# Electromagnetic and mechanical governing equations and constitutive laws .

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## Simplified analytical method in static regime

The simplified analytical method presented here is a 0D-modelling that has been used in  (denoted as the matrix resolution method) to investigate the output voltage of the magnetoelectric composite in LT and TT modes shown in Fig.2.3. In L-T mode, the magnetostrictive material is magnetized along the longitudinal direction whereas the piezoelectric material is polarized along the transversal direction. In TT-mode, the magnetostrictive material and piezoelectric material are magnetized and polarized along the transversal direction.

### Equivalent circuit method

The equivalent circuit method for the magnetoelectric composite uses an extension of the well-known 1D electrical Mason’s model of piezoelectric material in combining the magnetic-mechanical coupling with the mechanical-electrical coupling of the piezoelectric. It was recently employed in literatures [40, 41, 49] to investigate the voltage magnetoelectric coefficient for different modes in static and dynamic regimes. The procedures based for the LT and TT modes are presented in Figures 2.6 and 2.7.
Considering the 3D composite structure, if the length of the composite bulk is much longer than the
width and thickness, we can approximately only take into account the components in z-direction of the mechanical variables(stress and strain) and that of the electric displacement in polarization direction(direction ‘3’ in local coordinates).
If an infinitesimal longitudinal length ’Δz’ is taken into consideration that is as much smaller than the
total length of the bulk, then the continuous mechanical equilibrium equation in dynamic regime (2-2- 10) for the LT mode can be rewritten as: 𝜕2𝑢 𝜕𝑡2 (2Δ𝑚1 + Δ𝑚2) = Δ𝑇33 𝑚 ∙ 2𝐴1 + Δ𝑇11 𝑝 ∙ 𝐴2 = 𝜕2𝑢 𝜕𝑡2 (2𝜌𝑚𝐴1Δ𝑧 + 𝜌𝑝𝐴2Δ𝑧).

#### FEM modelling of the field problem in 2D

Due to the complexity of the partial differential constitutive equations of the studied material and of the geometry, it is necessity to introduce appropriate numerical resolution methods. The finite element method (FEM) is employed to perform the discretization of the solution domain for the studied ME composite problem. In this section we expose the 2D finite element approach to investigate the ME energy transducer presented in Figure 2.12. It is a trilayer magnetostrictive/piezoelectric laminated composite including an electrical load representing the conditioning circuit connected to the electrodes of the piezoelectric layer. The magnetostrictive material is magnetized along the longitudinal direction whereas the piezoelectric material is polarized along the transversal direction (i.e. L-T mode). It is to be noticed that the configuration of figure 2.12 is only given in title of example and the FEM formulation presented in this section is general and valid for any configuration modes and more complicated structures.

Establishment of the 2D tensors

The symmetry of the laminate structures problem satisfies the plane theory of elasticity that reduces
the problem in a 2D problem. For that, two conditions are suitable: Either 2D plane stress conditions or 2D plane strain conditions. In the 2D plane stress, the geometry of the structure is essentially that of a plate with one dimension much smaller than the other, whereas for the 2D plane strain conditions the direction (z-coordinate) of the structure in one direction is very large in comparison with in other directions (x and y-coordinate axes). In both cases, the stiffness tensors can be written .

Nonlinear static case

As previously stated, the magnetoelectric composites operate under a composite external magnetic field excitation: a static biasing field Hdc and a small amplitude alternative field Hac. Due to the nonlinear property of magnetostrictive materials, the change of static biasing field allows determining the optimal operation point for which the magnetoelectric coupling coefficient for the small signal field maximizes. In order to determine the physical properties of the materials around the operation point, the so called incremental characteristics, the system equation (2-4-28) needs to be solved in nonlinear regime for static biasing magnetic excitation.

