Coupling effects between elastic and electromagnetic fields from the perspective of conservation of energy

Coupling effects among different physical fields reflect the conversion of energies from one field into another substantially.For simple physical processes,their governing or constitutive equations all satisfy the law of conservation of energy(LCE).Then,an analysis is extended to the coupling effects.First,for the linear direct and converse piezoelectric and piezomagnetic effects,their constitutive equations guarantee that the total energy is conserved during the process of energy conversion between the elastic and electromagnetic fields.However,the energies are converted via the work terms,(βijkEi),kvj and(γijkHi),kvj,rather than via the energy terms,βijkEiejk andγijkHiejk.Second,for the generalized Villari effects,the electromagnetic energy can be treated as an extra contribution to the generalized elastic energy.Third,for electrostriction and magnetostriction,both effects are induced by the Maxwell stress.Moreover,their energies are purely electromagnetic and thus both have no converse effects.During these processes,the energies can be converted in three different ways,i.e.,via the non-potential forces,via the cross-dependence of the energy terms,and directly via the electromagnetic interactions of ions and electrons.In the end,the general coupling processes which involve elastic,electromagnetic fields and diffusion are also analyzed.The advantages of using this energy formulation are that it facilitates discussion of the conversion of energies and provides better physical insights into the mechanisms of these coupling effects.

WAVE PROPAGATION IN PIEZOELECTRIC/PIEZOMAGNETIC LAYERED PERIODIC COMPOSITES

This paper is concerned with the dynamic behaviors of wave propagation in layered periodic composites consisting of piezoelectric and piezomagnetic phases. The dispersion relations of Lamb waves axe derived. Dispersion curves and displacement fields are calculated with different piezoelectric volume fractions. Numerical results for BaTiO3/CoFe2O4 composites show that the dispersion curves resemble the symmetric Lamb waves in a plate. Exchange between the longitudinal （i.e. thickness） mode and coupled mode takes place at the crossover point between dispersion curves of the first two branches. With the increase of BaTiO3 volume fraction, the crossover point appears at a lower wave number and wave velocity is higher. These findings are useful for magnetoelectric transducer applications.

SCATTERING OF HARMONIC ANTI-PLANE SHEAR STRESS WAVES BY A CRACK IN FUNCTIONALLY GRADED PIEZOELECTRIC/PIEZOMAGNETIC MATERIALS

In this paper, the dynamic behavior of a permeable crack in functionally graded piezoelectric/piezomagnetic materials is investigated. To make the analysis tractable, it is assumed that the material properties vary exponentially with the coordinate parallel to the crack. By using the Fourier transform, the problem can be solved with the help of a pair of dual integral equations in which the unknown is the jump of displacements across the crack surfaces. These equations are solved to obtain the relations between the electric filed, the magnetic flux field and the dynamic stress field near the crack tips using the Schmidt method. Numerical examples are provided to show the effect of the functionally graded parameter and the circular frequency of the incident waves upon the stress, the electric displacement and the magnetic flux intensity factors of the crack.

PROPAGATION BEHAVIORS OF SHEAR HORIZONTAL WAVES IN PIEZOELECTRIC-PIEZOMAGNETIC PERIODICALLY LAYERED STRUCTURES

This paper investigates shear horizontal （SH） waves propagating in a periodically layered structure that consists of piezoelectric （PE） layers perfectly bonded with piezomagnetic （PM） layers alternately. The explicit dispersion relations are derived for the two cases when the propagation directions of SH waves are normal to the interface and parallel to the interface, respectively. The asymptotic expressions for dispersion relations are also given when the wave number is extremely small. Numerical results for stop band effect and phase velocity are presented for a periodic system of alternating BaTiO3 and Terfenol-D layers. The influence of volume fraction on stop band effect and dispersion behaviors is discussed and revealed.

Piezomagnetic In-situ Stress Monitoring and its Application in the Longmenshan Fault Zone

The relative change of in-situ stress is an inevitable outcome of differential movement among the crust plates. Conversely, changes of in-situ stress can also lead to deformation and instability of crustal rock mass, trigger activity of faults, and induce earthquakes. Hence, monitoring real-time change of in-situ stress is of great significance. Piezomagnetic in-situ stress monitoring has good and longtime applications in large engineering constructions and geoscience study fields in China. In this paper, the new piezomagnetic in-situ stress monitoring system is introduced and it not only has overall improvements in measuring cell＇s structure and property, stressing and orienting way, but also enhances integration and intelligence of control and data transmission system, in general, which greatly promotes installing efficiency of measuring probe and quality of monitoring data. This paper also discusses the responses of new piezomagnetic system in large earthquake events of in-situ stress monitoring station at Qiaoqi of Baoxing and Wenxian of Gansu. The monitoring data reflect adjustments and changes of tectonic stress field at the southwestern segment of and the northern area near the Longmenshan fault zone, which shows that the new system has a good performance and application prospect in the geoscience field. Data of the Qiaoqi stress-monitoring station manifest that the Lushan Earthquake did not release stress of the southwestern segment of the Longmenshan fault zone adequately and there still probably exists seismic risk in this region in the future. Combined with absolute in-situ stress measurement, carrying out long-term in-situ stress monitoring in typical tectonic position of important regions is of great importance for researchers to assess and study regional crust stability.

