Measurements of the Static Friction Coefficient

Rebecca D. Ibrahim Dickey Robert L. Jackson George T. Flowers Department of Mechanical Engineering, Auburn University, Auburn, AL 36830

Measurements of the Static Friction Coefficient Between Tin Surfaces and Comparison to a Theoretical Model A new experimental apparatus is used to measure the static friction between tin surfaces under various loads. After the data is collected it is then compared to an existing theoretical model. The experiment uses the classical physics technique of increasing the incline of a plane and block until the block slides. The angle at the initiation of sliding is used to find the static friction coefficient. The experiment utilizes an automated apparatus to minimize human error. The finite element based statistical rough surface contact model for static friction under full stick by Li, Etsion, and Talke (2010, “Contact Area and Static Friction of Rough Surfaces with High Plasticity Index,” ASME Journal of Tribology, 132(3), p. 031401) is used to make predictions of the friction coefficient using surface profile data from the experiment. Comparison of the computational and experimental methods shows similar qualitative trends, and even some quantitative agreement. After adjusting the results for the possible effect of the native tin oxide film, the theoretical and experimental results can be brought into reasonable qualitative and quantitative agreement. [DOI: 10.1115/1.4004338] Keywords: static friction coefficient, elastic plastic contact, surface roughness, statistical, experimental method

Introduction In the 15th and 16th centuries, Leonardo Da Vinci may have been the first to scientifically investigate friction. Da Vinci is often credited with discovering that the friction force is independent of the (apparent) area of contact and that it is proportional to load. However, these “laws of friction” are often also linked to Guillaume Amontons, who made similar findings over 100 years later. Later, Charles-Augustin de Coulomb also attributed friction to rough surfaces dragging and interlocking against each other and also found that the friction force appeared to be mostly independent of velocity. Therefore, the laws of friction provided by Da Vinci, Amonton, and Coulomb are 1. The force of friction is directly proportional to the applied load (amontons’ 1st law). 2. The force of friction is independent of the apparent area of contact (amontons’ 2nd law). 3. Kinetic friction is independent of the sliding velocity (coulomb’s law). The equation which summarizes these laws of friction is then often credited to Leonhard Euler and given as F ¼ lN

(1)

Euler is also the first person to postulate that friction behaves different for two surfaces initially at rest (static friction) then two surfaces in relative sliding motion (dynamic friction). Euler also examined the now classic block on an inclined plane problem; and found that one could measure the static friction coefficient by tilting the incline until the block begins to slide (see Fig. 1). At Contributed by the Tribology Division of ASME for publication in the JOURNAL TRIBOLOGY. Manuscript received August 24, 2010; final manuscript received June 2, 2011; published online July 28, 2011. Assoc. Editor: Andreas A. Polycarpou.

OF

Journal of Tribology

this angle h Euler provides the following equation to predict the static friction coefficient: l ¼ tan h

(2)

Bowden and Tabor [1] later made a critical insight into the cause of friction and the physical reason behind the laws. Bowden and Tabor were perhaps the first to theorize about the real area of contact between surfaces. “Even the smoothest surfaces are rough on the atomic scale, and placing them together is rather like turning Switzerland upside down and putting it on top of Austria.” – Bowden and Tabor [1]. Therefore when two rough surfaces are pressed together only isolated asperities on the surface are in contact. It was then concluded that adhesion could occur between contacting asperities so that for tangential motion to occur, the shear strength of the contact (ss ) must be overcome by the shear traction. This concept is used by later researchers to create a model meant for predicting static friction. In 1988 Chang, Etsion, and Bogy [2] used the principles Tabor and Bowden [1] outlined to develop an elastic-plastic statistical theory of static friction and what became known as the CEB model. The CEB model used a statistical representation of surface roughness to calculate the static friction force, while accounting for normal preloading. Chang, Etsion, and Bogy [2,3] also created one of the first elastic-plastic asperity contact models and included it in a statistical model of rough surface contact (Greenwood and Williamson [4]). The model by Chang, Etsion, and Bogy [2] also predicted the static friction coefficient based on the real area of contact and the shear strength at the contacts. In their model the inception of slip occurred at the initiation of any yielding of the material in the contact (considered shear and normal load). The expression to quantify the elements outlined by Bowden and Tabor is given in Chang, Etsion, and Bogy [2] by

