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Continuum Mechanics Modeling of Adhesion and Friction† K. L. Johnson Cambridge University, Cambridge CB11EY, England Received October 17, 1995X Fundamental studies of friction between solid surfaces are bedevilled by their inevitable roughness. However recent developments in nanotribology through molecular dynamics, the surface force apparatus, and the atomic force microscope have made possible direct examination of adhesion and friction at a single asperity in which the real and apparent contact areas coincide. In this paper the continuum theories of straight adhesion at a spherically tipped asperity are reviewed and a “map” presented which indicates the conditions under which the different theories apply. The interaction between friction and adhesion is then considered from the point of view of “fracture mechanics”. This approach has the advantage of clearly separating the roles of continuum mechanics and physics. The former is used to calculate the “elastic energy release rate” G as the adhesion is broken; the work of adhesion Gc can only be found from physical considerations.
Introduction I interpret my role at this workshop to be that of reviewing and assessing the contribution which continuum mechanics can make to the study of friction on the nanometer scale. Although the action of lubricant films is a subject of the workshop, discussion will be restricted to “dry” contact in which the solid surfaces interact directly with each other. The fundamental study of friction on the macro scale is bedevilled by the problem of surface roughness, particularly since there are no reliable methods for measuring the real area of contact in situ or predicting it from measurements of surface topography. Accordingly the usual expedient will be followed of considering a single asperity contact in which the real and apparent contact areas coincide, modeled by a sphere in contact with a plane, a situation more likely to be realized on the micro rather than the macro scale. Experimental realization of this state of affairs has been achieved or approximated by crossed cylinders of cleaved mica or fine nylon threads, by sharp pointed probes and soft rubber spheres in contact with glass or Perspex. The association of friction with adhesion is an old one, generally attributed to Desaguliers in the early years of the 18th century, but it was Bowden and Tabor1 who made it a main plank of their “plastic junction” theory of friction between clean and dry surfaces. The fact that most surfaces do not continue to adhere when the load is removed gave rise to considerable scepticism. Although this common observation can be explained by the progressive breaking of the junctions by the elastic recovery of a multijunction contact, the relation between adhesion and friction remains a question of debate to which the modern techniques of nanotribology have much to contribute. It provides the main theme of this paper. The whole question of the extent to which continuum mechanics can model processes on the nanometer scale is one on which I hope that my presentation at the workshop will stimulate discussion. My approach is strongly influenced by the analogous problem of fracture. The processes at a crack tip are inescapably governed by intermolecular forces, microstructure, and dislocations, but much has been learned about fracture through the † Presented at the Workshop on Physical and Chemical Mechanisms in Tribology, Bar Harbor, ME, August 27 to September 15, 1996. X Abstract published in Advance ACS Abstracts, Sept. 15, 1996.
(1) Bowden, F. P.; Tabor, D. Friction & Lubrication of Solids, Oxford: Oxford, 1950.
S0743-7463(95)00889-4 CCC: $12.00
continuum theory of fracture mechanics. A deformable sphere pressed into contact with a plane to which it adheres corresponds to two solids joined by a circular ligament which is surrounded by an external interface crack. Removing the compressive load and applying a tensile force, which attempts to pull the solids apart, produces a situation which would be described in the jargon of fracture mechanics as a “mode I crack”. As in the fracture of a single solid, separation occurs by propagation of the crack, which can be in a “brittle” manner, with little damage to the surfaces, or “ductile”, with extensive plastic deformation of one or both surfaces. If the same junction is subjected to a tangential force tending to slide one solid on the other, it produces the situation of a “mode II crack”. The junction is then subjected to “mixed mode” loading. The approach to modeling adhesion and friction in this contribution will be through the concepts of fracture mechanics. Although modeling of straight adhesion along these lines is fairly well established in the literature, the present state of the art will be reviewed briefly to provide a basis for the discussion of friction. Adhesion as Mode I Fracture The effect on the contact of rigid spheres (or a sphere and a plane) of surface forces having a Lennard-Jones potential, of surface energy γ and equilibrium separation z0, was examined by Bradley.2 If the spheres are approximated in the region of significant interaction by paraboloids, the relationship between the force P exerted between the spheres (repulsion positive) and their separation h is given by the expression
P)
2πRw 3
[( ) ( ) ] z0 h
8
-4
z0 h
2
where R is the radius of the sphere and w is the “work of adhesion” ) 2γ. The surfaces pull apart when P reaches its maximum (-ve.) value Pc ) -2πRw at h ) z0. Models of elastic spheres in contact with adhesion were put forward independently by Johnson, Kendall, and Roberts3 (JKR) in 1971 and by Derjaguin, Muller, and Toporov4 (DMT) in 1975. Starting from different assumptions about the deformed shapes, DMT obtained the same pull-off force Pc as Bradley at h ) z0, whereas JKR (2) Bradley, R. S. Philos. Mag. 1932, 13, 853. (3) Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R. Soc. London, Ser. A 1971, 324, 301. (4) Derjaguin, B. V.; Muller, V. M.; Toporov, Y. P. J. Colloid & Interface Sci. 1975, 67, 378.
