6. Experimental evaluation of differential hardness and trends in numerical simulation

An experiment to certify the existence of the differential hardness as predicted by Tabor is not a simple task. Jamari and Schipper [22] made an important attempt for that. These researchers used SiC as rigid plane, in which copper or aluminum balls were pressed against it.

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 by Tabor (2.5).

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 increasing of sphere hardness, which was relatively insignificant.

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 contact.

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 differential hardness:

$$Y^\* = {}^{\chi\_{\mathcal{B}}}/\_{\chi\_{\mathcal{M}}} \tag{3}$$

In addition, the hardening and friction effects on the sphere-plane system can be more

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

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

55

explored through both numerical simulations and experimental arrangements.

The author thanks CNPq, through Project 312385/2014-5.

Acknowledgements

Nomenclature

a Contact radius

C Constraint factor D Sphere diameter E Elastic modulus

n Work-hardening exponent

Y\* Ratio between yield stress of sphere and plane

Address all correspondence to: giuseppepintaude@gmail.com

UTFPR – Federal University of Technology, Paraná, Brazil

pM Mean contact pressure

YB Yield stress of sphere YM Yield stress of metal z Depth from the surface

δ Deflected depth

ν Poisson's ratio

Author details

Giuseppe Pintaude

τmax Maximum shear stress

H Hardness

R Sphere radius

Y Yield stress

c Radius of plastically affected zone

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 1.7, 32% lower than that predicted by Tabor.
