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Effect of precipitate shape on slip and twinning in magnesium alloys
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Acta Materialia 59 (2011) 1945–1956 www.elsevier.com/locate/actamat
Effect of precipitate shape on slip and twinning in magnesium alloys J.D. Robson a,⇑, N. Stanford b, M.R. Barnett b b
a Manchester Materials Science Centre, Grosvenor Street, Manchester M1 7HS, UK Centre for Material and Fibre Innovation, Deakin University, Pigdons Road, Geelong 3217, Australia
Received 15 September 2010; received in revised form 24 November 2010; accepted 25 November 2010 Available online 31 December 2010
Abstract The predicted strengthening effect of precipitates of different shape and habit on the basal, prismatic and f1 0 1 2g twinning deformation systems in magnesium has been calculated. In parent material, rod precipitates parallel to the c-axis are predicted to be more effective than plates parallel to the basal plane in hardening the basal and prismatic slip systems. However, in twinned material, nonsheared basal plates are highly effective in inhibiting the basal slip necessary to relieve incompatibility stresses. The predictions suggest basal plates will reduce asymmetry in strongly textured extrusions by preferentially hardening against twin growth compared to prismatic slip, whereas c-axis rods can have the opposite effect. The predictions have been compared with the measured asymmetry for two magnesium alloys that form either c-axis rods (Z5) or basal plates (AZ91). In agreement with the model, it is shown that precipitation in Z5 leads to an increase in asymmetry, whereas in AZ91 precipitation reduces asymmetry. These results suggest that designed precipitation may provide a useful tool for reducing asymmetry in wrought magnesium alloys. Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Magnesium alloys; Twinning; Extrusion; Modelling
1. Introduction Precipitate particles are exploited for strengthening in many commercially important magnesium alloys [1]. The precipitates that typically form in magnesium alloys take the form of rods or plates lying on particular crystallographic planes, with the precipitate morphology and habit depending on the alloy system and thermal treatment [2]. It is known that different precipitate types lead to markedly different strengthening responses; for example, in randomly textured as-cast material the basal plates formed in Mg–Al–Zn (AZ) alloys generally give poor strengthening compared to the prismatic plates that form in Mg–Y–RE (rare earth) alloys, such as WE54. Previous work has shown that this difference in strengthening may be at least partly attributed to the different Orowan hardening of basal slip (the most easily activated deformation mode in ⇑ Corresponding author.
E-mail addresses: [email protected], manchester.ac.uk (J.D. Robson).
joseph.robson@
magnesium) due to the different precipitate shapes and habits [2,3]. In addition to suggesting desirable precipitate characteristics for maximum strength, the prediction that different precipitate shapes and habits lead to different strengthening effects might be exploited in another important way. One key problem with wrought magnesium alloys are the high levels of mechanical asymmetry that are often observed [1,4]. This is the result of the strong textures typically produced after thermomechanical processing coupled with the unidirectional nature of twinning [5]. Since particles of different shape and habit will strengthen the different deformation modes to different extents, it may be possible to exploit this to reduce anisotropy and asymmetry. In particular, if the deformation modes with a low critical resolved shear stress (CRSS), i.e. basal slip and f1 0 1 2g c-axis tension twinning, can be strengthened more strongly than those which already have a high CRSS (e.g. prismatic slip), then more isotropic properties will result and a reduction in asymmetry will be possible. There is evidence from recent studies of both cast and extruded AZ alloys that precipita-
1359-6454/$36.00 Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2010.11.060
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J.D. Robson et al. / Acta Materialia 59 (2011) 1945–1956
tion leads to a marked reduction in asymmetry due to this effect, in particular the suppression of f1 0 1 2g twinning [6,7]. In this paper, the effect of particle shape in strengthening the different deformation systems in magnesium is predicted, and the likely consequences on mechanical asymmetry are discussed. In the case of slip, strengthening by the Orowan mechanism has been well studied, and although the effect of particle shape and habit is less well understood, previous work by Nie and Muddle [2,3] has shown how the situation of rationally oriented rod- and plate-shaped precipitates may be treated. In the case of twinning, the influence of particles remains much less clear. Several studies have shown a strong influence of precipitates on twinning in magnesium alloys [8–11]. In some alloys an increased number of smaller twins is commonly observed when particles are present [11]. This paper discusses how particles may influence twin growth, and the effect of particle shape and habit. 2. Hardening against slip Nie [2] has previously presented an analysis of the Orowan strengthening effect of different precipitate shapes and habits in magnesium alloys for basal slip. In this work, a similar approach is followed and extended to also consider prismatic slip and twinning. Attention is paid to prismatic slip because it is considered to be a critical deformation mode in controlling the yield of strongly textured magnesium alloy extrusions when subject to tensile stress along the extrusion axis [12]. This is important because the ease of activating this mode compared to f1 0 1 2g twinning (which controls yield in axial compression) is important in determining the tension/compression asymmetry in extrusions.
Three particle habit/morphology combinations are considered in this study. These are basal plates, c-axis rods and randomly distributed spheres. Fig. 1 illustrates the first two of these schematically and also shows the particle cross-section when cut by the basal and prismatic slip planes. These morphologies were chosen because basal plates form in precipitation-strengthened AZ alloys (e.g. Mg–9 wt.% Al– 1 wt.% Zn, AZ91 [13]) and c-axis rods form in Mg–Zn alloys (e.g. Mg–5 wt.% Zn, Z5 [8,11]). As well as forming precipitates of different shape and habit, the AZ and ZK (Mg–Zn–Zr) classes of alloy are of great practical importance. The precipitates formed in the magnesium alloys of interest in this work are too large and insufficiently coherent to be cut by dislocations. The influence of precipitate shape and habit on strength is therefore assumed to depend on the Orowan stress required to bow dislocations around the particles. The general form of the Orowan equation used in this work is the same as used by Nie and Muddle [2,3], and can be written [14]: Gb 1 Dp pffiffiffiffiffiffiffiffiffiffiffi Ds ¼ log ð1Þ r0 2p 1 m k where Ds is the increase in CRSS due to precipitate strengthening, G is the shear modulus, b the Burgers vector of the gliding dislocations, m the Poisson ratio, k the effective interparticle spacing on the slip plane, Dp the mean planar diameter of the particles on the slip plane and r0 the dislocation core radius. 2.1. Basal slip Calculations for basal slip have been previously presented by Nie [2] (Section 2.3), and this analysis is not repeated here. A minor modification has been made in
(a)
(b)
Fig. 1. A schematic showing the (a) basal plate and (b) c-axis rod precipitate morphologies considered in the present study, and the cross-sections produced when these particles are cut by basal and prismatic slip planes.
J.D. Robson et al. / Acta Materialia 59 (2011) 1945–1956
1 p k ¼ pffiffiffiffiffiffi d t NA 4
the present work; Nie assumed that the precipitates have a triangular arrangement when cut by the slip plane. However, since the analysis, which is based on the work of Fullman [15], considers the probability of cutting randomly distributed rods, plates or spheres, there appears no reason to assume a triangular arrangement of particles on the slip plane. Therefore, in this work, a regular array of precipitates on the slip plane is assumed, with the mean centerpffiffiffiffiffiffi to-center distance simply given by 1= N A , where NA is the number of particles per unit area on the slip plane. This simplifies the mathematics of the analysis and the use of a more complex geometry is not considered to be justified given the highly approximate nature of a treatment based on average spacings.
The same approach can be applied to prismatic slip by considering the new geometries involved (Fig. 1). The analysis will be restricted to prismatic edge dislocations, since it is assumed that prismatic screw dislocations will be able to cross-slip onto another prism plane to avoid obstacles in their path. Note that for randomly distributed spheres, the effective particle spacing on the prismatic and basal planes is identical, since the probability of cutting a spherical particle and the planar diameter is the same for both planes. Therefore, it is not necessary to reproduce the analysis for spherical particles. The probability of a single basal plate in a unit volume being cut by a prismatic plane is equal to the plate diameter dt (for consistency, we follow the nomenclature of Nie [2]). Therefore, the number of basal plates cut per unit area of slip plane(NA) is equal to dtNV, where NV is the number of plates per unit volume. Random intersection of the plates will produce a rectangular cross-section on the slip plane of mean length pdt/4 and width tt (the plate thickness). Assuming a regular array, the effective spacing can be written as:
2
2
10
10
(a)
1
Δτ Orowan (MPa)
ð2Þ
The analysis for rods is similar, except the length of the rectangular cross-section produced by random intersection of the rods is lt (rod length), and this replaces the final term in Eq. (2) to calculate the spacing for rods. Note that basal plates and c-axis rods are variants of the same cylindrical shape with different aspect ratios. Fig. 2 shows the calculated Orowan stress for (a) basal and (b) prismatic slip for basal plates, c-axis rods and spherical particles. This calculation was performed assuming a precipitate volume fraction of 5%, with the aspect ratio of plates defined as tt/dt = 0.1 and the aspect ratio of rods lt/dt = 10. These represent typical values for an age-hardening magnesium alloy and are used throughout the paper to highlight the important trends. Application to some specific alloys is discussed later. In this, and subsequent plots, the x-axis gives the effective particle dimension, which is defined as V p1=3 , where Vp is the particle volume. This means that a given x-value corresponds to the same particle volume for plates, rods and spheres. It is plotted this way to facilitate comparison between particles of different shape at a constant particle volume. The calculations confirm that basal plates are poor strengtheners against basal slip. This is because the probability of such a plate being intersected by a given basal plane is very low, leading to a large interparticle spacing on the plane. c-Axis rods are predicted to be more effective strengtheners for both basal and prismatic slip compared to basal plates or spheres. The ability of c-axis rods to strengthen strongly against prismatic slip is particularly important. This is because prismatic slip usually controls the tensile yield strength of extrusions [12]. Changing aspect ratio was found to follow expected behavior; as the aspect ratio of the rods is increased and they become more “rod like”, their predicted strengthening effect increases. Similarly, as the basal plates become thinner but broader (more “plate like”) their predicted strengthening decreases.
2.2. Prismatic slip
(b)
plates rods spheres
1
10
10
0
0
10
10
−1
10
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−1
0
2
Particle dimension (μm)
4
10
0
2
4
Particle dimension (μm)
Fig. 2. A calculation of the Orowan stress required to bypass plate, rod and spherical precipitates for (a) basal slip and (b) prismatic slip. Particle volume fraction = 5%, aspect ratio of plates = 0.1, aspect ratio of rods = 10.
