Critical Resolved Shear Stress Ratios Research Articles

In-situ investigation of tensile anisotropy mechanism in an advanced Ti2AlNb-based alloy associated with CRSS ratio and damage model

Combined with in-situ tensile test and electron backscatter diffraction technology, the critical resolved shear stress (CRSS) ratio of O phase and the damage model of grain boundaries were proposed to reveal the tensile anisotropy mechanism of Ti–22Al–25Nb alloy isothermally forged in B2 phase region. In-situ observation suggests that the single slip mode of lamellar O phase is the major deformation behavior, and its preferential slip dominates the yield of the material. Based on the types, number, Schmid factor (SF) and orientation of slip activation, the CRSS ratio of basal <a>, prism <a>, first-order and second-order pyramidal <c+a> slips for O phase is estimated to be 1.19: 1: 1.37: 1.39. Meanwhile, basal and prism <a> slips mainly appear the radial and axial directions (RD and AD) samples with [012] and [100] texture, while the [001], [010] and [212] texture of OD (45° to RD) sample shows an increase in pyramidal <c+a> slips. Thus, the slip activation of O phase depends on the grain orientation and CRSS, resulting in significant strength anisotropy. Furthermore, the microcracks and secondary cracks at grain boundaries are significantly affected by the slip accumulation of lamellar O phase at the O/O and O/B2 interfaces. Considering the stress state of the grain boundary, a damage model about crack nucleation and propagation is proposed to explain the ductility anisotropy. From this, the RD sample has a higher crack nucleation and propagation resistance than the AD and OD samples due to the flattened morphology of B2 grains.

The mechanism for Li-addition induced homogeneous deformation in Mg-4.5wt.% Li alloy

The mechanism by which Li additions induce a more homogeneous deformation was systematically investigated in this study by comparing the deformation behavior of pure Mg and Mg-4.5 wt.% Li using experiments, crystal plasticity simulations, and theoretical calculations. The results of the digital image correlation measurements showed that under tension along the transverse direction (TD), strong localized deformation bands with an angle of approximately 20° with respect to the TD were observed in pure Mg, but were absent in Mg-4.5 wt.% Li. Slip trace analyses indicated that basal slip was the predominant deformation mode for pure Mg, whereas that for Mg-4.5 wt.% Li was prismatic slip. A high relative activity of pyramidal 〈c+a〉 slip, approximately 17 %, was observed in Mg-4.5 wt.% Li at a high strain of 10 %. Contraction or double twins were hardly detected during the entire tension process. Crystal plasticity simulations revealed that the 4.5 wt.% Li addition dramatically reduced the critical resolved shear stress (CRSS) ratio of pyramidal 〈c+a〉 slip: prismatic slip: basal slip, which was 31.5:21:1 for pure Mg and 4.3:1.3:1 for Mg-4.5 wt.% Li. Theoretical calculations using the CRSS ratios determined by simulations revealed that easy and uniform deformation propagation induced by a dramatically reduced CRSS ratio of prismatic slip to basal slip played a more important role in improving the deformation homogeneity, in addition to the enhanced activity of pyramidal 〈c+a〉 slip. The mechanisms for how Li additions improve the ductility of Mg are also revisited in this paper.

Establishment and Improvement of Prediction System for CRSS ratio in Polycrystalline Metals

Critical resolved shear stress (CRSS) determines the ease of slip deformation of metals and the values are different among slip systems. The CRSS ratio among slip systems is an important factor governing the plastic deformation mechanism. Thus, estimation methods for easily evaluating the CRSS ratio will play an important role to understand the deformation mechanisms. While our laboratory has developed a prediction system for CRSS ratio, it has not been sufficiently verified. In this study, this prediction system was verified using the results obtained by and input parameters employed in crystal plasticity analysis, and we showed that CRSS ratios were successfully predicted when two or more slip systems were activated. However, the prediction failed when only one slip system was dominantly activated. The method to improve the prediction accuracy was also discussed.

