Ciencias Naturales e Ingenierías
Magneto-electric coupling constants in piezoelectric / piezomaganetic layered composite
Constantes de acoplamiento magnetoeléctricas en compuestos estratificados piezoeléctricos / piezomagnéticos
*Instituto de Física, Universidad Nacional Autónoma de México, México.
**Escuela de Ingeniería y Ciencias, Tecnológico de Monterrey, Ciudad de México, México.
***Escuela de Ingeniería y Ciencias, Tecnológico de Monterrey, Estado de México, México.
Abstract:
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 BaTiO3 and CoFe2O4 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.
Keywords: electromagnetic elasticity (electroelasticity, magnetoelasticity); magnetoelectric effects; piezoelectric constants; composite materials
PACS: 46.25.Hf; 75.85.+t; 77.65.Bn; 77.84.Lf
Resumen:
Durante los últimos años se han estudiado los composites piezoeléctricos / piezomagnéticos debido a las numerosas aplicaciones relacionadas con el acoplamiento entre estos materiales y los campos. En el presente trabajo se presentan dos modelos teóricos para el cálculo del factor de acoplamiento magneto / eléctrico del composite con conectividad 2-2. Utilizando el método de homogeneización asintótica, los coeficientes efectivos de un compuesto estratificado magneto-electro-elástico periódico se pueden obtener en forma de matriz. Mediante el uso de esta matriz se estudia un compuesto de dos capas formado por BaTiO3 y CoFe2O4 y se obtienen expresiones para los coeficientes efectivos. El factor de acoplamiento magneto / eléctrico efectivo en función de la fracción volumétrica piezoeléctrica se encuentra a partir de estos coeficientes particulares. Además, se analiza un modelo dinámico del composite piezoeléctrico / piezomagnético multicapa. El modelo dinámico se ha utilizado para determinar las constantes de acoplamiento magnetoeléctrico.
Palabras clave: elasticidad electromagnética (electroelasticidad, magnetoelasticidad); efectos magnetoeléctricos; constantes piezoeléctricas; materiales compuestos
PACS: 46.25.Hf; 75,85 + t; 77.65.Bn; 77.84.Lf
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, 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. Composites with piezoelectric and piezomagnetic phases exhibit magnetoelectric properties due to the coupling of phases, and several researchers investigate the ME effect in these composites. (Kuo & Hsin 2018, 1503) investigated fibrous composites while (Shi 2018, 474), (Praveen 2018, 392) and (Hohenberger 2018, 184002) have studied laminate composites.
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 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 electro-elastic coupling factor (kt) 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 x1, which is known as a composite material of the type 2-2. The coordinate system is chosen such that the x3 axis is along the polarization direction of the piezoelectric and the piezomagnetic medium, the x1 axis is perpendicular to the interfaces; therefore the discontinuity direction is in the x1 direction and the x2 axis is a long the plane of the plate.
For a typical 2-2 composite, the dimensions are much larger than the period d and thickness. In this approximation, they can be considered as infinite. Thus, the problem is independent of the x2 direction. The governing equations for the dynamic heterogeneous plates are
∂σ1∂x1+∂σ5∂x3=ρ∂2u1∂t2,∂σ5∂x1+∂σ3∂x3=ρ∂2u3∂t2∂D1∂x1+∂D3∂x3=0,∂B1∂x1+∂B3∂x3=0,
(1)
where σI (we use the Voigt notation) are the components of stress tensor, ui are the components of displacement vector, Di are the components of electric displacement vector and Bi are the components of magnetic field vector.
The constitutive equations, which relate σI, Di Bi the components of the strain tensor SI where SI=12∂ui∂xj+∂uj∂xi, the component of electric field intensity vectorEi and the components of magnetic field intensity vector Hi are given by:
σ1=c11S1+c13S3-e31E3-q31H3,σ3=c13S1+c33S3-e33E3-q33H3,σ5=2c55S5-e15E1+q15H1,D1=2e15S5-E11E1-λ11H1,D3=e33S3+e31S1-ε33E3-λ33H3,B1=2q15S5-λ11E1-μ11H1,B3=q33S3+q31S1-λ33E3-μ33H3.
