An approach to the theory of compression of nonwoven fabrics

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Indian Journal of Fibre & Textile Research Vol. 21, December 1996. pp. 235-243

An approach to the theory of compression of nonwoven fabrics

V K Kothari&ADas

Department ofTextile Technology,Indian Institute ofTechnology, New Delhi 110016, India Received 8 February 1996; revised received 23 May 1996; accepted 1 July 1996

A theoretical analysis of the compressional behaviour of nonwoven fabrics has been proposed based on bending of fibre elements. A fibre segment, which is assumed to be bonded at two neighbouring contact points, has been chosen as the unit bending element. An interactive algorithm, in which the system geometry is updated on successive increments of load, is used to cope with large and non-linear deformation. A comparison between the experimental results and the theoretical predictions shows reason- able agreement. Spunbonded heat-sealed fabrics show a good agreement with the theory, while the spunbonded needle-punched fabrics show deviation, mainJy due to fibre-to-fibre slippage.

Keywords: Compressional behaviour, Nonwoven fabrics, Spunbonded heat-sealed fabrics, Spunbonded needle-punched fabrics, Theory of compression

1 Introduction

The compressional behaviour of a fibre assembly, e.g. loose fibre mass, yarn and fabric, is important as it affects the properties of fibre assembly. Nonwoven fabrics are often used in various technical and industrial' applications where these fabrics are subjected to compression. A survey of literature indicates that most of the reported work has been on the deformation of fibre mass due to application of compressional load. Van Wyk

I,

in an early paper, theoretically analyzed the compressional behaviour of a loose fibre mass. In spite of many deficiencies, his approach is useful for the theoretical analysis of the compressional behaviour of fibre assemblies. Some thirty years later, Komori and Makashima2 developed a model to find the number of fibre-to-fibre contact points in a fibre assembly with arbitrary distributions of orientation. The estimation of the number of fibre- to-fibre contact points is of fundamental import- ance in the determination of the mechanical behaviour of fibre assemblies, as these contacts determine the average free element length, which, in turn, will bend during the application of compressional force and also at these contact points the frictional interactions take place. Kallmes and Corte3 derived the expression for the number of fibre intersections for a two-dimensional fibre ne- twork, e.g. paper. They assumed that all the fibres lay in a plane with their direction perfectly random and assuming the thickness of the assembly

to be twice the diameter of the fibres. Their results cannot be applied if the fibres form a three- dimensional structure. Stearn4 modified Van Wyk's approach by taking into account the change in orientation of fibres in the assembly during compression. Again, the application of his theory for nonwoven fabrics seems to be restricted, because it was assumed that the orientation of the fibres was random before the mass was compressed. By means of the radioactive tracer technique TaylorS determined the number of fibre-to-fibre contacts in twistless slivers and found that at low densities (up to 0.08 for the ratio of volume of wool to total volume) the number of contacts increased linearly with the density, whereas at higher densities the number of contacts increased at a steadily in- creasing rate. Curiskis and Carnaby6 proposed a model to predict the mechanical behaviour of a fibre bundle as a continuum. Lee and Lee7 analyzed the initial compressional behaviour of fibre assembly. They derived the compressional modulus of the general fibre assembly from the bending deformation of the fibre segments using an orientation density function of the fibres in a random fibre assembly. Carnaby and PanR derived the theory of the compressional hysteresis of fibrous assembly by extending the theory of Lee and Lee7• In their theory, they have taken into account the effect of fibre slippage and succeeded for the first time in theoretically reproducing the compression hysteresis of fibrous masses. Following the same line of

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"1 II 11'1'1 IN -TIt! UI

236 INDIAN 1. FIBRE TEXT. RES., DECEMBER 1996

"

thought, Pan and CarnabyY developed a theory of shear deformation of fibre masses. Komori and Itoh10, in their approach on the theory of compression of fibre assemblies, took the length of bending element to be orientation dependent.

They did not take into account the effect of fibre slippage for simplifying the presentation.

