To gain insight into inelastic deformation mechanisms for shocked hexagonal close-packed (hcp) metals, particularly the role of crystal anisotropy, magnesium (Mg) single crystals were subjected to shock compression and release along the a-axis to 3.0 and 4.8 GPa elastic impact stresses. Wave profiles measured at several thicknesses, using laser interferometry, show a sharply peaked elastic wave followed by the plastic wave. Additionally, a smooth and featureless release wave is observed following peak compression. When compared with wave profiles measured previously for c-axis Mg [Winey et al., J. Appl. Phys. 117, 105903 (2015)], the elastic wave amplitudes for a-axis Mg are lower for the same propagation distance, and less attenuation of elastic wave amplitude is observed for a given peak stress. The featureless release wave for a-axis Mg is in marked contrast to the structured features observed for c-axis unloading. Numerical simulations, using a time-dependent anisotropic modeling framework, showed that the wave profiles calculated using prismatic slip or (101¯2) twinning, individually, do not match the measured compression profiles for a-axis Mg. However, a combination of slip and twinning provides a good overall match to the measured compression profiles. In contrast to compression, prismatic slip alone provides a reasonable match to the measured release wave profiles; (101¯2) twinning due to its uni-directionality is not activated during release. The experimental results and wave profile simulations for a-axis Mg presented here are quite different from the previously published c-axis results, demonstrating the important role of crystal anisotropy in the time-dependent inelastic deformation of Mg single crystals under shock compression and release.

Plane shock wave experiments are well suited for understanding inelastic deformation in solids subjected to high stresses and high loading rates. Single crystal studies are particularly useful, because the deformation mechanisms can be examined selectively through shock propagation along different crystal orientations. To date, cubic single crystals have been studied most extensively under shock wave loading as seen in Refs. 1–14 and references cited therein, and understanding the shock response of non-cubic single crystals constitutes an important scientific need. Hexagonal close packed (hcp) metals, due to a lower crystal symmetry than cubic metals, display a more complex inelastic deformation response.15–17 Therefore, understanding and modelling shock wave induced inelastic deformation in hcp metals is challenging. A limited number of exploratory shock wave studies18–24 have been reported on hcp metal single crystals. A recent, comprehensive study on beryllium (Be) single crystals25 has been particularly useful in demonstrating that rigorous analysis of wave profiles measured along different orientations can provide significant insight into inelastic deformation mechanisms operative in shock compressed hcp metals. Since, no release wave profiles were measured for shock loaded Be, the unloading response was not examined in Ref. 25.

To broaden the understanding of inelastic deformation in shocked hcp metals, shock compression and release wave profiles were recently measured and analyzed for magnesium (Mg) single crystals oriented along the c-axis.26 Numerical simulations showed that the operation of pyramidal slip and (101¯2) twinning provided a good match to both compression and release wave profiles. In particular, (101¯2) twinning (not operative in compression) was required to match the unusual features observed during c-axis unloading. These deformation mechanisms, motivated by quasi-static loading results,15,27,28 demonstrated consistency between two very different loading conditions.

For shock wave propagation along the c-axis, prismatic and basal slips are not activated. Also, the Be work25 and quasi-static studies on Mg single crystals27 suggest that loading along different orientations can activate these other inelastic mechanisms resulting in a better understanding of the deformation response. Toward this objective, we have examined the response of Mg single crystals, shocked along the a-axis. Specifically, we wanted to address the following key questions: (1) What role does crystal orientation play during shock and release of Mg single crystals? (2) Do the quasi-statically determined inelastic deformation systems (prismatic slip and (101¯2) twinning)27,29 govern deformation for shocked a-axis Mg? (3) Can we determine the relative contributions of competing inelastic mechanisms to the overall deformation response?

To address these questions, shock wave profiles were measured for a-axis Mg using plate impact experiments. Similar to the c-axis Mg work,26 both compression and release profiles were measured to evaluate the relative roles of twinning and slip. The measured wave profiles in combination with numerical simulations, using an anisotropic modelling framework,25,30 were used to gain insight into the inelastic deformation mechanisms for shocked a-axis Mg. The a-axis results were compared with the previous c-axis Mg results26 to gain insight into the role of crystal anisotropy.

