Vibration characteristics of composite doubly curved shells reinforced with graphene platelets with arbitrary edge supports

The present study deals with the free vibration analysis of composite laminated doubly curved shells which are reinforced with graphene platelets. The weight fraction of graphene platelets in each layer of the composite medium may be different which leads to a functionally graded graphene platelet-reinforced medium. The elastic modulus of the composite medium is obtained using the Halpin-Tsai approach which captures the size effects of the reinforcements. Using the first-order shear deformation shell theory and Sanders kinematic assumptions, the total strain and potential energies of the shell are established. The shape functions of the essential variables are estimated by means of Chebyshev polynomials which are suitable for arbitrary combinations of in-plane and out-of-plane boundary conditions. The results of the present study are well-validated with the available data in the open literature. After that, novel results are given to discuss the effects of different parameters on the frequencies and mode shapes of functionally graded graphene platelet-reinforced composite shells. It is shown that increasing the amount of graphene platelets in the shell results in higher frequencies. Also, among the introduced functionally graded patterns, the FG-X pattern leads to higher frequencies.