Calculating shock Hugoniots with LAMMPScradpdf.drdc-rddc.gc.ca/PDFS/unc244/p804516_A1b.pdf ·...

Calculating shock Hugoniots with LAMMPS

Grant W.J. McIntosh DRDC – Valcartier Research Centre

Defence Research and Development Canada Scientific Report DRDC-RDDC-2016-R152 August 2016

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© Her Majesty the Queen in Right of Canada, as represented by the Minister of National Defence, 2016

© Sa Majesté la Reine (en droit du Canada), telle que représentée par le ministre de la Défense nationale, 2016

Abstract

The molecular modelling program LAMMPS was used to predict the shock Hugoniot relationship (pressure-volume) of aluminum and two explosives, RDX and PETN. The method worked successfully for aluminum but uncertainties in the intermolecular potential and simplifications in crystal model yielded results of the right form and order of magnitude but inaccurate for the two explosives.

Significance to defence and security

High-speed impacts that happen, for example, when a bullet strikes a bulletproof vest or when a shaped charge jet strikes an armoured vehicle, are of great interest to armed forces everywhere. For predicting and understanding the effects of such impacts, the physical response always involves a shock wave whose properties are determined by what is called a shock Hugoniot relation. The Hugoniot relation can be measured experimentally or, as demonstrated in this report, can be calculated theoretically (with care) based on fundamental atomic or molecular properties. For new armour materials, small-scale experiments can generate the parameters for these fundamental properties and hence, via careful calculations, a shock Hugoniot can be determined to predict large-scale behaviour during impacts without the need for large-scale expensive trials.

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Résumé

Le programme de modélisation moléculaire LAMMPS a été utilisé pour prédire la relation de choc (pression-volume) d’Hugoniot de l’aluminium et de deux explosifs, le RDX et la PETN. La méthode a bien réussi pour l’aluminium. Toutefois, des incertitudes dans le potentiel intermoléculaire et des simplifications du modèle de cristal ont donné des résultats sous une forme correcte et dans le bon ordre de grandeur, mais imprécis dans le cas des deux explosifs.

Importance pour la défense et la sécurité

Des impacts à haute vitesse qui se produisent, par exemple quand une balle frappe une veste pare-balle ou quand le jet d’une charge creuse frappe un véhicule blindé, sont partout d’un grand intérêt pour les forces armées. Pour pouvoir prédire et comprendre les effets de tels impacts, une réponse physique doit toujours comporter la production d’une onde de choc dont les caractéristiques seront déterminées par ce qu’il est convenu d’appeler la relation de choc d’Hugoniot. On peut mesurer cette relation expérimentalement ou théoriquement (en procédant avec soin), comme il est démontré dans ce rapport, à partir des propriétés fondamentales à l’échelle atomique ou moléculaire. Pour de nouveaux matériaux de blindage, des expériences à petite échelle peuvent fournir les paramètres des propriétés fondamentales. Par des calculs minutieux, on peut ensuite déterminer la relation de choc d’Hugoniot pour prédire le comportement à grande échelle au moment de l’impact sans que des expériences coûteuses et à grande échelle soient nécessaires.

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Table of contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i Significance to defence and security . . . . . . . . . . . . . . . . . . . . . . i Résumé . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Importance pour la défense et la sécurité . . . . . . . . . . . . . . . . . . . . ii Table of contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv List of tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . vi 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3 Modelling a shock wave . . . . . . . . . . . . . . . . . . . . . . . . 3 4 Comments on the method and approach . . . . . . . . . . . . . . . . . . 10 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 References/Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 13

LAMMPS input files . . . . . . . . . . . . . . . . . . . . . . . 15 Annex AList of symbols/abbreviations/acronyms/initialisms . . . . . . . . . . . . . . . . 21

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List of figures

Figure 1: Shock wave in Al at 500 m/s impact (eam fs model). . . . . . . . . . . . 4

Figure 2: Shock wave in Al at 500 m/ s impact (eam alloy model). . . . . . . . . . 4

Figure 3: Shock wave at 500 m/s impact (lj model). . . . . . . . . . . . . . . . 5

Figure 4: Shock wave in Al at 3000 m/s impact (eam fs model). . . . . . . . . . . 5

Figure 5: Shock wave in Al at 3000 m/s impact (eam alloy model). . . . . . . . . . 6

Figure 6: Shock wave in Al at 3000 m/s impact (lj model). . . . . . . . . . . . . 6

Figure 7: Pressure – velocity relations for Al. . . . . . . . . . . . . . . . . . . 7

Figure 8: Pressure – Volume for PETN. . . . . . . . . . . . . . . . . . . . . 8

Figure 9: Pressure – Volume for RDX. . . . . . . . . . . . . . . . . . . . . 8

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List of tables

Table 1: Shock pressures generated by various models. . . . . . . . . . . . . . 7

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Acknowledgements

The author wishes to thank Dr. Dennis Nandlall for inspiring this work and moral support during its course.

