Nonlocal beam model for nonlinear analysis of carbon nanotubes on elastomeric substrates

Postbuckling, nonlinear bending and nonlinear vibration analyses are presented for single-wall carbon nanotubes (SWCNTs) resting on a two-parameter elastic foundation in thermal environments. The SWCNT is modeled as a nonlocal nanobeam which contains small scale effects. The elastomeric substrate with finite depth is modeled as a two-parameter elastic foundation. The thermal effects are included and the material properties of both SWCNTs and the substrate are assumed to be temperature-dependent. The governing equation that includes beam–foundation interaction is solved by a two-step perturbation technique. The numerical results reveal that the small scale parameter e0a reduces the postbuckling equilibrium paths, the static large deflections and natural frequencies of SWCNTs resting on an elastic foundation. The results also reveal that the effect of the small scale parameter is significant for compressive buckling, but less pronounced for static bending and marginal for free vibration of SWCNTs resting on an elastic foundation.

Research highlights
► The small scale parameter e0a reduces the postbuckling equilibrium paths, the static large deflections and natural frequencies of SWCNTs resting on an elastic foundation. In contrast, it increases and reduces the nonlinear to linear frequency ratios slightly for the nanobeam with immovable and movable end conditions, respectively.
► The effect of the small scale parameter is significant for compressive buckling, but less pronounced for static bending and marginal for free vibration of SWCNTs resting on an elastic foundation.