Introduction

Nanoindentation is increasingly becoming the technique of choice for the assessment of mechanical properties in submicron-sized volumes of material, because of its load and depth resolution in the μN and nm range, respectively [1–3]. While it is still primarily linked to room temperature hardness and elastic modulus measurements, much effort has been dedicated toward extending its use to elevated temperatures [4], as well as to the measurement of time-dependent properties, such as creep and strain rate sensitivity (m).

Three methods have proven especially popular in this field. The first, proposed by Mayo and Nix [5], is the constant rate of loading (CRL) method, in which a steady loading rate is used until the tip displacement rate becomes nearly constant. This allows for a simple calculation of strain rate, and has produced good correlations to conventional values in tests on Sn and Pb [5]. Another technique, the constant strain rate (CSR) method, proposed by Lucas and Oliver [6, 7], uses an exponential load–time function to produce a steady strain rate. It too was found to produce good correlations on a range of different samples [6, 7], and was also shown to be free from intrinsic “size effects” on hardness values (in most materials, hardness reaches a constant value that is independent of depth) [2, 6–8]. However, both of these methods require multiple indents to achieve the necessary strain rate—hardness pairs needed to calculate an m value (either because of the need for multiple loading rates or multiple strain rates). This implies that a large area of the sample, several hundred square microns or more, must be averaged out to compute a single value of m. These methods are, therefore, not easily applicable to the measurement of values from individual features or defects, or from samples in which large variations in properties are observed in adjacent locations.

A few methods have been suggested to overcome this weakness, including the strain rate jump tests, presented recently in [9, 10], which uses a series of exponential loading rates to generate several strain rate and hardness pairs. The most popular technique, however, remains the constant load (CL) method of Mayo et al. [11, 12], which uses a hold segment at steady load to achieve continually changing strain rate and hardness pairs. This allows the calculation of strain rate sensitivity from a single indent, and therefore from a much smaller area of the sample, potentially in the submicron range. Additionally, a number of studies [7, 8, 13–16] have found that the CL method is capable of producing good matches to conventional values. That being said, many authors have reported conflicting size effects in CL results. Most have observed decreasing m with load in metals [17–23], ceramics [17, 24], and metallic glasses [25], but others have found the opposite trend in metals [26] and ceramics [27], or no trend at all in both metals [16] and ceramics [27].

A number of different phenomena have been proposed to explain this effect, including surface effects [17–20, 22], strain gradients or changes in the apparent hardness [24, 25], as well as changes in the dominant deformation mechanisms [17, 19, 20, 22]. However, many of the results were taken at depths where surface effects should be relatively negligible, and in many cases, there was no correlation with any perceived changes in hardness [17–19, 24]. The lack of understanding of this trend reduces the reliability of m measurements assessed by CL nanoindentation and warrants additional investigation.

The purpose of this study is therefore to address this poorly understood load and depth dependence by measuring and analyzing the strain rate sensitivity of a range of materials, namely, zinc, aluminum, copper, titanium, and fused quartz, using the CL method at a number of different loads. These results are then compared with CSR results, as well as values reported in the literature, and disparities are discussed. Finally, a model is presented and simulations are carried out to supplement the experimental results and confirm the trends observed.

Experimental procedure

Materials

Pure samples of zinc (HCP structure; T M = 693 K), aluminum (FCC structure; T M = 933 K), copper (FCC structure; T M = 1,358 K), titanium (HCP structure; T M = 1,941 K), and fused quartz (amorphous; T M = 1,943 K) were used in this study. For the metals, as-cast materials with large grain size, when compared with the average indent width (1–10 μm), were used to minimize microstructural effects (e.g., grain boundaries) on deformation.

In preparation for nanoindentation, all metallic samples were mechanically polished with a series of diamond suspensions followed by 0.05 μm colloidal silica. Final polishing was performed by vibratory polishing in a bath of colloidal silica until a surface roughness of <10 nm was achieved (RMS value measured in situ by “contact mode” imaging in the nanoindenter). The fused quartz specimen used was a standard sample for calibration of the instrument and was used as-received.

Nanoindentation testing

The indentation experiments were performed on a Hysitron Ubi3 instrumented indenter (Minneapolis, MN) with a load and displacement resolution of 0.1 μN and 0.2 nm, respectively. A diamond Berkovich indenter (measured tip defect radius <150 nm) was used and the tip area function was calibrated on the fused quartz sample using the standard procedures outlined by the instrument’s manufacturer and based on [28].

