Effect of loading/unloading rate on instrumented indentation measurements in alloy Cu-2%Be

The Cu-2Be (wt%) samples homogenized for 10 min at 800 °C consist of an α phase matrix with some annealing twins and exhibit a grain size of around 40 μm (figure 1(a)). After an aging treatment at 400 °C, Vickers microhardness increases from around 100 HV to a maximum of 364 HV after 1 h of aging (figure 1(b)). For longer aging times, Vickers microhardness decreases slightly and reaches a value of 314 HV after 10 h of aging.

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In a previous study of the same alloy [11], and based on calorimetric studies, the homogenized sample was found to consist of an α solid solution phase. This was concluded since after heating the homogenized sample from room temperature to around 550 °C, the presence of exothermic reactions associated with the formation of GP zones, γ'', γ' and γ phase was observed [11]. Therefore, the homogenized samples (M1) correspond to an α phase matrix.

After 1 h of aging, the maximum Vickers microhardness was reached (figure 1(b)). The formation of γ' precipitates has been reported to produce the most effective hardening and strengthening among metastable precipitates in Cu-Be alloys [4, 9, 31]. On the other hand, the calorimetric results of a previous work on a similar alloy indicated the presence of γ' phase for the maximum value of Vickers microhardness and elastic modulus for an aging time in the range of 0.75 to 1 h at 400 °C [11]. Therefore, the sample aged for 1 h at 400 °C (M2) would correspond to an α matrix with γ' metastable nanoprecipitates.

To estimate the size of the precipitates formed after aging for 0.75 h at 400 °C, SAXS measurements were performed. Likewise, measurements were made on a homogenized sample to corroborate the absence of nanoprecipitates. In figure 2, the Kratky plots (I_c ·q² versus q, where I_c is the corrected intensity) are shown for M1 and M2. In M2, a peak is observed at values of q close to 0.018 Å⁻¹, which would be associated with the scattering due to the presence of γ' nanoprecipitates. The Guinier radius (R_g ) can be estimated from q, where the maximum value of I_c ·q² (q_max) is reached [29, 30]:

$$\begin{eqnarray}&&{R}_{g}=\sqrt{3}/{q}_{{\rm{\max }}}\end{eqnarray} \tag{ 4 }$$

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The value obtained for R_g was 98 Å in the sample M2. R_g is of the order of the radius of the precipitate (R): for monodisperse spherical precipitates R_g = (3/5)^1/2 R [29]. In the case of ellipsoids with aspect ratios between 0.5 and 2, R_g obtained from equation (4) provides a good estimation of the precipitate size. For larger aspect ratios, R_g should be used to describe, with caution, the average precipitate size [29]. The area under the curve in the Kratky plot is related to the volume fraction of the precipitates. Therefore, the nanoprecipitates present in sample M2 would have sizes of the order of 10 nm. In contrast, the presence of precipitates was not observed in M1 (figure 2).

Longer aging times induce a softening of the material (figure 1(b)) due to an increase in the misfit angle between the γ' metastable nanoprecipitates and the matrix, and the increase in the length and width of the nanoprecipitates [6, 9]. As the aging time increases, the γ' nanoprecipitates transform into γ phase [4, 9]. Therefore, the sample aged for 10 h at 400 °C (M3) would correspond to an α matrix with precipitates close to the γ phase.

Figure 3 shows representative P-h indentation curves at different loading/unloading rates of samples with different microstructures. The indentation curves of the sample in α phase (M1) were influenced by the loading/unloading rate (figure 3(a)). h_max for a load of 9 mN decreases slightly from 431 nm to 421 nm as the rate increases from 0.15 to 1 mN s⁻¹ (figure 4(a)). However, the final depth (depth at the end of the unloading, h_f ) remains almost constant with the rate, with a mean value of 369 nm and a standard deviation of 5 nm (figure 4(b)). The decrease of h_max with the rate is similar to the behavior reported for a polycrystalline copper sample subjected to different loading rates between 0.05 and 50 mN s⁻¹ at a maximum depth of 10 mN [26] and to that found in a copper (110) single crystal subjected to loading rates between 2 and 25 mN s⁻¹ at a maximum depth of 100 mN [27]. The indentation curves of the samples aged at 400 °C show a different behavior. In the samples with precipitates (M2 and M3), h_max increases very slightly with the rate (figurex 3(b) and 4(a)), while an increase in h_f is observed between the lowest rate (0.15 mN s⁻¹) and the highest rates (figures 3(b) and 4(b)).

