Physics of Composites for Low-Frequency Magnetoelectric Devices

The article discusses the physical foundations of the application of the linear magnetoelectric (ME) effect in composites for devices in the low-frequency range, including the electromechanical resonance (EMR) region. The main theoretical expressions for the ME voltage coefficients in the case of a symmetric and asymmetric composite structure in the quasi-static and resonant modes are given. The area of EMR considered here includes longitudinal, bending, longitudinal shear, and torsional modes. Explanations are given for finding the main resonant frequencies of the modes under study. Comparison of theory and experimental results for some composites is given.


Introduction
At present, magnetoelectric (ME) composites are extensively studied [1][2][3]. Researchers pay main attention to layered composites because of the possibility of obtaining the maximum ME effect on their basis. Because of the content of magnetic and electrical (piezoelectric or ferroelectric) components in its structure, the ME composite is a multifunctional material and considerably interests developers of ME devices compared to conventional magnetic and electrical materials. The presence of a magnetic component makes it possible to change the magnetic properties of the composite by applying an external electric field, and the electrical properties change when exposed to an external magnetic field. Depending on the external applied fields in ME composites, the direct and inverse ME effects are distinguished. In the case of the direct effect, electric polarization is induced in the composite when it is exposed to a magnetic field, whereas, in the case of an inverse effect, magnetization occurs when exposed to an electric field. The main characteristic of the ME composite in the case of a direct effect is the ME voltage coefficient, which is the ratio of the induced electric field to the alternating magnetic field acting on the composite. There are numerous works devoted to the calculation of individual characteristics of ME composites [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] and examples of the development of various ME devices: sensors [21][22][23][24][25][26][27][28], gyrators [29,30], harvesters [31][32][33], antennas [34,35], and microwave devices [3,36]. The information in the literature concerns the calculations of the ME effect for individual electromechanical (EMR) regimes [6][7][8][9][10][11][12][13][14][15][16][17][18][19], then further in the article we will carry the comparison of the obtained theoretical results with these data out.
As an example of a developed perspective ME device, we can discuss the ME magnetic field sensor [22]. The great interest in this device is because, having a simple three-layer Metglas-PMN-PT-Metglas structure, in the near future it can replace a complex electronic device such as a SQUID operating at helium temperature, since comparable values for giant ME voltage coefficient of 4.26 × 10 4 V cm −1 Oe −1 and the equivalent magnetic noise of 2.89 fT Hz −1/2 at EMR frequency on this structure have already been achieved. If we consider its small weight and size parameters and operation at room temperature, then we can expect widespread use Magnetic fields are parallel to each other and lie in the plane of structure, the electric field is perpendicular to the plane of structure, and p t and m t are the thicknesses of piezoelectric and magnetostrictive layers.
We consider small mechanical oscillations in a composite under the influence of a small external variable magnetic field. In the presence of a constant magnetic field, the strengths of both fields are directed along the l (x) axis: where h1 (t) is an external variable magnetic field, and ω is a cyclic frequency of the external alternating magnetic field. The material Equation for the piezoelectric layer is given by: where p S1 is the strain tensor component of piezoelectric phase; d31 is piezoelectric coefficient; E3 is component of the vector of the electric field; p s11 is compliance tensor component of the piezoelectric phase; p T1 is the stress tensor component of the piezoelectric phase.
The longitudinal component of the stress tensor in a piezoelectric phase can be expressed as: The longitudinal component of the stress tensor of the magnetostrictive phase is given by: where m T1 is the stress tensor component of the magnetostrictive phase; m Y is their Young's module; m S1 is the strain tensor component of magnetostrictive layer; q11 is piezomagnetic coefficient; 1 h  is the intensity of the alternating magnetic field inside the ferromagnet.
The constitutive Equations for the ferromagnetic phase are given by:  The material Equation for the piezoelectric layer is given by: where p S 1 is the strain tensor component of piezoelectric phase; d 31 is piezoelectric coefficient; E 3 is component of the vector of the electric field; p s 11 is compliance tensor component of the piezoelectric phase; p T 1 is the stress tensor component of the piezoelectric phase. The longitudinal component of the stress tensor in a piezoelectric phase can be expressed as: The longitudinal component of the stress tensor of the magnetostrictive phase is given by: where m T 1 is the stress tensor component of the magnetostrictive phase; m Y is their Young's module; m S 1 is the strain tensor component of magnetostrictive layer; q 11 is piezomagnetic coefficient; h 1 is the intensity of the alternating magnetic field inside the ferromagnet. The constitutive Equations for the ferromagnetic phase are given by: where B 1 is magnetic induction; µ is magnetic permeability of an isotropic medium; µ 0 is magnetic constant; h 1 is the intensity of an external alternating magnetic field away from the ferromagnet. Express h 1 from Equation (6): Substituting Equation (7) in Equation (4), we get: Express m T 1 from Equation (8): where: (10) where m Y B is the Young's modulus under constant magnetic induction; m K 11 is the coefficient of magnetomechanical coupling. The square of the coefficient of magnetomechanical coupling is: Since the length of the composite is much greater than its width and height, longitudinal vibrations arise in it.
In accordance with the condition of the problem: The longitudinal component of the composite stress tensor is: where volume fractions of the p ν piezoelectric and m ν magnetostrictive phases are: where p t and m t are the thicknesses of piezoelectric and magnetostrictive layers and effective composite stiffness coefficient: Composite effective density can be obtained from: where p ρ, m ρ are density of the piezoelectric and magnetostrictive phases, respectively. The longitudinal component of strain tensor is: where Ux is longitudinal component of the strain vector. Consider the Equation of motion for deformations: Substituting Equation (13) in Equation (18), we get: The solution of this Equation is obtained as: where the wave number is: A, B are unknown constants. Then: To obtain the constant A and B, we use the equilibrium conditions for a free sample: where l is length of the ME structure. Substituting Equation (23) in Equation (24): where: we get: The transverse component of the electric displacement vector can be obtained from: where ε is dielectric permittivity of the medium; ε 0 is electrical constant. The transverse component of the electric field strength vector can be found from the condition that the electric induction flux through the interface between the upper layer of the magnetostrictive phase and the piezoelectric is equal to zero: Substituting Equation (28) in Equation (29): and substituting Equation (27) in Equation (30): from Equation (31), E 3 is obtained as: As the electric field exists only in the piezoelectric phase, the voltage is given by the following equation: Average electric field strength in ME composite is: Then, the ME voltage coefficient is obtained as: Below, Figure 2 shows the dependence of the ME voltage coefficient on the frequency of the alternating magnetic field for two cases, when PZT and a cut of lithium niobate y + 128 • [13,19] are taken as the piezoelectric phase. Metglas is taken as the magnetostrictive phase. For the calculation, the following thicknesses of Metglas m t = 29 µm and piezoelectric p t = 0.5 mm are taken, and the length of ME composite is l = 10 mm. To take into account losses in the calculation, it is assumed: ω = 2π(1 + (1/2Q)i)f, where Q is the quality factor of the resonant system. For this calculation, the value of the quality factor Q = 130 was taken. Below, Figure 2 shows the dependence of the ME voltage coefficient on the frequency of the alternating magnetic field for two cases, when PZT and a cut of lithium niobate y + 128° [13,19] are taken as the piezoelectric phase. Metglas is taken as the magnetostrictive phase. For the calculation, the following thicknesses of Metglas m t = 29 µm and piezoelectric p t = 0.5 mm are taken, and the length of ME composite is l = 10 mm. To take into account losses in the calculation, it is assumed: where Q is the quality factor of the resonant system. For this calculation, the value of the quality factor Q = 130 was taken.
The fundamental resonant frequency for this case is:  In [7,8], the corresponding theory for the longitudinal mode of the ME effect in the EMR region was used, and it showed its good agreement with the experiment.

