To gain insights into the relative contributions of different plastic deformation mechanisms, particularly basal slip, for shocked hexagonal close-packed (hcp) metals, magnesium (Mg) single crystals were subjected to shock compression and release along a low-symmetry (LS) orientation to 1.9 and 4.8 GPa elastic impact stresses. LS-axis is a “nonspecific” direction resulting in propagation of quasilongitudinal and quasishear waves. Wave profiles, measured using laser interferometry, show a small elastic wave followed by two plastic waves in compression; release wave profiles exhibited a structured response for the higher stress and a smooth response for the lower stress. The LS-axis wave profiles are significantly different than profiles published previously for c- and a-axes, demonstrating that Mg single crystals exhibit strong anisotropy under shock compression/release. Numerical simulations, using a time-dependent anisotropic modeling framework, show that shock wave loading along the LS-axis involves the simultaneous operation of multiple deformation mechanisms. Shock compression along LS-axis is dominated by basal slip while prismatic slip and pyramidal I {101¯1}112¯3 slip play a smaller role; coupling between longitudinal and shear deformations was observed. The unloading response is dominated by basal slip with some contribution from prismatic slip; pyramidal I slip is not activated. The present results, unlike results obtained for c- and a-axes, show that the deformation mechanism observed under quasistatic loading conditions along LS-axis is not sufficient to determine the shock response along this orientation. Although requiring numerical simulations for wave analysis, shock propagation along a LS-orientation provides new insights into the plastic deformation response of hcp metal single crystals.

Hexagonal close-packed (hcp) materials require multiple plastic deformation mechanisms to accommodate arbitrary deformation, displaying a complex deformation response.1–4 The plastic deformation behavior is highly anisotropic due to dislocation slip along the basal plane,3,4 and additional anisotropy is also observed due to the tendency for polycrystalline hcp metals to form twins during rolling or extrusion process leading to significant texture development.2–4 Therefore, understanding and modeling shock wave induced plastic deformation in hcp metals is challenging and constitutes an important scientific need. Toward this objective, single crystal studies are inherently beneficial because different deformation systems can be selectively activated and examined by shock loading along different crystal orientations—while avoiding complications arising from dislocation/grain-boundary interactions.5 For hexagonal close-packed (hcp) metals like magnesium (Mg), this approach is particularly useful for separating multiple deformation mechanisms that are operative simultaneously and for developing accurate material models. Past shock studies on metal single crystals have focused almost exclusively on examining plastic deformation response along high-symmetry crystallographic directions.5–18 In such experiments, a purely longitudinal wave is propagated into the material resulting in uniaxial strain, which simplifies the wave propagation analysis considerably.

Magnesium (Mg), because of its light weight and high strength properties, is an attractive candidate for structural applications.19,20 To gain insights into the shock-induced plastic deformation response of Mg single crystals, samples oriented along the c- and a-axes were subjected to shock compression/release.12,14 The focus of the earlier work was to identify and understand the operative plastic deformation mechanisms (dislocation slip and/or deformation twinning) under shock loading conditions. These experiments along the high-symmetry c- and a-axes provided significant insights into the role of pyramidal slip, prismatic slip, and (101¯2) twins toward shock-induced plastic deformation.12,14 However, shock loading along these high-symmetry directions cannot activate the close-packed plane (basal slip) for hcp metals. Basal slip is an important and dominant deformation mechanism because it has the highest atomic density and the lowest critical resolved shear stress (RSS). When loaded quasistatically along a low-symmetry (LS)-axis, basal slip is the only operative system governing deformation.2,21–23 To understand the role of basal slip under shock loading, experiments along low-symmetry (LS) directions are needed. In this work, the following questions were addressed:

  1. What role does basal slip play during shock compression and release of Mg single crystals?

  2. How do the plastic deformation mechanisms for Mg shocked and released along LS-axis differ from those determined previously under quasistatic compression?

  3. What are the plastic deformation mechanisms that govern the response of shock compression and release in Mg single crystals along a LS direction, and how to determine the relative contributions of competing plastic mechanisms to the overall deformation response?

In this work, Mg single crystals oriented along a LS-axis [normal to (101¯2) twin plane] of varying sample thicknesses were subjected to shock compression and release. For completeness, we note that free surface profiles were reported at ∼45° to the c-axis in Ref. 18. However, in that work, the measured wave profiles were analyzed under the uniaxial strain assumption and the profiles beyond the elastic limit were not analyzed. As discussed in the present work, the uniaxial strain assumption is incorrect for shock loading along LS directions. Because the LS direction is not a specific direction, wave propagation analysis required a computational approach using a time-dependent anisotropic modeling framework incorporating dislocation slip and deformation twinning, similar to that used in the Be work.11 Through these simulations, the deformation mechanisms were identified, and their relative contributions were examined. This work, along with our previous studies on shocked Mg single crystals,12,14 provides a comprehensive understanding of plastic deformation mechanisms in Mg single crystals under shock wave loading.

The remainder of the paper is organized as follows: The experimental methods are summarized in Sec. II and the measured wave profiles are presented in Sec. III. Orientation dependence of the elastic wave amplitude, numerical simulations of the measured LS-axis Mg wave profiles, and a discussion of the results are presented in Sec. IV. The main findings of this study are summarized in Sec. V.