(2)
The constitutive equations, which relate σI Di, Bi with, ui the electric potential, φ and the magnetic potential Ψ are given by:
σ1=c11∂u1∂x1+c13∂u3∂x3+e31∂φ∂x3+q31∂ψ∂x3,σ3=c13∂u1∂x1+c33∂u3∂x3+e33∂φ∂x3+q33∂ψ∂x3,σ5=c55∂u1∂x3+∂u3∂x1+e15∂φ∂x1+q15∂ψ∂x1,D1=e15∂u1∂x3+∂u3∂x1-E11∂φ∂x1-λ11∂ψ∂x1,D3=e33∂u3∂x3+e31∂u1∂x1-E33∂φ∂x3-λ33∂ψ∂x3B1=q15∂u1∂x3+∂u3∂x1-λ11∂φ∂x1-μ11∂ψ∂x1,B3=q33∂u3∂x3+q31∂u1∂x1-λ33∂φ∂x3-μ33∂ψ∂x3,
(3)
Where we have used the quasi-static approximation of the fields. The symbols c
ij
εijμij λij, eij and qij represent the elasticity, dielectric permittivity, magnetic permittivity, magnetoelectric, piezoelectric, and piezomagnetic tensors, respectively.
Homogeneous asymptotic method
The double-scale asymptotic homogenization (MHA) method introduces two spatial coordinate systems. The position of the body is denoted by the Cartesian coordinate system
X= (x 1 ,x2 ,x3 ).. We introduce the local variable Y= (y 1 ,y2,y3) whose components are given by yi =xi /α ; with α<<1. The material functions are periodic respect to Y.
An asymptotic double scale development around the parameter α for the displacement and for the potentials is proposed as follows
u1x1,x3,y1,t=u10x1,x3,t+αu11x1,x3,y1,t+…u3x1,x3,y1,t=u30x1,x3,t+αu31x1,x3,y1,t+…φx1,x3,y1,t=φ0x1,x3,t+αφ1x1,x3,y1,t+…ψx1,x3,y1,t=ψ0x1,x3,t+αψ1x1,x3,y1,t+…
(4)
Substituting eqs (4) into eqs (3), the constitutive equations take the following form:
σ1x1,x3,y1,t=σ10x1,x3,y1,t+ασ11x1,x3,y1,t+…σ3x1,x3,y1,t=σ30x1,x3,y1,t+ασ31x1,x3,y1,t+…σ5x1,x3,y1,t=σ50x1,x3,y1,t+ασ51x1,x3,y1,t+…D1x1,x3,y1,t=D11x1,x3,y1,t+αD11x1,x3,y1,t+…B1x1,x3,y1,t=B11x1,x3,y1,t+αB11x1,x3,y1,t+…
(5)
Where,
σ1k=c11∂u1k∂x1+∂u1k+1∂y1+c13∂u3k∂x3+e31∂φk∂x3+q31∂ψk∂x3,σ3k=c13∂u1k∂x1+∂u1k+1∂y1+c33∂u3k∂x3+e33∂φk∂x3+q33∂ψk∂x3,σ5k=c55∂u1k∂x3+∂u3k∂x1+∂u3k+1∂y1+e15∂φk∂x1+∂φk+1∂y1+q15∂ψk∂x1+∂ψk+1∂y1.D1k=e15∂u1k∂x3+∂u3k∂x1+∂u3k+1∂y1-E11∂φk∂x1+∂φk+1∂y1-λ11∂ψk∂x1+∂ψk+1∂y1,D3k=e33∂u3k∂x3+e31∂u1k∂x1+∂u1k+1∂y1-E33∂φk∂x3-λ33∂ψk∂x3,B1k=q15∂u1k∂x3+∂u3k∂x1+∂u3k+1∂y1-λ11∂φk∂x1+∂φk+1∂y1-μ11∂ψk∂x1+∂ψk+1∂y1B3k=q33∂u3k∂x3+q31∂u1k∂x1+∂u1k+1∂y1-λ33∂φk∂x3-μ33∂ψk∂x3.
(6)
Substituting eqs (5) and (6) in eqs (1) and rearranging terms of equal potential of α we have:
for α-1
∂σ10∂y1=0,∂D10∂y1=0,∂B10∂y1=0.
(7)
For α=0
∂σ10∂x1+∂σ50∂x3+∂σ11∂y1=ρ∂2u10∂t2,∂σ50∂x1+∂σ30∂x3+∂σ51∂y1=ρ∂2u30∂t2,∂D10∂x1+∂D30∂x3+∂D11∂y1=0,∂B10∂x1+∂B30∂x3+∂B11∂y1=0.
