Molecular dynamics simulations of shock-induced plasticity in tantalum

lable at ScienceDirect

High Energy Density Physics 10 (2014) 9e15

Contents lists avai

High Energy Density Physics

journal homepage wwwelsevier comlocatehedp

Molecular dynamics simulations of shock-induced plasticityin tantalum

Diego Tramontina ab Paul Erhart ck Timothy Germann d James Hawreliak cAndrew Higginbothame Nigel Park f Ramoacuten Ravelo dg Alexander Stukowski hMathew Suggit e Yizhe Tang i Justin Wark e Eduardo Bringa bj

aAgencia Nacional de Promocioacuten Cientiacutefica y Tecnoloacutegica CABA C1054AAH Argentinab Instituto de Ciencias Baacutesicas Universidad Nacional de Cuyo Mendoza M5502JMA Argentinac Lawrence Livermore National Laboratory Livermore CA 94550 USAd Los Alamos National Laboratory Los Alamos NM 87545 USAeDepartment of Physics Clarendon Laboratory University of Oxford Parks Road Oxford OX1 3PU UKfMaterials Modeling Group AWE Aldermaston Reading Berkshire RG7 4PR UKg Physics Department and Materials Research Institute University of Texas El Paso TX 79968 USAhDarmstadt University of Technology Darmstadt 64289 Germanyi Johns Hopkins University Baltimore MD 21218 USAjConsejo Nacional de Investigaciones Cientiacuteficas y Teacutecnicas ArgentinakChalmer University of Technology Department of Applied Physics Gothenburg 41296 Sweden

a r t i c l e i n f o

Article historyReceived 10 October 2013Accepted 16 October 2013Available online 31 October 2013

KeywordsTantalumMolecular dynamicsShocks

Corresponding author Instituto de Ciencias BaacutesiCuyo Mendoza M5502JMA Argentina

E-mail address ebringayahoocom (E Bringa)URL httpsitesgooglecomsitesimafweb

1574-1818$ e see front matter 2013 Elsevier BV Ahttpdxdoiorg101016jhedp201310007

a b s t r a c t

We present Non-Equilibrium Molecular Dynamics (NEMD) simulations of shock wave compression alongthe [001] direction in monocrystalline Tantalum including pre-existing defects which act as dislocationsources We use a new Embedded Atom Model (EAM) potential and study the nucleation and evolutionof dislocations as a function of shock pressure and loading rise time We find that the flow stress anddislocation density behind the shock front depend on strain rate We find excellent agreement withrecent experimental results on strength and recovered microstructure which goes from dislocations to amixture of dislocations and twins to twinning dominated response as the shock pressure increases

2013 Elsevier BV All rights reserved

1 Introduction

Shock compression of condensed matter allows the study ofmaterials under extreme conditions [1] Improving experimentaland simulation techniques allow detailed studies of the shock-induced microstructure which can lead to large changes in me-chanical properties A large amount of work has been recentlycarried out for Face-Centered Cubic Metals (FCC) [2e13] and Body-Centered Cubic metals (BCC) [214e20] However most of the workon BCC metals has focused on Fe due to the large number oftechnological applications for Fe for instance as part of structuralmaterials and also due to the role of Fe properties in Earthrsquos interiormechanics Fe displays a solidesolid phase transformation near15 GPa which makes dislocation plasticity difficult to identify insimulations [21e23]

cas Universidad Nacional de

ll rights reserved

Atomistic simulations of high strain rate loading of BCC metals[324e26] are sparse mostly due to the lack of interatomic poten-tials which are reliable at high pressures In particular it has beenshown for Nb [18] and Ta [27] that many potentials display anartificial phase transition from BCC to Hexagonal Close-Packed(HCP)

Among BCC metals Ta has several technological applicationsand no phase transitions are thermodynamically present up tofairly high pressures and temperatures [28] Shock-loaded Ta hasbeen studied using both gas-gun [29] and laser-driven shocks[30e32] These experiments show a rich behavior including highstrength [3233] dislocations [29e31] twinning above a criticalpressure around 40 GPa [30313435] and the presence of u-phase [3031] in some recovered samples shocked above w70GPa

Recent Non-Equilibrium Molecular Dynamics (NEMD) simula-tions from Cuesta-Lopez and Perlado [36] subjected Ta W and Femonocrystals to particle velocities Up ranging from 01 to25 km s1 spanning the elastic to shock-melting response Theydid not observe dislocation activity but instead found nucleation

D Tramontina et al High Energy Density Physics 10 (2014) 9e1510

of a close-packed phase above a critical shock pressure Howeverthis observed BCC close-packed phase transition in Ta is actuallyan artifact of the potential by Li et al [37] used in their studyAnother group recently reported NEMD simulations of Ta spall [38]using the Johnson Embedded Atom Model (EAM) potential [39]without focusing on shock-induced plasticity

Ravelo et al recently presented a new EAM potential [27] for Taspecifically developed for high-pressure shock loading environ-ments They showed that several often-used Ta potentials displayan artificial BCC HCP transition below 100 GPa Using their newpotential they calculated the Hugoniot along different directions[001 011] and [111] finding an excellent agreement with experi-mental data up to several Mbars Above a velocity threshold(Upfrac14 088 km s1 Pw 70GPa for the [001] direction) they observedhomogeneous nucleation of twins for shocks along all the studieddirections Above that threshold dislocations could be nucleatedfrom twin boundaries Rudd et al [25] studied the plastic relaxationrates of compressed Ta samples using the Model GeneralizedPseudopotential Theory (MGPT) potential and homogeneouscompression finding dislocations homogeneously nucleated abovew65 GPa

