A dislocation-based model for twin growth within and across grains

Abstract

A computational method is presented for representing twins via two-dimensional dislocation statics in an isotropic elastic solid. The method is compared with analytical approximations of twin shape and is used to study how twins evolve within grains subjected to an arbitrary external shear stress. Twin transfer across grains is then studied using the same computational method. The dislocation-based model for twin growth gives the following dependencies: twin thickness increases linearly with grain size and external stress, and increases substantially as the grain is able to traverse multiple grain boundaries with low misorientation angles; the model also predicts that twin transfer becomes less prominent across grain boundaries with high misorientation angles. These predictions are consistent with experimentally measured extension twin growth in magnesium polycrystals. This study suggests that representing twins via discrete dislocations provides a physically reasonable approximation of twinning that can be naturally incorporated into existing dislocation statics and dynamics codes.

1. Introduction

Deformation twinning is a deformation mechanism that provides shear strain by reorienting a finite portion of the crystal lattice. The amount of shear, γ₀, that the twin provides depends on the twinning plane and direction, as well as other details of the crystal structure [1]. Like dislocation slip, deformation twinning proceeds in order to minimize the work done in a material in response to an externally imposed deformation or stress [2]. Dislocations can cross-glide, entangle and form dense bands to minimize their long-range stress fields, whereas twins must form fine, low-energy structures and sequences to shield their long-range stress fields, similar to complex structures formed during phase transformations [3,4]. These interactions are inherently non-local and arise within and across the length scale of grains within a polycrystal. Even under homogeneous loading conditions complex twinning structures are experimentally observed, as seen in figure 1.

Figure 1.

Electron backscatter diffraction scan of rolled magnesium AZ31B specimens loaded to 6% strain in (a) compression along the rolling direction and (b) tension along the normal direction. (Online version in colour.)

Because of its importance in dictating mechanical anisotropy, and the large amount of experimental work performed to characterize its behaviour, this work focuses on $\{10\overline{1}2\}⟨10\overline{1}1⟩$ extension twinning in pure magnesium and its alloys. Several empirical observations have been made of extension twinning in magnesium and its dependence on grain size and orientation: the Hall–Petch coefficient for twinning, k_y,twin, is greater than that for slip, k_y,slip [5]; an abrupt yield ‘plateau’ occurs when twinning causes yield, and this plateau elongates as grain size is decreased [6]; the likelihood of twins crossing grain boundaries (forming twin pairs) decreases with increasing grain boundary misorientation [7]; and twins that have a twin pair across a boundary are often thicker than twins that terminate at a grain boundary [7]. These empirical observations have been included in crystal models with varying complexity [8–10], but lack a unified framework with which they can be studied.

In this work, a two-dimensional dislocation statics framework is used to model the nucleation and growth of extension twins in magnesium polycrystals. Describing twins using dislocations is not new [11–16]. Dislocation-based methods use analytical functions to approximate the equilibrium configuration of a continuous distribution of twinning dislocations based on the assumption that the width of the twin is small compared with its length. This implies that the solution for the position of dislocations in a pile-up derived by Eshelby et al. [17] can be used to approximate the shape of the twinned region [18]. Analytical dislocation-based methods can describe elastic twin growth for homogeneous and point loads in the case where the twin grows freely in pristine material, is nucleated at a surface and when it is halted at obstacles. However, it is rather difficult to introduce twin boundary interactions or alter the governing equations to admit ‘inelastic’ features that lead to hysteresis, such as friction stress, immobilization at boundaries and interaction with other dislocations that causes multiplication, immobilization or annihilation. A notable exception is the work of Yoo & Wei [19], who used anisotropic elasticity solutions of discrete dislocations to describe twin growth in zinc at bi-crystal interfaces. However, this work only gave upper bounds to stable tip apertures at the bi-crystal interface (less than or equal to 22° for $\{10\overline{1}2\}$ twins in zinc), which greatly exceed those computed via homogenized dislocation theory (less than 1°) [18].

