Introduction

Extruded magnesium alloys exhibit low ductility at room temperature caused by the limited number of slip systems. The principal slip system at room temperature for magnesium alloy is the dislocation slip on the basal (0001) plane along the <a> direction, due to its lowest critical resolved shear stress (CRSS).1,2,3 Non-basal slip systems, such as dislocation slip along the <a> direction on prismatic {10 \(\overline{1}\) 0} planes or dislocation slip along the <a+c> direction on pyramidal {11 \(\overline{2}\) 2} planes, have been also reported, although CRSSs are two orders of magnitude higher than for basal slip.2,3,4,5,6,7,8,9,10,11,12 In addition, twinning is an important deformation system in magnesium alloys, because it can also contribute to fulfill the Von Mises criterion.13,14

Dislocation slip or twinning is activated in each grain, depending on its crystallographic texture. During the extrusion process, magnesium grains orient their basal planes parallel to the extrusion direction, developing a strong fiber texture.15,16 Tensile twinning is only activated when strain induces an extension of the c-axis, for example, during compression along the extrusion direction.3,15 However, microstructural aspects, such as alloying elements, texture, or grain size, or experimental conditions, such as temperature or strain rate, also have influence over the competition between different deformation systems.15,16,17,18,19,20,21,22,23,24

A refinement in the grain size induces the rise of the mechanical strength and a change in the deformation mechanisms from twinning to dislocation slip.25 The transition grain size at room temperature, calculated using discrete dislocation dynamics simulations, is around 2.7 μm,26 simulating experimental data obtained in equal channel angular pressed Mg by Li et al.27 Experimentally, the transition grain size at 423 K has also been reported to be between 4 µm and 5 µm.16 The stress for the activation of tensile twinning increases with the decrease in the grain size more rapidly than the stress required for activation of dislocation slip.28 This means that the Hall–Petch slope, i.e., slope of the linear dependence of the yield stress as a function of the d-1/2, is higher when plastic deformation is controlled by twinning than when it is controlled by basal slip.16,28,29 The calculation of Hall–Petch parameters for dislocation slip or tensile twinning is measured assuming that one of these deformation systems controls plasticity. However, it is well known using crystal plasticity simulations that all deformation systems can contribute with different weight to macroscopic strain.

The use of diffraction experiments during in situ mechanical tests can univocally identify the activation of different deformation systems in multicomponent metallic alloys and metal matrix composites, as a function of the applied stress or macroscopic strains.30,31,32,33 Moreover, these results can be combined with crystal plasticity simulation to support the experimental results. Neutrons or synchrotron radiation are the incident source to perform in situ experiments in large facilities, such a neutron reactor, spallation sources, or synchrotrons. Synchrotron radiation is faster than neutrons to obtain a reliable diffraction peaks avoiding the plastic relaxation process, especially at high temperature, that could disturb the analysis of diffraction peaks. This paper analyzes the Hall–Petch parameters for basal slip and tensile twinning systems using synchrotron radiation diffraction (SRD) during in situ compression tests in two AZ31 alloys with different grain sizes. Compression tests were carried out from room temperature up to 473 K to analyze their temperature dependence.

Experimental Procedure

The cast AZ31 alloy (designated IM-AZ31) supplied by Magnesium Elektron was machined to obtain a cylinder of 0.041 m in diameter, and was then extruded at 623 K with an extrusion ratio of 25:1 and an extrusion rate of 10-3 m s-1. Rapid-solidified powders were fabricated from the cast AZ311 alloy using the electrode induction-melting gas atomization technique by TLS-Technik, with a powder diameter below 100 µm. AZ31 powders were uniaxially cold-compacted up to 340 MPa in a special die designed to obtain a cylindrical compact of 0.041 m in diameter. The compact was degassed and then extruded in the same experimental conditions as the cast AZ31 alloy. The material produced using a powder metallurgy route has been designated PM-AZ31.