(8)
Taking the average per unit length <F>=1Y∫Fdy of the expressions given by eqs(8) and using the periodicity of σ(1) , D(1) and B(1) with respect to Y the dynamic expressions of the homogeneous problem is obtained as follows:
∂σ-10∂x1+∂σ-50∂x3=ρ-∂2u10∂t2,∂σ-50∂x1+∂σ-30∂x3=ρ-∂2u30∂t2,∂D-10∂x1+∂D-30∂x3=0,∂B-10∂x1+∂B-30∂x3=0,
(9)
Where F- is the average of F (global valor of F ). Let N, W, S, Φ,Θ,Ξ,ψ,Ω and Υ be auxiliary functions that only depend on the local variables (local functions) and periodicty in Y We can write u11,u31,φ1 and ψ1 in terms of the local functions as follows:
un1=NnkIy1∂uk0∂xI+WnIy1∂φ0∂xI+SnIy1∂ψ0∂xI,φ1=ΦkIy1∂uk0∂xI+ΘIy1∂φ0∂xI+ΞIy1∂ψ0∂xIψ1=ΨkIy1∂uk0∂xI+ΩIy1∂φ0∂xI+γIy1∂ψ0∂xI
(10)
Where Eintein’s sum has been used.
Substituting the expressions given by eqs (9) in eqs (5) and taking the average, seven equations are obtained which relate the averages of zero order of the fields, with the derivatives of the components of zero order of the displacements and the potentials. These equations are shown in appendix A and they are numerated as (A1). Comparing the equations in Annex (A1) with the contiguous equations, it can be seen that the coefficients that multiply the derivatives are the effective coefficients. These expressions are the constitutive equations of the homogeneous problem. The coefficients that appear in equations (A1) must be independent of Y, i.e. their derivatives with respect to y1 must be equal to zero. From this condition, three systems of equations are obtained as follows:
L1=dσ1Nnkk,Φkk,Ψkkdy1=-dc1kdy1dσ5Nn13,Φ13,Ψ13dy1=-dc55dy1dD1Nn13,Φ13,Ψ13dy1=-de15dy1dB1Nn13,Φ13,Ψ13dy1=-dq15dy1L2=dσ1Wn3,Θ3,Ω3dy1=-de31dy1dσ5Wn1,Θ1,Ω1dy1=-de15dy1dD1Wn1,Θ1,Ω1dy1=dE11dy1dB1Wn1,Θ1,Ω1dy1=dλ11dy1L3=dσ1sn3,Ξ3,Υ3dy1=-dq31dy1dσ5sn1,Ξ1,Υ1dy1=-dq15dy1dD1sn1,Ξ1,Υ1dy1=dλ11dy1dB1sn1,Ξ1,Υ1dy1=dμ11dy1
(11)
Wich can be solved by using the periodicity of the local functions. In this way the effective coefficients can be obtained through the following relations:
Elasticc-11=c11-1-1,c-13=c11-1c13c11-1-1,c-33=c33+c11-1c132-c11-1c132c11-1-1,c-55=M11.
(12)
Piezoelectrice-31=e31c11-1c11-1-1,e-33=e33+e31c11-1c11-1c13c11-1-1-e31c13c11-1,e-55=M12.
(13)
Piezomagmeticq-31=q31c11-1c11-1-1,q-33=q33+q31c11-1c11-1c13c11-1-1-q31c13c11-1,q-55=M13.
(14)
DielectricE-11=-M22,E-33=E33-e312c11-1+e31c11-12c11-1-1.
(15)
Diamagneticμ-11=-M33,μ-33=μ33-q312c11-1+q31c11-12c11-1-1.
(16)
Magnetoelectricλ-11=-M23,λ-33=λ33-q31e31c11-1+q31c11-1e31c11-1c11-1-1.
(17)
Where M=N-1-1 and N=c55e15q15e15-ε11-λ11q15-λ11-μ11.
Fig. 2 shows the magnetoelectric coefficients, which have the most interesting behavior.
Coupling magnetoelectric constant
The thermodynamic potential W is defined from the internal energy U (Pérez-Fernández, 2009, 343):
W=U-EiDi-HiBi.
