Full Length Article

# Pressure waves induced by the bcc-hcp phase transition in dynamically loaded single crystal iron

N. Amadou<sup>a,b</sup>, T. de Rességuier<sup>b,*</sup>

<figure><img src="image_1.png" /></figure>

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## ARTICLE INFO

**Keywords:**

Phase transition

Molecular dynamics

Shock physics

Wave propagation

Iron

## ABSTRACT

Molecular Dynamics simulations have been used to investigate the dynamic response of single crystal iron to ramp compression along the [001] crystallographic direction, with a focus on the coupling between the propagation and interaction of pressure waves and the phase transformation process. In particular, we report an original observation at the atomic level of release and recompression waves specifically induced by the phase transition. We provide a straightforward, physically-based explanation of the origin of these waves based on impedance mismatch between the parent, daughter, and mixed-phase region, and show how they depend directly on the kinetics of the transformation. This analysis may be generalized to other time-dependent phase transformations in other materials subjected to dynamic loading, and it is probably still valid at the much larger space and time scales encountered in experiments, which could have practical implications.

## 1. Introduction

Dynamic compression techniques enable exploring the thermo-mechanical behavior of materials over ranges of extreme pressure, temperature and ultrahigh strain rates, relevant in a variety of applications for both basic science (e.g. planetology, meteoritic impacts, materials science) and engineering (e.g. protection against high velocity debris, safety, armor design, crashworthiness) [1-7]. Solid-solid phase transformations are one of the key aspects of this dynamic behavior, which has motivated extensive research work for many decades. Due to its crucial importance in both geophysics and industrial applications, iron, which undergoes a well-known transition from its body-centered-cubic (bcc) structure at ambient conditions to a hexagonal-close-packed (hcp) phase at high pressure, has probably been the most widely studied material in this context, e.g. [8-12]. However, major questions remain highly debated, such as the kinetics of this phase transformation [13-21] and its coupling with plasticity [22-30].

In recent years, we have used Molecular Dynamics (MD) simulations to investigate at the atomic scale the coupling between plastic deformation and the bcc-hcp phase transition in single crystal iron [25,28,30-32] using the modified Ackland potential reported by Gunkelmann et al. [33]. In [001]-oriented crystals, ramp compression above the yield strength was shown to produce plastic deformation in the form

of twins, then further compression was observed to induce a hardening-like behavior where the twins were progressively removed while no dislocation activity was detected, leading to a strong increase of the shear stress. In a perfect crystal, such post-yield stiffening of the bcc matrix was shown to inhibit the bcc-hcp phase transformation, which onset pressure was then shifted to very high values (on the order of 100 GPa). For the phase transition to be activated at moderate pressure, more consistent with experimental data, one must either introduce pre-existing defects such as microvoids within the crystal [28,34] or let the ramp wave steepen during its propagation to evolve into a shock wave, due to the increase of sound velocity with pressure [25,31]. In this paper, we study this latter scenario with a focus on the coupling between the propagation and interaction of pressure waves and the phase transformation process. In particular, we report an original observation at atomic level of release and recompression waves specifically induced by the phase transition and we show how these waves depend directly on the kinetics of the transformation, which may have practical implications for further studies of a variety of materials.

## 2. Method and computational details

MD simulations were performed with the Lammps code [35] where the atomic equation of motion was integrated using the Verlet algorithm

<footnote><sup>a</sup> Département de Physique, Université Abdou Moumouni de Niamey, BP. 10662, Niamey, Niger</footnote>

<footnote><sup>b</sup> Institut Pprime, CNRS, ISAE-ENSMA, Université de Poitiers, France</footnote>