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. A comparison of a- and c-axes Mg wave profiles,26 numerical simulations of the measured a-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 high quality (99.999% pure) Mg single crystal boules obtained from Metal Crystals and Oxides, Ltd. (Cambridge, UK). Samples were oriented to within 2° of the a-axis using Laue x-ray diffraction and cut using a slow speed saw. The cut surfaces were ground flat and then hand polished on paper using diamond suspension having particles down to 1 μm size. Ambient densities and longitudinal and shear sound speeds were measured for all the samples. The average values were 1.738 ± 0.002 g/cm3 for density, 5.94 ± 0.03 mm/μs for longitudinal sound speed, and 3.1 ± 0.02 mm/μs for shear sound speed.

A schematic view of the experimental setup is shown in Figure 1(a). The overall approach used in this work was similar to that published previously for c-axis Mg.26 The target assembly consisted of a z-cut quartz buffer (∼3.192 mm thick) bonded to the Mg sample, backed by a fused silica window (∼9.545 mm thick). The sample thicknesses are listed in Table I. Similar to the c-axis Mg work,26 an optical window was used to obtain release wave measurements without spall. The lateral dimensions used in this work permitted complete data acquisition during uniaxial strain (before the arrival of edge waves). Prior to bonding the various target components, the rear surface of the buffer and the front surface of the optical window were vapor plated to enable laser interferometry (VISAR)31 measurements.

FIG. 1.

(a) Overall configuration for shock compression and release experiments on a-axis Mg single crystals. Wave profiles were measured at the Mg/window interface using velocity interferometry (VISAR). (b) Time (t) vs Lagrangian distance (X) showing the propagation of the shock compression and release waves in a-axis sapphire impactor, z-cut quartz buffer, and magnesium single crystal samples.

FIG. 1.

(a) Overall configuration for shock compression and release experiments on a-axis Mg single crystals. Wave profiles were measured at the Mg/window interface using velocity interferometry (VISAR). (b) Time (t) vs Lagrangian distance (X) showing the propagation of the shock compression and release waves in a-axis sapphire impactor, z-cut quartz buffer, and magnesium single crystal samples.

