A reliability model for friction and wear experimental data

Wear 269 (2010) 213–223

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A reliability model for friction and wear experimental data A. Ramalho ∗ CEMUC – Department of Mechanical Engineering, University of Coimbra, Portugal

a r t i c l e

i n f o

Article history: Received 2 September 2009 Received in revised form 22 March 2010 Accepted 24 March 2010 Available online 30 March 2010 Keywords: Friction Wear Tribodata Reliability

a b s t r a c t Tribology experimental research has been a considerable improvement during the last years, leading to a significant amount of results and consequently an increasing number of papers appear every month. Unfortunately, in spite of this development, the obtained results usually are characterized by big scatter and significant discrepancies can be founded for the same materials if tested by different research teams. The scatter found in the data has been frequently attributed to many variables involved in the experiments, namely: environment (especially humidity), contaminant layers, differences on test conditions, uncertainty on the results evaluation and rarely on the experimental equipment response. The control of the test variables, the atmosphere and also the test procedure can play an important role on the results reliability but accurate equipments and careful laboratory practices certainly could improve the quality of the results. However, the basic problem remains and even taking into account all that statements, the results could have significant scatter. This work aims to discuss several sources of imprecision which lead to scatter of the experimental tribology results. The usual experimental procedure used to calculate performance parameters, namely wear and friction coefficients will be compared to other solutions, including energy approaches. A reliability method will be proposed to characterize friction and wear data. Experimental results obtained by unidirectional sliding and by micro-abrasion will be used to support the discussion. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Wear is a failure mode often identified as the main source of drawbacks of modern machinery. The response of materials science to surpass this problem includes composite materials, nanostructured metal alloys, new ceramics and a wide range of coatings. The increase in new solutions offered for applications simultaneously subjected to high contact stresses and tangential relative motion, leads to an appreciable raise in the amount of new papers specially focusing the wear and friction of unlubricated sliding contacts. The development in this research area also provokes a spread out of the parameters which are used to quantify the materials wear resistance. Nevertheless, the mass lost during the test, the change in the area evaluated transversally to the sliding direction and the volume of the removed material are in the basis of the most used quantitative parameters, let us call these as primary wear quantities. These primary quantities are in general used to calculate new parameters with the objective to let the results less dependent of the test equipment and of the experimental variables. Therefore the primary wear quantities usually appear normalized to the sliding distance, the time, the normal load, the speed, the contact pressure,

∗ Tel.: +351 239 790 700; fax: +351 239 790 701. E-mail address: [email protected]. 0043-1648/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.wear.2010.03.023

or simultaneously several of these variables. Let us call these new derived parameters as design-using parameters. The deep difference in the parameters generally used leads hard the comparison between the published results and most of them difficulty could be compared with the performances of new trial products. In some cases, due to scarceness experience description, even the main scientific principle of the repeatability of the experiments by other laboratories could very difficultly be applied. Concerning the experimental approach on tribology, the target results are in general the friction (friction force or friction coefficient), the wear amount and the investigation of the wear mechanisms. The fundamental aspects outlined previously center on the reliable prediction of friction and wear behavior. To date, there exist no widely accepted friction and wear models; consequently, tribologically based design rules are available only for specific applications and for a tiny range of conditions. In general the full domain parameters are not yet accurately understood, limiting the validity of tribological measurements to industrial applications. Therefore, the current friction and wear models for dry sliding are inadequate to support contemporary product design strategies. The present paper aims to systematically discuss the general aspects of the experimental approach in tribology. The source of errors in experimental approaches will be discussed considering the suitability of statistical models to analyze the tribodata. The Archard’s wear equation will be analyzed and the effects of system

