Polynomial decay rate in extensible Timoshenko–Boltzmann beam

In this article we derive the equations that form the nonlinear mathematical model of the Timoshenko–Boltzmann extensible system with memories. The nonlinear governing equations are derived by applying the Hamiltonian principle to full von Kármán equations in the framework of Euler–Bernoulli beam and Boltzmann theories for viscoelastic materials. The model takes into account the effects of extensibility where the dissipations are entirely contributed by memories. Based on semigroup theory, we establish existence and uniqueness of weak and strong solutions to the derived problem with two memory kernels. Using a resolvent criterion due to Borichev and Tomilov and eliminating the extensibility effect and one of the memory kernels, we prove the optimality of the polynomial decay rate for the corresponding problem under certain conditions on the physical coefficients. Moreover, by an approach based on the Gearhart–Herbst–Prüss–Huang theorem, we show the non-exponential stability of each problem, even if the memory kernel functions are of exponential type. Finally we show the strong stability of the semigroup corresponding to the derived model without the extensibility by a result due to Arendt–Batty.