More recently, Lee and Carnaby" and Lee et

al.12 derived mathematical theories of the

compression of the random fibre assembly by energy method. Leaf and OxenhamU.14 also analyzed the compressional behaviour of yarn as energy function form. The nonwoven fabrics are now widely used in various technical and industrial fields, e.g. filtration, geotextiles, etc. In these applications, nonwoven fabrics are subjected to compressional load, which, in turn, affects the mechanical and hydraulic properties. Theoretical analysis of the compressional behaviour is essen- tial in predicting the fabric characteristics, knowing the fibre and fabric parameters. In this paper, a model has been proposed to predict the compressional behaviour of different types of nonwoven fabric. The theory has been evaluated for a seri- es of needle-punched and heat-sealed spunbonded nonwoven fabrics.

2 Theory of Structural Deformation

Various complex deformations of the structure take place during compression of nonwoven fabrics. However, in the present analysis the following assumptions have been made to simplify the model:

• All the fibres have identical properties. They are homogeneous, linearly elastic and have cir- cular cross-section.

• The fibres are assumed to be continuous, i.e. no free ends are present within the nonwoven fabrics.

• The fibre diameter is infinitesimal, so that even a small element of the fibre assembly may be regarded as composed of a very long length.

• Torsional, compressional and tensile deformations of the fibres are so small that these can be neglected.

• Each fibre element between two neighbouring contact points is equal in length and straight before loading.

• Continuity of the curvature at contact points during bending of fibre element is neglected, so

the rotation of the segments can occur freely around them, i.e. each element bends independ- ently.

• No new contact points are formed during the incremental increase in the load.

• The magnitude of the loads acting on all the contact points is equal to each other and direction of them is identical with that of compressional force acting on the fibre assembly.

• Effect of vertical pegs of fibres on compression of needle-punched nonwovens is neglected.

2.1 Geometry of the Structure of Nonwoven Fabrics

In the spunbonded nonwoven fabrics, fibres are often randomly oriented in the plane of the fabrics as the deposition of filaments from the spinnerette is random in most cases. In some cases, preferen- tial orientation may also be there due to tension in the fibre web during processing or due to the nature of production process. In the present study, the fibres are assumed to be randomly oriented in planner direction. Fig. 1 shows the section of an arbitrarily chosen fibre OA, whose direction with respect to spherical co-ordinate system is defined by polar angle ^(j, i.e. the angle between the fibre and Z-axis, and the azimuthal angle ¢, i.e. the angle between the X-axis and the projection of fibre on XY plane, OB. As it has been assumed that the planner orientation of fibres is random, the range of ^¢ is between 0 and x. But so far as the vertical angle of orientation () is concerned, the fibres are distributed within a very small zone, i.e. from angle ()I to ()2" The vertical angle of orientation (()) is different for different fabrics, particularly in the case of needle-punched nonwoven fabrics. To find out this angle, the fabrics were cut into slices and mounted on glass .slides.

The cross-sections of the fahrics were observed under a projection microscope. The extreme angle

z

y

x

Fig. I--Fibre orientation in spherical coordinate system

'I ^1'1 ^'11 ^'I 1II I, I'I' ~ 11,II I' ;'I ~II l'lllnl'~lw,,' r' '" ,,! 'I" tI "'11'; I"

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KomARI &DAS : THEORY OF COMPRESSION OF NONWOVEN FABRICS 237

Let the probability of finding the orientation of a fibre in the infinitesimal range of angle () to (() + d()) and ~ to (~+ d~) be Q((), ~) sin () d() d~, where Q( (), ~)sin () is a direction density function of fibres. According to the definition of probability density function, the following equation must be satisfied

of deflection of fibres from the fabric plane were measured for different fabrics and the values of the vertical angle of orientation are given in Table

1.