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dynamics on friction will be discussed. The main basis of the energetic approach of wear will be revisited and afterwards an analysis of experimental results applying different methods will be used to discuss the accuracy of the design-using parameters and to support a new method which allows the achievement of a stochastic assessment of tribological design using parameters. 2. Error sources in experimental approach Uncertainty or error is defined as the difference between a measured or estimated value for a quantity and its true value, and is inherent in all measurements. Knowledge of the type and degree of error likely to be present is essential if data are to be used widely. Making measurements and analyzing them is a key part of the engineering process, namely in the tribology field. Reported experimental results should always include a realistic estimate of their error, either explicitly, as plus/minus an error value, or implicitly, using the appropriate number of significant figures. Furthermore, scatter is an inherent characteristic of mechanical properties of materials. This also applies to tribological properties. Friction and wear are mainly material surface phenomena, therefore are strongly dependent on the material surface conditions. Tribological properties of alloys are sensitive to both average and extreme of microstructure therefore depend on the local microstructure features. Therefore an inherent variability needs to be accounted for in experimental friction and wear studies. Errors may be divided into two categories: systematic and random. On one hand, systematic error in a measurement is a consistent and repeatable bias or offset from the true value. This is typically the result of miscalibration of the test equipment, or problems with the experimental procedure. On the other hand, variations between successive measurements made under apparently identical experimental conditions are called random errors. Random variations can occur in the physical quantity being measured, the measurement process, or both. We will outline some statistical procedures for handling this type of error. In reporting experimental results, a distinction should be made between “accuracy” and “precision.” Accuracy is a measure of how close the measured value is to the true value. A highly accurate measurement has a very small error associated with it. Note that in experimental work the true value is often not known, and thus what is reported must be an estimated accuracy or error. Precision is a measure of the repeatability and resolution of a measurement—the smallest change in the measured quantity that can be detected reliably. Highly precise experimental equipment can consistently measure very small differences in a physical quantity. Note that a highly precise measurement may, nevertheless, be quite inaccurate. High precision in a measurement is a necessary but insufficient condition for high accuracy. Systematic error can be difficult to identify and correct. Given a particular experimental procedure and setup, it does not matter how many times you repeat and average your measurements, the systematic error remains unchanged. No statistical analysis of the data set will eliminate a systematic error. Systematic error can be located and minimized with careful analysis and design of the test conditions and procedure; by comparing your results to other results obtained independently, using different equipment or techniques; or by trying out a calibration by an experimental procedure on a known reference value, and adjusting the procedure until the desired result is obtained. Blau and Budinski [1] describe some of the main advantages of ASTM standard wear test methods, namely: the test methods have been rigorously evaluated, by previous interlaboratory exercises, and the procedures carefully documented; the repeatability and reproducibility of results tends to be better documented and understood than for specialized or one-of-a-kind types of wear testing machines.

It is unusual to make a direct measurement of the quantity we are interested in. Most often, we make measurements of a related physical quantity, often several times removed, and at each stage some kind of assumption must be made about the relationship between the data you obtain and the quantity you are actually trying to measure. The quality of tribology experimental data is affected by both systematic and random errors. 3. Statistical considerations—a review 3.1. General aspects Random errors can usually be estimated and minimized through statistical analysis of repeated measurements. When the experimental work allows several results, being n, by repeated trials or measurements, the simplest data analysis is the calculus of the sample average (xa ) and standard deviation (xstd ), Eqs. (1) and (2) 1 xi n n

xa =

(1)

i=1

  n  1  xstd =  (xi − xa )2 n−1

(2)

i=1

Although there is no single accepted standard, one commonly used way of estimating the value of x reporting the error, is as displayed by Eq. (3). A normal distribution of the measurements was usually assumed and the coefficient ˛ is a function of the desired probability that the true value lies within the upper and lower error limits x = xa ± ˛xstd

(3)

In order to obtain reliable estimation from a set of experimental data, ideally all the different statistical effects have to be considered, namely: • • • •

variance of data; variance of the mean value; difference of the population and the distribution of the sample; deviation from the assumed Gaussian distribution.

The coefficient ˛ of Eq. (3), which multiplies the standard deviation to obtain the error, depends mostly of the reliability desired, however, the way used to calculate this value could be successively improved to take in consideration all the statistical effects before presented. In many applications only the effect of the variance of the data and the difference of the population and the distribution of the sample was considered, therefore the value of the coefficient ˛ can be calculated as the confidence interval by Eq. (4). In this equation ˚ is the distribution function of the Gaussian normal distribution, R is the value of the reliability (the probability that the true value lies within the upper and the lower error limits) and the superscript (−1) indicates inverse function ˛=

˚−1 ((R + 1)/2) √ n

(4)

From Eq. (4), the value of ˛ can be calculated. Assuming a sample of 10 tests the values of 0.46, 0.52 and 0.62 are obtained for reliability of 85%, 90% and 95% respectively. If all the statistical effects have to be considered, a more complicated approach must be used. Eq. (5) is a suitable possibility and

A. Ramalho / Wear 269 (2010) 213–223

215

Table 1 Values of ˛ for several number of trials and different values of reliability. R (%)

N, number of trials

85 90 95

5

9

11

13

15

17

19

21

25

30

2.48 2.92 3.58

1.92 2.27 2.80

1.80 2.14 2.64

1.72 2.05 2.53

1.66 1.98 2.45

1.61 1.92 2.39

1.57 1.88 2.34

1.54 1.84 2.29

1.49 1.79 2.23

1.45 1.74 2.17

assumes similar considerations as proposed for fatigue test data [2]



˛=

t(1 − c, n − 1) + ˚−1 (R) √ n

n−1 2 (((c + 1)/2), n − 1)

(5)

In this equation the first fraction considers the variance of the mean value and the difference between the population and the tested sample (Gaussian versus t distribution). The square root is to cover a possible deviation of the test data from the assumed Gaussian normal distribution, chi-square correct this effect. • for the t-distribution: c is the confidence of the average and N − 1 degrees of freedom; • for the 2 distribution: c is the confidence of the average and N − 1 degrees of freedom. Table 1 includes the ˛ values calculated by applying Eq. (5) to different sample dimensions assuming a confidence of 87.5% for the average and considering reliabilities of 85%, 90% and 95%. When the results to be analyzed include more than one random variable, more complex methods should be considered.

