Decay in full von Kármán beam with temperature and microtemperatures effects

Abstract

In this article we derive the equations that constitute the mathematical model of the full von K\'{a}rm\'{a}n beam with  temperature and microtemperatures effects. The nonlinear governing equations are derived by using Hamilton principle in the framework of Euler–Bernoulli beam theory.   Under quite general assumptions on nonlinear damping function acting on the transversal component and based on nonlinear semigroups and the theory of monotone operators, we establish existence and uniqueness of weak and strong solutions to the derived problem.  Then using the  multiplier method, we show that solutions decay exponentially. Finally we consider the case  of zero thermal conductivity and we show that the dissipation given only by the microtemperatures is strong enough to produce  exponential stability.