(18)
Internal energy is defined as
U=12σISI+12EiDi+12HiBi.
(19)
Substituting eqs (19) and the constitutive equations (3), into (18) the following expression are obtained:
U=12SIcIJSI-12EiεijEj-12HiμijHj-SIeIjEj-SIqIjHj-EiλijHj,=Wc-We-Wq-2Wce-2Wcq-2weq,
(20)
Where Wc=12SIcIJSI is the elastic energy density, We=12EiεijEj is the electric energy density, Wq=12HiμijHj is the magnetic energy density, Wce=12SIeIjEj is the piezoelectric energy density, Wcq=12SIqIjHj is the piezomagetic energy density and Weq=12EiλijHj is the magnetoeectric energy density.
The coupling ME constant in terms of this potential can written as
Ot=WceWcWe.
(21)
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 u
1, u
3 and the electric potential φ 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.
u3=Aexpik1x1+k3x3-ωt,u1=Bexpik1x1+k3x3-ωt,φ=Cexpik1x1+k3x3-ωt,ψ=Dexpik1x1+k3x3-ωt,
(22)
Where, k
i
are the components of the wave vector, w is angular frequency and A, B, C and D are indeterminate constants. Let's work first in the piezoelectric medium.
Substituting (22) into the system we obtain three homogeneous equations with three unknown independent constants A, B and C. These equations can been writer in matrix form:
QCoT=0,
(23)
Where
Co=A, B, C
(24)
And
Q=c33k32+c44k12-ρω2c13+c44k1k3e33k32+e15k120c13+c44k1k3c11k12+c44k32-ρω2e15+e44k1k30e33k32+e15k12e15+e44k1k3-E11k12+E33k320000-μ11k12+μ33k32
(25)
The condition for a nontrivial solution is that the determinant of the coefficients vanished. In the piezoelectric medium this determinant can be write as:
W=0.
(26)
The expression (26) is an implicit function of w k
1
and k
3
For each pair (w,k
3
) four k
1
values are obtained, which correspond to the quasi-longitudinal, quasi-shear, quasi-piezoelectric and quasi-piezomagnetic wave. In the piezoelectric material the quasi-piezomagnetic wave is such that
k1=μ33μ11=ik3.
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.
u3e=∑i=13Riecosk1iex1sink3x3,u1e=∑i=13fieRiesink1iex1cosk3x3,φe=∑i=13gieRiecosk1iex1sink3x3,ψe=R4ecoshk3x3sink3x3,
(27)
where index i refers to each of the values of k1ie index e indicate piezoelectric medium, fie,gie are obtained from the relationship between A,B and C. Since the system is not determined, its solution is indeterminate in at least one constant, Rie are this constant. Substituting (27) into (1) the form of the fields are obtained.
A similar development is carried out in the case of the piezomagnetic medium and similar solutions are obtained (28).
u3q=∑i=13Riqcosk1iqx1sink3x3,u1q=∑i=13fiqRiqsink1iqx1cosk3x3,φq=R4qcoshE33E11k3x3sink3x3,ψq=∑i=13giqRiqcosk1iqx1sink3x3,
(28)
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).
u1e=u1q,σ1e=σ1q,u3e=u3q,σ5e=σ5q,φe=φq,D1e=D1q,ψe=ψq,B1e=B1q.
(29)
These conditions are evaluated at the interphases x1=γd2where γ is the volumetric fraction of BaTiO3. They are eight homogenous equations with eight indeterminate constants (Rie and Riqwhith i=1,2,3,4) 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 k
3
and wThese functions are the dispersion curves for the composite (Fig. 3).
Coupling magnetoelectric constant
To determine the coupling factor ME O
t
using this model, have been used the definition of the coupling factor ME from the relationship between the wave velocity at E and H constants (v
EH
) and wave velocity at D and B constants (v
DB
).
vEHvDB2=1-Ot2.
(30)
For a single piezoelectric material, the velocity (v
EH
) 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 k
3
→0 as discussed in (Zhang & Geng 1994, 614). Similarly, (v
DB
).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 (v
EH
). and (v
DB
). 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 k
t
.
The results of these two methods show a very good agreement at low/high volumetric fractions of γBaTiO3 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 co-presence 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.
Aknowledments
The National Scholarship Program of CONACyT for financing this work.