can be tested. As a general method a correlation factor can be used to quantify the strength of the linear association between the two variables. In fact linear correlation means to go together in a straight line. The correlation coefficient, also known as the Pearson product–moment correlation coefficient, Eq. (14) [9], is a number that summarizes the direction and degree (closeness) of linear relations between the two variables. If the test program is planed including replicated values for i values of the independent variable, at least for i = 3, a statistical test for linearity can be made based on the F distribution, Eq. (15). F(1 − R, n1 , n2 ) is the inverse of the F-distribution for a probability (1 − R), n1 (n1 = l − 2) and n2 (n2 = n − l) are freedom degrees, l is the number of different tested values of x and rpi is the number of repetitions for the i level. This methodology is well described for fatigue tests [5]

   n mstd = 

n−2

n ×

− y0 − mxi )2

i=1 (yi

n

x2 i=1 i

n

−

1 n−2

CIm = m ± t(1 − R, n − 2) ×

n

(7)

[(xi − xa )(yi − ya )]

n

i=1

n S2 = r=

i=1

(yi − ya )2 − m ×

i=1

y0 = ya − mxa

(xi − xa )2

(8)

n

n−2

i=1

×

1 + n

n (x i=1 i

− xa )2

F(1 − R, n1 , n2 ) <



n (y i=1 i

(xp − xa )2

n

i=1

(xi − xa )2

(xi − xa )(yi − ya )

l i=1

(12)

(13) (14)

− ya )2

l rp (y i=1 i 0



(11)

(xi − xa )

n (xi − xa )(yi − ya )

 i=1 

(6)

Assuming a linear dependency between y(x) and x, the problem is fitting to data a straight line not necessarily passing through the origin to allow the value yˆ of an estimate of the dependent variable, Eq. (7). After the calculus of the average values of the available n values (x, y), the m and y0 values can be achieved by the least square method, Eqs. (8) and (9)

m=

CI = y(xp ) ± t(1 − R, n − 2) ×

(10)

i=1



S2

2

n (y − y0 − mxi )2 i=1 i n 2



This section aims to review linear fitting of experimental data values and is based mainly on the adaptation to tribometry of the concepts well described in the literature for fatigue tests [3–8]. Being two variables, the independent variable x essentially without error and y(x) the experimentally observed value of the dependent variable for each value of x. The error ε of y(x) can be calculated by Eq. (6) as a function of the difference between the trial value and its real value 

yˆ = m(x) + y0

x i=1 i



3.2. Linear fitting between two variables

y(x) −  = ε

n



+ mxi − yia )2 /(l − 2)

rpi



(y − yia )2 /(n − l) j=1 ij

(15)

4. Substantiation of the wear energetic approach For a sliding contact, the wear test has as output the wear amount of a pair of materials tested under fixed conditions. The Archard’s equation is surely the most appropriate to calculate a design using parameter. Based on the experiments Archard’s [10] derived the following conclusions:

(9)

The value m is an estimate whose standard error can be calculated by Eq. (10). For each specific value of x, being xp the corresponding value yp can be estimated applying Eq. (7). Letting R the desired reliability, the confidence limits for m can be achieved by Eq. (11). Additionally, the confidence bands can be established for a known domain of the independent variable applying Eq. (12) where S2 is the variance, Eq. (13). t(1−R,n−2) is the inverse of the two sided Student’s t-distribution for a probability (1 − R) and (n − 2) freedom degrees. In the methodology before presented it was assumed that a linear model is valid. However, the adequacy of the linear model

• the material volume removed by wear is proportional to the sliding distance; • the material volume removed by wear is proportional to the normal applied load; • the materials display an wear amount inversely proportional to their hardness. Taking into account the above conditions, the widely known Archard’s equation can be easily established, Eq. (16) V =K

Nx H

(16)

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A. Ramalho / Wear 269 (2010) 213–223

where V is the wear volume, N is the normal load, x is the sliding distance, H is the hardness of the material and K is a nondimensional wear coefficient. Czichos [11], proposed a new form for the Archard’s equation, Eq. (17), where the material influences, H and K are grouped in a new parameter k, which is generally called wear coefficient or specific wear rate V = kNx

(17)

