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Kinematic and dynamic characterization of plastic instabilities occurring in nano- and microindentation tests
Materials Science and Engineering A 409 (2005) 100–107
Review
Kinematic and dynamic characterization of plastic instabilities occurring in nano- and microindentation tests Nguyen Q. Chinh ∗ , Gy. Horv´ath, Zs. Kov´acs, A. Juh´asz, Gy. B´erces, J. Lendvai Department of General Physics, E¨otv¨os University, 1117 Budapest, P´azm´any P. S´et´any 1/A., Hungary Received in revised form 22 March 2005; accepted 14 April 2005 Dedicated to Professor James C.M. Li on the occasion of his 80th birthday.
Abstract This paper surveys the phenomenon of plastic instabilities occurring in depth sensing indentation measurements, during which a stepwise increase has been observed in the indentation depth–load (F–h) curves measured in constant loading rate mode. In these measurements, plastic instability – serrated indentation – manifests itself as hardness oscillations around a nearly constant value of the conventional dynamic microhardness. Investigations presented here are focused on both kinematic and dynamic characterization of Portevin–Le Chˆatelier type indentation instabilities. The effect of important factors such as solute concentration and polycrystalline grain size is described and discussed. Taking into account the experimental observations, a macro-mechanical dynamic model was also proposed for the characterization of indentation instabilities. © 2005 Elsevier B.V. All rights reserved. Keywords: Plastic instabilities; Serrated indentation; Hardness oscillation; Solute concentration; Grain size; Dynamic model
Contents 1. 2. 3.
4.
5.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The phenomenon of plastic instabilities in depth sensing indentation tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kinematic characterization of the indentation plastic instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Effect of solute concentration on the indentation plastic instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Effect of grain size on plastic deformation processes during indentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamic characterization of the indentation plastic instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. The correlation between hardness oscillation and indentation velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Dynamic modeling of the indentation instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Introduction Plastic instabilities as the phenomenon of discontinuous yielding during plastic deformation have been widely observed, and discussed in the literature under various names like serrated yielding, Portevin–Le Chˆatelier (PLC) effect, jerky flow, ∗
Corresponding author. Tel.: +36 1 372 2845; fax: +36 1 372 2811. E-mail address: [email protected] (N.Q. Chinh).
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100 101 102 102 103 104 104 105 106 107 107
etc. [1–5]. In unidirectional deformation, the most characteristic feature is the appearance of stress drops or stress jumps, σ, beginning from a critical strain, εc , on the originally smooth stress–strain curves. Therefore, several works focused both theoretically [6–9] and experimentally [4,10,11] on the interpretation and determination of the parameter σ. The physical basis for the appearance of the PLC effect is a negative strain rate sensitivity (SRS) originating mainly from the interaction between mobile dislocations and diffusing solute atoms during plastic deformation, a phenomenon called dynamic strain ageing
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Fig. 1. Schematic of the N-shaped stress–strain rate (σ–˙ε) function of alloys showing PLC plastic instabilities.
(DSA) [12–15]. Depending on the testing conditions, negative SRS may occur at a certain part of the N-shaped stress–strain rate (σ–˙ε) function proposed by Penning [5], shown schematically in Fig. 1. Several models have been developed both on the solute–dislocation interaction [16–19] and on the explicit expression of the mentioned N-shaped σ–˙ε function [20,21]. Although the PLC effect has been observed and investigated mainly by uniaxial tensile and compression tests [6,11,17], it was also studied in other modes of deformation [22]. Recently, the phenomenon of PLC plastic instabilities in metals was investigated also by depth sensing indentation (DSI) tests [23–34], which are nowadays increasingly used for the investigation of complex mechanical properties of materials [35–38]. 2. The phenomenon of plastic instabilities in depth sensing indentation tests
101
the steps depend both on the loading rate and on the composition of the alloy [23,24]. Analyzing the depth–load curves containing steps, the indentation process can be physically regarded as a series of two subsequent local processes, in which firstly the load increases by Fp at essentially negligible increase in the indentation depth, and then the deformation (indentation depth) increases rapidly by Dp at practically no increase in load (see also the inset in Fig. 2). It has been shown earlier [24] that in the case of stable solid solutions, such as Al–Mg alloys, the instability steps are relatively regular and reproducible. Furthermore, analyzing the development of the steps at a given (h; F) point on the indentation curve, it has been shown that both Dp h
and
Fp F
(1)
ratios are approximately constant, that is, the step height, Dp , and the load period, Fp , characterizing instability steps are proportional to the total indentation depth and the total load, respectively. The value of these ratios, however, is slightly increasing with decreasing loading rate [24]. Fig. 3 shows the dynamic Vickers microhardness, HV , calculated by the usual equation HV = 38.9
F h2
(2)
where F is given in N, h in m and HV in MPa, and the indentation velocity, v = h˙ from the experimental data presented in Fig. 2 as function of the indentation depth, h. In the case of pure Al, HV is a smooth function of the indentation depth. Consid-
During DSI, the load, F, is increased at constant loading rate, µ, so that at a moment, t, the load F(t) = µt. Some typical depth–load (h–F) curves obtained on different samples are shown in Fig. 2. Two types of the indentation depth–load curves can be distinguished: (i) for instance, on pure Al a smoothly changing curve appeared and (ii) for several solid solution alloys – such as Al–Mg and Al–Cu, which can be usually regarded as model materials showing plastic instabilities in tensile measurements – from a certain critical load, Fc (and a corresponding critical depth, hc ), characteristic steps appear in the h–F curves. It has been shown that the occurrence and the development of
Fig. 2. Typical indentation depth–load curves obtained in dynamic microhardness tests [33].
Fig. 3. Changes of the microhardness, HV [33] (a) and of the indentation velocity, v [32] (b) during indentation; obtained from F–h curves shown in Fig. 2.
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ering the step-containing indentation curves, however, HV and v oscillate quasi-periodically. This phenomenon corresponds to the serrated yielding (Portevin–Le Chˆatelier effect) observed in tensile tests. In both types of measurements, the phenomenon manifests itself in the occurrence of the successive low (denoted as AB in Figs. 1–3) and high (denoted as BC) rate regimes. The oscillation of both HV and v, as well as the correlation between these quantities, will be investigated in detail in the next sections. 3. Kinematic characterization of the indentation plastic instabilities 3.1. Effect of solute concentration on the indentation plastic instabilities
Fig. 5. The dependence of global microhardness, HV2N , of Al–Mg alloys on the solute concentration, C [34].