Quasi-Static Mode
Assuming in Equation (35) the frequency f is equal to zero, we obtain: 11 Below, Figure 3 shows the dependence of the ME voltage coefficient on the volume The fundamental resonant frequency for this case is: In [7,8], the corresponding theory for the longitudinal mode of the ME effect in the EMR region was used, and it showed its good agreement with the experiment.

Quasi-Static Mode
Assuming in Equation (35) the frequency f is equal to zero, we obtain: Below, Figure 3 shows the dependence of the ME voltage coefficient on the volume fraction of the piezoelectric for two cases, when PZT and a cut of lithium niobate y + 128 • are taken as the piezoelectric phase. Metglas is taken as the magnetostrictive phase.  In [9,10], the corresponding theory for the longitudinal mode of the ME effect in the quasi-static regime was applied, and it showed good agreement with the experiment.

Resonance Regime of the Longitudinal Mode
For an asymmetric ME structure in the resonant mode of the longitudinal ME mode, the voltage coefficient can be found from Equation (35), and only in Equation (14) is it necessary to remove the number 2 before m t. The fundamental resonant frequency for this case can be found in Equation (36). The ME structure shown at Figure 4. We consider bending oscillations in a two-layer magnetostrictive-piezoelectric structure. We assume that the sample has the form of a thin bar, whose thickness and width are much less than the length. In this case, we can consider only one component of the stress and strain tensor. In [9,10], the corresponding theory for the longitudinal mode of the ME effect in the quasi-static regime was applied, and it showed good agreement with the experiment.

Resonance Regime of the Longitudinal Mode
For an asymmetric ME structure in the resonant mode of the longitudinal ME mode, the voltage coefficient can be found from Equation (35), and only in Equation (14) is it necessary to remove the number 2 before m t. The fundamental resonant frequency for this case can be found in Equation (36). The ME structure shown at Figure 4.  In [9,10], the corresponding theory for the longitudinal mode of the ME effect in the quasi-static regime was applied, and it showed good agreement with the experiment.

