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The effect of deformation twinning on stress localization in a three dimensional TWIP steel microstructure

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Modelling Simul. Mater. Sci. Eng. 23 (2015) 045010 (18pp) doi:10.1088/0965-0393/23/4/045010

The effect of deformation twinning on stress localization in a three dimensional TWIP steel microstructure

Vahid Tari1, Anthony D Rollett2, Haitham El Kadiri134, Hossein Beladi5, Andrew L Oppedal1 and Roger L King1,6

1 Center for Advanced Vehicular Systems, Mississippi State University, Mississippi State, MS 39762, USA

2 Materials Science and Engineering Department, Carnegie Mellon University, Pittsburgh, PA 15213, USA

3 Department of Mechanical Engineering, Mississippi State University, Mississippi State, MS 39762, USA

4 Adjunct Faculty, Université International de Rabat, Parc Technopolis Rabat-Shore, Rocade, Rabat-Salé, 11100 Sala El Jadida, Morocco

5 Institute for Frontier Materials, Deakin University, Geelong, VIC 3216, Australia

6 Department of Electrical and Computer Engineering, Mississippi State University, Mississippi State, MS 39762, USA

E-mail: vahidtari@gmail.com, vt152@msstate.edu

Received 25 August 2014, revised 11 March 2015 Accepted for publication 23 March 2015 Published 23 April 2015

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Abstract

We present an investigation of the effect of deformation twinning on the visco-plastic response and stress localization in a low stacking fault energy twinning-induced plasticity (TWIP) steel under uniaxial tension loading. The three-dimensional full field response was simulated using the fast Fourier transform method. The initial microstructure was obtained from a three dimensional serial section using electron backscatter diffraction. Twin volume fraction evolution upon strain was measured so the hardening parameters of the simple Voce model could be identified to fit both the stress-strain behavior and twinning activity. General trends of texture evolution were acceptably predicted including the typical sharpening and balance between the (1 1 1) fiber and the (1 0 0) fiber. Twinning was found to nucleate preferentially at grain boundaries although the predominant twin reorientation scheme did not allow spatial propagation to be captured. Hot spots in stress correlated with the boundaries of twinned voxel domains, which either impeded or enhanced twinning based on which deformation modes were active locally.

Keywords: TWIP steel, twinning, viscoplastic, electron backscatter diffraction

(Some figures may appear in colour only in the online journal)

0965-0393/15/045010+18$33.00 © 2015 IOP Publishing Ltd Printed in the UK

1. Introduction

Twinning-induced plasticity (TWIP) steels have high strength and ductility and are finding use in demanding applications in the automotive industry. The high manganese (Mn) content in TWIP steel (typically ~ 20%) reduces stacking fault energy and increases the formation of mechanical twins [1]. Several investigations have been performed to study the effect of various factors such as grain size [2], chemical composition [3] and grain orientation [4] on the formation of mechanical twinning.

During plastic deformation, the orientation of material deformed by dislocation slip changes modestly, whereas material undergoing deformation by twinning causes a large discrete change in orientation, such as the ~60° for the {111} < 112) twin [5]. These orientation changes cause homogeneous and localized deformation regions, respectively. Several computational twinning models have been developed for efficient capture of the crystallographic reorientation caused by deformation twinning in polycrystals [6, 7].

The predominant twin reorientation (PTR) scheme [7] was implemented in the visco-plastic self-consistent (VPSC) polycrystal simulation code to predict twin volume fraction and grain reorientation due to deformation twinning. In previous research efforts, the VPSC code was used to predict texture evolution in HCP [8] and FCC [9] metals undergoing deformation twinning during plastic deformation. The VPSC method is a 'mean-field' approach, which is based on an Eshelby-like interaction of each grain with a homogenized medium. The mean-field approach allows the strain (increment) in each grain to deviate from the average while determining the response of each grain. The grain based approach means that there is no information on the local stress and strain in specific regions such as the interior of a grain or near boundaries.

Because of this limitation and to gain better insight into the spatial distribution of mechanical twinning, full-field calculations such as the crystal plasticity finite element method (CPFEM) have been used. CPFEM, based on a non-homogenization scheme, is a full-field solution of crystal plasticity [10]. Some researchers implemented a constitutive model to compute both texture and twin volume fraction [11]. Kalidindi [12, 13] also proposed a model based on the total Lagrangian approach to predict twin volume faction and texture evolution. Several examples are available of the use of the PTR model in CPFEM to calculate the deformation twin volume fraction and stress distribution in hexagonal metals [14-16]. Use of the FE method requires a mesh; it is easy to generate a pixelated mesh, which leads to large numbers of degrees of freedom in the calculation. Less dense meshes that conform to the grain boundary network can also be generated but the process is complicated and time-consuming.

