High temperature deformation and microstructural instability in AZ31 magnesium alloy

Materials Science & Engineering A 570 (2013) 135–148

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Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea

High temperature deformation and microstructural instability in AZ31 magnesium alloy S. Spigarelli a,n, O.A. Ruano b, M. El Mehtedi a, J.A. del Valle b a b

DIISM, Universita Politecnica delle Marche, Ancona, Italy Department of Physical Metallurgy, CENIM, CSIC Madrid, Spain

a r t i c l e i n f o

abstract

Article history: Received 23 November 2012 Received in revised form 24 January 2013 Accepted 25 January 2013 Available online 4 February 2013

The high temperature deformation behaviour of AZ31 magnesium alloy is analysed by comparing numerous investigations, including work by these authors and by other researchers. Three main deformation mechanisms are observed, i.e grain boundary sliding, solute drag creep and climbcontrolled dislocation creep. A combined set of constitutive equations, which takes into account the concurring effect of these different deformation mechanisms, is proposed. Grain boundary sliding is observed to cause a superplastic behaviour in fine-grained materials, but grain growth due to excessively prolonged high temperature exposure invariably results in a transition to either viscous glide or dislocation climb as a rate-controlling mechanism. On the basis of these considerations, the differences observed by testing the same material under constant strain rate or by strain rate change experiments are rationalised by quantifying the effect of static and dynamic grain growth and dynamic recrystallisation. This procedure provides a unitary description of the high temperature deformation of AZ31 in a wide range of strain rates and temperatures. & 2013 Elsevier B.V. All rights reserved.

Keywords: Creep Magnesium alloys Creep mechanisms Microstructure

1. Introduction Magnesium alloys fabricated by mechanical forming, are of considerable interest because they provide higher ductility and specific strength than castings. However, it is necessary to develop processes with competitive productivity and performance to ensure a profitable use of these alloys. High temperature forming (300–450 1C) is especially important both as a primary process (rolling, extrusion) to produce long products for secondary manufacturing and as a direct secondary process such as forging. High temperature plastic deformation of materials and creep both occur at temperatures above approximately 0.5 T/Tm where Tm is the absolute melting point. Constitutive equations, which have been developed for predicting the behaviour of materials, are essential for design and control. Generally, the creep rate or strain rate, e_ , has been observed to be related to the absolute temperature, T, and the stress, s, by the following equation:

e_ ¼ k1 expðQ =RTÞðs=EÞn

ð1Þ

where k1 is a function of structure, E is Young’s modulus, n is the stress exponent, Q is the activation energy for plastic flow and R is the universal gas constant. Although creep is usually correctly n Correspondence to: DIISM, Universita Politecnica delle Marche, via Brecce Bianche, 60131 Ancona, Italy. Tel.: þ 39 071 2204746. E-mail address: [email protected] (S. Spigarelli).

0921-5093/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.msea.2013.01.060

described by a diffusion-controlled dislocation creep equation, other mechanisms may be important. Grain boundary sliding (GBS) and diffusional flow may in fact dominate deformation especially in fine-grained materials. Each of these mechanisms is considered to operate independently of the others and has a particular dependence on creep rate with stress, temperature and grain size. Over a certain temperature and strain rate range, when two mechanisms operate in parallel, their contribution to the total strain will be additive, and if one is much faster than the other, it will dominate the deformation process. By contrast, when the mechanisms operate in sequence, the slowest will become rate-controlling. However, great attention should also be paid to the influence of the microstructure on the creep mechanism because the microstructure of the AZ31 alloy is prone to evolve dynamically at high temperatures. As is well-known, grain size is of central importance for the grain boundary sliding mechanism. Del Valle et al. [1–3], for example, analysed the behaviour of the AZ31 alloy with a grain size of 17 mm, observing that the low thermal stability of this material leads to noticeable grain growth during tensile tests, and a subsequent increase in flow stress, which affects the GBS mechanism. In such circumstances, the analysis of the deformation mechanism using measurements of the maximum stress could lead, erroneously, to the conclusion that a creep mechanism occurs involving an n ¼3 exponent, interpreted as evidence of viscous glide-controlled deformation. Therefore, the experimental technique used is of great importance when trying to study the deformation mechanisms

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The dislocation creep models include an equation with n ¼3 corresponding to a solute drag dislocation glide mechanism which may take place in materials that contain alloying elements in solid solution [5–7], i.e.

in this alloy. While strain rate change (SRC) experiments clearly highlight the existence of a well-defined regime of GBS-controlled deformation where n ¼2 [1–3], in the case of constant strain rate (CSR) or constant stress (CS, i.e. creep) tests the stress exponent is sometimes higher than 3 [4]. In addition, the microstructure could evolve by dynamic recrystallisation (DRX) which is the mechanism that in most systems has the major role in controlling the evolution of the structure during high temperature straining. Therefore, DRX operates as a dynamic grain refiner which opposes the effect of dynamic grain growth. A necessary condition for DRX is the formation of a dislocation substructure; therefore DRX is mainly operative at high strain rates where the deformation mechanism involved is dislocation creep. The aim of the present paper is, on one hand, to quantify the effect of grain growth and DRX on the high temperature mechanical properties. On the other hand, an extensive recompilation of literature data regarding creep in the AZ31 alloy has allowed a set of creep laws to be introduced, which, in addition to the microstructure evolution equations, are suitable for describing the entire range of the strain rate–stress–temperature map in AZ31 alloy, providing a unitary model.

e_ ¼ k6 ðDs =b2 Þðs=EÞ3

ð6Þ

where the solute atom diffusivity is again reported in Table 1, which also reports the k3, k4, k5 and k6 values calculated as illustrated in the following sections. Young’s modulus is given by E(MPa)¼43000[1  5.3  10–4(T 300)] [8]. While grain boundary sliding and dislocation creep operate independently (although a certain degree of dislocation activity in the grain mantle is essential to accommodate GBS), the movement of the dislocations is a succession of glide and climb, i.e. these two mechanisms operate in sequence, and this will be reflected in the appropriate choice of the constitutive equation. The accuracy of the different constitutive models will be analysed in the following by considering a wide collection of experimental results [9–40] in conjunction with the data reported in [3], completed by additional unpublished results obtained by the same authors. Table 2 reports all the literature sources of the data which will be analysed in detail in the next sections.

2. Analysis of mechanical data at high temperatures

2.1. Grain boundary sliding mechanism (n ¼2)

In this section we review creep power laws in order to characterise the different deformation mechanisms that occur in the AZ31 alloy, i.e. the constitutive laws (Eqs. (2)–(6)) for creep corresponding to the three modes of deformation found for this material. The GBS-controlled deformation is usually described by one of the following equations, i.e.

