A 2D rough surfaces contact analysis.

Cananau, Sorin

In the present paper is analyzed the problem of two-dimensional (2D) elastic contact techniques regarding the contact between two interfering elastic surfaces. The material for the bodies in contact is modeled as elastic plastic and follows the von Misses yield criterion. The main results of such analyses are shown to include the detailed interface geometry and the subsequent contact pressure distribution involved. Methods of defining the resulting subsurface stresses created by this contact pressure distribution can be further developed for static normal loading, and for the case of normal load in the presence of a frictional surface shear. Key words: contact, deformation, hertz

1. INTRODUCTION

The cases for elastic and elastic-plastic contacts have been analyzed in detail in the last four decades. Predominantly considering normal loading only, a wide array of works have analyzed the contact of rough surfaces (Whithehouse & Archard, 1970), (Sayles & Bailey, 1987). Concerning the characterization of the surfaces the probabilistic theory is one of the latest development in the area. Statistical and random process methods require the rough surface to be specified in terms of height and spatial parameters. These parameters are usually found from measured roughness profiles, after some form of high-pass filtering is performed to refer the surface geometry to a datum in the form of a mean-plane or mean-line. The position of the datum is defined by arranging equal quantities of solid material above the mean-plane or mean-line to void below it. Such a definition is rather arbitrary and as a result some care is often needed in selecting the appropriate filtering to suit the contact problem in hand.

Many applications of numerical contact methods are reported in the scientific papares, with very important and notable early contributions by KaIker (Kalker & VanRanden, 1972). This numerical approach involves minimizing the elastic strain energy for a frictionless interfacial contact subject to the condition of all contact pressures being compressive and therefore greater than zero, other than for regions of no-contact.

There are other methods and applications related to the work reviewed in this paper, such as Johnson et a1. (Johnson et a1., 1985) who examined the contact of wavy surfaces in relation to some relatively simple-to-apply asymptotic solutions to the real-area/mean-pressure relationship for sinusoidal waviness.

2. NUMERICAL CONTACT METHOD DEVELOPED FOR ELASTIC DISPLACEMENTS OF A SURFACE

2.1 Hypothesis

Following are the assumptions that are used to simplify the problem:

(1) The contact model of Hertz is taken into account (Green, 2005)

(2) We can define the relative surface deformation of an elastic half-space, anywhere on the surface.

(3) We take into account simple loading element, in terms of the element load (Johnson et al., 1985)

(4)We used digitized roughness profiles with imposed intervals across the surface to simulate the elastic deformations

(5) We set the width of the pressure field according the width of sample imposed intervals.

(6) Sliding is not taken into account. In the elastic domain and up to the limit of plasticity, the

Hertzian solution (Nosonovsky & Adams, 2000) provides critical values of load, contact half-width, and strain energy. The hardness is not an unique material property as it varies with the deformation itself as well as with other material properties such as yield strength or the elastic modulus.

The model proposed for a 2D contact, one or two other conditions need addressing as we can see in Fig. 1. Also it is often convenient, but not essential, to add the topographies of the contacting bodies and consider the contact of a smooth body with a composite rough surface (Johnson & Moon, 2006)

[FIGURE 1 OMITTED]

The expression relating the magnitude of the pressure "element" (on the indenter) to the relative vertical displacement of the half-space (roughness surface) at distance x' from the point of application of the pressure is given by Equation ( 1): The relation for interference is given as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where

[delta]--vertical displacement

E--modulus of elasticity

v--Poisson's coefficient

p--equivalent pressure applied

[DELTA]x--small interval across the surface

K--arbitrary constant representing the displacement at some value of x', and from where thesome value of x', and from where the relative displacement can be defined.

2.2 Method and algorithm

The first step in a rough--smooth surface contact solution is to impose an arbitrary overlap of the contact surfaces. This is equivalent to imposing the same arbitrary displacement of points at the measured distantce within the contacting bodies, i.e. a rigid body displacement. Figure 2 show a model view of one of the profile area represented in Fig.1 the effect of an arbitrary overlap in terms of the nomenclature involved. The resolution of the subsequent analysis is obvioualy related to the digitized profile sampling interval Ax, which is typically in the range of 1 to 5 um for the examples used in this paper.