From the nature of the interaction between mobile dislocations and diffusing solute atoms, it can be expected that the solute concentration must exceed a certain minimal value, C0 , for Portevin–Le Chˆatelier type plastic instabilities to appear. Fig. 4 shows some typical indentation depth–load (h–F) curves taken on Al–Mg alloys with different Mg concentrations. It can be seen that the Mg content influences strongly both global and local behaviors of the indentation curves. On the one hand, the addition of Mg increases the strength of the materials (as the indentation depth decreases with increasing Mg content). On the other hand, at small Mg concentration (0.45% Mg), the measured curve is smoothly changing, while on the F–h curves measured on the samples with higher Mg concentrations characteristic plastic instability steps appear [23,29,34]. Taking the depth values belonging to the maximum load (Fmax = 2 N), the Vickers microhardness, HV2N , can be obtained. This amount can be considered as global behavior, and then obviously depends on the solute content of the materials. Fig. 5 shows the global hardness, HV2N , as a function of the Mg concentration for which a power law function
of the solute atmosphere around the dislocations in analyzing the dislocation–solute interaction. According to these models, the strength-increment of a solid solution hardened material is a power law function of the solute concentration with an exponent of 2/3. It is worth to mention that taking into account only the effect of the nearest solutes (being in the core of dislocation), the value of the exponent is 1/2, as it has been shown theoretically by Friedel [41]. In order to characterize the occurrence and development of instability steps, the smoothly changing part of the F–h curves – which corresponds to the dynamically stable (or global) behavior – is considered as a baseline which has been subtracted from the original Fmeasured curves. The baseline could be very accurately fitted with a
HV2N = HV0 + A · Cn
F = Fmeasured − Ffit
(3)
can be fitted well with the fitting parameters shown also in Fig. 4. The exponent n was found to be 0.63. This suggests that the global strengthening effect of solute atoms can be described by the statistical models developed by Labusch and Nabarro [39,40], who have taken into account also the long-range effect
Fig. 4. Indentation depth–load curves of different concentration Al–Mg alloys [23].
Ffit = Bhr
(4)
function where r is about 1.7–1.8 and B is a concentration dependent coefficient. The deviation of Fmeasured from the baseline (stable behavior, i.e. Ffit ), (5)
can then be transformed into microhardness changes according to HV = 38.9
F . h2
(6)
Fig. 6 shows characteristic HV –h curves of the indentation processes for different Mg concentrations. It can be seen that at a given concentration at the critical depth, hc , microhardness oscillations set in, the amplitude of which, HV0 – following a fast transient increase – is near constant. The HV versus h curves in Fig. 6 clearly show that HV0 decreases with decreasing Mg content. It can also be seen that the onset point of oscillations is shifted to higher hc values with decreasing solute content. In the case of 0.45 wt.% Mg (Fig. 6c), the value of HV0 is practically zero in the whole indentation process. Therefore, it should be concluded that in this sample under the present experimental conditions plastic instability does not occur. This result is in agreement with the physical basis of the DSA process for which a critical solute concentration is necessary. The existence of the critical solute concentration has also
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Fig. 7. The dependence of the amplitude of hardness drops, HV0 , on the solute concentration [34].
Fig. 6. The HV –h curves for: (a) 0.45%, (b) 0.95% and (c) 2.7% Mg concentrations [34].
been shown in another paper [23], on the basis of the solute concentration dependence of the critical depth, hc . In order to determine the dependence of HV0 on the Mg concentration, the values of HV0 have been plotted as a function of the nominal concentration, C, as shown in Fig. 7. Analyses show that the HV0 –C data points can be fitted with a HV0 = A∗ (C − C0 )m ,
(7)
type power law function where the value of the exponent, m, is also close to 2/3. The value of critical concentration, C0 , is about 0.76 wt.% Mg and A* is a constant. According to Eq. (7), the increase of HV0 with increasing solute content is a consequence of the higher number of solute atoms taking part in the DSA effect. Furthermore, if the Mg content decreases to the value of 0.76 wt.% (critical concentration), HV0 tends to zero. This means that if the Mg content, C ≤ 0.76 wt.% under the present experimental conditions plastic instabilities do not
occur. This result is in good agreement with both experiment and theoretical data reported before [23]. In a recent paper [34], these experimental results obtained by DSI were also compared with the results of numerical analysis of current literature models. It was shown that the term describing the effect of dynamic strain ageing in serrated flow is not proportional, but rather a power expression of the local solute concentration, Cs , on the dislocation line with the exponent of 1/2. The dependence on the local solute concentration, Cs , as a power law function with the exponent of 1/2 suggests that during plastic instabilities, the effect of dynamic strain ageing may arise dominantly from the interaction between dislocation and nearest solutes (being in the core of dislocation) by the Friedel mechanism [41], while – as it was mentioned above – the global strengthening effect of solute atoms arises from the long-range interaction between the solute atmosphere and the dislocations by the Labusch and Nabarro mechanism [39,40]. Furthermore, the analysis of the results of DSI experiments also leads to the conclusion that the kinetics of solute segregation during DSA is controlled by the pipe diffusion. Above, some important aspects characterizing indentation instabilities of Al–Mg alloys were discussed. It has been shown earlier [24] that these alloys are stable solid solutions (at least at room temperature), in which the instability steps are relatively regular and reproducible. In the case of age-hardenable alloys, like Al–Cu or Al–Zn–Mg alloys [27,29,42], indentation instabilities were also observed in the early stage of the decomposition of these alloys due to natural ageing. It has been shown that in these alloys, the characteristics of the instability steps are influenced also by Guinier–Preston zone formation. On the basis of the analysis of indentation instability, it was concluded that the addition of Cu retards the initial part of the decomposition process at room temperature in Al–Zn–Mg alloys [29,42]. 3.2. Effect of grain size on plastic deformation processes during indentation Recently, by applying the so-called Equal Channel Angular Pressing (ECAP) technique [43–45], the grain size of annealed Al–3% Mg alloy was reduced from ∼400 m to ∼300 nm [30]. It
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Fig. 8. The load–depth (F–h) and velocity–depth (v–h) obtained on the ECAP, ultrafine-grained Al–3%Mg alloy [30].