Fig. 1 shows an overview of a large collection of creep data for AZ31 provided by various researchers. In these works, published datasets usually cover regions, in the graph e_ vs s, involving different creep mechanisms. When building Fig. 1, data e_  s were taken from regions with n¼2 and n ¼3. The line shown in Fig. 1a corresponds to Eq. (3) for grain boundary sliding controlled by Dgb, with n ¼2, but with a preexponential constant that is 7 times lower than that of Eq. (3) typical of high stacking fault energy materials. In the case of the AZ31 magnesium alloy with a fine grain size, there is direct evidence of a superplastic response [3,9–27]; even when the stress exponent n is slightly higher than 2, the reported activation energy is close to the Q gb value considered to correspond to a mechanism of grain boundary sliding (see Table 1). Lower data scattering is however obtained if the grain size compensation is performed using an exponent of p ¼2 instead of p¼ 3 (Fig. 1b). Thus, these data follow neither Eq. (2) nor Eq. (3), but the best fit is obtained with a hybrid form (Eq. (4)). Recently, Figueiredo and Langdon [24] studied GBS in an AZ31 processed by ECAP using a set of bibliographic data and found a dependency similar to that given in Eq. (4). It must in any case be noted that, even with the most favourable description, the use of different methods for

e_ ¼ k2 ðDL =d2 Þðs=EÞ2

ð2Þ

and

e_ ¼ k3 ðDgb =d3 Þðs=EÞ2

ð3Þ

 10

with b¼3.21  10 m (the expressions for bulk and grain boundary diffusivities, DL and Dgb respectively, and the relevant values of the activation energies are listed in Table 1). An hybrid formulation of Eqs. (2) and (3) assumes the form

e_ ¼ k3 ðDgb =d2 Þðs=EÞ2

ð4Þ

Climb-controlled dislocation creep can be described by the relationship

e_ ¼ k5 ðDL =b2 Þðs=EÞ5

ð5Þ

Table 1 Constitutive equations for creep in AZ31; Eqs. (2)–(5) describe GBS controlled and climb controlled (slip creep) deformation respectively. Eq. (6) is the constitutive law for viscous glide controlled creep. R is the gas constant, d is the true grain size, b is the Burgers vector and cnAl is the amount of Al in solid solution which effectively acts as a solid solution strengthener, forming atmospheres of solute atoms around dislocations. Creep process

Equation

k-value This work, AZ31

Grain boundary sliding

2. e_ ¼ k2 ðDL =d Þðs=EÞ2

Grain boundary sliding

3. e_ ¼ k3 ðDgb b=d Þðs=EÞ2

2

3

Grain boundary sliding

2

4. e_ ¼ k4 ðDgb =d Þðs=EÞ2 2

Slip creep (climb)

5. e_ ¼ k5 ðDL =b Þðs=EÞ

Slip creep (solute drag)

5

2 6. e_ ¼ k6 ðDs =b Þðs=EÞ3 DL ¼ 1  10–4 exp(  QL/RT) [m2/s], with: QL ¼ 135 [kJ/mol]; b¼ 3.21  10–10 [m] Dgb ¼ D0,gb exp(  Qgb/RT) [m2/s], with: D0,gb ¼ 5  10–12/2b; QGB ¼ 92 [kJ/mol]

– 8  107 1.5  103 3.7  108 22.4

Ds ¼ 1.2  10–3 exp(  QS/RT) [m2/s], with: QS ¼143 [kJ/mol] and k6 ¼ k6 /cnAl , cnAl ¼0.022a n

a n cAl is the amount of Al in solid solution forming the solute atoms atmospheres around dislocations; with cnAl ¼0.022 it is supposed that all the atoms are still clustering around dislocations, i.e. that dislocations have not yet been ‘‘torn away’’ by these atmospheres.

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Table 2 Literature data on the AZ31 alloy. Data were collected in the low stress-exponent regime for drawing Figs. 1 and 4; and in the high stress-exponent regime for drawing Figs. 5and 6. For each data set the grain size, the testing temperature and the type of stress measurement method are reported, with the literature source. Grain size[mm] 1 1.6 2.8 2.8 3 4 4.3 4.5 4.5 5 6.6 8 8.5 8.5 9.6 10 10.4 11 11.6 12 14.1 16.8 17 17.8 18 19.1 25 25 25 25 27 28 30 40 40 56 60 85 98 130 150 180 220 300 300 450 n

Symbol

v

n



& 1 1 2 S S .

/ R c C . ’ A c % K m

Temperatures of data on Figs. 1 and 4

Temperatures of data on Figs. 6 and 7

Stress measurement

Reference

423 450 523, 573 573 523 573 623, 598, 573, 523, 573 525 523, – 573, 623, 573, – 673 – 623, 573, 623,

– – 523, 573 – – 573, 623, 498, 573, 523, 373, 525 473, 523 523, – 473, 643 473, 673 – 573, – 623, 673 – – 423, 473, 673 473, 673 723 573, 473, 473, – 423, – 573, – – 673, – 473,

e ¼0.1* e ¼0.1* e ¼0.5 e ¼0.1

9 10 11 12 13 14 15 16 17 18 19 20 21 22 34 23 24 25 34 26 35 27 3, this work 24 36 35 28 21 37 38 29 25 35 29 3 39 26 30 31 32 40 32 32 33 32 26

– 623, 525 – – 673 573, – 723 573, – 473, 573, 423, 598, – 648 648 673, 648 473,

573

673 623 598, 623 623, 673

573 623, 673 673 673

673, 723 608, 628, 648, 673, 698,723, 773 673

673, 723

673

648, 723 573, 623, 448, 623,

673 673, 723 473 648, 673

723, 773 573, 673

573

573, 623, 598, 598, 623, 473,

673 673 623, 648, 673, 698, 723 623 673 573

523,573 573, 623, 673 573, 673 573, 673

608, 628, 648, 673, 698, 723 723

473, 523, 573 523, 573, 623 573, 673

648, 723 523, 573, 723 573, 673 448, 473 598, 623, 648

723, 773 573, 673

Yield stress Peak stress* Peak stress e ¼0.2 Yield stress e ¼0.1 SRC Mim. creep rate SRC SRC Peak stress SRC e ¼0.1 SRC Peak stress Peak stress Peak stress* e ¼0.1 SRC e ¼0.1 SRC Peak stress* e ¼0.15 SRC e ¼0.1* SRC e ¼0.1 SRC Peak stress* e ¼0.1 SRC SRC Peak stress Mim. creep rate Mim. creep rate e ¼0.1 SRC e ¼0.1 e ¼0.1 SRC e ¼0.1 Peak stress

Determined from reported s–e curve.

estimating the strength (stress at 0.1 strain, at the peak of the flow curve or at yielding in continuous tests, testing stress in creep, or steady-state stress in SRC experiments) leads to a rather large scatter in the data. The experimental results lie between the curves with k4 ¼2.6  102 and 6.5  103, with the solid line in the Fig. representing an intermediate reference value of k4 ¼1.5  103. For the purpose of comparison, Fig. 1 also presents data for coarse grain sizes (d4 50 mm) where n is slightly higher than 2, going up to n ¼3. It can easily be observed that the data for coarse grain size having stress exponents of about 3 do not fit the predicted behaviour corresponding to the grain boundary equation. Furthermore, these data do not obey a relationship with grain size and the activation energy for grain boundary diffusion does not seem to be adequate to compensate the results. These data, which can be interpreted in terms of Eq. (6) in Table 1, corresponding to a solute-drag creep mechanism, will be considered in detail in Section 2.3.

