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3.4 Standing Waves in Lattices. Wave Number Space and Dispersion Law ... Value v0 is the phase velocity of the longest waves (at p 0). continuum. continuum ... – PowerPoint PPT presentation

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Title: Nano Mechanics and Materials: Theory, Multiscale Methods and Applications


1
Nano Mechanics and MaterialsTheory, Multiscale
Methods and Applications
  • by
  • Wing Kam Liu, Eduard G. Karpov, Harold S. Park

2
3. Lattice Mechanics
3
3.1 Elements of Lattice Symmetries
The term regular lattice structure refers to any
translation symmetric polymer or crystalline
lattice 1D lattices (one or several degrees of
freedom per lattice site) 2D lattices
n-2 n-1 n n1
n2
n-2 n-1 n n1 n2

n-2 n-1 n n1 n2
4
Regular Lattice Structures
The 14 Bravais lattices
3D lattices (Bravais crystal lattices) Bravais
lattices represent the existing basic symmetries
for one repetitive cell in regular crystalline
structures. The lattice symmetry implies
existence of resonant lattice vibration
modes. These vibrations transport energy and are
important in the thermal conductivity of
non-metals, and in the heat capacity of solids.
5
3.2 Equation of Motion of a Regular Lattice
n-2 n-1 n n1
n2
Equation of motion is identical for all
repetitive cells n
Introduce the stiffness operator K
6
Periodic Lattice Structure Equation of Motion
n-2 n-1 n n1
n2
Equation of motion is identical for all
repetitive cells n
Introduce the stiffness operator K
7
3.3 Transforms
Recall first A function f assigns to every
element x (a number or a vector) from set X a
unique element y from set Y. Function f
establishes a rule to map set X to Y
Examples yxn ysin x yB x
A functional operator A assigns to every
function f from domain Xf a unique function F
from domain XF . Operator A establishes a
transform between domains Xf and XF
Examples
8
Functional Operators (Transforms)
Inverse operator A-1 maps the transform domain XF
back to the original domain Xf
fA-1F
f
F
Xf
XF
Linear operators are of particular importance
Examples
9
Integral Transforms
Laplace transform (real t, complex s)
Fourier transform (real x and p)
Linear convolution with a kernel function K(x)
Important properties
10
Laplace Transform Illustration
Laplace transform gives a powerful tool for
solving ODE Example
Solution Apply Laplace transform to both sides
of this equation, accounting for linearity of LT
and using the property
y(t)
t
11
Discrete Fourier Transform (DFT)
Motivation discrete Fourier transform (DFT)
reduce solution of a large repetitive structure
to the analysis of one representative cell only.
Discrete functional sequences
DFT of infinite sequences
p wavenumber, a real value between p and p
DFT of periodic sequences
Here, p integer value between N/2 and N/2
Discrete convolution
12
DFT Illustration
Transform p-sequence
Original n-sequence
13
3.4 Standing Waves in Lattices
14
Wave Number Space and Dispersion Law
? 10d, p ?p/5
Wave number p is defined through the inverse wave
length ? (d interatomic distance) The waves
are physical only in the Brillouin zone
(range), The dispersion law shows dependence of
frequency on the wave number
? 4d, p ?p/2
? 2d, p ?p
? 10/11d, p ?11p/5 (NOT PHYSICAL)
15
Phase Velocity of Waves
The phase velocity, with which the waves
propagate, is given by Dependence on the
wave number Value v0 is the phase velocity
of the longest waves (at p ? 0).
16
3.5 Greens Function Methods
17
Periodic Structure Response (Greens) Function
n-2 n-1 n n1
n2
Dynamic response function Gn(t) is a basic
structural characteristic. G describes
lattice motion due to an external, unit momentum,
pulse
18
Lattice Dynamics Greens Function Example
Assume first neighbor interaction only
n-2 n-1 n n1
n2
Displacements
Velocities
Illustration (transfer of a unit pulse due to
collision)
19
Time History Kernel (THK)
The time history kernel shows the dependence of
dynamics in two distinct cells. Any time history
kernel is related to the response function.
f(t)
-2 -1 0 1
2
20
Elimination of Degrees of Freedom
Domain of interest
-2 -1 0 1
2
Equations for atoms nr1 are no longer required
21
3.6 Quasistatic Approximation
  • Miultiscale boundary conditions
  • Applications
  • Conclusions