The orientation of the fibre OA, as shown in Fig. 1, can be defined uniquely by ^{(0, ~).} The mean projections on the axes X, Y, Z are given as follows:

From Komori and Makashima2, under these conditions,

fH:d()f\d~^H, ⁽⁾ Q((),~) sin()=

I

Px= OA'sinO'cos~, pz= OA·cosO

1= V/2 DLI and

where

Py= OA'sinO'sin~ and

... (1)

... (2)

where

Kz=^JX^H,²d0JX2d~⁽⁾ Q(0, ~) sinO^cosO

2.2 Mechanism of Deformation

Compressional deformation of the internal structure of spun bonded nonwoven fabrics is assumed to be mainly due to bending of free elements between two neighbouring contact points.

Slippage between two adjacent fibres also takes place in case of needle-punched structures but in the present analysis the slippage between fibres is neglected. Extension of fibres and twisting of fibres may also affect the compression, but their ef- fects are negligible.

Let us consider a single initially straight section of one fibre (AB) (Fig. 2) that is supported by two

Table I-Fabric particulars Sample Fabric Initial Extreme Fabric

No. weight volume fibre angle type g/m" fraction with respect

(V,) to fabric plane (8)

deg

At, 120 0.0600 6 Polyester

Af2 167 0.0812 7 spunbonded

Af3 189 0.0819 13 needle-punched

AC1 220 0.0674 to nonwoven fabrics

AC2 355 0.0754 13

I=fO'd0, oJx0d~J(0, ~),Q(0, ~)sinf),

f02 fX

J(0, ~)= dO' d~' Q(0',~')sinO'[l-{cos ^{O'cos 0'}

0, 0

+

sinO'sinO'· cos( ~ - ~')}l)1 /2,

90 Polypropylene0.0875 6 180

0.106910 spunbonded 280

0.1150II needle-punched 350

0.124715 surface-calendered nonwoVen fabrics 115

0.34353 Polypropylene 198

0.43453 spun bonded 136

0.31793 heat-sealed 170

0.37313 nonwoven fabrics

I is the mean free element length between two contact points; D, the fibre diameter;

L,

the total length of fibre in the volume V; V,the volume of the fabric; and Nv, the number of contact points in volume V.

Fibres in the fabric are oriented within a nar- row polar angle region, Le. from 0t to ^O2, But so far as the azimuthal angles are concerned, the fibres are distributed randomly in all directions.

From Lee and Lee^7, the mean projection length of any fibre on the Z-axis, ^Iz, is

/ /

,'ilz /

AL

--".,..".../

>/

z

)-v

Iz= 2V

DLI 'Kz ... (3) Fig. 2-Deformation of a fibre element due to force Fz at the

contact point

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238 [NOlAN 1. FrBRE TEXT. RES., DECEMBER] 996

contact points A and B at a distance 2/ apart. Let a third contact point C fall at the middle of AB.

The fibre section AB will act as a beam, supported at two points A and B. Let the mean vertical deflection of the mid point C by the downward load ^Fz be ^ozz. The deflection is calculated as follows:

... (4)

where B is the flexural rigidity of fibre and rnzz is given by Lee and Lee⁷as:

T

1

T

·t

Fig. 3-Elements of fabric of unit area of thickness Tand the unit cell of thickness I,

... (9) ... (8)

... (10) From Eqs (3), (5) and (9) we have

where ^Fz is the average vertical downward force per contact point.

During compression of the nonwoven fabric of volume V in the Z-direction, there will be some expansion in X- and Y-directions. The Poisson's ratio,

iw

is given as follows:

distributed evenly in all the contact points and transmitted through the various contact points.

The external force per unit area is as follows:

where ^ozx is the expansion in X-direction; and ^Ix, the projection of any element on X-axis.

... (5)

... (6) Ii is given as E( x' D4/64, where ^Ef is the initial modulus and ^D,the fibre diameter.

An incremental approach has been used in the same way as by Carnaby and Pan~ to accommod- ate the large deformations of the fabrics during the application of the compressional loads. The incremental form of the^{Eg.( 4)} hecomes

Van Wyk^1, Lee and Lee^7, and Carnaby and Panx related this deflection of the mid-point of such bending element in the vertical direction

(~ozz) to the general compression of a layer of such bending units of thickness 'L' Such a layer of thickness 'Lwould, on an average, change in thickness by an amount equal to ~^Ozz with the application of force ~FL per contact point. The same assumption has also been used in the pre~ent model.

