Nonlocal Reddy Third-Order Shear Deformation Analysis of Hollow Microtubule-Like Nanocylinders: Thermal Buckling and Vibration with a Differential-Quadrature Solution for Four Boundary Conditions

Abstract

A nonlocal third-order shear-deformation (Reddy) beam model is developed for the thermal buckling and free vibration of a hollow, microtubule-like nanocylinder subjected to a uniform temperature rise. The aim is to determine how far third-order shear kinematics and Eringen nonlocality each alter the thermal stability and free vibration of a microtubule relative to first-order and classical beam models, and to delimit the parameter regime in which each correction is quantitatively necessary rather than merely formal. Eringen’s differential nonlocal constitutive law is combined with the Reddy kinematics, so that the transverse-shear strain varies parabolically through the annular wall and vanishes at the outer surface without recourse to a shear-correction factor, and the temperature rise enters as a compressive thermal axial force. The coupled governing equations are reduced in closed form to the nonlocal Timoshenko and Euler–Bernoulli limits. An exact Navier solution is derived for the simply-supported tube, and a non-dimensional differential-quadrature method (DQM) is constructed for the four classical boundary conditions (S–S, C–C, C–S, C–F); the non-S–S results, which have no published closed form, are cross-validated against an independent finite-difference solver. With the thermal-expansion coefficient calibrated to the flexural-rigidity measurements of Kawaguchi and Yamaguchi [2010, “Temperature dependence rigidity of non-taxol stabilized single microtubules,” Biochemical and Biophysical Research Communications 402(1), 66–69], the model reproduces the closed-form Euler–Bernoulli critical temperatures to within [Formula: see text]% and a simply-supported buckling force of 3.29[Formula: see text]pN consistent with the optical-trap measurement of Kurachi et al. [1995, “Buckling of a single microtubule by optical trapping forces: direct measurement of microtubule rigidity,” Cell Motility and the Cytoskeleton 30(3), 221–228]. The central finding is that transverse shear is decisive for microtubules through their anomalously low shear-to-extensional modulus ratio rather than through slenderness: the simply-supported critical temperature is softened by 0.78%, 7.3% and 44% (first-order Timoshenko) at ratios of [Formula: see text], [Formula: see text] and [Formula: see text], and the Reddy–Timoshenko spread grows to several percent. The nonlocal small-scale effect is negligible (below 0.1%) for fundamental modes of micron-scale microtubules. The clamped-free configuration is the thermomechanically most sensitive of the four, with both its critical temperature and its steeply softening fundamental frequency falling inside the physiological window — a regime that only a shear-deformable, higher-order description resolves correctly.