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[19] include the effect of junction growth during sliding (the contact area increases). However, the theoretical equation for static friction in Cohen et al. [19] is not valid for plasticity indices above 8. Li, Etsion, and Talke [20] extended the model to plasticity indices past 8 by incorporating the FEM results of spherical contact farther into the elastic-plastic and fully plastic regimes by employing the FEM results of Jackson and Green [21]. Very recently, Lee et al. [9] also examined the effect of non-Gaussian or asymmetric asperity height distributions on the statistical static friction models and compared the predictions to experimental measurements. Using a more representative distribution appeared to improve the predictions greatly. Although the current work presents measurements of static friction and a comparison to a fundamentally based theoretical model, the motivation behind the work is to characterize the static friction between the contacts of a tin plated electrical connector because it will influence the fretting failure mechanism. The apparatus was utilized to test how normal load affects the static friction coefficient at room temperature. The experimental data obtained from that experiment was then compared to the theoretical finite element based model derived by Li et al. [20].

Fig. 1 Schematic of the classic block on an incline plane problem

l¼

Qmax Qmax ¼ F P Fs

(3)

where l is the static friction coefficient, F is the external force, P is the real contact load, Fs is the adhesion force in the normal direction (which is also known as the intermolecular force of attraction), and Qmax is the tangential force needed to shear the surfaces apart. They found that the static friction coefficient decreases as both the plasticity index and the dimensionless external force increase. Experimentally, Gassenfeit and Soom [5] measured the static friction coefficient of planar contacts with aim to apply the findings to the start-up conditions of machines and engines. Static friction is also important for other applications such as in tires [6,7], valves [8], constant velocity (CV) joints [9], mechanical seals, electrical contacts, metal working [10], magnetic read-write heads in computer hard-drives [11], and in micro-electrical mechanical systems (MEMS) devices [12] where the importance of friction is amplified because as the scale decreases the surface forces become dominant over the body forces. In the work by Polycarpou and Etsion [11] the effect of a thin boundary layer of fluid was also included in the model of static friction. Kogut and Etsion [13,14] improved this model by incorporating the results of a finite element model (FEM). In their model the inception of sliding occurred once contact was surrounded by a volume of yielded material. Jang and Komvopoulos [15] also extended this concept using a fractal-based contact model rather than a statistical one. The effect of skewness and kurtosis on the original Chang et al. statistical static friction model were investigated by Tayebi and Polycarpou [16]. In 2007 Lee and Polycarpou [17] sought to verify the statistical based static friction models by using a precision experimental apparatus. They made comparison to the measurements using the static friction model derived by Kogut and Etsion [14], which also includes the effects of adhesion. For some cases the results compared well, but for other cases they deviated significantly. This appeared to be due to time dependent creep and contamination effects. To improve the statistical models of static friction, Brizmer, Kligerman, and Etsion [18] and Cohen, Kligerman, and Etsion 031408-2 / Vol. 133, JULY 2011

Theoretical Methodology. An experiment to be discussed in a later section was performed to test the effect of pressure on the static friction coefficient. The results are then compared to an existing model of static friction given by Li, Etsion, and Talke [20]. A theoretical prediction of the static friction coefficient can be derived by using Eq. (2), if a reasonable relationship between Qmax and F is known. Chang et al. [2] initially used an analytical model of a single elastic-plastic asperity contact for this purpose, but Kogut and Etsion [13,14] later used the finite element method (FEM) to find more accurate empirical equations. In 2008 Cohen, Kligerman, and Etsion [19] published an equation that was fit to improved theoretical results by incorporating a full stick condition and junction growth. However, for high plasticity indices the equation predicts an increasing friction coefficient with load and is no longer valid. In 2010 Li, Etsion, and Talke modified Cohen et al. [19] by using the results of Jackson and Green [21] to extend the model for higher plasticity indices. As noted above, the statistical rough surface contact models used to predict the static friction require a single asperity model (often modeled as an elastic-plastic sphere in contact with a flat). Jackson and Green [21] conducted a FEM elasto-plastic contact model for the contact between a deformable sphere and a flat rigid surface. Their work finds that both the hardness and the fully plastic average contact pressure actually vary with the deforming geometry of the sphere. Chaudri et al. [22] also found this experimentally. When the contact pressure is plotted against the contact radius, a limit appears to emerge for the average pressure during fully plastic contact. According to Jackson and Green, as a/R increases, the limiting average pressure to yield strength ratio must change from Tabor’s [23] predicted value of approximately 2.8 to a theoretical value of 1 when a ¼ R (nearly equivalent to the case of a compressed cylinder). By fitting a function to their FEM results, Jackson and Green [21] provide the following formula: a 0:7 P ¼ 2:84 1 exp 0:82 Sy R