References
Bakhvalov and Panasenko (1989). Averaging processes in periodic media. Kluwer, Dordrecht.
[ Links ]
Cabanas, J. H. Otero, J. A. Bravo-Castillero, J. Rodríguez-Ramos, R. and Monsivais, G. (2010). Laminados magneto-electro-elásticos con variaciones en la orientación de la magnetización. Nova Scientia 2 (4) 58-76. https://doi.org/10.21640/ns.v2i4.210
[ Links ]
Fang F., Zhou Y., Xu Y. T., Jing W. Q. and Yang W. (2013) Magnetoelectric coupling of multiferroic composites under combined magnetic and mechanical loadings. Smart Materials and Structures (22) 7075009. https://doi.org/10.1088/0964-1726/22/7/075009
[ Links ]
Fu, J. Santa Rosa, W. M'Peko, J. C. Algueró, M. and Veneta, M. (2016). Magnetoelectric coupling in lead-free piezoelectric Lix(K0.5Na0.5)1 − xNb1 − yTayO3 and magnetostrictive CoFe2O4 laminated composites. Physics Letters A 380 (20) 1788-1792. https://doi.org/10.1016/j.physleta.2016.03.024
[ Links ]
Hohenberger, S.; Lazenka, V.; Temst, K.; Selle, S.; Patzig, C.; Höche, T.; Grundmann, C. and Lorenz, M. (2018) Effect of double layer thickness on magnetoelectric coupling in multiferroic BaTiO3 -Bi 0.95Gd 0.05 FeO 3 multilayers. Journal of Physics D: Applied Physics. 51 (18) 184002. https://doi.org/10.1088/1361-6463/aab8c9
[ Links ]
Kuo, HY. and Hsin, K. C. (2018) Functionally graded piezoelectric-piezomagnetic fibrous composites. Acta Mechanic 229 (4) 1503-1516. https://doi.org/10.1007/s00707-017-2065-3
[ Links ]
Pérez-Fernández, L. D., Bravo-Castillero, J., Rodríguez-Ramos, R. and Sabina, F. J. (2009). On the constitutive relations and energy potentials of linear thermo-magneto-electro-elasticity. Mechanics Research Communications 36, 343-350. https://doi.org/10.1016/j.mechrescom.2008.10.003
[ Links ]
Pobedrya, B. E. (1984). Mechanics of composite materials. Moscow State University Press, Moscow
[ Links ]
Praveen, J. Reddy, V. Chandrakala, E. and Indla, S. Dineshkumar, S. Subramanian, V. and Das, D. (2018) Enhanced magnetoelectric coupling in Ti and Ce substituted lead free CFO-BCZT laminate composites. Journal of Alloys and Compounds. 750 392-400. https://doi.org/10.1016/j.jallcom.2018.04.026
[ Links ]
Qiu, J. Wen, Y. Li, P. and Chen, H. (2014). Magnetoelectric coupling characteristics of five-phase laminate composite transducers based on nanocrystalline soft magnetic alloy. Applied Physics Letters 104 (11) 112401. https://doi.org/10.1063/1.4868983
[ Links ]
