The effects of rheological characteristics of lubricant and surface roughness on the load capacity of a hydrodynamic slider bearing

Wear, I46 (1991)

165

165-177

The effects of rheological characteristics of lubricant and surface roughness on the load capacity of a hydrodynamic slider bearing Jianming Wang Department of Mechanical Chicago, IL 60616 (U.S.A.)

and Aerospace Engineering,

Illkois

Institute of Technology,

BiaoIin Li Department of Mechanical Engineering, shanghai phina)

CoUege of Engineering,

Shanghai University,

(Received August 30, 1990; revised November 6, 1990; accepted November 19, 1990)

Abstract Using the approximate analytical Reynolds equation for a bearing with a smooth surface as the basis for an analysis, an endeavour is made to provide a non-Newtonian theory for rough surfaces. Generalized forms of one-dimensional Reynolds equations are established for two types of roughness arrangements: transverse and longitudinal. These equations are subsequently used to determine the optimum design for a Rayleigh slider bearing, as an example of an application in engineering. Various bearing characteristics are obtained and compared with those of a smooth bearing with the same nominal f&n thickness. The calculated results show that both lubricant rheological behaviour and surface roughness have an important infhrence on the load capacity and friction drag at the surfaces of the bearing.

1. Introduction

The importance of using non-Newtonian fluids such as solutions of high polymer additives, synthetic lubricants, silicon fluids, greases and the natural lubricating fluids which appear in animal joints, as lubricants has been emphasized in various investigations [l-l 1 ] during the past two decades. Recently, practical applications have also been considered [ 12-l 5 1. However, ail these studies implicitly assume that the surfaces of the contact bodies are smooth. Owing to machining Iimitations, the actual surfaces of contact bodies are invariably rough. So, it is meaningful to study the effects of the interaction of surface asperities and the non-Newtonian nature of lubricants on bearing characteristics under certain optimized geometric parameters. In particuiar, such investigations are necessary for the design of bearings which operate under conditions of high load. In such situations, the film thickness on the bearing can be of the same order of magnitude as the average height of surface asperities. The effects of surface roughness and the rheological behaviour of the lubricants then becomes important. 0043-1648/91/$3.50

Q Elsevier Sequoia/Printed in The Netherlands

166

Recognizing these facts, an endeavour is made in this paper to describe an optimum procedure for the design of slider bearings, combining the effects of the lubricant’s rheological character and the surface roughness of the bearing. Using previous results [ 12] for the smooth surfaces as the basis for the analysis, a non-Newtonian theory for surface roughness is developed using Christensen’s assumptions [ 161 of a stochastic rough surface. General forms of one-dimensional Reynolds equations are derived for two types of roughness ~gemen~: transverse and longitu~al. Subsequently, these are used to optimize the Rayleigh slider bearing as one example of a practical application. The optimum design procedure described in the present paper is not restricted to the Rayleigh slider bearing. It can also be readily applied to other types of hydrodynamic bearings with other non-Newtonian models. 2. Theoretical analysis

The power law model is chosen as the constitutive equation for the nonNewtonian lubricants. The model has been widely used and research has shown that it is a suitable approach to studying the stress-strain rel~io~~p for polymer-thickened oils, greases, synthetic lubricants and joint fluids [ I 7-201. The equation is

au

n * au

()

T=may=??G-J

(I)

where u is the velocity ~~bution of the fluid and g is the vertical coordinate (see Fig. 1 and Appendix A). m and n are the rheological constants, q* is the so-called equivalent viscosity n-1

rt*=m (G1 au

Fig. 1. The conf@umtion of the Rayleigh slider bearing and the pressure distribution.

(2)

167

The rheological index n reflects the fluid characteristic and its deviation from unity determines the non-Newtonian behaviour of the lubricants. For n = 1, the fluid is Newtonian, n < 1 signihes pseudoplastic behaviour, whereas n > 1 characterizes dilatant fluids. Theoretically, 0
&[(&)$]=(:)g+v

(3)

where V=ahbt and 77;=m(Ulh)“-‘. In ref. 12, it was shown that the above approximate equation is accurate enough to be used in engineering with the maximum error not exceeding lo%, compared with the exact equation. The slider bearing configuration to be investigated is shown in Fig. 1. The film thickness h is the sum of the nominal thickness hN and the surface roughness amplitudes S1, 8, measured from the nominal level. Thus h=h,(~,2/,t)+6,(It;-U,t,y)+~(L2;-U2t,Y)

