Stiffness of Contacts between Rough Surfaces

Abstract

The effect of self-affine roughness on solid contact is examined with molecular dynamics and continuum calculations. The contact area and normal and lateral stiffnesses rise linearly with the applied load, and the load rises exponentially with decreasing separation between surfaces. Results for a wide range of roughnesses, system sizes and Poisson ratios can be collapsed using Persson's contact theory for continuous elastic media. The atomic scale response at the interface between solids has little affect on the area or normal stiffness, but can greatly reduce the lateral stiffness. The scaling of this effect with system size and roughness is discussed. PACS numbers: 46.55.+d, 62.20.Qp, 81.40.Pq The presence of roughness on a wide range of length scales has profound effects on contact and friction between experimental surfaces. Under a broad range of conditions [1][2][3][4][5][6], the area of intimate contact between rough surfaces A c is orders of magnitude smaller than the apparent surface area A 0 . As discussed below, this provides the most common explanation for Amontons' laws that friction is proportional to load and independent of A 0 . Because A c is small, the interfacial region is very compliant. In a range of applications the interfacial compliance can significantly reduce the stiffness of macroscopic joints formed by holding two components together under pressure [1,7].In this paper, we examine the effect of surface roughness on the normal and lateral stiffness of contacts between elastic solids using molecular dynamics (MD) and continuum calculations. The results provide a numerical test of recent continuum theories [8,9] and their applicability to real solids. The contact area and normal stiffness approach continuum predictions rapidly as system size increases. Continuum theory also captures the internal deformations in the solid under tangential load, but the total lateral stiffness may be greatly reduced by atomic scale displacements between contacting atoms on the opposing surfaces. This makes it a sensitive probe of the forces underlying friction and may help to explain unexpectedly small experimental results [10].The topography of many surfaces can be described as a self-affine fractal [2,11]. Over a wide range of lengths, the root mean squared (rms) change in height dh over a lateral distance ℓ scales as a power law: dh ∼ ℓ H , where the roughness or Hurst exponent H is typically between 0.5 and 0.9. Greenwood and Williamson (GW) considered the peaks of rough landscapes as independent asperities and found that A c rose linearly with normal load F N for nonadhesive surfaces [2]. This explains Amontons's laws if there is a constant shear stress at the interface. A linear scaling of area with load is also obtained from Persson's scaling theory, which includes elastic coupling between contacts approximately [12,13].Dimensional analysis implies that the linear relation between load and area must have the form( 1) where a modulus like the contact modulus E ′ is the only dimensional quantity char...

Contact between rough surfaces and a criterion for macroscopic adhesion

“…The normal contact stiffness defined as k N = dN=du is typically found to rise linearly with external load for nonadhesive surfaces (35,36). In the regime where surfaces are not sticky, we find that the relation between surface separation and N rep is nearly unchanged, just as the relation between N rep and A rep is nearly the same (Fig.…”

Contact area of rough spheres: Large scale simulations and simple scaling laws

“…The prevalent theory of contact of a rough sphere on a flat was presented by Greenwood & Tripp [25]. Their analysis is based on the traditional Greenwood-analytical [7] and numerical [8][9][10][11] works have revealed important limitations of GW results for total contact area [13] and contact geometry [8,27,28] motivating us to revisit the contact of rough spheres.…”

“…Roughness enters only through the dimensionless root mean square (rms) slope of the surface h rms ≡ |∇h| 2 1/2 , where h( r) is the height as a function of position r in the plane. Numerical studies find p rough ≡ h rms E * /κ, where the dimensionless constant 1/κ ≈ 1/2 in the continuum, hard-wall limit [8][9][10][11][12]. The constant 1/κ becomes smaller for finite range soft repulsion and adhesive interactions.…”

“…At low loads and contact areas, recent results for rough flat surfaces can be applied to rough spheres [7][8][9][10][11][12][13][14]. Smooth sphere theories [15][16][17][18][19][20] can be used to fit experimental data in the limit of full contact, i.e.…”

“…As in flat surface studies [8][9][10][11][12], h has a self-affine fractal form from lower wavelength λ s to upper wavelength λ L with Hurst exponent H [31].…”