Application of Asymptotic Expansion Homogenization to Atomic Scale

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Application of Asymptotic Expansion
Homogenization to Atomic Scale
N Chandra and S Namilae
Department of Mechanical Engineering FAMU-FSU
College of Engineering Florida State University
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Why link atoms and continuum ?

• Atomic details (structural and Material) have
  profound influence
• on properties
• -Thermomechanical
• -Physical, electrical, magnetic
• However computational problems
• -100 nm cube of Si billion atoms

• Macroscopic phenomenon effected by atomic scale
  details

Grain boundaries
Creep /SP ?
Fracture ? Crack tip
Nanotechnology
Plasticity ? Dislocation
Materials by design
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Problems in Atomic scale domain

• Grain boundaries play a important role in the
  strengthening and deformation of metallic
  materials.
• Some problems involving grain boundaries
• Grain Boundary Structure
• Grain boundary Energy
• Grain Boundary Sliding
• Effect of Impurity atoms

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Equilibrium Grain Boundary Structures
110?3 and 110?11 are low energy boundaries,
001?5 and 110?9 are high energy boundaries
GB
GB
110?3 (1,1,1)
001?5(2,1,0)
GB
GB
110?9(2,21)
110?11(1,1,3)
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Grain Boundary Energy Computation
GBE ? (Eatoms in GB configuration) N ? Eeq(of
single atom)
Calculation
Experimental Results1
1 Proceeding Symposium on grain boundary
structure and related phenomenon, 1986 p789
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Grain Boundary Sliding Simulation
Y
X
Simulation cell contains about 14000 to 15000
atoms
A state of shear stress is applied
T 450K
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Sliding Results
Grain boundary sliding is more in the boundary,
which has higher grain boundary energy
Monzen et al1 observed a similar variation of
energy and tendency to slide by measuring
nanometer scale sliding in copper
Reversing the direction of sliding changes the
magnitude of sliding
1
Monzen, R Futakuchi, M Suzuki, T Scr. Met.
Mater., 32, No. 8, pp. 1277, (1995) Monzen, R
Sumi, Y Phil. Mag. A, 70, No. 5, 805,
(1994) Monzen, R Sumi, Y Kitagawa, K Mori, T
Acta Met. Mater. 38, No. 12, 2553 (1990)
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Mg Segregation in Al Grain Boundaries
Variation of grain boundary energy in presence of
Mg atom
Segregation of Mg atoms to particular locations
in grain boundary is based on size effect and
hydrostatic pressure
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Hydrostatic Stress and Segregation Energy
Distribution of atoms around impurity atom in ?9
STGB
Grain boundary energy and segregation are
influenced by changes in coordination of atoms at
grain boundary
Effect of Mg on sliding
Simulation results also indicate that there is an
increase In grain boundary sliding when Mg atoms
are present
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Problems in macroscopic domain influenced by
atomic scale

• MD provides useful insights into phenomenon like
  grain boundary sliding
• Problems in real materials have thousands of
  grains in different orientations
• Multiscale continuum atomic methods required

A possible approach is to use Asymptotic
Expansion Homogenization theory with strong math
basis, as a tool to link the atomic scale to
predict the macroscopic behavior
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Continuum-Atomics linking

• Sinclair (1975) Hoagland et.al (1976)
• Mullins (1982)
• Gumbusch et.al. (1991)
• Tadmor et.al. (1996), Shenoy et.al. (1999)

• Flexible Border Technique
• Finite element Atomistic method
• FE-At method
• Quasicontinuum method

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Continuum-Atomics linking

• Rafii Tabor (1998)
• Broughton et.al. (2000)
• Lidorkis et. al. (2001)
• Friesecke and James

• Three scale model
• Coarse grained molecular dynamics
• Handshaking methods -CLS
• Multiscale scheme

Other efforts CZM based, description of
continuum in atomic Regions, lipid membranes etc
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Homogenization methods for Heterogeneous Materials