Resonance Regime of the Longitudinal Mode
For an asymmetric ME structure in the resonant mode of the longitudinal ME mode, the voltage coefficient can be found from Equation (35), and only in Equation (14) is it necessary to remove the number 2 before m t. The fundamental resonant frequency for this case can be found in Equation (36). The ME structure shown at Figure 4. We consider bending oscillations in a two-layer magnetostrictive-piezoelectric structure. We assume that the sample has the form of a thin bar, whose thickness and width are much less than the length. In this case, we can consider only one component of the stress and strain tensor.

Resonant Mode of the Bending Mode
We consider bending oscillations in a two-layer magnetostrictive-piezoelectric structure. We assume that the sample has the form of a thin bar, whose thickness and width are much less than the length. In this case, we can consider only one component of the stress and strain tensor.
The full thickness of the composite: The volume fractions of the piezoelectric and magnetostrictive phases are: The X axis will be drawn along the neutral line of the ME composite ( Figure 5). The X axis will be drawn along the neutral line of the ME composite ( Figure 5). In the case of rigid connection between the components of the composite, we have: where w is the transverse displacement. The longitudinal component of the stress tensor and the third component of the electric stress vector of a piezoelectric phase are given by:  In the case of rigid connection between the components of the composite, we have: where w is the transverse displacement. The longitudinal component of the stress tensor and the third component of the electric stress vector of a piezoelectric phase are given by: where c D 11 is longitudinal component of the stiffness tensor at a constant electrical displacement; h 31 is piezoelectric coefficient at a constant longitudinal component of the strain tensor; β S 33 is inverse permittivity at a constant longitudinal component of the strain tensor: Substituting Equation (40) in Equation (9), we get: The torque is: where b is the sample width, z 0 is position of the boundary between the piezoelectric and magnetostrictive phases relative to the neutral line, and: Then, the voltage across the piezoelectric phase is: From Equation (48) we obtain the electric displacement in the piezoelectric phase: Substituting the resulting expression in Equation (45): where: The position of the boundary between the piezoelectric and magnetostrictive phases relative to the neutral line z 0 is determined from the minimum condition c 11 : The shear force is: The equation of bending vibrations can be written as: Substituting Equation (53) in Equation (54), we obtain: Given that the time dependence of the shift is harmonic w ∼ e iωt , the equation of bending vibrations can be written as: The general solution of the motion equation is: where C 1 , C 2 , C 3 , C 4 are unknown constants. The open circuit condition is: Integrating Equation (48) over x, we obtain: where: Free Clamping of Both Ends of the ME Composite.
The boundary conditions for free ends of the beam are given by: Combining Equation (61) with Equation (59), we obtain a linear system of five inhomogeneous algebraic equations for five unknowns C 1 , C 2 , C 3 , C 4 , U: We solve this system by considering the fact that: The voltage across the piezoelectric is given by the following equation: As a result, the ME voltage coefficient is obtained in the form: Below, Figure 6 shows the dependence of the ME voltage coefficient on the frequency of the alternating magnetic field for two cases, when PZT and a cut of lithium niobate y + 128 • are taken as the piezoelectric phase. Metglas is taken as the magnetostrictive phase. For the calculation, the following thicknesses of Metglas m t = 29 µm and piezoelectric p t = 0.5 mm are taken, and the length of ME composite is l = 10 mm. To take into account losses in the calculation, it is assumed: ω = 2π(1 + (1/2Q)i)f, where Q is the quality factor of the resonant system. The value of the quality factor was taken to be the same as for the longitudinal mode, Q = 130. Cantilever Clamping of ME Composite. The boundary conditions for this case: The general solution of the equation of motion is: The linear system of five inhomogeneous algebraic equations for five unknowns C1, C2, C3, C4, U: The fundamental resonant frequency for this case is: Cantilever Clamping of ME Composite.
The boundary conditions for this case: The general solution of the equation of motion is: The linear system of five inhomogeneous algebraic equations for five unknowns C 1 , C 2 , C 3 , C 4 , U: We solve this system, considering the fact that: The voltage across the piezoelectric is given by: As a result, the ME voltage coefficient is obtained as: Below, Figure 7 shows the dependence of the ME voltage coefficient on the frequency of the alternating magnetic field for two cases, when PZT and a cut of lithium niobate y + 128 • are taken as the piezoelectric phase. Metglas is taken as the magnetostrictive phase. For the calculation, the following thicknesses of Metglas m t = 29 µm and piezoelectric p t = 0.5 mm are taken, length of ME composite is l = 10 mm. To take into account losses in the calculation, it is assumed: ω = 2π(1 + (1/2Q)i)f, where Q is the quality factor of the resonant system. The value of the quality factor was taken the same as for the longitudinal mode Q = 130.
Below, Figure 7 shows the dependence of the ME voltage coefficient on the frequency of the alternating magnetic field for two cases, when PZT and a cut of lithium niobate y + 128° are taken as the piezoelectric phase. Metglas is taken as the magnetostrictive phase. For the calculation, the following thicknesses of Metglas m t = 29 µm and piezoelectric p t = 0.5 mm are taken, length of ME composite is l = 10 mm. To take into account losses in the calculation, it is assumed: ω = 2π(1 + (1/2Q)i)f, where Q is the quality factor of the resonant system. The value of the quality factor was taken the same as for the longitudinal mode Q = 130.
The fundamental resonant frequency for this case is: (73) Figure 7. Theoretical dependence of the ME voltage coefficient on the frequency of the alternating magnetic field. Black color of the line is PZT, blue is LN cut y + 128°.
In [11,12], the theory of the bending mode of the ME effect in the EMR region was considered, based on the hypothesis that the electric field strength in the piezoelectric phase is independent of the coordinate along the thickness of an asymmetric magnetostrictive-piezoelectric composite, and its satisfactory agreement with experimental data was shown. The theory of the same phenomenon based on a more plausible hypothesis of independence of the electric displacement in the piezoelectric phase from the coordinate along the thickness of an asymmetric magnetostrictive piezoelectric composite was considered in [13], and it showed good agreement with the experiment. However, in this work, the corresponding theory is presented very briefly. In our article, we describe this In [11,12], the theory of the bending mode of the ME effect in the EMR region was considered, based on the hypothesis that the electric field strength in the piezoelectric phase is independent of the coordinate along the thickness of an asymmetric magnetostrictivepiezoelectric composite, and its satisfactory agreement with experimental data was shown. The theory of the same phenomenon based on a more plausible hypothesis of independence of the electric displacement in the piezoelectric phase from the coordinate along the thickness of an asymmetric magnetostrictive piezoelectric composite was considered in [13], and it showed good agreement with the experiment. However, in this work, the corresponding theory is presented very briefly. In our article, we describe this theory in as much detail as possible for a better understanding and ease of application, if necessary.