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3. Hardening against twinning Experimental studies have clearly shown that particles can have a strong effect on twinning in magnesium [8– 11]. However, in contrast to slip, the theory to determine what hardening effect particles have against twinning has not yet received any attention. Nevertheless, since twinning is such an important deformation mode in magnesium, an understanding of the likely effect of particles on strength, anisotropy and asymmetry must necessarily include a consideration of the influence of particles on twinning. There is now sufficient experimental evidence of the interaction of twinning and particles in magnesium alloys to enable a qualitative, if not yet fully quantitative, analysis of how particle shape is likely to influence hardening against twinning. Twins can act with particles in a number of ways, all of which have been reported in experimental studies of magnesium alloys depending on particle characteristics [8–11]. Firstly, particles of the size commonly obtained in magnesium alloys after ageing never appear to be effective in completely suppressing twin nucleation. Indeed, in some alloy systems (e.g. Z5) the presence of particles leads to an increase in the number of twins [11], and it has been argued that this is due to an increase in stress required for twin growth [16]. Therefore, in this work, we focus on the effect particles have on twin growth. Furthermore, the analysis is limited to f1 0 1 2g (c-axis tension) twinning since this is by far the dominant mode in magnesium and is also the critical mode that controls the compressive yield strength of extrusions. Particles have been observed (i) to arrest twin growth completely, often with twinning continuing by nucleation of a new twin in the matrix on the far side of the particle [10]; (ii) to become fully engulfed by a growing twin being rotated but not sheared [11,10]; and (iii) to become fully engulfed and sheared by the twin [8]. The type of behavior observed appears to depend on precipitate size and shape, with the largest precipitates showing type (i) behavior, intermediate-sized precipitates showing type (ii) behavior, and one study [8] reporting small rod-shaped precipitates
showing type (iii) behaviour (although only type (ii) behaviour was observed in another study of the same system [11], so it remains unclear whether type (iii) behaviour ever occurs in practice). For the size of particles that achieves a maximum in the hardening response during ageing, type (ii) behaviour has been reported for both basal plate- and rod-shaped precipitates. Fig. 3 shows example images from (a) AZ91 and (b) Z5 in the peak-aged state, demonstrating what happens to the particles when they are engulfed by a twin. For both basal plates (AZ91) and c-axis rods (Z5) it can be seen that when inside the twin, a small rotation of the particle occurs (’4°). This is consistent with the particles undergoing rigid body rotation (RBR), but not shearing, as demonstrated by Gharghouri et al. [10]. It is also interesting to note an apparent deflection of the twin habit plane (indicated by the dashed black line) close to the particles, demonstrating that the particles do inhibit twin growth. In addition, and consistent with previous work, evidence for a large number of basal dislocation loops in the twin can be observed. These dislocations are evidence of the plastic relaxation of the incompatibility stresses between particles and twinned matrix, as discussed later [11,10]. A forthcoming publication will discuss these observations in more detail, but for the purposes of the present work the key point is that the precipitates are not sheared in the twin, whereas the matrix is, and this creates a strain discontinuity that has to be accommodated. 3.1. Back-stress The observation that twins can engulf particles without shearing the particles themselves leads to several important consequences that we now explore. Although twinning is a distinctly different deformation mode from slip, understanding how a twin interacts with a non-shearing particles has much in common with understanding the role of nondeforming particles on slip, a topic that has received much experimental and theoretical attention (e.g. [17–19]). The material within a f1 0 1 2g twin is deformed by simple shear with a simple shear strain tw = 0.13 [10]. This
Fig. 3. Interaction of a f1 0 1 2g twin and particles in peak-aged material (a) AZ91, showing a basal plate interacting with the twin boundary. A small rotation of the particle in the twin (T) can be seen, as evidenced by the deviation between the particle interface in the twin and the superimposed white line which shows the extrapolation of the interface in the untwinned matrix (M). (b) Z5, showing a c-axis rod-shaped precipitate crossing the twin boundary and evidence of a similar small rotation of the particle in the twin (T). The twin boundary has been highlighted with a dashed line for clarity.
J.D. Robson et al. / Acta Materialia 59 (2011) 1945–1956
rij ¼ 2lV f cijkl pkl
ð3Þ
where l is the shear modulus of the matrix, Vf is the particle volume fraction, cijkl the accommodation tensor, and pkl the plastic strain discontinuity tensor. For slip deformation, the case of different particle shapes has been treated by Brown and Clarke [19] and the same approach is followed here. Unlike in the case of slip, where the plastic strain discontinuity increases progressively as the applied strain increases, in the case of twinning the full plastic strain discontinuity is imposed immediately on twinning and determined by the fixed twinning shear. To calculate this plastic strain discontinuity the simple shear of twinning is partitioned into its pure shear and RBR components. It is the difference in the pure shear between the particle and matrix that must be accommodated to prevent a strain discontinuity [20]. The required transformation is given by Gharghouri et al. [10]. To calculate the back-stress it is necessary also to rotate the axes of the stress tensor to those that correspond to the particle coordinate system, and Gharghouri et al. describe how this is done, with the same method used here [10]. The appropriate values for the accommodation factor (cijkl) can then readily be calculated from the tabulated values for Sijkl reported in Brown and Clarke (Table 1 [19]) for fibres (used to represent c-axis rods), discs (used to represent basal plates) and spheres using a Poisson ratio value of 0.35 [21]. The effect of the back-stress on basal slip in the twin is likely to be particularly important, as discussed later. Therefore, to enable a simple comparison to be made between the effect of particles of different morphology on Table 1 Measured compressive and tensile 0.2% offset stress for (rC0:2 and rT0:2 , respectively) and asymmetry ratio as-extruded and after ageing. Alloy AZ91 AZ91 AZ91 Z5 Z5
Ageing treatment None 18 h at 150 °C 8 h at 350 °C None 192 h at 150 °C
rC0:2 (MPa) 160 249 174 91 153
rT0:2 (MPa) 232 263 233 150 286
rC0:2 =rT0:2 0.69 0.95 0.75 0.61 0.53
the influence of back-stress on basal slip, the back-stress tensor was resolved to calculate the shear stress component acting on the basal plane in the twin. This calculation gives: rb ¼ lV f ð0:10Þ basal plates rb ¼ lV f ð0:05Þ c-axis rods
ð4Þ ð5Þ
rb ¼ lV f ð0:07Þ spheres
ð6Þ
The calculated resolved back-stress values for basal plates, c-axis rods and spheres (l = 17 GPa [21]) are shown in Fig. 4 as a function of volume fraction. Looking first at the relative back-stress for the different particle shapes, it can be seen that basal plates are predicted to lead to a much higher back-stress than c-axis rods or spheres. The magnitude of the predicted back-stresses become very large for the precipitate fractions typically formed in age-hardening magnesium alloys. For example, in the Mg–7.7 wt.% Al alloy studied by Gharghouri et al. [10], the volume fraction of basal plates was reported as 10%. If the strain incompatibility around these particles was accommodated entirely elastically, the back-stress calculation predicts this precipitate fraction would generate stress levels approaching 170 MPa in the twin, which are far in excess of the CRSS for basal slip [12]. This clearly demonstrates that plastic deformation by slip must play an important role in moderating the strain discontinuity around particles in the twin, thereby reducing the back-stress. Indeed evidence of the large number of dislocations that produce this deformation are commonly observed in the twin [10]. This calculation therefore shows that for the precipitate fractions of practical interest in age-hardening magnesium alloys (typically 3–15%), plastic deformation in the twin is required to enable further twin growth without particle shear. Note that this requirement for slip in the twin to relieve the back-stress caused by particles is additional to the slip processes in the twin and matrix that are required 350 plates rods
300
Resolved backstress (MPa)
simple shear can be divided into a RBR and pure shear component. When the twinned region contains a particle which remains unsheared, application of the twinning shear alone will result in a plastic strain discontinuity at the particle–matrix interface. To moderate this strain discontinuity to produce a continuous strain field, elastic and/or plastic deformation must occur. The case where accommodation is entirely elastic has been treated for slip by Brown and Stobbs [17,18]. They demonstrate that the requirement for strain continuity results in the generation of a uniform mean stress in the matrix that opposes the stress causing deformation (and is therefore commonly called the back-stress). The back-stress depends on the particle volume fraction, shape and orientation (but not size or spacing). The back-stress (rij) is given by [20]:
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spheres 250 200 150 100 50 0
0
0.05
0.1
0.15
0.2
Volume fraction precipitate Fig. 4. A calculation of the back-stress (resolved shear stress on the basal plane) produced by accommodation of non-deforming particles of different shape in a f1 0 1 2g twin, assuming accommodation occurs entirely elastically.
J.D. Robson et al. / Acta Materialia 59 (2011) 1945–1956
to accommodate the twinning shear regardless of the presence of precipitates. Plastic deformation will itself be inhibited by the presence of particles. Since the matrix is reoriented by twinning (in particular, the basal plane is rotated by 86°), the effectiveness of particles in strengthening against slip in the twin will be different to that in the parent material. The level of back-stress due to the precipitates that can be sustained in the twin will therefore be controlled by the level of slip inhibition. If the particles in the twin are highly effective in inhibiting slip, then it is to be expected that less plastic relaxation can occur, and hence a higher back-stress is sustained. This back-stress acts to oppose the stress applied to grow the twin and hence would be expected to increase the stress required for continued twin growth. 3.2. Basal slip in the twin It is likely that basal slip is responsible for the majority of the plastic deformation necessary to moderate the strain incompatibility between non-shearing particles and the sheared matrix in the twin. This is because the CRSS for basal slip is far less than that for other slip modes, and furthermore the basal planes in the twin are well oriented for relief of the back-stress by basal slip (the angle between the direction of maximum back-stress, which is in the twinning direction, and the basal plane in the twin is 43.2°). Therefore, it is reasonable to consider that it is suppression of basal slip in the twin that will have the greatest effect on the ability to relieve the back-stress by slip. The Orowan calculation for slip can now be repeated for basal slip in the twin, using the assumption that the particles do not deform but do undergo the observed rigid body rotation (which for full rigid body rotation is 3.7° about h1 1 2 0i [10]). The effect of the reorientation of the lattice by twinning and the rigid body rotation is that rods that were parallel to the c-axis in the parent become perpendicular to the c-axis in the twin, and lie in the basal plane. Plates that were perpendicular to the c-axis in the parent (parent–basal) become parallel to the c-axis in the twin (and thus have a “prismatic plate”-type orientation [2]). This profoundly influences the Orowan hardening contribution of these precipitate morphologies, as shown in Fig. 5. In particular, parent–basal plates that are poor strengtheners against slip in the parent become highly effective strengtheners in the twin because they are in their best possible orientation to inhibit slip. For particles less than 1 lm in size, the Orowan stress required to bypass parent–basal plates in the twin is predicted to be over an order of magnitude greater that in the parent. Meanwhile, rods oriented parallel to the c-axis in the parent (parent–c-axis) become far less effective strengtheners in the twin because lying in the basal plane in the twin means they are poorly oriented to inhibit basal slip (with dislocations able to bow in the relatively large gap perpendicular to the long axis of the rods).
2
10
plate (parent) rod (parent) sphere plate (twin)
Δτ Orowan (MPa)
1950
1
rod (twin)
10
0
10
−1
10
0
1
2
3
4
Particle dimension (μm) Fig. 5. A calculation of the Orowan stress inhibiting basal slip for plate, rod and spherical shaped precipitates in both parent and twinning material. Particle volume fraction = 5%, aspect ratio of plates = 0.1, aspect ratio of rods = 10.
