To further understand and eventually better simulate the heterogeneous deformation processes in polycrystalline α-titanium, this research project is focused on both detailed experimental characterization and the CPFE modeling of heterogeneous deformation using a phenomenological CPFE constitutive model. Several experimental characterization techniques, including atomic force microscopy (AFM), confocal microscopy, three-dimensional x-ray diffraction (3D-XRD), and nano- indentation, were used to quantitatively measure the active deformation systems in grains with different orientations. The critical resolved shear stresses (CRSS) of prismatic, basal, and pyramidal slip are important constitutive parameters for the CPFE model. These values were identified by optimizing the CRSS values in simulations of the topographic pile-ups surrounding conical nano-indentations in grain interiors using the same phenomenological CPFE model. By comparing experimental results with CPFE simulations from the same grain patches, along with careful study of dislocation interactions at grain boundaries using 3D-XRD, the accuracy of the phenomenological model was assessed. From this, developments required to improve modeling strategies have been identified. See the sidebar for experimental details.

Figure 1 shows a backscattered electron micrograph with a superposed orientation map that illustrates a patch of microstructure after 1.5% plastic strain. Two overlaid frames identify regions that were examined using AFM (dashed lines) and simulated with CPFE (solid line). The free surface topography shows slip traces that vary from grain to grain, due to the differences in crystal orientation (most visible in grains 0, 1, 9; higher resolution images clearly show traces parallel to the colored lines that identify slip traces). The activated deformation systems were identified using trace analysis,²¹ based on the backscattered electron (BSE) images and electron backscattered diffraction (EBSD) to determine grain orientation. Prismatic, basal, and pyramidal slip, as well as T1 twinning, were found in this patch, as denoted by the trace colors in Figure 1.

The mesh for the CPFE model was generated based on the two-dimensional geometry EBSD map of the undeformed patch. A 3D mesh was generated by extending the two-dimensional geometry into a five-element thick slab, so that the grain boundaries were perpendicular to the surface. A pan-like rim, with an orientation that is at the center of the dominant macro-texture component, provided a somewhat realistic bulk constraint to the modeled region. Deformation was imposed by constraining the left side of the pan to zero displacement, and putting a face load to the right.

The local shear distribution in the microstructure patch from all dislocation slip systems at 1.5% plastic strain is quite heterogeneous, as shown in Figure 2. The CPFE simulations showed that the highest shear occurred in grain 14, caused by prismatic slip. Prism slip was dominant in grains 1, 3, and 9, with the local shear ranging from 0.05 to 0.07. Basal slip was observed in grains 5, 7, and 10, but with smaller magnitudes, from 0.005 to 0.02. No significant pyramidal slip occurred in the simulation of this microstructure patch. T1 twinning activity dominated the deformation in the lower part of grain 2.

To quantitatively assess the accuracy of the simulation results, comparisons were made with direct experimental measurements of the local shears arising from dislocation activity in the microstructure patch. A technique combining AFM and EBSD-based trace analysis was recently developed to quantitatively measure the local dislocation shear activity associated with activated deformation systems in different grains.¹⁴ Figure 3 displays a high magnification BSE image of an example of dislocation slip lines, as well as an AFM image of the same area measuring the surface height change due to the slip lines.

Based on the measurement of surface height change (h), the number of slip/twinning dislocations shearing a given volume of material can be calculated as Equation 6 where Na is the number of dislocations, ba is the Burgers vector of the identified deformation system, a, using trace analysis, and ez is the normal to the sample surface. The microstructural patch was subdivided into a 25 × 25 array of 10 μm tiles indicated by indices m,n, each of which was scanned by AFM. AFM section lines similar to that shown in Figure 3 were collected along the centerline of each tile, and the local shear (ga mn) associated with each deformation system α was calculated as Equation 7 where nα is the plane normal of deformation system α and Xmn is a vector pointing along the scan line with a constant length of 10 mm.