In this work we investigate a Ta sample with pre-existing de-fects which act as dislocation sources shock loaded along the [001]crystallographic direction We focus on pressures below 70 GPa thethreshold for twin nucleation to study the influence of shockstrength (particle velocity Up) and shock rise time tr on the resultingmicrostructures

2 Methodology

We use the LAMMPS molecular dynamics simulation code [40]to model perfect single crystal samples in which nanovoids havebeen added to act as dislocation sources [1041e44] A few nano-voids can be simple to introduce and relax and have been exten-sively studied regarding dislocation emission under loading[152545e47] They lead to results similar to the introduction ofdislocation loops [4448] and have an activation stress roughlyproportional to the inverse of their radius compared to dislocationloop Frank-Read-type sources which have a nucleation stressproportional to the inverse of their length [49] We use nanovoidswith a radius of 3 nm which leads to an activation stress of 12 GPaunder homogeneous uniaxial compression [50] This allows us tostudy shock-plasticity well below the homogeneous twin nucle-ation limit

We use the new EAM potential by Ravelo et al [27] and theExtended Finnis Sinclair (EFS) potential [51] These potentialsshould work well below 70 GPa and dislocation activity from voidsis qualitatively similar for both of them [50]

We tested different sample sizes with cross-sections reaching300 300 BCC cells to investigate possible size effects and foundthat a cross-section of 50 50 cells was large enough to obtainsmooth shock profiles The length of the sample varied between400 and 2000 BCC cells along the [001] direction Periodicboundary conditions are applied in the transverse directions andthe back region allows for free surface release Most simulationswere run at low temperatures (10 K) to simplify defect detectionbut the few room temperature simulations performed also dis-played the same behavior

There are several ways to run non-equilibrium shock simula-tions [45253] and we use a simple rigid piston moving at animposed particle velocity Up [5455] An ideal shock wave is typi-cally applied as a perfect square wave with zero rise time Exper-imental shocks on the other hand are typically applied with risetimes as long as a nanosecond For this reason we use a linearvelocity ramp for the piston which can lead to a large change in the

resulting microstructure due to the dynamics of dislocation pro-duction in the pre-existing sources [4856] Of course the shockwave will eventually develop a steady state depending on pressureand material properties We investigate piston velocities in therange Up frac14 025e09 km s1 and rise times in the range tr frac14 01e50 ps As a guide the volumetric compression reached forUp frac14 075 km s1 is 12

The deviatoric shear stress was calculated using [4]

s1 frac14 12

szz 1

2sxx thorn syy

(1)

while for the full von Mises stress we use

s2vm frac14 3J2 frac14 12Sn thorn 6Ss (2)

Sn frac14 sxx syy

2 thorn syy szz

2 thorn ethsxx szzTHORN2 (3a)

Ss frac14 s2xy thorn s2yz thorn s2xz (3b)

where J2 is the second invariant of the stress deviator [13]Tracking of defects was performed by use of the Dislocation

Extraction Algorithm (DXA) [57] In the DXA defects are identifiedvia Common Neighbor Analysis (CNA) [58] and a geometricdescription of dislocation lines is generated All remaining crystaldefects which cannot be represented by dislocation lines areidentified as point defects or surfaces which in our specific casewould apply for vacancies twin boundaries or void surfaces Thisgeometric representation is ideal for visualization of complexdefect structures We also used the Crystal Analysis Tool (CAT) [59]which allows for strain and structure-type calculations includingtwin detection Although twin identification in unstrained latticesis fairly straightforward using CNA or other methods [60] twinningin a lattice with large uniaxial compression can be challenging CAT[59] is able to detect twins up to high strains and a newmethod fortwin identification has also been recently presented [61] We useOvito [62] VMD [63] and ParaView [64] to visualize defectstructures

3 Results

Shock loading leads to emission of dislocations from pre-existing sources as expected Initially as described in detail forhomogeneous loading simulations [46] one observes mostly rapidedge segments advancing while screw segments remain mostlysessile as shown in Fig 1a for Up frac14 035 km s1 tr frac14 15 ps Shocksimulations using the EFS interatomic potential by Dai et al [51]show the same features Although there are some quantitativedifferences in the activation threshold for dislocation emissionbetween the EFS and Ravelo potentials dislocation structures for3 nm voids are qualitatively the same under homogeneouscompression [50] After some time a dislocation forest develops asin the back of Fig 1a similar to the case of higher porosity andhomogeneous uniaxial loading [47] with a large fraction of straightscrew segments as seen in some recovered samples ForUp lt 04 km s1 (29 GPa) only dislocations are nucleated from thepre-existing defects and the situation evolves similarly to Fig 1aFor Upfrac14 05 km s1 twinning appears as shown in Fig 1b We find atwinning threshold between 25 and 30 GPa in agreement withexperiments [31] Twinning appears alongside dislocations pro-ducing a mixed structure as observed in Fig 1b obtained with CAT[59] This has also been seen in gas-gun [65] and laser-driven shockexperiments [31] In our simulations twins are only few nm wide

Fig 1 Shock-induced microstructure for the Ravelo potential [27] with shock fronts moving approximately from left to right (Left) Up frac14 035 km s1 and tr frac14 15 ps showingdislocations 60 ps after piston started (Center) Up frac14 050 km s1 and tr frac14 15 ps showing twinning and dislocations 70 ps after shock started Blue dislocations teal twins greentwin boundaries (Right) Up frac14 088 km s1 tr frac14 15 ps Behind dislocations emitted from the void twins are nucleated homogeneously Color represents depth (For interpretation ofthe references to color in this figure legend the reader is referred to the web version of this article)

