Free vibrations of cracked functionally graded graphene platelets reinforced Timoshenko beams based on Hu-Washizu-Barr variational method

Abstract

In this paper, a new theoretical approach is presented for modelling the free vibrations of functionally graded (FG) graphene platelets (GPL) reinforced thick beams with a single-edge crack. To model the single-edge crack in the GPL reinforced thick beams, strain-displacement, strain-stress, velocity-momentum, and dynamic equilibrium equations are presented in the framework of the Hu-Washizu-Barr variational method, in which three complex coupled motion equations are obtained. The beam is assumed to be continuous and thick, and the shear effects are considered by employing the Timoshenko beam theory. GPL are distributed uniformly or non-uniformly through the depth of the cracked Timoshenko beam and their reinforcement effects are formulated using a two-dimensional Halpin-Tsai model of micromechanics and a rule of mixtures. The coupled motion equations are solved using a modal decomposition scheme. The current methodology is verified by modelling simplified versions of the present model and by comparing the results with those from literature and a finite element software. Detailed case studies are presented to understand the impact of a single-edge crack on the vibration behaviour of the GPL reinforced beams. It has shown that the presence of the single-edge crack and its position can significantly alter the free vibrations of the reinforced Timoshenko beam. The distribution pattern and weight fraction of GPL also have a great significance in the free vibrations of the cracked beam, affecting the natural frequencies. The results and conclusions drawn from this paper, which uses the Hu-Washizu-Barr variational method, can be useful in guiding optimal GPL distributions and concentrations for different mechanical designs that might foresee a single-edge crack in structures.