Computational Materials Science 135 (2017) 141–151

Contents lists available at ScienceDirect

Computational Materials Science

journal homepage: www.elsevier .com/locate /commatsci

Shock induced deformation response of single crystal copper: Effect ofcrystallographic orientation

http://dx.doi.org/10.1016/j.commatsci.2017.04.0090927-0256/� 2017 Elsevier B.V. All rights reserved.

⇑ Corresponding author.E-mail addresses: anupamneogi@atdc.iitkgp.ernet.in (A. Neogi), nilanjan@civil.

iitkgp.ernet.in (N. Mitra).

Anupam Neogi a,⇑, Nilanjan Mitra b

aAdvanced Technology Development Center, Indian Institute of Technology Kharagpur, Kharagpur 721302, IndiabDepartment of Civil Engineering and Center for Theoretical Studies, Indian Institute of Technology Kharagpur, Kharagpur 721302, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 14 December 2016Received in revised form 7 April 2017Accepted 8 April 2017

Keywords:Shock waveMolecular dynamicsCrystallographic orientationDeformationTemperature profilesMaxwell-Boltzmann distribution

We have carried out multimillion atom non-equilibrium molecular dynamics simulations for investigat-ing the effect of crystallographic orientation over the evolution of deformation pathway of single crystalcopper under shock compression. Based on symmetry, three different crystallographic directions,h100i; h110i and h111i are selected and taken as shock directions. Shock Hugoniot points has been cal-culated and compared among these different directions up to �450 GPa of shock pressure i.e. pistonvelocity of 3.0 km/s. Orientational anisotropy has been observed for the bulk Cu single crystals shockloaded along these three different directions. Even though this feature may not show up explicitly inexperimental investigations which typically measures shock-velocity and density Hugoniot curve, it isapparent from large scale atomistic simulations which measures the temperature Hugoniot curve quiteaccurately. Differences are observed in the von-Mises strain and stress plot distributions for shock load-ing of different intensities along the three directions. Large directional dependency is also evident in theevolution mechanism of deformation. Temperature profiles at different piston velocities for the shockfront and the shock equilibrated regions shows significantly different and interesting patterns alongthe three orientations. Maxwell-Boltzmann distribution is observed in the atomic velocities (therebythe temperature profiles also) for both the shock front region as well as the shock equilibrated regionfor shock loading along all the three directions.

� 2017 Elsevier B.V. All rights reserved.

1. Introduction

Application of mechanical load to a crystal structure results inaccumulation of strains and also global deformation of the mate-rial. After a certain specified value of strain for the material,microstructural deformations in the crystal structure originateand thereby results in formation of dislocations and/or twinswhich globally is manifested as development of plasticity in thematerial. If more load is applied then there are possibilities ofstructural or conventional phase transition in the material. Thecharacteristics of deformation microstructure depends on crystalstructure, orientation, stacking fault energy along with the type,intensity and duration of the load. Deformation mechanism of met-als have been studied for many decades and is of considerableinterest to scientific community through out the globe due to it’sapplication in various fields, such as auto-mobile, defence & secu-rity, aero-industry and nuclear power plant. The manuscript deals

with microstructural deformation mechanisms of bulk face-centered cubic (FCC) metal single crystal Cu under very high strainrate loading situations which are of importance in different situa-tions such as inertial confinement fusion (ICF) chamber, defenceaircraft and spacecraft. Typically these high strain situations inwhich the metal is subjected to very high temperature and pres-sure within a very short span of time can be experimentallyattained through shock-tube and/or laser-ablation studies whereasnumerically can be attained through atomistic simulation studies.Large scale molecular dynamic studies has been carried out in thismanuscript to develop a comprehensive understanding of theeffect of crystallographic orientation for single crystal Cu subjectedto different shock load intensities.

The sensitivity of crystal orientation over the dislocation microstructures of a recovered specimen, shocked under laser ablatedultra fast pulses has been reported [1–5] and threshold shock pres-sure at which deformation mechanism changes from glide to twin-ing has also been reported [4–6]. Enhancement of dislocationdensity with shock pressure has been reported [4,7,8] in the exist-ing literature. Observance of dislocation cells, microbands, defor-mation twin [4,9,10] have also been reported in the shock

142 A. Neogi, N. Mitra / Computational Materials Science 135 (2017) 141–151

experiment recovered samples. The above mentioned researchesare primarily experimental in nature and thereby do not necessar-ily describe the associated residual strains and their evolutionleading to the formation of different microstructural defects inthe shocked recovered samples, which this manuscript aims toachieve. Moreover, being experimental in nature, these works alsodo not comment on the effect of lattice orientations over the shockinduced temperature rise when single crystal Cu is subjected toshock load along different crystallographic directions at differentlevel of shock intensities.

Multimillion atom molecular dynamics (MD) simulation [11]has been done to investigate plasticity of high symmetry h100isingle crystal FCC metal; however the work is limited to shockpressure of �48 GPa (i.e. piston velocity of 1 km/s). They alsopointed out the need of studying the lower symmetry directions(like, h110i and h111i) to reveal the orientational sensitivity ofshock induced plasticity of FCC perfect crystal. Appearance of elas-tic precursor wave structure which separates out the shock frontfrom the plastic region in case of low symmetry (e.g. h110i andh111i) directions has been reportedly identified through multimil-lion molecular dynamics study using different potentials -Lennard-Jones (LJ) [?,12] and embedded atom method (EAM)[13]. Cao et al. [14] carried out atomistic simulation studies todemonstrate orientational anisotropy in shock loaded Cu for[221] orientation primarily looking into the difference of the wavevelocities for the plastic wave and its elastic predecessor. Apartfrom identification of different velocities for the plastic wavesand the elastic waves, there are no studies in existing literaturewhich demonstrates the evolution of strains and stresses alongwith temperature in the samples which this manuscript highlights.It should be noted that the scope of this manuscript is only limitedto global spatially averaged variables without taking deeper looksinto the type of dislocation interactions along different directions[15].

In this comprehensive numerical study, different crystal orien-tations of FCC-Cu have been considered which is subjected to rangeof shock intensities to identify the different microstructural defor-mations along with its relation to von-Mises strain, stress and alsotemperature evolution in the samples. Details elucidation of thedeformation path-way from a global mechanistic perspective isthe primary objective of this work.

Table 1Necessary details of the initial configurations of the NEMD-samples with desired orientatipressure.

Sample No. No. of crystal lattice units Orientation

X Y

1. 500� 100� 100 h100i h010i2. 350� 80� 70 h110i h001i3. 280� 60� 60 h111i h1 �10i

Table 2Necessary details of the initial configurations of the MSST-samples with desired orientatiopressure.

