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A512 (2009) 39.

CHAPTER 5 Indentation Behaviour*

5.1. Introduction One of the objectives of this study was to examine two important properties, namely the hardness and Young’s modulus of nanocrystalline Ni with varying grain sizes and nanocrystalline Ni-Fe alloys as a function of Fe concentration and/or grain size. The study will also consider the effects of other important factors that are known to influence Young’s modulus, especially preferred crystallographic orientation, or texture. A better understanding of the effect of grain size on the Young’s modulus of nanocrystalline Ni-Fe alloys is of particular interest because of the unusual effect of composition observed for these alloys in the polycrystalline form. In contrast to polycrystalline Ni-Cu alloys, which show a relatively linear decrease in Young’s modulus with increasing Cu concentration [Guy (1972)], polycrystalline Ni-Fe alloys show a deep minimum at about 60wt.% Fe (as shown in Fig. 5.1) even though pure Fe and pure Ni have almost the same Young’s modulus values [Ledbetter and Reed (1973)].

It should also be pointed out that although a general trend in Fig. 5.1 is obvious, for any given composition including the pure materials, there are significant variations in the data presented in different studies. The Young’s modulus trend for the Ni-Fe alloys may be explained based on the fact that at greater than 60wt.%Fe, there is a transition from the fcc phase to the bcc phase. On the other hand, the Ni-Cu alloys do not show this behaviour because it remains in the fcc phase over the entire compositional range.


The key findings presented in this chapter were previously published in the following refereed journal article:

J.D. Giallonardo, U. Erb, K.T. Aust and G. Palumbo, “The influence of grain size and texture on the Young’s modulus of nanocrystalline nickel and nickel-iron alloys”, Philosophical Magazine, 91 (2011) 4594.

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5.2. Results The hardness and Young’s modulus for each sample were determined using nanoindentation as described in Chapter 3 (Section 3.6). The last elastic unloading curves were analyzed to determine the Young’s modulus using the procedure outlined by Oliver and Pharr (1992). An example of a series of load-unload curves in a force-depth graph is shown

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in Fig. 5.2 for five different loads (150, 130, 110, 90 and 70 mN) performed at a loading rate of 13.3 mN/s. Table 5.1 summarizes the hardness and Young’s modulus results for each of the samples. The errors reported in the table and figures correspond to one standard deviation. For discussion purposes, the orientation indices that were determined in Chapter 4 (Table 4.5) are also provided in Table 5.1. As expected, the hardness values gradually increase with decreasing grain size. The measured Young’s modulus values, Em, have some noticeable differences when compared to the literature values, E0, for polycrystalline materials. These observations are discussed in the subsequent sections.

5.3. Effect of Grain Size on Hardness A plot of the hardness values with the inverse square root of the average grain size for all materials produced (Fig. 5.3) shows a transition from regular to inverse Hall-Petch behaviour which is in agreement with previous studies [El-Sherik et al. (1992), Erb et al.

(1996), Cheung et al. (1995), Ebrahimi et al. (1999), and Li and Ebrahimi (2003)]. The slope for the region representing regular Hall-Petch behaviour was determined to be approximately 24 GPa/nm-1/2 which is similar to 28 GPa/nm-1/2 reported by Hughes et al. (1986) for electrodeposited nanocrystalline Ni with grain sizes in the range of 12 to 12,500 nm. It should be noted that solid solution hardening in the Ni-Fe alloys was previously found to be insignificant compared to grain size hardening [Erb et al. (1996), Cheung et al. (1995)].

Several interpretations have been given to explain deviations from the regular HallPetch behaviour. When the grain size is sufficiently small, i.e., less than 20 nm, dislocation slip is no longer the dominant deformation mechanism. As a result, deviations from the Hall

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Figure 5.3.

Hall-Petch plot for the series of nanocrystalline Ni and Ni-Fe alloys.

Petch relationship are observed and other deformation mechanisms begin to operate.

Chokshi et al. (1989) observed a similar behaviour for nanocrystalline Cu and Pd produced using the inert gas condensation technique and suggested that room temperature Coble (Nabarro-Herring) creep occurs and is possible at this temperature because of the relatively

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nanocrystalline Ni-P alloys, Palumbo et al. (1990b) owed deviations from regular Hall-Petch behaviour to the fact that triple lines account for a significant fraction of the bulk volume, especially when the grain size is less than 20 nm. The deformation of electrodeposited nanocrystalline Ni over a broad grain size range of 40 to 6 nm was analyzed by Wang et al.

