Analyticity for Double Wall Carbon Nanotubes Modeled as Timoshenko Beams with Kelvin-Voigt and Intermediate Damping

TL;DR

The paper addresses the analytic regularity and stability of a nanotechnological model in which double-walled carbon nanotubes are represented as two coupled Timoshenko beams linked by Van der Waals forces and damped by Kelvin-Voigt and fractional mechanisms with parameters $({\alpha},{\beta})\in [0,1]^2$. By formulating the dynamics as a linear evolution problem in a suitable Hilbert space and applying semigroup theory, the authors prove well-posedness, exponential stability for all damping exponents, and analyticity of the generated semigroup across the entire parameter square. They further establish new analytic results for Timoshenko systems in decoupled and fractional-damping regimes, broadening the understanding of regularity for coupled beam models. These results provide a rigorous, parameter-unrestricted foundation for stable numerical simulation and potential control design in nanoscale DWCNT applications.

Abstract

This manuscript studies a model of double-walled carbon nanotubes using two Timoshenko beams which are coupled by the Van der Walls force $(y-u)$. Kelvin-Voigt type dampings $(u_x-v)_{xt}$ and $(y_x-z)_{xt}$ and fractional dampings $(-\partial_{xx})^αv_t$ and $(-\partial_{xx})^βz_t$ in both beams have been considered. We show that our proposed model is well established and that the semigroup associated is exponentially stable and analytical for any $(α, β) \in [0, 1]^2$. As a consequence of this, a result on the analyticity of a Timoshenko System is obtained.

Figures (2)

• Figure 1: Nanotube structure DaSilva
• Figure 2: 2D and 3D Representations of the Double Wall Carbono Nanotubes Model Ramos2023CNTs

Theorems & Definitions (13)

• Theorem 2: Lions' Interpolation, see EN2000, Theorem 5.34
• Theorem 3: see LiuZ, Theorem 1.2.4
• Theorem 4
• Theorem 5: See LiuZ, Theorem 1.3.2
• Remark 6
• Lemma 7
• Lemma 8
• Lemma 9
• Theorem 10
• Theorem 11: See LiuZ