Sample No. No. of crystal lattice units Orientation

X Y

1. 20� 20� 20 h100i h010i h02. 15� 20� 15 h110i h001i h13. 12� 15� 12 h111i h1 �10i h1

2. Simulation methodologies

A series of multi-million atom non-equilibrium moleculardynamics (NEMD) simulations has been performed utilizing aquantum-mechanical many-body potential, embedded atom-method (EAM) potential for FCC single crystal copper (parameter-ized by Mishin et al. [16]) with different crystal orientations of[100]. The interatomic interactions of copper has been describedthrough the well studied Mishin-EAM [16] potential which is cap-able to reproduce the thermodynamic and mechanical propertiesat ambient condition [17,18] and at high pressure situation [19]as well. In addition, this Mishin-EAM [16] potential has been usedto predict the equation of state (EoS) up to 300 GPa [20] withnoticeable accuracy. In this current work, we have produced theshock Hugoniots up to 450 GPa which shows reasonable agree-ment with available experimental shock data [21].

However, for simulating shock wave propagation through thetarget samples of single crystal copper two different types ofmethodology, direct-shock method i.e. NEMD and multiscale shocktechnique (MSST) [22] has also been utilized in this current work.Direct-shock or NEMD simulations are used to explore the femto-second time resolved evolution of plasticity behind the shock front.Whereas, MSST algorithm has been employed for large timescalesimulations (for this work, up to 10 ns) for investigating shockequilibrated (after post-shock relaxation) thermodynamical statevariables at a significant distance (�100s of microns) behind theleading shock front. MSST is a simulation technique based on theNavierStokes equations for compressible flow and follows aLagrangian point through the shock wave which is accomplishedby time evolving equations of motion for the atoms, as well as vol-ume of the computational cell to constrain the stress in the shockpropagation direction to the Rayleigh line and the energy of thesystem to the Hugoniot energy condition. This method has beentested for simulating a unidirectional planar shock propagationfor various class of materials [23–25] with sufficient accuracy.While performing the MSST simulations, special care has beentaken to ensure that there is no significant drift in energy at anytime instance with the chosen values of ‘q’ and ‘tscale’ (parametersrequired in MSST simulation as implemented in LAMMPS [26]framework) for a particular shock intensity. Although, it shouldbe noted here that any choice of ‘q’ (chosen as 1 for current simu-

ons. The mass density of all the samples are 8.806 g/cm3 at ambient temperature and

Size (nm) No. of atoms

Z Shock dirn. X Y Z

h001i 180.75 36.15 36.15 20;000 � 103

h1 �10i 178.93 28.92 35.79 15;680� 103

h11 �2i 175.32 30.67 35.42 16;128� 103

ns. The mass density of all the samples are 8.806 g/cm3 at ambient temperature and

Size (nm) No. of atoms

Z Shock dirn. X Y Z

01i 7.23 7.23 7.23 32� 103

�10i 7.7708 7.3023 7.77082 36� 103

1 �2i 7.6389 7.7708 7.2012 34:56� 103

0 50 100 150 200 250 300 350 400 4509

10

11

12

13

14

15

16 <100> <110> <111> Expt. (Ref. 21)

Den

sity

(g/c

m3 )

Pressure (GPa)

Fig. 1. Calculated Hugoniot points, density and pressure, for shock propagationalong h100i; h110i and h111i has been plotted and compared with the availableexperimental [21] shock Hugoniot data of poly-crystalline copper. The representedshock Hugoniot points in this figure are from MSST simulations, although NEMDsimulations has been carried out to obtain this shock Hugoniot data. The differencesbetween NEMD and MSST results are below 1%, which indicates the absence ofprofound post-shock relaxation of thermodynamical state variables up to 10 ns.

A. Neogi, N. Mitra / Computational Materials Science 135 (2017) 141–151 143

lations) and ‘tscale’ (taken 0.01 for current simulations) does notaffect the essential generality of the background physics i.e. typi-cally does not affect the obtained results from the simulations.

The initial structure of single crystal copper is generated in sucha way that the shock wave propagation directions are h100i; h110iand h111i, whereas, other two orthogonal directions has beentaken as h010ih001i; h001i h1 �10i and h1 �10i h11 �2 i respectively.The details of the target samples, NEMD and MSST has been givenin Tables 1 and 2 respectively. The NEMD sample dimensions aretaken sufficient enough so that it can contain the entire wave-structure in lateral directions. Computational cell size for MSST-simulation has been so chosen such that stress, mass density andenergy density do not vary appreciably along the length of thecomputational domain (a requirement for MSST in which the stressand energy of a molecular dynamic simulation are constrained toobey the momentum and energy Hugoniot relations such thatthe simulation proceeds through the same thermodynamic statesas would occur in a steady shock) as well as not too small such thatboundary effects plays a role in the simulations.

The initial configuration of the samples (maintaining periodicboundary condition, PBC, along all three orthogonal directions)has been equilibrated at ambient temperature and pressure byusing isothermal isobaric, NPT ensemble in conjunction with stan-dard velocity-Verlet scheme of integration for �100 ps with thetimestep size of 1 fs.

To generate a uni-directional planar shock wave in direct-shock/NEMD simulations, so-called standard ‘momentum-mirror’method (details can be found elsewhere [27,11]) is applied overthe equilibrated initial configurations conjointly with micro-canonical ensemble, NVE, to achieve the conservation of energyduring the dynamics of shock propagation. We have explored a ser-ies of shock intensity, comparatively mild (shock wave velocity of�5 km/s; pressure and temperature of �20.45 GPa and �406.87 K,respectively) to plastically over-driven, strong shocks (e.g. shockvelocity of �11 km/s; pressure and temperature of �458.6 GPaand �16957.8 K, respectively) intensity. The size of the timestepduring all shock simulation is 1 fs unless otherwise mentioned.Please note that all the results reported in this work are calcu-lated/measured for the progressive shocks, no reflection or spalla-tion has been considered. All MD simulations has beenaccomplished by Large-scale Atomic/Molecular Massively ParallelSimulator (LAMMPS) [26] open-source package.

For analyzing the deformation pathway of the samples undershock compression in NEMD simulations, we have calculatedatomic strains [28,29] (decomposed in elastic and plastic compo-nent), centro-symmetry parameter (CSP) and also thermodynami-cal state variables (pressure, temperature and density) of thedeformed samples at various time instances during the passageof shocks. For atomic strain calculations, adaptive-common neigh-bor analysis (a-CNA), visualization of MD trajectories and for ren-dering purpose, the open source visualization tool (OVITO [30])has been utilized. For resolving spatially averaged profile of phys-ical properties, like, density, stress tensor, von-Mises stress, parti-cle velocity and temperature, we have done 1D and 2D binninganalyses with a bin size of 10 Åalong the shock propagation direc-tion. Temperature of the sample has been calculated by adoptingthe well-known equipartition theorem, i.e.

