Rough spheres in elastic contact

While spheres might look smooth and round to the naked eye, their surfaces are far from it. Even a ball bearing, when held up to intense scrutiny, is riddled with peaks and valleys. These inconsistencies change how the spheres interact with the world and can cause problems for everything from surface measurements to the strength of electrical currents. A transatlantic team of researchers explain the creation of a simulation model that can help scientists mathematically correct for any errors related to a sphere's roughness this week in Applied Physics Letters, from AIP Publishing.

The team's calculations are designed to tell a scientist when they should worry about surface roughness, which should make measurements more accurate, Pastewka explained. Pastewka and Robbins looked at the surface of spheres on the atomic level. They studied how the roughness -- those irregular peaks and valleys -- interacted mechanically with the surfaces they were pushed against to form areas of intimate atomic contact.

Knowing how to correct the imperfections mathematically is the cheapest and most plausible way to tackle this problem.

This could be exploited to minimize friction that sphere will create when sliding on a surface. For spheres meant to conduct electricity, scientists would most likely want lower peaks so more of the sphere is in contact with the medium. So far, the team has only looked at how the peaks and valleys react to elastic and adhesive surfaces. Elastic surfaces are like balloons, you can poke them and they bounce back to their original shape. The next step is to focus on plastic surfaces, which will change shape permanently when under pressure.

Applied Physics Letters features concise, rapid reports on significant new findings in applied physics. The journal covers new experimental and theoretical research on applications of physics phenomena related to all branches of science, engineering, and modern technology.

Skip to main content. Scientists have developed a simulation model to take the guesswork out of calculating the influence of roughness on mechanical properties of a sphere. From the Journal:. Applied Physics Letters. Article title:. Contact area of rough spheres: Large scale simulations and simple scaling laws.

Calculating the Mechanics of a Rough Sphere

Author affiliations:. About the journal:.Roman Pohrt, Valentin L. JuniBerlin, Germany. We investigate the normal contact stiffness in a contact of a rough sphere with an elastic half-space using 3D boundary element calculations. A new analytical model is derived describing the contact behavior at any force.

rough spheres in elastic contact

Since Bowden and Tabor [ 1 ], it has been known that surface roughness plays a decisive role in contact, adhesion, friction, and wear. The main understanding of the contact mechanics of nominally flat rough surfaces was achieved in the middle of the 20th century due to works by Archard [ 2 ] and Greenwood and Williamson [ 3 ].

In the last years, contact mechanics of rough surfaces has once again become a hot topic [ 4 — 6 ]. Most of the previous work was devoted to investigation of nominally flat surfaces. For many tribological applications, however, the contact properties of rough bodies with macroscopically curved surfaces are of great interest. A first analysis of the contact problem including a curved but rough surface was given by Greenwood and Tripp [ 7 ]. In this model, the roughness can be seen as an additional compressible layer.

They calculated the mean pressures as a function of the radius and for low loads found a reduction in the maximum pressure and an enlargement of the apparent area of contact.

For high loads, the indentation behavior found was Hertzian. In the present paper, we will investigate the indentation of a rough sphere into an elastic half-space without the restrictions stemming from the GW model. We calculate the incremental normal stiffness of the contact, which not only determines the dynamic properties of a tribological system, but also its electrical and thermal conductivity [ 89 ].

In the interest of purity of results, we assume the bodies to be elastic at all scales and confine ourselves to self-affine roughness without cutoff.

Contact Mechanics of Rough Spheres: Crossover from Fractal to Hertzian Behavior

Contact stiffness of such surfaces has been recently studied in detail numerically and analytically [ 1011 ]. We show that there is a pronounced crossover from the behavior which is typical for fractal surfaces [ 1213 ] to Hertz-like behavior [ 14 ], similar to GT [ 7 ]. Furthermore we derive an analytical approximation for the entire range of forces. We consider a rigid rough spherical indenter with the radiuswhich is approximated by a superposition of a parabolic shape and a nominally flat random self-affine roughness with the Hurst Exponent Figure 1.

The power spectrum of the randomly rough self-affine statistically isotropic surface has the form where is the system size and is the absolute value of the wave vector. The surface topography was calculated with the help of the power spectrum according to with and the phaseswhich are randomly distributed on the interval.

