Model based phenomenological and experimental investigation of nanoindentation creep in pure Mg and AZ61 alloy

Materials and Design 105 (2016) 142–151

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Model based phenomenological and experimental investigation of nanoindentation creep in pure Mg and AZ61 alloy Kumar Sambhava a, Pranjal Nautiyal b,1, Jayant Jain b,⁎ a b

Department of Mechanical Engineering, Indian Institute of Technology, Delhi 110016, India Department of Applied Mechanics, Indian Institute of Technology, Delhi 110016, India

a r t i c l e

i n f o

Article history: Received 9 March 2016 Received in revised form 7 May 2016 Accepted 11 May 2016 Available online 20 May 2016 Keywords: Creep mechanisms Nanoindentation Magnesium Strain rate Stress exponent

a b s t r a c t In this study, nanoindentation induced creep in Mg metal system is investigated theoretically as well as experimentally. Analytical equations relating indentation creep rate to penetration depth for different creep deformation mechanisms were derived. The theoretical model was fitted with experimental results obtained for pure magnesium and AZ61 alloy. Dislocation glide was found to be the predominant indentation induced creep mechanism for both the materials at room temperature, for the entire range of load (from 50 to 150 mN). To gain insight into kinetics of deformation mechanism, activation energy and 0 K flow stress were determined for the two materials by fitting the experimental curves with the derived equations. Notable enhancement was found in the value of 0 K flow stress due to alloying, signifying highly effective solid solution strengthening on addition of 6 wt% Al and 1 wt% Zn. Stress exponents exhibited size effect, showing an increasing trend with increase in the value of indentation load or penetration depth; however, transition in the values of stress exponent did not correspond to transition in major creep mechanism (the theoretical results indicate prominence of glide throughout the investigated range). This leads to an important conclusion: unlike uniaxial tests, in nanoindentation creep tests, deformation mechanisms cannot be deduced based on stress exponent values. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction With the advent of nanoscience and nanotechnology, considerable attention has been focused towards the investigation of nano-scale mechanical properties of materials. Nanoindentation technique is a prominent tool for nanomechanical characterization of materials [1–8]. In nanoindentation, penetration depth is recorded with increasing load. Loads applied are in micro-milli Newton range and depths probed are typically in nanometer-micrometer range. In indentation experiments, increase in penetration depth has been observed even after load becomes constant [9]. In fact, many materials are found to exhibit this creep behavior during nanoindentation at temperatures well below half their melting points [10–20]. This phenomenon of nanoindentation induced creep makes it a very useful tool to study time dependent plastic flow of materials. Numerous studies have been carried out in recent years on nanoindentation creep in metals and metal alloys [15–23]. Research on lightweight metals has been at the center of materials research, for

⁎ Corresponding author. E-mail address: [email protected] (J. Jain). 1 Current affiliation: Department of Mechanical and Materials Engineering, Florida International University, Miami, FL 33174, USA.

http://dx.doi.org/10.1016/j.matdes.2016.05.036 0264-1275/© 2016 Elsevier Ltd. All rights reserved.

automobiles and aerospace applications. Magnesium and its alloys are particularly the most promising candidates for lightweight metal industry. Magnesium, being an hcp metal, has complex deformation characteristics because of the availability of different slip and twin systems [24]. Mechanical behavior of magnesium can be significantly varied, based on the interplay of material texture and microstructural variables [25–27]. Probing localized mechanical deformation behavior can enhance the understanding of plasticity mechanisms in Mg and its alloys. Considering the criticality of target applications in aerospace industry, it is important to have insight into localized mechanical response, to engineer alloys with superior mechanical performance. Nanoindentation is a powerful technique to study localized mechanical behavior, which can help in examining individual effect of different parameters, such as texture, precipitate particles and grain boundaries in the material. Nautiyal et al. [17] recently studied nanoindentation induced creep deformation in AZ61 magnesium alloy as a function of loading paths and microstructure, and probed twinning and slip mechanisms by microscopic examinations. Somekawa and Mukai [18] examined creep response of grain boundary in pure magnesium, and suggested grain boundary sliding (GBS) to be the prominent mechanism on the basis of stress exponent calculations (n ~ 2). In another study, Nautiyal et al. [19] probed creep response of grain boundary, and compared it with grain interior in pure magnesium and AZ61 alloy. They too attested GBS

