On the problem of modeling piezoelectric/piezomagnetic plates from pre-stressed functionally gradient materials

An approach to modeling composite magnetoelectroelastic structures from prestressed inhomogeneous piezoelectric and piezomagnetic layers is proposed. Magnetoelectroelastic plates are considered as a structure, the outer surfaces of which in the absence of mechanical stresses freely contact with vacuum, and the adhesion conditions are satisfied at the interface between the layers. Wave propagation in the structure is caused by the action of a remote source of harmonic oscillations. The layers are made of functionally graded materials, the properties of which vary quite arbitrarily across the thickness. Each of the layers contains initial deformations induced by the action of initial mechanical stresses. The initial strain state of the plate layers is assumed to be uniform, the inhomogeneity of the initial stress state is due to the inhomogeneity of the material properties. The presence of electric and magnetic fields in the layers is caused by the action of mechanical stresses, is associated with the manifestation of piezoelectric and piezomagnetic effects and depends on the piezoelectric and piezomagnetic properties of materials. The proposed approach is based, on the one hand, on the use of linearized in the Lagrangian coordinate system associated with the natural state of the material, constitutive relations and equations of motion of a prestressed magnetoelectroelastic medium. On the other hand, it is reduced by means of the integral Fourier transform of systems of second-order partial differential equations with variable coefficients to a system of ordinary second-order differential equations with variable coefficients. The solution of the latter is based on the solution of a system of Cauchy problems with initial boundary conditions. The use of numerical methods allows one to quickly and efficiently restore the values of unknown functions of stresses, displacements, electric and magnetic potentials across the thickness of layers, construct an integral representation describing the motion of an arbitrary point of the structure, and a dispersion equation, the solution of which determines the main characteristics of surface acoustic waves