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AFM images are shown in figure 5 for samples M1 and M2 after the indentations at different loading/unloading rates. The sample without precipitates (M1) does not exhibit the presence of pile-up (figures 5(a) and 5(b)), while the samples with precipitates (M2 and M3) exhibit the presence of pile-up (figures 5(c) and 5(d)), which agrees with the results reported in [18] on samples with similar heat treatments. Differences in the size of the indentations after unloading can also be observed between the samples with and without precipitates, which may be related to the differences in hardness (figure 1(b)) and the final depth obtained (figures 3 and 4(b)). However, no differences were observed with the loading/unloading rate.

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For a Berkovich indenter, the indentation strain rate (SR) during loading can be obtained using the following equation [26, 32]:

$$\begin{eqnarray}&&SR=\,\frac{\dot{h}}{h}\,\end{eqnarray} \tag{ 5 }$$

where $\dot{h}$ is dh/dt, h is the depth and t is the time. The variation of SR with h for samples with different microstructures is shown in figure 6. At the initial contact, SR exhibits a high value and decreases rapidly at greater depths. The levels of SR are influenced by the loading rate. For sample M1 and at a rate of 1 mN s⁻¹, SR reaches a value greater than 50 s⁻¹ at a depth of 2 nm, decreasing to ∼11 s⁻¹ at 10 nm and then to ∼5 s⁻¹ at 20 nm (figure 6(a)). On the other hand, at a rate of 0.15 mN s⁻¹, SR exhibits a value of only ∼10 s⁻¹ at the depth of 2 nm, 1.5 s⁻¹ at 10 nm and 0.8 s⁻¹ at 20 nm. A similar behavior has been reported by Chen et al [26] in a polycrystalline copper sample. For the samples with precipitates, there is also an increase in SR with the loading rate for a given depth (figure 6(b)). Figure 7 shows the values of SR obtained for different loading rates from representative curves in figure 6 at depths of 2 nm (figure 7(a)) and 10 nm (figure 7(b)). SR shows an almost linear increase as a function of the loading rate for samples M1 and M2. On the other hand, SR experienced a decelerated increase with the loading rate in sample M3.

It has been reported that in polycrystalline copper samples, the most probable deformation mechanism is slip of independent slip systems at strain rates lower than 1 s⁻¹ [26, 33]. In our case, these rates were reached for the lowest loading rate (0.15 mN s⁻¹) and depths greater than 10 nm (figure 6(a)). For higher rates, higher strain rates are achieved, and cross slip and dislocation nucleation can occur [26, 33]. In the homogenized Cu-2Be samples studied in the present work, the interactions of the dislocations with the grain boundaries can be neglected since the grain size of the samples is around 40 μm, significantly greater than the size of the indentations, which are in the range of 1 to 2 μm (figure 5).

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E_r and H were obtained from the P-h curves following the Oliver and Pharr methodology [16, 17]. As was observed in figure 5, and from the images obtained by AFM of the surfaces of the samples after the indentations, the samples with precipitates exhibit pile-up (M2 and M3), while the homogenized samples do not (M1). Therefore, the correction proposed by Loubet et al for h_.c. was applied to the samples that exhibit pile-up, M2 and M3, for the calculation of H [18, 20]:

$$\begin{eqnarray}&&{h}_{c}=1.2\left({h}_{\max }-{P}_{\max }/S\right)\end{eqnarray} \tag{ 6 }$$

The variation of H with the loading/unloading rate is shown in figure 8. It can be observed that for the sample M1, H remains almost constant with the rate at a value around 2.4 GPa. Choi et al [27] found a similar behavior of H with the loading rate in a conventional copper single crystal over a wider range of loading rates from 2 to 25 mN s⁻¹. Chen et al [26] reported a slight increase in H with the loading rate for a polycrystalline copper sample in the range of 0.05 to 50 mN s⁻¹. For higher loading rates, higher strain rates are reached (figures 6 and 7), resulting in a higher strain gradient, and a high density of geometrically necessary dislocations near the surface would be expected according to the results of Chen et al [26]. However, according to our results, the influence of the loading rate on hardness would not be important in the range of rates studied in the present work, between 0.15 and 1 mN s⁻¹.