<footnote>* Corresponding author.
E-mail address: resseguier@ensma.fr (T. de Rességuier).</footnote>
with a time step of 1 fs. Atomic interactions were treated in the embedded atom model formalism [36,37] using the modified version of the Ackland potential [33]. Indeed, this potential was successfully used in the past to account for both plasticity and phase transition in iron [22,29,33], providing predictions in correct overall agreement with a variety of experimental data. Samples containing about 28 million atoms were single crystal iron of 100 × 100 $a_0^2$ section and 1400 $a_0$ length along a Z direction, where $a_0 = 2.87$ Å is the equilibrium lattice

parameter. They were first thermalized to 50 K and zero pressure in the NPT ensemble, like in previous work [25,28,31], as low temperature initial conditions are commonly used in MD simulations to minimize thermal noise and optimize structure identification. Then they were compressed in the NVE ensemble by driving an effective infinite-mass wall piston with an imposed velocity along Z (shrinking boundary) while periodic boundary conditions were applied in the transverse directions X and Y. Here, the Z direction was chosen to coincide with the

<figure><img src="image_2.png" /><figcaption>**Fig. 1.** Spatial profiles of the longitudinal pressure $P_z$ (a) and particle velocity $U_z$ (b) in [001]-oriented single crystal iron ramp-compressed to a maximum piston velocity of 1000 m/s. (c) selected snapshots of atomic configuration where bcc atoms are colored according to their $P_z V_a$ value ($V_a$ is the atomic volume), twins are colored in brown, and hcp atoms are colored in pink regardless of the pressure.</figcaption></figure>
[001] crystal orientation. The piston velocity was increased linearly from 0 to a maximum ranging between 600 m/s and 1000 m/s during 30 ps then was held constant afterwards. The resulting load is a ramp compression to about 30 to 70 GPa with a strain rate of about 10⁹ s⁻¹. Local thermodynamic and mechanical variables such as longitudinal pressure Pz, particle velocity Uz and temperature, were evaluated within a planar bin of 3a₀ width perpendicular to the loading direction Z, in the same manner as in previous work [25]. Finally, local structural analysis was performed using adaptative Common Neighbor Analysis (CNA) and Centrosymmetry as implemented in the OVITO software [38].

### 3. Results and discussion

Fig. 1 shows the response of a defect-free [001]-oriented iron crystal ramp-compressed during 30 ps to a maximum piston velocity of 1000 m/s. After 30 ps, the initially linear pressure ramp has split into an elastic wave, up to a yield strength of about 11 GPa, followed by a second compression wave gradually steepening (from its lower portion), due to the increase of sound velocity with increasing pressure. In the corresponding snapshot (Fig. 1c, top), this second compression beyond the yield strength is observed to induce plastic deformation in the form of twins, then these twins are annihilated upon further compression (blue regions in Fig. 1c). At about 50 ps, the ramp has evolved into a shock front (Fig. 1a). This triggers the bcc-hcp phase transformation, as reported in more details elsewhere [25,31], so that hcp phase is observed to grow from then on (Fig. 1c). Meanwhile, the final pressure decreases from about 67 GPa to about 60 GPa, and both pressure (Fig. 1a) and particle velocity (Fig. 1b) exhibit complex, non-monotonous variations behind the compression front. They might be partially attributed to the change of behavior from the isentrope to the Hugoniot upon shock occurrence, due to the entropy jump in the shock front [39]. However, in the present case where shock formation coincides with the onset of the phase transformation, these variations are directly related to the structural phase transition as discussed in details next.

A schematic description of the dynamic behavior of ramp-compressed iron is proposed in Fig. 2, based on classical shock physics combining a time-distance diagram (Fig. 2a) with a pressure-particle velocity plot (Fig. 2b). State 0 is the initial state at ambient (negligible) pressure and null velocity. Ramp loading on the left surface can be regarded as a continuous succession of elementary compression waves (green triangle in Fig. 2a) taking bcc iron to state 1, located on the bcc isentrope (green solid line in Fig. 2b). Because the sound speed increases

