Nanomechanics: Atomistic Modeling Introduction Energy Kinetics Kinematics Damage Mechanics Outline Mark F. Horstemeyer, PhD CAVS Chair Professor in Computational Solid Mechanics Mechanical Engineering Mississippi State University

METHODS USED DEPEND ON THE ENTITY BEING MODELED MethodEntityExamples first principleselectrondensity functional theory (DFT) quantum chemistry (QC) semi-empiricalatomembedded atom method (EAM) modified EAM (MEAM) N-body, glue empiricalatom, grain, Lennard-Jones (LJ), Potts dislocationdislocation dynamics (DD) phenomenacontinuumFicks Law fieldelasticity/plasticity

Energy: Embedded Atom Method (EAM) Total energy E F i : embedding energy of atom i  i : electronic density of atom i r ij : separation distance between atom i and j  ij : pair potential of atom i and j 1.Molecular Dynamics (f=ma, finite temperatures) 2.Molecular Statics (rate independent, absolute zero) 3.Monte Carlo Simulations (quasi-static, finite temperatures)

Determination of Atomic Stress Tensor (Daw and Baskes, 1984, Phys. Rev) Embedded Atom Method (EAM) and Modified Embedded Atom Method (MEAM) potentials Local force determined from energy Dipole Force Tensor (virial stress) is determined from local forces Note: the difference between EAM and MEAM is an added degree of angular rotations that affect the electron density cloud. For EAM, this quantity is simply a scalar, but for MEAM it includes three terms that are physically motivated:

Calc. Expt. Lattice Constant (Å) Cohesive Energy (eV) Vacancy Formation Energy (eV) Elastic Constants (GPa) C C C Surface Energy (mJ/m 2 ) Stacking Fault Energy (mJ/m 2 )85125 EAM Represents the properties of Nickel extremely well

CURRENTLY DEVELOPED MEAM FUNCTIONS COVER MOST OF THE PERIODIC TABLE Mn ThU He B Sn Zn In Ga S Cd P Hg Ba Sr Ca Bi Be La As Sb Mg LiCN Si Ti Sc H O Pr Na K Al CoVFe Zr Cr Ni Ru Y Hf Ge Cu Rh MoNb PbRe Nd TbGd Tl Au Ag DyHoEr Pt Pd IrWTa Impurities BCCFCC DIA CUB Pu HCP

Atomistic Stress Clausius, Maxwell Viral Theorem, 1870 Maxwell Tensor form of Virial, 1874 Rayleigh, 1905 Irving-Kirkwood, 1949 Generalized Continuum Theories Cosserat 1909 Truesdell, Toupin, Mindlin, Eringen, Green-Naghdi 1960s Kinetics: Historical Background

atoms follow Newton’s second law temperature maintained at 300K x y z GEOMETRY FOR ATOMISTIC CALCULATIONS atoms moved at constant velocity fixed atoms periodic in z periodic, free surface, or GB in x

(a) (b) Shear stress-strain curve of material blocks at an applied strain rate of 2.4e8/sec with (a) ten thousand atoms and (b) ten million atoms showing microyield 1 at the proportional limit, microyield 2 at 0.2% offset strain, and macroyield atoms  yield

shear stress (GPa) shear strain EVOLUTION OF DISLOCATION SUBSTRUCTURE Strain rate 10 8 /sec T = 300K

10 6 atoms 0.16  m x 0.08  m 10 7 atoms 0.5  m x 0.25  m shear stress (GPa) e8/sec 6.58e9/sec 1.53e10/sec 5.26e10/sec shear stress (GPa) shear strain e8/sec 1.53e10/sec shear strain LARGER SAMPLES HAVE CONSIDERABLY LESS STRUCTURE IN THE STRESS/STRAIN CURVES

single slip double slip quadruple slip octal slip pseudopolycrystal shear stress (GPa) shear strain resolved shear stress (GPa) shear strain Schmid type plasticity observed at nanoscale Horstemeyer, M.F. Baskes, M.I., Hughes, D.A., and Godfrey, A. "Orientation Effects on the Stress State of Molecular Dynamics Large Deformation Simulations," Int. J. Plasticity, Vol. 18, pp , Strain rate /sec T = 300K

Yield Stress Depends on Specimen Size Because of Dislocation Nucleation Dominance Strain rate 2.4 x 10 8 /sec T= 300K Horstemeyer, M.F., Plimpton, S.J., and Baskes, M.I."Size Scale and Strain Rate Effects on Yield and Plasticity of Metals," Acta Mater., Vol. 49, pp , 2001.