A full field solution based on visco-plastic fast Fourier transforms (vpFFT) was applied to find the local stress and strain rate inside of a grain or near to grain boundaries. The simulation domain for vpFFT is a three-dimensional image of the microstructure of interest. Originally, the full field solution for stress and strain rate using Fast Fourier transform (FFT) was developed to compute the elastic and inelastic, effective and local response of composites [17]. In further developments, Lebensohn [18] and his collaborators [19, 20] used FFT to compute the full field solution of a visco-plastic polycrystalline aggregate. The vpFFT simulations on both synthetic and measured microstructures showed that highly stressed regions ('hot spots' [21]) tend to occur near microstructural features such as grain boundaries.

In this paper we propose an approach for the incorporation of twinning in an actual three dimensional TWIP steel microstructure with vpFFT. The actual three dimensional TWIP steel microstructure was obtained by three dimensional serial sectioning electron backscatter diffraction (EBSD). An approach that we introduce as 'three dimensional PTR' is used in this work and is based on the predominant twin reorientation (PTR) scheme. This approach does

not capture the shape or growth of a twinned region after nucleation, so the typical lamellar, or plate like, morphology of twins is not accounted for. However, the twin volume fraction is controlled based upon considering an admissible accumulated shear of the predominant twin system (PTS). Thus, the approach should perform better for the case of random textures where deviations from the average strain rates play an important role in twin nucleation. We benchmark the results of twinning contribution to strain localization with the behavior of a three-dimensional (3D) microstructure of a classical high Mn-containing TWIP steel.

2. Methods

2.1. 3D microstructure

A hot rolled Fe-0.6C-18Mn-1.5Al (wt%) TWIP steel [4] was loaded under uniaxial tension along the transverse direction (TD) until about 0.4 total strain. Electron backscatter diffraction (EBSD) was performed on the pre-deformed state and after 0.4 strain in an effort to instantiate the numerical simulations by the final deformation texture and twin volume fraction. The sample preparation procedure and EBSD characterization were described by Beladi et al [4] and the reader is referred to this work for more details.

To generate a realistic 3D grain orientation microstructure for use by the FFT model, a rectangular specimen was extracted from the middle of the hot rolled plate perpendicular to TD. Then, two sides of the sample were mechanically ground to make parallel sides normal to TD, which were then subjected to the 3D EBSD serial sectioning technique explained in detail by Beladi and Rohrer [22]. The experimental data was then processed using the Dream.3D (Digital Representation Environment for Analyzing Microstructure) software package [23, 24] to generate the 3D orientation map shown in figure 1.

Because of poor reconstruction at the edges, 50 voxels were removed at the edges in the x and y directions and a subset with dimensions 128 x 128 x 100 was extracted from the original 3D image which had dimensions of 343 x 267 x 100 in the x, y and z directions, respectively. Buffer layers (28) were added in the z dimension to make a cell with power-of-two dimensions. By inserting a buffer layer, a free surface is obtained on the top and bottom of the structure. In the Dream.3D clean-up procedure, any grains larger than 50 voxels were considered as a grain and smaller regions were absorbed into their majority neighbor. This resulted in a total of 713 grains in the FFT input microstructure. The FFT input texture (figure 2(b)) was close to the experimental one (figure 2(a)) despite the small number of grains in the domain.

Since the simulation is periodic, but the actual microstructure is not, there is a concern about artifacts near the edges. However, in work by Rollett et al [21] this was shown to not be a concern.

2.2. FFT method and hardening rule

The description of the FFT method for simulating the visco-plastic behavior of 3D polycrystal-line materials has already been described in detail [18-20, 25]. The key feature of the method is the use of a Green's function in the solution, which leads to a convolution integral that is replaced via the FFT with a local tensor product. We emphasize, however, that because the perturbation (polarization) field of the local strain at each grid-point is not a priori known, an iterative convergence criterion must be employed to compute a compatible strain rate field that fulfills the equilibrium condition. Since the latter is sensitive to the contrast in local properties [18, 21], the augmented Lagrangian algorithm [20] was used wherein the compatible local

Figure 1. Three-dimensional inverse pole figure (IPF) grain orientation map reconstructed by electron backscatter diffraction (EBSD). The colors in the IPF map correspond to the orientations in the transverse direction.