The microstructures of both materials have been analyzed using optical and scanning electron microscopy. The crystallographic texture of the alloy was evaluated using electron backscattered diffraction (EBSD) technique and by Rietveld analysis of the synchrotron diffraction patterns. EBSD measurements were performed in a direction parallel to the extrusion direction.

SRD experiments during in situ compression tests were performed at the P07 beamline of PETRA III, at the Deutsches-Elektronen-Synchrotron. The compression samples were cylinders of 5–10-3 m diameter and 10-2 m length machined from the extruded bar along the extrusion direction. In situ compression tests at room temperature, 373 K, 423 K, and 473 K at a strain rate of 10-3 s-1 were performed on both materials, IM- and PM-AZ31. LaB6 was used to calibrate the experimental set-up, and 2θ diffraction patterns were obtained by azimuthal integration of the 2D diffraction rings along the axial and radial directions. Diffraction peaks were individually fitted with FIT2D software34 using a Voigt function. The elastic strain for each individual diffraction peak during the compression test is given by:15

$$ \varepsilon_{hkl} = \frac{{d_{hkl} - d_{0,hkl} }}{{d_{0,hkl} }} $$
(1)

where dhkl and d0,hkl are the interplanar distances of the hkl plane in the stressed and stress-free lattice, respectively. Using Bragg´s equation:

$$ 2d_{hkl} sin_{hkl} = \lambda $$
(2)

where θhkl are the diffraction angle of the hkl plane y and λ is the wavelength of the radiation, the elastic strain can be expressed as a function of the diffraction angle which is obtained in a diffraction pattern:

$$ \varepsilon_{hkl} = \frac{{sin_{0,hkl} - sin_{hkl} }}{{sin_{hkl} }} $$
(3)

The macroscopic texture was evaluated using a Rietveld texture analysis in MAUD.35 The 2D detector image has been converted to a set of diffraction patterns, using the ImageJ plugin in the software,36,37 fitting the α-Mg (P63/mmc). The best fit to the experimental data was achieved using the E-WIMV (the expanded method of the Williams,38 Imhof,39 and Matthies and Vinel40) algorithm with 5º resolution. This procedure was repeated along the compression test, obtaining the evolution of the texture as a function of the compressive strain.

Results

Microstructural and Mechanical Characterization

The microstructure and orientation image map of both materials are shown in Fig. 1a–d. The microstructure is fully recrystallized after extrusion at 623 K, and Fig. 1e shows the grain histogram. The grain size is finer in the alloy fabricated using a powder metallurgy route than in the IM-AZ31 material. The average grain sizes are 3 and 7 μm for the PM-AZ31 and IM-AZ31 materials, respectively. It is important to note that the interval of grain sizes in the IM-AZ31 alloy is wider than in the case of the PM-AZ31 alloy. Thus, the maximum grain size in IM-AZ31 alloy is about 21 μm, while in the PM-AZ31 alloy, all grains are below 8 μm.

Fig. 1
figure 1

Grain microstructure along extrusion direction of (a) IM-AZ31 and (b) PM-AZ31. Orientation image mapping along the extrusion direction of (c) IM-AZ31 and (d) PM-AZ31. (e) Grain histograms of IM-AZ31 and PM-AZ31.

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The macroscopic texture was evaluated using the Rietveld texture analysis of the diffraction pattern. Figure 2a, b shows Debye–Scherrer rings obtained before the compression test at room temperature for IM-AZ31 and PM-AZ31 materials. In Fig. 2a, diffraction rings for each family of plains are indicated. The intensity of the diffraction rings changes along the azimuthal angle due to the crystallographic texture. Figure 3a–d shows the Rietveld fitting of diffraction patterns in the axial and radial directions for both materials in the as-extruded condition. In the PM-AZ31 alloy, additional diffraction peaks from MgO can also be observed. MgO particles come from fragmentation of the oxide film covering the surface of RS powders during the extrusion process. The volume fraction of the MgO phase, estimated according to Rietveld analysis was around 0.2%.

