2D-to-3D Deformation Gradient: In-plane stretch: 2D Green-Lagrange Strain Tensor:

1. k q3 i q2 ijk q4 q1 jil j l 1.367 X2 1.391 q4 q3 1.412 1.398 x3 q2 1.425 q1 x2 X1 1.425 1.425 1.420 x1 1.421 1.420 1.420 C 1.420 X2 (n,n): armchair B X1 D  (n,0): zigzag A Nonlinear Mechanics of Graphene-Based Materials Qiang Lu, Wei Gao and Rui Huang Department of Aerospace Engineering and Engineering Mechanics The University of Texas at Austin Grant Information Uniaxial Stretch of Monolayer Graphene Graphene Nanoribbon (GNR) • Grant Title: Nonlinear Mechancis of Graphene-Based Materials • Grant Number: 0926851 • NSF Program: Mechanics of Materials • PI Name: Rui Huang Excess Edge Energy and Edge Force Zigzag edge: Armchair edge: fZ fZ Introduction Anisotropic Tangent Moduli • 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. Edge Buckling Zigzag GNR fA fA Graphene is linear and isotropic under infinitesimal deformation, but becomes nonlinear and anisotropic under finite strain. Intrinsic wavelength ~ 6.2 nm Armchair GNR Fracture Strength Nonlinear Continuum Model of Graphene Intrinsic wavelength ~ 8.0 nm GNRs under Uniaxial Tension Fracture occurs as a result of intrinsic instability of the homogeneous deformation. • 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): Zigzag GNRs Armchair GNRs Graphene Under Uniaxial Tension Initial Young’s modulus Bending of Monolayer Graphene Disagreement: REBO potential underestimates the initial Young’s modulus Agreement: Fracture stress/strain is higher in the zigzag direction than in the armchair direction Fracture Strength coupling between tension and bending Zigzag GNR: Homogeneous nucleation Atomistic Modeling Method The intrinsic bending stiffness of monolayer graphene results from multi-body interatomic interactions (second and third nearest neighbors). Armchair GNR: Edge-controlled heterogeneous nucleation References • 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 • 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). • D = 0.83 eV by REBO-1 • D = 1.4 eV by REBO-2 • D = 1.5 eV by first principle calculations