Figure 2. Inverse pole figure diagram of (a) measured texture with EBSD at strain 0%and (b) FFT input microstructure via 3D serial sectioning EBSD technique with dimension 128 X 128 X 100.

strain rate field and the equilibrated stress field are simultaneously updated. The compatible strain-rate field still satisfies the following classical constitutive equation at every grid-point:

e(xd) = £ mjx)f(x) = f £ mfx)[ ^¿¿f*)) s8n №): okl(x)) (1)

Here summation of N = Ns + Nt terms includes all Ns slip and Nt twin systems, Xs, ms and js are the current critical resolved shear stress, Schmid tensor and local shear rate associated with each deformation system, respectively; e(xd) and a'(x) are the strain-rate and deviatoric stress

tensors at grid-point x; y0 is a normalization factor which, in practice, is a constant value in relation to the normalized imposed macroscopic strain rate and n is the rate-sensitivity exponent, which was set equal to 12.

The hardening rule used in the simulations is the extended phenomenological Voce model, as defined for the VPSC model [26]. The stress-strain data used was a single quasi-static test and therefore the Voce equation was a sufficient description. In this model, hardening is described by:

Ts = T§ + (t{ + 0{r)-{ 1 - exp | -r-0 jj (2)

where the evolution of the slip resistance for each slip system, 5, is represented by phenomenological parameters reflecting a purely curve fitting approach: t0s, rf, 9{] and Of. The r term is the accumulated shear strain in each grid-point. Latent hardening can be included in this scheme but was not used here because of the lack of multi-axial test data.

3. Three dimensional PTR scheme

The three dimensional PTR scheme consists in following the accumulated shear strain of the most active twin system upon strain in each grid-point, xd until it reaches a threshold twin volume fraction F th,mode for reorientation. The most active twin system is detected by following the accumulation of the shear strain yu n at each twin system t for a given increment n. Thus, a virtual volume fraction F at each grid-point can be calculated as:

n n t,n

Fmode(t, xd) = 2 AF mode(t, xd) = 2 — (3)

1 1 So

where S0 represents the characteristic shear of the twin mode. Whenever a grid-point satisfies Fmode^p) _ fth,mode for a predominant twin system p, the subroutine allows it to reorient that grid-point and updates the effective twin volume fraction, F eff, which becomes:

Feff = N (4)

where NR and N are the total reoriented grid-points by twinning and total grid-points in the unit cell, respectively.

Naturally, the threshold will be a function of both Feii and the sum of accumulated Fmode of all twin systems averaged over all grid-points:

Fth = d + C2 — (5)

xd t (6)

where c\ defines the incubation strain for the onset of twinning and c2 is a constant that will be tuned to the observed resistance to twin propagation.

Following [27], the reorientation of the grid-points is monitored in such a way that F eff never exceeds Facc. However, as mentioned already, this purely local approach does not capture the lamellar, or thin plate, growth often observed in mechanical twinning [5]. This limitation is discussed in the interpretation of the simulation results.

Strain Strain

(a) (b)

Figure 3. Plot of the (a) measured stress-strain response of the TWIP steel showing strong strain hardening, overlaid with the simulated curve based on the Voce model and (b) the plot of experimental and simulation strain hardening versus strain.

4. Results and discussions

4.1. Overall behavior

Figure 3(a) compares the experimental and modeling results of the stress-strain behavior. The rate sensitive constitutive formulation of equation (1) assumed all the 24 bi-directional close-packed {111} <110) slip systems in addition to all 12 uni-directional {111} <112) twinning systems to accommodate the imposed strain. The polarity of twinning was captured by setting the negative values of to as <x>. Following transmission electron microscopy (TEM) observations [4], nearly 4%strain of pure slip is needed before twinning triggers within favorably oriented grains. Generally, authors argue that multiple slip systems must be actively reacting to produce stable twins in an FCC lattice [28, 29]. Following these common hypotheses about twin nucleation, we set c1 = 0.15 in our simulations.