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mechanism in pure magnesium (n ~ 2), but not in the case of alloys (n ~ 1). This difference was ascribed to solid solution strengthening. Han and co-workers [21] studied the effect of cooling rate during casting on nano-creep behavior of AC52 magnesium alloy, and suggested dislocation and twinning as the dominant creep mechanisms. Wang et al. [22] compared uniaxial and nanoindentation creep in pure magnesium, and reported similar stress exponent values, concluding identical creep mechanisms. However, all these studies are experimental in nature, and conclusion on creep mechanisms is based on stress exponent calculations. It is noteworthy that there is skepticism on the validity of stress exponents calculated in nanoindentation creep [16,28]. Recently, indentation size effect on stress exponent values for pure magnesium was reported, where stress exponent values were found to increase with increase in peak holding load [23]. Also, poor reproducibility of nanoindentation stress exponent values raises credibility on the validity of purely experimental approach for studying nanoindentation creep phenomenon [28]. There is a need for rigorous theoretical study of deformation mechanisms to gain deeper insight into nanoindentation creep. In this study, mathematical modeling of nanoindentation creep in magnesium and its alloy is attempted to develop better understanding of experimental results. Frost and Ashby [29] developed strain rate equations for different plasticity mechanisms. Based on Frost and Ashby's model, Li et al. [30] derived hardness-rate equations for different creep mechanisms, specifically for ‘indentation creep’. The developed model was compared with experimental results for a range of materials: Pb, SiC, ZrO2, MgO and diamond. It was found to be in agreement with the experimental output. In this study, we have modeled Li et al.'s hardness rate equations in terms of steady state strain rate equations or creep rate equations as a function of indenter penetration depth for different creep deformation mechanisms. The strain rate equations derived in this study differ from Frost and Ashby's model in terms of mathematical definition of strain rate. Unlike uniaxial tests, strain rate fields developed around indenter tip are complex in nature. The equations derived in this study are based on Mayo and Nix's representative strain rate formula for conical indentation [31], discussed in the subsequent section. Strain rate is more intuitive than hardness rate, since it directly represents material flow triggered due to indenter penetration. In this study nanoindentation creep experiments are performed on wrought magnesium and AZ61 alloy across varying loads for systematic understanding. Experimental results are then compared with the model predictions. Dominant creep mechanisms for pure magnesium and AZ61 alloy are determined based on the developed equations, to gain insight into the effect of alloying elements on deformation behavior. Moreover, the model derived here facilitates the determination of key thermodynamic and kinetic variables associated with deformation mechanisms, whose estimation otherwise is not readily available in the literature. This will also result in enhanced understanding of plastic deformation in HCP metal systems in general, and magnesium alloys in particular. 2. Background 2.1. Nanoindentation creep Nanoindentation creep experiments consist of three stages: loading, dwell at the peak load and unloading. The indenter tip penetrates the material during loading, creeps the material during dwell, and finally retracts during unloading. The hold at the peak load could vary from a few seconds to a few hours (depending on the material). Marsh [32] and Johnson [33] proposed that deformation zone (material beneath the indenter) during indentation is analogous to expanding hemispherical cavity, subjected to internal hydrostatic pressure. The stress developed around the indenter is variable; it is very high in the vicinity of the indenter tip, and

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progressively decreases as the distance from the tip increases. The effective stress during indentation is assumed as [34]. ð1Þ

σ ¼ F=Ap

where F is indentation load and Ap is projected area of contact. Characteristic indentation creep rate is given by the following relationship [31]: ε_ ¼ −

1 dh h dt

ð2Þ

where h is the penetration depth. Use of Eqs. (1) and (2) for determining representative stress and strain rate is supported by the theoretical calculations of Bower et al. [35]. Typically, creep rate is assumed to follow secondary creep equation, given as: ε_ ¼ Aσ n

