Nonlinear vibrations of carbon-nanotube-reinforced doubly curved shells: Effects of hygrothermal environments

This paper investigates the free vibration, nonlinear resonance, and low-velocity impact behavior of carbon nanotubereinforced composite (CNTRC) doubly-curved shells under hygrothermal environmental conditions. The primary research methodology and objectives are outlined as follows: (1) Dynamic modeling: Effective material properties of the composite, with carbon nanotubes as the reinforcement phase, are derived using composite inclusion theory. The displacement field of the doubly-curved shell is represented using Reddy's higher-order shell theory, establishing the dynamic model for the CNTRC doubly-curved shells. (2) Nonlinear free vibration: The Galerkin method is then employed to discretize these differential equations, enabling the investigation of the shell's free vibration characteristics. (3) Nonlinear main resonance: Incorporating external excitation, the governing equations for forced nonlinear vibration are formulated. These equations are solved using the modified Lindstedt-Poincaré (MLP) method to obtain a second-order approximate solution characterizing the amplitudefrequency relationship. A parametric study examines the influence of various factors on the shell's nonlinear primary resonance. (4) Low-velocity impact response: The impact force expression is derived based on the nonlinear Hertzian contact law. The governing equations for low-velocity impact are subsequently obtained using the Euler-Lagrange principle. These equations are simplified via the Galerkin method and solved numerically using the Runge-Kutta method. The influence of key parameters on the low-velocity impact behavior is analyzed and discussed.