Nonuniform laminated beam of Lord–Shulman type

Abstract

The nonuniform thermoelastic laminated beam of the Lord–Shulman type is considered. The model is a two‐layered beam with structural damping due to the interfacial slip. The well‐posedness is proved by the semigroup theory of linear operators approach together with the Lumer–Phillips theorem. The stability results presented in this paper depend on the nature of a stability function χfalse(xfalse)$\chi (x)$, which we define in (12). We first prove the lack of exponential stability of the system if χfalse(xfalse)≠0$\chi (x)\ne 0$, x∈false(0,1false)$x\in (0,1)$. And then, we establish the exponential stability for χfalse(xfalse)≡0$\chi (x) \equiv 0$ and polynomial decay with rate t−12$t^{-\frac{1}{2}}$ provided χfalse(xfalse)≠0$\chi (x)\ne 0$, x∈false(0,1false)$x\in (0,1)$. The result is new, and it is the first time that the nonuniform laminated beam is considered.