PNi¼1Ti ¼

� 13kB

PNi¼1mi v i2

x þ v i2y þ v i2

z

� �; where, N = total number of particles,

m = atomic mass and kB = Boltzmann constant. Per atom stress ten-sor (rij) has been computed by using the following expression,

rij ¼ � mv iv j þ 12

PNpn¼1 r1 iF1j þ r2iF2j

� � þ 12

PNbn¼1 r1iF1j þ r2iF2j

� �hþ

13

PNan¼1 r1iF1j þ r2iF2j þ r3iF3j

� �þ 14

PNdn¼1 r1iF1j þ r2iF2j þ r3iF3j þ r4iF4j

� �þKspace ria; Fib

� ��, where, m, v, r and F denotes atomic mass, velocityof atoms, position vector of atoms and force on each atoms, respec-

tively. During computation of shock temperature and pressure,center-of-mass translational velocity components(vCOM

i , i = x, yand z) has been subtracted off from the velocity vectors of eachparticle to avoid any additional contribution which occurs due tothe drift of the center-of-mass of the target sample during shockcompression. In the post processing he atomic velocity distribution(which eventually gives us the temperature distribution) has beenobserved by considering a thin 3D slice of the sample and observ-ing the temporal variation of atomic velocity distribution in thatthin slice. The instant of time at which the temperature starts todecrease after reaching the peak temperature is taken as theinstant in which the shock front has exited from the thin slice.Atomic velocity distribution at a much later instant of time (inwhich the temperature distribution is observed to be flat) is takenas the shock equilibrated state.

3. Results and discussion

Fig. 1 shows no significant differences between the shockpressure-density Hugoniot curve for single crystal Cu shock loadedalong three different crystal orientations: h100i; h110i and h111iup to �450 GPa. However, a general statement of absence of direc-tional anisotropy (as suggested by Chau et al. [31]., based on Us-up

relationship) cannot be suggested since Fig. 2 does show differencein the temperature-shock velocity Hugoniot curve for loadingalong the different crystal orientations. Moreover, differences inthe HEL point (Hugoniot elastic limit point indicating originationof dislocations and thereby initiation of plasticity) has also beenreported by previous researchers, in which 0.65 km/s has been pre-scribed for h100i direction [13] and 0.95 km/s prescribed for h111idirection [32].

This apparent difference in temperature, observed in Fig. 2,without any associated change in density can be attributed tothe orientation of the slip planes and the associated dislocation for-mation and interactions for the different crystal orientations.Energy imparted by the shock wave is utilized in activation ofthe dislocations which typically depends on many factors such asthe orientation of the crystal slip planes with the loading direc-tions; more energy is required for activation of dislocations alongthe slip planes for the h110i and the h111i direction in comparison

5 6 7 8 9 10 110.0

3.0x103

6.0x103

9.0x103

1.2x104

1.5x104

1.8x104

5.0 5.5 6.0 6.5 7.0

500

1000

1500

2000

2500 <100> <110> <111>

Tem

pera

ture

(K)

Shock velocity (km/s)

Fig. 2. Comparison has been made between the measured shock temperature forthe shocks along h100i; h110i and h111i-FCC as a function of shock velocity up to11.0 km/s. In the inset shock velocity up to 7.0 km/s has been zoomed for the visualclarity of the shock data points. The represented shock Hugoniot points in thisfigure are from MSST simulations, although NEMD simulations has been carried outto obtain this shock Hugoniot data. The differences between NEMD and MSSTresults are below 1%, which indicates the absence of profound post-shock relaxationof thermodynamical state variables up to 10 ns.

1.0 1.5 2.0 2.5

4

5

6

7

8

9

Piston velocity (km/s)

<100> <110> <111>

Vol

. stra

in (%

)

Fig. 3. Measured shock equilibrated volumetric strain of the shock-induceddeformed samples for shocks along h100i; h110i and h111i direction has beenplotted and compared as a function of piston velocity.

1.0 1.5 2.0 2.5

11

12

13

14

15

16

17

Piston velocity (km/s)

<100> <110> <111>

von-

Mis

es st

rain

(%)

Fig. 4. Measured shock equilibrated von-Mises strain of the shock-induceddeformed samples for shocks along h100i; h110i and h111i direction has beenplotted and compared as a function of piston velocity.

144 A. Neogi, N. Mitra / Computational Materials Science 135 (2017) 141–151

to the h100i direction. As a result, within the shock velocityregimes of 5.5–6.5 km/s in Fig. 2 the shock equilibrated tempera-ture of h110i and h111i direction is observed to be lower than thatof the h100i direction. In the regime between 8 and 9 km/s shockspeed, shock induced melting has been observed [19] resulting indirectional anisotropy. Differences in temperature has also beenobserved in this regime primarily because of reported zero super-heating along h100i direction compared to pronounced premelting(15–20%) along the h110i and h111i direction [19,33]. The natureof phase transition can also be pointed out to account for differ-ences in the shock-velocity-temperature Hugoniot curve [19]:melting along h100i is first order compared to being quasi-continuous along h110i and h111i direction.

At a global level, prior to melting and above the Hugoniot Elas-tic limit, these different plasticity events are manifested throughvariation of strains and stresses in the crystal which this manu-script primarily highlights. The plot for the shock equilibrated vol-umetric strain (expressed as ev ¼ 1

3 fexx þ eyy þ ezzg) in Fig. 3 shows

an increasing trend with increase in piston velocity. Apart fromthat, there are differences in volume compression when singlecrystal Cu is loaded along three different directions for the sameshock loading intensity. Infact the volume compression for theh111i direction is lowest compared to the other two directionsand for the h100i direction it is the highest (Maximum and mini-mum difference between [100] and [111] direction is �21.6%and �3.3% respectively; whereas maximum and minimum differ-ence between [110] and [111] direction is �13.48% and �0.35%respectively for the range of piston velocity of 1.0–2.5 km/s). Thistypically happens because of the availability of easy operative slipplanes for the h100i direction by virtue of simple geometric orien-tations of highly dense atomic planes. Since less energy is dissi-pated for slippage to occur, much of the energy is dissipated involume compression of the material eventually demonstratinghigh volumetric strain for the h100i direction. On the other hand,relatively large energy is required to activate slippage for the h111ishock loading direction and thereby volume compression is low forthis particular direction in comparison to the other directions. Vol-umetric strain plot (see Fig. 3) for h110i is observed to be inbetween the plots for h100i and h111i, however at higher shockintensity (e.g. piston velocity of 2.0 and 2.5 km/s) these strainsbecome equal to that of h111i direction.