This analysis has important practical implications. It has been demonstrated that compared to spherical or c-axis rod-shaped particles, parent–basal plates lead to (i) the highest incompatibility stresses between non-shearing particles and the twinned matrix and (ii) the maximum inhibition of the slip necessary to accommodate the strain incompatibility in the twin. 3.3. Orowan stress on the twinning dislocation In addition to the back-stress resulting from the strain incompatibility between the sheared matrix and nonsheared precipitates, there will also be an inhibition of twin growth due to the Orowan stress required to bow the twinning dislocation around the precipitates. The twinning dislocations are partial dislocations characterized by a step in the twin interface and a small Burgers vector. Twin growth involves the nucleation and glide of these partial dislocations across the twin plane. For the specific case of a ð1 0 1 2Þ twin in magnesium, the twinning dislocation has a Burgers vector btw ¼ 0:0694½1 0 1 1 and the twin step has a height equal to 2d ð1012Þ , where d ð1012Þ is the spacing of the ð1 0 1 2Þ planes [22,23]. These characteristics are required to shear atoms in the plane a distance 2d ð1012Þ into their correct position, with atoms in the intermediate plane having to shuffle to restore the perfect hexagonal closepacked structure [23]. Assuming the passage of the twinning dislocation leaves the particle unsheared, then it is necessary for the twinning dislocation to bow around the particle. The mechanism by which this bowing occurs is not yet clear, and the exact form of the Orowan equation that describes the extra stress required to perform bowing of the twinning partial is not yet known. Nevertheless, as for slip dislocations, an inverse relationship to the particle spacing is expected, and as a
J.D. Robson et al. / Acta Materialia 59 (2011) 1945–1956
first approximation it seems reasonable to apply the standard Orowan law (1) to calculate the relative magnitude of the Orowan stress required to bow precipitates of different shape. The same approach is used to calculate the effective particle spacing on the twin plane as previously described for slip. The angle between the twin plane normal and the c-axis is / = 43.2°. The probability of the twin plane cutting a single plate and a single rod in a unit volume are therefore given by Eqs. (7) and (8), respectively [24]: d t sin / þ tt cos / basal plates d t sin / þ lt cos / c-axis rods
ð7Þ ð8Þ
A basal plate cut by the twin plane will have a cross-section with a rectangular shape with mean length equal to pdt/4. A c-axis rod cut by the twin plane will have an elliptical cross-section with major axis length dt/cos /. With this information, the effective precipitate spacing on the twin plane may be calculated following an identical procedure as for slip, and Eq. (1) may be applied to predict the Orowan stress, using the appropriate values of k and b = btw. The results of this calculation are shown in Fig. 6. The first thing to note is that for all particle shapes the maximum Orowan stress on the twinning dislocation is small, and is an order of magnitude less than that predicted on the slip dislocations. This is principally due to the very small Burgers vector of the twinning partial dislocation. Plates and rods are predicted to be more effective than spheres in inhibiting the twinning partial dislocation. The difference between plates and rods is small for a volume fraction of 5% (shown here). It was found that as volume fraction increased, plates were predicted to be increasingly effective obstacles, and for a volume fraction of 15% the Orowan stress for plates was up to twice that for c-axis rods. Nevertheless, even for this very high volume fraction, the magnitude of the Orowan stress for the twinning partial
1
Orowan hardening for twin (MPa)
10
basal plates c−axis rods spheres
0
10
−1
10
0
1
2
3
4
Particle dimension (μm) Fig. 6. A calculation of the Orowan stress inhibiting motion of the f1 0 1 2g twinning partial dislocation for plate, rod and spherical particles. Particle volume fraction = 5%, aspect ratio of plates = 0.1, aspect ratio of rods = 10.
1951
dislocation remained an order of magnitude below that for slip dislocations. The Orowan stress on the twinning dislocation is also far less than the back-stress calculated in the previous section. This suggests that the back-stress (and the ease with which this can be relieved by slip) is likely to be more important in determining the increase in stress required to grow the twin in the presence of particles than the Orowan stress on the twinning dislocation. 4. Effect of particle shape on tension–compression asymmetry in extrusions It has been demonstrated that particle shape has an important influence in determining the potency of strengthening of different deformation modes in magnesium. It is technologically useful to investigate whether this effect can be exploited to favourably alter the balance in strength of deformation modes, for example to reduce the mechanical asymmetry of extrusions. As already discussed, the mechanical asymmetry in a magnesium alloy extrusion with strong basal texture is expected to be largely determined by the relative ease with which f1 0 1 2g twinning occurs (controlling yield in axial compression) compared to prismatic slip (controlling yield in axial tension). The compressive strength of magnesium alloy extrusions can be as little as half the tensile strength due to the relative ease of activating f1 0 1 2g twinning compared to prismatic slip. By strengthening against f1 0 1 2g twinning more strongly than prismatic slip, a suitable precipitate distribution should be capable of reducing, or even eliminating, tension–compression asymmetry. Even an equal strengthening of deformation modes will lead to a reduction in asymmetry, since it will reduce the relative difference between the CRSSs, as discussed by Hutchinson and Barnett [25]. A simple calculation can be performed to explore this effect in more detail, and determine the relative levels of strengthening required to theoretically eliminate asymmetry. In pure magnesium single crystals, the CRSS for f1 0 1 2g twinning is approximately 4 MPa, and that for prismatic slip is approximately 40 MPa [26,27]. Applying the appropriate Schmid factors, the theoretical compressive to tensile yield stress ratio can easily be calculated for a single crystal deformed along h1 0 1 0i, assuming prismatic slip controls tensile yield and f1 0 1 2g twinning controls compressive yield. Clearly, in a magnesium alloy extrusion compared to a pure magnesium single crystal there will be additional strength contributions arising from the presence of grain boundaries, solute elements and precipitate particles. These contributions may strengthen all deformation systems equally, but more typically will strengthen twinning and slip by different amounts. Indeed, in some cases, such as the effect of solute elements on prismatic slip, softening rather than hardening may be induced [28]. Fig. 7a shows the theoretical effect that strengthening of f1 0 1 2g twinning and prismatic slip will have on the ratio of the yield stress in compression to that in tension (the
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J.D. Robson et al. / Acta Materialia 59 (2011) 1945–1956 2.5 t
50
(a)
ΔCRSS = ΔCRSS
p
(b)
ΔCRSS =2ΔCRSS t
2
ΔCRSS (twin)/ΔCRSS (prismatic)
t
σY (compression)/σy (tension)
p
ΔCRSS =3ΔCRSS
p
ΔCRSS =0.5ΔCRSS t
p
1.5
1
0.5
0 0
50
100
150
ΔCRSS prismatic (MPa)
40
30 σY(T)=σY(C)
20
10
0 −20
0
20
40
60
80
100
ΔCRSS prismatic (MPa)
Fig. 7. (a) Predicted compressive to tensile yield strength ratio as a function of the increase in CRSS for prismatic slip, showing the effect of increasing the CRSS for f1 0 1 2g twinning by 0.5, 1, 2 and 3 times the increase for prismatic slip. (b) Relative increase in CRSS for twinning compared to the increase in CRSS for prismatic slip predicted to lead a compressive to tensile yield strength ratio of 1 (elimination of asymmetry).
yield asymmetry). To produce this plot, it was assumed the prismatic system was strengthened by an amount given by the value plotted as x, and the strengthening of the f1 0 1 2g twinning system was scaled in proportion (assumed to be either 0.5, 1, 2 or 3 times greater, as shown by the four curves in Fig. 7a). This is not intended to be physically accurate; indeed, it is expected that the strengthening of these two systems will not scale in proportion. However, it serves to illustrate an important point, which is that even if twinning is strengthened less than prismatic slip (e.g. half as much, as shown in Fig. 7), asymmetry will still be reduced. Indeed, this simple calculation predicts that asymmetry will only be made worse if f1 0 1 2g twinning is strengthened by less than 1/10 of the strengthening due to prismatic slip. This number is simply determined by the original ratio of the CRSS for twin growth compared to prismatic slip, as demonstrated below. Consider the CRSS for prismatic slip is increased by Ds and the CRSS for f1 0 1 2g twinning is increased by kDs, where k is a constant. Yield will be less asymmetric if the ratio of the stress to activate f1 0 1 2g twinning compared to that for prismatic slip is increased. This condition may be written as: S t ðst þ kDsÞ S t st > S p ðsp þ DsÞ S p sp
ð9Þ
where St and st are the Schmid factor and initial CRSS for f1 0 1 2g twinning, respectively, and Sp and sp are the Schmid factor and initial CRSS for prismatic slip. Rearrangement of this equation leads to: k>
st sp
ð10Þ
Therefore, as long as the CRSS for f1 0 1 2g twinning is increased more than k times the increase in CRSS for prismatic slip, the influence of any strengthening feature in the microstructure will be to reduce asymmetry. Note that this analysis can be applied not only to consider the influence of strengthening a pure magnesium single crystal, but can also be applied to any individual strengthening mechanism (such as precipitation) to determine whether asymmetry is likely to be increased or decreased (given the appropriate values of sp and st prior to the strength increment). This analysis predicts that an equal strengthening of f1 0 1 2g twinning and prismatic slip will lead to marked reduction in asymmetry (the solid line in Fig. 7a). However, elimination of asymmetry (rY (tension) = rY (compression)) can only be achieved by strengthening against f1 0 1 2g twinning more strongly than against prismatic slip. As Fig. 7a shows, the greater the additional strengthening of f1 0 1 2g twinning relative to prismatic slip, the smaller the strength increment required to eliminate asymmetry. This last point is more clearly illustrated in Fig. 7b. This plot shows the locus of points that leads to equality of the tensile and compressive yield strengths (i.e. elimination of asymmetry). The plot demonstrates that if the strengthening of the prismatic system is low, then the relative strengthening against f1 0 1 2g twinning has to be large to eliminate asymmetry. However, as the prismatic system is further strengthened, the relative strengthening of f1 0 1 2g twinning required to eliminate asymmetry decreases (although the absolute value of strengthening needed increases). If the prismatic system is softened (i.e. by small solute additions), then the relative increase in strengthening against f1 0 1 2g twinning needed to
J.D. Robson et al. / Acta Materialia 59 (2011) 1945–1956
ΔCRSS (twin growth)/ΔCRSS (prismatic)
eliminate asymmetry falls sharply, and the absolute value also reduces. It is now possible to investigate in detail the predicted effect that the formation of precipitates with different morphologies and habits will have on the yield asymmetry. The inference from the analysis of both the back-stress and the Orowan stress for twinning is that the dominant factor that determines how particles influence twin growth is the effectiveness of slip in the twin (and mainly basal slip) in relieving the back-stress. Parent–basal plate-shaped precipitates in particular are predicted to lead to a large increase in the stress required to accommodate basal slip in the twin. When present, these precipitates will therefore enable a higher back-stress to be sustained in the twin, which is assumed to directly increase the stress required for twin growth. For this analysis we assume: (i) the Orowan stresses already calculated for slip describe the increase in CRSS expected for the prismatic and basal slip modes; (ii) the increase in CRSS for twin growth is given by the Orowan stress inhibiting the required basal slip in twin resolved on the twin plane in the twinning direction; and (iii) other slip modes (e.g. hc + ai slip) have negligible influence, which is reasonable at room temperature in the magnesium alloys considered here [12]. Although this is clearly an oversimplification, in particular of the multiple slip events that are required to accommodate the twin shear around particles, as will be shown it does provide a useful qualitative tool for determining the likely effect of a particular particle morphology and habit on yield asymmetry. Fig. 8 shows the predicted ratio of the increase in the CRSS for twin growth (as defined above) to the increase
1953
in CRSS for prismatic slip. This is referred to as the CRSS increment ratio. The initial CRSS values prior to precipitation for a typical age-hardenable magnesium alloy extrusion are approximately 18 MPa for f1 0 1 2g twinning, and 54 MPa for prismatic slip [11]. Referring to Eq. (10), this gives k = 1/3. The analysis above predicts that if the precipitates lead to a CRSS increment ratio greater than this value (k > 1/3), asymmetry will be reduced. If they lead to a CRSS increment ratio less than this value (k < 1/3), then asymmetry is expected to increase. Fig. 8 shows that basal plates lead to a maximum value in the CRSS increment ratio, implying that they are most effective in reducing the asymmetry for a given strength increment. Spherical particles are also predicted to result in an asymmetry reduction since they provide equal strengthening against slip in the parent and twin. However, c-axis rods (of aspect ratio 10) lead to a CRSS increment ratio that is less than the critical value to reduce asymmetry, i.e. the precipitation of such rods would be expected to lead to an increase in asymmetry. These calculations obviously depend on the aspect ratio chosen for the particles. If the aspect ratio of the rods is further increased, then their effect on increasing asymmetry is also predicted to strengthen. An aspect ratio of 10 is a conservative estimate of that typically reported for rod-containing magnesium alloys. For example, in magnesium alloy Z5, the reported aspect ratio for rods is in the range 5–40 [11,8]. Therefore, these predictions suggest that for a Mg–Zn alloys in which rod-shaped precipitates form parallel to the c-axis, precipitation is likely to be accompanied by an increase in extrusion asymmetry. However, in alloys in which plate-shaped precipitates form on the basal plane (e.g. AZ91), a marked reduction in asymmetry after precipitation is expected.
