Grain size constraints on twin expansion in hexagonal close packed crystals

Transcription

1 Grain size constraints on twin expansion in hexagonal close packed crystals M. Arul Kumar 1 *, I.J. Beyerlein 2, C. N. Tomé 1 1 Materials Science and Technology Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA 2 Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Abstract Deformation twins are stress-induced transformed domains of lamellar shape that form when polycrystalline hexagonal close packed (HCP) metals, like Mg, are strained. Several studies have reported that the propensity of deformation twinning reduces as the grain size decreases. Here we use a 3D crystal plasticity based micromechanics model to calculate the effect of grain size on the driving forces responsible for expanding twin lamellae. The calculations reveal that constraints from the neighboring grain where the grain boundary and twin lamella meet induce a stress reversal in the twin lamella. A pronounced grain size effect arises as reductions in grain size cause these stress-reversal fields from twin/grain boundary junctions to affect twin growth. We further show that the severity of this neighboring grain constraint depends on the crystallographic orientation and plastic response of the neighboring grain. We show that these stress-reversal fields from twin/grain boundary junctions will affect twin growth, below a critical parent grain size. These results reveal an unconventional yet influential role that grain size and grain neighbors can play on deformation twinning. Keywords: polycrystal, magnesium, micromechanics, plasticity, grain size 1. Introduction Polycrystalline hexagonal close packed (HCP) metals can plastically deform by a combination of dislocation slip and deformation twinning [Akhtar, 1971; Partridge, 1967; Reed- Hill, 1964; Yoo, 1981; Yoo and Lee, 1991; Yoo et al., 2001]. Their relative contributions to accommodating the imposed strain depend on several internal microstructural variables, most notably the size and crystallographic orientation of the grains for a given strain rate and temperature [Beyerlein and Tomé, 2008; McCabe et al., 2006; 2009; Proust et al., 2007; Tomé et al., 2001]. The connection between microstructure and slip is usually either made directly in 1

2 microscopic analysis of slip traces or indirectly inferred via polycrystal modeling of texture and stress-strain evolution [Kaschner et al., 2006; Proust et al., 2007; Tomé et al., 2001]. Compared to slip, the microstructure-twinning connection in post-mortem microscopic analysis can be straightforward because, unlike slip, deformation twins manifest as lamellar shaped domains inside the crystals. Within the twin domains the lattice has been significantly reoriented and sheared [Beyerlein et al., 2014; Christian and Mahajan, 1991; Partridge 1967; Yoo, 1981]. Consequently, in orientation mapping such as EBSD or precession electron diffraction (PED), twin lamellae that are even a small fraction (1%) of the grain size in dimension are visible [Barnett el al., 2004a; 2012; Beyerlein et al., 2010; Capolungo et al., 2009a; Carpenter et al., 2012; Econ and Ralph, 1983; Fan et al., 2016; Ghaderi and Barnett, 2011; Jonas et al., 2011; Khosravani et al., 2015; McCabe et al., 2009; Morrow et al., 2013a]. From such analyses, the location, variant, number per grain, and thicknesses of the twin lamellae can be determined [Beyerlein et al., 2010; Capolungo et al., 2009a]. A strong microstructural effect on twinning reported in most of these investigations on deformed HCP metals, as well as polycrystalline metals of other crystal structures, is the grain size effect. Deformation twins are more likely to appear in the larger grains than the smaller ones for similar crystal orientation [Agnew, 2003; Armstrong et al., 1962; Barnett et al., 2004b; Barnett, 2008; Beyerlein et al., 2010; Dobron et al., 2011; Jain et al., 2008; Okazaki and Conrad, 1973]. The effect has been used successfully on several occasions to suppress or lower the occurrence of twinning via grain refinement [Ecob and Ralph 1983; Fan et al., 2016; Jain et al., 2008; Lentz et al., 2014; Ghaderi and Barnett, 2011; Wang and Choo, 2014]. In order to complement the empirical evidence and to facilitate microstructural design of structures using HCP metals and its alloys, a better understanding of the grain size-twin relationship is required. Toward this end, it is important to note that most characterization studies involved EBSD and/or optical methods and, hence, uncovered grain size effects on grain-scale twins [Barnett et al., 2004b; Beyerlein et al., 2010; Capolungo et al., 2009a]. These twins have already crossed the grain and are bounded either by free surface (single crystal, Fig. 1a) or neighboring grains (in a polycrystal, Fig. 1b). Many smaller scale processes for twin nucleation and propagation occurred before the twin could reach grain-scale dimensions [Beyerlein et al., 2011; Niezgoda et 2

