2d-to-3d deformation gradient: in-plane stretch: 2d green-lagrange strain tensor: bending: 2d...
TRANSCRIPT
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
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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
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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:
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Zigzag GNR
Armchair GNR
Intrinsic wavelength ~ 6.2 nm
Intrinsic wavelength ~ 8.0 nm
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GNRs under Uniaxial Tension
Fracture Strength
Zigzag GNR: Homogeneous nucleation
Armchair GNR: Edge-controlled heterogeneous nucleation
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(n,n): armchair
(n,0): zigzag
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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
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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
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odul
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N/m
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zigzag (MM/REBO)armchair (MM/REBO)zigzag (Wei et al., 2009)armchair (Wei et al., 2009)
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• 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).
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