Prior to testing, instrumental stability was assured by bringing the tip in contact with the specimen surface with a load of 2 μN and allowing over an hour for equilibrium to be reached. The tip was not retracted from the surface until all testing had been finalized. This method allowed drift rates to be consistently brought to <0.1 nm/s. Additionally, a 40-s drift assessment at 80 % unload was used to determine if any large changes to the drift rate had occurred (>0.05 nm/s), and such tests were discarded.

Individual indents were separated from one another by at least 25 μm (several times larger than the average indent width), and each test was performed 15 times per specimen. An average value was calculated, and the error bars shown in this study represent the standard deviation obtained. For each material, strain rate sensitivity was determined using both the constant strain rate and CL methods, described in detail in the following.

Strain rate sensitivity

The strain rate (\( \dot{\varepsilon} \)) and the hardness (H) during indentation are obtained from the following relationships

$$ \dot{\varepsilon } = \frac{{\dot{h}}}{h} $$
(1)
$$ H = \frac{P}{A} $$
(2)

in which h is the tip depth, \( \dot{h} \) is the displacement rate, P is the applied load, and A is the projected area of the indent. Note that the projected area depends on the depth according to the tip geometry. For instance, an ideal Berkovich tip will have an area function A = 24.5 h 2. Additionally, the instantaneous area (determined from the instantaneous total depth, rather than the contact depth) is used in this study to monitor hardness throughout the duration of the test. This differs from nanoindentation studies of static properties, but is relatively common in the measurement of rate-dependent properties (see, for instance, [5–7, 17, 18]).

The flow strain rate (\( \dot{\varepsilon }_{F} \)) and stress (σ F ) are linked to one another by the following well-known power law relationship, in which both K and m are temperature-dependent material constants, with the latter defined as the strain rate sensitivity.

$$ \sigma_{F} = K\dot{\varepsilon }_{F}^{m} . $$
(3)

By taking the indentation strain rate and hardness as proportional to flow strain rate and stress, respectively, it follows that the strain rate sensitivity (at a fixed temperature) can be defined as

$$ m = \left( {\frac{\partial \ln H}{{\partial \ln \dot{\varepsilon }}}} \right)_{T} . $$
(4)

Constant strain rate (CSR) tests

Following the procedure outlined in [6, 7], the indentation strain rate can be held constant as long as the ratio of loading rate (\( \dot{P} \)) to load (P) is kept at a constant value k. This implies an exponential loading curve of the following form, in which P 0 is the load at time t = 0.

$$ P = P_{0} \exp \left( {kt} \right) $$
(5)

In this study, the CSR tests were performed at \( \dot{P}/P \) ratios of 0.01, 0.03, 0.1, and 0.3, resulting in constant strain rates spanning slightly more than an order of magnitude. Both the preload P 0 (500 μN) and the peak load (10 mN) were the same for all the tests.

When analyzing the results, the strain rate was calculated by taking an exponential fit of the depth data and using the time derivative of the fit as the displacement rate in Eq. (1). To discard the early instability caused by the sudden preloading to 500 μN, the strain rate was taken as the average of the second half of the test, and the hardness was taken as the value reached at peak load. Note that the hardness was indeed constant (barring small fluctuations) throughout the duration of the test for all materials tested, as was expected.

Constant load (CL) tests

CL tests were carried out with peak loads ranging from 1 mN to the apparatus’ upper limit of 25 mN. In all cases, the loading rate was set to reach the peak load in 1 s, and the hold duration was 100 s.

This hold duration was long enough to generate two distinct regions in the depth–time curve: an initial rapid increase in the indentation depth (termed Stage I in this study) and a slower seemingly linear increase in depth (Stage II). These two regimes (shown in Fig. 1) have been observed in a large number of CL studies [17–21, 24–27, 29–33], and are sometimes explicitly referred to as a transient (Stage I) followed by a steady-state (Stage II) [19, 20, 30–32]. It is worth noting that because Stage II is presumed to be more stable, and is reminiscent of secondary, or steady-state, creep behavior, measured m values are almost exclusively taken in this regime, and Stage I results are often discarded.

