Nano Mechanics and Materials:Theory, Multiscale Methods and Applications

byWing Kam Liu, Eduard G. Karpov, Harold S. Park

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

Role of Computational Methods

Nano- and micro-structureElectronic structure Molecular mechanicsContinuum mechanics

PotentialsConst. lawsPlasticity

Multiscale methods

Manufacturing platformFunction

Performance Reliability

Computations and design

Prediction

Validation

• Structural and material design

• Optimization

• Prediction and validation

Examples of Multi-Scale Phenomena in Solids

17:1

250:1250:1

200 m

Shear bands Mechanics of carbon nanotubes

Figures: D. Qian, E. Karpov, NU Shaofan Li, UC-BerkeleyMovie: Michael Griebel, Universität Bonn

Why Multiscale Methods?

Limitations of industrial simulations today:

a) 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

b) 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

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

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).

MAAD: Concurrent Coupling of Length Scales

/

/

, , , ,

, ,

,

Tot FE FE MD

MD MD TB

TB

H H H

H H

H

u u u u r r

r r r r

r r

• 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

Nakano et al, Comput. In Sci. and Eng., 3(4) (2001).

Broughton, et al, PRB 60(4) (1999).

Handshake at MD/FE interface

Quasicontinuum Method

Tadmor and Phillips, Langmuir 12, 1996

• 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

T

W

P

F

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

Typical Issues

1. True coarse scale discretization and coupling between the scales

2. Handling interfaces where small and large scales intersect; handshake is expensive and non-physical; spurious wave reflection

3. Double counting of the strain energy

4. Implementation: usage of existing MD and continuum codes is hard; parallel computing

5. Dynamic mesh refinement/enrichment

6. Finite temperatures

7. 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

The Bridging Scale Method

( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( )

x x xx xx x x x x

u u uu Puu u Pu I P u Qu

= +

• 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:

( )xu ( )xu ( )xu

BS projectionT

1 T

A

A

M N M NQ I NM N M

Within the bridging scale method, the MD and FE formulation exist simultaneously over the entire computational domain:

+ =

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

' u u u Nd Qq

T

T

( )( )A

U

Md N f u

fuM q Q f u

Bridging-Scale Equations of Motion

MD, q FEM, d MD + FE, (q, d)

T T T1 1( , , , ) ( , )2 2 AL U d d q q d Md q Q M q d q

Impedance Boundary Conditions / MD Domain Reduction

MD degrees of freedom outside the localized domain are solved implicitly

+

Due to atomistic nature of the model, the structural impedance is evaluated computed at the atomic scale.

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:

FE + Reduced MD + Impedance BC

MD FE

T

0

( )( ) ( ) ( ) ( )

t

A t d

Md N f uM q f u Θ q u

1D Illustration: Non-Reflecting MD/FE InterfaceImpedance 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 1:10

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.

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 MDFEM FEM

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).

-1 -0.5 0.5 1

0.5

1

1.5

2

2.5

3

/p

0/ continuum

vp

-6 -4 -2 2 4 6

0.2

0.4

0.6

0.8

1

/p

continuum

0/v v

0 0

12sin sin2 2p v p

v p

Incompatible Dispersion Properties of Lattices and Continua

lattice structure

lattice structure

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

Selected References

Quasicontinuum Method E. Tadmor, M. Ortiz and R. Phillips, Philosophical Magazine A 1996;

73:1529-1563 Coupled Atomistic/Discrete Dislocation method (CADD)

L. Shilkrot, R.E. Miller and W.A. Curtin, Journal of the Mechanics and Physics of Solids 2004; 52:755-787

Bridging Domain method S.P. Xiao and T. Belytschko, Computer Methods in Applied

Mechanics and Engineering 2004; 193:1645-1669 Review articles:

W.A. Curtin and R.E. Miller, Modelling and Simulation in Materials Science and Engineering 2003; 11:R33-R68

W.K. Liu, E.G. Karpov. S. Zhang and H.S. Park, Computer Methods in Applied Mechanics and Engineering 2004; 193:1529-1578