3 al., 2014; Wang J et al., 2010; 2013b; Wang L et al., 2010]. The widely reported grain size effects on twinning have predominantly addressed the later stages of twin expansion or thickening, rather than the earlier stage of twin nucleation. Within the grain-scale twin domain, the lattice has been sheared and reoriented and the reaction of the immediate surroundings to the deformation can affect further growth of this domain. If the twin resided in a single crystal and spanned from one free surface to another, the shearing action would be unconstrained. The resulting shape change imposed by the twin domain would manifest on the free surfaces as shown in Fig 1(a). This kind of twin can be very often seen in micro-pillar experiments, for instance [Ye et al., 2011]. If the twin resides in a grain belonging to a polycrystalline aggregate, the twin shear would be constrained by the surrounding material, comprised of neighboring grains (Fig. 1(b)). (a) (b) c c c c c c Neigh. grain Parent grain Neigh. grain Figure 1: Schematics of a twin lamella with different surroundings (a) free surface (constraint free) and (b) neighboring grains. (a) Represents a single crystal case under which the twin experiences little constraint, whereas (b) represents a crystal within a polycrystal wherein the twin shear is constrained by its neighboring grains. The constraint by neighboring grains on the twin shear results in localized stress and strain states that develop most intensely at the twin tip/gb junction and extend into the parent and neighboring grains. These local stress states have been calculated previously using spatially resolved crystal plasticity based micromechanics techniques [Abdolvand and Daymond 2012; 3

4 2013; Ardeljan et al. 2015; Arul Kumar et al, 2015; Kanjarla et al., 2012; Lebensohn et al., 2012]. It was found that the neighboring grain constraint produces a stress concentration at the twin tips and a stress reversal on the twin lamella. The latter resists the driving force for twin boundary migration needed to thicken the twin. It can be envisioned that as the grain size decreases these twin tip/gb stress fields could start to interact and affect the degree of stress reversal in the twin lamella. Such a grain size constraint would resist thickening and would increase with decreases in grain size. Accordingly smaller grains would require more additional applied stress to expand the same twin lamella. In this work, we use micromechanical CP-FFT simulations (Arul Kumar et al., 2015) to investigate grain size effects on the driving forces for expanding (growing) grain-scale twins, where the tips of the twin are connected to the grain boundaries flanking the parent grain. The twin expansion depends on several microstructural features like orientation, grain size, neighboring grain orientation and so on. In our calculations, the parent and twin orientation and the loading direction are fixed, while the neighboring grain orientation and size are varied. We show that for the same twin smaller grains have associated more constraint than larger ones. We also find that the grain size constraint increases for thicker twins, indicating that twin growth is self-suppressing. Grain neighbors that are plastically hard (activating few yet harder slip systems) provide an even higher grain size constraint than plastically soft ones (activating many, easier slip systems). 2. Numerical Method The crystal plasticity Fast Fourier Transform (CP-FFT) micromechanics model we employ here was recently advanced to account for the characteristic shear transformation strain within the finite twin domains (Arul Kumar et al., 2015). As such, it is well suited for studying the characteristics of the non-uniform fields generated where twin lamellae and grain boundaries meet. In our calculations, we consider Mg as a model material and the {10-12} tensile twin mode, the most commonly observed twin type in this metal [Partridge, 1967; Yoo and Lee, 1991; Beyerlein et al., 2010]. Deformation at each material point is carried by a combination of 4

5 anisotropic elasticity and crystallographic slip. The anisotropic elastic constants we use are those measured for Mg at room temperature: C11 = 59.75; C12 = 23.24; C13 = 21.70; C33 = 61.70; C44 = GPa [Simmons and Wang, 1971]. For plasticity we make available three slip modes and their associated slip systems: basal <a> slip, prismatic <a> slip, and pyramidal <c+a> slip. The corresponding initial critical resolved shear stresses (CRSS) to activate these slip modes are: 3.3 MPa for basal, 35.7 MPa for prismatic and 86.2 MPa for pyramidal sip [Beyerlein et al., 2011]. Possible strain hardening in these CRSS values is not taken into account in the present calculations, since our interests lie in the driving forces that would further their growth but not in modeling the growth process itself. Fig. 2 shows the 3D model geometry of the tri-crystal studied here. It is comprised of a central grain of size D and two neighboring grains of size D N. Also shown is a buffer region, surrounding the tri-crystal, possessing a random texture, intended to represent the bulk material response. The crystallographic orientation of the central grain is (0 o, 0 o, 0 o ), with its c-axis aligned with the z-direction. The two neighboring grains have the same crystallographic orientation and will be varied in simulation. The size of the tri-crystal simulation cell is made sufficiently large such that periodic boundary conditions have negligible effects on the results reported here. A uniform compression stress is applied along the y-direction, which produces a uniform stress state in each crystal. When this stress state reaches a critical resolved shear stress (CRSS) of 20.0 MPa on the {10-12}<0-111> twin variant, we embed inside the central grain a single twin lamella of this variant, which spans across the grain and is arrested at both grain boundaries. The inclination of the {10-12} twin boundary in Mg for the parent grain orientation is 43.1 o with respect to the y-direction. Over several fine increments, the characteristic twin shear (12.6% for Mg) is enforced uniformly throughout the twin lamella [Arul Kumar et al., 2015]. The shape of the embedded twin is parallelepipedic, instead of the elliptical one usually adopted. Similar FFT calculations performed with elliptical shaped twins require more refined discretization to avoid local numerical fluctuations and increase computational requirements. However the results are very similar to the ones reported here using parallelepipedic twins. 5