In this work, twin growth within a grain is modelled first; then the case of twin transfer across grain boundaries is analysed. For thin twins, if the macroscopic shear stress τ is small enough so that the repulsive force due to the head dislocation is greater than the combination of τ and the forces due to piled-up dislocations, the twin shape will be ellipsoidal and no inelastic features arise. However, as τ increases, more dislocations are generated and the combination of τ and the forces due to piled-up dislocations overcome the repulsive force of the head dislocation, and the dislocations glide to the grain boundary. The way these dislocations interact with the grain boundary and with neighbouring grains governs the shape of the original twin. Wherever possible the simplest rules for inelastic behaviour are adopted, although care is taken to discuss when simplifying approximations may become troublesome. Despite its simplicity, this dislocation-based description of twinning gives qualitative and quantitative agreement with several of the aforementioned empirical observations of deformation due to twinning.

2. Material and methods

Dislocations in this work are modelled using isotropic elasticity and small-strain dislocation statics following classical discrete dislocation studies [20,21]. Two-dimensional, plane strain solutions for the Cauchy stress of infinite, straight edge dislocations in their local frame, σ′, are mapped to global stresses via a passive rotation σ=R^T⋅σ′⋅R [22]. After a dislocation is introduced into the simulation, glissile dislocations are allowed to glide at a velocity linearly proportional to the local shear stress until all dislocations are in equilibrium, e.g. until the maximum force on all glissile dislocations falls below a critical threshold. Dislocation motion is restricted to the glide plane, which is equivalent to assuming climb does not occur. For all of the simulations, a homogeneous, macroscopic shear stress τ is imposed on the infinite domain.

Grain boundaries are introduced as impenetrable barriers to dislocation motion. If a glissile dislocation enters a boundary, it is immobilized and remains sessile unless otherwise noted. In the case where multiple dislocations enter the boundary in a single time step, the dislocation that has the greatest displacement beyond the boundary is identified, immobilized at the boundary and then the step is re-computed. This is done iteratively until no additional dislocations bypass the boundary.

Embryonic twins are constructed by introducing a sessile stacking fault spanning two grain boundaries separated by a distance d. Twinning dislocation dipoles are sequentially introduced on atomic planes separated by a distance h above and below the initial sessile stacking fault, which produces an atomically faceted twin boundary. The Burgers vector for $\{10\overline{1}2\}$ twinning dislocations in hcp crystals with $c/a<\sqrt{3}$ is expressed as ${\mathbf{\text{b}}}_t=(3-{\gamma }^2)/(3+{\gamma }^2)⟨\overline{1}011⟩$, where γ=c/a and the magnitude of $⟨\overline{1}011⟩$ is $\sqrt{3a^2-c^2}/2$ [23]. For magnesium, a=3.21 Å , γ=1.624 and γ_t=b_t/h=0.129, which gives the magnitude of the Burgers vector as b_t=0.245 Å and the spacing between dislocations as h=1.90 Å [24]. After the dislocations are introduced, the equilibrium solution for the positions of these dislocations as well as all other glissile dislocations in the simulation are found. This process is repeated until the equilibrium twin configuration is found, which occurs when, instead of the inserted dislocation dipole spreading to the twin boundary, it self-annihilates due to the back stress of the twinning dislocations that constitute the twin boundary. This process is illustrated schematically in figure 2. Introducing twins using this method is consistent with previous work on elastic twin growth [18] and relies on a hypothetical infinite source of twinning dislocations.

Figure 2.

Process of creating an equilibrium embryonic twin constrained between two grain boundaries: (a) initial partial dislocations immobilized at a grain boundary of width d; (b) dislocation dipoles inserted at ±h glide to equilibrium configuration; (c) equilibrium twin shape is formed; and (d) insertion of additional twinning dislocations self-annihilate, indicating that the final twin shape has reached equilibrium. Grey dislocations indicate inserted positions; black dislocations indicate equilibrium configuration. (Online version in colour.)