(4)

Note that in Eq. (4) P is used instead of H (see Ref. [10]) as the symbol for the average pressure during fully plastic contact. This is used here to emphasize that the P predicted by Eq. (4) varies with the deformation of the sphere, whereas the traditional value of H does not. This deformation dependent hardness is important when modeling heavily deforming contacts, which will be the case in the current work due to the high prediction of the plasticity index [14]. The FEM based model also provides predictions of the contact force during elastic-plastic contact as Transactions of the ASME


F ¼ Fc

(

"

5 #) 32 1 x 12 x exp 4 xc xc ( " 5 #) 4P 1 x 9 x þ 1 exp CSy 25 xc xc

Li, Etsion, and Talke [20] then fit empirical equations to the results and gave them as (5)

Jackson and Green [21] also formulated on predictions of the contact radius during elastic and elastic- plastic contact. Jackson and Green predicted the ratio of contact radius to the original radius as a ¼ R where

rffiffiffiffi B=2 x x R 1:9xc

(6)

Sy B ¼ 0:14 exp 23 0 E

(7)

1 ð1 v21 Þ ð1 v22 Þ ¼ þ 0 E E1 E2

(8)

C ¼ 1:295 expð0:736vÞ

(9)

Quicksall et al. [24] also validated these equations for a wider range of material properties. Due to the limited range of FEM cases used to find Eqs. (4)–(9), these models are only valid for normalized contact radii of 0 < a/R 0.412. Although not used by Li, Etsion, and Talke [20] or in the current work, it should be noted that Wadwalkar et al. [25] recently extended the predictions of Jackson and Green for larger values of a/R and farther into the elastic-plastic regime. To find F and Qmax, Li, Etsion, and Talke [20] used the same underlying statistical rough surface contact model derived by Chang, Etsion, and Bogy [2] and based on the original elastic Greenwood and Williamson microcontact model [4] (which included several assumptions). First, it is assumed that the surface peaks are distributed in height by a Gaussian distribution curve: " 2 # 1 z (10) /ðzÞ ¼ pffiffiffiffiffiffi exp 0:5 rs 2prs

0:4 l ¼ 0:26 þ 0:32 exp 0:34w1:19 ðF Þ expð1:9w Þ rffiffiffiffiffi 2E0 rs w¼ pCSy ð1 v2 Þ R F ¼

F An Sy

(14) (15) (16)

where w is the plasticity index, F* is the dimensionless normal load, l is the static friction coefficient, E0 is the equivalent modulus of elasticity (Eq. (8)), Sy is the yield strength, C is a constant dependent on Poisson’s Ratio (t) (Eq. (9)), rs is the standard deviation of asperity heights, An is the nominal or apparent area of contact, F is the load, and R is the average asperity radius. Further details of the static friction model can be found in Li, Etsion, and Talke [20]. This model will be used in the current work to be compared to the experimental measurements and to explain the observed trends. The static friction model given by Eqs. (10)–(16) requires that several surface profile parameters be calculated from surface data. To obtain the required values the raw surface heights were gathered using a Veeco Dektak 150 stylus profilometer as shown in Fig. 2. Raw surface heights were taken of 20 of the 126 samples of tin plated copper that are used in the static friction experiment. Figure 3 is an example of one of the raw surface profiles. It can be seen that the raw surface heights are still curved at the macroscale. During testing this curvature is flattened out because the samples are thin and adhered to a much more thick and rigid flat surface. To remove the curvature, a second order polynomial was fit to the surface data and then removed. Figure 4 shows the same sample after the second order polynomial is removed as discussed. The surface data for 20 samples were measured and the results averaged for use in the theoretical model. The method used to calculate the necessary statistical surface quantities are explained next. The surface height distributions are not perfectly Gaussian (skewness 1, kurtosis 4), but they are not bimodal. Note that no surface is ever perfectly Gaussian. Since there is not a closed form model of static friction for a non-Gaussian asymmetric surface, the model by Li et al. [20] is still used even though it assumes that the surface height distributions are Gaussian.