The local shear maps at 1.5% strain generated from AFM data (Figure 4) indicate that the simulation successfully predicted the presence and magnitude of most of the active dislocation slip and twinning systems. The highest shear value in the simulation is about the same as that measured using AFM for most of the grains (2, 3, 6, 8, 9, 10, and 13) in the center of the patch. However, the spatial distribution of simulated local shear is frequently different from the experimental measurement. In grain 3, the CPFE model showed the highest shear in the lower right of grain 14 rather than to the left of center. Also, near the boundary between grains 3 and 8, the shear caused by prismatic slip in grain 3 is about 0.05 to 0.07, which is lower in the simulation. In contrast, the basal shear activity in grains 5 and 7 varied from 0.005 to 0.02, which is modeled accurately in terms of magnitude and distribution. For grain 10, the CPFE simulation successfully captured the basal activity both spatially and quantitatively, but did not predict the pyramidal slip activity in the right side of this grain. Since the shear contribution of twinning was only simulated as a homogeneous unidirectional slip system, the localized twins in grain 2 could not be captured properly. Thus, the shears caused by twinning are diffuse rather than spatially concentrated, so the magnitude can only be semi-quantitatively compared to the experiment. Given this, the twin shear was distributed in a spatially similar way along the lower left grain boundary, but did not extend into the grain interior.

In a concurrent study, the use of scanning laser confocal microscopy to measure quantities similar to that obtained by AFM is under investigation with characterization following ≈6% global strain. At this strain, the twins grew sufficiently thick to nearly merge with each other on the side next to grain 1, but remained tapered on the side next to grain 3. Figure 5 shows a topographic representation of the neighborhood of grains 1, 2, 3 that indicates how the harder grain 2 resisted deformation (it has the highest topographic elevation) while grains 1 and 3 sunk due to being more highly strained. The twin topography is also evident, as the upper side has a higher elevation than the lower side in each twin. Clearly the influence of deformation in grains 0 and 1 affect deformation in grain 2, as the upper part of grain 2 has a depression that indicates that a greater amount of local strain has occurred. Figure 5b illustrates in higher magnification the region where the twins nucleated at the boundary between grains 1 and 2, which shows an additional depression along the grain boundary that may be the beginning of a crack. Further deformation will be imposed and the continuing evolution of deformation and damage in this region will be reported in a future paper.

An efficient way to improve the existing CPFE model is to more accurately determine the critical resolved shear stress (CRSS, or sa) and hardening parameters of deformation systems using single crystal experiments. However, for hexagonal metals it is difficult to use conventional uniaxial tensile tests of single crystals to measure the CRSS. This is because the wide range of CRSS values for the different deformation system types makes it difficult to isolate specific systems without activating other systems with lower CRSS. Nano-indentation experiments combined with CPFE simulations provide an alternative opportunity to study the behavior of single crystals in a polycrystalline environment. Because in most cases the grain size is much larger than the size of the indentations, nanoindention can be treated as the deformation of a constrained single crystal. Such experiments allow the separation of the influence of intrinsic grain properties, such as grain orientations, from the influence of polycrystallinity such as interfaces and neighboring grain orientations. A study of the anisotropic nano-indentation response of a-titanium was conducted to quantitatively identify the CRSS for different slip systems.^19,20

Based on a large-area EBSD scan, a suitable microstructure patch in another specimen from the same plate was chosen to provide a variety of crystallographic orientations for nanoindentation, as shown in Figure 6. These indentations were carried out using a spheroconical diamond tip with a nominal tip radius of 1 μm and a nominal cone angle of 90°. Load-controlled indentations were performed with a maximum load of 6 mN. Residual surface topography of selected indents in the middle of each grain (to avoid grain boundary effects) were measured by AFM. Figure 7a displays the residual pile-up topographies as measured by AFM positioned on an inverse pole figure of the indentation direction. The indentation data and the indent sizes show that the [0001] indentation axis is the hardest direction, as the residual impressions are shallower than in other orientations. For indentation axes away from the [0001] direction, two dominant pile-up hillocks are always formed on opposite sides of the impression. No twins were found in EBSD scans after indentation, and the AFM topographies showed no twin-shaped surface features.

As shown in Figure 7b, corresponding CPFE simulations using a constitutive model similar to that used in the previous section (but with latent hardening included) predicted pile-up patterns in good agreement with the experimental measurements. The CRSS values for prismatic, basal, and pyramidal slip of the CPFE model were identified by optimizing the simulation results (load–displacement and residual pile-up pattern) of the indentation process in different grain orientations. Non-linear optimization was conducted by applying a custom implementation of the downhill simplex method after Nelder and Mead.^22,23 The calculated values of CRSS were (150 ± 4) MPa for prismatic slip, (349 ± 10) MPa for basal slip, and (1107 ± 39) MPa for pyramidal slip, respectively. The CRSS value for prismatic slip is expected to have better accuracy using this optimization process than the lesser active slip systems. These values imply that basal and slip are more difficult to activate than the values used in the prior model, where the ratios used were 1:2:3 for prism:basal: CRSS values. However, increasing these ratios resulted in negligible basal and slip, which is inconsistent with the experimental observations. Introduction of latent hardening using the 1:2:3 CRSS ratio made only minor differences, indicating that this highly tuned phenomenological model based upon indentation did not significantly improve the simulation. Two possible reasons for the poorer fidelity with experiment are that the loading conditions for indentation contain significant hydrostatic compression, which may affect slip resistance by non-Schmid stress components that affect slip activation.²⁴ Secondly, the lack of grain boundaries may also frustrate dislocation nucleation processes for nonprism slip.²⁵ Furthermore, the CPFE simulation does not contain any form of additional slip resistance across a grain boundary.¹³ Simulation of the sub-surface grain geometry with an accurate 3D mesh may also positively affect model accuracy. Improvement in modeling slip behavior near grain boundaries appears to be necessary to improve the agreement between experiment and simulation.