The proportionality of the wear volume with the sliding distance is easily accepted because most of systems reach a steady-state regime after a more-or-less small running in period. Thus, the wear appears as a cumulative phenomenon linearly proportional to the sliding distance. However, for short tests, when the running-in is a significant fraction of the total test time, the linear evolution is difficult to be reached. The effect of the normal load is harder to explain. Actually the normal load is perpendicular to the sliding direction during all the relative motion; therefore, the work done by the normal load is zero during all the sliding action. Thus, this seems to be an inconsistency of the Archard’s equation because this model establishes the direct proportionality between the wear volume and the normal load which do a nil work during the motion. Nevertheless, Archard’s equation has a well established empirical base which fit the results obtained in most of the experimental studies. The reason for this is related to the fact that the normal load determines the value of the friction force if a constant friction coefficient, , is assumed. Even so, when the friction force varies significantly along the test, or when the tests are not long enough to reach a friction steady state regime, the approach obtained from Archard’s model can be very rough. Considering the Amonton’s friction Eq. (18), the tangential force appears proportional to the normal load F = N

(18)

Using Eqs. (17) and (18) to eliminate N, a new Eq. (19) could be derived V=

k 

Fx = k Fx

(19)

where k = (k/) is the specific wear rate as a function of the friction force. This new equation is physically consistent because it establishes a proportionality between the wear amount, which is an output of the sliding process, and the work done by the friction force which is the main energy input in the tribosystem. Therefore, this new equation, Eq. (19), establishes proportionality between the wear volume and the work done by the friction force along the sliding. Eqs. (17) and (19) agree more or less similarly with the results if a constant friction regime is attained early in the initial phase of the sliding. However, in a large number of cases, as listed by Blau [12], the friction force varies significantly at least in running-in and several times changes along the entire test and the Archard’s equation no more fit the results. In these cases, it is difficult to calculate a reasonable value for the friction coefficient; therefore Eq. (19) needs to consider the variation of the friction with the time arriving in Eqs. (20) and (21)





V = k

F(x)dx = k v x

V = k

 i

F(t)dt

(20)

t



¯ i xi = k v Fx

¯ i ti Ft

(21)

i

where F(x) and F(t) are the instantaneous values of the friction force ¯ i and Ft ¯ i are average for the sliding distance x or the instant t. Fx values of the friction force during xi or ti .

These concepts can be easily applied to the analysis of tribological experiments because it involves only the value of the friction force along the sliding distance (or along the time). Usually discrete values Fi of the friction were acquired during the test with a period t between acquisitions; therefore, Eq. (21) can be directly applied. Alternatively the arithmetical average of the friction force can be calculated by applying the resultant value in Eq. (19) leading to the final value of the specific wear rate function of friction (k ). 5. Linear models to estimate the friction coefficient from experimental data As established by the Amontons–Coulomb friction model, Eq. (18), there is a linear proportionality between the friction force and the normal applied load. Eq. (18) should be considered as the equation of a straight line, being the friction coefficient the slope. Therefore, in order to characterize the behavior of a pair of contacting materials under a specified contact conditions, the study should be carried out defining a domain of the normal load, centered in the target value, and an increment to scan the domain. Decreasing the increment gives a better discrimination of the domain; however, the number of tests could increase to a number excessively large. One test should be carried out for each value of the normal load, leading as output the average value of the friction force measured in steady state regime. Finally, the friction coefficient should be calculated as the slope of a straight line obtained using a linear correlation of the data points. Using this methodology further minimizing the effect of the test equipment on the friction coefficient, the validity of the linear Amontons–Coulomb model is also verified. Habitually the value of friction coefficient is calculated from the value of the tangential force measured on a single test, or using the average value of several tests carried out for the same experimental condition. Frequently, this approach leads to an evolution of the friction such that the value obtained for low loads is significantly higher than the value for high loads. This fact was on the basis of a logarithmic model sometimes used to justify the evolution of the experimental results [13–16]. For example, Fig. 1(a) displays some results obtained for MoSx thin films [13]. Applying to the same results a more suitable analysis, a very linear evolution could be observed, Fig. 1(b). Carefully analyzing the results of Fig. 1(b) the reason for the divergence of the results could be clearly identified. In fact the straight line fitted to the data points do not pass at the origin, i.e. the linear relationship displays an intercept non-nil. Achanta et al. [17] identified this problem in a recent publication. The reason for this fact could be one of two possibilities, Fig. 2: • an offset of the normal load due to internal adhesion forces, unbalanced masses on the loading system, or even unbalance of the measurement system; • an offset of the friction force due to unbalance of the measurement system. Applying the linear fitting procedure before presented, to the data points of Fig. 1(b), and using Eq. (11), assuming a reliability of 90% and a probability of 87.5% for the average value, the resultant friction coefficient is 0.047 ± 0.005. The complete analysis of Fig. 1(b) can be done by a linear model as described by Eqs. (7)–(11) allowing the relationship between friction coefficient and reliability as summarized in Table 2. The application of a linear reliability analysis to the Grosseau-Poussard et al. [13] results leads to constant value of the friction coefficient which is in the interval of [0.043, 0.051] for a reliability of 90%. This result is very different from the original values that pointed

A. Ramalho / Wear 269 (2010) 213–223

Fig. 1. (a) Results obtained for MoSx thin films by Grosseau-Poussard et al. [13]. (b) The same results applying a linearization analysis.

