Stability of the Rao-Nakra sandwich beam with a dissipation of fractional derivative type: theoretical and numerical study

TL;DR

This work analyzes the stability of a Rao–Nakra sandwich beam with domain-embedded damping modeled by a generalized Caputo fractional derivative with exponential weight. By introducing a diffusion-based augmentation in an auxiliary variable $\xi$, the authors obtain a dissipative semigroup formulation and prove existence, uniqueness, and polynomial energy decay dependent on the fractional parameters $\alpha$ and $\eta$. They establish stability results via semigroup theory: non-uniform stability for $\eta=0$ with $t^{-1/2}$ decay, and polynomial decay for $\eta>0$ with rate $(1+t)^{-1/(1-\alpha)}$, along with a fully discrete, energy-conserving numerical scheme. Numerical experiments validate the predicted polynomial decay and show that the Mbodje-augmented model preserves energy dissipation more robustly than Grünwald–Letnikov discretizations, highlighting a practical approach for simulating fractional damping in multilayer beams.

Abstract

This paper is devoted to the solution and stability of a one-dimensional model depicting Rao--Nakra sandwich beams, incorporating damping terms characterized by fractional derivative types within the domain, specifically a generalized Caputo derivative with exponential weight. To address existence, uniqueness, stability, and numerical results, fractional derivatives are substituted by diffusion equations relative to a new independent variable, $ξ$, resulting in an augmented model with a dissipative semigroup operator. Polynomial decay of energy is achieved, with a decay rate depending on the fractional derivative parameters. Both the polynomial decay and its dependency on the parameters of the generalized Caputo derivative are numerically validated. To this end, an energy-conserving finite difference numerical scheme is employed.

Figures (4)

• Figure 1: Comparison between the Grünwald–Letnikov derivative approximation and the discretized Mbodje augmented model. At the top left: $E^{GL}_\Delta(t)=\dfrac{1}{2} \dot{\mathbf{U}}^T \mathbf{M} \dot{\mathbf{U}} + \dfrac{1}{2} {\mathbf{U}}^T \mathbf{K} {\mathbf{U}}$; At the top right: $E^{GL,\Phi}_\Delta(t)=E^{GL}_\Delta(t) + \dfrac{\mathfrak{C}}{2} \sum_{\ell=1}^M {\mu}_\ell \left|\mathbf{\Phi}_\ell\right|^2$; At the bottom: $E^{Mbodje}_\Delta(t)$ defined in \ref{['ener_Mbodje']}.
• Figure 2: Simulation of the displacements. On the left: the displacements at $t=2$. On the right: the evolution in time of the transverse displacements.
• Figure 3: Behavior of the auxiliary variables $\phi$, $\varphi$ and $\psi$ of the augmented model \ref{['213']}
• Figure 4: Energy with different values of $\eta$ and $\alpha$ in loglog scale.

Theorems & Definitions (8)

• Theorem 2.1
• Lemma 2.2
• Proposition 3.1
• Proposition 3.2
• Theorem 4.1: BCT
• Proposition 4.2
• Proposition 4.3
• Remark 5.1