has been shown that the occurrence and development of instability indentation steps in the case of this ultrafine-grained material depends strongly on the indentation size. Fig. 8 shows a typical indentation depth–load curve obtained on ECAP samples. Indentation instabilities were observed only at small indentation sizes with indentation depths of <1.5 m. In the microindentation regime with indentation depths of >1.5 m, in contrast to the behavior of the annealed, coarse-grained sample, the ultrafinegrained sample did not show any traces of the plastic instabilities. It was also shown early [30] that when increasing the grain size by applying different annealing treatments above 200 ◦ C plastic instabilities enhancedly appear again during depth sensing indentation. The appearance of the PLC plastic instabilities in the ultrafine-grained material deformed by sub-micrometer indent sizes indicates that in these conditions intragranular dislocation motion plays a decisive role. At large indentation sizes, however, dislocation motion within the grains makes only a limited contribution to the plastic deformation and rather grain boundary processes become rate-controlling. The transition from dislocation motion at small indentations to a predominantly grain boundary mechanism at large indentations was estimated to take place when the deformation volume during indentation extends to the volume of approximately 1000 grains in the fine-grained sample [30]. The effect of individual grain boundaries on the PLC effect in depth sensing Vickers indentation has been also investigated [31]. As the result of the influence of grain boundaries on the indentation instabilities, two kinds of phenomena were found. The first, so-called small-step event, when irregular and smaller steps appeared besides the regularly developing steps, is attributed to PLC stepping with nearly constant phase shift in neighboring grains. The other is so-called large-step event, when unexpectedly large steps occurred on the indentation curves as shown in Fig. 9. The large-step even (see Fig. 9a), which leads to the large – “abnormal” – hardness oscillation (see Fig. 9b) is the direct consequence of the formation of dislocation pileups at the grain boundary and the overshooting of the plastic deformation after the pile-up dislocations are released by the boundary.
Fig. 9. Large-step event (a) and the corresponding hardness (b) due to grain boundaries during instability indentation [31].
4. Dynamic characterization of the indentation plastic instabilities 4.1. The correlation between hardness oscillation and indentation velocity As it has been shown, during instability at a given solute concentration, after the initial transient, both the quantity, HV , ˙ are oscillating between two and the indentation velocity, v = h, approximately constant limiting values. The typical correlation between the v indentation velocity and the HV oscillations is shown in Fig. 10 which is essentially a phase field representation showing the evolution of the t → [v(t), HV (t)] points. This representation shows that initially, after a rapid decrease in the velocity, the points get in the middle of the ellipsis-like contour, then following a few small amplitude oscillations they approximate the nearly elliptical limit cycle where they stay for longer, while the amplitudes of HV and v oscillations are hardly changing. The evolution in this phase field representation suggests an N-shape function relation between the HV hardness oscillations and the v indentation velocities. This N-shape HV → v function is analogous to the stress–strain rate relationship proposed by Penning [5] (shown in Fig. 1) and applied in some simulations [46,47]. The instability sets in at a point where the velocity of the indenter head, i.e. the corresponding deformation rate attains a critical value. Subsequently, the development of the instability process is governed by the spe-
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is reasonable to describe the dynamics of the formation of ˙ (or HV (v)) function – as instability steps [32]. The HV (h) mentioned above – is N-shaped in the present case. Introducing a βh˙ term to account for a rate dependent damping (without which there would be no static solution) in the (9) equation of motion, we arrive at Meff h¨ = µt − βh˙ − a0 h − a1 h2 −
˙ h2 HV (h) 38.9
(11)
The mechanical model corresponding to this motion equation can be seen in Fig. 11 where a body of mass Meff is accelerated by a force increasing linearly in time and is decelerated, besides the forces acting from a non-linear spring and from a viscous member, by the instability-resulting force, ˙ = h2 HV (h)/38.9, ˙ ˙ = f (h, h) denoted in the figure as f (h, h) 2 ∗ ∗ ˙ ˙ ˙ h HV (h), where HV (h) = HV (h)/38.9. In this simulation, the initial values for the parameters a0 and a1 were estimated on the basis of the fitting F–h curve to Eq. (8), the value for Meff was varied between 0.01 and 1 kg. Furthermore, the N-shaped HV (v) function was defined as Fig. 10. Phase field representation of the correlation between the v indentation velocity and the HV oscillations obtained at loading rate of 35 mN s−1 in the whole measured time [32] (a) and for only one period (b).
cial properties of the HV (v) relation. Consequently, to follow the dynamics of the phenomenon, the rate dependence of HV must be considered. 4.2. Dynamic modeling of the indentation instabilities During indentation, the external load and the force from the sample surface are acting on the indenter head and then determine its motion. The resulting force at the sample surface in case of a smooth indentation curve (without stair formation) is described by the power law function (4) with the exponent of about 1.7–1.8. In a recent paper [32], it has been shown that this function can be reasonably simplified to a second-order polynomial F = a0 h + a1 h2
(8)
where a0 and a1 are constants. The equation of motion accounting for the stepwise motion of the indenter head, under a continuously increasing external load, then has the following form: ˙ Meff h¨ = µt − (a0 h + a1 h2 ) − f (h, h)
HV (v) = δB(v − v∗ ),
if v < v∗
HV (v) = δ(v − v∗ ){(v − v∗ )2 − A}, if v∗ < v
(12)
where A, B, δ and v∗ are constant parameters which must be chosen as to be consistent with the experimental results in different cases. In the case of Al–3%Mg samples deformed by indentation with load rate, vl = 35 mN s−1 , for instance, the HV = HV0 + HV (v) function corresponding to Eqs. (11) and (12) is shown in Fig. 12. In the indentation experiments, the ˙ is highest at the beginning of the indenindentation velocity, h, tation and subsequently it is monotonously decreasing. The oscillations set in when h˙ arrives at the value which corresponds to the minimum of, because that part of the curve having negative slope is unstable. At this point, the system is jumping on the branch of the curve at low indentation velocities. The reverse jump takes place at the (v∗ ; HV (v∗ )) point. A typical solution of the equation of simulation for loading rate, µ = 35 mN s−1 , is shown in Fig. 13, where both the simulated load–depth (F–h) and indentation velocity–time (v–t) curves are compared with experimental ones. Preliminary results
(9)
where Meff is the mass of the moving part of the hardness tester ˙ denotes the rate dependent part of the force, i.e. the and f (h, h) ˙ force determines deviation from the smooth curve. The f (h, h) the dynamics of the step formation. We have found that follow˙ ing the onset of the oscillations, the amplitude of the HV (h, h) function is nearly constant with the progress of indentation which means that the assumption ˙ = f (h, h)
˙ ˙ h2 HV (h) h2 HV (h, h) = 38.9 38.9
(10)
Fig. 11. Mechanical model corresponding to the equation of motion described by Eq. (11) [32].
