Size-dependent bending and vibration analysis of piezoelectric nanobeam based on fractional-order kinematic relations

• Published in: Applied mathematics and mechanics Vol. 46; no. 7; pp. 1261 - 1272
• Main Authors: Fan, Zhiwen, Qing, Hai
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.07.2025; Springer Nature B.V
• Edition: English ed.
• Subjects: Applications of Mathematics; Bending; Boundary conditions; Boundary value problems; Cantilever beams; Classical Mechanics; Differential equations; Euler-Bernoulli beams; Finite element method; Fluid- and Aerodynamics; Fractional calculus; Free vibration; Kinematics; Kinetic energy; Mathematical analysis; Mathematical Modeling and Industrial Mathematics; Mathematical models; Mathematics; Mathematics and Statistics; Partial Differential Equations; Strain energy; Variational principles; Vibration analysis; Vibration response; scale effect; Riesz-Caputo fractional-order derivative; Euler-Bernoulli piezoelectric beam; fractional-order finite element method (FEM); O342; 26A33
• ISSN: 0253-4827; 1573-2754
• DOI: 10.1007/s10483-025-3274-9

In this paper, a fractional-order kinematic model is utilized to capture the size-dependent static bending and free vibration responses of piezoelectric nanobeams. The general nonlocal strains in the Euler-Bernoulli piezoelectric beam are defined by a frame-invariant and dimensionally consistent Riesz-Caputo fractional-order derivatives. The strain energy, the work done by external loads, and the kinetic energy based on the fractional-order kinematic model are derived and expressed in explicit forms. The boundary conditions for the nonlocal Euler-Bernoulli beam are derived through variational principles. Furthermore, a finite element model for the fractional-order system is developed in order to obtain the numerical solutions to the integro-differential equations. The effects of the fractional order and the vibration order on the static bending and vibration responses of the Euler-Bernoulli piezoelectric beams are investigated numerically. The results from the present model are validated against the existing results in the literature, and it is demonstrated that they are theoretically consistent. Although this fractional finite element method (FEM) is presented in the context of a one-dimensional (1D) beam, it can be extended to higher dimensional fractional-order boundary value problems.