Micromechanical analysis of magneto-electro-thermo-elastic composite materials with applications to multilayered structures

The method of asymptotic homogenization was used to analyze a periodic magnetoelectric smart composite structure consisting of piezoelectric and piezomagnetic phases. The asymptotic homogenization model is derived, the governing equations are determined and subsequently general expressions called unit-cell problems that can be used to determine the effective elastic, piezoelectric, piezomagnetic, thermal expansion, dielectric, magnetic permeability, magnetoelectric, pyroelectric and pyromagnetic coefficients are presented. The latter three sets of coefficients are particularly interesting in the sense that they represent product or cross-properties; they are generated in the macroscopic composite via the interaction of the different phases, but may be absent from the constituents themselves. The derived expressions pertaining to the unit-cell problems and the resultant effective coefficients are very general and are valid for any 3-D geometry of the unit cell. The model is illustrated by means of longitudinally-layered smart composites consisting of piezoelectric (Barium Titanate) and piezomagnetic (Cobalt Ferrite) constituents. Closed-form expressions for the effective properties are derived and the results are plotted vs. the volume fraction of the piezoelectric phase. Pertaining to the product properties of this particular magnetoelectric laminate, it is observed that the effective pyroelectric and pyromagnetic coefficients attain a maximum value at a BaTiO₃ volume fraction of 0.5 and maximum values for the magnetoelectric coefficients at a BaTiO₃ volume fraction of 0.4. Likewise, the maximum value of a magnetoelectric figure of merit (characterizing efficiency of energy conversion in longitudinal direction) is also attained at a volume fraction of 0.4.     

Effective properties of three-phase electro-magneto-elastic composites

Coupling between the electric field, magnetic field, and strain of composite materials is achieved when electro-elastic (piezoelectric) and magneto-elastic (piezomagnetic) particles are joined by an elastic matrix. Although the matrix is neither piezoelectric nor piezomagnetic, the strain field in the matrix couples the electric field of the piezoelectric phase to the magnetic field of the piezomagnetic phase. This three-phase electro-magneto-elastic composite should have greater ductility and formability than a two-phase composite in which the electric field and the magnetic field are coupled by directly bonding two brittle materials. A finite element analysis (FEA) and micromechanics based averaging of a representative volume element (RVE) are performed in this work to determine the effective dielectric, magnetic, mechanical, and coupled-field properties of an elastic matrix reinforced with piezoelectric and piezomagnetic fibers as functions of the phase volume fractions, the fiber arrangements in the RVE, and the fiber material properties with special emphasis on the poling directions of the piezoelectric and piezomagnetic fibers. The effective magneto-electric moduli of this three-phase composite are found to be less than the effective magneto-electric moduli of a two-phase piezoelectric/piezomagnetic composite, because the elastic matrix is not stiff enough to transfer significant strains between the piezomagnetic and piezoelectric fibers.     

Effect of piezoelectric grain size on magnetoelectric coefficient of Pb(Zr0.52Ti0.48)O3–Ni0.8Zn0.2Fe2O4 particulate composites

This study investigates the variation of magnetoelectric (ME) coefficient as a function of the piezoelectric grain size in the composite system of 0.8 Pb(Zr_0.52Ti_0.48)O₃–0.2 Ni_0.8Zn_0.2Fe₂O₄. It was found that as the piezoelectric-phase grain size increases the overall resistivity, piezoelectric, dielectric, and ferroelectric property of the composite increases and saturates above 600 nm. Below 200 nm average grain size, piezoelectric and dielectric properties decrease rapidly. The ferroelectric Curie temperature was found to decrease from 377 to 356 °C as the average grain size decreases from 830 to 111 nm. ME coefficient of the composite showed a rapid change below grain size of 200 nm and was found to saturate above 600 nm to a value of 155 mV/cm.Oe.     