(4)

where U, and U, are the motion velocities of the upper and lower plates respectively. For the Rayleigh bearing, the nominal film thickness is

hN= ho = constant

(O
@a)

hN= hl = constant

(l,
(5b)

The representation of the bearing motion given above is pure sliding with the constant velocity U, = U at the upper surface and VI =0 at the lower plate. &, 8, are regarded as randomly varying quantities of zero mean. Since the upper surface is moving Sz, in general, it is time dependent. In the following section the stochastic form of the Reynolds equation (eqn. (3)) apphcable to rough surfaces will be derived using the approach of Christensen [ 161. To achieve this, the average (expected) value of various quantities may be de6ned as follows

(6) where 6= a1+ 8, and j(8) is the probability density distribution function of the random variable 6. In particular, for the ergodic stochastic process, we may have E(S,) =E(&) = 0 and E(h) = hN. 2.1. Traverse

sueme roughness In this case, the asperities on both lubricated surfaces are assumed to

run perpendicular to the direction of sliding in the form of long narrow

168

ridges and furrows. Equation (4) for the film shape in this case can be reduced to h=h,(x,t)+S,(x)+~(~-Ut)

(7) Differentiating with respect to eqn. (7) for t and x respectively and substituting the results into eqn. (3) yields

]= y$ 2 [(122;:-1,) 2,- ;(hN+s1-s2) Since ah,lat represents the normal squeezed velocity of two lubricated bodies, it should be zero for stationary flow. We now denote the bracketed term in eqn. (8) by

It should be noted that il4 is the expression of flow flux in the presence of roughness, and it can be regarded as a stochastic variable with zero variance using Christensen’s assumptions [ 161. Rearranging eqn. (9) properly and taking the expected value leads to

By substituting the above equation into eqn. (8), the stochastic Reynolds equation for the transverse roughness can be obtained -d dx

l 12mnU”-‘E[h-(“+2)]

expected values between two difference S is the where {E[S, h- (n+2)]-E[S2h- (n+2)]}. If both surfaces are rough and have a similar roughness distribution (with different standard deviations), then the difference vanishes. However, if only one of the contact surfaces is considered as rough, then the expected value of the smooth surface is zero and the result becomes + E[ 8hh-@‘+2)].The sign depends on which surface is smooth. So, the parameter s has the following values: S = 0, both surfaces are rough and have identical roughness distribution (with different standard deviations); S= 1, only one surface is rough with the smooth surface moving (S= S1, S2= 0); S = - 1, only one surface is rough with the rough surface moving (S, = 0, S= S,). 2.2. Lmzgitudiml su@zce roughness In this case the asperity ridges are parallel to the direction of sliding. Thus, the film thickness components Sr, S2 arising from roughness

169

considerations are independent of x, U and t. So, the pressure gradient dp/ dx can be regarded as a variabIe with zero variance [ 161. We can take the expected value of eqn. (3) directly with the corresponding stochastic Reynolds equation for the longitudinal roughness surface

2.3. Roqlnwss distribution jimtion To evaluate the expected values of the various film thickness functions appearing in eqns. (11) and (1 Z), a detailed knowledge of the roughness ~~bution is necessary. A convenient choice is the polynomial form, approximating to the gaussian distribution function [ 151 f(s> i= ;5(c 2 - S2)3/32c7

(-c
(13)

where the standard deviation P= *c/3. Let p= c/h, be the non-dimensional parameter of roughness amplitude measured from the average film thickness. The value of p is taken so as to avoid direct asperity-asperity contact. Clearly, the roughness effects are signifkant when the maximum asperity height G is comparable with the ~~ nominal film thickness, A detailed discussion of this parameter is given in refs. 22 and 23. Here it suffices to say that 0 G p< 1. BiTerent values of p are taken to study the effects of surface roughness on the load capacity, friction drag and optimized geometric parameters of the bearing.

