Cyclic Nanoindentation for Local High Cycle Fatigue Investigations: A Methodological Approach Accounting for Thermal Drift

Cyclic nanoindentation allows characterizing the influence of single phases and their interactions on fatigue mechanisms. Herein, a method for high cycle fatigue testing by nanoindentation is presented. By combining high‐ and low‐frequency indentation modes, high cycle numbers are achieved while obtaining sufficient data points to reconstruct force–displacement hysteresis loops. A challenge is the stochastic course of thermal drift which is addressed by measuring drift rate in regular low‐force holding segments. Drift rates are used to correct the displacement values, yielding reproducible cyclic deformation data as it is shown for two very different materials, a ductile metal and a brittle ceramic.


Introduction
Many components are exposed to cyclic loading, occurring in a variety of forms and under different conditions. This may, eventually, lead to material fatigue. Fatigue damage mechanisms rely on complex microstructural processes which need to be well understood to allow safe design of components, avoiding premature failure and the corresponding economical and personal damage. Hence, profound knowledge of the material fatigue behavior and mechanisms is essential under consideration of the desired application and number of cycles. Traditional macrofatigue testing provides detailed information on the cyclic deformation behavior and fatigue limits under a wide variety of loading conditions. However, a high number of experiments and standardized, usually relatively large samples are necessary, which is time-consuming and cost-intensive.
Cyclic indentation tests are an alternative, offering several advantages over standard fatigue tests: first, experiment duration and specimen numbers may be reduced drastically because multiple indents may be performed within a small area; second, the experimental setup is comparably simple and the only requirements on sample geometry are flat surfaces with sufficiently low roughness. An, increasing number of studies address the use of micro-and nanoindentation for fatigue investigations of and the correlation between behavior and damage mechanisms under uniaxial macrofatigue loading.
Cyclic nanoindentation offers answers to these scientific questions. The very small indents allow individual characterization of each phase, and even the smallest and most fragile samples, e.g., walls of closed-cell engineering foams or nanostructures can be tested. [10,11,14] For thin films and coatings, changes in hardness and Young's modulus were correlated with the fatigue-induced degradation of the elastic properties. In this manner the evolution of the overall elastic-plastic cyclic material behavior was investigated. [15][16][17][18][19] Most experiments were performed in the low cycle fatigue range. Occasionally, tests were extended to high cycles by using the high-frequency options that many nanoindentation devices provide for continuous contact stiffness measurements (CSM) or for nanodynamic mechanical analysis (nanoDMA). CSM allows determination of material properties complementary to those from quasistatic testing, e.g., contact stiffness is measured by superimposing a small dynamic load amplitude to a constant or increasing quasistatic load. [20] NanoDMA is typically used to continuously measure material properties of viscoelastic materials, e.g., storage and loss modulus, as a function of indentation depth by frequency or amplitude sweeps. These methods have been frequently used to determine the fatigue behavior of thin films by analyzing the evolution of contact or storage stiffness over time as measures for fatigue damage. [21][22][23][24][25] The applied frequencies span 30-300 Hz, yielding 2 Â 10 4 and 1.5 Â 10 6 cycles, respectively. [23][24][25] For the investigated thin films, the applied loads were relatively low, ranging from P = 10-22 μN for the constant mean load and P = 4-10 μN for the loading amplitude. As an exception, Weikert et al. used a mean load and loading amplitude of P = 10 mN to investigate carbon coatings. [25] In each case, the superimposed dynamic load was a sinusoidal oscillation.
For bulk materials, a common approach has been multicycle nanoindentation with an incremental load. [26][27][28][29][30][31] The tests were performed with very low maximum numbers of cycles in the range of 10-1000. A dynamic mechanical analysis mode was used to further increase the number of cycles up to 8 Â 10 5 . [32] Changes in the displacement amplitude over the number of cycles were interpreted as a fatigue measure. However, information on the force-displacement hysteresis could not be obtained due to the high testing frequency. To overcome this problem, Schmahl et al. proposed interspersing interspersed low-frequency "measuring" cycles between blocks of high-frequency "loading" cycles to characterize the fatigue behavor of single struts extracted from an open-cell aluminum alloy foam. [33] Thus, high numbers of cycles up to N = 10 5 could be achieved while collecting sufficient data points during the measuring cycles to evaluate the force-displacement hysteresis loops. This information was used to describe the deformation behavior of the material and to investigate the influence of the microstructure on the fatigue mechanisms during indentation. The same approach was applied to fatigue tests on Mg-SiC nanocomposites with. [34] In summary, cyclic nanoindentation is a promising method to characterize the local fatigue behavior of a wide variety of materials. The main advantages of the nanoindentation method are the precision and the high resolution of the force-depth signals in the range of micro-Newtons and nanometers. Unfortunately, this also increases the impact of potential errors or measuring inaccuracies on the results. The force-displacement data and the subsequently calculated mechanical properties are influenced by a variety of material, environmental, and mechanical factors. [35][36][37] A very important source of error is the thermal drift of the indenter which directly results from the mechanics of the testing device. Thermal drift is a well-known phenomenon also during quasistatic nanoindentation. [38,39] Therefore, many devices provide built-in options for drift correction. These are usually based on monitoring the displacement for a certain period of time before the actual test while maintaining a constant, very low load. The drift rate is calculated from the change of the displacement over time, assuming a linear relationship, and the measured displacement data are subsequently corrected by this calculated value.
Several authors have adopted this "quasistatic" approach. They measured the drift prior to their experiments at a low load and used the obtained value for the drift correction. [26,29] Others adjusted the method slightly by extending the holding segment until the thermal drift fell below a certain threshold before starting the cyclic indentation process. [25] Alternatively, drift monitoring segments were placed at the end of the fatigue test, following cyclic loading of the material, or both approaches were combined and the drift was measured at the beginning and the end. [19,30,40] A common assumption of these methods is that the changes in drift over time follow a linear equation. In contrast to the findings reported by Cairney et al. that support this assumption, other investigations on drift during quasistatic experiments have shown that it may not always be true. [19,39,41] For example, a quadratic relationship was observed between drift and time for long-term measurements and the authors therefore proposed that the holding time prior to the experiment should equal the duration of the indentation experiment. [39,41] A different approach is to use so-called reference segments at the beginning of the test during which initial values are determined, e.g., for the indent displacement in creep tests or for contact modulus in dynamic measurements. [32,38,42] Based on these values, which are presumably unaffected by significant thermal drift, the indent or contact area is continuously calculated throughout the measurement. [32,38] Thus, the influence of thermal drift on the results is reduced because they are independent of the actual measurement of the indentation depth during the experiment.
Overall, studies show that the force-displacement hysteresis loops obtained during repeated nanoindentation provide information about the evolution of the material fatigue behavior. However, that the use of cyclic nanoindentation for a detailed characterization of the fatigue behavior is still challenging, especially in the high cycle fatigue regime. It is difficult to reach high numbers of cycles while simultaneously obtaining sufficient data points to reconstruct the force-displacement data. This information, however, is essential to evaluate the elastic-plastic behavior of the material. In addition, the results are strongly influenced by thermal drift of the testing device. Reliable hysteresis data are, however, a prerequisite to use the measured local cyclic deformation data as input for material fatigue simulations aiming at the prediction of fatigue limits and behavior on the macroscale. Thus, the phenomena during cyclic nanoindentation need to be better understood, specifically during high and very high cycle fatigue tests. So far, no ultimate solution has been presented to overcome the reported problems, mainly, as profound knowledge of how drift changes over time is still missing.
Here, we propose a cyclic nanoindentation methodology for high cycle testing tackling the challenges described above. High-and low-frequency measurements can be combined in tests up to 10 5 cycles and the load function is complemented by holding segments at low forces. Using this approach, we actively measure changes in the drift rate during the long-term experiments which we then use to correct the displacement data. The proposed methodology has been validated for metals and ceramics and yielded reliable cyclic deformation curves.

