Experimental verification of a fatigue wear equation

wear, 79 (1982)

241

241 - 253

EXPER~ENTAL

VERIFICATION

OF A FATIGUE

WEAR EQUATION*

V. K. JAIN Mechanical Engineering

Department,

University of Dayton, Dayton, OH 45469

(U.S.A.)

S. BAHADUR Mechanical Engineering Department, University, Ames, IA 50011 (U.S.A.)

Engineering

Research

institute,

Iowa State

(Received November 23,198l)

Summary It is demonstrated that the wear equation based on the concept of repetitive loading of surface asperities in sliding can be used to estimate the wear of polymeric materials. Wear and surface topography data for poly{methyl methac~la~), polyfvinyl chloride) and high density polyethylene pins sliding against a hardened and ground AISI 4340 steel disk are reported. The fatigue data for these polymers were obtained by using notched cylindrical specimens in a rotating beam fatigue-testing machine and were fitted to the Wiihlers equation N = (S,/S)‘. The wear equation was solved by assuming the gaussian distribution for asperity heights and involves a parabolic cylinder function in its solution. The procedure to compute the wear rate from the fatigue wear equation is described and the wear rates computed from the fatigue and surface topography data are reported. It is found that the computed steady state wear rates are in excellent agreement with the experimental wear rates for the three polymers tested.

1. Introduction There are four basic mechanisms commonly used to explain the wear of materials. These are adhesion, abrasion, corrosion and surface fatigue. Of these, the adhesive and abrasive mechanisms have most often been applied to polymer wear. Fatigue wear is usually associated with rolling but localized fatigue on an asperity scale is being increasingly recognized now as an important factor in sliding [ 11. The separation of adhesive and fatigue *Paper presented at the International Conference on Wear of Materials 1981, San Francisco, CA, U.S.A., March 30 - April 1, 1981. 0043-1648182/0000-0000/$02.75

@ Elsevier Sequoia/Printed

in The Netherlands

242

processes is almost impossible; there are grounds to believe that the same wear processes previously categorized as adhesive may involve a large contribution from fatigue [ 1, 2] . In fact, the well-known Archard equation [3] derived from adhesive wear considerations involves a proportionality constant which, in an obscure way, implies the probability of producing a wear particle per asperity encounter. This may be construed to indicate that repetitive loading may be needed, as in fatigue, for the separation of a wear particle. Fatigue as a likely mechanism for the wear of polymers has been advocated for some time in the Russian literature [ 41. Using the concept of fatigue, Kraghelsky and Nepomny~hchi [5] derived a wear equation for rubbers. Later Halling [6] developed a fatigue wear equation for metals. The limitations of the models serving as the basis for the above equations were discussed by the present authors in their earlier paper and a fatigue wear equation was derived in the light of more realistic assumptions [ 7 3 . The equation is for fatigue failure during sliding from the interactions between two rough surfaces having asperities with spherical tips. Here the deformation in discrete contact zones, which are assumed to act independently of each other, has been considered to be of an elastic nature. These assumptions led to the following wear equation: 1 - 2V

V=

(* + v)n + --3---

(1)

where

(2) (3) 1 - VI2 1 _=-..----+ E’ El PlPZ p=-.._---

01 + Pz (I

=

(U12+ Q2p2

1 - us2 (4) E2

(5) (6) (7) (3) (9)

243

where the symbols have the meaning given in the Nomenclature subscripts 1 and 2 refer to the two surfaces in contact.

