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Controlling mechanisms of deformation of AA5052 aluminium alloy at small strains under hot working conditions
Materials Science and Engineering A 393 (2005) 157–163
Controlling mechanisms of deformation of AA5052 aluminium alloy at small strains under hot working conditions R. Carmona∗ , Q. Zhu1 , C.M. Sellars, J.H. Beynon IMMPETUS—Institute for Microstructural and Mechanical Process Engineering: The University of Sheffield, Mappin Street, Sheffield S13JD, UK Received 5 April 2004; received in revised form 4 October 2004; accepted 2 November 2004
Abstract The objective of the present work was to investigate the thermomechanical behaviour of a commercial Al–Mg alloy and to understand the deformation mechanisms taking place at small strains (<0.2) and high temperatures (≥300 ◦ C) at strain rates relevant to hot working conditions (0.01–10 s−1 ). The experimental approach addresses different effects of testing machines on the load–displacement measurements. The results show distinct sorts of behaviour depending on the Zener–Hollomon parameter (Z): normal continuous increase in the stress with strain for high values of Z and a rise followed by a drop in stress with increasing strain for low values of Z. For steady-state behaviour, the relationship between stress and strain rate follows the typical power law with an exponent of 3 at low values of Z and an exponential relationship at higher values. These observations are similar to those of creep. The difference in stress response may be explained by different controlling mechanisms, namely climb of dislocations for the normal hardening behaviour and solute drag in the case where a drop of stress is observed in the stress–strain curve. © 2004 Elsevier B.V. All rights reserved. Keywords: Hot working; Small strain deformation; Aluminium alloys; Flow stress; Deformation mechanism; Zener–Hollomon parameter
1. Introduction To date, interest in thermomechanical processing has been mainly focused on large strain plastic deformation. Recently, however, the small strain behaviour has received attention, because some regions in the hot formed workpiece may undergo very little deformation during certain stages of processing [1]. Such is the case for large section rolling or forging, where localised regions of the sections may receive very little or no deformation in certain passes. As a result, the mechanical properties in these localised regions of products may be poor, because the deformation may not ∗ Corresponding author. Present address: EADS-CASA, Ensayos Estructurales, Paseo John Lennon, s/n, E-28906 Getafe, Madrid, Spain Tel.: +34 659 68 5627. E-mail address: [email protected] (R. Carmona). 1 Present address: Holset Engineering, St. Andrew’s Road, Huddersfield HD16RA, UK
0921-5093/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2004.11.010
be enough to trigger recrystallisation. Mechanical equations of state are available for aluminium alloys for large strain plasticity [2] but these equations cannot be extrapolated with confidence to small plastic strains. The processes undergone by the material when the strains are small have not yet been addressed and there is a knowledge gap in this area of study of high temperature deformation at hot working strain rates. Creep studies of Al–Mg alloys show that, under some deformation conditions, this type of alloy presents special features, such as an exponent of 3 in the stress–strain rate relationship instead of 5 usually found for pure Al or other Al alloys [3,4]. Yavari and Langdon [3] and Zhu et al. [4] carried out their tests not only under traditional creep conditions (constant stress and low strain rate), but also under constant cross-head displacement rate (close to constant strain rate). Al–Mg alloys showed a transition between the exponents 3 and 5, which was interpreted as a breakaway of the dislocations from their solute atmospheres, with a changeover in
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R. Carmona et al. / Materials Science and Engineering A 393 (2005) 157–163
Nomenclature b E D Do fv G kB M n n
Burgers vector Young’s modulus diff diffusion coefficient: D = Do exp −Q RT frequency factor volume fraction of second phase particles shear modulus Boltzmann’s constant: 1.381 × 10−23 J/K Taylor factor (∼ =3 for fcc polycrystals) exponent relating strain rate and stress exponent relating velocity of dislocations and effective stress Qdef , Qdiff activation energies for deformation and diffusion, respectively R universal gas constant: 8.314 J/(mol K) r second phase particle radius T absolute temperature Wm binding energy between solute atoms and dislocations def Z Zener–Hollomon parameter Z = ε˙ exp QRT δ subgrain size ε strain ε˙ strain rate ρi internal dislocation density θ subgrain misorientation ν velocity of moving dislocations σ equivalent tensile stress α, β, C material constants
the acting mechanisms for the control of dislocation motion, from viscous drag controlled glide to high temperature climb or cross-slip. However, that investigation was focused only on the steady-state behaviour. In fact, much earlier, Jonas et al. [5] reviewed the whole spectrum of conditions in creep and hot working, and established the interdependence of stress and strain rate by a hyperbolic sine function, which becomes a power law at lower stresses and an exponential law at higher stresses. It is clear that there is a need to understand fully the whole deformation behaviour, including transient stress–strain curves, and the microstructural processes taking place at small strains in thermomechanical processing. Current knowledge of creep or large plastic deformation under hot working conditions does not give sufficient insight into the deformation behaviour at low strain. In the present study, the deformation behaviour at strains up to about 0.15 for Al–Mg alloys under hot working conditions is investigated in detail. Aluminium alloy AA5052 was used because it represents a group of aluminium alloys widely used in the transportation, marine and automotive industries. The controlling mechanisms of hot deformation behaviour are discussed using inter-
nal state variables, i.e. the internal dislocation density (ρI ) the subgrain size (δ) and the misorientation between subgrains (θ).