Quasi-Static Mode
In [14,15], the theory of the longitudinal and bending modes of the ME effect in the quasi-static mode for an asymmetric magnetostrictive-piezoelectric structure was considered. Separate expressions are found for the contributions of the planar and bending modes, and then the full expression. In our article, we start immediately from general expressions that consider the planar and flexural modes and obtain the result. We do this in as much detail as possible to facilitate understanding and ease of application of this theory if necessary.
In the quasi-static mode, Equation (18) is given by: This means that T 1 must not depend on x. It is obvious that S 1 must not depend on x either. Since both the longitudinal and bending modes are excited in the asymmetric ME structure in the quasi-static mode: where A, B are unknown constants. Substituting Equation (75) in Equation (9) and Equation (41), also considering that due to the open circuit condition D 3 = 0, we obtain: The first condition for the static equilibrium of the ME composite is the equality to zero of the total longitudinal force is given by: Substituting Equations (76) and (77) in Equation (78), we obtain: The second condition for the static equilibrium of the ME composite is the zero total moment is given by: Substituting Equations (76) and (77) in Equation (80), we obtain: The Equations (79) and (81) form a linear inhomogeneous system of two equations with two unknowns A, B. Solving them, we obtain A and B as: Substituting Equation (75) in Equation (42), and considering that due to the open circuit condition D 3 = 0, we get E 3 : Then, the voltage across the piezoelectric is: Substituting Equation (82) in Equation (84), we obtain: From Equation (85) we find the ME voltage coefficient as: Below, Figure 8 shows the dependence of the ME voltage coefficient on the volume fraction of the piezoelectric for two cases, when PZT and a cut of lithium niobate y + 128 • are taken as the piezoelectric phase. Metglas is taken as the magnetostrictive phase.
Substituting Equation (75) in Equation (42), and considering that due to the open circuit condition D3 = 0, we get E3: Then, the voltage across the piezoelectric is: Below, Figure 8 shows the dependence of the ME voltage coefficient on the volume fraction of the piezoelectric for two cases, when PZT and a cut of lithium niobate y + 128° are taken as the piezoelectric phase. Metglas is taken as the magnetostrictive phase.

Symmetrical ME Structure
In a symmetric ME structure, excitation of the torsional mode of the ME effect is impossible. Therefore, we first consider the general case of a longitudinal-shear mode for an arbitrary frequency of an alternating magnetic field, which also includes the resonant

Symmetrical ME Structure
In a symmetric ME structure, excitation of the torsional mode of the ME effect is impossible. Therefore, we first consider the general case of a longitudinal-shear mode for an arbitrary frequency of an alternating magnetic field, which also includes the resonant mode, and the expression for the ME voltage coefficient for the quasi-static case will be obtained from the general expression, assuming the frequency f is equal to zero.

