Nano Mechanics and Materials: Theory, Multiscale Methods and Applications

Nano Mechanics and Materials:Theory, Multiscale Methods and Applications

byWing Kam Liu, Eduard G. Karpov, Harold S. Park

7. Bridging Scale Numerical Examples

7.1 Comments on Time History Kernel

1D harmonic lattice

where indicates a second order Bessel function, is the spring stiffness, and the frequency , where m is the atomic mass

Spring stiffness utilizing LJ 6-12 potential

where k is evaluated about the equilibrium lattice separation distance

22( ) (2 )kt J tt

2J k/k m

12 6

14 8

624 168kr r

eqr

Truncation Time History Kernel

Impedance force

due to salient feature of , an approximation can be made by setting the later components to zero

0( )( )( )

timp t d f q u

( )t

• Plots of time history kernel (full) • Plots of time history kernel (truncated)

Comparison between time history integrals (full and truncated)

• Plot of time history kernel (full and truncated)

7.2 1D Bridging Scale Numerical Examples – Lennard-Jones

• Initial MD & FEM displacements

• MD impedance force (applied correctly)

• MD impedance force (not applied correctly)

1D Wave Propagation

• 111 atoms in bridging scale MD system• 40 finite elements• 10 atoms per finite element• tfe = 50tmd

• Lennard-Jones 6-12 potential

• Initial MD & FEM displacements

1D Wave Propagation - Energy Transfer

• 99.97% of total energy transferred from MD domain• Only 9.4% of total energy transferred without impedance force

7.3 2D/3D Bridging Scale Numerical Examples

• Lennard-Jones (LJ) 6-12 potential

• Potential parameters ==1

• Nearest neighbor interactions

• Hexagonal lattice structure; (111) plane of FCC lattice

• Impedance force calculated numerically for hexagonal lattice, LJ potential

• Hexagonal lattice with nearest neighbors

7.4 Two-Dimensional Wave Propagation

MD region given initial displacements with both high and low frequencies similar to 1D example

30000 bridging scale atoms, 90000 full MD atoms 1920 finite elements (600 in coupled MD/FE region) 50 atoms per finite element

• Initial MD displacements

2D Wave Propagation

• Snapshots of wave propagation

2D Wave Propagation

• Final displacements in MD region if MD impedance force is applied.

• Final displacements in MD region if MD impedance force is not applied.

2D Wave Propagation

Energy Transfer Rates:• No BC: 35.47%• nc = 0: 90.94%• nc = 4: 95.27%• Full MD: 100%

• nc = 0: 0 neighbors• nc = 1: 3 neighbors• nc = 2: 5 neighbors

n n+1 n+2n-1n-2

7.5 Dynamic Crack Propagation in Two Dimensions

Problem Description:• 90000 atoms, 1800 finite elements (900 in coupled region)• Full MD = 180,000 atoms• 100 atoms per finite element• tFE=40tMD

• Ramp velocity BC on FEM

Time

Velocity

t1

Vmax

2D Dynamic Crack Propagation

• Beginning of crack opening • Crack propagation just before complete rupture of specimen


• Bridging scale potential energy • Full MD potential energy


• Crack tip velocity/position comparison

• Full domain = 601 atoms• Multiscale 1 = 301 atoms • Multiscale 2 = 201 atoms• Multiscale 3 = 101 atoms

Zoom in of Cracked Edge

• FEM deformation as a response to MD crack propagation

7.6 Dynamic Crack Propagation in Three Dimensions

• 3D FCC lattice• Lennard Jones 6-12 potential• Each FEM = 200 atoms• 1000 FEM, 117000 atoms• Fracture initially along (001) plane

Time

Velocity

t1

Vmax

FEM

FEM

MD+FEMPre-crack

V(t)

V(t)

Initial Configuration

• Velocity BC applied out of plane (z-direction)• All non-equilibrium atoms shown

MD/Bridging Scale Comparison

• Full MD • Bridging Scale







7.7 Virtual Atom Cluster Numerical Examples – Bending of CNT

Global buckling pattern is capture by the meshfree method

Local buckling captured by molecular dynamics simulation

VAC coupling with tight binding

Comparison of the average twisting energy between VAC model and tight-binding model

Meshfree discretization of a (9,0) single-walled carbon nanotube

Nano Mechanics and Materials: Theory, Multiscale Methods and Applications

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