Methodology Development
We used the TI 950 TriboIndenter for purposes different from those that it was originally designed for. This required modifying the standard testing and evaluation methods. These modifications resulted in a variety of challenges: we needed 1) to overcome the limited number of data points that can be collected, and 2) to consider the thermal drift and drift correction of the data.

Methodological Approach: High Cycle Nanofatigue Testing
With the high possible frequencies in the nanoDMA mode, we can reach significant numbers of indentation cycles fast. In the following, we refer to these cycles as "loading cycles".
However, data acquisition rates sufficient to fully reconstruct the force-displacement hysteresis loops would result in an immense number of data points swiftly exceeding the softwaregiven limit. Consequently, we used intermittent, relatively slow quasistatic "measuring cycles" allowing the collection of sufficient data points to fully reconstruct the force-displacement hysteresis loops without loss of information on the fatigue behavior. The deformation behavior in these "measuring cycles" is assumed to represent the cyclic deformation behavior of the preceding "loading cycles". Figure 1 displays the load function we used for the tests on the α-Al matrix: the first ten cycles were performed with a low frequency of 0.1 Hz. Subsequently, batches of fast loading cycles, performed with 201 Hz, were interrupted after 100, 1000, and then after every 10 000th cycle to perform three measuring cycles (frequency f = 0.1 Hz). Indentation was performed with closedloop force control. For both frequencies, the minimum force was P min = 75 μN and the maximum force P max = 965 μN (force range ΔP = 890 μN).
For the tests on zirconia, the minimum and maximum forces were P min = 0.35 mN and P max = 10 mN, respectively. Loading and measuring cycles were executed with a frequency of 201 and %0.5 Hz (corresponding to a loading/unloading rate of 0.1 mN s À1 ), respectively. Besides the different force and, consequently, frequency settings, the overall load function was the same as for the α-Al matrix.