ð3Þ

where A is temperature dependent constant, and n is stress exponent, which is an important creep parameter. Most of the studies on nanoindentation creep give prime emphasis to determination of stress exponent (n) in order to predict active creep deformation mechanism [17– 23]. Value of stress exponent ~ 1 implies diffusion mechanism, 2 for grain boundary sliding, 4–6 for dislocation/power law creep and higher values of n are believed to be due to power law breakdown mechanism [20,36,37]. The reliability of nanoindentation stress exponents to determine creep mechanisms is debatable, and is discussed in detail in Section 7. 2.2. Indentation creep mechanisms It is noteworthy that plastic flow is a kinetic process [29]. Glide and climb of dislocations, diffusive flow of individual atoms, relative motion of grains along boundaries etc. are the key determinants of plastic phenomena in the material. Based on these atomistic processes, the mechanisms possible for creep could be: dislocation glide, power law creep, power law breakdown, dislocation climb and diffusion. The details of these mechanisms are mentioned in [38]. One or more mechanisms can contribute to deformation simultaneously. The one with the maximum contribution is said to be the dominant mechanism. Ashby and Frost [29] developed strain rate equations for these mechanisms in terms of stress and thermodynamic parameters. Based on these equations, they constructed deformation mechanism maps. However, stress and strain rate fields created during indentation in the elastic-plastic zone are complex in nature. Using the concept of projected contact area of indentation, Li and co-workers [30] developed hardness rate equations for various indentation creep mechanisms. The developed equations were then used to fit experimental data for a range of materials. Dislocation glide, with following hardness rate equations, was found to dominate for all materials at room temperature [30]: 8 3 < ΔF ðkr þ 1Þ  C 21 p _ exp − H 1 ¼ −γ_ p : kT 6  μ2

9
34 !43 = C1 H3 1− H ; τp


 3 ðkr þ 1Þ ΔF C1 H_ 2 ¼ −γ_ o exp − H 1− ̂ H 6 kT τ

ð4Þ

ð5Þ

where Eqs. (4) and (5) are hardness rate equations for dislocation glide obstructed by lattice resistance and discrete obstacles, respectively. All the parameters are defined in Table 2 in the Appendix A.

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3. Experimentation Pure magnesium (with 99.9% purity) and AZ61 alloy (Mg-6 wt% Al1 wt% Zn) specimens used in this study were cut out from extruded rods obtained from Good Fellow, UK. Specimens were heat treated: magnesium was treated at 250 °C for 30 min to relieve residual stresses and AZ61 was treated at 410 °C for 10 h, followed by water quenching to solutionize the material. Specimens were then prepared for nanoindentation by grinding the surface with SiC paper from P1200 to P4000, followed by cloth polishing in colloidal silica. The surface was etched with Acetic-Picral. The grains were large enough to ensure that anomalous grain boundary phenomena arising due to aggregation of impurities, defects and extra solute atoms do not interfere in the study. Details of initial microstructure and texture can be found in our earlier works [17,19]. Nanoindentation creep tests were performed using ASMEC's Universal Nanomechanical Tester (Bautzner Landstraße 45, Germany) in a closed loop mode. Data points recorded in closed loop testing involves very short dwell at each point for better control and accuracy of measurement. A Berkovich diamond indenter, with a tip radius of 180 nm, was used to make the indents. The tip was calibrated with a standard fused silica sample, and the contact area was fitted with a polynomial function of the contact depth [5]. Indentation creep tests were conducted for peak normal loads varying from 50 to 150 mN. The maximum load was maintained for 900 s to obtain creep data. Constant loadingunloading rate of 0.25 mN/s was programmed for all the tests. The indenter was held for 60 s at 10% of maximum load during the unloading cycle for the purpose of thermal drift correction. Indents were made on the sample surface for which the indentation axis (or the loading path) was perpendicular to the extrusion axis. All the tests were performed at room temperature (~25 °C). A minimum of 10 indents were made for each condition, and the data with standard deviation less than 5% were averaged. 4. Derivation of analytical equations for nanoindentation creep strain rate The equations developed by Li et al. [30] for indentation creep are hardness rate equations. However, steady state strain rate, or creep rate is a more intuitive parameter and is of direct physical significance for the study of creep flow. Creep rate equations are derived for all the mechanisms, in terms of indenter penetration depth. As discussed in the previous section, Li et al. developed Eqs. (4) and (5) for dislocation glide, obstructed by lattice resistance and discrete obstacles, respectively. Indentation hardness for Berkovich indenter for indentation load, ‘W’ and penetration depth, ‘h’, is given as [5]:   2 H ¼ W= 24:5h :

