Multi-scale Modeling of Nanocrystalline Materials

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Multi-scale Modeling of NanocrystallineMaterials
N Chandra and S Namilae
Department of Mechanical Engineering FAMU-FSU
College of Engineering Florida State
University Tallahassee, FL 32312 USA Presented
at ICSAM2003, Oxford, UK, July 28, 03
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Nano-crystalline materials and Nanotechnology ?

• Richard Feynman in 1959 predicted that
• There is a lot of room below
• Ijima in 1991 discovered carbon nanotubes that
• conduct heat more than Copper
• conduct electricity more than diamond
• has stiffness much more than steel
• has strength more than Titanium
• is lighter than feather
• can be a insulator or conductor just based on
  geometry
• Nano refers to m (about a few atoms in
  1-D)
• It is not a miniaturization issue but finding
  new science, nano-science-new phenomena
• At this scale, mechanical, thermal, electrical,
  magnetic, optical and electronic effects interact
  and manifest differently
• The role of grain boundaries increases
  significantly in nano-crystalline materials.
  

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Mechanics at atomic scale
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Stress at atomic scale

• Definition of stress at a point in continuum
  mechanics assumes that homogeneous state of
  stress exists in infinitesimal volume surrounding
  the point
• In atomic simulation we need to identify a volume
  inside which all atoms have same stress
• In this context different stresses- e.g. virial
  stress, atomic stress, Lutsko stress,Yip stress

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Virial Stress
Stress defined for whole system
For Brenner potential
Includes bonded and non-bonded interactions
(foces due to stretching,bond angle, torsion
effects)
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BDT (Atomic) Stresses
Based on the assumption that the definition of
bulk stress would be valid for a small volume ??
around atom ?
- Used for inhomogeneous systems
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Lutsko Stress
- fraction of the length of ?-? bond lying inside
the averaging volume

• Based on concept of local stress in
• statistical mechanics
• used for inhomogeneous systems
• Linear momentum conserved

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Strain calculation

• Displacements of atoms known
• Lattice with defects such as GBs meshed as
  tetrahedrons
• Strain calculated using displacements and
  derivatives of shape functions
• Borrowing from FEM
• Strain at an atom evaluated as weighted average
  of strains in all tetrahedrons in its vicinity
• Updated lagarangian scheme used for MD

GB
Mesh of tetrahedrons
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GB as atomic scale defect

• 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
• We need to model GB for its thermo-mechanical
  (elastic and inelastic) properties possibly using
  molecular dynamics and statics.

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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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Elastic Deformation-Strain profiles
?9(2 2 1) Grain boundary Subject to in plane
deformation
Strain intensification observed At the grain
boundary
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Stress profile

• Stress Calculated in various regions calculated
  using lutsko stress
• Stress Concentration observed at the grain
  boundary
• Stress concentration present at 0 strain
  indicating residual stress due to formation of
  grain boundary

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Stress-Strain response of GB

• Stress Strain response of bicrystal bulk and at
  grain boundary
• Grain boundary exhibits lower modulus than bulk

GB
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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
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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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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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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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Three Scale models to link disparate scales

• Conventional AEH approach fails when strong
  stress or strain localizations occur (as in crack
  problem)
• molecular dynamics in the region of localization
• Conventional non-linear/linear FEM for macroscale
• Displacements, energies and forces are
  discontinuous across the interface connecting two
  descriptions.
• Handshaking method handshaking methods to join
  the two regions

A three scale modeling approach using non-linear
FEM with or without AEH to model macroscale and
MD to model nano scale and a handshaking method
to model the transition between macro to nano
scale.
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AEH idea
Overall problem decoupled into Micro Y scale
problem and Macro X scale problem
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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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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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Computational Procedure

• Create an atomically informed 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

• Y scale as polycrystal with 7
• grains as shown above (50A)
• Grain boundary 2A thick
• Elastic constants informed
• from MD
• E for GB 63GPA
• Homogenized E71 GPA

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Summary

• Nanoscience based nanotechnology offers a great
  challenge and opportunity.
• Combining superplastic deformation with other
  physical phenomena in the design/manufacture/use
  of nanoscale devices (not necessarily large
  structures) should be explored.
• MD/MS based simulation can be used to understand
  the mechanics (static and flow) of interfaces,
  surfaces and defects including GBs.
• 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.

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