To produce high-quality property maps, consideration of the stress field underneath the indenter tip is critical. Not only is there potential for the damage zones from individual tests to overlap and invalidate the results, but the “resolution” of the nanoindentation test is relevant when testing near boundaries of features, such as grain and phase boundaries, weld zones, composite interfaces, and material gradients, for damage or composition. The indenter resolution needs to be carefully defined, as the stress field occurs in three dimensions and consists of separately sized elastic and plastic zones. A second consideration involves the necessity of high loading rates, which may induce strain rate sensitivity changes in the measured hardness. Both subjects will be covered in the following two sections, “Indentation Spacing and Resolution” and “Strain Rate Sensitivity”.
Indentation Spacing and Resolution
When mapping surface properties, the in-plane spatial resolution is of primary concern; however, defining this requires consideration of the full three-dimensional shape of the indentation stress field or the volume of material being tested. Since the stress field decays continuously as a function of distance from the contact zone, boundaries can only be defined by a specific stress or strain value. The most important is the subdivision into a purely elastic25 and an elastic–plastic zone with the boundary set by the yield criterion of the material,26 as illustrated in Fig. 1. Thus, the entire elastic zone contributes to the modulus measurement, while only the elastic–plastic zone contributes to the hardness, H, measurement. In terms of a more practical definition for defining the indentation resolution, one can define an acceptable relative change of properties in proximity to a feature, such as a microstructural boundary or a previous indent.
Some indenters have a geometry that can be described as self-similar, which is simply a tip shape with a constant ratio of the contact area to depth versus load. This property is maintained by common pyramidal indenters, including Berkovich and cube corners, as well as conical tips, and implies that the measured properties will not change as a function of indentation load. Notably, spherical tips are an exception. Thus, for a self-similar indenter, all-important geometrical parameters can be expressed as a function of the contact radius, a. The contact radius is defined as the radius of a circle of estimated equivalent area to that of the actual contact, thus allowing pyramidal probes to be described by the same parameter as spherical and conical.
Regarding issues with indent-to-indent spacing, there are several effects. If the second indent overlaps with the residual impression or pileup from the previous, this clearly invalidates the semi-infinite half-space assumption and the actual contact radius will deviate significantly from the assumed value. Subtler is the overlap of plastic zones, which extend further in the sub-surface of the testing plane. The residual plastic zone could be considered cold-worked, thereby elevated hardness, but the exact interaction of plastically nucleated defects could also produce a softening effect by providing dislocation sources. The radius of the plastic zone, \( R_{\text{p}} \), in relationship to the contact radius is material specific because of differences in plasticity mechanisms. For metals, this ratio can range from 3a to 6a.27,28,29 For a sharp Berkovich tip with 50-nm radius of curvature, reliable hardness measurements can be achieved at a depth of at least 15 nm as shown in Fig. 1 (note that the modulus was constant for the three tips over the entire depth range). This corresponds to a maximum contact radius of 57 nm, thus requiring a 315-nm indent spacing for a soft metal with \( R_{\text{p}} \sim6a \). This situation improves for a cube corner that has a steeper contact radius to a depth ratio of 0.7 compared to 3.5 for a Berkovich. Since these tips are self-similar, the plastic zone size is proportional to the contact radius and is reduced by approximately a factor of 5 as well. As previously discussed, the exact plastic zone size is specific to a given tip-material-depth combination, so experimental evaluation is required to determine this size precisely. This can be illustrated by two case studies: (1) the influence of one indentation on its neighbor and (2) the influence of interfaces in the proximity of an indent.
To illustrate this effect, the indent spacing, d, was varied on an Al sample as tested with a Berkovich tip, as shown in Fig. 2a. Interestingly, reducing d results in a corresponding decrease in hardness rather than an increase as would be expected for work hardening. Additionally, no effect on modulus would be expected from plastic zone overlap. Therefore, the effect on modulus at the smaller values of d indicates the invalidation of the area function due to pileup. This pileup-affected zone begins at d = 750 nm, slightly less than the recommended distance of 5.6 times the contact radius, or 840 nm for a maximum displacement of 50 nm. Clearly, a trade-off occurs between lateral resolution and hardness measurement accuracy, as using a sharper tip at smaller depths gives a reduced elastic–plastic zone. An additional tradeoff can be made by sacrificing the hardness measurement altogether and limiting testing to a purely elastic regime. The absence of a plastic zone and residual displacement and thus no pileup in a purely elastic indent allow for the contact radii to overlap between individual indents.
