Dynamic responses of functionally graded magneto-electro-elastic shells with open-circuit surface conditions

Three-dimensional (3D) free-vibration analysis of simply supported, doubly curved functionally graded (FG) magneto-electro-elastic shells with open-circuit surface conditions is studied using an asymptotic approach. The material properties of the FG shells are regarded as heterogeneous through the thickness coordinate. The 29 basic equations of 3D magneto-electro-elasticity are firstly reduced to a system of 10 state space vector equations in terms of 10 primary variables in elastic, electric and magnetic fields. Apart from the regular asymptotic expansion in the early paper on static analysis, the method of multiple time scales is used to eliminate the secular terms and to make the asymptotic expansion feasible. Through a straightforward derivation, we finally decompose the present 3D problem as recursive sets of two-dimensional (2D) problems with motion equations of the coupled classical shell theory (CCST). The orthonormality and solvability conditions for various order problems are derived. With these conditions, it is shown that the 3D asymptotic solutions can be obtained by repeatedly solving the CCST-type motion equations order-by-order in a systematic and hierarchic manner. The influence of the gradient index of material properties on the natural frequencies and their corresponding modal field variables of various FG piezoelectric and magneto-electro-elastic shells is presented.