Fig. 2
figure 2

Synchrotron radiation diffraction pattern recorded on the 2D flat-panel detector (a, b) before compression and (c, d) after compression testing at room temperature of (a, c) IM-AZ31 and (b, d) PM-AZ31; plane indexes are indicated in (a).

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Fig. 3
figure 3

Rietveld analysis of diffraction patterns of (a, b) IM-AZ31 and (c, d) PM-AZ31 in the (a, c) axial and (b, d) radial directions.

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Figure 4 shows the \(\left\{ {10\overline{1}0} \right\}\) and (0002) pole figures obtained from the Rietveld texture analysis (a, b) and from the EBSD analysis (c, d). Both techniques present similar results, and therefore Rietveld texture analysis has been used to follow the evolution of the crystallographic texture during plastic deformation. Independently of the processing route, extruded materials exhibit a fiber texture with the basal plane parallel to the extrusion direction. However, the fiber intensity of the alloy using the powder metallurgy route is lower than that of the IM-AZ31 alloy.

Fig. 4
figure 4

(0002) and {10 \(\overline{1}\) 0} pole figures obtained from (a, b) EBSD and (c, d) Rietveld analysis for (a, c) IM-AZ31 and (b, d) PM-AZ31. The extrusion (ED) and radial (RD) directions are indicated in the first pole figure of each calculation method.

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Figure 5a and b shows compression curves tested for both materials from room temperature to 473 K during in situ diffraction experiments in the beamline. At room temperature, the shape of compression stress–strain curves is similar for IM-AZ31 and PM-AZ31. The sigmoidal or “S-type” concave shape of compression curves is indicative of \(\left\{ {10\overline{1}2} \right\}<\overline{1}011\) extension twinning activation.16 Two main differences are clearly observed in the compressive curves. On the one hand, the yield stress is higher for the PM-AZ31 due to its finer grain size. On the other hand, a stress plateau takes place after yielding, which extends for higher strains in the PM-AZ31 alloy (1% and 3% for IM-AZ31 and PM-AZ31, respectively). Figure 5c shows the evolution of the yield stress as a function of d-1/2. For each processing route, the yield stress decreases as the test temperature increases, but the yield stress is higher for the PM-AZ31 alloy, except at 473 K, where the opposite behavior is observed. The slope of Hall–Petch fitting, i.e., the k parameter, calculated for both materials is plotted in Fig. 5d. The k parameter decreases linearly as the compression temperature increases.

Fig. 5
figure 5

(a) Compressive true stress–true strain curves for (a) IM-AZ31 and (b) PM-AZ31 from room temperature to 473 K. The legend in (a) also refers to (b). (c) Grain dependence of yield stress of the AZ31 alloy from room temperature to 473 K. (d) Temperature dependence of the macroscopic Hall–Petch parameter from room temperature to 473 K.

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Synchrotron Radiation Diffraction During In Situ Compression Tests

Figure 2c and d shows Debye–Scherrer rings obtained after compression tests at room temperature for the IM-AZ31 and PM-AZ31 materials, respectively. After integration along the axial and radial directions, diffraction patterns as a function of 2θ are obtained for all the studied temperatures. As an example, Fig. 6 shows diffraction patterns along the axial direction for the IM-AZ31 and PM-AZ31 alloys before and after compressive deformation at room temperature and 473 K as extreme cases. The diffraction peak corresponding to the \(\left\{ {10\overline{1}0} \right\}\) planes exhibits the highest intensity, due to the fiber texture of the alloy with the basal plane parallel to the extrusion direction. After compression tests, the intensity of the (0002) diffraction peak increases strongly. As the compression temperature increases, simultaneous changes in the intensity between the \(\left\{ {10\overline{1}0} \right\}\) and (0002) diffraction peaks disappear, especially for the PM-AZ31 alloy.

Fig. 6
figure 6

Axial diffraction patterns as a function of 2θ before compression testing and after a compressive strain of 8% at (a, b) room temperature and (c, d) 473 K for (a, c) IM-AZ31 and (b, d) PM-AZ31.