Basic solutions of multiple parallel symmetric mode-III cracks in functionally graded piezoelectric/piezomagnetic material plane

The Schmidt method is adopted to investigate the fracture problem of multiple parallel symmetric and permeable finite length mode-III cracks in a functionally graded piezoelectric/piezomagnetic material plane. This problem is formulated into dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces. In order to obtain the dual integral equations, the displacement jumps across the crack surfaces are directly expanded as a series of Jacobi polynomials. The results show that the stress, the electric displacement, and the magnetic flux intensity factors of cracks depend on the crack length, the functionally graded parameter, and the distance among the multiple parallel cracks. The crack shielding effect is also obviously presented in a functionally graded piezoelectric/piezomagnetic material plane with mul- tiple parallel symmetric mode-III cracks.

Electromagnetic effect stimulated by strain wave in rocks

When strain wave passes through rock, a changing magnetic field due to piezomagnetic effect will propagate at the velocity of sound. The changing magnetic field will stimulate electromagnetic ef fect in the surrounding area. This paper reports the experimental results which proves the existence of such phenomena. And the synthetic piezomagnetic coefficients of rock are measured from experimental results. The piezomagnetic effect can be used to explain some electromagnetic phenomena produced in the process of earthquakes.

Point-piezomagnetic Properties of the FeCuNbSiB Alloy Strips

The piezomagnetic properties of rapidly quenched Fe73.5Cu1Nb3Si13.5B9 alloy strips were investigated in as-quenched state and after annealing for 2 h in vacuum in the temperature range of 100-300 ℃. The impedance of amorphous strips increases with frequency, and sensitively de-creases with stress increasing, especially in the frequency range of 10-100 MHz. The impedance in-creases approximately linearly with frequency below the critical frequency, but begins to decrease non-linearly with frequency above the critical frequency. The higher the point compressive stress is, the smaller the critical frequency will be. The stability of piezomagnetic properties is very excellent. The impedance of amorphous strips after annealing, especially at 300 ℃, decreases very strongly. The impedance and the absolute values of the sensitive degree of couple layers＇ amorphous strips are lower than those of single layer strips.

Solution of two-dimensional scattering problem in piezoelectric/piezomagnetic media using a polarization method

Using a polarization method, the scattering problem for a two-dimensional inclusion embedded in infinite piezoelectric/piezomagnetic matrices is investigated. To achieve the purpose, the polarization method for a two-dimensional piezoelectric/piezomagnetic ＂comparison body＂ is formulated. For simple harmonic motion, kernel of the polarization method reduces to a 2-D time-harmonic Green＇s function, which is obtained using the Radon transform. The expression is further simplified under conditions of low frequency of the incident wave and small diameter of the inclusion. Some analytical expressions are obtained. The analytical solutions for generalized piezoelectric/piezomagnetic anisotropic composites are given followed by simplified results for piezoelectric composites. Based on the latter results, two numerical results are provided for an elliptical cylindrical inclusion in a PZT-5H-matrix, showing the effect of different factors including size, shape, material properties, and piezoelectricity on the scattering cross-section.

Dynamic behavior of two collinear interface cracks between two dissimilar functionally graded piezoelectric/piezomagnetic material strips

The dynamic interaction of two collinear interface cracks between two dissimilar functionally graded piezoelectric/piezomagnetic material strips subjected to the anti-plane shear harmonic stress waves was investigated. By using the Fourier transform, the problem can be solved with the help of a pair of triple integral equations in which the unknown variable is jump of displacement across the crack surfaces. These equations are solved using the Schmidt method. Numerical examples are provided to show the effect of the functionally graded parameter, the circular frequency of the incident waves and the thickness of the strip upon stress, electric displacement and magnetic flux intensity factors of cracks.