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 (mm/μs)Plastic shock velocitya (mm/μs)At peak of the elastic wave
Interface particle velocity (mm/μs)In-material particle velocity (mm/μs)In-material longitudinal stress (GPa)
01 (15-511) 311 3.0 0.505 … … 0.05 ± 0.004 0.057 ± 0.004 0.61 ± 0.06 
02 (16-504) 311 3.0 0.510 … … 0.03 ± 0.004 0.033 ± 0.004 0.36 ± 0.06 
03 (15-513) 310 3.0 1.888 6.02 ± 0.05 4.79 ± 0.02 0.035 ± 0.003 0.038 ± 0.003 0.41 ± 0.04 
04 (15-509) 312 3.0 2.005 5.98 ± 0.04 4.78 ± 0.03 0.028 ± 0.003 0.031 ± 0.003 0.33 ± 0.03 
05 (16-503) 310 3.0 2.008 5.96 ± 0.06 4.78 ± 0.03 0.029 ± 0.001 0.032 ± 0.001 0.34 ± 0.03 
06 (15-510) 312 3.0 4.008 5.96 ± 0.05 4.80 ± 0.02 0.026 ± 0.002 0.029 ± 0.002 0.31 ± 0.05 
07 (15-506) 483 4.8 0.508 … … 0.052 ± 0.004 0.058 ± 0.004 0.62 ± 0.06 
08 (14-520) 483 4.8 2.005 6.00 ± 0.03 5.01 ± 0.01 0.035 ± 0.002 0.039 ± 0.002 0.41 ± 0.04 
09 (14-521) 485 4.8 2.008 6.02 ± 0.04 5.03 ± 0.02 0.032 ± 0.001 0.036 ± 0.001 0.38 ± 0.03 
10 (15-501) 483 4.8 4.004 5.96 ± 0.02 5.02 ± 0.01 0.038 ± 0.002 0.042 ± 0.002 0.45 ± 0.06 
Expt. No.Impact velocity (m/s)Elastic impact stress (GPa)Mg sample thickness (mm)Elastic shock velocity (mm/μs)Plastic shock velocitya (mm/μs)At peak of the elastic wave
Interface particle velocity (mm/μs)In-material particle velocity (mm/μs)In-material longitudinal stress (GPa)
01 (15-511) 311 3.0 0.505 … … 0.05 ± 0.004 0.057 ± 0.004 0.61 ± 0.06 
02 (16-504) 311 3.0 0.510 … … 0.03 ± 0.004 0.033 ± 0.004 0.36 ± 0.06 
03 (15-513) 310 3.0 1.888 6.02 ± 0.05 4.79 ± 0.02 0.035 ± 0.003 0.038 ± 0.003 0.41 ± 0.04 
04 (15-509) 312 3.0 2.005 5.98 ± 0.04 4.78 ± 0.03 0.028 ± 0.003 0.031 ± 0.003 0.33 ± 0.03 
05 (16-503) 310 3.0 2.008 5.96 ± 0.06 4.78 ± 0.03 0.029 ± 0.001 0.032 ± 0.001 0.34 ± 0.03 
06 (15-510) 312 3.0 4.008 5.96 ± 0.05 4.80 ± 0.02 0.026 ± 0.002 0.029 ± 0.002 0.31 ± 0.05 
07 (15-506) 483 4.8 0.508 … … 0.052 ± 0.004 0.058 ± 0.004 0.62 ± 0.06 
08 (14-520) 483 4.8 2.005 6.00 ± 0.03 5.01 ± 0.01 0.035 ± 0.002 0.039 ± 0.002 0.41 ± 0.04 
09 (14-521) 485 4.8 2.008 6.02 ± 0.04 5.03 ± 0.02 0.032 ± 0.001 0.036 ± 0.001 0.38 ± 0.03 
10 (15-501) 483 4.8 4.004 5.96 ± 0.02 5.02 ± 0.01 0.038 ± 0.002 0.042 ± 0.002 0.45 ± 0.06 
a

Plastic shock speed is Eulerian shock speed.

Using a gas gun, a-axis sapphire impactors (∼3.192 mm thick) were launched onto the target assembly. The measured projectile velocities for all of the experiments are shown in Table I. Upon impact, elastic shock waves were propagated into both the quartz buffer and the sapphire impactor as shown in the time-distance plot in Figure 1(b). The shock wave propagating 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 as a release wave and propagates back into the target assembly. Due to the impedance mismatch between the impactor and the target materials, the sample experiences only a partial release from its peak shocked state.

As shown in Figure 1(a), particle velocity histories were measured at the Mg sample/window interface (green arrows) and at the back of the buffer (red arrows). The particle velocity histories at the sample/window interface were determined unambiguously, using a dual velocity per fringe (VPF) arrangement.32 

To ensure high precision shock wave velocity measurements in the Mg samples, shock wave arrival times at the quartz buffer free surface were measured at three locations (Figure 1(a)-red arrows). These three measurements were used to measure the impact tilt (<0.8 mrad) and determine the shock wave arrival times at the buffer/sample interface location that was laterally coincident with the center probe (green arrow). By knowing the sample thicknesses and wave arrival times at both interfaces, shock wave velocities were determined precisely. Due to the lower precision associated with thin samples (0.5 mm), shock velocity measurements were not obtained for the thin samples.

A total of ten experiments, including experiments to determine experimental reproducibility, were performed in this study. These experiments examined the Mg response at two peak elastic impact stresses (∼3.0 GPa and ∼4.8 GPa) and a range of sample thicknesses (0.5–4.0 mm). The experimental parameters and results are summarized in Table I.