• Heterogeneous Materials e.g. composites, porous
  materials
• Two natural scales, scale of second phase (micro)
  and scale of overall structure (macro)
• Computationally expensive to model the whole
  structure including fibers etc
• Asymptotic Expansion Homogenization (AEH)

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AEH idea
Overall problem decoupled into Micro Y scale
problem and Macro X scale problem
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AEH literature

• Functional analysis
• Bensoussan et.al. (1978), Sanchez Palencia
  (1980)
• Elasticity well established
• Kikuchi et. al. (1990) Adaptive mesh refinement
• Hollister et. al. (1991) Biomechanics Application
• Ghosh et. al. (1996),(2001) AEH combined with
  VCFEM
• Buannic et. al. (2000) Beam theory with AEH
• Inelastic Problems
• Fish et.al. (2000) Plasticity
• Chung et.al. (2001) Viscoplasticty
• Transport Problems in Porous media

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Formulation

• Let the material consist of two scales, (1) a
  micro Y scale described by atoms interacting
  through a potential and (2)a macro X scale
  described by continuum constitutive relations.
• Periodic Y scale can consist of inhomogeneities
  like dislocations impurity atoms etc
• Y scale is Scales related through ?
• Field equations for overall material given
  by

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Contd
The basic concept in AEH is to expand the primary
variables as an asymptotic series. Using the
expansion for displacement u
The functions u(i) (x,y) are Y periodic in
variable y. and are independent of the scaling
parameter ?.
From the definition of the scaling parameter, for
any g(x,y)
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Hierarchical Equations
Strain can be expanded in an asymptotic expansion
Substituting in equilibrium equation ,
constitutive equation and separating the
coefficients of the powers of ? three
hierarchical equations are obtained as shown
below.
Micro equation
Macro equation
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Microscale Equation
Using the following transformation
Micro equation can be solved as
In Variational form

• ? corrector term in macro scale due to
  microscale perturbations.
• series of vectors

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Microscale Equation

• The Y scale here is composed of atoms
  interacting through an
• interatomic potential.
• If we consider a finite element mesh refined to
  atomic scale in the Y
• region then, would denote the atomic level
  stiffness matrix
• W is the total strain energy density of the Y
  scale and q dente the
• displacements of individual atoms.
• Micro equation can be solved as

q ? Atomic displacements
Cloc ? Local elastic constants determined from
MD
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Macroscale equation
Given by
apply the mean operator on this equation, by
virtue of Y-periodicity of u(2) equation reduces
to
(A)
C H is the homogenized elasticity matrix for the
overall region given by

Equation (A) solved by FEM with appropriate BC
gives solution corrected for atomic scale effects
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Local Elastic Constants
Based on Kluge et al J. of App. Phy. (1990)
Knowing local strain and local stress in a small
region V of MD Simulation local elastic constants
system of N interacting atoms in a parallelepiped
whose edges are described by vectors a, b and c
with H(a,b,c)
Constant strain application HHo to HHo? Ho
(Parinello Rahman Variable cell MD)
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Local Elastic constants
Local stress in small area defined as

• ? volume ,
• rij ? distance between ith and jth atoms,
• U ? interatomic potential function
• ?unit step function
• ? Dirac delta function
• Rij ? center of mass of particles i and j

This Method has been applied to grain boundaries
using EAM and pair potentials
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Computational Procedure

• Create an atomic model of microscopic Y scale
• Use molecular dynamics to obtain the material
  properties at various defects such as GB,
  dislocations etc. Form the ? matrix and
  homogenized material properties
• Make an FEM model of the overall (X scale)
  macroscopic structure and solve for it using the
  homogenized equations and atomic scale properties

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Summary

• Incorporating atomic-scale effects in determining
  the material behavior is important in a number of
  engineering applications.
• Grain boundaries structure and deformation
  characteristics can be studied at atomic scale.
  Using Molecular Dynamics it has been shown that
  extent of grain boundary sliding is related to
  grain boundary energy
• The formulation for AEH to link atomic to macro
  scales has been proposed with detailed derivation
  and implementation schemes.
• Work is underway to implement the computational
  methodology.