Resonance Mode
We consider a magnetoelectric composite as a thin narrow plate. Layers of the magnetostrictive phase of the same thickness are above and below the piezoelectric layer. The ME structure created in this way is symmetrical. The X axis is directed along the length of the plate, and the Z axis is perpendicular to the sample plane, as in Figure 9. mode, and the expression for the ME voltage coefficient for the quasi-static case will obtained from the general expression, assuming the frequency f is equal to zero.

Resonance Mode
We consider a magnetoelectric composite as a thin narrow plate. Layers of the m netostrictive phase of the same thickness are above and below the piezoelectric layer. T ME structure created in this way is symmetrical. The X axis is directed along the length the plate, and the Z axis is perpendicular to the sample plane, as in Figure 9. We consider small longitudinal-shear mechanical oscillations in a composite un the influence of a small external variable magnetic field. The AC magnetic field is direc along the X axis, and the DC magnetic field is directed along the Y axis, then: The material equation for the piezoelectric layer is given by:  We consider small longitudinal-shear mechanical oscillations in a composite under the influence of a small external variable magnetic field. The AC magnetic field is directed along the X axis, and the DC magnetic field is directed along the Y axis, then: The material equation for the piezoelectric layer is given by: where p S 6 is shear strain tensor component of piezoelectric phase; d 36 is piezoelectric coefficient; p s 66 is shear compliance tensor component of the piezoelectric phase; p T 6 is the shear stress tensor component of the piezoelectric phase. The shear component of the stress tensor in a piezoelectric can be expressed as: The shear component of the stress tensor in the magnetostrictive phase has the form: where m S 6 is shear strain tensor component of magnetostrictive phase, m G is shift modulus in the magnetostrictive phase, and q 16 is corresponding pseudo-piezomagnetic coefficient.
In accordance with the condition of the problem for longitudinal-shear mode is: Shear component of the composite stress tensor is: where volume fractions of the piezoelectric and magnetostrictive phases are: The effective shear composite stiffness coefficient is: Composite effective density is given by: Consider the motion equation for deformations is: Substituting Equation (90) in Equation (94), we get: The solution of this equation is obtained as: where the wave number is: and A, B are unknown constants. Then: To obtain the constants A and B, we use the equilibrium conditions for a free sample: Substituting Equation (103) in Equation (102), we get: where: As a result, we get: The transverse component of the electric displacement vector can be obtained from: We can find the transverse component of the electric field strength vector from the condition that the electric induction flux through the interface between the upper layer of the magnetostrictive phase and the piezoelectric are equal to zero: Substituting Equation (104) As the electric field exists only in the piezoelectric phase, the voltage is given by the following equation: Average electric field strength in ME composite is: Then, the ME voltage coefficient is obtained as: 16  .
The fundamental resonant frequency for this case is: Figure 10. Theoretical dependence of the ME voltage coefficient on the frequency of the alternating magnetic field for symmetrical ME structure Metglas/GaAs of the longitudinal-shear mode.

Quasi-Static Mode
Assuming in Equation (114) the frequency f equal to zero, we obtain: 16 (116) Figure 10. Theoretical dependence of the ME voltage coefficient on the frequency of the alternating magnetic field for symmetrical ME structure Metglas/GaAs of the longitudinal-shear mode.
The fundamental resonant frequency for this case is:

Quasi-Static Mode
Assuming in Equation (114) the frequency f equal to zero, we obtain: 16  Below, Figure 11 shows the dependence of the ME voltage coefficient on the volume fraction of the piezoelectric, when GaAs are taken as the piezoelectric phase. Metglas is taken as the magnetostrictive phase. Below, Figure 11 shows the dependence of the ME voltage coefficient on the volume fraction of the piezoelectric, when GaAs are taken as the piezoelectric phase. Metglas is taken as the magnetostrictive phase. Figure 11. Theoretical dependence of the ME voltage coefficient on the volume fraction of the piezoelectric material for ME structure Metglas/GaAs of the longitudinal-shear mode.

Resonance Regime for the Longitudinal-Shear Mode
For an asymmetric ME structure in the resonant mode of the longitudinal-shear ME mode, the voltage coefficient can be found from Equation (114). Only in Equation (94) is it necessary to remove the number 2 before m t. The fundamental resonant frequency for this case can be found in Equation (115). The ME structure shown at Figure 12.

Resonance Regime for the Longitudinal-Shear Mode
For an asymmetric ME structure in the resonant mode of the longitudinal-shear ME mode, the voltage coefficient can be found from Equation (114). Only in Equation (94) is it necessary to remove the number 2 before m t. The fundamental resonant frequency for this case can be found in Equation (115). The ME structure shown at Figure 12. Figure 11. Theoretical dependence of the ME voltage coefficient on the volume fraction of the piezoelectric material for ME structure Metglas/GaAs of the longitudinal-shear mode.