Shi Y. (2018) Modeling of nonlinear magnetoelectric coupling in layered magnetoelectric nanocomposites with surface effect. Composite Structures 185 474 - 482. https://doi.org/10.1016/j.compstruct.2017.11.019
[ Links ]
Zhang, Q. M. and Geng, X. (1994). Dynamic modeling of piezoceramic polymer composite with 2-2 connectivity. Journal of Applied Physics 76 6014-6016. https://doi.org/10.1063/1.358354
[ Links ]
Zeng, Y. Bao, Yi, G. J. Zhang, G. and Jiang, S. (2015). Study on electronic structures and mechanical properties of new predicted orthorhombic Mg2SiO4 under high pressure Journal of Alloys and Compounds (630) 11-22. https://doi.org/10.1016/j.jallcom.2014.10.201
[ Links ]
Zhou, C. Shen, L. Liu, M. Gao, C. Jia, Ch. and Jiang, Ch. (2017). Strong Nonvolatile Magnon-Driven Magnetoelectric Coupling in Single-Crystal. Physical Review Applied. 9 (1) 014006-014014. https://doi.org/10.1103/PhysRevApplied.9.014006
[ Links ]
Appendix A
σ-10=c11+τ11∂u10∂x1+c13+τ13∂u30∂x3+e31+d31∂φ0∂x3+q31+h31∂ψ0∂x3σ-30=c13+τ13∂u10∂x1+c33+τ33∂u30∂x3+e33+d33∂φ0∂x3+q33+h33∂ψ0∂x3σ-50=c55+τ55∂u10∂x3+∂u30∂x1+e15+d15∂φ0∂x1+q15+h15∂ψ0∂x1D-10=e15+ζ15∂u10∂x3+∂u30∂x1-E11-δ11∂φ0∂x1-λ11-β11∂ψ0∂x1D-30=e33+ζ33∂u30∂x3+e31+ζ31∂u10∂x1+ε33-δ33∂φ0∂x3-λ33-β33∂ψ0∂x3B-10=q15+ξ15∂u10∂x3+∂u30∂x1-λ11-κ11∂φ0∂x1-μ11-χ11∂ψ0∂x1B-30=q33+ξ33∂u30∂x3+q31+ξ31∂u10∂x1+λ33-κ33∂φ0∂x3-μ33-χ33∂ψ0∂x3
(A1)
Where
τ11=c11∂N111∂y1+c13∂N311∂y3+e31∂Φ11∂y3+q31∂Ψ11∂y3τ33=c13∂N133∂y1+c33∂N333∂y3+e33∂Φ33∂y3+q33∂Ψ33∂y3τ13=c11∂N133∂y1+c13∂N333∂y3+e31∂Φ33∂y3+q31∂Ψ33∂y3τ55=c55∂N333∂y1+∂N113∂y3+e15∂Φ13∂y1+q15∂Ψ13∂y1d15=e15∂N313∂y1+∂N113∂y3-E11∂Φ13∂y1-λ11∂Ψ13∂y1d31=e31∂N111∂y1+e33∂N311∂y3-E33∂Φ11∂y3-λ33∂Ψ11∂y3d33=e31∂N133∂y1+e33∂N333∂y3-E33∂Φ33∂y3-λ33∂Ψ33∂y3h15=q15∂N313∂y1+∂N113∂y3-λ11∂Φ13∂y1-μ11∂Ψ13∂y1h31=q31∂N111∂y1+q33∂N311∂y3-λ33∂Φ11∂y3-μ33∂Ψ11∂y3h33=q31∂N133∂y1+q33∂N333∂y3-λ33∂Φ33∂y3-μ33∂Ψ33∂y3
(A2)
ζ31=c11∂W13∂y1+c13∂W33∂y3+e31∂Θ11∂y3+q31∂Ω3∂y3ζ33=c13∂W13∂y1+c33∂W33∂y3+e33∂Θ3∂y3+q31∂Ω3∂y3ζ15=c55∂W31∂y1+∂W11∂y3+e15∂Θ1∂y1+q15∂Ω1∂y1δ11=e15∂W31∂y1+∂W11∂y3-E11∂Θ1∂y1-λ11∂Ω1∂y1δ33=e31∂W13∂y1+e33∂W33∂y3-E33∂Θ3∂y3-λ33∂Ω3∂y3β11=q15∂W31∂y1+∂W11∂y3-λ11∂Θ1∂y1-μ11∂Ω13∂y1β33=q31∂W13∂y1+q33∂W33∂y3-λ33∂Θ3∂y3-μ33∂Ω3∂y3
(A3)
ξ31=c11∂S13∂y1+c13∂S33∂y3+e31∂Ξ3∂y3+q31∂Υ3∂y3ξ33=c13∂S13∂y1+c33∂S33∂y3+e33∂Ξ3∂y3+q33∂Υ3∂y3ξ15=c55∂S31∂y1+∂S11∂y3+e15∂Ξ1∂y1+q15∂Υ1∂y1k11=e15∂S31∂y1+∂S11∂y3-E11∂Ξ1∂y1-λ11∂Υ1∂y1k33=e31∂S13∂y1+e33∂S33∂y3-E33∂Ξ3∂y3-λ33∂Υ3∂y3X11=q15∂S31∂y1+∂S11∂y3-λ11∂Ξ1∂y1-μ11∂Υ13∂y1X33=q31∂S13∂y1+q33∂S33∂y3-λ33∂Ξ3∂y3-μ33∂Υ3∂y3
(A4)
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