217

Fig. 3. Crossed-cylinder wear test equipment.

derive the value of the design-using parameter from the analysis of test results could induces significant differences. An experimental study will be used to compare different approaches. 6.1. Unidirectional sliding

Fig. 2. Intercept non-nil.

Table 2 Application of a linear reliability analysis to the results of Fig. 4(a). Average

Std

Conf. interval (R = 90%)

r2

0.047

0.002

0.043–0.051

0.997

a decrease of the friction with the rise of the normal load, from 0.13 to 0.052.

6.1.1. Experimental work A set of tests were done using a sliding tribometer with unlubricated crossed cylinder contact, Fig. 3. The equipment includes a rotating cylinder (3) and a stationary specimen (5) with cylindrical shape also. The normal load is applied by means of a spindle/spring system (4) and is measured by a load cell (1). The stationary specimen, whose diameter is 10 mm, is supported by a free rotating system, which is equilibrated by a second load cell (2) used to measure the friction force. This experimental study was planned to compare the efficiency of different methods which could be used to calculate the value of the specific wear rate of materials under sliding contacts. Table 3 summarizes the materials and the contact conditions. The normal load ranged from 1 to 5 N and for each value of the normal load the test sliding distance was varied from 285 to 1710 m in five steps, therefore 25 tests were done. Table 3 Test conditions.

6. Methods of data analysis for sliding wear tests Assuming the specific wear rate as a suitable design-using parameter to express the wear performance of the materials, an important question remains: what can be done to assure a high reliability of the experimentally achieved value of the wear rate? Besides the effect of the quality of the test equipment, the care of the test procedures and the accuracy of the measurement techniques play important role on the reliability, the method used to

Rotating cylinder

Fixed cylinder

Hardness (MPa)

57 ␣ Brass (67%Cu 33%Zn) 1710

10 Steel 34CrNiMo6 3650

Sliding speed (m/s) Normal load (N) Sliding distance (m)

0.5 1, 2, 3, 4 and 5 285, 570, 855, 1140 and 1710

Diameter (mm) Material

218

A. Ramalho / Wear 269 (2010) 213–223

Table 4 (a) Brass-cylinder wear volume (mm3 ). (b) Brass-cylinder specific wear rate (mm3 /N m). (a) Normal load (N)

Sliding distance (m)

285 570 855 1140 1710

1

2

3

4

5

0.251 0.822 0.909 1.381 2.438

0.922 1.223 2.383 3.088 4.405

0.776 2.133 3.533 4.857 8.414

1.454 2.710 4.963 5.528 9.534

1.417 2.805 4.362 7.077 11.439

8.82 × 10−4 1.44 × 10−3 1.06 × 10−3 1.21 × 10−3 1.43 × 10−3

3.23 × 10−3 2.14 × 10−3 2.79 × 10−3 2.71 × 10−3 2.57 × 10−3

2.72 × 10−4 3.74 × 10−3 4.13 × 10−3 4.26 × 10−3 4.92 × 10−3

5.10 × 10−3 4.75 × 10−3 5.80 × 10−3 4.48 × 10−3 5.03 × 10−3

(b) 285 570 855 1140 1710

4.97 × 10−4 4.92 × 10−3 5.10 × 10−4 6.21 × 10−3 6.69 × 10−3

Table 5 (a) Steel-pin wear volume (mm3 ). (b) Steel-pin specific wear rate (mm3 /N m). (a) Normal load (N)

Sliding distance (m)

285 570 855 1140 1710

1

2

3

4

5

0.002 0.016 0.009 0.012 0.038

0.021 0.034 0.031 0.043 0.055

0.034 0.049 0.053 0.063 0.096

0.032 0.055 0.074 0.073 0.108

0.034 0.057 0.062 0.084 0.132

7.02 × 10−6 2.81 × 10−5 1.05 × 10−5 1.05 × 10−5 2.22 × 10−5

7.36 × 10−5 5.96 × 10−5 3.62 × 10−5 3.77 × 10−5 3.21 × 10−5

1.19 × 10−4 8.59 × 10−5 6.20 × 10−5 5.52 × 10−5 5.61 × 10−5

1.12 × 10−4 9.64 × 10−5 8.65 × 10−5 6.40 × 10−5 6.31 × 10−5

1.19 × 10−4 1.00 × 10−4 7.24 × 10−5 7.37 × 10−5 7.72 × 10−5

(b) 285 570 855 1140 1710

Before testing, the specimens were ultrasonically cleaned with ethyl alcohol. During the test, the friction force value was acquired periodically, at time intervals of t. In each acquisition, a set of several thousand of values were collected, corresponding to an acquisition time greater than the rotation period. Therefore, the ¯ calculated from the acquired average value of the friction force, F, friction force data, corresponds to the average of the friction during a rotation. The volume of the wear scar can be calculated assuming an imposed wear shape using the approximate expression (22) derived by Ramalho [18]. This very simple equation is very accurate with errors of less than 0.2% [18] V=

