Variational Formulations and Isogeometric Analysis of Timoshenko–Ehrenfest Microbeam Using a Reformulated Strain Gradient Elasticity Theory

Abstract

This paper presents a novel non-classical Timoshenko–Ehrenfest beam model based on a reformulated strain gradient elasticity theory. The strain gradient effect, couple stress effect, and velocity gradient effect for vibration are included in the new model by only one material length scale parameter for each. The variational formulation and Hamilton’s principle are applied to derive the governing equations and boundary conditions. Both an analytical solution and an isogeometric analysis approach are proposed for static bending and free vibration of the microbeam. A non-uniform rational B-splines (NURBS) isogeometric analysis with high-order continuity can effectively fulfill the higher derivatives of the displacement variables in the reformulated gradient beam model. Convergence studies and comparisons to the corresponding analytical solutions verify the model’s performance and accuracy. Finally, different boundary conditions, material length scale parameters, and beam thicknesses are investigated in order to certify the applicability of the proposed approach.

Keywords

• isogeometric analysis
• Timoshenko–Ehrenfest beam
• size effect
• strain gradient theory
• couple stress theory