3. The optimum design of the Rayleigh slider bearing 3.1. I”ransverse suflbce roughness The analysis assumes that the width of the bearing is much greater than its length, so side leakage can be neglected. The geometric parameters of the bearing are dete~ed to carry a rn~~ unit load for a specified film thickness, contact length and non-Newtonian rheological characteristic index under different roughness parameters. According to eqn. (1 l>j it can be seen that the average pressure gradient is constant for each part of the stepped bearing because the nominal film thickness and the expected values are constants in the same stepped part (see Fig. 1). This means that the expected pressure distribution is stepwise linear. let E&&l be the maximm average pressure, then the average pressure gradient can be written as

( dx )

-df%P> =-- Ex?hRx) 0

10

(OGZGl*)

co,cx~lo)

Wa)

Integrating eqn. (11) and applying the continuity principle that the rate of flow flux across the sections in the film should be equal everywhere, the averaged maximum pressure can be obtained by combining eqns. (5) and (14) for each stepped part of the bearing. First, the following eon-tensional qu~ti~es are introduced X=~ll~,H=hih,,H,=h,lh~,L=Il~,L~=I~l~, P=ph,@

&=S/&, @=c&

‘~lmU”l,, ?= 7honlmUn, W, = wRhocn+“Imu”

c;w, = [E(WR) -

2

(15)

Wsl iw,, CFR= VW-R)--f’s] I&

The subscripts R and S denote the values of the optimal quantities for rough and smooth bearings respectively. W,, FR represent the dimensionless load capacity and friction force for a rough bearing and their expected values are given by eqns. (17), (19) and (30) respectively, The values of these quantities for a smooth bearing, W, and F,, are found simply by letting p = 0 in these equations. Thus, the ratios CW, and CFx determine the changes in the load capacity and friction traction for a rough bearing surface compared with a smooth bearing surface. Thus, the dimensionless form of the averaged maximum pressure is

(16) The dimensionless load capacity is expressed by L E(W,)=

s

E(P) dX-

0

(~,-~)~[(l+~)-‘“+2’]-S{E[~(1+~)-~”+2~]-E[~(II,~~)-‘“+2’~,~ (1 +-&)(I&

+J&)

I1 7) where E 1=E[(l

+ a)- (“+%]/E[H, + j&W,]

3.2. L~~~~~~a~ surrface r~~h~ss Integrating eqn. (12) and using a method similar to that used for transverse roughness, the dimensionless equations of the average maximum pressure and the load capacity can be determined as

171

E(pmd= 6nL1 E[(l

HI--l + 8)“+2](L, +E[(H, + 8)“+2]/E[(l+

&)=+a]}

(13)

H1-1

(1 +IQ).E[(1 +S)n+2]{L1 +E[(H~ -td)n+a]/E[(l +li)n+21}

(19) 3.3. The ~t~rn~rn design procedure The optimum design problem can then be expressed as the foIlowing map

@WG@G,

&I II

WI

The variables of eqn. (20) are bounded within ranges according to their physical conditions l
and O
(21) Therefore, eqns. (20) and (21) define a non-linear optimization problem with two restricted variables. A similar optimization method and computational procedure to those used in ref. 12 are adopted to obtain optimized geometric parameters of the bearing for both transverse and longitudinal roughness striations (detailed descriptions can be found in ref. 12). 4. Calculation of friction traction When the relative surface velocity of two contact bodies is high (e.g. for the pure sliding condition considered here), the initial assumption can be made that the fluid in the contact area is a Couette-dominated highly non-Newtonian fluid. So, the equivalent viscosity of the lubricant expressed by eqn. (2) can be approximated by q* = 7: = m( Ulh)n- l

(22)

Therefore, the expression of shear stress is 7=m

u -

0 h

n-1

au

(231

G

The velocity gradient can be expressed as [ 121

l1

au u 1 -_=-+ (2y-h) E (24) Bnm(Ulh)“ag h Then, the average value of the shear stress on the upper surface y = h is (25) 4.1. Transverse sm$zce chap Let the bracketed term in eqn. (25) be

(26) By comparing the above equation with eqn. (9), another form of eqn. (26) can be derived in which D can be expressed in terms of M M

D=6 ,,+3-$+3$$ uk

+$ (

1

(27)

Now, taking the expected value for eqn. (27) and using eqn. (10) for the expression M, the result is then substituted into eqn. (25) and the dimensionless form of the average shear stress is obtained

(W

+E(H -“)

Here, eqns. (14) and (16) must be used for the average pressure gradient in each part of the stepped bearing. 4.2. Longitudinal sueace roughness In this case, since dpldx is a variable with zero variance, we can take the expected value for eqn. (25) directly with the result, in dimensionless form