Fig. 4 shows that the shock equilibrated von-Mises strain(which is the deviatoric component of the strain energy and

expressed in strains as evm ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie2xyþe2xzþe2yzð Þþðexx�eyyÞ2þðexx�ezzÞ2þðeyy�ezzÞ2

6

q)

also demonstrates an increasing trend for all the shock loadeddirections. However in contrast to that of the volumetric strainplot, the h100i direction shows a minimum value of von-Misesstrain compared to that of the other two directions (maximumand minimum difference between [100] and [111] direction is�19.22% and �7.39% respectively; whereas maximum and mini-mum difference between [110] and [111] direction is �0.73%and �0.44% respectively for the range of piston velocity of 1.0–2.5 km/s). This essentially highlights the fact that there are easyoperative slip systems for the h100i direction by virtue of simplegeometric orientations of highly dense atomic planes. Based ongeometric orientation of the atomic planes, h110i directiondemonstrates much easier propensity to form glissile slips planesin comparison to that of the h111i direction. However due to com-plex arrangement of the slip planes for the two low symmetrydirections, sessile junctions are formed and thereby at higher shock

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

0

10.5

0

0

0

0

12.5

13

15

15.5

Fig. 5. Figure (a)–(e) represents deformation microstructure (as obtained from adaptive-CNA analysis) of the shocked samples for the shocks along h100i, with pistonvelocity of 1.0, 1.2, 1.5, 1.8 and 2.0 km/s. Color, red, green, blue and white indicates hcp, fcc, bcc and non-definitive local crystal ordering of the deformed lattices respectively.Figure (f)–(j) demonstrates the corresponding distribution of von Mises strain of the shocked samples over the deformation matrix, for the shocks along h100i, with pistonvelocity of 1.0, 1.2, 1.5, 1.8 and 2.0 km/s. The color bar blue!white! red signifies lowest! highest strain value. For all the figures shock travels from left! right direction.(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 6. Figure represents zoomed one-atomic layer configuration of the formation ofstacking faults (SFs) in the deformation matrix of the shocked sample for the shocksalong h100i as observed from the perpendicular direction of the shock. Color, greenand red indicates fcc and hcp local lattice ordering of the lattices as obtained fromadaptive-CNA analysis. Red colored arrow indicates the SFs and Burger vectors ofthe faults has been mentioned. Pink colored dashed boxes indicates the core of thedislocations. (For interpretation of the references to colour in this figure legend, thereader is referred to the web version of this article.)

A. Neogi, N. Mitra / Computational Materials Science 135 (2017) 141–151 145

intensities the difference in von-Mises strain between the two lowsymmetry directions are reduced.

These differences between volumetric and von-Mises strain val-ues resembles differences in the inherent deformation mechanismswhich eventually give rise to different types of defects (e.g. disloca-tions, twining, shear band formation, stacking fault tetrahedron,etc. as discussed by the authors in other publication [15]). It shouldbe noted that since in these situations we have both slip as well assignificant volume change (as determinant of the deformation gra-dient is not equal to a value of 1), standard plasticity models suchas J2 plasticity or even Drucker-Prager plasticity may not be appli-cable as both these types of plasticity formulations relies on iso-choric behaviour (no volume change) of the material subjected toa particular loading situation. The spatially averaged temporalstrain plots demonstrate similar information as presented in previ-ous literature [34–36]: plastic waves do not have an elastic prede-cessor for shock loading along the h100i direction; plastic wave dohave an elastic predecessor along the h110i and h111i direction forlow intensity shocks and these difference between the two veloc-ities diminish at higher shock intensities. Detailed discussion onthe spatially averaged temporal evolution of von-Mises stressand strain along all the three directions of shock loading has beenpresented as supplementary information.

Typically the strain energy in a material is composed of two dif-ferent parts - volumetric/dilatational and distortional/deviatoric.The deviatoric part of the strain energy if expressed in strain vari-ables is referred to as the von-Mises strain; whereas if expressed instress components is referred to as von-Mises stress. The abovetwo Figs. 3 and 4 demonstrate the shock equilibrated volumetricand von-Mises strains respectively. The spatially-averaged tempo-ral evolution of these stress strain quantities for shock loading ofparticular intensities in different directions have been presentedin the following subsections.

3.1. Shock along h100i

Upto a shock pressure of �60 GPa (corresponding to a pistonvelocity of 1.2 km/s) deformation takes place through onset of slipsand associated thickening along with volume compression andvon-Mises strain of �6.5% and �11.5% respectively. Typically thestrain concentrations are observed to occur along the activated slipplanes. Increase in shock intensity results in gradual thickening ofthe dislocation activity along the activated slip planes which even-tually manifests as thickening of shear concentration regions (evi-dent from Fig. 5). The regions of highly localized strains results inorigination of stacking faults which eventually interact to formstable locks at the intersection points (see Fig. 6). These observa-

146 A. Neogi, N. Mitra / Computational Materials Science 135 (2017) 141–151

tions from numerical simulations matches well with patternsobserved in experimental investigations by Meyers et al. [4]. Thisphenomena of strain localization along the slip planes indicatesthat there is an increased possibility of formation of shear bandswhich has also been observed in the experimental investigationsof Meyers et al. [4].

Above �60 GPa of shock pressure when the sample exceeds vol-ume and shear strain of �6.75% and �12% respectively mode ofdeformation changes from slip! twining to accommodate theadditional strain. Fig. 5(d) shows that at a piston velocity of1.5 km/s, significant thickening of strain localized bands could beobserved along with formation of lock regions which eventuallyresults in formation and elongation of shear band like lath (elon-gated grain like regions). Twinning mechanism has been observedwithin these grain regions (average grain size of 20–25 nm) at aplastic strain of around 22%. In twinning mechanism the slip/glideof a number of atomic layers eventually creates a mirror likeboundary separating two crystals sharing some lattice points (seeFig. 7). Our analysis of cross sections perpendicular to shock direc-tion shows twin formation at a piston velocity of 1.2 km/s. Detailedanalysis of nucleation and evolution of the twins in shock loadedCu single crystals has been presented elsewhere by the authors.Rapid decrease of the temperature of the material just behind

Fig. 7. Figure on the right side represents a slice of the deformation microstructure forepresents the zoomed one-atomic layer configuration of a portion of the right sided slicecolored hypothetical lines are the guide to eye to visualize twin configuration and twin plreferences to colour in this figure legend, the reader is referred to the web version of th

1.0 1.5 2.0 2.5

1000

2000

3000

4000

5000

6000

0 5 10 15 20 25 30

400

600

800

1000

1200

Tshock frontTshock equilibrated

Tem

pera

ture

(K)

Piston velocity (km/s)

Tem

pera

ture

(K)

MD time (ps)

<100> shockPiston velocity = 1.0 km/s

(a)

Fig. 8. (a) Measured temperature at shock-front and shock equilibrated condition has bcharacteristic profile of temperature at a slice of 10 nm in the target sample has been shshock along h100i. (b) Atomic velocity distribution at shock front and at shock equilibrplotted and fitted with Maxwell-Boltzmann distribution function.

the shock front creates a rapid quench like situation (shock tem-perature 2221.4 K relaxed to the shock equilibriated temperatureof 1188.41 K within approximately 10 ps; rate of cool-ing = 103.3 K/ps) which may also assist in slip to twin mechanismformation. Slip to twin transition is typically observed at 60 GPapressure from these simulations, which is quite close to pressuresof 55 GPa obtained from experimental observations by Meyerset al. [4] under similar loading situations.