All samples were generated on a grid with discrete, evenly spaced points. A typical example of a self-affine surface with the Hurst exponent is shown in Figure 1while Figure 2 shows a sample of the superposition of both sphere and roughness. The indenter was pressed into an elastic half-space with the normal force. The indentation depth and the configuration of the contact were calculated using the boundary element method with an iterative multilevel algorithm similar to [ 1516 ]. The incremental normal stiffness was calculated by evaluating the differential quotient of force and indentation depth.

All values were obtained by ensemble averaging of 50 surface realizations having the same power spectrum. Figure 3 shows the resulting dependency red points. For small normal forces, the system is dominated by the roughness and the stiffness approaches the asymptotic dependence with the slope characteristic for nominally flat fractal surfaces [ 12 ]. In [ 12 ], it was shown that, for randomly rough self-affine surfaces, the contact stiffness is a power function of the normal force of the form with.

In [ 17 ], it was shown that this relation remains valid for Hurst exponents and that the only property needed for the validity of the scaling law 4 is the self-affinity of the surface which means that the surface appears undistinguishable from the original when viewed under an arbitrary magnification [ 15 ]: Here it is not important whether the surface is randomly rough or just a simple axisymmetric profile with the same scaling properties [ 18 ]: Thus, with respect to the contact stiffness, the randomly rough, fractal, and self-affine surface can be replaced equivalently with a simple axisymmetric form 6.

Of course, this equivalence only holds for the average values of multiple random rough realizations. In [ 1517 ], it was shown that, for the indentation of a rigid indenter having the shape 6 into an elastic half-space, the following relation of the normal contact stiffness and the normal force is valid: with For example, the Hertzian contact is one particular case of self-affine surfaces with andgiving the classical result [ 14 ]: or, resolved with respect to the force, The same is true for fractal rough surfaces [ 1215 ]: or The constant must be until now found from numerical calculations.

Latest studies suggest that Comparison of 7 and 11 shows that the contact properties of a self-affine roughness and a corresponding axis-symmetric profile can be made identical if the following prefactor in 6 is chosen: Using this analogy between randomly rough surfaces and axis-symmetric profiles, we suggest the rough sphere to be modeled as a superposition of the parabolic shape with the equivalent rotationally symmetric shape : Figure 4 c shows a cut through the resulting three-dimensional, rotationally symmetric indenter shapes.

As the contact stiffness depends only on the current contact radius [ 17 ], the superposition principle is valid for forces 10 and 12 at the given contact stiffness: For small values of the normal force or the contact stiffness we expect to see behavior according to 11while at higher forces the Hertzian behavior 9 is predicted. The crossover is expected to take place at the intersection of both asymptotic curves: The dependence 16 is shown in Figure 3 together with the numerical results of the Boundary Element Method.This article is part of the thematic issue "Biological and biomimetic surfaces: adhesion, friction and wetting phenomena".

Guest Editor: L. Heepe Beilstein J. The adhesive contact between a rough brush-like structure and an elastic half-space is numerically simulated using the fast Fourier transform FFT -based boundary element method and the mesh-dependent detachment criterion of Pohrt and Popov. The problem is of interest in light of the discussion of the role of contact splitting in the adhesion strength of gecko feet and structured biomimetic materials. For rigid brushes, the contact splitting does not enhance adhesion even if all pillars of the brush are positioned at the same height.

Introducing statistical scatter of height leads to a further decrease of the maximum adhesive strength. At the same time, the pull-off force becomes dependent on the previously applied compression force and disappears completely at some critical roughness. For roughness with a subcritical value, the pressure dependence of the pull-off force qualitatively follows the known theory of Fuller and Tabor with moderate modification due to finite size effect of the brush.

rough spheres in elastic contact

Keywords: adhesion; brushes; contact splitting; pressure sensitive adhesion; roughness. The study of adhesive contacts has been largely enhanced by studies of the extremely effective adhesion pads of geckos [1]. For example, the adhesion can be optimized by controlling the size and shape of the fiber cap [2,3] ; this mushroom-shaped microstructure can provide a stronger adhesive performance than the flat punch [4,5].

