4. Full plasticity

The full plasticity regime associated with the contact between a sphere and a plane as defined by Tabor [5] presents no variations in the mean contact pressure; once this regime has been reached, this pressure has a defined value in relation to the yield stress of material. Tabor empirically found that the relationship between the hardness and the yield stress during a Brinell hardness test is 2.8; the value used by him to calculate the differential hardness ensures the plasticity of the material tested.

The use of a constant value to the constraint factor independent of system properties is a simplification in many ways, since a set of mechanical properties of materials dictates the behavior during the mechanical loading. Therefore, it is necessary to understand in a section, which occurs during this loading process, until the full plasticity is established for relatively large deformations/depths.

Figure 3 shows the definition of a plastic zone developed along an indentation process.

The contour of the plastic zone (c/a), shown in Figure 3, was modeled by Bishop et al. [11], which defines the limits of elastic-plastic deformation, being the same proportional to the ratio between the elastic modulus and the yield stress (E/Y). Thus, the same authors indicated that, by similarity, the constraint factor is also proportional to this ratio.

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 Poisson's ratio of 0.3.

It is found that SK and MM models differ more significantly for materials with relatively low values of E/Y and them approaching to 2.8 (used by Tabor) as E/Y increases.

Changes in Poisson's ratio are sufficient to change both the magnitude of ratio τmax/pm and the

Figure 2. Distribution of normalized shear stress along normalized depth (z/a) under sphere-plane contact considering

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

The full plasticity regime associated with the contact between a sphere and a plane as defined by Tabor [5] presents no variations in the mean contact pressure; once this regime has been reached, this pressure has a defined value in relation to the yield stress of material. Tabor empirically found that the relationship between the hardness and the yield stress during a Brinell hardness test is 2.8; the value used by him to calculate the differential hardness ensures

The use of a constant value to the constraint factor independent of system properties is a simplification in many ways, since a set of mechanical properties of materials dictates the behavior during the mechanical loading. Therefore, it is necessary to understand in a section, which occurs during this loading process, until the full plasticity is established for relatively

Figure 3 shows the definition of a plastic zone developed along an indentation process.

The contour of the plastic zone (c/a), shown in Figure 3, was modeled by Bishop et al. [11], which defines the limits of elastic-plastic deformation, being the same proportional to the ratio

location of this maximum value, as shown in Figure 2.

utmost importance in the next section.

different values of Poisson's ratio (Adapted from [8]).

the plasticity of the material tested.

large deformations/depths.

4. Full plasticity

48 Contact and Fracture Mechanics

Figure 3. Plastic zone definition during an indentation process. Contours indicate the limits for deformation regimes (Adapted from [10]). Caption: a = contact radius and c = radius of plastically affected zone.

Figure 4. Variation of constraint factor with ratio E/Y following models SK [12] and MM [13], for a material with Poisson's ratio of 0.3.

The work-hardening effects could be added to the results of Figure 4, but before checking it is important to discuss the failure of Figure 1 for describing the behavior of a material at high loads, which produce consequently large contact radii.

Alcalá and Esqué-de los Ojos [15] consider a friction coefficient value of 0.07 as ideal for property extraction from a spherical indentation test. The effect of friction described in [14]

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Moreover, Alcalá and Esqué-de los Ojos [15] commented on the difficulty of any experimental support existence to prove the decay in C. According to these authors, the reasons for this are

i. Metals with low work-hardening exponent generally have sufficiently high yield stress, such that the increase occurring in C takes place within a broad range, which limits the

ii. The frictional effects are significant enough to lower work-hardening exponent values,

While this experimental evidence is not presented, the simulation results generated a series of equations for region 2 of Figure 5 (the decay). Figure 7 shows a comparison among some of

The last regime inserted into the full plasticity of a perfectly plastic material can be defined as a physical limit for the mechanical contact existence. This phenomenon can be treated as a

Figure 8A shows the geometry of the sphere-plane contact, indicating a depth δ that varies in conjunction with the contact radius for a same radius R or diameter D. It is possible to relate the variation of the a/D ratio with δ/A by means of simple geometry, and this variation is shown in Figure 8B for a ball diameter with D = 3 mm. One can see that δ/a ≈ a/D for a certain

Figure 6. Variation of constraint factor with ratio a/R for a material with E/Y = 10,000 and Poisson's ratio of 0.3, with and

such that an increase in hardness occurs, while the decay is not observed.

"decoupling" of the contact. Figure 8 helps to explain better the phenomenon.

can be seen in Figure 6.

related to the strain hardening, being:

these equations, for the variation of C with a/R.

without friction (Adapted from [14]).

occurrence of hardness drop with high values of a/R

A pioneer investigation for comprehension of the full plasticity is due to Mesarovic and Fleck [14]. These authors defined two regimes for the full plasticity, finite deformation, and plastic similarity. The transition among them depends on the ratio E/Y, as previously demonstrated in Figure 4. A more detailed description can be seen in the work done by Alcalá and Esqué-de los Ojos [15]. For the current purpose, only a general description will be presented (Figure 5), for a perfectly plastic material (n ! 0) with specific properties.