√  × h2 × r1 × r2 2

(22)

where r1 is radius of the stationary specimen; r2 is radius of the rotating specimen; h is depth of the scar. Each scar is measured by taking the larger, a, and the smaller, b, dimensions of the wear surface. The value of scar depth can be calculated by Eq. (23)



h = r1 −

r12 −

 a 2 2



= r2 −

r22 −

 b 2 2

6.1.2. Data analysis method A—each test considered separately This is the method more frequently used. For each test, considered separately, the wear volume measured at the end of the test is used to calculate the value of the specific wear rate applying Eq. (17). Applying this method, the current study originated a total of 25 different values for the specific wear rate of each one the pin and the disc. Tables 4 and 5 show the results for the brass rotating cylinder and the steel fixed pin respectively. Table 6 summarizes the average of the results obtained for the 25 test conditions and also the maximum and minimum values as well as the standard deviation. Taking into account the statistic considerations before presented, Eqs. (3) and (4) can be used to achieve the values of the specific wear rate considering the error. Assuming a reliability of 95% and using only the confidence interval to characterize the values result: kbrass cylinder = (1.25 ± 0.081) × 10−3 mm3 /N m ksteel pin = (2.07 ± 0.308) × 10−5 mm3 /N m

(23)

The sliding produces a wear track on the rotating cylinder; the assessment of the removed material was done by measuring the area of the transversal profile of the track. For each wear track five measurements were done in different radial positions, and the wear volume was calculated multiplying the average value of the area by the track length.

Table 6 Values obtained using the results of each test separately. Specific wear rate (mm3 /N m)

Brass cylinder Steel pin

Average

Maximum

Minimum

Stdev

1.25 × 10−3 2.07 × 10−5

1.64 × 10−3 3.98 × 10−5

8.82 × 10−4 7.02 × 10−6

2.05 × 10−4 7.87 × 10−6

A. Ramalho / Wear 269 (2010) 213–223

219

Table 7 Specific wear rate calculated by the method B. Specific wear rate, k (mm3 /N m)

Brass cylinder Steel pin

r2

Average

STD

Conf. interval 95%

1.37 × 10−3 1.44 × 10−5

5.20 × 10−5 8.41 × 10−7

1.26 × 10−3 to 1.48 × 10−3 1.26 × 10−5 to 1.61 × 10−5

0.97 0.93

For brass and steel, an uncertainty of respectively 6.48% and 14.9% of the average was obtained. However, if all the statistical effects are considered, Eq. (5), with a confidence of 87.5% for the average and a reliability 0f 95%, the uncertainties increase to 36.5% and 84.5% of the average, respectively for brass and steel. The signal plus should be used to design any mechanical component to resist wear. The signal minus is less used in engineering applications; however, should be used for example to evaluate the wear volume produced by a finishing process. 6.1.3. Data analysis method B—analysis involving simultaneously all the available experimental data Eq. (17) represents a linear evolution of V as a function of the product Nx. Neglecting dynamic effects, the normal load times sliding distance is a deterministic variable whereas the wear volume V is a random variable. The major problem of the method A is the underlying premise that the common subject which correlates all the data points is the origin point. In fact in the method A, the n data points allow n values of specific wear rates as the slopes of the family of straight lines passing in the origin and in each one of data point. The subjacent premise is incorrect because if we assume that the specific wear rate is a characteristic of a contacting pair of materials, all the results should be verified by only one value of the specific wear rate. Therefore, the data points must be analyzed considering that the set of data points define only one straight line, whose slope is the average value of the specific wear rate, being ka . The linear fitting method, described in Section 3, can be applied to calculate the specific wear rate considering simultaneously all the experimental data points. Fig. 4 displays the linear evolution fitted to the experimental data of both brass and steel. Table 7 summarizes the results of the reliability analysis applying the concepts before presented (Eqs. (8)–(11) and (14)). The high values obtained for the linear correlation coefficient means that the linear model between the product Nx and the wear volume strongly agree the 25 data points. A similar conclusion could be obtained by the statistical test for linearity described by Eq. (15). The brass points agree the linearity criteria with reliability higher than 99.99%. Further the reliability analysis before presented the proposed methodology allows to calculate the confidence bands for the wear volume. The maximum and minimum expected values can be calculated by applying Eq. (12) to a specific domain of the independent variable. Fig. 5 shows the results obtained for a range of Nx from 200 to 8600 N m. 6.1.4. Data analysis method C—energetic approach Taking into account the definitions previously introduced and resumed in Eqs. (19)–(21), the energy dissipated by friction, Ef , can be calculated by knowing the evolution of the friction force along the test, Eq. (24) Ef =



¯ i xi Fx

Fig. 4. Assess the specific wear rate considering simultaneously all the experimental data. (a) Brass cylinder. (b) Steel pin.

corresponding to all the tests. The value of the specific wear rate function of the friction force, k , can be calculated applying a method alike the used in B, plotting the wear volume against the dissipated energy (Fig. 7). However, the application of a complete reliability analysis is now more difficult. In fact it should be considered that in this case there are two random variables, the wear volume and the dissipated energy. In fact the random character of the dissipated energy arises from the value of the friction force. Table 9 summarizes the average values and the standard deviation of the specific wears rate function of the friction for both the brass cylinder and the steel pin. However, the standard deviation values are underestimated because considers only the random character of the wear volume.