2.2. Debate about the action of GBS (n¼2) or solute drag (n¼3) in fine-grained samples As mentioned in the introduction, the measurement technique becomes important when trying to study the deformation mechanisms in this alloy; while strain rate change (SRC) experiments clearly highlight the existence of a well-defined regime of GBS-controlled deformation where n¼2, in the case of constant strain rate (CSR) or constant stress (CS, i.e. creep) tests, the stress exponent is invariably higher, even when the same materials and experimental range are considered. In Table 2 the method used by the various authors for the determination of the flow stress is given. Most reports, which confirm an n¼2 exponent [3,16–19,21,22,24,25,27,29] in fine-grained AZ31, are based on SRC measurements, or on the determination of the stress at e ¼0.1 or even at yield (these differences partly account for the high scatter in Fig. 1, as already mentioned). On the contrary, reports of values n¼2.5–3 [11,12,15,28] are in general based on the determination of

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Fig. 1. Diffusion and grain size compensated strain rate as a function of modulus-compensated stress for AZ31. (a) A grain exponent of 3 has been used. (b) A grain exponent of 2 has been used.

Fig. 2. Stress–strain curves obtained at 648 K by CSR testing (del Valle et al. [3]).

maximum stresses in dynamic tests or the measurements of minimum strain rate in creep tests. Del Valle et al. [3] investigated the high temperature deformation of an AZ31 alloy using SCR and CSR tests. Fig. 2 re-plots the representative flow curves obtained by testing the alloy in tension (CSR experiments) at 648 K. A notable, and indeed quite unusual, feature of the curves presented in Fig. 2 is the shift of the strain for peak stress towards higher values as the strain rate is decreased. The SRC tests complemented the CRS experiments. Fig. 3 shows the variation in the stress with strain rate, as obtained by CSR (peak stress) and SRC experiments. While the two techniques give overlapping results in the high strain rate regime, in the low strain rate region a significant deviation is observed. The stress exponent for CSR data in this regime is close to 4, while in the case of the SCR experiments it has a lower value (n ¼1.7 and 2.5 for initial grain size d ¼17 and 40 mm respectively) [3]. The activation energy for high temperature deformation Q varies between 91 and 106kJ/mol. Fig. 3 clearly shows that GBS accommodated by grain boundary diffusion has an important role in the low strain rate regime and slip creep governs deformation at high strain rates. Since GBS becomes more and more inefficient as the grains grow during high temperature exposure, there is a progressive transition to slip creep during CSR tests. The importance of intragranular

dislocation activity was revealed by the formation of high-angle boundaries at relatively low strains (e 40.55). Fig. 3a also shows a set of creep data [4] obtained by testing an AZ31 with an initial grain size of 8 mm. It is interesting to observe that, although the alloy tested using the CS test had a finer grain size than the material investigated by del Valle et al., it exhibited lower creep rates. This behaviour cannot be explained easily, unless the effect of grain growth, which in the long creep experiments should be stronger, is invoked. In this regard the observed trend is fully compatible with the theory which interprets an increase in the low-stress regime stress exponent as an effect of grain growth. It can thus be concluded that at low strain rates and high temperatures the fine-grained AZ31 alloy deforms by GBS. Microstructural instability, especially at very high temperatures, may lead to an increase in the flow stress predicted by the GBS mechanism, and finally the deformation of the alloy evolves away from the GBS region. The analysis of the deformation mechanism using measurements of the maximum stress, could thus imply the occurrence of a creep mechanism involving an n ¼3 exponent, interpreted as evidence of viscous glidecontrolled deformation in fine-grained AZ31, which by contrast mainly deforms as a result of GBS. 2.3. Solute drag mechanism (n¼3) in samples with a coarse grain size The existence of very large elongation to failure (superplasticity) is closely related to a low stress exponent, which is not exclusive of the GBS deformation mechanism. For materials undergoing viscous glide-controlled deformation, the stress exponent is n¼3, equivalent to a strain rate sensitivity m ¼1/n¼0.33, and this leads to fairly high elongations of more than 300%. This behaviour was first noted by Mohamed [41] and has been widely documented in Al–Mg alloys [5], in which a solid solution of Mg is present in an Al matrix. In the case of AZ31, we have a symmetrical situation, since Al is in solid solution in the Mg matrix. The main characteristics of samples deforming by solute drag dislocation creep are: (i) the presence of an inverse transient creep, especially in the very low-stress regime; (ii) a delayed subgrain formation; (iii) the elongation of the grains, typical of a dislocation mechanism, and (iv) the absence of a strong inverse dependence of the creep rate on grain size. As mentioned in Section 2.1, there is a large amount of data for coarse-grained AZ31 alloy showing low stress exponent values

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Fig. 4. Diffusion and grain size compensated strain rate as a function of moduluscompensated stress for AZ31 for samples with large grain size in the region of n¼ 3 (the significance of the symbols is presented in Table 2).

Fig. 5. Diffusion compensated strain rate as a function of modulus-compensated stress for pure Mg.

dependence of the strain rate on the grain size, were indeed observed when analysing a limited number of datasets showing an n ¼3 regime, without taking into account the role of GBS and microstructural instabilities [4]. 2.4. Climb-controlled dislocation creep mechanism (n¼5)

Fig. 3. Strain rate change tests (SRC) and creep tests (CS) for AZ31. [3,4]. In (a) the data obtained by creep tests (CS) at 573 K are also presented for comparison purposes.

that do not follow the grain boundary sliding equation. These data are represented in Fig. 4 as normalised strain rate versus Young’s modulus compensated stress. The line given in the graph corresponds to Eq. (6). As can be seen, most of the data are very well described by Eq. (6), with the exception of the results provided by Kim and Kim [31]. The reason for this discrepancy is not clear; it has been argued that their data were measured at temperatures of 473 K and below, and taking into account the phase diagram there is a possible lack of solute due to second phase precipitation. Nonetheless the difference observed in Fig. 4 cannot be easily explained in these terms. In fact, Fig. 1 shows that, as stress decreases, the Kim and Kim data tend to rapidly converge towards the scatter band describing the GBS-controlled regime. This hypothesis is supported by the activation energy of 101 kJ/mol, in itself not consistent with the solute drag or GBS mechanisms, but intermediate between the corresponding values of 92 and 143 kJ/mol. Similar values of the activation energy, and a weak