1.6
5. Comparison with experiment
1.4
5.1. Experimental method
1.2
To test these predictions, extrusions have been studied where it has been possible to precipitate either basal plates or c-axis rods through choice of alloy composition. Basal plates were produced by age-hardening AZ91 alloy; this leads to the formation of non-coherent equilibrium b– Mg17Al12 plates, predominantly on the basal plane [13]. c-Axis rods were produced by age-hardening Z5, which leads to a transition phase (MgZn) [11]. The experimental procedure used to produce the Z5 alloy is described in detail elsewhere [11] and the same procedure was followed for AZ91. In brief, this involved giving the cast billets an 8 h solution treatment at 420 °C, followed by extrusion at 400 °C with a ram speed of 1 mm s1 and an extrusion ratio 30. This produced dynamically recrystallized microstructures with a strong basal texture (texture strength at least 5.3 times random) and a grain size of 18 lm in the case of AZ91 and 30 lm in Z5. Selected samples from the Z5 extrusion were aged to peak hardness prior to testing by a heat treatment of 8 days at 150 °C.
1 plates rods
0.8
spheres 0.6 asymmetry reduced
0.4
asymmetry increased
0.2 0
0
2
4
6
8
10
Particle dimension (μm) Fig. 8. A calculation of the ratio of the increase in CRSS for f1 0 1 2g twin growth compared to the increase in CRSS for prismatic slip for plate, rod and spherical particles. A value of this ratio <1/3 is predicted to lead to an increase in asymmetry on precipitation, a value >1/3 is predicted to lead to a reduction in asymmetry. Particle volume fraction = 5%, aspect ratio of plates = 0.1, aspect ratio of rods = 10.
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Ageing treatments for AZ91 followed those used by Braszczyn´ska-Malik [29]. Ageing to peak hardness was performed at 150 °C for 18 h. In this case, the microstructure is dominated by discontinuous precipitates, as shown in Ref. [29]. The theory developed above cannot be directly applied to the case of discontinuous precipitation, and so ageing was also performed for 8 h at 350°, which gave fully continuous precipitation (see [29]). Precipitate size and spacing were measured using transmission electron microscopy for Z5 (described elsewhere [11]) and field emission gun scanning electron microscopy for AZ91, where the precipitates are larger. Specimens from both alloys in the aged and as-extruded condition were tested in compression and tension under identical conditions following the procedure detailed elsewhere [11]. 5.2. Experimental results The stress–strain curves for AZ91 and Z5 in tension and compression in the peak-aged condition are shown in Fig. 9. From these, the tension–compression asymmetry values (compressive 0.2% offset stress/tensile 0.2% offset stress) have been calculated, and are reported in Table 1. It can be seen that in the as-extruded case, the asymmetry values for both alloys are quite similar, and the compressive strength is between 0.6 and 0.7 that of the tensile strength. On ageing, the asymmetry of the AZ91 extrusion is reduced greatly, and examination of the stress–strain curves reveals that this in primarily due to an increase in the compressive yield strength. This effect is most marked in the peak-aged condition where discontinuous precipitates dominate, but is also the reason for the reduced anisotropy after ageing at 350 °C, where only continuous precipitation occurs. In Z5, however, the opposite is observed, and ageing leads to an increase in asymmetry. Examination of the
(a)
stress–strain curves demonstrates that this is primarily due to a strong increase in the tensile yield strength, with the compressive yield strength being increased less markedly. 6. Discussion The observed change in asymmetry on ageing AZ91 and Z5 is qualitatively in agreement with what we have predicted using the Orowan models. The basal plates that precipitate is AZ91 lead to a reduction in asymmetry, whereas the c-axis rods that form in Z5 lead to increased asymmetry. For Z5, the mean rod diameter measured after ageing at 150 °C for 192 h was 20 nm, with a length of 120 nm and volume fraction of 2.3% [11]. For AZ91 after ageing at 350 °C for 8 h (continuous precipitation) the plates had a mean diameter of 2.8 lm and thickness of 0.4 lm with a volume fraction of 2.8%. For these particular particle fractions and sizes, the calculated Orowan strengthening of the various deformation systems is presented in Table 2. The key differences between the predicted strengthening in the two materials are: (i) the greater Orowan hardening in Z5, particularly for the basal and prismatic systems, which is consistent with the strong measured increase in tensile strength of this material on ageing and (ii) the relatively large hardening for basal slip in AZ91 twins compared to that of the other systems. The calculated CRSS increment ratios for twin growth/prismatic slip as defined earlier are also shown. Recall that for extrusions a value of this ratio greater than 1/3 (as shown here for AZ91) was predicted to lead to a reduction in asymmetry, whereas a value less than 1/3 (as shown here for Z5) was predicted to lead to an increase in asymmetry. This matches the observations and supports the idea that the CRSS increment ratio gives a useful indication of whether any
(b)
Fig. 9. Mechanical properties of the AZ91 and Z5 alloys studied here. (a) AZ91 in the peak-aged (18 h at 150 °C) and as-extruded condition. (b) Z5 in the peak-aged and as-extruded condition.
J.D. Robson et al. / Acta Materialia 59 (2011) 1945–1956
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Table 2 Calculated Orowan stress (MPa) for basal and prismatic slip, the twinning partial dislocation, and basal slip in twinned material for the measured particle parameters in AZ91 aged at 350 °C for 8 h and peak-aged Z5 along with calculation of the CRSS increment ratio (defined in text).
AZ91 (aged 350 °C) Z5
Basal
Prismatic
Twin
Basal (twin)
DCRSS ratio
0.7 43
1.4 34
0.3 7
2.5 15
0.89 0.22
strengthening effect is likely to lead to increased or reduced asymmetry. Although the analysis presented here is undoubtedly a great oversimplification, and does not consider in detail the mechanism by which twins and particles interact, it does appear to provide important insights into the influence of particle shape that are consistent with experimental measurements. In particular, it is suggested that whilst basal plates are not very effective strengtheners since they do not effectively block basal slip, they can be useful in increasing the stress required for twin growth (i.e. strengthening against twinning). It is clear that there are many unresolved issues; in particular the mechanism by which the twinning dislocation negotiates the particles is less well understood than the interaction of slip dislocations with precipitates. Furthermore, direct suppression of twin nucleation by particles is a possibility that is not considered in the present work. Finally, better quantitative prediction of strengthening will only be possible with a more realistic treatment of the distribution of particle spacings and a better understanding of how dislocations bypass high-aspectratio particles. 7. Conclusion The effect of precipitate shape and habit on the strength of magnesium alloys has been considered by calculating the Orowan stress for the deformation systems of most importance at room temperature. These calculations have been used to assess the likely effect of different precipitate types on the asymmetry of strongly textured extrusions. The predictions of the model have been tested against mechanical property measurements made for alloys with two distinctly different precipitate shapes and habits. The following conclusions can be drawn from this work: For an identical volume fraction and particle size, c-axis rods are more effective in hardening against basal slip and prismatic slip compared to spheres and basal plates. Basal plates are particularly poor at hardening against basal slip due to the low fraction of precipitates that will be cut by a given basal plane. When a growing twin engulfs a particle without shearing it, a back-stress is necessarily generated as a result of the need to mitigate the strain discontinuity at the particle– matrix interface. This back-stress is predicted to reach unrealistically large values in the twin without plastic accommodation around the particles. However, the particles will also inhibit this plastic accommodation by acting as barriers to slip in the twin.
Due to the lattice rotation in the twin, basal plates in the parent become oriented perpendicular to the basal plane in the twin and become very effective barriers to preventing basal slip in twinned material. c-Axis rods in the parent provide poor obstacles to prevent slip in the twin. It is postulated that a high resistance to basal slip in the twin will increase the stress required for twin growth by allowing a higher back-stress to be sustained in the twin. The Orowan stress on the twinning partial dislocation is an order of magnitude less than that for slip due to the small Burgers vector of the twinning partial. It is therefore likely that this stress is never sufficient to prevent twin growth once sufficient stress has been applied to nucleate twins. It is suggested that the ratio of the increase in CRSS for twin growth to that for prismatic slip (named the CRSS increment ratio) is the key parameter determining whether a strengthening effect will increase or reduce asymmetry of strongly textured magnesium alloy extrusions. Below a critical value of this ratio, a strengthening effect will increase asymmetry; otherwise it will reduce it. Using the CRSS ratio above, it has been predicted that basal plates will lead to reduced asymmetry, whereas c-axis rods can increase asymmetry. The predictions of the model have been tested against mechanical data for alloys Z5 and AZ91 aged to peak strength and deformed in tension and compression. The presence of c-axis rods in Z5 leads to an increase in asymmetry, whereas the precipitation of basal plates in AZ91 reduced asymmetry. This is in agreement with predictions made on the basis of the CRSS increment ratio as defined above.
Acknowledgements J.D.R. thanks the UK Royal Academy of Engineering for funding this collaboration between the University of Manchester and Deakin University through the Global Research Award Scheme. Dr J. Geng of Monash University is thanked for provision of Fig. 3. References [1] Polmear IJ. Light alloys. 3rd ed. London: Arnold; 1995. [2] Nie JF. Scripta Mater 2003;48:1009–15. [3] Nie JF, Muddle BC, Polmear IJ. Mater Sci Forum 1996;217– 222:1257–62. [4] Bettles C, Gibson M. JOM 2007;57:46–9. [5] Beck A. The technology of magnesium and its alloys. F.A. Hughes & Co.; 1941.
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[6] Jain J, Poole WJ, Sinclair CW, Gharghouri MA. Scripta Mater 2010;62:301–4. [7] Kleiner S, Uggowitzer PJ. In: Proceedings of the 1st international light metal technology conference; 2003. p. 327–32. [8] Clark JB. Acta Metall 1965;13:1281–9. [9] Clark JB. Acta Metall 1968;16:141–51. [10] Gharghouri MA, Weatherly GC, Embury JD. Philos Mag A 1998;78:1137–49. [11] Stanford N, Barnett MR. Mater Sci Eng A 2009;516:226–34. [12] Agnew S, Brown DW, Tome´ CN. Acta Mater 2006;54:4841–52. [13] Celotto S. Acta Mater 2000;48:1775–87. [14] Embury JD, Lloyd DJ. Aluminium alloys: contemporary research and applications. In: Vasudevan AK, Doherty RD, editors. Treatise on materials science and technology, vol. 31. New York: Academic Press; 1989. p. 579–601. [15] Fullman RL. Trans AIME 1953;197:447–50. [16] Robson JD, Stanford N, Barnett MR. Scripta Mater 2010;63:823–6.
