Stability of solution for rao-nakra sandwich beam motel with Kelvin-Voigt damping and time delay

Abstract

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 𝜌1ℎ1𝑢𝑡𝑡 − 𝐸1ℎ1𝑢𝑥𝑥 − 𝜅(−𝑢 + 𝑣 + 𝛼𝑤𝑥) − 𝑎𝑢𝑥𝑥𝑡 − 𝜇𝑢𝑥𝑥𝑡( ・ , 𝑡 − 𝜏) = 0, 𝜌3ℎ3𝑣𝑡𝑡 − 𝐸3ℎ3𝑣𝑥𝑥 + 𝜅(−𝑢 + 𝑣 + 𝛼𝑤𝑥) − 𝑏𝑣𝑥𝑥𝑡 = 0, 𝜌ℎ𝑤𝑡𝑡 + 𝐸𝐼𝑤𝑥𝑥𝑥𝑥 − 𝜅𝛼(−𝑢 + 𝑣 + 𝛼𝑤𝑥)𝑥 − 𝑐𝑤𝑥𝑥𝑡 = 0. 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.