**10. Roughness anisotropy**

Above-mentioned roughness characteristics have implicitly assumed that surface roughness is isotropic phenomenon. This assumption can be accepted in the cases when surfaces have the same micro geometric properties no matter what direction they are investigated in. Majority of textiles structures have anisotropic nature. Surface of woven fabric is clearly patterned due to nearly regular arrangements of weft and warp yarns. The special non-random patterns are visible on knitted structures as well. It is well known that anisotropy of mechanical and geometrical properties of textile fabrics are caused by the pattern and non-isotropic arrangement of fibrous mass. Periodic fluctuations of surface heights can be spatially dependent due to arrangements of yarns. Non-periodic complexity spatial dependence is subtler. The roughness characteristics computed from SHV trace are therefore dependent on the direction of measurements i.e. angle of transect line according to fabric cross direction (perpendicular to machine direction). In KES system, it is possible anisotropy treated by averaging of roughness parameters in weft and warp directions only. This approach is generally over simplified and can lead to under or over estimation of surface roughness.

For anisotropic surfaces the so called surface spectral moments mp,q can be used (Longuet-Higgins, 1957)

$$m\_{p,q} = \bigcap \{ \alpha \: \: \: \: \: \: \: \: \: \: \: \: S(\alpha\_1, \alpha\_2) \text{ d}\alpha\_1 \text{ d}\alpha\_2 \} \tag{36}$$

where 1 2 *S*(,) is bivariate power spectral density of surface. Necessary condition for the case of degenerated spectrum (one dimensional) is

$$\mathcal{Z}(m\_{2,0}, m\_{0,2} - m\_{1,1}^2) = 0 \tag{37}$$

For degeneration to more dimensions, similar conditions can be derived (Longuet-Higgins, 1957). The profile spectral moment ( ) *mr* in the direction defined by eqn. (27) is connected with surface moments mp,q by relation

$$m\_r(\theta) = m\_{r,0} \cos^r \theta + \binom{n}{1} m\_{r-1,1} \cos^{r-1} \theta \ \sin \theta + ... + m\_{0,r} \sin^r \theta \tag{38}$$

The second profile spectral moment <sup>2</sup> *m* ( ) , which is equal to the variance of profile slope PS2, is function of three surface moments m2,0, m1,1 and m0,2 only. This dependence has the simple form derived directly from eqn. (36).

1 <sup>1</sup> ( ) (( ) ( ) 2 () *M h*

where *M*(h) is the number of pairs of observations separated by lag *h*. Problems of bias in this estimate when the stationarity hypothesis becomes locally invalid have led to the proposal of more robust estimators. It can be summarized that simple statistical characteristics are able to identify the periodicities in data but the reconstruction of "clean"

Above-mentioned roughness characteristics have implicitly assumed that surface roughness is isotropic phenomenon. This assumption can be accepted in the cases when surfaces have the same micro geometric properties no matter what direction they are investigated in. Majority of textiles structures have anisotropic nature. Surface of woven fabric is clearly patterned due to nearly regular arrangements of weft and warp yarns. The special non-random patterns are visible on knitted structures as well. It is well known that anisotropy of mechanical and geometrical properties of textile fabrics are caused by the pattern and non-isotropic arrangement of fibrous mass. Periodic fluctuations of surface heights can be spatially dependent due to arrangements of yarns. Non-periodic complexity spatial dependence is subtler. The roughness characteristics computed from SHV trace are therefore dependent on the direction of measurements i.e. angle of transect line according to fabric cross direction (perpendicular to machine direction). In KES system, it is possible anisotropy treated by averaging of roughness parameters in weft and warp directions only. This approach is generally over simplified and can lead to under or over estimation of surface roughness.

For anisotropic surfaces the so called surface spectral moments mp,q can be used (Longuet-

, 1 2 12 1 2 ( , ) d d *p q m S p q*

For degeneration to more dimensions, similar conditions can be derived (Longuet-Higgins,

,0 1,1 0, ( ) cos cos sin .. sin

PS2, is function of three surface moments m2,0, m1,1 and m0,2 only. This dependence has the

*rr r r r <sup>r</sup> <sup>r</sup>*

1

*n*

  

 (36)

2,0 0,2 1,1 2( ) 0 *mm m* (37)

, which is equal to the variance of profile slope

defined by eqn. (27) is

(38)

 

is bivariate power spectral density of surface. Necessary condition for the

in the direction

1

*<sup>m</sup>*

2

case of degenerated spectrum (one dimensional) is

1957). The profile spectral moment ( ) *mr*

The second profile spectral moment <sup>2</sup> *m* ( )

simple form derived directly from eqn. (36).

connected with surface moments mp,q by relation

*mm m*

*M h* 

*j G h Rd Rd*

dependence is more complicated.