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On the other hand, the samples with γ' and γ phase exhibit higher H values at 0.15 mN s⁻¹, while H remains approximately constant for the other rates, with values of 4.3 GPa (M2) and 4.1 GPa (M3) (figure 8). As found with Vickers microhardness measurements (figure 1(b)), M1 is much softer, while the presence of γ' nanoprecipitates (sample M2) increases the hardness of the material, reaching the maximum value. The sample aged for a longer time (sample M3) has a slightly lower hardness compared to M2.

Huang et al [14] studied the contributions of different types of pinning effect on dislocations due to precipitation in a Cu-2Be alloy aged at 320 °C. They found that the strengthening effect consists of the dislocation by-shearing and by-passing mechanisms. During the initial stage of precipitation, the dislocation by-shearing would be the main strengthening mechanism, while the dislocation by-passing (Orowan mechanism) would gain importance for longer aging times (longer than 4 h at 320 °C, where γ' nanoprecipitates exhibit a size around 30 nm) as the fraction and dimension of the precipitates grow continuously [14]. Other authors have proposed that the by-passing mechanism would be the main strengthening mechanism in Cu-2Be alloys with γ' nanoprecipitates, and this mechanism does not change for longer aging times in which γ' transforms into γ phase [1, 9]. In this study, γ' nanoprecipitates with a size around 10 nm were found in sample M2. Based on the literature, the predominant strengthening mechanism in this type of sample would be expected to be the dislocation by-passing mechanism.

On the other hand, it is known that the dislocation movement is a kinetic process, and the stress necessary for deformation depends on the strain rate [26]. As observed in figure 8, a higher value of H is obtained for the samples with precipitates for the lowest loading/unloading rate, which would suggest that the plastic deformation process in the samples with precipitates would be hindered at the lowest rate.

The variation of E_r with the loading/unloading rate is shown in figure 9(a). E_r increases with the loading/unloading rate and this increase is more pronounced for the sample with γ precipitates (M3). Previous works have reported higher values of the elastic modulus for Cu-2Be samples with precipitates [10, 11, 34]. This effect can be clearly observed for the sample tested at 0.3 mN s⁻¹.

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According to equation (2), E_r calculated from the Oliver and Pharr methodology depends on S and A_c . It was found that S increases with the loading rate, while A_c remains approximately constant, with a value around 3.5·10⁶ nm² for the sample M1, and around 1.7·10⁶ nm² for the samples M2 and M3. Therefore, the increase in E_r calculated from the Oliver and Pharr methodology with the loading rate is due to the increase in S with the loading rate.

In previous studies, Cu-2Be samples with γ' precipitates have been reported to have low free energy, which includes elastic strain energy, interfacial energy between matrix and precipitates, and stored energy due to dislocations, which produces a higher elastic modulus [10, 34]. The transition from γ' precipitates to γ phase would be associated with an elastic strain accommodation [34]. In sample M3 with γ precipitates, the misfit angle between the precipitates and the matrix is greater and the precipitates would be longer and wider than γ' precipitates [6, 9]. The different behavior of sample M3 with respect to M2 would be associated with the differences between γ' and γ phases. It can be seen in figure 7, that for a given depth and loading rate, sample M3 exhibits lower values of strain rate with respect to M1 and M2, and this difference is more important for higher loading rates.

The sensitivity of E_r to the loading/unloading rate (m) was obtained as the slope of the plot of ln E_r versus ln loading/unloading rate (figure 9(b)). Two ranges can be observed, when the loading rate was between 0.15 and 0.3 mN s⁻¹ and the range between 0.3 and 1 mN s⁻¹. Sample M1 presents lower values of m (0.17 and 0.07) than sample M3 (0.27 and 0.10) for the first and second ranges, respectively.