Nonlinear Parametric Instability of Functionally Graded Rectangular Plates in Thermal Environments

Abstract

Geometrically nonlinear parametric instability of functionally graded (FG) rectangular plates in thermal environments is investigated via multi-modal energy approach. Nonlinear higher-order shear deformation theory is used and the nonlinear response to in-plane static and harmonic excitation in the frequency neighbourhood of twice the fundamental frequency is investigated. The boundary conditions are assumed to be simply supported movable. The plate displacements and rotations are expanded in terms of double series trigonometric functions and Lagrange equations are used to reduce the energy functional to a system of infinite nonlinear ordinary differential equations with time varying coefficients, and quadratic and cubic nonlinearities. In order to obtain the complete dynamic scenario, numerical analyses are carried out by means of pseudo arc length continuation and collocation technique to obtain frequency-amplitude and force-amplitude relations in the presence of temperature variation in the thickness direction. The effect of volume fraction exponent as well as temperature variation on the on-set of instability for both static and periodic in-plane excitation are fully discussed and the post-critical nonlinear responses are obtained.