Figures 2(a) and 2(b) show representative wave profiles at the sample/window interface for a-axis Mg samples shocked to elastic impact stresses of 3.0 GPa and 4.8 GPa, respectively. Sample thicknesses are shown in the figures. Elastic impact stress is defined as the longitudinal stress that would be attained if the sample were to remain elastic during compression. Because the a-axis has twofold rotational symmetry, the wave profiles in Figure 2 resulted from the propagation of pure longitudinal waves.33 In all the profiles, the reference time (t = 0) corresponds to the impactor/buffer impact. The measured profiles for a-axis Mg showed a two wave structure: sharply peaked elastic wave followed by the plastic wave. The elastic wave amplitude was observed to decrease with propagation distance; however, no significant attenuation was seen beyond 2 mm propagation distance. The sharply peaked elastic wave, relaxation behind the elastic wave, and the elastic wave attenuation with propagation distance are a consequence of time-dependent, elastic-inelastic deformation in shocked a-axis Mg.

FIG. 2.

Measured velocity histories (black solid curves) at the sample-window interface for a-axis Mg crystals shocked to 3.0 GPa (a) and 4.8 GPa (b) elastic impact stress and then released. t = 0 μs corresponds to the moment of impact.

FIG. 2.

Measured velocity histories (black solid curves) at the sample-window interface for a-axis Mg crystals shocked to 3.0 GPa (a) and 4.8 GPa (b) elastic impact stress and then released. t = 0 μs corresponds to the moment of impact.

Close modal

The following observations are noteworthy regarding a-axis Mg shocked to elastic impact stresses of 3.0 GPa and 4.8 GPa. Within experimental scatter, the elastic wave amplitudes for both stresses were observed to decay to the same longitudinal stress. From Figure 2 and Table I, the plastic shock speed for the 4.8 GPa experiments, as expected, was faster than for the 3.0 GPa experiments. Additional experiments were conducted to evaluate the reproducibility of the measured wave profiles. Good reproducibility was observed in the 4.8 GPa experiments; at 3.0 GPa, some variation was observed from experiment to experiment as discussed in the  Appendix.

We note that the measured release wave profiles, following peak compression, were smooth and featureless for a-axis Mg, typical of the unloading response for metals.34,35

To evaluate the role of crystal anisotropy, wave profiles measured at 2 mm propagation distance for a- and c-axes Mg26 are compared in Figure 3. For a given propagation distance, the measured elastic wave amplitude is lower for a-axis Mg than for c-axis Mg. Interestingly, the featureless and smooth release wave observed during a-axis unloading is in marked contrast to the structured feature observed for c-axis unloading. This difference is discussed later in this section. As shown in Figure 4, the observed elastic wave attenuation with propagation distance for a-axis is less compared to c-axis Mg for the propagation distances examined. The elastic limit, the near constant elastic wave amplitude at larger propagation distance, for a-axis loading does not depend on the elastic impact stress. This response is different from that observed for crystals shocked along the c-axis. The a- and c-axes results presented in Figures 3 and 4 demonstrate that the shock loading/unloading response of Mg single crystals displays significant anisotropy.

FIG. 3.

Comparison of measured velocity histories at the sample-window interface for shock compression and release of c- and a-axes Mg crystals (4.8 GPa elastic impact stress).

FIG. 3.

Comparison of measured velocity histories at the sample-window interface for shock compression and release of c- and a-axes Mg crystals (4.8 GPa elastic impact stress).

Close modal
FIG. 4.

Elastic wave amplitude vs thickness for all c- and a-axes Mg experiments. Solid horizontal lines represent the elastic impact stresses and dashed lines are curve fits for visual clarity.

FIG. 4.

Elastic wave amplitude vs thickness for all c- and a-axes Mg experiments. Solid horizontal lines represent the elastic impact stresses and dashed lines are curve fits for visual clarity.