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1 Introduction

The understanding of the physical world can be approached on many levels, from detailed quantum mechanical calculations at the subatomic level (for a few atoms) through macroscopic bulk property descriptions at the engineering level. As one examines a system at coarser and coarser scales, the more detailed and presumably more precise descriptions are replaced by approximations. This is normal as the mathematical complexity required to model a large system at a detailed level becomes unfeasible. This is still the situation today in most practical cases.

The arrival of powerful computational capacities of modern computers has modified this situation. A full-scale calculation of say 1023 atoms is still impossible but simulations involving 106 to 109 + atoms on massively parallel clusters of computers are now possible. With this number of atoms, bulk properties can be calculated.

Within this work, we will examine a limited case of a detailed calculation to determine a bulk property. This will be done using a freely available computer program for molecular modelling called LAMMPS, literature descriptions of the atomic structure of a few materials and the interatomic potential between the atoms of the structure (the potential being either found in the literature or based upon reasonable estimates). The particular bulk property to be determined is the shock response of the material, commonly referred to as the shock Hugoniot relation.

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2 Background

Molecular modelling has been around since the 1950’s and by 1996, was being identified as an advance in material science of interest to the defence community (Porter, 1996). The field of molecular modelling is quite vast and general surveys of the field exist as textbooks, for example, Leach (2001). The aim in the present work is modest. It is simply to see whether an understanding of a material at the atomic or molecular level can yield a useful result for engineering or macroscopic purposes. Within classical (as opposed to quantum mechanical) molecular modelling, given an interatomic potential energy relation, it is in principle possible to generate elastic constants (Ashcroft, 1972, Chap 22). Alternatively, given the same interatomic potential, again it is in principle possible to generate the partition function and hence p-v relationships (Pathria, 1972, Chap 3). In practice, it is not possible to do the necessary calculations analytically except in very limited cases.

The development of faster computers with more memory as well as general purpose simulation programs for molecular modelling has made the numerical calculation of the elastic constants or the partition function possible for a large enough sample of material to yield potentially accurate results. The key to calculation is an accurate interatomic potential. The most accurate potentials are generally those between two interacting items that are relatively independent (e.g., in a low-density gas). As the boundaries between the two interacting items start to blur (e.g., in metals), the simple potentials become increasing less accurate.

One computer program that is now widely available at no cost is called LAMMPS. The best succinct description of what LAMMPS does is found in its documentation: “In the most general sense, LAMMPS integrates Newton’s equations of motion for collections of atoms, molecules, or macroscopic particles that interact via short- or long-range forces with a variety of initial and/or boundary conditions.” The numerical algorithms at the core of LAMMPS are described in Plimpton (1995).

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3 Modelling a shock wave

Modelling a shock wave in a material is relatively straightforward and an approach closely based upon Guerrero-Miramontes (2013) was used. The material is modelled as a crystal using an experimentally determined structure (e.g., a face centered cubic lattice) with atoms (in the case of an element) or molecules (in the case of a molecular solid) referenced to the lattice points. A thin layer at one surface is then moved in a sustained piston-like fashion towards the bulk of the material. The material compresses and a shock wave is generated in response. The pressure in the bulk material is calculated by standard statistical mechanics and thermodynamic considerations.