Micro-compression of high oxygen doped single-crystal titanium along different orientations

To explore the deformation mechanism of α-Ti with high oxygen doping, micro-pillars of single crystal with orientations of near 0°, 45°, and 90° inclined to c-axis were fabricated via focused ion beam (FIB) milling from a high oxygen content commercial purity Ti (CP–Ti). The micro-pillars were loaded under quasi-static compression. Post-mortem analysis reveals that only prismatic slip activated for 45° and 90° oriented samples with their critical resolved shear stress (CRSS) values of about 457 MPa and 596 MPa, and only {10–11}<c+a> the first-order pyramidal slip activated on the 0° sample with a CRSS of around 1019 MPa. The results suggest that the oxygen addition remarkably reduces the CRSS ratio between non-prismatic and prismatic slips of α-Ti. In addition, the results indicate that the Schmid law may not be applicable for α-Ti single crystal at micro-size level.

Manipulating deformation mechanisms with Y alloying of Mg

The effect of Y concentration on the slip and twinning mechanisms in binary Mg–Y alloys are investigated using transmission electron microscopy, electron backscattered diffraction, and visco-plastic self-consistent polycrystal constitutive modeling. Four concentrations of Y are studied in hot-rolled and recrystallized sheet material. The materials were deformed in tension and compression in the rolling direction and compression in the normal direction in order to invoke distinct proportions of slip and twin mechanisms with each test. Within the single crystal hardening model used in polycrystal modeling, a slip-twin interaction law is introduced to account for dislocation density reductions due to dislocation absorption during twin boundary migration. We show that increasing Y concentration reduces the intensities of both the initial and deformation textures. During deformation, the plastic anisotropy in yield stress, the tension-compression asymmetry, and amount of 101‾2⟨1‾011⟩ twinning is shown to decrease with increasing Y. For each alloy, the model identifies a single set of material parameters that successfully reproduced all measured stress-strain curves and achieved agreement with measured deformation textures and twin area fractions. Transcending texture effects, the model interpretation of the flow responses suggests that increased concentrations of Y increase the critical resolved shear stress for basal slip but have negligible effects on the other slip modes. The reduced plastic anisotropy with increases in Y is explained by a concomitant decrease in the prismatic-to-pyramidal slip critical resolved shear stress ratio. The model suggests that their nearly equivalent critical resolved shear stress values lead to the enhanced non-basal activity of Mg–Y alloys, which was confirmed by transmission electron microscopy. The calculations suggest that beyond any texture differences, this reduction in twinning can be attributed to a slightly increased resistance for 101‾2⟨1‾011⟩ twin propagation, particularly in the binary with the highest Y content.

Tuning critical resolved shear stress ratios for bcc-titanium Ti21S via an automated data analysis approach

We introduce a novel method for automated slip trace detection and analysis based on the Hough transform algorithm and apply it to electron channeling contrast imaging micrographs. This is further augmented with an automation procedure for the determination of slip-trace crystallography in conjunction with orientation data acquired via electron backscattered diffraction. Automation is required for faster indexation of the slip traces and for more reliable statistical studies. The automation procedure was applied to different regions of interest on a β-Ti21S sample loaded in situ in tension. β-Ti21S is a bcc alloy with 48 slip systems available to accommodate plastic deformation, including all complexities associated with pencil glide. Multiple regions of interest were analyzed using the automation procedure. The acquired slip distribution statistics reveals that the majority of the slip traces belong to the {112} and {123} slip families. The deformation response of the observed regions of interest was also simulated using a full-field crystal plasticity model implemented in DAMASK, based on a phenomenological power law based constitutive formulation, incorporating all potentially active 48 slip systems. The slip system activity distribution from modelling is compared with the slip distribution statistics observed experimentally. The plasticity parameters for β-Ti21S were taken from the literature and the critically resolved shear stress (CRSS) values were adjusted to match the experimentally observed yield stress value. We begin with uniform CRSS ratios for all three potential slip system families and tune the CRSS ratios to match the slip-distribution statistics experimentally, keeping the average CRSS value the same for all cases. Thus, a method has been introduced to tune average CRSS values and ratios by considering both the macroscopic stress–strain response and the locally observed slip-distribution statistics, obtained via automated slip trace detection procedure.