[17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
Brown LM, Stobbs WM. Philos Mag 1971;23:1185–200. Brown LM, Stobbs WM. Philos Mag 1971;23:1201–32. Brown LM, Clarke DR. Acta Metall 1975;23:821–30. Bate P, Roberts WT, Wilson DV. Acta Metall 1981;29:1797–814. Avedesian M, Baker H, editors. Magnesium and magnesium alloys. Materials Park (OH): ASM; 1998. Wang J, Beyerlein IJ, Tome´ CN. Scripta Mater 2010. doi:10.1016/ j.scriptamat.2010.01.047. Serra A, Bacon DJ. Philos Mag A 1996;73:333–43. Fullman RL. Trans AIME 1953;197:1267–8. Hutchinson WB, Barnett MR. Scripta Mater 2010;63:737–40. Kelley EW, Hosford WF. Trans AIME 1968;242:5–13. Wonseiwicz BC, Backofen WA. Trans Am Inst Min Eng 1967;239:1422. Akhtar A, Teghtsoonian E. Acta Metall 1969;17:1351–6. Braszczyn´ska-Malik KN. J Alloys Compd 2009;477:870–6.
Acta Materialia 59 (2011) 1945–1956 www.elsevier.com/locate/actamat
Effect of precipitate shape on slip and twinning in magnesium alloys J.D. Robson a,⇑, N. Stanford b, M.R. Barnett b b
a Manchester Materials Science Centre, Grosvenor Street, Manchester M1 7HS, UK Centre for Material and Fibre Innovation, Deakin University, Pigdons Road, Geelong 3217, Australia
Received 15 September 2010; received in revised form 24 November 2010; accepted 25 November 2010 Available online 31 December 2010
Abstract The predicted strengthening effect of precipitates of different shape and habit on the basal, prismatic and f1 0 1 2g twinning deformation systems in magnesium has been calculated. In parent material, rod precipitates parallel to the c-axis are predicted to be more effective than plates parallel to the basal plane in hardening the basal and prismatic slip systems. However, in twinned material, nonsheared basal plates are highly effective in inhibiting the basal slip necessary to relieve incompatibility stresses. The predictions suggest basal plates will reduce asymmetry in strongly textured extrusions by preferentially hardening against twin growth compared to prismatic slip, whereas c-axis rods can have the opposite effect. The predictions have been compared with the measured asymmetry for two magnesium alloys that form either c-axis rods (Z5) or basal plates (AZ91). In agreement with the model, it is shown that precipitation in Z5 leads to an increase in asymmetry, whereas in AZ91 precipitation reduces asymmetry. These results suggest that designed precipitation may provide a useful tool for reducing asymmetry in wrought magnesium alloys. Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Magnesium alloys; Twinning; Extrusion; Modelling
1. Introduction Precipitate particles are exploited for strengthening in many commercially important magnesium alloys [1]. The precipitates that typically form in magnesium alloys take the form of rods or plates lying on particular crystallographic planes, with the precipitate morphology and habit depending on the alloy system and thermal treatment [2]. It is known that different precipitate types lead to markedly different strengthening responses; for example, in randomly textured as-cast material the basal plates formed in Mg–Al–Zn (AZ) alloys generally give poor strengthening compared to the prismatic plates that form in Mg–Y–RE (rare earth) alloys, such as WE54. Previous work has shown that this difference in strengthening may be at least partly attributed to the different Orowan hardening of basal slip (the most easily activated deformation mode in ⇑ Corresponding author.
E-mail addresses: [email protected], manchester.ac.uk (J.D. Robson).
joseph.robson@
magnesium) due to the different precipitate shapes and habits [2,3]. In addition to suggesting desirable precipitate characteristics for maximum strength, the prediction that different precipitate shapes and habits lead to different strengthening effects might be exploited in another important way. One key problem with wrought magnesium alloys are the high levels of mechanical asymmetry that are often observed [1,4]. This is the result of the strong textures typically produced after thermomechanical processing coupled with the unidirectional nature of twinning [5]. Since particles of different shape and habit will strengthen the different deformation modes to different extents, it may be possible to exploit this to reduce anisotropy and asymmetry. In particular, if the deformation modes with a low critical resolved shear stress (CRSS), i.e. basal slip and f1 0 1 2g c-axis tension twinning, can be strengthened more strongly than those which already have a high CRSS (e.g. prismatic slip), then more isotropic properties will result and a reduction in asymmetry will be possible. There is evidence from recent studies of both cast and extruded AZ alloys that precipita-
1359-6454/$36.00 Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2010.11.060
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tion leads to a marked reduction in asymmetry due to this effect, in particular the suppression of f1 0 1 2g twinning [6,7]. In this paper, the effect of particle shape in strengthening the different deformation systems in magnesium is predicted, and the likely consequences on mechanical asymmetry are discussed. In the case of slip, strengthening by the Orowan mechanism has been well studied, and although the effect of particle shape and habit is less well understood, previous work by Nie and Muddle [2,3] has shown how the situation of rationally oriented rod- and plate-shaped precipitates may be treated. In the case of twinning, the influence of particles remains much less clear. Several studies have shown a strong influence of precipitates on twinning in magnesium alloys [8–11]. In some alloys an increased number of smaller twins is commonly observed when particles are present [11]. This paper discusses how particles may influence twin growth, and the effect of particle shape and habit. 2. Hardening against slip Nie [2] has previously presented an analysis of the Orowan strengthening effect of different precipitate shapes and habits in magnesium alloys for basal slip. In this work, a similar approach is followed and extended to also consider prismatic slip and twinning. Attention is paid to prismatic slip because it is considered to be a critical deformation mode in controlling the yield of strongly textured magnesium alloy extrusions when subject to tensile stress along the extrusion axis [12]. This is important because the ease of activating this mode compared to f1 0 1 2g twinning (which controls yield in axial compression) is important in determining the tension/compression asymmetry in extrusions.
Three particle habit/morphology combinations are considered in this study. These are basal plates, c-axis rods and randomly distributed spheres. Fig. 1 illustrates the first two of these schematically and also shows the particle cross-section when cut by the basal and prismatic slip planes. These morphologies were chosen because basal plates form in precipitation-strengthened AZ alloys (e.g. Mg–9 wt.% Al– 1 wt.% Zn, AZ91 [13]) and c-axis rods form in Mg–Zn alloys (e.g. Mg–5 wt.% Zn, Z5 [8,11]). As well as forming precipitates of different shape and habit, the AZ and ZK (Mg–Zn–Zr) classes of alloy are of great practical importance. The precipitates formed in the magnesium alloys of interest in this work are too large and insufficiently coherent to be cut by dislocations. The influence of precipitate shape and habit on strength is therefore assumed to depend on the Orowan stress required to bow dislocations around the particles. The general form of the Orowan equation used in this work is the same as used by Nie and Muddle [2,3], and can be written [14]: Gb 1 Dp pffiffiffiffiffiffiffiffiffiffiffi Ds ¼ log ð1Þ r0 2p 1 m k where Ds is the increase in CRSS due to precipitate strengthening, G is the shear modulus, b the Burgers vector of the gliding dislocations, m the Poisson ratio, k the effective interparticle spacing on the slip plane, Dp the mean planar diameter of the particles on the slip plane and r0 the dislocation core radius. 2.1. Basal slip Calculations for basal slip have been previously presented by Nie [2] (Section 2.3), and this analysis is not repeated here. A minor modification has been made in
(a)
(b)
Fig. 1. A schematic showing the (a) basal plate and (b) c-axis rod precipitate morphologies considered in the present study, and the cross-sections produced when these particles are cut by basal and prismatic slip planes.
J.D. Robson et al. / Acta Materialia 59 (2011) 1945–1956
1 p k ¼ pffiffiffiffiffiffi d t NA 4
the present work; Nie assumed that the precipitates have a triangular arrangement when cut by the slip plane. However, since the analysis, which is based on the work of Fullman [15], considers the probability of cutting randomly distributed rods, plates or spheres, there appears no reason to assume a triangular arrangement of particles on the slip plane. Therefore, in this work, a regular array of precipitates on the slip plane is assumed, with the mean centerpffiffiffiffiffiffi to-center distance simply given by 1= N A , where NA is the number of particles per unit area on the slip plane. This simplifies the mathematics of the analysis and the use of a more complex geometry is not considered to be justified given the highly approximate nature of a treatment based on average spacings.
The same approach can be applied to prismatic slip by considering the new geometries involved (Fig. 1). The analysis will be restricted to prismatic edge dislocations, since it is assumed that prismatic screw dislocations will be able to cross-slip onto another prism plane to avoid obstacles in their path. Note that for randomly distributed spheres, the effective particle spacing on the prismatic and basal planes is identical, since the probability of cutting a spherical particle and the planar diameter is the same for both planes. Therefore, it is not necessary to reproduce the analysis for spherical particles. The probability of a single basal plate in a unit volume being cut by a prismatic plane is equal to the plate diameter dt (for consistency, we follow the nomenclature of Nie [2]). Therefore, the number of basal plates cut per unit area of slip plane(NA) is equal to dtNV, where NV is the number of plates per unit volume. Random intersection of the plates will produce a rectangular cross-section on the slip plane of mean length pdt/4 and width tt (the plate thickness). Assuming a regular array, the effective spacing can be written as:
2
2
10
10
(a)
1
Δτ Orowan (MPa)
ð2Þ
The analysis for rods is similar, except the length of the rectangular cross-section produced by random intersection of the rods is lt (rod length), and this replaces the final term in Eq. (2) to calculate the spacing for rods. Note that basal plates and c-axis rods are variants of the same cylindrical shape with different aspect ratios. Fig. 2 shows the calculated Orowan stress for (a) basal and (b) prismatic slip for basal plates, c-axis rods and spherical particles. This calculation was performed assuming a precipitate volume fraction of 5%, with the aspect ratio of plates defined as tt/dt = 0.1 and the aspect ratio of rods lt/dt = 10. These represent typical values for an age-hardening magnesium alloy and are used throughout the paper to highlight the important trends. Application to some specific alloys is discussed later. In this, and subsequent plots, the x-axis gives the effective particle dimension, which is defined as V p1=3 , where Vp is the particle volume. This means that a given x-value corresponds to the same particle volume for plates, rods and spheres. It is plotted this way to facilitate comparison between particles of different shape at a constant particle volume. The calculations confirm that basal plates are poor strengtheners against basal slip. This is because the probability of such a plate being intersected by a given basal plane is very low, leading to a large interparticle spacing on the plane. c-Axis rods are predicted to be more effective strengtheners for both basal and prismatic slip compared to basal plates or spheres. The ability of c-axis rods to strengthen strongly against prismatic slip is particularly important. This is because prismatic slip usually controls the tensile yield strength of extrusions [12]. Changing aspect ratio was found to follow expected behavior; as the aspect ratio of the rods is increased and they become more “rod like”, their predicted strengthening effect increases. Similarly, as the basal plates become thinner but broader (more “plate like”) their predicted strengthening decreases.