The simplest materials to model are crystals of a pure element. As an initial study, aluminum was chosen as it is a commonly used material and is also well characterised for shock wave studies. The crystal structure for aluminum is a face centered cubic with a lattice spacing of 4.05 Å. A choice of interatomic potential has to be made. Standard references like Leach (2001) can be used to narrow the choice. The best potentials for a metal are based upon the embedded atom method (eam) which is calculated using quantum mechanical considerations. A couple of eam potentials were found and used, namely, Mishin (1999) for the eam alloy model and Mendelev (2008) for the eam Finnis-Sinclair (eam fs) model (parameters found at http://www.ctcms.nist.gov/potentials/Al.html). Out of curiosity, a third potential, the Lennard-Jones (lj), was also tried for comparison although it is certainly not appropriate for metals. The Lennard-Jones potential has the convenience of being analytically simple and requires just two parameters, the binding energy (ε) and potential well zero crossing distance (σ). For the binding energy, 3.34 eV was used as this was the energy used by Avnic (1999) and 2.55 Å was used for the zero crossing distance based upon the nearest neighbour distance in a fcc cubic crystal with a lattice spacing of 4.05 Å. The distance between a corner atom and the atom at the center of the face is 4.05 Å / √2 i.e., 2.86 Å (an approximate distance for the potential minimum). The conversion to the zero crossing distance in the lj potential is then trivial by a simple multiplication of 2-1/6. The resulting shock for impacts (actually piston pushes) at 500 m/s and 3000 m/s for each of the models are shown in Figures 1–6. Impacts at other velocities were performed and the results of the Pressure-Impact velocity are summarized in Table 1. For comparison, the experimental shock pressure P can be calculated from measured shock velocity versus particle (piston) velocity relation, usually given to a good approximation by us = c + s up, so that:

𝑃𝑃 = 𝜌𝜌𝑢𝑢𝑝𝑝 𝑢𝑢𝑠𝑠 = 𝜌𝜌 𝑢𝑢𝑝𝑝 (𝑐𝑐 + 𝑠𝑠 𝑢𝑢𝑝𝑝). (1)

The parameters for a typical aluminum alloy (6061) can be found in Marsh (1980), namely the density ρ 2703 kg/m3, c 5350 m/s and s 1.34 (unitless). For reference, the input file for LAMMPS is included in Annex A.


http://www.ctcms.nist.gov/potentials/Al.html

Figure 1: Shock wave in Al at 500 m/s impact (eam fs model).

Figure 2: Shock wave in Al at 500 m/ s impact (eam alloy model).


Figure 3: Shock wave at 500 m/s impact (lj model).

Figure 4: Shock wave in Al at 3000 m/s impact (eam fs model).


Figure 5: Shock wave in Al at 3000 m/s impact (eam alloy model).

Figure 6: Shock wave in Al at 3000 m/s impact (lj model).


Some comments on the results can be made. Firstly, it is clear that the method of generating a shock does indeed generate a shock wave as seen in Figure 1 to Figure 6. Secondly, examining Table 1 or Figure 7, it is equally clear that the Lennard-Jones does not generate the right results whereas the embedded atom methods generate much closer fits to the experimental results with the eam alloy parameters produced by Mendelev (2008) generating a very close fit. The cause of the pressure overshoot in Figure 1 is unknown. In Figure 4 and Figure 6, effects of the shock unloading (smearing or reversal of direction) at the right hand boundary can be seen. These effects are not seen in Figure 5 as this was for the slowest shock speed (hence lowest pressure) and the shock did not reach the right hand boundary in 5ns.

Table 1: Shock pressures generated by various models.

Figure 7: Pressure – velocity relations for Al.

Calculations were also performed for the explosives PETN and RDX. A different approach was used to model these. A crystal structure was created and then simply isotopically compressed to a high pressure. The resulting pressure-volume relationship was then determined by LAMMPS (Figure 8 and Figure 9). For these materials, the Lennard-Jones potential is much more

Piston Velocity Pressure (GPa) (m/s) Eam fs model Eam alloy model LJ model Experimental

500 7 6.8 36 8.1 1000 12 16 76 18.1 1500 30 31 120 29.8 2000 50 44 170 43.4 2500 78 61 220 58.8 3000 110 80 270 76.0


appropriate than for metals as they are both molecular (weakly interacting) solids. A very crude model of the crystals was used. The molecules which make up the unit cell (2 for PETN and 8 for RDX) are considered as lumped units and it is these lumped units which interact via a Lennard-Jones potential. For PETN, an ε of 0.06 eV and a σ of 4.84 Å were used from Gee (2006). A PETN crystal has a tetragonal lattice 9.38 Å by 9.38 Å by 6.71 Å from Miller (2001). For RDX, an ε of 0.06 eV and a σ of 4.84 Å (same as for PETN in the absence of better parameters) were used but its structure is an orthorhombic lattice 13.18 Å by 11.57 Å by 10.71 Å from Miller (2001). For reference, the LAMMPS input file for explosives is included in Annex A.