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The elastic strains were evaluated from Eq. 1 for the \( \left\{ {10\overline{1}0} \right\}\), (0002), \(\left\{ {10\overline{1}1} \right\}\),\( \left\{ {10\overline{1}2} \right\}, \) and \(\left\{ {11\overline{2}0} \right\}\) diffraction peaks of the magnesium phase in the axial and radial directions for the IM-AZ31 and PM-AZ31 samples compressed at room temperature (Fig. 7b and e). Engineering stress–strain compression curves obtained during the in situ diffraction experiment are also plotted (Fig. 7a and d). Compression curves allow an easy comparison of overall sample strains and specific-planar compressive strains. For the magnesium phase, the relative integrated intensity along the axial direction for the \(\left\{ {10\overline{1}0} \right\}\) and {0002} diffracted peaks as a function of the applied stress in logarithmic scale are plotted in Fig. 7c and f. Changes in the intensity of these diffraction peaks provide valuable information about the initialization and magnitude of the tensile twinning process.

Fig. 7
figure 7

Compressive true stress–true strain curves for the extruded (a) IM-AZ31 and (d) PM-AZ31. Axial and radial internal strains as a function of the applied stress during the in situ compression test at room temperature for the {10 \(\overline{1}\) 0}, (0002), {10 \(\overline{1}\) 1} and {11 \(\overline{2}\) 0} diffraction peaks for the extruded (b) IM-AZ31 and (e) PM-AZ31. Evolution of the integrated intensity in the axial direction of the {10 \(\overline{1}\) 0} and (0002) diffracted peaks during the compression test at room temperature for the extruded (c) IM-AZ31 and (f) PM-AZ31.

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The macroscopic yield stresses at room temperature were 195 MPa and 275 MPa for the IM-AZ31 and PM-AZ31 materials, respectively. Below the yield stress, both alloys behave purely elastically. Internal strains for all the studied diffraction peaks show the same linear slope (Fig. 7b and e) in the initial stage of the elastic regime, in agreement with isotropic behavior of magnesium in the elastic field. The slope of the elastic strain as a function of the applied stress normally represents Young’s modulus. Experimental values at room temperature are similar for both materials, close to 42 GPa, which coincides with that of pure magnesium.41

The majority of grains are oriented with their \(\left\{ {10\overline{1}0} \right\}\) planes perpendicular to the compression/extrusion axis. The evolution of elastic strains of these grains shows a continuous linear elastic behavior over the entire compression test. Schmid´s factor of this lattice orientation for the slip of the <a> dislocations along the basal plane is zero, and, therefore, these grains are not favorably oriented for dislocation slip. Schmid´s factor of grains oriented with the {11 \(\overline{2}\) 0} plane perpendicular to the compression axis is also zero for the slip of the <a> dislocations along the basal plane. However, in contrast to grains oriented with their \(\left\{ {10\overline{1}0} \right\}\) planes perpendicular to axial direction, internal strains lose their linearity above the macroscopic yield stress, at 250 MPa and 325 MPa for IM-AZ3 and PM-AZ31, respectively.

In contrast to both previous families of grains, grains oriented with their \(\left\{ {10\overline{1}1} \right\}\) planes perpendicular to the compression axis are the first to lose their linearity below the macroscopic yield stress (see blue arrow). These grains are favorably oriented for the activation of the basal slip system, which is the main deformation system in magnesium alloys, because of its lowest CRSS. This “microyielding” has been previously reported in the same alloy deformed under tension or compression along the extrusion direction.15 However, the volume fraction of these grains is very low and, therefore, they do not control macroscopic plasticity.

