Strong and Polynomial Stability in Extensible Timoshenko Microbeam with Memories Based on the Modified Couple Stress Theory

Abstract

In this article we derive the equations that constitute the nonlinear mathematical model of extensible Timoshenko microbeam with memories based on the modified couple stress theory. The nonlinear governing equations are derived by applying the Hamilton principle to full von Kármán equations together with Boltzmann theory for viscoelastic materials. The model takes into account the effects of extensibility, where the dissipation is entirely contributed by memories. Based on semigroups theory, we establish existence and uniqueness of weak and strong solutions to the derived problem. By using a resolvent criterion, developed by Borichev and Tomilov, we prove the optimality of the polynomial decay rate of the derived equations without extensibility when the viscoelastic law acts only on the shear force under the condition (4.10). In particular, we show that the considered problem is not exponentially stable. Moreover, by following a result due to Arendt-Batty, we show that the derived problem (without extensibility) is strongly stable.