D Tramontina et al High Energy Density Physics 10 (2014) 9e15 11

and few tens of nm long after only 50e80 ps and they couldcontinue growing The microscopy work by McNaney et al showedthat twins in recovered Ta (001) samples shocked at 55 GPa weretens of nm wide and hundreds of nm long [35] Recent work forcompressive loading of Ta at 104 s1 also shows thin twins [66] Thecase when the twin nucleation threshold has already been reachedis shown in Fig 1c Since we are using ramp loading the early partof the ramp is enough to trigger dislocation emission from a voidwhile the peak pressure reached at the top of the ramp inducestwinning This should be compared to the FCC loading shown inRefs [4856] where loops were used instead of voids and homo-geneous nucleation of shear loops replaced twinning

Twinning in BCC metals is complex [67e69] but there areseveral atomistic studies for BCC metals describing twin growth[7071] and twin nucleation from grain boundaries [7273] In oursimulations twinning is closely related to dislocation emissionfrom voids and their reactions [46] As expected from compressivestrain shock-induced twins are in the eth121THORNfrac12111 system [46] Fig 2shows the close-up view of a twin from the simulation shown inFig 1b

A view of the shock-induced dislocation forest produced by asingle source can be seen in Fig 3 together with pressure and shearstress profiles The von Mises stress has an average value ofw12 GPa over the region with dislocations which is nearly iden-tical to its value throughout the entire shock-compressed regionThe shear stress goes down to values of 3 GPa less than half itsvalue in the elastically compressed region

To understand the large differences between the von Mises andshear stress we consider one particular case In Fig 5 the profilesfor the longitudinal stress tensor component in the shock directionszz vonMises stress svm shear s1 and the orthogonal shear stressessxy sxz and syz are plotted for a shocked sample at Up frac14 05 km s1

Fig 2 Region of the sample for Up frac14 050 km s1 and tr frac14 15 ps showing twinninginduced by the pre-existing sources below the homogeneous twin nucleationthreshold 70 ps after the shock started Top surface perpendicular to the (112) plane

with a linear ramp of tr frac14 10 ps A void at z w 66 nm serves as adislocation source It can be seen that szz maintains its value atw36 GPa in the region with dislocations between 55 and 90 nmbut we sometimes observe perturbations near voids especially forlonger rise-times The von Mises stress has a mean value of 13 GPawith a drop of only about 5 near the center of the dislocation zoneOn the other hand as the shear stress is only computed from theprincipal stress directions it shows a marked drop over the sameregion where a particular complex state of stress evinced by thenon-zero values of the non-diagonal shear stress components di-minishes the ability of the von Mises calculation to reflect devia-toric stress relaxation due to dislocation motion

Fig 4 shows profiles for samples with 3 dislocation sourcesalong their length Pressure profiles show the expected steepeningof the wave and kinks in the profile indicate plastic activityDislocation activity increases with time and leads to a roughlyhomogeneous dislocation density Shear profiles display increasingrelaxation due to that dislocation motion as expected

Fig 6a shows the evolution of dislocation density vonMises andshear stress versus Up Measurements for this figure were taken atthe same depth when the shock front is reaching the end of thesample The dislocation density grows with increasing particlevelocity Up as expected It increases linearly with Up for rise timesup to 15 ps but when the ramp is as long as 35 ps and Up is at075 km s1 the dislocation density no longer follows this trendbut instead shows a lower value At longer rise times dislocationmotion dominates dislocation emission lowering the dislocationdensity needed to relax the volumetric strain Ref [48] The large

Fig 3 Snapshot for Up frac14 050 km s1 tr frac14 15 ps taken 35 ps after the piston startedfor a sample with an initial void in the center (top) Dislocation lines (center) Longi-tudinal stress (solid line) shear stress (10 dashed line) and von Mises stress (dottedline) all in GPa (bottom) Temperature (102 K solid line) and dislocation density(1017 m2 dashed line)

2

4

6

50 100 150 200 250

ρ d[10

17m

-2]

20 ps30 ps40 ps50 ps60 ps70 ps

50 100 150 200 250Depth [nm]

50 100 150 200 250

2

4

6

8

10

σ 1 [G

Pa]

20

40

60

Pzz[

Gpa

]

Fig 4 Selected profiles at distinct shock velocities and rise times First column Up frac14 075 km s1 tr frac14 50 ps Second column Up frac14 075 km s1 tr frac14 25 ps Third columnUp frac14 050 km s1 tr frac14 25 ps


dislocation densities might be partly due to particularly largemultiplication rates for (001) loading [34] The von Mises stressdepends weakly and the shear stress strongly on both Up and tr overthe entire range shown here

In Fig 6b the shear stress s1 von Mises stress svm and dislo-cation density rd have been plotted as a function of the simulationtime in order to observe the evolution at distinct ramp times In allcases there is an initial rise stage up to a maximum value followedby a decay at later times All curves shown in this figure are forUp frac14 05 km s1 so the maximum shear stress in this condition isabout 12 w 15 GPa This is enough to trigger dislocation emissionfrom pre-existing dislocation sources and lead to relaxation of theshear stress [2548] As expected the dislocation density is greatestfor the shorter rise time

The shear stress von Mises stress and dislocation density as afunction of the rise time are shown in Fig 7 for Up frac14 05 km s1Whereas the von Mises stress shows the same trend for all

Fig 5 Stress profiles at t frac14 30 ps for Up frac14 05 km s1 and tr frac14 10 ps Shear differsdepending of the criteria employed specially in the dislocation zone between 55 and90 nm

simulated rise times the shear stress and dislocation densitypresent a local minimum at tr frac14 5 ps due to the competition be-tween dislocation emission and motion We note that dislocationdensities go down to w1016 m2 for the longest simulated risetime as also shown in Fig 4 The longest rise times presented heremight still be shorter than typical experimental rise timespointing to possible further reduction of transient dislocationdensities in experiments