The evolution of internal strains of the (0002) diffraction peak is more complex. Initially, the intensity of (0002) diffraction peaks is quite low (see Fig. 7c and f), although they can be fitted. This fact explains the dispersive variation of elastic strain in the elastic regime below the macroscopic yield stress. At the macroscopic yield stress, the peak intensity increases rapidly, more markedly in the PM-AZ31 alloy, due to the nucleation, propagation, and growth of twins. Therefore, the beginning of the macroscopic plasticity is caused by the activation of tensile twinning. It is well known that twins are developed preferably in grains oriented with the \(\left\{ {10\overline{1}0} \right\}\) planes perpendicular to the compression axis, inducing a lattice rotation of 86º with respect to the initial \(\left\{ {10\overline{1}0} \right\} \) parent grain orientation.15,42,43 Consequently, the evolution of the internal strains of the (0002) diffraction peak reveals information about the internal strain within twins. At the yield stress, the internal strain exhibits the lowest value (in absolute values), independently of the fabrication route. After the stress plateau observed after yielding, the internal strain decreases rapidly linearly. The slope defined by \(\frac{{{\Delta }\sigma_{app} }}{{{\Delta }\varepsilon_{0002} }}\) is higher for the PM-AZ31 alloy than for the IM-AZ31 alloy (0.31 and 0.25 MPa/µ-strains, respectively). This hardening was classically attributed to the crystal rotation of the twinned area from a soft grain orientation ({10 \(\overline{1}\) 0} planes perpendicular to the compression axis) to a hard grain orientation ({0002} planes perpendicular to the compression axis), or with the dislocation-twin interaction by considering twin boundaries as obstacles to slip or twinning deformation.44,45 However, the rapid increase in the dislocation density within twins with a higher hardening rate than that observed in the parent grains could also account for the linear increase in the elastic strain within the twins.46

Finally, it is important to note that, just above the yield point, a sudden inflection in the elastic strains occurs during the stress plateau for all the diffraction peaks. This variation is most noticeable for the \(\left\{ {11\overline{2}0} \right\}\) diffraction peak (around 800 and 600 µ-strains in absolute values for IM-AZ31 and PM-AZ31, respectively), although it can also be observed for the \(\left\{ {10\overline{1}0} \right\}\), \(\left\{ {10\overline{1}1} \right\}\) and \(\left\{ {10\overline{1}2} \right\}\) diffraction peaks. Such behavior in the stress plateau has been reported for the extruded AZ31 alloy.42 The relaxation phenomenon in the vicinity of twins explains this rapid strain evolution. The fast acquisition time of the P07 beamline allows following this relaxation phenomenon better than using neutron diffraction.

Similar to Figs. 7 and 8a–f shows the elastic strains for the \(\left\{ {10\overline{1}0} \right\}\), (0002), \(\left\{ {10\overline{1}1} \right\}\),\( \left\{ {10\overline{1}2} \right\} \) and \(\left\{ {11\overline{2}0} \right\}\) diffraction peaks of the magnesium phase in the axial and radial directions for in situ compression tests at 373 K, 423 K, and 473 K for the IM-AZ31 and PM-AZ31 alloys. In both materials, as the compression temperature increases, the macroscopic yield stress decreases, as was observed in Fig. 5. Similar to room temperature, grains oriented with their \(\left\{ {10\overline{1}1} \right\}\) plane perpendicular to the compression axis yield at an applied stress below the macroscopic yield stress. These microyielding values, associated with slip activation of the <a> dislocations along the basal plane, decrease as the compression temperature increases. It is important to note that the values are slightly higher for the PM-AZ31 alloy.

Fig. 8
figure 8

Compressive true stress–true strain curves for the extruded, axial and radial internal strains as a function of the applied stress during the in situ compression test at room temperature for the {10 \(\overline{1}\) 0}, (0002), {10 \(\overline{1}\) 1} and {11 \(\overline{2}\) 0} diffraction peaks. Evolution of the integrated intensity as a function of applied stress in the axial direction of the {10 \(\overline{1}\) 0} and (0002) diffracted peaks for extruded IM-AZ31 and PM-AZ31 at (a, b) 373 K, (c, d) 423 K, and (e, f) 473 K. (a, c, e) IM-AZ31 and (b, d, f) PM-AZ31.