with pressure, the later waves progressively catch up with the earlier ones, so that the ramp profile steepens until forming a single shock front at time $t_i$. Then, as reported elsewhere [25,31], the bulk driving force becomes high enough to initiate the bcc-hcp phase transformation. Since this transformation is not instantaneous, a mixed-phase region is created where bcc and hcp phases are assumed to coexist until full completion at time $t_c$. The influence of such transition to a higher density phase on the shock behavior is well-known, e.g. [40,41]. It includes a kink in the Hugoniot at the transition onset pressure, then a region of higher compressibility corresponding to the gradual transformation, up to a second kink at completion when switching to the Hugoniot of the high-pressure phase. The shock velocity drops at the beginning of the transformation then it remains lower in the mixed phase region than in both the parent and daughter phases [42]. Consequently, in the pressure-particle velocity diagram, the shock adiabat of the mixture (like the Rayleigh line, which slope is the product of initial density and shock velocity) flattens below those of the pure phases before going back up to that of the high pressure phase. This continuous evolution during the transformation is represented schematically by the thick shaded arrows in Fig. 2b. Note that the thin dotted line illustrates the shock adiabat of one particular mixture with given proportions (as there is no single curve representative of the entire mixed phase region). Now, considering that the onset of the phase transformation creates an interface within the material at time $t_i$, impedance mismatch across this interface generates a shock (0 → 2) transmitted in the bcc-hcp mixture and a release wave (1 → 2) reflected within bcc iron, the location of state 2 in the pressure-velocity plane being classically determined at the intersection of shock adiabats and release isentropes [43]. At time $t_c$, the phase transformation is assumed to be fully completed, which creates a second interface between the mixed-phase region and pure hcp iron. Again, impedance mismatch across this interface takes iron to a new state labelled 3, through a transmitted shock (0 → 3) and a reflected wave (2 → 3), which is found to be a recompression from the pressure-particle velocity plot. Hence, the simple hydrodynamic analysis above predicts that a non-instantaneous phase transformation triggered within the bulk of the sample will give rise to two waves propagating backwards (i.e. to the left), namely a release (1 → 2) followed by a recompression (2 → 3). Besides, the amplitudes of these two waves and the time delay between them will be directly conditioned by the kinetics of the transformation, which governs the thickness and phase proportions of the mixed-phase region (or equivalently the completion time $t_c-t_i$) as well as the gradual changes between the different curves in the pressure-velocity plot (thick shaded arrows in Fig. 2b).

<figure><img src="image_3.png" /><figcaption>Fig. 2. Schematic description in a space-time diagram (a) and in a pressure-particle velocity plot (b) of the successive states (bold numbers) induced in ramp-compressed iron, assuming a mixed region where bcc and hcp phases coexist between times $t_i$ and $t_c$, during a gradual transformation represented by the thick shaded arrows.</figcaption></figure>
Because time $t_i$ also corresponds to a change from the isentrope (ramp compression) to the Hugoniot (shock compression), the generation of the release wave (1 → 2) might also be partially attributed to this change, without necessarily invoking the phase transformation. However, (i) it would not explain the later recompression, and (ii) similar simulations at lower piston velocities [34], where the ramp still steepens into a shock but where shock pressure is too low to trigger the phase transition, do not show the left-propagating waves predicted above and observed below.

Fig. 3 shows the pressure distribution in time and space (i.e. distance from the loaded surface along the Z direction) extracted from the simulation presented in Fig. 1, for a ramp-compression to a 1000 m/s maximum piston velocity. It is consistent with the phenomenological analysis in Fig. 2a, with the same successive states induced by the ramp compression of bcc-iron (1), the reflection of a release wave after shock formation at $t_i \approx 50$ ps (2) and the recompression after full completion of the bcc-hcp transition at $t_c \approx 54$ ps (3). Because the simulation captures the detailed evolution of the material state and phase proportions during the transformation, its quantitative influence on the pressure waves, and the modification of the wave fronts during their propagation, the numerical picture is obviously more complex and richer than in the simplified analysis. In particular, both the amplitude and duration of the release pulse (expected state 2) are actually observed to evolve, which is a direct consequence of the kinetics of the phase transition.