Predicting Shear Transmission Across Grain Boundaries with an Iterative Stress Relief Model

A new model is proposed to quantitatively describe the accommodation of shear from multiple deformation systems at a grain boundary. The model uses an iterative approach to sequentially determine the accommodating slip systems and their relative shear. The outcome of this iterative stress relief model is mainly controlled by the continuity of Burgers vector in the grain boundary and the evolution of the impinging stress tensor at the grain boundary. The model was tested by comparing predictions with observations of shear accommodation in a -titanium quantified using orientation informed slip trace analysis and quantitative atomic force microscopy. Similar comparisons were conducted between tangential continuity model predictions and the experimental observations. These comparisons show that the iterative stress relief model agrees much better with the observations when used to predict the accommodating deformation system(s) compared to the tangential continuity model. Critical resolved shear stress ratios used in this iterative stress relief were optimized by maximizing the accuracy of the model predictions. The ranges of the optimized critical resolved shear stress ratios were determined to be basal 〈a〉: prism 〈a〉: pyramidal 〈a〉: pyramidal 〈c+a〉: T1 twin = 1.0 : (0.8–1.0) : (1.0–1.3) : (1.6–9.5) : (3.0–9.5).

Effect of heat treatment on the tensile behavior of selective laser melted Ti-6Al-4V by in situ X-ray characterization

Selective Laser Melted Ti-6Al-4V (as-SLMed) exhibits decreased yield strength, increased work hardening, and increased ductility after heat treatment at 730 °C (HT-730) or 900 °C (HT-900) for 2 h. To understand the change of mechanical properties, in situ high energy X-ray diffraction (HEXRD) is used to examine the phase composition, load partitioning, slip system activity, and dislocation density evolution in all three specimens. The as-SLMed specimen is dominated by martensitic α΄. After heat treatment, α΄ partly or fully decomposes into α+β, reducing the yield strength. In HT-730, β precipitates with confined size show much higher lattice strain than the α΄/α matrix during deformation; in HT-900, the lattice strain difference is mostly eliminated. This is a key reason for the increased ductility in HT-900. From the anisotropic lattice strain development, basal slip is identified as the easiest slip system in α΄/α. Using an elasto-plastic self-consistent (EPSC) model, the critical resolved shear stress ratio between prismatic slip and basal slip (CRSSprism/CRSSbasal) is estimated to be 1.31 and 1.16 in the as-SLMed and the HT-900 specimens, respectively. The α phase in HT-900 is able to activate multiple slip systems and accumulate more dislocations during plastic deformation. This explains why HT-900 has better ductility and higher work hardening rate than the other two specimens.

Effect of Al content on the critical resolved shear stress for twin nucleation and growth in Mg alloys

The effect of Al atoms in solid solution on the critical resolved shear stress for twin nucleation and growth was analyzed by means of the combination of diffusion couples with compression tests in micropillars oriented for twinning. The critical resolved shear stress for twin nucleation was higher than that for twin growth and both increased by the same amount with the Al content. Nevertheless, the increase was small (≈ 10 MPa) for 4 at%Al but large (up to 60–70 MPa) for 9 at%Al. These results were in agreement with Labusch-models based on first principles calculations in the dilute regime (< 5 at%Al). Comparison with recent data in the literature showed that Al atoms are more effective in increasing the critical resolved shear stresses for twin nucleation and growth than for basal slip. Finally, compression tests in micropillars oriented along [0001] showed the critical shear stress for pyramidal slip increased rapidly with the Al content from 98 MPa in pure Mg to 250 MPa in Mg-9 at%Al. Overall, the presence of Al atoms decreased slightly the plastic anisotropy of Mg (understood as the critical resolved shear stress ratios between non-basal and basal plastic deformation mechanisms).

Investigation of strain redistribution mechanism in α titanium by image-based crystal plasticity analysis

Mechanisms of strain localization and localized activation of slip systems in α titanium were investigated using a crystal plasticity finite element (CPFE) method. A microscopic image of polycrystalline α titanium was obtained by electron back scatter diffraction (EBSD), and the data was converted from the microscopic image into the geometric model for the CPFE analysis. The uniaxial tensile deformation of the model was numerically reproduced by the CPFE method employing a dislocation density based constitutive equation. The results showed that the strain distribution corresponds well with that obtained by the experiment when the ratio of critical resolved shear stress (CRSS) employed in the numerical simulation is basal:prismatic ⟨a⟩:1st-pyramidal ⟨a⟩:1st-pyramidal ⟨c + a⟩:2nd-pyramidal ⟨c + a⟩ = 1.0:1.0:1.3:2.0:2.0. Next, numerical simulations were performed by changing the ratio of CRSS among the slip systems but keeping all other conditions the same as those of the above uniaxial tensile analysis. The results showed that strain redistribution typically occurs between hard and soft regions with high and low CRSSs for the primary slip systems; this redistribution resulted in a localized higher strain and activation of slip systems. However, localized activation of slip systems was observed even in slip systems with higher CRSS; the mechanism could be explained by the strain redistribution in the tensile direction.

