To understand crystal anisotropy effects on shock-induced elastic-plastic deformation of molybdenum (Mo), results from high-purity single crystals shocked along [110] and [111] orientations to an elastic impact stress of 12.5 GPa were obtained and compared with the [100] results previously reported [A. Mandal and Y. M Gupta, J. Appl. Phys. 121, 045903 (2017)]. Measured wave profiles showed a time-dependent response, and strong anisotropy was observed in the elastic wave attenuation with the propagation distance, elastic limits, shock speeds, and overall structure of the wave profiles. Resolved shear stresses on {110}〈111〉 and {112}〈111〉 slip systems provided insight into the observed anisotropy in elastic wave attenuation and elastic limits and showed that shear stresses, and not longitudinal stresses, are a better measure of strength in shocked single crystals. Under shock compression, resolved shear stresses at elastic limits were comparable to the Peierls stress of screw dislocations in Mo. Elastic wave attenuation was rapid when shear stresses were larger than the Peierls stress. Large differences in the elastic limits under shock and quasi-static loading are likely a consequence of the large Peierls stress value for Mo. Numerically simulated wave profiles, obtained using the dislocation-based plasticity model described in the [100] work, showed good agreement with all measured wave profiles but could not differentiate between the {110}〈111〉 and {112}〈111〉 slip systems. Overall, experimental results and corresponding numerical simulations for the three crystal orientations have provided a comprehensive insight into shock-induced elastic-plastic deformation of Mo single crystals, including the development of a continuum material model.

Shock wave loading of single crystals provides valuable insights into the fundamental deformation mechanisms operative under high stress dynamic loading, because slip and/or twin systems determined from quasi-static studies serve as a good starting point and the absence of dislocation/grain-boundary interaction simplifies data analysis.1 A comprehensive understanding of shock-induced deformation mechanisms requires that shock propagation along different crystal orientations be examined in detail.2–15 

In contrast to the references cited above, elastic-plastic wave profiles along different crystal orientations have been minimally studied for body-centered cubic (BCC) metals. As such, the deformation response of shock compressed BCC single crystals is not well understood. As pointed out in detail elsewhere,16 earlier studies on tungsten (W),17 molybdenum (Mo),18 and tantalum (Ta)19 single crystals represent a start but do not constitute a comprehensive study of shock-induced deformation mechanisms. To achieve a detailed understanding of shock-induced deformation in a BCC single crystal and to develop a continuum material model, we examined the effects of crystal anisotropy on the elastic-plastic deformation of shock compressed Mo single crystals.

Our previous effort16 reported elastic-plastic wave profile measurements in Mo single crystals shocked along the [100] orientation to 12.5 GPa elastic impact stress, defined as the in-material longitudinal stress in the shocked sample if it remains elastic. In this study, we examined the deformation response of Mo single crystals shock compressed to the same elastic impact stress along [110] and [111] orientations to address the following questions:

  1. What is the role of crystal anisotropy on the shock-induced elastic-plastic deformation response?

  2. Can the observed anisotropy effects be explained in terms of {110}〈111〉 and/or {112}〈111〉 slip systems that operate under room temperature quasi-static loading?20–26 Can we differentiate between the two systems?

  3. How do the elastic limits under shock loading compare with those reported under quasi-static loading?

  4. Is the Peierls stress for screw dislocations important for understanding the deformation response of shock compressed Mo single crystals?

  5. Is the dislocation generation mechanism, which was required to simulate the measured [100] wave profiles,16 also operative along [110] and [111] orientations? If so, can a physical justification for this mechanism be obtained?

To address these questions, plate-impact experiments were performed to measure shock wave profiles in [110] and [111]-oriented Mo samples of thicknesses ranging between 0.2 and 3.1 mm. Crystal anisotropy effects were examined by comparing [110] and [111] results with the [100] results from Ref. 16. Both analytic and numerical methods were used to understand the anisotropy effects and to identify the operative dislocation mechanisms. The analytic method involved calculation of resolved shear stress (RSS) values on the operative {110}〈111〉 and {112}〈111〉 slip systems. Numerical simulations of the measured wave profiles were carried out using the dislocation-based continuum model described in the [100] work.16 

In the following, experimental details are briefly summarized in Sec. II, and the measured wave profiles along [110] and [111] orientations are described in Sec. III. Section IV presents a comparison of the measured wave profiles for all three orientations, along with a detailed analysis and discussion of our results. The key findings of this work are specified in Sec. V.

High-purity Mo single crystal samples (Accumet Materials Co., NY), oriented along the desired [110] and [111] loading axes, were characterized by measuring the ambient density, longitudinal and shear sound speeds, and dislocation density. These are listed in Table I along with the corresponding [100] values from Ref. 16 for comparison.

TABLE I.

Ambient physical properties and dislocation density of Mo single crystal samples.

OrientationDensity (g/cm3)Longitudinal sound speed (mm/μs)Shear sound speed (mm/μs)Average dislocation density (cm−2)
[100]a 10.23 ± 0.01 6.76 ± 0.01 3.269 ± 0.006 7 × 108 
[110]b 6.47 ± 0.02 3.28 ± 0.01, 3.858 ± 0.009 
[111] 6.36 ± 0.01 3.66 ± 0.01 
OrientationDensity (g/cm3)Longitudinal sound speed (mm/μs)Shear sound speed (mm/μs)Average dislocation density (cm−2)
[100]a 10.23 ± 0.01 6.76 ± 0.01 3.269 ± 0.006 7 × 108 
[110]b 6.47 ± 0.02 3.28 ± 0.01, 3.858 ± 0.009 
[111] 6.36 ± 0.01 3.66 ± 0.01 
a

Values obtained from Ref. 16.

b

Shear sound speeds are non-degenerate for [110] propagation directions.

