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

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5. Introduction to Multiple Scale Modeling

• Motivation for multiple scale methods
• Coupling of length scales
• Bridging scale concurrent method
• Molecular dynamics (MD) boundary condition
• Numerical examples
• 1D wave propagation
• 2D dynamic crack propagation
• 3D dynamic crack propagation
• Discussion/Areas of improvement for bridging
  scale
• Conclusions and future research

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Role of Computational Methods

• Structural and material design
• Optimization
• Prediction and validation

Nano- and micro-structure Nano- and micro-structure
Electronic structure Molecular mechanics Continuum mechanics Potentials Const. laws Plasticity
Manufacturing platform Function Performance
Reliability
Prediction Validation
Computations and design
Multiscale methods
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Examples of Multi-Scale Phenomena in Solids
Shear bands
Mechanics of carbon nanotubes
Figures D. Qian, E. Karpov, NU
Shaofan Li, UC-Berkeley Movie Michael Griebel,
Universität Bonn
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Why Multiscale Methods?

• Limitations of industrial simulations today
• Continuum models are good, but not always
  adequate
• Problems in fracture and failure of solids
  require improved constitutive models to describe
  material behavior
• Macroscopic material properties of new materials
  and composites are not readily available, while
  they are needed in simulation-based design
• Detailed atomistic information is required in
  regions of high deformation or discontinuity
• Molecular dynamics simulations
• Limited to small domains (106-108 atoms) and
  small time frames (nanoseconds)
• Experiments, even on nano-systems, involve much
  larger systems over longer times
• Opportunities 1) Obtain material properties by
  subscale (multiscale) simulation
• 2) Enrich information
  about material/structural performance across
  scales
• via concurrent
  multiscale methodologies

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Hierarchical vs. Concurrent

• Hierarchical approach
• Use known information at one scale to
  generate model for larger scale
• Information passing typically through some sort
  of averaging process
• Example bonding models/potentials,
  constitutive laws
• Concurrent approach
• Perform simulations at different length
  scales simultaneously
• Relationships between length scales are dynamic
• Classic example heat bath techniques

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Macroscopic, Atomistic, Ab Initio Dynamics (MAAD)

• Finite elements (FE), molecular dynamics (MD),
  and tight binding (TB) all used in a single
  calculation (MAAD)
• MAAD macroscopic, atomistic, ab initio dynamics
• Atomistics used to resolve features of interest
  (crack)
• Continuum used to extend size of domain
• Developed by Abraham (IBM), Broughton (NRL), and
  co-workers

From Nakano et al, Comput. In Sci. and Eng., 3(4)
(2001).
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MAAD Concurrent Coupling of Length Scales

• Scales are coupled in handshake regions
• Finite element mesh graded down to atomic
  lattice in the overlap region
• Total Hamiltonian is energy in each domain,
  plus overlap regions

Broughton, et al, PRB 60(4) (1999).
Handshake at MD/FE interface
Nakano et al, Comput. In Sci. and Eng., 3(4)
(2001).
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Quasicontinuum Method

• Developed by Ortiz, Phillips and coworkers in
  1996.
• Deformation is represented on a triangulation
  of a subset of lattice points points in between
  are interpolated using shape functions and
  summation rules
• Adaptivity criteria used to reselect
  representative lattice points in regions of high
  deformation
• Applications to dislocations, grain boundary
  interactions, nanoindentation, and fracture
  (quasistatic modeling)
• Cauchy-Born rule assumes 1) continuum energy
  density can be derived from the atomic potential
  2) deformation gradient F describes deformation
  at both continuum and atomic scales, and
  therefore serves as the link. Thus, atomic
  deformation has to be homogeneous
• Issues non-local interaction, long
  dislocations/ill conditioning, separation of
  scales, finite temperatures, universal scenarios
• Later improved by Arroyo, Belytschko, 2004, in
  application to CNT PRB 69, article 115415

Tadmor and Phillips, Langmuir 12, 1996
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Challenges

• Large number of degrees of freedom at the
  atomic scale
• Interfaces mismatch of dynamic properties,
  and other issues
• Consistent and accurate representation of
  meso-, micro- nano- level behavior within
  continuum models
• Multiple time scales
• Potentials
• Interdisciplinary nature of multiscale
  methods
• - continuum mechanics
• - classical particle dynamics (MD), and lattice
  mechanics
• - quantum mechanics and quantum chemistry
• - thermodynamics and statistical physics
• Atomic scale plasticity lattice dislocations
  
