Analytical Solution for Free Vibration of Multilayered Circular Plate

An analytical solution is deduced for a three-dimensional transversely isotropic axisymmetric multilayered circular plate made of piezoelectric (PE) and piezomagnetic (PM) under simply supported boundary condition. The state space vector, finite Hankel transform and propagation matrix methods are utilized together to obtain the full-field solutions for the multilayered circle plate. Numerical examples for five-layered PE/PM composites with dimensionless frequencies of the multilayered plate under simple-supported lateral boundary conditions are presented. The frequencies increase with the ratio of the thickness to radius.


Introduction
Magnetoelectroelastic coupling effect exists in multiphase materials which possess the ability of transforming energy from one form to another (among elastic, electric and magnetic forms). Intensive studies on the physical and mechanical properties of the structures were carried out by means of analytical, numerical and experimental methods. Among those structures, the composites made of piezoelectric (PE) and piezomagnetic (PM) layers were most frequently considered and some full-field exact solutions of these structures under certain boundary conditions were obtained. Pan and colleagues [1,2] derived the static and free vibration solutions for multilayered rectangular plates under simply-supported boundary conditions by using the Stroh formulism and propagation matrix methods. By applying the state vector approach and propagation matrix method, Wang et al [3] derived the exact solution of the multilayered plate under static deformation, Chen et al [4] extended the static solution to the vibration case, and Chen et al [5] discussed the modal analysis of the multilayered plates. Combining the discrete layer approach and Ritz method, Ramirez et al [6] derived an approximate solution for the free vibration problem of two-dimensional laminate under both simply supported and fixed boundary conditions.
In this paper, starting from the equilibrium equations for each homogeneous layer and making use of the geometric and constitutive equations. By use of finite Hankel transform and let the free terms derived from the transform be zero. Base on the solutions of the state equations, the frequency equations are derived using the propagation matrix method and the boundary conditions on the bottom and top surfaces of the plates.

Basic equations
For a transversely isotropic coupling solid with its material axis parallel to z-axis, the general basic equation can be expressed as: where σ, D and B are vectors of elastic stress, electric displacement and magnetic induction; γ, E and H are elastic strain, electric field and magnetic field; C, e, q, ε, µ and α, are respectively matrices of elastic stiffness, PE coefficients, PM coefficients, permittivity coefficients, permeability coefficients and magnetoelectric (ME) coefficients, where u, w, φ and ψ are, respectively, the displacement in rand z-directions, and the electric and magnetic potentials; ρ is the density of the material.
The multilayered circular plate model is shown in Fig. 1

General Solutions
Following [7], the dimensionless variables are defined as follows ( where h and a are, respectively, total thickness and radius of the plate; respectively, the elastic stiffness, electric permittivity and magnetic permeability constants of the first layer material; Ω is the dimensionless frequency of the multilayered circular plate. By choosing the primary variables as the state space vector and rearranging Eqs. (1), (2) and (3), we arrive at the following state equation The related parameters are list in Appendix. Define the following finite Hankel transform where ( ) J k µ ξ is the first kind of Bessel function of µth order.
Applying the Hankel transform to both sides of Eq. (5), we obtain And eliminate Q according to boundary condition.

Load boundary conditions
We now consider the simply supported lateral boundary condition at ξ=1, i.e., The frequency equation is transcendental with respect to i Ω and gives infinite number of frequencies for each k. plate. The total thickness is assumed to be 20mm with each layer at 4mm. The BFBFB and FBFBF five dimensionless frequencies and the corresponding k1 , k2 and k3 for the simply supported lateral boundary condition with different thickness-to-radius ratio s are shown in Table 2 and 3. It shows that the dimensionless frequency increase with the thickness-to-radius ratio s for the simply supported lateral boundary condition.

Conclusions
We derived an analytical solution for three-dimensional transversely isotropic axisymmetric multilayered circular plates made of piezoelectric (PE) and piezomagnetic (PM) under simply supported lateral boundary condition. The finite Hankel transform and propagation matrix methods are utilized to find the full-field solutions for the multilayered circle plate. Numerical examples for five-layered PE/PM composites with dimensionless frequencies of the multilayered plate under simple-supported lateral boundary conditions are presented and discussed. The frequencies increase with the raitos of thickness to radius. These results can be served as benchmark solutions for future numerical analyses of multilayered circle plates.  [1] (C ij : elastic constants in GPa; e ij : piezoelectric coefficients in N/(V·m); q ij : piezomagnetic coefficients in N/(A·m); ε ij : permittivity coefficients in 10 -9 C/(V·m); and µ ij : permeability coefficients in 10 -6 Wb/(A·m).