The overall experimental configuration for the present study, shown schematically in Fig. 1, was similar to that used in the [100] study.16 In all experiments, the target assembly, which consisted of a Mo sample bonded to an a-axis sapphire front buffer and a c-axis sapphire optical window, was impacted by a 7075 aluminum (Al) impactor to shock compress the Mo sample. The relevant experimental parameters are listed in Table II.

FIG. 1.

A schematic view of the overall experimental configuration.

FIG. 1.

A schematic view of the overall experimental configuration.

Close modal
TABLE II.

Relevant experimental parameters and measurements. The impactor material in all experiments was 7075-T651 aluminum.

Exp. no.OrientationSample thickness (mm)Buffer thickness (mm)Window thickness (mm)Impact velocity (mm/μs)Elastic impact stress (GPa)Particle velocity at the HELa (mm/μs)Elastic shock velocity (mm/μs)Lagrangian plastic shock velocityb (mm/μs)
1 (14-005) [110] 0.259 2.513 4.999 0.832 12.58 0.063 ± 0.003 6.9 ± 0.2 5.49 ± 0.11 
2 (14-008) [110] 0.276 2.512 5.009 0.828 12.51 0.066 ± 0.003 6.68 ± 0.15 5.34 ± 0.09 
3 (13-048) [110] 0.458 2.512 4.999 0.831 12.56 0.065 ± 0.002 6.19 ± 0.08 5.04 ± 0.06 
4 (12-045) [110] 1.295 2.515 4.999 0.838 12.68 0.063 ± 0.001 6.58 ± 0.03 5.31 ± 0.02 
5 (12-042) [110] 1.299 2.515 4.999 0.834 12.61 0.062 ± 0.001 6.52 ± 0.03 5.24 ± 0.02 
6 (15-006) [110] 2.289 2.512 5.009 0.831 12.56 0.065 ± 0.001 6.53 ± 0.02 5.234 ± 0.012 
7 (15-002) [110] 3.070 2.512 5.009 0.832 12.58 0.063 ± 0.001 6.502 ± 0.012 5.269 ± 0.009 
8 (14-015) [111] 0.211 2.512 5.009 0.829 12.50 0.113 ± 0.002 7.1 ± 0.2 5.01 ± 0.12 
9 (14-017) [111] 0.259 2.512 5.009 0.830 12.51 0.132 ± 0.002 7.0 ± 0.2 4.67 ± 0.09 
10 (13-043) [111] 0.480 2.479 4.998 0.831 12.53 0.126 ± 0.002 6.34 ± 0.08 4.57 ± 0.05 
11 (13-039) [111] 0.505 2.515 4.999 0.825 12.43 0.122 ± 0.002 6.46 ± 0.08 4.62 ± 0.04 
12 (12-043) [111] 1.257 2.522 4.997 0.829 12.50 0.104 ± 0.002 6.60 ± 0.03 4.90 ± 0.02 
13 (12-044) [111] 1.290 2.515 4.997 0.832 12.55 0.100 ± 0.002 6.62 ± 0.03 4.88 ± 0.02 
14 (13-037) [111] 2.355 2.512 4.996 0.829 12.50 0.094 ± 0.002 6.55 ± 0.02 4.94 ± 0.01 
15 (16-015) [111] 3.090 2.512 5.009 0.825 12.43 0.092 ± 0.001 6.603 ± 0.013 5.001 ± 0.008 
Exp. no.OrientationSample thickness (mm)Buffer thickness (mm)Window thickness (mm)Impact velocity (mm/μs)Elastic impact stress (GPa)Particle velocity at the HELa (mm/μs)Elastic shock velocity (mm/μs)Lagrangian plastic shock velocityb (mm/μs)
1 (14-005) [110] 0.259 2.513 4.999 0.832 12.58 0.063 ± 0.003 6.9 ± 0.2 5.49 ± 0.11 
2 (14-008) [110] 0.276 2.512 5.009 0.828 12.51 0.066 ± 0.003 6.68 ± 0.15 5.34 ± 0.09 
3 (13-048) [110] 0.458 2.512 4.999 0.831 12.56 0.065 ± 0.002 6.19 ± 0.08 5.04 ± 0.06 
4 (12-045) [110] 1.295 2.515 4.999 0.838 12.68 0.063 ± 0.001 6.58 ± 0.03 5.31 ± 0.02 
5 (12-042) [110] 1.299 2.515 4.999 0.834 12.61 0.062 ± 0.001 6.52 ± 0.03 5.24 ± 0.02 
6 (15-006) [110] 2.289 2.512 5.009 0.831 12.56 0.065 ± 0.001 6.53 ± 0.02 5.234 ± 0.012 
7 (15-002) [110] 3.070 2.512 5.009 0.832 12.58 0.063 ± 0.001 6.502 ± 0.012 5.269 ± 0.009 
8 (14-015) [111] 0.211 2.512 5.009 0.829 12.50 0.113 ± 0.002 7.1 ± 0.2 5.01 ± 0.12 
9 (14-017) [111] 0.259 2.512 5.009 0.830 12.51 0.132 ± 0.002 7.0 ± 0.2 4.67 ± 0.09 
10 (13-043) [111] 0.480 2.479 4.998 0.831 12.53 0.126 ± 0.002 6.34 ± 0.08 4.57 ± 0.05 
11 (13-039) [111] 0.505 2.515 4.999 0.825 12.43 0.122 ± 0.002 6.46 ± 0.08 4.62 ± 0.04 
12 (12-043) [111] 1.257 2.522 4.997 0.829 12.50 0.104 ± 0.002 6.60 ± 0.03 4.90 ± 0.02 
13 (12-044) [111] 1.290 2.515 4.997 0.832 12.55 0.100 ± 0.002 6.62 ± 0.03 4.88 ± 0.02 
14 (13-037) [111] 2.355 2.512 4.996 0.829 12.50 0.094 ± 0.002 6.55 ± 0.02 4.94 ± 0.01 
15 (16-015) [111] 3.090 2.512 5.009 0.825 12.43 0.092 ± 0.001 6.603 ± 0.013 5.001 ± 0.008 
a

The measured particle velocities at the sample/window interface that correspond to the elastic wave peak.

b

Calculated at the midpoint of the shock.