To better assess how local dislocation content and orientation gradients are affected by grain boundaries, slip transfer phenomena were investigated in this specimen.^26,27 Evidence for slip transfer is apparent in the boundary between grains 1 and 2, where the twin in grain 2 developed due to slip transfer from the active prism slip system in grain 1. In this case, the geometrical alignment between the prism slip system in grain 1 and the twinning system in grain 2 was quite high. This alignment is described by the slip transfer parameter m' = cos ψ · cos κ= 0.94, where ψ is the angle between the prism slip plane normal in grain 1 and the twinning plane normal in grain 2 and similarly, κ is the angle between the prism slip Burgers vector and the twinning Burgers vector. In contrast, slip transfer from the twin system in grain 2 to the active prism slip system in grain 3 has m' = 0.73, a significantly lower value. A statistical comparison of 26 boundaries with similar geometrical relationships for active prism slip in one grain and high Schmid factors for twinning (under uniaxial tension) in the neighboring grain showed that only boundaries with m' values > 0.88 exhibited slip transfer. Because two boundaries with m' > 0.95 did not show slip transfer stimulated twinning, this parameter is apparently necessary but not sufficient, indicating that local stress tensors must be considered, or additional criteria must be satisfied. An important outcome of this analysis Figure 9. A section of the DAXM volume scan near the boundary between grains 1 and 2 at 2 mm beneath the surface. Grain 1 and the twin are represented by cyan and turquoise color, respectively. The 3 pixels surrounded by thick boxes show moderately streaked peaks, while the other 47 pixels all show sharp peaks. Laue patterns of 4 pixels are shown. In pattern 3, the black dotted arrows near three indexed peaks represent the theoretical peak streak direction caused by ( )[ ] _ _ 1010 1210 edge dislocations. Figure 10. Three sections of the DAXM area scan near the boundary between grains 2 and 3 show how the grain boundary and twin are inclined to the surface. Selected Laue patterns from numbered voxels on the top scan are shown. Laue patterns from grain 3 display significantly streaked peaks. Pattern 6 uses an intensity threshold that is lower than the rest of the patterns to make the peaks more visible. was that the activated twin system only sometimes had the highest Schmid factor (and in a couple of cases the activated twin system had the lowest Schmid factor based upon the global stress state), indicating that nucleation is the critical part of the process for activating mechanical twins (see also Reference 28 for a similar outcome based upon a statistical study, and Reference 29 that shows how dislocations entering a grain boundary can facilitate twin nucleation). These two boundaries were further investigated using 3D differential aperture x-ray microscopy (DAXM30–32) to characterize subsurface microstructure and GNDs at two ends of a twin in grain 2 to gain further understanding of slip transfer and the twin nucleation process. Quantified GND content is useful for making detailed comparisons between measured and simulated GND content.

The experimental results indicate that a CPFE model with phenomenological hardening can simulate the heterogeneous deformation process in commercial purity titanium at a level that is roughly consistent with experiments. The spatial distribution of deformation, however, shows some differences from the experimental measurement. Optimized simulation of the nano-indentation process constrained by experimental measurements led to refined constitutive parameters, but use of these parameters did not significantly improve the simulation. The nanoindentation approach evaluated slip behavior in hydrostatic compression states that may not be equivalent to the predominantly tensile stresses present in the bent polycrystalline sample. To further improve the accuracy of CPFE simulation, a deeper understanding of how deformation is influenced at and across grain boundaries needs to be cast into the constitutive description of crystal plasticity.

This research is supported by a Materials World Network grant (NSF DMR-0710570 and DFG EI 681/2-1). Use of the Advanced Photon Source was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357. R.B. is supported by the Materials Sciences and Engineering Division, Office of Basic Energy Sciences, U.S. Department of Energy.

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