Several physical processes have been proposed for the creation of twinning dislocations during twin growth, and often depend on the particular crystal structure or twin system studied. Sleeswyk & Verbraak [25] showed that for bcc metals dissociation of intersecting edge dislocations due to slip into twinning dislocations caused twin growth. However, Ogawa [26] showed that, instead of dissociating into a single dislocation, it was energetically favourable for edge dislocations to dissociate into a triplet of twinning dislocations, and this process was used to rationalize the occurrence of twinning instead of slip at low temperatures and high loading rates. The creation of twinning dislocations via a pole mechanism, in which a pinned point continuously generates excess dislocations, has been proposed fcc, bcc and hcp metals, and is summarized by Hirth & Lothe [22]. For $\{10\overline{1}2\}$ twinning in hcp metals, Serra & Bacon [27] proposed that, instead of the pole mechanism, intersecting basal dislocations decompose into interface defects, which, under appropriate stress conditions, have been observed using transmission electron microscopy to affect twin growth [28]. The nucleation process and equilibrium twin shape of the embryonic twin have been examined using density functional theory and molecular statics by Wang et al. [29]. Because each nucleation process will introduce different local stresses at the twin interface, which will alter the computed equilibrium twin shape, the simple infinite source model is used with the caveat that more realistic models to introduce twinning dislocations may give more physically relevant behaviour.

3. Twinning within a single grain

(a). Analytical expressions for twin growth between two grain boundaries

A derivation of the twin thickness as a function of position is based on the equilibrium solution for the density of dislocations piled up against a boundary [17,18,22]. For dislocations nucleated from x=0 against grain boundaries located at x=±d/2, the density of dislocations is expressed as

$$\rho (x)=\frac{2(1-v)\tau }{\mu b}\frac{x}{\sqrt{{(d/2)}^2-x^2}},$$ 3.1

where, for the schematic shown in figure 2, b is positive for x>0 and negative for x<0. In the limit that the twin thickness is much less than the grain boundary dimension, i.e. t≪d, the positions found in the above relation correspond to the x position of twinning dislocations, noting that each pair is approximated by a single dislocation, so b=2b_t. For simplicity consider x>0. The vertical position of the twin interface y is the integrated density from the tip to the source multiplied by the vertical spacing between twinning dislocations, i.e.

$$y(x)=h{\int }_x^{d/2}\rho (z)\hspace{0.17em}\mathrm{d}z=\frac{h(1-\nu )\tau }{\mu b_t}\sqrt{{\left(\frac{d}{2}\right)}^2-x^2}.$$ 3.2

This expression will be referred to as the ‘pile-up’ approximation. The maximum thickness that the twin achieved is at t=2y(x=0), which is expressed as

$$t=(1-\nu )\frac{h\tau d}{\mu b_t}=(1-\nu )\frac{\tau d}{\mu {\gamma }_t}.$$ 3.3

This implies that, for large stresses, twin thickness is proportional to applied stress and grain size, but inversely proportional to twinning shear γ_t.

If τ is large and the grain boundary is assumed to be an obstacle with infinite strength, dislocations overcome the repulsive stress of the sessile head dislocation and glide into the grain boundary. As an approximation, if all of the dislocations glide to the boundary, the twin will thicken until the external stress at the nucleation source is overcome by the shear relaxation due to n dislocations above and below the head dislocation at the two boundaries. The twin is stable for thicknesses t=2nh, and will grow to the maximum value of n for which

$$2K\sum _{i=-n}^n{x}_i\frac{x_i^2-y_i^2}{{(x_i^2+y_i^2)}^2}\le \tau ,$$ 3.4

where K=b_tμ/(2π(1−ν)), x_i=d/2 and y_i=(n−i)h. This representation of a twin will be referred to as a ‘twin lamella’ approximation, as has been done elsewhere [30].

For the twin lamella approximation, if the twins are assumed to be thin, i.e. t≪d, then (3.4) simplifies to

$$t\approx \frac{\tau dh}{4K}=(1-\nu )\frac{\pi }{2}\frac{\tau d}{\mu {\gamma }_t}.$$ 3.5

This expression is referred to as the ‘super-dislocation’ approximation, and is akin to representing the twins in the boundary as a single dipole, where the net Burgers vector is related to the height of the twin. Although it is an oversimplification of the twinning process, the super-dislocation approximation thickness is π/2 multiplied by the pile-up approximation thickness, and does not need to be solved iteratively as in the twin lamella approximation.

Based on the three simplified models for twin shape introduced above, it is simple to show that the pile-up approximation represents a lower bound in t, whereas the lamella approximation represents the upper bound. The estimate of t given by the super-dislocation approximation is slightly less than the lamella approximation because it assumes that all dislocations are a vertical distance t/2 away from the twin dislocation nucleation site. Despite their differences, all three expressions give a reasonable and similar approximation of purely elastic twin growth. In the following section, they are contrasted with computed twin shapes. Owing to the symmetry of the problems examined, quarter symmetry is used in the rest of the computations in §3, but is suppressed in §4.