A SIMPLE MODEL TO EXPLAIN STRAIN RATE DEPENDENCE Dislocation is nucleated at a distance x 0 from the free surface at a critical stress  * T Dislocation accelerates according to classical law Effective stress depends on image stress, Peierls barrier, and orientation At nucleation there is no net stress on the dislocation Yield is defined when  b x = 0.2% T

Experimental data examining yield stress versus applied strain rate from Follansbee (1988) and Edington (1969) for copper.

Experiment yield stress/elastic modulus strain rate (1/sec) 1332 atoms 2e4 atoms 1e5 atoms 10 6 atoms 10 7 atoms 10 8 atoms (1.6 microns) exp (Maloy et al. 1995) Calculations PREDICTED STRAIN RATE DEPENDENCE OF YIELD STRENGTH IS CONSISTENT WITH EXPERIMENTS

a b c d e f ghi Strain Rate and Size Scale Effects

Size Scale Dependence Observed in Copper Also

Size Scale Effect Observed in Tension Also

Volume averaged stress is a function of volume per surface area large scale experiments EAM calculations indentation and torsion experiments interfacial force microscopy experiments  Conventional theory predicts that yield stress is independent of sample size  Because of their small size, properties of materials to be used in nano-devices are predicted to be vastly different than the properties of materials used in conventional devices Horstemeyer, M.F. and Baskes, M.I., “Atomistic Finite Deformation Simulations: A Discussion on Length Scale Effects in Relation to Mechanical Stresses,” J. Eng.Matls. Techn. Trans. ASME, Vol. 121, pp , Horstemeyer, M.F., Plimpton, S.J., and Baskes, M.I."Size Scale and Strain Rate Effects on Yield and Plasticity of Metals," Acta Mater., Vol. 49, pp , 2001.

Size Scale is related to Dislocation Nucleation (volume per surface area) and strain gradients EAM Ni Expt. various fcc metals Free surface BC T = 300 K normalized resolved yield stress not converged rst strain rate Gerberich, W.W., Tymak, N.I., Grunlan, J.C., Horstemeyer, M.F., and Baskes, M.I., “Interpretations of Indentation Size Effects,” J. Applied Mechanics, Vol. 69, No. 4, pp , 2002

length scale 1: nanoscale length scale 2: submicron scale length scale 3: microscale length scale 4: macroscale stress strain Schematic showing the stress-strain responses at four different size scales.

Determination of Stress Dislocations are fundamental defects related to plasticity and damage Edge dislocation

distance for 12 stress component distance for 21 stress component Distance range is 4-48 Angstroms from center of dislocation sig12 sig21 Comparison of stress asymmetry in a simple shear simulation at 300K with one dislocation

distance for 12 stress component distance for 21 stress component Distance range is Angstroms from center of three dislocations sig12 sig21 Comparison of stress asymmetry in a simple shear simulation at 300K with four dislocations

distance for 12 stress component distance for 21 stress component Distance range is 4-48 Angstroms from center of dislocation sig12 sig21 Comparison of stress asymmetry in a simple shear simulation at 10K with one dislocation

distance for 12 stress component distance for 21 stress component Distance range is Angstroms from center of three dislocations sig12 sig21 Comparison of stress asymmetry in a simple shear simulation at 10K with four dislocations

distance for 12 stress component distance for 21 stress component Distance range is Angstroms from center of three dislocations -sig12 -sig21 Comparison of stress asymmetry in a pure shear simulation at 10K with four dislocations

distance for 12 stress component distance for 21 stress component Distance range is Angstroms from center of three dislocations -sig12 -sig21 Comparison of stress asymmetry in a pure shear simulation at 10K with one dislocation

distance for 12 stress component distance for 21 stress component Distance range is Angstroms from center of three dislocations -sig12 -sig21 Comparison of stress asymmetry in a pure shear simulation at 10K with four dislocations

distance for 12 stress component distance for 21 stress component Distance range is Angstroms from center of three dislocations -sig12 -sig21 Comparison of stress asymmetry in a pure shear simulation at 10K with one dislocation