First Evidence of Surface SH-Wave Propagation in Cubic Piezomagnetics

This theoretical work provides with results of characteristics calculation of the ultrasonic surface Zakharenko waves (USZWs) existing in piezomagnetic cubic monocrystals of class m3m that can be readily used for non-destructive testing. The piezomagnetic waves propagate in direction [101] corresponding to relatively easy magnetization for the following piezomagnetics: Galfenol, Terfenol-D, and CoFe2O4 with cubic structures. The phase velocities of the USZW-waves and the coefficient of magnetomechanical coupling (CMMC) K2 were calculated for the crystals. It was found that the coefficient K2 for piezomagnetics with Km2 > 1/3 and Km2 >> 1/3 is about 8% to 9%, where K2 = 2 (VUSZW,o – VUSZW,c)/VUSZW,o and Km2 = h2/(Cμ). Knowledge of piezomagnetic properties of cubic crystals makes possible the use of them in new products utilizing the phenomenon called the magnetoelectric effect. Also, this study is useful for possible application of cubic piezomagnetics in composite structures consisting of piezoelectric and (or) piezomagnetic materials and in the microwave technology. This broadens choice of possible piezomagnetic materials for utilization in various technical devices.

Optimizing magnetoelastic properties by machine learning and high-throughput micromagnetic simulation

Magnetoelastic couplings in giant magnetostrictive materials(GMMs)attract significant interests due to their extensive applications in the fields of spintronics and energy harvesting devices.Understanding the role of the selection of materials and the response to external fields is essential for attaining desired functionality of a GMM.Herein,machine learning(ML)models are conducted to predict saturation magnetostrictions(λ_(s))in RFe_(2)-type(R=rare earth)GMMs with different compositions.According to ML-predicted composition–λsrelations,it is discovered that the values ofλshigher than1100×10^(-6)are almost situated in the composition space surrounded by 0.26≤x≤0.60 and 1.90≤y≤2.00 for the ternary compounds of Tb_(x)Dy_(1-x)Fe_(y).Assisted by ML predictions,the compositions are further narrowed down to the space surrounded by 0.26≤x≤0.32 and 1.92≤y≤1.97 for the excellent piezomagnetic(PM)performance in the Tb_(x)Dy_(1-x)Fe_(y)based PM device through our developed high-throughput(HTP)micromagnetic simulation(MMS)algorithm.Accordingly,high sensitivities up to10.22-13.61 m T·MPa^(-1)are observed in the optimized range within which the available experimental data fall well.This work not only provides valuable insights toward understanding the mechanism of magnetoelastic couplings,but also paves the way for designing and optimizing highperformance magnetostrictive materials and PM sensing devices.

Bending of a Saturated Ferromagnetoelastic Plate Under a Local Mechanical Load

We study the bending of a magnetically saturated ferromagnetoelastic plate.The plate is rectangular and simply-supported along its edges.It is under a local distribution of normal mechanical load on its top surface,simulating a mechanical probe or manipulation of the magnetization field.The three-dimensional equations of saturated ferromagnetoelasticity for small fields superposed on finite biasing fields due to spontaneous magnetization are used.The plate is effectively piezomagnetic under the biasing fields.A trigonometric series solution is obtained.The perturbation of the magnetization field by the mechanical load is calculated and examined.It is found that the magnetization is sensitive to the mechanical load,particularly near the loading area.The perturbation of the magnetization is found to be associated with the transverse shear stresses in bending.

Electrical Response of a Multiferroic Composite Semiconductor Fiber Under a Local Magnetic Field

We study the electrical response of a multiferroic composite semiconductor fiber consisting of a piezoelectric semiconductor layer and two piezomagnetic layers under a transverse magnetic field applied locally to a finite part of the fiber.The phenomenological theory of piezomagnetic-piezoelectric semiconductors is employed.A one-dimensional model is derived for magnetically induced extension of the fiber.For open-circuit boundary conditions at the two ends of the fiber,an analytical solution is obtained from the model linearized for small carrier perturbations.The solution shows a local electric polarization and a pair of local electric potential barrier-well.When the two ends of the fiber are under a voltage,a nonlinear numerical solution shows that the potential barrier and well forbid the passage of currents when the voltage is low.The results have potential applications in piezotronic devices when magnetic fields are involved for manipulating the devices or sensing and transduction.

THE DIRECTION DEPENDENCE OF SURFACE WAVE SPEED AT THE SURFACE OF MAGNETO-ELECTRO-ELASTIC HALF-SPACE

The direction dependence of surface wave speed and the influence of electrically and magnetically short/open circuit are investigated in this paper. First, the elastic, piezoelectric and piezomagnetic coefficients in the considered ordinate system are obtained by Bonde transformation from those in the crystal axis ordinate system. Then, the secular equation of surface wave is derived from the free traction condition on the surface of half space with consideration of short/open circuit case. Some numerical examples are given. The direction dependence of surface wave speed and the influences of short/open circuit are shown graphically and discussed based on the numerical results.