Table 8 Energy dissipate by friction, Ef (J). Sliding distance (m)

(24)

i

For all tests the friction force was acquired along the test (Fig. 6). Table 8 shows the values of the energy dissipated by friction along the test calculated applying Eq. (24) to the friction force values

285 570 855 1140 1710

Normal load (N) 1

2

3

4

5

54.0 313.1 361.7 577.0 1010.2

246.7 506.1 896.9 1100.1 2080.4

444.7 960.0 1594.2 2069.7 3276.4

673.0 1414.3 2211.4 2645.5 4342.4

788.0 1681.7 2673.7 3652.8 5825.1

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Fig. 5. Average curve and confidence bands for a reliability of 95%. (a) Brass cylinder. (b) Steel pin.

Fig. 6. Evolution of the friction along the test (normal load: 4 N, sliding distance: 1710 m).

6.1.5. Discussion The methods A and B use the same wear design-using parameter although calculated by different ways. Comparing the methods it can be concluded that the method B allows the lower uncertainty of the results. In fact, for the same parameters, the method B leads to lower scatter values respectively of 4 and 9.4 times for brass and steel. Focusing on the results obtained by method A, it can be observed that, in general, the scatter of the specific wear rate is much higher for the tests done with the lower sliding distances. Also it can be Table 9 Specific wear rate function of the friction, k . Specific wear rate, k (mm3 /J)

Brass cylinder Steel pin

Average

STD

2.02 × 10−3 2.13 × 10−5

4.28 × 10−4 4.59 × 10−6

r2

0.97 0.93

Fig. 7. Wear volume as a function of the energy dissipated by friction. (a) Brass cylinder. (b) Steel pin.

observed that the steel pins shows a decreasing of average value of the specific wear rate when the sliding distance rise from 285 to 1710 m, on the contrary the brass cylinders reveals a increasing tendency. These behaviours reveal a dependence of the wear rate with the test duration which affects the two materials by different ways. It is widely accepted that there is a typically evolution of the wear with the time that includes 3 regimes. Czichos [19] identified a first regime with the wear growing proportional to the square root of the time, followed by a steady state regime and finally a regime with an exponential grow of the wear. Thus, assuming this evolution, the wear is proportional to the time, or the sliding distance under constant speed, only during the second regime. Therefore, assuming a linear evolution from the origin could induce significantly errors in the value of the specific wear rate. The wear mechanisms identified by the observation of the wear scar by scanning electron microscope allow clarifying the different trends observed for the steel and the brass. Fig. 8 shows the wear track observed for the 4 N to 1140 m test. The wear is governed by a transfer of brass to the steel surface, zone II of the picture of Fig. 8(b). The amount of material transferred increases with the test severity, therefore increases with the rise of both normal load and sliding distance. This explain the tendency before presented, for tests with higher duration the wear of the steel decreases since the amount of brass rises and acts as a protector layer of the steel surface, and by the contrary the brass wear increases as the adhesion effect becomes more active. Besides the effect on the scatter, the method used to analyze the results also affect significantly the average value of the specific wear rate. When compared to the method A, the method B allows average results which are 9% higher for the brass and 44% lower for the steel. Moreover the lower scatter and higher reliable average values, the method B allows establishing a complete reliability analysis which is absolutely necessary to apply as design parameters. The energetic approach could be an alternative to calculate a reliable parameter to be used on the design of mechanical com-

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Fig. 8. Morphology of the steel cylinder wear scar (a) and (b) (test conditions: 4 N; 1140 m). Analysis EDS of zone I (c) and zone II (d).

ponents. However one of the difficulties to apply the energetic approach in design is the fact that this methodology requires two parameters: one to characterize the friction and the other to quantify the relationship between the wear and the energy dissipated by friction. Nevertheless an advantage of the energy approach is the fact that the evolution of the wear as a function of the energy dissipated by friction is always linear passing on the origin. In the present work the use of energetic approach leads to values of wear parameters with reasonably low scatter. 6.2. Micro-abrasion The methodology before presented can be applied to all kinds of friction and wear tests. Some early published results [20] will be revisited to demonstrate the ability of the methodology to analyze micro-abrasion results and to discriminate the transition of wear mechanisms. 6.2.1. Experimental work In the fixed ball equipments, the ball is directly connected to the driven shaft and the specimen is placed in a pivoted holder and the normal load is applied by dead weights. The abrasive slurry, continuously agitated by a magnetic stirrer, is gravity feed onto the rotating ball. A AISI 52100 steel ball bearing with 25.4 mm diameter was used with a relative tangential speed of 0.1 m/s. The tested material was a hard tool steel, AISI D2, quenched and tempered with hardness of 750 HV1. The abrasive medium was a slurry of SiC particles P2500 grade with a median particle size of 4 ␮m in distilled water, in a concentration of 20 vol%.