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It is possible to evaluate the effect of the grain size in the activation of the basal system, representing the microyielding stress as a function of d-1/2 (Fig. 9a). At the macroscopic yield stress, the internal strains of grains oriented with the (0002) plane perpendicular to the compression axis lose their linear behavior. At 373 K, yielding is associated with the appearance of a stress plateau, as found at room temperature. The stress plateau disappears above 423 K. In any case, the beginning of plasticity is always linked with the decrease (increase in absolute values) in the evolution of the elastic strain as a function of the applied stress. This effect, associated with twinning, is strongly influenced by the test temperature and the grain size. As the test temperature increases, twinning tends to decrease. In the same way, by decreasing the grain size, this also tends to decrease deformation by twinning. Even at 473 K, its influence is minimal in the case of the PM-AZ31 alloy.

Fig. 9
figure 9

Grain dependence of the activation stress obtained from Figs. 8 and 10 for the (a) basal slip and (b) twinning systems for IM-AZ31 and PM-AZ31.

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This can be clearly observed in the \(\left\{ {10\overline{1}0} \right\}\) and (0002) pole figures after compression tests (Fig. 10). Initially, the texture of the extruded bar showed that basal planes were parallel to the extrusion direction (see Fig. 4). The \(\left\{ {10\overline{1}0} \right\}\) planes, originally perpendicular to the extrusion direction, are, after the compression test, almost parallel to the extrusion direction, as observed in Fig. 10. Therefore, the maximum intensity is now observed in the radial and extrusion directions for the \(\left\{ {10\overline{1}0} \right\}\) and (0002) pole figures, respectively. As the compression temperature increases, the deformed texture shows the maximum along the radial direction in the \(\left\{ {10\overline{1}0} \right\}\) pole figure, but with a lower intensity. This tendency is more pronounced in the case of the PM-AZ31 alloy, in which the fiber texture is almost totally lost at 423 K (Fig. 10c). IM-AZ31 still shows a small deformed component in the radial direction at 473 K, together with the intense initial fiber texture.

Fig. 10
figure 10

(0002) and {10 \(\overline{1}\) 0} pole figures after compression tests of IM-AZ31 and PM-AZ31 at (a) room temperature, (b) 373 K, (c) 423 K, and (d) 473 K. Extrusion and radial directions are indicated in the first pole figure.

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From the (0002) pole figures obtained during plastic deformation, the twin volume fraction can be calculated following the procedure described in.42,47 The volume fraction is given by:

$$ f_{tw} = \mathop \smallint \limits_{\varphi = 0}^{\varphi \approx 45^\circ } \left[ {I\left( \varphi \right) - I_{0} } \right] {\text{sin}}\varphi {\text{d}}\varphi $$
(4)

where φ is the angle of tilt between the (0002) poles and the compression axis, I(φ) is the intensity at that tilt angle φ, and I0 is the intensity in the undeformed condition.

The twin volume fraction of the IM-AZ31 alloy is higher than that of the PM-AZ31 alloy (Fig. 11), due to its coarser grain size. In the IM-AZ31 alloy, a small twin volume fraction is still observed in the alloy even deformed at 473 K. In contrast, in the PM-AZ31 alloy, the twin volume fraction is negligible above 373 K.

Fig. 11
figure 11

Evolution of the twin volume fraction as a function of the true strain for the extruded (a) IM-AZ31 and (b) PM-AZ31 from room temperature to 473 K.

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It is also possible to evaluate the influence of the grain size on the activation of the \(\left\{ {10\overline{1} 2} \right\}\left\langle {10\overline{1} \overline{1} } \right\rangle\) twinning system representing the applied stress, when the intensity of the (0002) diffraction peak begins to rise as a function of the d-1/2 (Fig. 9b). It is important to note that only twin growth contributes to the increase in the (0002) diffraction peak. However, we can assume that twin nucleation is too fast, and that there are no large differences between the two stress values.

Finally, grains oriented with \(\left\{ {10\overline{1}0} \right\}\) and \(\left\{ {11\overline{2}0} \right\}\) planes perpendicular to the loading direction show an elastic behavior up to the macroscopic yield stress. After macroscopic yielding, the evolution of the elastic strain shows a low deviation from the linearity with an asymptotic behavior due to the activation of non-basal slip systems.