Enhanced magnetoelectric effect in multiferroic composite beams due to flexoelectricity and transverse deformations

This paper is concerned with the exploration of the role of transverse normal and shear deformations on enhancing the magnetoelectric (ME) coefficient of multiferroic bilayer composite beams composed of a piezoelectric layer and a piezomagnetic layer. Analytical models have been derived based on the displacement field which accounts for both the transverse normal and shear deformations, Timoshenko beam theory and Euler Bernoulli beam theory. The induced flexoelectricity in the piezoelectric layer due to axial strain gradient and transverse shear strain gradient has also been taken into consideration for estimating the ME coefficient. It has been found that the contribution of transverse normal strain in the piezoelectric layer for enhancing the ME coefficient is significantly larger than that due to axial strain, transverse shear strain and flexoelectricity. For the particular values of the thicknesses of the piezoelectric layer and the piezomagnetic layer, the ME coefficient remains invariant for both thick and thin multiferroic composite beams.     

含不完美界面的磁电复合材料倾斜裂纹问题

研究了含非完美界面的双层压电/压磁复合材料中压电相存在一个倾斜于界面的Ⅲ型裂纹问题。采用弹簧型耦合界面模型模拟非完美界面,运用Fourier积分变换法将裂纹面条件转化为奇异积分方程,并使用Lobatto-Chebyshev方法数值求解了裂纹尖端应力强度因子（SIF）。详细地研究了裂纹尖端SIF与界面参数、压电/压磁材料参数和材料的层厚、裂纹的倾斜角、裂纹与界面的距离等几何参数的关系。结果表明：力学不完美性可以独立地增大SIF,而磁学、电学不完美性只有与力学不完美性耦合时才会减小SIF;力学-电学、力学-磁学不完美性的耦合会减小SIF,而磁学-电学不完美性的耦合不会影响SIF;磁场作用下,增大压磁层弹性模量会减小SIF,而增大压电层压电系数,减小压电层弹性模量和介电常数,均会减小SIF;界面不完美性会影响SIF随裂纹倾斜角度或裂纹与界面之间距离的变化规律;在一定范围内增加压电层或压磁层厚度可以减小SIF。     

压电、压磁和电磁各向异性弹性介质二维问题的Green函数

基于Dirac-delta函数的积分表示和Cauchy留数定理,导出了压电、压磁和电磁各向异性弹性介质二维问题的Green函数。所得Green函数的主要特征为：(1) 其数学表达式是以封闭形式给出的;(2) 适用于平面、反平面以及平面和反平面相互耦合问题;(3) 对于“退化材料”的情形也是有效的。     

Piezoelectric and magnetic properties of PZT-(Ni<Subscript>0.284</Subscript>Zn<Subscript>0.549</Subscript>Cu<Subscript>0.183</Subscript>)Fe<Subscript>1.984</Subscript>O<Subscript>4</Subscript> composites

The piezoelectric/piezomagnetic composite, PZT/Ni_0.284Zn_0.549Cu_0.183Fe_1.984O₄, was fabricated by the mixed oxide method. The phase assemblage, piezoelectric strain constant and saturation magnetization were investigated. The results indicate that the PZT phase is compatible with Ni_0.284Zn_0.549Cu_0.183Fe_1.984O₄ phase, and dense diphasic ceramic composites were obtained. It is found that piezoelectric strain constant decreases exponentially as the amount of doped piezomagnetic materials in the composite increases. Correspondingly, saturation magnetization of the composite also decreases with the increasing weight fraction of piezoelectric materials. Three reasons cause the results. First, the grain growth of piezomagnetic phase at the co-sintering temperature reduces grain size and continuity of the piezoelectric phase. Second, the pore size and porosity in composite increase dramatically with increasing amount of piezomagnetic phase. Third, the low resistivity of the composite prevents the poling process and reduces the piezoelectric strain constant. The tailoring of microstructure to achieve a high performance piezoelectric/piezomagnetic composite is proposed based on the analysis.     