The role of sample interfaces, important to high-resolution mapping of compositional and phase varied materials, is illustrated by mapping with a metal-ceramic cross-sectional sample. Here, indents are placed near a material interface in a sample, specifically, a 750-nm-wide Ti layer sandwiched between two extremely hard (H ~ 25 GPa) Ti-N layers. The indenter was carefully placed in the center of the Ti layer using in situ SPM imaging. A profile of hardness and modulus as a function of depth is shown in Fig. 2b, where correct hardness and reduced modulus values for Ti were only measured at 15–30-nm depth. At larger depths, the indentation stress field increasingly interacts with the TiN layers resulting in increasing modulus and hardness values. In some mapping scenarios, the indentation grid is generated with a predetermined spacing that will place indents at varying distances from, or on top of, a phase boundary or interface in the sample. This can produce measurement errors if plasticity mechanisms are affected by the presence of the boundary, such as providing defect sources, sinks, and barriers. These data points can usually be filtered during analysis by examining the statistical distribution of measured properties and removing outliers. These boundary effects can change the ideal indent spacing for mapping. Therefore, examining the effect near sample boundaries to determine the best spacing value is recommended.
To summarize this section, the achievable limits of nanoindentation resolution depend strongly on the tip shape, material being tested, and, crucially, the deformation regime. Following Jakes et al.,30,31 three dimensionless parameters can be defined that control indent resolution: the contact area relative to the distance to a feature, √A/d, and two material-dependent parameters, the ratio E/H and the Poisson’s ratio. One can define a maximum modulus or hardness change versus d using these parameters. Therefore, the smallest achievable d values are found at the lowest indentation depths in the elastic regime. However, if hardness measurements are desired, testing in the elastic–plastic regime is necessary and a balance between the accuracy of hardness and lateral resolution must be chosen.
Strain Rate Sensitivity
One of the drawbacks of high-speed nanoindentation mapping is a loss of flexibility in the load function, where high loading rates are needed for increased mapping speed. These high loading rates can influence the measured hardness, but this depends again on the material type, tip shape, and several other variables. The hardness from a nanoindentation test is strain rate dependent and fit by a power law relationship by a characterizing parameter, \( m\sim\frac{\partial \ln H}{{\partial \ln \dot{\varepsilon }}} \), where \( \dot{\varepsilon } \) is the strain rate. The strain rate for indentation is defined proportionally as the displacement rate over the total displacement \( \dot{h}/h \) or \( \frac{1}{2}\dot{P}/P \)24 correspondingly for loading rate over total load for materials that do not have depth dependence to their response, and for self-similar indenters. Since this relationship is fit by a power law, one can describe it as an order of magnitude effect. As previously discussed, high-speed nanoindentation techniques can run about two orders of magnitude faster than standard indentation techniques. Since typical strain rate sensitivity parameter m values range from 0.001 to 0.1 in crystalline materials, this corresponds to a hardness value shift between 0.4% and 37% compared to standard speed indentation. However, one must look at the bigger picture, which is that strain rate sensitivity is determined by the predominant deformation mechanism and is strongly affected by variables that aid or hinder operation of these mechanisms. These variables most notably include temperature, but also crystalline orientation and grain size. Special cases, such as nanocrystalline or ultrafine grain materials, can possess high strain rate sensitivity values32 because of the dominance of grain boundary diffusion mechanisms or, in the case of glasses, unusual behavior due to shear transformation zones.33,34 Some literature data are presented in Table I, which shows how much of a hardness shift would be expected by increasing the strain rate by two orders of magnitude as discussed above for some of the more interesting scenarios.
Table I Hardness change expected for two orders of magnitude increase in loading rate
Thus, the origins of strain rate sensitivity are complex and require considerations of many subtleties. However, for several classes of materials the effect is essentially marginal. The best approach is to directly measure the strain rate sensitivity for the materials of interest; these techniques have recently been reviewed by Maier-Kiener and Durst.39 As a final point, the role of indentation depth should be acknowledged, as shallower indents are typical for indentation mapping and deeper indents for strain rate sensitivity measurements. Ideally, this should not affect results but as real tips are blunt, shallower indents are increasingly dominated by a spherical-like contact.
Summary and Complimentary Nanoindentation Validation
In the absence of sophisticated analysis and/or modeling, one can simply advocate the approach of exploring the effects of the parameter space on the measured properties of interest whenever possible, specifically, the combination of loading rate, indentation depth, and indentation spacing. Thus, a typical complimentary approach to validate a high-speed indentation map would include:
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1.
Measurement of depth sensitivity: Depth profiling is already frequently done to calibrate tip area functions. A variety of methods can be used, including a varying depth indent arrays, partial unload load functions, or dynamic methods.40,41
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2.
Measurement of spacing sensitivity: With the depth dependence established, the user can chose their desired depth for the indentation map. Next, the spacing effects at that desired depth can be studied with indent arrays. When high spatial resolution is unnecessary or impractical because of the desire to map a larger area by nanoindentation, conservatively large spacing could be used freely.
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3.
Measurement of rate sensitivity: Either through grids with varying indent speed or characterizing the strain rate sensitivity coefficient.39
There could also be a desire to define the desired indent spacing first, i.e., resolution of the map, then what maximum depth can be used needs to be determined, thereby reversing steps 1 and 2.