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to the N-shaped relationship between the stress, σ, and the strain rate, ε˙ , in tensile tests – is the reason of the occurrence of Portevin–Le-Chˆatelier effect during plastic (indentation or tensile) deformation. It is clear (see Fig. 11) that in the parameters of Eqs. (11) and (12), the characterizations of the measuring equipment and of the material are involved, the separation of which needs further experiments. Further analyses are also needed to develop the present dynamic model so that numerical fitting should add more physical insight. 5. Summary and conclusions Fig. 12. The HV (v) function used in the present simulations containing the Nshaped HV (v) function defined by Eq. (12) [32].
show that with the following set of parameters: Meff = 0.2 kg,
β = 8.0 mN s m−1 ,
δ = 3 GPa (m s)−3 , B = 2.0 m2 s2 ,
A = 0.4 m2 s−2 , v∗ = 0.06 m s−1
good agreement was found between the experimental and the simulated curves. It can be seen that the simulation correctly reflects many features of the experiments. Considering the equations of the dynamic model, it can be seen that Eq. (12) is a simple mathematical representation of the N-shape function relation, HV (v), between the HV hardness oscillations and the v indentation velocities, which – similarly
Fig. 13. Comparison of the measured and simulated load–depth (F–h) curves (a) and indentation velocity–time (v–t) curves obtained at loading rate of 35 mN s−1 (b) (the simulated curve in (b) is shifted by 1 m s−1 relative to its original position) [32].
In this paper, the phenomenon of plastic instability occurring in depth sensing indentation measurements was reviewed. Investigations were focused on the kinematic and dynamic characterization of Portevin–Le Chˆatelier type indentation instabilities observed in typical cases, for which the main results can be summarized as follows: (i) It was shown that the load–depth indentation curves obtained for different alloys are not always smoothly changing functions, but may contain characteristic steps. This means that dynamically the local behavior of the material deviates from the global one, leading to the occurrence of plastic instabilities during indentation. The smooth background component of the indentation curves could be accurately fitted with the Ffit = Khr power law function. The indentation instability can be characterized with the momentary hardness changes, HV , calculated from the amount of load deviation, F = Fmeasured − Ffit . (ii) Experimental results show that in the case of Al–Mg solid solutions, both the global hardness, HV2N , and the amplitude of hardness oscillation (HV0 ) change with the solute Mg content according to simple power law functions. Considering the solute concentration dependence of the hardness oscillation, it was also demonstrated that plastic instabilities occur only if the solute concentration is higher than a critical concentration, C0 . Under the present experimental conditions, for Al–Mg alloys around room temperature C0 was found to be 0.76 wt.% Mg. (iii) It was also shown that the occurrence and development of instability indentation steps in the case of ultrafine-grained materials depends strongly on the indent size. The appearance of the indentation instabilities under sub-micrometer indent sizes indicates that under these conditions intragranular dislocation motion plays a decisive role. At large indentation sizes, however, dislocation motion within the grains makes only a limited contribution to the plastic deformation and grain boundary processes become rate-controlling. The effect of individual grain boundaries on the motion of dislocations – the formation of dislocation pile-ups at the grain boundary – can be also investigated by studying the PLC effect in depth sensing Vickers indentation. (iv) Analysis has shown that the hardness oscillation, HV , depends sensitively on the indentation velocity, v, by an N-shaped HV –v function. Taking into account the experimental observations, a macro-mechanical dynamic model
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was proposed for the characterization of indentation instabilities. Acknowledgement This work was supported by the Hungarian National Scientific Research Fund (OTKA) under contract numbers T-038048, T-043247, D-048461 and T-046990. References [1] H. Neuh¨auser, C. Schwink, in: R.W. Cahn, P. Haasen, E.J. Kramer, H. Mughrabi (Eds.), Materials Science and Technology, vol. 6, VCH, Weinheim, Germany, 1993, p. 191. [2] Y.J.M. Br´echet, Key Eng. Mater. 103 (1995) 21. [3] P.G. McCormick, Scripta Met. 15 (1982) 441. [4] D. Thevenet, M. Mliha-Touati, A. Zeghloul, Mater. Sci. Eng. A 266 (1999) 175. [5] P. Penning, Acta Metall. 20 (1972) 1169. [6] P.G. McCormick, Acta Metall. 20 (1972) 351. [7] L.P. Kubin, Y. Estrin, Acta Metall. 38 (1990) 697. [8] Y. Br´echet, Y. Estrin, Acta Metall. 43 (1995) 955. [9] S. Kok, M.S. Bharathi, A.J. Beaudoin, C. Fressengeas, G. Ananthakrishna, L.P. Kubin, M. Lebyodkin, Acta Mater. 51 (2003) 3651. [10] R. Kr´al, P. Luk´acˇ , M. Janeˇcek, Mater. Sci. Forum 217–222 (1996) 1025. [11] E. Pink, A. Grinberg, Acta Metall. 30 (1982) 2153. [12] A.H. Cottrell, Philos. Mag. 44 (1953) 829. [13] A. Van den Beukel, Phys. Stat. Sol. (a) 30 (1975) 197. [14] J.M. Robinson, M.P. Shaw, Int. Mater. Rev. 39 (1994) 113. [15] J.M. Robinson, Int. Mater. Rev. 39 (1994) 217. [16] L.P. Kubin, K. Chihab, Y. Estrin, Acta Metall. 36 (1988) 2707. [17] P.G. McCormick, S. Venkadesan, C.P. Ling, Scripta Met. 29 (1993) 1159. [18] P.G. McCormick, C.P. Ling, Acta Metall. Mater. 43 (1995) 1969. [19] P. Luk´acˇ , J. Bal´ık, F. Chmel´ık, Mater. Sci. Eng. A 234 (1997) 45. [20] H.M. Zbib, E.C. Aifantis, Scripta Metall. 22 (1988) 1331. [21] P. H¨ahner, Mater. Sci. Eng. A 164 (1993) 23. [22] P.G. McCormick, Acta Metall. 30 (1982) 2079.
