Nonlocal shear deformable shell model for postbuckling of axially compressed microtubules embedded in an elastic medium

Abstract

Buckling and postbuckling analysis is presented for axially compressed microtubules (MTs) embedded in an elastic matrix of cytoplasm. The microtubule is modeled as a nonlocal shear deformable cylindrical shell which contains small scale effects. The surrounding elastic medium is modeled as a Pasternak foundation. The governing equations are based on higher order shear deformation shell theory with a von Kármán-Donnell-type of kinematic nonlinearity and include the extension-twist and flexural-twist couplings. The thermal effects are also included and the material properties are assumed to be temperature-dependent. The small scale parameter e (0) a is estimated by matching the buckling load from their vibrational behavior of MTs with the numerical results obtained from the nonlocal shear deformable shell model. The numerical results show that buckling load and postbuckling behavior of MTs are very sensitive to the small scale parameter e (0) a. The results reveal that the MTs under axial compressive loading condition have an unstable postbuckling path, and the lateral constraint has a significant effect on the postbuckling response of a microtubule when the foundation stiffness is sufficiently large.

Application of Nonlocal Shell Models to Microtubule Buckling in Living Cells

“…carbon nanotubes [20][21][22] , single layer graphene sheets [23] and MTs [24,25] , lies in that the latter structure has a small scale effect. Although the classical continuum models are efficient in MTs buckling analysis through their relatively simple formulae, the small scale effects on the buckling and postbuckling behavior of MTs have been ignored.…”

“…(9.9)-(9.12) are successfully used in solving postbuckling problems of MTs subjected to axial compression or external pressure [24,25] .…”

“…Consequently, it is reasonable to state that the curvature of MTs should be greater than or equal to 0.4 rad/μm, but must be less than 1.5 rad/μm when buckling occurs. The wavelength is about 0.07 μm which is smaller than that of 0.11μm in the case of axial compression [24] . It is worthy to note that the wavelength in Euler buckling pattern [9] is much greater than that in shell buckling pattern [17] for compressive buckling of MTs.…”

“…Small scale effects on the mechanical behaviors of protein microtubules were produced analytically by assuming a range of e 0 a values since its actual value is not known [29][30][31] . It has been shown that the small scale parameter e 0 a will have different values under different loading cases [22,24,25] . In the present study, we give the estimation of parameter e 0 a by matching the buckling curvature of MTs observed from measurements [47] with the numerical results obtained from the nonlocal shear deformable shell model.…”

Analysis of Free Torsional Vibration in Carbon Nanotubes Embedded in a Viscoelastic Medium

“…Shen presented buckling and postbuckling analysis for axially compressed microtubules embedded in an elastic matrix of cytoplasm. He modelled the microtubule as a nonlocal shear deformable cylindrical shell which contains small scale effects and the surrounding elastic medium as a Pasternak foundation [4]. Soltani et al developed the transverse vibrational model of a viscous-fluidconveying SWCNT embedded in biological soft tissue.…”