2. Description of the stress as a function of the internal state variables The flow stress of the material (σ) can be described as [6]: σ = σe + σρi + σδ + σd + σp
(1)
where σ e is the effective stress (also known as friction stresses, σ f ) σρi the athermal stress due to interactions of dislocations within the subgrains, σ δ and σ d are the long range athermal stresses from the subgrain boundaries and from the grain boundaries, respectively, and σ p is the particle hardening term. The athermal stress arising from interaction of internal dislocations can be expressed as [7]: √ σρi = αMGb ρi , (2) where α is a material parameter, M the Taylor factor, G the shear modulus, b the Burgers vector and ρi is the internal dislocation density. The stress arising from the grain boundaries (σ d ) is negligible when the grain size is large in comparison with the size. The value of the stress arising from the subgrain boundaries (σ δ ) depends mainly on the subgrain size (δ) but it is also a function of the Taylor factor, the shear modulus and the Burgers vector [8]. The precipitation hardening stress (σ p ) is given by a relationship between different constants, such as the shear modulus and the Burgers vector, the volume fraction of precipitates and the particle radius [9]. The strain rate (˙ε) is related to the dislocation density and to the velocity of dislocations (ν) as [7]: ε˙ =
b ρi ν, M
where ν is given by [10]: Qdef C 2 σe ν = C1 exp − , sinh kB T T
(3)
(4)
where Qdef is the deformation activation energy, and C1 and C2 are material constants. Eq. (4) can be re-written as a power law [11]:
ν = βσen ,
(5)
where n is an exponent reflecting the sensitivity of the relationship between velocity of dislocation motion and applied stress, which is dependent on the magnitude of the stress. At low stresses, n = 1 for Al–Mg and other strongly solute hardened alloys and n = 3 for pure Al and other weakly solute hardened alloys. At high stresses, n increases with increasing stress.
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159
Dislocation density in Eq. (3) is determined by the combination of strain hardening and recovery [12]. At steady-state, ρi ∝ σ 2 [18], and Eq. (3) can then be re-written as: ε˙ =
b Cβσ 2 σen , M
(6)
where C is a material constant. From Eqs. (5) and (6), n = n−2, where n is the stress exponent relating strain rate and stress. From Eqs. (3) and (5), the friction or effective stress can be expressed as: M ε˙ 1/n σe = . (7) βbρi Combining Eqs. (1), (2) and (7), the flow stress will be given by: M ε˙ 1/n √ σ= + αMGb ρi + σδ + σp . (8) βbρi Yavari and Langdon [3] analysed the stress required to change the controlling mechanism of deformation for Al-Mg alloys, as the necessary stress for the dislocations to breakaway from the solute cloud: σb =
Wm2 c 3
5b kT
,
(9)
where Wm is the binding energy between solute atoms and the dislocation and c is the solute concentration. During transient deformation, the relationship of dislocation density and stress may differ from that at steady state. Unfortunately, this relationship is unclear up to now. After the results of evolution of dislocation density with increasing strain, the relationship between internal dislocation density and stress during transient deformation should be similar to that at steady state [13], and the relationship at steady state will be used for transient deformation in the present paper.
3. Experimental procedure The material used in this research was AA5052 of composition shown in Table 1. The as-received structure was fully recrystallised in all cases, with grain sizes of 88 ± 14 m (Fig. 1). TEM analysis showed a particle size of r = 0.12 ± 0.02 m and a volume fraction of fv = 0.032. For Al–Mg alloys, there is a high work hardening and a low recovery rate at small strains [14] and, thus, the stress change with strain is also high. An accurate procedure is required to obtain mechanical equations of state at small strains. To esTable 1 Composition of Al–Mg alloy (wt%, balance Al)
AA5052
Mg
Cr
Si
Fe
Mn
Cu
Ti
Zn
2.70
0.18
0.14
0.34
0.054
0.008
0.008
0.010
Fig. 1. Polarised optical micrograph of the as-received material (electropolished and etched in Barker’s solution).
tablish this accurately, different mechanical testing machines were used in this research: 1. screw-driven tension-compression testing machine, 2. servo-hydraulic tension testing machine, 3. Sheffield plane strain compression testing machine (for academic research), and 4. CORUS plane strain compression testing machine (for industrial practical application). The use of different test machines had a second objective, namely to investigate a wide range of conditions. Machine 1 is a uniaxial testing machine. It allows both axisymmetric tension and compression tests. It is precise because it is screw-driven and has two extensometers to measure the displacement only in the gauge length of the specimen. Machine 2 is a tensile machine that allows testing over a wider range of strain rates and temperatures. Both machine 1 and 2 were manufactured by Mayes. In the case of machine 3 (University of Sheffield, Department of Engineering Materials) and machine 4 (CORUS, RD&T, IJmuiden), they perform plane strain compression tests (PSC) on lubricated flat specimens. The strain rates of machines 3 and 4 were the highest of the four machines and the deformation conditions were the closest to those of industrial hot rolling. However, they have the drawback of a less uniform strain compared with the other two testing machines [15]. The results from the different test machines were corrected taking into account machine and specimen geometry effects (the former by measuring the machine compliance and the latter by finite element modelling). The curves are plotted for plastic strain, removing the elastic component of strain. Besides this, temperature rises and changes in the strain rate have also been removed from the experimental results, so that the curves obtained are isothermal and constant strain rate. Details of these corrections have been given elsewhere [16].
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The test conditions were in the ranges: temperatures between 300 and 550 ◦ C, strain rates between 0.01 and 10 s−1 and strains up to 0.15.
4. Results Fig. 2 shows the stress–strain curves for AA5052 tested at 400 ◦ C and different strain rates in different machines. The results were corrected for strain rate and temperature changes that occur during testing so that they are for truly isothermal and constant strain rate conditions [16]. The agreement between the results from different machines for the same conditions of strain rate and temperature is good. The same behaviour is observed in machines 1 and 2 at the lowest strain rate (0.01 s−1 ) with a peak followed by a decay in stress. At higher strain rates, however, the flow stress increases steadily with strain to a steady-state value. The stress–strain curves for a wide range of temperature and strain rate conditions are shown in Fig. 3. These curves have also been corrected for machine and geometry effects, as well as, for the strain rate change and temperature rise that may take place in some tests. This figure indicates that
Fig. 2. Stress–strain curves for AA5052 at 400 ◦ C and different strain rates from machine 1—screw-driven tension testing machine, machine 2—servohydraulic tension machine, machine 3—Sheffield plane strain compression machine and machine 4—CORUS plane strain compression machine.
