New Analytical Solutions for Free Vibration of Cracked Functionally Graded Rectangular Plates

Abstract

In aerospace engineering, the existence of cracks highly alters the mechanical behaviors of structural components. As a new class of key structural components, functionally graded material (FGM) rectangular plates with an interior/side crack are focused on in this study. New analytical free vibration solutions are put forward for such cracked FGM plates. First, a cracked plate is decomposed into several subplates with the necessary continuity conditions to address the physical discontinuity issue caused by the crack. The challenge then becomes deriving analytical free vibration solutions for each subplate. To accomplish this task, a symplectic superposition method (SSM) is developed to deal with FGM rectangular plates with various boundary conditions, including the complex continuity conditions. The SSM eliminates the need for a predefined solution form, and its solution procedure, including variable separation and eigen expansion, is mathematically rigorous. Afterward, the analytical solutions obtained via the SSM are integrated to serve as the final free vibration solution of the cracked FGM plate. Comprehensive results based on the obtained analytical solutions, including natural frequencies and vibration modes, are presented and validated. The solutions are further used to investigate the effects of some important parameters, such as crack length and crack location.