Particle velocity histories (wave profiles) were measured unequivocally at the sample/window interface (probe 1) using laser velocimetry (VISAR27) with a dual velocity-per-fringe (VPF) setup.28 Measurement of shock wave arrival times at three different locations on the back of the buffer using probes 2, 3, and 4 permitted calculation of the impact tilt (<1.5 mrad) and the elastic and plastic shock wave velocities through the sample.15,16

The experimental parameters for the seven plate impact experiments conducted on [110]-orientated samples are listed in Table II. The measured wave profiles showed excellent reproducibility. Five representative profiles, corrected for refractive index change in the c-sapphire window,29 are shown in Fig. 2, and the corresponding experiment numbers and sample thicknesses are indicated.

FIG. 2.

[110] wave profiles measured at the sample/window interface. The corresponding experiment numbers and propagation distances are indicated. Red arrows point to the kinks in the plastic wave. Time is relative to the loading of the sample front surface.

FIG. 2.

[110] wave profiles measured at the sample/window interface. The corresponding experiment numbers and propagation distances are indicated. Red arrows point to the kinks in the plastic wave. Time is relative to the loading of the sample front surface.

Close modal

Measured wave profiles consisted of a near-constant amplitude elastic wave followed by a plastic wave. Measured particle velocities at the sample/window interface corresponding to the elastic wave peak (HEL) are listed in Table II, along with the elastic and Lagrangian plastic shock velocities. A yield drop behind the elastic wave was present in the profiles measured at 2.3 mm or larger propagation distances. A close inspection of the measured wave profiles showed a kink (shown with the red arrows) in the plastic wave profiles that is characterized by a difference in the plastic wave slope before and after the kink. Similar kinks in the plastic wave profile were reported for shock19 and ramp30 compression of BCC Ta single crystals along the [110] orientation. It will be shown in Sec. IV that the kink likely resulted from successive yielding on nonequivalent slip systems.

The experimental parameters relevant to the eight plate impact experiments conducted on [111]-oriented Mo samples are listed in Table II. Five representative, window-corrected wave profiles are shown in Fig. 3, and the corresponding experiment numbers and sample thicknesses are indicated.

FIG. 3.

[111] wave profiles measured at the sample/window interface. The corresponding experiment numbers and propagation distances are indicated. Green arrows point to the distinct features observed between the elastic and plastic waves. Time is relative to the loading of the sample front surface.

FIG. 3.

[111] wave profiles measured at the sample/window interface. The corresponding experiment numbers and propagation distances are indicated. Green arrows point to the distinct features observed between the elastic and plastic waves. Time is relative to the loading of the sample front surface.

Close modal

For [111] loading, the elastic wave amplitude showed measurable attenuation for the propagation distances examined. Measured particle velocities corresponding to the elastic wave peak at the sample/window interface, along with the elastic and Lagrangian plastic shock velocities, are listed in Table II. A yield drop—largest at the 0.5 mm distance—was present in all measured profiles. For propagation distances of 1.3 mm or larger, a distinct structure appeared between the elastic and the plastic waves (shown with the green arrows).

Experiments 9, 10, and 13 were repeated (repeat experiments are 8, 11, and 12, respectively) to check for consistency. Wave profiles at 0.5 mm showed excellent reproducibility, and those measured at 1.3 mm were quite consistent except for the structure observed between elastic and plastic waves. In contrast, the elastic amplitude at 0.21 mm obtained from experiment 8 was measurably smaller than that obtained from experiment 9 conducted on a sample of similar thickness (0.26 mm). Reasons behind this difference are not clearly understood, but a smaller elastic wave amplitude in experiment 8 is inconsistent with amplitudes measured at somewhat larger thicknesses (experiments 9, 10, and 11). Hence, the result obtained from experiment 8 is not considered in the remaining discussion.

To evaluate crystal anisotropy effects on the measured shock response of Mo, we have compared the measured [100] (blue solid lines, from Ref. 16), [110] (red dashed lines), and [111] (green dash-dotted lines) wave profiles in Fig. 4. In this figure, the wave profiles measured at comparable propagation distances have been time-shifted and grouped together for ease of comparison, and the corresponding propagation distances are indicated at the top.

FIG. 4.

A comparison of the experimentally measured [100], [110], and [111] wave profiles. The profiles measured at comparable propagation distances were time-shifted and grouped together for clarity. The [100] data are reproduced from Ref. 16.

FIG. 4.

A comparison of the experimentally measured [100], [110], and [111] wave profiles. The profiles measured at comparable propagation distances were time-shifted and grouped together for clarity. The [100] data are reproduced from Ref. 16.

Close modal

For the propagation distances examined in our work, the elastic wave amplitudes were nearly the same along the [100] and [110] orientations but decreased measurably along [111]. For a given propagation distance, the elastic wave amplitude was the largest along the [111] orientation. The plastic shock speed was considerably faster along [110] than the other two orientations. The overall structure of the measured profiles differed significantly as well. A yield drop behind the elastic wave was present in all [111] wave profiles, but not all [100] and [110] profiles exhibited a yield drop. The [110] plastic wave profiles had kinks, while the [111] profiles were distinctly structured between the elastic and plastic waves at propagation distances ≥1.3 mm. Overall, the wave profiles presented in Fig. 4 clearly demonstrate that the shock-induced deformation response of Mo single crystals is both time-dependent and strongly anisotropic.

