Application of the modified Wakashima–Tsukamoto model on nonlocal torsional vibration analysis of functionally graded Timoshenko–Gere microbeam

Abstract

The objective of this study is to assess the nonlocal torsional dynamic characteristics of functionally graded (FG) microscale beams based on Timoshenko–Gere’s theory. In this contribution, the modified Wakashima–Tsukamoto homogenization model is applied to calculate the effective material properties of FG microstructure, while the properties vary in the microbeam thickness direction. To consider the small-scale effect, nonlocal strain gradient theory is developed for the Timoshenko–Gere microbeam. Also, it is assumed that the FG microbeam possesses a rectangular cross section. The governing equation of torsional vibration of FG microbeam is derived with the help of the virtual work’s principle and analytically solved via Galerkin’s method. The numerical comparison between the findings of this study and the former investigation is performed to validate the accuracy of the methodology. Furthermore, two boundary conditions including clamped–free and clamped–clamped are regarded. Finally, the influence of different considerable variants including gradient index, microbeam cross-section geometry parameters, nonlocal and length scale parameters, and length of microbeam on the natural torsional vibration behavior is explored and shown in a group of illustrations and tables.