Introduction

Compression may be defined as a decrease in intrinsic thickness with an appropriate increase in pressure. Intrinsic thickness is the thickness of the space occupied by a fabric subjected to barely perceptible pressure. Compression is one of the important properties of fabric, in addition to friction, bending, tension and shear. In garment automation, for instance, compressibility can be a crucial property for successfully separating plies from a stack. With the growing need for better modelling material for simulation purposes, objective measurements of fabric compression will become increasingly important, since static compression gives an indication of the material‘s mechanical ”springiness‘. A fabric that compresses easily is likely to be judged as soft, possessing a low compression modulus or high compression. Any surface changes in the fabric such as singeing, milling, or pressing, which are generally used to improve the hand, will therefore have an essential impact on the compressibility.

Modelling the compressional behaviour of a woven fabric under pressure would basically involve understanding the relative change in thickness under applied pressure. The compressive force applied allows the yarn to undergo deformation non-linearly, resulting in a change in thickness of the fabric. This change depends on a number of factors which need to be investigated in detail.

The low-load compression behaviour of woven fabrics is very important in terms of handle and comfort. In transforming fabrics into clothing articles, one has to know, as well as the manner of processing, how the fabric will behave during particular manufacturing processes and when exposed to various strains. The answers to these questions may be obtained by investigating fabric mechanics, such as non-linear mechanical fabric properties at lower strains, which is the case in transforming fabrics into garments. The area of structural deformation during processing is quite wide.

Fabrics are exposed to various strains when processed into clothes and behave in different ways. Fabrics are stretched when spread, they are exposed to compression during the cutting operation because of the vacuum present, and in sewing they are transformed from two-dimensional structures into three-dimensional articles of clothing. In surgical compression, the garment's softness of fabric is important for two principal reasons. The fabric should minimise skin irritation during prolonged wear, and has to maximise the chances for the patient to comply with the extended plastic surgery recovery period. This is also the case with non-woven diaper cloths, where compression is very important, apart from the properties of the inner fabrics. The creation of a fabric which is soft, but still meets compression needs, was one of the main objectives behind the design of ComfortWeave. The softness of a fabric can be objectively measured by the Kawabata Evaluation System (KES). This ES is a weighted-regression analysis combining 17 factors measured in five separate tests, including compression, shear, tensile, bending and surface tests. ComfortWeave possess KES values similar to those of knitted silk. PowerWeave, the generic fabric used in most other compression garments, ranks close to emery cloth, a fine grade of sandpaper.

The low mechanical stress properties of fabric also influence the clothing manufacturing process. Seam pucker, pattern matching, long seam sewing, shape retention and pressing are known to be influenced by tensile properties (LT, WT, ET, and RT). Formability and drape is known to be altered by changes in shear and bending properties (G, 2HG, 2HG5, B, 2HB). Garment hand and appearance is influenced by surface (MIU, MMU, SMD) and compressional properties (LC, WC, RC).

Primary hand includes components of total fabric hand (THV) and represents characteristics of stiffness, smoothness, etc. Kawabata et al. have developed an evaluation system for fabric hand, relating subjective standardised hand values obtained by ranking the fabric‘s experimentally-measured mechanical and surface properties.

Theory on compression of textiles

Compression has been the subject of many studies, but a great number of Investigations have been empirical. The simplest but most realistic theory seems to be that of van Wyk, which is based on the assumption that fibres simply bend as cylindrical rods. The equation relates the applied pressure ”P‘ to the inverse cube of the volume ”v‘ and the intrinsic volume ”vo‘ as follows:

⎛m ⎞³  11 

P = k1 Y − (1)

^ 33 ^

⎝ρ⎠_⎣⎢() ()v_{o ⎦}⎥

v

The equation is independent of fibre diameter or elasticity, but includes Young‘s modulus ”Y‘, the mass of the fibres ”m‘ and the bulk density ”ρ‘ at low pressure. In addition, the equation comprises a dimensionless constant ”k1‘ characterising the fibres. This constant, which is generally of the order of 0.01, will vary with the fibre orientation and crimp, and can only be determined if Young‘s modulus is known independently. This constant ‘k1 ‘ and Young‘s modulus are the dominant parameters in distinguishing the compression of different kinds of fibres in bulk.

From the point of view of engineering mechanics, the non-linear relationship is a clear indication of elasto-plastic deformation, and the cross-section deforms in both the elastic and plastic ranges. Fabric compression involves the movement of fibres and yarns within the diameter axis to which the fabric is oriented. This behaviour is accounted for by studying the fabric‘s internal non-linear structure, the visco-elastic nature of the fibres themselves, and to some extent the friction between fibres and yarns.

