An upper-bound solution for elevated temperature wire drawing

Journal of Mechanical Working Technology, 18 (1989) 33-51

33

Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands

A N U P P E R - B O U N D S O L U T I O N FOR E L E V A T E D T E M P E R A T U R E WIRE D R A W I N G

N.H. LOH

School o[ Mechanical and Production Engineering, Nanyang Technological Institute, Nanyang Avenue (Singapore 2263) and D.H. SANSOME

Technoform Sonics Ltd., Enterprise Trading Estate, Brierley Hill, West Midlands (Great Britain) (Received February 17, 1988; accepted May 27, 1988)

Industrial Summary Although the elevated-temperature drawing of wire has been employed in industry for a number of years, the process has not been investigated systematically and many facets of the process are not yet understood. Generally, exploration of the mechanics of wire drawing has been confined to cold working, for which the effects of strain-rate and temperature on the flow stress can be neglected: it is believed that no attention has been paid to the mechanics of wire drawing at elevated temperatures. The theory proposed in this paper takes into consideration the combined effects of temperature and strain-rate on the flow stress and it can be used to predict the draw stress taking into account the draw temperature, the speed and the area of reduction if the constant friction factor is known. The strain-ageing phenomenon and the effect of draw speed on the strain-ageing region are also revealed. The theory can be used to schedule the drawing process effectively on bull-blocks.

Nomenclature B

N; No P~ Pf P

Q S T

TN Tm V

constant n u m b e r o f e q u a l l y - s p a c e d arc divisions n u m b e r of e q u a l l y - s p a c e d a n g u l a r divisions t o t a l i n t e r n a l p o w e r of d e f o r m a t i o n t o t a l p o w e r o f friction t o t a l p o w e r of s h e a r i n g at t h e s h e a r surface c o n v e r s i o n f a c t o r b e t w e e n m e c h a n i c a l a n d t h e r m a l energies surface area drawing temperature new t e m p e r a t u r e after c o n s i d e r i n g t h e t e m p e r a t u r e rise velocity-modified temperature linear velocity

0378-3804/89/$03.50

© 1989 Elsevier Science Publishers B.V.

34

Ym c

d m r~

%

OL

0 P

q G

~T(J)

~o ~v ~uf ~T ~0 ~t

volume of an element at the die-wire interface mean yield stress mean yield stress of a flow line exponential constant density of wire constant used in the velocity-modified temperature equation constant friction factor strain-hardening index specific heat of wire radial velocity in the deformation zone at radius p from the apex radial velocity in the deformation zone at the shear surface die semi-angle inclination of the shear plane to the draw axis (measured in the clockwise direction) inclination of a radial line to the draw axis radial distance from the virtual apex to any point in the deformation zone angle between the shear plane and a line perpendicular to the radius p shear stress flow stress equivalent strain equivalent strain at shear surface F2 standard strain rate used in the velocity-modified temperature Tm equivalent strain rate strain rate component velocity discontinuity at the shear surface velocity discontinuity at the conical die surface temperature rise due to deformational work temperature rise due to frictional work time interval

Subscripts I and 2 refer to conditions at shear surfaces F1 and F2, respectively. 1. Introduction

Elevated-temperature wire drawing, that is, wire drawing above the normal ambient temperature, is a relatively unexploitated process, but one with numerous advantages [ 1 ]. The process is employed in various countries, mainly for the drawing of materials that are difficult to cold draw. Loh and Sansome [2,3] did extensive work on the drawing of mild-, medium- and "M2 high speed"-steel wires at temperatures of up to about 700 ° C, the reduction of area achievable being in the region of 40-45%. The art of cold drawing dates from before 1350 BC but no serious thought

35 was given to the mechanics of cold drawing until 1900. The earliest papers on wire drawing are generally qualitative and empirical in nature. In 1927, Sachs [4] used the "slab" method to give a logical analysis of the drawing process. After the proposal of Sachs' equation, the theoretical development of cold drawing grew very rapidly and various methods of analysis were used to improve the theory and to advance knowledge of the process. With the development of plasticity theory, upper-bound solutions were formulated by Avitzur [5], Kobayashi [6] and others. In brief, extensive investigations into the cold drawing of metals have already been undertaken by various researchers. Generally, the theories of cold wire-drawing assumed a flow stress that is independent of strain-rate and temperature but is dependent on work-hardening. This can be tolerated for cold deformation, but at elevated temperatures the flow stress of metals is affected significantly by temperature and strain-rate. It is believed that no attention has been paid to the effects of temperature and strain-rate in studies on the mechanics of wire drawing at elevated temperatures: the purpose of this paper is to enhance theoretical and practical knowledge in this area. 2. The upper-bound solution

Upper-bound model The postulated model of the velocity field is shown in Fig. 1. The inlet and outlet shear surfaces (F1 and F2, respectively) were assumed to be exponential in shape, where Pl follows the equation:

ZONE

3

APEX

Fig. 1. The model of the velocity flow (p0=OA).