Fig. 1
figure 1

Example of the a depth–time curve and b log–log plot for FQ at 5 mN, revealing Stages I and II in both cases

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When analyzing the results, the depth–time curves from the hold segment were fitted to the following empirical function:

$$ h\left( t \right) = h_{0} + a\left( {t - t_{0} } \right)^{P} + kt. $$
(6)

This is the most commonly used equation for analyzing CL results [17, 19–26, 31–35] and consists of a term related to the curvature (a(t − t 0)P), as well as a linear term (kt) which fits the data at the end of the hold segment.

The least-squares method was used to fit the depth–time curves (shown as the line in Fig. 1a). The instantaneous displacement rates were obtained from the derivative of this fitted curve, and the instantaneous strain rates were then calculated from Eq. (1). The hardness was then calculated with Eq. (2) from the average load during the hold segment and the projected area (calculated from the depth–time fit). An example of a log–log plot of hardness and strain rate is shown in Fig. 1b, revealing two m values, for Stages I and II.

Results and discussion

Comparison of strain rate sensitivity results

The CSR hardness and strain rate pairs are plotted in Fig. 2, revealing each material’s m value as the slope of the line between them. The results from the CL tests are shown in Fig. 3, with both Stage I and II plotted as a function of the peak load used, as well as the CSR value (dashed line) for comparison. It was observed that CL Stage I results matched considerably well with CSR results, as was observed in a similar study on tin [33]. The Stage I results also matched the range of values obtained by conventional methods, although small disagreement are expected because of differences in the testing conditions and materials used in the studies referenced.

Fig. 2
figure 2

Results from the CSR tests on a log–log plot, revealing the m of each sample

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Fig. 3
figure 3

Comparison of CL Stage I (square) and CL Stage II (diamond) results at all peak loads with CSR values (dashed line), for a Zn, b Al, c Cu, d Ti, and e FQ. The inset figures show the percentage of total displacement during the hold segment accounted by the linear term kt of the fitting equation used

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Indeed, in the case of zinc, a value of 0.085 ± 0.01 was obtained by the CSR method, which matches the 0.076–0.094 found in CL Stage I results (Fig. 3a). Values of this magnitude are expected, because Zn should be approaching a dislocation creep regime (m > 0.1), at room temperature (0.43 T M). Indeed, they are comparable to the 0.06–0.095 observed by conventional tensile and strain rate jump tests [36, 37].

For aluminum, a value of 0.022 ± 0.001 was found by CSR, matching the CL Stage I results of 0.019–0.021 (Fig. 3b). This finding is similar to the 0.01–0.032 found in coarse grained pure Al by tensile and strain rate jump tests [38, 39]. Additionally, these values are comparable, but expectedly lower, to those found in ultrafine grained and nanocrystalline Al of 0.03–0.06 by tensile [39] and strain rate jump tests [40–42], and 0.012–0.026 found in compression [43], and similar to the 0.008–0.027 found in certain aluminum alloys [44, 45]. These results are also comparable to previous nanoindentation CL stage I and CSR results of 0.018–0.023 [33].

The copper results were 0.018 ± 0.004 for CSR and 0.011–0.015 for CL Stage I (Fig. 3c). This was comparable to literature values of 0.006–0.037 (depending on grain size) found with tensile [38, 46], strain rate jump tests [38, 46, 47], compression [48], and shear compression tests [49]. These values also correlated to previous nanoindentation results of 0.005–0.036 in ultrafine grained copper by the CRL method [50], as well as 0.006–0.038 in nanocrystalline copper by the CSR method [51].

Titanium strain rate sensitivity was measured as 0.030 ± 0.004 by CSR and 0.025–0.028 in CL Stage I results (Fig. 3d). Once again, this was found to be comparable to values of 0.019–0.026 calculated from compression test data found in [52–54], and 0.02–0.04 calculated from tensile test data found in [55, 56] (both in pure titanium). These are also similar to the 0.01–0.019 found in Ti-6Al by compression and strain rate jump tests [57].