Fig. 5 shows literature data for pure Mg [36,42,43]. It is worth noting that the slip creep equation (Eq. (5) in Table 1, with k5 ¼1  1011 ) predicts reasonably well the low stress creep behaviour of high purity Mg, a major discrepancy being nevertheless observed in the data obtained by Vagarali and Langdon [42]: a possible explanation for this anomalous behaviour could lie in a different purity of the material, but no definitive conclusions can be drawn on this subject on the basis of the scant evidence available. However, in all the reported datasets, the slope of the straight line describing the experimental result is close to 5 only in the low stress region, although the activation energy corresponds to that of self-diffusion. We thus obtain the classical ‘‘class M’’ (pure metal) behaviour until the region of power-law breakdown is reached. A more complex response is observed in AZ31; as strain rate or stress increases, the deformation mechanism changes and n increases from 3 to 5. This transition is usually attributed to the break-away of dislocations from the solute atom atmospheres which considerably hinder glide [5]. In the transition region, the dislocation progressively breaks away from the solute atom atmospheres, i.e. cnAl (the amount of aluminium atoms effectively clustering around dislocation cores, see Table 1), is

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Fig. 6. Diffusion compensated strain rate as a function of modulus-compensated stress for AZ31. Only data for n43 were included (symbols in Table 2).

reduced and viscous glide becomes increasingly rapid, until climb becomes the slower mechanism, and replaces the viscous glide in its rate-controlling role. In this case the experimental data are expected to align on the curves of slope n ¼5 (Eq. (5)). Fig. 6 shows tensile data for rolled and extruded AZ31 in a broad range of grain sizes. In the referenced works, published datasets usually cover regions having diverse creep mechanism in the e_ vs s graph. When building Fig. 6, data was taken from regions with n43. Since in most cases experimental data come from tensile tests on flat specimens obtained from rolled sheets, the tensile axis is almost perpendicular to the c-axis, i.e. is roughly parallel to the basal planes. This configuration strongly reduces the potential effect of the differences in texture which affect the slip creep behaviour [44]; flow stress in the dislocationcontrolled regime has been indeed observed to depend on the orientation of the basal planes to the tensile axes, albeit the texture influence disappears at 648 K. In addition, twinning effects will be neglected because, as will be described in the following, twinning is not favoured with this experimental conditions. An analysis of Fig. 6 demonstrates that these AZ31 data, at low strain rates, can be well correlated with Eq. (5). The difference in k5 can be at least in part attributed to the substantial effect of Al in reducing the stacking fault energy (27.8 mJ m  2) in AZ31 [27], versus the 78 mJ m  2 of pure Mg [43,45,46]. It can be observed in Fig. 5 that the power-law breakdown for AZ31 starts at about e_ b2/ DL ¼10  6. This corresponds to e_ /DL ¼1013 m  2 which is similar to the value found for aluminium and other metals [47]. The dislocation creep model used to describe climb-controlled high temperature deformation in the whole envelope of experimental conditions is based on the Garofalo relationship, in the form

e_ v ¼ k7 fsinh½aðs=EÞgn ðDL =b2 Þ

ð7Þ

where e_ v is the minimum creep rate (or the testing strain rate in the case of constant strain rate tests) produced by intragranular dislocation slip, and a and k7 are material parameters. Since creep is controlled by climb, Eq. (7) reduces to a power law for sufficiently low stresses, and n ¼5. The a constant was found to vary with the amount of Al in solid solution [4,48]; in the case of AZ31 tested above 473 K, Al is completely in solid solution, and a ¼714 when using E for normalisation of the stress s, rather than G as in [4,48]. Eq. (7) has no grain size effect included. The grain size effect on the dislocation creep properties of the AZ31 magnesium alloy has been investigated in [4] by comparing data obtained by testing materials of similar chemical composition but different initial microstructure [3,30,31]. The results of this analysis confirmed

Fig. 7. Diffusion and grain size compensated strain rate as a function of the hyperbolic sine of the modulus-compensated stress for AZ31. Symbols in Table 2.

the presence of a weak but not negligible effect of the grain size in the climb-controlled regime. Eq. (7) was thus modified as follows:

e_ v ¼ k8 fsinh½aðs=EÞgn ðDL =b2 Þðb=dÞp

ð8Þ

where d is the grain size. In the previous work, which considered a limited population of literature data, the authors report a value of p ¼0.6 [4]. In the present work, Eq. (8) was fitted to the more complete recompilation dataset of Fig. 6. As is shown in Fig. 7, the parameter p ¼0.4 gives the best correlation, with the parameter k8 ¼1.3  10–4 71  10  4 being obtained from the fit. 2.5. Phenomenological constitutive model for high temperature deformation of magnesium alloys The above discussion clearly demonstrates the existence of three different creep regimes:

i. A high strain rate regime, where creep is climb-controlled and n ¼5, but most of the experiments are carried out at above the power-law breakdown; ii. A low strain rate regime, with n¼ 3, typical of alloys in which deformation is controlled by the viscous glide of dislocations in an atmosphere of solute Al atoms; iii. A low strain rate regime, which could partially overlap the n ¼3 region, in which grain boundary sliding (GBS) is responsible for a large fraction of the total strain, in particular in the alloys with fine grain sizes. On the basis of these considerations, it is reasonable to conclude that the contribution of GBS and solute drag must be included in the model Eq. (8), in order to properly describe the low stress–strain rate range. Taking into account that the GBS and intragranular slip are independent, they merely add as follows:

e_ ¼ k4 ðDgb =d2 Þðs=EÞ2 þ e_ v

ð9aÞ

where

e_ v ¼ ðkn6 =cn6 ÞðDs =b2 Þðs=EÞ3

ð9bÞ

provided that the dislocations glide in atmospheres of Al atoms. As dislocations break away from the solute atoms, i.e. cnAl decreases, there is a rapid acceleration of the glide phase of dislocation motion, which eventually become faster than the climb phase. Above this transition

e_ v ¼ k8 fsinh½aðs=EÞg5 ðDL =b2 Þðb=dÞ0:4

ð9cÞ

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Fig. 8. Comparative analysis of the model curves Eq. (9) and the experimental data from [3, this study] (a), [30,40,49] (b) and [31] (c). Data in (c) were obtained by testing in shear.