and the

2. Experimental details The sliding experiments were performed on a pin-on-disk type of wear machine which has been described elsewhere in detail [8]. The sliding system consisted of a hardened and ground AISI 4340 steel disk and a cylindrical polymer pin. The cylindrical polymer pin was secured to a vertical arm that carried strain gauges for measuring the friction force. A linear differential transformer was used to monitor continuously the reduction in length of the polymer specimen; this provided an estimate of the wear volume through a calibration process. The average temperature rise in the sliding contact zone was measured by using two iron-constant thermocouple junctions embedded in the metal disk and a slip ring-and-brush assembly. Thus the sliding experiments provided data for the wear volume, coefficient of friction and temperature rise in the contact zone for varying loads, sliding speeds and times. Three polymeric materials, i.e. poly(me~y1 methac~la~), poly(viny1 chloride) and high density polye~ylene, were selected for this investigation. These represented an amorphous glassy polymer, an amorphous rubbery polymer and a highly crystalline polymer respectively. The polymer specimens in the form of cylinders 6,5 mm diameter and 25 mm long were machined from sheets 9 - 12 mm thick supplied by Cadillac Plastics and Chemical Co. One of the circular ends of each of these cylindrical pins was finished by abrasion against 600 grade emery paper under running water. The specimens were washed in distilled water and methanol and then dried and stored overnight in a desiccator before being tested. The changes produced in the surface topography of the metal disk as a result of rubbing were investigated by evaluating surface profiles for sliding times of 4 and 6 h which co~esponded to the steady state wear conb;ition. After every interruption in sliding for the purpose of surface analysis, the wear test was repeated from the beginning with a new polymer sample and the cleaned metal disk. The surface profiles were analyzed by using a Bendix profilometer with a diamond stylus of tip radius 2.5 pm. The profile ordinates at 2 pm intervals over a length of 2 mm were obtained using a data acquisition system. A number of surface parameters pertinent to the wear process were then calculated using a FORTRAN IV program written for an IBM 360 computer. The fatigue tests were performed at 30 Hz on a Krouse rotating cantilever beam fatigue-testing machine according to the ANSI-ASTM D 671-71 recommendation [ 91. The notched fatigue samples were used so as to obtain the fatigue data independent of the frequency of loading [ 10, II]. Optical microscopy examination of the fatigue fracture surfaces revealed that the polymer samples did not suffer from thermal softening caused by loading in the reversed bending mode.

244

3. Results and discussion 3.1. Polymer wear Thevariation in polymer wear volume with sliding time for poly(methy1 methacrylate), poly(viny1 chloride) and high density polyethylene pins sliding against the metal disk is shown in Figs. 1 - 3 respectively. The wear data for the first two materials were obtained by using a linear differential transformer. Since the wear rate for high density polyethylene was very small, the wear data for this material were obtained by weighing. As the wear test was started from the beginning after every interruption in sliding for weighing, which was very time-consuming, the data here were collected for one sliding speed only. For all the materials, two data points were obtained for every sliding condition. Whereas the initial portion of the curve representing the unsteady state wear was drawn free hand, the later portion representing the steady state wear was obtained by fitting a straight line through the data points by the method of least squares. It should be noted that the correlation coefficients which are given for every regression line are close to unity in almost all cases. The wear rate behavior for poly(methy1 methacrylate) and high density polyethylene was normal in that the wear rate decreased with time in the unsteady state, later exhibiting a constant value to represent the steady state condition. The decreasing wear rate in the unsteady state condition is attributed to a continuing modification of the coun~rface topo~aphy from the transfer of polymer to the metal disk periphery. The steady state wear rate is restored only after a stable thickness of the polymer film builds up on the disk surface. Contrary to the above, the wear rate for poly(viny1 chloride)

0

0

120

240 SLIDING

360

480

600

TIME (mln)

Fig. 1. Variation in wear volume with time for a poly(methy1 methacrylate) pin sliding against a steel disk at various sliding speeds (normal load, 3.33 N): V, 2.6 m s-l (f = 0.36; correlation coefficient, 0.94); 0, 2.0 m s-l (f = 0.36; correlation coefficient, 0.90); A, 1.0 m s-l (f= 0.35; correlation coefficient, 0.97); n, 0.5 m s-l (f= 0.35; correlation coefficient, 0.95).

245

0

120

480

360

240 SLIDING

61

TIBE (mini

Fig. 2. Variation in wear volume with time for a poly(viny1 chloride) pin sliding against a steel disk at various sliding speeds (normal load, 8.83 N): 0, 2.5 m s-l (f = 0.33; correla0.99); 0, 1.6 m s-l tion coefficient, 0.99); v, 2.0 m s-l (f = 0.33; c orrelation coefficient, (f = 0.35; correlation coefficient, 0.99); A, 1.0 m s-l (f = 0.35; correlation coefficient, 0.99); 0, 0.5 m s-l (f = 0.36; correlation coefficient, 0.98).

