1.  2D-to-3D Deformation Gradient:  In-plane stretch: 2D Green-Lagrange Strain Tensor:  Bending: 2D Curvature Tensor:  2 nd 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 GrapheneGraphene 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: 2 nd generation REBO potential X1X1 X2X2 coupling between tension and bending  Graphene is a one-atom-thick planar sheet of sp 2 –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 modulus Agreement: Fracture stress/strain is higher in the zigzag direction than in the armchair direction i j k l  ijk  jil  q4  q2  q1  q3  q3  q4  q1  q2 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 fZfZ fZfZ Armchair edge: 1.367 1.412 1.425 1.398 1.425 1.420 1.421 1.420 fAfA fAfA Edge Buckling Zigzag GNR Armchair GNR Intrinsic wavelength ~ 6.2 nm Intrinsic wavelength ~ 8.0 nm GNRs under Uniaxial Tension Fracture Strength Zigzag GNR: Homogeneous nucleation Armchair GNR: Edge-controlled heterogeneous nucleation A C D B X2X2 (n,n): armchair (n,0): zigzag X1X1  Graphene Under Uniaxial Tension D = 0.83 eV by REBO-1 D = 1.4 eV by REBO-2 D = 1.5 eV by first principle calculations Zigzag GNRsArmchair 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).Nonlinear mechanics of single-atomic-layer graphene sheets Q. Lu, M. Arroyo, R. Huang, Elastic bending modulus of monolayer graphene. J. Phys. D: Appl. Phys. 42, 102002 (2009).Elastic bending modulus of monolayer graphene 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).Effects of mismatch strain and substrate surface corrugation on morphology of supported monolayer graphene 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). x1x1 x3x3 x2x2