Thermal Drift
After considering the specifications and mechanics of the testing device, the authors concluded that the data may be influenced by the thermal drift of the nanoindenter. The thermal drift of the testing device originates from the axial movement of the piezo attached to the transducer and cannot be avoided. Drifting of the piezo during indentation causes a constant readjustment of the force signal and a consequential deflection of the transducer's middle plate. This deflection interferes with the actual indentation process and results in a faulty displacement signal. Therefore, before each indentation test the software allows to perform an automated drift measurement. After approaching the sample, the drift rate is determined by measuring the displacement variation at a constant small force. After indenting, the displacement data are corrected using this specific drift rate. This procedure is precise and reliable for short-term measurements because drift changes are approximately linear with time over short periods. However, over longer time spans, drift rate changes; thus, we cannot estimate the drift from the initial drift measurement for our nanofatigue experiments that take hours as compared to minutes of a standard quasistatic or nanoDMA test.
To understand the exact progression of thermal drift over time, we measured the displacement on quartz, an Al sample and zirconia for a constant force of 10 μN held for 1 and 2 h, which is equal to and twice the duration of the cyclic nanoindentation tests, respectively. The low load was chosen to ensure contact between sample and indenter while excluding any potential influence of creep. Typical results are shown in Figure 2.
Measurements performed on quartz, Al, and zirconia are marked by the prefixes Q-, Al-, and Z-, respectively. Tests Q-1 and Q-2 were performed for 1 h, while the displacements for all other tests were monitored for 2 h. In all experiments, except for Q-1, we observe a steady increase in displacement with time that levels off over the course of the test. Neither the material nor creep seems to influence the measurements, as the inclinations and the maximally reached values decrease in a stochastic series Figure 1. Schematic load function showing the first ten low-frequency loading/measuring cycles, followed by blocks of high-frequency loading cycles (blue) and low-frequency measuring cycles (black). The frequency and force values refer to the test on the α-Al matrix. www.advancedsciencenews.com www.aem-journal.com Z-1, Q-2, Al-1, Q-3, Al-2, Z-2, where no material influence is observed. For Q-1, the displacement progression deviates from the other experiments showing a wave-like pattern with turning points at around 450 and 1900 s. Thus, the long-term holding experiments show that the drift rate can change stochastically over the relatively long measuring times of our fatigue experiments. These changes cannot be approximated/forecasted based on a value obtained at the beginning of the experiment, as it is common practice -and applicable -for short-term experiments. Hence, a frequent evaluation of thermal drift is necessary to detect any potential variations of the drift rate. Figure 2b shows a magnified view of the first peak of experiment Q-1. The progression of the displacement is approximated by linear functions for a span of 50 (blue) and 200 s (red). The linear fit for the 50 s stretch fits the slope of the displacement versus time curve perfectly. With small deviations, the curve progression can also be approximated by linear regression for a time span of 200 s.
In our cyclic tests, all but the initial two dynamic loading segments lasted 50 s; 200 s is the duration of the initial ten slow measuring cycles and this is the longest period between two consecutive drift correction segments.