Discussion

The compression test temperature, the crystallographic texture, and the grain size have a strong influence on the deformation of extruded AZ31 alloys. The stress plateau after yielding was more extended in strain in the PM-AZ31 and IM-AZ31 materials. After yielding, a catalytic twin cascade has been reported that induces the stress plateau. The stress generated by a twin in neighbor-grains promotes the formation of a new twin, even when the misorientation between them is low (10–20º).48 For angles between 30º and 70º, the stress generated by the twin produces massive dislocation slip, while the activation of slip or twinning systems is highly restricted for angles between 80 and 90°, resulting in a high accumulation stress. This increase in the local stress in neighbor-grains oriented with \(\left\{ {10\overline{1}0} \right\}\) and \(\left\{ {11\overline{2}0} \right\}\) planes perpendicular to the loading direction (misorientation 80–90º) induces a rapid change in the elastic strain in the stress plateau. As the compression test temperature increases, this change in the elastic strain tends to disappear due to the activation of non-basal slip system. This is confirmed by the loss of linearity in the evolution of the elastic strains above 373 K in grains oriented with \(\left\{ {10\overline{1}0} \right\}\) and \(\left\{ {11\overline{2}0} \right\}\) planes perpendicular to the loading direction.

The evolution of the CRSS as a function of the temperature for the AZ31 alloy obtained from equations reported by Ref. 49 were plotted for <a> basal slip, <a> prismatic slip, <a+c> pyramidal slip, and tensile twinning (Fig. 12). CRSSs for the <a> basal and tensile twinning are constants in the whole temperature range. From the evolution of the internal strains during the compression tests (Figs. 7 and 8), the applied stresses for the activation of the <a> basal slip system and tensile twinning system can be estimated. The product of these stress values and the Schmid factors for the <a> basal slip system in grains oriented with the \(\left\{ {10\overline{1}1} \right\}\) planes perpendicular to the applied stress (m = 0.43), and for tensile twinning system in grains oriented with the \(\left\{ {10\overline{1}0} \right\}\) planes perpendicular to the applied stress (m = 0.499), give an idea of the temperature dependence of CRSS for the deformation systems (Fig. 13) for both PM-AZ31 and IM-AZ31 materials. For both deformation systems, stress values are not constant, as in Ref. 49, but they decrease as the compression temperature increases. Stress values are higher in the alloy fabricated using a PM route, but they tend to approach each other at 473 K.

Fig. 12
figure 12

Temperature dependence of the critical resolved shear stress (CRSS) for dislocation slip in the <a> direction on the basal (0001) plane, dislocation slip in the <a> direction on {10 \(\overline{1}\) 0} planes, dislocation slip along the <a+c> direction on {11 \(\overline{2}\) 2} planes, and tensile twinning in the AZ31 alloy from equations reported by Ref. 49.

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Fig. 13
figure 13

Product of the activation stress for the basal slip and twinning systems, and the Schmid´s factor as a function of compression temperature.

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The applied stresses for the activation of the <a> basal slip system and the tensile twinning system were plotted as a function of the d-1/2 (Fig. 9). The slope gives the Hall–Petch parameter for each individual deformation system. Figure 14 shows the evolution of the Hall–Petch parameter as a function of the compression temperature for the <a> basal slip system and the tensile twinning system. The twinning system is more sensitive to the decrease in grain size than the basal slip. The k value for extension twinning is 0.57 MPa m−1/2 at room temperature, and it decreases rapidly to almost zero at 473 K. The k value for basal slip is 0.25 MPa m–1/2 at room temperature. It remains almost constant at 373 K, and then decreases to 0.06 MPa m−1/2 at 473 K. At 473 K, unlike at room temperature, the Hall–Petch parameter for extension twinning is lower than for the <a> basal slip system. Comparing these values with those obtained from the macroscopic yield stress, it is interesting to note that, at room temperature, the macroscopic value is between the value obtained between the basal slip and the tensile twinning. However, for the other temperatures, the macroscopic Hall–Petch parameter is smaller compared with those for individual deformation systems. The contribution to the macroscopic plasticity of other deformation systems at high temperature decreases the grain sensibility of the AZ31 alloy. The increase in the deformation to failure associated with super-plasticity has been reported in magnesium alloys with a refined grain size.50,51,52,53 Super-plasticity is controlled by the activation of the grain boundary sliding mechanism, resulting in a marked decrease in the yield stress, especially at 473 K. At this temperature, the k value is negative, since the yield stress is smaller for PM-AZ31 than for IM-AZ31.

