Asymptotic behavior of laminated beams with Kelvin-Voigt damping

This work considers a one-dimensional system consisting of two identical Timoshenko beams. The model considers that an adhesive layer of small thickness joins the two surfaces, thus producing an interfacial slip under homogeneous mixed Neumann-Dirichlet-Dirichlet boundary conditions. We introduce a Kelvin-Voigt type damping into the rotation equation, and we study the well-posedness of the problem and the asymptotic behavior of the solutions using techniques from the semigroup theory of linear operators and the frequency domain method. When the wave’s propagation speeds are equal in both beams, we show that the Kelvin-Voigt dissipative term acting on the rotation equation is sufficient to obtain the exponential decay of the solutions while maintaining the structural dissipation characteristic of the model. When these propagation speeds differ, we show the lack of exponential decay and prove that the solutions decay polynomially with a decay rate of t-12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t^{-\frac{1}{2}}$$\end{document}. We prove, finally, that this decay rate is optimal.