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Friction under elastic contacts J. Malzbender *, G. de With Laboratory of Solid State and Materials Chemistry, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands Received 29 July 1999; accepted in revised form 20 October 1999

Abstract The friction coefficients and shear stresses for a sliding contact at low loads are calculated on the basis of the elastic contact and adhesion beneath a sliding spherical indenter and compared to experimental data. The contribution of the adhesion appeared to be negligible. The method allows either the critical shear stress to be assessed from experimentally determined friction coefficient and normal load or the friction coefficient to be calculated as a function of normal load for a material. As model materials, different hybrid organic-inorganic coatings as well as hot-pressed alumina and float glass are used. © 2000 Elsevier Science S.A. All rights reserved. Keywords: Elastic contact; Friction coefficients; Shear stresses; Sliding contact

1. Introduction The determination of the critical shear stress as well as the prediction of the friction coefficient is of interest for bulk materials as well as coatings, but there is still a need for a simple methodology to determine these materials properties. The current work suggests a procedure that uses the results of low load scratch testing. Sol–gel coatings are widely used to modify the functional behavior of glass component or to protect the substrate from environmental influences such as particle impact and moisture [1–6 ]. The sol–gel process requires a final curing, which results in shrinkage of the coating and can produce a residual stress. The probability of relaxing the stress by cracking usually increases with thickness, and, therefore, non-cracked coatings have a critical thickness [7]. An increase in this critical thickness has been obtained by the addition of silica particles [8]. Scratch testing is widely used to access the mechanical properties of coatings qualitatively [9]. In a previous paper, the effect of plowing of the coating by a spherical asperity was quantified, and the observed maximum of the friction coefficient was related to the fracture of the coating [10]. In the present paper, an attempt is made to determine the shear stress and to predict the friction * Corresponding author. Tel.: +31-40-247-3059; fax: +31-40-2445619. E-mail address: [email protected] (J. Malzbender)

coefficient using low-load scratch testing for bulk materials and sol-gel coatings.

2. Experimental The experiments were carried out using hot-pressed alumina (grain size ~1 mm, relative density 99.9%, no additives), float glass and float glass that was coated with organic–inorganic hybrid coatings (see Table 1). The details of the preparation of the coatings have been reported elsewhere [11,12]. The coating fluid (A) contained methyltrimethoxysilane (MTMS) and polymerized colloidal silica in a ratio of 1:1 with approximately 50% of water and resulted in crack-free coatings with a maximum thickness of 4 mm [11]. Coating fluid (B) contained approximately 30% (weight) solid components, being equal weight amounts of methyltrimethoxysilane (MTMS ) and Ludox © (average particle size 20 nm), and 70% solvents (2% water, 32% methanol, 1% propanol, 35% glycol ), which allowed deposition of thicker crack free coatings up to 20 mm [8,12]. The scratch experiments were performed on a scratch tester that was designed and constructed at Philips Research Laboratories, Eindhoven. Spherical sapphire indenters with a radius of 150 and 500 mm were used in the experiments. The load was increased at a constant rate from 0 to 1000 mN, while the indenter was moved at a constant displacement rate from the relative position

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J. Malzbender, G. de With / Surface and Coatings Technology 124 (2000) 66–69 Table 1 Material parameters and shear stresses Material

R (mm)

H (GPa)

E (GPa)

n

Al O 2 3 Glass Coating (A) Coating (B) Coating (A)

150 150 150 150 500

20 7 1 1.2 1

406 70 2 10 2

0.24 0.25 0.225 0.225 0.225

[22] [11] [12] [11] [12]

[23] [11] [12] [11] [12]

[23] [11] [12] [12] [12]

k (GPa)

t (GPa)a

t (GPa)

Dc(J m−2)

3.3 1.2 0.17 0.2 0.17

0.94±0.13 0.24±0.03 0.024±0.003 0.085±0.014 0.020±0.004

1.03±0.14 0.26±0.03 0.026±0.003 0.094±0.015 0.026±0.005

0.1 0.1 0.1 0.1 0.2

a For Dc=0 J m−2.

x=0 to 1000 mm. The position, depth as well as the corresponding forces were recorded. In all experiments, the load and scratch length were increased simultaneously at rates of 10 mN/s and 10 mm/s, respectively.

can be calculated after Hamilton [15] by: 1

t=

앀3

S

(1−2n)2

×

If a sphere with a normal load P is in contact with a plane elastic body, the elastic material will deform to make contact over a circle of the radius a given by Hertz’s equation [13]: a3=

4 E

,

(1)

E

=

4

(16−4n+7n2)m2p2

+

64

A

B

1−n2 1−n2 1+ 2 . E E 1 2

(2)

Here, the suffices 1 and 2 refer to the sphere and the plane elastic body, respectively. Johnson et al. [14] used a Griffith energy approach to introduce the effect of the surface energy into the Hertz equation, which leads to an apparent load P∞ that can be defined as: P∞=P+3DcpR+앀6DcpRP+(3DcpR)2.

