A Finite Element Study of Elasto-plastic Hemispherical Contact

This work presents a finite element study of elasto-plastic hemispherical contact. The results are normalized such that they are valid for macro contacts (e.g., rolling element bearings) and micro contacts (e.g., asperity contact). The material is modeled as elastic-perfectly plastic. The numerical results are compared to other existing models of spherical contact, including the fully plastic case (known as the Abbott and Firestone model) and the perfectly elastic case (known as the Hertz contact). At the same interference, the area of contact is shown to be larger for the elasto-plastic model than that of the elastic model. It is also shown, that at the same interference, the load carrying capacity of the elasto-plastic modeled sphere is less than that for the Hertzian solution. This work finds that the fully plastic average contact pressure, or hardness, commonly approximated to be a constant factor (about three) times the yield strength, actually varies with the deformed contact geometry, which in turn is dependant upon the material properties (e.g., yield strength). The results are fit by empirical formulations for a wide range of interferences and materials for use in other applications. NOMENCLATURE A = area of contact C = critical yield stress coefficient E = elastic modulus H = hardness HG = hardness geometric limit K = hardness factor P = contact force R = radius of hemispherical asperity Sy = yield strength a = radius of the area of contact ey = uniaxial yield strain, Sy/E k = mean contact pressure factor po = maximum contact pressure z = axis of symmetry for hemisphere ω = interference between hemisphere and surface ν = Poisson’s ratio Subscripts E = elastic regime F = fit to current FEM data c = critical value at onset of plastic deformation o = maximum t = transitional value from elastic to elasto-plastic behavior Superscripts ′ = equivalent * = dimensionless. INTRODUCTION The modeling of elasto-plastic hemispheres in contact with a rigid surface is important in contact mechanics on both the macro and micro scales. This work presents a dimensionless model that is valid for both scales. In the former, e.g., rolling element bearings, load may be high and the deformations excessive. In the latter, e.g., asperity contact, a model on the micro-scale is of great interest to those investigating friction and wear. In addition, the real area of contact of such asperities will affect the heat and electrical conduction between surfaces. In either scale contact is often modeled as a hemisphere against a rigid flat. One of the earliest models of elastic asperity contact is that of Greenwood and Williamson [1]. This model (GW) uses the solution of the contact of an elastic hemisphere and a rigid flat plane, otherwise known as the Hertzian solution, to stochastically model an entire contacting surface of asperities with a postulated Gaussian height distribution. The GW model also assumes that the asperities do not interfere with adjacent asperities. The Gaussian distribution is often approximated by an exponential distribution to allow for an analytical solution, although Green [2] has analytically solved the integrals using the complete Gaussian height distribution.