All samples used in this study were from high quality (99.999% pure) Mg single crystal boules obtained from Metal Crystals and Oxides, Ltd. (Cambridge, UK). The boules were oriented to within 2° of the desired orientation [normal to the (101¯2) twin plane] using x-ray Laue diffraction and then cut using a slow speed saw. The direction of the normal to the (101¯2) twin plane is denoted by the azimuthal angle ϕ = 30° and the polar angle θ = 46.85°, as shown in Fig. 1. Using this nomenclature, the shock wave propagation direction is described by the azimuthal angle ϕ = 29.2° and the polar angle θ = 45.2°.

FIG. 1.

(101¯2) twinning plane (shaded) showing the Burgers vector b and the normal vector n^. The direction of the normal to the twin plane, with respect to the crystal axes, is described by the azimuthal angle ϕ = 30° and the polar angle θ = 46.85°. Direction of shock wave propagation along the low-symmetry (LS) direction is described by the azimuthal angle ϕ = 29.2° and the polar angle θ = 45.2°.

FIG. 1.

(101¯2) twinning plane (shaded) showing the Burgers vector b and the normal vector n^. The direction of the normal to the twin plane, with respect to the crystal axes, is described by the azimuthal angle ϕ = 30° and the polar angle θ = 46.85°. Direction of shock wave propagation along the low-symmetry (LS) direction is described by the azimuthal angle ϕ = 29.2° and the polar angle θ = 45.2°.

Close modal

The overall experimental approach in this work was similar to that published previously for the other orientations.12,14 Samples were prepared by using progressively finer sanding papers (320 grit, 400 grit, and 600 grit) to remove ∼300 μm from each cut surface. After checking the orientations, ambient density was measured using the Archimedean method and sound speed measurements were measured using the pulse-echo technique for all the samples. The average value for ambient density was 1738 ± 2 kg/m3, for longitudinal sound speed was 5.81 ± 0.05 km/s, and 3.07 ± 0.03 km/s and 3.36 ± 0.02 km/s for the two shear sound speeds. After grinding, stepwise polishing was done by hand and the polishing process removed ∼145–150 μm of additional material from each ground surface.

A schematic view of the experimental setup is shown in Fig. 2. A typical target, similar to that used for the other orientations,12,14 consisted of a z-cut quartz buffer (∼3.192 mm thick) that is bonded to a Mg sample, which was then bonded to a fused silica window (∼9.545 mm thick). The sample thicknesses are listed in Table I. The optical window provided release wave measurements without spall. The lateral dimensions used in this work permitted complete data acquisition before the arrival of edge waves. Prior to bonding the various target components, a thin aluminum mirror was vapor deposited onto the front surface of the fused silica window. Except for the thin (∼0.5 mm) sample experiment, the back surface of the z-quartz was also vapor plated with aluminum, which served as a reflector for laser interferometry (VISAR-Velocity Interferometer System for Any Reflector)24 measurements.

FIG. 2.

Overall configuration for shock compression and release experiments on LS-axis Mg single crystals. Wave profiles were measured at the Mg sample/FS window interface using velocity interferometry.

FIG. 2.

Overall configuration for shock compression and release experiments on LS-axis Mg single crystals. Wave profiles were measured at the Mg sample/FS window interface using velocity interferometry.

Close modal
TABLE I.

Summary of experimental results.

Expt. No.Impact velocity (m/s)Elastic impact stress (GPa)Mg sample thickness (mm)Elastic shock velocity (km/s)Elastic wave peak
Interface particle velocity (km/s)In-material particle velocity (km/s)In-material longitudinal stress (GPa)
01 (16-511)a 485 4.74 0.512 … 0.006 ± 0.002 0.007 ± 0.002 0.079 ± 0.02 
02 (16-509) 485 4.74 2.005 5.72 ± 0.08 0.002 ± 0.001 0.002 ± 0.001 0.026 ± 0.01 
03 (15-506) 486 4.75 2.007 5.73 ± 0.06 0.002 ± 0.001 0.002 ± 0.001 0.026 ± 0.01 
04 (15-510) 486 4.75 4.003 5.76 ± 0.07 0.001 ± 0.001 0.001 ± 0.001 0.013 ± 0.01 
05 (17-509) 201 1.89 1.941 5.76 ± 0.07 0.001 ± 0.001 0.001 ± 0.001 0.013 ± 0.01 
06 (17-505) 200 1.88 4.007 5.76 ± 0.07 0.001 ± 0.001 0.001 ± 0.001 0.013 ± 0.01 
07 (17-513) 201 1.89 4.007 5.80 ± 0.06 0.001 ± 0.001 0.001 ± 0.001 0.013 ± 0.01 
Expt. No.Impact velocity (m/s)Elastic impact stress (GPa)Mg sample thickness (mm)Elastic shock velocity (km/s)Elastic wave peak
Interface particle velocity (km/s)In-material particle velocity (km/s)In-material longitudinal stress (GPa)
01 (16-511)a 485 4.74 0.512 … 0.006 ± 0.002 0.007 ± 0.002 0.079 ± 0.02 
02 (16-509) 485 4.74 2.005 5.72 ± 0.08 0.002 ± 0.001 0.002 ± 0.001 0.026 ± 0.01 
03 (15-506) 486 4.75 2.007 5.73 ± 0.06 0.002 ± 0.001 0.002 ± 0.001 0.026 ± 0.01 
04 (15-510) 486 4.75 4.003 5.76 ± 0.07 0.001 ± 0.001 0.001 ± 0.001 0.013 ± 0.01 
05 (17-509) 201 1.89 1.941 5.76 ± 0.07 0.001 ± 0.001 0.001 ± 0.001 0.013 ± 0.01 
06 (17-505) 200 1.88 4.007 5.76 ± 0.07 0.001 ± 0.001 0.001 ± 0.001 0.013 ± 0.01 
07 (17-513) 201 1.89 4.007 5.80 ± 0.06 0.001 ± 0.001 0.001 ± 0.001 0.013 ± 0.01 
a