0

36

(i)

Pl =Po exp [c(O- a) ]

and where c is a variable used to minimise the draw stress. In reality, the shear surface is more complex (as can be seen in visioplasticity test patterns), but an exponential surface was chosen to facilitate theoretical analysis. It is assumed that under a steady flow, a particle enters the inlet shear surface F1 with a uniform linear velocity V1, parallel to the draw axis. The velocity field in deformation zone 2 is assumed to be directed towards the virtual apex of the die. The equivalent strain rate ~ is expressed as: 2

~ = - - [~p +~o + ~o + 2(~o + ~oo+~p)] ~v p2 =---1 ~ exp [2e(O-o~)] [(cos O+e sin O) 2 +~

1

(3c cos O+ ( 2 c 2 - 1 ) sin O)Z] 1/2

(2)

The equivalent strain ~ is given by: ~=

~ dt

where dt = dplHp, from which ~=2 In pl P

[

1+~

\

cco 0+ c9

c~ssO~-~cs~n0

(3)

Distribution of temperature and flow stress within the deformation zone It is assumed that adiabatic conditions prevail and that both plastic and frictional work degenerate into heat. For the purpose of computation, the deformation zone is divided into (Np+ 1 ) and No equally spaced arcs and angles, respectively. Thus, the whole deformation zone can be identified by co-ordinates (I, J ) , as shown in Fig. 2. For the medium-carbon steel investigated, it is assumed that the true stressstrain curve obeys the power relationship: a=B~n

(4)

The opposing effects of temperature and strain rate are assumed to be related to a single parameter known as the velocity-modified temperature Tm [7], ex-

37

------/'--

/

~

~.

/

(2,1)

~

I~ ~

~

(N

+ 1,I)

0

APEX

Fig. 2. Dimensioning of the deformation zone.

pressed as:

Tm=T[1-•ln ({/{o) ] Using the compressive stress-strain data of Oyane et al. [8], the above equation was modified to take into consideration the temperature rise AT that occurs in the compression testing. Thus, Tm is expressed as: Tm=

(T+AT)[1-Kln (e/{o)]

Before using eqn. (4), a suitable value of ~c was found by trial and error, using a computer sub-program which is basically a graph-plotting program. The following information from the compressive stress-strain data were fed into the sub-program: (i) test temperatures, strain rates, flow stresses and corresponding strains; (ii) constant B and strain-hardening index n, which were found from the plot of log a against log e. For a particular value of ~c, the program performs two main operations: (i) it computes the values of Tm, taking into consideration the temperature rise in the compression testing; (ii) it plots the curves of flow stress against Tm for various values of strain. The computations and graph plotting were repeated for different values of ~cby using a do-loop in the computer program. The various graphs were studied visually, and a K value of 0.08 (which gave curves of similar shape for the various strains) was used in the Tm equation. Figure 3 shows the relationships between B, n and T~, expressed mathematically by polynomials, using the data of Oyane et al. [8]. It will be noted that it is necessary to determine similar relationships for different types of

38

i

i

1

f

~

l

e

900

~e

< = 0.08

800

700

i

0.3 q~ c

600

m

o.2~

500

/

400

o.,

\° O.l-I

t)

.m ¢5

300 200

0.0~

100 0

I 0

1O0

z

I

200

300

Velocity-modified

l

i

~

400

500

600

Temperature

Tm

i

i

!