4 Discussion of results and comparison with experiments

Experimental single crystal metal samples always contain alevel of pre-existing defects including vacancies impuritiesdislocation loops and dislocation networks The amount andstructure of these defects can determine the Hugoniot Elastic Limit(HEL) and the plastic shock response at low pressures For instancedislocation loops which are near micron-sized as often found inexperimental samples would lead to dislocation multiplication atstresses of tens of kbars as shown in FEM-DD (Finite ElementMethod-Dislocation Dynamics) of shocks in Ta [56] The role ofdislocation networks in shock plasticity is more difficult to asses Inatomistic simulations one is typically limited by dislocation sourceswith activation stresses above 1 GPa Because of this and the highstrain-rate in shock compression there is an immense productionof dislocations reaching densities of w1016e1017 m2 Similardensities are found in homogeneous compression of defective Tasamples [47] and are comparable to densities in shocked FCCmetals [485674] The dislocation densities we find in our simula-tions are similar to the predictions of the Multi-Scale Strength(MTS) model which predicts a saturation dislocation density ofw1016 m2 at a strain rate of 109 s1 [75] There are experimentswhere dislocation densities in shock-loaded and recovered Tasamples have been estimated to be larger than 1016 m2 [65]

Fig 6 (left) Evolution of local dislocation density resolved shear stress and von Mises stress versus piston velocity for selected rise times Quantities are evaluated 60 nm aheadof the piston when the shock wave is reaching the far back of the sample (right) Same parameters evaluated as a function of time for Up frac14 05 km s1 Symbol size is indicativeof error


Lu et al [31] discuss two semi-analytical models of dislocationproduction One of them assumes homogeneous nucleation (HN) ofdislocations and another assumes dislocation emission fromsources sometimes referred as heterogeneous nucleation (HetN)The first one leads to dislocation densities of 1016e1017 m2 andthe second one leads to densities of w1014 m2 Dislocation den-sities in recovered samples were around 1014 m2 and therefore itwas concluded that HN was not relevant but HetN was dominantHowever HN was suggested again to explain the results by Comleyet al [32] based on the simulations by Rudd et al [25] In this workwe found high dislocation densities due to source emission and donot observe HN neither in shocks nor in homogeneous uniaxialcompression We thus agree with Lu et al [31] regarding the crucialrole of HetN

The large discrepancy between our simulated values and theones reported by Lu et al [31] could be explained by the differencebetween densities during loading and after recovery For FCCmetals it was shown that there could be several orders of magni-tude decrease between the two values [76] For BCC metals onewould expect that recovery might play a lesser role due to reduceddislocation mobilities However preliminary results for simulated

recovery show a dislocation density decrease by a factor of 10within few tens of ps This points to the possibility of furtherreduction of the dislocation density during macroscopic timescales Recent experiments looking at perturbation growth duringTa loading [24] also suggest that the transient dislocation densitiesreached during loading are much larger than those found inrecovered samples

Hammel et al [33] found flow stresses of 2e3 GPa for shocks inTa up to 50 GPa at a strain rate of 107 s1 Our simulations arecarried out at higher strain rates and a larger flow stress is ex-pected In fact the values of strength in the simulations as given bythe von Mises stress are consistent with X-Ray Diffraction-basedobservations of Ta strength obtained by Comley et al [32] atsimilar pressures and strain rates This would also support the largedislocation densities we find in our simulations during loading

Based on our results one could distinguish three different re-gimes in the shock-induced microstructure At relatively lowpressures there are only dislocations which upon recovery wouldarrange into dislocation cells Above 30 GPa there is a mixture ofdislocations and twins up to w70 GPa when massive twinnucleation would dominate the resulting microstructure This

Fig 7 Dislocation density shear and von Mises profiles versus rise time forUp frac14 05 km s1 Symbol size is indicative of error


hierarchy of shock-induced microstructures agrees very well withexperimental results by Hsiung [65] and Lu [31] The work byFlorando et al [34] for shocks in Ta (001) showed twin fractions inrecovered samples which are negligible for 25 GPa shocks and ofup to few percent for 55 GPa shocks We have 0 twin volumefraction in Fig 1a for 25 GPa shocks (Up frac14 035 km s1) 35 twinvolume fraction in Fig 1b for 37 GPa shocks (Up frac14 05 km s1) and6 twin volume fraction in Fig 1c for 67 GPa shocks(Up frac14 088 km s1)

These results point out the need for constitutive models thatinclude twinning as the ones recently presented for HCP [77] andBCC metals [3478] The recent MTS model by Barton et al [75]which agrees well with experimental strength up to 200 GPa [32]only includes dislocation plasticity Therefore its applicability tomonocrystals shocked above w50 GPa where recovered samplesand simulations display twinning should be takenwith care as theauthors themselves point out The newly presented constitutivemodel by Florando et al [34] includes both slip and twinning andreproduces trends seen in experiments for Ta single crystals Itcontains dislocation slip as MTS and focuses on twin growth andinteraction between slip and twinning without including twinnucleation explicitly In our simulations twins are nucleated neardislocation sources and the twinning fractions we find areconsistent with results from this new model

Regarding the nucleation of the u-phase seen in recoveredsamples [3165] we do not find evidence of such nucleation in ourloading simulations but phase-stability studies are in progress forthe interatomic potentials used here It could certainly happen thatthe kinetics of u-phase nucleation is much longer than thew100 psscale covered in our simulations or that recovery plays a role in theformation of the omega phase

5 Summary and conclusions

Molecular Dynamics simulations of shocks in (001) Ta singlecrystals with pre-existing defects show dislocation and twin pro-duction which lead to shear stress relaxation Stress relaxation isless noticeable if one uses the von Mises stress because the off-diagonal stress components increase the values of the shearstress Dislocation densities and flow stress decrease withincreasing rise time of the applied load The simulated Ta strengthagrees well with recent experimental values [32]