Figure 8: Pressure – Volume for PETN.

Figure 9: Pressure – Volume for RDX.


The experimental unreacted Hugoniots for both PETN and RDX are easily calculated. The pressure P can be found from Zeldovich (1967) (p.710) simply as P=ρoc2η / (1-sη)2 where η= (1-ρo/ρ) using the Us-up relationships for PETN, ρo 1700 kg/m3, c 2600 m/s, s 1.6 found in Cooper (1996) and for RDX, ρo 1800 kg/m3, c 2870 m/s, s 1.61 found in Dobratz (1985).

Rather surprisingly, given the crudeness and approximations used, reasonable order of magnitude pressure-volume relationships were obtained. For both explosives, the experimental p-v relationship is above the calculated curve. The lj calculations were performed using dimensionless lj units available in LAMMPS. This means amongst other things that the pressure as calculated internally is simply multiplied by a factor ε/σ3 to convert to actual pressures for a given material. Changes or uncertainties in the parameters of the lj potential thus have a direct impact on the final results. For example for PETN, if ε is 0.09 eV instead of 0.06 eV or σ is 4.23 Å instead of 4.84 Å, the lj calculations match the experimental curves.

As a result of the work with the explosives, two possible scenarios can be envisaged. First, if the correct Lennard-Jones parameters are known, the calculated Hugoniot curve can be compared with the experimental results, thereby confirming (or not) the use of the lj model as an accurate representation of the intermolecular forces in an explosive crystal. Conversely, given the experimentally determined Hugoniot curve, the lj parameters can be fitted and the resulting lj potential can be used to determine other properties of the explosives. Once again, it should be mentioned that the explosive crystals were uniformly compressed and hence crystal orientation was not considered. It should also be noted that pressures above 10 GPa, while calculable, are generally not realisable as crystals will react at such pressures and a non-reactive Hugoniot curve cannot be observed experimentally.


4 Comments on the method and approach

In a classical molecular dynamics calculation with an accurate interatomic potential, the shock Hugoniot (response) to a piston driven impact can be calculated with sufficient precision for engineering purposes. The devil is in the details. The Hugoniot can almost always be calculated for a given potential, but the problem is to know whether or not a priori the potential is accurate. For metals, even as simple as pure aluminum, multiple interatomic potentials exist, all of which have some physical basis. For the two embedded atom potentials that were tried in this study, different, but similar, Hugoniot curves were found, one of which was quite good, within +19% to -5% depending on what point the comparison is made. Further work is necessary to say definitely that the “best” potential really does represent the underlying physical properties in all cases.

Similarly, an even more moderated statement must be made about the use of molecular modelling to calculate the Hugoniot for a molecular material such as an explosive. While it is clear that a simplified model of the crystal structure and a fitted Lennard-Jones potential can reproduce an experimentally determined Hugoniot, one cannot then claim that the simulation is a true representation of the physical description. An explosive molecule is not a “point” but is an extended assemblage of several types of atoms and atomic bonds. How the molecules are oriented within a crystal structure and also how the atoms of one molecule interact with the atoms of neighbouring molecules are certainly important but not investigated within this limited study. Caveat emptor.


5 Conclusions

Molecular modelling can be used to calculate the shock Hugoniot for various materials. The accuracy of the final result depends upon accurate descriptions of both the locations of the atoms with the solid and the interatomic energy potentials between the atoms. It is difficult to know a priori whether a given interatomic potential is sufficiently accurate and a given atomic geometry is descriptive enough to generate correct results. Molecular modelling is interesting and of potentially great use but great care must also be exercised when using it.

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References/Bibliography

Ashcroft, N.W. and Mermin, N.D., (1972), Solid State Physics, Holt, Rinehart & Winston, New York.

Avinc, A. and Dimitrov, V.I., (1999), Effective Lennard-Jones potential for cubic metals in the frame of embedded atom model, Comp. Mat. Sci. 13, 211–217.

Cooper, P.W., (1996), Explosives Engineering, Wiley-VCH, New York.

Dobratz, B.M. and Crawford, P.C., (1985), LLNL Explosives Handbook, Lawrence Livermore National Laboratory Report UCRL-52997-Change 2.

