair.res.in/handle/123456789/406http://www.ellisdev.co.uk/fabric.htmlhttp://www.intechopen.com/download/pdf/12251http://www.amazon.com/EFFECTS-FABRIC-GEOMETRY...PERMEABILIhttp://www.intechopen.com/download/pdf/12245http://www.intechopen.com/download/pdf/12243http://www.intechopen.com/download/pdf/36900http://www.texeng.co.uk/papers/Modelling_Fabric_Mechanics.pdf

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