(b). Computed twin growth between two grain boundaries

The computed twin shapes for a d=1 μm grain subjected to macroscopic shear stress levels τ of 10, 30, 50 and 100 MPa are compared with the analytical approximations in figure 3. At τ=10 MPa, none of the dislocations have reached the critical stress required to overcome the repulsive stress of the sessile head dislocations, so their positions closely resemble the analytical positions predicted by the pile-up approximation, which confirms that the numerical scheme calculations are accurate. The low stress creates a very fine twin, with an aspect ratio of approximately 3:1000. At τ=30 MPa, a few dislocations have overcome the repulsive stress of the head dislocation and are rendered sessile in the grain boundary, but the majority are piled up against dislocations on adjacent planes. Because few dislocations are in the grain boundary, the pile-up solution is again by far the most accurate approximation. As τ is increased from 50 to 100 MPa, more dislocations enter the grain boundary. At 100 MPa, the shape of the twin goes from ellipsoidal to a combination where half of the twin is ellipsoidal and the other half is rectangular. Additionally, the twin shape near the grain boundary becomes more linear as the number of dislocations in the boundary is increased. At this higher stress, the maximum thickness starts to exceed that predicted by the pile-up approximation. By increasing the τ from 10 to 100 MPa, the aspect ratio also increases nearly by a factor of 10 to 35:1000.

Figure 3.

Equilibrium twin configuration in a d=1 μm grain for four values of τ calculated using discrete twinning dislocations and various analytical approximations. Note all dimensions are in micrometres, and the vertical axes are scaled differently from the horizontal axes. (a) τ=10 MPa, (b) τ=30 MPa, (c) τ=50 MPa and (d) τ=100 MPa. (Online version in colour.)

The twin shape at τ=10 MPa is fundamentally different from that at τ=100 MPa, so one may expect that the macroscopic stress field around the dislocation also differs. In figure 4, the total shear stress field is plotted around the twin and near the twin tip for τ=10 and 100 MPa. From figure 4a,b, it is clear that the main features of the macroscopic stress field due to the twin are nearly identical if they are scaled appropriately. One difference is that because τ=10 MPa contains only nine dipole pairs above and below the twin mid-plane, whereas τ=100 MPa contains 94, the thinner twin has more discrete stress concentrations about individual twinning dislocations and the twin itself is not in a fully stress-free state.

Figure 4.

Shear stress contours in a d=1 μm grain calculated using discrete twinning dislocations for applied shear stresses of τ=10 and 100 MPa. All dimensions are in micrometres and stresses in MPa, and the grain boundary is at 0.5 μm. (a) τ=10 MPa, (b) τ=100 MPa, (c) τ=10 MPa and (d) τ=100 MPa. (Online version in colour.)

The stress fields at the twin tip, plotted in figure 4c,d, differ between the two cases. For τ=10 MPa, within the twin there are regions where the shear stress goes from positive to negative because the local stress concentration near an individual twinning dislocation exceeds the back stress from piled-up dislocations near the grain boundary. Ahead of the twin tip, there is a small region concentrated near the twin tip that is stressed nearly 10 times the externally imposed shear load. For τ=100 MPa, there is a large back stress near the flat twin tip, but the majority of the twin is stress free and stresses are smoothly varying. Beyond the twin tip there is also a region that is stressed nearly 10 times the externally imposed shear load, which is due to the increased thickness of the twin.

Work by Paudel et al. [31] has shown that the stress field due to ellipsoidal twins is directly related to their shape, which is consistent with the observation that the stress concentration scales with the twin thickness. Using any of the aforementioned approximations, the twin thickness scales linearly with resolved stress, and so does the magnitude of the stress field due to the twin. Similarly, if the applied shear stress is held constant and the grain size is increased, the thickness of the twin increases linearly with grain size, and so does the stress concentration in the adjacent grain.

