Existence And Stabilization Of Solutions For The Thermoelastic Laminated Beam With Fourier Law

In recent years,laminated composite materials have been widely applied in various fields of modern technology and engineering because of their good mechanical properties and designabil-ity,and thus receive the attention of a large number of scholars.Composite laminated beam is one of the most widely used components in mechanical and structural engineering.In practical applications,there exists a structure of two-layered beams of uniform thickness fasten together through a thin and light adhesive.Such a structure is called a laminated beam.This paper mainly discusses a kind of laminated beam system with structural damping,thermal effect,or memory term.The well-posedness and stability of the solution of the system are obtained.The main contents are given as follows:In the first chapter,we review the background and some development of the related prob-lems and briefly describe the main work of this paper.In Chapter 2,we study the well-posedness and asymptotic behavior of a one-dimensional laminated beam,where the heat conduction is given by Fourier's law effective in the rotation angle displacements.We show that the system is well-posed by using the semigroup theory.For the system with structural damping,we prove the exponential stability if and only if the wave speeds are equal,and also prove the polynomial stability in case of non-equal speeds.For the system without structural damping,we prove that the system is exponentially stable if and only if the wave speeds are equal.In Chapter 3,we study the well-posedness and asymptotic stability of a thermoelastic laminated beam with Fourier law and past history.The well-posedness of the system is obtained by using the semigroup theory.For the system with structural damping,without any restriction on the wave speeds,we prove the exponential and polynomial stabilities by using the perturbed energy method,where exponential and polynomial stabilities depend on the behavior of the kernel function of the history term.For the system without structural damping,we prove the exponential and polynomial stabilities in case of equal speeds by using the perturbed energy method and lack of exponential stability in case of non-equal speeds by using Gearhart-Herbst-Pruss-Huang Theorem.In the last chapter,we summarize the research contents of this paper and give some prob-lems which could be studied in the future.