Stabilization in extensible thermoelastic Timoshenko microbeam based on modified couple stress theory

In this article we derive the equations that constitute the nonlinear model of extensible thermoelastic Timoshenko microbeam. The constructed mathematical model is based on the modified couple stress theory which implies prediction of size dependent effects in microbeam resonators, by applying the Hamilton principle to full von Karman equations. This takes account of the effects of extensibility where the dissipations are entirely contributed by temperature. Based on semigroups theory, we establish existence and uniqueness of weak and strong solutions to the derived problem. By an approach based on the Gearhart-Herbst-Pruss-Huang theorem, we prove that the associated linear semigroup (without extensibility) is not analytic in general. In the absence of additional mechanical dissipations, the system is often not highly stable. Then by adding a damping frictional function to the first equation of the derived model and using the multiplier method, we show that the solutions decay exponentially under a condition on the physical coefficients.