2.2. Prismatic slip
(b)
plates rods spheres
1
10
10
0
0
10
10
−1
10
1947
−1
0
2
Particle dimension (μm)
4
10
0
2
4
Particle dimension (μm)
Fig. 2. A calculation of the Orowan stress required to bypass plate, rod and spherical precipitates for (a) basal slip and (b) prismatic slip. Particle volume fraction = 5%, aspect ratio of plates = 0.1, aspect ratio of rods = 10.
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J.D. Robson et al. / Acta Materialia 59 (2011) 1945–1956
3. Hardening against twinning Experimental studies have clearly shown that particles can have a strong effect on twinning in magnesium [8– 11]. However, in contrast to slip, the theory to determine what hardening effect particles have against twinning has not yet received any attention. Nevertheless, since twinning is such an important deformation mode in magnesium, an understanding of the likely effect of particles on strength, anisotropy and asymmetry must necessarily include a consideration of the influence of particles on twinning. There is now sufficient experimental evidence of the interaction of twinning and particles in magnesium alloys to enable a qualitative, if not yet fully quantitative, analysis of how particle shape is likely to influence hardening against twinning. Twins can act with particles in a number of ways, all of which have been reported in experimental studies of magnesium alloys depending on particle characteristics [8–11]. Firstly, particles of the size commonly obtained in magnesium alloys after ageing never appear to be effective in completely suppressing twin nucleation. Indeed, in some alloy systems (e.g. Z5) the presence of particles leads to an increase in the number of twins [11], and it has been argued that this is due to an increase in stress required for twin growth [16]. Therefore, in this work, we focus on the effect particles have on twin growth. Furthermore, the analysis is limited to f1 0 1 2g (c-axis tension) twinning since this is by far the dominant mode in magnesium and is also the critical mode that controls the compressive yield strength of extrusions. Particles have been observed (i) to arrest twin growth completely, often with twinning continuing by nucleation of a new twin in the matrix on the far side of the particle [10]; (ii) to become fully engulfed by a growing twin being rotated but not sheared [11,10]; and (iii) to become fully engulfed and sheared by the twin [8]. The type of behavior observed appears to depend on precipitate size and shape, with the largest precipitates showing type (i) behavior, intermediate-sized precipitates showing type (ii) behavior, and one study [8] reporting small rod-shaped precipitates
showing type (iii) behaviour (although only type (ii) behaviour was observed in another study of the same system [11], so it remains unclear whether type (iii) behaviour ever occurs in practice). For the size of particles that achieves a maximum in the hardening response during ageing, type (ii) behaviour has been reported for both basal plate- and rod-shaped precipitates. Fig. 3 shows example images from (a) AZ91 and (b) Z5 in the peak-aged state, demonstrating what happens to the particles when they are engulfed by a twin. For both basal plates (AZ91) and c-axis rods (Z5) it can be seen that when inside the twin, a small rotation of the particle occurs (’4°). This is consistent with the particles undergoing rigid body rotation (RBR), but not shearing, as demonstrated by Gharghouri et al. [10]. It is also interesting to note an apparent deflection of the twin habit plane (indicated by the dashed black line) close to the particles, demonstrating that the particles do inhibit twin growth. In addition, and consistent with previous work, evidence for a large number of basal dislocation loops in the twin can be observed. These dislocations are evidence of the plastic relaxation of the incompatibility stresses between particles and twinned matrix, as discussed later [11,10]. A forthcoming publication will discuss these observations in more detail, but for the purposes of the present work the key point is that the precipitates are not sheared in the twin, whereas the matrix is, and this creates a strain discontinuity that has to be accommodated. 3.1. Back-stress The observation that twins can engulf particles without shearing the particles themselves leads to several important consequences that we now explore. Although twinning is a distinctly different deformation mode from slip, understanding how a twin interacts with a non-shearing particles has much in common with understanding the role of nondeforming particles on slip, a topic that has received much experimental and theoretical attention (e.g. [17–19]). The material within a f1 0 1 2g twin is deformed by simple shear with a simple shear strain tw = 0.13 [10]. This
Fig. 3. Interaction of a f1 0 1 2g twin and particles in peak-aged material (a) AZ91, showing a basal plate interacting with the twin boundary. A small rotation of the particle in the twin (T) can be seen, as evidenced by the deviation between the particle interface in the twin and the superimposed white line which shows the extrapolation of the interface in the untwinned matrix (M). (b) Z5, showing a c-axis rod-shaped precipitate crossing the twin boundary and evidence of a similar small rotation of the particle in the twin (T). The twin boundary has been highlighted with a dashed line for clarity.
J.D. Robson et al. / Acta Materialia 59 (2011) 1945–1956
rij ¼ 2lV f cijkl pkl
ð3Þ
where l is the shear modulus of the matrix, Vf is the particle volume fraction, cijkl the accommodation tensor, and pkl the plastic strain discontinuity tensor. For slip deformation, the case of different particle shapes has been treated by Brown and Clarke [19] and the same approach is followed here. Unlike in the case of slip, where the plastic strain discontinuity increases progressively as the applied strain increases, in the case of twinning the full plastic strain discontinuity is imposed immediately on twinning and determined by the fixed twinning shear. To calculate this plastic strain discontinuity the simple shear of twinning is partitioned into its pure shear and RBR components. It is the difference in the pure shear between the particle and matrix that must be accommodated to prevent a strain discontinuity [20]. The required transformation is given by Gharghouri et al. [10]. To calculate the back-stress it is necessary also to rotate the axes of the stress tensor to those that correspond to the particle coordinate system, and Gharghouri et al. describe how this is done, with the same method used here [10]. The appropriate values for the accommodation factor (cijkl) can then readily be calculated from the tabulated values for Sijkl reported in Brown and Clarke (Table 1 [19]) for fibres (used to represent c-axis rods), discs (used to represent basal plates) and spheres using a Poisson ratio value of 0.35 [21]. The effect of the back-stress on basal slip in the twin is likely to be particularly important, as discussed later. Therefore, to enable a simple comparison to be made between the effect of particles of different morphology on Table 1 Measured compressive and tensile 0.2% offset stress for (rC0:2 and rT0:2 , respectively) and asymmetry ratio as-extruded and after ageing. Alloy AZ91 AZ91 AZ91 Z5 Z5
Ageing treatment None 18 h at 150 °C 8 h at 350 °C None 192 h at 150 °C
rC0:2 (MPa) 160 249 174 91 153
rT0:2 (MPa) 232 263 233 150 286
rC0:2 =rT0:2 0.69 0.95 0.75 0.61 0.53
the influence of back-stress on basal slip, the back-stress tensor was resolved to calculate the shear stress component acting on the basal plane in the twin. This calculation gives: rb ¼ lV f ð0:10Þ basal plates rb ¼ lV f ð0:05Þ c-axis rods
ð4Þ ð5Þ
rb ¼ lV f ð0:07Þ spheres
ð6Þ
The calculated resolved back-stress values for basal plates, c-axis rods and spheres (l = 17 GPa [21]) are shown in Fig. 4 as a function of volume fraction. Looking first at the relative back-stress for the different particle shapes, it can be seen that basal plates are predicted to lead to a much higher back-stress than c-axis rods or spheres. The magnitude of the predicted back-stresses become very large for the precipitate fractions typically formed in age-hardening magnesium alloys. For example, in the Mg–7.7 wt.% Al alloy studied by Gharghouri et al. [10], the volume fraction of basal plates was reported as 10%. If the strain incompatibility around these particles was accommodated entirely elastically, the back-stress calculation predicts this precipitate fraction would generate stress levels approaching 170 MPa in the twin, which are far in excess of the CRSS for basal slip [12]. This clearly demonstrates that plastic deformation by slip must play an important role in moderating the strain discontinuity around particles in the twin, thereby reducing the back-stress. Indeed evidence of the large number of dislocations that produce this deformation are commonly observed in the twin [10]. This calculation therefore shows that for the precipitate fractions of practical interest in age-hardening magnesium alloys (typically 3–15%), plastic deformation in the twin is required to enable further twin growth without particle shear. Note that this requirement for slip in the twin to relieve the back-stress caused by particles is additional to the slip processes in the twin and matrix that are required 350 plates rods
300
Resolved backstress (MPa)
simple shear can be divided into a RBR and pure shear component. When the twinned region contains a particle which remains unsheared, application of the twinning shear alone will result in a plastic strain discontinuity at the particle–matrix interface. To moderate this strain discontinuity to produce a continuous strain field, elastic and/or plastic deformation must occur. The case where accommodation is entirely elastic has been treated for slip by Brown and Stobbs [17,18]. They demonstrate that the requirement for strain continuity results in the generation of a uniform mean stress in the matrix that opposes the stress causing deformation (and is therefore commonly called the back-stress). The back-stress depends on the particle volume fraction, shape and orientation (but not size or spacing). The back-stress (rij) is given by [20]:
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spheres 250 200 150 100 50 0
0
0.05
0.1
0.15
0.2
Volume fraction precipitate Fig. 4. A calculation of the back-stress (resolved shear stress on the basal plane) produced by accommodation of non-deforming particles of different shape in a f1 0 1 2g twin, assuming accommodation occurs entirely elastically.
J.D. Robson et al. / Acta Materialia 59 (2011) 1945–1956
to accommodate the twinning shear regardless of the presence of precipitates. Plastic deformation will itself be inhibited by the presence of particles. Since the matrix is reoriented by twinning (in particular, the basal plane is rotated by 86°), the effectiveness of particles in strengthening against slip in the twin will be different to that in the parent material. The level of back-stress due to the precipitates that can be sustained in the twin will therefore be controlled by the level of slip inhibition. If the particles in the twin are highly effective in inhibiting slip, then it is to be expected that less plastic relaxation can occur, and hence a higher back-stress is sustained. This back-stress acts to oppose the stress applied to grow the twin and hence would be expected to increase the stress required for continued twin growth. 3.2. Basal slip in the twin It is likely that basal slip is responsible for the majority of the plastic deformation necessary to moderate the strain incompatibility between non-shearing particles and the sheared matrix in the twin. This is because the CRSS for basal slip is far less than that for other slip modes, and furthermore the basal planes in the twin are well oriented for relief of the back-stress by basal slip (the angle between the direction of maximum back-stress, which is in the twinning direction, and the basal plane in the twin is 43.2°). Therefore, it is reasonable to consider that it is suppression of basal slip in the twin that will have the greatest effect on the ability to relieve the back-stress by slip. The Orowan calculation for slip can now be repeated for basal slip in the twin, using the assumption that the particles do not deform but do undergo the observed rigid body rotation (which for full rigid body rotation is 3.7° about h1 1 2 0i [10]). The effect of the reorientation of the lattice by twinning and the rigid body rotation is that rods that were parallel to the c-axis in the parent become perpendicular to the c-axis in the twin, and lie in the basal plane. Plates that were perpendicular to the c-axis in the parent (parent–basal) become parallel to the c-axis in the twin (and thus have a “prismatic plate”-type orientation [2]). This profoundly influences the Orowan hardening contribution of these precipitate morphologies, as shown in Fig. 5. In particular, parent–basal plates that are poor strengtheners against slip in the parent become highly effective strengtheners in the twin because they are in their best possible orientation to inhibit slip. For particles less than 1 lm in size, the Orowan stress required to bypass parent–basal plates in the twin is predicted to be over an order of magnitude greater that in the parent. Meanwhile, rods oriented parallel to the c-axis in the parent (parent–c-axis) become far less effective strengtheners in the twin because lying in the basal plane in the twin means they are poorly oriented to inhibit basal slip (with dislocations able to bow in the relatively large gap perpendicular to the long axis of the rods).
