Influence of homogenization schemes on vibration of functionally graded curved microbeams

Abstract

The influence of various homogenization models is studied on the free vibration behavior of a functionally graded material (FGM) curved microbeam which is modeled using the modified strain gradient theory of elasticity as well as the first-order shear deformation theory. Different homogenization models (such as Voigt, Reuss, Hashin-Shtrikman bounds, and cubic local representative volume elements (LRVE) schemes) are used to predict the effective material properties of a two-phase particle composite as a function of the volume fraction of particles which are continuously varying along the thickness of a functionally graded microbeam. Employing Hamilton’s principle, the governing equations of motion and boundary conditions are derived. Finally, the numerical results are presented to determine the effect of homogenization models on the vibrational behavior of FGM curved microbeams corresponding to different continuum models for over a wide range of material composition, opening angle, dimensionless length scale parameter and aspect ratio. It is shown that the common used Voigt model overestimates frequencies and the Mori-Tanaka model also overestimates frequencies but provides smaller error in comparison with the Voigt model. In terms of theoretical analysis, LRVE model creates a good compromise between estimation accuracy and the easy of implement, especially for the FGM with a relative high stiffness ratio. The difference of various homogenization models can be raised when considering the decrement in aspect ratio or the increment of opening angle and mode number.