In-material longitudinal stresses at the elastic wave peak (elastic wave amplitudes) for [110] and [111] loading were calculated from the corresponding particle velocities measured at the sample/window interface (listed in Table II) following Ref. 16. The following nonlinear elastic Hugoniot relations constructed for Mo single crystals from the published second-order31 and third-order32 isentropic elastic moduli values were used in these calculations

[110]shockcompression:Px=65.829up+17.759up2,
(1)
[111]shockcompression:Px=64.724up+21.319up2,
(2)

where longitudinal stress Px and particle velocity up are in GPa and mm/μs, respectively. It should be noted that anisotropic, nonlinear elastic stress states were obtained from the above calculations. The calculated elastic wave amplitudes, along with the calculated in-material particle velocities and densities at the HEL, are reported in Table III. The Eulerian plastic shock velocities, UplE=uHEL+ρ0ρHELUplL, calculated from the Lagrangian plastic shock velocities (UplL, Table II), and particle velocities (uHEL) and densities (ρHEL) at HEL (Table III) following Ref. 33 are also provided in Table III for comparison.

TABLE III.

Summary of in-material amplitudes at the peak of the elastic wave and the Eulerian plastic shock velocity.

Exp. no.OrientationHEL stateEulerian plastic shock velocity (mm/μs)
Particle velocity (mm/μs)Longitudinal stress (GPa)Density (g/cm3)
[110] 0.053 ± 0.003 3.5 ± 0.2 10.310 ± 0.005 5.50 ± 0.11 
[110] 0.055 ± 0.003 3.7 ± 0.2 10.311 ± 0.005 5.35 ± 0.09 
[110] 0.054 ± 0.002 3.61 ± 0.14 10.310 ± 0.003 5.06 ± 0.06 
[110] 0.053 ± 0.001 3.54 ± 0.07 10.309 ± 0.002 5.32 ± 0.02 
[110] 0.052 ± 0.001 3.47 ± 0.07 10.307 ± 0.002 5.25 ± 0.02 
[110] 0.054 ± 0.001 3.61 ± 0.07 10.310 ± 0.002 5.247 ± 0.017 
[110] 0.053 ± 0.001 3.54 ± 0.07 10.309 ± 0.002 5.282 ± 0.014 
[111] 0.111 ± 0.002 7.45 ± 0.14 10.401 ± 0.003 4.70 ± 0.09 
10 [111] 0.106 ± 0.002 7.10 ± 0.14 10.393 ± 0.003 4.60 ± 0.05 
11 [111] 0.102 ± 0.002 6.82 ± 0.14 10.387 ± 0.003 4.65 ± 0.04 
12 [111] 0.087 ± 0.002 5.79 ± 0.14 10.364 ± 0.003 4.92 ± 0.03 
13 [111] 0.084 ± 0.002 5.59 ± 0.14 10.359 ± 0.003 4.90 ± 0.03 
14 [111] 0.079 ± 0.002 5.25 ± 0.14 10.351 ± 0.003 4.961 ± 0.015 
15 [111] 0.077 ± 0.001 5.11 ± 0.07 10.348 ± 0.002 5.021 ± 0.013 
Exp. no.OrientationHEL stateEulerian plastic shock velocity (mm/μs)
Particle velocity (mm/μs)Longitudinal stress (GPa)Density (g/cm3)
[110] 0.053 ± 0.003 3.5 ± 0.2 10.310 ± 0.005 5.50 ± 0.11 
[110] 0.055 ± 0.003 3.7 ± 0.2 10.311 ± 0.005 5.35 ± 0.09 
[110] 0.054 ± 0.002 3.61 ± 0.14 10.310 ± 0.003 5.06 ± 0.06 
[110] 0.053 ± 0.001 3.54 ± 0.07 10.309 ± 0.002 5.32 ± 0.02 
[110] 0.052 ± 0.001 3.47 ± 0.07 10.307 ± 0.002 5.25 ± 0.02 
[110] 0.054 ± 0.001 3.61 ± 0.07 10.310 ± 0.002 5.247 ± 0.017 
[110] 0.053 ± 0.001 3.54 ± 0.07 10.309 ± 0.002 5.282 ± 0.014 
[111] 0.111 ± 0.002 7.45 ± 0.14 10.401 ± 0.003 4.70 ± 0.09 
10 [111] 0.106 ± 0.002 7.10 ± 0.14 10.393 ± 0.003 4.60 ± 0.05 
11 [111] 0.102 ± 0.002 6.82 ± 0.14 10.387 ± 0.003 4.65 ± 0.04 
12 [111] 0.087 ± 0.002 5.79 ± 0.14 10.364 ± 0.003 4.92 ± 0.03 
13 [111] 0.084 ± 0.002 5.59 ± 0.14 10.359 ± 0.003 4.90 ± 0.03 
14 [111] 0.079 ± 0.002 5.25 ± 0.14 10.351 ± 0.003 4.961 ± 0.015 
15 [111] 0.077 ± 0.001 5.11 ± 0.07 10.348 ± 0.002 5.021 ± 0.013 

In Fig. 5, the calculated elastic wave amplitudes are plotted against the propagation distance. In this plot, [100] data from Ref. 16 are shown with blue circles, and [110] and [111] data from Table III are shown with red squares and green triangles, respectively.

FIG. 5.

Elastic wave amplitudes at different propagation distances for the three orientations. The [100] data are taken from Ref. 16. The purple pentagon represents the elastic impact stress (12.5 GPa) for all three crystal orientations.

FIG. 5.

Elastic wave amplitudes at different propagation distances for the three orientations. The [100] data are taken from Ref. 16. The purple pentagon represents the elastic impact stress (12.5 GPa) for all three crystal orientations.

Close modal

For the propagation distances examined, the elastic wave amplitudes along the [100] and [110] orientations stayed nearly the same (∼3.6 GPa), similar to that observed in a pure LiF single crystal shocked along the [100] orientation.34,35 In contrast, the elastic wave amplitudes along the [111] orientation attenuated significantly with propagation distance before reaching a near-constant value of 5.1 GPa at distances larger than 2.5 mm.

