2D-to-3D Deformation Gradient:

In-plane stretch: 2D Green-Lagrange Strain Tensor:

Bending: 2D Curvature Tensor:

2nd Piola-Kirchoff Stress and Moment:

Tangent Modulus:

Incremental stress-strain relation (nonlinear and anisotropic):

Department of Aerospace Engineering and Engineering Mechanics The University of Texas at Austin

Qiang Lu, Wei Gao and Rui Huang

Nonlinear Mechanics of Graphene-Based Materials

Introduction

Grant Title: Nonlinear Mechancis of Graphene-Based Materials Grant Number: 0926851 NSF Program: Mechanics of Materials PI Name: Rui Huang

Nonlinear Continuum Model of Graphene

Uniaxial Stretch of Monolayer Graphene Graphene Nanoribbon (GNR)

References

Grant Information

Molecular Mechanics Minimize potential energy to simulate a static equilibrium state. Molecular Dynamics Study the dynamic process like fracture and temperature effects. Empirical Potential: 2nd generation REBO potential

d dx F XX1

X2

J

iiJ X

xF

JKiKiJJK FFE 2

1

JI

ii

J

iIiIJ XX

xn

X

Fn

2

KLIJKL

IJIJKL EEE

SC

2

KLIJKL

IJIJKL

MD

2

KLIJIJ

KL

KL

IJIJKL EE

MS

2

12

22

11

333231

232221

131211

12

22

11

333231

232221

131211

12

22

11

22 d

d

d

dE

dE

dE

CCC

CCC

CCC

dS

dS

dS

12

22

11

332313

322212

312111

12

22

11

333231

232221

131211

12

22

11

22 dE

dE

dE

d

d

d

DDD

DDD

DDD

dM

dM

dM

IJIJ

IJIJ M

ES

,

coupling between tension and bending

Graphene is a one-atom-thick planar sheet of sp2 –bonded carbon atoms that are densely packed in a honeycomb crystal lattice.

Motivation: Develop a theoretical framework to study mechanical properties of monolayer graphene and its derivatives.

Approach:- Develop a nonlinear continuum mechanics model for 2D sheets under

arbitrary deformation.- Conduct atomistic simulations to study the response of graphene

under different loading conditions.- Combine continuum and atomistic methods to obtain fundamental

mechanical properties.

Anisotropic Tangent Moduli

Graphene is linear and isotropic under infinitesimal deformation, but becomes nonlinear and anisotropic under finite strain.

Fracture Strength

Fracture occurs as a result of intrinsic instability of the homogeneous deformation.

Atomistic Modeling Method

Bending of Monolayer Graphene

Disagreement: REBO potential underestimates the initial Young’s modulusAgreement: Fracture stress/strain is higher in the zigzag direction than in the armchair direction

i

j

k

l

ijk

jil

q4

q2q1

q3

q3

q4q1

q2

)()( ijAijijRij rVbrVV

RCij

DHijjiijij bbbb )(2

1

),(,

20 )()(cos12 jilk

jlcikcijklDHij rfrf

Tb

ijljikijkl nn cos

The intrinsic bending stiffness of monolayer graphene results from multi-body interatomic interactions (second and third nearest neighbors).

Excess Edge Energy and Edge Force

Zigzag edge:

1.391

1.425

1.420

1.420

fZfZ

Armchair edge:

1.367

1.412

1.425

1.398

1.425

1.4201.421

1.420

fA fA

Edge Buckling

Zigzag GNR

Armchair GNR

Intrinsic wavelength ~ 6.2 nm

Intrinsic wavelength ~ 8.0 nm

W

EEE

edgebulk 000

2

GNRs under Uniaxial Tension

Fracture Strength

Zigzag GNR: Homogeneous nucleation

Armchair GNR: Edge-controlled heterogeneous nucleation

A

C

D

B

X2

(n,n): armchair

(n,0): zigzag

X1

Graphene Under Uniaxial Tension

0 0.05 0.1 0.15 0.2 0.25 0.30

10

20

30

40

Nominal strain

Nom

inal

2D

str

ess

(N/m

)

zigzag (MM/REBO)armchair (MM/REBO)zigzag (Wei et al., 2009)armchair (Wei et al., 2009)

0 0.05 0.1 0.15 0.2 0.25 0.30

100

200

300

400

Nominal strain

2D Y

oung

's m

odul

us (

N/m

)

zigzag (MM/REBO)armchair (MM/REBO)zigzag (Wei et al., 2009)armchair (Wei et al., 2009)

0 0( ) 14

2 3ijA

ijk

bV r TD

• D = 0.83 eV by REBO-1• D = 1.4 eV by REBO-2• D = 1.5 eV by first principle calculations

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

0

10

20

30

40

Nominal strain

Nom

inal

2D

str

ess

(N/m

)

W = 1.3 nmW = 2.6 nmW = 4.3 nmW = 8.5 nmbulk graphene

0 0.05 0.1 0.15 0.2-5

0

5

10

15

20

25

30

35

Nominal strain

Nom

inal

2D

str

ess

(N/m

)

W = 1.2 nmW = 2.5 nmW = 4.4 nmW = 8.9 nmbulk graphene

Zigzag GNRs Armchair GNRs

Initial Young’s modulus

• Q. Lu and R. Huang, Nonlinear mechanics of single-atomic-layer graphene sheets. Int. J. Applied Mechanics 1, 443-467 (2009).

• Q. Lu, M. Arroyo, R. Huang, Elastic bending modulus of monolayer graphene. J. Phys. D: Appl. Phys. 42, 102002 (2009).

• Q. Lu and R. Huang, Excess energy and deformation along free edges of graphene nanoribbons. Physical Review B 81, 155410 (2010).

• Q. Lu, W. Gao, and R. Huang, Atomistic Simulation and Continuum Modeling of Graphene Nanoribbons under Uniaxial Tension. Submitted, January 2011.

• Z.H. Aitken and R. Huang, Effects of mismatch strain and substrate surface corrugation on morphology of supported monolayer graphene. J. Appl. Phys. 107, 123531 (2010).

• J.H. Seol, I. Jo, A.L. Moore, L. Lindsay, Z.H. Aitken, M.T. Pettes, X. Li, Z. Yao, R. Huang, D. Broido, N. Mingo, R.S. Ruoff, and L. Shi, Two-dimensional phonon transport in supported graphene. Science 328, 213-216 (2010).

x1

x3 x2