Resonance Regime for the Longitudinal-Shear Mode
For an asymmetric ME structure in the resonant mode of the longitudinal-shear ME mode, the voltage coefficient can be found from Equation (114). Only in Equation (94) is it necessary to remove the number 2 before m t. The fundamental resonant frequency for this case can be found in Equation (115). The ME structure shown at Figure 12.  Figure 9.
Below, Figure 13 shows the dependence of the ME voltage coefficient on the frequency of the alternating magnetic field. To take into account losses in the calculation, it is assumed: ω = 2π(1 + (1/2Q)i)f, where Q is the quality factor of the resonant system. Figure 12. Asymmetric two-layer magnetoelectric composite for calculation of longitudinal-shear mode. All designations are the same as in Figure 9.
Below, Figure 13 shows the dependence of the ME voltage coefficient on the frequency of the alternating magnetic field. To take into account losses in the calculation, it is assumed: ω = 2π(1 + (1/2Q)i)f, where Q is the quality factor of the resonant system.

Resonant Regime for the Torsional Mode
In [16,17], the theory of the torsional mode of the ME effect in the EMR region was considered for an asymmetric magnetostrictive-piezoelectric structure. However, since the torsion of the structure was considered around the axis passing along the width of the structure, the numerical values of the ME stress coefficient turned out to be too small for the torsional mode to be seen against the background of a relatively large longitudinal shear mode. In this article, we consider the torsion of an ME structure around an axis running along the length of the structure. This made it possible to obtain relatively large values of the ME voltage coefficient.
Draw the X axis along the length of the sample in the corresponding plane of symmetry of the sample, and the Y axis along the axis of rotation of the composite beam during torsional vibrations in the direction of the sample width as in Figure 14.

Resonant Regime for the Torsional Mode
In [16,17], the theory of the torsional mode of the ME effect in the EMR region was considered for an asymmetric magnetostrictive-piezoelectric structure. However, since the torsion of the structure was considered around the axis passing along the width of the structure, the numerical values of the ME stress coefficient turned out to be too small for the torsional mode to be seen against the background of a relatively large longitudinal shear mode. In this article, we consider the torsion of an ME structure around an axis running along the length of the structure. This made it possible to obtain relatively large values of the ME voltage coefficient.
Draw the X axis along the length of the sample in the corresponding plane of symmetry of the sample, and the Y axis along the axis of rotation of the composite beam during torsional vibrations in the direction of the sample width as in Figure 14.
structure, the numerical values of the ME stress coefficient turned out to be too small for the torsional mode to be seen against the background of a relatively large longitudinal shear mode. In this article, we consider the torsion of an ME structure around an axis running along the length of the structure. This made it possible to obtain relatively large values of the ME voltage coefficient.
Draw the X axis along the length of the sample in the corresponding plane of symmetry of the sample, and the Y axis along the axis of rotation of the composite beam during torsional vibrations in the direction of the sample width as in Figure 14. The AC magnetic field is directed along the X axis, and the DC magnetic field is directed along the Y axis.
The full thickness of the composite: Shear components of the strain tensor are: where θ-the twist angle. The AC magnetic field is directed along the X axis, and the DC magnetic field is directed along the Y axis.
The full thickness of the composite: Substituting Equation (133) in Equation (136), we get: The dependence of the twist angle on time is harmonic θ ∼ e iωt , therefore: where the wave number is: The general solution of the Equation (140) is: where A, B are unknown constants. The open circuit condition is: Then, we integrate Equation (135) over x: where: Boundary conditions for a free sample are: Combining boundary conditions Equation (146) with Equation (144), we obtain a linear system of three inhomogeneous algebraic equations with three unknowns, A, B, U: As a result, the ME voltage coefficient is obtained as: (151) Figure 15 shows the dependence of the ME voltage coefficient on the frequency of the alternating magnetic field. To take into account losses in the calculation, it is assumed: ω = 2π(1 + (1/2Q)i)f , where Q is the quality factor of the resonant system. In the calculation, the same material parameters were used as for the longitudinal-shear mode.

Quasi-Static Mode
In the quasi-static mode, there are no vibrations along the length of the composite. This means that S5 and S6 must not depend on x. Since both the longitudinal-shear and torsional modes are excited in the asymmetric ME structure in the quasistatic mode, then: The fundamental resonant frequency for this case is:

Quasi-Static Mode
In the quasi-static mode, there are no vibrations along the length of the composite. This means that S 5 and S 6 must not depend on x. Since both the longitudinal-shear and torsional modes are excited in the asymmetric ME structure in the quasistatic mode, then: where A, B are unknown constants. Substituting Equation (153) in Equations (120), (123), and (128), also considering that due to the open circuit condition D 3 = 0, we obtain: The Equations (153) and (155) form a linear inhomogeneous system of two equations with two unknowns A, B. Solving it, we find A and B: Substituting Equation (153) in Equation (126), and considering that due to the open circuit condition D 3 = 0, E 3 is obtained as: The voltage across the piezoelectric: Substituting Equation (156) in Equation (158), we get: From Equation (159) we find the ME voltage coefficient: Below, Figure 16 shows the dependence of the ME voltage coefficient on the volume fraction of the piezoelectric, when GaAs are taken as the piezoelectric phase. Metglas is taken as the magnetostrictive phase. The material parameters of the ME structure are the same as for the calculation of the longitudinal shear mode.  As can be seen from the comparison of Figures 13 and 15, the ME voltage coefficient in the EMR regime for the longitudinal-shear mode is several times larger than for the torsional mode. Therefore, it is quite natural that in the quasistatic mode the torsional mode does not make a very significant contribution, and the ME voltage coefficient in the quasistatic mode is mainly determined by the contribution of the longitudinal-shear mode.