Shock velocity for thin experiment assumed to be average of shock velocities from thicker samples

The quartz−Mg−fused silica sample assemblies were impacted with a-axis sapphire impactors (∼3.192 mm thick), launched using a light gas gun. The measured projectile velocities, having a typical precision of ∼0.5% for all of the experiments, are shown Table I. Upon impact, elastic shock waves propagated into both the quartz buffer and the sapphire impactor. The forward propagating shock wave in the buffer interacts with the buffer/sample interface, resulting in a transmitted wave through the Mg sample and a small reflected wave into the buffer. The propagating waves in the Mg undergo further transmission and reflection at the sample/window interface. The shock wave in the impactor gets reflected off the impactor free surface and propagated back through the sapphire−quartz−Mg−fused silica system as a release wave. Because the impedance of the impactor is higher than the target materials, the sample experiences only a partial release from its peak shocked state.

Similar to the work on c- and a-axes Mg,12,14 particle velocity histories were measured at the Mg sample/window interface (green arrows) and at the back of the buffer (red arrows), as shown in Fig. 1, using a VISAR. Additionally, by using a dual velocity per fringe (VPF#1: 0.09, VPF#2: 0.173) arrangement,25 the particle velocity histories at the sample/window interface were determined unambiguously. Similar to the previous Mg studies,12,14 precise shock wave velocity measurements in the Mg samples and the impact tilt (<1.3 mrad) were obtained. Due to the lower precision associated with thin samples (0.5 mm), shock velocity measurements were not obtained in the thin samples. The shock velocity for the thin experiment was assumed to be the average of shock velocities from the thicker samples.

In the most general case, planar loading in anisotropic solids results in three waves:26,27 one quasilongitudinal (QL) wave and two quasishear (QS) waves. The particle velocities associated with these three waves, though mutually orthogonal, are neither perpendicular nor parallel to the wave propagation direction; hence, the prefix “quasi.” Only for isotropic solids or along “specific directions” in anisotropic solids, the particle velocities are perpendicular and parallel to the wave propagation direction. For Mg single crystals, c- and a-axes are “specific directions,” resulting in the propagation of pure longitudinal waves. In contrast, shock compression along a LS direction results in the propagation of QL and QS waves.11,28 The combination of the LS direction and elastic-plastic deformation leads to multiple waves and results in a fairly complex loading/unloading response.

Four experiments, including an experiment to determine experimental reproducibility, were conducted on 0.5–4.0 mm thick LS-axis samples shocked to ∼4.8 GPa elastic impact stress. The experimental parameters relevant to these experiments are listed in Table I. Figure 3 shows representative wave profiles at the sample/window interface for these experiments. In all the profiles, the reference time (t = 0) corresponds to the impactor/buffer impact. Sample thicknesses are shown in the figures. It is important to note that, despite the presence of QL and QS waves, the particle velocities are measured only along a direction normal to the wave front (same as along the sample thickness direction).

FIG. 3.

(a) Particle velocity histories measured at the sample/window interface for shock and release of Mg single crystals along the 101¯2 LS-axis orientation (4.8 GPa); t = 0 is time of impact. The corresponding experiment number and propagation distances are noted. The particle velocity shown here represents velocity component normal to the propagating wave front. (b) Shock compression profile (magnified) for Expt. 03 showing the arrival of the second plastic wave. Inset shows the elastic wave amplitude (magnified).

FIG. 3.

(a) Particle velocity histories measured at the sample/window interface for shock and release of Mg single crystals along the 101¯2 LS-axis orientation (4.8 GPa); t = 0 is time of impact. The corresponding experiment number and propagation distances are noted. The particle velocity shown here represents velocity component normal to the propagating wave front. (b) Shock compression profile (magnified) for Expt. 03 showing the arrival of the second plastic wave. Inset shows the elastic wave amplitude (magnified).

Close modal

The compression profile in Fig. 3(a), measured at the sample/window interface, shows a two-wave structure, a very small amplitude elastic wave followed by the plastic wave. However, a closer examination of the plastic wave, by magnifying the wave profile [Fig. 3(b)], demonstrates a clearer structure in the plastic wave. Because the wave speeds of the two plastic waves are not very different for an elastic impact stress of 4.8 GPa, their separation is not easily observed.

To better resolve the wave structure observed in the 4.8 GPa experiments, the LS-axis Mg was shocked to a lower elastic impact stress. Three experiments, including an experiment to determine experimental reproducibility, were conducted on 2.0 and 4.0 mm thick LS-axis samples shocked to ∼1.9 GPa elastic impact stress. Figure 4 shows the wave profiles obtained at the sample/window interface for LS-axis Mg for the two sample thicknesses. The reference time (t = 0) corresponds to the impactor/buffer impact. Sample thicknesses are shown in the figure.