From these simulations it is shown that, for overall stress fields due to twins, there is not a significant difference between the macroscopic stress of ellipsoidal twins and those with flat tips that terminate at grain boundaries. Both types of twins are approximated reasonably well with the analytical solutions for pile-up and lamella twin approximations, and the macroscopic fields from these analytical solutions should be in close agreement with computed twin shapes. However, when the behaviour near the boundary is of interest, e.g. when the stress state dictates plastic deformation in an adjacent grain, correctly modelling the evolving geometry of the twin tip may be important. Although flat twin tips have been observed experimentally [19,32], the majority of flat-tipped twins are ones that cross into adjacent grains [7]. Whether the tip is actually flat or sharp before it crosses a grain boundary is not clear from experiments, and often exceeds the resolution of most commonly used methods such as electron backscatter diffraction.

(c). Coarse-grained representation

As twin thickness increases, computation of the equilibrium twin shape becomes increasingly burdensome. Not only does the number of computations scale as $\mathcal{O}(n^2)$, but as dislocations pile up near the head dislocation, the force on these dislocations decreases the maximum time step used in the problem. One method to speed up computations is to use ‘super-dislocations’ to represent a set of dislocations. These dislocations have a net Burgers vector and vertical spacing of b_k=kb_t and h_k=kh, respectively. To understand the effect of this coarse-graining, the twins in figure 3 are re-computed using k=5 and 10, and are plotted in figure 5.

Figure 5.

Coarse-grained simulation of equilibrium twin configuration in a d= 1 μm grain from figure 3. Black dislocations represent the full simulation (k=1), whereas blue and red representsuper-dislocations consisting of k=5 and k=10 dislocations, respectively. Dislocation size is scaled according to the number of dislocations. (a) τ=10 MPa, (b) τ=30 MPa, (c) τ=50 MPa and (d) τ=100 MPa. (Online version in colour.)

For cases where the number of twinning dislocations is of the order of k, such as for τ=10 MPa, the coarse-grained description does a poor job of predicting the width and the shape of the twin. For τ=30 MPa, where there are approximately 25 dislocation dipoles above and below the centre line, k=5 gives a close approximation, whereas k=10 does not. As τ increases it is clear that both approximations are reasonable and give the correct thickness, within ±kh. The only feature of the twin that is represented poorly is the sharp interface between dislocations in the grain boundary and those piled up next to the grain boundary; however, unless specific reactions at the grain boundary are of interest, this will not have a significant bearing on the overall shape or behaviour of the calculated twin. It appears that a reasonable rule of thumb is that super-dislocations representing k twinning dislocations will give an accurate approximation of twin shape if the total number of dipoles above the centre line divided by k is greater than 5. As a conservative estimate, (3.3) can be used to calculate the thickness in order to compute an appropriate value of k.

4. Twin transfer across grains

Several researchers have studied the effect of grain misorientation, denoted by α, on the propensity for twins to transfer across grain boundaries. Kacher & Minor [32] observed that, in rhenium, twin transfer only occurred in grain boundaries with α<25°, and Beyerlein et al. [7] observed that, in magnesium, the majority of twin transfer occurred with α<45°. Computationally, some methods have been proposed to explicitly calculate the stresses in adjacent grains, but have not gone so far as to model twin transfer itself. Barnett et al. [33] have described twin growth and interaction stresses using Eshelby’s elastic solution for ellipsoids, which should give average stresses of the order of those given by dislocation theory, but may be inaccurate for studying twin-tip behaviour required to model twin transfer between grains. Kumar et al. [30] calculated the stresses on neighbouring grains of twin lamellae extending across grains in an anisotropic elastic medium to understand stress states preceding twin transfer. Transmission of twins from a parent grain to adjacent grains is invariably controlled by several factors such as elastic anisotropy, concurrently operative inelastic mechanisms and the character of the grain boundary.