The introduction of the more complete Eq. (9), with the theoretical values of the diffusion coefficient, cnAl ¼ 2% and the constants ki determined in the present work over a broad data universe, results in a very good description of the SRC results, which present the advantage of obtaining different data from a single experiment with a limited grain growth. In Fig. 8a, SRC data AZ31 with d ¼17 mm already reported in [3] and new unpublished data are shown together with the model curves obtained by Eq. (9). Fig. 8b shows a comparison between the model curves and the data obtained by testing coarse-grained materials, which are intrinsically less affected by grain growth. By making only minor corrections to the magnitude of the k4 and k8 parameters an excellent description of the data can be obtained. Fig. 8c plots the data obtained by Kim and Kim by testing in shear [31] and the respective model curves. Only in the latter case is a significant deviation observed; the model curves describe well the curvature of the experimental trend and the high strain rate results, but overestimate the strain rate in the low stress regime (an excellent fitting could be obtained by taking d¼ 50 mm). The reason for this discrepancy, as already mentioned, is not clear, mainly because no

information is available regarding the microstructure apart from the average grain size. Since in most cases the constitutive model gives an excellent description of the SRC tests, it could be modified to take into account microstructural evolution during continuous testing. The following paragraphs will deal with the different concurring phenomena which control the microstructure.

3. Mechanisms driving microstructural instability 3.1. Grain growth in static annealing (SGG) High temperature processing, such as rolling or extrusion, usually brings about the partial or complete dynamic recrystallisation of the alloy. Subsequent annealing leads to static recrystallisation (SRX) of the deformed structure, and/or to static grain growth (SGG). Yang et al. studied this process by annealing, at 493, 503 and 513 K, different samples previously deformed in compression up to e ¼0.3 or 1.2 [50]. Grain growth was observed in all the

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annealing on samples deformed between 593 and 473 K by Equal Channel Angular Pressing (ECAP). Each ECAP pass led to dynamic recrystallisation (DRX) and the progressive refining of the structure, which exhibited slightly different annealing responses. In this case, the authors assumed that ng ¼2, obtaining particularly low values of the activation energy for grain growth (about 1/3 of the activation energy for grain boundary diffusion). In fact, the majority of the above mentioned data could be appropriately described by simply assuming that QSG is equivalent to the activation energy for grain boundary diffusion (Qgb) (Fig. 9). In particular, the use of the same activation energy allowed a very good description of the experimental behaviour, with ng decreasing from 10 to 6 with an increasing number of passes, i.e. decreasing grain size. Similarly, an excellent description of the data obtained by Alsagabi and Charit between 523 and 723 K is obtained with ng ¼6. A last example of an excellent description obtained by assuming the activation energy for grain growth equivalent to the activation energy for grain boundary diffusion, is given by the data reported in Kim and Kim [40], although in this case the growth exponent is lower (ng ¼ 4). In all the mentioned examples, even when the same growth exponents and values of the activation energy are used, a considerable latitude in the value of the growth constant KSG is observed. A reasonable conclusion is that the Burke and Turnbull’s equation is in fact capable of describing the annealing response of a variety of initial conditions, with an activation energy for grain growth very close to the activation energy for grain

Fig. 9. Grain size as a function of annealing temperature for AZ31; data from [40,54–56] and after high temperature deformation by ECAP [55,56].

investigated conditions, irrespective of the different initial microstructure, which was partially (e ¼0.3 ) or fully (e ¼1.2) recrystallised. The growth of the grains, after an initial incubation time in which the growth occurs only locally in limited portions of the sample, is at first very rapid and then becomes slower as the annealing time increases. Similar experiments (annealing at 723 K after compression at 703 K-10  4 s  1 up to e ¼1.4) were carried out by Abdessameud et al. [51]. These authors observed a grain growth which was described by the usual Burke and Turnbull’s equation [52]   Q n n d g ¼ d0 g þ kSG exp  SG t ð10Þ RT where d0 was the initial grain size, KSG was a constant, QSG was the activation energy of the process which controls grain growth, and ng was the growth exponent. Abdessameud et al., in particular, obtained a good fit of the experimental grain size vs time data by considering ng ¼ 14–15. Miao et al. [53], who investigated the annealing behaviour of a rolled AZ31, by contrast, obtained ng ¼4 while QSG ¼80 kJ/mol, a value which is reasonably close to the activation energy for grain boundary diffusion in Mg. A similar value of the activation energy for growth was found by Alsagabi and Charit [54], who investigated the annealing response of an AZ31-H24 sheet; according to these authors, a good fit of the experimental data was obtained with ng ¼4 and QSG ¼86 kJ/mol. Very different results were obtained by Kim and Kim [55,56], who investigated the effect of

Fig. 10. Grain growth during annealing (data from [57]). The solid lines represent the model curves obtained with QSG ¼Qgb and ng ¼ 6; the dotted curves were obtained by phase simulation by Wang et al. [58].

Fig. 11. Grain growth during annealing (data from [53]). The model curves were calculated by using different values of the growth exponent.

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boundary diffusion. The different initial state leads to different growth exponents and constants. In order to verify this conclusion, Fig. 10 plots the data obtained by Liu et al. [57], the model curves obtained by Eq. (6) with QSG ¼Qgb and ng ¼6, and the curves calculated by phase field simulation carried out by Wang et al. [58]. The agreement between the experimental data and the curves is relatively good for both models, which provide quite comparable results. Fig. 11 plots the experimental data obtained by Miao et al. [53] and the model curves calculated with QSG ¼Qgb, ng ¼4 and 6 respectively. It is easy to observe that a good fit of the data between 573 and 673 K is obtained in Fig. 10 with ng ¼6 and KSG ¼7  1011 mm6 s  1, while a notable underestimation of the grain size in the same range of temperatures is shown in Fig. 11 as a result of the use of a growth constant which is one order of magnitude higher. Therefore it is plainly evident that the different history and initial microstructure play a major part in determining the magnitude of the growth parameters, possibly as an effect of the superimposition of SGG and SRX. 3.2. Dynamic grain growth (DGG) A detailed analysis of the dependence of the grain growth rate on the specimen strain rate during high temperature deformation in superplastic conditions has been published by Perevezentsev et al. [59]. These authors, who based their model on an accurate analysis of the driving forces and kinetics of grain growth, proposed, for single phase alloys tested under low strain rates, an equation which, in simplified form, can be written as d ¼ da þ K 0DG se_ v t expðQ gb =RTÞ

ð11Þ

where da is the grain size in the undeformed part of the specimen, 0 which underwent static annealing, and the growth constant K DG depends on the grain boundary mobility. In the high strain rate regime, i.e. above a transitional intergranular strain rate e_ nv , the boundary mobility is defined by the mobility of the orientation misfit dislocations, and the growth law can be rewritten, again in simplified form, as 2

d ¼ ½da þ K 00DG st expðQ gb =RTÞ1=2 where

0 KDG

and

00 KDG

ð12Þ

assume different values.