After exceeding the shock pressure of �85 GPa, i.e. shear strainof �12.5%, the shock-wave energy has been observed to be accom-modated through structural phase transition before the shock-induced melting at the shock pressure regime of �180–190 GPawhich causes volumetric strain of �16.7%. Details of these phasetransition associated with shock loading have been discussed inother paper by the authors [37].

Fig. 5 demonstrates that the pattern of observance of strain dis-tribution matches with the patterns observed in the deformationmicrostructure plots. Typically the hcp regions in between thefcc regions denotes the slip planes and the strains are observedto be higher along the slip planes. In Fig. 5d) bcc regions areobserved with a middle region showing a conglomerate of fccand hcp structure. A closer look into that region demonstrates for-mation of twins as shown in Fig. 7. Similarly the strains are

r the shock along h100i with piston velocity of 1.2 km/s. Left sided boxed figure, which demonstrate the formation of twins at (111) atomic planes. Black and yellowanes respectively. Black arrow indicates the twin direction. (For interpretation of theis article.)

0.00 0.63 1.25 1.88 2.50 3.130.0

500.0

1.0k

1.5k

2.0k

2.5kShock equilibrated stateShock front Fit to MB distribution Fit to MB distribution

Num

ber o

f ato

ms

atomic velocity (km/s)

Piston velocity=2.0 km/sShock direction <100>

(b)

een plotted as a function of piston velocity for the shocks along h100i. In the insetown along with simulation time up to 30 ps for the piston velocity of 1.0 km/s andated section for the shocks along h100i with piston velocity of 2.0 km/s, has been

A. Neogi, N. Mitra / Computational Materials Science 135 (2017) 141–151 147

observed to be higher at the twin boundaries within the fcc-hcpconglomerate region compared to the bcc regions. It should benoted that adaptive CNA typically shows some regions as BCC, adetailed exposition of the possibility of structural phase transitionin shock loaded Cu (using both classical MD [37] and ab� initioMD) has been presented by the authors in another journal. In anutshell, it can thereby be demonstrated that behind the shockfront stress relaxation is primarily confined along the slip planesand no spatial diffusion of the relaxation process is observed. Withregards to the strain, the redistribution of strain if any is also con-fined within the slip planes and eventually the shock equilibratedvon-Mises strain increases with increase in piston velocity.

Fig. 8(a) shows temperature distribution at the shock front andalso the shock equilibrated temperature for different piston veloc-ities. Along this h100i direction it can be observed that as the lead-ing shock front enters a particular spatial location, the temperatureof that region is increased significantly due to thermal instabilitiesintroduced by the shock. This increase in temperature is primarilyassociated with increased inter-atomic vibrations of the atoms as ashock wave enters a particular spatial regime. However, as theshock wave crosses that spatial location, the temperature of thatparticular region decreases and eventually a shock equilibratedtemperature is obtained. The atomic velocity distribution has beenplotted in Fig. 8(b) and it has been observed to obey Maxwell-Boltzmann distribution in shock equilibrated state as well as inthe shock front as well. It should be noted in this regard that thereare previous literature reviews which demonstrates non-Maxwellian distribution of temperature at shock front. However,it needs to be pointed out here that all of these previous literature[38,39] considers poly-atomic molecules (such as gasses andmacromolecular polymers). In polyatomic molecules subjected tohigh energy phenomenon, significant amount of applied energyis consumed in vibration of the molecules. In fact, this vibrationdegrees of freedom may be the cause for non-Maxwellian distribu-

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

Fig. 9. Figure (a)–(e) represents deformation microstructure (as obtained from adaptivvelocity of 1.0, 1.5, 1.8, 2.0 and 2.5 km/s. Color, red, green, blue and white indicates hcp, fcFigure (f)–(j) demonstrates the corresponding distribution of von Mises strain of the shovelocity of 1.0, 1.5, 1.8, 2.0 and 2.5 km/s. The color bar blue!white! red signifies lowes(For interpretation of the references to colour in this figure legend, the reader is referre

tion of temperature. The type of bonds observed in macro-molecules and gasses are usually covalent in nature compared tometallic bonds in metals. Thereby, there exists a possibility thatvibration may not be the major degrees of freedom which con-sumes the applied energy to the system in an event of shock load.The absence of significant role played by vibration degrees of free-dommay be a reason for Maxwellian distribution of velocities evenat the shock front for metals under shock loads. The inset of theFig. 8(a) shows a representative temperature profile of a specificspatial bin for a piston velocity of 1.0 km/s. This relaxation of tem-perature (observed between the shock front and the region behindthe front) is accommodated by formation of different types of plas-ticity mechanisms, primarily, dislocation interactions and twining.It can also be observed that increase in piston velocity results in anincrease in difference between the shock front and shock equili-brated temperature.

3.2. Shock along h110i

For the shock loading direction of [110], deformation (seeFig. 9) starts with simple operation of the slip of the operative slipsystems (three slip systems, [�110](�1 �11), [�10 �1](�1 �11) and [�1 �10](�111)) up to 20 GPa, i.e. volumetric and von-Mises strain of�4.75% and �12.5% respectively. At higher shock intensity, e.g.Up = 1.5 km/s (volumetric and von-Mises strain of �4.45% and�14.49% respectively) and above but before melting (primarily,identified at Up = 3.5 km/s), significant accumulation of residualvon-Mises stress behind the shock front has been identified whicheventually helps the dislocations for climbing (resulting to jog for-mation) or form perfect dislocations, which glide in other slipplanes, i.e. cross-slip mechanism is observed. However, Fig. 9 ofthe contour of calculated atomic shear strain shows a complicatedlocalization of slips along the operative slip planes resulting in an

0

12

0

13

0

15

0

0

16

20

e-CNA analysis) of the shocked samples for the shocks along h110i, with pistonc, bcc and non-definitive local crystal ordering of the deformed lattices respectively.cked samples over the deformation matrix, for the shocks along h100i, with pistont! highest strain value. For all the figures shock travels from left! right direction.d to the web version of this article.)

