Qualitative Results on the Dynamics of a Berger Plate with Nonlinear Boundary Damping

TL;DR

This paper investigates the nonlinear dynamics of Berger plates with boundary damping, establishing well-posedness and the existence of a global attractor under certain boundary conditions, with implications for engineering and applied mathematics.

Contribution

It provides new results on well-posedness and attractors for nonlinear Berger plates with boundary damping, including explicit construction of absorbing sets and a comparative discussion of nonlinear models.

Findings

Well-posedness achieved via nonlinear boundary dissipation.

Existence of a compact global attractor under hinged boundary conditions.

Energy methods and trace results are effectively employed.

Abstract

The dynamics of a (nonlinear) Berger plate in the absence of rotational inertia are considered with inhomogenous boundary conditions. In our analysis, we consider boundary damping in two scenarios: (i) free plate boundary conditions, or (ii) hinged-type boundary conditions. In either situation, the nonlinearity gives rise to complicating boundary terms. In the case of free boundary conditions we show that well-posedness of finite-energy solutions can be obtained via highly non- linear boundary dissipation. Additionally, we show the existence of a compact global attractor for the dynamics in the presence of hinged-type boundary dissipation (assuming a geometric condition on the entire boundary [22]). To obtain the existence of the attractor we explicitly construct the absorbing set for the dynamics by employing energy methods that: (i) exploit the structure of the Berger nonlinearity, and (ii) utilize sharp trace results for the Euler-Bernoulli plate in [24]. We provide a parallel commentary (from a mathematical point of view) to the discussion of modeling with Berger versus von Karman nonlinearities: to wit, we describe the derivation of each nonlinear dynamics and a discussion of the validity of the Berger approximation. We believe this discussion to be of broad value across engineering and applied mathematics communities.