3.3. Dynamic recrystallisation (DRX) Dynamic recrystallisation is the mechanism which in principle has the most important role in controlling the evolution of the microstructure during high temperature testing. Tensile straining at 473–673 K, for example, was observed to produce a fully recrystallised microstructure, with grain sizes as low as 6.5–10 mm, under strain rates of 1  10  4, 1  10  3 and 1  10  2 s  1 [60]. The analysis of the microstructural evolution during the test carried out at 423 K under 1  10  4 s  1 revealed that in the sample deformed up to e ¼0.05, 18% of the total grain population consisted of fine grains, while, at e ¼0.2, DRX led to the presence of 31% of very fine grains. At e ¼0.69, the structure was almost completely recrystallised, and the grains exhibited high-angle boundaries which were capable of grain boundary sliding during a possible subsequent high temperature straining. According to Tan and Tan [60], the mechanism controlling grain refinement is dynamic continuous recrystallisation (CDRX), i.e. the progressive increase in misorientation of existing subboundaries, until they are converted into high-angle boundaries. At first, subgrains develop in the vicinity of serrated grain boundaries, although subgrain formation spreads over the whole grain volume as the deformation increases. The effect of tensile deformation at 323–473 K, in the strain rate range between 1.4  10  3 and 1.4  10  1 s  1, has been

143

investigated by Yin et al. [61]; these authors calculated the strain for the onset of DRX (ec) at 473 K, which decreased from 0.19 to 0.14 with increasing strain rate. A completely different result was obtained by Park et al. [62], who observed a marked decrease in ec with increasing temperature and decreasing strain rate. However, even in this case, the critical strain for the onset for DRX initiation in an annealed sample was found to vary between 0.1 and 0.2 in the testing range 423–573 K and 1  10  3 and 1  10  1 s  1. Serrated boundaries form at e ¼0.25 upon straining at 423 K, and new grains develop by CDRX along the serrated boundaries as the deformation increases. At higher temperatures, i.e. above 473 K, the interpretation of the microstructural evolution, given by Park et al., is somewhat different from the one described by Tan and Tan [60]. Grain boundary migration was considered to be dominant in the initial stages of deformation, up to e ¼0.05, and discontinuous dynamic recrystallisation (DDRX), i.e. the conventional nucleation of new grains at high-angle boundaries and their subsequent growth, led to the bulging of the original high-angle boundaries and to the nucleation of new grains at higher strains. Yang at al. [50] observed the microstructural evolution with straining at 573 K–1  10  3 s  1; after the sample was deformed up to e ¼0.3, new fine grains developed in colonies in proximity to the original grain boundaries (necklace recrystallisation), and subboundaries, which sometimes crossed each other, appeared in the grain interior. The regions of the grain fragmented by these subboundaries are finally bounded by high-angle boundaries as deformation proceeds, leading to a 90% recrystallised structure at e ¼1.2. While only 30% of the structure is recrystallised after deformation up to e ¼0.2 at 573 K–3  10  3 s  1 [50], Wang et al. [63], who investigated the microstructural evolution during compression at 573 K and 1  10  1 s  1 of a sample machined from a rolled sheet, observed that most of the structure was composed of fine recrystallised grains when the same level of deformation was reached. Del Valle and Ruano [21], in their study of the superplastic properties of AZ31, used tensile straining at 523 K–3  10  3 s  1 to produce a bimodal population of fine recrystallised and coarse unrecrystallised grains. A deformation up to e ¼0.22 caused 19% of the structure to be composed of fine recrystallised grains; this fraction increased up to 62, 83 and 100% as pre-straining increased up to 0.41, 0.56 and 1.1, respectively. These observations seem to indicate that the completion of DRX, if not its initiation, is somewhat delayed towards higher strains as the strain rate decreases. In fact, Fig. 12, which plots the recrystallised fraction in samples deformed at different temperatures and strain rates, demonstrates that the DRX behaviour of the alloy is not this simple. The analysis of Fig. 12, which collects

Fig. 12. Fraction of dynamically recrystallised microstructure (data from [21,50,60,64]).

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data from [21,50,60,64], suggests that the kinetics of DRX could be described by an Avrami-type equation in the form 00

f DRX ¼ 1expðkDRX t k Þ

ð13Þ

where the k’’ is a constant, and kDRX is a parameter which could be tentatively considered to depend on temperature and slipcontrolled strain rate by a relationship in the form kDRX ¼ k0 e_ m expðQ 0 =RTÞ

ð14Þ

It can be observed that in the high strain rate regime typical of the test in Fig. 12, deformation is controlled by dislocation motion that is, e_  e_ v . Since it is unlikely that GBS plays any significant part in controlling DRX, hereafter the strain rate in Eq. (14) will be assumed to be equivalent to the mere e_ v term. A multilinear regression analysis was carried out in order to quantify the value of the different parameters, giving k00 ¼ 1.5, m¼1.4 and Q0 ¼ 4.8 kJ/mol. Eq. (13), which is purely empirical in nature, could nevertheless be used to obtain an estimation of the DRX kinetics during high temperature tensile testing. Fig. 14. Schematic diagram illustrating the step-wise procedure for the estimation of the average grain size during DRX and DGG.

4. Constitutive model incorporating the effect of microstructural instability 4.1. Microstructural instability during CSR and SRC experiments The constitutive model described above contains two parameters which are specific expressions of the microstructure, namely the amount of Al in solid solution (through the a parameter) and the grain size. While the amount of Al in solid solution in AZ31 is substantially constant at above 473 K, the grain size can exhibit important variations due to the combined effect of DRX and DGG. Fig. 13 shows the variation in the grain size as measured by del Valle et al. [3] during the continuous test carried out at 623 K under a strain rate of 5  10  5 s  1. The sample tested by del Valle et al. should undergo DRX during the test; in parallel, the unrecrystallised portion of the sample undergoes DGG. The DRX kinetics, on the other hand, weakly depend on the strain rate, in particular through the e_ v term: the strain rate, for a given stress, varies with the average grain size, which should be calculated by considering the combined effect of DGG and DRX. This extremely complex behaviour has been modelled by describing the DRX evolution as a sort of step-wise phenomenon (Fig. 14). At the beginning of the test, at t ¼t1 and after a short incubation period, the structure is mainly composed of relatively coarse (d ¼d1) grains, which undergo DGG,

Fig. 13. Grain size evolution during a continuous test (data from [3]); the curve shows the model curve which takes into account DRX, and the conventional equations for SGG.

while DRX leads to the formation of the first new grains, with volume fraction fDRX,1. The grain size can be roughly calculated as a weighted average dav_1, i.e. dav,1 ¼ Sdi f i ¼ f DRX,1 dDRX þ ð1f DRX,1 Þd1

