Free Vibrational Behaviour of Multi-Directional Porous Functionally Graded Structures

Abstract

The free vibrational frequencies of a multi-directional functionally graded (FG) structure are investigated for the first time in this paper considering the influences of variable grading (power-law, sigmoid, exponential) and porosity distribution (even and uneven type). Also, the structural properties (Young’s modulus, density and Poisson's ratio) varied along with two different directions simultaneously, i.e., the longitudinal and transversional ones, respectively. The present multi-directional grading model of the FG structure is reconstructed mathematically for the numerical analysis by considering adequate state-space deformation kinematics with the help of higher-order displacement functions and shear stress continuity. The general motion equation of a multi-graded structure is expressed using Hamilton’s principle and the finite element method including the necessary porosity effect. Initially, model consistency is verified and the eigenvalues obtained in the analysis are compared with the ones found in the literature. The comparison also includes the directional grading effect on their frequencies. Further, the influence of different parameters, i.e., power exponents (nz and nx), aspect ratio, thickness ratio, geometry, end support conditions, curvature ratio, porosity index, porosity distribution and material grading patterns, on the vibration characteristics of the multi-directional FG structure is computed. The analysis of the numerical results confirms that material grading; porosity distribution pattern and other design parameters have a significant influence on the frequency response characteristics of a single/multi-directional porous FG structure.