A sequence of the successive pressure-velocity states induced at $Z = 600$ a₀ in the same simulation (Fig. 1 and Fig. 3) is plotted in Fig. 4. Again, a good consistency is found with the analytical prediction in Fig. 2b, with the same material states reported in bold numbers. Finally, Fig. 5 recalls the spatial profiles of pressure (blue) and particle velocity (red) computed at 65 ps in the same simulation (already shown in Fig. 1). They correspond to a horizontal section (fixed time) in the diagram in Fig. 3 (along the white dotted line). They clearly exhibit the reflected release wave propagating backwards within the bcc phase, which logically increases particle velocity (higher in state 2 than in state 1, see Fig. 2b), followed by the recompression of the bcc-hcp mixture, as confirmed by the corresponding snapshot at the bottom. This accounts for the non-monotonous variations mentioned earlier in the left part of the late profiles in Fig. 1. Quantitatively, pressure values in states 1, 2 and 3 are about 67 GPa, 45 GPa and 60 GPa, respectively. As observed in those late profiles, the release wave spreads during its propagation to the left, while the recompression wave steepens, again due to the variations

of sound velocity with pressure. According to the simulation, this recompression wave induces a temperature increase of about 230 K in iron.

Future work will include similar investigation in iron single crystals of different orientations, study of the dependence of the observed behavior on initial temperature, search for analogous pressure waves in other phase transitions in different metals, and the design of dedicated laser compression experiments to measure particle velocity profiles in iron samples, hoping to observe at a ~µm-scale the state 2 evidenced and discussed above (Figs. 3 and 5).

## 4. Summary and conclusions

In summary, MD simulations of the response of single crystal iron to ramp compression have been performed and fully interpreted on the basis of classical shock physics, including impedance mismatch analyses in the general framework of a time-dependent phase transformation. The results provide the first straightforward explanation of the origin of release and recompression waves induced by the phase transition, also observed within a finite deformation continuum framework combining nonlinear elasto-viscoplasticity with multivariant phase-field theory [26]. Furthermore, they show how these waves depend directly on the kinetics of the transformation. Thus, the detection and analysis of such waves in future experiments, over larger spatial and temporal scales, might be a promising and original way to investigate this kinetics, which is of major importance to interpret impact events and to improve multiphase equations of state used in hydrocodes. In addition, these waves may play a role in the processes that may occur after the phase transition, such as tensile loading due to later wave interactions and subsequent spall damage [44]. They may also affect to some extent the final microstructure since they induce significant temperature variations as mentioned above, which may be useful information in studies involving post-recovery observations of shock or ramp-compressed specimens. Finally, this generation of specific pressure waves accompanying a phase transformation is not limited to iron, so that its understanding may be helpful for other investigations of polymorphic transitions in a variety of materials under dynamic loading conditions.

### CRediT authorship contribution statement

N. Amadou: Writing – review & editing, Writing – original draft,

<figure><img src="image_4.png" /><figcaption>Fig. 3. Isocontours of pressure $P_z$ in a space-time diagram extracted from the simulation shown in Fig. 1 (maximum piston velocity of 1000 m/s), to be compared with the analytical prediction in Fig. 2a, with the same states labelled in bold numbers.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Fig. 4. Temporal evolution of pressure $P_z$ and particle velocity $U_z$ at $Z = 600$ $a_0$, extracted from the simulation shown in Fig. 1 (maximum piston velocity of 1000 m/s), to be compared with the analytical prediction in Fig. 2b, with the same states inserted in bold numbers.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 5. Spatial distribution of pressure $P_z$ (blue) and particle velocity $U_z$ (red) along the Z direction at time 65 ps already shown in Fig. 1, to be seen as an horizontal section along the white dotted line in Fig. 3 and to be compared with the analytical prediction in Fig. 2 with the same states inserted in bold numbers and corresponding snapshot below.</figcaption></figure>

Supervision, Methodology, Investigation, Formal analysis, Data cura-
tion, Conceptualization. **T. de Rességuier:** Writing – review & editing,
Writing – original draft, Validation, Supervision, Resources, Investiga-
tion, Funding acquisition, Formal analysis, Data curation,
Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence
the work reported in this paper.