6 Buffer Neighboring grain Parent grain Neighboring grain t Twin D D N Z X Y Figure 2. Tri-crystal simulation setup used to study the effect of parent grain size and grain neighbor orientation on driving stresses for twin growth. The twin lamella, thickness of t, is embedded inside a central parent grain of width D and its twin tips are arrested by the two grain boundaries on both sides formed by neighboring grains both of width D N. The foregoing model is used to calculate the effects of grain size on the driving forces relevant for twin expansion. In doing so, we are aware that the mechanisms involved in twin growth and hence the forces needed to activate them are not completely understood. It has been seen in atomic-scale simulations that the migration of twin boundaries is accommodated by the glide of twinning dislocations (or disconnections) along a coherent twin boundary [Serra and Bacon, 1996; Serra et al., 2010; Wang J et al., 2013a and b]. These twinning dislocations would be driven by a positive resolved shear stress on the twin plane in the direction of the twinning Burgers vector (T-RSS). Other possible mechanisms for twin expansion or twin boundary migration have been proposed, involving the glide and climb of disconnections and facets [Capolungo et al., 2009b; Serra and Bacon, 1996; Wang J et al., 2013a]. The FFT model 6

7 calculates all mechanical tensor fields and thus can provide for any relevant driving force depending on the migration mechanism. Here, in the interest of relating grain size and grain neighbor orientation to twin growth, we elect to use a positive T-RSS as a plausible driving force for thickening the twin. Later, we will address other driving forces, associated with the motion of facets, and show that they do not change the results obtained here. 3. Results 3.1. Grain size effects on driving forces for twin growth In order to study grain size effects on twinning, the parent grain size is varied while keeping the neighboring grain size fixed. The neighbor grain size is made sufficiently large (D N = 400 voxels) such that it will not affect the localized plastic stress field that develops at the twin and grain boundary junction. The size of the parent will be varied from D/D N = 0.3 (120 voxels) to D/D N = 1.5 (600 voxels). The dimension of the simulation cell in the vertical direction, i.e., z- direction, is taken as 1200 voxels, which is large enough to avoid the effects of periodic boundary conditions. We first consider the largest parent grain size D/D N = 1.5. In this case the orientation of the neighboring grains (Bunge angles) is (0 o, 5 o, 0 o ), corresponding to a slight mis-orientation of ~5 with respect to the central parent grain. This case represents a typical grain within a strongly textured material. The compression stress of ~40 MPa is applied to the tri-crystal, which generates a uniform T-RSS field in the center parent crystal of T-RSS 0 = MPa, which exceeds the CRSS = 20.0 MPa, presumably sufficient to produce a {10-12}<0-111> twin. Under this state, we insert a twin lamella spanning the central crystal, with the associated lattice reorientation and twin shear imposed as described earlier. The twin lamella is assumed to be relatively thin with thickness t=13 voxels, which yields a twin volume fraction of 1.5%. For the purpose of this calculation the field associated with the twin transformation should be independent of the periodic boundary conditions and the dimension of the neighbor. As a consequence, the latter fraction is only an indicator that guarantees such condition. We further 7

8 define an effective grain size, D* = D/t, by normalizing by the twin thickness, which for this first case is D*~46. Fig. 3(a) shows the T-RSS stress field due to this grain-scale twin after an equilibrated state is achieved. As seen, an inhomogeneous stress state develops in the parent grain, neighboring grains, and within the twin domain itself. It arises as a result of the shear transformation, which is concentrated within a finite band-like domain, and the constraints of the surrounding neighboring and parent grains. In Fig. 3(a) we see that at the twin tips the stress state is locally complex in the parent and in the neighboring grain, unlike the nearly uniform state farther away from the twin lamella. As a consequence of the twin shear a positive stress concentration (T-RSS/T-RSS 0 > 1) develops locally in the neighboring grain at the junction of the twin tip and grain boundary. As a reaction to the constraint imposed by the neighboring grain on this twin shear, a backstress where T-RSS < CRSS is generated in the twin domain, which would indicate that twin growth would be stunted under the current loading state. (a) T-RSS field (b) T-RSS profile along twin interface Before twinning D* = ~46: A er Twinning D* = ~15: A er Twinning TRSS [MPa] Backstress S max 70 Twin p of D* = 46 Twin p of D* = 15 Twin p of D* = Posi on along twin interface [Voxels] Figure 3 (a) Twin plane resolved shear stress (T-RSS) distribution after twinning with the effective grain size D* = D/t = 46. (b) T-RSS profile along twin interface before and after twinning with D* = 46 and 15. The neighboring grain orientation is (0 o, 5 o, 0 o ). 8