FIG. 4.

(a) Particle velocity histories measured at the sample/window interface for shock and release of Mg single crystals along the 101¯2 LS-axis orientation (1.9 GPa); t = 0 is time of impact. The particle velocity shown here represents velocity component normal to the propagating wave front. The corresponding experiment number and propagation distances are noted. (b) Shock compression profile (magnified) for Expt. 06 to better show the arrival of the three waves.

FIG. 4.

(a) Particle velocity histories measured at the sample/window interface for shock and release of Mg single crystals along the 101¯2 LS-axis orientation (1.9 GPa); t = 0 is time of impact. The particle velocity shown here represents velocity component normal to the propagating wave front. The corresponding experiment number and propagation distances are noted. (b) Shock compression profile (magnified) for Expt. 06 to better show the arrival of the three waves.

Close modal

In contrast to the higher stress experiments, wave profiles at the lower stress show two distinct plastic waves following the very small elastic precursor. Interestingly, the peak of the first plastic wave decreases with increasing propagation distance. Also, the first plastic wave for the 2 mm thick sample shows a small drop behind the peak. Both of these observations are similar to the stress relaxation behind the elastic front and the decay of the elastic wave with propagation distance observed in the other orientations.12,14 They are characteristic of time-dependent, plastic response.

For a given elastic impact stress, the elastic wave amplitudes along this orientation, listed in Table I, are markedly smaller than the elastic wave amplitudes for c- and a-axes Mg.12,14 For both impact stresses, rapid attenuation of the elastic wave amplitudes likely occurs near the impact surface with no significant attenuation thereafter. In addition, the measured elastic wave amplitudes are comparable irrespective of the input stress.

The release wave profiles obtained along the LS-axis are also shown in Figs. 3 and 4. Similar to the compressive wave, the release wave also results in complex motions (QL and QS). The measured release wave profile along the LS-axis exhibits a structured unloading response when released from the higher peak stress, but a somewhat smooth release when released from lower stress.

Additional experiments were conducted to evaluate the reproducibility of the measured wave profiles. Good overall reproducibility was observed for both 1.9 GPa and 4.8 GPa experiments (Expt. 07 and Expt. 02, respectively), as discussed in  Appendix A.

To reiterate, planar impact along a nonspecific direction results in QL and QS waves. These, in combination with elastic-plastic deformation, result in complex loading/unloading response. Since the particle velocities are measured along a direction normal to the wave front, analysis of the wave profiles beyond the elastic wave is quite complex and numerical simulations are needed11 to relate the wave profiles to the material response.

To understand the role of crystal orientation on the elastic wave amplitude, the in-material longitudinal stresses corresponding to the elastic wave amplitude for shocked c-,12a-14 and LS-axes Mg are plotted vs the sample thickness in Fig. 5. The solid horizontal lines in Fig. 5 represent the elastic impact stress (σimp)—the longitudinal stress that would be attained if the sample were to remain elastic during shock compression. For a step input, this is also the elastic wave amplitude in the material at zero propagation distance. The dashed lines are curve fits shown for better clarity.

FIG. 5.

Elastic wave amplitude vs thickness for all Mg single crystal experiments. Solid horizontal lines represent the elastic impact stresses and dashed lines are fits for visual clarity. Experimental data are shown as circles, squares, and triangles for loading along the c-,12 a-,14 and LS-axes, respectively. Red points correspond to high stress experiments, while blue and green points correspond to low stress experiments. The green (not visible) are the same value as the red points for LS-axis Mg. Uncertainty in these measurements is smaller than the symbol size.

FIG. 5.

Elastic wave amplitude vs thickness for all Mg single crystal experiments. Solid horizontal lines represent the elastic impact stresses and dashed lines are fits for visual clarity. Experimental data are shown as circles, squares, and triangles for loading along the c-,12 a-,14 and LS-axes, respectively. Red points correspond to high stress experiments, while blue and green points correspond to low stress experiments. The green (not visible) are the same value as the red points for LS-axis Mg. Uncertainty in these measurements is smaller than the symbol size.

Close modal

Figure 5 shows that the elastic wave amplitudes are significantly different for different orientations, demonstrating considerable crystallographic orientation dependence.

The following observations from Fig. 5 are summarized:

  1. The measured elastic wave amplitude for shocked LS-axis is ∼13 MPa and is significantly lower when compared to shocked c-axis (0.9 GPa) and a-axis Mg (0.45 GPa).

  2. Comparing the elastic wave amplitudes at 0.5 mm thickness for the three orientations to the corresponding elastic impact stresses, it is clear that elastic wave attenuation is very rapid near the impact surface for the LS-orientation.

  3. The elastic limit, defined as the near constant elastic wave amplitude at larger propagation distance for LS- and a-axes loading, does not depend on the elastic impact stress in contrast to c-axis loading.

To relate the observations summarized above for the different orientations, numerical simulations are needed.

The material models, presented in the previous a-axis Mg study,14 were incorporated into a one-dimensional, Lagrangian wave propagation code (COPS)29 to numerically calculate the wave profiles and to compare them with the measured wave profiles. In the COPS code, the governing equations are of a generalized one-dimensional form30 and are not restricted to uniaxial strain. Because of the generalized one-dimensional equations, wave propagation along nonspecific directions can be simulated; and QL and QS wave propagation can be examined using anisotropic elastic-plastic material descriptions. The numerical approach in the COPS code is similar to a typical finite difference, artificial viscosity approach31 that uses explicit time integration to numerically solve the governing equations.