Close modal

In recent studies on hcp metals,25,26 it was shown that by incorporating dislocation slip and deformation twinning into a continuum modeling framework, numerical simulations of shock and release wave profiles provided considerable insight into the inelastic deformation response under shock loading. The modelling framework used in the present simulations is similar to that used previously for c-axis Mg26 and for beryllium,25 and detailed descriptions of this framework have been published previously.25,30 Similar to c-axis Mg,26 the elastic response of a-axis Mg single crystals was described using measured second-order and third-order elastic constants36 and calculated fourth-order elastic constants.37 To analyze the measured wave profiles for a-axis Mg, the prismatic slip and (101¯2) twinning systems observed in quasi-static experiments provide a good starting point. The inelastic strain was calculated by determining the rate of inelastic shear deformation γ̇α for each dislocation slip or twinning system α from the resolved shear stress (RSS) τα using micromechanical models.25,30 For completeness, relevant equations for the two inelastic deformation models are summarized below:

1. Dislocation-based plasticity

The Orowan equation38 was used to relate the rate of inelastic shear deformation γ̇α to dislocation motion for each dislocation slip system

γ̇αp=Nαmbv¯α,
(1)

where Nαm is the mobile dislocation density for the slip plane, b is the magnitude of the Burgers vector, and v¯α is the average dislocation velocity. To perform the numerical simulations, both Nαm and v¯α have to be specified. A model that incorporates dislocation multiplication was employed by relating the dislocation density to the accumulated inelastic shear strain γαp (Ref. 38)

Nαm=Nm0+Mαγp,
(2)

where Nm0 is the initial mobile dislocation density and M is the multiplication parameter. The model also incorporates a stress-dependent dislocation velocity38 

v¯α=v0exp[D/(τατ0)],
(3)

where D is the drag stress parameter, τα is the resolved shear stress, τ0 is a threshold stress for dislocation motion, and v0 is the shear wave speed.

The prismatic slip model also incorporated a back-stress mechanism that was activated during unloading, similar to that used for aluminum single crystals.34 For each slip system, the plastic strain rate due to reverse motion of pinned dislocation loops is given by

γ̇back=b2npB(τβ),
(4)

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

β̇=8μb2BL2(τβ),
(5)

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

In the calculations presented here, the following model parameters were used to describe prismatic slip

Nm0=1×108cm2;M=1×1012cm2;D=0.2GPa;τ0=0.01GPaBb2np=6.5×102Pas;BL2b2=8×103Pas.

2. Deformation Twinning

The approach by Johnson and Rhode39 for incorporating twinning was used here; the inelastic strain rate for a given twinning system α is

γ̇αtw=kαλ̇V,
(6)

where λ̇V is the twinned volume fraction and k is the twinning shear. The phenomenological model for twin growth was described in Ref. 39 as

λαV=[1TR0tv(τα)dt]m,
(7)

where TR is a characteristic time and m is an integer. The non-dimensional growth rate v is a function of the resolved shear stress τα

v(τα)={τατ01;τατ0=0;τατ0.
(8)

The (101¯2) twinning model used here is identical to that used in the c-axis Mg26 calculations; the following model parameters were used:

k=0.129,TR=2μs,τ0=6MPa,m=3.

3. Wave profiles

The deformation models summarized above were incorporated into a 1-dimensional wave propagation code (COPS)40 to simulate a-axis profiles and were then compared with the measured wave profiles, similar to our c-axis Mg work.26 

Figure 5 shows the calculated wave profiles at 2 mm propagation distance incorporating prismatic slip only (green dashed line), (101¯2) twinning only (blue dashed line), and a combination of prismatic slip and (101¯2) twinning (red dashed line). As seen in Figure 5, a good match to the measured wave profiles, particularly the elastic wave amplitude and the inelastic wave speed, was not obtained when prismatic slip and twinning mechanisms were considered separately. However, the combination of these two systems provided a good overall match to the measured wave profiles. Due to its uni-directionality,15,41(101¯2) twinning was not activated during release. Correspondingly, a purely elastic unloading was observed for calculations when only (101¯2) twinning was operating. However, incorporation of prismatic slip provided a reasonable match to the measured unloading wave profiles showing that prismatic slip is the likely governing mechanism of plastic deformation during unloading. The back-stress mechanism34 was incorporated into the prismatic slip model to match the initial onset of unloading.

FIG. 5.

Measured and calculated velocity histories at the sample-window interface for shock compression and release of 2 mm thick a-axis Mg crystal to 4.8 GPa elastic impact stress.(Expt. 09) The black solid curve is the measured velocity history. The remaining curves are calculations using the prismatic slip model (green dotted-dashed curves), the twinning model (blue dotted-dashed lines), and prismatic slip and twinning models (red dotted-dashed curves).

