1. Introduction

The contact mechanics, a branch of structural mechanics developed by the German physicist Heinrich Rudolf Hertz, describes the stresses and strains associated with a surface. His 1882 work, "Ueber die Berührung elastischer fester Körper" ("On the Contact Elastic Solid") is considered the starting point of this branch of science [1].

The solutions presented by Hertz do not involve friction; therefore, the bodies do not experience adhesion, and they are associated with purely elastic deformation field. The advancing in contact mechanics for modeling different deformation regimes depended on the development of computational simulation tools, especially the finite element method (FEM). In this sense, Mackerle [2] presented a summary on the use of FEM for indentation problems, a specificity of contact mechanics. In a period of 4 years, this author reported 187 references using FEM for a better understanding of indentation phenomena.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

distribution, and eproduction in any medium, provided the original work is properly cited.

An immediate finding of this list of articles is that the contact stress distribution depends on the geometry of the bodies and the contact between a sphere and a plane is always the first approach for that, given the amount of engineering systems that can be associated with this kind of system. One of them is the hardness testing, wherein different scales are composed of this geometry, such as Brinell [3] and Rockwell B [4].

On the other hand, if the sphere cannot experience plasticity, the deformation seen by the ratio of the mean contact pressure and yield stress must be restricted to a certain value. Thus, using the values for C shown in Figure 1, Tabor [5] calculated the differential hardness required for a

Deformation Regimes for Sphere-Plane Contact: Revisiting Tabor's Criteria for Differential Hardness

If the material hardness has equivalency to the yield stress, the sphere must be 2.5 times harder than the tested body. As an example, Tabor describes a sphere of quenched and tempered steel, with typically 900 HV; for this case, it could be used to test materials with a 400 HV maximum. The next items will be devoted for detailing the values used by Tabor for each regime beginning of elastic-plastic regime and full plasticity—and further advances in the literature,

The imminence of plastic yielding shall be described with the use of a suitable criterion. The criteria commonly used for metals are Tresca and von Mises, which are equivalent only in few specific conditions. Regardless of these conditions, the result of applying both criteria indicates that a metal yields by the action of shear forces, which makes the maximum shear stress significant to know. In a sphere-plane contact, the resulting shear stress can be described considering a normalized distance (z/a), where "z" is the depth below the surface and "a" is

The analytical equations necessary for calculating the maximum shear stress in the sphereplane contact will not be presented here, but it is known that these are dependent upon Poisson's ratio of the material (ν) [6]. To describe a single curve as shown in Figure 2, it is necessary to set a value for this property, in which Tabor used 0.3 as a convenient value for

The curve in Figure 2 with ν = 0.3 has the maximum value for the ratio τmax/pm of 0.468, which

Therefore, a relationship between the mean pressure and the yield stress of 1.07 is obtained, and this value was rounded to 1.1, as presented by Tabor for the formulation of the differential hardness, according to Eq. (1). For a material not yielding in a sphere-plane contact, the value of the applied load must correspond to a mean pressure not exceeding this value (see

<sup>2</sup>:ð Þ <sup>0</sup>:<sup>468</sup> <sup>¼</sup> <sup>1</sup>:07<sup>Y</sup> (2)

is normalized to a defined depth. In applying Tresca's criterion, one obtains [7]

τmax=pm <sup>¼</sup> <sup>Y</sup>

pm <sup>¼</sup> <sup>τ</sup>max

where YM is the yield stress of metal and YB is the yield stress of sphere.

provided by the numerical simulation techniques.

the contact radius established by Hertzian analytical solution.

pm ≈ 2:8YM ≥ 1:1YB∴YM ≥ 0:4YB (1)

http://dx.doi.org/10.5772/intechopen.72642

47

sphere as follows:

3. Yielding inception

most metals.

Figure 1).

A very important aspect of the reliability of the hardness testing is to ensure that the plastic deformation occurs in the tested body, while only small deformations are allowed on the sphere. This warranty is given by the difference in hardness between the sphere and the tested body, which can be called differential hardness. Tabor [5] described an analytical estimate of what would be necessary for differential hardness. For that purpose, Tabor made use of some nontrivial simplifications, deserving to be detailed.

In this context, this book chapter intends to present an update of the theories involving the problem of differential hardness from the presentation of the assumptions made by Tabor to the initial problem, involving concepts of elasticity and plasticity. Thus, the tribology student interested in issues directly affected by these formulations, such as contact area calculation and contact fatigue, can find a fundamental theoretical basis for conducting investigations in this field of knowledge.