Dc=c +c −c , 1 2 12

1

t=

앀3

S

(1−2n)2 3

A

+

(1−2n)(2−n)mp 4

+

(16−4n+7n2)m2p2 64

B

3 16P∞E2 1/3 . × 2p 9R2

(6)

Furthermore, a solution of this equation allows a calculation of the friction coefficient as a function of the shear stress, t, and the work of adhesion Dc. The only solution that yields positive friction coefficients is: m=

8(n−2) (2n−1) p(16−4n+7n2) +

(3)

This apparent load has to be inserted into Eq. (1) to take into account the work of adhesion Dc, which is equivalent to:

(5)

where n is the Poisson ratio of the elastic material. This equation can be combined with Eq. (1) to:

×

where R is the radius of the spherical asperity and E is the effective modulus, which is given by [13]: 1

(1−2n)(2−n)mp

3P∞ × , 2pa2

3. Theory

3 PR

3

+

+

S SAA B A B 8

(n−2)2(2n−1)2

p

(16−4n+7n2)2

3 1/3 4

R2

P∞E2

2/3

−

(2n−1)2 3(16−4n+7n2) p2t2

(16−4n+7n2)

B

.

(7)

(4)

where c and c are the surface energies of the two 1 2 surfaces, and c is the energy of the interface. 12 The maximum shear stress, t, at the surface of an elastic material under a sliding rigid spherical indenter

4. Results and discussion Scratches were made on bulk materials and coatings of different thickness of the compositions given in the experimental section. Typical graph showing the friction

68

J. Malzbender, G. de With / Surface and Coatings Technology 124 (2000) 66–69

coefficient as a function of the applied normal load are shown in Figs. 1 and 2. The variation of the friction coefficient with the normal load was independent of the coating thickness for coating (A) and (B) in the analyzed thickness range 0.5–4 and 2–10 mm, respectively. Assuming the existence of a critical shear stress t at the surface of a sliding contact, Eq. (6) allows a calculation of t as a function of load, friction coefficient and work of adhesion. The shear stress and work of adhesion were calculated for each load using the associated friction coefficient as determined experimentally. A typical graph is shown in Fig. 3. The average shear stresses and the sample standard deviation as well as the work of adhesion are given in Table 1. Furthermore, the shear stress assuming a work of adhesion of zero is also given. It is possible to fit the friction coefficients as a function of load using Eq. (7) and thus to make an estimate of the friction coefficient for any normal load given an experimental friction coefficient for a single normal load. A typical fit is shown along with the experimental data in Figs. 1 and 2 for a Dc of 0 and 0.1 J m−2, respectively.

Fig. 1. Friction coefficient as a function of normal load for a coating of composition (A) for a Dc of 0 J m−2.

Fig. 2. Friction coefficient as a function of normal load for a coating of composition (A) for a Dc of 0.1 J m−2.

Fig. 3. Shear stress as a function of load for coating of composition (A) for a Dc of 0.1 J m−2.