(29) Here, eqns. (14) and (18) must be used for the average pressure gradient in each part of the stepped bearing. 4.3. Fkiction calculation The average value of the dimensionless frictional force is given by integrating the shear stress distribution E(&(?)

dxtlj%(i) 0

dx

(30)

1

Substituting eqns. (28) and (29) into eqn. (30), the frictional drag for transverse and 1ongitudinaJsurface roughnesses can be solved for the optimized bearing parameters. A similar method can also be applied to find the friction drag at the lower plate surface (y =O). 5. Results and discussion A large number of calculations have been performed numerically using the so-called network method to discover the influence of the lubricant’s

173

rheological behaviour and the bearing roughness on the load capacity and friction force of the bearing under the optimized bearing geometry parameters. F’igure 2 shows the results of the optimally stepped height ratio HI and stepped location ratio L1. Figures2(a)-2(c) show that the transverse roughness decreases the height ratio, whereas the longitudinal roughness increases it (Fig. 2(d)), for all values of rheological index n, when compared with a smooth bearing. If the roughness parameter /3 is kept constant, the height ratio decreases when the non-Newtonian index n increases. Regarding the changes in the stepped location ratio, it should be noted that the parameter S has an important infhrence in the case of transverse roughness. When S = 0 or S = - 1, the ratioincreases with increasing roughness parameter /3 for a certain rheological index n. However, for S= 1, the situation is the contrary. Because the stepped location ratio reflects the location of the maximum pressure, the higher the value of the ratio, the further is the deviation of the maximum pressure from the centre of the lubricated area and thus the more serious the bearing leakage which leads to a dip in the load capacity. For longitudinal roughness, as shown in Fig. 2(d), the stepped location ratio decreases with increasing roughness parameter. In all these cases, a higher value of the rheological index n causes L1 to decrease for the same roughness parameter, except for the case S= - 1 and a high roughness parameter. When n = 1 and /3= 0, we have the solution for the classical ~e~o~~) Rayleigh stepped bearing with smooth surfaces, H,=1.866, &=~2.548, Ws = 0.206. The optimal load capacity ratio CWR is plotted against the roughness parameter j3 in Fig. 3. Comparisons with a smooth bearing and a Newtonian L,

ijzzk& 2~~ l::~

2.7

2.548

2.3

0.5 la1

0.99 8

O

0.5

0

0.99 B

@I

0.5

0

099

099

8

Fig. 2. Optimal geometic parameters HI, L, zts. roughness parameter /? under different values of rheofogical iudex ?a: (a) transverse (S=O), (b) txansverse (S= l), (e) transverse (S= - 1), (d) lon~~~~.

Fig. 3. Optimal load capacity ratio CWR vs. roughness parameter p under different values of rheological index n: (a) transverse, (b) lon~tu~~.

(4 Fig. 4. Friction ratio CF, vs. roughness parameter p under dierent index n: (a) transverse, (b) longitudinal.

values of rheological

fluid show that the rheological index n, the heights of the roughness asperities and the different roughness arrangements have important effects on the load capacity of the bearing. Generally, the load capacity increases for transverse roughness and this increase is more pronounced for a higher rheological index. For longitudinal roughness, the load capacity decreases for all values of n. This decrease, however, is not sign&ant. It is interesting to note that the results in both cases of transverse and longitudinal roughness indicate that the roughness effects only become conspicuous after p= 0.4 and that the dilatant lubricants (n> 1) are more effective when used with rough surfaces compared with the pseudoplastic lubricants (n < l), except in the case of longitudinal roughness. The roughness parameter p depends on the magnitude of the applied load and the machining processes. It is usually small and therefore the influence of roughness can be neglected. However, under a heavier load, it is possible that the depth of some of the asperities will be commensurate with the minimal film thickness on the bearing. Furthermore, with a high load and low sliding speed, the viscoelasticity of the fluid, which is also a property of some non-Newtonian fluids, may need to be taken into account, along with the temperature field in the lubricated region, because a high friction drag will produce higher viscous dissipation. Compared with a smooth surface moving against a stationary rough surface in the transverse case, which has the highest load capacity, the results of a rough surface sliding past a smooth surface are also quite interesting. It can be seen from F’ig. 3(a) that the load capacity of the latter (S= - 1) is much lower than that of the former (S= 1). This result may be