With reference to Fig 2, at any possible contact point we know that the total elastic displacement of the surfaces needed at that point to satisfy the imposed rigid body displacement and this one must be equal to the overlap of the rigid body profiles.

That means we superposed the acumulated surface displacements of all contacting points (elements) of profile within the contact interface.

[FIGURE 2 OMITTED]

In terms of given materials properties, Equation (1) can be expressed in terms of an influence coefficient C which defines the surface displacement, resulting from a unit pressure element at any distance x', i.e.:

1/C = p/[delta] (2)

We can define an array of influence coefficients C. These coefficients could be calculated covering every possible value of x' for the nominal contact geometry, material properties, and surface profile topographie being used, i.e.:

[[delta].sub.i] = [C.sub.ij][p.sub.j] (3)

where [C.sub.ij] the influence coefficient, represents the increment of surface displacement at x, due to a unit pressure at i.e. at "j", i.e. at distance x' from [x.sub.i].

For example, in Fig.2 we can see that the displacemets are due to the action of three elemental pressures with different values but with the same length of action [DELTA]x. We can find the values of these uniform pressures steps pj with the use of inflence coefficients [C.sub.ij].

If now we know the all [summation][[delta].sub.i]i displacements and [C.sub.ij] values for the coefficients we can calculate the total load involved with the overlap as is shown in Eq.4:

[W.sub.i] = [summation] [p.sub.i] [DELTA]x (4)

With these values we can build a numerical model to find an equilibrium load-displacements using an iterative algorithm. The solution method described up to this point is for pure elastic contact. For the situation when the calculated pressure values for some of the contact regions are excessive m relation to the surface strength then some form of plasticity criterion must be employed.

3. ANALYSIS AND RESULTS

An example of elastic contact between a real profile measured for a 50 mm diameter cylinder on a ground surface, both having modulii and Poisons ratio's equivalent to common steel, is shown in Fig 3. For each value of high pressure the computer method suppose there could be singular points charged with elemental pressure. At the same time the neighbouring points could be considered free of load.

A reasonable supposition regarding the initial values and the degree of overlap must be made. Again some iteration is required to arrive at a solution.

In this way we can simulate the surface contact for a riugh surface.

[FIGURE 3 OMITTED]

The position of the maximum pressure on the surface of the cylinder is in correlation with the 2D roughness profile as is shown in Fig.1

4. CONCLUSION

The paper presents a numerical method to simulate the contact between two interfering elastic surface. The rough surface is synthetic surface with a 2D profile.

For most engineering topographies, we can consider that the scale of asperities are generally measured (around 1 to 8 um sampling interval. In these conditions the slopes are not usually excessive and it would seem that using influence coefficients method and a model with individual response of each contact point would be reasonable providing isolated asperities are involved. The numerical method can be used to calculate the subsurface stresses created by this contact pressure distribution.

5. REFERENCES

Green, I. (2005). Poisson ratio effects and critical values in spherical and cylindrical Hertzian contacts, Int. J. Appl. Mech. 10 (3) (2005) 451-462

Johnson K.L.; Greenwood J.A. & Higginson, I.G. (1985). The contact of elastic wavy surfaces. Int. .I. Mech. Sci., (1985), 27, 6. 383-396

Johnson, J.A. & Moon, F.C. (2006). Elastic waves and solid armature contact pressure in electromagnetic launchers. IEEE Trans. Magn. 42 (3) (2006), 422-429

Kalker J.J. & VanRanden Y. (1972) A minimum principle for frictionless elastic contact with application to non-Hertzian halfspace contact problems. J. Eng Maths. (1972), 6, 193

Nosonovsky, M. & Adams, G.G. (2006). Steady-state frictional sliding of two elastic bodies with a wavy contact interface. J. Tribol. Trans. ASME. 122 (3) (2000), 490-499

Sayles, R.S. & Bailey, D.M. (1987). The modelling of asperity contacts. Tribology in Particulate Technology (Eds. Brisco, B.J. & Adams, M.J.) Adam Hilger, Bristol, (1987)

Whitehouse D.J. & Archard, J.F. (1970). The properties of random surfaces of significance in their contact Pro.Roy.Soc., A316. (1970), 97-121