Hence, the compressional strain of the fabric of thickness 'zis given by:

The fabric of thickness T will also be having the same compressional strain value.

Fig. 3 shows a hexahedron section of a non- woven fabric of volume V of a unit cross-section.

The total number of contact points in the unit cell of infinitesimal volume in a section of thickness ^Iz is calculated as follows:

... (7)

From Lee and Lee^7,we have

JXI2 JXI2

rnzx= d() d~sin()cos()cos~Q((),~)sin()

R] 0

and

Express:ions for mzz and Kz have been given The external force, Pz' on the unit cell will be earlier.

I I 'Ii _t_II _t _I'

I" ~Il'¹¹¹¹

(5)

...~

KOTHARI &DAS : THEORY OF COMPRESSION OF NONWOVEN FABRICS 239

100 250

Pressure, ^IrPa

50 2.0

o

Fig. 5-A typical non-linear nature of compression curve Updating Pz, T ,

( JV andAI e , ~)

E 3.5 E

4.0

••••

~c

......

•..:c

Fig. 4-Flowchart of the software developed for the present model to update the geometry of the fabric after each succes-

sive increment of load on fabric 2.3 Change in Geometry During Compression

An incremental method has been adopted to cope with the large deformation during compression of nonwoven fabrics. Initially the fibres are assumed to be evenly distributed within the zone, as stated above. A very small load ^!!"Fz per contact point is applied, which causes the fibre element to deflect. This deflection is mainly responsible for the compression of nonwoven fabrics. Keeping all other parameters same, as the free element length

(n increases, the compression will also increase. In the present model, the parameters like volume fraction Vf (Vf

=

fibre volume/fabric volume), distribution of fibres within the fabric and fibre diameter ^(D) determine the element length, ^I.From Eq. (5), it is also clear that a fabric made of the fibres with higher flexural rigidity ^(B) will have lower deflection, which results in lower compression of fabric.

The range of vertical orientation angle (()^{1- () 2)} of fibres within fabric has considerable effect on the compression of fabrics. The angle ± (), as given in Table 1, represents the angle of deviation from the plane of the fabric, which means the range of polar angle within which the fibres are oriented randomly is ()^{1 (}

=

^{xl2 -} 6) to 62 (

=

^x/

2+ 6).

In the first step of the incremental application of load, the parameters I, Kz, Kx, rnzz, mxz and ^N1 are calculated after calculating the relevant geom~

etrical parameters. In this incremental method, it is necessary to recalculate all the geometrical parameters on the completion of each increment. A very small load ^!!..^Fz per contact point is applied and the value of compressional load ^I:1Pz is calculated from the Eq. (8). Before starting the next iteration, it is necessary to update the geometrical parameters like orientation distribution function, Q(6, rp), and the volume fraction (Vf) by using the new values of ez and fzx. The derivations for gett- ing the updated values of the above parameters are given in the Appendix 1. The flow diagram of the algorithm which has been used to update the system geometry on successive increment of loads is shown in Fig. 4. The resultant non-linear behaviour of a typical nonwoven fabric during compression, as predicted by the above/model, is given in Fig. 5.

3 Evaluation of the Theory

To evaluate the above theoretical model, a seri- es of spunbonded nonwoven fabrics was taken The fabric particulars are given in Table 1. Table 2 gives the details of fibre properties in these fabrics. Knowing these fabric and fibre· parameters,

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240 INDIAN J. FIBRE TEXT. RES., DECEMBER 1996

--

-- Theoretical Curve D Experimental Points

3.(0 )

one would be able to predict the compressional behaviour of these fabrics. The experiments were carried out on Essdiel thickness tester in the same way as done earlierls, and the experimental results were compared with the corresponding theoretical curves (Figs. 6-10).