ð6Þ

Differentiating Eq. (6) and substituting strain rate from Eq (2): dH 2 W dϵ  : ¼−  H_ ¼ dt 24:5 h2 dt

ð7Þ

Using Eqs.(4)–(7), creep rate equation for dislocation glide mechanism is obtained as: W 2 γ_ p ðkr þ 1Þ ϵ_ 1 ¼ 1200:5 h4 6

3

ϵ_ 2 ¼

8
2 < ΔF C1 p exp − : kT μ

9
34 !43 = C1 W ð8Þ 1− ̂ ; τ p 24:5h2


 ΔF C1 W γ_ o 3 ðkr þ 1Þ exp − 1− ̂ 2 12 kT τ 24:5h

ð9Þ

where Eqs. (8) and (9) hold true for dislocation glide obstructed by lattice resistance and discrete obstacles, respectively. γ_ p and γ_ o are

the pre-exponential factors. Taking a cue from [29], these were chosen as constants with values: 1011 and 10 6 , respectively. k r is the ratio of the radius of the elastic-plastic core to the hydrostatic core. It generally lies in the range 2–3.5 [30]. Higher the value of kr, greater is the susceptibility of the material for plastic deformation. C1 is the ratio of the hardness to the average shear stress. It has a widely accepted value of 0.2 [30]. μ is the shear modulus of the material, which in our case, is 1.66 × 1010 Pa. b is the magnitude of the Burgers vector. For the slip mechanism, the Burgers vector is given by b = c + a, where c and a are the lattice constants of Mg HCP structure. The chemistry of the material is captured in the values for ΔFp and τp. ΔFp can be understood as the energy required to overcome the obstacle without any external stress aid. The creep rate is highly sensitive to the value of ΔF p because it occurs in the exponent. A wide range of ΔF p is reported in the literature: 0.05 to 2μb3 [29]. This makes it even more important as a fitting parameter while modeling creep data. τp is the shear strength of material without any thermal vibration, that is, athermal flow strength or the flow strength at 0 K. [29] specifies a rough threshold for the value of τp which is μb/ l; the value is less than the threshold for weak obstacles and vice versa. τp also lies in the exponent, and hence the creep rate is very sensitive to it too. These are the two major parameters for fitting the theoretical strain rate to the experimental data. These parameters also have physical significance, which will be discussed later. ΔF and τ are similar to ΔFp and τp respectively, having similar physical significance and similar range of values. The indentation strain rate equations have been derived for all the creep mechanisms in Section A.1 in the Appendix A. Based on the derived strain rate equations, relative contribution of each mechanism towards overall creep in pure magnesium and AZ61 alloy is plotted in Figs. 1 and 2, respectively. Creep rate as a function of dwell time is plotted in these figures, for five load conditions. The values of parameters used and their sources are given in the Appendix A (Table 2). There is no prior report on theoretical modeling and fitting of experimental creep results for magnesium and its alloys. As a result, values of many of these parameters cannot be found in the literature. The choice of values for such parameters is discussed later. An iterative approach was adopted: initial value was assigned first, and then refined and established. It is noteworthy that the final 500 s of nanoindentation creep experiments were chosen for fitting (Figs. 1 and 2). The derived equations are valid for steady state creep; therefore, strain rate in primary and transient creep regimes (which were observed for roughly first 400 s [17, 19]) are not plotted. The mathematical analysis carried out in this study is only for secondary creep. It can be seen from the figures that dislocation glide is the dominant mechanism for both pure magnesium and AZ61 alloy for all load conditions at room temperature. This is in corroboration with the finding of Li et al. [30]. Also, it is clearly evident that dislocation glide obstructed by lattice resistance dominated in pure magnesium, whereas glide obstructed by discrete obstacles is prevalent in AZ61 alloy. It is also observed that the contribution of stress induced diffusion creep to the total creep rate is much lesser than other mechanisms. It is almost 20 orders of magnitude lower than other creep mechanisms, which lie in the range of 1 × 10 − 4 s − 1 to 1 × 10− 12 s− 1. The strain rate due to dislocation climb is even lesser, of the order of 1 × 10− 55 s− 1. This is so because recovery is a thermally activated process, and hence, is unlikely to play an important role at room temperature. In summary, the theoretical model predicts dislocation glide to be the dominating deformation mechanism for both pure magnesium and AZ61 alloy. The validity of the mathematical model is examined by comparison with experimental results in subsequent sections. Dislocation glide mechanism is inspected in detail with respect to experimental data obtained, and values of key kinetics parameters are determined by fitting.