2
10
plate (parent) rod (parent) sphere plate (twin)
Δτ Orowan (MPa)
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1
rod (twin)
10
0
10
−1
10
0
1
2
3
4
Particle dimension (μm) Fig. 5. A calculation of the Orowan stress inhibiting basal slip for plate, rod and spherical shaped precipitates in both parent and twinning material. Particle volume fraction = 5%, aspect ratio of plates = 0.1, aspect ratio of rods = 10.
This analysis has important practical implications. It has been demonstrated that compared to spherical or c-axis rod-shaped particles, parent–basal plates lead to (i) the highest incompatibility stresses between non-shearing particles and the twinned matrix and (ii) the maximum inhibition of the slip necessary to accommodate the strain incompatibility in the twin. 3.3. Orowan stress on the twinning dislocation In addition to the back-stress resulting from the strain incompatibility between the sheared matrix and nonsheared precipitates, there will also be an inhibition of twin growth due to the Orowan stress required to bow the twinning dislocation around the precipitates. The twinning dislocations are partial dislocations characterized by a step in the twin interface and a small Burgers vector. Twin growth involves the nucleation and glide of these partial dislocations across the twin plane. For the specific case of a ð1 0 1 2Þ twin in magnesium, the twinning dislocation has a Burgers vector btw ¼ 0:0694½1 0 1 1 and the twin step has a height equal to 2d ð1012Þ , where d ð1012Þ is the spacing of the ð1 0 1 2Þ planes [22,23]. These characteristics are required to shear atoms in the plane a distance 2d ð1012Þ into their correct position, with atoms in the intermediate plane having to shuffle to restore the perfect hexagonal closepacked structure [23]. Assuming the passage of the twinning dislocation leaves the particle unsheared, then it is necessary for the twinning dislocation to bow around the particle. The mechanism by which this bowing occurs is not yet clear, and the exact form of the Orowan equation that describes the extra stress required to perform bowing of the twinning partial is not yet known. Nevertheless, as for slip dislocations, an inverse relationship to the particle spacing is expected, and as a
J.D. Robson et al. / Acta Materialia 59 (2011) 1945–1956
first approximation it seems reasonable to apply the standard Orowan law (1) to calculate the relative magnitude of the Orowan stress required to bow precipitates of different shape. The same approach is used to calculate the effective particle spacing on the twin plane as previously described for slip. The angle between the twin plane normal and the c-axis is / = 43.2°. The probability of the twin plane cutting a single plate and a single rod in a unit volume are therefore given by Eqs. (7) and (8), respectively [24]: d t sin / þ tt cos / basal plates d t sin / þ lt cos / c-axis rods
ð7Þ ð8Þ
A basal plate cut by the twin plane will have a cross-section with a rectangular shape with mean length equal to pdt/4. A c-axis rod cut by the twin plane will have an elliptical cross-section with major axis length dt/cos /. With this information, the effective precipitate spacing on the twin plane may be calculated following an identical procedure as for slip, and Eq. (1) may be applied to predict the Orowan stress, using the appropriate values of k and b = btw. The results of this calculation are shown in Fig. 6. The first thing to note is that for all particle shapes the maximum Orowan stress on the twinning dislocation is small, and is an order of magnitude less than that predicted on the slip dislocations. This is principally due to the very small Burgers vector of the twinning partial dislocation. Plates and rods are predicted to be more effective than spheres in inhibiting the twinning partial dislocation. The difference between plates and rods is small for a volume fraction of 5% (shown here). It was found that as volume fraction increased, plates were predicted to be increasingly effective obstacles, and for a volume fraction of 15% the Orowan stress for plates was up to twice that for c-axis rods. Nevertheless, even for this very high volume fraction, the magnitude of the Orowan stress for the twinning partial
1
Orowan hardening for twin (MPa)
10
basal plates c−axis rods spheres
0
10
−1
10
0
1
2
3
4
Particle dimension (μm) Fig. 6. A calculation of the Orowan stress inhibiting motion of the f1 0 1 2g twinning partial dislocation for plate, rod and spherical particles. Particle volume fraction = 5%, aspect ratio of plates = 0.1, aspect ratio of rods = 10.
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dislocation remained an order of magnitude below that for slip dislocations. The Orowan stress on the twinning dislocation is also far less than the back-stress calculated in the previous section. This suggests that the back-stress (and the ease with which this can be relieved by slip) is likely to be more important in determining the increase in stress required to grow the twin in the presence of particles than the Orowan stress on the twinning dislocation. 4. Effect of particle shape on tension–compression asymmetry in extrusions It has been demonstrated that particle shape has an important influence in determining the potency of strengthening of different deformation modes in magnesium. It is technologically useful to investigate whether this effect can be exploited to favourably alter the balance in strength of deformation modes, for example to reduce the mechanical asymmetry of extrusions. As already discussed, the mechanical asymmetry in a magnesium alloy extrusion with strong basal texture is expected to be largely determined by the relative ease with which f1 0 1 2g twinning occurs (controlling yield in axial compression) compared to prismatic slip (controlling yield in axial tension). The compressive strength of magnesium alloy extrusions can be as little as half the tensile strength due to the relative ease of activating f1 0 1 2g twinning compared to prismatic slip. By strengthening against f1 0 1 2g twinning more strongly than prismatic slip, a suitable precipitate distribution should be capable of reducing, or even eliminating, tension–compression asymmetry. Even an equal strengthening of deformation modes will lead to a reduction in asymmetry, since it will reduce the relative difference between the CRSSs, as discussed by Hutchinson and Barnett [25]. A simple calculation can be performed to explore this effect in more detail, and determine the relative levels of strengthening required to theoretically eliminate asymmetry. In pure magnesium single crystals, the CRSS for f1 0 1 2g twinning is approximately 4 MPa, and that for prismatic slip is approximately 40 MPa [26,27]. Applying the appropriate Schmid factors, the theoretical compressive to tensile yield stress ratio can easily be calculated for a single crystal deformed along h1 0 1 0i, assuming prismatic slip controls tensile yield and f1 0 1 2g twinning controls compressive yield. Clearly, in a magnesium alloy extrusion compared to a pure magnesium single crystal there will be additional strength contributions arising from the presence of grain boundaries, solute elements and precipitate particles. These contributions may strengthen all deformation systems equally, but more typically will strengthen twinning and slip by different amounts. Indeed, in some cases, such as the effect of solute elements on prismatic slip, softening rather than hardening may be induced [28]. Fig. 7a shows the theoretical effect that strengthening of f1 0 1 2g twinning and prismatic slip will have on the ratio of the yield stress in compression to that in tension (the
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J.D. Robson et al. / Acta Materialia 59 (2011) 1945–1956 2.5 t
50
(a)
ΔCRSS = ΔCRSS
p
(b)
ΔCRSS =2ΔCRSS t
2
ΔCRSS (twin)/ΔCRSS (prismatic)
t
σY (compression)/σy (tension)
p
ΔCRSS =3ΔCRSS
p
ΔCRSS =0.5ΔCRSS t
p
1.5
1
0.5
0 0
50
100
150
ΔCRSS prismatic (MPa)
40
30 σY(T)=σY(C)
20
10
0 −20
0
20
40
60
80
100
ΔCRSS prismatic (MPa)
Fig. 7. (a) Predicted compressive to tensile yield strength ratio as a function of the increase in CRSS for prismatic slip, showing the effect of increasing the CRSS for f1 0 1 2g twinning by 0.5, 1, 2 and 3 times the increase for prismatic slip. (b) Relative increase in CRSS for twinning compared to the increase in CRSS for prismatic slip predicted to lead a compressive to tensile yield strength ratio of 1 (elimination of asymmetry).
yield asymmetry). To produce this plot, it was assumed the prismatic system was strengthened by an amount given by the value plotted as x, and the strengthening of the f1 0 1 2g twinning system was scaled in proportion (assumed to be either 0.5, 1, 2 or 3 times greater, as shown by the four curves in Fig. 7a). This is not intended to be physically accurate; indeed, it is expected that the strengthening of these two systems will not scale in proportion. However, it serves to illustrate an important point, which is that even if twinning is strengthened less than prismatic slip (e.g. half as much, as shown in Fig. 7), asymmetry will still be reduced. Indeed, this simple calculation predicts that asymmetry will only be made worse if f1 0 1 2g twinning is strengthened by less than 1/10 of the strengthening due to prismatic slip. This number is simply determined by the original ratio of the CRSS for twin growth compared to prismatic slip, as demonstrated below. Consider the CRSS for prismatic slip is increased by Ds and the CRSS for f1 0 1 2g twinning is increased by kDs, where k is a constant. Yield will be less asymmetric if the ratio of the stress to activate f1 0 1 2g twinning compared to that for prismatic slip is increased. This condition may be written as: S t ðst þ kDsÞ S t st > S p ðsp þ DsÞ S p sp
ð9Þ
where St and st are the Schmid factor and initial CRSS for f1 0 1 2g twinning, respectively, and Sp and sp are the Schmid factor and initial CRSS for prismatic slip. Rearrangement of this equation leads to: k>
st sp
ð10Þ
Therefore, as long as the CRSS for f1 0 1 2g twinning is increased more than k times the increase in CRSS for prismatic slip, the influence of any strengthening feature in the microstructure will be to reduce asymmetry. Note that this analysis can be applied not only to consider the influence of strengthening a pure magnesium single crystal, but can also be applied to any individual strengthening mechanism (such as precipitation) to determine whether asymmetry is likely to be increased or decreased (given the appropriate values of sp and st prior to the strength increment). This analysis predicts that an equal strengthening of f1 0 1 2g twinning and prismatic slip will lead to marked reduction in asymmetry (the solid line in Fig. 7a). However, elimination of asymmetry (rY (tension) = rY (compression)) can only be achieved by strengthening against f1 0 1 2g twinning more strongly than against prismatic slip. As Fig. 7a shows, the greater the additional strengthening of f1 0 1 2g twinning relative to prismatic slip, the smaller the strength increment required to eliminate asymmetry. This last point is more clearly illustrated in Fig. 7b. This plot shows the locus of points that leads to equality of the tensile and compressive yield strengths (i.e. elimination of asymmetry). The plot demonstrates that if the strengthening of the prismatic system is low, then the relative strengthening against f1 0 1 2g twinning has to be large to eliminate asymmetry. However, as the prismatic system is further strengthened, the relative strengthening of f1 0 1 2g twinning required to eliminate asymmetry decreases (although the absolute value of strengthening needed increases). If the prismatic system is softened (i.e. by small solute additions), then the relative increase in strengthening against f1 0 1 2g twinning needed to
J.D. Robson et al. / Acta Materialia 59 (2011) 1945–1956
ΔCRSS (twin growth)/ΔCRSS (prismatic)
eliminate asymmetry falls sharply, and the absolute value also reduces. It is now possible to investigate in detail the predicted effect that the formation of precipitates with different morphologies and habits will have on the yield asymmetry. The inference from the analysis of both the back-stress and the Orowan stress for twinning is that the dominant factor that determines how particles influence twin growth is the effectiveness of slip in the twin (and mainly basal slip) in relieving the back-stress. Parent–basal plate-shaped precipitates in particular are predicted to lead to a large increase in the stress required to accommodate basal slip in the twin. When present, these precipitates will therefore enable a higher back-stress to be sustained in the twin, which is assumed to directly increase the stress required for twin growth. For this analysis we assume: (i) the Orowan stresses already calculated for slip describe the increase in CRSS expected for the prismatic and basal slip modes; (ii) the increase in CRSS for twin growth is given by the Orowan stress inhibiting the required basal slip in twin resolved on the twin plane in the twinning direction; and (iii) other slip modes (e.g. hc + ai slip) have negligible influence, which is reasonable at room temperature in the magnesium alloys considered here [12]. Although this is clearly an oversimplification, in particular of the multiple slip events that are required to accommodate the twin shear around particles, as will be shown it does provide a useful qualitative tool for determining the likely effect of a particular particle morphology and habit on yield asymmetry. Fig. 8 shows the predicted ratio of the increase in the CRSS for twin growth (as defined above) to the increase
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in CRSS for prismatic slip. This is referred to as the CRSS increment ratio. The initial CRSS values prior to precipitation for a typical age-hardenable magnesium alloy extrusion are approximately 18 MPa for f1 0 1 2g twinning, and 54 MPa for prismatic slip [11]. Referring to Eq. (10), this gives k = 1/3. The analysis above predicts that if the precipitates lead to a CRSS increment ratio greater than this value (k > 1/3), asymmetry will be reduced. If they lead to a CRSS increment ratio less than this value (k < 1/3), then asymmetry is expected to increase. Fig. 8 shows that basal plates lead to a maximum value in the CRSS increment ratio, implying that they are most effective in reducing the asymmetry for a given strength increment. Spherical particles are also predicted to result in an asymmetry reduction since they provide equal strengthening against slip in the parent and twin. However, c-axis rods (of aspect ratio 10) lead to a CRSS increment ratio that is less than the critical value to reduce asymmetry, i.e. the precipitation of such rods would be expected to lead to an increase in asymmetry. These calculations obviously depend on the aspect ratio chosen for the particles. If the aspect ratio of the rods is further increased, then their effect on increasing asymmetry is also predicted to strengthen. An aspect ratio of 10 is a conservative estimate of that typically reported for rod-containing magnesium alloys. For example, in magnesium alloy Z5, the reported aspect ratio for rods is in the range 5–40 [11,8]. Therefore, these predictions suggest that for a Mg–Zn alloys in which rod-shaped precipitates form parallel to the c-axis, precipitation is likely to be accompanied by an increase in extrusion asymmetry. However, in alloys in which plate-shaped precipitates form on the basal plane (e.g. AZ91), a marked reduction in asymmetry after precipitation is expected.