Because the elastic impact stress (Pimp)—the expected elastic wave amplitude at the impact surface of the Mo sample—is 12.5 GPa (purple pentagon), data presented in Fig. 5 suggest that elastic wave amplitudes along all three orientations attenuated rapidly with propagation distance close to the impact surface. The attenuation is more rapid along [100] and [110] orientations, where the elastic limit is (Pel) reached within ∼0.2 mm of propagation, compared to the [111] orientation. Overall, the data presented here are similar to that reported by Michaels for W single crystals17 and suggest that the elastic wave attenuation near the impact surface and the elastic limit depend strongly on the crystal orientation. Such anisotropy effects were not observed by Kanel and co-workers18 in their work on Mo single crystals, likely due to the large scatter in their data.

To gain insight into the observed anisotropy in the elastic wave attenuation and the measured elastic limits, resolved shear stresses on the {110}〈111〉 and {112}〈111〉 slip systems at the peak of the elastic wave were calculated from corresponding elastic wave amplitudes (in-material longitudinal stress at the elastic wave peak, listed in Table III) following the method described in Ref. 1. The calculated resolved shear stress values on the {110} and {112} systems corresponding to both the elastic impact stress (τimp) and the elastic limit (τel) are listed in Table IV, and they are shown pictorially in Figs. 6(a) and 6(b), respectively. In these figures, the calculated τimp and τel values are shown with solid and dashed lines, respectively, for shock loading along [100] (blue), [110] (red), and [111] (green) orientations.

FIG. 6.

Resolved shear stresses (τ) on the operative (a) {110}〈111〉 and (b) {112}〈111〉 slip systems for shock compression along [100], [110], and [111] orientations corresponding to the elastic impact stress (solid lines) and the measured elastic limits (dashed lines).

FIG. 6.

Resolved shear stresses (τ) on the operative (a) {110}〈111〉 and (b) {112}〈111〉 slip systems for shock compression along [100], [110], and [111] orientations corresponding to the elastic impact stress (solid lines) and the measured elastic limits (dashed lines).

Close modal
TABLE IV.

Resolved shear stresses (RSS) on the operative slip systems.

Exp. typeStress stateLongitudinal stress (GPa)RSS on {110}〈111〉 (GPa)RSS on {112}〈111〉 (GPa)
[100][110][111][100][110][111][100][110][111]
Shock wave compression (uniaxial strain) Elastic impact stress 12.5 12.5 12.5 3.33 3.15 1.81 3.84 3.64 2.10 
Elastic limit 3.61 3.58 5.11 0.96 0.90 0.74 1.11 1.04 0.86 
Quasi-static compressiona (uniaxial stress) Elastic limit 0.05 0.20 0.18 0.02 0.08 0.05 0.02 0.09 0.06 
Exp. typeStress stateLongitudinal stress (GPa)RSS on {110}〈111〉 (GPa)RSS on {112}〈111〉 (GPa)
[100][110][111][100][110][111][100][110][111]
Shock wave compression (uniaxial strain) Elastic impact stress 12.5 12.5 12.5 3.33 3.15 1.81 3.84 3.64 2.10 
Elastic limit 3.61 3.58 5.11 0.96 0.90 0.74 1.11 1.04 0.86 
Quasi-static compressiona (uniaxial stress) Elastic limit 0.05 0.20 0.18 0.02 0.08 0.05 0.02 0.09 0.06 
a

Data obtained from Ref. 25; compressive strain rate 9 × 10−5 s−1.

Data presented in Figs. 6(a) and 6(b) show that the resolved shear stress values (τel) at the elastic limit under shock loading are somewhat comparable for all three orientations, although the corresponding longitudinal stresses were significantly different (3.6 GPa for [100] and [110] loading and 5.1 GPa for [111] loading). The τel values listed in Table IV are also comparable to the reported Peierls stresses (τPscrew) of ∼0.9 and ∼0.7 GPa, respectively, of screw dislocations on {110} and {112} planes36 but significantly larger than the Peierls stress of edge dislocations (τPedge ≈ 25 MPa)37 in Mo single crystals.

In contrast to the measured elastic limits, the longitudinal stresses at impact were the same (∼12.5 GPa) for all three orientations, but the corresponding resolved shear stresses (τimp) on the operative slip systems are quite different. In particular, the τimp value for [111] loading is significantly smaller than τimp values reached for [100] and [110] loading. Consequently, differences between the resolved shear stress values at impact and at the elastic limit, i.e., τimpτel, are almost twice as large along [100] and [110] orientations compared to the [111] orientation. Since elastic wave attenuation was more rapid along the [100] and [110] orientations compared to the [111] orientation (Fig. 5), the elastic wave decay rate near the impact surface is likely proportional to the shear stress difference (τimpτel). Examination of the shock response along multiple crystal orientations helped establish this finding.

The longitudinal stresses corresponding to the elastic limits measured from shock (this work) and quasi-static25 loading experiments on [100], [110], and [111]-oriented Mo single crystals are presented in Table IV. The values of the corresponding resolved shear stresses on the operative {110}〈111〉 and {112}〈111〉 slip systems for both types of experiment are also listed.

The longitudinal stresses at the elastic limit under shock loading are more than an order of magnitude larger compared to those measured under quasi-static compression along all three orientations. However, due to differences in the loading conditions between shock (uniaxial strain) and quasi-static (uniaxial stress) experiments, the corresponding resolved shear stress values provide a more meaningful comparison. Our data show that the resolved shear stresses at the elastic limits under shock loading are significantly larger (∼10 to 50 times) than the shear stress values at quasi-static elastic limits. Later in this section, we discuss potential reasons for this large difference.