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[23] N.Q. Chinh, F. Csikor, Zs. Kov´acs, J. Lendvai, J. Mater. Res. 15 (2000) 1037. [24] G. B´erces, N.Q. Chinh, A. Juh´asz, J. Lendvai, Acta Mater. 46 (1998) 2029. [25] G. B´erces, N.Q. Chinh, A. Juh´asz, J. Lendvai, J. Mater. Res. 13 (1998) 1411. [26] N.Q. Chinh, F. Csikor, Gy. B´erces, J. Lendvai, Alum. Alloys 2 (1998) 979. [27] N.Q. Chinh, F. Csikor, J. Lendvai, Mater. Sci. Forum 331–337 (2000) 1007. [28] Zs. Kov´acs, N.Q. Chinh, J. Lendvai, J. Mater. Res. 16 (2001) 1171. [29] N.Q. Chinh, Gy. Horv´ath, Zs. Kov´acs, J. Lendvai, Mater. Sci. Eng. A 324 (2002) 219. [30] Zs. Kov´acs, N.Q. Chinh, J. Lendvai, Z. Horita, T.G. Langdon, Mater. Sci. Forum 396–402 (2002) 1073. [31] Zs. Kov´acs, D. F´atay, K. Nyilas, J. Lendvai, J. Eng. Mater. Technol. 124 (2002) 23. [32] G. B´erces, J. Lendvai, A. Juh´asz, N.Q. Chinh, J. Mater. Res. 18 (2003) 2874. [33] N.Q. Chinh, J. Gubicza, Zs. Kov´acs, J. Lendvai, J. Mater. Res. 19 (2004) 31. [34] Gy. Horv´ath, N.Q. Chinh, J. Lendvai, J. Mater. Res. 20 (2005) 331. [35] W.C. Oliver, G.M. Pharr, J. Mater. Res. 7 (1992) 1564. [36] D.S. Stone, K.B. Yoder, J. Mater. Res. 9 (1994) 2524. [37] J. Gubicza, F. W´eber, Mater. Sci. Eng. A 263 (1999) 101. [38] A.C. Fischer-Cripps, Nanoindentation, second ed., Springer-Verlag, New York, 2004. [39] R. Labusch, Acta Metall. 20 (1972) 917. [40] F.R.N. Nabarro, Philos. Mag. 35 (1977) 613. [41] J. Friedel, Dislocations, Pergamon Press, New York, 1964, p. 644. [42] N.Q. Chinh, J. Lendvai, D.H. Ping, K. Hono, J. Alloys Compd. 378 (2004) 52. [43] Y. Iwahashi, J. Wang, Z. Horita, M. Nemoto, T.G. Langdon, Scripta Mater. 35 (1996) 143. [44] M. Furukawa, Y. Iwahashi, Z. Horita, M. Nemoto, T.G. Langdon, Mater. Sci. Eng. A257 (1998) 328. [45] M. Furukawa, Z. Horita, M. Nemoto, T.G. Langdon, J. Mater. Sci. 36 (2001) 2835. [46] M. Lebyodkin, Y. Brechet, Y. Estrin, L.P. Kubin, Acta Mater. 44 (1996) 4531. [47] M. Lebyodkin, Y. Brechet, Y. Estrin, L.P. Kubin, Phys. Rev. Lett. 74 (1995) 4758.
Review
Kinematic and dynamic characterization of plastic instabilities occurring in nano- and microindentation tests Nguyen Q. Chinh ∗ , Gy. Horv´ath, Zs. Kov´acs, A. Juh´asz, Gy. B´erces, J. Lendvai Department of General Physics, E¨otv¨os University, 1117 Budapest, P´azm´any P. S´et´any 1/A., Hungary Received in revised form 22 March 2005; accepted 14 April 2005 Dedicated to Professor James C.M. Li on the occasion of his 80th birthday.
Abstract This paper surveys the phenomenon of plastic instabilities occurring in depth sensing indentation measurements, during which a stepwise increase has been observed in the indentation depth–load (F–h) curves measured in constant loading rate mode. In these measurements, plastic instability – serrated indentation – manifests itself as hardness oscillations around a nearly constant value of the conventional dynamic microhardness. Investigations presented here are focused on both kinematic and dynamic characterization of Portevin–Le Chˆatelier type indentation instabilities. The effect of important factors such as solute concentration and polycrystalline grain size is described and discussed. Taking into account the experimental observations, a macro-mechanical dynamic model was also proposed for the characterization of indentation instabilities. © 2005 Elsevier B.V. All rights reserved. Keywords: Plastic instabilities; Serrated indentation; Hardness oscillation; Solute concentration; Grain size; Dynamic model
Contents 1. 2. 3.
4.
5.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The phenomenon of plastic instabilities in depth sensing indentation tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kinematic characterization of the indentation plastic instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Effect of solute concentration on the indentation plastic instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Effect of grain size on plastic deformation processes during indentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamic characterization of the indentation plastic instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. The correlation between hardness oscillation and indentation velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Dynamic modeling of the indentation instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Introduction Plastic instabilities as the phenomenon of discontinuous yielding during plastic deformation have been widely observed, and discussed in the literature under various names like serrated yielding, Portevin–Le Chˆatelier (PLC) effect, jerky flow, ∗
Corresponding author. Tel.: +36 1 372 2845; fax: +36 1 372 2811. E-mail address: [email protected] (N.Q. Chinh).
0921-5093/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2005.04.057
100 101 102 102 103 104 104 105 106 107 107
etc. [1–5]. In unidirectional deformation, the most characteristic feature is the appearance of stress drops or stress jumps, σ, beginning from a critical strain, εc , on the originally smooth stress–strain curves. Therefore, several works focused both theoretically [6–9] and experimentally [4,10,11] on the interpretation and determination of the parameter σ. The physical basis for the appearance of the PLC effect is a negative strain rate sensitivity (SRS) originating mainly from the interaction between mobile dislocations and diffusing solute atoms during plastic deformation, a phenomenon called dynamic strain ageing
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Fig. 1. Schematic of the N-shaped stress–strain rate (σ–˙ε) function of alloys showing PLC plastic instabilities.