ð15Þ

where di and fi are the size and the volume fraction of each grain type. This population of ‘‘old’’ grains undergoes DGG during the next time interval t2  t1, i.e., taking ng ¼6, 6

d2 ¼ ½d1 þ K SG expðQ gb =RTÞðt2 t 1 Þ1=6 þ K 0DG se_ v expðQ gb =RTÞðt 2 t 1 Þ

ð16Þ until, at t ¼t2, DRX causes the formation of a new population of fine recrystallised grains, whose volume fraction is DfDRX,2 (properly calculated by Eqs. (13) and (14), with e_ v from Eqs. (6) or (8), with the real flow stress and grain size). The average grain size becomes dav,2 ¼ Df DRX,2 dDRX þð1Df DRX,2 Þd2

ð17Þ

This multi-step calculation, where dDRX, for the sake of simplicity, was taken ¼1 mm, gave the curve presented in Fig. 13, where 0 the KSG and KDG parameters were estimated by a best-fitting procedure of the experimental grain sizes. The calculated value of the static growth constant (KSG ¼1  1012 mm6 s  1) is indeed substantially in line with those obtained by interpolating the static annealing literature data (7  1011–8  1012 mm6 s  1, see above). 0 The computed value of the KDG constant was 1.8  108 mm MPa  1. Fig. 13 also shows, for the purpose of comparison, two examples of SGG curves, as calculated for a material with the same in grain size. Although the curves overlap for a sufficiently long testing time, Eqs. (15) and (16) provide growth curves which depend on the experimental conditions, while the SGG equation does not imply any dependence of the kinetics of grain growth on the strain rate or stress. At 648 K–5  10–5 s  1 [3], e_ v  10–5 s  1 when the flow stress reaches 8.5 MPa; Eqs. (15) and (16) thus predict that a mere 3% of the structure is recrystallised at e ¼0.5 (Fig. 13). The analysis of the microstructure did in fact reveal an equiaxed fine-grained structure, in substantial agreement with the prediction of the model, at e ffi 0.7 (5% of the structure underwent DRX). A similar picture is observed when considering the results of strain rate change experiments carried out at the same temperature and illustrated in Fig. 2; in this case a pre-straining, typically at 5  10–3 s  1, up to e ¼0.1, was followed by a sharp decrease in

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strain rate down to 2  10–5 s  1. Subsequently the strain rate was increased in successive jumps up to the initial strain rate, which is reached at e E0.25. According to del Valle et al. [3], this technique allowed steady state vs stress data to be obtained before significant grain growth altered the microstructure of the alloy. In fact, the use of Eqs. (15) and (16) suggests that a minor but not negligible fraction (about 10%) of the initial microstructure could undergo DRX during the pre-straining. However, subsequent deformation steps at lower strain rates do not lead to significant grain growth, due to the short time of exposure, but might increase the volume fraction of the recrystallised structure, maintaining the grain size very fine, possibly even finer than the initial one. This behaviour can thus qualitatively explain the marked differences observed when comparing CSR and SRC results obtained on the same alloy and at the same temperature. In the latter case, the occurrence of DRX and the relative brevity of the test maintain a fine grain size, while in the former case grain growth significantly reduces the contribution of grain boundary sliding to overall deformation, and, in some cases, even determines the transition to slipcontrolled deformation. 4.2. Strain rate dependence on stress in CSR and CS tests

Fig. 16. Strain rate vs strain curves obtained by constant load experiments at 573 K, after correction to take into account the reduction in area of the sample. The Fig. also plots the curves representing the calculated fraction of DRX structure for the tests carried out under the lowest and highest loads;e_ v was estimated in 5  10  7 s  1 Eq. (5) and 2  10  4 s  1 (Eq. (8)), at e ¼0.05, under 10 and 40 MPa respectively.

strain rate regime, the same equation should assume the form

For each of the experiments carried out by del Valle et al. [3], a curve similar to the one shown in Fig. 13 was obtained, and the strain corresponding to the peak stress was calculated by an approximate relationship, obtained from the data in Fig. 2, in the form

ep ¼ 0:0017e_ 0:65

145

ð18Þ

although a limiting value of the peak strain was considered for the low strain rate tests (the identification of this limiting strain, 0.8 and 1.8 at 573 and 648 K, was based on the rupture strains given in [3]). This simple relationship permitted a rough estimation of the time of exposure at the peak, tp (a 10 min pre-heating was considered to estimate the static growth), and the subsequent estimate of the grain size. Eq. (16), according to Perevezentsev et al. [59], is formally valid for e_ v o e_ nv , i.e. in the low strain rate regime. In the high

Fig. 15. Description of the CSR tests using the models illustrated in the text. The broken lines represent the calculation for a constant grain size.

1=2

6

d2 ¼ f½d1 þ K SG expðQ gb =RTÞðt 2 t 1 Þ1=3 þ K 00DG s expðQ gb =RTÞðt 2 t1 Þg

ð19Þ 00

The values of KDG and e_ nv were estimated by assuming that the grain size at the peak monothonically decreases with increasing strain rate in the whole range of experimental conditions, 00 obtaining KDG¼2  106 mm2 MPa  1 s  1 and e_ nv ffi 2  10–4 s  1. Once the grain size for each test, calculated by Eqs. (16) or (19), is incorporated in the equations describing the strain rate dependence on stress and temperature, with the same values of the parameters which gave an excellent description of the SRC tests, the curves presented in Fig. 15 are obtained. The description of the experimental data is again excellent. It can be specified here that the model illustrated above was obtained in an attempt to base the constitutive equations on a physically reliable growth law, which takes into account the presence of a relatively important fraction of DRX grains. Fig. 16 plots some representative constant load creep curves, corrected to take into account the sample elongation, and the calculated trend of the fraction recrystallised during the test. The low strain rates associated to low initial applied stress result in a partial DRX at rupture, and the parallel DGG causes the formation of a relatively inhomogeneous structure, with different populations of very coarse unrecrystallised, recrystallised/dynamically grown and fine freshly recrystallised grains, in excellent agreement with the microstructural observations. However, the minimum creep rate is generally encountered at low strains (e E0.05), i.e. when DRX is almost negligible. The only exceptions are the tests carried out under the lower stresses, since the corresponding long time of exposure leads to a marked grain growth, which in turn causes a progressive decrease in the strain rate. Accordingly, even in this case an overlapping of different mechanisms can make the interpretation of the data quite complex. Therefore, in order to simplify the calculations of the minimum strain rate dependence on applied stress, it has been assumed that DRX has a negligible role. The model curve is calculated at t ¼tm (time corresponding to minimum creep rate) by supposing that the grain size grows according to Eqs. (16) or (19), with d0 ¼8 mm. Since all the parameters which appear in the different equations are now available, this simple analysis permits the curves shown in Fig. 17 to be obtained directly. The agreement between the

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temperatures), Fig. 18. According to this model, in the investigated range of grain sizes reported in Table 2, twinning assumes a significant role only in the case of the creep tests carried out at 373 K, or, at higher temperatures (473 K), in the very high strain rate regime. As a matter of fact, the recent study carried out by Boehlert et al. on a rolled AZ31 with 13 mm grain size [65], unambiguously demonstrated that, in tension, twinning plays a significant role only at 323 K, while it disappears above this temperature. Plasticity at 423 K is thus controlled by a combination of basal and prismatic slip, although limited GBS has also been observed by in-situ experiments between 423 and 523 K. On these bases, it could be safely inferred that the assumption of a negligible, if any, rate-controlling role of twinning at high temperature appears more than reasonable.