Fig. 3. Stress–strain curves.
Fig. 4. Dependence of slope of the stress–strain curve with the Zener–Hollomon parameter for different values of plastic strain (AA5052).
AA5052 shows a continuous increase of stress with strain at lower temperatures (300 and 400 ◦ C) and higher strain rates (1 and 10 s−1 ). With increase in temperature and decrease in strain rate, the curves are almost flat for T ≥ 400 ◦ C and ε˙ ≤ 0.1 s−1 . At higher temperatures and lower strain rates the curves show a peak in stress, followed by a decrease. Fig. 4 shows the dependence of strain hardening rate on the Zener–Hollomon parameter at different values of strain. The initial slope of the stress–strain curve, i.e. at ε = 0.000, is very high, but is still smaller than M·G (i.e. Taylor factor · shear modulus) at the highest value of Z. At higher values of strain, e.g. ε = 0.002, the slope becomes negative for the lower Z, because the peak in stress was reached at a lower value of strain, but steady state has not yet been reached. At ε = 0.004 the curves which showed negative slope have now reached steady state (dσ/dε = 0) and at higher values of Z the stress is still increasing, which means that steady state is not yet reached. Fig. 5 shows the relationship between log˙ε and the steady state stress σ, for different values of temperature. At higher
Fig. 5. Dependence of steady-state stress on strain rate at different temperatures.
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tion behaviour can be investigated for extensive strain rate and temperature ranges, without machine effects, as reported elsewhere [16]. The stress–strain behaviour of AA5052 shows two different regimes: at lower temperatures and higher strain rates, i.e. high values of Z, there is strain hardening to steady state, whereas at lower values of Z the strain hardening is small, and there is a decrease in flow stress after an initial stage of work-hardening. This agrees with the findings of Chen et al. [19], who studied AA5182 under similar conditions of temperature and strain rate to large strains and found the same types of behaviour, with two regimes depending on Z. These differences in behaviour are explained in terms of the controlling mechanisms of dislocation movement during plastic deformation:
Fig. 6. Normalised strain rate as a function of normalised stress for pure aluminium and for Al–5% Mg.
values of stress, i.e. lower temperature or higher strain rate, there is a linear relationship, which means an exponential relationship exists between strain rate and stress. This is consistent with the findings of Jonas et al. [5]. However, at lower values of stress, the slope is higher and decreases with increasing stress. This indicates that the exponential relationship is no longer valid. Fig. 6 shows a log–log plot of stress normalised by the shear modulus and temperature compensated strain rate, for both pure Al and Al–Mg alloys. Fig. 6 includes data from the literature [17]. The normalisation has been carried out using the following values of the parameters, for consistency with previous data: G = 29484−13.6T MPa, Do = 1.7 × 10−4 m2 s−1 , QD = 142000 J/mol, and b = 0.286 nm [18]. In this plot, all the results corresponding to the same alloy show a continuous trend. These results show a varying slope with normalised strain rate: it starts with a slope of three for Al–Mg alloys and five for pure Al, in the power law regime, and changes towards an exponential relationship for higher stresses in both cases. The present experimental results of AA5052 show a similar behaviour to Al–5% Mg alloys, but with slightly lower stresses than those of Al–5 Mg alloys at higher values of temperature compensated strain rate. This arises from the lower magnesium content in the AA5052 (2.7%; Table 1).
5. Discussion The agreement between the results from the diverse machines shown in Fig. 2 is very good allowing the use of any machine to investigate deformation behaviour at small strains, as long as appropriate corrections are applied to the raw data. Each of the mechanical testing machines, however, has its own advantages, in terms of strain, strain rate and temperature ranges. The results show that hot deforma-
- At higher values of Z, deformation is controlled by climb or cross slip of dislocations, as for pure Al. In this case n , in Eqs. (5) and (6), is higher than 3, and the increase of dislocation density with strain will lead to a monotonic increase of stress with strain based on Eq. (8). The result is conventional strain hardening behaviour, as shown in the stress–strain curves (Fig. 3). This is also shown in the normalised stress–strain rate plots (Fig. 6), where n is higher than 5 for higher normalised strain rates. Therefore, the exponent n (n = n−2) is higher than 3. - At lower values of Z, the dislocation density is relatively low at the beginning of deformation and the movement of dislocations is faster than that of solute atoms. Thus, deformation is still controlled by climb of dislocations at small strains, and strain hardening can be observed. With an increase in strain, however, when the dislocation density rises as a consequence of deformation, the velocity of dislocations slows down (Eq. (3)). Under these conditions, the migration rate of solute atoms has the same order of magnitude as that of climb or cross slip of dislocations and solutes can gather round dislocations to form a so-called “solute cloud”. The solute clouds will move with the dislocations and now viscous glide of dislocations controls the deformation behaviour. For glide control, the value of n in Eq. (8) is unity. Thus, the first term dominates the flow curves, i.e. with increasing strain, dislocation density increases and thus the flow stress decreases after the initial deformation, up to a steady state value of dislocation density. This gives a peak in the flow stress after initial deformation, as shown in Fig. 3 at lower Z. This sort of behaviour is also shown in Fig. 6, where, at lower normalised strain rates, n is approximately 3, giving n = 1 as a result. - The change in deformation mechanism can also be described by the variation of the strain hardening rate with the Zener–Hollomon parameter (Z) as shown in Fig. 4. For all deformation conditions, the initial strain hardening rate is high and increases with increasing Z. With increasing strain, e.g. a strain of 0.002, however, the strain hardening rate is negative at the lower values of Z, but increases to become positive at higher values of Z. This results in a change
