Nomenclature

Whereas deformation profiles plotted after the test, it was shown that the hardness difference for the case of Cu would be 1.33 and 1.39 for the case of Al, much lower than that established

Perhaps, for this reason, the results were the scenes of further discussion. Jackson and Green [23] criticized especially the method of measuring the deformation by profilometry after removal of the load and the effect of hardening. Jamari and Schipper [24] argued explaining the measuring method for the profile of spheres before the test and the approximation of the profile thereof after deformation. About hardening, they provided values showing the increas-

The experimental demonstrations for differential hardness remain scarce, especially for the boundary conditions given by Alcalá and Esqué-de los Ojos [15] in the abovementioned. These can be considered as inherent challenges to the contact mechanics in its present state of the art. In this line, the work presented by Ghaednia et al. [19] sheds light to the theories discussed here, with a new numerical limit for the occurrence of differential hardness in the sphere-plane

The first important question raised by these authors, also discussed by Jamari and Schipper [22], is the effect to consider if the load is being applied either on the plan or on the sphere. For that, Ghaednia et al. [19] make it clear what equation to adopt for each case, being adopted the Jackson-Green expression [16] for plane hardness and the Jackson et al. one [18] for the ball hardness.

From this, each equation is then used to calculate a stress ratio (Y\*), equivalent to the differen-

By selecting different combinations of properties for sphere and plane, these authors got simulated for what value the constraint factor presents with no further changes, within the full plasticity regime for the plane. The found value for the differential hardness is equivalent to

The main contributions in the field of contact mechanics were demonstrated along this chapter applied for an important specific system, sphere-plane contact, in which several hardness tests

Clearly, there is a lack in the experimental demonstration of theories presented. This is an interesting challenge, as the numerical simulation increases at much higher speed than the

An updated value obtained through numerical simulation for the differential hardness is 1.7, different from that predicted by Tabor (2.5). Certainly, an experimental demonstration could be

<sup>Y</sup><sup>∗</sup> <sup>¼</sup> YB <sup>=</sup>YM (3)

ing of sphere hardness, which was relatively insignificant.

by Tabor (2.5).

54 Contact and Fracture Mechanics

contact.

tial hardness:

are performed.

experimental results.

elucidating this difference.

1.7, 32% lower than that predicted by Tabor.

7. Conclusions and final remarks