IL

I

I

d

lo-

“E E aY 3 6-

0

120

360 240 SLIDING TIME (min)

480

600

Fig. 3. Variation in wear vofume with time for a high density polyethylene pin sliding against a steel disk (normal load, 47 N; sliding speed, 1.75 m s-’ ; f = 0.38; correlation coefficient, 0.96).

in the steady state is higher than in the unsteady state (Fig. 2). This uncommon behavior was further verified by another sliding test where the wear was obtained by weighing the polymer pin at regular intervals. The investigation of this seemingly anomalous behavior was pursued in terms of the change in counterface topography resulting from the transfer of polymer film. It was observed that the transferred material was in the form of thin films which could be seen adhering to the disk. It contributed to an increase in the counterface roughness which is believed to be responsible for the increased wear rate in the steady state.

24ti

The wear rates for poly(methy1 methacrylate) and poly(viny1 chloride) increased with increasing sliding speeds, which agrees with previous experimental observations. The variation in temperature rise in the contact zone with sliding time for a poly(methy1 methacrylate) pin rubbing against the steel disk at different speeds is shown in Fig. 4. Similar behavior was observed for the other two materials and so is not reported here. In all cases, the temperature increased initially with sliding time but after about 2 h of sliding it assumed a fairly constant value. The steady state temperature rise was always proportional to the sliding velocity. The maximum temperatures observed during sliding, within the range of sliding conditions used in this investigation, were well below the melting points of the respective polymers, thus precluding any possibility of polymer melting in the wear tests. The friction force between the polymer pins and the metal disk was measured using strain gauges. The average values of the coefficient of friction in the steady state wear condition are indicated within parentheses for the wear curves in Figs. 1 - 3. It is noted that the friction coefficient for a material remains almost unchanged in the range of sliding speeds used.

0

0

120

240 360 SLIDING TIME (min)

480

600

Fig. 4. Variation in average disk surface temperature rise with time for sliding between a poly(methy1 methacrylate) pin and steel disk at various sliding speeds (sliding conditions as for Fig. 1): G’, 2.5 m SC’;a, 2.0 m s-l;A, 1.0 m s-l; 0, 0.5 m s-l.

3.2. Polymer fatigue The fatigue data for the polymeric materials were obtained by testing the cylindrical notched specimens in the reversed bending mode. The size and shape of the notch and the type of loading used in testing the specimens resulted in an elastic stress concentration factor of 3.2 [ 121. The notched specimens were chosen on the basis that a conventional fatigue failure independent of the loading frequency was to be obtained. It was further concluded through experiments that the conventional fatigue behavior for

247

high density polyethylene could not be obtained using unnotched specimens because of its low modulus of elasticity. In the fatigue tests, the crack always initiated at the notch and progressed normal to the direction of principal stress. There was no sign of thermal failure. In order to plot the fatigue curves, the stress values were obtained by multiplying the maximum nominal bending stress by the elastic stress concentration factor. The stress amplitude uersus the number of cycles to failure (S-N) curves were plotted on log-log axes as shown typically for poly(methy1 methacrylate) in Fig. 5. The values of the correlation coefficients for the straight lines fitted to the data by the method of least squares indicated that the scatter in the data was small. These plots provided the values of Se and t (Table 1) in the Wohlers equation N = (S,/5’)f where N is the number of cycles to failure and S (N mmP2) the stress amplitude.

10

103

I 104

I lo5

I 106

NUMBER PF CYCLES-TO-FAILURE

Fig. 5. Reversed bending fatigue failure curve for notched specimens of poly(methy1 methacrylate) (b = -0.0296; b,, = -0.0423 (5% significance level); bmin = -0.0169 (5% significance level); correlation coefficient, 0.78). TABLE 1 80 and t values in the Wiihlers equation for polymeric materials Polymer

so (N rnrnm2)

t

Poly(methy1 methac~late) Poly(viny1 chloride) High density polyethylene

135 1245 496

20.44 3.16 4.61

3.3. Computation of wear from the fatigue wear equation 3.3.1. Evaluation of the inte~al F,(h) In order to determine the values of Fe(h), F,(h) and F3,2(h) in eqn. (2), the solution of the integral F,(h) is required. Assuming a gaussian distribution for asperity heights, as confirmed by surface analysis studies [ 8, 131, the integral F,(h) may be expressed as