Fig. 17. Minimum creep rate dependence on applied stress; the broken lines represent the model curve calculated by assuming a constant grain size, while the solid lines were obtained by considering the effect of DGG.

Fig. 18. Critical grain size for transition between slip and twinning dominated flow, according to Barnett et al. [66], for different testing temperature, as a function of strain rate.

model curves and the experimental data is very good, in particular considering that they were obtained by estimating only one mechanical parameter, i.e. the time corresponding to the minimum strain rate. A complete overlapping of the curves with the experimental data could in fact be obtained by k4 ¼500. 4.3. On the role of twinning and textures in high temperature deformation Many different deformation modes could be operative in Mg alloys; in particular, detailed in-situ experiments (see [65] for an extensive report on these investigations) demonstrated that deformation can be controlled by slip occurring in basal, prismatic or pyramidal planes, and by twinning. In this sense, the model proposed in this study could be penalised by the implied assumption that creep is controlled only by slip and/or GBS. Barnett et al. [66] identified a lower limiting grain size for the transition from twinning dominated flow (low temperatures/high strain rates) to slip dominated deformation (low strain rates/high

Fig. 19. Strain rate against stress for an AM60 alloy tested in tension at 473 and 573 K with different initial textures [44]; for each condition the grain size and the Schmid factor are reported (the datum denoted by n was calculated by Eq. (22)). The model curves were obtained by Eqs. (9a) and (21), with k4 ¼ 2000, k6 ¼ 22.4, k8 ¼ 1  10–4; the Al content in solid solution was supposed to be equivalent to the equilibrium one at 473 K, i.e. roughly 0.03, giving, for a sample with tensile axis parallel to the basal planes, a ¼636.

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The presence of a textural effect is typical of hcp metals in the high stress regime (dislocation creep), since textures play a significant role in determining the magnitude of the flow stress under a given strain rate, without altering the stress exponent in the power-law region [44,67]. In this context del Valle and Ruano [44] proposed a modified version of the power-law Eq. (5), in the form

e_ =ms,b pðDL =b2 Þðsms,b =EÞn

ð20Þ

where ms,b is the Schmid factor for basal slip, and n ranges from 7.9 at 573 K to 10 at 473 K. For orientations with basal planes close to the ideal orientation (451), the flow stress assumes its minimum value (ms,b ¼0.5). In case of randomly textured or rolled samples, the Schmid factor is lower, intermediate between 0.3 and 0.5 [44]. Eq. (20) could be thus used to predict the strain rate dependence on applied stress in the case of samples which have different textures; yet, the high stress exponents observed in [44] clearly indicate that the considered data lie above power-law breakdown, so Eq. (5) should be replaced by Eq. (9c). By analogy with Eq. (20), the effect of texture could be in this case expressed as

e_ =ms,b ¼ k8 fsinh½ao ms,b ðs=EÞg5 ðDL =b2 Þðb=dÞ0:4

ð21Þ

In Eq. (21), a0 is the composition-dependent term of a [4,48]. Fig. 19 illustrates the data obtained by SRC experiments at 473 and 573 K for the AM60 alloy with different initial textures [44] (in the case of the AZ31, similar textural effects have been observed by testing in tension the alloy in as cast, extruded and Equal Channel Angular Pressed conditions, Fig. 20 [67]). The data in Fig. 19 were modelled by combining Eqs. (9a) and (21), with cAl ¼0.03%, roughly corresponding to the equilibrium concentration of Al in solid solution at 473 K, giving, according to [4], a ¼646 for a sample with the tensile axis oriented parallel to the basal planes, the condition which, as clearly stated in Section 2.4, should be valid for all data in Fig. 6. An excellent correlation between Eq. (21) and the data presented in Fig. 19 is obtained by modifying the expression for a given in [4], which assumes the form

a ¼ 210ms,b ½2ð1 þ nÞc0:37  Al

ð22Þ

147

where n is Poissons’ modulus and cAl the total amount of Al in solid solution. The same approach was used to obtain the curves in Fig. 20; in the case of extruded material, the basal planes are parallel to the extrusion direction, and the behaviour is similar to that of the rolled condition (ms,b ¼ 0.33). The ECAPed material is again the softer, and as in the case of the AM60, it was considered to have the maximum Schmid factor [44], i.e. ms,b ¼0.50; the cast alloy has an intermediate behaviour, described as in Fig. 19 by ms,b ¼ 0.43. The accuracy of these descriptions (a significant deviation is observed only in the case of the AM60 with lower Schmid factor at the 573 K) shows that this revised formulation is in principle able to take into account also the effect of textures in tension.

5. Conclusions The high temperature deformation of an AZ31 magnesium alloy was analysed by comparing the results of numerous investigations. Three main deformation mechanisms were observed: 1. grain boundary sliding, which causes a superplastic behaviour in fine-grained materials; grain growth due to prolonged high temperature exposure leads to a reduction in the creep rate and eventually to the transition to either viscous glide or dislocation climb as the rate-controlling mechanism; 2. deformation controlled by solute drag of dislocations in Al-atom atmospheres; this mechanism is frequently obscured by the effects of grain boundary sliding, and becomes fully apparent only in coarse-grained alloys; 3. climb-controlled deformation, in the high stress region. A combined set of constitutive equations, which takes into account the concurring effect of these different deformation mechanisms, was proposed; the model was properly modified to include the effects of static and dynamic grain growth, dynamic recrystallisation and textures. On the basis of these considerations, the differences observed by testing the same material with different techniques were rationalised and an unitary description of the high temperature deformation of AZ31 in a wide range of strain rates and temperatures was obtained.

Acknowledgements Financial support from CICYT, Spain, Project MAT2012-38962 is gratefully acknowledged. References

Fig. 20. Strain rate against stress at e ¼ 0.1 for an AZ31 alloy tested in tension at 523 K with different initial textures [67]; for each condition the grain size is reported. The model curves were obtained by Eqs. (9a) and (21), with k4 ¼1200, k6 ¼ 22.4, k8 ¼ 1  10–4; the Al content in solid solution was supposed to be 0.025. The Schmid factor was 0.5, 0.43 and 0.33 for the ECAPed, cast and extruded conditions respectively.

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