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in deformation mechanism from glide control, where the stress decreases with increasing strain, to climb control, where the stress increases with increasing strain. At higher strain, e.g. a strain of 0.004, strain hardening is balanced by recovery, giving a steady state stress with strain at lower Z, but still a high rate of strain hardening at higher Z. The results shown in Figs. 2–4 and the above discussion, also indicate that the constitutive equations of hot deformation for finite element modelling are different for higher Z and for lower Z deformation. Constitutive equations developed based on higher Z tests, for large strain plasticity, are no longer valid for lower Z conditions and new ones need to be developed. A change in the controlling mechanism of dislocation movement can be confirmed by the steady-state stress–strain relationship, as shown in Figs. 4 and 5. As normally presented for experimental results for hot working, there is a linear relationship between log˙ε and σ at higher Z, i.e. an exponential relationship between strain rate and stress. At lower Z, however, the log ε˙ –σ curves depart from the linear relationship (Fig. 4). The log˙ε–log plot, which is normalised in terms of G, D and T to compensate for the temperature effect, as normally presented for creep results [3,17,20,21] shows a linear relationship at lower stresses (lower Z), i.e. a power law relationship between strain rate and stress, and an exponential function at higher Z. The slope, i.e. the stress exponent (n) is about 3 for the intermediate stress regime for Al–Mg alloys. At higher stresses, n increases with increasing stress. This agrees with the findings by Yavari and Langdon [3], Zhu et al. [4] and Horita et al. [20,21]. Yavari and Langdon [3] showed that this transition from a power law with the exponent of 3 to an exponential function with increasing stress arises from the change in dislocation control mechanisms, i.e. from glide control to climb control of dislocation motion. The transition stress of the deformation controlling mechanism from glide control to climb control can be calculated using the solute drag theory. The transition stresses that are the sum of the transition effective stress calculated by the equation developed by Horita and Langdon [20], σ e , and the athermal stress arising from the dislocation interaction (σρi ) is approximately 60 MPa for AA5052. This value is consistent with the critical stress for change in shape of the flow stress–strain curves, i.e. with and without a peak in stress with strain.
6. Conclusions The deformation behaviour of AA5052 at small strain under hot working conditions has been characterised as follows: • Two regimes of deformation exist in the stress–strain behaviour at small strain under hot working conditions, i.e. work-hardening and almost flat stress–strain curves. At higher temperatures and lower strain rates, the stress–strain curves show a peak followed by a drop in stress. It is pro-
posed that this arises from a different deformation mechanism from normal deformation behaviour, i.e. viscous glide control of dislocation motion under these conditions and climb or cross-glide control of dislocation motion at lower temperatures and higher strain rates. • Initial work hardening has been observed in all cases, independent of the appearance of a peak at higher strains. The strain to reach steady state and the stress peak increase with Z. • Depending on Z, there will be different constitutive equations. It is not acceptable to extrapolate the equations available for large strain plasticity. • For steady state, the relationships between stress and strain rate are in reasonable agreement with the data obtained from previous studies in hot working and creep. The stress exponent changes from 3 at low stresses, corresponding to control by viscous glide of dislocations, to an exponential function, where climb or cross-slip of dislocations dominates the deformation behaviour.
Acknowledgements The authors would like to thank Dr. M. van der Winden for use of the CORUS Development and Technology centre (The Netherlands) testing facilities and for useful discussions. The assistance of Mr. J. Tan (CORUS) and Mr. A.J. Lacey (University of Sheffield) in the plane strain compression testing is appreciated. The National Physical Laboratory and CORUS are acknowledged for the provision of the material. EPSRC is acknowledged for the financial support (grant GR/L50198).
References [1] R. Carmona, Q. Zhu, C.M. Sellars, J.H. Beynon, in: N. Hansen, et al. (Eds.), Proceedings of 21st Risø International Symposium on Materials Science, Roskilde, Denmark, 2000, pp. 283–289. [2] H. Shi, A.J. McLaren, C.M. Sellars, R. Shahani, R. Bolingbroke, Mater. Sci. Technol. 13 (1997) 210–216. [3] P. Yavari, T.G. Langdon, Acta Metall. 30 (1982) 2181–2196. [4] Q. Zhu, W. Blum, H.J. McQueen, in: J.H. Drivers, et al. (Eds.), Aluminium Alloys: Their Physical and Mechanical Properties (ICAA 5), Transtec Publications, Switzerland, 1996, pp. 1169–1174. [5] J.J. Jonas, C.M. Sellars, W.J. McTegart, Met. Rev. 14 (1969) 1–24. [6] Q. Zhu, H.R. Shercliff, C.M. Sellars, in: T. Chandra, T. Sakai (Eds.), Thermec 97, The Minerals, Metals and Materials Society, 1997, pp. 2039–2046. [7] G.E. Dieter, Mechanical Metallurgy, McGraw-Hill, London, 1998. [8] S.V. Raj, G.M. Pharr, Mater. Sci. Eng. 81 (1986) 217–237. [9] M.F. Ashby, Proceedings of Second Bolton Landing Conference on Oxide Dispersion Strengthening, Bolton Landing, New York, 1966, pp. 143–205. [10] E. Nes, Prog. Mater. Sci. 41 (1998) 129–193. [11] D. Hull, D.J. Bacon, Introduction to Dislocations, Pergamon Press, Oxford, 1984. [12] C.M. Sellars, Q. Zhu, Mater. Sci. Eng., A 280 (2000) 1–7. [13] Q. Zhu, C.M. Sellars, in: J.H. Beynon, et al. (Eds.), Proceedings of Second International Conference on Modelling of Metal Rolling Processes, London, December, 1996, pp. 158–166.