Gee, R., Wu, C. and Maiti, A., (2006), A coarse-grained model for PETN crystals, Lawrence Livermore National Laboratory UCRL-JRNL-219077.

Oscar Guerrero-Miramontes A beginner’s guide to the modeling of shock/uniaxial/quasi-isentropic compression using the LAMMPS molecular dynamics simulator, http://www.researchgate.net/publication/259644293 January 2013, (Access date: 15 Oct 2015).

Leach, A.R., (2001), Molecular modelling: Principles and Applications, Pearson Prentice Hall, Harlow, England.

Marsh, S.P., (1980), LASL Shock Hugoniot Data, University of California Press, Berkeley.

Mendelev, M.I., Kramer, M.J., Becker, C.A., and Asta, M., (2008), Analysis of semi-empirical interatomic potentials appropriate for simulation of crystalline and liquid Al and Cu, Phil. Mag. 88:12, 1723–1750.

Miller, G.R. and Garroway, A.N., (2001), A Review of Crystal Structures of Common Explosives Part I, Naval Research Laboratory report NRL/MR/6120-01-8585.

Mishin, Y., Farkas, D., Mehl, M.J. and Papaconstantopoulos, D.A., (1999), Interatomic potentials for monoatomic metals from experimental data and ab initio calculations, Phys. Rev., B 59, 3393.

Pathria, R.K., (1972), Statistical Mechanics, Pergamon, Oxford.

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

Porter, D., (1996), Advances in materials modelling, J. Defence Science, Vol. 1, No 3.

Zeldovich, Y.B. and Raizer, Y.P., (1967), Physics of Shock waves and high Temperature Hydrodynamic Phenomena, Academic Press, New York.


http://www.researchgate.net/publication/259644293%20January%202013



LAMMPS input files Annex A

Input file used for Al simulations # SCRIPT MADE BY: OSCAR GUERRERO # modified by GMc for fcc Aluminum in Oct 2015 # ---------- Initialize Simulation --------------------- clear units metal dimension 3 boundary p p p atom_style atomic atom_modify map array # ---------- define variables --------------------- variable stemperature equal 300 # temperature in kelvin variable alattice equal 4.05 # lattice constant (unit A) variable myseed equal 12345 # the value seed for the velocity variable xmax equal 10 # size in the x-direction variable ymax equal 10 # size in the y-direction # zmax should be long enough that shock does not arrive at boundary by time_shock (140 eam, 420 lj) variable zmax equal 420 # size in the z-direction variable time_step equal 0.001 # time step in pico seconds variable time_eq equal 100 # time steps for the equilibration part variable time_shock equal 5000 # time steps for the piston variable vpiston equal 0.100 # piston speed in (km/s) multiply by ten to obtain (A/ps) variable Nevery equal 10 # use input values every this many timesteps variable Nrepeat equal 5 # number of times to use input values for calculating variable Nfreq equal 1000 # calculate averages every this many timesteps variable deltaz equal 20 # thickness of spatial bins in dim (distance units) variable atomrate equal 100 # the rate in timestep that atoms are dump as CFG variable tdamp equal "v_time_step*100" # DO NOT CHANGE variable pdamp equal "v_time_step*1000" # DO NOT CHANGE # DO NOT CHANGE variable Up equal "10*v_vpiston" timestep ${time_step} # ---------- Create Atoms --------------------- lattice fcc ${alattice} origin 0.0 0.0 0.0 orient x 1 0 0 orient y 0 1 0 orient z 0 0 1 #tmd 27g/NA 6e23 *4 atoms/cell/(4.05e-8cm)**3 = 2.7 g/cm**3 # the create box commands create_box natoms idname, where natoms is the number of atoms type # in the simulation box for this case we only have one type of atoms # you can use the command "set" to set the different region of atoms to different Ntype # the command "set" is very useful examples: "set group piston type 1" # define size of the simulation box region sim_box block 0 ${xmax} 0 ${ymax} 0 ${zmax} # units for region are lattice (default)