Numerical simulations of the measured [110] and [111] wave profiles were performed using a one-dimensional wave propagation code (COPS)38 by incorporating dislocation-based crystal plasticity into a continuum modeling framework.8,16 Details of the dislocation model used to describe plastic deformation of Mo single crystals can be found in Ref. 16, and only the relevant equations are summarized next. The plastic shear strain rate αγ˙p on a slip system α is given by

αγ˙p=bαNmαv¯,
(3)

where b is the Burgers vector, αNm is the mobile dislocation density, and αv¯ is the average dislocation velocity on the corresponding slip system.The following relations were used in our dislocation model to describe the evolution of the mobile dislocation density αNm and the average dislocation velocity αv¯ on slip system α:

αN˙m=Mαγ˙pforατ<τc,Mαγ˙pmultiplicationterm+A(αττc)ατ˙nucleationtermforαττc,
(4)
αv¯=v0exp[D/(αττ0)].
(5)

In Eq. (4), the “multiplication term” represents an increase in αNm with the accumulated plastic shear strain αγp due to regenerative multiplication, and the “nucleation term” accounts for an additional increase in αNm when the resolved shear stresses (ατ) on the operative slip planes exceed a critical valueτc. In Eq. (5), v0 is shear sound speed on the slip plane, D is the drag stress associated with dislocation motion, and τ0 is the threshold shear stress required to move the dislocations.

To differentiate between the {110}〈111〉 and {112}〈111〉 slip systems, we considered them separately in our simulations. The dislocation model parameter values used to describe {110} and {112} slip were the same as those used in the [100] work,16 except for the nucleation constant (A). Values of 5 × 10−10 (cm/dyn)2 and 5 × 10−12 (cm/dyn)2 for A in Eq. (4) provided a good overall match with the measured [110] and [111] wave profiles, respectively. These are an order of magnitude different from A= 2 × 10−11 (cm/dyn)2 used to simulate the [100] profiles.16 

1. Shock compression along [110]

The measured (black solid lines) and calculated (red dashed lines for {110} slip and blue dotted-dashed lines for {112} slip) [110] wave profiles presented in Fig. 7 show a good overall agreement, including the kink in the plastic wave, irrespective of the slip systems considered. The resolved shear stress and plastic shear strain histories on the operative slip systems, calculated following Ref. 16, provide insight into the observed kink in the plastic wave profiles and, therefore, they are discussed below. For slip system indices α in the following, refer to Table III of Ref. 16.

FIG. 7.

A comparison of the measured [110] wave profiles (black solid lines) and the [110] wave profiles calculated assuming either {110}〈111〉 (red dashed lines) or {112}〈111〉 slip systems (blue dotted-dashed line) to be operative. Time is relative to the loading of the buffer front surface.

FIG. 7.

A comparison of the measured [110] wave profiles (black solid lines) and the [110] wave profiles calculated assuming either {110}〈111〉 (red dashed lines) or {112}〈111〉 slip systems (blue dotted-dashed line) to be operative. Time is relative to the loading of the buffer front surface.

Close modal
a. {110} slip

The measured wave profile (black solid line) at a propagation distance of 1.299 mm and the corresponding resolved shear stress (ατ) and plastic shear strain (αγp) histories on the operative {110}〈111〉 slip systems are shown in Fig. 8. The {110} systems with indices α = 1–4 do not contribute to plastic deformation (ατ = 0), and, therefore, they are not shown. Resolved shear stresses on slip systems α = 5, 8, 9, and 11 (blue solid line) and α = 6, 7, 10, and 12 (red solid line) are related to the stress differences (PyyPzz) and (PxxPzz), respectively. The in-material longitudinal stress component Pxx and the lateral stress components Pyy and Pzz are shown in Fig. 9, where x, y, and z are defined along [110], [1¯10], and [001] directions, respectively. Since PyyPzz, the blue and the red systems are nonequivalent under [110] loading. Plastic shear strains accumulated on the blue (α = 5, 8, 9, and 11) and red (α = 6, 7, 10, and 12) slip systems are shown with the blue and red dashed lines, respectively.

FIG. 8.

Measured [110] wave profile (black solid line) at a propagation distance of 1.299 mm (Exp. 5) and the corresponding calculated resolved shear stress (red and blue solid lines) and plastic shear strain (red and blue dashed line) histories on the operative {110}〈111〉 slip systems (see Table III of Ref. 16 for slip system indices α). Time is relative to the loading of the buffer front surface.

FIG. 8.

Measured [110] wave profile (black solid line) at a propagation distance of 1.299 mm (Exp. 5) and the corresponding calculated resolved shear stress (red and blue solid lines) and plastic shear strain (red and blue dashed line) histories on the operative {110}〈111〉 slip systems (see Table III of Ref. 16 for slip system indices α). Time is relative to the loading of the buffer front surface.

Close modal
FIG. 9.

Calculated in-material stress histories for Mo single crystals shocked along the [110] orientation at the propagation distance of 1.299 mm for the operation of {110}〈111〉 slip systems. Here, x, y′, and z′ axes are defined along the [110], [1¯10], and [001] crystallographic orientations, respectively. Time is relative to the loading of the buffer front surface.

FIG. 9.

Calculated in-material stress histories for Mo single crystals shocked along the [110] orientation at the propagation distance of 1.299 mm for the operation of {110}〈111〉 slip systems. Here, x, y′, and z′ axes are defined along the [110], [1¯10], and [001] crystallographic orientations, respectively. Time is relative to the loading of the buffer front surface.