R. Carmona et al. / Materials Science and Engineering A 393 (2005) 157–163 [14] C.M. Sellars, in: N. Hansen, et al. (Eds.), Proceedings of 17th Risø International Symposium on Materials Science, Roskilde, Denmark, 1986, pp. 167–187. [15] J.H. Beynon, C.M. Sellars, J. Test. Eval. 13 (1) (1985) 28–38. [16] R. Carmona, A.J. Lacey, J.H. Beynon, C.M. Sellars, J. Test. Eval. 29 (6) (2001) 520–528. [17] Q. Zhu, PhD Thesis. Universit¨at Erlangen-Nurnberg, Germany, 1994. [18] H.J. Frost, M.F. Ashby, Deformation Maps, Pergamon Press, Oxford, 1982. [19] S.R. Chen, M.G. Stout, U.F. Kocks, S.R. MacEwen, A.J. Beaudoin, in: T.R. Bieler, et al. (Eds.), Proceedings of International Conference
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on Hot Deformation of Aluminium Alloys, The Minerals, Metals and Materials Society, 1998, pp. 205–216. [20] Z. Horita, T.G. Langdon, in: H.J. McQueen, et al. (Eds.), Proceedings of International Conference on Strength of Metals and Alloys (ICSMA 6), Pergamon Press, Oxford, UK, 1985. [21] Z. Horita, T. Shimada, T. Endo, T.G. Langdon, in: B. Wilshire, R.W. Evans (Eds.), Proceedings of International Conference on Creep and Fracture of Engineering Materials and Structures, Swansea, UK, 1987.
Controlling mechanisms of deformation of AA5052 aluminium alloy at small strains under hot working conditions R. Carmona∗ , Q. Zhu1 , C.M. Sellars, J.H. Beynon IMMPETUS—Institute for Microstructural and Mechanical Process Engineering: The University of Sheffield, Mappin Street, Sheffield S13JD, UK Received 5 April 2004; received in revised form 4 October 2004; accepted 2 November 2004
Abstract The objective of the present work was to investigate the thermomechanical behaviour of a commercial Al–Mg alloy and to understand the deformation mechanisms taking place at small strains (<0.2) and high temperatures (≥300 ◦ C) at strain rates relevant to hot working conditions (0.01–10 s−1 ). The experimental approach addresses different effects of testing machines on the load–displacement measurements. The results show distinct sorts of behaviour depending on the Zener–Hollomon parameter (Z): normal continuous increase in the stress with strain for high values of Z and a rise followed by a drop in stress with increasing strain for low values of Z. For steady-state behaviour, the relationship between stress and strain rate follows the typical power law with an exponent of 3 at low values of Z and an exponential relationship at higher values. These observations are similar to those of creep. The difference in stress response may be explained by different controlling mechanisms, namely climb of dislocations for the normal hardening behaviour and solute drag in the case where a drop of stress is observed in the stress–strain curve. © 2004 Elsevier B.V. All rights reserved. Keywords: Hot working; Small strain deformation; Aluminium alloys; Flow stress; Deformation mechanism; Zener–Hollomon parameter
1. Introduction To date, interest in thermomechanical processing has been mainly focused on large strain plastic deformation. Recently, however, the small strain behaviour has received attention, because some regions in the hot formed workpiece may undergo very little deformation during certain stages of processing [1]. Such is the case for large section rolling or forging, where localised regions of the sections may receive very little or no deformation in certain passes. As a result, the mechanical properties in these localised regions of products may be poor, because the deformation may not ∗ Corresponding author. Present address: EADS-CASA, Ensayos Estructurales, Paseo John Lennon, s/n, E-28906 Getafe, Madrid, Spain Tel.: +34 659 68 5627. E-mail address: [email protected] (R. Carmona). 1 Present address: Holset Engineering, St. Andrew’s Road, Huddersfield HD16RA, UK
0921-5093/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2004.11.010
be enough to trigger recrystallisation. Mechanical equations of state are available for aluminium alloys for large strain plasticity [2] but these equations cannot be extrapolated with confidence to small plastic strains. The processes undergone by the material when the strains are small have not yet been addressed and there is a knowledge gap in this area of study of high temperature deformation at hot working strain rates. Creep studies of Al–Mg alloys show that, under some deformation conditions, this type of alloy presents special features, such as an exponent of 3 in the stress–strain rate relationship instead of 5 usually found for pure Al or other Al alloys [3,4]. Yavari and Langdon [3] and Zhu et al. [4] carried out their tests not only under traditional creep conditions (constant stress and low strain rate), but also under constant cross-head displacement rate (close to constant strain rate). Al–Mg alloys showed a transition between the exponents 3 and 5, which was interpreted as a breakaway of the dislocations from their solute atmospheres, with a changeover in
158
R. Carmona et al. / Materials Science and Engineering A 393 (2005) 157–163
Nomenclature b E D Do fv G kB M n n
Burgers vector Young’s modulus diff diffusion coefficient: D = Do exp −Q RT frequency factor volume fraction of second phase particles shear modulus Boltzmann’s constant: 1.381 × 10−23 J/K Taylor factor (∼ =3 for fcc polycrystals) exponent relating strain rate and stress exponent relating velocity of dislocations and effective stress Qdef , Qdiff activation energies for deformation and diffusion, respectively R universal gas constant: 8.314 J/(mol K) r second phase particle radius T absolute temperature Wm binding energy between solute atoms and dislocations def Z Zener–Hollomon parameter Z = ε˙ exp QRT δ subgrain size ε strain ε˙ strain rate ρi internal dislocation density θ subgrain misorientation ν velocity of moving dislocations σ equivalent tensile stress α, β, C material constants
the acting mechanisms for the control of dislocation motion, from viscous drag controlled glide to high temperature climb or cross-slip. However, that investigation was focused only on the steady-state behaviour. In fact, much earlier, Jonas et al. [5] reviewed the whole spectrum of conditions in creep and hot working, and established the interdependence of stress and strain rate by a hyperbolic sine function, which becomes a power law at lower stresses and an exponential law at higher stresses. It is clear that there is a need to understand fully the whole deformation behaviour, including transient stress–strain curves, and the microstructural processes taking place at small strains in thermomechanical processing. Current knowledge of creep or large plastic deformation under hot working conditions does not give sufficient insight into the deformation behaviour at low strain. In the present study, the deformation behaviour at strains up to about 0.15 for Al–Mg alloys under hot working conditions is investigated in detail. Aluminium alloy AA5052 was used because it represents a group of aluminium alloys widely used in the transportation, marine and automotive industries. The controlling mechanisms of hot deformation behaviour are discussed using inter-
nal state variables, i.e. the internal dislocation density (ρI ) the subgrain size (δ) and the misorientation between subgrains (θ).