Reflection and refraction of plane waves at the interface between piezoelectric and piezomagnetic media

The paper analyzes the reflection and refraction of a plane wave incidence obliquely at the interface between piezoelectric and piezomagnetic media. The materials are assumed to be transversely isotropic. Numerical calculations are performed for BaTiO₃/CoFe₂O₄ material combination. Four cases, incidence of the coupled quasi-pressure (QP) and quasi-shear vertical (QSV) wave from BaTiO₃ or CoFe₂O₄ media, are discussed. The reflection and transmission coefficients and energy coefficients varying with the incident angle are examined. Calculated results are verified by considering the energy conservation. Results show that the reflected and transmitted wave fields in the sagittal plane consist of six kinds of waves, i.e. the coupled QP and QSV waves, evanescent electroacoustic (EA) and magnetic potential (MP) waves in the piezoelectric medium (BaTiO₃), evanescent magnetoacoustic (MA) and electric potential (EP) waves in the piezomagnetic medium (CoFe₂O₄), among which the EA, MA, MP and EP waves propagate along the interface. The most amount of the incident energy goes with the waves that are the same type as the incident wave, while the energy arising from wave mode conversion occupies a less part of the incident energy. The electric energy in BaTiO₃ is higher than the magnetic energy in CoFe₂O₄; they both attain their maximum values at/before the critical angle. Critical angles have little effect on evanescent waves except when the total reflection takes place. These results would provide useful complementary information for magnetoelectric composite materials.     

Multicoated elliptic fibrous composites of piezoelectric and piezomagnetic phases

A theoretical framework is developed to investigate the magnetoelectroelastic potential in a multicoated elliptic fibrous composite with piezoelectric and piezomagnetic phases. We generalize the classic work of Rayleigh (1892) to obtain the electrostatic potential in ordered conductive composites and its extension to a disordered system ( [Kuo, 2010] and [Kuo and Chen, 2008]) to the current coupled magnetoelectroelastic multicoated elliptic composites. We combine the methods of complex potentials with a re-expansion formulae and the generalized Rayleigh’s formulation to obtain a complete solution of the multi-field many-inclusion problem. It is shown that the coefficients of field expansions can be written in the form of an infinite set of linear algebraic equations. Numerical results are presented for several configurations. We use this method to study BaTiO₃-CoFe₂O₄ composites and find that, with appropriate coating, the effective magnetoelectric voltage coefficient can be enhanced with one order of magnitude compared to their non-coating counterpart.     

An in situ grown eutectic magnetoelectric composite material

The phase diagram of the quinary system Fe-Co-Ti-Ba-O contains a region of compositions with the property that unidirectional solidification of liquids with these compositions results in an aligned two phase composite material. One of the phases is a piezomagnetic spinel and the other one a piezoelectric perovskite. The resulting composite is a magnetoelectric material, which can convert magnetic fields into electric fields and vice versa. This paper deals with the determination of the above-mentioned composition region and with the unidirectional solidification process. Some results of magnetoelectric measurements are reported.     

Micro-Mechanical Behavior of a Unidirectional Composite Subjected to Transverse Shear Loading

The effects of interphase between fibers and matrix on the micro-and macro-mechanical behaviors of fiber-reinforced composite lamina subjected to transverse shear load at remote distance have been studied. The interphase has been modeled by the compliant spring-layers that are linearly related to the normal and tangential tractions. Numerical analyses on composite basic cells have been carried out using the boundary element method. For undamaged composites the micro-level stresses at the matrix side of the interphase and effective shear modulus have been calculated as a function of the fiber volume fraction and the interphase stiffness. Results are presented for various interphase stiffnesses from perfect bonding to total debonding. For a square array composite results show that for a high interphase stiffness k > 10, an increase in a fiber volume fraction results in a higher effective transverse shear modulus. For a relatively low interphase stiffness k < 1, it is shown that an increase in the fiber volume fraction causes a decrease in the effective transverse shear modulus. For perfect bonding, the effective shear modulus for a hexagonal array composite is slightly larger than that for a square array composite. Also for the damaged composite with partially debonded interphase, local stress fields and effective shear moduli are calculated and a decrease in the effective shear modulus has been observed.     