Shock-induced microstructure consists of dislocations at rela-tively low pressures followed by a combination of dislocations andtwins starting at w30 GPa and finally a predominance of twinsabove 70 GPa This succession of microstructures and the shockpressures at which they occur agree well with experimental results

[3134] However we find dislocation densities which are muchhigher than the ones in recovered samples [31] which could beexplained by experimental unloading and recovery during macro-scopic time At the end of our loading simulations we typicallyobserve a dislocation forest with a large fraction of screw segmentsRecovered samples also include many screw segments alongdislocation loops and dislocation cells [31] However it is difficult tocompare our dislocation structure during loading to structuresgenerated after unloading and long thermal processing Dynamicdiffraction experiments might offer a window to study dislocationplasticity during loading [4879]

There are several aspects of simulations of shocks in Ta singlecrystals which need to be explored further For instance shocks insingle crystals with different orientation such as [011] analysis oftwinning [5961] detailed elasticeplastic strain calculations [59]together with simulated diffraction patterns which will allow adirect comparison with experimental results [32486179] Shocksin Ta polycrystals would also offer rich microstructures

Plasticity in Ta offers a scenario to test our knowledge of thebehavior of solids at high strain rates The increasingly betterstudies of microstructure evolution during and after shock loadingwill lead to improved understanding of materials performance andthe possible design of improved materials for use under extremeconditions [80]

Acknowledgements

D Tramontina and EM Bringa were funded by projectsPICT2008-1325 from the ANCyT and 06M035 from SecTyP-UNCuyoWe thank R Rudd B Remington MA Meyers BL Holian andCJ Ruestes for useful and stimulating discussions A Higginbothamacknowledges support from AWE M Suggit and JS Warkacknowledge support from EPSRC under grant PJ0172561R Ravelo acknowledges support from the Air Force Office of Sci-entific Research under Award FA9550-12-1-0476 Work at LosAlamos was performed under the auspices of the US Departmentof Energy (DOE) under Contract No DE-AC52-06NA25396 P Erhartacknowledges support from the Swedish Research Council (VR) andthe Area of Advanced Materials at Chalmers

References

[1] E Gamaly Phys Rep 508 (4) (2011) 91 httpdxdoiorg101016jphysrep201107002

[2] MA Meyers C Taylor Aimone Prog Mater Sci 28 (1) (1983) 1 httpdxdoiorg1010160079-6425(83)90003-8

[3] BA Remington P Allen EM Bringa J Hawreliak D Ho KT LorenzH Lorenzana JM McNaney MA Meyers SW Pollaine K RosolankovaB Sadik MS Schneider D Swift J Wark B Yaakobi Mater Sci Technol 22(4) (2006) 474 httpdxdoiorg101179174328406X91069

[4] BL Holian Science 280 (5372) (1998) 2085 httpdxdoiorg101126science28053722085

[5] RW Armstrong SM Walley Int Mater Rev 53 (3) (2008) 105 httpdxdoiorg101179174328008X277795

[6] MA Meyers H Jarmakani EM Bringa BA Remington in JP Hirth L Kubin(Eds) Dislocations in Solids Elsevier 2009 pp 91e197 httpdxdoiorg101016S1572-4859(09)01502-2

[7] V Lubarda M Schneider D Kalantar B Remington M Meyers Acta Mater 52(6) (2004) 1397 httpdxdoiorg101016jactamat200311022

[8] S Traiviratana EM Bringa DJ Benson MA Meyers Acta Mater 56 (15)(2008) 3874 httpdxdoiorg101016jactamat200803047

[9] EM Bringa S Traiviratana MA Meyers Acta Mater 58 (13) (2010) 4458httpdxdoiorg101016jactamat201004043

[10] X Deng W Zhu Y Zhang H He F Jing Comput Mater Sci 50 (1) (2010) 234httpdxdoiorg101016jcommatsci201008008

[11] AS Pohjonen F Djurabekova K Nordlund A Kuronen SP Fitzgerald J ApplPhys 110 (2) (2011) 023509 httpdxdoiorg10106313606582

[12] EM Bringa VA Lubarda MA Meyers 63 (1) (2010) 148 httpdxdoiorg101016jscriptamat201002038

[13] MA Meyers H Jarmakani BY Cao CT Wei B Kad BA Remington EMBringa B Maddox D Kalantar D Eder A Koniges vol 2 EDP Sciences 2009p 999 httpdxdoiorg101051dymat2009140


[14] RE Rudd JF Belak Comput Mater Sci 24 (1e2) (2002) 148 httpdxdoiorg101016S0927-0256(02)00181-7

[15] J Marian J Knap G Campbell Acta Mater 56 (10) (2008) 2389 httpdxdoiorg101016jactamat200801050

[16] H Jarmakani B Maddox C Wei D Kalantar M Meyers Acta Mater 58 (14)(2010) 4604 httpdxdoiorg101016jactamat201004027

[17] S-N Luo AIP Conf Proc 1426 (2012) 1259 httpdxdoiorg10106313686509

[18] RF Zhang J Wang IJ Beyerlein TC Germann Philos Mag Lett 91 (12)(2011) 731 httpdxdoiorg101080095008392011615348

[19] SV Razorenov G Garkushin GI Kanel ON Ignatova AIP Conf Proc (2012)991 httpdxdoiorg10106313686444

[20] Q An R Ravelo TC Germann IWA Goddard AIP Conf Proc 1426 (2012)1259 httpdxdoiorg10106313686509

[21] K Kadau Science 296 (5573) (2002) 1681 httpdxdoiorg101126science1070375

[22] N Gunkelmann H Ledbetter HM Urbassek Acta Mater 60 (12) (2012) 4901httpdxdoiorg101016jactamat201205038

