A new method of evaluating the abrasion resistance of wires. Part I: Presentation of the device and mathematical background

Wear, 171

99

(1994) 99-107

A new method of evaluating the abrasion resistance of wires. Part I: Presentation of the device and mathematical background S. Pensaert,

A. Van Peteghem

and F. Debuyck

Laboratorium voor Non-Ferro Metalen en Elektrometallurgie, Technologiepark 9, B-9052 Zwijnuarde, Gent (Belgium) (Received

March 31, 1993; accepted

September

9, 1993)

Abstract In this first paper a new device is presented to measure the abrasive wear on thin wires. The advantage of the device is the fact that wear measurements can be performed on a very small spot, and that coatings on wires can be tested. The principle of the measurement is purely based on a mathematical model, which will be derived in this paper. In addition, the paper evaluates all mechanical deviations from the ideal setup. The accuracy of the device presented will be tested. In a second paper the wear rate of several materials will be evaluated experimentally and compared. Furthermore, wear experiments on coated wires are discussed. A model will be proposed to predict the wear rate, independently of the geometry of device and wires.

1. Introduction In the literature and in different standards [l-11] many methods are described for the evaluation of the abrasive wear resistance of flat surfaces. There are several reasons for this variety: abrasive wear is very material-dependent, and it is difficult to obtain reproducible and comparable results. However, when it comes to testing the abrasive wear resistance of small and non-flat surfaces, such as wires, there is a great lack of test methods.

2. Presentation of the wear testing device for wires The difficulties in reproducibility and comparability imply the need for a new method to measure the abrasive wear resistance on wires themselves. Therefore a new device was constructed which makes it possible to test metallic, non-metallic, coated and uncoated wires, with a diameter range between 0.2 and 5 mm. This device consists basically of two main parts (Fig. 1): a roll coated with abrasive paper of a known coarseness, and perpendicular to it a support that holds the tested wire. The wire, clamped on the support, rests on the abrasive roll with a known and constant force, dependent on the support geometry, its mass and its position. The roll rotates with a known velocity. In this way the wire will be worn locally. The support is frequently shifted along the axis of the abrasive roll

0043-1648l94R07.00 0 1994 Elsevier Sequoia. All rights reserved SSDI 0043-1648(93)06348-8

Fig. 1. Schematic

representation

of the wear device.

to ensure contact of the wire with fresh abrasive paper. After a known time of testing the wire can be removed, and the volume of abraded material, which is often too small to weigh, can be determined by measuring the length 2 of the wear spot (Fig. 2). Fourteen supports are provided on the device, which make it possible to perform 14 tests at once and allow the results to be averaged.

3. Theoretical background The application of the described test method is based on knowledge of the cross-section of two perpendicular cylinders. The cylinder with the highest diameter rep-

of this coordinate system is not completely arbitrary because it makes the problem symmetrical and simplifies the calculation. 3.1. ~~~cul~~io~ of the area of the wear surjace According to Fig. 3 the position in the direction of the z-axis of a general point P on the wear surface is Fig. 2. Definition

measured

wear

spotlength l-

of the wear spot length 1.

z,=R

sin yp

VPE arc AB

(la)

and

.z,=Rsin yB

xl

(lb)

%e value of zB is of great importance, since zg is half of the wear spot length I= IAF3[, which will be measured in the described test method. All the important geometrical parameters will be calculated as a function of zBr r (radius of the wire) and R (radius of the abrasive roll). The equation of the wire is xZ+y2=?

(2a)

For point P this yields x~=r2-y2,

PI

where (xp, yp, zp) are the coordinates With yp= r-

of point P.

R(cos yp-cosye)

(3)

and eqns. (la) and (lb) taken into account, this gives yp= r-

(R*-z$)~~+(R~-z~)~~

Relation _x$=r2-

Yl Fig. 3. Projection and y-r-planes.