FIG. 5.

Measured and calculated velocity histories at the sample-window interface for shock compression and release of 2 mm thick a-axis Mg crystal to 4.8 GPa elastic impact stress.(Expt. 09) The black solid curve is the measured velocity history. The remaining curves are calculations using the prismatic slip model (green dotted-dashed curves), the twinning model (blue dotted-dashed lines), and prismatic slip and twinning models (red dotted-dashed curves).

Close modal

Figure 6 shows a comprehensive comparison between experimentally measured wave profiles at several locations and the corresponding numerical simulations for samples shocked to 3.0 GPa and 4.8 GPa elastic impact stress. Although the release profiles are somewhat offset at larger thickness, the overall match between the simulations and the measured profiles is good. The combination of prismatic slip and (101¯2) twinning provided a good match to the compressive part of the measured wave profiles, while prismatic slip alone was sufficient to match the release. Given that the twinning shear for the (101¯2) system is 0.129,15 the twinned volume fraction at the peak state was estimated from these calculations to be approximately 32% at 3.0 GPa elastic impact stress and 43% for 4.8 GPa elastic impact stress.

FIG. 6.

Representative measured and calculated velocity histories at the sample-window interface for shock compression and release of a-axis Mg crystals. The black solid curves are the measured velocity histories. The red dotted-dashed curves are calculations using the prismatic slip and twinning models. t = 0 μs corresponds to the moment of impact. The wave profiles for the 4.8 GPa elastic impact stress are all shifted 0.13 μs to the left for visual clarity.

FIG. 6.

Representative measured and calculated velocity histories at the sample-window interface for shock compression and release of a-axis Mg crystals. The black solid curves are the measured velocity histories. The red dotted-dashed curves are calculations using the prismatic slip and twinning models. t = 0 μs corresponds to the moment of impact. The wave profiles for the 4.8 GPa elastic impact stress are all shifted 0.13 μs to the left for visual clarity.

Close modal

Examination of the resolved shear stresses (RSS) and the accumulated inelastic shear strains for relevant dislocation slip and twinning systems is useful for gaining insight into the material response under shock loading. The RSS on a given plane is related to the difference between the longitudinal and lateral stress components. Unlike longitudinal stress, lateral stresses cannot be determined from longitudinal wave profiles. Therefore, additional calculations, where the window material in the experimental setup was replaced with Mg crystals to eliminate wave reflections, were performed to obtain these in-material quantities. Calculated in-material stress histories obtained for a 2 mm thick a-axis Mg sample shocked to 4.8 GPa elastic impact stress are shown in Figure 7, where the longitudinal stress (σx) is along 112¯0 and the lateral stresses are along 101¯0 and 0001 directions. Since Mg crystals possess only a two-fold rotational symmetry along the a-axis, the two lateral stresses are not equal (σyσz) in contrast to c-axis loading where the lateral stresses are equal.26 The relationship between the resolved shear stresses on individual slip and twinning planes and the stress differences are given by

τprism={34(σxσy);4slipplanes0;2slipplanes,
(9)
τtwin={12(σxσz)18(σxσy);4twinplanes12(σxσz)12(σxσy);2twinplanes.
(10)

The above equations show that slip on prismatic planes (τprism) depends only on the stress difference σxσy while twinning (τtwin) depends on stress differences σxσy and σxσz. Because the stress difference σxσy appears in both equations, the operation of the twin systems is influenced by the activation of the prismatic slip systems and vice versa.

FIG. 7.

Calculated in-material stress histories at 2 mm propagation distance for shock compression and release of a-axis Mg crystals shocked to 4.8 GPa elastic impact stress (Expt. 09). The calculation uses a combination of both prismatic slip and twinning models. The solid curve denotes the longitudinal stress along the shock propagation direction while the dotted-dashed curves are the lateral stresses (σz representing lateral stress along c-axis).

FIG. 7.