175

useful in bearing rn~~a~e. To obtain bigher load capacities for hydrodynamic bearings, the surface of the rotating parts (usually a shaft) should be smoother than that of the stationary parts (bushing or bearing). These results agree very well with those obtained in refs. 23 and 24. The only difference is that those results show a larger dip in the load capacity when the parameter fl is larger than 0.8 for the case S- - 1. This is attributed to consideration of the non-Newtonian behaviour of the lubricants. When a Newtonian fluid is considered, it can be seen from Fig. 3(a) that the dip is small. Figure 4 shows the results of friction traction at the upper surface. The effects of rougbness and rheological index on friction traction are similar to the effects on load capacity except in the case of lon~~~~ roughness in which the friction drag increases with increasing roughness parameter 6 and rheological index n in spite of the load capacity decreasing compared with a smooth bearing. 6. Conclusions The combined effects of surface roughness and rheological behaviour of the lubricant on the optimum design of a hydrodynamic sliding bearing have been studied. Various bearing characteristics have been obtained and compared with those of a smooth bearing. The following resultswere obtained. (1) The types of roughness have a strong infhrence on the bearing characteristics. In general, transverse roughness increases the load capacity, while longitudinal roughness decreases it. The effects of transverse roughness are more significant for rough surfaces, compared with the effects of longitudinal roughness. The roughness effects only become conspicuous for p>o.4. (2) There are quantitative differences in the various bearing characteristics caused by the lubricant non-Newtonian behaviour. Dilatant lubricants (n > 1) are more effective in increasing the load capacity of rough surfaces than pseudoplastic Iubricants (n < l), except for longitudinal roughness. (3) The parameter S for transverse roughness has sn important influence on the load capacity of the bearing. A smooth surface moving against a stationary rough surface has the highest load capacity. Conversely, when a rough surface moves past a smooth surface, the effect of the roughness is to reduce severely the load capacity for the dilatant fluids under a higher roughness parameter. (4) In all these cases, an increase in roughness parameter and rheological index results in higher values of friction traction for a rough bearing than for a smooth bearing. References 1 H. H. Horowitz and F. E. Steidler, Calculated jo~~-~~ thickened lubricants, ASLZ lYcm.s., 4 (1961) 257.