22
Quasistatic MSBC
All excitations propagate with infinite
velocities in the quasistatic case. Provided
that effect of peripheral boundary conditions,
ua, is taken into account by lattice methods, the
continuum model can be omitted
Multiscale boundary conditions
Standard hybrid method
The MSBC involve no handshake domain with ghost
atoms. Positions of the interface atoms are
computed based on the boundary condition
operators T and ?. The issue of double counting
of the potential energy within the handshake
domain does not arise.
23
1D Illustration
1D Periodic lattice Solution for atom 0
can be found without solving the entire domain,
by using the dependence This the 1D
multiscale boundary condition
24
Application Nanoindentation
Problem description
R
C - Au L-J Potential
Diamond Tip
Au
FCC
Au - Au Morse Potential
25
Face centered cubic crystal
Bravais lattice
Numbering of equilibrium atomic positions (n,m,l)
in two adjacent planes with l0 and l1.
(Interplanar distance is exaggerated).
26
Atomic Potential and FCC Kernel Matrices
Morse potential K-matrices Fourier
transform in space Inverse Fourier transform
for r (evaluated numerically for all p,q and l)

27
Atomic Potential and FCC Kernel Matrices
Boundary condition operator in the transform
domain is assembled from the parametric matrices
G (a coarse scale parameter) Inverse
Fourier transform for p and q Final form of
the boundary conditions
Qn,m , element (1,1)
redundant block, if
This sum can be truncated, because T decays
quickly with the growth of n and m (see the plot).
28
Method Validation
a
1/4
Karpov, Yu, et al., 2005.
29
Compound Interfaces
Problem description
Edge assumption
30
Performance of Multiscale Boundary Conditions
Back half-domains are shown
Radius of Diamond Tip 1 nm Full MD domain size
36864 atoms Reduced domain size 3600 atoms For
1/18 of the original volume Computation time
has been reduced from 73 hours to 1 hour
Lattice deformation pattern is similar for the
benchmark and the multiscale simulations
1/18 of the original volume
31
MSBC Twisting of Carbon Nanotubes
The study of twisting performance of carbon
nanotubes is important for nanodevices. The MSBC
treatment predicts u1 well at moderate
deformation range. Efforts on computation for
all DOFs in the range between l 0 and a are
saved.
Load
a 20
(13,0) zigzag
Fixed edge
Large deformation
MSBC
l a
l 0
Qian, Karpov, et al., 2005
32
MSBC Bending of Carbon Nanotubes
The study of bending performance of carbon
nanotubes is important for nanodevices. The MSBC
treatment predicts u1 well at moderate
deformation range. Efforts on computation for
all DOFs in the range between l 0 and a are
saved.
Computational scheme
l a
Qian, Karpov, et al., 2005
l 0
33
MSBC Deformation of Graphene Monolayers
The MSBC perform well for the reduced domain MD
simulations of graphene monolayers Problem
description red fine grain, blue coarse
grain. Coarse grain DoF are eliminated by
applying the MSBC along the hexagonal interface
Indenting load
Tersoff-Brenner potential
Medyanik, Karpov, et al., 2005
34
MSBC Deformation of Graphene Nanomembranes
Shown is the reduced domain simulations with MSBC
parameter a10 the true aspect ration image
(non-exaggerated). Error is still less than 3.
Deformation
Comparison (red MSBC, blue benchmark)
Shown vertical displacements of the atoms
35
Conclusions on the MSBC
  • We have discussed
  • MSBC a simple alternative to hybrid methods
    for quasistatic problems
  • Applications to nanoindentation, CNTs, and
    graphene monolayers
  • Attractive features of the MSBC
  • SIMPLICITY
  • no handshake issues
    (strain energy, interfacial mesh)
  • in many applications, continuum model
    is not required
  • performance does not
    depend on the size of coarse scale domain
  • implementation for
    an available MD code is easy
  • Future directions
  • Dynamic extension
  • Passage of dislocations through the interface
  • Finite temperatures
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