The use of the larger diameter sphere for coating (A) leads to no change in the shear stress, t. We conclude that this shear stress at the surface is constant for the load range considered corroborating the initial assumption. As can be seen in Table 1 the work of adhesion is independent of the material used and only slightly larger for the larger sphere. This observation and the fact that the work of adhesion is close to the surface energy of liquid water (0.119 J m−2 [16 ]) suggest that the work of adhesion determined in these experiments is influenced by the adsorption of water on the surface. The effect of the work of adhesion on the shear stress and the critical load can be assessed using Eq. (3). Eq. (3) can be simplified for the considered R, P and Dc range as: P∞=P+앀6DcpRP. The effect of Dc can be determined using the ratio of P∞/P. In the load range considered, a Dc of 0.1 J m−2 will result in an average increase of this ratio by approximately 10%. A Dc of 1 J m−2 will result in an average increase of this ratio by approximately 35%. However, it is not possible to obtain a reasonable fit to the data using a Dc value larger than about 0.1 J m−2. The material will yield when the shear stress at the surface, t, is equivalent to the shear yield strength, k ( Table 1), which is estimated as one-sixth of the hardness of the material [17]. It can be seen from Table 1 that the shear stress. t, as calculated from Eq. (4), is a factor of 2–7 smaller than the theoretical yield stress, k, for all materials and for the different spheres, which again indicates the importance of the effect of the adsorption of water. The sliding contact is largely elastic and avoids plowing. If plowing occurs, the above description is not valid. One objection could be that Hamilton’s analysis is strictly only applicable to bulk material. Generally, it is considered that for small indentation depths (i.e. 10% of the coating thickness, the elastic modulus will be dominated by the coatings properties [18]. The load range used in these experiments corresponds to an indentation depth of approximately 0.05 mm, which is

J. Malzbender, G. de With / Surface and Coatings Technology 124 (2000) 66–69

much smaller than the maximum analyzed coating thickness of 4 and 10 mm for coating (A) and (B), respectively. Thus, the coating can be considered as a bulk material. Furthermore, King et al. [19] have shown that the pressure under a sliding contact will only be slightly altered due to the mismatch in moduli for the considered indentation depths. This is in agreement with investigations by Olveira and Bower [20] as well as Ogilvy [21] on the modification of the contact radius by the mismatch in elastic moduli. Finally, the dependence of the friction coefficient/shear stress on the load was not affected by a variation in coating thickness, indicating a negligible influence of the substrate.

5. Conclusion A method is shown to evaluate the critical shear stress and the friction coefficient of a material beneath a sliding spherical contact. The method allows either the shear stress at the surface to be assessed from experimental data or the friction coefficient to be estimated as a function of normal load given an experimental friction coefficient for a single normal load. The friction coefficients estimated in this way are in good agreement with the available experimental data.

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A. Paul, Chemistry of Glasses, Chapman & Hall, New York, 1990. H. Bach, D. Krause, Thin Films on Glass, Springer, Berlin, 1997. G.L. Smay, Glass Technol. 26 (1985) 46. F.H. Wang, X.M. Chen, B. Ellis, R.J. Hand, A.B. Seddon, J. Mater. Sci. Technol. 13 (1997) 163. [5] T.H. Wang, P.F. James, J. Mater. Sci. 26 (1991) 354. [6 ] H. Schmidt, J. Sol-Gel Sci. Technol. 1 (1994) 217. [7] A. Atkinson, R.M. Guppy, J. Mater. Sci. 26 (1991) 3869. [8] R.H. Brzesowsky, G.de With, S. van de Cruijsem, I.J.M. SnijkersHendrickx, W.A.M. Wolter, J.G. van Lierop, J. Non-Cryst. Sol. 241 (1998) 27. [9] S.J. Bull, Surf. Coat. Technol. 50 (1991) 25. [10] J. Malzbender, G. de With, Wear 236 (1999) 355. [11] J. Malzbender, J.M.J. den Toonder, G. de With, Thin Solid Films, accepted for publication. [12] J. Malzbender, G. de With, J. Non-Crystalline Sol. submitted for publication. [13] H. Hertz, Miscellaneous Papers, MacMillan, London, 1896. [14] K.L. Johnson, K. Kendall, A.D. Roberts, Proc. R. Soc. Lond. A 324 (1971) 301. [15] G.M. Hamilton, Proc. Inst. Mech. Eng. 197C (1983) 53. [16 ] R.K. Iler, The Chemistry of Silica, Wiley-Interscience, Chichester, UK, 1979. [17] K.L. Johnson, Contact Mechanics, Cambridge University Press, Cambridge, 1985. [18] T.Y. Tsui, C.A. Ross, G.M. Pharr, Mater. Res. Soc. Symp. Proc. 473 (1997) 57. [19] R.B. King, T.C. O’Sullivan, Int. J. Solids Struc. 23 (1987) 581. [20] S.A.G. Oliveira, A.F. Bower, Wear 198 (1996) 15. [21] J.A. Ogilvy, J. Phys. D 26 (1993) 2123. [22] E. Doerre, H. Huebner, Alumina, Springer, Berlin, 1984. [23] G. de With, J. Mater. Sci. 19 (1984) 2195.

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