Methodological Approach: Drift Correction
To allow drift correction, holding segments were introduced into the load function ( Figure 3) to measure the displacement at a constant low force before and after the measuring cycles, allowing reconstruction of the drift over time function. These "drift correction segments" (DCS) were performed at 10 μN (α-Al matrix)/350 μN (zirconia) to maintain constant contact between tip and sample without causing creep; the force was held for 60 s. The evolution of the displacement over time is due to thermal drift and subsequently used to individually calculate the drift rate for each segment. A high number of relatively long DCS was used to detect any changes in the drift rate in between the slow measurement cycles, and between successive measurement segments, and to eliminate the influence of momentary fluctuations and noise in the force or displacement signal. Figure 4 shows the strategy for drift correction for the tests on the α-Al matrix. As the time between the two DCS before and after a measurement cycle (Figure 4a,b) or of the loading segments is very short, the evolution of the drift between successive DCS is assumed to be linear, based on our drift measurements at constant low force (compare Figure 2b). First, the drift rates for two successive DCS are calculated by applying a linear fit to the displacement-over-time data (Figure 4c). Then, a linear regression through the two resulting drift rates is performed providing drift rate as a function of time for the two DCS and the embedded measuring cycles (Figure 4d), or between one and the next measuring cycle. Subsequently, the drift rate and the resulting drift-induced error of the displacement are calculated for each data point enclosed by the two DCS. The same strategy was applied to calculate the drift during the loading cycles, interpolating between the DCS after one measuring cycle, and the DCS before the successive measuring cycle (i.e., after the next loading segment). Thus, during this process, the influence of the drift measured in all previous DCS is taken into consideration. As a result, the displacement data from the measuring cycles are corrected by the value of the corresponding displacement caused by the drift accumulated over the whole preceding course of the experiment, resulting in true displacement values that exclude the influence of the thermal drift of the piezo. This process is then repeated for each of the analysis segments before the data were further processed and evaluated to investigate the nanoscale cyclic deformation behavior of the material.
Several nanofatigue tests were performed in the α-Al-matrix of the A365 open-cell Al foam. Figure 5 shows the progression of the displacement for each of the 26 DCS and the resulting calculated drift rates for a typical dataset. Cyclic nanoindentation causes fatigue of the material which, among other effects, results in an increase in penetration depth with the number of cycles. The repeated indentation between the DCS leads to different initial displacement values for each DCS. The displacement data of each DCS were normalized by subtracting the corresponding initial displacement value. Therefore, the changes in displacement displayed in Figure 5 are solely due to the drift of the piezo. Note that the normalization was only performed for clarity of this graph, and that, naturally, the data progression and slope remain unaltered. During holding, the displacement increases in all DCS but the inclination varies between the segments. It is steepest in DCS-1. Within every single DCS, displacement increases comparably steady with a constant inclination for most segments; others, however, e.g., DCS-6, 9, 24, and 25, display fluctuations in the displacement data. This is particularly pronounced for DCS-10 which shows a distinctive peak. However, the overall inclination of each curve is still predictable. The observations for the displacement progression during the drift correction segments show that the segments are long enough to perform a reliable prediction of the displacement progression which is unaffected by local fluctuation and noises of the displacement signal. Hence, the results from the drift correction segments can be used to calculate the drift rate by applying a linear regression through the displacement data. The results for each segment are shown in Figure 5 above the corresponding displacement progression. The highest drift rate is calculated for the initial DCS-1 (0.14 nm s À1 ) but decreases significantly Figure 3. Schematic modified load function with additional DCS: the initial load function with low-frequency measuring cycles (black) and highfrequency loading cycles (blue) was extended by 26 DCS (red lines and arrows). During these segments, the force was maintained at 10 μN (α-Al matrix)/350 μN (zirconia) for 60 s and the displacement was constantly monitored.
www.advancedsciencenews.com www.aem-journal.com to approximately 0.053 nm s À1 in DCS-2. It subsequently increases or decreases reaching a minimum of 0.035 nm s À1 at DCS-8. The changes in the drift rate between each segment occur in a stochastic manner but correlate to the inclination of the displacement data in the corresponding drift correction segment. The differences resulting from the drift correction on the displacement values are shown for the progressions of the maximum displacement at maximum force D max and the minimum displacement after unloading D min versus the number of cycles in Figure 6. The data of the first N = 9 measuring cycles and subsequently the second cycle of each cluster of N = 3 measuring cycles in between the loading cycles were included in the evaluation. Cycle 10 was not considered because D min could not be detected reliably due to the transition into the first 100 highfrequency loading cycles.
For both, the α-Al matrix and zirconia, the progressions of the minimum and the maximum displacement are very similar. The values for D min and D max steadily increase over the number of cycles, independent of the drift. However, the drift corrected values are all lower and the inclination of the curves is not as steep compared to the noncorrected data. The discrepancy between the noncorrected and corrected curves is small for the first cycles but increases throughout the experiment. This is further illustrated by the progression of Δ drift D max , representing the change in maximum displacement due to drift by calculating the difference between the corrected and the noncorrected value for each evaluated cycle, over N.
The differences are much higher for the tests on the α-Al matrix as compared to those on zirconia. For zirconia, D min and D max increase for the first nine cycles followed by a saturation-like state up to N = 1000 and a subsequent secondary inclination with a rising slope until the end of the experiment. For both materials, the differences increase with the number of cycles.
The magnified view of the progression of D min over the first 100 cycles (Figure 7a) shows that the uncorrected and the corrected curve intersect twice. This specific progression results from a change of sign of the drift rate and is explained as follows (Figure 7b,c): drift rates calculated from DCS-1 and DCS-2 are used to correct the displacement data of the first ten measuring cycles (M 10 ). The drift rate is positive for DCS-1 but negative for DCS-2 ( Figure 7b); because an individual drift rate is calculated for each hysteresis loop with relative portions of the drift rate measured in DCS-1 and DCS-2 (see Figure 4), this change in signs results in a first crossover of the curves after approximately four cycles. The noncorrected data, therefore, is too high at the beginning, and too low at the end of this segment, and correction leads to a flattening of the curve. After performing the next 100 loading cycles (L 100 ), DCS-3 and DCS-4 are considered to correct the measuring cycles M 100 . During these cycles, the drift rate changes sign again, from À0.0034 to þ0.0094 nm s À1 , hence the curve for the noncorrected raw data crosses the corrected curve again. Subsequently, the drift rate is positive throughout the entire experiment and the noncorrected curve progresses above the corrected data. This data is thus an example of drift