Resonant Regime for the Torsional Mode
Draw the X axis along the axis of rotation of the composite beam during torsional vibrations in the direction of the length of the sample, and the Y axis along the width of the sample in the corresponding plane of symmetry of the sample, as in Figure 17. As can be seen from the comparison of Figures 13 and 15, the ME voltage coefficient in the EMR regime for the longitudinal-shear mode is several times larger than for the torsional mode. Therefore, it is quite natural that in the quasistatic mode the torsional mode does not make a very significant contribution, and the ME voltage coefficient in the quasistatic mode is mainly determined by the contribution of the longitudinal-shear mode. Draw the X axis along the axis of rotation of the composite beam during torsional vibrations in the direction of the length of the sample, and the Y axis along the width of the sample in the corresponding plane of symmetry of the sample, as in Figure 17.  The AC magnetic field is directed along the X axis, and the DC magnetic field is directed along the Y axis.
The shear components of the strain tensor are given by: where α-the twist angle. Material equations for a ferromagnetic phase: Boundary conditions for this case: Then, we integrate Equation (179) over x: Combining boundary conditions with Equation (195), we obtain a linear system of three inhomogeneous algebraic equations with three unknowns, A, B, U: Solving this system, the voltage across the piezoelectric can be obtained in the form: The average electric field strength in the composite is: As a result, the ME voltage coefficient is obtained as: Below, Figure 18 shows the dependence of the ME voltage coefficient on the frequency of the alternating magnetic field for case, when bimorph LiNbO 3 Zyl + 45 • are taken as the piezoelectric phase. Metglas is taken as the magnetostrictive phase. The length of ME composite was l = 23 mm, and width was b = 0.5 mm. In the calculation, the following material parameters of the initial components were used: for Metglas: m ρ = 7180 kg/m 3 , m G = 3.85·10 10 Pa, q 16 = 1.0·10 −9 m/A, m t = 29 µm; for LiNbO 3 Zyl + 45 • : p ρ = 4647 kg/m 3 , p c E 55 = 6.75·10 10 Pa, p c E 56 = 6.75·10 10 Pa, p c E 55 = 7.5·10 9 Pa, ε 33 = 36.5, p1 e 35 = − p21 e 35 = 2.5 C/m 2 , p1 e 36 = − p21 e 36 = 2.5 C/m 2 , p t = 0.4 mm. To take into account losses in the calculation, it is assumed: ω = 2π(1 + (1/2Q r )i)f, where Q r is the quality factor of the resonant system. The value of the quality factor was taken the same as for the longitudinal mode Q r = 100. ω = 2π(1 + (1/2Qr)i)f, where Qr is the quality factor of the resonant system. The value of the quality factor was taken the same as for the longitudinal mode Qr = 100.
The fundamental resonant frequency for this case is: (200) Figure 18. Theoretical dependence of the ME voltage coefficient on the frequency of the alternating magnetic field for ME structure Metglas/LiNbO3 Zyl + 45° in case of torsional mode. The fundamental resonant frequency for this case is:

Quasi-Static Mode
In a magnetostrictive-piezoelectric structure based on bimorph lithium niobate, the longitudinal-shear mode is not excited due to oppositely directed polarization in the layers of lithium niobate. Therefore, the expression for the ME voltage coefficient in the quasistatic mode can be obtained from Equation (199) by assuming the frequency f equal to zero: Below, Figure 19 shows the dependence of the ME voltage coefficient on the volume fraction of the piezoelectric material for the asymmetric ME structure Metglas/LiNbO 3 Zyl + 45 • for the quasi-static torsional mode.

Quasi-Static Mode
In a magnetostrictive-piezoelectric structure based on bimorph lithium niobate, the longitudinal-shear mode is not excited due to oppositely directed polarization in the layers of lithium niobate. Therefore, the expression for the ME voltage coefficient in the quasistatic mode can be obtained from Equation (199) Below, Figure 19 shows the dependence of the ME voltage coefficient on the volume fraction of the piezoelectric material for the asymmetric ME structure Metglas/LiNbO3 Zyl + 45° for the quasi-static torsional mode. Figure 19. Theoretical dependence of the ME voltage coefficient on the volume fraction of the piezoelectric material for the asymmetric ME structure Metglas/LiNbO3 Zyl + 45° for the quasi-static torsional mode.