248

Substituting

s - h = x so that ds = dx, we obtain -1

= ----& The integral 3.462.1

expi-

2 (

X+/Q2

dx t

Gjfin

exp(-

g

-hx)

(10)

dx

in ref. 14 is

m X “-l

s 0

exp(-_Px2 - yx) dx 2

(2@)-“‘2r(i9

=

for@> O,v> 0. The comparison V

=Vl+l

p

=

0 i

exp &

between

D-v

Y

(2p)‘/2

(11)

the integrals (10) and (11) provides

l/2

y=h so that the integral F,,(h) is reduced

to

1

(12)

F,(h) = (27#‘2

where&.+l,( o+,+,,(h)

h 1 is related to the parabolic

cylindrical

function

by

= V(n + l/2, h)

The value of U(n + l/2, h) has been tabulated by Miller [ 151 for different values of n and h. The various forms of the integral &(h) involved in eqn. (2) correspond to the three values of ft, i.e. 3/2,1 and 0. The substitution of these values in eqn. (12) gives F,,,(h)

= 0.5303 exp(-h2/4)U(2,

h)

(13)

F,(h) = 0.3989 exp(-h2/4)

U( 1.5, h)

(14)

F,-,(h) = 0.3989 exp(-h2/4)

U(0.5, h)

(15)

The values of these three integrals and of Fs,2(h)/FI(h) and F,(h)/F,(h) have been calculated for h = 0 to h = 2 and they are tabulated in Appendix

A.

249

For any sliding situation, F,,s(h) is calculated from eqn. (7) because all other parameters relevant to the contact situation are known. The value of h corresponding to the calculated value of B’s,&) is obtained from Appendix A. For this particular value of h, the other integrals can then be determined. 3.3.2. Asperity density and standard deviation for agaussian distribution of asperity heights A FORTRAN IV computer program was written to reduce the population of asperity heights to a gaussian distribution. Using this program, the asperity density and the standard deviation of asperity heights for the pin and disk surfaces were computed. The means of the asperity densities and the standard deviations corresponding to sliding times of 4 and 6 h are given in Table 2. These are used later in the c~culation of the wear rates. TABLE2 Surface data for the pin and the disk sliding surfaces sliding conditions

for the given

Polymeric mater~l in the steel disk-polymer pin combination High density polyethylene

Poly(methy1 methacrylate)

Poly(viny1 chloride)

1.75 47

1.0 8.83

1.0 8.83

2.0 8.83

k (dW(w)

49.00 0.38

46.67 0.35

47.00 0.36

49.60 0.33

P2 (Pin)

40.00 0.61

48.08 0.39

40.00 0.62

34.98 0.53

Sliding conditions u(ms-l) p (W Surface data

01 (dW

firm)

@m)

0.86

0.55

0.63

0.99

7)1 (disk) (mm-l)

51

0 (disk) (mmm2)

4488

37 3441

44 3982

46 4186

02 (pin)

(Ctm)

3.3.3. Volume of the ~earpartic~e The wear particles were assumed to have a shape similar to a flattened sphere (special case of an ellipsoid) where the radius of the spherical portions of the particle was taken equal to the radius of a discrete contact zone. Thus the volume of a wear particle is given by VP = $ 7ra2c

(16)

where a is the radius of a discrete contact zone and c the thickness of the wear particle. The latter was calculated from the following expression [ 161:

250

c=

6Ey(l

+ 3f2)

(17)

s,

where f is the coefficient of friction and E, y and S, are the modulus of elasticity, surface energy and yield strength of the particle material respectively.