Evaluated Parameters
During cyclic loading, fatigue mechanisms cause changes in the material. Plastic deformation can be displayed by the incremental  www.advancedsciencenews.com www.aem-journal.com change of D min (ΔD min ) between consecutive measurement cycles N x and N xþ1 . To eliminate the influence of small fluctuations in the applied force, the minimum displacement is scaled by the minimum force of the corresponding measurement cycles and the resulting difference Δ is further normalized by the number of cycles over which the change occurred Further, the ratio of minimum to maximum displacement for one measuring cycle provides information about the amount of plastic deformation during each cycle. Increasing values of this ratio indicate an overall more elastic behavior with little plasticity. The more plastic deformation is seen, the lower the ratio becomes. Thus, over the course of cyclic loading, decreasing values indicate cyclic softening, while increasing values indicate cyclic hardening.

α-Al Matrix
Loading was performed on A356 foam sections prepared as described below (see Section 6) with the developed load function (see Figure 3). The minimum and maximum force in the loading and measurement cycles were P min = 75 μN and P max = 965 μN. The load in the DCS was set to P DCS = 10 μN. The frequency of the loading and measurement cycles was 201 Hz and 0.1 Hz, respectively, and the holding time for the DCS was 60 s. The results are displayed in Figure 8.
Within the first two cycles, the material exhibits plastic deformation leading to pronounced hardening and resulting in a decrease in ΔD min_norm . Subsequently, the values increase again, until the curve reaches a peak at N = 9. For further cycling, the value drops drastically again until N = 100 before the curve declines with a constant slope until the end of the experiment. The differences in the corrected and noncorrected curves are especially apparent at the beginning of the measurement, due to the initially higher drift rates. The values for the original data are twice as high compared to the corrected values and the inclination following the initial plastic deformation is less steep. Correcting the data also reduces the secondary decline between N = 9 and 100 but does affect the curve progression from N = 10 000 ongoing to a much lesser extent with still higher values for ΔD min_norm in the uncorrected data set.
This observation also applies to the curves of the ratio D min / D max . However, apart from that, their progression is very similar, only showing small deviations in the slope of the curve after 100 cycles. The material exhibits a higher proportion of elastic deformation per cycle in the first two cycles (strain hardening) followed by a higher proportion of plasticity up to cycle 5 (softening). From N = 6, the curves increase constantly except for two small peaks at N = 8 and N = 80 000, indicating less and less additionally stored plastic deformation per cycle and increasing cyclic hardening.

Zirconia
Typical results of tests performed on a zirconia section prepared as described below (Section 6) are summarized in Figure 8. The minimum and maximum forces were P min = 350 μN and P max = 10 000 μN, respectively. The DCS were performed at P min . Loading and measuring cycles were executed with a frequency of 201 Hz and %0.5 Hz (corresponding to a loading/ unloading rate of 0.1 mN s À1 ), respectively. Besides the different force settings, the overall load function was the same as for the α-Al matrix. Besides small fluctuations, the curves for ΔD min_norm constantly decrease until N = 1000 and progress with nearly constant values from N = 10 000 to the end of the test. The evolution of D min /D max over the number of cycles is very similar compared to the progression of the D min and D max curves (compare Figure 6 and 7).
From N = 100, the overall courses of the corrected and noncorrected curves are very similar. In accordance with the positive Figure 7. Influence of the sign of the drift rate on the curves for Zr0 2 : a) magnified view of the progression of D min over N for the first 100 cycles with (red) and without (black) drift correction (DC). The curves overlap after 4 and between 9 and 100 cycles (black arrows); b) calculated drift rate for each DCS throughout the experiment; the discussed values are numbered consecutively; c) first section of the load function showing the DCS (red arrows and numbers), the measuring cycles M x and the corresponding loading segments L x (blue arrows).
www.advancedsciencenews.com www.aem-journal.com drift rates seen in the DCS (Figure 7b), the noncorrected data values are higher. However, a different observation was made within the first 100 cycles: after cycle 4, the corrected curve is higher than the noncorrected curve, before crossing it between N = 9 and N = 100. This is seen for the ΔD min_norm curve (Figure 8), and it is also visible in the magnified view of the D min curve (compare Figure 7). As described above for the D min and D max data the intersection of the curves with and without drift correction is due to the change of sign of the drift rate during the first 100 cycles (see Figure 7). The differences between the noncorrected and the corrected data are very small for all evaluated parameters, but they are higher for higher cycle numbers of N = 10 000 and above.

Discussion
We report a method for the characterization of the nanoscale fatigue behavior under consideration of thermal drift and data acquisition issues in high cycle nanoindentation fatigue tests. Our findings highlight the importance of thermal drift correction in these long-term tests. In the following, the proposed method for drift correction and introduction of low-frequency measuring cycles, and the influence of thermal drift on the measured displacement parameters are discussed in detail.