© 1996 American Chemical Society
Continuum Mechanics
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Boussinesq pressure has a singularity in tension at the edge of the contact which corresponds to a mode I stress intensity factor KI given by
KI )
Pa 2axπa
Equating the elastic energy release rate G to the work of fracture by Irwin’s relationship
G)
Figure 1. The pull-off force Pc for an adhesive contact of spheres, showing the transition from the rigid-body limit (Bradley) to the elastic-body limit (JKR) as a function of the parameter µ: (s) Muller, Yushenko, and Derjaguin (ref 5); (×) Greenwood (ref 8); (- - -) Derjaguin, Muller, and Toporov (refs 4 and 6); (- ‚ - ‚) Johnson, Kendall, and Roberts (ref 3). 3/ 4
obtained of this value at a separation h > z0. A public debate followed in which almost became acrimonious until it was recognized on both sides that the two theories applied to the opposite ends of the spectrum of a nondimensional parameter
( ) 2
µ)
Rw E*2z03
1/3
where the combined elastic modulus E* ) [(1 - ν12)/E1 + (1 - ν22)/E2]-1. The parameter µ is a measure of the ratio of elastic deformation to the effective range of surface forces. A small value of µ corresponds to spheres which are small and stiff; a large value corresponds to large and compliant spheres. By obtaining elastic deformations of the solids which were compatible with the magnitude of Lennard-Jones forces, Muller, Yushenko, and Derjaguin5 (MYD) showed that the pull-off force varied continuously from the DMT value for µ < 0.1 to the JKR value for µ > 5 (see Figure 1). It was discovered later that the original DMT calculations were in error except at the point of separation. When correctedsMuller, Yushenko, and Derjaguin6 and Pashley7sthe DMT calculations predicted an increase rather than a fall in pull-off force with increasing µ, as indicated in Figure 1. It would now seem more correct to regard the rigid-solid analysis of Bradley, rather than DMT, as the limit of the transition shown in Figure 1 as µ f 0. More complete numerical calculations by Greenwood8 of compliance in the transition region have quantified the irreversible “jump” phenomena, whereby surfaces snap into or out of contact. The JKR theory was formulated using the Griffiths concept of brittle fracture in which the rate of release of elastic strain energy was balanced against the increase in surface energy. We shall now indicate how the JKR results can be obtained neatly by the use of fracture mechanics concepts, as was first done by Maugis & Barquins.9 Under equilibrium conditions, with a contact area of radius a, the pressure distribution comprises the superposition of a Hertz pressure of load P1 and a Boussinesq pressure of load -Pa such that the net load is P1 - Pa, where Pa can be looked upon as the force of adhesion. The (5) Muller, V. M.; Yushenko, V. S.; Derjaguin, B. V. J. Colloid Interface Sci. 1980, 73, 294. (6) Muller, V. M.; Yushenko, V. S.; Derjaguin, B. V. J. Colloid Interface Sci. 1983, 92, 92. (7) Pashley, M. D. Colloids Surf. 1984, 12, 69. (8) Greenwood, J. A. To be submitted for publication. (9) Maugis, D.; Barquins, M. J. Phys. D, Appl. Phys. 1978, 11, 1989.
KI2 )w 2 E*
w Pa ) x6πwRP1 Normalize by writing
P ha ≡
(
2P1 Pa ) 3πwR 3πwR
)
1/2
≡ (2P h 1)1/2
and
a j≡
a (9wR /4E*)1/3 2
By Hertz, the normalized contact area
A h ) πa j 2 ) πP h 12/3 so that the normalized load
P h )P h 1 - x2P h1 ) a j 3 - x2a j3 which is the JKR relationship between contact size a and load P. It has been substantiated by many experiments. The maximum tensile stress σˆ in a Lennard-Jones potential is related to the surface energy by σˆ ) 1.03w/z0, which permits µ to be expressed in an alternative form used by Maugis,10 i.e.