[23] K Wang S Xiao M Liu H Deng W Zhu W Hu Proc Eng 61 (2013) 122httpdxdoiorg101016jproeng201307104

[24] H-S Park N Barton JL Belof KJM Blobaum RM Cavallo AJ ComleyB Maddox MJ May SM Pollaine ST Prisbrey B Remington RE RuddDW Swift RJ Wallace MJ Wilson A Nikroo E Giraldez AIP Conf Proc1426 (2012) 1371 httpdxdoiorg10106313686536

[25] RE Rudd AJ Comley J Hawreliak B Maddox H-S Park BA Remington AIPConf Proc (2012) 1379 httpdxdoiorg10106313686538

[26] TC Germann BL Holian PS Lomdahl JR Ravelo Phys Rev Lett 84 (23)(2000) 5351 httpdxdoiorg101103PhysRevLett845351

[27] R Ravelo TC Germann O Guerrero Q An BL Holian Phys Rev B 88 (2013)134101 httpdxdoiorg101103PhysRevB88134101

[28] L Burakovsky SP Chen DL Preston AB Belonoshko A RosengrenAS Mikhaylushkin SI Simak JA Moriarty Phys Rev Lett 104 (2010)255702 httpdxdoiorg101103PhysRevLett104255702

[29] L Murr M Meyers C-S Niou Y Chen S Pappu C Kennedy Acta Mater 45(1) (1997) 157 httpdxdoiorg101016S1359-6454(96)00145-0

[30] JK Zhou LL Hsiung R Chau CK Saw M Elert MD Furnish R ChauN Holmes J Nguyen AIP Conf Proc 955 (2007) 677 httpdxdoiorg10106312833192

[31] C Lu B Remington B Maddox B Kad H Park S Prisbrey M Meyers ActaMater 60 (19) (2012) 6601 httpdxdoiorg101016jactamat201208026

[32] AJ Comley BR Maddox RE Rudd ST Prisbrey JA HawreliakDA Orlikowski SC Peterson JH Satcher AJ Elsholz H-S ParkBA Remington N Bazin JM Foster P Graham N Park PA RosenSR Rothman A Higginbotham M Suggit JS Wark Phys Rev Lett 110 (11)(2013) 115501 httpdxdoiorg101103PhysRevLett110115501

[33] B Hammel D Swift B El-Dasher M Kumar GW Collins J Florando AIPConf Proc (2012) 931 httpdxdoiorg10106313686430

[34] JN Florando NR Barton BS El-Dasher JM McNaney M Kumar J ApplPhys 113 (8) (2013) 083522 httpdxdoiorg10106314792227

[35] JM McNaney B Torralva KT Lorenz BA Remington M Wall M KumarM Elert MD Furnish WW Anderson WG Proud WT Butler AIP Conf Proc(2009) 677 httpdxdoiorg10106313295230

[36] S Cuesta-Lopez JM Perlado 60 (2) (2011) 590[37] Y Li DJ Siegel JB Adams X-Y Liu Phys Rev B 67 (2003) 125101 http

dxdoiorg101103PhysRevB67125101[38] J-P Cuq-Lelandais M Boustie L Soulard L Berthe J Bontaz-Carion T

d Resseguier AIP Conf Proc 1426 (2012) 1167 httpdxdoiorg10106313686487

[39] RA Johnson DJ Oh J Mater Res 4 (05) (1989) 1195 httpdxdoiorg101557JMR19891195

[40] S Plimpton J Comp Phys 117 (1) (1995) 1 httpdxdoiorg101006jcph19951039

[41] LP Davila P Erhart EM Bringa MA Meyers VA Lubarda MS SchneiderR Becker M Kumar Appl Phys Lett 86 (16) (2005) 161902 httpdxdoiorg10106311906307

[42] P Erhart EM Bringa M Kumar K Albe Phys Rev B 72 (2005) 052104httpdxdoiorg101103PhysRevB72052104

[43] A Kubota M-J Caturla S Payne T Diaz de la Rubia J Latkowski J NuclMater 307e311 (2002) 891 httpdxdoiorg101016S0022-3115(02)01008-5

[44] A Kubota DB Reisman WG Wolfer Appl Phys Lett 88 (24) (2006) 241924httpdxdoiorg10106312210799

[45] RE Rudd Philos Mag 89 (34) (2009) 3133 httpdxdoiorg10108014786430903222529

[46] Y Tang EM Bringa MA Meyers Acta Mater 60 (12) (2012) 4856 httpdxdoiorg101016jactamat201205030

[47] C Ruestes E Bringa A Stukowski J Rodriacuteguez Nieva G Bertolino Y TangM Meyers Scr Mater 68 (10) (2013) 817 httpdxdoiorg101016jscriptamat201301035

[48] EM Bringa K Rosolankova RE Rudd BA Remington JS WarkM Duchaineau DH Kalantar J Hawreliak J Belak Nat Mater 5 (10) (2006)805 httpdxdoiorg101038nmat1735

[49] LM Dupuy EB Tadmor RE Miller R Phillips Phys Rev Lett 95 (2005)060202 httpdxdoiorg101103PhysRevLett95060202

[50] DR Tramontina CJ Ruestes Y Tang EM Bringa (submitted for publication)[51] XD Dai Y Kong JH Li BX Liu J Phys Condens Mater 18 (19) (2006) 4527

httpdxdoiorg1010880953-89841819008[52] B Holian Phys Rev A 37 (7) (1988) 2562 httpdxdoiorg101103

PhysRevA372562[53] MM Budzevich VV Zhakhovsky CT White II Oleynik Phys Rev Lett 109