9 The T-RSS reversal within the twin domain induced by twin formation has been detected experimentally via x-ray synchrotron diffraction experiments [Aydiner et al., 2009]. In this test the authors measured the average T-RSS in the incipient (less than 1% of grain volume) twin lamella, finding that it was negative. Using FFT we can determine from the calculated stress fields (such as Fig. 3a) the average backstress S B in the twin lamella, which we define here as the difference in the averaged T-RSS in the domain of the twin lamellae, S ave, before and after twinning under the same external applied strain. S B S Before After ave S ave (1) Because the former is positive and the latter usually negative (see Fig 3b) a higher value of S B indicates that a higher applied stress would be needed to reverse the transformationinduced stress and potentially thicken the twin. Figure 4(a) shows the calculated variation in S B with D* = D/t. As shown, S B increases non-linearly as D* decreases: S B = 34.2 MPa for D*:46 and S B = 47.7 MPa when for D*:9. In these calculations the enhancement in backstress that could hinder twin expansion is solely due to grain size, since the orientations of the parent, neighbor and twin remained the same. Therefore, these results suggest that decreasing grain size makes it harder to grow a twin since the twin-suppressing backstress field has increased. Back stress, S B [MPa] D* = D/t 50 (a) (b) (c) Back stress, S B [MPa] (D*) 1/2 Applied stress to grow twin [MPa] (D*) 1/2 Figure 4 (a) Relationship between the average backstress, S B, in the twin lamella and the effective grain size D* = D/t; (b) a Hall-Petch plot of the backstress and (D*) -1/2 and (c) a Hall- Petch plot of the applied stress needed to grow the twin and (D*) -1/2. The orientation of the neighbor is (0 o, 5 o, 0 o ). One expectation based on experimental observation is that twinnability, a general probability for twinning, follows a Hall-Petch effect [Barnett et al., 2015; Cepeda-Jimenez et al., 9

10 2016; Dobron et al., 2011; Ghaderi and Barnett, 2011; Jain et al., 2008]. Figure 4(b) shows the same results as Fig. 4(a) but on a Hall Petch plot, where a linear relationship would imply a Hall Petch scaling law. As it turns out, we find that a linear scaling indeed emerges between S B and (D*) -1/2. The backstress in the twin S B, although measurable [Aydiner et al., 2009; Balogh et al., 2013], would be extremely difficult to assess experimentally for the same twin inside different parent grain sizes. Numerically, we accomplish that in Fig. 4(a). Also it would be experimentally hard to measure the far field applied stress that is required to grow the same twin inside different parent grain sizes. For this reason, we calculate the required applied stress such that the average T-RSS in the twin, S ave, exceeds the CRSS. The applied stress required to get S ave greater than CRSS is 58 MPa for D*=46 and 81.0 MPa for D*=9. The required applied stress for twin expansion as a function of all D* on a Hall Petch plot is shown in Fig. 4(c). To grow the twin, the T-RSS that matters is the one acting at the twin boundary, rather than the average stress in the twin lamella. With this in mind, Fig. 3(b) shows the T-RSS profile for the top twin boundary before and after twinning transformation for two grain sizes: D* =15 and 46 (the T-RSS profile along the bottom boundary is inversely symmetric). As before, we analyze the backstress, defined in this case as the difference in the T-RSS along the boundary before and after twinning under the same applied strain (see Fig. 3(b)). For D* = 46, the backstress is positive, resisting twin growth, along the entire boundary. It is most severe at the twin/gb junction and decays along the twin boundary toward the center of the grain. An additional applied stress would be required to overcome the backstress and locally increase the T-RSS stress along the twin interface to the value of the CRSS. From Fig. 3b, we see that the twin boundary located at the center of the grain experiences the lowest value of the backstress or, equivalently, the highest CRSS S max, and hence is the most likely part of the twin interface to reach a suitable value for twin growth when more stress is applied. Using CP-FFT simulation, we raise the applied stress further and recalculate the stress state, finding that an increase from 40 MPa to 56.0 MPa locally increased S max to the twin CRSS of ~20 MPa. 10

11 For comparison, Fig. 3b also shows the corresponding T-RSS profile along the twin boundary after twinning in a smaller grain D* = 15. The shape of the T-RSS profile is similar to the previous larger grain case of D* = 46, with the most severe backstress generated at the twin tip/grain boundary junction and the least backstress S max occurring in the center of the twin. With the reduction in grain size, the former changes in value only slightly; however, S max has decreased (becoming more negative) significantly. As before, we incremented the applied stress further to determine the critical value to raise S max to the twin CRSS. We find that it is 71.5 MPa, higher than the stress needed to grow the twin in the larger grain D* = 46. Thus far the calculations have shown that reducing the grain size increases the average backstress in the lamella and also the local backstress along the twin boundary in the center of the twin. Thus, local backstress along the twin boundary in the center of the twin is the smallest for the largest grain size. To further examine grain size effects, we calculate S D, which is the difference in S max for different grain sizes D* from 9 to 46 with respect to S max for the largest grain size studied D * D * R 46. Specifically, S D is defined as (2) Here the S max is calculated over the central region of the twin boundary marked as Σ in Fig. 5(b). A high value of S D suggests that the particular case experiences larger backstresses compared to the reference case (D* = 46). Figure 5 shows that S D increases with reductions in grain size. These finer calculations, however, indicate the existence of a critical D* below which grain size effects become significant. For the particular case studied in Fig. 4, two well-defined rectilinear slopes can be identified, each one associated with strong and weak dependence of S max with grain size (see Fig. 5a). We choose to define the critical point at the intersection of the slopes, namely, at D * C

12 (a) S D S max DR * S max D* (b) S D [MPa] 6 Σ 4 D C * Twin D* = D/t Figure 5: The difference in S max for different D* with respect to the reference case grain size, i.e., D * D * R 46. The S max is calculated by averaging the T-RSS over the region Σ (see (b)). The size of the region is six-voxels along the twin boundary in the middle of the twin lamella. The orientation of the neighbor is (0 o, 5 o, 0 o ). The intersection of the slopes indicated in Fig 5a suggest a qualitative change in behavior past D * C Interaction of twin tips To understand how grain size can affect the value of backstresses within the twin, we return to the entire T-RSS profile along the twin boundary from the twin/gb junction to the center of the parent grain. Fig. 6 shows the corresponding T-RSS profiles for different D*. Over this span of grain sizes, the peak backstress generated at the twin tip/gb junction changes only slightly with grain size reduction; however, the value of S max in the center of the twin increases (becoming more negative) with smaller grain size. 12

