Anisotropy and Strain Localization in Dynamic Impact Experiments of Tantalum Single Crystals

Deformation mechanisms in bcc metals, especially in dynamic regimes, show unusual complexity, which complicates their use in high-reliability applications. Here, we employ novel, high-velocity cylinder impact experiments to explore plastic anisotropy in single crystal specimens under high-rate loading. The bcc tantalum single crystals exhibit unusually high deformation localization and strong plastic anisotropy when compared to polycrystalline samples. Several impact orientations - [100], [110], [111] and [1¯49\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{1}49$$\end{document}] - are characterized over a range of impact velocities to examine orientation-dependent mechanical behavior versus strain rate. Moreover, the anisotropy and localized plastic strain seen in the recovered cylinders exhibit strong axial symmetries which differed according to lattice orientation. Two-, three-, and four-fold symmetries are observed. We propose a simple crystallographic argument, based on the Schmid law, to understand the observed symmetries. These tests are the first to explore the role of single-crystal orientation in Taylor impact tests and they clearly demonstrate the importance of crystallography in high strain rate and temperature deformation regimes. These results provide critical data to allow dramatically improved high-rate crystal plasticity models and will spur renewed interest in the role of crystallography to deformation in dynamics regimes.

Most metals, when used in a technologically relevant application, are polycrystalline aggregates comprised at the micro-scale of many distinct regions of different crystallographic orientations, i.e. grains. Each of these grains consists (mostly) of a single crystallographic character that behaves, e.g. during deformation, approximately like a single crystal of material subjected to the complex constraints and boundary conditions imposed by the grains around it. Therefore, an accurate understanding of single crystal behavior is critical to predicting the bulk properties of polycrystals that arise from the collective behavior of their individual grains 1,2 . Recent high fidelity computational approaches at various length scales, i.e. molecular dynamics (MD) 3,4 and crystal plasticity-finite element (CP-FE) models [5][6][7] , explicitly resolve individual grains, and use grain-scale material properties to parameterize and/or validate continuum-scale models. Although mechanical behaviors and plastic anisotropy of tantalum single crystals are well characterized in quasi-static regimes [8][9][10] , their dynamic behaviors at higher strain rates are less well-known.
In this report, we show that high-rate single crystal response is substantially more anisotropic, i.e. dependent on crystallographic orientation, than either quasi-static experiments on single crystals or high-rate experiments on polycrystals would suggest. This information is crucial for not only building a physics based, microstructure-aware model which does not require empirical fitting for every polycrystalline microstructure and texture, but also for validating predictions of microstructure-scale high-rate behavior.
Taylor cylinder impact testing provides a simple, yet robust approach for subjecting a single specimen to a wide range of strain rates across its length 11 . In these, well known tests, a cylindrical projectile is shot into a rigid anvil at velocities in the range of 100 to 200 m/s. See the schematic description in Fig. 1, and impact movie in the online Supplementary Information. The leading end of the projectiles undergo high strain rates and temperatures while the trailing end experiences little or no deformation. The test creates large gradients of stresses and strains, which are recorded in the final shape of the deformed cylinder. Thermo-mechanical finite element simulations of polycrystalline tantalum Taylor impact experiments at 175 m/s showed that the local region with the maximum deformation can reach strain rate of 5 × 10 4 s −1 12 while the heat generation caused by the mechanical dissipation associated with plastic straining predict average and maximum temperatures of 405 K and 1007 K, respectively. The maximum strain rate and temperature are associated with regions of highly localized plastic deformation. Taylor impact tests provide dynamic material responses in strain-rate regimes between Split-Hopkinson pressure bar (10 2 -10 4 s −1 ) and laser driven shock experiments (10 7 -10 9 s −1 ). Deformation behaviors in these two regimes are generally described by dislocation plasticity theory and equation of state (EOS) theories, respectively. Taylor impact experiments are generally non-destructive, even at high-rates. Therefore, the recovered samples are especially useful in validating material models in dynamic finite element simulations [12][13][14][15][16][17] . Plastic anisotropy of the material is also captured in the cross-section of the impacted surface, i.e. the foot shape of the deformed specimen. For example, isotropic polycrystals exhibit circular foot shapes, while textured polycrystals develop oval foot shapes. Taylor impact tests have been conducted with various polycrystalline metals, i.e. tantalum 14 , copper 13 , steel alloys 13,18,19 , aluminum alloys 18 as well as various hexagonal close-packed (hcp) metals [20][21][22] . However, Taylor impact tests using single crystals have not been reported previously. This work demonstrates characteristics of high rate deformation behaviors in single crystals, severe strain localization and plastic anisotropy.

