1. Multi-scale Modeling of Nanocrystalline Materials
N Chandra and S Namilae
Department of Mechanical Engineering
FAMU-FSU College of Engineering
Florida State University
Tallahassee, FL USA
Presented at
ICSAM2003, Oxford, UK, July 28, 03

2. 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.

3. Mechanics at atomic scale
Molecular Dynamics
-Fundamental quantities (F,u,v)
Compute Continuum quantities
-Kinetics (,P,P’ )
-Kinematics (,F)
-Energetics
Use Continuum Knowledge
- Failure criterion, damage etc

4. 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

5. Virial Stress Stress defined for whole system For Brenner potential:
Includes bonded and non-bonded interactions
(foces due to stretching,bond angle, torsion effects)

6. 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

7. 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

8. 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

9. 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.

10. 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)

11. 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

12. Elastic Deformation-Strain profiles
9(2 2 1) Grain boundary
Subject to in plane deformation
Strain intensification observed
At the grain boundary

13. 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

14. Stress-Strain response of GB
Stress Strain response of bicrystal bulk and at grain boundary
Grain boundary exhibits lower modulus than bulk GB

15. Grain Boundary Sliding Simulation
Generation of crystal for simulation of sliding. Free boundary conditions in X and Y directions, periodic boundary condition in Z direction. Y’ X’
Simulation cell contains about to atoms
BC for sliding are shown. Experimental studies are performed on bicrystals oriented at 45 deg. Similar BC were used and shear state of stress was applied. Temp of 450K .
The crystals contained about atoms and 5000 iterations were done.
A state of shear stress is applied
T = 450K
Grain boundaries studied: 3(1 1 1), 9(2 2 1),
11 ( ), 17 ( ), 43 (5 5 6 ) and 51 (5 5 1)

16. Sliding Results
Grain boundary sliding is more in the boundary, which has higher grain boundary energy
Fig.6 Extent of sliding and Grain boundary energy Vs misorientation angle
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)

17. 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

18. 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)
Schematic of macro and micro scales

19. 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.

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

21. 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

22. 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

23. 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 E=71 GPA

24. 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.