13 TRSS [MPa] D*: L eff for D* = Posi on along twin interface from twin center [Voxels] Figure 6: T-RSS profile along twin boundary from the twin center to the twin tip for different D* = D/t from ~9 to 46. The orientation of the neighbor is (0 o, 5 o, 0 o ). In the largest size grain, D* = 46, the reaction stresses at the twin tip/gb junction extend into the parent grain a finite distance and eventually reach a plateau value S max. As the grain size shrinks below D* = 46, the stress fields from the twin tips on the opposing sides of the grain are brought closer together. When they become sufficiently close, they can interact. We see from Fig. 6 that interaction causes the value of S max to decrease perceptibly when D * C 21. To be more quantitative, we define an effective region L eff from the twin/gb junction to the point where the T-RSS reaches 90% of S max. As long as D > 2L eff, the stress fields emanating from the twin tips are not interacting and the maximum value S max does not change. However, when D < 2L eff, the twin tip stress fields interact. Thus, D = 2L eff becomes an approximate critical value below which S max is significantly affected. For instance, for D* = 46, the critical grain size D * C = 2L eff /t, is 17.8, which agrees well with 21, the grain size below which grain size effect is seen to strengthen significantly. The 90% cut-off value on which L eff is defined is clearly 13

14 arbitrary, but nonetheless, this analysis indicates that the grain size constraint on twin growth strengthens substantially when the grain size is sufficiently small to cause the twin tip/gb reaction stress fields to strongly interact Nearest-neighbor effects The foregoing result suggests that the grain size constraint on twin growth ought to depend on the characteristics of the stress fields at the twin tip/gb intersection. Characteristics such as the severity of the backstress and its extent could depend on the deformation response of the neighboring grain on the other side of the GB. The neighboring grain reacts to the twin shear at the twin tip/gb intersection predominantly by relaxing stress via plastic deformation. For an HCP grain like Mg the plastic behavior can depend sensitively on its crystallographic orientation. Thus, a change in the neighboring grain orientation could easily convert it from one that is plastically hard to one that is plastically soft. If the plastically hard neighboring grain were to indeed impose the greater constraint on the same twin and its parent, thereby enhancing the severity and/or extent of the backstresses in the twin, then the neighbor orientation could favor twin growth. Neighboring grain could affect not only twin growth in the parent but also the effect of parent grain size on twin growth. To explore nearest neighbor effects on grain size constraints, we carry out the simulations on the same parent and twin but with different neighbors, ideally a few that represent extreme cases of plastically soft and plastically hard. However, this choice cannot be made a priori based on a Schmid factor analysis of the neighboring and parent grain orientations. In general, plastic accommodation of the complex stress states created in the parent and neighbor in the vicinity of the twin/grain boundary junction in an HCP would need to involve slip on multiple slip systems. For Mg, the easiest basal <a> slip (~3 MPa) only provides two independent slip systems. Thus, basal slip alone would be insufficient and in general prismatic <a> slip and/or pyramidal <c+a> slip would also have to be activated to accommodate the complex stress state at the twin/grain boundary junction. How easily an adequate number of slip systems can be activated would be one way to determine whether a neighboring grain orientation is plastically soft or plastically hard. 14

15 To obtain an idea of which slip systems could be activated at the twin tip/gb junction, we consider the following geometric quantity m : m ' (b T.b s )(n T.n s ), (3) which is a measure of the relative orientation of a slip system (b s,n s ) in the neighboring grain to the twin variant in the parent (b T,n T ). If the value of m is 1.0, then the shear imposed by the twin variant in the parent is well aligned to cause shear on this particular slip system in the neighbor. Using this measure, a plastically soft neighbor would be oriented such that m is not only nonzero on many slip systems, but predominantly on the relatively easier ones. A plastically hard neighbor, on the other hand, may also have a non-zero m on many slip systems which pertain to the harder slip modes. Based on this notion, two neighboring grain orientations were selected, one plastically hard neighbor with (90 o, 86.3 o, 90 o ) and one plastically soft neighbor with (0 o, 43 o, 45 o ). Table 2 shows the m values for these two cases as well as those for the 5 misoriented grain neighbor studied earlier. Grain neighbor Orientation Maximum value of m for each deformation mode (Bunge angles) Basal <a> Prismatic <a> Pyramidal <c+a> 5 o misoriented (0 o, 5 o, 0 o ) Hard neighbor (90 o, 86.3 o, -90 o ) Soft neighbor (0 o, 43 o, -45 o ) Table 1. The three different neighboring grain orientations studied here and its crystallographic details The grain selected here as the plastically hard one has its highest values of m for prismatic <a> slip followed by pyramidal <c+a> slip, two of the harder to activate slip modes in Mg. The soft neighbor grain, in contrast, has its higher m values for the easiest basal <a> slip system in Mg. By the same criterion, the earlier case we presented with the 5 misoriented 15