(
λ ) 2σˆ
)
9R 16πwE*2
1/3
) 1.16µ
For the material at the “crack tip” to remain elastic the maximum adhesive stress σˆ should not exceed the yield stress σy. If it does, then a plastic zone forms at the crack tip with a maximum cohesive stress ) cσy, where c ≈ 3. We now redefine λ with σˆ replaced by 3σy. i.e.
Λ ) 6σy
(
)
9R 16πwE*2
1/3
()
)3
σy λ σˆ
We have seen that for the JKR analysis to be acceptable, µ should be greater than 5, i.e., Λ > 5.8 for which (σy/σ)λ > 5.8/3 ) 1.9. These are the conditions of linear elastic fracture mechanics in which the plastic zone is small compared with the size of the contact. Under these conditions we would expect the JKR analysis to still apply, but the effective work of adhesion wf to be scaled up by a factor k which is governed by dissipation in the plastic zone. Thus
KI2 ) wf ) kw 2E* Calculations of k for an elastic-plastic solid have been made by Tvergaard and Hutchinson.11 As the yield stress is reduced, the plastic zone increases in size until is covers the whole of the contact area. This is the fully plastic state in which the adhered ligament is plastically extended in a fully ductile way. For this to (10) Maugis, D. J. Colloid Interface Sci. 1992, 150, 243. (11) Tvergaard, V.; Hutchinson, J. W. J. Mech. Phys. Solids 1992, 40, 1377.
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Johnson
Figure 3. Peeling of an adhesive contact by a tangential force. The normalized contact load P h is maintained constant at 2.0. With monotonically increasing tangential force T h the contact radius a j decreases from its initial value a j 0 at a rate which depends on the parameter R.
Figure 2. Adhesion map with coordinates µ and (σy/σˆ ). For (σy/σˆ ) < 1/3 a plastic zone exists. For λ(σy/σˆ ) > 6/3 the plastic zone size , a (LEFM regime). For λ(σy/σˆ ) < 0.24 fully ductile separation occurs.
occur
3σy
1; a plastic zone develops if (σy/σˆ ) < 1/3. The map comprises five regions: I. The Bradley regime: λ < 0.01. This regime applies to small effectively rigid particles as in a colloidal suspension. II. The JKR regime: elastic, λ > 5. Elastic solids with strong adhesion; realized in experiments with crossed cylinders of mica or polymer threads and compliant rubber. III. The MYD regime: 0.01 < λ < 5. Transition between regimes I and II. IV. The LEFM regime: λ(σy/σˆ ) > 2. The JKR results can be used in this regime with an appropriately modified work of adhesion to take into account plastic or viscoelastic dissipation at the crack tip. This regime displays “adhesion hysteresis” in a load-unload cycle. Applies to rapid peeling tests with rubber. V. The ductile regime: λ (σy/σˆ ) < 0.24. Fully plastic deformation as in experiments and molecular dynamics simulations of sharp tips. Before leaving the subject of straight adhesion attention should be drawn to two aspects in which the continuum treatment might have to take into account nanoscale effects: (i) The peeling and healing processes in the elastic continuum treatment are perfectly reversible, but where these processes take place by the separation of discrete atoms, this may no longer be the case. (ii) The yield stress of a metal on the macroscale is the consequence of the presence of dislocations in the crystal lattice; on the nanoscale the yield strength could approach the ideal strength of the material.
KII )
(2 - ν) T T = 2(1 - ν) 2a xπa 2a xπa
There is no agreement as to what happens at this juncture. Savkoor & Briggs13 predicted that the surfaces would peel apart with increasing tangential force, stably at first and then unstably at a critical force Tc, until the Hertz contact radius was reached. But their experiments with rubber did not support the predictions: only a marginal reduction in contact area was observed even for T > Tc. Israelachvili’s experiments with the mica cylinder apparatus seemed to show that the JKR radius remained unchanged even during full sliding with a tensile load. In an attempt to resolve this question, an appeal is made to the current approach to mixed mode interfacial fracture mechanics. With similar materials in contact the elastic energy release rate G may be written
G)
1 1 [Pa2 + T2] [K 2 + KII2] = 3 2E* I 8πE*a
We now follow Hutchinson14 in writing
Gc ) w f(KII/KI) and choose a simple linear function
Gc ) w[1 + R(KII2/KI2)] ) w [1 + R(T2/Pa2)] where the parameter R can vary from 0 to 1.0. The relationship between the contact radius a and Pa for a specified value of load P is given by JKR above. Equating G to Gc gives the variation of a with T. Results of this calculation for the example of P ) 2.0 and varying values of R are presented in normalized form in Figure 3. The (12) An alternative treatment of this problem by the author has been submitted to the Procedure of the Royal Society, London. (13) Savkoor, A. R.; Briggs, G. A. D. Proc. R. Soc. London, Ser. A 1977, 356, 103. (14) Hutchinson, J. W. Scripa. Metall. 1989.
