Evaluating Continuum Mechanical Quantities at the ... - Lammps

2013 LAMMPS Users’ Workshop and SymposiumAlbuquerque, New MexicoAugust 6-8, 2013Γ1Photos placed in horizontal position0.8with even amount of white space0.6between photos and 0.4 headerJ / J nucl1.61.41.2R o =28R o =56R o =84theory,-.*,,!/0123!%#&!%#%!%#$!%!"#'!"#&7 8) 91:)9; ?@AB4.42C0.2!"#%00 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4(K I /2µ) 2Evalua&ng Con&nuum Mechanical Quan&&es at the Atomic Scale Jonathan Zimmerman !"#$!"!( !$((((( !%((((( !&((((( !'((((( !)*+(&-45*,-*6Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed MartinCorporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. SAND NO. 2011-XXXXP

Motivation: Continuum mechanics at the nanoscaleMaterial deformation and failure occurs at various length scales:• Brittle and ductile fracture• Dislocation activity• Grain boundary sliding• Stiction, friction and wear(Pferner; 1999)(Bulatov, Tang& Zbib; 2001 )(Boyce; 2012 )Continuum mechanics-based models to predict failure have been veryeffective at various length scales (10 -4 – 10 2 m) but can/should they work atthe nanoscale?Our premise:• Define consistent fields from molecular simulation• Continuum theorynanoscaleresultsFigure 7. A cross-sectional SEM image of several voids and cavities at different length scales in a single neckedtensile bar. Contrast is due to electron channeling, which provides details on the subgrain morphology. Toobtain this image, a backscattered electron detector was used with a 15 keV electron beam at a working distanceI.e. first connect atomistics to continuum fields in a manner of 4 consistent mm. withbalance laws, then use continuum theory to analyze the process.12

!Atomistic-Continuum formulation of R.J. Hardy!Hardy (Journal of Chemical Physics, 1982) -( ) = m # $ x # % x" x,tN&#=1( )( ) = m " v " # x " $ xp x,t( )N% 1(E 0( x,t) = -&2 m" v " # v " + $ " )'* + ( x" , x)"=1Applying these expressions!to the spatial forms of the balance laws, e.g.v ˆ" x,t!yields an expression for the Cauchy stress tensor:!N%"=1"p"t = " "x # $ % &v ! ' v( )and separating continuum-scale momentum flux from atomic motion,!( ) = # +1 , 2" x,t*,-( ) = v " ( t) # v( x,t)x " = 0["] ~ 1 V c" > 0v " p/#N NN!((x $% & f $% B $%( x)+ ( m $ v ˆ$ & v ˆ$ ) x $ # x$=1 % '$$=1linear momentum ( )Thermal variables (heat flux, temperature) are also part of this formulation.Now that we have these tools, what can we do with them… ?.,/0 ,

Deformation of nanocrystalline Ta• Objective: Characterize the defects that form during uniaxial tension of aNC-Ta thin film, and use continuum fields to connect their generation tolocally “measured” critical stresses.• Simulation details:• EAM by Li et al. (Phys. Rev. B, 2003)• Simulations done with LAMMPS (http://lammps.sandia.gov)• Geometry: thin film of dimensions 34.5 nm x 34.5 nm x 7 nm• 3 columnar grains of size 20 nm with random orientations• Approximately 464,000 atoms• Equilibration via heating to 1700K for 200 ps,then cooling to 300K for 200 ps,using a Nosé-Hoover thermostat• Uniaxial stress applied at strain rates 10 5 to 10 9 sec -1

Material frame version of Hardy formulationMaterial frame balance laws:Densities:Substitute densities into balance laws and do a lot of math…

Resulting variables

J-integral analysis for quasi-static crack initiationJ =Quasi-static analysis shows:"• path independency of J integral0• agreement with LEFM for Harmonic material• fracture occurs when J = 2γ for materials offinite strength# S ! Nds = # ( W 0N $ F T ! P ! N)ds" 0J 1 /2!1.81.61.41.210.80.60.40.20-0.2Lennard-Jones: loop 0Lennard-Jones: loop 1Lennard-Jones: loop 2Lennard-Jones: loop 3Lennard-Jones: loop 4Harmonic: loop 0Harmonic: loop 1Harmonic: loop 2Harmonic: loop 3Harmonic: loop 40 0.2 0.4 0.6 0.8 1 1.2K I /K Ic