1.0 1.5 2.0 2.5

1000

2000

3000

4000

0 5 10 15 20 25 30

400

600

800

1000

1200

Tshock frontTshock equilibrated

Tem

pera

ture

(K)

Piston velocity (km/s)

Tem

pera

ture

(K)

MD time (ps)

<110> shockPiston velocity = 1.0 km/s

0.0 0.5 1.0 1.5 2.0 2.50.0

500.0

1.0k

1.5k

2.0k

2.5k

Piston velocity=2.0 km/sShock direction <110>

Shock equilibrated stateShock front Fit to MB distribution Fit to MB distribution

Num

ber o

f ato

ms

atomic velocity (km/s)

(b)

(a)

Fig. 10. Measured temperature at shock-front and shock equilibrated condition has been plotted as a function of piston velocity for the shocks along h110i. In the insetcharacteristic profile of temperature at a slice of 10 nm in the target sample has been shown along with simulation time up to 30 ps for the piston velocity of 1.0 km/s andshock along h110i. (b) Atomic velocity distribution at shock front and at shock equilibrated section for the shocks along h110i with piston velocity of 2.0 km/s, has beenplotted and fitted with Maxwell-Boltzmann distribution function.

148 A. Neogi, N. Mitra / Computational Materials Science 135 (2017) 141–151

approximate spatially uniform strain concentration over the entireshock equilibrated region. It should be noted that the strain pat-terns observed are not similar to that of the deformationmicrostructure. This observation is in stark contradiction to thepatterns observed for shock induced deformation of Cu crystalsin h100i direction. In the h100i direction the strains at one spatiallocation are observed to be primarily concentrated along the zonesof slippage whereas for the h110i direction there occurs a reorien-tation/delocalization of strain distributions at one spatial locationand the strains are not necessarily concentrated along the opera-tive slip planes. This delocalized nature of the strains at the shockequilibrated regions hints at formation of complex dislocationactivities in the samples thereby indicating extensive work-hardening in the sample with less probability of strain localizedbands. It should be pointed out that for shock loading along thisparticular direction, based on the complexity of the geometricalorientation only two slip planes are found to be operative in whichthree slip systems are activated intersecting the [110] surface par-allel to each other along the h1�11i axis. The complicated geometri-cal distribution of operative slip planes eventually results information of multiple cross-slips and Jogs (refer experimentalobservations by Meyers et al. [4] and also numerical simulationresults by the authors, currently unpublished), which eventuallyresults in relaxation of von-Mises stress behind the shock front.

Temperature distribution at the shock front and at the shockequilibrated region for shock loading along h110i direction isshown in Fig. 10(a) for different piston velocities. The atomic veloc-ity distribution has been plotted in Fig. 10(b) and it has beenobserved to obey Maxwell–Boltzmann distribution in shock equili-brated state and at shock front as well similar to the observation inthe h100i direction. An interesting difference is observed on com-parison of this plot with that of a similar plot for the h100i direc-tion. For the h100i direction, the shock front temperature is higherthan the shock equilibrated temperature. On the other hand for theh110i direction since the elastic wave travels faster than the plas-tic wave (for low shock intensities) the shock front temperature issimilar to the temperature in the elastic wave regime; whereas theshock equilibrated temperature is similar to the temperature of theplastic wave regime. Interestingly instead of a lower shock equili-brated temperature compared to the shock front temperature asobserved for the h100i direction; the shock equilibrated tempera-ture is observed to be higher than the shock front temperature. Asthe shock wave crosses a particular spatial regime, there is anincreased vibration of the atoms due to which there is a rise intemperature (please see the inset of Fig. 10(a)) but still at that

instant the dislocation activities have not initiated in that regime.As the dislocation mechanisms are initiated (identified by the plas-tic wave front) the temperature of the spatial region is furtherincreased. At higher piston velocities, the difference between theshock front temperature and the shock equilibrated temperatureis reduced, the reason for which may be traced to the reductionin velocity differences between the elastic and the plastic wave.An inset in the figure shows a representative temperature profileof a specific spatial bin for a particular shock intensity.

3.3. Shock along h111i

From Fig. 11 it is apparent that the plastic wave front in the caseof h111i direction is not very discrete in comparison to what isobserved for the h110i direction. Thereby the plateau formed todenote separation distance between the elastic and plastic wavesfronts is not entirely smooth or flat. Typically it is known thath111i direction exhibits symmetry with three possible slip planes,namely (�111), (�1 �11) and (1 �11). As the first two planes within theavailable slip planes are inclined at an angle of 60� and the thirdone is parallel with the [111] surface, hence the observed slipplanes can be observed to make a triangular arrangement alongthe view directions of h1�10i and h11 �2 i (as observed in Fig. 11).In fact, this type of triangular arrangement of slip planes is alsoresponsible for a non-discrete nature of the plastic wave front,demonstrated in the deformation matrix contour of calculatedatomic von-Mises strain. Due to insufficiency of favourable slipplanes in such complex geometric orientation like h110i (unlikeh100i direction), comparatively lesser slip operation (manifestedby regions of high strains) can be visualized on the outer surfaceof deformed microstructure along h111i direction in Fig. 11.

Quite different from the h110i direction, strain concentrationregions are not uniform over the entire shock equilibrated regionbut can be distinctively identified at the junctions or the point ofdislocation reactions. The patterns of strain concentrationsobserved in the samples also differs from that of the h100i direc-tion since strains are not entirely concentrated on slip planes asobserved from the deformation microstructure plots. Therebythere is spatial strain delocalization of the sample in comparisonto that of no spatial delocalization for the h100i direction but theamount is smaller in comparison to the h110i direction. The trian-gular distribution of strains eventually result in complex disloca-tion interactions such as the stacking fault tetrahedron [15]eventually leading to stress relaxation in the samples. Because ofthe complex orientation of the slip planes for shock loading in this

1.0 1.5 2.0 2.51000

2000

3000

4000

5000

0 5 10 15 20 25 30300

600

900

1200

0 5 10 15 20 25 30300

600

900

1200

1500

1800

0 5 10 15 20 25

600

1200

1800

2400

3000

Tshock frontTshock equilibrated

Tem

pera

ture

(K)

Piston velocity (km/s)

Tem

pera

ture

(K)

MD time (ps)

Up = 1.0 km/s Up = 1.5 km/s

MD time (ps)

Up = 2.0 km/s

MD time (ps)

0.0 0.5 1.0 1.5 2.0 2.50.0

500.0

1.0k

1.5k

2.0k

2.5k

Piston velocity=2.0 km/sShock direction <111>

Shock equilibrated stateShock front Fit to MB distribution Fit to MB distribution

Num

ber o

f ato

ms

atomic velocity (km/s)

(b)

(a)

Fig. 12. Measured temperature at shock-front and shock equilibrated condition has been plotted as a function of piston velocity for the shocks along h111i. In the insetcharacteristic profile of temperature at a slice of 10 nm in the target sample has been shown along with simulation time up to 30 ps for the piston velocity of 1.0 km/s andshock along h111i. (b) Atomic velocity distribution at shock front and at shock equilibrated section for the shocks along h111i with piston velocity of 2.0 km/s, has beenplotted and fitted with Maxwell-Boltzmann distribution function.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