The steady state wear rates for the three polymers were estimated from eqn. (1) for different combinations of sliding speeds and normal loads. Table 3 lists the properties of the polymeric materials needed for calculations. In addition, the fatigue parameters SO and t, as reported earlier in Table 1, were used. The procedure for cumulating the wear rates is as follows. First the values of p and CJare calculated from eqns. (5) and (6). The value of E’,,,(h) is then calculated from eqn. (7). The corresponding values of h, F,(h) and F,(h) are read from Table Al. This leads to the calculation of K from eqn. (2). The volume VP of a wear particle is determined by using eqns. (8), (9), (16) and (17). The wear rates are then estimated from eqn. (I). The wear rates for different sliding combinations were calculated according to the procedure described above and are given in Table 4. The corresponding experimental wear rate values are also listed in Table 4 for the sake of comparison. It should be noted that the estimated wear rates are within permissible limits according to the error analysis performed with allowed variations of I 1% in the measured qu~tities L, SO and t, 5% in Q, 0 and 0, and 7% in f. Some discrepancy between the wear rates is inevitable for the following reasons. (1) The wear rate is affected by many variables that are hard to control; as a result of this a scatter of about 25% in the experimental wear data is frequently observed.

TABLE 3 Properties of polymeric materials Property

High density polyethylene

Poly~methy~ methacryiate~

Poly~vinyl chlorite}

Reference

Surface energy (NmmM2)

33.45 x10-3

34.43 x10-3

34.00 x 10-a

[171

Poisson’s ratio

0.47

0.40

0.42

iI71

Modulus of elasticity (N mm-2)

412.00

1825.00

2413.00

PSI

Yield strength (N mm -2)

22.00

57.50

41.00

WI

251 TABLE 4 Comparison of estimated and experimental steady state wear rates for the steel disk and polymer pin sliding system Pin material

High density polyethylene Poly(methy1 methacryiate) Polyfvinyl chloride) Poly(viny1 chloride)

Sliding speed (m s-l )

Load (N)

1.75 1.0 1.0 2.0

47 8.83 8.83 8.83

Wear rate ( mm3 h-’ ) Estimateda

Experimental

0.62 2.75 3.74 9.25

0.90 3.00 4.60 8.70

+ 0.28 f 1.20 + 1.07 f 2.67

Fl%e error range given was determined by the error analysis.

(2) In the compu~tion of wear rates, the properties of bulk polymers have been used. However, the wear process is controlled by thin layers of the transferred polymeric material. The properties of these thin layers are hitherto unknown. (3) During the sliding process, a moderate increase in temperature occurred. This could have changed the adhesional characteristics at the sliding interface. (4) The mechanical properties used correspond to the normal slow rates of testing whereas in the sliding process very high strain rates are involved. The sliding speed-strain rate dependence is not yet known. (5) The characterization of the topography of the steel disk surface with transferred films of the polymer on it has a considerable potential for error because of the likelihood of deformation and penetration of the soft polymeric material by the profilometer stylus. (6) The fatigue properties of discrete contacts may be different from the fatigue properties of the bulk polymeric material. (7) The modelling of the asperities on the sliding surfaces under adhesive contacts resulting in progressive fracture is at best approximate.

4. Conclusion There is excellent agreement between the steady state wear rates estimated from the fatigue wear equation and the experimental wear rates for the case of polymer-metal sliding. It provides support for the concept that the repetitive loading of surface asperities in sliding is responsible for the separation of wear particles.

Acknowledgment The research was supported Iowa State University.

by the Engineering

Research

Institute

of

252

Nomenclature

1, Ao i f L n0

N P S

SY So t V

V ? Y 7) 771 v u

radius of a discrete contact zone real area of contact nominal area of contact thickness of a wear particle Young’s modulus of elasticity coefficient of sliding friction sliding distance number of discrete contact zones number of cycles to failure normal load between the sliding surfaces engineering stress yield strength failure stress for a single cycle power exponent of the fatigue curve sliding speed wear volume average volume of a wear particle radius of curvature of asperities surface energy surface density of asperities line density of asperities on the moving surface Poisson’s ratio standard deviation of asperity heights