Oxygen effects on crystal plasticity of Titanium: A multiscale calibration and validation framework

This work investigates macroscopic plastic behavior in polycrystalline Ti with varying oxygen concentrations using both experiments and a crystal plasticity finite element framework applied to a representative volume element (RVE) of the microstructure. The proposed multiscale framework makes use of parameters such as critical resolved shear stress (CRSS) ratios obtained from lower length scale first-principle calculations and strain hardening rates obtained from experiments. Generalized stacking fault energy calculations (GSFE) in combination with the Peierls-Nabarro model and a temperature dependent phenomenological description of CRSS were utilized to compute the CRSS ratios for different slip systems in Ti with and without oxygen. Experimentally measured stress-strain responses along the rolling direction for polycrystalline Ti with different oxygen concentrations were used to obtain the strain hardening rates. The crystal plasticity framework was then calibrated using the computed CRSS ratios and the strain-hardening rate for Ti with various oxygen concentrations. The calibrated model was then used to predict the macroscopic response of Ti under different loading conditions and orientations with different oxygen concentrations. Without tuning the fitting parameters for the crystal plasticity framework, we show that the model is able to simulate the experimental responses within the experimental uncertainty. Thus, a systematic calibration procedure is presented to capture the macroscopic homogeneous responses of Ti with various oxygen concentrations.

Influence of critical resolved shear stress ratios on the response of a commercially pure titanium oligocrystal: crystal plasticity simulations and experiment

Direct comparisons between crystal plasticity simulations and experiments dedicated to test a sample containing a small number of grains (oligocrystal), instrumented using full field measurement techniques (DIC, grid method), have demonstrated their efficiency to reproduce the behavior of BCC and FCC metals at the grain scale. The extension of this approach to HCP metals, specifically for a grade 2 commercially pure titanium (CP-Ti) sample, is the objective of the present work. The influence of the critical resolved shear stress ratios on the numerical predictions of strain field and slip activity during uniaxial tension of the titanium oligocrystal was investigated. Simulation results were compared with their experimental counterparts, showing that a small variation of the critical resolved shear stress ratios identified by the authors in a previous study dedicated to grade 1 CP-Ti [Hama et al., Int. J. Plast., 91(2017), 77.] reproduced the experimental deformation behavior satisfactorily.

Back-stresses and geometrical hardening as competing mechanisms enhancing ductility in HCP metals

By recourse of computational mechanics, back-stresses are unveiled as a major source for the increase in work hardening during forming of hexagonal close-packaged (hcp) metals. Polycrystalline visco-plastic self-consistent (VPSC) and crystal plasticity finite element modelling (CPFEM) simulations of tensile uniaxial experiments were used along with experimental texture information. Simulations took into account the analogous variation in the critical resolved shear stress (CRSS) values of each slip family that could result from an increase in the test temperature. As the CRSS ratio between secondary and primary slip families increased, two different contributions to the variation of the work hardening rate were observed depending on the simulation framework: (i) a decrease in the work hardening rate in VPSC simulations attributed to texture evolution or geometrical hardening and (ii) an increase in the work hardening rate in CPFEM simulations due to back-stresses. While geometrical hardening is present in both simulation frameworks, only CPFEM is able to capture the influence of back-stresses on the increase of the work hardening rate with temperature. The results provided here contribute to a better understanding of the deformation mechanisms present in warm forming of hcp metals, showing also that CPFEM is a better simulation framework to study warm forming of hcp metals.

Analysis of the Deformation Behavior in Tension and Tension-Creep of Ti-3Al-2.5V (wt pct) at 296 K and 728 K (23 °C and 455 °C) Using In Situ SEM Experiments