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Fig. 1. Log of strain rate vs. time plots for all the mechanisms of indentation creep for pure Mg. Subplots (a) to (e) represent W in increasing order, 50 mN, 75 mN, 100 mN, 125 mN and 150 mN.2 2 The values used for the parameters are given in Table 2 (the values of ΔFp and τ p , ΔF and τ for Mg AZ61 alloy were not available in the literature. The authors assumed the reasonable values of 0.3μb3 and 2.6μ for ΔFp and τ p , and 0.62μb3 and 0.01μ for ΔF and τ, keeping in mind that the criterion in [25] is satisfied and each of the values is greater than the corresponding quantities for pure Mg which should be the case).

5. Evaluation of model parameters The theoretical development in the previous section can only give us the form of equation. Fitting of experimental data is necessary to establish the value of constants for a particular material. This approach is called model-based phenomenology [29], and is a useful technique for modeling and understanding plasticity, which is a complex phenomenon. It is stressed that fitting of experimental results should conform to the underlying physics, and should not be purely mathematical in approach. As observed in the previous section, dislocation glide is the dominating mechanism for both pure magnesium and AZ61 alloy. In case of pure magnesium, dislocation glide would be obstructed by lattice resistance (Fig. 1). On the other hand, in AZ61, due to the presence of foreign solute atoms (Al and Zn) in magnesium matrix, glide would be obstructed by discrete obstacles (Fig. 2). This was clearly seen in the figures. Therefore, Eq. (8) was fitted with experimental data obtained for pure magnesium, and Eq. (9) was fitted with experimental data obtained for AZ61. The five degree polynomial equations simulating experimental displacement-time data for each load and both the materials are provided in the Supplementary information. Values of ΔFp and τ p for pure Mg, and ΔF and τ for AZ61 alloy were obtained by fitting these experimental polynomial equations with the analytical equations derived in Section 4 (Eqs.

(8) and (9)). As discussed in Section 4, creep rate is very sensitive to the values of parameters: ΔFp, τ p , ΔF and τ. The range of values that these parameters can adopt is very wide. Fitting facilitates the determination of their approximate values, which is otherwise not easy to predict. ΔFp, 3

τp , ΔF and τ can be expressed as: ΔFp = t1μ0b3, τp ¼ t 2 μ 0 b ,ΔF = t3μ0b3 and τ ¼ t 4 μ 0 where the constants t1, t2, t3 and t4 are fitting parameters. Creep rates in Eqs. (8) and (9) were obtained using Eq. (2). Other constants were used from Frost and Ashby [29]. The best fit was found for every pair of (t1, t2) when fitting to ε1, and (t3, t4) when fitting to ε2 for all the load values. For measuring fitting, root mean square of the error, rmse was calculated as:

rmse ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xn

2ffi ϵ_ −_ϵexperimental i¼1 theoretical n

ð10Þ

where ‘i’ spans the data points and ‘n’ is the total number of data points. The code written in MATLAB calculated the rmse for each pair of (t1, t2) and (t3, t4), in a very wide range (0.02 b t1 b 2; 0.01 b t2 b 1.0) and (0.02 b t3 b 2; 0.01 b t4 b 1.0), for all the sets of indentation loads. The pair of values with minimum total rmse was chosen, such that the theoretical prediction closely matches with the experimental curve.

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Fig. 2. Log of strain rate vs. time plots for all the mechanisms of indentation creep for AZ61 alloy. Subplots (a) to (e) represent W in increasing order, 50 mN, 75 mN, 100 mN, 125 mN and 150 mN.3 3 The values used for the parameters are given in Table 2 (the values of ΔFp and τ p , ΔF and τ for Mg AZ61 alloy were not available in the literature. The authors assumed the reasonable values of 0.3μb3 and 2.6μ for ΔFp and τ p , and 0.62μb3 and 0.01μ for ΔF and τ, keeping in mind that the criterion in [25] is satisfied and each of the values is greater than the corresponding quantities for pure Mg which should be the case).

6. Results Nanoindentation creep tests were performed on Mg and AZ61 specimens, having average grain sizes 100 μm and 30 μm, respectively (Fig.