700

808

900

(°K)

Fig. 3. Relationship between B, n and Tin,based on the experimentaldata of Oyane et al. [8 ]. steel, as not all steels exhibit the same properties. The curve of B against Tm was divided into suitable sections for curve fitting by a least-squares polynomial approximation (up to 3 degrees). The same procedure was repeated for representing Tm and the strain-hardening index n by polynomials. Thus, for a particular value of Tin, the values of B and n can be evaluated. Consequently, the flow stress pertaining to a particular equivalent strain can be calculated by eqn. (4). The initial temperature of a point in the deformation zone was assumed to be that of the drawing temperature, denoted by T. For a particular flow line, e.g. at point (I, J), the equivalent strain, equivalent strain-rate and temperature are known, therefore the flow stress can be calculated and represented by a0{I, J). When the particle travels from (/, J) to ( I + 1, J) the new temperature (taking into consideration the rise due to frictional work A0 and deformational work AT) is given by: TN(I+I,

J ) -~"

T(I, J )

"JcA0(I+

1, j )

+AT(~+I, j)

The new flow stress o'(i + 1, J) can be calculated as before, but the temperature is now TN(~+ 1, J). The temperature rises AT and AO due to deformationat and frictional work respectively are expressed as:

39

AT(I+I,j)-

( ~ z ~ ( I + 1, J )

Qcpd

ma#pSdt d0(i+l, a) --x/~ Q%dVa where a = ½ (ao(i+1, j) +a(i, j) ).

Total power of work Internal power of deformation The internal power of deformation Pa is obtained by integrating the equation over the entire volume of the deformation zone:

Pa = Ya Iv 4dV

(5)

where dV=p 2 sin 0 d0 d¢~dp

(6)

Substituting eqns. (2) and (6) into eqn. (5) and integrating:

Pd=47cV1Ydp~ln Pl

sinOexp[2c(O-ol)]FdO

(7)

F = [(cos 0+c sin 0)2+ 1 (3c cos 0+ (2c2 - 1 ) s i n 0)2] 1/2 The mean yield stress Ym(J) for that particular flow line is given by: ,

(7(1 , j )

d~(I, j )

Ym(J) -~T(J) -- ~(1, J )

The yield stress Yd used in eqn. (7) is given by:

f:r(;)) Ym (J) d~T(j) Yd--

~T(1) - -

~T(J)

(8)

4O 500

0J 49 ~8

y U t , eoret ca ~

Experi!ental

100 m=0.3 m=O. 2 m=0.1 m=0.0

0

i00

200

300

Drawing

400

Temperature

500

600

700

800

(°C)

Fig. 4. Comparison of experimental and theoretical results at various ~mperatures ~ r 2 0 % re-

duction in area.

Power losses in shearing at the inlet and outlet shear surfaces It is assumed that the shearing of material occurs on both the inlet and exit shear surfaces and that the power losses P1 and P2 are given by: ,, 2gYd 2 p 1 :/-'2 :------/~--~P

ju

1"~ V1 Jo exp[2c(O-o~)]sin20 ( l + c 2) dO

(9)

Power losses in friction at the interface of the die and the wire The power losses at the die-wire interface are: 2~ Pf =----~m Yd V1 p~ l n ( P l ~ cos a sin a x/3 \P2 /

(10)

41 700

oo

~"

400

4J

~ ~ S

300

~

~

~'-T~e°reti"al Exper nent al

~ ~" m=0.3 m=0.2 m=0.1 m=0.C

i00

0

i00 200 300 400 500 600 700 800 DrawingTemperature(°C)

Fig. 5. Comparison of experimental and theoretical results at various temperatures for 30% reduction in area.

The total power of deformation = Pd + P1 + P2 + Pf. The total power was minimised with respect to c. 3. Experimental w o r k A detailed account of the art of drawing wire and the experimental procedure are presented in Refs. [ 2,3 ]. Medium-carbon steel wires of 6 m m diameter were

42

drawn at a speed of 6 m m i n - 1. The lubricant used was a suspension of graphite and molybdenium disulphide. The die was made from "Syalon" material with a die semi-angle of 9 o and a small land to maintain dimensional stability. The die casing and the wire emerging from the die were cooled with water. The draw load and draw speed were measured by a specially designed loadcell [9 ] and a tachometer, respectively. The temperature of the wire just before entry to the die was measured by a non-contact infrared thermometer. All the apparatus used for measurement was calibrated accurately before use. 4. D i s c u s s i o n

Typical trends of the theoretical results for medium-carbon steel are shown in Figs. 4 and 5. The experimental results are superimposed onto the theoretical results to show clearly three different regions: gradual softening, strainageing and rapid softening. It is evident from the figures that the predicted curves follow the shape of the experimental curves. The theoretical response 6oo

I I

1

500

/

Q) 49 Ul

300

///~ /~

200

I00 i0

-Theoretica]

m=0.1 m=0.03 ~ f m=O.O

15

20

--=x~er~men+a l ~ ~~ ~ ~

25

Reduction

30 of

Area

35

40

45

50

(%)

Fig. 6. Comparison of experimental and theoretical results ( 100 ° C) for various reductions in area.