In spite of its usefulness, van Wyk‘s model has some limitations. The real physical meaning of the ”k1‘ constant is undetermined, and the model does not take into account the hysteresis caused by fibre slippage and friction during every compression and decompression cycle. The equation suggested by van Wyk was only suitable for moderate compressive pressures. At larger pressures, the incompressible volume ” ν′‘ was neglected. In order to fit van Wyk‘s equation to the experimental data, some researchers reduced the order of the formula to 2.5 instead of 3.

Van Wyk himself has made corrections to the equation, and suggested a more elaborate one to include the incompressible volume, this bringing about a new equation, as follows:

 m ⎞³  11 

P = k1 Y − (2)

^ 33 ^

⎝ρ _⎣⎢(v − v')(v_o − v') ⎥_

Van Wyk‘s equation can be implemented in different forms, First, by using the original Equation 1 which was derived directly from the compression of wool wads, the data would fit close to experimental curves, though deviations may exist. This may be accounted to the omission of the incompressible volume ” ν′ ‘‘ which becomes significant when compressing fibre assemblies to a small enough volume(P > 0.5cN/cm²). Either vo, which is the volume at zero pressure, is relatively high compared to ”v‘ and can therefore be ignored, or v, v‘ and vo have the same order of magnitude and so cannot be neglected.

In addition to van Wyk‘s equation, there are numerous empirical equations relating the thickness of a fabric to the pressure applied. Hyperbolic functions give the best empirical results, but exponential smoothed curves are also credible.

From these equations, compression energy can be calculated from ”0‘ to ”P‘. If Vo is higher than V and V‘, then

1  m ⎞³

P ≈a with a = KY⎜ (3)(v − v')³  P 

pv

a

W = −_∫ p.dv = -₎dv (4)

_{0 vo} ^∫(v = v')³

a 11 

= − (5)

⎢22 ⎥

2 _⎣(v − v') (vo − v') _

Still supposing that Vo is high in relation to V,

a  1  p

W = _⎜ _{2 ⎟} = (v − v') (6)

2 _ (v − v') _ 2

or

2w 8w³

v'=v − and a = (7)Pp²

As far as we are concerned, we do not believe that vo is very high compared with v and v‘ , so

 11 ⎤

P= a_⎢3 − _{3 ⎥} . (8)

_⎣(v − v') (vo − v') _

^va ^v 1

W = −_{∫ 3} dv +_{∫ 3} dv (9)

(v − v') (vo − v')

vo vo

a  11  v − vo

= ₂ − _{2 ⎥}+ a ₃ (10)

⎢

2 _⎣(v − v') (vo − v') _ (vo − v')

or

p (v − vo)

W =(v − v')+1.5a (11)

2(vo − v')³

As most instruments measure ”W‘ more easily than V, it is interesting to calculate the values of a, Vo and V‘ from the curve W=f(V). Smoothing both curves gives very close values.

Interpreting the van Wyk parameters V’ and a’

In the analysis, we refer to v‘ as the —incompressible volume“, though the physical significance cannot just be interpreted as the volume of the fibres in the fabric excluding the air. V‘ must be defined as the volume of the inner core of the fabric, which is relatively incompressible even for pressures up to 50gf/cm² (which is the maximum Kawabata pressure).

Knowing the fibre density ”ρ‘ and area density ”ma‘ of the fabric, we can define the packing fraction ‘α‘ of the fibres for this volume V‘(or core thickness T‘ per unit area) as follows.

^ma

α= (12)

ρT '

The second constant in van Wyk‘s equation, which needs more clarification, is ”a‘. Grouping together Young‘s modulus ”Y‘, the density ”ρ‘ and the mass ”m‘, if the core layer of the fabric is not compressed and surface hairs in the outer layers [*], fibre mass can be calculated using

3

m = ^a ρ (13)

KY

To arrive at this stage, we use the measured value of the KES-FB3 results. Fibre mass in van Wyk‘s equation is only a small percentage of the total fabric mass. De Jong et al. measured a mass of surface fibres within the range 3-20g/m² for a swatch of woven fabric between 167-323g/m². However, there are no calculations involved for geometries of woven materials; the value of ”k1‘ is not precisely known, and the influence of the pressure on the constants is still unknown. Here we attempt to improve compressional studies using the geometric thickness of fabric, yarn and fibre structural parameters.

Modelling compression behaviour from fabric construction parameters

Fabric properties

Mukhopadyhay et al. [11] carried out exhaustive work on the thickness and compressional characteristics of air-jet textured yarn woven fabrics, which concludes that compressibility, thickness recovery, compressional energy, recovered energy and resiliency are influenced by fabric constructional parameters. Most of the results are based on the compression-recovery graph obtained from Instron.