Next, the summits of the asperities are spherically capped and the asperity radius of curvature is the same for all asperities, no matter their height. While loading, the contact load P, the adhesion force F, and the static force Qmax of each individual asperity depend only on its own interference, as if there is no interaction taking place between the asperities. The interference x is defined as (where z is the height of an asperity measured from the mean of asperity height and d is the distance between the mean of asperity heights and the smooth rigid surface): x¼zd

(11)

Each term in Eq. (11) is often nondimensionalized by the root mean square (RMS) (standard deviation) of the surface heights rs and then written as x ¼ z d . The method used to calculate rs from the surface data will be discussed later. Using two of the same governing equations in the Greenwood and Williamson model [4], the load (F) and maximum friction force (Qmax)can be expressed as ð1 (12) Fðx Þ/ ðz Þdz F ¼ gAn d

Qmax ¼ gAn

ð1 d


Qmax ðx Þ/ ðz Þdz

(13)

Fig. 2 A picture of the profilometer used to obtain the raw surface data

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Table 1 Parameter m0 m2 m4 R Sy rs C E0 v w

Fig. 3 Sample of one set of surface data and the parabolic fit used to remove curvature

However, the very recent model by Lee et al. [9] could provide a better prediction by including this effect, but is not a closed form model and requires much more effort to implement. It is still a common assumption in the field to assume that a surface possesses a Gaussian height distribution when using most models that are descendents of the popular Greenwood and Williamson model. From the leveled surface data above (Fig. 4), the spectral moments outlined by McCool [26] can be found. The spectral moments m0, m2, and m4 are the variance of heights, mean square slope, and the mean square curvature, respectively, and are given by McCool as m0 ¼

N 1X

N

½zðxÞ2n

(17)

n¼1

N 2 1X dz m2 ¼ N n¼1 dx n N 2 2 1X d z m4 ¼ N n¼1 dx2 n

(18)

(19)

The spectral moments m2 and m4 are calculated using the center difference methods for each of the surfaces. The isotropic nature of the surfaces pffiffiffiffiffiare pffiffiffiffiffiffi the ratios of the RMS ffi checked by calculating roughness ( m0 ) and RMS slope ( m2 ) in several perpendicular directions for several surfaces. The ratios are fairly close to one with the average value of the RMS roughness ratio being 0.98 and the average of the RMS slope ratio being 1.08. Therefore it does

Fig. 4 A surface after the parabolic fit is removed from the raw surface data

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Numerical values of parameters for Eqs. (17)–(22) Units

Value 1.0721015 1.154105 1.045107 2.056104 9.000106 3.257108 1.649 2.4851010 0.33 15.06

2

m n/a m2 m N/m2 m n/a Pa n/a n/a

seem to be a reasonable assumption that the surfaces are isotropic. The top and bottom surfaces are taken from the same large sample of material and are therefore considered identical in the model. The average values of the required surface parameters are calculated from the 20 measured surface profiles. The methodology given in Ref. [26] is then followed to calculate the effective values of m0, m2, and m4 for two identical surfaces in contact. The effective values of the spectral moments will be represented by m0, m2, and m4 from here on. From the above effective spectral moments the following equations for the asperity radius of curvature, and the asperity height standard deviation can be predicted again from McCool [26] as 0:5 p R ¼ 0:375 m4 0:8968 0:5 0:5 m0 rs ¼ 1 a

(20) (21)

where a is the bandwidth parameter: a¼

m0 m4 m22

(22)

The statistical parameters obtained from Eqs. (17)–(22) and the material properties for tin were used to calculate the plasticity index (w) from Eq. (15). Table 1 summarizes all the values used in the static friction model by Li, Etsion, and Talke [20] including the spectral moments, the asperity radius of curvature, the summit height standard deviation, the surface standard deviation, and the resulting plasticity index.