J-integral for ductile crack propagation[110][00-1]FCC Au• EAM potential by Foiles et al.(1986)• Emission of type partialedge dislocations along {111}planes• Formation of stacking faults• J Rice = 1.27 < J c = 1.95 [J/m 2 ][01-1] BCC Fe[100]• EAM potential by Simonelli etal. (1993)• Emission of type fulledge dislocations along {211}planes• No stacking faults• J Rice = 6.56 > J c = 2.87 [J/m 2 ]10

J-integral for ductile crack propagationJ / J nucl1.61.41.210.80.60.40.20R o =28R o =56R o =84theory0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4(K I /2µ) 2FCC Au• Dislocation emission occurs at J = 1.25 J/m 2• After emission, J-integral exhibits a jump of0.29 J/m 2 , then recovers.• Second dislocation emission occurs at amarginally higher J and also displays nearimmediaterecovery.J/J nucl10.80.60.40.2Ro = 28Ro = 56Ro = 84theoryBCC Fe• Dislocation emission occurs at J = 3.42 J/m 2• After emission, J-integral exhibits a jump of0.20 J/m 2 ; J remains at it’s new, lower level.• Second dislocation emission occurs at asignificantly higher value of J = 3.78 J/m 2 .00 0.2 0.4 0.6 0.8 1 1.2 1.4(K I /2µ) 211

J-integral formulation at finite temperature At finite temperatures, stress and deformation are conjugates of the freeenergy. Thus, the J-integral must be defined as…J =% ( ! N " F T # P # N)ds = J = % ( ! N " H T # P # N)ds$ 0 Free energy Ψ is difficult to calculate directly. Instead, we choose toapproximate it with the Quasi-Harmonic (QH) approach:$ 0! " ! QH= # 0+ k BTV log%'&n$i=1!! ik BT We further use the Local-Harmonic (LH) simplification of QH that depends ona localized (diagonal) version of the dynamical matrix:! LH= " 0+ k BTV !##!log%%$ $ k BT&('3det( D LH )&('(*)D LH= 1 m! 2 "!u 0!u 0

J-Integral analysis for finite temperaturecrack initiation Using thermalization of a sequence of minimized states with far field loading,we obtain fields that show localization together with thermal noise:• Zero temperature annulus of atoms used to enforce LEFM BCs.• Fields are time-averaged over a period of 40 ps (10 5 samples).

J-Integral retains path independenceJ 1 /2 γ1.41.210.80.60.40.20-0.2T=100 loop 0T=100 loop 1T=100 loop 2T=100 loop 3T=100 loop 4T=300 loop 0T=300 loop 1T=300 loop 2T=300 loop 3T=300 loop 4LEFM0 0.2 0.4 0.6 0.8 1 1.2K I / K Ic• Path independence at both 100 and 300K.• J decreases with increasing T due to thermal compression reducingeffective stress intensity factor.

Concluding remarksThank you to:• Reese Jones*, Jeremy Templeton*• Jay Foulk, Alejandro Mota• Stephen Foiles, Aidan Thompson• Jay Oswald* (ASU), Ted Belytschko* (Northwestern)• Jeff Rickman (Lehigh)Publications:J.A. Zimmerman et al., Modelling Simul. Mater. Sci. Eng. (2004) 12 S319-S332J.A. Zimmerman, R.E. Jones and J.A. Templeton, J. Comput. Phys. (2010) 229 2364-2389R.E. Jones and J.A. Zimmerman, J. Mech. Phys. Solids (2010) 58 1318-1337R.E. Jones et al., J. Phys.: Condens. Matter (2011) 23 015002J.A. Zimmerman and R.E. Jones, J. Phys.: Condens. Matter (2013) 25 155402