Fig. 6 shows the theoretically predicted curves with the experimental points for the fabrics AfI' Af2 and Af3' These fabrics of different weight per unit area were made of fibres of same denier. The compressional behaviour of the fabric Af

I

shows that after initial steep drop in thickness, the curve almost flattens. This is due to the higher average initial free element length (n due to lower volume fraction of fabric Af!, which is responsible for higher deflection of mid-point. After the initial stage, the number of contact points increased very rapidly, which means a rapid reduction of the average free element length. As the average free element length becomes very small after the initial stage, the further deflection will be small with the application of loads. Thus the curve becomes flat- ter in the later stage of compres~ion. With the increase in initial volume fraction the compression in the initial stage becomes relatively gradual, as shown in Fig. 6 for fabric samples Af2 and Af3' A similar trend is also observed in the case of the fabric samples Ac! and AC2 (Fig. 7) which are made of coarser fibres.

Fig. 7-Comparison of thoretical compression curve with ex- perimental points of fabric samples (a) Ac, and (b) Acz

250

o o

200 D

D

150

-- Theoretical Curvt o Experimental Points -- Thltoretical CurYf

o Experimtntal Points

5.5 ²⁰²

5.5

230

,~ '

5.5

282 D

8.5

227 D

aD

8.5 215e

E

.

10.0 231= 0

10.0

277.~

10.0 .••4

235

~ ~bJ - Theoretical Curve

10.0

304 _D

Experimental Points

12.0

278

C",-

I

248

~.

D 12.0

259 2 21.0 D

271

D

21.0 ₁ 0 0

0

0 250

-- Thtoretlcal Curve

I

Pressure, k Pa

0

Experimfntal Points

o 0

o

100

Pressure , k Pa

Table 2-Fibre properties

Fibre Fibre Fibre initial modulus

type denier g/tex

B^J Polypropylene 8)

BJ B4

Cf, Polypropylene Cf)

CCI CC2

Af, Polyester Af)

Af3 Ac, Ac)' Sample

No.

2.0\ (o) 1.01.5

I0 0.5 0

2.01{a b) E

E

.

~ti^c: ^I ⁰

:i: I-

0.5 0 2.5 {el

2.01·5

D D

1.0 0

0.5 0

Fig. 6-Comparison of theoretical compression curve with experimental points of fabric samples (it) Afl• (b) Af)

and (c) AfJ

'I ^II' I II I' I,!', ~~ It II d1 ~I

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KOTHARI &DAS : THEORY OF COMPRESSION OF NONWOVEN FABRICS 241

a a

a

- Theordical Curve a Experimental Points

lS-~

-- Theordical Curve a Expuilll~ntal Paints -- Theordirol Curv~

a Experimental Points

- Theo~tkal Curve a Experi •• eotul Points a

Pressure , kPa 50 100

0·5 0.6.

(0)

0.3

0·4 0·4

0·3 0·4 0.5.(0)

0·2 o

:

•• 0.2

I

^o.6l<b)

'l5;;:

.-

I~

^0·2

5 0.6Hbl ...c:

~ 0.5

.-

iour of heat-sealed spunbonded fabrics of Group C. As most of the fibres are fused at intersections in heat-sealed spunbonded fabrics there will be virtually no slippage between the fibres when the fabrics are subjected to compression which fits well with the assumption made in the present model that there will be no slippage between the fibres during compression. This is a major reason for a very good agreement of the experimental da- ta with the theoretical curves. Table 3 shows that the deviation at the final thickness as predicted by theory at 200 kPa from the experimental value at the same pressure as a percentage of the experi-

Fig. 9-Comparison of theoretical compression curve with experimental points of fabric samples (a)Cfl and (b)Cf2

D

D D

D

D D

D

-- Thmretical Curve D Experimental Point

--Theoretical Curve D Experimental Points --Thmreticol Curv~

D Experimental Points

-- Theoretical Curve D Experimenlal Points

tl

D

tl

D 2

2.0I( OJ

1.5

1.01~

0.50

2S[

1.52.0 D DD 1·0~

0.5 E

E::l ^.><••^c:^.!:!~ 41(c03 J

~ _I- ., II~

I

1

D

0 4.(d)

Fabric ACt shows a more steep drop in thickness than fabric AC2 due to lesser volume fraction

(Vf) in case of fabric ACt. Fig. 8 shows the compressional behaviour of needle-punched surface- calendered spunbonded fabric samples Bll B2, B3 and B4' The nature of the curves is similar to that of Group A needle-punched spunbonded fabrics.