In this work, a simplistic model of twin transfer is borrowed from work on slip transfer [34] and is shown schematically in figure 6. When the shear stress on the twin system in an adjacent grain exceeds some critical value, denoted by τ_crit, a dipole pair is nucleated. This value may depend on the specific material of choice, its thermo-mechanical processing history, an intrinsic resistance to transfer due to the grain boundary and even the position of the nucleation site in the material [35], but is taken as a constant in this work. Because the stress is singular at the interface, the stress is evaluated at 10 nm from the interface. The negative dislocation from grain 2 combines with the sessile dislocation in grain 3 according to b₁+b₂→b_r. The mobile dislocation in grain 2 is allowed to glide until it reaches another grain boundary. For simplicity in the numerical calculations, all of the dislocations in the adjacent grains are assumed to glide into the boundary. As these dislocations are far away from the dislocations in the parent grain, their exact configuration has little bearing on the equilibrium solution in the parent grain. Additionally, the twin is assumed to have a constant length d, which implies that the width of adjacent grains is $d\cos \alpha$. This assumption is enforced so that twins in grains with large α do not approach infinite length.

Figure 6.

Process of twin transfer from grain 2 to grain 3: (a) an initial dislocation dipole is created in grain 3 when the shear stress on the dipole exceeds some critical value; (b) the boundary dislocations in grains 2 and 3 form a residual sessile dislocation with a smaller Burgers vector in the grain boundary while the dislocation in grain 3 glides to the next boundary; and (c) stresses at the grain boundary relax as additional twinning dipoles nucleate from the boundary into grain 3, which causes the twins in both grains 1 and 2 to thicken. Variable α denotes the misorientation between the grains on the left and right side. (Online version in colour.)

As the number of grains that twins can transfer across increases, computing twin thickness becomes computationally burdensome. An upper bound in twin thickness for increasing number of grains can be computed by setting τ_crit=0 and assuming that, upon transfer, all dislocations immediately glide to the appropriate grain boundary. This process continues until the pre-specified number of grains is traversed. A schematic of a twin with fixed thickness growing across multiple grains in this ‘lamella transfer’ approximation is shown in figure 7. As in the lamella approximation for a single grain, the thickness in the originating twin is calculated by adding dislocations on adjacent planes until the back stress at the nucleation site becomes negative.

Figure 7.

Twin lamella approximation for twin growth across multiple grains. The sign of α alternates between grains to keep the twin centred in the vertical direction. (a) Zero grain, (b) one grain, (c) two grains and (d) three grains. (Online version in colour.)

For twin transfer, the specific value of τ_crit will change the dependence of twin thickness on misorientation between grains. For a fixed macroscopic shear stress, the importance of τ_crit will also change; if τ_crit≫τ, twin transfer will not occur, whereas if τ_crit≪τ twin transfer will depend on whether dislocations can overcome the back stress of the head dislocation, and, if they can, the magnitude of back stress due to the transformed residual dislocations. To understand how τ_crit and the misorientation between grains affect computed twin width, the maximum twin width is plotted as a function of α for different values of τ_crit in figure 8.

Figure 8.

Computed twin thickness as a function of α and different values of τ_crit with τ=50 MPa, α, and d=1 μm. The simulations used a coarse-grained approximation by representing every 10 dislocations as a single super-dislocation. Simulations were computed by varying α in 5° increments. (Online version in colour.)

For all three sets of simulations, the maximum twin width is observed to vary strongly with α for values of α<20°. Twins that transfer across a pair of grain boundaries with α<20° are predicted to be two to four times as thick as twins that do not transfer across grain boundaries. The thickness of these twins is tied directly to the value of τ_crit. For τ_crit=100 MPa, i.e. when twin transfer is easier, twins are much thicker for very low misorientation angles. As more dislocations are transferred to the boundary, the repulsive force on piled-up dislocations within the twin in the parent grain decreases. This causes more dislocations to enter the boundary and eventually transfer. For τ_crit=200 MPa, i.e. when twin transfer is harder, even though dislocations can enter the boundary, fewer transfer across. The transformed dislocations exert the same back stress in both cases, but fewer dislocations are transformed, increasing the back stress on the piled-up dislocations within the twin in the parent grain. For all three values of τ_crit, there was a critical angle of α at which no twinning dislocations transferred across the boundary: for τ_crit=100 MPa no transfer occurred for α≥70°; for τ_crit=150 no transfer occurred for α≥60°; and for τ_crit=200 MPa no transfer occurred for α≥50°. Also, τ_crit=0 MPa and τ_crit=50 MPa gave identical results to τ_crit=100 MPa.