• Finite temperatures
• Entropic elasticity, soft materials
• Dynamics of infrequent events diffusion,
  protein dynamics
• Algorithmic issues in large scale coupled
  simulations

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Typical Issues

• True coarse scale discretization and coupling
  between the scales
• Handling interfaces where small and large scales
  intersect handshake is expensive and
  non-physical spurious wave reflection
• Double counting of the strain energy
• Implementation usage of existing MD and
  continuum codes
  is hard parallel computing
• Dynamic mesh refinement/enrichment
• Finite temperatures
• Multiple time scales and dynamics of infrequent
  events
• - BSM has resolved issues 1-2, and
  partially 3-6.
• - The alternative MSBC method,
  where issues 1-4 DO NOT ARISE

Typical interface model
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The Bridging Scale Method

• Two most important components
• - bridging scale projection
• - impedance boundary conditions
  applied MD/FE interface in the form of a
  time-history integral
• Assumes a single solution u(x) for the entire
  domain. This solution is decomposed into the fine
  and coarse scale fields

BS projection


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Bridging-Scale Equations of Motion
Within the bridging scale method, the MD and FE
formulation exist simultaneously over the entire
computational domain
MD FE, (q, d)
FEM, d
MD, q


The total displacement is a combination of the FE
and MD solutions Multiscale Lagrangian
Lagrangian formulation gives coupled, coarse
and fine scale, equations of motion
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Impedance Boundary Conditions / MD Domain
Reduction
The MD domain is too large to solve, so that we
eliminate the MD degrees of freedom outside the
localized domain of interest. Collective atomic
behavior of in the bulk material is represented
by an impedance force applied at the formal
MD/continuum interface
MD degrees of freedom outside the localized
domain are solved implicitly
FE Reduced MD Impedance BC
MD
FE

Due to atomistic nature of the model, the
structural impedance is evaluated computed at
the atomic scale.
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1D Illustration Non-Reflecting MD/FE Interface
Impedance boundary conditions allows
non-reflecting coupling of the fine and coarse
grain solutions within the bridging scale
method. Example Bridging scale simulation of a
wave propagation process ratio of the
characteristic lengths at fine and coarse scales
is 110 Direct coupling with continuum

Impedance BC are involved
Over 90 of the kinetic wave energy is reflected
back to the fine grain.
Less than 1 of the energy is reflected.
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Why is Multiscale Modeling Difficult?

• Wave reflection at MD/FE interface
• Larger length scales (FE) cannot represent wave
  lengths typically found at smaller length scales
  (MD)
• Also due to energy conserving formulations for
  both MD and FEM

MD
MD
FEM
FEM
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Incompatible Dispersion Properties of Lattices
and Continua
The phase velocity of progressive waves is given
by Dependence on the wave number Value
v0 is the phase velocity of the longest waves (at
p ? 0).
lattice structure
lattice structure
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Issues in Multiscale Modeling

• Preventing high frequency wave reflection
• - Need reduced MD system to behave like full MD
  system
• - Reflection of high frequency waves can lead
  to melting of the atomistic system
• Need for a dynamic, finite temperature multiple
  scale method
• True coarse scale representation
• - No meshing FEM down to MD lattice spacing
• - Different time steps for MD and FEM
  simulations
• Mathematically sound and physically motivated
  treatment of high frequency waves emitted from MD
  region at MD/FE interface

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Selected References

• Quasicontinuum Method
• E. Tadmor, M. Ortiz and R. Phillips,
  Philosophical Magazine A 1996 731529-1563
• Coupled Atomistic/Discrete Dislocation method
  (CADD)
• L. Shilkrot, R.E. Miller and W.A. Curtin, Journal
  of the Mechanics and Physics of Solids 2004
  52755-787
• Bridging Domain method
• S.P. Xiao and T. Belytschko, Computer Methods in
  Applied Mechanics and Engineering 2004
  1931645-1669
• Review articles
• W.A. Curtin and R.E. Miller, Modelling and
  Simulation in Materials Science and Engineering
  2003 11R33-R68
• W.K. Liu, E.G. Karpov. S. Zhang and H.S. Park,
  Computer Methods in Applied Mechanics and
  Engineering 2004 1931529-1578