16 neighbor fits into the category of a plastically hard neighbor since its highest m corresponds to pyramidal slip. 90 (a) 90 (b) Back stress, S B [MPa] 70 With 5deg misori Hard neighbor 60 Soft neighbor Applied stress to grow twin [MPa] With 5deg misori Hard neighbor 30 Soft neighbor (D*) 1/2 (D*) 1/2 S D [MPa] (c) * D S D S R D max * Smax with 5 deg misori Hard neighbor Soft neighbor D* = D/t Figure 7. Effect of neighboring grain orientation on the relationship between (a) the backstress, S B and grain size via a Hall-Petch scaling 1/(D*) 1/2 ;(b) the applied stress to grow twin, i.e., S ave > CRSS and grain size via a Hall-Petch scaling; and (c) the difference in S max compared to that for D * D * R 46 as a function of grain size. S max is calculated by averaging T-RSS in the region shown in Fig. 5. Observe that the behavior in the hard neighbor case (Fig 5c) is closer to a power law rather than the bilinear one used to define * C D

17 Figure 7a compares the average backstress, S B, in the twin lamella for the full range of grain sizes D* for the plastically hard and soft neighbor with the original case of a neighbor with a 5 o misorientation. As before, the twin backstress increases with decreasing D* for the two additional cases as well. In Fig. 7b, we present the Hall-Petch plot for the total applied stress needed to grow the twin further after the twin transformation. Again, the grain size scaling emerges in both cases, despite the disparate plastic responses of these two neighboring grain orientations. The results in Fig. 7 indicate that the orientation and plastic response of the neighbor influences the grain size effect for twinning. The plastically harder the neighboring grain, the more severe the grain size constraint is. Figure 7b illustrates best such effect, as it shows that the stress needed to grow the same twin in the same parent is higher if the neighbor is plastically harder. This neighbor-grain influence can be attributed to the differences in the stress field emanating from the twin/gb junction. We calculate L eff in each case using the T-RSS profile calculated for the largest grain size considered (D* = 46), finding that the effective region increases when going from the plastically harder to softer neighbor. These values would suggest that the critical grain sizes, 2L eff /t, are 27.5 and 15.0 for hard and soft neighbors, respectively. To compare this estimate with the calculation, Figure 7c shows the decrease in S max with respect to D* = 46. As shown, the grain size constraint effect takes hold at a larger critical value as the plastic response of the neighbor becomes harder. The calculated D C /D N values follow well the estimates, a consistency that simply confirms that grain size constraint begins to strengthen when the twin/gb stress fields begin to interact. Overall, these results demonstrate that plastically harder neighbors will provide a higher grain size constraint on twin growth than a plastically softer neighbor. Thus the grain size constraint on twin growth will not be the same for two grains of the same orientation and containing the same twin but with different neighboring grains. This grain neighbor effect could be one explanation for why in a polycrystal some grains twin and others do not, despite being of similar size and orientation [Beyerlein et al., 2010]. This finding could also provide insight into designing microstructures that hamper twinning for the sake of increasing formability. Grain 17

18 neighbors that are plastically hard provide an even higher grain size constraint that could make grain refinement even more effective on suppressing twinning Grain size effect on alternative driving stresses for twin growth Some studies have observed that the twin boundaries are faceted at the nanometric scale with a sequence of coherent twin boundary (CTB), basal-prismatic interface (B/P) and prismaticbasal interface (P/B) [Morrow et al., 2013b; Sun et al., 2014; 2015; Wang J et al., 2013a]. For such a boundary, migration would involve motion of these facets, and hence stress measures other than the T-RSS would be required to drive boundary migration. The B/P boundaries are parallel to basal planes in parent grain and prismatic planes in twin, and vice versa for the P/B boundaries. The basal and prismatic planes separation is 5.21 A o and 5.55A o, respectively, for Mg. Therefore, expanding the twin domain via motion of B/P steps implies increasing the plane spacing when transforming basal planes in the parent to prismatic planes in the twin. Therefore, a local tensile force acting normal to the B/P face would be required. Likewise, to expand the twin via motion of P/B boundaries requires transformation of prismatic planes to basal planes in the parent, requiring compressive force acting on the P/B boundary face. In all simulations presented thus far, tensorial fields of displacement gradients, stress, and strain are calculated and from these any scalar component can be analyzed. To determine the grain size constraint on twin growth via B/P and P/B step motion, we extract the corresponding driving forces: a tensile stress along the B/P boundary normal or a compressive stress along the P/B boundary normal. In the tri-crystal model, the orientation of the parent grain remains with its c-axis aligned along the z-direction. Thus, the prismatic planes in the parent grain and the basal planes in the twin are parallel to z-direction. The B/P interface normal is then the y-direction and the P/B interface normal the z-direction. So, for this particular orientation, expanding the twin requires a normal tensile stress acting along the z-direction (+S ZZ ) or a normal compressive stress along the y-direction ( S YY ). Figure 8 plots the normal stresses S YY and S ZZ along twin interface for Mg before and after twinning for a range of grain sizes D*. In this example, we consider the 5 o misoriented 18