Local Continuum Theory Equations Local Continuum Equations Where constants A, B incorporate elastic constants and boundary conditions

Nonlocal Continuum Theory Equations Nonlocal Continuum, Eringen 1977 where, a:internal characteristic length k:constant b:relative radial displacement

EVOLUTION OF MICROSTRUCTURE SHOWS TWIN FORMATION compression 5%2.3%4.8%4.3% tension slice through sample

Simulation Setup Single crystal 5800 atoms Low angle grain boundary 5860 atoms High angle grain boundary 5840 atoms 35Å 70Å 140Å [1 0 0] [0 1 1] [1 0 0] [0 1 1] [1 0 0] [0 1 1] [1 0 0] [0 1 1] [1 0 0] [10 1 1] [ ]

Boundary Conditions FixedPeriodic Fixed Prescribed velocity V=0.035 Å/ps Fixed ends Forward loading 300 K Periodic Flexible Prescribed velocity V=0.035 Å/ps Periodic Fixed Flexible ends Forward loading 300 K

Single Crystal (Fixed-end

Low Angle Grain Boundary (fixed-end)

High Angle Grain Boundary (fixed-end)

Bauschinger Effect Analysis Bauschinger Stress Parameter / Bauschinger Effect Parameter  f : stress in the forward load path at reverse point  r : yield stress in the reverse load path  y : initial yield stress in the forward load path Bauschinger effects indicated by BSP and BEP The larger the BSP and BEP are, the stronger the Bauschinger effect

BSP and BEP for fixed ends, reverse load at 9% strain BSP and BEP for flexible ends, reverse load at 8% strain Bauschinger Effect Results

Bauschinger Effects Summary Less constrained materials exhibit reduced yield stresses compared to highly constrained materials Single crystal material has the smallest Bauschinger effects while high angle GB material has the strongest for both fixed-end and flexible-end BC’s Dislocation nucleation occurs earlier in low angle GB than single crystal and high angle GB cases for both BC’s

Kinematics/Strain No size scale effects observed –Macroscale torsion experiments, crystal plasticity, and atomistics show the same plastic spin independent of size scale –Macroscale internal state variable theory, crystal plasticity, and atomistics show same strain contours independent of size scale –High rate plastic collapse of same geometry in macroscale experiments show identical result at atomistic results

Crystal plasticity/finite element simulation of torsion single crystal Cu peaks oscillation troughs max CCW rotation max CW rotation Plastic Spin Displ.

Condition/Methodwave amplitude ratio Torsion experiment (cm)0.02 finite elements (cm)0.05 molecular dynamics (solid cylinder) 0.06 molecular dynamics (hollow cylinder) Simple Shear finite elements (cm)0.25 molecular dynamics (nm)0.23 peaks troughs Plastic Spin is same throughout length scales A L wave amplitude ratio=A/L

STRAIN CONTOURS AT 30% STRAIN - 8x1 Aspect Ratio ISV atomistics crystal plasticity orientation angle - crystal plasticity

Average yield stress for a block of material of varying aspect ratios computed with different modeling methods: a) constant y, varying x; b) constant x, varying y. (a)(b)

60 Å 30 Å y x z 13 Å EAM Model, Single-crystal Copper (1 3 4), Periodic in z (4 unit cells) constant velocity and strain rate of 10 9 s -1

FEA 40  s ps Experiment al Copper (1 3 4) Partial Collapse Crack initiation point

EAM ps experimental  s (removed at 5.5  s) FEA Copper (1 3 4) Fully Collapse flow localization point anisotropic inelastic behavior

central curved lines inner boundary FEA  s (removed at 5.5  s) EAM Copper (1 3 4) Fully Collapse

EAM Experimental Size Scale Effect (30Å/18mm) Copper (1 3 4) EAM Experimental outer boundary inner boundary

10.0  s 12.5  s 15.0  s FEA ps ps Copper (0 0 1) flow localization

Macroscale Modeling of Size Scale Effects Observation 1: we observe a size scale effect on stress state Observation 2: we do not observe a size scale effect on the geometric, strain, or plastic spin states Question: where should the macroscale repository be for the size scale dependence? Answer: constitutive relation in dislocation density, not the strain tensor