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Application of nonlocal beam models to double-walled carbon nanotubes under a moving nanoparticle. I: Theoretical formulations. (English) Zbl 1398.74145
Summary: The current work suggests mathematical models for the vibration of double-walled carbon nanotubes (DWCNTs) subjected to a moving nanoparticle by using nonlocal classical and shear deformable beam theories. The van der Waals interaction forces between atoms of the innermost and outermost tubes are modeled by an elastic layer. The equations of motion are derived for the nonlocal double body Euler-Bernoulli, Timoshenko and higher-order beams connected by a flexible layer under excitation of a moving nanoparticle. Analytical solutions of the problem are provided for the aforementioned nonlocal beam models with simply supported boundary conditions. The dynamical deflections and nonlocal bending moments of the innermost and outermost tubes are then obtained during the courses of excitation and free vibration. Finally, the critical velocities of the moving nanoparticle associated with the nonlocal beam theories are expressed in terms of small-scale effect parameter, geometry, and material properties of DWCNTs. This article is lovingly dedicated to my father and mother, Amrullah Kiani and Kobra Ahmadi, whose love and encouragements I feel every day of my life.

##### MSC:
 74H45 Vibrations in dynamical problems in solid mechanics 74K10 Rods (beams, columns, shafts, arches, rings, etc.) 74M25 Micromechanics of solids
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