The deformation mechanisms observed during quasistatic compression/tension experiments2,32 provide a useful starting point to simulate the measured wave profiles for shocked and released Mg single crystals. Under quasistatic loading, LS-axis Mg deformation is solely governed by the basal slip. However, in the present calculations, contribution from prismatic, {101¯1} pyramidal I planes and {112¯2} pyramidal II planes, and deformation twinning on {101¯2} planes were also considered along with the dominant basal slip. Although {101¯1} twinning was also explored, no significant contribution was observed in the calculated wave profiles and was not included in the deformation response. Overall, the present approach is similar to that used for analyzing wave profiles in previous studies:12,14 plastic strain was calculated by determining the rate of plastic shear deformation αγ˙ for each operating dislocation slip or deformation twinning system α from the resolved shear stress (RSS) ατ using the dislocation-based plasticity model and deformation twinning model that were described in our previous Mg work.14 For completeness, the overall model and parameters used for nonbasal slip deformation mechanisms are very briefly summarized in  Appendix B.

As noted earlier, basal slip is not activated for c- and a-axes shock loading.12,14 Because basal slip plays an important role and causes the highly anisotropic plastic deformation response observed in Mg single crystals, examination of basal slip was the prime motivation for examining shock wave propagation along the LS-axis. To incorporate basal slip in the numerical simulations, the following model parameters were treated as adjustable parameters and their values are listed here:

{0001}112¯0 basal slip:

Nm0=8.32×1012m2;M=7.95×1014m2;D=0.65GPa;τ0=0.5MPa;H=1GPa,

where Nm0 is the initial mobile dislocation density, M is the multiplication parameter, D is the drag stress parameter, τ0 is a threshold stress for dislocation motion, and H is the hardening constant. Additionally, for the basal slip, a back-stress mechanism (see  Appendix B) was also incorporated during unloading, similar to that used for a-axis Mg14 and aluminum single crystals.10 The quantities Bb2np and BL2b2 were treated as adjustable parameters and their values are listed below,

Bb2np=3.5×102Pas;BL2b2=8×104Pas.

All the parameters used for {101¯0}112¯0 prismatic and {101¯1}1123 pyramidal I slip planes were obtained directly from simulations used for a-axis14 and c-axis12 as summarized in  Appendix B. However, the drag stress parameter for prismatic slip used in the LS-axis calculations is D=0.68GPa, instead of D=0.20GPa used to match the a-axis Mg wave profiles. Because the wave structure observed in the measured wave profiles is better resolved at the lower elastic impact stress, we focus first on a simulation of the wave profile for the 4.0 mm thick LS-axis crystal shocked to 1.9 GPa.

Figure 6 shows the calculated wave profiles at 4.0 mm propagation distance incorporating basal slip only (green dashed-dotted line), and a combination of basal slip, prismatic slip, and pyramidal I slip (red dashed-double dotted line). As seen in these simulations, basal slip alone (green dashed-dotted line) provides a reasonable match to the very small measured QL elastic precursor and the early plastic wave features. In this calculation, significant strain hardening [described by Eqs. (B3) and (B4)] was incorporated into the basal slip model to match the first plastic wave speed and the measured amplitude of the QL wave. However, the calculated profile, with only basal slip, does not show the multiple wave features (related to QS waves) observed during the experiments. The reason that no QS waves are observed in these calculations is because basal slip lacks the ability to couple longitudinal and shear deformations when a hcp material is shock compressed along a direction that is ∼45° to basal plane.33 

FIG. 6.

Particle velocity histories at the sample/window interface for 4.0 mm thick LS-axis single crystals shocked to 1.9 GPa elastic impact stress and released. The black solid curve is the measured velocity history (Expt. 06). The green dashed and red dotted-dashed curves are calculated profiles using the anisotropic modeling framework: basal slip only (green dashed-dotted); and combination of basal slip, prismatic slip, and pyramidal I slip (red dashed-double dotted). Time is relative to the moment of impact.

FIG. 6.

Particle velocity histories at the sample/window interface for 4.0 mm thick LS-axis single crystals shocked to 1.9 GPa elastic impact stress and released. The black solid curve is the measured velocity history (Expt. 06). The green dashed and red dotted-dashed curves are calculated profiles using the anisotropic modeling framework: basal slip only (green dashed-dotted); and combination of basal slip, prismatic slip, and pyramidal I slip (red dashed-double dotted). Time is relative to the moment of impact.

Close modal

The red dashed-double dotted curves in Fig. 6 are the wave profiles calculated by incorporating a combination of basal slip, prismatic slip, and pyramidal I slip. These calculated profiles provided a very good overall match to the measured wave profiles including the QS wave (second plastic wave). The good match suggests that prismatic slip and pyramidal I slip also play a role in the plastic deformation response along the LS-axis, and the coupling of longitudinal and shear deformations for shock compression along the LS-axis is due to the activation of prismatic or pyramidal I slip. We note that neither {112¯2}1123 pyramidal II slip nor {101¯2}101¯1 twinning contributes to the plastic deformation when shocked along this axis. This is because the resolved shear stress (RSS) along pyramidal II slip does not reach the required activation stress and the RSS along the (101¯2) twinning is not favorable for twinning to operate (it has the wrong sign).

