The elastic-plastic contact of a single asperity of a rough surface

A penetration of spherical asperity into the elastic-plastic hardening half-space is described. The elastic-plastic material properties correspond to Hollomon’s power law. In this case the empirical Meyer law relating a spherical indentation load with an indentation diameter d is used. Initially, the Meyer law is not related to the mechanical characteristics of the test material. The study used the relations between the strain hardening exponent n and the Meyer law constant obtained by S.I. Bulychev. The effects relating to elastic punching and plastic displacement of material are taked into account. It is shown that there is no need to define Meyer law constants. Expressions relating the value of the relative load magnitude to the relative indenter penetration magnitude are presented. The scope of application of the proposed equations is defined. A comparison of the obtained results with the experimental data and published data of the finite element analysis is given.


Introduction
Widespread use in tribology finds a discrete model of roughness, in which the asperities are presented as a set of bodies of regular geometric shapes, for which solutions of contact problems are available [1,2].In this case, the asperity model in the form of a spherical segment is considered to be optimal.
Problems of spherical asperity elastic-plastic penetration are not sufficiently studied and the solutions suggested require clarification and improvement [3].One of important problems to be taken into account is material hardening.The authors' approach to solving this problem is given in [4-6 et al.].The method consists in the use of the kinetic indentation load-displacement diagram and the similarity method of deformation characteristics.In this case, the notion "plastic hardness" is used as characteristics of material resistance to contact plastic strain.Plastic hardness has the form where y σ is the yield strength, ( ) is a parameter defined by the double indentation technique [4] using the results of the finite element analysis [7], y ε and n are the characteristics of Hollomon's elastic-plastic material.
In a number of works [8,9] the empirical Meyer law linking the spherical indentation load and an indenter diameter was used to allow for material hardening in solving the tribomechanics problems.In [9] an influence of some physical and mechanical properties of real materials on the features of contact elastic-plastic deformation formation is emphasized.However, the limitation of the method is that it does not explicitly take into account elastic-plastic characteristics of the hardened material.
The aim of this work is to use the above mentioned hardened material characteristics in the Meyer law application.

Problem solution
In describing elastic-plastic characteristics of the hardened material Hollovon's power law is widely used.According to this law the relation between the true stress S and the strain ε under uniaxial tension or compression is described by equations where E is an elastic modulus, n is a strain-hardening exponent.
The constant K is determined from the equality condition for σ at εy.Then the second expression in Eq. ( 1) can be written as where Taking into accord that the limiting uniform strain n u = ε , the strain hardening exponent can be defined from following Eq.as [10] ( ) MATEC Web of Conferences 129, 06017 (2017 where σu is the tensile strength.E. Meyer was the first to describe a material behavior in the elastic-plastic domain.He related the load P to the indentation diameter d as .
The empirical Meyer law is often written as where a is the radius of the contact area.
Using the maximum Meyer hardness concept we have  5) and ( 6) it follows that The maximum Meyer hardness relates to the Brinell hardness as [11] , From [12] it follows that , HB k u ⋅ = σ σ (10) where 333 .0 = σ k for carbon and pearlitic steel, for other materials the values of kσ are given in [13,14].As it shown in [14], in practice well justifies the dependence where u δ is the uniform elongation, ψ is the deformation in a print of a spherical indenter.
According to the data obtained by S.I.Bulychev [15], the limiting uniform strain u ε corresponding to σu is equal to ( ).
where e is base of natural logarithm.
where * E is reduced elastic modulus, ( , where c h is the depth at which the sphere contact with a half-space takes place, h is the depth of indentation from the initial surface level, we have The parameter h h c c = 2 can be defined based on the data obtained in [16] ( ) Equation ( 17) is in good agreement with the results of FE simulations [17].
A lower limit is R a a y y = corresponding to y ε .
According to [15] under spherical indentation, the strain is defined by the relation , where Or with regard to Eq. ( 12) it follows that Then we have , and the corresponding load is defined from Eq. ( 15).An upper limit is defined from the condition . With regard to Eq. ( 12) we have The corresponding load is defined from Eq. ( 16).

Comparison with experimental data and published data of the finite element analysis
For estimation of the obtained results, we will compare it with the experimental study given in [18].In this paper for calculating parameters of elastic-plastic contact, the authors used the generalized deformation curve and method of variable elasticity parameters.The experimental study were to determine the contact radius curvatur during indentation hardened steel ball (HRC 63…64) with a radius of curvature 2.5 mm into material samples specimens in the table.Mechanical properties of the material of specimens defined by results of tension.
To the analytical dependences for each material from the Eq. ( 3), the values of hardening exponent n were determined.The values of εy were determined by the formula where i ν , i E are Poisson's ratio and the elastic modulus of the indenter.The fig. 1 show the experimental data [18] are presented by points, the coordinates were have been "digitized" in the processing of the results.The lines denote the corresponding dependence, calculated by the Eq. ( 15).Let us also compare the results with the data of the finite element analysis [19].
m, A, A * are constants.A * has a dimension of strength.The equation in the left part is a mean contact area pressure referred to as the Meyer hardness , .

Fig. 1 .
Fig. 1.Dependence of the relative radius of contact of the load P.