create_box 2 sim_box # define atoms in sim_box region atom_box block 0 ${xmax} 0 ${ymax} 0 ${zmax} create_atoms 1 region atom_box # define a group for the atom_box region group atom_box region atom_box region piston block INF INF INF INF INF 4 region bulk block INF INF INF INF 4 INF group piston region piston group bulk region bulk set group piston type 1 set group bulk type 2 # ---------- Define Interatomic Potential --------------------- # for lj, epsilon (eV) = heat of sublimation/atom # for lj, sigma (ang)= 2**(-1/6)*nearest neighbour distance # epsilon is 3.34 eV (cohesive energy per atom from Avinc) # sigma is 2**(-1/6) * (4.05**2+4.05**2)**.5 /2 angstrom == 2.55 Angstrom * 2.5 == 6.38 pair_style lj/cut 6.38 pair_coeff * * 3.34 2.55 # al1.eam.fs is file available from http://www.ctcms.nist.gov/potentials # pair_style eam/fs # pair_coeff * * al1.eam.fs Al Al # al99.eam.alloy is file available from http://www.ctcms.nist.gov/potentials # pair_style eam/alloy # pair_coeff * * Al99.eam.alloy Al Al mass 1 27.0 mass 2 27.0 compute myCN bulk cna/atom 3.45708 compute myKE bulk ke/atom compute myPE bulk pe/atom compute myCOM bulk com compute peratom bulk stress/atom NULL compute vz bulk property/atom vz compute vorvol bulk voronoi/atom # ------------ Equilibrate --------------------------------------- reset_timestep 0 # Now, assign the initial velocities using Maxwell-Boltzmann distribution velocity atom_box create ${stemperature} ${myseed} rot yes dist gaussian fix equilibration bulk npt temp ${stemperature} ${stemperature} ${tdamp} iso 0 0 ${pdamp} drag 1 variable eq1 equal "step" variable eq2 equal "pxx/10000" variable eq3 equal "pyy/10000" variable eq4 equal "pzz/10000" variable eq5 equal "lx" variable eq6 equal "ly" variable eq7 equal "lz" variable eq8 equal "temp"


variable eq9 equal "etotal" fix output1 bulk print 10 "${eq1} ${eq2} ${eq3} ${eq4} ${eq5} ${eq6} ${eq7} ${eq8} ${eq9}" file run.out screen no thermo 10 thermo_style custom step pxx pyy pzz lx ly lz temp etotal run ${time_eq} unfix equilibration unfix output1 # -------------- Shock ------------------------------------------- change_box all boundary p p s reset_timestep 0 # WE CREATE A PISTON USING A FEW LAYERS OF ATOMS AND THEN WE GIVE IT # A CONSTANT POSTIVE SPEED. YOU COULD ALSO USE LAMMPS' FIX WALL/PISTON COMMAND fix 1 all nve velocity piston set 0 0 v_Up sum no units box fix 2 piston setforce 0.0 0.0 0.0 # WE CREATE BINS IN ORDER TO TRACK THE PASSING OF THE SHOCKWAVE fix density_profile bulk ave/spatial ${Nevery} ${Nrepeat} ${Nfreq} z lower ${deltaz} density/mass file denz.profile units box variable temp atom c_myKE/(1.5*8.61e-5) fix temp_profile bulk ave/spatial ${Nevery} ${Nrepeat} ${Nfreq} z lower ${deltaz} v_temp file temp.profile units box # meanpress was originally pressure *volume per atom (cubic distance units) # manual suggests use compute voronoi/atom to estimate atomic volume variable meanpress atom -(c_peratom[1]+c_peratom[2]+c_peratom[3])/3/c_vorvol[1] fix pressure_profile bulk ave/spatial ${Nevery} ${Nrepeat} ${Nfreq} z lower ${deltaz} v_meanpress units box file pressure.profile fix velZ_profile bulk ave/spatial ${Nevery} ${Nrepeat} ${Nfreq} z lower ${deltaz} c_vz units box file velocityZcomp.profile variable eq1 equal "step" variable eq2 equal "pxx/10000" variable eq3 equal "pyy/10000" variable eq4 equal "pzz/10000" variable eq5 equal "lx" variable eq6 equal "ly" variable eq7 equal "lz" variable eq8 equal "temp" variable eq9 equal "etotal" variable eq10 equal "c_myCOM[3]" fix shock bulk print 10 "${eq1} ${eq2} ${eq3} ${eq4} ${eq5} ${eq6} ${eq7} ${eq8} ${eq9} ${eq10}" file run.${stemperature}K.out screen no thermo 10 thermo_style custom step pxx pyy pzz lx ly lz temp etotal c_myCOM[3] #Use cfg(?) for AtomEye # dump 1 all custom ${atomrate} dump._*.cfg id type xs ys zs c_myPE c_myKE c_myCN # dump_modify 1 element Al # dump 2 all image 1000 image.*.jpg type type axes yes 0.8 0.02 view 120 -90


