The effect of Kelvin–Voigt damping on the stability of Timoshenko laminated beams system with history

This paper considers a model composed of two identical and uniform Timoshenko beams, one on top of the other, which are held together by an adhesive layer of negligible thickness, allowing interfacial slip between them. Using a semigroup approach and the frequency domain method, we study the system’s global well-posedness and asymptotic behavior under the influence of Kelvin–Voigt and memory-type dampings. It is well-known that if the wave propagation speeds are equal, a single memory damping on the rotation equation can exponentially drive the system to equilibrium. In this work, we study the impact of introducing Kelvin–Voigt damping on the displacement equation. We prove that the presence of Kelvin–Voigt damping destroys the exponential stability of the system achieved with memory damping. In light of the lack of exponential stability, we show that the system is polynomially stable.