Thermal Drift Correction
Most of our long-term holding experiments ( Figure 2) show a degressively increasing course of the displacement-versus-time curve for time spans equivalent to the duration of the nanofatigue tests. While this is in accordance with the observations of some authors, others assumed or measured a linear progression. [19,39,41] These inconsistencies may be due to different testing or loading conditions. In addition, environmental conditions, mainly room temperature, might influence the thermal drift. Under the testing conditions that we used, a linear evolution of the displacement-versus-time curve can certainly be excluded. Moreover, sometimes, thermal drift follows a sinusoidal waveform. Additionally to the external factors, the continuous tip movement during the nanofatigue experiments might influence changes in the drift rate. Consequently, drift correction methods based on a linear progression of the displacement are not suitable for our nanofatigue tests. This especially applies to very short drift correction segments at the beginning or the end of the actual experiment. While increasing the holding time improves the accuracy of this approach, it does not consider turning points in the thermal drift that may occur (see, e.g., holding experiment Q-1). The TI 950 Triboindenter provides a "reference mode" for drift correction, used, e.g., by Diehl et al. [32] This approach to drift correction works well for CSM in experiments solely performed www.advancedsciencenews.com www.aem-journal.com with the nanoDMA mode. However, for the proposed method, combining high-frequency loading cycles (performed in the nanoDMA mode) with slower measuring cycles, it is not suitable. The measuring cycles are necessary to collect sufficient data points for the evaluation of single hysteresis loops to characterize the elastic-plastic behavior of the material due to software limitations restricting the overall number of data points that can be acquired during one experiment. These slow measuring cycles are not performed in the nanoDMA mode, thus the "reference mode" cannot be used for drift correction. An additional issue is the duration of the nanofatigue tests which are much longer than classical nanoDMA tests. Consequently, we need to measure the drift actively, and we propose to do so in DCS before and after the measuring cycles. We showed that holding times of 60 s are long enough to exclude the influence of short-term events. In addition, the time between two subsequent DCS is short enough to approximate the changes in drift using linear regression (compare Figure 2). The authors are aware of the minor discrepancy between the fit over 200 s and the curve progression. However, the error for the example shown in Figure 2b is comparably small (6.5%). Hence, the approach is suitable if no turning point occurs within 200 s, which was only observed in one of the conducted long-term holding experiments. Such a sinusoidal progression of the displacement (experiment Q-1 and the zirconia example shown) is an extremely rare event in the testing times we used. However, for longer times in experiments reaching even higher maximum numbers of cycles, changes in the drift direction might become more frequent. The proposed method will yield reliable drift correction also in those cases. The data of the measuring cycles are corrected based on the preceding and subsequent DCS by applying a linear regression through the respective calculated drift rates. Considering both DCS instead of only one further increases the accuracy of the method because thereby changes occurring during the measuring cycles are not neglected. From the linear regression through the drift rates, the drift displacement for any point of the enclosed measuring cycle can be calculated, allowing correction of all corresponding displacement data. This way, we were able to extract only fatigue-induced changes in the displacement.

Influence of Drift Correction on the Evaluated Parameters
Al alloys and zirconia have highly diverging mechanical properties, hence the courses of the curves displaying the material fatigue behavior are very different. Generally, for all our tests, we observed higher drift rates and, consequently, bigger differences between the corrected and noncorrected curves for the Al material than for zirconia. Changes of the sign of the drift rate were observed very seldom, and if so, then for tests on zirconia. Both observations may be due to the number of indents performed directly after each other which was lower for Al than for zirconia, and the lower overall and holding force in the Al experiments. The stability of the testing device is expected to increase the more experiments are performed within one batch. Hence, the order of the tests correlates with the magnitude of the drift.
The drift correction impacts the overall curve progression and the data values to a different extent depending on the chosen evaluated parameter. The drift increasingly influences the values for the minimum and maximum displacement with ongoing cyclic loading but the curve progressions for D min and D max are very similar for both the corrected and the original dataset. This is to be expected because all displacement values for one hysteresis loop are corrected by the same drift correction value. Due to the positive drift throughout the test, all hysteresis loops are shifted to smaller displacement values. Minor changes of the drift rate between each DCS (compare Figure 5) still influence the course of the D min and D max curves (compare Figure 6), especially for higher numbers of cycles, despite the decreasing drift rate with time. This is due to the adding up of drift values throughout the test.
For noncorrected data, evaluating the D min /D max ratio minimizes the influence of thermal drift on the progression of the curve. Further, the course of the D min /D max ratio is less influenced by the drift and the drift correction than the ΔD min_norm graph. This latter parameter, showing the incremental change in plastic deformation, shows more pronounced variations between the values with and without drift correction especially within the first 100 cycles. The high drift rate at the beginning of the experiments leads to very different correction values for D(N x ) and D(N xþ1 ); with increasing number of cycles, drift rate decreases: thus, the changes in drift correction from one DCS to the next, also one decreases. Consequently, ΔD min_norm is less influenced by the drift correction with increasing number of cycles. Importantly, however, the general curve progression with significant spikes, especially up to N = 100, is not changed by the drift correction.
In conclusion, the ΔD min_norm and D min /D max data are less influenced by the drift compared to the directly measured parameters D min and D max . While the absolute values of all parameters change, the overall type of progression of the curves is not changed by the drift correction. [33] Thus, while also the noncorrected curve progressions reflect the temporal changes in fatigue processes, the drift has to be corrected to reliably compare different datasets and characterize the cyclic deformation behavior in a quantitative manner, e.g., for simulations of fatigue life predictions.
In summary, the results confirm that the proposed method is valid to characterize the nanoscale fatigue behavior by cyclic nanoindentation. The influence of the varying thermal drift is eliminated by actively measuring the displacement in drift correction segments throughout the experiments. Hence, the driftcorrected values and parameters are considered to reveal the "real" fatigue response of the material. Experiments on different materials show that the drift correction is valid, independent of the magnitude and sign of the drift rate or the loading conditions.