(2012) 125505 httpdxdoiorg101103PhysRevLett109125505[54] EM Bringa JU Cazamias P Erhart J Stolken N Tanushev BD Wirth

RE Rudd MJ Caturla J Appl Phys 96 (7) (2004) 3793 httpdxdoiorg10106311789266

[55] EM Bringa A Caro Y Wang M Victoria JM McNaney BA RemingtonRF Smith BR Torralva HV Swyngenhoven Science 309 (5742) (2005) 1838httpdxdoiorg101126science1116723

[56] MA Shehadeh EM Bringa HM Zbib JM McNaney BA Remington ApplPhys Lett 89 (17) (2006) 171918 httpdxdoiorg10106312364853

[57] A Stukowski K Albe Model Simul Mater Sci Eng 18 (8) (2010) 085001httpdxdoiorg1010880965-0393188085001

[58] H Tsuzuki PS Branicio JP Rino Comput Phys Commun 177 (6) (2007) 518httpdxdoiorg101016jcpc200705018

[59] A Stukowski A Arsenlis Model Simul Mater Sci Eng 20 (3) (2012) 035012httpdxdoiorg1010880965-0393203035012

[60] RE Rudd Mater Sci Forum 633 (2009) 3 httpdxdoiorg104028wwwscientificnetMSF633-6343

[61] A Higginbotham MJ Suggit EM Bringa P Erhart JA Hawreliak G MogniN Park BA Remington JS Wark Phys Rev B 88 (2013) 104105 httpdxdoiorg101103PhysRevB88104105

[62] A Stukowski Model Simul Mater Sci Eng 18 (1) (2010) 015012 httpdxdoiorg1010880965-0393181015012

[63] W Humphrey A Dalke K Schulten J Mol Graph 14 (1) (1996) 33 httpdxdoiorg1010160263-7855(96)00018-5

[64] AH Squillacote J Ahrens The ParaViewGuide Kitware Clifton Park NY 2006[65] LL Hsiung J Phys Condens Mater 22 (38) (2010) 385702 httpdxdoiorg

1010880953-89842238385702[66] C Chen G Hu J Florando M Kumar K Hemker K Ramesh Scr Mater 69

(10) (2013) 709 httpdxdoiorg101016jscriptamat201307010[67] K Lagerlof Acta Metall Mater 41 (7) (1993) 2143 httpdxdoiorg101016

0956-7151(93)90384-5[68] J Christian S Mahajan Prog Mater Sci 39 (1e2) (1995) 1 httpdxdoiorg

1010160079-6425(94)00007-7[69] Y Zhu X Liao X Wu Prog Mater Sci 57 (1) (2012) 1 httpdxdoiorg

101016jpmatsci201105001[70] Y Gu L-Q Chen TW Heo L Sandoval J Belak Scr Mater 68 (7) (2013) 451

httpdxdoiorg101016jscriptamat201211022[71] LA Sandoval MP Surh AA Chernov DF Richards J Appl Phys 114 (11)

(2013) 113511 httpdxdoiorg10106314821956[72] Y Zhang PC Millett M Tonks S Biner Acta Mater 60 (18) (2012) 6421

httpdxdoiorg101016jactamat201208029[73] Y Tang EM Bringa MA Meyers Mater Sci Eng A 580 (2013) 414 http

dxdoiorg101016jmsea201305024[74] J Hawreliak HE Lorenzana BA Remington S Lukezic JS Wark Rev Sci

Instum 78 (2007) 083908 httpdxdoiorg10106312772210[75] NR Barton JV Bernier R Becker A Arsenlis R Cavallo J Marian M Rhee

H-S Park BA Remington RT Olson J Appl Phys 109 (7) (2011) 073501httpdxdoiorg10106313553718

[76] B Cao EM Bringa MA Meyers Metall Mater Trans A 38 (11) (2007) 2681httpdxdoiorg101007s11661-007-9248-9

[77] DW Brown IJ Beyerlein TA Sisneros B Clausen CN Tomeacute J Int Plast 29(2012) 120 httpdxdoiorg101016jijplas201108006

[78] RW Armstrong FJ Zerilli J Phys D 43 (49) (2010) 492002 httpdxdoiorg1010880022-37274349492002

[79] MJ Suggit A Higginbotham JA Hawreliak G Mogni G Kimminau P DunneAJ Comley N Park BA Remington JS Wark Nat Commun 3 (2012) 1224httpdxdoiorg101038ncomms2225

[80] MA Meyers BA Remington B Maddox EM Bringa JOM 62 (1) (2010) 24httpdxdoiorg101007s11837-010-0006-x


of a close-packed phase above a critical shock pressure Howeverthis observed BCC close-packed phase transition in Ta is actuallyan artifact of the potential by Li et al [37] used in their studyAnother group recently reported NEMD simulations of Ta spall [38]using the Johnson Embedded Atom Model (EAM) potential [39]without focusing on shock-induced plasticity

Ravelo et al recently presented a new EAM potential [27] for Taspecifically developed for high-pressure shock loading environ-ments They showed that several often-used Ta potentials displayan artificial BCC HCP transition below 100 GPa Using their newpotential they calculated the Hugoniot along different directions[001 011] and [111] finding an excellent agreement with experi-mental data up to several Mbars Above a velocity threshold(Upfrac14 088 km s1 Pw 70GPa for the [001] direction) they observedhomogeneous nucleation of twins for shocks along all the studieddirections Above that threshold dislocations could be nucleatedfrom twin boundaries Rudd et al [25] studied the plastic relaxationrates of compressed Ta samples using the Model GeneralizedPseudopotential Theory (MGPT) potential and homogeneouscompression finding dislocations homogeneously nucleated abovew65 GPa