Figure 7 shows a comprehensive comparison between all experimentally measured wave profiles (along the LS-orientation) and the corresponding numerical simulations for samples shocked to 1.9 GPa and 4.8 GPa elastic impact stresses. The measured wave profiles are shown with black solid lines, and the wave profiles calculated using basal slip, prismatic slip, and pyramidal I slip model are shown in red dashed line. Although the compressive wave profiles are slightly offset for the higher stress experiments (4.8 GPa elastic impact stress), the overall match between the simulations and the measured profiles is very good.

FIG. 7.

Measured and calculated velocity histories at the sample-window interface for all LS-axis experiments. The black solid curves are the measured velocity histories. The red dot-dashed curves are calculations using the basal slip, prismatic slip, and pyramidal I slip models.

FIG. 7.

Measured and calculated velocity histories at the sample-window interface for all LS-axis experiments. The black solid curves are the measured velocity histories. The red dot-dashed curves are calculations using the basal slip, prismatic slip, and pyramidal I slip models.

Close modal

To better understand the results shown in Fig. 7 in terms of material deformation, calculated in-material particle velocities and stress histories—corresponding to the measured wave profile (Fig. 6) for a 4.0 mm thick LS-axis Mg sample shocked to 1.9 GPa elastic impact stress—are shown in Fig. 8; the longitudinal stress (σxx) is along the wave propagation direction and the lateral stresses (σyy) and (σzz) are approximately along 112¯0 and 101¯1 directions, respectively. As expected, the longitudinal stress history shows a three-wave structure similar to that observed in the longitudinal particle velocity profiles [Figs. 6 and 8(a)]. An interesting aspect of the calculated stress histories is the nonzero shear stress (σxz) associated with QL wave, due to the coupling between the longitudinal and shear deformations.

FIG. 8.

Calculated in-material (a) particle velocities and (b) stress histories at 4 mm propagation distance for shock compression and release of LS-axis crystals shocked to 1.9 GPa elastic impact stress (Expt. 06). The calculation used a combination of basal slip, prismatic slip, and pyramidal I slip models. Time is relative to moment of impact.

FIG. 8.

Calculated in-material (a) particle velocities and (b) stress histories at 4 mm propagation distance for shock compression and release of LS-axis crystals shocked to 1.9 GPa elastic impact stress (Expt. 06). The calculation used a combination of basal slip, prismatic slip, and pyramidal I slip models. Time is relative to moment of impact.

Close modal

Following the QL elastic wave and the arrival of the QL plastic wave, σxxσzz does not increase significantly because this stress difference is relaxed due to basal slip. Because strain hardening was incorporated into the basal slip model, the stress difference σxxσzz is higher at the peak of QL wave when compared to the peak of the elastic wave. However, the stress difference σxxσyy increases because this stress difference is not relaxed by basal slip. With the arrival of the QS wave, prismatic slip and pyramidal I slip are activated, which significantly reduce the stress difference σxxσyy.

To better understand the relative contribution of the different plastic deformation systems, the calculated cumulative plastic strains along different deformation systems and the resolved shear stress from only the contributing slip systems are shown in Figs. 9(a) and 9(b), respectively. Comparing the plots of cumulative plastic shear strain histories and the stress histories [Fig. 8(b)], it is seen that both prismatic slip and pyramidal I slip are activated upon the arrival of the QS wave, while basal slip is the only operative system during the propagation of the QL wave. This confirms the role of prismatic and pyramidal I slip in coupling longitudinal and shear deformations. These results also indicate that the basal slip is the dominant plastic deformation mechanism that governs shock deformation of LS-axis Mg. Although the model parameters used in these calculations are not able to fully capture the QL peak amplitude for 1.9 mm experiment [Fig. 7(a)], they do capture the decay of this amplitude with propagation distance. This phenomenon can be attributed to the activation of prismatic slip, which is confirmed to show time-dependent behavior, from a-axis experiments.

FIG. 9.

Calculated in-material histories at 4 mm propagation distance for shock compression and release of LS-axis crystals shocked to 1.9 GPa elastic impact stress (Expt. 06): (a) cumulative plastic shear strain from each slip system; (b) resolved shear stresses from only the contributing slip systems. The calculations used the basal slip, prismatic slip, and pyramidal I models.

FIG. 9.

Calculated in-material histories at 4 mm propagation distance for shock compression and release of LS-axis crystals shocked to 1.9 GPa elastic impact stress (Expt. 06): (a) cumulative plastic shear strain from each slip system; (b) resolved shear stresses from only the contributing slip systems. The calculations used the basal slip, prismatic slip, and pyramidal I models.

Close modal

The accumulated total plastic shear strain in Fig. 9(a) also provides insights into the deformation mechanisms during release. Both basal slip and prismatic slip govern the plastic deformation for LS-axis Mg upon release, and basal slip is again dominant during unloading. Because the RSS for pyramidal I slip due to the partial stress release in our experiments does not achieve the magnitude for activation, pyramidal I slip is not operative during unloading.