2. Description of the stress as a function of the internal state variables The flow stress of the material (σ) can be described as [6]: σ = σe + σρi + σδ + σd + σp
(1)
where σ e is the effective stress (also known as friction stresses, σ f ) σρi the athermal stress due to interactions of dislocations within the subgrains, σ δ and σ d are the long range athermal stresses from the subgrain boundaries and from the grain boundaries, respectively, and σ p is the particle hardening term. The athermal stress arising from interaction of internal dislocations can be expressed as [7]: √ σρi = αMGb ρi , (2) where α is a material parameter, M the Taylor factor, G the shear modulus, b the Burgers vector and ρi is the internal dislocation density. The stress arising from the grain boundaries (σ d ) is negligible when the grain size is large in comparison with the size. The value of the stress arising from the subgrain boundaries (σ δ ) depends mainly on the subgrain size (δ) but it is also a function of the Taylor factor, the shear modulus and the Burgers vector [8]. The precipitation hardening stress (σ p ) is given by a relationship between different constants, such as the shear modulus and the Burgers vector, the volume fraction of precipitates and the particle radius [9]. The strain rate (˙ε) is related to the dislocation density and to the velocity of dislocations (ν) as [7]: ε˙ =
b ρi ν, M
where ν is given by [10]: Qdef C 2 σe ν = C1 exp − , sinh kB T T
(3)
(4)
where Qdef is the deformation activation energy, and C1 and C2 are material constants. Eq. (4) can be re-written as a power law [11]:
ν = βσen ,
(5)
where n is an exponent reflecting the sensitivity of the relationship between velocity of dislocation motion and applied stress, which is dependent on the magnitude of the stress. At low stresses, n = 1 for Al–Mg and other strongly solute hardened alloys and n = 3 for pure Al and other weakly solute hardened alloys. At high stresses, n increases with increasing stress.
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Dislocation density in Eq. (3) is determined by the combination of strain hardening and recovery [12]. At steady-state, ρi ∝ σ 2 [18], and Eq. (3) can then be re-written as: ε˙ =
b Cβσ 2 σen , M
(6)
where C is a material constant. From Eqs. (5) and (6), n = n−2, where n is the stress exponent relating strain rate and stress. From Eqs. (3) and (5), the friction or effective stress can be expressed as: M ε˙ 1/n σe = . (7) βbρi Combining Eqs. (1), (2) and (7), the flow stress will be given by: M ε˙ 1/n √ σ= + αMGb ρi + σδ + σp . (8) βbρi Yavari and Langdon [3] analysed the stress required to change the controlling mechanism of deformation for Al-Mg alloys, as the necessary stress for the dislocations to breakaway from the solute cloud: σb =
Wm2 c 3
5b kT
,
(9)
where Wm is the binding energy between solute atoms and the dislocation and c is the solute concentration. During transient deformation, the relationship of dislocation density and stress may differ from that at steady state. Unfortunately, this relationship is unclear up to now. After the results of evolution of dislocation density with increasing strain, the relationship between internal dislocation density and stress during transient deformation should be similar to that at steady state [13], and the relationship at steady state will be used for transient deformation in the present paper.
3. Experimental procedure The material used in this research was AA5052 of composition shown in Table 1. The as-received structure was fully recrystallised in all cases, with grain sizes of 88 ± 14 m (Fig. 1). TEM analysis showed a particle size of r = 0.12 ± 0.02 m and a volume fraction of fv = 0.032. For Al–Mg alloys, there is a high work hardening and a low recovery rate at small strains [14] and, thus, the stress change with strain is also high. An accurate procedure is required to obtain mechanical equations of state at small strains. To esTable 1 Composition of Al–Mg alloy (wt%, balance Al)
AA5052
Mg
Cr
Si
Fe
Mn
Cu
Ti
Zn
2.70
0.18
0.14
0.34
0.054
0.008
0.008
0.010
Fig. 1. Polarised optical micrograph of the as-received material (electropolished and etched in Barker’s solution).
