Exponential matrix method for the solution of exact 3D equilibrium equations for free vibrations of functionally graded plates and shells

The present paper analyzes the convergence of the exponential matrix method in the solution of three-dimensional equilibrium equations for the free vibration analysis of functionally graded material structures. The three-dimensional equilibrium equations are written in general orthogonal curvilinear coordinates for one-layered and sandwich plates and shells embedding functionally graded material layers. The resulting system of second-order differential equations is reduced to a system of first-order differential equations redoubling the variables. This system is exactly solved using the exponential matrix method and harmonic displacement components. In the case of functionally graded material plates, the differential equations have variable coefficients because of the material properties which depend on the thickness coordinate z. For functionally graded material shells, the differential equations have variable coefficients because of both changing material properties and curvature terms. Several mathematical layers M can be introduced to approximate the curvature terms and the variable functionally graded material properties to obtain differential equations with constant coefficients. The exponential matrix is applied to solve the resulting system of partial differential equations with constant coefficients, where the used expansion has a very fast convergence ratio. The present work investigates the convergence of the proposed method related to the order N used for the expansion of the exponential matrix and to the number of mathematical layers M used for the approximation of curvature shell terms and variable functionally graded material properties. Both N and M values are analyzed for different geometries, thickness ratios, materials, functionally graded material laws, lamination sequences, imposed half-wave numbers, frequency orders, and vibration modes.