The Mathematics Models And Their Theories Of Viscoelastic Shells Systems

The dynamic systems of viscoelastic shells are very important models in both theory and applications. Many famous mathematicians worked in this area. This thesis includes two part. In the first part, applying the method of asymptotic analysis, we establish two-dimensional models of viscoelastic linear shells (membrane shell, flexural shell) from three-dimensional system, and show the convergence of the solutions of the three-dimensional problem and two-dimensional Koiter systems. In the second part, we prove the existence and uniqueness of solution to the mixed initial-boundary value problem for nonlinearly viscoelastic full Marguerre-von Karman shallow shell system, study the asymptotic behavior of the solution as t —>∞, and show the convergence of the solution, to solution of viscoelastic von Kdrmdn plate.Now we state our main results.(1) Applying the method of asymptotic analysis, we establish the two-dimensional model of linearly viscoelastic membrane shell from three-dimensional viscoelastic system, and then show that the solution of three-dimensional problem converges to one of the two-dimensional model, as the thickness tends to zero.(2) Under some assumptions, applying the method of asymptotic analysis, we establish the two-dimensional model of linearly viscoelastic flexural shell from three-dimensional viscoelastic system, and then show that the solution of three-dimensional problem converges to one of the flexural shell model, as the thickness tends to zero.(3) We prove the existence and uniqueness of the solution to the system of viscoelastic Koiter shell , and then show that its solution converges to ones of membrane and flexural shell model, respectively under different conditions, as parameter tends zero.(4) Applying Galerkin based on finite element approach, we prove the global existence and uniqueness to the initial-boundary value problem of nonlinearly viscoelastic full Marguerre von Karman shallow shell system. Then, we discuss the asymptotic behavior and show that, under suitable assumption, the solution decays exponentially as time goes to infinity.(5) We show the solution, to the system of nonlinearly viscoelastic full Marguerre von Karman shallow shell, converges to the solution to corresponding viscoelastic von Kdrmdn plate. This justifies the shallow shell model in a sense.