[r-

(4)

(2b) then becomes

(R*-z~)~'~+(R~-$J~~]~

=~~(R2-~~)~~-(Rz-z~)"*]-[(Rz-~)1~

of the system wire abrasive roll on the x-z-

resents the abrasive roll, while the one with the smallest diameter is the tested wire sample. As the test wire presses on the rotating abrasive roll, it wears off, and the abraded volume can be described as the crosssection of roll and wire. This implies that if, furthe~ore, the diameter of the roll and the wire are known, the abraded volume can be calculated from just a single parameter, this being the length Eof the abraded surface. Besides the wear volume, it is possible to determine the depth of wear and the area of the wear surface. Generally, the cross-section of two perpendicular cylinders can be calculated by use of the projection in Fig. 3. An appropriate orthogonal coordinate system is presented: the z-axis is the axis of the wire, the naxis is parallel to the abrasive roll’s axis, and the yaxis crosses the axis of the abrasive roll. The choice

(5) An infinitesimal part dw of the wear surface, as shown in the upper part of Fig. 3, can be written as do = (2r,)(R dr)

(6)

Since dy = d(arcsin(z,/R)) dz, = (R2 -z;)‘n

(7)

and together with eqn. (5), eqn. (6) yields dw-

~R{~T[(R*--z;)~~fR2-z;)1n][(R2-~$)'R

(8)

S, Pensaeti et al. / Evaluating the abmsion resistance of wires,Part I

The actual area of the abraded surface is given by =B S=2

s

dw

(9)

0

or S=4Rj&(R2-z;)‘“-

(R’-Y&)‘~]

- [(R2-&)1R

become noticeable. The combinations of IIR and rlR used in practice cause errors lower than about 0.5%. 3.2. Calculation of the wear volume The cross-section of the wire at an arbitrary point P is shown in Fig. 5. C, represents the area of this cross-section lost by wear; it can be calculated indirectly by subtracting the area of triangle ZPP’ from the area of the sector ZPP’. The area of the sector is equal to ps’. Because y, = r cm /3, j3 can be written as

(10) j3 =

This expression gives S as a function of z,, r and R. Because an elliptical integral is included, an analytical expression cannot be found. However, this equation is useful because it can be solved by numerical integration. On the other hand, S can easily be approximated in several ways. The real wear surface can be spread out on the X-Zplane. It will have a quasi-elliptical form. If it is assumed to be an ellipse, the first approximation for S is

arccos(yJr)

= arccos 5 [r- (R2-z$)lR+

pr2=r2

arccos i [r-(R2-zZp)1n+(R2-zi)1”z] I

(11)

A second approximation is obtained by the area of the projection of the real wear surface on the X-Zplane:

(13)

(14)

The area of the triangle ZPP’ is

=[r-

- [R - (R” -z;)‘“]‘)‘”

(R2-~i)1n]

which yields the area of the sector as

yprp= [r-

= n-R arcsin(z,/R){Lk[R - (R2 -zg)‘“]

101

(P-&)‘“+

(R'-z;)'qr'-y;)'n

(R2-z;)'R+(R2-&)1R]{~(R2-z;)1R

-(R2-.&)1~]-[(R2-z~)1R-(R2-z~)1~2)1R (15) Finally, X, becomes zP=P2-YpxP

=I* cos-1 : [r-(R2-z$)ln+(R2-z&)‘IL] I

Figure 4 presents the relative error of the first approximation as a function of 1IR and r/R. Only for very thin wires and long wear surfaces does the error

-[r-

(R'-z~)'~+(R~-,@~{~~[R~-z~)'~

-(R2-1,)"-+[(R2-z$)1R-(R2-~~)1~2)'R (16) The wear volume is obtained by integration of Z, in the z-axis direction between 0 and zB (zB on Fig. 3):

"0

0.04

0.08

0 12

0 16

02

024

028

r/R

Fig. 4. Relative error with the approximation of S by S’ as a function of rlR and l/R.

VY Fig. 5. Cross-section of the wire perpendicular to the z-axis at point P.

102

S. Pensaeri

et al. I Evaluating the abrasion

resistance

of wires, Part I

IF.3

v=2

s 0

cpdzp

-(RZ-z~)‘R]-[(R2-~~)‘n-(~2-~~)*R]Z)’n

cizp (18) 1 Again an elliptical integral is obtained, which can easily be calculated by numerical integration. An approximation for V, purely based on geometrical considerations, is not possible. 3.3. Calculation of the wear depth h The wear depth h can be defined as the distance on the y-axis between point M on the wear surface and the original wire contour (Fig. 3). According to Fig. 3 the wear depth h is given by h=r-y, = R - (R2 - &)‘n

(19)

3.4. Altemat~ve ~ppr~~~ of V Comparison of the ratio V/S, calculated by numerical integration, and the wear depth h yields that h is a very good approximation for 2v/s. This offers the opportunity to find an analytical approximation for 1/: V=*hS=thS’=V’

Pa)

so: V’=)hS’