MLPG Analysis of Layered Composites with Piezoelectric and Piezomagnetic Phases

A meshless method based on the local Petrov-Galerkin approach is proposed, to solve static and dynamic problems of two-layered magnetoelectroelastic composites with specific properties. One layer has pure piezoelectric properties and the second one is a pure piezomagnetic material. It is shown that the electric potential in the piezoelectric layer is induced by the magnetic potential in the piezomagnetic layer. The magnetoelectric effect is dependent on the ratio of the layer thicknesses. Functionally graded material properties of the piezoelectric layer and homogeneous properties of the piezomagnetic layer are considered too. The magnetoelectric composites are analyzed under a pure magnetic or combined magneto-mechanical load. Various boundary conditions and geometric parameters are considered to analyze their influence on the value of the electromagnetic parameter.     

Magneto stress analysis of a strip with a semi elliptical notch under uniform magnetic field

Two dimensional solutions of the magnetic field and magneto elastic stress are presented for a magnetic material of a thin strip with a semi-elliptical notch subjected to uniform magnetic field. The strip is a finite plate of a simply connected region. A linear constitutive equation is used for the stress analysis. According to the electro-magneto theory, only Maxwell stress is caused as a body force in a plate. Therefore, the magneto elastic stress is analyzed using Maxwell stress. In the present problem, as a result, the plane stress state does not arise, and the σ_z in the direction of the plate thickness and the shear deflection (anti-plane shear stress) arise for soft ferromagnetic material. The stress σ_z in the plate is strong compressive stress for a soft ferromagnetic material. A rational mapping function is used for the stress analysis, and the each solution is obtained as a closed form. No further assumption of the plane stress state that the plate is thin is made for the stress analysis, though Maxwell stress components are expressed by nonlinear terms. The rigorous boundary condition is completely satisfied without any linear assumptions on the boundary. The anti-plane shear stress causes Mode III stress intensity factor when the notch is a crack. Stress concentration values are investigated for a notch problem, of which expression is given. Figures of the anti-plane shear stress distribution, Mode III stress intensity factor, and stress concentration values are shown.     

Magnetoelectric behavior in in situ grown piezoelectric and piezomagnetic composite-phase system

A few model combinations of piezoelectric (PE), perovskite and piezomagnetic (PM), spinel phase has been chosen on BaO-TiO-FeO-CoO quinary oxides system for in situ preparation of magnetoelectric composite materials by solid-state route. The various phases, which appeared during in situ preparation in different model systems, have been characterized by X-ray diffraction (XRD). The magnetoelectric effect (ME) of such composites has been measured by dynamic method. The present paper deals with a qualitative comprehension of the ME property of such composites in relation to their proposed composition model.     

Piezomagnetic losses in bilayered magnetostrictive/piezoelectric composites: Effect of Terfenol-D content on the magnetoelectric susceptibility

This letter reports on the influence of piezomagnetic loss in magnetostrictive Terfenol-D (TeD)/polyurethane composite layer/piezoelectric PZT ceramic layer laminates on the phase shift between magnetoelectric current and ac magnetic field versus bias dc field. For a given low measurement frequency (1 kHz), piezomagnetic losses are predominant when the amount of TeD powder embedded in polyurethane is small (2 vol.%), inducing an imaginary magnetoelectric susceptibility and a 0° up to 180° variation of phase shift of magnetoelectric current. When TeD content is high enough (7 vol.%), piezomagnetic losses are almost null, which yields a real magnetoelectric susceptibility and a − 90° up to + 90° variation of phase angle of the current.     