Results
Tantalum is a refractory body-centered cubic (bcc) metal that is often used in extreme environments such as nuclear, and ballistic applications. In this work, Taylor Figure 1 shows a schematic drawing of initial sample dimensions, Taylor impact experimental setup and a representative deformed polycrystalline tantalum projectile. Figure 2 shows deformed single crystal samples after impact at various velocities. Red dashed lines represent the initial dimension of a specimen before the impact. Side profiles of deformed projectiles are significantly different from that of the polycrystalline projectile, Fig. 1. In contrast to the polycrystalline projectile that shows smooth and continuous deformation, single crystals exhibit more localized deformation, especially at higher velocities. For example, [110] single crystal at 101.7 m/s started to exhibit X-shaped shear bands and cusps -even forming 'shoulders' on the side of the projectile. At 137.5 m/s, subsequent deformation is strongly localized in two shoulders. In  Table 1.  In order to understand the observed symmetries in foot shapes of deformed single crystals, crystallographic slip systems and their Schmid factors were calculated. The Schmid law describes the relationship between the applied stress (σ) and the resolved shear stress (τ) using slip directions and slip plane normal directions as follows: where M is the Schmid factor, φ is the angle between the slip direction and the loading axis and χ is the angle between the slip plane normal and the loading axis. M can be obtained for each set of slip plane and slip direction.
The slip system(s) with maximum M value will be the most favored under uniaxial loading. While close-packed cubic metals follow well-defined slip systems (i.e. face-centered cubic (fcc) on twelve 〈111〉{110} slip systems), active slip planes in bcc metals are more controversial. While it is generally accepted that slip directions are along 〈111〉 directions, many atomistic simulations and experiments on tantalum support the theory of plastic slip on     planes and slip directions projected on the plane normal to the loading axis (foot plane). Active slip planes are obtained using Equation (1) Supplementary Information, Tables 2 and 3.
This simple crystallographic analysis qualitatively describes the source of the foot symmetries and the plastic anisotropy observed in single crystals. More accurate considerations of complex stress and strain states, crystal rotations, thermal and dynamic effects as well as advanced theories on plastic instability are required to better understand the deformation mechanisms of single crystals. In particular, understanding the locations of shear band formations and strain localization in single crystal is important since these local phenomena eventually lead to ductile fracture of polycrystalline materials 35,36 . Therefore, these experimental results present significant challenges, as well as future opportunities for the computational modeling community. A comprehensive computational model must incorporate thermo-mechanical coupling, temperature and strain rate dependent constitutive model as well as crystal plasticity framework to capture the plastic anisotropy and strain localizations. Most classical high temperature and strain rate dependent strength models are based on isotropic polycrystalline materials 13,[37][38][39][40] or considers collective anisotropic behaviors by utilizing continuum-scale anisotropic yield criteria 14,15,17,21 . A more recent model explicitly incorporates a crystal plasticity framework 41 .

Conclusions
This work has shown strong strain localization and plastic anisotropy in the dynamic response of tantalum single crystals. These non-uniform plastic deformations in single crystals are much more pronounced when compared to polycrystalline tantalum. Symmetries observed in the deformed foot profile agreed well with a crystallographic analysis that suggests the plastic deformation of tantalum single crystals at high strain rates and temperature regimes is dominated by dislocation slips along close-packed 〈111〉 directions. A more complete understanding of, and predictive capability for, the dynamic deformation behavior of single crystals, including plastic localization and anisotropy, will require crystal-level simulations with accurate temperature and strain rate dependent strength models.
We plan to conduct high-resolution experimental analyses in the near future which will allow subsequent analysis comparing different state variables from computational models, i.e. local strain fields and microstructural evolution.

Methods
Three single crystal specimens were made per orientations (a total of 12 specimens) from Princeton Scientific Corporation and machined to have a diameter of 6.35 mm and a length of 38.1 mm. Two Teflon spacers were attached to single crystal specimens to fit in a 30-caliber 0.3 inch (7.62 mm) smooth-bore launch tube. Taylor impact tests were conducted at the impact facility in Los Alamos National Laboratory. Single crystal projectiles were shot at various velocities, ranging from 78 to 151 m/s against a pneumatically-positioned AF1410 high-strength steel anvil ground to a mirror surface finish. All experiments were performed at room temperature and vacuum conditions, typically at 10 torr. The projectile velocities were measured using four laser beams located at various locations along the traveling path. The last laser beam was placed about 1 mm before the impact. High-speed photography was added to capture 128 frames of images to reveal the deformation process in real time.