Postle et al. [43] concludes that fabric bending, compression and surface characteristics are the three most important characteristics for predicting overall handle and associated quality attributes.

Mechanical and surface properties [44] were determined for different fabrics. Polyester fabric showed higher resistance to tensile deformation than cotton fabric, while the blends showed intermediate resistance. In contrast, the trend was reversed regarding bending and lateral deformation behaviour. Friction behaviour shows relatively little dependence on fibre type, but bending depends on yarn packing density. The tensile stresses studied were ??[[within the range of the magnitude deflections]] found in wear. Lateral compression behaviour indicated that the cotton fabric showed the highest resistance to compression. Compression, friction and contacting surface fibre counts are all related to the increased number of protruding fibres on the cotton and cotton-containing fabrics.

The classical interpretation of fabric friction [45] is viscoelastic, but its correlation with compression curves is poor. Measurements show that at low pressures, friction essentially depends as much on fabric hairiness as on compression. The limit of compressibility is a function of the yarn arrangement, the yarn structure itself being less important. Surface finishes of the fabric are known to influence the cloth‘s handle.

Yarn properties

Dupuis-D et al. [47] critically analyse the function of yarn structure on compressibility of fabric, and conclude that data analysis of a fabric‘s compression can only be done using a mathematical model to smooth the results. The best formula is van Wyk‘s, although the meaning of the physical parameter is very difficult to define for a fabric. On the basis of van Wyk‘s equation, a study of the compressibility of a fabric whose weft is made of multiple threads (classic ply yarns, open-end, combed or carded) shows that the structure of fabric has greater importance than that of the thread. The mechanical properties of a fabric are supposed to be essentially due to its construction, and to a lesser extent to the threads from which it is made.

Fibre properties

Matsudaira.M et al. [48] uses the crimp of fibres measured from various stages of the spinning process, and studies the effect of this fibre crimp on fabric quality. The following conclusions were drawn:

1. Fibre crimp and crimp recoverability decreases gradually with the progress of the spinning process, and the decrease is most marked during carding.
2. The crimp level of fibres correlates well with the lateral compressional energy of the fibre bundle.
3. There is a clear correlation between fibre crimp and fabric quality evaluated objectively from the mechanical properties of the fabric.
4. A fibre assembly with high crimp is more deformable in both tension and compression, which correlates well with the high quality or high fukurami (softness and fullness) of the fabric.

Existing models

Yung Jin-jeong and Tae Jin-kang [9] conclude that a set of equations should be solved which is based of a three-dimensional fabric geometry, and consists of seven differential equations that are severely coupled and have moving boundary conditions . The fabric geometry is also affected by weaving conditions. Using a unit cell model, an attempt has been made to determine the compressive force acting on the cell. Ttwo plates are used; the lower one is fixed, while the upper plate is movable. The upper plate compresses the fabric while it moves downward. Using these assumptions, the analysis of woven fabric deformation under compression forces has been carried out using the finite element package, ABAQUS FEM developed by Hibbit, Karlsson and Sorenson, Inc., (ABAQUS, 1989b). [9] is the latest research reported. Some of the empirical equations to describe the relationship between the thickness and compressive force by Kawabata et.al. 1978b) (Equation 14) was an exponential function, Samson (1972)‘s equation (Equation 15) was logarithmic function. Equation 16 was used by inversely proportional function (Holmes and Brown, 1981):

(T −To)

P = po exp − (14)

b

log T = log a œb log p (15)

 (b) 

T = a + (16)

⎢

_⎣( p + p_o ) _

Where: po = initial compressional load

To = initial thickness at po and

a, b = constant, determined from the experimental data for each fabric. However, these empirical approaches have some limitations in explaining the role of the fabric structure and yarn compressional characteristics in the compressional behaviour of fabric.

Poste [2] regards compression as a three-stage process, namely the flattening of the fibres that protrude from the surface of the fabric, the flattening of buckles in the fabric as well as those areas of the fabric that are thicker than average, and the compression of the main body of the fabric. The analysis of the pressure-thickness relationship [3] demonstrates a very prominent effect in terms of fabric construction and yarn structure. The fabric compressibility depends primarily on the fibre material.

De.Jong et al. (1986)[4]‘s experiments on wool and cotton fabrics suggests that the exponential function proposed by van- Wyk holds good for cotton fabrics as well as wool fabrics. Since the pressure-thickness curve of fabrics looks similar to what happens under tensile deformation, inspired by the success of tensile stress-strain curve modelling (Hu & Newton, 1993) and the comparison of two groups of curves in low-stress regions, an exponential curve was proposed. The fit of the equation governing the experimental curves may be improved by using a non-linear regression method:

αt −β

P= e −1 (17)

where P is the pressure and t the thickness; the two constants α and β must be estimated.