Close modal

As shown in Fig. 8, when the magnitudes of resolved shear stresses (ατ) on the red slip systems reach the critical resolved shear stress τc needed to activate the “nucleation term” in Eq. (4), the red systems yield. Following yielding of the red slip systems around t = 0.42 μs (Fig. 8), shear stresses on the red and blue slip systems remain nearly constant until accumulation of plastic shear strain (αγp) on the red slip systems leads to a rapid relaxation of shear stresses on all eight systems. Values of ατ on the blue systems eventually become negative at t = 0.45 μs, i.e., the slip direction is reversed. However, the magnitudes of ατ on these systems continue to grow and eventually reach τc. Subsequently, the blue systems yield, and shear stresses on these systems begin to relax. The kink in the plastic wave profile is coincident with the yielding of the blue systems near t = 0.47 μs, suggesting that it results from successive yielding of nonequivalent {110} slip systems (red and blue systems). Following the kink, the blue and the red systems contribute equally to plastic deformation, and the shear stresses continue to relax and reach a minimum behind the plastic wave. At the peak state, the resolved shear stresses on both sets of slip systems are equal in magnitude but have opposite signs.

b. {112} slip

As shown in Fig. 10, the two sets of main contributing non-equivalent {112}〈111〉 slip systems are (1) α = 15, 21 (shown with the red lines) and (2) α = 16, 22 (shown with the blue lines). Contributions from the red and blue {112} slip systems are equivalent to those discussed above for the red (α = 6, 7, 10, and 12) and the blue (α = 5, 8, 9, and 11) {110} slip systems. The yielding of the red {112} slip systems (α = 15 and 21), operative in the twinning sense,30 results in the observed elastic wave amplitude. The subsequent yielding of the blue {112} systems (α = 16 and 22) at t ≈ 0.47 μs, while operative in the antitwinning sense,30 leads to the observed kink on the plastic wave profile. Contributions from the remaining eight {112}〈111〉 systems toward plastic deformation are not significant.

FIG. 10.

Measured [110] wave profile (black solid line) at a propagation distance of 1.299 mm (Exp. 5) and the corresponding calculated resolved shear stress and plastic shear strain histories (shown with solid and dashed lines of the same color, respectively) on the operative {112}〈111〉 slip systems (see Table III of Ref. 16 for slip system indices α). Time is relative to the loading of the buffer front surface.

FIG. 10.

Measured [110] wave profile (black solid line) at a propagation distance of 1.299 mm (Exp. 5) and the corresponding calculated resolved shear stress and plastic shear strain histories (shown with solid and dashed lines of the same color, respectively) on the operative {112}〈111〉 slip systems (see Table III of Ref. 16 for slip system indices α). Time is relative to the loading of the buffer front surface.

Close modal

2. Shock compression along [111]

A comparison between the measured [111] wave profiles (black solid lines) and the corresponding calculated profiles (red dashed lines for {110} slip and blue dotted-dashed lines for {112} slip) is presented in Fig. 11. The calculated and measured profiles are in excellent agreement for propagation distances less than 1.3 mm. For distances ≥1.3 mm, the elastic and plastic wave profiles were captured adequately, but not the distinct structures observed behind the elastic wave. Although the observed structures between the elastic and plastic waves were not captured accurately, a plausible explanation for the appearance of these structures is provided below in terms of {110} slip.

FIG. 11.

A comparison of the measured [111] wave profiles (black solid lines) and the [111] wave profiles calculated assuming either {110}〈111〉 (red dashed lines) or {112}〈111〉 (blue dotted-dashed lines) slip systems to be operative. Time is relative to the loading of the buffer front surface.

FIG. 11.

A comparison of the measured [111] wave profiles (black solid lines) and the [111] wave profiles calculated assuming either {110}〈111〉 (red dashed lines) or {112}〈111〉 (blue dotted-dashed lines) slip systems to be operative. Time is relative to the loading of the buffer front surface.

Close modal

The calculated resolved shear stress (ατ) (red solid lines) and plastic shear strain (αγp) (red dashed lines) histories on the operative {110}〈111〉 slip systems at a propagation distance of 1.290 mm are shown in Fig. 12. In this figure, magnitudes of ατ on all six contributing slip systems (α = 1, 2, 5, 6, 9, and 10) at the peak of the elastic wave are lower than the critical stress required to activate the “nucleation term,” i.e., ατ<τc ( = 0.91 GPa). As a result, very little plastic shear strain is accumulated behind the elastic wave, which leads to an increase in ατ behind the elastic wave. When the shear stress magnitude exceeds τc near t = 0.45 μs, the “nucleation term” becomes operative. Operation of the “nucleation term” leads to the accumulation of measurable plastic shear strain and in turn, a rapid relaxation of ατ values, which reach a minimum behind the plastic wave. Calculated rise and subsequent rapid relaxation of shear stresses behind the elastic wave are coincident with and, therefore, likely responsible for the structures observed in the measured [111] profiles (shown with the black solid line in Fig. 12).

FIG. 12.

Measured [111] wave profile (black solid line) at a propagation distance of 1.290 mm (Exp. 13) and the corresponding calculated resolved shear stress (red solid lines) and plastic shear strain (red dashed line) histories on the operative {110}〈111〉 slip systems (see Table III of Ref. 16 for slip system indices α). Time is relative to the loading of the buffer front surface.

FIG. 12.

Measured [111] wave profile (black solid line) at a propagation distance of 1.290 mm (Exp. 13) and the corresponding calculated resolved shear stress (red solid lines) and plastic shear strain (red dashed line) histories on the operative {110}〈111〉 slip systems (see Table III of Ref. 16 for slip system indices α). Time is relative to the loading of the buffer front surface.

Close modal

The observed structures can also be explained similarly in terms of the operation of {112}〈111〉 slip systems, where the {112} slip systems with indices α = 17, 21, and 24 contribute similarly to the {110}〈111〉 systems with indices α = 1, 2, 5, 6, 9, and 10 toward the shock-induced deformation along [111].

The measured elastic-plastic wave profiles, presented in Fig. 4, suggest that Mo single crystals exhibit time-dependent, strongly anisotropic material response under compressive shock wave loading. The crystal anisotropy effects were manifested in (a) the measured elastic wave amplitudes and their attenuation with propagation distance (Fig. 5), (b) the overall structure of the measured wave profiles (Fig. 4), and (c) the measured elastic and plastic shock velocities (Table I in Ref. 16 and Tables II and III in this work).

