Constantes de acoplamiento magnetoeléctricas en compuestos estratificados piezoeléctricos / piezomagnéticos Magneto-electric coupling constants in piezoelectric / piezomaganetic layered composite

During the last few years, piezoelectric/piezomagnetic composites have been studied due to the numerous applications related to the coupling between these materials and the fields. In the present work, two theoretical models for calculating the magneto/electric coupling factor of the composite with 2-2 connectivity, are presented. Using the asymptotic homogenization method, the effective coefficients of a periodic magneto–electro–elastic layered composite can be obtained in matrix form. By using this matrix, a two-layered composite formed by BaTiO 3 and CoFe 2 O 4 are studied, and expressions for the effective coefficients are obtained. The effective magneto/electric coupling factor as a function of the piezoelectric volumetric fraction are found from these particular coefficients. In addition, a dynamic model of the multilayer piezoelectric/piezomagnetic composite is discussed. The dynamical model has been used to determinate the magnetoelectric coupling constants.


Introduction
The study of materials that exhibit magneto-electric (ME) coupling has attracted a lot of interest due to the multiple applications related to these materials. ME coupling of laminate composites has been investigated under combined magnetic and mechanical loadings (Fang et al. 2013, 075009).
In previous work, the ME effect in a three-phase ME composite is experimentally studied (Zeng et al. 2015, 11). In (Fua 2016(Fua , 1788, the authors analyzed the ME coupling in lead-free piezoelectric bilayer composites and ME phases. A five-phase laminate composite transducer based on nanocrystalline soft magnetic FeCuNbSiB alloy is presented; whose ME coupling characteristics have been investigated in (Qiu et al. 2014, 112401). (Zhou et al. 2017, 014016) where a strong ME coupling at the interface in a Co /[PbMg1/3Nb2/3O3]0.71[PbTiO3]0.29 bilayerd structure, was found.
There are several ways to determine the coupling factors between different fields. In this paper, we have used two ways to determine the ME coupling factor. The first method is the asymptotic homogenization method. The effective coefficients are determined through the formulation of (Cabanas et al. 2010, 58). The double-scale asymptotic homogenization (MHA) method introduces two spatial coordinate systems: the local coordinate which studies the problems at the microstructure level, and the global coordinate system which uses the global characteristics Cabanas,J. H. et al. Nº 26,Vol. 13 (2), 2021. ISSN 2007 -0705, pp.: 1 -21 -3 -of the composite. From these effective ME coefficients the coupling factor is obtained through the thermodynamic definition.
The second method was proposed in (Zhang & Geng 1994, 614) to determine the electroelastic coupling factor ( t k ) where a dynamic study of the laminate is performed. From the dispersion curves, the required parameters for calculating the ME coupling constant are calculated.
Vertically polarized waves (SV waves) that propagate in the polarization direction of the materials are studied in each phase. Using contour equations at the interfaces of the composite, these waves can be related and can be obtained to the dispersion curves.
Homogenization methods are the most common type of methods, used for the calculation of coupling factors (Cabanas et al. 2010, 58). The asymptotic homogenization method, formally developed by (Pobedrya 1984) and (Bakhvalov & Panasenko 1989) is one of the most robust. The dynamic method has been used in piezoelectric-polymer compounds to calculate electromechanical coupling factors, yielding results that are closer to the experimental values than those predicted by homogenization methods (Zhang & Geng 1994, 614). In this work, we propose the comparison results obtained from both methods.

System studied
Let us consider a heterogeneous piezoelectric/piezomagnetic material ( Fig. 1), made of alternating plates of piezoelectric and piezomagnetic materials, forming a parallel arrangement in the direction 1 x , which is known as a composite material of the type 2-2. The coordinate system is chosen such that the 3 x axis is along the polarization direction of the piezoelectric and the piezomagnetic medium, the 1 x axis is perpendicular to the interfaces; therefore the discontinuity direction is in the 1 x direction and the 2 x axis is a long the plane of the plate.   The governing equations for the dynamic heterogeneous plates are   2  2  5  5  3  3  1  1  2  2  1  3  1  3   3  3  1  1   1  3  1  3 , , where I  (we use the Voigt notation) are the components of stress tensor,  -, Where we have used the quasi-static approximation of the fields. The symbols , and ij q represent the elasticity, dielectric permittivity, magnetic permittivity, magnetoelectric, piezoelectric, and piezomagnetic tensors, respectively.
Internal energy is defined as In this way, the coupling ME constant is obtained by this method.