The present work has shown that Mg single crystals need to be shocked along different crystallographic orientations to activate different plastic deformation modes and to gain a comprehensive understanding of the deformation response. Because neither c-axis nor a-axis loading activated basal slip, experiments along the LS direction were needed. Along this direction, longitudinal and shear deformations are coupled, resulting in the propagation of QL and QS waves and complex wave features due to elastic-plastic deformation. Numerical simulations, incorporating different plastic deformation mechanisms, are needed to meaningfully analyze the wave profiles.

Measured wave profiles along the LS direction are significantly different than the profiles along c- and a-axes, demonstrating that Mg single crystals exhibit strong anisotropy under both shock compression and release. The role of crystal anisotropy was observed in the measured elastic wave amplitudes, in the decay of the elastic wave amplitudes, and in the compression and release wave profiles. In contrast to c- and a-axes results, the elastic wave amplitude along the LS direction was the lowest and the elastic wave amplitude decay near the impact surface with propagation distance was the fastest.

The results in Fig. 6 show that the deformation of shocked and released Mg along the LS-axis is due to the operation of multiple crystallographically distinct plastic deformation mechanisms. The small elastic wave amplitude for the LS direction is due to the activation of basal slip, which has the lowest critical resolved shear stress compared to other slip/twin systems governing Mg deformation. The numerical calculations show that basal slip, the deformation mechanism most commonly observed under quasistatic loading conditions, is the dominant mechanism governing shock wave compression along LS direction—∼46° angles from the c-axis. However, a secondary role from the prismatic slip and pyramidal slip I is required to match the measured wave profile beyond the QL wave. Both prismatic slip and pyramidal I slip are only activated upon the arrival of the quasishear wave, thereby providing a coupling between longitudinal and shear deformations. The unloading response is again dominated by basal slip; prismatic slip provides some contribution to the plastic deformation and pyramidal I slip is not activated during release. This suggests that, unlike the c- or a-axes shock results,12,14 the deformation mechanisms governing LS-axis under shock loading conditions are not solely governed by the mechanisms observed under quasistatic loading conditions. This is a significant finding and is a consequence of the lower crystal symmetry of hcp metals, compared to cubic metals.

Shock deformation of c-axis Mg was shown to be governed by either pyramidal I ( Appendix B) or pyramidal II slip12 systems and a clear distinction between these two systems was not achievable from the measured profiles. However, results from the present work show that pyramidal II slip does not contribute to the plastic deformation when shocked along LS-axis, suggesting that our models are self-consistent with the overall deformation mechanisms governed by pyramidal I slip (and not pyramidal II slip). Hence, the present results suggest that shock compression of c-axis Mg is most likely governed by the pyramidal I slip and not pyramidal II slip.

Overall, the present results show that analysis of shock wave propagation along LS directions helps one to provide new insights into the plastic deformation response of crystalline solids and can help evaluate the relative contributions of different and competing plastic deformation mechanisms.

To gain insights into the role of basal slip and to determine the relative contributions of the different plastic deformation mechanisms for shocked hexagonal close-packed (hcp) metals, magnesium (Mg) single crystals were subjected to shock compression and release along a low-symmetry (LS) direction [normal to (101¯2) twin plane] to 1.9 and 4.8 GPa elastic impact stresses. Numerical simulations, using a time-dependent anisotropic modeling framework, were needed to analyze these wave profiles (along a nonspecific direction) in terms of the deformation mechanisms. The main findings from this work are:

  1. The measured wave profiles along the LS-axis show significant differences (qualitatively and quantitatively) when compared to profiles along c-axis and a-axis in both compression and release demonstrating that shocked Mg single crystals exhibit strong anisotropy.

  2. Numerical simulations showed that the shock wave response of Mg single crystals along the LS-axis involves the simultaneous operation of multiple plastic deformation mechanisms and explain the observed anisotropy.

  3. Shock compression along the LS-axis is dominated by basal slip, while prismatic slip and pyramidal I slip play a somewhat smaller role. Both prismatic slip and pyramidal I slip are only activated upon the arrival of the quasishear wave, thereby providing a coupling between longitudinal and shear deformations.

  4. The unloading response is again governed primarily by basal slip; prismatic slip provides some contribution to plastic deformation and pyramidal I slip was not activated during release.

  5. Unlike c- and a-axis Mg results, the deformation system determined from quasistatic (uniaxial stress) loading is not solely applicable for shock wave loading along the LS-axis.

Overall, the plastic deformation mechanisms governing both shock compression and release of Mg single crystals can be understood in terms of dislocation slip on basal, prismatic, and pyramidal I planes, and deformation twinning along (101¯2) twinning planes—plastic deformation mechanisms that have been observed previously for Mg single crystals under quasistatic loading/unloading. However, shock experiments along different crystal orientations are needed to understand the contributions of different deformation mechanisms and to develop a comprehensive understanding of their applicability. Finally, we note that shock wave propagation along a nonspecific direction, though requiring numerical simulations to analyze/interpret the experimental results, constitutes a valuable approach to evaluate plastic deformation mechanisms under shock loading.

N. Arganbright, T. Eldredge, Y. Toyoda, and K. Zimmerman are thanked for their expert assistance with the plate impact experiments. Dr. J. M. Winey is thanked for many insightful discussions regarding this work. This work was supported by the Department of Energy/NNSA (Cooperative Agreement No. #DE-NA0002007) and by the Army Research Laboratory (Cooperative Agreement No. #W911NF-12-2-0022).