Finally, values of 0.0114 ± 0.0002 and 0.0097–0.011 were found for CSR and CL Stage I, respectively (Fig. 3e). While room temperature strain rate sensitivity values of ceramics are rarely measured (as it is assumed that m ≈ 0), the values obtained are similar to the CL results of Mayo et al. [11, 12] in other oxides. For instance, an m value of 0.01 was found in single crystal TiO2, as well as 0.016–0.018 and 0.018–0.021 observed in nanocrystalline TiO2 and ZnO, respectively (as long as the grain size was larger than 30 nm).

As for the Stage II results, they were found to suffer from a significant load dependence (of an order of magnitude from 1 to 25 mN), and were far less likely to correlate to CSR values, and therefore to m values in the literature. Both the change from Stage I to Stage II and the load dependence of the results were similar, however, to those obtained in previous Stage II CL studies. For instance, the change in m from Stage I to Stage II (from 0.016 to 0.038) found in Al at 4 mN in [29] is comparable to that obtained at 5 mN in the current results (0.018–0.061). Additionally, the load dependence on FQ (from 0.016 at 25 mN to 0.28 at 1 mN) is similar to the change of 0.0125–1 found in [17] and 0.008–0.2 obtained in [24], although the loads used were quite different in all three cases.

Finally, it is worth noting that the Stage II results, especially at low loads, reach values commonly associated with creep, even in samples with high melting points such as titanium and quartz which should not be undergoing creep at room temperature. Values of this kind have been noted in a number of previous studies. Indeed, in their 2006 study (which used the CL method and took results from a regime of nearly constant displacement rate, consistent with the current definition of Stage II behavior), Goodall and Clyne [58] found creep-like m values (0.08 ≤ m ≤ 1) in all 15 materials tested, including tungsten (T M = 3,695 K) and vanadium (T M = 2,183 K), for which room temperature creep is highly unlikely. Interestingly, it was observed that there were no correlations between their results and creep exponents reported in the literature. All in all, this suggests that, unlike the Stage I results, Stage II behavior is not depicting material properties, and is not the “secondary creep” or “steady-state” it is often made out to be.

Constant displacement rate and size effects

Despite being similar to results obtained in previous studies, the appearance of Stage II is puzzling. Indeed, to cause such a large change in strain rate sensitivity between Stage I and II, it would be expected that the strain rates experienced (>10−4) would have to be lower and that the homologous temperatures (<0.4 T M in most cases) would need to be higher [59]. Additionally, the significant load dependence of Stage II results cannot be explained by surface effects, because the depths involved (usually >100 nm, even at 1 mN) were large enough for such effects to be negligible. This load dependence is also not attributable to the indentation size effect (ISE) on hardness (see, for instance, [60]). In fact, Alkorta et al. [9] calculated that the maximum change in m caused by this ISE (from zero depth to infinite depth) was approximately 33 %, far less than the order of magnitude or more observed in our results, and in previous Stage II studies [17–22, 24, 26].

Instead, the Stage II results are found to correlate with an increase in the importance of the linear term (kt) of Eq. (6) (an increase in linearity causes an increase in the Stage II m value). This is shown in the inset of Fig. 3, in which the percentage of total displacement (during the hold segment) accounted by the linear term was plotted as a function of peak load.

The effect of this increased linearity can be rationalized by considering the effect of a constant displacement rate on the measured m value. Assuming that the hardness is directly proportional to P/h 2 (in other words: assuming an ideal area function), and that the load is constant throughout the test, m can be given by

$$ m = \frac{{\partial \left( {\ln P - 2\ln h} \right)}}{{\partial \left( {\ln \dot{h} - \ln h} \right)}} = - 2\frac{{\partial \left( {\ln h} \right)}}{{\partial \left( {\ln \dot{h} - \ln h} \right)}}. $$
(7)

If the displacement rate is given by a constant k, the previous equation can be reduced to

$$ m = - 2\frac{{\partial \left( {\ln h} \right)}}{{\partial \left( {\ln k - \ln h} \right)}} = - 2\frac{{\partial \left( {\ln h} \right)}}{{\partial \left( { - \ln h} \right)}} = 2. $$
(8)

The measured strain rate sensitivity would, therefore, be expected to increase (toward an ultimate value of 2), as the linearity of the curve (and the importance of the linear term of Eq. 6) increases. This result is consistent with the behavior observed.