performanceof

polymer-

176

2 Y. C. HSUand E. Saibel, Slider bearing performance with a non-newtonian lubricant, ASLE Trans., 8 (1965) 191. 3 R. I. Tanner, Study of anisotbermal short journal bearings with non-Neck lubricants, J. A&. Mech., 32 (1965) 781. 4 J. B. Shukla and J. Prakash, The rheostatic thurst bearing using a power law fluid as lubricant, Jpn. J. Appl. P&s., 8 (1969) 1567. 5 A. Dyson and A. R. Wilson, Pilm thickness in elastohydrodynamic lubrication by silicon fluids, Proc. Inst. Mech. Engt I80 (1965-1966) 97. 6 S. Bair and W. 0. Wmer, Shear strength measurements of lubricants at high pressure, J. L&r. Technot., 101 (1979) 251. 7 K. L. Johnson and J. L. Tevaarwerk, Shear behaviour of elsstohydrodynamic oil films, Proc. R. Sot. Lixwbn, Ser. A., 356 (1977) 215, 8 R. B. Bird, W. E. Stewart and E. N. Lightfoot, Transport Phmwmena, Wiley, New York, 1960. 9 S. T. Swamy, B. S. Prahbu and B. V. Rao, Calculated load capacity of non-Newtonian lubricants in finite width journal bearing, Weur, 31 (1975) 277. 10 E. G. Tracbman and H. S. Cheng, Thermal and non-Noon effect on traction in elastohydrodynamic contacts, Proc. 2nd Smp. on ~~toh~~~~~~ L&r&atInstitution of Mechanical Engineers, London, 1972, p. 142. 11 A. A. Milen, Theory of hydrodynamic lubrication for a Maxwell liquid, Proc. Con;f. on Lubrication and Wear, Institution of Mechanical Engineers, London, 1967. 12 J. Wang and G. Jin, The optimum design of the Raylelgh slider bearing with a power law fluid, Wear, 129 (1989) 1. 13 I. K. Dien and H. G. Elrod, A generalized steady-state Reynolds equation for non-Ne~o~~ fluids witb application to journal bearings, J. Lubr. Techrwk, 105 (1983) 385. 14 S. H. Wang and H. H. Zhang, The surface strength of gears and the rheological characteristics of lubricants, Wear, 127 (1988) 1. 15 J. Prakash and H. Peeken, The combined effects of surface roughness and elastic deformation in the hydrodynamic slider bearing problem, ASIX mns., 28 (1985) 69. 16 H. Christensen, Stochastic models for hydrodynamic lubrication of rough surfaces, F%c. Inst. Me&. Eng., 184 (1969-1970) 1013. 17 S. Wada and H. Hayashi, Hydrodynamic lubrication of journal bearing by pseudo-plastic lubricants, BuU Jpn. Sot. Mech. Eng., 14 (1971) 268-278; 279-286. 18 A. I. Nakorchevskii and S. T. Andalojue, Rheological characteristics of greases and the main principles of their movement ln tubes, National Technical Seruice BuZZetin AD757471, U.S.S.R., November 3, 1973 (National Bureau of Standards). 19 W. Machidori, T. Moriucbi and H. Kagegama, Grease lubrication in e~~~~od~~c contacts, 50th Neal L&&a&g Grease Inst&u& Ann Meet., Part 2, 1983. 20 P. A. Marnell, A theoretical evaluation of the persistence of hyrodynamic lubrication in the hip joint during walking, D.Sc. Th&s, Department of Mechanical Engineering, Columbia University, 1983. 21 P. Sinha, J. B. Shuka, C. Singh and K. P, Prssad, Non-Newtonian lubrication theory for rough surfaces: application to rigid and elastic rollers, J. Mech. Eng. SC& 24 (1982) 147. 22 H. Christensen and K. Tonder, Tribology of rough surfaces: parametric study and Comparison of lubrication models, SINTEF, Res. Rep 22/69-18, US-A., 1969. 23 L. S. H. Chow and H. S. Cheng, The effect of surface rougbness on the average fflm thickness between lubricated rollers, J. L&r. Technd., 98 (1976) 117. 24 S. K. Rhow and H. G. Elrod, The effects on bearing load capacity of two slider striated roughness, J. Lubr. Technd., 96 (1974) 116.

177

Appendix rl: Nomenclature C

CFR C-W, D E FR Fs

h ho hl Hl 1 b, 4 L1 Ii n P Plwa P P max s t U VI9

G_?z

V WR

maximum magnitude of asperity amplitude (m) friction ratio parameter, CF, = [E(px) - Fs ]/Fs load capacity ratio parameter, CWk = [E(WR) - Ws] /w, defmed by eqn. (26) expectancy operator dimensionless friction force of a rough bearing surface (eqn. (30)) dimensionless friction force of a smooth bearing surface film thickness (m) outlet f&n thickness (m) inlet film thickness (m) stepped height ratio, HI = hr /ho bearing length, 1=l,,+lr (m) parts of stepped bearing length (m) stepped location ratio, L1 = 1r/lo viscosity constant for a power law fluid (eqn. (1)) defined by eqn. (9) non-Newtonian rheological characteristic index (eqn. (1)) pressure (N m-“) maximum pressure (N ma2) dimensionless pressure, P =pho@+ ~~~rnU~~~ ~mensionless m~um pressure P =p,, ho@+ ‘)/rnU~Z~ parameter for transverse roughness, defined in text time (s) velocity component (m s- ‘) sliding velocities of surfaces 1 and 2 (m s-‘) normal squeezed velocity, V=ah/at (m s-l) dimensionless load capacity of a rough bearing, wR = zu,h,(“+ ‘I/ mU”1 2

ws

dimensionless load capacity of a smooth bearing Cartesian coordinates (m) dimensionless coordinates, X= xl&,, Y= ashy

Greek symbols B a

61, &? s 9*

77: CT 7 7

roughness parameter p = c Iho composite roughness amplitude S= Sr+ S, (m) roughness amplitudes measured from the nominal level of surfaces 1 and 2 (m) dimensionless roughness amplitude 8 = 6/h, equivalent viscosity of lubricant (eqn. (2)) (N s mb2) approach of equivalent viscosity (eqn.(22)) (N s mF2) standard deviation a= fc/3 (m) shear stress (N m-‘) dimensionless shear stress, ?= ~~n/mU~