Acknowledgments

Computations were performed on the supercomputer facilities of the
Mésocentre de calcul de Poitou Charentes (France). This work was
partially funded by the French program CNRS-Africa.

Data availability

Data will be made available on request.

References

[1] P.S. de Carli, J.C. Jamieson, Formation of diamond by explosive shock, Science 133 (1961) 1821.

[2] D.J. Milton, P.S. de Carli, Maskelynite: Formation by explosive shock, Science 140 (1963) 670.

[3] R.L. Kovach, D.L. Anderson, The interiors of the terrestrial planets, J. Geophys. Res. 70 (1965) 2873.

[4] J.R. Asay, The use of shock-structure methods for evaluating high-pressure material properties, Int. J. Impact Eng. 20 (1-5) (1997) 27-61.

[5] R.E. Rudd, T.C. Germann, B.A. Remington, J.S. Wark, Metal deformation and phase transitions at extremely high strain rates, MRS Bull. 35 (2010) 999-1006.

[6] N. Amadou, E. Brambrink, T. Vinci, A. Benuzzi-Mounaix, G. Huser, S. Brygoo, G. Morard, F. Guyot, T. de Rességuier, S. Mazevet, K. Miyanishi, N. Ozaki, R. Kodama, O. Henry, D. Raffestin, Probing iron at Super-Earth core conditions, Phys. Plasmas 22 (2015) 022705.
[7] B.A. Remington, R.E. Rudd, J.S. Wark, From microjoules to megajoules and kilobars to gigabars: probing matter at extreme states of deformation, Phys. Plasmas 22 (2015) 090501.

[8] D. Bancroft, E.L. Peterson, S. Minshall, Polymorphism of iron at high pressure, J. Appl. Phys. 27 (1956) 291.

[9] D.H. Kalantar, J.F. Belak, G.W. Collins, J.D. Colvin, H.M. Davies, J.H. Eggert, T. C. Germann, J. Hawreliak, B.L. Holian, K. Kadau, P.S. Lomdahl, H.E. Lorenzana, M. A. Meyers, K. Rosolankova, M.S. Schneider, J. Sheppard, J.S. Stolken, J.S. Wark, Direct Observation of the α-ε Transition in Shock-Compressed Iron Via Nanosecond X-Ray Diffraction, Phys. Rev. Lett. 95 (2005) 075502.

[10] T. de Rességuier, M. Hallouin, Effects of the α-ε phase transition on wave propagation and spallation in laser shockloaded iron, Phys. Rev. B 77 (2008) 174107.

[11] J.C. Crowhurst, B.W. Reed, M.R. Armstrong, H.B. Radousky, J.A. Carter, D.C. Swift, J.M. Zaug, R.W. Minich, N.E. Teslich, M. Kumar, The phase transition in iron at strain rates up to 10⁹ s⁻¹, J. Appl. Phys. 115 (2014) 113506.

[12] H. Hwang, E. Galtier, H. Cynn, I. Eom, S.H. Chun, Y. Bang, G.C. Hwang, J. Choi, T. Kim, M. Kong, et al., Subnanosecond phase transition dynamics in laser-shocked iron, Sci. Adv. 6 (2020) 5132.

[13] L.M. Barker, R.E. Hollenbach, Shock wave study of the α-ε phase transition in iron, J. Appl. Phys. 45 (1974) 4872.

[14] K. Kadau, T.C. Germann, P.S. Lomdahl, B.L. Holian, Microscopic View of Structural Phase Transitions Induced by Shock Waves, Science 296 (2002) 1681.

