Long-Time Behavior of a Nonlinearly-Damped Three-Layer Rao-Nakra Sandwich Beam

In this paper, a three-layer Rao-Nakra sandwich beam is considered where the core viscoelastic layer is constrained by the purely elastic or piezoelectric outer layers. In the model, uniform bending motions of the overall laminate are coupled to the longitudinal motions of the outer layers, and the shear of the middle layer contributes to the overall motion. Together with nonlinear damping injection and nonlinear source terms, the existence and uniqueness of local and global weak solutions are obtained by the nonlinear semigroup theory and the theory of monotone operators. The global existence of potential well solutions and the uniform energy decay rates of such a solution, given as a solution to a certain nonlinear ODE, are shown are proved under certain assumptions of the parameters and by the Nehari manifold. Finally, the existence of a smooth global attractor with finite fractal dimension, which is characterized as an unstable manifold of the set of stationary solutions, and exponential attractors for the associated dynamical system are proved. The present paper extends the linear analysis of the stability of the Rao-Nakra sandwich beam to nonlinear analysis in the existing literature.

Automated summary

In this paper, a three-layer Rao-Nakra sandwich beam with viscoelastic core is investigated. The existence and uniqueness of local and global weak solutions are proven by utilizing nonlinear semigroup theory and the theory of monotone operators. Under certain assumptions, the global existence of potential well solutions and the uniform energy decay rates are shown to be solutions to a specific nonlinear ODE. Additionally, the existence of a smooth global attractor with finite fractal dimension and exponential attractors for the associated dynamical system are also established. This paper extends the existing linear analysis to nonlinear analysis of the stability of the Rao-Nakra sandwich beam.