Free vibration analysis of functionally graded conical, cylindrical shell and annular plate structures with a four-parameter power-law distribution

Based on the First-order Shear Deformation Theory (FSDT) this paper focuses on the dynamic behavior of moderately thick functionally graded conical, cylindrical shells and annular plates. The last two structures are obtained as special cases of the conical shell formulation. The treatment is developed within the theory of linear elasticity, when materials are assumed to be isotropic and inhomogeneous through the thickness direction. The two-constituent functionally graded shell consists of ceramic and metal. These constituents are graded through the thickness, from one surface of the shell to the other. A generalization of the power-law distribution presented in literature is proposed. Two different four-parameter power-law distributions are considered for the ceramic volume fraction. Some material profiles through the functionally graded shell thickness are illustrated by varying the four parameters of power-law distributions. For the first power-law distribution, the bottom surface of the structure is ceramic rich, whereas the top surface can be metal rich, ceramic rich or made of a mixture of the two constituents and on the contrary for the second one. Symmetric and asymmetric volume fraction profiles are presented in this paper. The homogeneous isotropic material can be inferred as a special case of functionally graded materials (FGM). The governing equations of motion are expressed as functions of five kinematic parameters, by using the constitutive and kinematic relationships. The solution is given in terms of generalized displacement components of the points lying on the middle surface of the shell. The discretization of the system equations by means of the Generalized Differential Quadrature (GDQ) method leads to a standard linear eigenvalue problem, where two independent variables are involved without using the Fourier modal expansion methodology. Numerical results concerning six types of shell structures illustrate the influence of the power-law exponent, of the power-law distribution and of the choice of the four parameters on the mechanical behaviour of shell structures considered.