Figure 10 shows the repeat experiments for 4 mm thick LS-axis Mg shocked to 1.9 GPa elastic impact stress and 2 mm thick LS-axis Mg shocked to 4.8 GPa elastic impact stress. The wave profiles show excellent reproducibility, including the structure denoting the arrival of quasishear waves.

FIG. 10.

Shows repeat experiments for (a) 4 mm thick low-symmetry axis Mg shocked to 1.9 GPa elastic impact stress and (b) 2 mm thick low-symmetry axis Mg shocked to 4.8 GPa elastic impact stress. The overall wave profile was well reproduced including the structure denoting the arrival of quasishear waves.

FIG. 10.

Shows repeat experiments for (a) 4 mm thick low-symmetry axis Mg shocked to 1.9 GPa elastic impact stress and (b) 2 mm thick low-symmetry axis Mg shocked to 4.8 GPa elastic impact stress. The overall wave profile was well reproduced including the structure denoting the arrival of quasishear waves.

Close modal

The material model parameters used for incorporating deformation mechanisms—other than basal and pyramidal slip—in numerical simulations for c-axis and a-axis have been published previously.14 A brief summary of the overall approach and relevant model parameters [prismatic slip, pyramidal I and II slips, and (101¯2) twinning] used in the LS-axis simulations are presented below.

The Orowan equation was used to relate the plastic shear deformation rate, αγ˙, to dislocation motion for each dislocation slip system,

αγ˙p=αNmbαv¯,
(B1)

where αNmis the mobile dislocation density for the slip plane, b is the magnitude of the Burgers vector, and αv¯is the average dislocation velocity. A model that incorporates dislocation multiplication was employed by relating the dislocation density to the accumulated plastic shear strain αγp,

αNm=Nm0+Mαγp,
(B2)

where Nm0 is the initial mobile dislocation density and M is the multiplication parameter. Strain hardening behavior was incorporated by a phenomenological model that considers the threshold stress for dislocation motion as a function of the shear strain on the slip plane. The stress-dependent dislocation velocity is written as

αv¯=v0exp[D/(αττlevel)],
(B3)
τlevel=τ0+Hγp,
(B4)

where H is the hardening constant and τ0 is the initial threshold stress for dislocation motion on the slip plane. τlevel increases linearly with plastic shear strain in this model.

The basal and prismatic slip model also incorporated a back-stress mechanism that was activated during unloading, similar to that used for aluminum single crystals.10 For each slip system, the plastic strain rate due to reverse motion of pinned dislocation loops is given by

γ˙back=b2npB(τβ),
(B5)

where np is the line length of pinned dislocation loops per unit volume, β is the backstress, and B is the damping parameter for movement of pinned loops. The backstress evolves in time according to

β˙=8μb2BL2(τβ),
(B6)

where L is the average distance between pinning sites. The quantities Bb2np and BL2b2 were treated as adjustable parameters.

{101¯0}112¯0 prismatic slip:

Nm0=1.0×1012m2;M=1.0×1016m2;D=0.68GPa;τ0=0.01GPa,
Bb2np=6.5×102Pas;BL2b2=8×103Pas.

{101¯1}112¯3 pyramidal I slip:

Nm0=1.55×1011m2;M=8.7×1014m2;D=0.47GPa;τ0=0.05GPa;H=5GPa.

{112¯2}112¯3 pyramidal II slip:

Nm0=3.50×1011m2;M=3.5×1014m2;D=0.30GPa;τ0=0.04GPa;H=4GPa.

The approach by Johnson and Rhode34 for incorporating twinning was used here; the plastic strain rate for a given twinning system α is

αγ˙tw=kαλ˙V,
(B7)

where λ˙V is the twinned volume fraction and k is the twinning shear. The phenomenological model for twin growth was described as

αλV=[1TR0tv(ατ)dt]m,
(B8)
v(ατ)={αττ01;αττ0,=0;αττ0,
(B9)

where TR is a characteristic time, m is an integer. v is the nondimensional growth rate given as a function of the resolved shear stress ατ.

{101¯2}101¯1 twinning:

k=0.129,TR=2×106s,τ0=6MPa,m=3.

Numerical simulations using the above model parameters provided a good overall match to the measured profiles for shock propagation along c-axis12 and a-axis.14 As a representative example, Fig. 11 shows the numerically simulated and measured profiles for c-axis. The combination of the model parameters listed above and the basal slip parameters in Sec. IV completes the material model description for shocked Mg single crystals.

FIG. 11.

Measured and calculated velocity histories at the sample-window interface for shock compression and release of c-axis Mg crystals. The black solid curves are the measured velocity histories. The remaining curves are calculations using the pyramidal slip and {101¯2} twinning models. Blue dotted-dashed curves are calculations using {101¯1} pyramidal I planes and red dotted-dashed curves are calculations using {112¯2} pyramidal II planes.12 

FIG. 11.

Measured and calculated velocity histories at the sample-window interface for shock compression and release of c-axis Mg crystals. The black solid curves are the measured velocity histories. The remaining curves are calculations using the pyramidal slip and {101¯2} twinning models. Blue dotted-dashed curves are calculations using {101¯1} pyramidal I planes and red dotted-dashed curves are calculations using {112¯2} pyramidal II planes.12 

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