Alternate approach for calculating hardness based on residual indentation depth: Comparison with experiments

It is well known that plastic deformation is a highly nonlinear dissipative irreversible phenomenon of considerable complexity. As a consequence, little progress has been made in modeling some well-known size-dependent properties of plastic deformation, for instance, calculating hardness as a function of indentation depth independently. Here, we devise a method of calculating hardness by calculating the residual indentation depth and then calculate the hardness as the ratio of the load to the residual imprint area. Recognizing the fact that dislocations are the basic defects controlling the plastic component of the indentation depth, we set up a system of coupled nonlinear time evolution equations for the mobile, forest, and geometrically necessary dislocation densities. Within our approach, we consider the geometrically necessary dislocations to be immobile since they contribute to additional hardness. The model includes dislocation multiplication, storage, and recovery mechanisms. The growth of the geometrically necessary dislocation density is controlled by the number of loops that can be activated under the contact area and the mean strain gradient. The equations are then coupled to the load rate equation. Our approach has the ability to adopt experimental parameters such as the indentation rates, the geometrical parameters defining the Berkovich indenter, including the nominal tip radius. The residual indentation depth is obtained by integrating the Orowan expression for the plastic strain rate, which is then used to calculate the hardness. Consistent with the experimental observations, the increasing hardness with decreasing indentation depth in our model arises from limited dislocation sources at small indentation depths and therefore avoids divergence in the limit of small depths reported in the Nix-Gao model. We demonstrate that for a range of parameter values that physically represent different materials, the model predicts the three characteristic features of hardness, namely, increase in the hardness with decreasing indentation depth, and the linear relation between the square of the hardness and the inverse of the indentation depth, for all but 150 nm, deviating for smaller depths. In addition, we also show that it is straightforward to obtain optimized parameter values that give good fit to the hardness data for polycrystalline cold worked copper and single crystals of silver.

Figure 1

Schematic diagram showing (a) relevant variables during indentation, (b) the geometry of an ideal Berkovich indenter, and (c) the effective radius of blunted Berkovich indenter.

Figure 2

Influence of ${\sigma }_m$ on the model $F-z$ curve for ${\sigma }_m=E^{*}/900$ (dotted), $E^{*}/700$ (dash-dotted), $E^{*}/500$ (dashed), and $E^{*}/300$ (continuous) for $m=1.5$.

Figure 3

(a) Loading and several unloading curves for ${\sigma }_m=E^{*}/500$ and $m=1.5$. (b) Influence of ${\sigma }_m$ on $H$ as a function of $z$ keeping $m=1.5$ for ${\sigma }_m=E^{*}/900\left(\blacksquare \right),E^{*}/700(\square ),E^{*}/500(•)$, and $E^{*}/300(\circ )$.

Figure 4

Influence of ${\sigma }_m$ on the characteristic features of hardness keeping $m=1.5$ for ${\sigma }_m=E^{*}/900\left(\blacksquare \right),E^{*}/700(\square ),E^{*}/500(•),$ and $E^{*}/300(\circ )$. (a) Plots of $H^2$ as a function of $1/z$ for $z\ge 200\text{n}\text{m}$. (b) The corresponding plots of $H^2$ verses $1/z$ for $z$ down to $50\text{n}\text{m}$. All other parameters are as in Table 2.

Figure 5

(a) The influence of the velocity exponent m on the model $F-z$ curves for $m=1.0$ (continuous), 1.5 (dashed), 2.0 (dot-dashed), and 2.5 (dotted). (b) Plots of $H$ as a function of $z$ for $m=1(\circ ),m=1.5(•),m=2(\square )$, and $m=2.5\left(\blacksquare \right)$. Other parameters are as in Table 2.

Figure 6

(a) Plots of $H^2$ as a function of $1/z$ for $z\ge 200\text{n}\text{m}$ for $m=1(\circ ),m=1.5(•),m=2(\square )$, and $m=2.5\left(\blacksquare \right)$ keeping ${\sigma }_m=E^{*}/500\text{G}\text{Pa}$. The inset shows $H^2$ as a function of $1/z$ for $m=1$ on an expanded scale. (b) Plots of $H^2$ verses $1/z$ for $m=1(\circ ),m=1.5(•),m=2(\square ),$ and $m=2.5\left(\blacksquare \right)$ for $z$ down to $50\text{n}\text{m}$. The inset shows the plot of $H^2$ verses $1/z$ for $m=1$ on an expanded scale. Other parameters are as in Table 2.

Figure 7

(a) Comparison of model area function (continuous line) with the experimental points ($\square$). (b) The model $F-z$ plot showing the loading, holding and unloading parts (continuous) obtained using $C=0.164,q=1.7$, and $s=1$. The experimental points from Fig. 3(b) of Ref. [9] are shown by $\square$. Optimized parameters used are given in Table 3.

Figure 8

Comparison of model hardness features with experiment [9]. The symbols $•$ and $\circ$ correspond to the model predicted and the area corrected model $H$ values, respectively. $\square$ corresponds to the area corrected experimental points. Continuous lines are a guide to the eye. (a) Comparative plots of $H$ as a function of $z$. (b) Plots of ${[H/H_0]}^2$ as a function of $1/z$ for $z\ge 150\text{n}\text{m}$. Parameter values used are listed in Table 3.

Figure 9

Comparison of the model predicted hardness features with experimental values [9]. Plots of ${[H/H_0]}^2$ as function of $1/z$ predicted by the model ($\circ$) along with the experimental points ($\square$) for the extended range $z\ge 50\text{nm}$. The symbol ($\circ$) corresponds to the area corrected model value and $\square$ to the area corrected experimental value. Continuous lines are a guide to the eye. Parameter values used are as in Table 3.

Figure 10

(a) Plot of ${\rho }_m$ as a function of $z$ using ${\sigma }_m=198\text{M}\text{Pa}$ and $m=1.25$. (b) The corresponding plots of ${\rho }_f$ (equivalently, ${\rho }_s$ in standard notation) and ${\rho }_G$ as a function of $z$. Other parameters are as in Table 3.

Figure 11

(a) Comparison of the model area function (Continuous line) with the experimental area points $\square$ (extracted from Fig. 7 of Ref. [7]). Note that since the $x$ axis is the total depth $z, s=1$. (b) Comparison of the load-displacement plot using $C=10.72,q=1.7$ showing both the loading and unloading parts of the $F-z$ curve (continuous line) along with the experimental points ($\square$) extracted from Fig. 2 of Ref. [7].

Figure 12

Comparison of model predicted features of ISE with the experiment on single crystal of Ag for (110) orientation [7]. Parameter values used are listed in Table 4. The symbol $\circ$ corresponds to the model prediction and $\square$ to the experimental points. Continuous lines are guide to the eye. (a) Comparative plots of the hardness as a function of $z$. (b) Plots of ${[H/H_0]}^2$ as function of $1/z$ predicted by the model and the experimental points for $z\ge 150\text{n}\text{m}$.