Gob)

or with eqn. (II): V’ = ~TR arcsin(z,/R) X(2r[R-(R2-rZ,)‘R]-[R-(R2-~)1/2]~’/2 x [R - (R” -&)lR]

(2Oc)

Figure 6 presents the error due to the approximation of V by V’ as a function of l/R and r/R. V’ will always be an underestimate of V. 3.5. Relation between the dimensionless parameters VIR3, SIR’, i/R and rlR The system of two perpendicular crossing cylinders suggests the intr~uction of a dimensionless approach. Simplifying eqn. (12) by considering tB Q=R, and approximation of the Taylor series of the square root by its first two terms, gives the following equation:

Fig. 6. Relative error with the approximation function of riR and VI?.

of V by V’ as a

(211 An analogous relation can be found for V/R3. As V=(1/2)hS, it can be found that V -=_ R3

Tr -1 4 r Ii2 64R00 i

(22)

This relationship may look somewhat surprising, because of the fourth power dependence of V/R3 on l/R.

4. Errors due to the device or to measurements 4.1. Wear device related errors The theoretical considerations of the former paragraph are only valid for the ideal case in which wire and abrasive roll are pe~endicular to each other, and their positions remain fixed. In practice however, several deviations from the ideal can occur. Deviation of the section of wire and roll from the ideal circular shape. This deviation will be statistically compensated if several tests are performed at the same time. Therefore it will not be taken into account. Rotations of the wire around its length axis. This can be caused by an unstable fixation of the wire support. It leads to higher wear rates for the same wear spot length 1. Eccentricity of the abrasive roll. Besides the real eccentricity of the roll this effect includes other phenomena: small vibrations of the wire in the zdirection and ditviation of the roll from ideal circularity. The wire can be out of the perpendicular to the abrasive roll.

S. Pensaeri et al. i EvaIaa~~ the

ahsian

resistance of

wires,Parr f

103

Fig. 7. Cross-section of the wire perpendicular to the r-axis at point P in the case of rotation of the wire around the z-axis.

The extended calculations following sections.

will be left aside in the

4.1.1. Rotation of the wire around the z-axis If the maximum rotation angle of the wire is equal to 8, the cross-section of the wire can be drawn as in Fig. 7. The rotation is completely orientated clockwise, which simplifies the calculations, but does not effect the result. If this rotation is taken into account, the area of the wear surface becomes S,=S+2RB([r+(RZ-z~)‘R]

arcsin(z,/R)-2,)

(23)

with S the ideal wear surface area (when no deviations occur, as calculated above) and S, the surface area when rotation occurs. As Fig. 7 shows, V, can be calculated from EPe:

Fig. 8. Projection of the system wire+abrasive roll on the x-zand y-z-pfane in the case of eccentricity.

rg-(l/2)-e

S=4R

s

{‘&[(R2-zz,)1” - (I? -&‘“]

0

It can also be found that with V the wear volume in the case of ideal conditions, as calculated in Section 3.2. Note that V,>V. 4.1.2. ~cce~~ci~ of the abrasive roll Figure 8 shows the result of the eccentricity of the abrasive roll. The eccentricity is defined as the maximum distance between the real axis of the roll and the theoretical axis. It is obvious that the eccentricity, denoted by e, can vary along the length of the roll. By analogy with Section 2.1, it can be written that Sc=S~4e(2r[R-fR2-z~))‘~]-fR-(Rz-~)1~~1n

Again V is the wear volume in the case of ideal conditions (calculated with zB = (1/2)-e). Note that v,>v.

G.9

where S is the wear surface for ideal conditions. Note that this time zB is not equaI to half of 2, but to (112)-e. This means that S has to be calculated by

4.1.3. ~~~-~~~&~~~~ of wire vemis roll Figure 9 shows the situation when the wire is not completely perpendicular to the abrasive roll. This effect

104

S. Pensaeti et al. i E~a~uai~g the abrasion resistance of wires, Pan I

perposed on the two other effects. The combination of these leads to new expressions for the wear surface area and the wear volume:

Se, 8,

l

=

$&

S

+4e{2r[R -

- [R - (R2 -z;)~~]‘)“”

f

I

m@ [r-t (R2-Z;)'"]

X sin- 1 ?!z - zB + 2e@0R

(R’ -z;)“‘]

[R - (R2 -z;)l’*]}

(31)

I

with S the wear surface in the case of ideal conditions. So S has to be calculated with z,=(Z/2) COS(E)-e. In an analogous way, the non-ideal wear volume v ,,,B,r can be calculated:

with

(32b)

