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Compressive behavior and buckling response of carbon nanotubes (CNTs) Aswath Narayanan R Dianyun Zhang

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Introduction –Buckling problem of carbon nanotube –Literature review Approach –Mathematical model –Simulation GULP Abaqus Future work Conclusion Outline 2

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What’s carbon nanotubes (CNTs) 3 Building blocks – beyond molecules ME 599 (Nanomaufecturing) lecture notes, Fall 2009, Intstructor: A.J. Hart, University of Michigan

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Exceptional properties of CNTs National Academy of Sciences report (2005), http://www.nap.edu/catalog/11268.html and many other sources High Young’s modulus ~1 TPa 4

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CNTs kink like straws 5 Yakobson et al., Physical Review B 76 (14), 1996. High recoverable strains and reversible kinking Kink shape develops! Seiji et al., Japan Society of Applied Physics, 45 (6B): 5586-9, 2006.

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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 Buckling problem of CNTs 6

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From a recent research paper… 7 Seiji et al., Japan Journal of Applied Physics, 44(34): L1097-9, 2005. E ~ 0.8 TPa (a) 20 Shells d outer = 14.7 nm d inner = 1.3 nm L = 1.19 µm F cr = 24.5 nN (b) 6 Shells d outer = 14.7 nm d inner = 10.3 nm L = 1.07 µm F cr = 24.0 nN Euler-type buckling! Boundary Condition: Clamp – free

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Something interesting… 8 Motoyuki et al., Mater. Res. Symp. Proc. 1081:13-05, 2008 Poncharal et al., 283:1513, 1999. Ripple – like distortions Outer wall Inner wall Multi-wall carbon nanotubes (MWCNTs)

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Two-DOF model 9 0 0 0 0 P K t1 K r1 K t2 L/2 (L- R) cos(θ) u θ P Initial ConfigurationDeformed Configuration Inner wall: k t1, k 1 Outer wall: k t2

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Total potential energy Non-dimensional form where Equilibrium condition Two-DOF model cont. 10 Inner wall Outer wall

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Force – displacement curve 11 k 1 = 1, k 2 = 0 (no outer wall) Trifurcation θ = 0 k 1 = 1, k 2 = 1 Outer wall increases the slope of post-buckling curve Snapback behavior

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12 k 2 = 1 k 2 = 0.8 k 2 = 1.2 k 2 = 1.5 k 2 = 0.5 Force – displacement curve cont. k 1 = 1, vary k 2 Initial slope = 4 (k 1 +2 k 2 ) Snapback behaviors are observed when k 1 = 1 Trifurcation point is based on both k 1 and k 2

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Compared with the experimental data 13 Experimental data k 1 = 0.99, k 2 = 1.1 Trifurcation Snapback

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GULP simulation of 6,6 CNT (Armchair) 14

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Minimization of the potential of the multi atom system Takes into account various multi body potentials NON LOCAL interactions (twisting, three body moments) General Utility Lattice Program (GULP) 15

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What are non local interactions? 16 Ref. C. Li et al, Int J Sol & Str

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Force –displacement curve for 6,6 CNT Force – displacement curve 17 INTERNAL ENERGY - DISPLACEMENTFORCE - DISPLACEMENT X X F E F=dE/dX

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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 Parameters used in simulation 18

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Frame-like structure Primary bonds between two nearest- neighboring atoms act like load-bearing beam members Individual atom acts as the joint of the related load-bearing beam members FEA using Abaqus 19

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Buckling mode 20 12 34 5

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Mathematical model –Imperfection sensitivity –Non-linear springs Post-buckling analysis using Abaqus –Figure out parameters in the model –Implement rotational springs in the joints Future work 21

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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 Conclusion 22

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NASA Video on MWCNTs 23

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Thank You! Questions? 24

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