The best fit of the TD tensile behavior in figure 3(a) corresponded to the Voce hardening model parameters for slip and twinning that are listed in table 1. A good fit did not require self-latent hardening between the slip systems nor between the twin variants. In general, assigning parameters for latent hardening requires multiaxial tests, which were not available. Similar values of self and latent hardening parameters were used by Beyerlein et al [9] in their simulations of twinning effects on the behavior of silver-copper cast eutectic nanocomposites. Table 1 shows that the critical resolved shear stress for twinning is more than that for slip. The t0 parameters reported for slip and twinning in other literature on TWIP steel [29] are, however, consistent with our results.

4.2. Relative activities

The simulated stress-strain behavior includes both slip and twinning and figure 4(a) shows the variation in relative activity as a function of strain. The results show that twinning decreases during the course of deformation, which agrees with the relative activities reported in other work on the simulation of TWIP steels [30, 31]. Slip is significantly more active than twinning, which is directly related to the notably higher hardening rate parameters assigned to the

Table 1. Simulation parameters identified for the best fit of the stress-strain behavior.

Parameter

Twinning

T0 (MPa) t1 (MPa) e0 (MPa) 01 (MPa)

215 330 410 225

260 460 510 275 1

hss Slip hss Twin

C1 C2 n

0.15 0.15 12

twinning modes, compared to those for slip. The underlying pseudo-slip approach for twinning may not be adequate to describe the fast progress of twin lamellae expected in grains undergoing a fairly homogeneous and favorable stress state distribution. However, twin propagation may be obstructed by twin-twin interactions as the 12 twin variants allow for multiple nucleation events in any given grain. Thus, the effect of twin-twin hardening on the overall plastic behavior can be considered to be implicitly included in the relatively higher values of 80 and t1 (for twinning).

Figure 4(b) shows measured and calculated twin volume fraction. The volume fraction of twins was obtained from Beladi et al [4] using EBSD technique. It is worth mentioning that the spatial resolution of EBSD measurement (i.e. step size) is lower than twin size (i.e. 20 nm), which makes it difficult to resolve each individual twin. Therefore, the EBSD measurement only shows the twinned area, which consists of twin and grain matrix. In addition, most grains do not display uniform twinning. Considering these limitations arising from the EBSD technique and microstructure complexity, the volume fraction of twinned area measured by EBSD was divided by a factor of 3 [4] to estimate the volume fraction of twins at different true strains. Overall, the evolution of twin volume fractions was satisfactorily predicted by the vpFFT simulations, as shown in figure 4(b). The stress hot spot analysis aggregates points with high twin activity which avoids assigning significance to individual points. As the twin distribution on the exterior surfaces of the simulation volume reveals, figure 6, only a few agglomerations of highly twinned material are larger than the typical grain size in the volume.

4.3. Orientation change

Figure 5 shows change of misorientation of each voxel deformed by slip or twinning with respect to initial condition during deformation. The orientation of voxels deformed by twinning undergoes a jump to the new twinned orientation, while other voxels deformed by slip gradually reorient in each step. Note that slip-induced lattice rotation is more pronounced around twins, indicating the important contribution of twins to localization. Field et al [32] reported the existence of orientation gradients near twins which disappeared, caused by twin boundary migration, after performing channel die deformation in copper. Their results show that twins disappear at high misorientation regions (e.g 10°) in the parent grains. Recently, using orientation microscopy by TEM confirms the presence of high geometrically necessary dislocation densities in the regions with high local misorientation [33].

Figure 4. Results of FFT simulations showing (a) the relative activity of slip and twinning during deformation and (b) the simulated evolution of the twin volume fractions in the unit cell as compared to that measured experimentally by Beladi et al [4].

4.4. Twin initiation at high stress regions

Twin nucleation events in low SFE FCC metals were demonstrated to follow a pseudo-slip mechanism, i.e. they trigger in regions where the resolved shear stress is greater than the critical resolve shear stress for twinning [34]. Figure 10(a) clearly shows that most of the highly stressed voxels are located in the vicinity of grain boundaries and the associated resolved shear stress on twinning systems is also high. Correlated with this, both figures 6(a) and (b) indicate that the twins primarily nucleated at regions with high local stress values (e.g. grain boundaries) and then grew inside the grains. Although the typical lamellar shape of the twins was not captured for obvious reasons inherent to the three dimensional PTR twinning model, prediction of preferential nucleation at GBs is an encouraging feature of the FFT simulations. Twins were observed to mainly nucleate at low-angle boundaries [35, 36]. Interface defects are known to mediate twin propagation in both FCC and HCP metals in some cases [37, 38], the stress surrounding the twin is mainly relaxed by the motion of such defects thus accommodating twin edgewise thickening. In the following, more detailed analysis shows that twinned points tend to be also high stress points.