The normal load ranged from 0.2 to 1.5 N in five steps and for each value of the normal load five durations were considered, with the number of rotations varied from 150 to 600; therefore 25 tests were done. Table 10 summarizes the materials and the test conditions. Before and after testing, the samples were ultrasonically cleaned in acetone to remove all traces of contaminants. A toolmaker’s microscope Mitutoyo was used to assess the crater dimensions and a Philips XL30-TMP scanning electron microscope (SEM) was used to observe the morphology of the wear surfaces. The total volume removed by wear can be calculated as a function of the crater dimensions and ball diameter using suitable equation [20]. 6.2.2. Micro-abrasion results and discussion To quantify the wear behavior of materials by micro-abrasion it is usually applied Eq. (17) to achieve a characteristic value of specific wear rate. Therefore, the methodology before presented as method B should be applicable. Thus, Fig. 9 shows a graphic of the evolution of the wear volume against the product Nx. Table 10 Micro-abrasion test conditions. Tested material Ball Abrasive particles Particles content Normal load (N) Rotations Tangential speed (m/s)

AISI D2 steel; 750 HV1 ␾25.4 mm; AISI 52100 steel SiC P2500 grade median size of 4 ␮m 20 vol% in distilled water 0.2; 0.35; 0.5; 1; 1.5 150; 200; 300; 400; 600 0.1

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Fig. 9. Evolution of the wear volume as a function of the product normal load by the sliding distance.

Observing the graphic in Fig. 9 the data points can be organized in two different linear trends. In fact the normal load value of 0.5 N establishes a boundary between the two observed behaviors. Therefore the tests were grouped in two sets: one corresponding to the 10 tests carried out for normal load of 0.2 and 0.35 and the second for the other 15 tests corresponding to normal loads of 0.5, 1 and 1.5 N. The linearization method, before presented as method B, was applied to each group (Fig. 10). To understand the different behaviour observed by the two sets of test conditions, the wear scars were observed by scanning electron microscope to identify the wear mechanisms. The micrographs of Fig. 11 allow the identification of two different wear mechanisms. For the normal load of 0.2 and 0.35 N the wear mechanism was predominantly a 3-body or rolling, while a 2-body or grooving abrasion

Fig. 10. Assess the specific wear rate considering simultaneously several experimental data. (a) Normal load of 0.2 and 0.35 N. (b) Normal load of 0.5, 1 and 1.5 N.

Fig. 11. (a) and (c) Normal load of 0.2 N, 200 rotations: 3-body abrasion or rolling wear mechanism. (b) and (d) Normal load of 1 N, 200 rotations: 2-body abrasion or grooving wear mechanism.

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Table 11 Results of the linearization analyis applied to the cases of 2-body and 3-body abrasion. Specific wear rate, k (mm3 /N m)

3-Body 2-Body

r2

Average

STD

Conf. interval 95%

1.58 × 10−3 5.35 × 10−4

9.05 × 10−5 2.21 × 10−5

1.37 × 10−3 to 1.79 × 10−3 4.87 × 10−4 to 5.83 × 10−4

mechanism occurs when the normal load is equal or greater 0.5 N. This kind of transition mechanism function of the normal load has been discussed by different authors [21,22]. A complete reliability analysis can be applied to the two sets of tests as summarized in Table 11. These results allow concluding that the reliability method proposed permit the identification of the transition between different wear mechanisms and surely should be a suitable tool to establish the boundaries in wear maps. 7. Concluding remarks The present paper discussed systematically the general aspects of the experimental approach in tribology. The source of errors in experimental approaches were analyzed considering the suitability of statistical models to be applied to tribodata. The Archard’s wear equation has been analyzed and the main basis of the energetic approach of wear was revisited. Based on experimental results a complete method to calculate the wear design-using parameters were exposed and the results were analyzed comparing the new model with the traditional procedures. The proposed method is a general approach that can be applied to characterize results from different experimental techniques. The ability of the proposed method to identify the transition between different wear mechanisms was demonstrated for micro-abrasion results. The new method to characterise the test results need to be applied more widely to compare their efficiency to characterize the wear behaviour of materials. However, the obtained results show that it should be a promising possibility to obtain reliable design-using parameters based on experimental studies. References [1] P.J. Blau, K. Budinski, Development and use of ASTM standards for wear testing, Wear 225–229 (1999) 1159–1170. [2] A. Hobbacher, Fatigue Design of Welded Joints and Components, The International Institute of Welding, 1996.

0.97 0.98

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