run ${time_shock} Input file for explosive simulations units lj boundary p p p atom_style atomic # For PETN # tetragonal lattice with a 2 molecule basis unit (bc like) lattice custom 0.9 a1 9.38 0. 0. a2 0. 9.38 0. a3 0 0 6.71 basis 0. 0. 0. basis 0.5 0.5 0.5 # tmd 1.785 g/cm3 (MW 316.2, 2 molecules per unit cell volume 9.38 x 9.38 x 6.71 Angstroms3) # For RDX #orthrohombic lattice with 8 molecules (diamond like) #lattice custom 0.9 a1 13.18 0. 0. a2 0. 11.57 0. a3 0 0 10.71 basis 0 0 0 basis 0 0.5 0.5 basis 0.5 0 0.5 basis 0.5 0.5 0 basis 0.75 0.75 0.75 basis 0.75 0.25 0.25 basis 0.25 0.75 0.25 basis 0.25 0.25 0.75 # tmd 1.8067 g/cm3 (MW 222.13, 8 molecules per unit cell volume 13.18 x 11.57 x 10.71 Angstroms3) region mycell block 0.0 10.0 0.0 10.0 0.0 14.0 units lattice create_box 1 mycell mass * 1.0 create_atoms 1 box # Using units of Rmin, so sigma = 2^-1/6 = 0.8908987 pair coeffs p. 1115 eps, sigma, r cutoff pair_style lj/cut 2.5 pair_coeff * * 1.0 0.8908987 2.5 # Relax box dimensions - program will squeeze/expand box to attain equilibrium with an external applied pressure fix 3 all box/relax iso 0. vmax 1.0e-02 nreset 10 thermo 100 thermo_style custom step temp pe etotal pxx pyy pzz lx ly lz press vol min_modify line quadratic minimize 0.0 1.e-06 10000 100000


fix 3 all box/relax iso 500. vmax 1.0e-4 nreset 100 # control frequency (in timesteps) and variables to output thermo 100 thermo_style custom step temp pe etotal pxx pyy pzz lx ly lz press vol min_modify line quadratic minimize 0.0 0. 10000 100000 # Define initial velocities (Maxwell-Boltzmann) velocity all create 0.1 87287 mom yes rot yes dist gaussian write_restart restart.equil # Start Run log log.nodrag clear read_restart restart.equil neighbor 0.2 bin neigh_modify every 1 delay 0 check yes timestep 0.001 reset_timestep 0 # Pzz = 40.0, drag/damping term off fix myhug all nphug temp 1.0 1.0 10.0 z 60000.0 60000.0 70.0 drag 0.0 tchain 1 pchain 0 # Specify reference state from paper, times 1000 atoms # what are e0 and v0 ? # fix_modify myhug e0 -6334.0 p0 0.0 v0 680.73519 # Add fix energy to output etotal # fix_modify myhug energy yes # Define output variable dele equal f_myhug[1] # energy delta [temperature] variable us equal f_myhug[2] # shock velocity [distance/time] variable up equal f_myhug[3] # particle velocity [distance/time] variable pzz equal pzz # axial stress variable tau equal 0.5*(pzz-0.5*(pxx+pyy)) # shear stress


variable time equal dt*step thermo 1000 thermo_style custom step temp ke epair etotal pzz v_tau lz f_myhug v_dele v_up v_us press vol fix stress all print 10 "${time} ${pzz} ${tau} " screen no append stress_vs_t.dat title '#time pzz tau (no drag)' #dump id all atom 500 dump.hugoniostat dump 2 all image 500 image.*.jpg type type & axes yes 0.8 0.02 view 60 -30 dump_modify 2 pad 5 run 100


List of symbols/abbreviations/acronyms/initialisms

ε binding energy for the Lennard-Jones potential

η reduced density (1- ρo/ρ)

ρ mass density

ρo initial mass density

σ energy potential zero crossing distance for the Lennard-Jones potential

c sound speed in linear us-up relation

eam embedded atom method

fs Finnis-Sinclair (potential)

lj Lennard-Jones (potential)

P pressure

s slope in a linear us-up relation

up particle speed after the passage of a shock

us shock speed




DOCUMENT CONTROL DATA (Security markings for the title, abstract and indexing annotation must be entered when the document is Classified or Designated)

1. ORIGINATOR (The name and address of the organization preparing the document. Organizations for whom the document was prepared, e.g., Centre sponsoring a contractor's report, or tasking agency, are entered in Section 8.) DRDC – Valcartier Research Centre Defence Research and Development Canada 2459 route de la Bravoure Quebec (Quebec) G3J 1X5 Canada

2a. SECURITY MARKING (Overall security marking of the document including special supplemental markings if applicable.)

