The volume ratio between piezoelectric and magnetostrictive phases is an important parameter for magneto-electric (ME) composites. The ME voltage coefficients can be enhanced greatly at optimum volume ratio. However, no previous report has focused on the study of the volume ratio effect on ferromagnetic-ferroelectric-substrate multilayer composites. We consider an arbitrary laminated structure of length 2L and N layers. In this case, there is no middle plane of the bar that can serve as a plane of symmetry. For simplicity we assume that the multilayer structure is two dimensional (i.e. bar structure), and the field functions depending only on the space coordinates X1, X2. In the Cartesian system of coordinates the X1 axis is directed along the bar length, the X2 -across the width, and X3 is orthogonal to X1 and X2. It is assumed that piezoelectric layers are poled in the X1 direction (L-L mode). It should be mentioned that the proposed theory can be successfully applied to multilayer structures when the polarization direction of the piezoelectric layers is along the X2 direction, or when some of them are along the X1, and others along X2, or X3 directions. An averaging method is used for deriving effective material parameters of composites. We consider only (symmetric) extensional deformation in this model and at first ignore any (asymmetric) flexural deformations of the layers that would lead to a position dependent elastic constants and the need for other methods to be applied. Using the continuity conditions for magnetic and electric fields, as well as the open and closed circuit conditions, one obtains the analytic expression for longitudinal ME voltage coefficient, which depends on electro-mechanical material properties and thicknesses of the layers. Analytical expression for ME coefficient allows us to find the optimal volume ratio of layers, for which the ME coefficient approaches to its maximum value.

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