Fig. 14
figure 14

Temperature dependence of the Hall–Petch parameters for (a) basal slip and (b) twinning systems from room temperature to 473 K.

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Table I compares the Hall–Petch parameters obtained in this study with those obtained by Barnett et al.16 and Garces et al.54 The Hall–Petch values calculated by Barnett et al.16 for the dislocation slip are smaller than those obtained in this study, but they decrease following the same tendency. The behavior for the twinning system is totally different. While values of this study decrease sharply from a high value, values obtained from16 remain almost constant (with a difference of 0.03 MPa m-1/2) independently of the compression temperature. It is important to point out that stress for the calculation of the Hall–Petch parameter is different. Barnett et al.16 use the macroscopic yield stress assuming that twinning or slip is the dominant deformation system, while this study uses the applied stress at which the deformation system is activated. In the first case, the sum of different deformation systems can contribute to the macroscopic yield stress.

Table I Hall–Petch parameters for the basal slip and twinning systems obtained in this study from room temperature to 200 K compared with other values in the same alloy reported in Ref. 16 and 54
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Finally, it is important to point out that the Hall–Petch parameters were measured with only two grain size values. We understand that this is a weakness of the analysis of this study. We have calculated Hall–Petch values for basal slip and twinning systems at room temperature using stress values also reported in the bibliography for coarse-grained wrought AZ31 alloy (10 µm15 and 20 µm55), obtained using in situ neutron and x-ray diffraction experiments. Figure 15 shows the grain dependence of the activation stress for the basal slip and twinning systems. The evolution of the activation stress for tensile twinning follows the same tendency with a correlation value of 0.999. On the other hand, the correlation of the evolution of the activation stress for basal slip as a function of d-1/2 is 0.97, and the Hall–Petch value is slightly higher compared with our initial value (0.27 MPa m-1/2). Therefore, our approach can be considered appropriate to estimate the evolution of Hall–Petch parameters with temperature. The use of in situ diffraction allows the determining of the stress at which each deformation system starts individually, allowing the evaluation of the individual hardening of each system, in contrast to an average behavior obtained from traditional uniaxial tensile or compression tests. This process could be further improved by combining these results with simulations using crystal plasticity techniques.

Fig. 15
figure 15

Hall–Petch parameters for basal slip and twinning systems at room temperature using the stress values of this study,15 and Ref. 55. In Ref. 55, the stress for the activation of the basal slip is calculated using the (10 \(\overline{1}\) 2) diffraction peak.

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Conclusion

The grain dependence of basal slip and twinning systems has been studied using synchrotron radiation diffraction during compression tests from room temperature to 473 K, in an extruded AZ31 alloy fabricated using two processing routes to obtain two grain sizes. The following conclusions can be drawn:

  1. 1.

    At room temperature, the activation of twinning controls the onset of plasticity regardless of the grain size. However, the twin volume fraction decreases with the increase in the compression test temperature, faster in the alloy fabricated using a powder metallurgy route. Above 373 K, dislocation slip controls plasticity in PM-AZ31.

  2. 2.

    Grain size dependence of twinning system is much higher than that for the basal slip system at room temperature. As the compression test temperature increases, the difference between Hall–Petch values for the two deformation system decreases. Furthermore, at 473 K, this behavior is inverted and the Hall–Petch value for the basal slip system is higher than the value for twinning system.