Figs 9 and 10 show the compressional behav-

0.3

Fig. 8-Comparison of theoretical compression curve with experimental points of fabric samples (a) Bl' (b) B2•(c)B)and

(d) B4

Fig. lO-Comparison of theoretical compression curve with experimental points of fabric samples (a) CCI and (b)CC2

P rlSsure ,kPII

m

244 100

0·2 o 250

Pressure ,k Pa 100

50

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242 INDIAN1.FIBRETEXT.RES.,DECEMBER1996

Table 3-Comparison of experimental and theoretically predicted values of fabric thickness at maximum pressure

Sample Experimental Experimental Theoretical Percentage No. initialthicknessfinalthickness finalthickness deviationof

(To) (Tf) (Tf) finalthickness

mmmmmm Af,

1.45+ 28.900.380.80

Af2

1.49+ 23.500.530.86

Af3

1.72+30.810.621.15 ACj

2.36+ 26.270.791041

AC2

3041+ 29.322.201.20

8,

1.10+ 14.5400450.61

82

1.80+ 23.880.751.18

83

2.611.17+ 22.311.75 84

3.02+26.002.231.45

~.

Cfj

0.36+5.550.250.27

Cf2

0.49+3.260.370.39

Cc,

0.46+0.070.330.33

CC2

0.49+ 1.220.360.37

--

2S0 Initial modulus

--100

^g/tex

',:-.. .••••• 200 J/

"~,::,: -- - 300 "

---- -=:"' -'-:- ~ ':-=_~:==-==--=--==-=

EE

o o 4

~ 2

.J<c

...

:EI- 1

250 Vertical orientation angle

,\ __ .n tf

t10·

'~:::---_ --- t11·

...'"::...-=-~-:==:.=--=:--=.:= -"'=--=--=---= -~ .•

4

-

E ^'"E

"

²

'" ^.J<~

!

^...

00

PressUrE, kPa Pressure, k Po

Fig. ll-J::ffect of vertical orientation angle (0) on compres-

sional behaviour Fig. 12-Effect of initial modulus of fibres on compressional behaviour

PI'l!SSUl't , k Po

Fig. 13-Effect of fibre denier on compressional behaviour

Keeping all other parameters same, the compressional behaviour of a nonwoven fabric is affected by vertical orientation angle, fibre initial modulus and fibre denier when these are varied at a time in the theoretical model as shown in Figs 11, 12 and 13 respectively. The extent of changes in the compressional behaviour due to changes in mental value of the final thickness. In case of nee-

dle-punched nonwovens the deviation is very high whereas this is very little in case of heat-sealed nonwovens. The main cause of deviation of the theoretical curves from the experimental points in case of needle-punched fabrics is the slippage between fibres during the application of loads. There are some other factors which are also responsible for deviation, e.g. the distance between two neighbouring contact points is not always the same and the contact points do not always alternate regular- ly. It has also been assumed that the deformation mechanism of the fibres in the nonwoven fabrics is analogous to bending of straight rods and each rod bends separately, which is not the case in ac- tual situation. There may also be a chance of vis- coelastic extension of fibres during compression of fabrics. Non-uniformity in the fabric mass per unit area, fibre crimp and curl can also affect the compressional behaviour of needle-punched fabrics.

E 3 E"'

'"

~ 2

.J<c

...