Experimental Methodology An automated tests apparatus was designed and fabricated to measure the static friction between two nominally flat surfaces using Euler’s inclined plane theory (Eq. (2)). A linear actuated stepper motor was used to drive a shaft to slowly lift the plane while steadily increasing its inclination angle (h). Figures 5, 6, and 7 show the experimental setup and the close up of the inclined plane. A proximity sensor is placed to initially be close to and excited by the weight set fastened to one of the test surfaces resting on the inclined plane (see Fig. 7). As the inclination increases, eventually the tangential force from gravity will overcome the static friction and cause the mass to slide away from the proximity sensor. Once the sensor senses motion of the weight set, it sends a signal to the electrical driver to stop the motor. A digital angle meter is then used to determine the tilt (h) of the inclined plane (see Fig. 1). Using the angle meter is more accurate than measuring the projected height and length of the incline and using trigonometry to find h. Here Eq. (2) is used to relate the static friction coefficient to the measured value of the incline angle when sliding begins to occur. Transactions of the ASME


Fig. 5

A labeled picture of the entire experimental setup

In the current work the desired static friction coefficient was for that of tin sliding against tin in “normal” atmospheric conditions so that the static coefficient can be predicted for electrical contact insertion and pullout forces. Therefore the same tin plated copper used fabricate the electrical connectors in a stamping and forming process is used here; however many different types of surfaces and materials could be used. The tin plated copper metal samples were used and cut into precise square shapes with a surface area of 2.89 cm2. Figure 8 shows how the sample is adhered to the masses, for both experiments. It can also see in Fig. 7 how the opposing tin sample was clamped to the inclined plane. The experiment procedure used to measure the static friction coefficient is described in the following sequence. A total of 126 tin plated copper samples were fabricated and fixed on the apparatus and test mass (63 for each). Before each test the surfaces were cleaned using acetone. The test mass was varied from 50 to 250 g in 10 g increments; each load resulting in a new test with a new sample. Each load case was repeated using three different samples to capture the uncertainty of the measurements. Next the results of the experiment will be discussed in more detail and compared to the theoretical predictions produced by the previously discussed model.

Fig. 6 A labeled picture of the inclined plane of the experimental setup


Fig. 7

A picture of the sample clipped on inclined plane

Results and Discussion Using the average surface parameters from the tin plated surfaces and the material properties of tin, the predicted plasticity index is 15 (see Table 1). This is fairly high because tin is a relatively soft material and its strength to stiffness ratio (Y/E0 ) is very small. As can be seen in a previous source [5], the adhesion parameter (h) is negligible if the plasiticity index (W) is greater than two, and therefore does not need to be included in the static friction predictions. The predictions of the model derived by Li, Etsion and Talke [20] (Eq. (14)) and the measured static friction coefficients for the tin on tin contacts are shown in Fig. 9. The error bars are also shown using the standard deviation of the data (note that three data points are included for each point). Figure 9 shows that both theory and experiments predict that as the normal load increases, the static friction coefficient decreases. This is very significant because the experimental data confirms the often disputed trend of

Fig. 8 mass

A picture of one of the samples used in adhered to the

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Fig. 9 Comparison between the theoretical model and experimental measurements

Fig. 10 Comparison between a modified analytical model and the experimental measurements

