Free vibrations of functionally graded piezoceramic hollow spheres with radial polarization

By introducing displacement functions as well as stress functions, two independent state equations with variable coefficients are established from the three-dimensional equations of a radially inhomogeneous spherically isotropic piezoelastic medium. By virtue of the laminated approximation methods the state equations are then transformed into the ones with constant variables in each layer, and the state variable solutions are presented. Based on the solutions, linear algebraic equations about the state variables only at the inner and outer spherical surfaces are derived by utilizing the continuity conditions at each interface. Frequency equations corresponding to two independent classes of vibrations are finally obtained from the free surface conditions. Numerical calculations are presented and the effect of the material gradient index on natural frequencies is discussed. (C) 2002 Elsevier Science Ltd.