(1997). They demonstrated that at high stress levels, grain boundary sliding is essentially the main room temperature deformation mechanism, although creep mechanisms can contribute

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significantly at small grain sizes. Wang et al. (1997) proposed that the deviations from the Hall-Petch relationship are caused by a dynamic creep process due to diffusion mechanisms.

5.4. Effect of Grain Size on Young’s Modulus Table 5.1 summarizes the Young’s modulus measurement data for each of the samples. Depending on the load and sample, indentation depths ranged between approximately 0.8 and 1.5 µm, i.e., ≤ 3% of the sample thickness. The Young’s modulus for the Ni samples was measured and the values obtained were found to be similar to conventional (randomly oriented) polycrystalline Ni, ~207 GPa [Davis (1990)], down to an average grain size of about 20 nm. The Young’s modulus for the nanocrystalline Ni-Fe alloys was also measured; however, for comparative purposes, values for conventional (randomly oriented) polycrystalline Ni-Fe counterparts with the exact same compositions were not available. Instead, a comprehensive collection of data for the complete range of conventional (randomly oriented) polycrystalline Ni-Fe alloys was compiled earlier by Ledbetter and Reed (1973), shown in Fig. 5.1. For the fcc phase polycrystalline Ni-Fe alloys, there is a notable variation and distinct trend with increasing Fe concentration. From about 0 to 20 wt.%Fe, the Young’s modulus is relatively constant. This is then followed by a decrease down to a minimum average value of ~140 GPa at around 60wt.%Fe. Increasing the Fe concentration in the nanocrystalline Ni-Fe alloys shows a similar trend (Table 5.1). In order to compare the values obtained for nanocrystalline Ni-Fe alloys to their conventional (randomly oriented) polycrystalline counterparts, an average of the Young’s modulus values, was taken from the collection of data [Ledbetter and Reed (1973)] for compositions close to the four alloys obtained in the current study. The measured values were then normalized,

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Em / E0, where Em is the measured value and E0 is the respective conventional (randomly oriented) polycrystalline counterpart value, and then plotted as a function of grain size (Fig.

5.4). In the case of Ni, E0 is the value of its conventional polycrystalline counterpart value, ~207 GPa [Davis (1990)].

1.2 1.1

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Figure 5.4.

Normalized Young’s modulus values as a function of grain size. The horizontal solid line represents Em / E0 = 1, where Em is the measured value and E0 is the respective conventional (randomly oriented) polycrystalline counterpart value. The dashed lines correspond to the values predicted by the upper bound composite model (Eq. 5-1). The solid lines correspond to the values predicted by the lower bound (Eq. 5-2) composite model.

The effect of decreasing grain size on Young’s modulus was previously investigated by Shen et al. (1995) for nanocrystalline Fe produced by mechanical attrition. Zhou et al.

(2003a), Zhou et al. (2003b), and Zhou et al. (2009) also investigated the effect of grain size on the Young’s modulus of electrodeposited Ni-2.5wt.%P. In both cases, the influence of the various structural components (grain interiors, grain boundaries, and triple junctions) in nanocrystalline materials [Palumbo et al. (1990a)] was estimated according to a simple rule

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of mixtures by considering their volume fractions in a composite model and taking into consideration both upper (Eq. 5-1) and lower (Eq. 5-2) bound solutions [Callister (2005)],

–  –  –

where, Em is the measured Young’s modulus of the material, f g ( E g ), f gb ( E gb ) and f tj ( Etj ) the volume fractions (average Young’s moduli) for grain interiors, grain boundaries and triple junctions, respectively. Note that the upper bound solution (Eq. 5-1) is based on the isostrain condition while the lower bound is based on the isostress condition. Volume fractions were calculated by using a reasonable range for grain boundary thicknesses of 0.7 and 1.1 nm [Kirchheim et al. (1988)] and assuming a three-dimensional tetrakaidecahedral model for the crystal shape [Palumbo et al. (1990a)] using Eq. 2-1, 2-2, 2-3, and,

–  –  –

where, f ic is the total intercrystal volume fraction. The value for E g is considered to be the same as that of the conventional (randomly oriented) polycrystalline counterparts. The values for E gb and Etj are taken to be about 76% and 73%, respectively, of the value for the conventional polycrystalline counterparts [Zhou et al. (2003b)]. Normalized values, for Young’s modulus were then plotted in Fig. 5.4 as a function of grain size along with the composite model predictions for the upper bound (dashed lines) and lower bound (solid lines) solutions using both grain boundary thicknesses.

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