Figure 8 shows that the lamella approximation of twin thickness is only somewhat accurate for low values of α and low values of τ_crit. As α increases, the approximation that all dislocations are transferred to the adjacent grain becomes increasingly inaccurate. Despite its limitations, it still correctly predicts that, as α increases, the twin thickness should decrease. Also, it gives a rapid method to calculate the upper bound twin thickness for fixed τ, d and α due to twin transfer alone.

To understand how twin thickness changes as the number of grains a twin can traverse increases, the lamella approximation is applied for twin transfer in up to five grains, as shown in figure 9. Grains are assumed to have alternating orientation so that the twin remains somewhat centred vertically. As anticipated, when the number of grains that the twin can transfer across increases, so does the twin thickness. Interestingly, as the number of grains the twin transfers across increases, the twin thickness decreases more rapidly for increasing values of α. Physically, as the twin traverses more grains, the ratio of the number of residual twinning dislocations in grain boundaries to the total number of dislocations increases. The magnitude of the residual Burgers vector is directly proportional to the magnitude of the back stress that these dislocations exert on twinning dislocations in the parent grain. For the tilt boundaries used in this work, it can be shown that $b_r\propto \sqrt{2(1-\cos \alpha )}$, which is approximately linear for small values of α. The nearly linear increase in b_r with increasing α explains the rapid drop in twin thickness with increasing α for twins traversing several grains. In twins that traverse fewer grains, the ratio of residual twinning dislocations in the boundary to total dislocations is lower, and the predicted thickness is less sensitive to α. However, it is reiterated that the lamella transfer approximation is an upper bound and does not correctly account for cases where a large fraction of twins in the grain boundary do not transfer to adjacent grains, as happens for cases where α is large and where τ_crit≫τ.

Figure 9.

Twin thickness calculated as a function of α for varying numbers of grains that the twin transfers across. The calculations use the lamella transfer approximation with τ=50 MPa and d=1 μm. The grain misorientation angle alternates signs as indicated in figure 7. (Online version in colour.)

5. Discussion

This work uses dislocation statics within an isotropic elastic framework to describe the growth of twins within and across grain boundaries. For twin growth within a grain, computed twin equilibrium shapes are compared with three analytical expressions—the dislocation pile-up, the twin lamella and the super-dislocation approximations. The growth of twins via twin transfer was then examined computationally and with the twin lamella approximation to reveal how twin growth is affected by the shape and orientation of neighbouring grains. Model predictions are compared in more detail with experimental observations in this section.

Several researchers have suggested that yield via twinning is more sensitive to grain size than dislocation-mediated plasticity [5,36,37]. This suggests in the Hall–Petch relationship, σ_y=σ₀+k_y⋅d⁻ⁿ, either k_y is greater for twinning than for slip or n is not equal to $\frac{1}{2}$, which is what is conventionally used to describe slip-based yielding. The dislocation-based model in this work uses an idealized infinite source model for the nucleation of twinning dislocations, so the stress at which the first twinning event begins cannot be captured; however, macroscopic yielding in polycrystals requires some finite deformation to provide polycrystalline compatibility, so approximating yield by the stresses at which twins grow gives a first-order approximation of yield behaviour. The analytical and computational approximations in §3 both predict that n=1, which is consistent with greater dependence of grain size for twinning than for slip.

It is difficult to experimentally determine how post-yield behaviour is influenced by grain size, because it is impossible to make two polycrystalline materials with different grain sizes but identical microstructural features such as initial dislocation density, texture, grain morphology, etc. Barnett et al. [6] carried out the closest approximation to date by performing different thermo-mechanical extrusion and annealing steps to AZ31B bars to get grain sizes in the range of 5.1–55 μm with nominally similar macroscopic textures. Their work showed that the post-yield elongation, e.g. the amount of strain in the low work-hardening region following yield, increased with decreasing grain size. Post-yield elongation is generally attributed to a deformation mechanism that causes large stress concentrations that set off sequential yielding in adjacent grains. At first glance, it appears that the grain size dependence on post-yield elongation is the opposite of what is expected from twinning dislocation analysis. For fixed τ, increasing d causes an increase in the magnitude of the peak stress ahead of the twin tip as well as the area over which some critical stress is reached. However, two things are not included in this rationale: the stress at yield and the number of twins that appear in the grain.