A first-principles study on the structural, elastic, electronic, and optical properties of CdRh2O4

The structural, elastic, electronic, and optical properties of CdRh₂O₄ with cubic $ (Fd\overline{ 3} m) $ and orthorhombic (Pnma) structures have been investigated using a pseudopotential plane wave (PP-PW) method within the local density approximation (LDA). The calculated lattice parameters agree reasonably with the experimental values. The single-crystal elastic stiffness constants C _ij s of the cubic and orthorhombic phases are investigated using the stress–strain method. In addition, the polycrystalline elastic properties including bulk modulus, shear modulus, Young’s modulus, bulk modulus–shear modulus ratio, Poisson’s ratio, and elastic anisotropy ratio are determined based on Voigt–Reuss–Hill approach. The use of the hybrid functional sX-LDA leads to considerably improved electronic properties compared to standard LDA approach. On the other hand, the dielectric function, refraction index, reflectivity, conductivity function, and energy-loss spectra were obtained and analyzed on the basis of electronic band structures and density of states.     

In-plane shear properties of unidirectional glass fiber (U)/random glass fiber (R)/epoxy hybrid and non-hybrid composites

The in-plane shear properties (shear strength τ_xy and shear modulus G_xy) of unidirectional glass fiber (U)/random glass fiber (R)/epoxy hybrid and non-hybrid composites have been investigated experimentally and theoretically. The effect of stacking sequence and random fiber relative volume fraction (V_fR/V_fT) in hybrid composites were reported. Laminates were fabricated by hand lay-up technique with a total of 5 plies, by varying the number and position of random glass layers so as to obtain four different hybrid laminates; i.e. [0.5R/U/U]_S, [U/0.5R/U]_S, [U/U/0.5R]_S, and [U/R/U/R/U]. All unidirectional fiber laminate [U]₅ and another of all random fiber laminate [R]₅ were also fabricated for comparison purpose. The average thickness of the manufactured laminates is 5.5 ± 0.2 mm and the total fiber volume fraction (V_fT) is 37%. Failure modes of all specimens were investigated. Results indicated that the in-plane shear properties (shear strength τ_xy and shear modulus G_xy) of unidirectional fiber composite can be considerably improved by incorporation of random glass fiber and forming hybrid composites.     

Impact response of an anti-plane crack in a FGPM bonded to a homogeneous piezoelectric strip

A finite crack under transient anti-plane shear loads in a functionally graded piezoelectric material (FGPM) bonded to a homogeneous piezoelectric strip is considered. It is assumed that the electroelastic material properties of the FGPM vary continuously according to exponential functions along the thickness of the strip, and that the two layered strips is under combined anti-plane shear mechanical and in-plane electrical impact loads. The analysis is conducted on the electrically unified crack boundary condition. Laplace and Fourier transforms are used to reduce the mixed boundary value problems to Fredholm integral equations of the second kind in the Laplace transform domain. Then, a numerical Laplace inversion is performed and the dynamic intensities are obtained as functions of time and geometric parameters, which are displayed graphically.     

Influence of powder surface area on the magnetoelectric effect in barium lead zirconate titanate/nickel ferrite composite ceramics

This paper examines the influence of powder surface area on the magnetoelectric properties of a ferrite + piezoelectric composite. The use of fine powders with a specific surface area in the range 3.1–7.6 m²/g improves the magnetoelectric parameters of bulk composites by 17 to 44% in comparison with a coarse powder with a specific surface area of 0.8 m²/g. This is due to the formation of a more homogeneous composite material, improvement of its dielectric properties, increase in contact area between phases, reduction in internal demagnetizing factor, and increase in the density of the material.     

Effect of Al<Superscript>3+</Superscript> modification on cobalt ferrite and its impact on the magnetoelectric effect in BCZT–CFO multiferroic composites

One of the methods to enhance the functional properties of two-phase multiferroic magnetoelectrics is to increase magnetostriction of the ferrite phase. Al³⁺-modified cobalt ferrite Co(Al_0.5Fe_1.5)O₄ shows better magnetostriction than unmodified cobalt ferrite. It is used in combination with (Ba,Ca)(Zr,Ti)O₃ which has very good piezoelectric properties, to form a multiferroic composite. The composite shows good magnetoelectric characteristics, both macroscopically (converse magnetoelectric coefficient of 11 ps/m) and microscopically. Al³⁺ proves to be the best non-magnetic dopant to enhance magnetostriction in CoFe₂O₄ and thus the magnetoelectric coefficient.