4.5. Predictions of deformation texture

The evolution of texture after 40%strain is mainly characterized by the stabilization and strengthening of the (1 1 1)|| TD and (0 0 1)|| TD components. A comparison between the initial state in figures 2(b) and 7(b) suggests that the (1 1 1)|| TD fiber strengthened at the expense of the orientation around it.

It is clear that there is over-prediction of (0 0 1 )| | TD in the FFT simulation compared to the experimental results. De Cooman et al [39] showed the similar over- and under-prediction of the (0 0 1) and (1 1 1) fibers in Al-added TWIP steel by VPSC. They mention that the Brasstype texture in Al-added high SFE TWIP steel is stronger than that in Al-free low SFE TWIP steel. This property increases the intensity of Goss orientation relative to Brass orientation in Al-added TWIP steel, in which the Goss component is the preferred orientation for slip and micro shear banding [31]. Both the Cu-texture {1 1 2}(1 1 1) and Goss-texture {1 1 0} (0 0 1) in

(c) (d)

Figure 5. Misorientation angle of each voxel with respect to initial condition (strain 0%) in three-dimensional (a) 10%, (b) 20%, (c) 30% and (d) 40% total tensile strain along the transverse direction. Blue regions are slip dominated, whereas red ones are twin dominated.

the t fiber gradually increase with increasing strain in Al-added TWIP steel [39]. Therefore, the (0 0 1)|| TD in figure 7 mostly contains twin free grains with (0 0 1) texture. Moreover, the higher fraction of (0 0 1) texture in the FFT input (figure 2) causes a stronger (0 0 1) FFT prediction compared to experimental one.

4.6. Effect of linearization scheme on texture prediction

Different methods are available to linearize the stress and strain relation in a grain scale and the selected linearization method influences the accuracy of the predicted texture. Tangent, secant, affine and neff are linearization methods implemented in the VPSC code [20]. To investigate the effect of these linearization methods on the output texture and grains rotation, VPSC simulations were performed under uniaxial tension to a strain of 0.4. Figure 8 shows the resulting VPSC simulations taking FCC random texture as an input. The results reveal that grains rotate toward (1 1 1) or (0 0 1) direction and their rotation angles are a function of the linearization scheme. The general trend of each grain rotation is consistent with 3DXRD

(a) (b)

Figure 6. Results of FFT simulations showing (a) twin distribution (in red) on the exterior surface of the simulation volume and (b) twin distribution on a middle plane. To ensure that the twinned regions (red) are apparent in the figure, the grain ID-based color scale was adjusted such that non-twinned points were in the blue-white range.

texture measurement in copper under tension [40]. Interestingly, the fraction of grains, which tend to align in a direction (1 1 1) or (0 0 1), strongly depends on the linearization approach. Moreover, figure 8 reveals that the intensity of (0 0 1) and (1 1 1) in predicted textures changes with the linearization methods. According to the full constraint (FC) Taylor model (compatibility) [41], tension on FCC metals develops texture with stronger (1 1 1) fiber and weaker (0 0 1) fiber aligning along the tensile axis [41]. Therefore, the secant linearization scheme calculates a more realistic texture, while the other linearization schemes (tangent, affine and neff) predict stronger (0 0 1) and (0 0 1)-(1 1 1) fibers compared to experimental ones. Lebensohn et al [20] reported that when n (rate-sensitivity exponent) gets higher values, the tangent approximation predicts a uniform stress-state like Sachs equilibrium based approximation (lower bound). On the other hand, the secant estimation is stiffer and predicts a uniform strain-rate similar to Taylor based compatibility approximation (upper bound). However, the vpFFT formulation uses the tangent method to linearize local stress and strain relation in each Fourier point (in equation (1)), which causes stronger predicted (0 0 1) fiber compared to experimental one.