Calculated in-material stress histories at 2 mm propagation distance for shock compression and release of a-axis Mg crystals shocked to 4.8 GPa elastic impact stress (Expt. 09). The calculation uses a combination of both prismatic slip and twinning models. The solid curve denotes the longitudinal stress along the shock propagation direction while the dotted-dashed curves are the lateral stresses (σz representing lateral stress along c-axis).

Close modal

Resolved shear stress (RSS) and the accumulated inelastic shear strain histories for relevant dislocation slip and twinning systems of shock compressed and released a-axis Mg are shown in Figure 8. From the RSS calculations, it is observed that at the peak of the elastic wave amplitude, both τprism and τtwin are large because of large stress differences. Immediately behind the elastic wave, both τprism and τtwin are seen to relax through twinning, since this is the only operative mechanism at this point. Subsequently, at the onset of the inelastic wave, τprism increases and the critical stress for prismatic slip is reached. Prismatic slip is then activated and τprism begins to relax, which reduces σxσy (Eq. (9)). However, σxσz is not reduced by prismatic slip, and hence, τtwin increases. (101¯2) twinning is then activated, which relaxes τtwin, demonstrating that prismatic slip and (101¯2) twinning are influenced by each other. Prismatic slip and (101¯2) twinning during shock compression lead to a rapid reduction of all the stress differences as the peak state is reached.

FIG. 8.

Calculated in-material histories at 2 mm propagation distance for shock compression and release of a-axis Mg crystals shocked to 4.8 GPa elastic impact stress (Expt. 09): (a) resolved shear stress (compare with Eqs. (1) and (2)); (b) inelastic shear strain. The calculations used the prismatic slip and (101¯2) twinning models.

FIG. 8.

Calculated in-material histories at 2 mm propagation distance for shock compression and release of a-axis Mg crystals shocked to 4.8 GPa elastic impact stress (Expt. 09): (a) resolved shear stress (compare with Eqs. (1) and (2)); (b) inelastic shear strain. The calculations used the prismatic slip and (101¯2) twinning models.

Close modal

As indicated previously, the unloading response is governed entirely by prismatic slip, and (101¯2) twinning does not operate during unloading. This lack of twinning deformation does not reduce σxσz (see Eq. (10)), and this stress difference increases during unloading as shown in Figure 7.

Overall, the deformation response of shock compressed a-axis Mg is governed by the operation of both prismatic slip and (101¯2) twinning. As shown in Figure 8(b), the compressive wave profile can be divided into three regions: (1) twin dominated region (peak of the elastic wave to the onset of the plastic wave), (2) slip dominated region (onset of the plastic wave to near the midpoint of the plastic wave), and (3) slip and twin region (remainder of the compressive wave profile). In contrast, prismatic slip alone dominates the unloading response following the shock compression.

The results presented in Figure 3 show that the measured shocked and released wave profiles for c- and a-axes Mg are markedly different. The differences observed in the compressive part of the Mg wave profiles, specifically the amplitude of the elastic wave and the plastic wave speed, are similar to the differences observed in Be single crystals shocked along c- and a-directions.25 The results from our numerical simulations showed that by incorporating both prismatic slip and (101¯2) twinning, along planes determined from quasi-static studies, a good match was achieved to the compressive part of the measured wave profiles. However, prismatic slip alone likely governs the unloading response. Because (101¯2) twinning is not activated during unloading, additional insight into the prismatic slip (such as inclusion of back stress model) was gained by analyzing the unloading response along the a-axis.

Recently, the work on shocked and released c-axis Mg26 exploited the same uni-directionality of (101¯2) twinning to examine the relative roles of dislocation slip and deformation twinning. That work showed that the deformation during shock compression of c-axis Mg was governed by pyramidal slip alone, while the unloading was governed only by (101¯2) twinning (tension twin for Mg under quasi-static loading15). Thus, the deformation mechanisms during shock compression and release in Mg single crystals differ substantially for loading along the a- and c-axes. Accordingly, the observed orientation dependence can be attributed to the activation of different inelastic deformation systems during shock compression and release along different orientations. For example, the differences observed in the measured elastic wave amplitudes between the different orientations is likely because a-axis loading is governed by relatively easier deformation mechanisms, prismatic slip and (101¯2) twinning, compared to c-axis Mg, which is governed by the pyramidal slip.42 Furthermore, the difference in the release wave response between the a- and c-axes can be attributed to the absence of or the activation of the (101¯2) twinning, respectively, during unloading from the peak state. We point out that shock compression along the c-axis and a-axis does not result in basal slip. Experiments along a low-symmetry direction are required to examine basal slip.

