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Stability of Discontinuous Elastic Rods with Applications to Nanotube Junctions

Lane, DT; (2012) Stability of Discontinuous Elastic Rods with Applications to Nanotube Junctions. Masters thesis, UCL (University College London). Green open access

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Abstract

Buckling and post-buckling stability of elastic rods with discontinuities in bending stiffness and curvature, as well as rods lying on an interactive surface is investigated using the theory of conjugate points. Second order matching conditions at points of discontinuity are formulated, which allow the classic Jacobi condition to be extended to incorporate calculus of variations problems with discontinuous integrands. Static equilibrium equations for intrinsically straight, inextensible and unshearable discontinuous rods are formulated. Conjugate points are found by numerically solving the Jacobi equation as an initial value problem. For the case of a rod interacting with a surface, an external force potential is added to the energy functional, causing the Jacobi operator to be an integrodifferential operator. Morse index theory is used to find expressions for critical buckling values of load pa- rameters with respect to parameters measuring jumps in bending stiffness, or parameters measuring the strength of the rod-surface interaction. Bifurcation diagrams of buckled rod solutions are presented, with the Morse stability index calculated for each solution branch. These are found to be consistent with the theory of stability exchange at folds for distinguished diagrams. The presence of a jump in bending stiffness is shown, in some cases, to cause an extra stable solution branch. Numerical continuation of folds in two parameters is used to find the parameter space for which these stable branches exist. The rod equilibrium equations are solved numerically using parameter continua- tion for a boundary value problem. Clamped boundary conditions are considered, as well as pinned boundary conditions, which require a more robust adaptation of the classic Jacobi condition. The theory is applied to the modelling of carbon nanotube intramolecular junctions, in which the bonding of two or more carbon nanotubes causes a jump in the diameter, chirality or cross-section shape of the resulting tube, as well as a possible kink (jump in the centreline curvature) in the tube. The effects of van der Waals forces between a nanotube undergoing compression, and a substrate are modelled.

Type:Thesis (Masters)
Title:Stability of Discontinuous Elastic Rods with Applications to Nanotube Junctions
Open access status:An open access version is available from UCL Discovery
Language:English
UCL classification:UCL > School of BEAMS > Faculty of Engineering Science > Civil, Environmental and Geomatic Engineering

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