Dynamical method
The second method, which we have called the dynamic method, is to study the behavior of the compound before the propagation of vertically polarized shear waves (SV). First the dispersion curves are obtained, and from them the coupling factor.
Combining (1) and (2) we have four differential equations of second order, which describe the behavior of the elastic displacements 13 , uu and the electric potential  for the composite.
The solution of these systems must be solved in each medium independently. The solution of the system is propose as plane wave for each medium, i.e.
Where, i k are the components of the wave vector,  is angular frequency and A, B, C and D are indeterminate constants. Let's work first in the piezoelectric medium. The condition for a nontrivial solution is that the determinant of the coefficients vanished. In the piezoelectric medium this determinant can be write as: The expression (26) Due to the symmetry of the system (1), the solution for the case of the piezoelectric material can be written as (27) which is one of the two modes of the Lamb wave.
Where index q indicate piezomagnetic medium.
The contact conditions give the conditions to be able to solve the system. We consider condition ideal contact in the interphases, that is to say conditions of continuity, as shown in (29).
Magneto-electric coupling constants in piezoelectric/piezomaganetic layered composite Nº 26, Vol. 13 (2) The condition for a nontrivial solution is that the determinant of the matrix associated to the system vanished. This condition gives a family of implicit functions of 3 k and .  These functions are the dispersion curves for the composite (Fig. 3).

Coupling magnetoelectric constant
To determine the coupling factor ME For a single piezoelectric material, the velocity EH v can be obtained from the dispersion curves as the slope of the first mode. For a composite material, this method is valid in the limit 3 0 k → as discussed in (Zhang & Geng 1994, 614). Similarly, DB v is obtained but now only using the elastic equations. The procedure is also discussed in (Zhang & Geng 1994, 614).

Results
By means of the first method (the asymptotic homogeneous method) the ME coupling factor of piezoelectric/piezomagnetic composites with layered of BaTiO3 and CoFe2O4 was computed starting from the effective coefficients of the composite. While the second method (dynamical method) uses the IEEE definition and determines the ME coupling factor trough the EH v and DB v obtained from the dispersion curves. Figure 4 shows the results obtained.
In Fig. 4 the solid line represents the result obtained from the asymptotic homogenization method. As follows from the formulation of the method, this results is an analytical function. While for the dynamical method we obtained a discrete plot because the calculations have been made for each volumetric fraction (the results are shown by black squares). This is a disadvantage of this method; however, it has the advantage of making the calculations directly from the phase constants and not from the effective constants. In (Zhang & Geng 1994, 614) it is shown that this second method is closer to experimental results than an homogenization method for the calculation of t k .  4. ME coupling factor as a function of piezoelectric volumetric fraction obtained for both methods. Note that the continues line passes by the points (0,0) y (1,0), as it should be. Furthermore, there is a square in each of these two points, but they are indistinguishable because they overlap with the axes.
The results of these two methods show a very good agreement at low/high volumetric fractions of ( ) 3 BaTiO .
 Both approach to zero in the limit cases when one of the phases is not present. The ME effect is a second order effect that appears in the compound through the interaction of both phases. However, in the center part of the interval the results are different although they show a similar behavior. This result shows that both methods provide a guide for the manufacture of laminated materials showing a ME effect. This mismatch is also obtained by (Zhang & Gheng 1994, 614) in the calculation of an electromechanical coupling factor. They also obtained a greater copresence in compounds with a larger amount of piezoelectric. They also demonstrated that the dynamic method out performs the results obtained through the homogenization methods when compared with the experimental results.
Homogenization methods constitute an approximation for modering heterogeneous materials as homogeneous materials. In order to make this approximation, strong conditions are required on the wavelengths which are used. The dynamical method has a better performance; however it may present numerical instabilities.

Conclusions
In this paper two methods to determine the ME coupling factor of piezoelectric-piezomagnetic multilaminates were used. The homogeneization method is based on calculations of the effective properties of the composite and from this method the effective coupling factor can be determined.
The dynamic method, in which the ME coupling factor is obtained from determining the slope of the dispersion curves, was described. Despite the difference between both methods; a similar trend is observed in both calculations. These results provide a valid guide for building a device with ME properties.