Discussion
It is necessary to discuss the accuracy of the above formulas for the fundamental resonant frequencies for various modes. In the case of a longitudinal mode, we turn to the expression for the ME voltage coefficient Equation (35). Obviously, the resonant frequency Figure 19. Theoretical dependence of the ME voltage coefficient on the volume fraction of the piezoelectric material for the asymmetric ME structure Metglas/LiNbO 3 Zyl + 45 • for the quasi-static torsional mode.

Discussion
It is necessary to discuss the accuracy of the above formulas for the fundamental resonant frequencies for various modes. In the case of a longitudinal mode, we turn to the expression for the ME voltage coefficient Equation (35). Obviously, the resonant frequency should vanish from the denominator of this expression. However, this denominator consists of the main term and term proportional to the dimensionless quantity of d 2 31 /(εε 0 p S 11 ). If this dimensionless quantity is small compared to unity, then the corresponding term in the denominator can be neglected, and then Equation (36) is obtained for the fundamental resonant frequency for the longitudinal mode.
If we estimate the value of the dimensionless quantity of d 2 31 /(εε 0 p S 11 ), then for PZT we get 0.13, and for LN y + 128 • 0.24. In this case, the main resonant frequency, determined from the exact plot for PZT Figure 2, is 158 kHz, determined according to the approximate Equation 36 of 151 kHz. Additionally, the main resonant frequency, determined from the exact graph for LN y + 128 • Figure 2, is of 296 kHz, and determined from the approximate Equation 36, is 267 kHz. Obviously, the larger the value of the dimensionless quantity of d 2 31 /(εε 0 p S 11 ), the more the exact fundamental resonant frequency differs from the fundamental resonant frequency. To discuss this issue for the bending mode in the case of a sample with free ends, we turn to the expression for the ME stress coefficient Equation (65). Likewise, the resonant frequency must vanish from the denominator of this expression. Similarly, the denominator consists of a principal term and a term proportional to the dimensionless quantity of p t 3 h 31 2 / t 3 c 11 β S 33 . If one estimates the value of the dimensionless quantity of p t 3 h 31 2 / t 3 c 11 β S 33 , then for PZT it will be 0.0023, and for LN y + 128 • it will be 0.0011. In this case, the main resonant frequency, determined from the exact graph for PZT Figure 6, is of 17,512 Hz, and determined from the approximate Equation (66) is of 17,498 Hz. Additionally, the main resonant frequency, determined from the exact graph for LN y + 128 • Figure 6, is of 33,098 Hz, and determined from the approximate Equation (66) of 33,086 Hz. Since the values of the dimensionless quantity are much less than unity for PZT and LN y + 128 • , the differences between the exact fundamental resonant frequencies and those determined by the approximate formula are negligible.
For the case of a bending mode for a sample with an ME cantilever, the voltage coefficient is determined by Equation (72), and the approximate formula for the fundamental resonant frequency is Equation (73). Obviously, the results for this case will be completely similar to the results for the case for the bending mode for a sample with free ends. To discuss this issue for the torsional mode, we turn to the expression for the ME voltage coefficient Equation (151). Likewise, the resonant frequency must vanish from the denominator of this expression. Similarly, the denominator consists of a principal term and a term, proportional to the dimensionless quantity of p t 3 h 31 2 / t 3 c 11 β S 33 . The value of this dimensionless coefficient is much less than unity. Therefore, the main resonant frequency, determined by the exact graph Figure  In addition, the correspondence of the described theory to the experimental data should be noted. The longitudinal and bending modes of the ME effect have been fairly well studied experimentally. The obtained experimental results are in good agreement with the stated theory [12,13]. The experimental study of the torsional mode of the ME effect is just beginning, so it is not yet possible to draw reasonable conclusions about the correspondence of the theory presented to the experimental data.

Conclusions
The article considers the theory of low-frequency direct ME effect in symmetric and asymmetric magnetostrictive-piezoelectric structures in longitudinal and bending, as well as longitudinal-shear and torsional modes. Expressions are obtained for the ME voltage coefficients in the quasi-static and EMR modes. Additionally, for the EMR mode, approximate formulas for the main resonant frequencies were obtained and their accuracy was investigated. For the torsion mode, the advantages of using a bimorph piezoelectric material are shown, which led to a significant increase in the ME voltage coefficient. A comparison of the obtained theoretical results with known data from the literature and experiment for the GaAs-Metglas and LiNbO 3 -Metglas structures showed satisfactory agreement. The value of the study, according to the authors, lies in the fact that within the framework of a unified approach, the main relationships for the ME voltage coefficients for all modes of low-frequency direct ME effect were obtained. The results obtained can be used for choosing ME composites that can create new ME devices in the low-frequency range. In terms of further research, it is of interest to carry out a similar calculation of the inverse low-frequency ME effect and compare the results obtained.