Dynamic responses of functionally graded magneto-electro-elastic shells with closed-circuit surface conditions using the method of multiple scales

A three-dimensional (3D) free vibration analysis of simply supported, doubly curved functionally graded (FG) magneto-electro-elastic shells with closed-circuit surface conditions is presented using the method of perturbation. By means of the direct elimination, we firstly reduce the twenty-nine basic equations of 3D magneto-electro-elasticity to ten differential equations in terms of ten primary variables of magnetic, electric and elastic fields. The method of multiple scales is introduced to eliminate the secular terms in various order problems of the present formulation so that the present asymptotic expansion to the primary field variables leads to be uniform and feasible. Through the mathematical manipulation of nondimensionalization, asymptotic expansion and successive integration, we finally obtain recurrent sets of governing equations for various order problems. The coupled classical shell theory (CST) is derived as a first-order approximation to the 3D magneto-electro-elasticity. Higher-order modifications can be further determined by considering the solvability and orthonormality conditions in a systematic and consistent way. Some benchmark solutions for the free vibration analysis of FG elastic and piezoelectric plates are used to validate the performance of the present asymptotic formulation. The influence of the material-property gradient index on the natural frequencies and corresponding modal field variables of the FG shells is mainly concerned.