computation of spatial kernel of carbon nanotubes in non-local elasticity theory

MULTI-SCALE STRUCTURAL SIMULATIONS LABORATORY

Computation of Spatial Kernel of Carbon Nanotubes in Non-Local

Elasticity Theory

Veera Sundararaghavan

Assistant Professor of Aerospace Engineering

Anthony Waas

Felix Pawlowski Collegiate Professor of Aerospace Engineering

University of Michigan, Ann Arbor


Non-local nature at nano-scales

Source: Dr. Taner, NIST

-Charge distribution around an atom depends on atoms over a region of influence.

- Molecular mechanics approaches employ force fields that are non-local in nature.

Quantum mechanical simulations: Charge densities

Molecular mechanics: Class II Force Fields


Non-local fields at nano-scale

-Nanomechanics integrates solid mechanics with atomistic simulations

-Molecular mechanics is inherently non-local.

-Conventional continuum mechanics assumes local-fields.

Potential cutoff

Non-local interactions between atoms


Local versus Non-local theory

• Non-local theory incorporates long range interactions between points in a continuum model.

• Stress at a point depends on the strain in a region near that point (Works of Eringen, Aifantis, Kunin)

- Non-local elasticity

- Local elasticity

Non-local kernel

Local kernel (delta function)


Implications of non-local elasticity

• An integrated approach to study phenomena at continuum- and nano- scales

• Advantages include: – 1) Contains internal length

scale to capture size effects – 2) No singularities at crack

tips and dislocation cores – 3) Correctly predicts

energetics of point defects.– 4) Predicts non-linear wave

dispersion

Kernel properties:1. Kernel is normalized with

respect to the volume

2. Kernel reverts to a delta function as the zone of influence vanishes (leading to the local elasticity formulation).

Non-local kernel

Local kernel (delta function)


Example: Computation of dispersion curves

is the Fourier transformed kernel

- Non-local elasticity

- Local elasticity

- 1D Dispersion curve (Non-local elasticity)

- 1D Dispersion curve is linear (local elasticity)2

2

u

x t

2

2

s u

x t


Approaches to construct the non-local kernel

• For pairwise potentials, a unique kernel can be derived (Picu, 2000).

• Direct fitting with atomistic dispersion curves (Eringen) works for all interatomic potentials. The form of the kernel is assumed (usually Gaussian)

• Simple forms of kernel (stress- and strain- gradient theories) use a single parameter that is fitted to a property of interest (eg. Critical buckling strain (Zhang, 2005), Elastic modulus (Wang, 2008))


Atomistic simulations – Force field

• Force Field model MD Simulation of phonon vibrations at 300 K

Walther et al (2001)


Atomistic simulations to construct dispersion curves

The nanotube structure is obtained from a graphene sheet by rolling it up along a straight line connecting two lattice points (with translation vector (L1,L2)) into a seamless cylinder in such a way that the two points coincide.


Helical Symmetry Lattice dynamics of SWNT

The nanotube can be considered as a crystal lattice with a two atoms unit cell and the entire nanotube may be constructed using screw operators.

Helical symmetry analysis (Popov et al 2000):

Wave-like solution with helical symmetry

Equations of motion:

Rotational boundary condition and translational periodicity constraints of the nanotube

Here, l is an integer number (l = 0; ..;Nc-1, where Nc is the number of atomic pairs in the translational unit cell of the tube), and the integers N1 and N2 define the primitive translation vector of the tube.

Advantages:- Gives 4 acoustic branches without correction to potentials (Saito PRB 1998)- Computation time is for 2 atoms independent of chirality


Atomistic simulations – Helical symmetry approach

Eigenvalues:

Solve for:

The equations of motion described above yield the eigenvalues (ql) where l labels the modes with a given wave number q in the one-dimensional Brillouin zone.