3). The solution-treated microstructure is free of any secondary particles as can be seen in Fig. 3. Indents were made within the grain, away from the grain boundaries, to avoid artifacts arising due to impurities and excessive solute segregation at the boundaries. Steady state creep

Fig. 3. Optical micrographs showing the microstructures of (a) pure Mg and (b) solution-treated AZ61 alloy [19] (note: black spots in (b) are the etching artifacts).

K. Sambhava et al. / Materials and Design 105 (2016) 142–151

was achieved for the chosen dwell time of 900 s, which is elucidated in our earlier experimental studies [17,19,23]. The plots of theoretical and experimental creep rates in pure magnesium and AZ61 for five different load conditions are shown in Figs. 4 and 5, respectively. As seen from Section 4, contribution of dislocation glide dominates. Therefore, the theoretical creep rate curves in Figs. 4 and 5 were plotted using Eqs. (8) and (9). The plots span the last 500 s of the test, that is, the steady state creep regime of nanoindentation tests. The theoretical plots are in good agreement with experimental results. Less than 15 percent errors were obtained for each fit which should be assumed reasonable considering the complex nature of nanoindentation creep. Table 1 lists the values found. The values of fitting parameters t1, t2, t3 and t4 for both pure Mg and AZ61, and for all load conditions lie well within the range specified by Frost and Ashby [29]. 7. Discussion The pre-dominance of dislocation glide can be understood from the deformation mechanism map given by Frost and Ashby [29]. These maps operate on normalized shear stress, which is given by: σn = σ/μ. For the nanoindentation creep tests conducted in this study, the normalized shear stress was found to vary from 10−2–10−1 (based on Eq. (1)). A look at the deformation mechanism map given in Chapter 6 of Ref [29] shows the dominant mechanism is dislocation glide plasticity for temperature around 298 K (room temperature). In nanoindentation, due to very low contact area (the indenter tip radius ~184 nm), overall stresses developed are very high. The stresses involved in nanoindentation are, in fact, much higher than those developed during uniaxial

147

tensile/compressive tests [22]. This enables dislocations to glide through the lattice and obstacles by either shearing them, or by-passing them. Hence, dislocation glide dominates for all load conditions at room temperature. Activation energy and 0 K flow stress values obtained by fitting experimental curves, were higher for AZ61 alloy. The addition of alloying elements results in ~3.5 times increase in athermal shear strength, while only marginal increase in activation energy for discrete obstacles. The presence of solute atoms introduces a friction like resistance to moving dislocation. The higher value of athermal strength indicates the effective strengthening resulting from the addition of Al and Zn solutes. This is consistent with the earlier reports [39,40]. On the other hand, the marginal increase in activation energy can be attributed to ineffectiveness of solute elements in creating the strong obstacles to the dislocation motion. Determination of stress exponent, n, is central in creep studies. Creep mechanisms are often predicted, based on the stress exponent obtained [38]. There are numerous nano-creep studies available in literature, that report stress exponent and relate it to dominant creep mechanism [17– 23,28,36]. Stress exponents were obtained for pure magnesium and AZ61 alloy in this study by linear fitting of double logarithmic strain rate-stress plots, obtained by employing using Eqs. (1)–(3). For detailed methodology, readers are referred to [19]. Pile-up of material around the indent was not significant enough to have any major influence on the calculations. The trend in stress exponent with respect to peak indentation creep load is shown in Fig. 6. Stress exponent values for certain loads is not shown here due to poor reproducibility, a concern raised by Goodall and Clyne [28] in their work on extraction of creep parameters from nanoindentation data.

Fig. 4. Theoretical and experimental creep rate vs. time plots for pure Mg. Subplots (a) to (e) represent W in increasing order: 50 mN, 75 mN, 100 mN, 125 mN and 150 mN.

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Fig. 5. Theoretical and experimental creep rate vs. time plots for AZ61. Subplots (a) to (e) represent W in increasing order: 50 mN, 75 mN, 100 mN, 125 mN and 150 mN.