4.7. Grain scale deformation texture

Development of dislocation pile-ups, which cause a local stress concentration [42] and activation of multiple slip systems [43] are thought to be necessary for initiation of mechanical twinning. TEM investigation in a single grain of Hadfield steel shows that mechanical twinning does not occur in the orientation (123) || TD, even though this orientation has a high Schmid factor for twinning. The (123) || TD orientation is less favorable for multiple slip compared with more symmetric orientations, which is essential to mechanical twinning nucle-ation. TEM observations show mechanical twinning is activated in grains with low Schmid factor while twinning can fail to appear in grains with high Schmid factor [44, 45].

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3,5 4.0 4.5 5.0 5,5 6.0

Multiples ol Random

Figure 7. Comparison of TD-mapped inverse pole figures of (a) measured and (b) FFT simulated after 40% total tensile strain along the transverse direction. (c) IPF map measured by EBSD after 40% total strain along TD. Note that the tensile axis was aligned with the original TD as shown in figure 6.

In the literature, several authors performed a Schmid factor analysis to explain the frequency of mechanical twinning as a function of orientation [46, 47]. However, Schmid's law is only reasonable for single crystals with an isostress condition leading to single slip and it is not able to capture grain interaction and strain compatibility in the grain boundaries with neighboring grains [48]. Other simulation work shows better prediction of the active twinning mode with the Taylor model compared to the Sachs model. The multiple slip approach results in a different stress state in each grain and can identify mechanical twinning formation at small strains [49].

To identify the role of twinning on texture, figure 9 shows the simulated IPFs for voxels in twins, voxels near twins and voxels in twin-free grains, all of which can be compared with the experimental result, figure 7(a). Figure 9(a) indicates that most of the strengthening of the (0 0 1)|| TD is due to deformation twinning which has primarily subtracted from the (1 1 1)|| TD component. Twinning of this component is substantiated by the EBSD inverse pole figure map, figure 7(c) after 40% strain deformation. In this figure, blue grains, showing (1 1 1)|| TD, contain twins whereas the red grains, (1 0 0)|| TD, may be fully twinned or twin-free.

To understand the origin of texture component, the Taylor factor of an FCC metal subjected to uniaxial tension with {1 1 1}(1 1 0) was calculated for different orientations in the inverse pole figure triangle (figure 9(d)) [48]. The comparison of the IPF for voxels near twins and voxels in twin-free grains reveals that they have high (M < 2.6) and low Taylor factors (M >3.4), respectively. These results are consistent with experimental work performed on the same material [4]. However, the IPF of voxels near twins reveals additional, but weak

Figure 8. Inverse pole figures of output texture and rotation maps of 500 grains along Tensile direction, after applying 40% tensile elongation in VPSC with linearization scheme, with arrows pointing from the location prior to deformation to that at 40%strain. (a) Secant Grain Rotation. (b) Secant. (c) Full Constraint Grain Rotation. (d) Full Constraint. (e) Tangent Grain Rotation. f) Tangent. (g) Affine Grain Rotation. (h) Affine. (i) neff Grain Rotation. (j) neff.

Figure 9. Results of FFT simulations showing inverse pole figures (IPFs) corresponding to (a) twinned voxels, (b) twin-free grain voxels and (c) voxels near twins after 40% total tensile strain along the transverse direction. (d) Inverse pole figure along tensile axis direction showing Taylor factor values, the dotted line correspond to the Taylor factor value for each specific orientation in triangle, the values were calculated for slip system {1 1 1 }(1 1 0) in FCC randomly textured deformed under uniaxial tension [48].

texture components between (0 0 1 )| | TD and (1 1 1 )| | TD fibers, which are not present in the experimental IPF. Similar results were reported in other simulations on TWIP steel with no clear explanation [39]. Since these components do not appear in twinned voxels, it is likely that they originate from regions undergoing localization, such that the active slip systems vary considerably. This explanation correlates well with the high Taylor factors associated with the (1 1 1)|| TD fiber. In general, grains with a high Taylor factor require higher stresses for plastic deformation. The active slip systems vary to satisfy strain compatibility and stress equilibrium in the near-grain boundaries regions. Therefore, the voxels near twins with a high Taylor factor, e.g. (1 1 1)|| TD, are well oriented for multiple slip as well as mechanical twinning. In contrast, the voxels in twin free grains with low Taylor factor can deform at lower stresses. In other experimental work, Miura et al [42] reported that the (0 0 1 )| | TD orientation promotes cross-slip, causing relief of stress concentration that might otherwise promote twinning.

