5. Hardening effects

value, there is a small deviation thereafter. Thus, increments above a certain value mean a geometric inconsistency, in which the ball cannot make a suitable contact with the plane in the axial direction of loading, representing a "decoupling" of the contact, which would entail artificially greater contact pressures (proportionally smaller radius values) as shown in

Figure 8. (A) Detailed geometry of sphere-plane contact and (B) variation of normalized depth with the normalized

Figure 7. Equations to describe the finite deformation regime for a perfectly plastic material: Jackson-Green [16], Alcalá

contact radius (Adapted from [12]).

et al. [17], and Jackson et al. [18] (Adapted from [19]).

Figure 5.

52 Contact and Fracture Mechanics

The hardening of metals was incorporated into Tabor's concept, through work-hardening exponent (n). To the full plasticity regime, empirical equations can be derived to compute the effect of this property in the constraint factor. This type of relationship was shown, for example, by Matthews [20] and subsequently reviewed by Sundararajan and Tirupataiah [21], who demonstrated it for a wider range of experimental points.

Although these formulations may be useful from a practical point of view, here it is relevant to present in what deformation regime during the mechanical contact the work-hardening can alter the behavior. The example presented by Komvopoulos and Song [12] for a material with E/Y = 11 (Figure 9) makes it interesting for that.

It is noticed that the work-hardening exponent changes with great intensity of the C values in the full plasticity regime. Thus, the differential hardness value will be affected by the work hardening, as will be discussed in the next section.

Figure 9. Variation of constraint factor with normalized depth for a material with E/Y = 11, for different work-hardening exponents (Adapted from [12]).