UNCLASSIFIED

2b. CONTROLLED GOODS

(NON-CONTROLLED GOODS) DMC A REVIEW: GCEC DECEMBER 2013

3. TITLE (The complete document title as indicated on the title page. Its classification should be indicated by the appropriate abbreviation (S, C or U) in

parentheses after the title.) Calculating shock Hugoniots with LAMMPS

4. AUTHORS (last name, followed by initials – ranks, titles, etc., not to be used) McIntosh, G.W.J.

5. DATE OF PUBLICATION (Month and year of publication of document.) August 2016

6a. NO. OF PAGES (Total containing information, including Annexes, Appendices, etc.)

32

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e.g., interim, progress, summary, annual or final. Give the inclusive dates when a specific reporting period is covered.) Scientific Report

8. SPONSORING ACTIVITY (The name of the department project office or laboratory sponsoring the research and development – include address.) DRDC – Valcartier Research Centre Defence Research and Development Canada 2459 route de la Bravoure Quebec (Quebec) G3J 1X5 Canada

9a. PROJECT OR GRANT NO. (If appropriate, the applicable research and development project or grant number under which the document was written. Please specify whether project or grant.)

9b. CONTRACT NO. (If appropriate, the applicable number under which the document was written.)

10a. ORIGINATOR’S DOCUMENT NUMBER (The official document number by which the document is identified by the originating activity. This number must be unique to this document.) DRDC-RDDC-2016-R152

10b. OTHER DOCUMENT NO(s). (Any other numbers which may be assigned this document either by the originator or by the sponsor.)

11. DOCUMENT AVAILABILITY (Any limitations on further dissemination of the document, other than those imposed by security classification.)

Unlimited

12. DOCUMENT ANNOUNCEMENT (Any limitation to the bibliographic announcement of this document. This will normally correspond to the Document Availability (11). However, where further distribution (beyond the audience specified in (11) is possible, a wider announcement audience may be selected.)) Unlimited

13. ABSTRACT (A brief and factual summary of the document. It may also appear elsewhere in the body of the document itself. It is highly desirable that the abstract of classified documents be unclassified. Each paragraph of the abstract shall begin with an indication of the security classification of the information in the paragraph (unless the document itself is unclassified) represented as (S), (C), (R), or (U). It is not necessary to include here abstracts in both official languages unless the text is bilingual.)

The molecular modelling program LAMMPS was used to predict the shock Hugoniot relationship (pressure-volume) of aluminum and two explosives, RDX and PETN. The method worked successfully for aluminum but uncertainties in the intermolecular potential and simplifications in crystal model yielded results of the right form and order of magnitude but inaccurate for the two explosives.

---------------------------------------------------------------------------------------------------------------

Le programme de modélisation moléculaire LAMMPS a été utilisé pour prédire la relation Hugoniot de choc (pression-volume) de l’aluminium et deux explosifs, RDX et PETN. La méthode a bien réussi pour l’aluminium mais des incertitudes dans le potentiel intermoléculaire et des simplifications dans le modèle de cristal ont donné les résultats de la forme correcte et du bon ordre de magnitude mais imprécis pour les deux explosifs.

14. KEYWORDS, DESCRIPTORS or IDENTIFIERS (Technically meaningful terms or short phrases that characterize a document and could be helpful in cataloguing the document. They should be selected so that no security classification is required. Identifiers, such as equipment model designation, trade name, military project code name, geographic location may also be included. If possible keywords should be selected from a published thesaurus, e.g., Thesaurus of Engineering and Scientific Terms (TEST) and that thesaurus identified. If it is not possible to select indexing terms which are Unclassified, the classification of each should be indicated as with the title.) Molecular modelling: shock; hugoniot.

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