0

13

0

13

0

15

0

15

0

16.5

Fig. 11. Figure (a)–(e) represents deformation microstructure (as obtained from adaptive-CNA analysis) of the shocked samples for the shocks along h111i, with pistonvelocity of 1.0, 1.5, 1.8, 2.0 and 2.5 km/s. Color, red, green, blue and white indicates hcp, fcc, bcc and non-definitive local crystal ordering of the deformed lattices respectively.Figure (f)–(j) demonstrates the corresponding distribution of von Mises strain of the shocked samples over the deformation matrix, for the shocks along h100i, with pistonvelocity of 1.0, 1.5, 1.8, 2.0 and 2.5 km/s. The color bar blue!white! red signifies lowest! highest strain value. For all the figures shock travels from left! right direction.(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

A. Neogi, N. Mitra / Computational Materials Science 135 (2017) 141–151 149

direction (h111i) and thereby the possibilities of complex disloca-tion interactions, the relaxation of stress occurs over a small spatialregion in comparison to that for the h110i direction. Highest val-ues of maximum von-Mises stresses are obtained for this particulardirection of shock loading in comparison to the other two direc-tions h100i and h110i.

The temperature distribution at the shock front and at the shockequilibrated region for shock loading along the h111i direction isshown in Fig. 12(a) for different piston velocities. In Fig. 12(b),the atomic velocity distribution has been shown for the shockalong h111i with piston velocity of 2.0 km/s and the distribution

has been observed to obey Maxwell-Boltzmann in both the shockequilibrated state as well as at the shock front. Quite contrary tothe behaviour observed in the h100i and the h110i directions,the shock front temperature curve intersects the shock equili-brated temperature curve in between 1.8 and 2.0 km/s pistonvelocity. At and below of piston velocity of 1.8 km/s, in this partic-ular shock loading orientation, the plastic wave front lags its elasticpredecessor (similar to that of the h110i directions) the shock fronttemperature is taken as the temperature of the elastic wave andthe shock equilibrated temperature is taken as the temperatureof the plastic wave. AS shown in the inset of Fig. 12(a), as soon

150 A. Neogi, N. Mitra / Computational Materials Science 135 (2017) 141–151

as the elastic wave passes through a particular spatial regime, thetemperature is increased. This increase in temperature is followedby a time instant in which it is decreased. As the dislocation activ-ities are initiated with the arrival of the plastic wave the tempera-ture in that spatial regime is again increased. However it isinteresting to note (quite contrary to the observations in theh110i direction) that the differences in temperatures at twoinstants are not very significant. For piston velocity of 1.0 km/sthere is no difference; for 1.5 km/s there is a difference which againis not very significant for 1.8 km/s. However, the pattern observedis similar to that observed for the h110i direction. For piston veloc-ity above 1.8 km/s the shock front temperature becomes more thanthe shock equilibrated temperature. Interestingly, this nature issimilar to the patterns observed for the h100i direction. At this pis-ton velocity the von-Mises strains also do not show any lag inspeed between the plastic wave-front and it’s elastic predecessor.The behavioural difference between the temperature plots for thetwo low symmetry direction is primarily because of complex ori-entation of the slip planes and associated dislocation reactions inthese two directions. The two insets in the figure shows represen-tative temperature profiles of a specific spatial bin at 1 km/s(where the shock equilibrated temperature is slightly above theshock front temperature) and 2 km/s (where the shock equili-brated temperature is lower than the shock front temperature).

4. Conclusions

Shock loading of Cu in different directions (namely h100i; h110iand h111i) presents directional anisotropy. It may not be apparentfrom the pressure-density Shock Hugoniot profile which matchwith experimental data of polycrystals; but distinct differencesare observed in the temperature-shock velocity Hugoniot curvefor large scale atomistic simulations along the three directions con-sidered in this manuscript. The distribution of strains in the sam-ples are observed to be confined along the slip planes for theh100i directions whereas no such strain localization could beobserved for the h110i direction even though a complex patternof stacking faults are observed in that direction. Along the h111idirection the strains are localized along a triangular region. Tem-perature measurements of the shock front region and the shockequilibrated region also shows interesting patterns for the threeshock loading directions considered in this manuscript. Quite con-trary to the behavior of polyatomic gasses, atomic velocity distri-bution at the shock front and at the shock equilibrated region areobserved to follow Maxwell-Boltzmann distribution for all thethree different directions of shock loading considered in thismanuscript.

Appendix A. Supplementary material

Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.commatsci.2017.04.009.

References

[1] D.H. Kalantar, J. Belak, E. Bringa, K. Budil, M. Caturla, J. Colvin, M. Kumar, K.Lorenz, R. Rudd, J. Stölken, high-strain-rate lattice response of shockedmaterials, Phys. Plasmas 10 (5) (2003) 1569–1576.

[2] D.H. Kalantar, J. Belak, J. Colvin, A. Lubcke, S. Pollaine, B. Remington, A. Allen, A.Loveridge, J. Wark, M. Meyers, high strain rate materials experiments, Bull. Am.Phys. Soc. (USA) 46 (4) (2001) 69.

[3] A. Loveridge-Smith, A. Allen, J. Belak, T. Boehly, A. Hauer, B. Holian, D. Kalantar,G. Kyrala, R. Lee, P. Lomdahl, Anomalous elastic response of silicon to uniaxialshock compression on nanosecond time scales, Phys. Rev. Lett. 86 (11) (2001)2349.

[4] M.A. Meyers, F. Gregori, B. Kad, M. Schneider, D. Kalantar, B. Remington, G.Ravichandran, T. Boehly, J. Wark, Laser-induced shock compression of

monocrystalline copper: characterization and analysis, Acta Mater. 51 (5)(2003) 1211–1228.

[5] M. Meyers, D. Benson, O. Vöhringer, B. Kad, Q. Xue, H.-H. Fu, Constitutivedescription of dynamic deformation: physically-based mechanisms, Mater. Sci.Eng.: A 322 (1) (2002) 194–216.

[6] M. Schneider, B. Kad, M. Meyers, F. Gregori, D. Kalantar, B. Remington, Laser-induced shock compression of copper: orientation and pressure decay effects,Metall. Mater. Trans. A 35 (9) (2004) 2633–2646.

[7] L. Murr, Residual microstructure-mechanical property relationships in shock-loaded metals and alloys, in: ShockWaves and High-Strain-Rate Phenomena inMetals, Springer, 1981, pp. 607–673.

[8] R. Wright, D. Mikkola, High strain rate deformation of mo and Mo-33Re byshock loading: Part II. Rates of defect generation and accumulation of plasticstrain, Metall. Trans. A 16 (5) (1985) 891–895.