References 1 J. K. Lancaster, Friction and wear. In A. D. Jenkins (ed.), Polymer Science, Elsevier, New York, 1972, pp. 989 - 1046. 2 J. R. Atkinson, K. J. Brown and D. Dowson, The wear of high molecular weight polyethylene, J. Lubr. Technol., 100 (2) (1978) 208 - 218. 3 J. F. Archard, Contact and rubbing of flat surfaces, J. Appl. Phys., 24 (8) (1953) 981- 988. 4 I. V. Kraghelsky, Friction and Wear, Butterworths, London, 1965. 5 I. V. Kraghelsky and E. F. Nepomnyashchi, Fatigue wear under elastic contact conditions, Wear, 8 (4) (1965) 303 - 319. 6 J. Halling, A contribution to the theory of mechanical wear, Wear, 34 (3) (1975) 239 - 249. 7 V. K. Jain and S. Bahadur, Development of a wear equation for polymer-metal sliding in terms of the fatigue and topography of the sliding surfaces, Wear, 60 (1) (1980) 237 - 248. 8 V. K. Jain and S. Bahadur, Surface topography changes in polymer-metal sliding: I, J. Lubr. Technol., 102 (4) (1980) 520 - 525. 9 ANSI-ASTM Stand. D 671-71, 1971, reapproved 1978, in 1979 Annual Book of ASTM Standards, Part 35, Plastics -General Test Methods; Nomenclature, ASTM, Philadelphia, PA, 1979. 10 R. J. Crawford and P. P. Benham, Some fatigue characteristics of thermoplastics, Polymer, 16 (12) (1975) 908 - 914. 11 R. J. Crawford and P. P. Benham, Fatigue and creep rupture of an acetal copolymer, J. Mech. Eng. Sci., 16 (3) (1976) 178 - 191. 12 R. E. Peterson, Stress Concentration Factors for Design, Wiley, New York, 1974. 13 V. K. Jain, Fatigue effects in the wear of polymers, Ph.D. Thesis, Iowa State University, Ames, 1980.

14 I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, Academic Press, New York, 1965, pp. 337 - 343 (translated by Scripta Technica Inc.; edited by A. Jeffrey). 16 J. C. P. Miller, Parabolic cylinder functions, Handbook of Mathematical Functions, National Bureau of Standards, Washington, DC, 1964, pp. 686 - 720. 16 M. K. Kar and S. Bahadur, Estimation of wear particle thickness in polymer-metal sliding, Wear, 63 (1) (1980) 105 - 112. 17 D. W. Vankrevelen and P. J. Hoftyzer, Properties of Polymers, Elsevier, New York, 1976. Vol. 48, McGraw-Hill, New York, 1971 - 1972. 18 Modern P&tics Encyclopedia,

Appendix

A

Values of the integrat ~~~~~ Values of the integral

for n = 3/2, 1 and 0 are given in Table Al. TABLE Al h

@‘z&h)

F,(h)

Fo(h 1

F3/z(h)/Fl(h~

Fo(h)/Fl(h)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

0.42999

0.39890

0.37151 0.31915 0.27256 0.23137 0.19519 0.16363 0.13630 0.11278 0.09269 0.07567 0.06133 0.04937 0.03945 0.03129 0.02464 0.01925 0.01493 0.01149 0.00877 0.00665

0.35090 0.30686 0.26673 0.23042 0.19778 0.16866 0.14286 0.12020 0.10042 0.08331 0.06861 0.05610 0.04552 0.03662 0.02930 0.02324 0.01829 0.01427 0.01106 0.00849

0.49994 0.46014 0;42068 0.38205 0.34454 0.30851 0.27422 0.24194 0.21183 0.18404 0.15864 0.13565 0.11506 0.09679 0.08075 0.06680 0.05479 0.04456 0.03593 0.02871 0.02275

1.07794 1.05873 1.04005 1.02186 1.00412 0.98690 0.97018 0.95408 0.93827 0.92302 0.90829 0.89389 0.88004 0.86665 0.85446 0.84096 0.82831 0.81629 0.80519 0.79367 0.78327

1.2533 1.31131 1.37092 1.43236 1.49527 1.55986 1.62587 1.69365 1.76231 1.83270 1.90421 1.97712 2.05098 2.12632 2.20508 2.27986 2.35757 2.43630 2.51787 2.59819 2.67962