Nonlinear vibration analysis of a generally orthotropic cracked micro-plate with variable thickness considering fiber effects and fluid-structure interaction

Abstract

This research investigates the nonlinear vibration response of partially cracked, generally orthotropic, rectangular micro-plates featuring non-uniform thicknesses coupled with a fluid medium. The traditional classical plate theory (CPT) in conjunction with a modified couple stress theory (MCST) is used to derive the governing equation of a generally orthotropic rectangular plate of varying thickness. The presence of a partial crack at the plate’s center is modeled using a simplified line spring model (LSM), and in-plane forces and bending moment resulting from a partial crack are incorporated into the equation. Furthermore, the interaction with a fluid medium is considered, with the introduction of virtual added mass via the velocity potential function and Bernoulli’s equations, considering both horizontal and vertical plate immersion scenarios. The study also explores thickness variations in one and two dimensions along the X and Y axes. The methodology employs Galerkin’s approach to convert governing equations into time-dependent modal coordinates, and Berger’s method introduces nonlinearity into these coordinates. The approximation solution, a method of multiple scales, is utilised to solve the plate’s nonlinear governing equations. Various parameters are considered in the analysis, including crack length, size effect, thickness variations, fiber orientations, fluid levels, and immersion depth. The study explores the impact of these factors on the vibration behaviour of submerged partially cracked, generally, orthotropic non-uniform thickness micro-plates under two different boundary conditions, presenting novel findings. Additionally, an illustration is provided to visually depict the bending-hardening and bending-softening phenomenon concerning the taper constant and depth of submergence.