It is important to note, however, that the linear term of Eq. (6) does not make sense physically. Indeed, as the tip sinks into the material, the projected area increases and the stress therefore decreases. Because the stress serves as the driving force for deformation, such a decrease should reduce the displacement rate until the stress is too low to push the tip any further into the sample. This implies that the displacement rate should decrease until it reaches an ultimate value of zero, rather than a constant value. In fact, this is the behavior observed in microindentation tests [61].

Drift-induced errors

The appearance of a constant displacement rate at the end of the depth–time curves is, therefore, not expected. Note, however, that this displacement rate is often comparable to that of the drift rate, especially at low loads (for example, in Fig. 1a, the nearly constant displacement rate at the end of the curve is approximately 0.02–0.03 nm/s). Consequently, it was important to assess the effect of drift-induced errors (differences between the drift rate measured and corrected for at the start of the test and the actual drift rate occurring during the indentation). This was achieved by simulating data (with the model presented in the Appendix) with varying drift rates superimposed. The model parameters H 0 and m were set to 2,000 MPa and 0.02, respectively (these are roughly in the middle of the range of values observed experimentally), and the load functions were unchanged from the experimental procedure, with a peak load of 10 mN.

The effect of drift on the log–log plot is shown in Fig. 4a. It was observed that with no drift errors, the log–log plot was completely straight, with a single slope equivalent to that of Stage I (this line directly follows from the input parameters, with a slope of 0.02 and a hardness of 2,000 MPa at a strain rate of 100). However, as the drift rate was increased, the end of the log–log curve begins to curl, generating a second, and larger, slope (Stage II). This is consistent with both the expected decrease in displacement rate over time and the effect of increased linearity in the depth–time data. Indeed, as the true displacement rate decreases, the contribution of the drift error increases. This causes the end of the depth–time curve to deviate from its expected behavior and become progressively more linear, which, in turn, increases the m value calculated in this regime, as expected from the result of Eq. (8).

Fig. 4
figure 4

Simulated results with different drift rates, revealing a the generation of a second slope in the log–log plot and b the load dependence of the Stage II results with nonzero drift. The model parameters H 0 and m were set to 2,000 MPa and 0.02, respectively

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As for the load dependence, it is shown in Fig. 4b, in which Stage II results are plotted as a function of load for varying drift rates. The strain rate sensitivity of Stage II is found to increase with decreasing load, as long as the drift rate is nonzero, in a similar manner (especially at a drift rate of 0.05 nm/s) as in the experimental results. This observed load dependence is to be expected, however, because lower loads create lower displacement rates throughout the test. This implies that the contribution of the drift error will be more significant as the load is decreased and, again, this leads to more linear depth–time plots and increased m values.

From this analysis, it appears clear that the appearance of Stage II is caused by the increased importance of drift throughout the test. This finding not only explains the larger slopes commonly encountered in this regime but also the lack of consistency between different CL Stage II studies (for instance, fused quartz results at 5 mN of 0.1 in [17], 0.008 in [24], and 0.05 in this study, because the drift rates were undoubtedly different), the nonexistent correlations between Stage II and literature results (as exemplified by [58]) and the absence of the Stage II phenomenon from studies of light metals at high loads, in which displacement rates were large throughout the duration of the test (such as a 20–100 mN CL study of Sn [16], which found no load dependence and straight log–log plots).

Finally, although only observed in a fraction of the indents conducted for this study, a negative drift rate leads to a smaller slope during Stage II, as shown in Fig. 4a. Such a result is also rarer in the literature, but consistent with the findings of [26, 27].

Recommendations

To reduce the occurrence of the previously described drift errors on CL measurements, it is of utmost importance to maintain large displacement rates, through the use of higher loads, and to use stricter drift limitations and corrections. However, there are limits imposed on the maximum possible load (instrument capabilities or depth limits in thin films, for instance). Furthermore, because of the extended nature of the test and the fact that very small amounts of unaccounted drift are needed for Stage II to occur (an error of 0.01 nm/s already causes a difference in Fig. 4a), it is unlikely that drift can ever be fully accounted for. In fact, correcting the results according to the 80 % unload drift measurement was not sufficient in preventing the occurrence of Stage II. That being said, a number of other steps can be taken to reduce the effects of drift on the measured m values.

