Solution of nonlinear free vibration problem of composite shallow shells reinforced with carbon nanotubes

In the thesis study, the solution of the nonlinear vibration problem of double-curved shallow shells reinforced by carbon nanotubes (CNTs) for simply-supported boundary conditions is presented. Nonlinear partial differential equations of double-curved shallow shells reinforced by CNTs are derived after creating mathematical models of nanocomposites by considering uniform and functionally graded patterns of CNT distributions through the thickness of the shell. The nonlinear theory of shells is used to derive the basic equations that take into account the nonlinearity of von-Karman type. The aforementioned differential equations are reduced to second order nonlinear ordinary differential equations by applying Galerkin method. The formulas for frequencies of linear and nonlinear vibrations are obtained by solving the nonlinear ordinary differential equation with the semi-inverse method. The aforementioned formulas are minimized according to the transverse and longitudinal wave numbers and the minimum values of linear and non-linear frequencies of double-curved shallow shells reinforced by CNTs are obtained numerically for fixed values of amplitude-thickness ratio. The reliability of the results is verified by comparison with numerical-analytical solutions in the literature. The influences of changes of shallow spherical and hyperbolic-paraboloidal (hypar) shell characteristics, volume fraction of CNTs, uniform and functionally graded CNT patterns on linear and nonlinear frequency parameters are investigated in detail with unique numerical calculations and analyzes.