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Modelling of nonlinear piezomagnetic coupling

When considering the case of linear elastic property (constant elastic coefficients) and nonlinear magnetic property, the expression of the stress tensor is: 𝑇 = 𝑐𝐵(𝑆 − 𝑆𝜇(𝑩)) = 𝑐𝐵𝑆 − 𝑇𝜇 (𝑩) (2-4-36) where 𝑆𝜇 (𝑩) and 𝑇𝜇 (𝑩) are respectively the “coercitive” strain and stress tensor induced by 𝑩 The expression of the “coercitive” strain tensor 𝑆𝜇(𝑩) of a single cubic crystal is the Maxwell magnetic stress tensor with six independent components and can be described by a quadratic model of the magnetic induction 𝑩 as : 𝑆𝜇 (𝑩) = 𝑆𝑠𝑎𝑡 𝜇 ( 𝑩 𝑩𝒔𝒂𝒕 ).

General introduction
Chapter 1. General context – State of Art
1.1 Magnetoelectric effect and materials
1.1.1 Single-phase ME materials
1.1.2 Two-phase ME composite materials
1.1.2.1 Bulk 0-3 composites
1.1.2.2 Laminated 2-2 composites
1.1.2.3 Rod matrix 1-3 type composites
1.2 Magneto-mechanical effect and material
1.2.1 Magneto-mechanical coupling
1.2.2 Magnetostrictive material
1.3 Electro-mechanical effect and materials
1.3.1 Electro-mechanical coupling
1.3.2 Piezoelectric material
1.4 Magnetoelectric effect applications
1.4.1 Magnetic field sensors
1.4.1.1 Static magnetic field sensor
1.4.1.2 Dynamic magnetic field sensor
1.4.2 Energy harvesting applications
1.5 Modelling and Characterizations of magnetoelectric materials and devices
1.5.1 Theoretical modelling methods
1.5.2 Experimental Characterization
1.6 Conclusion
Chapter 2. Analytical and Numerical Modelling of Magnetoelectric Composites
2.1 Introduction
2.2 Electromagnetic and mechanical governing equations and constitutive laws .
2.3 Analytical methods
2.3.1 Simplified analytical method in static regime
2.3.2 Equivalent circuit method
2.4 FEM modelling of the field problem in 2D
2.4.1 Establishment of the 2D tensors
2.4.2 FEM formulation
2.4.2.1 Fundamental formulations
2.4.2.2 Boundary conditions
2.4.2.3 FEM Simulation results
2.4.3 Nonlinear static case
2.4.3.1 Modelling of nonlinear piezomagnetic coupling
2.4.3.1.1 Hirsinger model
2.4.3.2 Modelling of magnetic nonlinearity
2.4.4 Dynamic small signal regime
2.4.4.1.Coupling with the electric circuit load equation
2.4.4.2 Effect of the complex impedance on the damping losses
2.4.4.3 Simulation results
2.5 Conclusion
Chapter 3. Assessment of ME composite performances
3.1 Introduction
3.2 Performances of a ME composite as energy transducer
3.2.1 Output deliverable power under different modes
3.2.2 Electrical equivalent circuit model
3.2.3 Establishment of the optimal electrical load
3.2.4 Transient dynamic response with a non-linear electrical load
3.3 Multilayer ME composite materials
3.3.1 Multilayer ME composite problems description and FEM modelling
3.3.2 Equivalent circuit method for multilayer ME composite problems
3.3.3 Results and comparisons
3.4 Conclusion
Chapter 4. Prospective application of ME composites as energy transducer
4.1 Introduction
4.2 Potential application in biomedical domain
4.2.1 Ferrite solenoid as external reader
4.2.2 Exposition limits of the magnetic field
4.3 Measurement of a bilayer ME composite
4.3.1 Measurement bench set-up
4.3.2 Static and dynamic measure responses
4.3.3 Deliverable output power
4.4 Conclusion
General conclusion
Appendix A. Characteristics of utilized materials
Appendix B. Different magnetostrictive nonlinear models
Appendix C. Modified Newton-Raphson method
Appendix D. Multilayer analytical modelling
References

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