While pure magnesium exhibits two distinct stress exponent regimes in secondary creep region (Fig. 6(a)), AZ61 was found to exhibit only one stress exponent regime (Fig. 6(b)). Stress exponents are exhibiting nanoindentation size effect, that is, with increasing value of penetration depth (or indentation load), value of n also increases. This is in corroboration with reports available in literature [23,41]. This stress exponent based size effect can be correlated with hardness size effect. It is well known that hardness value decreases with indentation load [42, 43]. In the determination of stress exponent by Eq. (3), stress term, which is mathematically analogous to hardness, would appear in the denominator. This is the reason why the two size effects show opposite trends (while hardness values decrease, the stress exponent increases with indentation load). We propose the following equation, correlating the two indentation size effects: n ¼ K ISE ∂

logε̇ Þ_ ∂ð logH Þ

ð11Þ

8. Conclusion

where KISE is an indentation size effect correlation constant.

Table 1 Proposed value of the material constants. Model parameters ΔFp τp ΔF τ

In the reports on indentation creep size effect [23,41], transition in the values of stress exponent is related to transition of creep mechanism. However, based on the findings of previous section, it seems unlikely that there is any transition of dominant mechanism. Dislocation glide clearly remains as the major deformation mechanism for entire range of loads investigated in this study (Figs. 3 and 4). This leads to an important conclusion: stress exponent is not a direct indicator of plasticity mechanism in case of nanoindentation induced creep. This is because the increasing n value with load is essentially due to hardness based ISE (Eq. 11), and not due to the inherent plasticity of the material. Similar concerns on the validity of stress exponent based creep mechanism analysis has been raised in literature [16,28]. Therefore, nanoindentation ‘power law’ stress exponent is a geometric factor, and its value depends on tip-material contact area. It does not convey meaningful information about intrinsic plastic flow of the material during indentation creep.

Pure Mg

AZ61 alloy 3

0.177μ0b 0.6894μ0 – –

– – 0.1829μ0b3 2.413μ0

Analytical equations for different creep deformation mechanisms were derived, that modeled nanoindentation creep rate as a function of indenter penetration depth. Theoretical curves based on mathematical equations were compared with the experimental results obtained for pure magnesium and AZ61 alloy, over a range of loads. Following conclusions were drawn: (a) Dislocation glide is the dominating deformation mechanism in pure magnesium and AZ61 alloy at room temperature, for all values of loads.

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Fig. 6. Trend in stress exponent values with respect to peak indentation load during creep in: (a) pure magnesium, and (b) AZ61 alloy.

(b) Insight was developed into kinetics of deformation mechanisms. Values for activation energy and 0 K flow stress were determined, which were obtained to be uniform across the indentation loads. However, alloying resulted in change in these parameters. Both activation energy required for dislocation glide as well as 0 K flow stress were higher for AZ61 alloy. The enhancement was more pronounced for 0 K flow stress, which signifies strong solid solution strengthening in AZ61 alloy. (c) Values of stress exponent exhibit size effect, with progressively increasing value as indentation load increases. However, deformation mechanism was mathematically found to be glide for entire load range. This suggests stress exponent is not indicative of deformation mechanisms in nanoindentation induced creep. Although, the derived model was used for fitting and analysis of pure magnesium and AZ61 magnesium alloy, it can be used for the study of indentation induced creep in a variety of materials. The model gives creep rate equations for different mechanisms, and hence, provides direct insight into deformation behavior.

For Mg, power law exponent, n is typically 5 [29]. Using Eq. (7), Creep rate equation for power law creep was determined as: " #
2 3 _̇ ð1 þ kr Þ C1 W ac Dc W 5 5 AbDv C 1 þ 10 ϵ3 ¼ 1 μb 24:5h2 Dv h10 kTμ 4 1:06  108

ð13Þ

where A is Dorn's constant, the value of which is taken as 1.2 × 106 [29]. ac is the cross-sectional area of the dislocation core. Dcis the diffusion coefficient of dislocation core. Dvis the lattice diffusion coefficient, and is highly temperature dependent. Since the temperature at which experiments were performed was room temperature (298 K), the value of Dv is very low, making the creep rate low too. Overall, this renders the contribution of this mechanism to the actual creep lower than creep due to glide. A.1.2. Power law breakdown Li et al. [30] derived the following hardness rate equation for power law breakdown mechanism: " #

 3 ð1 þ kr Þ A bμDv C 1 H 2 ac Dc α0C 1 H n _ sinh 1 þ 10 H: H4 ¼ − 6 Dv α0n kT μb μ

Acknowledgements Jayant Jain would like to acknowledge the financial support received from DST-SERB, India (Project No.: RP02797).