this theory that the static friction coefficient will decrease with increasing load. From the static friction theory originally predicted by Chang, Etsion, and Bogy [2] the friction is derived from how much shear stress the asperities can resist before they plastically deform and shear. Therefore, as the normal load increases, the surface becomes more plastically deformed and cannot resist as much tangential load before shearing as it did when it was lightly loaded. Although the qualitative trend is similar, the quantitative results are quite different. For this reason the plasticity index was artificially varied from the predicted value of 15.06 for the tin contact down to a value of 5 (see Fig. 9). As the plasticity index increases, the static friction coefficient decreases, and the qualitative trend of the curve appears to come into better agreement with the experimental data even though the quantitative agreement degrades. Note that this causes the friction coefficient to change by a value of 0.1. The noted small change in the friction coefficient is also seen experimentally since for different metals under similar situations (i.e., air, temperatures) the friction coefficient does not vary much (Rabinowicz [27] suggests a range of 0.18 to 0.38 for poorly lubricated metal contacts). Also, the statistical rough surface contact models do tend to average the effects of different height asperities and possible material variations which may cause the friction coefficient to reach more extreme values in some cases. The differences could also be due to the surface height distribution not actually being Gaussian as is assumed by the employed model [9,20]. This suggests that the reason for the differences between the model and the experiment may be twofold, as will be discussed. First, all tin surfaces have a naturally occurring oxide film that forms on the surface. It is very difficult to predict and control, but should effectively increase the hardness of the asperities and therefore decrease the plasticity index. However, the presence of the film will also have a second effect. The oxide will decrease the adhesion that would occur if clean tin surfaces were in contact. Since the Li, Etsion, and Talke model [20] effectively assumes perfect bonding between the surfaces, the oxide layer should cause the measured static friction coefficient to be much lower than the model. Therefore, a possible simplistic method to account for the oxide layer would be to decrease the plasticity index and lower the theoretical static friction coefficient by a constant value, or in equation form:

loxide ¼ ls ðwoxide Þ lfit

031408-6 / Vol. 133, JULY 2011

(23)

where loxide is the friction coefficient including the effect of the tin oxide, woxide is the plasticity index using the material properties of the tine oxide and the roughness of the surface, and lfit is the adjustment made for the tin oxide effectively reducing the overall value of the friction coefficient. The adjustments suggested by Eq. (23) implemented and compared to the experimental results in the following paragraph. To find the plasticity index of tin oxide, and more specifically tin dioxide, also known as the naturally occurring mineral Cassiterite, the elastic modulus and yield strength are needed. A typical elastic modulus appears to be 213.3 GPa [28] and the tensile strength is approximately 80 and 400 MPa in compression. Therefore, if the previously calculated geometric properties (Table 1) are used with the material properties of tin oxide, the plasticity index for the tin oxide layer (woxide ) would vary between approximately 2 and 8. Therefore a plasticity index (woxide ) of 2 will be assumed here. Then if the friction prediction for a plasticity index of 2 is shifted down by lfit ¼ 0:39, by using Eq. (23), then the experimental and theoretical results appear to be in reasonable qualitative and quantitative agreement as shown in Fig. 10. Please note that the fitting parameter is important and this may mean that the employed model may not be capturing the physics and mechanics entirely.

Conclusions The static friction between two tin plated surfaces in normal laboratory conditions was measured over a range of loads. The static friction coefficient showed a repeated trend of decreasing as the load was increasing. This is in qualitative agreement with the static friction theory first proposed by Chang, Etsion, and Bogy, with the most recent refinement being by Li, Etsion, and Talke. Therefore the most recent model was also directly compared to the experimental results. However, since the models are not in good qualitative agreement, the possible effect of the native tin oxide on the surface is also investigated. This may also be due to a difference in the measured surface height distribution from the assumed Gaussion distribution. By using the material properties of tin and shifting the entire curve, the model and experiments appear to now be in reasonable agreement. Transactions of the ASME


Nomenclature *¼ a¼ An ¼ d¼ E’ ¼ E1,E2 ¼ F¼ F ¼ g¼ m¼ m0 ¼ m2 ¼ m4 ¼ N¼ P¼ Qmax ¼ R¼ Sy ¼ z¼ a¼ l¼ rs ¼ h¼ ss ¼ t¼ w¼ x¼ xc ¼

nondimensionalized contact radius nominal or apparent area of contact distance between the mean asperity heights and the smooth rigid surface Hertz effective modulus of elasticity Young’s(elastic)) modulus of the top and bottom sample, respectively rough surface contact force single asperity contact force gravitational acceleration constant combined mass of weights and sample variance of heights mean square slope mean square curvature normal force average contact pressure during fully plastic deformation tangential force needed to shear the surface average asperity radius yield strength heights of summits bandwidth static friction coefficient standard deviation of the asperity heights angle between horizon and inclined plane shear strength of the contact Poisson’s ratio plasticity index interference critical interference

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