The deformation behavior of a Ti-3Al-2.5V (wt pct) near-α alloy was investigated during in situ deformation inside a scanning electron microscopy (SEM). Two plates with distinct textures were examined. Tensile experiments were performed at 296 K and 728 K (455 °C) (~0.4T m), while a tensile-creep experiment was performed at 728 K (455 °C) and 180 MPa (σ/σ ys = 0.72). The active deformation systems were identified in the α phase using electron backscattered diffraction based slip-trace analysis and SEM images of the surface. Prismatic slip deformation was the dominant slip mode observed for all the experiments in both plates, which was supported by a critical resolved shear stress (CRSS) ratio analysis. However, due to the texture of plate 1, which strongly favored the activation of prismatic slip, the percentages of prismatic slip activity for specimens from plate 1 tested at 296 K and 728 K (23 °C and 455 °C) were higher than the specimens from plate 2 under the same testing conditions. T1 twinning was an active deformation mode at both 296 K and 728 K (23 °C and 455 °C), but the extent of twinning activity decreased with increased temperature. T1 twinning was more frequently observed in specimens from plate 2, which exhibited a higher fraction of twinning systems favoring activation at both 296 K and 728 K (23 °C and 455 °C). The tension-creep experiment revealed less slip and more grain boundary sliding than in the higher strain rate tensile experiments. Using a previously demonstrated bootstrapping statistical analysis methodology, the relative CRSS ratios of prismatic, pyramidal 〈a〉, pyramidal 〈c+a〉, and T1 twinning deformation systems compared with basal slip were calculated and discussed in light of similar measurements made on CP Ti and Ti-5Al-2.5Sn (wt pct).

The Effect of Nd on the Tension and Compression Deformation Behavior of Extruded Mg-1Mn (wt pct) at Temperatures Between 298 K and 523 K (25 °C and 250 °C)

The tension and compression deformation behavior of extruded magnesium-1 wt pct manganese alloys with nominally 0.3 wt pct (MN10) and 1 wt pct neodymium (MN11) was studied over the temperature range of 298 K to 523 K (25 °C to 250 °C). Nd additions to Mg alloys tend to reduce the strong basal texture exhibited by conventional wrought Mg alloys and this work was intended to study the effect of Nd on the deformation behavior of Mg alloys. Insitu tensile and compressive experiments were performed using a scanning electron microscopy, and electron backscatter diffraction was performed both before and after the deformation. A slip trace analysis technique was used to identify the distribution of the deformation systems as a function of strain, and based on this analysis and the texture of the undeformed samples, the critical resolved shear stress ratios between the deformation systems were estimated. In the case of MN11, the deformation behavior under tension at all temperatures was dominated by slip, while in compression, extension twinning was the major deformation mode. In tension at 323 K (50 °C), extension twinning, basal, prismatic 〈a〉, and pyramidal 〈c + a〉 slip were active in MN11. Much less extension twinning was observed at 423 K (150 °C), while basal slip and prismatic 〈a〉 slip were dominant and presented similar relative activities. At 523 K (250 °C), twinning was not observed, and basal slip controlled the deformation. With the reduction of Nd content, less slip deformation and more twinning were observed during the tensile deformation. However, like for MN11, the extent of twinning in MN10 decreased with increasing temperature and basal slip was the primary deformation mode at elevated temperatures. Extension twinning was the major deformation mode under compression for all test temperatures in MN10 and MN11. The tensile strength decreased with increasing temperature for both alloys, where MN10 was slightly stronger than MN11 at 323 K (50 °C), which was expected to be a result of the stronger basal texture exhibited by MN10 due to its lower Nd content. However, MN11 maintained its strength more at elevated temperatures compared with MN10, and this was explained to be a result of the greater Nd content.

Methodology for estimating the critical resolved shear stress ratios of α-phase Ti using EBSD-based trace analysis

Methodology for estimating the critical resolved shear stress ratios of α-phase Ti using EBSD-based trace analysis

Methodology for estimating the critical resolved shear stress ratios of α-phase Ti using EBSD-based trace analysis

A novel method for calculating the critical resolved shear stress (CRSS) ratios of different deformation system types in polycrystalline non-cubic metals has been developed. The mean CRSS ratios between different deformation systems were calculated for both commercially pure (CP) Ti and Ti–5Al–2.5Sn (wt.%) tensile deformed at ambient temperature and 455°C using an in situ scanning electron microscope-based testing technique combined with electron backscattered diffraction. It was found that the relative activity of the different deformation systems changes as a function of alloying composition and deformation temperature. Prismatic slip was the most active deformation mode for CP Ti. CP Ti exhibited a lower resistance to prismatic slip at both ambient and elevated temperatures compared with Ti–5Al–2.5Sn. For Ti–5Al–2.5Sn, prismatic slip was the most active deformation system at ambient temperature although the basal slip activity significantly increased compared to CP Ti, mostly likely due to an increased c/a ratio resulting in a closer packed basal plane. At 455°C, basal slip exhibited a lower CRSS than prismatic slip for Ti–5Al–2.5Sn. The relative activity of other deformation systems was also affected by alloying and temperature. The statistical resampling technique of bootstrapping was used to generate multiple equivalent data sets from which mean CRSS ratios between different deformation systems, and associated confidence intervals, could be deduced. It was found that the mean CRSS ratios at low and high strains varied slightly for the same testing conditions. Moreover, lesser activated slip systems resulted in relatively larger confidence intervals for the CRSS means. This variability may be attributed to a number of potential factors, including measurement errors, rotations of grains during deformation, local stress state variations, and work hardening. The analysis further suggests that awareness of the intrinsic statistical variability in CRSS ratios should be considered when formulating crystal plasticity constitutive models.