Fig. 9. View of the system wire abrasive roll in the case of non~~e~~culari~.

is quantified by the parameter E, which is the angle of rotation between the z-axis and the wire axis. By the introduction of a new coordinate system x’yz’ it is possible to calculate the wear surface and volume as a function of E. S, becomes

s,= -

Mls

(28)

COSE

&=(;)‘cos-‘(l;

[R-(R2-&)‘nl)I~

-1

+[I-(%)2]11(2; {I-[l+qlJ -{~-[l_~~)lllji)‘a

(32~)

with rB-(VLCOSC S=4R

s

{2r[R2 - zy

- (R2 -&)‘“]

0

- [(R* -z$)‘“-

(R’ -z6)1n]2]1n

(Rz f&~/z

with this time zB= (i/2) cos e, because 1 is measured alongside the z’-axis. V, is given by v, = v/cos E

(324

(29)

(30)

but once more z,= (t/2) cos E has to be used in the formula (18) for K 4.1.4. Combination of all deviations The effects of the rotation of the wire and the eccentricity of the abrasive rolI are synergistic. The non-perpendicularity of the wire, however, can be su-

Once more I/ represents the ideal wear volume, calculated with zB = (f/2) cos(~) -e. 4.1.5. Numerical evaluation of the errors on S and V due to mechanical deviations

Via numerical integration of eqns. (31)-(32d) the errors in S and V were calculated. For R and r values of respectively 15.8 mm and 1 mm were chosen, as these were the most common values used in practice. The errors in S and V were determined as a function of e and 8, and this for different values of I (2, 3, 4 and 5 mm).

10s

S. Pensaert et al. / Evaluating the abrasion resistance of wires, Part I

The following approximate error in S(%) = -

laws could be derived:

-0.23164 -

-0.00109

e

(33)

4.2. Deviations due to measurement errors Besides the “mechanical” errors that can occur, which are solely a result of the imperfections of the new wear device, measurement errors will also occur. In this section we shall investigate how errors due to the measurement of r, R and 1 will affect the calculated values of S and V. 4.2.1. Propagation of an error in a calculation

error in V(%) = ~

(34)

-0.00057

with 1 in mm, 8 in degrees, and e in pm. These laws show that the relative errors in S and V are inversely proportional to the wear length 1.Together with rather large values of 8 and e, a small wear length 1 can lead to very high errors for S and V. Figure 10 illustrates this; the relative error of V is shown as a function of 8 and e for 1= 2 mm. The mechanical deviations of the proposed wear device were measured. This gave the following results: e
Before any actual calculations are done, it is necessary to investigate how the error of several variables propagates to a function that depends on those variables. From statistics [12] it can be found that the absolute error of a function f(x, y, z, . . .) can be expressed as

(35) The relative error is given by

It is quite obvious that, if the errors for S and V would be calculated by the use of eqns. (10) and (18), these expressions would become very complex, due to the large and unintegrated expressions for S and V. Even formulae (12) and (20) would yield expressions that are too complex. Nevertheless, to get an idea about the errors on S and V due to measurement errors, the dimensionless eqns. (21) and (22) will be used. This includes of course, that errors due to mechanical deviations, as discussed above, are neglected. Besides, any error analysis that takes both “mechanical” and measurement errors into account at the same time would get too complex. 4.2.2. Errors in S induced by errors in r, R and 1 Combining eqns. (21), (35) and (36) yields b-2 A-W)

Fig. 10. Error

in V (%) due to mechanical

deviations (1=2

= S

mm;

WW12+ $7 WW)l* + $ IAWlZ ( 1 $

R = 15.8 mm; r= 1 mm).

(37a) l/2

TABLE I

1. Maximum errors Error

of S and V as a function of 1 in S

Error

(mm)

(%)

(%)

2 3 4 S

17.5 11.5 8.5 6.7

23.7 15.5 11.5 9.1

WS)

=

(

s

WW12+ 5

WV)12 + $ WW12 Ii-2

in V

4[l?E(f)]*+

i [k??(R)]‘+

t [RE(r)]*

(37b)

4.2.3. Errors in V induced by errors in r, R and 1 From eqns. (22), (35) and (36) we obtain

S. Pensaeri et al. I Evaluating the abrasion resistance of wires, Pan I

106

l/2

AE(V) =

F

WWl*+ G2 WW91*+ $ [A-WI2

(384

F WWl* +G2[AE(R)]’+ -$l&E(r)]’

l/2

If2

16[RE(1)]2+ 3 [RE(R)]2+ $ [ZL5(r)]2

(38b)