[15] K. Kadau, T.C. Germann, P.S. Lomdahl, B.L. Holian, Atomistic simulations of shock-induced transformations and their orientation dependence in bcc Fe single crystal, Phys. Rev. B 72 (2005) 064120.

[16] K. Kadau, T.C. Germann, P.S. Lomdahl, R.C. Albers, J.S. Wark, A. Higginbotham, B. L. Holian, Shock Waves in Polycrystalline Iron, Phys. Rev. Lett. 98 (2007) 135701.

[17] M. Bastea, S. Bastea, R. Becke, High pressure phase transformation in iron under fast compression, Appl. Phys. Lett. 95 (2009) 241911.

[18] R.F. Smith, J.H. Eggert, D.C. Swift, J. Wang, T.S. Duffy, D.G. Braun, R.E. Rudd, D. B. Reisman, J.-P. Davis, M.D. Knudson, G.W. Collins, Time dependence of the alpha to epsilon phase transformation in iron, J. Appl. Phys. 114 (2013) 223507.

[19] N. Amadou, E. Brambrink, A. Benuzzi-Mounaix, G. Huser, F. Guyot, S. Mazevet, G. Morard, T. de Rességuier, T. Vinci, K. Myanishi, N. Ozaki, R. Kodama, T. Boehly, O. Henry, D. Raffestin, M. Koenig, Kinetics of the iron α-ε phase transition at high-strain rates: Experiment and model, Phys. Rev. B 93 (2016) 214108.

[20] N. Amadou, E. Brambrink, T. de Rességuier, A.O. Manga, A. Aboubacar, B. Born, A. Molineri, Laser-driven ramp compression to investigate and model dynamic response of iron at high strain rates, Metals 6 (2016) 320.

[21] J.L. Shao, W. He, T. Xi, J. Xin, Microscopic insight into the structural transition of single crystal iron under the ramp wave loading, Comput. Mater. Sci. 182 (2020) 109772.

[22] N. Gunkelmann, E.M. Bringa, D.R. Tramontina, C.J. Ruestes, M.J. Suggit, A. Higginbotham, J.S. Wark, H.M. Urbassek, Shock waves in polycrystalline iron: Plasticity and phase transitions, Phys. Rev. B 89 (2014) 140102.

[23] K. Wang, S. Xiao, H. Deng, W. Zhu, Hu. Wangyu, An atomic study on the shock-induced plasticity and phase transition for iron-based single crystals, Inter. J. Plast. 59 (2014) 180–198.

[24] A. Vattré, C. Denoual, Polymorphism of iron at high pressure: A 3D phase-field model for displacive transitions with finite elastoplastic deformations, J. Mech. Phys. Solids 92 (2016) 1.

[25] N. Amadou, T. de Rességuier, A. Dragon, E. Brambrink, Coupling between plasticity and phase transition in shock- and ramp-compressed single-crystal iron, Phys. Rev. B 98 (2018) 024104.

[26] A. Vattré, C. Denoual, Continuum nonlinear dynamics of unstable shock waves induced by structural phase transformations in iron, J. Mech. Phys. Solids 131 (2019) 387.

[27] Y. Huang, Y. Xiong, P. Li, X. Li, S. Xiao, H. Deng, W. Zhu, Hu. Wangyu, Atomistic studies of shock-induced plasticity and phase transition in iron-based single crystal with edge dislocation, Int. J. Plast. 114 (2019) 215–226.

[28] N. Amadou, T. de Rességuier, A. Dragon, E. Brambrink, Effects of orientation, lattice defects and temperature on plasticity and phase transition in ramp-compressed single crystal iron, Comput. Mater. Sci. 172 (2020) 109318.

