A Gurtin–Murdoch surface elasticity framework for hollow cylindrical nanostructures: Application in the vibration analysis of microtubules

This study presents a novel extension of the Gurtin–Murdoch surface elasticity theory for hollow cylindrical nanostructures and establishes a unified higher-order framework for analyzing the vibration of anisotropic microtubules (MTs) immersed in cytosolic fluid under external electric fields, thereby providing enhanced physical insight into their dynamic behavior. Unlike conventional approaches, the proposed framework rigorously reformulates the original Gurtin–Murdoch relations to satisfy all surface equilibrium and compatibility conditions for hollow circular geometries, provides a physically consistent representation of both the inner and outer surface effects. Furthermore, a novel stepwise computational approach is introduced for the first direct evaluation of the surface layer density of microtubules <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>ρ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>s</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> , offering a reliable reference for future biomechanical modeling of nanostructures. The vibrational behavior of MTs is investigated using the extended surface elasticity and nonlocal strain gradient theory (NSGT), incorporating viscoelastic effects of both the MT and its surrounding medium. The governing equations and boundary conditions are systematically derived from Hamilton’s principle and solved numerically via the differential Quadrature (DQ) method. The effects of key parameters on the natural frequencies are comprehensively analyzed. The results demonstrate that the conventional Laplace–Young-based surface model significantly overestimates the natural frequencies of hollow nanostructures as demonstrated through the quantitative comparisons presented in the results section. The proposed model establishes a reliable theoretical framework for accurate prediction of the mechanical behavior of microtubules and similar hollow nanostructures and provides a foundation for future nanoscale modeling and characterization of surface-related effects in bio- and nanomechanical systems.