First, because the true displacement rate decreases throughout the test, longer hold durations would be expected to result in lower displacement rates and thus higher contributions from drift errors. Simulations (carried out with the same parameters and load functions, but with a peak load of 5 mN) reveal that this is indeed the case, as shown in Fig. 5. Note that for hold durations of 300 s, even a low drift error of 0.01 nm/s would produce a significant difference in the calculated strain rate sensitivity. Many CL studies use even longer hold durations, however, spanning several hundreds or even thousands of seconds. Under such conditions, spectacularly accurate drift corrections would be required to calculate the true m value from CL results.

Fig. 5
figure 5

Simulated results with varying drift rates and hold durations. The model parameters H 0 and m were set to 2,000 MPa and 0.02, respectively, and the peak load was 5 mN

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Crucially, because Stage I is not a transient, but rather a period in which the displacement rate far exceeds the drift rate, such long holds, required to generate a “steady-state,” are not necessary. In fact, hold durations of 20 s or less may be more than enough to generate the strain rate and hardness pairs needed, without the increased potential for drift-induced errors.

Additionally, the same behavior is expected with longer loading times, which would promote lower displacement rates. Indeed, CL Stage II studies on both metals [31] and metallic glasses [32] have revealed lower displacement rates with longer loading times (or lower loading rates), which led to an increase in the measured m values. Again, this effect can be attributed to drift errors and lower loading times (such as the 1 s used in this study) are adequate.

Finally, as noted in [35], the fitting equation used can have a significant effect on the measured strain rates and m values. It is therefore imperative to select a fitting procedure that does not exaggerate the effect of drift-induced errors. This can be achieved by simply removing the linear term (kt) from Eq. (6), and an example of the difference caused is shown in Fig. 6. The change in the appearance between the two fits (Fig. 6a) is minimal (except near the end of the curve, where the drift-induced linearity cannot be accurately followed without a linear term), however, the difference in the log–log plot (Fig. 6b) is significant. Indeed, by simply removing the linear term, a single unambiguous value of m is revealed.

Fig. 6
figure 6

Example of the effect of fitting of data (Ti at 5 mN) with and without a linear term kt revealing a a minor effect of the appearance of the fitted depth–time curve and b a significant change to the log–log plot

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To showcase the effect of these changes, the data were reanalyzed with no linear term in the fitting equation, and taking only the first 20 s of the hold segment. A comparison between the single m produced from this analysis and the previous Stage I results is shown in Fig. 7. In all cases, the standard deviations were found to be reduced, and for most samples and loads, an improvement in the correlation with the CSR results was also observed. These improvements were most significant for Al, Cu, and, to a lesser extent, Ti (Fig. 7b–d). In the case of Zn, the displacement rate was high throughout the test, and the effect of drift was small. The slope of Stage I is, therefore, relatively unaffected by the presence of Stage II, and the improvements from the previous recommendations are minimal. At the other extreme, the large contribution of drift to the FQ results prevents the adequate fitting of the first 20 s of data without the introduction of a linear term. Thus, while the effect of the drift error is lowered, it is replaced by a new error caused by the lack of an adequate fit. Nevertheless, the benefits of a single unambiguous slope are substantial.

Fig. 7
figure 7

Comparison of the CL Stage I results obtained from a 100 s hold segment fitted with a linear term kt (triangle) and from a 20 s hold segment without kt (circle) for all peak loads, with CSR (dashed line) results for a Zn, b Al, c Cu, d Ti, and e FQ

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Conclusions

To better understand the load dependence of CL nanoindentation strain rate sensitivity results, an experimental study of five different materials, supported by simulations, was undertaken. It was observed that:

  • CL results were split into two distinct regimes, but these were not the transient and steady-states usually reported. Instead, they consisted of a stage in which displacement rate far exceeded the drift rate (Stage I) and a stage in which drift errors were significant (Stage II).

  • Values of m calculated from Stage I were found to be similar to both constant strain rate (CSR) results and conventional values reported in the literature. Additionally, these did not depend on load.

  • On the other hand, Stage II results did not correlate with literature values and suffered from a significant dependence on applied load.

  • A set of recommendations were made to limit the occurrence of drift errors (Stage II). These included using higher loads and stricter drift limitations, as well as shorter hold durations (20 s), loading times (1 s), and physically sound fitting methods (with no linear term). This framework was shown to produce unambiguous results, with good correlations to conventional values and reduced standard deviations.