MULTIPLE PARALLEL SYMMETRIC PERMEABLE MODEL-Ⅲ CRACKS IN A PIEZOELECTRIC/PIEZOMAGNETIC COMPOSITE MATERIAL PLANE

In this paper, the interactions of multiple parallel symmetric and permeable finite length cracks in a piezoelectric/piezomagnetic material plane subjected to anti-plane shear stress loading are studied by the Schmidt method.The problem is formulated through Fourier transform into dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces.To solve the dual integral equations, the displacement jumps across the crack surfaces are directly expanded as a series of Jacobi polynomials.Finally, the relation between the electric field, the magnetic flux field and the stress field near the crack tips is obtained.The results show that the stress, the electric displacement and the magnetic flux intensity factors at the crack tips depend on the length and spacing of the cracks.It is also revealed that the crack shielding effect presents in piezoelectric/piezomagnetic materials.

TRANSIENT RESPONSE OF A BI-LAYERED MULTIFERROIC COMPOSITE PLATE

In this paper, a three-dimensional finite-element formulation for the multiferroic composite is developed and implemented into the commercial software ABAQUS for its transient analysis. First, a special three-dimensional eight-node solid element is designed to handle the multiferroic composite made of elastic, piezoelectric, and piezomagnetic materials. Second, a userdefined subroutine for this newly developed element is implemented into ABAQUS. Finally, the transient responses of a bi-layered multiferroic composite are calculated by using the direct time integration method. Two typical magnetic potential signals, Gauss and Ricker pulses, are applied to the composite with various time durations of excitation. The induced electric field shows that the transient response can be substantially influenced by the input signal, which could be tuned for the strongest electric output.

An Analysis of Piezomagnetic-Piezoelectric Semiconductor Unimorphs in Coupled Bending and Extension under a Transverse Magnetic Field

We study the response of a composite beam(unimorph)with a piezomagnetic layer and a piezoelectric semiconductor layer under a transverse magnetic field.A one-dimensional model for coupled bending and extension of the beam is derived from the three-dimensional theory.A simple and analytical solution of the model is obtained,showing that various electromechanical fields develop in the beam through piezomagnetic and piezoelectric couplings as well as semiconduction.In particular,the mobile charges in the semiconductor layer redistribute themselves under the magnetic field.A coupling coefficient is defined to describe the strength of this magneto-semiconduction coupling.The effects of physical and geometric parameters on the fields and the coupling coefficient are examined.The results are fundamental to piezotronics when magnetic effects are involved.

THE BASIC SOLUTION OF A 3-D RECTANGULAR PERMEABLE CRACK IN A PIEZOELECTRIC/PIEZOMAGNETIC COMPOSITE MATERIAL

The solution of a 3-D rectangular permeable crack in a piezoelectric/piezomagnetic composite material was investigated by using the generalized Almansi＇s theorem and the Schmidt method.The problem was formulated through Fourier transform into three pairs of dual integral equations,in which the unknown variables are the displacement jumps across the crack surfaces.To solve the dual integral equations,the displacement jumps across the crack surfaces were directly expanded as a series of Jacobi polynomials.Finally,the relations between the electric filed,the magnetic flux field and the stress field near the crack edges were obtained and the effects of the shape of the rectangular crack on the stress,the electric displacement and magnetic flux intensity factors in a piezoelectric/piezomagnetic composite material were analyzed.

Mechanical and magnetic hysteresis as indicators of the origin and inception of fatigue damage in steel

N2 and N3 are known as the transition points of the three principal stages of fatigue: initial accommodation, accretion of damage and terminal fatigue. Many experiments show that the ratios of N2/Nf and N3/Nf tend to be stable even though the specific N2 and N3 values may fluctuate widely. The primary goal of this research is to study the piezomagnetic field surrounding AISI 1018 steel specimen under repeated loads and to find the ratio values of N2/Nf and N3/Nf by analyzing 11 sets of low-cycle fatigue data. An MTS-810 testing system with a peak capacity of 222 kN was used to obtain the data which consisted of stress, strain, and piezomagnetic field. A computer program was constructed to track the evolution of the piezomagnetic field and re- gression analysis was carried out to determine N2 and N3 values. It was observed that there exists a consistent relationship between N2 and Nf. The apparent invariance of the ratio N2/Nf implies that N2 may be identified as an index of performance in the early loading response of a specimen that forecasts its fatigue life, Nf. It has been demonstrated that measurements of the magnetic and mechanical hysteresis can yield significant insights into the various stages of the development of a fatigue critical microstructure which culminates in complete rupture of the material.