19 grain neighbors. Before twinning, S YY is compressive 46 MPa and nearly constant along twin interface, while the S ZZ component is almost zero because zero stress is imposed along the z- direction. After twinning, the grain size constraint adds a backstress to S YY and the compressive S YY decreases in value along twin interfaces for the wide range of D* selected. The backstress is more severe at the twin/gb junction and less negative along the boundary towards the twin center. This variation is similar to that of the T-RSS component analyzed earlier. Thus, just as for the T-RSS driving force, this result implies that twin expansion by the migration of P/B interfaces would also be suppressed and an increase in the applied deformation is needed to drive twin growth by this mechanism. Likewise, after twinning S ZZ becomes negative, which is in the opposite sense needed to migrate the boundary. So, twin expansion by the migration of B/P interfaces is also suppressed and an increase in the applied deformation would also be required to overcome it and expand the twin. To summarize, for three-twin boundary migration modes, namely twinning dislocation glide along the twin boundary, and B/P and P/B step motion, the grain neighbor constrains the expansion of a twin lamella. Further, twin expansion (increase in thickness) would be a self-suppressing event. By applying macroscopic load, the stresses in the center of the twin boundary would be the most favorable to initiate the expansion process. 40 (a) S YY 20 D*: S YY [MPa] Before twinning Twin center Posi on along TB from twin center 19

20 (b) S ZZ Before twinning S ZZ [MPa] D*: Twin center Posi on along TB from twin center Figure 8. (a) S YY and (b) S ZZ profile along the twin boundary from twin center to twin tip before twinning and after twinning for different D* = D/t. The orientation of the neighbor is (0 o, 5 o, 0 o ). While we find that the trends in S YY and S ZZ are similar to T-RSS, they exhibit different grain size constraint effects. Their grain size sensitivities are compared directly in Fig. 9, which shows the difference in the S YY, S ZZ, and T-RSS stresses in the region Σ, for different grain sizes with respect to D * R 46. The decline in the compressive stress that could migrate the P/B interface is similar to that in the T-RSS associated with migrating the CTB. However, the decrease in the tensile stress in B/P interface is relatively small. These results indicate that the grain size constraint for twin expansion is significant for the migration of CTB and P/B interface, not for the migration of B/P interface. A potential consequence of this result is that, if one assumes that BP and PB steps nucleated by slip dislocation reactions at the CTB [Serra et al, 1999], then the twin will show a tendency to expand more away than towards the tip, which may partly explain the spindle shape of not fully expanded twins 20

21 14 X D X max * DR X max DR Difference in driving stresses, X D, [MPa] X : TRSS X : SYY X : SZZ D* = D/t Figure 9. The difference in the T-RSS for migrating the twin boundary, S YY for moving the P/B interface and S ZZ for moving the B/P interface for different grain sizes with respect to D * D R * 46. The stresses are averaged in the region Σ as shown in Fig. 5. The orientation of the neighbor is (0 o, 5 o, 0 o ). 4. Discussion Deformation twinning can be visualized as a sequential process of nucleation, propagation and growth, and grain size plays an important role in all of these three stages of twinning. The dependence of grain size on the twin nucleation has been experimentally studied (Beyerlein et al., 2010; Capolungo et al., 2009a) and statistically represented in continuum scale predictive models (Niezgoda et al., 2014). In this work, we focus on the effect of grain size on twin growth (expansion) of a fully developed grain-scale twin. To do so we analyze the local stresses at the twin-matrix interface, since we know that the latter drives twin growth. The twinning shear transformation, which is constrained by the surrounding parent and neighboring grains, significantly alters the distribution of local stresses at the twin interface. The current work 21

22 provides an understanding on the dependence of grain size on the twin local stresses, which is responsible for twin growth. The CP-FFT based calculations show that: (a) the decrease in the grain size increases the average backstress in the twin lamella and the applied stress required to start the twin expansion; (b) the grain size dependency is severe in the smaller grains and almost negligible for grains larger than a critical grain size; (c) the grain size dependency is enhanced by the presence of hard neighboring grains (activating a few, harder slip systems) and less in the case of soft neighboring grains (activating many, easier slip systems); (d) the effect of grain size on the alternative driving stresses for twin growth (normal stresses at BP or PB facets) follows similar trends as those for the TRSS that drives growth of CTBs. In what follows we compare and discuss the current FFT based results with the available experimental observations of grain size effects on twinning. First we focus on the finding that the applied stress required for twin growth increases with decrease in grain size. In tests performed on AZ31 Mg alloy with four different average grain sizes, Barnett et al. (2012) showed that the twin aspect ratio increases with applied stress as the grain size increases (See Fig. 12 of Barnett et al., 2012). From that result we can deduce that the applied stress required to thicken the twins in larger grains is less compared to the one in the smaller grains. Although our FFT calculations do not simulate twin growth, they indicate that the applied stress required to reach the critical TRSS would be smaller for larger grains, which is in agreement with the experimental result. Next we focus on the saturation of the grain size effect on twinning. The present study shows that the effect of grain size on twinning is severe in grains whose size is less than a critical grain size approximately 30 times the twin thickness (see Fig. 5). EBSD statistical analysis done by Beyerlein et al. (2010) in pure Mg reports fraction of grains, fraction of twinned grains, and twinning frequency as a function of grain size, in the 5 to 80 m range (see Fig. 10). The observation is that the frequency of twinning is increases with grain sizes. But the tendency is not monotonic over the entire grain size regime. The twinning frequency increases rapidly for smaller grains, up to 30 m in size, and tends to saturate for grains larger than 30 m. The two regimes of different grain size dependencies on twinning are marked in the Fig. 10. Our critical size estimate would predict that in grains larger than 30 m activation of twins of 1 m would not depend on grain size (D/t~30). This observation of grain size dependency on the twinning 22