(DSA) [12–15]. Depending on the testing conditions, negative SRS may occur at a certain part of the N-shaped stress–strain rate (σ–˙ε) function proposed by Penning [5], shown schematically in Fig. 1. Several models have been developed both on the solute–dislocation interaction [16–19] and on the explicit expression of the mentioned N-shaped σ–˙ε function [20,21]. Although the PLC effect has been observed and investigated mainly by uniaxial tensile and compression tests [6,11,17], it was also studied in other modes of deformation [22]. Recently, the phenomenon of PLC plastic instabilities in metals was investigated also by depth sensing indentation (DSI) tests [23–34], which are nowadays increasingly used for the investigation of complex mechanical properties of materials [35–38]. 2. The phenomenon of plastic instabilities in depth sensing indentation tests
101
the steps depend both on the loading rate and on the composition of the alloy [23,24]. Analyzing the depth–load curves containing steps, the indentation process can be physically regarded as a series of two subsequent local processes, in which firstly the load increases by Fp at essentially negligible increase in the indentation depth, and then the deformation (indentation depth) increases rapidly by Dp at practically no increase in load (see also the inset in Fig. 2). It has been shown earlier [24] that in the case of stable solid solutions, such as Al–Mg alloys, the instability steps are relatively regular and reproducible. Furthermore, analyzing the development of the steps at a given (h; F) point on the indentation curve, it has been shown that both Dp h
and
Fp F
(1)
ratios are approximately constant, that is, the step height, Dp , and the load period, Fp , characterizing instability steps are proportional to the total indentation depth and the total load, respectively. The value of these ratios, however, is slightly increasing with decreasing loading rate [24]. Fig. 3 shows the dynamic Vickers microhardness, HV , calculated by the usual equation HV = 38.9
F h2
(2)
where F is given in N, h in m and HV in MPa, and the indentation velocity, v = h˙ from the experimental data presented in Fig. 2 as function of the indentation depth, h. In the case of pure Al, HV is a smooth function of the indentation depth. Consid-
During DSI, the load, F, is increased at constant loading rate, µ, so that at a moment, t, the load F(t) = µt. Some typical depth–load (h–F) curves obtained on different samples are shown in Fig. 2. Two types of the indentation depth–load curves can be distinguished: (i) for instance, on pure Al a smoothly changing curve appeared and (ii) for several solid solution alloys – such as Al–Mg and Al–Cu, which can be usually regarded as model materials showing plastic instabilities in tensile measurements – from a certain critical load, Fc (and a corresponding critical depth, hc ), characteristic steps appear in the h–F curves. It has been shown that the occurrence and the development of
Fig. 2. Typical indentation depth–load curves obtained in dynamic microhardness tests [33].
Fig. 3. Changes of the microhardness, HV [33] (a) and of the indentation velocity, v [32] (b) during indentation; obtained from F–h curves shown in Fig. 2.
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ering the step-containing indentation curves, however, HV and v oscillate quasi-periodically. This phenomenon corresponds to the serrated yielding (Portevin–Le Chˆatelier effect) observed in tensile tests. In both types of measurements, the phenomenon manifests itself in the occurrence of the successive low (denoted as AB in Figs. 1–3) and high (denoted as BC) rate regimes. The oscillation of both HV and v, as well as the correlation between these quantities, will be investigated in detail in the next sections. 3. Kinematic characterization of the indentation plastic instabilities 3.1. Effect of solute concentration on the indentation plastic instabilities
Fig. 5. The dependence of global microhardness, HV2N , of Al–Mg alloys on the solute concentration, C [34].
From the nature of the interaction between mobile dislocations and diffusing solute atoms, it can be expected that the solute concentration must exceed a certain minimal value, C0 , for Portevin–Le Chˆatelier type plastic instabilities to appear. Fig. 4 shows some typical indentation depth–load (h–F) curves taken on Al–Mg alloys with different Mg concentrations. It can be seen that the Mg content influences strongly both global and local behaviors of the indentation curves. On the one hand, the addition of Mg increases the strength of the materials (as the indentation depth decreases with increasing Mg content). On the other hand, at small Mg concentration (0.45% Mg), the measured curve is smoothly changing, while on the F–h curves measured on the samples with higher Mg concentrations characteristic plastic instability steps appear [23,29,34]. Taking the depth values belonging to the maximum load (Fmax = 2 N), the Vickers microhardness, HV2N , can be obtained. This amount can be considered as global behavior, and then obviously depends on the solute content of the materials. Fig. 5 shows the global hardness, HV2N , as a function of the Mg concentration for which a power law function
of the solute atmosphere around the dislocations in analyzing the dislocation–solute interaction. According to these models, the strength-increment of a solid solution hardened material is a power law function of the solute concentration with an exponent of 2/3. It is worth to mention that taking into account only the effect of the nearest solutes (being in the core of dislocation), the value of the exponent is 1/2, as it has been shown theoretically by Friedel [41]. In order to characterize the occurrence and development of instability steps, the smoothly changing part of the F–h curves – which corresponds to the dynamically stable (or global) behavior – is considered as a baseline which has been subtracted from the original Fmeasured curves. The baseline could be very accurately fitted with a
HV2N = HV0 + A · Cn
F = Fmeasured − Ffit
(3)
can be fitted well with the fitting parameters shown also in Fig. 4. The exponent n was found to be 0.63. This suggests that the global strengthening effect of solute atoms can be described by the statistical models developed by Labusch and Nabarro [39,40], who have taken into account also the long-range effect
Fig. 4. Indentation depth–load curves of different concentration Al–Mg alloys [23].
Ffit = Bhr
(4)
function where r is about 1.7–1.8 and B is a concentration dependent coefficient. The deviation of Fmeasured from the baseline (stable behavior, i.e. Ffit ), (5)
can then be transformed into microhardness changes according to HV = 38.9
F . h2
(6)
Fig. 6 shows characteristic HV –h curves of the indentation processes for different Mg concentrations. It can be seen that at a given concentration at the critical depth, hc , microhardness oscillations set in, the amplitude of which, HV0 – following a fast transient increase – is near constant. The HV versus h curves in Fig. 6 clearly show that HV0 decreases with decreasing Mg content. It can also be seen that the onset point of oscillations is shifted to higher hc values with decreasing solute content. In the case of 0.45 wt.% Mg (Fig. 6c), the value of HV0 is practically zero in the whole indentation process. Therefore, it should be concluded that in this sample under the present experimental conditions plastic instability does not occur. This result is in agreement with the physical basis of the DSA process for which a critical solute concentration is necessary. The existence of the critical solute concentration has also
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Fig. 7. The dependence of the amplitude of hardness drops, HV0 , on the solute concentration [34].