In this work we investigate a Ta sample with pre-existing de-fects which act as dislocation sources shock loaded along the [001]crystallographic direction We focus on pressures below 70 GPa thethreshold for twin nucleation to study the influence of shockstrength (particle velocity Up) and shock rise time tr on the resultingmicrostructures

2 Methodology

We use the LAMMPS molecular dynamics simulation code [40]to model perfect single crystal samples in which nanovoids havebeen added to act as dislocation sources [1041e44] A few nano-voids can be simple to introduce and relax and have been exten-sively studied regarding dislocation emission under loading[152545e47] They lead to results similar to the introduction ofdislocation loops [4448] and have an activation stress roughlyproportional to the inverse of their radius compared to dislocationloop Frank-Read-type sources which have a nucleation stressproportional to the inverse of their length [49] We use nanovoidswith a radius of 3 nm which leads to an activation stress of 12 GPaunder homogeneous uniaxial compression [50] This allows us tostudy shock-plasticity well below the homogeneous twin nucle-ation limit

We use the new EAM potential by Ravelo et al [27] and theExtended Finnis Sinclair (EFS) potential [51] These potentialsshould work well below 70 GPa and dislocation activity from voidsis qualitatively similar for both of them [50]

We tested different sample sizes with cross-sections reaching300 300 BCC cells to investigate possible size effects and foundthat a cross-section of 50 50 cells was large enough to obtainsmooth shock profiles The length of the sample varied between400 and 2000 BCC cells along the [001] direction Periodicboundary conditions are applied in the transverse directions andthe back region allows for free surface release Most simulationswere run at low temperatures (10 K) to simplify defect detectionbut the few room temperature simulations performed also dis-played the same behavior

There are several ways to run non-equilibrium shock simula-tions [45253] and we use a simple rigid piston moving at animposed particle velocity Up [5455] An ideal shock wave is typi-cally applied as a perfect square wave with zero rise time Exper-imental shocks on the other hand are typically applied with risetimes as long as a nanosecond For this reason we use a linearvelocity ramp for the piston which can lead to a large change in the

resulting microstructure due to the dynamics of dislocation pro-duction in the pre-existing sources [4856] Of course the shockwave will eventually develop a steady state depending on pressureand material properties We investigate piston velocities in therange Up frac14 025e09 km s1 and rise times in the range tr frac14 01e50 ps As a guide the volumetric compression reached forUp frac14 075 km s1 is 12

The deviatoric shear stress was calculated using [4]

s1 frac14 12

szz 1

2sxx thorn syy

(1)

while for the full von Mises stress we use

s2vm frac14 3J2 frac14 12Sn thorn 6Ss (2)

Sn frac14 sxx syy

2 thorn syy szz

2 thorn ethsxx szzTHORN2 (3a)

Ss frac14 s2xy thorn s2yz thorn s2xz (3b)

where J2 is the second invariant of the stress deviator [13]Tracking of defects was performed by use of the Dislocation

Extraction Algorithm (DXA) [57] In the DXA defects are identifiedvia Common Neighbor Analysis (CNA) [58] and a geometricdescription of dislocation lines is generated All remaining crystaldefects which cannot be represented by dislocation lines areidentified as point defects or surfaces which in our specific casewould apply for vacancies twin boundaries or void surfaces Thisgeometric representation is ideal for visualization of complexdefect structures We also used the Crystal Analysis Tool (CAT) [59]which allows for strain and structure-type calculations includingtwin detection Although twin identification in unstrained latticesis fairly straightforward using CNA or other methods [60] twinningin a lattice with large uniaxial compression can be challenging CAT[59] is able to detect twins up to high strains and a newmethod fortwin identification has also been recently presented [61] We useOvito [62] VMD [63] and ParaView [64] to visualize defectstructures

3 Results

Shock loading leads to emission of dislocations from pre-existing sources as expected Initially as described in detail forhomogeneous loading simulations [46] one observes mostly rapidedge segments advancing while screw segments remain mostlysessile as shown in Fig 1a for Up frac14 035 km s1 tr frac14 15 ps Shocksimulations using the EFS interatomic potential by Dai et al [51]show the same features Although there are some quantitativedifferences in the activation threshold for dislocation emissionbetween the EFS and Ravelo potentials dislocation structures for3 nm voids are qualitatively the same under homogeneouscompression [50] After some time a dislocation forest develops asin the back of Fig 1a similar to the case of higher porosity andhomogeneous uniaxial loading [47] with a large fraction of straightscrew segments as seen in some recovered samples ForUp lt 04 km s1 (29 GPa) only dislocations are nucleated from thepre-existing defects and the situation evolves similarly to Fig 1aFor Upfrac14 05 km s1 twinning appears as shown in Fig 1b We find atwinning threshold between 25 and 30 GPa in agreement withexperiments [31] Twinning appears alongside dislocations pro-ducing a mixed structure as observed in Fig 1b obtained with CAT[59] This has also been seen in gas-gun [65] and laser-driven shockexperiments [31] In our simulations twins are only few nm wide












2

4

6

50 100 150 200 250

ρ d[10

17m

-2]


50 100 150 200 250Depth [nm]

50 100 150 200 250

2

4

6

8

10

σ 1 [G

Pa]

20

40

60

Pzz[

Gpa

]




























Acknowledgements


References





























































































2

4

6

50 100 150 200 250

ρ d[10

17m

-2]


50 100 150 200 250Depth [nm]

50 100 150 200 250

2

4

6

8

10

σ 1 [G

Pa]

20

40

60

Pzz[

Gpa

]




























Acknowledgements


References




































































































Acknowledgements


References


















































































Molecular dynamics simulations of shock-induced plasticity in tantalum

Documents

Transcript of Molecular dynamics simulations of shock-induced plasticity in tantalum