23 frequency is consistent with the FFT based grain size dependency on the twin backstress. Although twin number frequency cannot be directly compared with the local driving stress for twin growth, it is reasonable to presume that they would be strongly correlated Grain number frac on Twinned grains All grains Twinning frequency Twinning frequency Grain size (μm) Figure 10. Total number fraction of all the grains and those grains that contain at least one twin as a function of grain size for Mg after 3% compressive strain. The relative twinning frequency (defined as the ratio between the two previous values) is reported in the second vertical axis as a function of grain size (Beyerlein et al., 2010). Two clearly defined regimes (shaded boxes) are apparent: a rapid increase for smaller grains and a trend to saturate for larger grains. In this work we analyze the effect of parent grain size on the deformation twin growth and considered the neighboring grain size to be large enough to encompass the entire localized plastic fields within it. In such a case the neighboring grain size does not play a role on the twin growth process. However in contrast, the neighboring grain size plays an important role on the formation of twin chains. If the neighboring grain size is smaller than the region of localized 23

24 plastic field associated with the twin, then the plastic field will alter the stress states in the noncontiguous grain boundary and may favor the formation of twin chains. Future work will study the role of both parent and neighboring grain sizes on twin chain formation and involve implementation this dependency into our polycrystal plasticity visco-plastic self-consistent (VPSC) model. 5. Summary In summary, we show that reductions in grain size can increase the threshold stress that needs to be applied for growing grain-scale twin lamellae. The constraint from the surrounding neighboring grains generates a backstress on the twin lamella that could hinder further growth. The backstress fields are strong where the grain boundary and twin meet and decay into the grain. Reducing the grain size effectively brings these backstress fields closer and for grain sizes that are small enough the magnitude of the backstress is driven up significantly. Consequently a higher increment in applied stress is required to thicken a twin in a smaller grain. For the same grain size, the constraint on twin thickening is stronger for a plastically hard (less easy slip systems) neighboring grain than a plastically soft (more easy slip systems) one. These results elucidate distinctive and effective roles of grain size and grain neighbor properties that could aid in the microstructural design of HCP metals. Acknowledgments This work is fully funded by the U.S. Dept. of Energy, Office of Basic Energy Sciences Project FWP 06SCPE401. The authors are grateful to Dr. R. A. Lebensohn for making available the FFT-EVPSC code used here for the simulations. The authors would like to thank Dr. R. J. McCabe for providing the data used in Fig. 10. References [1]. H. Abdolvand and M.R. Daymond, Internal strain and texture development during twinning: comparing neutron diffraction measurements with crystal plasticity finite element approaches, Acta Mater 60 (2012) [2]. H. Abdolvand and M.R. Daymond, Multi scale modeling and experimental study of twin inception and propagation in HCP materials using a crystal plasticity finite element approach Part I: average behavior, J Mech Phys Solids 61 (2013) [3]. S.R. Agnew, C.N. Tomé, D.W.Brown, T.M. Holden, S.C. Vogel, "Study of slip 24

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29 c c

30 c c c c Neigh. grain Parent grain Neigh. grain

31 Buffer Neighboring grain Parent grain Neighboring grain t Twin D D N Z X Y

32

33 80 Before twinning D* = ~46: A<er Twinning D* = ~15: A<er Twinning 50 TRSS [MPa] Backstress S max Twin tip of D* = 46 Twin tip of D* = 15 Twin tip of D* = Posi-on along twin interface [Voxels]

34 50 (a) Back stress, SB [MPa] D* = D/t

35 50 (b) Back stress, SB [MPa] (D*)-1/2

36 Applied stress to grow twin [MPa] 85 (c) (D*)-1/2

37 (a) S D = S max DR * S max D* SD [MPa] 6 4 D C * D* = D/t

38 (b) Σ Twin

39 10 TRSS [MPa] D*: L eff for D* = Posi-on along twin interface from twin center [Voxels]

40 Back stress, SB [MPa] (a) With 5deg misori Hard neighbor So= neighbor (D*)-1/2

41 Applied stress to grow twin [MPa] (b) 40 With 5deg misori Hard neighbor 30 So= neighbor (D*)-1/2

42 SD [MPa] (c) * D S D = S R D max * Smax with 5 deg misori Hard neighbor So8 neighbor D* = D/t

43 40 (a) S YY SYY [MPa] D*: Before twinning Twin center Posi,on along TB from twin center

44 (b) S ZZ Before twinning SZZ [MPa] D*: Twin center Posi,on along TB from twin center

45 Difference in driving stresses, XD, [MPa] X D = X max * DR X max DR X : TRSS X : SYY X : SZZ D* = D/t

46 Grain number frac2on Twinned grains All grains Twinning frequency Twinning frequency Grain size (μm)