The combination of c-axis results26 and the present a-axis results, along with the corresponding numerical simulations, has demonstrated that the shock induced deformation response of a hcp single crystal like Mg is strongly anisotropic. To gain a comprehensive insight into the different mechanisms governing inelastic deformation in shocked Mg, compression and release profiles along different crystal orientations are required. Based on all c- and a-axes results to date, the slip and twin systems determined from quasi-static studies appear to be also valid for shock wave loading.

To examine the effect of crystal anisotropy on the inelastic response of Mg single crystals under shock compression/release, wave profiles along the a-axis were measured/analyzed and compared with previously obtained results for c-axis Mg. Numerical simulations incorporating both dislocation slip and deformation twinning in a time-dependent anisotropic modelling framework were used to gain insight into the deformation mechanisms. The main findings from this work are summarized below:

  1. Measured wave profiles showed significant differences between a-axis and c-axis loading: for a given propagation distance, the elastic wave amplitude for the a-axis was lower than that for the c-axis; for the same peak stress, less attenuation with propagation distance was observed along the a-axis; and finally, the featureless release wave for the a-axis is in marked contrast to the structured features observed for c-axis unloading.

  2. Numerical simulations showed that the combination of prismatic slip and the (101¯2) twinning was needed to provide a good overall match to the compressive part of the measured wave profiles. These mechanisms, when considered independently, did not provide a good match to the data.

  3. Due to the uni-directionality of (101¯2) twinning, it was not activated during a-axis release and prismatic slip alone was sufficient to provide a reasonable match to the measured unloading wave profiles.

  4. Based on both a-axis and c-axis results, the deformation systems determined from quasi-static (uniaxial stress) loading appear to be applicable under shock wave or uniaxial strain loading.

The present experimental results and analysis have demonstrated that Mg single crystals exhibit strong anisotropy under both shock compression and release. The observed orientation dependence arises from the activation of different slip and/or twin systems during shock compression and unloading. Thus, a comprehensive understanding of the inelastic response of hcp crystals under shock loading requires wave profile measurements and their analysis along different crystal orientations. Finally, we point out that neither c-axis nor a-axis loading results in the activation of basal slip.

N. Arganbright, B. Williams, and K. Zimmerman are thanked for their expert assistance with the plate impact experiments. 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).

There was some variation in experiments performed with 0.5 and 2.0 mm thick a-axis Mg samples at 3.0 GPa elastic input stress. Figure 9 shows the measured wave profiles for these experiments. For 0.5 mm thick Mg samples, the elastic wave amplitude varied measurably and the onset of the plastic wave displayed structure (Figure 9(a)). For 2 mm thick Mg samples, the structure in the plastic wave was observed in one experiment but not in the other two (Figure 9(b)). We are unable to explain this experimental variation and have used only the featureless wave profiles (Expt.01 and Expt.03/Expt.05) in our analysis.

FIG. 9.

Wave profiles measured for (a) 0.5 mm (b) 2.0 mm thick a-axis Mg shocked to 3.0 GPa elastic impact stress. t = 0 μs corresponds to the moment of impact. Expt. 01 is shifted 0.05 μs to the left with respect to Expt. 02 for visual clarity. Expt. 03 and Expt. 05 are shifted 0.1 μs to the left and right, respectively, with respect to Expt. 04 for visual clarity.

FIG. 9.

Wave profiles measured for (a) 0.5 mm (b) 2.0 mm thick a-axis Mg shocked to 3.0 GPa elastic impact stress. t = 0 μs corresponds to the moment of impact. Expt. 01 is shifted 0.05 μs to the left with respect to Expt. 02 for visual clarity. Expt. 03 and Expt. 05 are shifted 0.1 μs to the left and right, respectively, with respect to Expt. 04 for visual clarity.

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