Influence of Testing Conditions
There is concern whether the change of frequency between the measuring and the loading cycles affects the results and whether the values measured under low-frequency loading conditions indeed represent the fatigue damage accumulated during the preceding high-frequency loading. The influence of loading frequency has previously been investigated for macrofatigue tests in the very high cycle fatigue (VHCF) regime (frequency range 50 Hz to 20 kHz). Higher frequencies may affect the fatigue processes due to strain rate or testing environment. [43] In addition, inadvertent heating of the sample due to plastic deformation needs to be prevented. [43] For fcc materials such as aluminum, the influence of strain rate is usually negligible and the results are predominantly affected by the environment. [43][44][45] While the influence of strain rate is negligible during macroscale fatigue testing of fcc materials, it is of concern in highfrequency nanoindentation CSM. [46][47][48] Especially materials with a high elastic modulus-to-hardness ratio (E/H ratio), such as copper, exhibit strain rate sensitivity even under small oscillation amplitudes in the range of a few nanometers. [47,48] There are two different causes for this strain rate sensitivity: one is the onset of plastic deformation during the dynamic unloading of the material. [48] As a result and as compared with quasistatic results, hardness is overestimated and modulus is underestimated which is less observed in materials with lower E/H ratio, at higher frequencies, and with increasing indentation depth. [46][47][48] A second cause is related to lock in amplifier errors leading to the underestimation of stiffness, as shown by Merle et al. [46] Increasing the testing frequency solves this problem. For materials with E/H ratios of 100 or below, these authors showed that the stiffness is underestimated by 5% or less at a frequency of 40 Hz. From the above we conclude that testing an Al material with a very low E/R ratio of 83 at a frequency of more than 200 Hz is insensitive to the strain rate such that the lowfrequency measuring cycles well characterize the behavior of the loading cycles. Furthermore, we evaluate the forcedisplacement hysteresis in the low-frequency measuring cycles which are very similar to quasistatic nanoindentation testing such that the machine-related error is of no concern. Moreover, plasticity-induced heating is of less concern in the test reported in the present article because the plastically deformed volume is very small as compared to the unaffected volume of the sample, and the thermal conductivity, especially for the Al sample, is high. Nevertheless, possible frequency influences are part of ongoing and yet unpublished investigations, especially for the zirconia material.
Another possible concern is creep that might occur during the drift correction segments. To exclude this, the load was maintained at a low value. In our long-term holding experiments, we tested three materials with very different deformation behavior and sensitivity to creep: hard and brittle inorganic materials (zirconia, quartz) and a very ductile metal (aluminum). The comparable curve progressions are an indicator that creep can be excluded. In addition, the inclination of the curves varied significantly, even within one material group, and for constant loading parameters and environment. Moreover, results for Q-1 and the drift rate within the DCS during the nanofatigue experiments on zirconia show a change of sign, which is not typical for creep behavior. Moreover, for fused quartz, investigations of others have shown that the behavior is fully elastic below a holding load of 400 μN. [49] Hence, we are convinced that any displacement changes during the long-term holding experiments and, thus, during the DCS result solely from thermal drift.
In nanoindentation tests, machine stiffness may influence the displacement measurements. Therefore, a constant machine stiffness is predefined in the data processing software, and used to correct the displacement values in all nanoindentation tests.
For testing with spherical indenters, Li et al. report variations of machine stiffness with increasing force and indenter tip radius. [50] For small indenters, deviation was negligible; for larger indenters, significant influence was observed for loads above 500 mN. As the radius of the Berkovich indenter tip we used is much smaller than the smallest indenter Li et al. investigated and because our loads are much lower, we are confident that the influence of a nonconstant machine stiffness with increasing load is negligible in our tests.