ð14Þ

Based on Eqs. (7) and (14) the creep rate equation for this mechanism was obtained as: Appendix A " #
2
 5 3 ð1 þ kr Þ AbμDv C1 W ac Dc α0C 1 W sinh 1 þ 10 ϵ_ 4 ¼ 12 α05 kT μb 24:5h2 Dv μ 24:5h2

A.1. Strain rate equations for different creep mechanism A.1.1. Power law creep Hardness rate equation for power law creep, as developed by Li et al. [30] is given as: " #
 3 _̇ ð1 þ kr Þ n AbDv C 1 H 2 ac Dc nþ1 H : C1 1 þ 10 H3 ¼ − 6 Dv μb kTμ n−1

ð12Þ

ð15Þ where α′ is a constant for power-law breakdown mechanism. Compared to other strain rate equations, in this equation, strain rate is proportional to shear modulus (which is a very large quantity). As a result, the value of strain rate computed using this equation is high, second to dislocation glide mechanism.

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A.1.3. Dislocation climb The hardness rate equation for indentation creep by climb mechanism is [30]:   " # 3 C 21 kr −1 ΩDv 40 8C 3 μ H3 : C H− H_ 5 ¼ − 2 2 3 kT 9 k −1 ðμbÞ

ð16Þ

r

The Creep rate equation for this mechanism, in terms of indentation depth is found to be:   " # 3 2 1 C 1 kr −1 ΩDv W 8C 3 μ W 2 C2 − 3 : ϵ_ 5 ¼ 2 4 kT 270:11 ðμbÞ2 24:5 h k −1 h

ð17Þ

r

The strain rate is directly proportional to atomic volume (Ω) and lattice diffusion coefficient (Dv) (both of which are very low at room temperature). As a result creep rate due to climb is the least among all mechanisms. A.1.4. Diffusion The rate equation for diffusion, as developed by Li et al. [30]: " #
 C 1 H 2 ac Dc πδ Db 2 3 C 1 ΩDv _ H : 1 þ 10 þ H 6 ¼ −7ð1 þ kr Þ 2 Dv μb d Dv kTd

ð18Þ

The creep rate equation for diffusion mechanism was obtained as: " #
2 3 ð1 þ kr Þ C 1 ΩDv C1 W ac Dc πδ Db W 1 þ 10 þ : ϵ_ 6 ¼ − 2 7 μb 24:5h2 Dv d D v h2 kTd

ð19Þ

Similar to climb mechanism, here too creep rate is proportional to atomic volume and lattice diffusion coefficient; as a result overall contribution of this mechanism is also low. A.2. Modeling parameters

Table 2 Nomenclature and values of modeling parameters for pure Mg. Symbol Parameter description

Values used

γp

1 × 1011 s−1 [30] 2.5 [30]

k μ C1 ΔFp τp γo ΔF τ

Pre exponential factor (for dislocation glide obstructed by lattice resistance) Ratio of the radius of the elastic plastic core to the hydrostatic core Shear modulus

1.66 × 1010

N/m2 C1 = hardness/average shear stress 0.2 Activation energy for lattice resistance controlled glide Found by fitting Flow stress at 0 K for lattice controlled glide Found by fitting Pre exponential factor (for dislocation glide obstructed 1 × 106 s−1 [30] by discrete obstacles) Activation energy for glide controlled by discrete 0.5μb3J [29] obstacles Flow stress at 0 K for glide controlled by discrete 6.7 × 10−3 obstacles N/m2 [29]

A n acDc

Dorn's constant Power law creep exponent Diffusion coefficient for core diffusion

1.2 × 106 [29] 5 [29] 2.24 × 10−39

Dv

Diffusion coefficient for lattice diffusion

m4/s [29] 2.16 × 10−28

Ω

Atomic volume of Mg

m2/s [29] 2.33 × 10−28

C2 C3 d δDb

Constant defined in the paper by W·Li Constant defined in the paper by W·Li Grain diameter Diffusion coefficient for grain boundary diffusion

m3 [29] 0.2 [29] 1 [29] 10−4 m 3.73 × 10−28 m3/s [29]

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