4.8. Predictions of stress localization

Figures 10(a)-(b) show the stress and strain rate distribution on the exterior surface of simulation volume after 40% strain deformation. The local variations in stress (figure 10(c)) and strain rate (figure 10(d)) exhibit different features in the two fields; mainly, the stress histogram has

Stress/<Stress> Strain/<Strain>

(c) (d)

Figure 10. von Mises (a) stress and (b) strain rate fields on the surface of the simulation volume and ((c)-(d)) their corresponding histograms, respectively.

two peaks and the strain histogram exhibits a long upper tail. The latter indicates that a small fraction of grid-points experienced substantially high strain rates. This may be visible in the 3D strain rate plot of figure 10(b). Figure 10 shows the distribution of local strain and stress on the surface of the simulation volume which is changing with grain structure. The variations of local stress by grain structure is more visible than that of local strain rate. Similar features were reported by other researchers [21, 50, 51].

To find a relation between local stress and orientations of twinned voxels, IPFs generated by orientations of twinned voxels which experienced a stress higher than, versus less than 90% of the peak stress are shown in figure 11, respectively. These IPFs clearly indicate that the most highly stressed points (relative to the peak stress) correspond to an increasingly sharp (0 0 1)|| TD texture. This suggests that most of the twinning happens in the voxels with high local stress, even though the criterion for twinning is strain-based. Correlation of hot spots in stress to twinning was previously reported with measurements of backstress effects in a low stacking fault energy steel [39].

Figure 11. Inverse pole figures revealing textures of twinned voxels in the regions with stresses less than (a) 0.9 of the peak stress and (b) greater than 0.9.

o o cn

o o oo

o o lo

Distance to Twinned Voxels[voxel]

Figure 12. Plot of the average distances for the twinned voxels in each stress class.

In order to quantify the relationship between twin voxels and local stress, the Euclidean distance map for twinned voxels was computed. In the grains containing twinned voxels, the minimum distance between untwinned voxels and twinned voxels was computed. This distance was binned based on the stress value and the distance values averaged over the points in each bin. Figure 12 shows the calculated plot of average distance to twinned voxels. In this graph, each point depicts the average distance to the closest twinned voxel and the vertical and horizontal axes were normalized by the global average distance and average stress, respectively. Figure 12 indicates that higher local stresses are found adjacent to twinned voxels, which suggests that twins induce hot spots close to their boundaries.

5. Conclusions

A PTR method was used to incorporate deformation twinning in a full-field, three-dimensional framework that uses the fast-Fourier transform modeling technique to compute the visco-plastic response of a face-centered cubic (FCC) steel with a Mn-induced low stacking fault energy (SFE). Hardening was described through the Voce model which treats twinning as a pseudo-slip mechanism. The occurrence of twinning at any given grid point is controlled by a criterion based on a threshold in accumulated slip that controls when a jump in lattice orientation is applied. The scheme does not contain rules for lengthwise versus edgewise growth of twin lamellae so the final microstructural morphologies and grain boundary texture likely contain artifacts from the sequence of grid-point interrogation. However, the successful reproduction of the observed evolution of the twin volume fractions and texture upon strain provide some confidence in the approach. The main results are as follows.

(a) The strengthening and stabilization of sharp and homogeneous (1 1 1)|| TD and (0 0 1)|| TD and fibers typical of low SFE FCC metals was satisfactorily reproduced. The strengthening of the (1 1 1)|| TD was mainly because of slip and marginally as a consequence of twinning of the (0 0 1)|| TD oriented grains. The development of the (0 1 1)|| TD was largely caused by twinning of the developed (1 1 1)|| TD fiber.

(b) Both twinning and slip evolve the texture to balance between the two major fibers, so one may expect that twin re-orientation and segmentation cause a radical change and evolution in orientation boundary distribution which greatly influences localization.

(c) A plot of local stress value versus distance to twin boundaries reveals that local stress increases with decreasing distance to twin boundaries and the value of the local stress in the voxels close to the twinned voxels is highest.

Acknowledgments

The authors also would like to recognize the National Science Foundation which supported this work under the DMREF (Designing Materials to Revolutionize and Engineer our Future) program with the award number: CMMI-1235009. ADR acknowledges the support under award number DESC0002001 of US Department of Energy, Office of Basic Energy Science, Division of Materials Science and Engineering. The work at Deakin University was supported through grants provided by the Australian Research Council.

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