2. Differential hardness: Tabor's model

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

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

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

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 sphere as follows:

$$p\_m \approx 2.8\,\mathrm{Y\_M} \ge 1.1\,\mathrm{Y\_B} \colon \mathrm{Y\_M} \ge 0.4\,\mathrm{Y\_B} \tag{1}$$

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

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, provided by the numerical simulation techniques.