43

600

/

500

400 / / / /

~o

/ I

~

300

xperi~ ental

Theoretical

m=0.1

r~ 200

i00 I0

m=0.05' m=0.0 ¢

15

20

25

30

35

40

45

50

Reduction of Area (%) Fig. 7. Comparison of experimental and theoretical results (300oC ) for various reductions in area. of the material to temperature is similar to that found experimentally, and although the strain-ageing regions do not coincide exactly, they are a reasonably close approximation. It will be noted that the experimentally derived values of B and n are used to account for strain-ageing effects. Tm is dependent mainly on temperature and strain-rate, the latter being dependent on drawing speed. A change in the drawing speed will thus cause a change in Tm and consequently in B and n, thus taking care of the strain-ageing phenomenon. As the constant friction factor is increased, the theoretical draw stress increases also. From Figs. 4 and 5, it can be seen that an increase in the value of the constant friction factor m does not shift the strain-ageing region. This is because an increase in m causes a temperature rise at the die-wire interface only, as adiabatic conditions are assumed. In general, an increase in m does not change significantly the Tm values of nodal points within the deformation zone and thus the values of B and n are not affected, which is why m does not affect the strain-ageing region.

44 From Figs. 6 to 9, it can be seen that for the range of temperatures and reductions of area studied, the predicted draw-stress curves follow the shape of the experimental curves. At a draw temperature of 100 °C (see Fig. 6), good correlation is found between the experimental and the theoretical results for m = 0.03. In the strain-ageing region of 300 ° C, the experimental results generally fall below the theoretical results (see Fig. 7), except for reductions of area of 40% and above: this is because the predicted strain-ageing regions do not coincide exactly with the experimental strain-ageing regions. As the draw-temperature increases, a good correlationship is found between the experimental and the theoretical results, for higher values of the constant friction factor (see Figs. 8 and 9). The increase is more significant for reductions of area of 40% and above, which is probably due to a greater increase in temperature at the die-wire interface causing breakdown of the lubricant. At a temperature of 700°C the value of the constant friction factor increases rapidly. The presence of shiney drawn wire and of wear found on the dies for temperatures above 400 ° C suggests that friction at the die-wire interface increases

500

400

Exp6rimental/ Theoretical ~

300

4J

200

m=0.3!~

~

~

j

/

,J

m=0.2i

r~

m=0.1 m=0.0 /

I00

0

I0

15

20

25

Reduction

30 of

35

40

45

50

Area (%)

Fig. 8. eompansonofexperimentalandtheoreticalresults(500°C) ~rvariousreductionsinarea.

45 400

300 Experimental~ Theoretical--~~ 200 4J In

100 m=0 4

I0

15

20

25

30

35

40

45

50

Reduction of Area (%) Fig. 9. Comparisonof experimentaland theoretical results (700°C ) for various reductionsin area. at draw temperatures above 400 oC, resulting in a higher value of the constant friction factor. This supports the deduction that the constant friction factor increases with increasing temperature. The theoretical effects of the draw speed, with the mean strain-rates calculated using Atkins' equation [10], are indicated in Fig. 10. The strain-ageing region shifts to higher temperatures as the strain-rate increases, i.e. as the draw speed increases. Since strain-ageing is a time-dependent phenomenon, the " h u m p " can be shifted to a higher temperature with increasing strain-rate. W h e n the draw speed is changed (i.e. a change in strain-rate), the effect on Tm is felt throughout the deformation zone, and thus the values of B and n are changed, which accounts for the shift of the strain-ageing region. This shift was also found by Oyane [8], Manjoine [11] and Samanta [12]. The strainageing region for medium-carbon steel for a strain-rate of 2 s - 1 is also in reasonable agreement with that of Thomason et al. [ 13 ] and Glen [ 14 ]. The strainageing region extends from about 200 °C to 300 oC. The trend of the theoretical results obtained from the upper-bound solutions generally follows that of the experimental results for the present medium-car-