**10. Roughness anisotropy** 

Higgins, 1957)

where 1 2 *S*(,) 

( ) <sup>2</sup>

*j jh*

(35)

$$m\_2(\theta) = m\_{2,0} \cos^2 \theta + 2 \, m\_{1,1} \cos \theta \, \sin \theta + m\_{0,2} \sin^2 \theta \tag{39}$$

The surface moments play central role in description of surfaces topography. The parameters m2,0 and m0,2, which are the 2nd surface spectral moments, denote the variance of slope in two vertical directions along cross direction and machine direction. The parameter m1,1 represents the association-variance of slope in these two directions. These parameters are generally dependent of the selected coordinate system. From the known values of 2( ) *m <sup>i</sup>* for selected set of directions *<sup>i</sup>* i = 1,..n it is possible to estimate the surface moments m2,0, m1,1 and m0,2 by using of linear regression. The maxima and minima of eqn. (37) are

$$\mathbb{E}\left(m\_{2\max}, m\_{2\min}\right) = 0, \mathbb{E}^\*\left[\left(m\_{2,0} + m\_{0,2}\right) \pm \sqrt{\left(\left(m\_{2,0} - m\_{0,2}\right)^2 + 4m\_{1,1}^2\right)}\right] \tag{40}$$

These occur in the angle *<sup>p</sup>* called principal direction given by relation

$$\tan\ \theta\_{\text{p}} = \frac{2m\_{1,1}}{m\_{2,0} - m\_{0,2}}\tag{41}$$

As one measure of anisotropy the so-called long-crestedness 1/g has been proposed (Longuet-Higgins, 1957), where

$$\mathbf{g} = \sqrt{\frac{m\_{2\min}}{m\_{2\max}}}\tag{42}$$

For an isotropic surface is *g =* 1 and for degenerated one-dimensional spectrum *g* = 0. The better criterion of anisotropy has been proposed in the form (Thomas et. al, 1999)

$$AN = 1 - \frac{2 \ast \sqrt{m\_{2,0} \ast m\_{0,2} - m\_{1,1}^2}}{m\_{2,0} + m\_{0,2}} \tag{43}$$

For *AN* = 0 surface perfectly is isotropic and for *AN* = 1 surface is anisotropic. Lower *AN*  characteristic indicates low degree of anisotropy.

For investigation of surface roughness anisotropy the twill fabric (see fig. 3) and Krull fabric were selected. The *R(d)* traces have been obtained by means of KES apparatus in the following directions:*<sup>í</sup>* 0o (weft direction), 30o, 45o, 60o and 90o (warp direction). The Kawabata SMD of individual profiles of twill fabric at chosen directions *<sup>í</sup>* is plotted as polar graph in fig. 32 and for Krull fabric in fig. 33.

The surface moments m2,0, m0,2 and m1,1 were computed from eqn. (37) by using of linear least squares regression. Estimated surface moments and AN anisotropy measure are given in the table 1.

The proposed technique is capable to estimate roughness characteristics of anisotropic surfaces typical for textile structures. Beside the anisotropy measure *AN* the direction *<sup>p</sup>* and values of

m2max will be probably necessary for deeper description of textiles surface roughness.

Woven Fabrics Surface Quantification 151

90

variance of slope

 0.05 0.1 0.15 0.2 0.25

180 0

270

0.5

1

90

variance of slope

270

180 0

60

300

210

Fig. 34. Angular dependence of profile slope variance for twill fabric

210

150

240

Fig. 35. Angular dependence of profile slope variance for Krull fabric

120

150

240

120

30

330

30

330

60

300

Fig. 32. Kawabata SMD of individual profiles at chosen directions *<sup>í</sup>* for twill fabric

Fig. 33. Kawabata SMD of individual profiles at chosen directions *<sup>í</sup>* for Krull fabric

The angular dependence of profile slope variance 2 *m* ( ) and experimental points for twill are shown on the fig. 34 and for Krull are shown in fig. 35.

90

Kawabata SMD

 0.01 0.02 0.03 0.04 0.05

180 0

270

90

Kawabata SMD

270

180 0

 0.02 0.04 0.06 0.08

210

Fig. 32. Kawabata SMD of individual profiles at chosen directions

210

Fig. 33. Kawabata SMD of individual profiles at chosen directions

The angular dependence of profile slope variance 2 *m* ( )

are shown on the fig. 34 and for Krull are shown in fig. 35.

150

240

120

150

240

120

30

330

30

330

*<sup>í</sup>* for Krull fabric

and experimental points for twill

*<sup>í</sup>* for twill fabric

60

300

60

300

Fig. 34. Angular dependence of profile slope variance for twill fabric

Fig. 35. Angular dependence of profile slope variance for Krull fabric

Woven Fabrics Surface Quantification 153

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4189


Table 1. Surface moments and anisotropy of samples