4.2.4. Numerical evaluation of the influence of the measurement emors on S and V Figures 11 and 12 show the relative errors in S and V as a function of the relative errors in I, R and r. On each graph four curves are shown, which all represent

Fig. 11. Relative error in S (%) as a function measuring errors in I, R and r.

of the relative

5. Conclusions

L

-

> .-i i_

x

, Fig. 12. Relative measuring errors

2 KE(1)

3 III

4

95

error in V (%) as a function in I, R and r.

the error in S (or, respectively, v> as a function of R!?(f), for certain combinations of R/Z(R) and R,?(r). For relatively high errors in 1 (k!?(I)> 3%) the influence of R/Z(R) and K!?(r) becomes negligible. At low errors in 1 (&Y(I) < 3%) RE(R) and k!?(r) are obviously important. Note that the impact of E(I) is much larger than the ones of RE(R) or RI?(r). To get a realistic idea about &Y(S) and f&!?(v), the maximum relative errors in R, r and I that can occur with the introduced wear device must be taken into account. These maximum errors depend on several factors: The maximum accuracy of the devices used to measure R and r (sliding callipers, maximum accuracy 0.01 mm) and I (microscope with calibrated ocular, maximum accuracy about 0.01 mm). The deviations of the abrasive roll diameter due to variations in the abrasive paper and the glue layer between the roll and paper. This deviation is kept within a limit of about 0.3 mm. The deviations of the wire diameter. For most industrially produced wires or rods these deviations are within 1%. Last, but not least, the ends of the elliptical wear surface are in some cases quite difhrse, which can lead to considerable errors in the measured wear spot length. With these thoughts in mind the following maximum relative errors were estimated: RE(I) < 5%, E(R) < 2%, RE(r) < 2%. This situation is represented by curve c on Figs. 11 and 12. The maximum relative errors for S are situated between 1.5 and lo%, and those for V between 3.2 and 20%, both depending on RE(I).

of the relative

5

The wire abrasion tester developed, together with the described theoretical model, offers a very simple and accurate way to measure the wear volume loss of the wires. By a quite simple measurement of the wear spot length I, the wear volume can be evaluated as a function of time. Mechanical deviations, however, can have a very severe impact on the errors of S and V. In particular, 0 and e have a large impact. Very accurate construction of the wire device, together with long period wear tests (which yield high values of I) can decrease these errors. When it comes to measurement errors, the error in 1 has a very large impact on the relative errors of S and V. This emphasizes the need for very accurate measurement of I. The problem however is that the “ends” of the elliptical wear surface are sometimes hard to define, which can lead to relative errors in 1

S. Pensaert et al. I Evaluating the abrasion resistance of wires, Part I

of up to 5%. The only possibility of decreasing this error is to increase the number of samples per wear test.

References 8 ASTM standards section 3: Metals Test Methods and Analytical Procedures. Volume 03.02 Wear and Erosion; Metal Corrosion, 1992. DIN-Taschenbuch 56: Materialprtlfnormen ftir metal&he Werkstoffe 2, 1987, DIN 50320-50322. L.E. Murr, Industrial Materials Science and Engineering, Marcel Dekker, New York, 1984. T.F.J. Quinn, Physical Analysis for Tribology, Cambridge University Press, Cambridge, 1991.

9 10 11 12

107

E.P. Polushkin, Defects and Failures of Metak, Elsevier, Amsterdam, 1956. E. Rabinowin, Friction and Wear of Materials, Wiley, New York, 1965. W.G. Schmitt-Thomas, Th. Happle and P. Steppe, Untersuchung der Strahlverschleissbesttindigkeit von Werkstoffen und Beschichtungen mit Hilfe eines Wirbelbett-Testverfahrens, Werkstoffe und Komsion, 41 (1990) 623-634. S. Jacobson, P. Wall& and S. Hogmark, Fundamental aspects of abrasive wear studied by a new numerical simulation model, Wear, 223 (1988) 207-223. A.A. Torrance, A three-dimensional cutting criterion for abrasion, Wear, 123 (1988) 87-96. K.H. Zum Gahr, Modelling of two-body abrasive wear, Wear, 124 (1988) 87-103. D.E. Kim and N.P. Suh, On microscopic mechanisms of friction and wear, Wear, 149 (1991) 198-208. M.R. Spiegel, Theory and Problems of Statistics, McGraw-Hill, New York, 1961.