46

400

300

%

200

% %%% •

%,

43 5q

254s-1 [52s-I %% %

~"'

~

50S-I

"-<.\

i00

lOs-I 2s_i 0

I00

200

300

400

500

Drawing Temperature

600

700

800

(°C)

F i g . 10. Theoretical results for various draw speeds and temperatures (20% reduction in area and constant friction factor of 0.1 ).

bon steel. The experimental results can be relied on because they were obtained with reliable instruments, which were accurately re-calibrated using established procedures. It can thus be concluded that the derived upper-bound solution can be used to predict the draw stress with reasonable accuracy, if the value of the constant friction factor is known. 5. Conclusions A theoretical study of the drawing of wire at elevated temperatures has been made for a range of reductions of area. The theoretical draw-stress curves have been found to follow the shape of the experimental curves. A strain-ageing region was found theoretically in the temperature range of 100-300 ° C, agreeing well with that observed experimentally in the temperature range of 150350 ° C. The theoretical model also predicts the shifting of the strain-ageing region, the predicted regions agreeing well with the experimental results of other researchers.

47

Acknowledgements T h e a u t h o r s a c k n o w l e d g e : t h e s u p p o r t of t h e Science a n d E n g i n e e r i n g R e s e a r c h Council ( U . K . ) a n d t h e U n i v e r s i t y of A s t o n ( U . K . ) in t h i s r e s e a r c h p r o g r a m m e ; t h e i n t e r e s t a n d s u p p o r t of A c h e s o n Colloids, L u c a s C o o k s o n Sya l o n a n d t h e P e t e r S t u b s c o m p a n i e s for t h e s u p p l y of l u b r i c a n t s , dies a n d wire m a t e r i a l s ; t h e a s s i s t a n c e r e n d e r e d b y M e s s r s G.M. J o n e s , T. R u d g e a n d P. M c G u i r e in t h e e x p e r i m e n t a l work; a n d lastly b u t n o t least, M r s T a n L a y T i n for t y p i n g t h e m a n u s c r i p t .

References 1 N.H. Loh and D.H. Sansome, A review of the warm drawing of wire, Fine Wires Int. Conf., 30-31 Oct. 1980, Aachen, F.R.G., Int. Wire and Mach. Assoc., Surrey, 1980. 2 N.H. Loh and D.H. Sansome, Drawing of wire at elevated temperatures, Wire Ind., (1983) 148. 3 N.H. Loh and D.H. Sansome, Further work on the drawing of wire at elevated temperatures, Scan Wire 85, 14-15 May 1985, Copenhagen, Denmark, Int. Wire and Mach. Assoc., Leamington Spa, 1985. 4 G. Sachs, Z. Ang. Math. Mech., I (1927) 235. 5 B. Avitzur, Analysis of wire drawing and extrusion through conical dies of small cone angle, Trans. ASME, J. Eng. Ind., 85 (1963) 89. 6 S. Kobayashi, Upper bound solutions of axisymmetric forming problems - II, Trans. ASME, J. Eng. Ind., 86 (1964) 326. 7 C.W. MacGregor and C.J. Fisher, A velocity-modified temperature for the plastic flow of metals, J. Appl. Mech., (1946) All. 8 M. Oyane, F. Takashima, K. Osakada and H. Tanaka, The behaviour of some steels under dynamic compression, in: Proc. 10th Japan Congress on Testing Materials, 1967, p. 72. 9 B.B. Basily, The mechanics of section drawing, Ph.D. Thesis, University of Aston, U.K., 1976. 10 A.G. Atkins, Consequences of high strain rates in cold drawing, J. Inst. Met., 97 (1969) 289. 11 M.J. Manjoine, Influence of rate of strain and temperature on yield stresses of mild steel, J. Appl. Mech., (1944) A211. 12 S. Samanta, Resistance to dynamic compression of low carbon steel and alloy steels at elevated temperatures and at high strain-rates, Int. J. Mech. Sci., 10 (1968) 613. 13 P.E. Thomason, B. Fogg and A.W.J. Chisholm, The effect of temperature and strain-rate on the cold and warm drawing characteristics of alloy steels, Proc. 14th Int. Mach. Tool Des. Res. Conf., Manchester, Sept. 1973, Macmillan, London, 1974, pp. 791-797. 14 J. Glen, Effect of alloying elements on the high temperature tensile strength of normalised low carbon steel, J. Iron Steel Inst., 186 (1957) 21.

