Microindentation data analyses
The general method followed for converting indentation load-displacement data into indentation stress-strain curves can be found in references [
4,
29]. It is emphasized that this is the first use of these protocols to study polycrystalline material volumes in the primary indentation zone. For microindentation testing equipment used in this work, there is no CSM as noted earlier, this is only available in most modern nanoindenters [
30]. As demonstrated in prior work [
4,
29], the measurement of unloading (elastic) stiffness is central to a reliable estimation of contact radius needed in the computation of indentation stress and indentation strain measures. Consequently, in the present work, we rely on superimposing intermittent unloading-loading cycles on the desired loading history. Note that each unload results in the estimation of one data point on the ISS curve (i.e., one value of indentation stress and indentation strain; [
29]). Several challenges arise in using these protocols for microindentation. First, it is difficult to capture the initial elastic loading with an instrumented microindenter. Nanoindentation systems have adequate load and depth sensing resolutions needed to capture the initial elastic loading; however, instrumented microindenters are not well designed for capturing the initial elastic loading segment. The higher loads required generally for mesoscale measurements (10
2–10
3 N) significantly reduce the load resolution during the early stages of loading, while the best displacement resolution currently available is around 20 nm. Second, the higher loads present a significant challenge in the sample alignment and mounting. For a successful test, the top and back surfaces of the sample must be polished parallel to each other and set directly on a rigid surface (hardened steel plate or directly on the stage); otherwise, misalignment, sample rotation, or compliance issues will produce erroneous results. Third, the combination of higher loads and less rigid indenter tips means that the elastic displacement of the tip may be significant. For example, the elastic displacement of the tip (usually diamond with a modulus >1000 GPa) is typically negligible in nanoindentation, especially with the low loads (<500 mN). The same assumption is likely to be a poor assumption for tungsten carbide tips (around half the modulus of diamond) at the higher loads seen in microindentation. Therefore, a suitable correction is needed in the analysis to account for the tip displacement based on the elastic properties of the tip material. Below, we briefly review the main details of the indentation data analyses protocols used in this study.
Hertz theory (Eqs. (
1)-(
4)) describes the frictionless, elastic contact between two isotropic, homogenous bodies with parabolic surfaces [
31].
$$ P=\frac{4}{3}{E}_{\mathrm{e}\mathrm{ff}}{R}_{\mathrm{e}\mathrm{ff}}^{1/2}{h}_{\mathrm{e}}^{3/2} $$
(1)
$$ a=\sqrt{R_{\mathrm{e}\mathrm{ff}}{h}_{\mathrm{e}}} $$
(2)
$$ \frac{1}{E_{\mathrm{eff}}}=\frac{1-{v}_{\mathrm{i}}^2}{E_{\mathrm{i}}}+\frac{1-{v}_{\mathrm{s}}^2}{E_{\mathrm{s}}} $$
(3)
$$ \frac{1}{R_{\mathrm{eff}}}=\frac{1}{R_{\mathrm{i}}}+\frac{1}{R_{\mathrm{s}}} $$
(4)
In the above equations, P and h
e denote the indentation load and displacement, respectively. R
eff and E
eff denote the effective radius and modulus of the combined indenter-sample system, while a denotes the contact radius. The subscripts i and s denote that the variables are associated with the indenter and sample, respectively.
During the early stage of loading on a flat sample surface, prior to any permanent deformation, the effective radius is equal to the indenter radius (i.e.,
R
eff =
R
i). In this initial elastic regime, it is relatively easy to extract a value of the effective modulus,
E
eff using Eq. (
1) and standard regression techniques. The sample Young’s Modulus,
E
s, can then be determined from Eq. (
3), provided the sample Poisson ratio,
v
s, and the indenter elastic properties are known. It is important to note that our treatment of the sample as elastically isotropic (reasonable for the weak texture and very little anisotropy observed in tensile tests) is not a limitation of the analysis. Many authors have shown that the Eq. (
1) extends to elastically anisotropic materials with only slight modifications to Eq. (
3) [
32‐
38]. The contact area becomes elliptical instead of circular; however, the error associated with treating it as circular is very small [
34]. The contact radius can be interpreted as an effective contact radius for elastically anisotropic materials.
In nanoindentation tests, the identification of the elastic loading segment involves determining the zero-point corrections for both load and displacement [
4,
29]. In microindentation, the indentation stress-strain curve is far less sensitive to surface and tip disparities because of the large tip radii, and in many cases, there is no need for load correction (see Fig.
3). The analysis in this study systematically examined different load corrections and their corresponding displacement corrections following the approach described in earlier work [
29]. Based on this exploration, it was decided to select the optimal load correction as the one that minimizes the log of the average absolute residual of the linear regression fit for the elastic segment (prior to any detected residual deformation in the sample); the displacement correction is also automatically identified in this process.
After plasticity occurs, the effective radius is unknown, and the total displacement is now the sum of the elastic displacement and the permanent displacement or residual height,
h
r. However, unloading is primarily elastic, and therefore the contact radius,
a, can be determined by applying Hertz’s equations to the unloading data using standard regression techniques and the following equations:
$$ h=k{P}^{\frac{2}{3}}+{h}_{\mathrm{r}} $$
(5)
$$ k={\left(\frac{3}{4}\right)}^{\frac{2}{3}}{E}_{\mathrm{eff}}^{-\frac{2}{3}}{R}_{\mathrm{eff}}^{-\frac{1}{3}} $$
(6)
In Eq. (
6),
R
eff is the only unknown, because
E
eff was already determined from the initial loading data and is assumed to be the same even after the sample experiences plastic deformation. This assumption is reasonable because the effective (average) plastic deformation in the indentation zone is quite small in these experiments. Unloading data between 95 and 50 % of the peak load was used for each unload in this analysis. The contact radius at the point of unloading is determined from Eq. (
2). After the contact radius is determined, indentation stress and strain are calculated using the following set of equations, where the subscript max represents the peak load and displacement for each unload.
$$ {\sigma}_{\mathrm{ind}}=\frac{P_{\max }}{\pi {a}^2} $$
(7)
$$ {\varepsilon}_{\mathrm{ind}}=\frac{4}{3\pi}\frac{h_{\mathrm{s}, \max }}{a} $$
(8)
$$ {h}_{\mathrm{s}}=h-{h}_i $$
(9)
$$ {h}_{\mathrm{i}}=\frac{3\left(1-{v}_{\mathrm{i}}^2\right)P}{4{E}_{\mathrm{i}}a} $$
(10)
In the above set of protocols, care was afforded to subtract the elastic displacement of the indenter,
h
i, in order to use only the displacement of the sample,
h
s, in the computation of the indentation strain in the sample,
ε
ind. This is accomplished using Eqs. (
9) and (
10), where the elastic displacement of the indenter,
h
i, is calculated using the Hertz’s theory (for the displacement of the indenter tip pressed into a rigid flat surface in the absence of a sample). We assume that this is a good approximation for the indenter displacement during our tests. It can be seen from Eq. (
10) that the correction will be higher for indenter tips with lower moduli. In this study, a Young’s modulus and Poisson ratio of 640 GPa and 0.21 were used for the indenter, based on values reported in literature for tungsten carbide [
39]. In this work, the sample modulus was computed assuming a Poisson ratio of 0.3, and the determination of indentation yield strength was made using a 0.2 % indentation plastic strain offset on the indentation stress-strain curve using an indentation modulus of
\( \frac{E_{\mathrm{s}}}{1-{v}_{\mathrm{s}}^2} \) (see Fig.
3).