Vangheluwe & Kiekens [20] modelled the relaxation behaviour of yarn using a non-linear spring placed in parallel with Maxwell elements, called the extended nonlinear Maxwell model. The formula obtained from studying the stress relaxation behaviour of yarns was fitted with experimental relaxation curves using a Marquardt algorithm for non-linear regression. The relaxation curves were classified into ordinary relaxation, mixed relaxation and inverse relaxation curves. Becker used Equation 18 to describe relaxation, and Equation 19 for describing inverse relaxation after unloading the yarn following relaxation testing.

P=C(t+a)ⁿ (18)

P=C3 .tⁿ³ œ C5(t-T3)ⁿ⁵ (19)

Where: P - force,

t - time

T3 - starting time of inverse relaxation

n, n3,n5,C,C3,C5,a: parameters

For the extended nonlinear Maxwell model represented by a linear spring in parallel with Maxwell elements to model relaxation and inverse relaxation behaviour, the set of equations from 20 to 22 governs the relationship between F(cN),strain ε, and time t of the extended nonlinear model.

2

F = E ε (20)

na bn1

∂ε 1 ∂FF

i 1 i

=+ (21)

∂t E_i ∂t η_i

ε =ε =ε , for all i- values, i = 1 to P.

n1 i p

F = F +_∑ F (22)

n1 I i=1

where: F and ε - force (in CN) and strain of the nonlinear spring,

n1 n1

Eb - spring constant of the nonlinear spring,

Fi and ε _i - force (in cN) and strain of Maxwell element number i,

Ei - spring constant (in cN) of the spring in Maxwell element number i,

η_i - viscosity (cNfls) in Maxwell element number i.

Equation 20 was proposed for the nonlinear spring, whereas the differential equation 21 is valid for the Maxwell elements in the model. A set of equations from 20 to 22 must be solved to establish the relaxation behaviour of yarns.

Modelling from fabric geometry

Assumptions

For the compressional deformation of woven fabric, the following assumptions would be appropriate to facilitate the analytical model:

1. The yarns in the fabric are assumed to be elastic and isotropic, although they have 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); therefore, the resistance of the fine hairs on the yarn surface is ignored.
4. The difference caused by different fibres and fibre arrangements arising from different spinning technologies is assumed to be captured in the material constant, and is not considered for modelling.
5. The fabric is assumed to be completely relaxed; it 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 (de-sizing, bleaching, softening, dyeing, reducing weight, etc.,) will not be taken into consideration.

Initial fabric configuration

By applying geometrical theory, the fabric thickness ”t‘ can be calculated as the sum of crimp height and diameter. Let the crimp height be ”h‘ and the yarn diameter be ”d‘. The fabric thickness ”t‘ is given by

t = h1 +d1 or h2 + d2 (23)

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

t= max (t1, t2) (24)

According to Pierce‘s geometry:

4

h1 =		3		p2		C1	(25)
h2 =		3 4		p1		C2	(26)
C1 and C2 are in fraction
d1	=		1 1 28 N				(27)
d2	=		28		2 1 N		(28)
Clothing setting and integral cloth stiffness

If the fibres were all quite free to move independently of each other, the cloth stiffness per thread would be equal to the sum of the stiffness of all the 'n' fibres in the thread. Then we refer to it as the integral fibre stiffness. However, if the fibres have no freedom of movement, it can be shown that cloth stiffness per thread will be dependent on n² times the fibre stiffness.

The relative freedom of the fibres depends on the closeness of cloth construction as well as the finish. We would also expect the ratio of cloth stiffness per thread to fibre stiffness to tend to be low for the more open cloths, and to be much greater for the close cloths.

100

The correction of integral fibre stiffness, multiplying by and by cos²θ , where ” θ ” is the

(100 + c%)

twist angle

n coverfactor(Kc) = (29)

N

then modified cover factor

coverfactor(Kc^') = ^n' (30)N

ρ

where N = yarn number in tex, ρ= fibre density, n= threads per inch. At any given cover factor, cloths woven from coarser yarns tend to be stiffer than those woven from finer yarns, which would be expected if cloth stiffness depends on some power of 'n' greater than unity.

Heat setting and post treatment operations may materially reduce the stiffness of a fabric by reducing the mutual pressures between the yarns; other finishing process produce effects by changing inter-fibre friction and adhesion.

Limits of fabric compression

On application of pressure, the crimp of fabric, count of yarn and cloth setting all change as a result of the flattening of the yarns. By introducing a flattening factor ”e‘, which is a function of crimp of yarn, cloth setting, and count, the thickness using flattening factor is given by

⎛⎞

C% C% 11

12

+

= 0.28 e

^⎜ (31)

+ nn  NN 

21 12

 