Geometrical criterion for the activation of prismatic slip in AZ61 Mg alloy sheets deformed at room temperature

AZ61 Mg alloy was hot-forged and subsequently hot-rolled to a thickness of 1 mm. Tensile anisotropy was investigated at room temperature within the plane of the rolled sheets. When the basal planes were tilted by more than 16.5° towards the tensile axis, basal 〈 a〉 slip was a dominant deformation mechanism and a small elongation of 8% was obtained. Otherwise, prismatic 〈 a〉 slip became a dominant deformation mechanism and a large elongation of more than 20% was obtained. The basal-plane tilt angle was considered as a geometrical criterion for the activation of the prismatic 〈 a〉 slip at room temperature. The corresponding Schmid factor ratio, or equivalently the critical resolved shear stress ratio of prismatic to basal slip, was found to be 1.5–2.0.

X-ray strain analysis at high pressure: Effect of plastic deformation in MgO

The factors that control the stress–strain state of a polycrystal under differential stress depend on whether or not plastic deformation has occurred in the solid. If not, then the elastic properties with the constraints of the Reuss–Voigt bounds limit this relationship. If plastic deformation becomes important then the Taylor and Sachs models are relevant. These models assume that the plastic process is enabled by dislocation flow on specific lattice planes and specific Burger’s vectors. Then, the relationship between stress and strain is controlled by the orientation of an individual grain with respect to the stress field, von Mises criterion, and the critical resolved stress on the dislocation that is necessary for flow. We use a self-consistent model to predict the flow stress during the plastic deformation of polycrystalline MgO with a slip system of {110}〈11̄0〉, {111}〈11̄0〉, and {100}〈011〉 at different critical resolved shear stress ratios for the different slip systems. The prediction of the models is correlated with the results of x-ray diffraction measurements. Uniaxial deformation experiments on polycrystalline and single-crystal MgO samples were conducted in situ using white x-ray diffraction with a multielement detector and multianvil high-pressure apparatus at a pressure up to 6 GPa and a temperature of 500 °C. A deformation DIA was used to generate pressure and control at a constant deformation rate. Elastic strains and plastic strains were monitored using x-ray diffraction spectra and x-ray imaging techniques, respectively. The correlation of the data and models suggests that the plastic models need to be used to describe the stress–strain observations with the presence of plasticity, while the Reuss and Voigt models are appropriate for the elastic region of deformation, before the onset of plastic deformation. The similarity of elastic strains among different lattice planes suggests that the {111} slip system is the most significant slip system in MgO at high pressure and high temperature.

INVESTIGATION OF CRYSTALLOGRAPHIC SLIP IN POLYCRYSTALLINE Fe3Al USING SLIP TRACE MEASUREMENT AND MICROTEXTURE DETERMINATION

An intermetallic Fe72Al28 alloy (doped with Cr, Zr, Mo, and C) with an imperfectly ordered B2 crystal structure was rolled at 830–860 K to ϵ = 20%. To investigate crystallographic slip an etching technique was developed which allowed slip traces to be determined in grain interiors rather than at the sample surface. To derive the prevalent glide systems both the slip traces and the corresponding orientations were determined in grain scale. Three types of slip systems were identified, namely {110}〈111〉, {112}〈111〉, and {123}〈111〉. However, the slip traces produced by {123}〈111〉 systems appeared wavy and were interpreted in terms of macroscopic or effective rather than crystallographic slip. The critical resolved shear stress ratio of the slip systems involved was fitted from experiment using a Relaxed Constraints Taylor model. The best correspondence between predicted and experimentally observed slip systems was attained for a critical resolved shear stress ratio of τ{110}/τ{112} = 1.05/1.0. © 1997 Acta Metallurgica Inc.