[9] J. Rivas, S. Quinones, L. Murr, Hypervelocity impact cratering: microstructuralcharacterization, Scripta Metall. Mater. 33 (1) (1995) 101–107.

[10] M. Mogilevskii, L. Bushnev, Deformation structure development in al and cusingle crystal on shock-wave loading up to 50 and 100 GPa, Combust., Explos.,Shock Waves 26 (2) (1990) 215–220.

[11] B.L. Holian, P.S. Lomdahl, Plasticity induced by shock waves in nonequilibriummolecular-dynamics simulations, Science 280 (5372) (1998) 2085–2088.

[12] T.C. Germann, B.L. Holian, P.S. Lomdahl, D. Tanguy, M. Mareschal, R. Ravelo,Dislocation structure behind a shock front in fcc perfect crystals: atomisticsimulation results, Metall. Mater. Trans. A 35 (9) (2004) 2609–2615.

[13] E. Bringa, J. Cazamias, P. Erhart, J. Stölken, N. Tanushev, B. Wirth, R. Rudd, M.Caturla, Atomistic shock Hugoniot simulation of single-crystal copper, J. Appl.Phys. 96 (7) (2004) 3793–3799.

[14] B. Cao, E.M. Bringa, M.A. Meyers, Shock compression of monocrystallinecopper: atomistic simulations, Metall. Mater. Trans. A 38 (11) (2007) 2681–2688.

[15] A. Neogi, N. Mitra, Evolution of dislocation mechanisms in single-crystal cuunder shock loading in different directions, Modell. Simul. Mater. Sci. Eng. 25(2) (2017) 025013.

[16] Y. Mishin, M. Mehl, D. Papaconstantopoulos, A. Voter, J. Kress, Structuralstability and lattice defects in copper: ab initio, tight-binding, and embedded-atom calculations, Phys. Rev. B 63 (22) (2001) 224106.

[17] S. Ogata, J. Li, S. Yip, Ideal pure shear strength of aluminum and copper, Science298 (5594) (2002) 807–811.

[18] D. Jang, X. Li, H. Gao, J.R. Greer, Deformation mechanisms in nanotwinnedmetal nanopillars, Nat. Nanotechnol. 7 (9) (2012) 594–601.

[19] Q. An, S.-N. Luo, L.-B. Han, L. Zheng, O. Tschauner, Melting of cu underhydrostatic and shock wave loading to high pressures, J. Phys.: Condens.Matter 20 (9) (2008) 095220.

[20] A. He, S. Duan, J. Shao, P. Wang, S. Luo, Local and bulk melting of shockedcolumnar nanocrystalline Cu: dynamics, anisotropy, premelting, superheating,supercooling, and re-crystallization, J. Chem. Phys. 139 (7) (2013) 074502.

[21] J. Asay, C. Hall, K. Holland, M. Bernard, W. Stygar, R. Spielman, S. Rosenthal, D.McDaniel, D. Hayes, M.D. Furnish, et al., Isentropic compression of iron withthe z accelerator, AIP Conf. Proc., vol. 505, AIP, 2000, pp. 1151–1154.

[22] E.J. Reed, L.E. Fried, J. Joannopoulos, A method for tractable dynamical studiesof single and double shock compression, Phys. Rev. Lett. 90 (23) (2003)235503.

[23] A. Neogi, N. Mitra, Shock induced phase transition of water: moleculardynamics investigation, Phys. Fluids 28 (2) (2016) 027104.

[24] A. Neogi, N. Mitra, Shock compression of polyvinyl chloride, J. Appl. Phys. 119(16) (2016) 165903.

[25] T.-R. Shan, R.R. Wixom, A.E. Mattsson, A.P. Thompson, Atomistic simulation oforientation dependence in shock-induced initiation of pentaerythritoltetranitrate, J. Phys. Chem. B 117 (3) (2013) 928–936.

[26] S. Plimpton, Fast parallel algorithms for short-range molecular dynamics, J.Comput. Phys. 117 (1) (1995) 1–19.

[27] A. Neogi, N. Mitra, On shock response of nano-void closed/open cell coppermaterial: non-equilibrium molecular dynamic simulations, J. Appl. Phys. 115(1) (2014) 013504.

[28] M. Falk, J. Langer, Dynamics of viscoplastic deformation in amorphous solids,Phys. Rev. E 57 (6) (1998) 7192.

[29] F. Shimizu, S. Ogata, J. Li, Theory of shear banding in metallic glasses andmolecular dynamics calculations, Mater. Trans. 48 (11) (2007) 2923–2927.

[30] A. Stukowski, Visualization and analysis of atomistic simulation data withOvito–the open visualization tool, Modell. Simul. Mater. Sci. Eng. 18 (1) (2009)015012.

[31] R. Chau, J. Stölken, P. Asoka-Kumar, M. Kumar, N. Holmes, Shock Hugoniot ofsingle crystal copper, J. Appl. Phys. 107 (2) (2010) 023506.

[32] D. Seif, G. Po, R. Crum, V. Gupta, N.M. Ghoniem, Shock-induced plasticity andthe Hugoniot elastic limit in copper nano films and rods, J. Appl. Phys. 115 (5)(2014) 054301.

[33] A. He, P. Wang, J. Shao, S. Duan, F. Zhao, S. Luo, Solid-liquid phase transitions insingle crystal Cu under shock and release conditions, J. Appl. Phys. 115 (14)(2014) 143503.

[34] T.C. Germann, B.L. Holian, P.S. Lomdahl, R. Ravelo, Orientation dependence inmolecular dynamics simulations of shocked single crystals, Phys. Rev. Lett. 84(23) (2000) 5351.

[35] D. Tanguy, M. Mareschal, P.S. Lomdahl, T.C. Germann, B.L. Holian, R. Ravelo,Dislocation nucleation induced by a shock wave in a perfect crystal: moleculardynamics simulations and elastic calculations, Phys. Rev. B 68 (14) (2003)144111.

A. Neogi, N. Mitra / Computational Materials Science 135 (2017) 141–151 151

[36] T.C. Germann, B.L. Holian, P.S. Lomdahl, D. Tanguy, M. Mareschal, R. Ravelo,Dislocation structure behind a shock front in fcc perfect crystals: atomisticsimulation results, Metall. Mater. Trans. A 35 (9) (2004) 2609–2615.

[37] A. Neogi, N. Mitra, Body-Centered Phase of Shock Loaded Cu, Also Available at:arXiv:1607.06202.

[38] H.K. Sen, The non-Maxwellian distribution in a shock front and the anomaly ofthe chromospheric temperature, Phys. Rev. 92 (4) (1953) 861.

[39] J. Raymond, P.A. Isenberg, J. Laming, Non-Maxwellian proton velocitydistributions in nonradiative shocks, Astrophys. J. 682 (1) (2008) 408.