Fig. 6. The HV –h curves for: (a) 0.45%, (b) 0.95% and (c) 2.7% Mg concentrations [34].
been shown in another paper [23], on the basis of the solute concentration dependence of the critical depth, hc . In order to determine the dependence of HV0 on the Mg concentration, the values of HV0 have been plotted as a function of the nominal concentration, C, as shown in Fig. 7. Analyses show that the HV0 –C data points can be fitted with a HV0 = A∗ (C − C0 )m ,
(7)
type power law function where the value of the exponent, m, is also close to 2/3. The value of critical concentration, C0 , is about 0.76 wt.% Mg and A* is a constant. According to Eq. (7), the increase of HV0 with increasing solute content is a consequence of the higher number of solute atoms taking part in the DSA effect. Furthermore, if the Mg content decreases to the value of 0.76 wt.% (critical concentration), HV0 tends to zero. This means that if the Mg content, C ≤ 0.76 wt.% under the present experimental conditions plastic instabilities do not
occur. This result is in good agreement with both experiment and theoretical data reported before [23]. In a recent paper [34], these experimental results obtained by DSI were also compared with the results of numerical analysis of current literature models. It was shown that the term describing the effect of dynamic strain ageing in serrated flow is not proportional, but rather a power expression of the local solute concentration, Cs , on the dislocation line with the exponent of 1/2. The dependence on the local solute concentration, Cs , as a power law function with the exponent of 1/2 suggests that during plastic instabilities, the effect of dynamic strain ageing may arise dominantly from the interaction between dislocation and nearest solutes (being in the core of dislocation) by the Friedel mechanism [41], while – as it was mentioned above – the global strengthening effect of solute atoms arises from the long-range interaction between the solute atmosphere and the dislocations by the Labusch and Nabarro mechanism [39,40]. Furthermore, the analysis of the results of DSI experiments also leads to the conclusion that the kinetics of solute segregation during DSA is controlled by the pipe diffusion. Above, some important aspects characterizing indentation instabilities of Al–Mg alloys were discussed. It has been shown earlier [24] that these alloys are stable solid solutions (at least at room temperature), in which the instability steps are relatively regular and reproducible. In the case of age-hardenable alloys, like Al–Cu or Al–Zn–Mg alloys [27,29,42], indentation instabilities were also observed in the early stage of the decomposition of these alloys due to natural ageing. It has been shown that in these alloys, the characteristics of the instability steps are influenced also by Guinier–Preston zone formation. On the basis of the analysis of indentation instability, it was concluded that the addition of Cu retards the initial part of the decomposition process at room temperature in Al–Zn–Mg alloys [29,42]. 3.2. Effect of grain size on plastic deformation processes during indentation Recently, by applying the so-called Equal Channel Angular Pressing (ECAP) technique [43–45], the grain size of annealed Al–3% Mg alloy was reduced from ∼400 m to ∼300 nm [30]. It
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Fig. 8. The load–depth (F–h) and velocity–depth (v–h) obtained on the ECAP, ultrafine-grained Al–3%Mg alloy [30].
has been shown that the occurrence and development of instability indentation steps in the case of this ultrafine-grained material depends strongly on the indentation size. Fig. 8 shows a typical indentation depth–load curve obtained on ECAP samples. Indentation instabilities were observed only at small indentation sizes with indentation depths of <1.5 m. In the microindentation regime with indentation depths of >1.5 m, in contrast to the behavior of the annealed, coarse-grained sample, the ultrafinegrained sample did not show any traces of the plastic instabilities. It was also shown early [30] that when increasing the grain size by applying different annealing treatments above 200 ◦ C plastic instabilities enhancedly appear again during depth sensing indentation. The appearance of the PLC plastic instabilities in the ultrafine-grained material deformed by sub-micrometer indent sizes indicates that in these conditions intragranular dislocation motion plays a decisive role. At large indentation sizes, however, dislocation motion within the grains makes only a limited contribution to the plastic deformation and rather grain boundary processes become rate-controlling. The transition from dislocation motion at small indentations to a predominantly grain boundary mechanism at large indentations was estimated to take place when the deformation volume during indentation extends to the volume of approximately 1000 grains in the fine-grained sample [30]. The effect of individual grain boundaries on the PLC effect in depth sensing Vickers indentation has been also investigated [31]. As the result of the influence of grain boundaries on the indentation instabilities, two kinds of phenomena were found. The first, so-called small-step event, when irregular and smaller steps appeared besides the regularly developing steps, is attributed to PLC stepping with nearly constant phase shift in neighboring grains. The other is so-called large-step event, when unexpectedly large steps occurred on the indentation curves as shown in Fig. 9. The large-step even (see Fig. 9a), which leads to the large – “abnormal” – hardness oscillation (see Fig. 9b) is the direct consequence of the formation of dislocation pileups at the grain boundary and the overshooting of the plastic deformation after the pile-up dislocations are released by the boundary.
Fig. 9. Large-step event (a) and the corresponding hardness (b) due to grain boundaries during instability indentation [31].
4. Dynamic characterization of the indentation plastic instabilities 4.1. The correlation between hardness oscillation and indentation velocity As it has been shown, during instability at a given solute concentration, after the initial transient, both the quantity, HV , ˙ are oscillating between two and the indentation velocity, v = h, approximately constant limiting values. The typical correlation between the v indentation velocity and the HV oscillations is shown in Fig. 10 which is essentially a phase field representation showing the evolution of the t → [v(t), HV (t)] points. This representation shows that initially, after a rapid decrease in the velocity, the points get in the middle of the ellipsis-like contour, then following a few small amplitude oscillations they approximate the nearly elliptical limit cycle where they stay for longer, while the amplitudes of HV and v oscillations are hardly changing. The evolution in this phase field representation suggests an N-shape function relation between the HV hardness oscillations and the v indentation velocities. This N-shape HV → v function is analogous to the stress–strain rate relationship proposed by Penning [5] (shown in Fig. 1) and applied in some simulations [46,47]. The instability sets in at a point where the velocity of the indenter head, i.e. the corresponding deformation rate attains a critical value. Subsequently, the development of the instability process is governed by the spe-
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is reasonable to describe the dynamics of the formation of ˙ (or HV (v)) function – as instability steps [32]. The HV (h) mentioned above – is N-shaped in the present case. Introducing a βh˙ term to account for a rate dependent damping (without which there would be no static solution) in the (9) equation of motion, we arrive at Meff h¨ = µt − βh˙ − a0 h − a1 h2 −
˙ h2 HV (h) 38.9
(11)
The mechanical model corresponding to this motion equation can be seen in Fig. 11 where a body of mass Meff is accelerated by a force increasing linearly in time and is decelerated, besides the forces acting from a non-linear spring and from a viscous member, by the instability-resulting force, ˙ = h2 HV (h)/38.9, ˙ ˙ = f (h, h) denoted in the figure as f (h, h) 2 ∗ ∗ ˙ ˙ ˙ h HV (h), where HV (h) = HV (h)/38.9. In this simulation, the initial values for the parameters a0 and a1 were estimated on the basis of the fitting F–h curve to Eq. (8), the value for Meff was varied between 0.01 and 1 kg. Furthermore, the N-shaped HV (v) function was defined as Fig. 10. Phase field representation of the correlation between the v indentation velocity and the HV oscillations obtained at loading rate of 35 mN s−1 in the whole measured time [32] (a) and for only one period (b).