[29] R.J. Hoang-Thien Luu, M. Ravelo, E.M. Rudolph, T.C. Bringa, D.R. Germann, N. Gunkelmann, Shock-induced plasticity in nanocrystalline iron: Large-scale molecular dynamics simulations, Phys. Rev. B 102 (2020) 020102.

[30] A. Batouré, N. Amadou, M.A. Nassirou Hassan, A. Moussa Hassane, I. Adamou, Plastic deformation of [001]-oriented single-crystal iron under shock compression: Effects of void size, J. Appl. Phys. 136 (2024) 145904.

[31] N. Amadou, A.R.A. Abdoulaye, T. de Rességuier, A. Dragon, Strain-rate dependence of plasticity and phase transition in [001]-oriented single-crystal iron, Crystals 13 (2023) 250.

[32] N. Amadou, T. de Rességuier, A. Dragon, Coupling between plasticity and phase transition in single crystal iron at ultra-high strain rate, in: AIP Conference Proceedings, Vol. 2272, No. 1, AIP Publishing, 2020 070001.

[33] N. Gunkelmann, E.M. Bringa, K. Kang, G.J. Ackland, C.J. Ruestes, H.M. Urbassek, Polycrystalline iron under compression: Plasticity and phase transitions, Phys. Rev. B 86 (2012) 144111.

[34] M.A. Nassirou Hassan, N. Amadou, A. Moussa Hassane, A. Batouré, I. Adamou, T. de Rességuier, Kinetics of the bcc to hcp phase transition in [001]-oriented single crystal iron under ramp compression, Comput. Condens. Matter. 41 (2024) e00965.

[35] A.P. Thompson, H. Metin Aktulga, R. Berger, D.S. Bolintineanu, W.M. Brown, P. S. Crozier, P.J. in't Veld, A. Kohlmeyer, S.G. Moore, T. Dac Nguyen, R. Shan, M. J. Stevens, J. Tranchida, C. Trott, S.J. Plimpton, Lammgs – a flexible simulation tool for particle-based materials modeling at the atomic, meso, and continuum scales, Comput. Phys. Commun. 271 (2022) 108171.

[36] M.S. Daw, M.I. Baskes, Semiempirical, quantum mechanical calculation of hydrogen embrittlement in metals, Phys. Rev. Lett. 50 (1983) 1285–1288.

[37] S.M. Foiles, M.I. Baskes, M.S. Daw, Embedded-atom-method functions for the fcc metals Cu, Ag, Au, Ni, Pd, Pt, and their alloys, Phys. Rev. B 33 (1986) 7983–7991.

[38] A. Stukowski, Visualization and analysis of atomistic simulation data with OVITO—the open visualization tool, Model. Simul. Mater. Sci. Eng. 18 (2009) 015012.

[39] D.C. Swift, R.G. Kraus, E.N. Loomis, D.G. Hicks, J.M. McNaney, R.P. Johnson, Shock formation and the ideal shape of ramp compression waves, Physical Review E—Statistical, Nonlinear, and Soft Matter Physics 78 (6) (2008) 066115.

[40] G.E. Duvall, R.A. Graham, Phase transitions under shock wave loading, Rev. Mod. Phys. 49 (3) (1977) 523–581.

[41] G.I. Kanel, S.V. Razorenov, V.E. Fortov, Polymorphic Transformations and Phase Transitions in Shock-Compressed Solids, in: Shock-Wave Phenomena and the Properties of Condensed Matter, Springer-Verlag, New York, 2004.

[42] B.J. Jensen, G.T. Gray III, R.S. Hixson, Direct measurements of the α-ε transition stress and kinetics for shocked iron, J. Appl. Phys. 105 (2009) 103502.

[43] Y.B. Zel'Dovich, Y.P. Raizer, Physics of shock waves and high-temperature hydrodynamic phenomena, Courier Corporation, 2002.

[44] N. Amadou, T. de Rességuier, Phase transformations and plasticity in single-crystal iron from shock compression to spall fracture, Phys. Rev. B 108 (2023) 174109.