Final equations of motion:


Atomistic simulations to construct dispersion curves


Gradient theories

• Stress gradient theory

Can be derived by assuming that thekernel is of a special form that satisfies:

This leads to the Constitutive equation for stress gradient theory

‘c’ is a single parameter that is fitted. The data is fitted by matching dispersion curves at the end of the Brillouin zone (ka=).The parameter c is a product of a material specific parameter (eo) and an internal (eg. lattice) parameter.

Dispersion curve


Gradient theories

In strain gradient theory, that above equation is written in terms of the local stress and higher powers of c are neglected:

Constitutive equation for stress gradient theory

1D rod model comparison of gradient theories

Strain gradient theoryStress gradient theory

22

2

( )( ) ( )

d s xs x c E x

dx

22

2

( )( ) ( ( ) )

d xs x E x c

dx


Comparison of dispersion curves

Results from Literature:

e0 = 0.82 (Zhang 2005) for stress gradient theory using critical buckling strain (From molecular statics result of Sears and Batra 2004)

e0 = 0.288 for strain gradient theory (Wang and Hu 2005) using MD calculations0 0.5 1 1.5 2 2.5 3 3.5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

ka

/

o

Atomistic data CNT (10,10)Stress gradient (e

o = 0.98)

Strain gradient (eo = 0.304)


Reconstructed FT Kernel Comparison

0 0.5 1 1.5 2 2.5 3 3.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ka

(k)

Atomistic dataStress gradient modelStrain gradient model


Development of 3D kernels for nanotubes

Short (6,6) nanotube (L=14.77A, Lc=13.52 A)

-Kernels for shell-type models need to be constructed and validated by comparing the critical buckling strains for CNTs of different chiralities and lengths with atomistic simulations.

-Molecular statics simulation to compute critical buckling strain. BFGS scheme used to equilibrate positions of atoms after application of longitudinal strain

(5,5) nanotube (L=24.62 A, Lc=23.1 A)


Flugge’s shell theory – Kinematics

Non-local forces and moments

Local shell theory

From force constants and Hu (2008)

Stress gradient version in Wang and Varadan (2007)


Flugge’s shell theory – Dispersion relations

Axisymmetric modes are modeled:


Validation with MD results for torsional waves

Torsion equation is decoupled from the other two equations

Dispersion equation in torsional mode


Kernel construction in Non-local shell theory

Fourier transform of the non-local kernel ((k)) can be reconstructed by plugging in the dispersion data ( versus k) obtained from atomistic simulations directly in the following expression (where |.| is the matrix determinant):

Dispersion relation for radial and longitudinal waves:

Dispersion relation for torsional waves:


Comparison with gradient theories


Displacement controlled tests

Tension Torsion

Energy changes using non-local theory with kernel

Atomistic testing


Atomic simulation vs Non-local theory

Size effect in Young’s modulus and shear modulus


New Gaussian Kernel

New kernel predicts both dispersion and shear modulus variation adequately


Reconstructured non-local kernel – (10,10) SWNT

Negative kernel at larger distances (also observed by Picu (JMPS 2002))

“kernel should change sign close to the inflection point of the interatomic potential”


Perturbation analysis to find spring constantsThe second layer interaction energies are negative as predicted by the calibrated kernel!

Energies computed from a perturbation analysis


Conclusions• The longitudinal, transverse and torsional

axisymmetric mode wave dispersions in single walled carbon nanotube (SWCNT) were studied in the context of nonlocal elasticity theory.

• Atomistic dispersion studies indicate that a Gaussian kernel is able to offer a better prediction for torsional wave dispersion in CNTs and the size effect than the non-local kernel from gradient theory.

• We postulated and confirmed that the fitted kernel changes sign close to the inflection point of the interatomic potential through an atomistic study of layer-by-layer interaction of atoms in a carbon nanotube.

Future Work• Development of a anisotropic non-local FE

approaches for modeling defect evolution (work in progress).

computation of spatial kernel of carbon nanotubes in non-local elasticity theory

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