cial properties of the HV (v) relation. Consequently, to follow the dynamics of the phenomenon, the rate dependence of HV must be considered. 4.2. Dynamic modeling of the indentation instabilities During indentation, the external load and the force from the sample surface are acting on the indenter head and then determine its motion. The resulting force at the sample surface in case of a smooth indentation curve (without stair formation) is described by the power law function (4) with the exponent of about 1.7–1.8. In a recent paper [32], it has been shown that this function can be reasonably simplified to a second-order polynomial F = a0 h + a1 h2
(8)
where a0 and a1 are constants. The equation of motion accounting for the stepwise motion of the indenter head, under a continuously increasing external load, then has the following form: ˙ Meff h¨ = µt − (a0 h + a1 h2 ) − f (h, h)
HV (v) = δB(v − v∗ ),
if v < v∗
HV (v) = δ(v − v∗ ){(v − v∗ )2 − A}, if v∗ < v
(12)
where A, B, δ and v∗ are constant parameters which must be chosen as to be consistent with the experimental results in different cases. In the case of Al–3%Mg samples deformed by indentation with load rate, vl = 35 mN s−1 , for instance, the HV = HV0 + HV (v) function corresponding to Eqs. (11) and (12) is shown in Fig. 12. In the indentation experiments, the ˙ is highest at the beginning of the indenindentation velocity, h, tation and subsequently it is monotonously decreasing. The oscillations set in when h˙ arrives at the value which corresponds to the minimum of, because that part of the curve having negative slope is unstable. At this point, the system is jumping on the branch of the curve at low indentation velocities. The reverse jump takes place at the (v∗ ; HV (v∗ )) point. A typical solution of the equation of simulation for loading rate, µ = 35 mN s−1 , is shown in Fig. 13, where both the simulated load–depth (F–h) and indentation velocity–time (v–t) curves are compared with experimental ones. Preliminary results
(9)
where Meff is the mass of the moving part of the hardness tester ˙ denotes the rate dependent part of the force, i.e. the and f (h, h) ˙ force determines deviation from the smooth curve. The f (h, h) the dynamics of the step formation. We have found that follow˙ ing the onset of the oscillations, the amplitude of the HV (h, h) function is nearly constant with the progress of indentation which means that the assumption ˙ = f (h, h)
˙ ˙ h2 HV (h) h2 HV (h, h) = 38.9 38.9
(10)
Fig. 11. Mechanical model corresponding to the equation of motion described by Eq. (11) [32].
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to the N-shaped relationship between the stress, σ, and the strain rate, ε˙ , in tensile tests – is the reason of the occurrence of Portevin–Le-Chˆatelier effect during plastic (indentation or tensile) deformation. It is clear (see Fig. 11) that in the parameters of Eqs. (11) and (12), the characterizations of the measuring equipment and of the material are involved, the separation of which needs further experiments. Further analyses are also needed to develop the present dynamic model so that numerical fitting should add more physical insight. 5. Summary and conclusions Fig. 12. The HV (v) function used in the present simulations containing the Nshaped HV (v) function defined by Eq. (12) [32].
show that with the following set of parameters: Meff = 0.2 kg,
β = 8.0 mN s m−1 ,
δ = 3 GPa (m s)−3 , B = 2.0 m2 s2 ,
A = 0.4 m2 s−2 , v∗ = 0.06 m s−1
good agreement was found between the experimental and the simulated curves. It can be seen that the simulation correctly reflects many features of the experiments. Considering the equations of the dynamic model, it can be seen that Eq. (12) is a simple mathematical representation of the N-shape function relation, HV (v), between the HV hardness oscillations and the v indentation velocities, which – similarly
Fig. 13. Comparison of the measured and simulated load–depth (F–h) curves (a) and indentation velocity–time (v–t) curves obtained at loading rate of 35 mN s−1 (b) (the simulated curve in (b) is shifted by 1 m s−1 relative to its original position) [32].
In this paper, the phenomenon of plastic instability occurring in depth sensing indentation measurements was reviewed. Investigations were focused on the kinematic and dynamic characterization of Portevin–Le Chˆatelier type indentation instabilities observed in typical cases, for which the main results can be summarized as follows: (i) It was shown that the load–depth indentation curves obtained for different alloys are not always smoothly changing functions, but may contain characteristic steps. This means that dynamically the local behavior of the material deviates from the global one, leading to the occurrence of plastic instabilities during indentation. The smooth background component of the indentation curves could be accurately fitted with the Ffit = Khr power law function. The indentation instability can be characterized with the momentary hardness changes, HV , calculated from the amount of load deviation, F = Fmeasured − Ffit . (ii) Experimental results show that in the case of Al–Mg solid solutions, both the global hardness, HV2N , and the amplitude of hardness oscillation (HV0 ) change with the solute Mg content according to simple power law functions. Considering the solute concentration dependence of the hardness oscillation, it was also demonstrated that plastic instabilities occur only if the solute concentration is higher than a critical concentration, C0 . Under the present experimental conditions, for Al–Mg alloys around room temperature C0 was found to be 0.76 wt.% Mg. (iii) It was also shown that the occurrence and development of instability indentation steps in the case of ultrafine-grained materials depends strongly on the indent size. The appearance of the indentation instabilities under sub-micrometer indent sizes indicates that under these conditions intragranular dislocation motion plays a decisive role. At large indentation sizes, however, dislocation motion within the grains makes only a limited contribution to the plastic deformation and grain boundary processes become rate-controlling. The effect of individual grain boundaries on the motion of dislocations – the formation of dislocation pile-ups at the grain boundary – can be also investigated by studying the PLC effect in depth sensing Vickers indentation. (iv) Analysis has shown that the hardness oscillation, HV , depends sensitively on the indentation velocity, v, by an N-shaped HV –v function. Taking into account the experimental observations, a macro-mechanical dynamic model
N.Q. Chinh et al. / Materials Science and Engineering A 409 (2005) 100–107
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