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Compressive behavior and buckling response of carbon nanotubes (CNTs) Aswath Narayanan R Dianyun Zhang Outline • Introduction – Buckling problem of carbon nanotube – Literature review • Approach – Mathematical model – Simulation • GULP • Abaqus • Future work • Conclusion 2 What’s carbon nanotubes (CNTs) Building blocks – beyond molecules ME 599 (Nanomaufecturing) lecture notes, Fall 2009, Intstructor: A.J. Hart, University of Michigan 3 Exceptional properties of CNTs High Young’s modulus ~1 TPa National Academy of Sciences report (2005), http://www.nap.edu/catalog/11268.html and many other sources 4 CNTs kink like straws High recoverable strains and reversible kinking Kink shape develops! Yakobson et al., Physical Review B 76 (14), 1996. Seiji et al., Japan Society of Applied Physics, 45 (6B): 5586-9, 2006. 5 Buckling problem of CNTs • Types of buckling of CNTs – Euler‐type buckling • general case – hollow cylinder – shell buckling • short or large‐diameter CNTs • We are interested in Euler-type buckling 6 From a recent research paper… E ~ 0.8 TPa (a) 20 Shells douter = 14.7 nm dinner = 1.3 nm L = 1.19 µm Fcr = 24.5 nN (b) 6 Shells douter = 14.7 nm dinner = 10.3 nm L = 1.07 µm Fcr = 24.0 nN Boundary Condition: Clamp – free Seiji et al., Japan Journal of Applied Physics, 44(34): L1097-9, 2005. Euler-type buckling! 7 Something interesting… Multi-wall carbon nanotubes (MWCNTs) Outer wall Inner wall Ripple – like distortions Motoyuki et al., Mater. Res. Symp. Proc. 1081:13-05, 2008 Poncharal et al., 283:1513, 1999. 8 Two-DOF model P P u Outer wall: kt2 Kt1 θ L/2 Kr1 Kt1 Kt2 0 Inner wall: kt1, k1 0 (L- R) cos(θ) L/2 L 2 Initial Configuration R 2 Deformed Configuration 9 Two-DOF model cont. • Total potential energy 1 2 k r (2 ) 2 1 2 k t1 ( R ) 2 2 2 1 2 k t 2 [ L ( L R ) C os ( )] 2 • Non-dimensional form 1 2 2 k1 r 4 k 2 [1 (1 r ) C os ( )] p (1 r ) C os ( ) p 2 where 2 2 r R p L 4kr P 4kr • Equilibrium condition r 0 k1 k t1 L 2 32 k r Inner wall k2 kt 2 L 2 32 k r Outer wall 0 10 Force – displacement curve k1 = 1, k2 = 0 (no outer wall) k1 = 1, k2 = 1 Force 4 Force 12 10 3 Trifurcation 8 Snapback behavior 2 6 4 1 θ=0 0.2 0.4 0.6 Outer wall increases the slope of post-buckling curve 2 0.8 1.0 Displacement 0.2 0.4 0.6 0.8 1.0 Displacement 11 Force – displacement curve cont. Force 12 k1 = 1, vary k2 • Initial slope = 4 (k1 +2 k2) 10 • Snapback behaviors are observed when k1 = 1 8 6 4 k2 = 1.5 • Trifurcation point is based on both k1 and k2 k2 = 1.2 k2 = 0.5 k2 = 0.8 k2 = 1 2 0.2 0.4 0.6 0.8 1.0 Displacement 12 Compared with the experimental data Force k1 = 0.99, k2 = 1.1 12 10 Trifurcation 8 Snapback 6 4 Experimental data 2 0.2 0.4 0.6 0.8 1.0 Displacement 13 GULP simulation of 6,6 CNT (Armchair) 14 General Utility Lattice Program (GULP) • Minimization of the potential of the multi atom system • Takes into account various multi body potentials • NON LOCAL interactions (twisting, three body moments) 15 What are non local interactions? Ref. C. Li et al, Int J 16 Sol & Str Force – displacement curve • Force –displacement curve for 6,6 CNT INTERNAL ENERGY - DISPLACEMENT FORCE - DISPLACEMENT 5 -2.36 x 10 8000 -2.38 6000 -2.4 4000 -2.42 2000 -2.44 E -2.46 F 0 -2.48 -2000 -2.5 -4000 -2.52 -2.54 -37 -36 -35 -34 -33 X -32 -31 -30 -29 -6000 1 2 F=dE/dX 3 4 5 6 7 X 17 Parameters used in simulation • Potential – it decides the way atoms interact with each other • Tersoff Potential is used for this simulation • It is a multi body potential, consisting of terms which depend on the angles between the atoms as well as on the distances between the corresponding atoms (bond order potential) • Selected due to its applicability to covalent molecules and faster speed of computation compared to other potentials 18 FEA using Abaqus • Frame-like structure • Primary bonds between two nearestneighboring atoms act like load-bearing beam members • Individual atom acts as the joint of the related load-bearing beam members 19 Buckling mode 1 2 3 4 5 20 Future work • Mathematical model – Imperfection sensitivity – Non-linear springs • Post-buckling analysis using Abaqus – Figure out parameters in the model – Implement rotational springs in the joints 21 Conclusion • 2-DOF model represents the Euler-type buckling of CNTs – Trifurcation – Snapback • GULP simulation – Minimization of potential energy – Force – displacement curve • Buckling analysis using Abaqus – Frame-like structure 22 NASA Video on MWCNTs 23 Thank You! Questions? 24