In Figure 5 it is possible to clearly observe the existence of three regions. Firstly, the C factor increases as the contact radius increases. This will happen up to a characteristic value, which depends on the ratio E/Y. The second region is one in which there is a drop in the C values, typical to simulate perfectly plastic materials, in which one would expect in fact a constant value. In [15] one can see that the plasticity theory applied to the simulation (deformation theory vs. flow plasticity theory) affects the description of this region (topic beyond the scope of this chapter). Finally, there is a third regime, in which the constraint factor backs to increase with increasing loading. This regime will be explained in detail further.

An effect that helps explain the fall in C factor with the a/D was demonstrated by Mesarovic and Fleck [14], verifying that there is lack of uniformity of the vertical speed at which the material experiences as the contact radius increases. In other words, this means that the friction between the sphere and the plane should be considered to provide a more realistic simulation.

Figure 5. Variation of constraint factor with ratio a/D (Adapted from [15]).

Alcalá and Esqué-de los Ojos [15] consider a friction coefficient value of 0.07 as ideal for property extraction from a spherical indentation test. The effect of friction described in [14] can be seen in Figure 6.

The work-hardening effects could be added to the results of Figure 4, but before checking it is important to discuss the failure of Figure 1 for describing the behavior of a material at high

A pioneer investigation for comprehension of the full plasticity is due to Mesarovic and Fleck [14]. These authors defined two regimes for the full plasticity, finite deformation, and plastic similarity. The transition among them depends on the ratio E/Y, as previously demonstrated in Figure 4. A more detailed description can be seen in the work done by Alcalá and Esqué-de los Ojos [15]. For the current purpose, only a general description will be presented (Figure 5), for a perfectly plastic material (n ! 0) with specific properties. In Figure 5 it is possible to clearly observe the existence of three regions. Firstly, the C factor increases as the contact radius increases. This will happen up to a characteristic value, which depends on the ratio E/Y. The second region is one in which there is a drop in the C values, typical to simulate perfectly plastic materials, in which one would expect in fact a constant value. In [15] one can see that the plasticity theory applied to the simulation (deformation theory vs. flow plasticity theory) affects the description of this region (topic beyond the scope of this chapter). Finally, there is a third regime, in which the constraint factor backs to increase with increasing loading. This regime will be explained in detail

An effect that helps explain the fall in C factor with the a/D was demonstrated by Mesarovic and Fleck [14], verifying that there is lack of uniformity of the vertical speed at which the material experiences as the contact radius increases. In other words, this means that the friction between the sphere and the plane should be considered to provide a more realistic simulation.

loads, which produce consequently large contact radii.

50 Contact and Fracture Mechanics

Figure 5. Variation of constraint factor with ratio a/D (Adapted from [15]).

further.

Moreover, Alcalá and Esqué-de los Ojos [15] commented on the difficulty of any experimental support existence to prove the decay in C. According to these authors, the reasons for this are related to the strain hardening, being:


While this experimental evidence is not presented, the simulation results generated a series of equations for region 2 of Figure 5 (the decay). Figure 7 shows a comparison among some of these equations, for the variation of C with a/R.

The last regime inserted into the full plasticity of a perfectly plastic material can be defined as a physical limit for the mechanical contact existence. This phenomenon can be treated as a "decoupling" of the contact. Figure 8 helps to explain better the phenomenon.

Figure 8A shows the geometry of the sphere-plane contact, indicating a depth δ that varies in conjunction with the contact radius for a same radius R or diameter D. It is possible to relate the variation of the a/D ratio with δ/A by means of simple geometry, and this variation is shown in Figure 8B for a ball diameter with D = 3 mm. One can see that δ/a ≈ a/D for a certain

Figure 6. Variation of constraint factor with ratio a/R for a material with E/Y = 10,000 and Poisson's ratio of 0.3, with and without friction (Adapted from [14]).

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 5.

5. Hardening effects

numerical simulation

exponents (Adapted from [12]).

demonstrated it for a wider range of experimental points.

E/Y = 11 (Figure 9) makes it interesting for that.

hardening, as will be discussed in the next section.

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

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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

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

6. Experimental evaluation of differential hardness and trends in

used SiC as rigid plane, in which copper or aluminum balls were pressed against it.

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

Figure 9. Variation of constraint factor with normalized depth for a material with E/Y = 11, for different work-hardening

Figure 7. Equations to describe the finite deformation regime for a perfectly plastic material: Jackson-Green [16], Alcalá et al. [17], and Jackson et al. [18] (Adapted from [19]).

Figure 8. (A) Detailed geometry of sphere-plane contact and (B) variation of normalized depth with the normalized contact radius (Adapted from [12]).
