New observations on transverse dynamics of microtubules based on nonlocal strain gradient theory

Abstract Based on nonlocal strain gradient theory, we present dynamical behaviors of a microtubule subjected to axial load, thermal load and variable transverse load simultaneously. The existing nonlocal strain gradient constitutive relation is adjusted from the perspective of dimensional analysis for better understanding and application, especially when establishing a multi-field coupling model. Subsequently, the strain potential, external potential and kinetic energies are obtained, and the governing equation of motion and corresponding classical and non-classical boundary conditions are derived via Hamilton’s principle, where the traditional nonlocality of strain and strain gradients, and higher-order gradients of nonlocal stress are involved. Variations of frequencies with respect to two intrinsic parameters and change of temperature are presented. The nonlocal effect indicates a softening mechanism while the strain gradient effect implies a hardening rule. By comparison, effect of the change of temperature is not as significant as two intrinsic parameters. In addition, it is observed for the first time that the inverse tendencies for nonlocal and strain gradient effects emerge with a relative large parameter. Such performances are inconsistent with the influence mechanism in nonlocal strain gradient theory. Therefore, the bounds of nonlocal scale and strain gradient parameters for transverse dynamics of microtubules are determined accordingly.