Appendix Derivation of strain-rates W i t h r e f e r e n c e to Fig. A. 1, s a t i s f y i n g t h e c o n t i n u i t y o f flow a t t h e inlet s h e a r s u r f a c e / ' 1 gives:

48 S HEAR

--~

PLANE\~NLET SHEAR

Ul ~

VI

B.

.

.

~p sin(B-O)

~ ~ ~B_O

up -~ _

up cos(6-Oi~~ 7;

-X

APEX 0

Fig.A.1. A viewalongthe inletshear surface.

0

Fig. A.2.Definitionof the variousangles.

~/1 - -

V1 sin fl sin (,6-0)

(A.1)

From Fig. A.2, since dO is very small, tan d0~ d0 and OB ~ OD. ~ED dpl tan t / = B D - ~ = c

Poexp[c(O_ol)] pl

and therefore tan t/=c

(A.2)

~ = 9 o - (,6-o)

(A.3)

,6=90--t/+0

49 and therefore li1 = V1 (cos O+c sin 0)

(A.4)

The velocity of the element at radial distance p (see Fig. 1) within the deformation zone can be calculated from the volume-constancy requirement, giving: liP = Vl \(Pl 7/

~2 (cos O+c sin O) 2

-~VI(~) exp[2c(O--a)](cosO+csinO) lie =

(A.5)

lie =0

The strain rates expressed as a function of the velocity components are given by:

J liP

(A.6)

1 Jlio + lip

(A.7)

7 1 .Jlio +lip +#e cot 0 p sin 0 J¢ p p

l[Jzo

(A.8)

~o ~_1 Jlip

(A.9)

p 1

1

J/to

1Jli~

~'o~=~ p si-n 0 j ¢ t p J0 1 5li¢ li~ t

Y~"=2 ~

p

I

cotO --tt¢ p

I

[

5#p

psine~¢

(A.10)

(A.11)

Appropriate substitution and simplication gives:

~z = - 2 VlP-~3exp[2c(O-a) ] (cos 0+c sin 0)

(A.12)

~e= ~z P~ exp[2c(0--a) ] (cos 0+c sin 0)

(A.13)

p

--1

P3

50 D

2

~0 = V1 ~ exp [2c ( 0 - c~) ] (cos 0+c sin 0)

(A.14)

7pe--~IiTfl~ -1 p3 (3 e x p [ 2 c ( 0 - ~ ) ] c o s 0 +

(A.15)

•

{2c2 exp[2c(O-~)]-exp[2c(O-~)]} sin 0) 7o~=0

(A.16)

~,p =0

(A.17)

Substituting the respective values from equations (A.12) to (A.17) into the expression ez + ee + ~ gives a value of zero. The velocity fields derived thus satisfy the incompressibility condition.

Power lossess in shearing at the inlet and outlet shear surfaces The power loss in shearing at the shear surface is given by:

P= fs TIAVl [dS

(A.18)

where AVl--=/2pC O S ( f l - - 0 ) -- 71 COS ]~ (see Fig. A.1). Substituting equations (A.2)- (A.4) into the above, and simplifying:

AVl =

gl

[sin 0 (1+c 2) cos t/]

r=pl sin 0 27rr dS = sin fldr ]sin 0 (1+c tan 0) d~ dS=27tp~ e x p [2c(0- O/) . . . . . . cos t/+ sin t/tan 0 The yon Mises yield criterion was assumed, for which T= YJx/~, so that p

2~zYa 2 =-~Po V1 ]o exp[2c(O--°l)lsin20(l+d)dO

(A.19)

Power losses in friction at the interface between the die and the wire The power losses at the die-wire interface are defined by:

Pf= f~l T[Avf[dSf

(A.20)

51 The velocity discontinuity Avf at the die-wire interface is defined by equation (A.5) and c = t a n ~/=0 2

Avf-~V1\-~/(P°~2exp[2c(O-a)](cosO-bcsinO):Vl(~)

cosa

(A.21)

dSf = 2n p sin a clp Assuming a constant friction factor m, the shear stress of the material z is given by:

Yd/ /3 It is postulated that m takes values of from 0 to 1, i.e. zero for the frictionless case and unity for sticking friction, so that

Pf=~33mYdVlp~ln(Pl~\P2 / cos a sin a

(A.22)