Stability of solution for Rao-Nakra sandwich beam model with Kelvin-Voigt damping and time delay

Victor R. Cabanillas, Carlos Alberto Raposo, Leyter Potenciano-Machado

This paper deals with stability of solution for a one-dimensional model of Rao--Nakra sandwich beam with Kelvin--Voigt damping and time delay given by \begin{gather*} \rho_1h_1u_{tt}-E_1h_1u_{xx}-ąppa(-u+v+lpha w_x)-au_{xxt}-\mu u_{xxt}(\,\cdot\,,t-au)=0, \rho_3h_3v_{tt}-E_3h_3v_{xx}+ąppa(-u+v+lpha w_x)-bv_{xxt}=0, \rho hw_{tt}+EIw_{xxxx}-ąppalpha(-u+v+lpha w_x)_x-cw_{xxt}=0. \end{gather*} A sandwich beam is an engineering model that consists of three layers: two stiff outer layers, bottom and top faces, and a more compliant inner layer called ``core layer''. Rao--Nakra system consists of three layers and the assumption is that there is no slip at the interface between contacts. The top and bottom layers are wave equations for the longitudinal displacements under Euler--Bernoulli beam assumptions. The core layer is one equation that describes the transverse displacement under Timoshenko beam assumptions. By using the semigroup theory, the well-posedness is given by applying the Lumer--Phillips Theorem. Exponential stability is proved by employing the Gearhart-Huang-Prüss' Theorem.