~

o o

Fib re Denier - 6 Denier ....• 9 "

---12 "

II I'!; !II III'I' I! III~ 11II I! r I ~" I'

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KOTHARI & DAS : THEORY OF COMPRESSION OF NONWOVEN FABRICS 243

Appendix-l

Following equations are used to update the volume fraction (Vlland Q(8, ;). The volume fraction is the ratio of fibre volume to the total volume (V)of fabric.

x/4·D2• L

Vf---

V

The equation to update the volume fraction is

where VI is the volume fraction before compression.

Now Jonsider a fibre segment before compression in the range of angle (J to ((J+d(J). When a vertically downward compression load is applied on the fabric, the fibre segment is assumed to move into the range of angle (}' to ((}'+d (}'). The

l

projection angle on XY plane is assumed to remain constant during compression. The probability that a fibre lies in that segment after compression will be Q(8',t/J) sin (}' d(}' d t/J.The probability before compression is Q( (},t/J)sin (}d(} dt/J.

From the Eq. (6), one would be able to update the orientation density function during every stage of compression.

The relationship between (} and (}' can be obtained from the amount of compression as follows:

Substituting the Eqs (3), (4) and (5) into Eq. (2), we have Q( (}',t/J)sin (}' d (}'d t/J

=Q((},t/J)sec3(}'.CZ(1+CZ·tan2(}')-3/2·sin(}' d(}' dt/J ", (6)

these parameters is important and their study helps in designing fabrics of required compressional behaviour.

4 Conclusions

A model for predicting the compressional behaviour of nonwoven fabrics has been proposed and evaluated for spunbonded nonwoven fabrics.

The model requires fibre denier, fibre initial modulus, fabric mass per unit area, initial thickness of fabric and vertical orientation of fibres and as- sumes that the compression of the nonwoven fabrics consists solely of the bending of constituent fibres.

Although the spunbonded heat-sealed nonwovens show a very good agreement with the present theory, the needle-punched fabrics deviate from the theory due to fibre-to-fibre slippage during compression. Results show that the compression of nonwoven fabric is affected by fibre denier, fibre initial modulus, vertical orientation of the fibres in the fabric and volume fraction.

References

1 Van Wyk C M, J Text Inst, 37 (1946) T-285-292.

2 Komori T & MakashimaK, Text ResJ, 47 (1977) 13-17.

3 Kallmes0 &CorteH, TAPPI,43 (1960) 737-752.

4 Steam A E,J Text Inst, 62 (1971) 353-360.

5 Taylor D S,J Text Ins!, 47 (1956)T141-T146.

6 Curiskis J I & Camaby G A, Text Res J, 55 (1985) 334- 344.

7 Lee D H & Lee J K, Objective measurement:Applications to product design and process contro~ edited by S Kawaba- ta, R Postle and M Niwa (Taxtile Machinery Society of Ja- pan, Osaka), 1985,613-622.

8 CamabyGA&PanN, TextResJ,59(1989)275-284.

9 Pan N & Camaby G A, Text ResJ, 59 (1989) 285-292 . 10 Komori T & Itoh M, Text Res J, 61 (1991) 420-428.

11 LeeD H & CamabyG A, Text ResJ, 62 (1992) 185-191.

12 Lee D H, Camaby G A & Tandon S K, Text Res J, 62 (1992) 258-265.

13 Leaf G A V & Oxenham W, J Text Inst, 72 (1981) 168- 175.

14 Leaf G A V & Oxenham W, J Text Inst, 72 (1981) 176- 182.

15 Kothari V K & Das A,JText Ins!, 84 (1993) 16-30.

Hence,

Q((}',t/J)sin(}' d(}' dt/J=Q((},t/J)sin(} d(} dt/J where

t/J=t/J'

tan 8=C tan 8'

where C is compression ratio and it is given by C=(l-e.)/(1 +^Tzx·e.)

From the differentiation of Eq. (3), we have

2 '

d(}= C'sec (} d(}' 1+C2·tan2(}

From the Eq. (3), we also have C·tan(}'

.sin(} jl +C2.tan '(}'

... (1)

'" (2)

... (3)

... (4)

'" (5)