Computational Study of Carbon Nanotubes under Compressive Loading
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Computational Study of Carbon Nanotubes under Compressive Loading Quasi-static reduced-order general continuum method with barycentric Interpolation. Yang Yang, William W. Liou Computational Engineering Physics Lab Western Michigan University Kalamazoo, Michigan.

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Yang Yang, William W. Liou

Computational Engineering Physics Lab

Western Michigan University

Kalamazoo, Michigan

36th Dayton-Cincinnati Aerospace Sciences Symposium

03/01/2011


Outline
Outline Loading

Introduction

Properties of carbon nanotubes

Applications of carbon nanotubes

Definition of carbon nanotubes

Numerical method

Overview

Reduced-order general continuum method

Simulation results

Model setup

Buckling patterns

Buckling patterns after barycentric conversion

Loading-unloading stress-strain curves for CNTs of

different types

Conclusions


Outline1
Outline Loading

Introduction

Properties of carbon nanotubes

Applications of carbon nanotubes

Definition of carbon nanotubes

Numerical method

Overview

Reduced-order general continuum method

Simulation results

Model setup

Buckling patterns

Buckling patterns after barycentric conversion

Loading-unloading stress-strain curves for CNTs of

different types

Conclusions


Introduction Loading

Properties of carbon nanotubes

  • Average diameter of SWNT

  • Carbon bond length

  • Density

  • Thermal conductivity

  • Young’s modulus of SWNT

  • Max. tensile strength


Introduction Loading

Properties of carbon nanotubes

  • Composed of all-carbon molecules in shell-like cylindrical

  • structure formed by strong covalent bonding of atoms

  • Tend to undergo buckling with compression or bending loads

  • One of the strongest materials known, both in terms of tensile

  • strength and elastic modulus


Introduction Loading

Applications of carbon nanotubes

  • Carbon nanotubes enhanced composite materials

  • Efficient heat remover composed of aligned structures and ribbons of CNTs

  • Drug delivery to prevent medicine from damaging healthy cells

  • Intrinsic tubule character of CNTs attributing to their very high surface area leads to the applications in energy storage material

  • Used as electrical conducting additives to producing conductive plastics

  • Flat panel CNT field emission display


Introduction Loading

Definition of carbon nanotubes


Outline2
Outline Loading

Introduction

Properties of carbon nanotubes

Applications of carbon nanotubes

Definition of carbon nanotubes

Numerical method

Overview

Reduced-order general continuum method

Simulation results

Model setup

Buckling patterns

Buckling patterns after barycentric conversion

Loading-unloading stress-strain curves for CNTs of

different types

Conclusions


Numerical method Loading

Overview

  • Classical molecular dynamics (MD)

Excels in modeling structural details of an atomic system by tracking

each atom

Computationally prohibitive for large systems; generally modeling a

system with the size up to a few hundred nanometers

  • Reduced-order general continuum method

Constitutive law is built based on an atomistic energy function by

intrinsic geometric quantities describing a deformation

No need for tracking individual atoms thus appropriate for modeling a

large system


Numerical method
Numerical method Loading

Reduced-order general continuum method

  • General idea

Every point in the continuum body

is described by a representative atom

embedded in a crystallite of radius

Finite elements discretizing the

continuum body


Numerical method1
Numerical method Loading

Reduced-order general continuum method

  • Cauchy-Born rule

  • Exponential map


Numerical method2
Numerical method Loading

Reduced-order general continuum method

  • REBO potential function for CNT

The repulsive pair:

The attractive pair:

The bond order term:


Numerical method3
Numerical method Loading

Reduced-order general continuum method

  • Lennard-Jones potential for long-range interaction


Numerical method4
Numerical method Loading

Reduced-order general continuum method

  • Atomic potential energies expressed in continuum variables

Interatomic energy density

Total interatomic energy over the CNT surface

Long-range Lennard-Jones energy for the CNT

  • Total energy of the CNT

  • Equilibrium state of the CNT correspondsMin ( )


Outline3
Outline Loading

Introduction

Properties of carbon nanotubes

Applications of carbon nanotubes

Definition of carbon nanotubes

Numerical method

Overview

Reduced-order general continuum method

Simulation results

Model setup

Buckling patterns

Buckling patterns after barycentric conversion

Loading-unloading stress-strain curves for CNTs of

different types

Conclusions


Simulation results
Simulation results Loading

Model setup

Fixed

end

Displacement

control B.C.

  • Buckling of different types of CNT under compressive loading

CNT cases studied

  • Displacement control method is used to apply the loading


Simulation results1
Simulation Results Loading

Buckling patterns

  • Van der Waals energy vs. strain Case 1

  • Total energy vs. strain Case 1

buckling


Simulation results2
Simulation Results Loading

Buckling patterns

Case 1

Case 2

Case 3

Case 4

before

after


Simulation results3
Simulation Results Loading

Buckling patterns


Simulation results4
Simulation Results Loading

Buckling patterns after barycentric conversion

  • Buckled state for Case 1

  • Incipient state for Case 1

  • Buckling events for Case 1


Simulation results5
Simulation Results Loading

Buckling patterns after barycentric conversion

  • Representative cells on the buckling surface of CNTs with different chiral angles.

(14, 0) CNT Case 1

(8, 8) CNT Case 4

(12, 3) CNT Case 2

(10, 5) CNT Case 3

  • The number of bonds that receives compressive load increases from Case 1 to Case 4

  • The bonds are compressed more uniformly in Case 4 than in Case 2 or Case 3


Outline4
Outline Loading

Introduction

Properties of carbon nanotubes

Applications of carbon nanotubes

Definition of carbon nanotubes

Numerical method

Overview

Reduced-order general continuum method

Simulation results

Model setup

Buckling patterns

Buckling patterns after barycentric conversion

Loading-unloading stress-strain curves for CNTs of

different types

Conclusions


Conclusions
Conclusions Loading

  • The reduced order general continuum method was used to study the behaviors of CNTs under compressive loading conditions.

  • Reverse mapping of the finite element results to the associated CNT lattice deformation using barycentric interpolation.

  • Different buckled configurations will be assumed by CNTs with different chiral angles.

  • The zigzag CNT has the most apparent buckling pattern.

  • The buckling strain increases with the increasing chiral angle.

  • The armchair CNT has the strongest resistance to the compressive loading.


Thank you
Thank Loading you!


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