Optimal polynomial stability for laminated beams with Kelvin–Voigt damping

Abstract

In this manuscript, we study the asymptotic behavior of two laminated beams systems formed by two identical layers of uniform thickness, whose surfaces are held together by an adhesive layer that allows interfacial sliding. We prove that when single Kelvin–Voigt damping acts on the transverse displacement equation, the system is not exponentially stable. When the system has two frictional dampings, one working on the transverse displacement and the other on the interface, the system is exponentially stable as long as the wave propagation speeds are equal. Then, one could think that if we replace the frictional damping (weak damping), acting on the transverse displacement, with Kelvin–Voigt damping (strong damping), the system should also be exponentially stable, at least when the wave speeds are equal. However, as we show in this work, Kelvin–Voigt damping causes the loss of exponential stability of the system, even when the system's waves' propagation speeds coincide. Therefore, we are in a situation where the weak damping (frictional) is stronger than the strong damping (Kelvin–Voigt). Due to these results of lack of exponential decay, we study the polynomial stability of each of these two systems by obtaining an optimal decay rate in each case.