tablish this accurately, different mechanical testing machines were used in this research: 1. screw-driven tension-compression testing machine, 2. servo-hydraulic tension testing machine, 3. Sheffield plane strain compression testing machine (for academic research), and 4. CORUS plane strain compression testing machine (for industrial practical application). The use of different test machines had a second objective, namely to investigate a wide range of conditions. Machine 1 is a uniaxial testing machine. It allows both axisymmetric tension and compression tests. It is precise because it is screw-driven and has two extensometers to measure the displacement only in the gauge length of the specimen. Machine 2 is a tensile machine that allows testing over a wider range of strain rates and temperatures. Both machine 1 and 2 were manufactured by Mayes. In the case of machine 3 (University of Sheffield, Department of Engineering Materials) and machine 4 (CORUS, RD&T, IJmuiden), they perform plane strain compression tests (PSC) on lubricated flat specimens. The strain rates of machines 3 and 4 were the highest of the four machines and the deformation conditions were the closest to those of industrial hot rolling. However, they have the drawback of a less uniform strain compared with the other two testing machines [15]. The results from the different test machines were corrected taking into account machine and specimen geometry effects (the former by measuring the machine compliance and the latter by finite element modelling). The curves are plotted for plastic strain, removing the elastic component of strain. Besides this, temperature rises and changes in the strain rate have also been removed from the experimental results, so that the curves obtained are isothermal and constant strain rate. Details of these corrections have been given elsewhere [16].
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The test conditions were in the ranges: temperatures between 300 and 550 ◦ C, strain rates between 0.01 and 10 s−1 and strains up to 0.15.
4. Results Fig. 2 shows the stress–strain curves for AA5052 tested at 400 ◦ C and different strain rates in different machines. The results were corrected for strain rate and temperature changes that occur during testing so that they are for truly isothermal and constant strain rate conditions [16]. The agreement between the results from different machines for the same conditions of strain rate and temperature is good. The same behaviour is observed in machines 1 and 2 at the lowest strain rate (0.01 s−1 ) with a peak followed by a decay in stress. At higher strain rates, however, the flow stress increases steadily with strain to a steady-state value. The stress–strain curves for a wide range of temperature and strain rate conditions are shown in Fig. 3. These curves have also been corrected for machine and geometry effects, as well as, for the strain rate change and temperature rise that may take place in some tests. This figure indicates that
Fig. 2. Stress–strain curves for AA5052 at 400 ◦ C and different strain rates from machine 1—screw-driven tension testing machine, machine 2—servohydraulic tension machine, machine 3—Sheffield plane strain compression machine and machine 4—CORUS plane strain compression machine.
Fig. 3. Stress–strain curves.
Fig. 4. Dependence of slope of the stress–strain curve with the Zener–Hollomon parameter for different values of plastic strain (AA5052).
AA5052 shows a continuous increase of stress with strain at lower temperatures (300 and 400 ◦ C) and higher strain rates (1 and 10 s−1 ). With increase in temperature and decrease in strain rate, the curves are almost flat for T ≥ 400 ◦ C and ε˙ ≤ 0.1 s−1 . At higher temperatures and lower strain rates the curves show a peak in stress, followed by a decrease. Fig. 4 shows the dependence of strain hardening rate on the Zener–Hollomon parameter at different values of strain. The initial slope of the stress–strain curve, i.e. at ε = 0.000, is very high, but is still smaller than M·G (i.e. Taylor factor · shear modulus) at the highest value of Z. At higher values of strain, e.g. ε = 0.002, the slope becomes negative for the lower Z, because the peak in stress was reached at a lower value of strain, but steady state has not yet been reached. At ε = 0.004 the curves which showed negative slope have now reached steady state (dσ/dε = 0) and at higher values of Z the stress is still increasing, which means that steady state is not yet reached. Fig. 5 shows the relationship between log˙ε and the steady state stress σ, for different values of temperature. At higher
Fig. 5. Dependence of steady-state stress on strain rate at different temperatures.
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tion behaviour can be investigated for extensive strain rate and temperature ranges, without machine effects, as reported elsewhere [16]. The stress–strain behaviour of AA5052 shows two different regimes: at lower temperatures and higher strain rates, i.e. high values of Z, there is strain hardening to steady state, whereas at lower values of Z the strain hardening is small, and there is a decrease in flow stress after an initial stage of work-hardening. This agrees with the findings of Chen et al. [19], who studied AA5182 under similar conditions of temperature and strain rate to large strains and found the same types of behaviour, with two regimes depending on Z. These differences in behaviour are explained in terms of the controlling mechanisms of dislocation movement during plastic deformation:
Fig. 6. Normalised strain rate as a function of normalised stress for pure aluminium and for Al–5% Mg.
values of stress, i.e. lower temperature or higher strain rate, there is a linear relationship, which means an exponential relationship exists between strain rate and stress. This is consistent with the findings of Jonas et al. [5]. However, at lower values of stress, the slope is higher and decreases with increasing stress. This indicates that the exponential relationship is no longer valid. Fig. 6 shows a log–log plot of stress normalised by the shear modulus and temperature compensated strain rate, for both pure Al and Al–Mg alloys. Fig. 6 includes data from the literature [17]. The normalisation has been carried out using the following values of the parameters, for consistency with previous data: G = 29484−13.6T MPa, Do = 1.7 × 10−4 m2 s−1 , QD = 142000 J/mol, and b = 0.286 nm [18]. In this plot, all the results corresponding to the same alloy show a continuous trend. These results show a varying slope with normalised strain rate: it starts with a slope of three for Al–Mg alloys and five for pure Al, in the power law regime, and changes towards an exponential relationship for higher stresses in both cases. The present experimental results of AA5052 show a similar behaviour to Al–5% Mg alloys, but with slightly lower stresses than those of Al–5 Mg alloys at higher values of temperature compensated strain rate. This arises from the lower magnesium content in the AA5052 (2.7%; Table 1).