Outlook
Summarizing, cyclic nanoindentation is a very suitable method to assess the local fatigue behavior of inhomogeneous materials, specifically the influence of microstructural inhomogeneities on the cyclic deformation behavior. We previously showed for specimens extracted from precision-cast A356 open cell foams that cyclic nanoindentation leads to dislocation structures similar to those observed in macrofatigue tests. [33] Silicon particles in the eutectic influence fatigue-induced microstructural changes in a volume of 5-10 μm below the indent, and the progression of the displacement parameters (D min /D max , ΔD min,norm ) correlates with the indent size and pile-up, dislocation structure below the indent, and the microstructure, that is the presence and orientation of brittle phases, in the indentation-affected zone. While the most prominent changes to the cyclic deformation occur at the beginning of the tests within the first 10-100 cycles, we observe further changes up to the maximum number of cycles. Interestingly, we see fluctuations in the cyclic deformation parameters that we attributed to residual stresses in the material. Ongoing work will show the temporal development of the fatigue-induced microstructure and the correlation with the microstructure in greater detail.
Cyclic nanoindentation is further a very suitable method to investigate the fatigue behavior of small structures and of materials that are otherwise very difficult to test in fatigue, such as brittle metal states or ceramics. Further, the influence of interfaces in composite structures can easily be assessed. The results shown for zirconia samples are one example. We are currently investigating the influence of transformation toughening during nanofatigue in layered structures, made by electrophoretic deposition from metastable, tetragonal zirconia partially stabilized with 3 mol% yttria and stable, cubic zirconia stabilized with 8 mol% yttria.

Conclusions
We proposed a modified method for cyclic nanoindentation up to 10 5 cycles under consideration of thermal drift, employing the high-frequency dynamic measurement mode of the TriboIndenter: 1) Sufficient numbers of data points to reconstruct the hysteresis loops were ensured by adding relatively slow "measuring cycles". 2) Holding experiments showed stochastic changes in thermal drift during the time of the cyclic nanoindentation tests. These variations can be detected by measuring the displacement at a constant low force during "drift correction segments". 3) Long-term development of thermal drift is considered by regularly interspersed DCS over the whole course of the cyclic nanoindentation experiment. Thus, also tests with higher maximum cycle numbers can be performed and reliably corrected. 4) Sixty seconds are long enough to calculate the drift rate from the DCS, without the influence of short-term fluctuations. Linear regression through the calculated drift rates of two subsequent DCS is sufficiently exact to correct the data of the enclosed measuring cycles. 5) Drift correction significantly influences the results of the cyclic nanoindentation tests, to a different extent for different evaluated parameters, and different materials.
Concluding, the method yields reliable results for nanofatigue testing of a broad variety of materials. In combination with 2D and 3D structural investigations, local fatigue responses, and how they are influenced by the microstructure, can be evaluated for metals and ceramics alike, e.g., [33] This approach paves the way for modeling the fatigue response.

Material and Sample Preparation:
We used an open-cell A356 (7 wt%Si, 0.3 wt%Mg) aluminum (Al) alloy foam fabricated by investment casting at the Foundry Institute of RWTH Aachen for the development of the method. [51] The foam structure was first divided into cuboids with an edge length of approximately 1.5-2 cm using an Accutom-50 saw (Struers, Ballerup, Denmark) with force set to "LOW" and feed rate set to 0.05 mm s À1 to minimize the induction of plastic deformation into the material. The small cuboids were subsequently embedded using EpoFix resin (Struers, Ballerup, Denmark) and ground with a series of silicon carbide papers followed by polishing with diamond suspension with a grain size of 3 μm and colloidal slurry with a particle size of 50 nm. Considering that results from nanoindentation are severely sensitive to the surface state and quality of the investigated area, utmost attention was paid to the specimen preparation process, ensuring a flat, smooth surface finish while avoiding plastic deformation of the surface layer.
The microstructure of the struts consists of an alpha aluminum matrix (α-Al matrix), eutectic silicon particles, and intermetallic phases. [33] Indentation was exclusively performed in the α-Al matrix for the development of the proposed nanofatigue method.
TI 950 TriboIndenter Testing Device: A TI 950 TriboIndenter (Bruker, Billerica, USA) equipped with a Berkovich tip and a capacitive three-plate transducer (normal force noise floor <30 nN, normal displacement noise floor of 0.2 nm) was used. It is attached to a piezo which is used for lateral positioning and sample approach. In the force-controlled mode, a force signal is applied and the resulting displacement is measured by the deflection of the center plate. All tests were performed in closed-loop force control, and force and displacement are recorded throughout the indentation tests. [53] In addition to standard quasistatic nanoindentation, the instrument allows nanoscale dynamic mechanical analysis (nanoDMA) which is typically used to continuously measure material properties of viscoelastic materials, e.g., storage and loss modulus, as a function of indentation depth by frequency or amplitude sweeps. To reach high numbers of cycles in the range of and above 10 5 with nanoindentation, we used this nanoDMA mode but slightly modified the standard method by maintaining a constant frequency and by loading with comparably high amplitudes (see Section 2.1).