The compliant fiber is known to increase the strength of adhesion [6,7]. Almost all works in this field are based on the idea that contact splitting is the sole reason for the enhanced adhesion [8,9]. In a previous work, we shared a contrary opinion [10] : the contact splitting alone does not lead to enhancement of adhesion. Related problems have been studied using a number of purely statistical models, which did not consider the elastic interactions between asperities.

Zhuravlev proposed a model originally published inwhereby the work was later translated into English consisting of asperities in the form of elastic spheres having the same radius but placed at various heights [11].

Kragelsky presented originally in an alternative model of a rough surface as a collection of elastic rods and assumed that the rod heights have a Gaussian distribution [12].

In the classical work of Greenwood and Williamson inthey considered both the exponential and Gaussian distribution of asperity heights [13]. A detailed review of hierarchical models of rough surfaces can be found in [14].

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A very similar problem was studied by Fuller and Tabor [15] as early as in Contrary to the above mentioned works, we consider the numerically exact solution of the adhesive contact problem using the boundary element method as described in [16] using the mesh-dependent detachment criterion [17]which later was extended to power-law-graded media [18] and extensively tested and validated experimentally in [19]. In this work, we find the dependence of the adhesive force on the size of the brush, the fill factor of pillars and the statistical distribution of the pillar heights simulating the relative roughness of surfaces in contact.

The brush is shown in blue while the green color map shows the surface deformation of the elastic half-space during pull-off. Figure 1: Simulated surface of a rough brush blue in adhesive contact with an elastic half-space green. Along the boundary of the square, one can see the pillars, whose heights are statistically distributed.

The elastic half-space is represented only by its surface. It is known that in the approximation of independent asperities, adhesion can be described in a most general and elegant way if the distribution of asperity heights is described by the exponential probability density. For easier comparison with existing theoretical predictions, we used this probability distribution throughout the paper.

Figure 2: The scheme of indenting and pull-off stages of an adhesive contact of exponentially distributed pillars. Figure 2: The scheme of indenting and pull-off stages of an adhesive contact of exponentially distributed pil The numeric experiment was carried out under conditions of displacement control.Hot Threads.

Featured Threads. Log in Register. Search titles only. Search Advanced search…. Log in. Contact us. Close Menu. JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding. Forums Engineering Materials and Chemical Engineering. Understanding the physics behind an elastic sphere. First off, I'm not a scientist or engineer and I apologize if I don't give a clear description of my question. I'm beginning a personal project and was hoping for some knowledge and assistance.

What I'm trying to achieve is to have a spherical object it will be at least twice as wide as it is tall that can maintain it's shape while also being able to rotate in any direction. If I were to put it on the ground, the top and bottom would be flattened into a circle and it's only a few inches tall, then for it to be moved along the ground, rotating in any direction while maintaining a flat surface and circular shape.

I began with a playground ball, to try and understand the relation of it's shape with it's internal pressure. Unfortunately the ball was too inelastic and as I let out air wouldn't flatten since the it couldn't stretch past it's diameter, if that makes sense. So what kind of materials would I want to look at?

I was thinking of using sand to fill it instead of air, so that if you were to step on it, it would retain its shape better. How could I gain a better understanding of how the internal pressure will affect the shape?Greenwood, J.

March 1, March ; 34 1 : — The Hertzian theory of elastic contact between spheres is extended by considering one of the spheres to be rough, so that contact occurs, as in practice, at a number of discrete microcontacts. It is found that the Hertzian results are valid at sufficiently high loads, but at lower loads the effective pressure distribution is much lower and extends much further than for smooth surfaces. The relevance to the physical-contact theory of friction and electric contact is considered.

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rough spheres in elastic contact

GreenwoodJ. This Site. Google Scholar. Tripp J. Author and Article Information. Mar34 1 : 7 pages. Published Online: March 1, Article history Received:. Views Icon Views. Cite Icon Cite. Issue Section:. You do not currently have access to this content.Macro Variables and the Components of Stock ReturnsArticleMar 2015J Empir FinanceViewShow abstract.

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