Nano Mechanics and Materials: Theory, Multiscale Methods and Applications

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Nano Mechanics and MaterialsTheory, Multiscale
Methods and Applications

• by
• Wing Kam Liu, Eduard G. Karpov, Harold S. Park

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7. Bridging Scale Numerical Examples
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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

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Truncation Time History Kernel

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

• Plots of time history kernel (full)

• Plots of time history kernel (truncated)

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Comparison between time history integrals (full
and truncated)

• Plot of time history kernel (full and truncated)

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7.2 1D Bridging Scale Numerical Examples
Lennard-Jones

• MD impedance force (applied correctly)

• Initial MD FEM displacements

• MD impedance force (not applied correctly)

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1D Wave Propagation

• 111 atoms in bridging scale MD system
• 40 finite elements
• 10 atoms per finite element
• ?tfe 50?tmd
• Lennard-Jones 6-12 potential

• Initial MD FEM displacements

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1D Wave Propagation - Energy Transfer

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

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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

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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

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2D Wave Propagation

• Snapshots of wave propagation

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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.

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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
n1
n2
n-1
n-2
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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
• ?tFE40?tMD
• Ramp velocity BC on FEM

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2D Dynamic Crack Propagation

• Beginning of crack opening

• Crack propagation just before complete rupture
  of specimen

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2D Dynamic Crack Propagation

• Bridging scale potential energy

• Full MD potential energy

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2D Dynamic Crack Propagation

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

• Crack tip velocity/position comparison

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Zoom in of Cracked Edge

• FEM deformation as a response to MD crack
  propagation

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7.6 Dynamic Crack Propagation in Three Dimensions
V(t)

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

Pre-crack
V(t)
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Initial Configuration

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

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MD/Bridging Scale Comparison

• Full MD

• Bridging Scale

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MD/Bridging Scale Comparison

• Full MD

• Bridging Scale

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MD/Bridging Scale Comparison

• Full MD

• Bridging Scale

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MD/Bridging Scale Comparison

• Full MD

• Bridging Scale

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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

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VAC coupling with tight binding

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

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