5. Discussion The agreement between the results from the diverse machines shown in Fig. 2 is very good allowing the use of any machine to investigate deformation behaviour at small strains, as long as appropriate corrections are applied to the raw data. Each of the mechanical testing machines, however, has its own advantages, in terms of strain, strain rate and temperature ranges. The results show that hot deforma-
- At higher values of Z, deformation is controlled by climb or cross slip of dislocations, as for pure Al. In this case n , in Eqs. (5) and (6), is higher than 3, and the increase of dislocation density with strain will lead to a monotonic increase of stress with strain based on Eq. (8). The result is conventional strain hardening behaviour, as shown in the stress–strain curves (Fig. 3). This is also shown in the normalised stress–strain rate plots (Fig. 6), where n is higher than 5 for higher normalised strain rates. Therefore, the exponent n (n = n−2) is higher than 3. - At lower values of Z, the dislocation density is relatively low at the beginning of deformation and the movement of dislocations is faster than that of solute atoms. Thus, deformation is still controlled by climb of dislocations at small strains, and strain hardening can be observed. With an increase in strain, however, when the dislocation density rises as a consequence of deformation, the velocity of dislocations slows down (Eq. (3)). Under these conditions, the migration rate of solute atoms has the same order of magnitude as that of climb or cross slip of dislocations and solutes can gather round dislocations to form a so-called “solute cloud”. The solute clouds will move with the dislocations and now viscous glide of dislocations controls the deformation behaviour. For glide control, the value of n in Eq. (8) is unity. Thus, the first term dominates the flow curves, i.e. with increasing strain, dislocation density increases and thus the flow stress decreases after the initial deformation, up to a steady state value of dislocation density. This gives a peak in the flow stress after initial deformation, as shown in Fig. 3 at lower Z. This sort of behaviour is also shown in Fig. 6, where, at lower normalised strain rates, n is approximately 3, giving n = 1 as a result. - The change in deformation mechanism can also be described by the variation of the strain hardening rate with the Zener–Hollomon parameter (Z) as shown in Fig. 4. For all deformation conditions, the initial strain hardening rate is high and increases with increasing Z. With increasing strain, e.g. a strain of 0.002, however, the strain hardening rate is negative at the lower values of Z, but increases to become positive at higher values of Z. This results in a change
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in deformation mechanism from glide control, where the stress decreases with increasing strain, to climb control, where the stress increases with increasing strain. At higher strain, e.g. a strain of 0.004, strain hardening is balanced by recovery, giving a steady state stress with strain at lower Z, but still a high rate of strain hardening at higher Z. The results shown in Figs. 2–4 and the above discussion, also indicate that the constitutive equations of hot deformation for finite element modelling are different for higher Z and for lower Z deformation. Constitutive equations developed based on higher Z tests, for large strain plasticity, are no longer valid for lower Z conditions and new ones need to be developed. A change in the controlling mechanism of dislocation movement can be confirmed by the steady-state stress–strain relationship, as shown in Figs. 4 and 5. As normally presented for experimental results for hot working, there is a linear relationship between log˙ε and σ at higher Z, i.e. an exponential relationship between strain rate and stress. At lower Z, however, the log ε˙ –σ curves depart from the linear relationship (Fig. 4). The log˙ε–log plot, which is normalised in terms of G, D and T to compensate for the temperature effect, as normally presented for creep results [3,17,20,21] shows a linear relationship at lower stresses (lower Z), i.e. a power law relationship between strain rate and stress, and an exponential function at higher Z. The slope, i.e. the stress exponent (n) is about 3 for the intermediate stress regime for Al–Mg alloys. At higher stresses, n increases with increasing stress. This agrees with the findings by Yavari and Langdon [3], Zhu et al. [4] and Horita et al. [20,21]. Yavari and Langdon [3] showed that this transition from a power law with the exponent of 3 to an exponential function with increasing stress arises from the change in dislocation control mechanisms, i.e. from glide control to climb control of dislocation motion. The transition stress of the deformation controlling mechanism from glide control to climb control can be calculated using the solute drag theory. The transition stresses that are the sum of the transition effective stress calculated by the equation developed by Horita and Langdon [20], σ e , and the athermal stress arising from the dislocation interaction (σρi ) is approximately 60 MPa for AA5052. This value is consistent with the critical stress for change in shape of the flow stress–strain curves, i.e. with and without a peak in stress with strain.
6. Conclusions The deformation behaviour of AA5052 at small strain under hot working conditions has been characterised as follows: • Two regimes of deformation exist in the stress–strain behaviour at small strain under hot working conditions, i.e. work-hardening and almost flat stress–strain curves. At higher temperatures and lower strain rates, the stress–strain curves show a peak followed by a drop in stress. It is pro-
posed that this arises from a different deformation mechanism from normal deformation behaviour, i.e. viscous glide control of dislocation motion under these conditions and climb or cross-glide control of dislocation motion at lower temperatures and higher strain rates. • Initial work hardening has been observed in all cases, independent of the appearance of a peak at higher strains. The strain to reach steady state and the stress peak increase with Z. • Depending on Z, there will be different constitutive equations. It is not acceptable to extrapolate the equations available for large strain plasticity. • For steady state, the relationships between stress and strain rate are in reasonable agreement with the data obtained from previous studies in hot working and creep. The stress exponent changes from 3 at low stresses, corresponding to control by viscous glide of dislocations, to an exponential function, where climb or cross-slip of dislocations dominates the deformation behaviour.
Acknowledgements The authors would like to thank Dr. M. van der Winden for use of the CORUS Development and Technology centre (The Netherlands) testing facilities and for useful discussions. The assistance of Mr. J. Tan (CORUS) and Mr. A.J. Lacey (University of Sheffield) in the plane strain compression testing is appreciated. The National Physical Laboratory and CORUS are acknowledged for the provision of the material. EPSRC is acknowledged for the financial support (grant GR/L50198).
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