1.6
5. Comparison with experiment
1.4
5.1. Experimental method
1.2
To test these predictions, extrusions have been studied where it has been possible to precipitate either basal plates or c-axis rods through choice of alloy composition. Basal plates were produced by age-hardening AZ91 alloy; this leads to the formation of non-coherent equilibrium b– Mg17Al12 plates, predominantly on the basal plane [13]. c-Axis rods were produced by age-hardening Z5, which leads to a transition phase (MgZn) [11]. The experimental procedure used to produce the Z5 alloy is described in detail elsewhere [11] and the same procedure was followed for AZ91. In brief, this involved giving the cast billets an 8 h solution treatment at 420 °C, followed by extrusion at 400 °C with a ram speed of 1 mm s1 and an extrusion ratio 30. This produced dynamically recrystallized microstructures with a strong basal texture (texture strength at least 5.3 times random) and a grain size of 18 lm in the case of AZ91 and 30 lm in Z5. Selected samples from the Z5 extrusion were aged to peak hardness prior to testing by a heat treatment of 8 days at 150 °C.
1 plates rods
0.8
spheres 0.6 asymmetry reduced
0.4
asymmetry increased
0.2 0
0
2
4
6
8
10
Particle dimension (μm) Fig. 8. A calculation of the ratio of the increase in CRSS for f1 0 1 2g twin growth compared to the increase in CRSS for prismatic slip for plate, rod and spherical particles. A value of this ratio <1/3 is predicted to lead to an increase in asymmetry on precipitation, a value >1/3 is predicted to lead to a reduction in asymmetry. Particle volume fraction = 5%, aspect ratio of plates = 0.1, aspect ratio of rods = 10.
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J.D. Robson et al. / Acta Materialia 59 (2011) 1945–1956
Ageing treatments for AZ91 followed those used by Braszczyn´ska-Malik [29]. Ageing to peak hardness was performed at 150 °C for 18 h. In this case, the microstructure is dominated by discontinuous precipitates, as shown in Ref. [29]. The theory developed above cannot be directly applied to the case of discontinuous precipitation, and so ageing was also performed for 8 h at 350°, which gave fully continuous precipitation (see [29]). Precipitate size and spacing were measured using transmission electron microscopy for Z5 (described elsewhere [11]) and field emission gun scanning electron microscopy for AZ91, where the precipitates are larger. Specimens from both alloys in the aged and as-extruded condition were tested in compression and tension under identical conditions following the procedure detailed elsewhere [11]. 5.2. Experimental results The stress–strain curves for AZ91 and Z5 in tension and compression in the peak-aged condition are shown in Fig. 9. From these, the tension–compression asymmetry values (compressive 0.2% offset stress/tensile 0.2% offset stress) have been calculated, and are reported in Table 1. It can be seen that in the as-extruded case, the asymmetry values for both alloys are quite similar, and the compressive strength is between 0.6 and 0.7 that of the tensile strength. On ageing, the asymmetry of the AZ91 extrusion is reduced greatly, and examination of the stress–strain curves reveals that this in primarily due to an increase in the compressive yield strength. This effect is most marked in the peak-aged condition where discontinuous precipitates dominate, but is also the reason for the reduced anisotropy after ageing at 350 °C, where only continuous precipitation occurs. In Z5, however, the opposite is observed, and ageing leads to an increase in asymmetry. Examination of the
(a)
stress–strain curves demonstrates that this is primarily due to a strong increase in the tensile yield strength, with the compressive yield strength being increased less markedly. 6. Discussion The observed change in asymmetry on ageing AZ91 and Z5 is qualitatively in agreement with what we have predicted using the Orowan models. The basal plates that precipitate is AZ91 lead to a reduction in asymmetry, whereas the c-axis rods that form in Z5 lead to increased asymmetry. For Z5, the mean rod diameter measured after ageing at 150 °C for 192 h was 20 nm, with a length of 120 nm and volume fraction of 2.3% [11]. For AZ91 after ageing at 350 °C for 8 h (continuous precipitation) the plates had a mean diameter of 2.8 lm and thickness of 0.4 lm with a volume fraction of 2.8%. For these particular particle fractions and sizes, the calculated Orowan strengthening of the various deformation systems is presented in Table 2. The key differences between the predicted strengthening in the two materials are: (i) the greater Orowan hardening in Z5, particularly for the basal and prismatic systems, which is consistent with the strong measured increase in tensile strength of this material on ageing and (ii) the relatively large hardening for basal slip in AZ91 twins compared to that of the other systems. The calculated CRSS increment ratios for twin growth/prismatic slip as defined earlier are also shown. Recall that for extrusions a value of this ratio greater than 1/3 (as shown here for AZ91) was predicted to lead to a reduction in asymmetry, whereas a value less than 1/3 (as shown here for Z5) was predicted to lead to an increase in asymmetry. This matches the observations and supports the idea that the CRSS increment ratio gives a useful indication of whether any
(b)
Fig. 9. Mechanical properties of the AZ91 and Z5 alloys studied here. (a) AZ91 in the peak-aged (18 h at 150 °C) and as-extruded condition. (b) Z5 in the peak-aged and as-extruded condition.
J.D. Robson et al. / Acta Materialia 59 (2011) 1945–1956
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Table 2 Calculated Orowan stress (MPa) for basal and prismatic slip, the twinning partial dislocation, and basal slip in twinned material for the measured particle parameters in AZ91 aged at 350 °C for 8 h and peak-aged Z5 along with calculation of the CRSS increment ratio (defined in text).
AZ91 (aged 350 °C) Z5
Basal
Prismatic
Twin
Basal (twin)
DCRSS ratio
0.7 43
1.4 34
0.3 7
2.5 15
0.89 0.22
strengthening effect is likely to lead to increased or reduced asymmetry. Although the analysis presented here is undoubtedly a great oversimplification, and does not consider in detail the mechanism by which twins and particles interact, it does appear to provide important insights into the influence of particle shape that are consistent with experimental measurements. In particular, it is suggested that whilst basal plates are not very effective strengtheners since they do not effectively block basal slip, they can be useful in increasing the stress required for twin growth (i.e. strengthening against twinning). It is clear that there are many unresolved issues; in particular the mechanism by which the twinning dislocation negotiates the particles is less well understood than the interaction of slip dislocations with precipitates. Furthermore, direct suppression of twin nucleation by particles is a possibility that is not considered in the present work. Finally, better quantitative prediction of strengthening will only be possible with a more realistic treatment of the distribution of particle spacings and a better understanding of how dislocations bypass high-aspectratio particles. 7. Conclusion The effect of precipitate shape and habit on the strength of magnesium alloys has been considered by calculating the Orowan stress for the deformation systems of most importance at room temperature. These calculations have been used to assess the likely effect of different precipitate types on the asymmetry of strongly textured extrusions. The predictions of the model have been tested against mechanical property measurements made for alloys with two distinctly different precipitate shapes and habits. The following conclusions can be drawn from this work: For an identical volume fraction and particle size, c-axis rods are more effective in hardening against basal slip and prismatic slip compared to spheres and basal plates. Basal plates are particularly poor at hardening against basal slip due to the low fraction of precipitates that will be cut by a given basal plane. When a growing twin engulfs a particle without shearing it, a back-stress is necessarily generated as a result of the need to mitigate the strain discontinuity at the particle– matrix interface. This back-stress is predicted to reach unrealistically large values in the twin without plastic accommodation around the particles. However, the particles will also inhibit this plastic accommodation by acting as barriers to slip in the twin.
Due to the lattice rotation in the twin, basal plates in the parent become oriented perpendicular to the basal plane in the twin and become very effective barriers to preventing basal slip in twinned material. c-Axis rods in the parent provide poor obstacles to prevent slip in the twin. It is postulated that a high resistance to basal slip in the twin will increase the stress required for twin growth by allowing a higher back-stress to be sustained in the twin. The Orowan stress on the twinning partial dislocation is an order of magnitude less than that for slip due to the small Burgers vector of the twinning partial. It is therefore likely that this stress is never sufficient to prevent twin growth once sufficient stress has been applied to nucleate twins. It is suggested that the ratio of the increase in CRSS for twin growth to that for prismatic slip (named the CRSS increment ratio) is the key parameter determining whether a strengthening effect will increase or reduce asymmetry of strongly textured magnesium alloy extrusions. Below a critical value of this ratio, a strengthening effect will increase asymmetry; otherwise it will reduce it. Using the CRSS ratio above, it has been predicted that basal plates will lead to reduced asymmetry, whereas c-axis rods can increase asymmetry. The predictions of the model have been tested against mechanical data for alloys Z5 and AZ91 aged to peak strength and deformed in tension and compression. The presence of c-axis rods in Z5 leads to an increase in asymmetry, whereas the precipitation of basal plates in AZ91 reduced asymmetry. This is in agreement with predictions made on the basis of the CRSS increment ratio as defined above.
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