Optimization and Uncertain Nonlinear Vibration of Pre/post-buckled In-Plane Functionally Graded Metal Nanocomposite Plates

Abstract

Abstract Purpose This paper studies the nonlinear free and forced vibration of in-plane bi-directional functionally graded (FG) metal nanocomposite plates considering uncertain material elastic properties in the pre/post buckling states. Initially, the distribution of the nano-reinforcement volume fraction is designed through an optimization process to minimize the amount of the reinforcement in case of simply supported and clamped plates. Methods The elastic modulus of the nanocomposite is modeled as a non-stationary random field using the Karhunen–Loève expansion (KLE) technique while the uncertain output variables are modeled using the polynomial chaos expansion (PCE). The considered plates are thin, so the classical plate theory with the von Kármán nonlinear strain field is used for the analysis. The harmonic balance method and the fourth-order Runge Kutta method are used to estimate the vibration responses. Results The in-plane optimization process of the nonreinforcement volume fraction distribution yielded a 14% and 70% saving in the reinforcement amount in the case of the simply supported plate and the clamped plate respectively. The uncertainty in the vibration amplitude in the pre-buckled state can be multiples of the uncertainty in the elastic modulus and follows near normal distributions. In the post-buckled state, the nature of the probability distribution depends on the excitation force and frequency. In general, the FG plates can have similar or more uncertainty levels compared to the equivalent homogenous plates. Conclusion The uncertainty in the nonlinear vibration of in-plane functionally graded plates depends on the boundary conditions, modeling definition of the input uncertainty, the excitation force and frequency.