3. Yielding inception

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

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

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.

As the force applied during a hardness test is increased, the tested material passes to experience different regimes of deformation. Initially, Tabor [5] identified three regimes: elastic,

By measuring the hardness of a material, one should ensure that it is subject to full plasticity. Therefore, load values should be sufficient for the ratio between the mean contact pressure (which may be equivalent to the hardness (H)) and the material yield stress (Y) which exceeds a typical value. The relationship between the mean pressure and the yield stress is defined as

elastic-plastic, and fully plastic. Figure 1 shows these regimes schematically.

Figure 1. Deformation regimes under sphere-plane contact (Adapted from [5]).

this geometry, such as Brinell [3] and Rockwell B [4].

46 Contact and Fracture Mechanics

nontrivial simplifications, deserving to be detailed.

2. Differential hardness: Tabor's model

the constraint factor (C (= H/Y)).

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 contact radius established by Hertzian analytical solution.

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 most metals.

The curve in Figure 2 with ν = 0.3 has the maximum value for the ratio τmax/pm of 0.468, which is normalized to a defined depth. In applying Tresca's criterion, one obtains [7]

$$p\_m = \frac{\tau\_{\max}}{\left(\tau\_{\max}/p\_m\right)} = \frac{Y}{2.(0.468)} = 1.07Y \tag{2}$$

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 Figure 1).

between the elastic modulus and the yield stress (E/Y). Thus, the same authors indicated that,

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

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

49

Since then, many models have been proposed to express the dependence of the factor C to the E/Y, and two currents are shown in Figure 4 for a perfectly plastic material. One of them is due to Song and Komvopoulos (SK model) [12], in which Poisson's ratio is implicit into the value of E (one can consider ν as fixed), while Megalingam and Mayuram (model MM) [13] made this coefficient in its equation explicit. Figure 4 compares the models for a fixed

It is found that SK and MM models differ more significantly for materials with relatively low

Figure 3. Plastic zone definition during an indentation process. Contours indicate the limits for deformation regimes

Figure 4. Variation of constraint factor with ratio E/Y following models SK [12] and MM [13], for a material with

(Adapted from [10]). Caption: a = contact radius and c = radius of plastically affected zone.

values of E/Y and them approaching to 2.8 (used by Tabor) as E/Y increases.

by similarity, the constraint factor is also proportional to this ratio.

Poisson's ratio of 0.3.

Poisson's ratio of 0.3.

Figure 2. Distribution of normalized shear stress along normalized depth (z/a) under sphere-plane contact considering different values of Poisson's ratio (Adapted from [8]).

Changes in Poisson's ratio are sufficient to change both the magnitude of ratio τmax/pm and the location of this maximum value, as shown in Figure 2.

It is logical to expect that the relationship of Eq. (2) presents variations with ν. A series of equations have been proposed in the literature for that, having been summarized by Pintaude [9]. The author showed that five equations of the literature did not show great differences and that they can be divided into just two groups, in which some difference begins to be considered significant. This difference is associated with the given equivalence between the yield stress and the hardness, i.e., the constraint factor previously defined, which will be treated with utmost importance in the next section.