Continuum Mechanics
Figure 4. Distribution of tangential traction τ with increasing tangential force (after Savkoor).
Figure 5. Dry sliding experiments with crossed cylinders of mica (Israelachvili, ref 14).
case of R ) 0 (i.e., Gc ) w) is the one analyzed by Savkoor. The contact radius a decreases from its JKR value a0 in a stable manner until T h )T h c, where upon the surfaces snap apart until a ) ah. This seems an unlikely scenario on physical grounds, since R ) 0 implies that there is no resistance to relative tangential displacement of the surfaces. At the other extreme, R ) 1.0 implies that the work done against surface forces in normal relative displacements and in tangential displacements are independent and do not interact. In these circumstances the application of a tangential force T will not affect the contact size which is controlled by Pa. It goes without saying that the value of R cannot be found by continuum mechanics. It would appear that the response of surface forces to a combination of normal and tangential forces is a basic problem in nanotribology. So far we have only considered the initial response to a tangential force, “static friction”; we should also consider the sliding situation, “kinetic friction”. To this end it is useful to reflect on the essential difference between normal and tangential relative displacements of two surfaces. In normal separation the attractive force increases to a maximum σˆ and then falls rapidly to a negligible value in a distance of order 2z0 say, corresponding to a finite work of adhesion w ) 2γ. A purely tangential displacement is rather different. If no separation occurs, the resisting shear stress might be expected to build up to a maximum value τˆ in a displacement of order z0 and to continue at the same value with increasing tangential
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displacement. There is no equivalent to surface energy. In the case of a circular contact area, tangential displacement (slip) starts at the periphery and spreads inward with increasing tangential force, in a stable way, until bulk sliding takes place at a force T ) πa2τˆ . The traction distributions during this process have been found by Savkoor (see Figure 4). We now speculate about the atomic processes underlying this simple continuum model of a uniform shear traction τˆ . The mode II stress intensity at the periphery of the contact is expected to nucleate glide dislocations there in the manner discussed by Rice.15 With increasing tangential force, these spread inward across the interface as in the Bilby-Cottrell-Swindon model of an external crack. The “smeared out” shear stress at the interface is that shown in Figure 4. The stress τˆ necessary to drive the dislocations is an effective Peierls stress. Support for this model is provided by the dry sliding experiments of Israelachvili,16 using the mica crossed cylinder apparatus, in which simultaneous measurements of friction and contact area were made at various loads (see Figure 5). The variation of contact area with load followed the JKR relationship17 and the friction was directly proportional to the contact area, including a negative (adhesive) load. These results imply that the interfacial shear stress s0 is a constant ()2.5 × 107 N/m2) independent of the load and sliding velocity, as assumed in our model. Cottrell suggests a value for the Peierls stress of about 2 × 10-4 × (shear modulus), which gives a stress of 7 × 106 N/m2: the same order as the measured stress. No peeling of the surfaces was observed when the tangential force was applied, which suggests no interaction between tangential and normal interfacial displacements (i.e., R ) 1.0 in the analysis above). Conclusion Continuum mechanics modeling of straight adhesion and separation of a single asperity contact now seems to be in good shape. A map has been presented which indicates the conditions under which different behaviors elastic or plastic; brittle or ductilesmight be expected and different models be appropriate. Further work on elasticplastic deformation under adhesive conditions would be useful. But interaction between adhesion and friction under both static and kinetic conditions is still an open question. An approach to continuum mechanics modeling of this interaction through mixed mode fracture mechanics has been suggested in this paper, but much remains to be done to establish the underlying physical processes. More needs to be known about the nature of the interface and the surface force response to combined normal and tangential displacements. The way in which dislocations move through such an interface would seem to be an essential feature of the frictional process. All this calls for a collaborative effort from workers in nanotribology, material science and continuum mechanics. LA950889A (15) Rice, J. R. J. Mech. Phys. Solids 1992, 40, 239. (16) Israelachvili, J. N. Wear 1990, 136, 65. (17) The JKR theory was established for two similar homogeneous solids in contact, whereas the experimental apparatus comprises thin sheets of high modulus mica attached to glass cylinders by a comparatively thick layer of adhesive. Modification to the JKR theory to take these features into account is presently being studied in Cambridge.