The observed anisotropies in the elastic wave attenuation and measured elastic limits are explained in terms of resolved shear stress values on the {110}〈111〉 and {112}〈111〉 slip systems, determined from quasi-static studies. For a given loading direction, the elastic wave decay rate was proportional to differences in the resolved shear stress values corresponding to the elastic impact stress and the elastic limit, i.e., (τimpτel). Furthermore, resolved shear stress values (τel) at the measured elastic limits along all three orientations were comparable, suggesting that resolved shear stresses, and not longitudinal stresses, are a better measure of the elastic limit or material strength in shocked anisotropic single crystals.

For all three crystal orientations, the resolved shear stresses at the elastic limit (τel) under shock loading are comparable to the Peierls stresses (the stress required to move a dislocation over the lattice barriers in the absence of any thermal activation) of screw dislocations (τPscrew), which govern the plastic deformation of Mo.37,39 This similarity further implies that the elastic wave amplitude attenuates rapidly for τel>τPscrew, but no significant attenuation is observed when τelτPscrew. Such transition in the elastic wave attenuation rate is similar to those observed by Zaretsky and Kanel in BCC tantalum (Ta) and vanadium (V) under planar shock loading, which they attributed to the transition of screw dislocation motion from being phonon drag-controlled to thermally activated.40 

Large differences (∼10–50 times) in the resolved shear stresses at the measured elastic limits under shock and quasi-static loading, discussed in Sec. IV D, are a likely consequence of both large τPscrew value in Mo and significant differences in the loading times/rates between the two types of experiment. Quasi-static conditions (slow loading rates) entail sufficient time for thermal activation and, therefore, very little externally applied shear stress is required to move screw dislocations over the lattice barrier to initiate yielding. In contrast, thermally activated processes are less important under shock loading (short times and high loading rates >106 s−1), and significantly large shear stresses—comparable to the τPscrew value—are needed for screw dislocations to overcome lattice barriers and initiate yielding under shock loading.

Numerical simulations of the measured [110] and [111] profiles provide further insight. Irrespective of the initial mobile dislocation density used in our simulations, we were not able to match the rapid elastic wave attenuation and the measured elastic-plastic wave profile, including the stress relaxation (yield drop) behind the elastic wave, simultaneously when only regenerative dislocation multiplication was considered. Similar to the [100] work,16 it was necessary to incorporate the “nucleation term,” operative when shear stresses are larger than τPscrew, into Eq. (4) of our dislocation model to obtain an adequate match to the measured wave profiles. The incorporation of the “nucleation term,” which represents an increase in the mobile dislocation density with shear stress, in our simulations was an ad hoc measure to account for an increase in plastic strain rate when shear stresses on the slip plane are larger than τPscrew as plastic strain rate is proportional to the mobile dislocation density [Eq. (3)]. Our simulations suggest that an increase in the plastic strain rate above the Peierls stress is responsible for the rapid elastic wave attenuation near the impact surface, which is consistent with conclusions drawn from previous shock studies on iron (Fe),41 and Ta and V40—all BCC metals.

After incorporation of the “nucleation term,” both [110] and [111] profiles were matched equally well in terms of either {110} or {112} slip systems; operation of either family of slip systems could explain the distinct features observed in the measured wave profiles. Therefore, based on the numerical simulations presented here and in Ref. 16, we conclude that {110} and/or {112} slip systems were operative during shock compression of Mo single crystals along all three orientations.

To understand the role of crystal anisotropy on the elastic-plastic deformation of shock compressed molybdenum (Mo), results from high-purity single crystals shocked along [110] and [111] orientations to an elastic impact stress of 12.5 GPa were obtained and compared with the [100] results reported previously.16 The main findings of this work are

  1. The deformation response of Mo is time-dependent and strongly anisotropic under shock loading. The anisotropy effects manifested in the measured elastic wave amplitudes and shock speeds and the overall structure of the measured wave profiles.

  2. Plasticity of shock compressed Mo is governed by screw dislocation slip on {110}〈111〉 and/or {112}〈111〉 slip systems; these slip systems could not be distinguished from our work.

  3. Elastic wave attenuation close to the impact surface is proportional to the difference in the resolved shear stresses on the operative slip systems corresponding to the elastic impact stress and the elastic limit, i.e., (τimpτel).

  4. Resolved shear stresses (τel) on the operative slip systems, and not longitudinal stresses, at the elastic limit are a better measure of the material strength in shocked single crystals.

  5. Shear stresses at the elastic limit (τel) under shock loading were comparable to the Peierls stress (τPscrew) of screw dislocations in Mo, which suggests that elastic wave amplitudes attenuate rapidly when shear stresses on the slip plane τ>τPscrew, but no measurable attenuation occurs when ττPscrew.

  6. Large differences (∼10–50 times) in the resolved shear stress values at the elastic limits under shock and quasi-static loading are a likely consequence of large Peierls stress of screw dislocations in Mo.

  7. The observed rapid elastic wave attenuation and the measured elastic-plastic wave profiles including the yield drop behind the elastic wave could not be simulated simultaneously using regenerative dislocation multiplication alone, irrespective of the initial mobile density considered. It was necessary to incorporate an ad hoc “nucleation term,” which accounted for an increase in the plastic strain rate above the Peierls stress, into our dislocation model to adequately match the measured elastic-plastic wave profiles along all three orientations.

Overall, our experimental results for the three crystal orientations and their analysis in terms of the resolved shear stresses on the operative slip planes and numerical simulations have provided a comprehensive insight into shock wave induced elastic and plastic deformation of Mo single crystals, including the development of a dislocation-based continuum-scale material model.

Nate Arganbright, Brendan Williams, Thomas Eldredge, Yoshi Toyoda, and Kurt Zimmerman are sincerely thanked for their help with the impact experiments. Dr. Michael Winey is thanked for his assistance with running the COPS code and for many useful discussions. Dr. Mukul Kumar of Lawrence Livermore National Laboratory is acknowledged for the dislocation density measurements. This publication is based upon work supported by the Department of Energy, National Nuclear Security Administration under Award No. DE-NA0002007.

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