From the twinning dislocation analysis t∝d, and if it is assumed that σ_y∝1/d is a reasonable approximation, the magnitude and area of the stress concentration ahead of the twin do not depend on grain size for a single twin. From the empirical observation that the number of twins per grain scales linearly with the grain area [7], or with d², it is expected that as d decreases so does the number of grains that have multiple twins. With its idealized nucleation criterion, the dislocation-based twinning analysis in this work cannot predict how many twins should exist in a grain of a given size. It can be used to show that, for a fixed volume fraction of twinning, as the number of twins per grain increases, the peak stress concentration in adjacent grains is inversely proportional to the number of twins. Therefore, smaller grains with fewer twins per grain provide a greater peak stress concentration ahead of the twin tips than larger grains with multiple twins, rendering them more effective at promoting deformation across multiple grains and increased yield elongation. Although consistent with the aforementioned findings, more detailed incorporation of nucleation mechanisms is needed to conclusively link post-yield elongation to grain size.

A more quantitative comparison can be made by comparing the predicted shape of twins yielded by polycrystalline specimens. Beyerlein et al. [7] performed the most exhaustive statistical analysis of twin shape in pure magnesium polycrystals at strains of approximately 3%. The average grain size was d≈32.8 μm and the average thickness was t_avg=2.08 μm. By setting τ=mσ with representative values of σ=100 MPa and m=0.4, the computed twin thickness is t=0.45 μm, which is between that predicted by the pile-up and super-dislocation approximations, t=0.42 μm and t=0.67 μm, respectively. In the magnesium sample Beyerlein et al. [7] tested, the majority of grains possessed a misorientation angle α between 10° and 45°. The frequency of grain boundaries containing twins that transferred across the boundary decreased almost linearly with increasing α, which is consistent with figures 8 and 9. These figures show that, for this range of orientations, a twin transferring across one set of grain boundaries may be approximately two to three times thicker than one that does not transfer, and crossing multiple grain boundaries will cause the twin to thicken even more. Beyerlein et al. [7] only measured that 8% of twins crossed twin boundaries, but estimated that, because the measurement is two-dimensional, the real value may be four to five times higher. If 40% of twins transfer across boundaries, calculated twin thicknesses give reasonable agreement with experiments.

Although twin transfer can account for the difference in the twin thickness observed experimentally and predicted via growth within a single grain, an alternative mechanism may cause significant thickening. A lower critical resolved shear stress is required to activate basal slip than twinning magnesium [38]. Intersecting basal dislocations may decompose at the twin interface, providing a pole mechanism for subsequent thickening [27,28]. The infinite source approximation accounts for the large number of dislocations that may be generated by intersecting basal dislocations, but does not account for stress relaxation due to glide of the basal dislocations within the twin, or in the surrounding matrix. If the back stress on the nucleation site is relaxed by the nucleation source, twin thickness will increase with increasing basal slip. Beyerlein et al. [7] found that the occurrence of twinning depended most heavily on the maximum Schmid factor for twinning, and not on the basal slip, but there was no investigation into the effect of the basal Schmid factor on the thickness of observed twins. Further analysis is needed to quantify how much thickening can be achieved via intersecting dislocation slip.

A preliminary study by Fan et al. [39] showed that sessile twins can be used within a dislocation dynamics framework to model cross-hardening between slip and twinning; however, this is for a constant twinning state. The work done in this paper shows that, if twins are modelled by discrete dislocations and allowed to evolve, the competition between slip and twinning processes should arise naturally in a dislocation dynamics framework [40]. Discrete processes that cause embryonic twins to nucleate can also be incorporated, such as the reaction given by Wang et al. [29], which implies that the entire competition between slip and twinning from nucleation and growth to coalescence can be captured in a single framework. Additionally, although it may not be feasible to model atomically faceted twins in some dislocation dynamics problems, this work showed that the coarse-grained representation either through analytical approximations or by substituting super-dislocations based on the thickness should give a reasonable approximation of fully discretized twin behaviour.

Data accessibility

This work does not have any experimental data. The source code for this work has been made available on GitHub at https://github.com/jlloydgt/twindislo.

Competing interests

The author has no competing interests.

Funding

This work was funded by the US Army Research Laboratory.