Green's functions of two-dimensional piezoelectric quasicrystal in half-space and bimaterials

In this paper,we obtain Green’s functions of two-dimensional(2D)piezoelectric quasicrystal(PQC)in half-space and bimaterials.Based on the elastic theory of QCs,the Stroh formalism is used to derive the general solutions of displacements and stresses.Then,we obtain the analytical solutions of half-space and bimaterial Green’s functions.Besides,the interfacial Green’s function for bimaterials is also obtained in the analytical form.Before numerical studies,a comparative study is carried out to validate the present solutions.Typical numerical examples are performed to investigate the effects of multi-physics loadings such as the line force,the line dislocation,the line charge,and the phason line force.As a result,the coupling effect among the phonon field,the phason field,and the electric field is prominent,and the butterfly-shaped contours are characteristic in 2D PQCs.In addition,the changes of material parameters cause variations in physical quantities to a certain degree.

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.

ANALYSIS OF CRACK-TIP SINGULARITIES FOR AN INTERFACIAL PERMEABLE CRACK IN METAL/PIEZOELECTRIC BIMATERIALS

By modeling metal as a special piezoelectric material with extremely small piezoelectricity and extremely large permittivity, we have obtained the analytical solutions for an interracial permeable crack in metal/piezoelectric bimaterials by means of the generalized Stroh formalism. The analysis shows that the stress fields near a permeable interracial crack tip are usually with three types of singularities： r^-1/2±iε and r^-1/2. Further numerical calculation on the oscillatory index ε are given for 28 types of metal/piezoelectric bimaterials combined by seven commercial piezoelectric materials： PZT-4, BaTiO3, PZT-5H, PZT-6B, PZT-7A, P-7 and PZT-PIC 151 and four metals： copper, silver, lead and aluminum, respectively. The explicit expressions of the crack tip energy release rate （ERR） and the crack tip generalized stress intensity factors （GSIF） are obtained. It is found that both the ERR and GSIF are independent of the electric displacement loading, although they seriously depends on the mechanical loadings.

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.

Effects of electric/magnetic impact on the transient fracture of interface crack in piezoelectric-piezomagnetic sandwich structure: anti-plane case

Due to the incompatibility of the interlaminar deformations,the interface debonding or cracking usually happens in a layered magnetoelectric(ME)structure under an applied load.In this paper,the transient responses of the anti-plane interface cracks in piezoelectric(PE)-piezomagnetic(PM)sandwich structures are studied by the standard methods of the integral transform and singular integral equation.Discussion on the numerical examples indicates that the PE-PM-PE structure under electric impact is more likely to fracture than the PM-PE-PM structure under a magnetic impact.The dynamic stress intensity factors(DSIFs)are more sensitive to the variation of the active layer thickness.The effects of the material constants on the DSIFs are dependent on the roles played by PE and PM media during the deformation process.

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.

CRACK KINKING IN PIEZOELECTRIC MATERIALS

The solution for a class of two-dimensional electro- elastic branched crack problems was presented. Explicit Green's function for an interface crack subjected to an edge dislocation was developed using the extended Stroh formulation, which allows the branched crack problem to be expressed in terms of coupled singular integral equations. The kink was modelled as the continuous distribution of edge dislocations and the dislocation density function was defined on the line of the branch crack only. Competition between crack extension along the interface and kinking into the substrate was investigated using the integral equations and the maximum energy release rate criterion. The numerical results have shown the effect of electric field on the path of crack extension. (Edited author abstract) 24 Refs.

压电/压磁复合楔形结构耦合场的奇异性研究:反平面问题

研究了反平面变形状态下压电/压磁复合楔形结构耦合场的奇异性.应用复变函数理论和本征函数展开方法,推导了16种机械、电学和磁学组合边界条件下关于奇异性指数本征方程的显函表达式.基于得到的本征方程,通过数值算例表明了楔角、边界条件和材料组合类型对耦合场奇异性的影响,发现压电效应和压磁效应之间的相互作用导致压电/压磁复合楔磁电弹场的奇异性比压电复合楔电弹场的奇异性更加复杂.

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.

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.

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.

Interaction between subinterface cracks and interface in metal/piezoelectric ceramic bimaterials

A class of problems concerning subinterface cracks interacting with the interface in metal/piezoelectric ceramic bimaterials are studied. The interaction problem is reduced to a system of integral equations with the aid of the pseudo-traction-electric-displacement method. The equations are solved numerically, and the stress, intensity factor, the electric displacement intensity factor and the mechanical strain energy release rate are evaluated. Numerical results of a Cu/PZT-4 bimaterial are given and shown in figures, in which three kinds of remote loading conditions are considered to make comparisions. It is found that the electric loading at infinity plays a quite important role in the present interaction problem. In addition, a conservation law of the first component of the J k-integral, vector is found; which does lead to a consistency check to confirm the effectiveness of the PTEDM as well as the numerical results derived in this paper.

Applied theory of bending vibration of the piezoelectric and piezomagnetic bimorph

Based on the variational principle,equations and boundary conditions for transverse steady vibrations of a bimorph consisting of a piezoelectric and piezomagnetic layers are obtained.The results of calculations of natural frequencies are compared with the finite element model of the device in ACELAN.

Magnetoelectric Properties of Piezoelectric—Piezomagnetic Composites with Elliptical Nanofibers

Based on the surface/interface theory,the effective magnetoelectric(ME)properties of piezoelectric(PE)-piezomagnetic(PM)composites with elliptical nanofibers are investigated analytically.Using the complex function theory,conformal mapping and the Mori-Tanaka mean-field method,the solutions of magneto-electro-elastic(MEE)fields are obtained,which satisfy both the interface and the single-valued conditions.The closed-form expression of the ME coefficient of the composites shows that the effect of ME coefficient is related to the interface parameters,elliptical aspect ratio,material properties of both fiber and matrix and the fiber volume fraction.Numerical calculation shows that by properly selecting some of these parameters,the effective ME coefficient could be substantially enhanced.

Non-local theory solution to two collinear limited-permeable mode-I cracks in a piezoelectric/piezomagnetic material plane

The non-local theory solution to two collinear limited-permeable mode-1 cracks in a piezoelectric/piezomagnetic medium was investigated by using the generalized Almansi＇s theorem and the Schmidt method in the present paper. The problem was for- mulated through Fourier transformation into two pairs of dual integral equations, in which the unknown variables are the dis- placement jumps across the crack surfaces. For solving the dual integral equations, the displacement jumps across the crack surfaces were directly expanded as a series of Jacobi polynomials. Numerical examples were provided to show the effects of the crack length, the distance between the two collinear cracks, the lattice parameter, the electric permittivity and the magnetic permeability of the air inside the crack on the stress fields, the electric displacement fields and the magnetic flux fields near the crack tips in a piezoeleetric/piezomaguetic medium. Different from the classical solutions, the present solution exhibits no stress, electric displacement and magnetic flux singularities at the crack tips in a piezoelectric/piezomagnetic medium.

一维六方压电准晶双材料界面共线裂纹问题

利用复变函数理论中的解析延拓、奇性主部分析和推广的Liouville定理,求解了一维六方压电准晶双材料在集中载荷作用下界面共线裂纹反平面弹性问题.导出了含有一条和两条有限长界面裂纹的封闭解,同时给出了裂纹尖端场强度因子(包含声子场和相位子场应力强度因子和电位移强度因子)的表达式.数值算例分析了外荷载与耦合系数之比对裂纹尖端场强度因子变化规律的影响.从数值结果中可以看出,当裂纹长度增加时,裂纹尖端场强度因子随之增加;应力强度因子随双材料耦合系数之比的增大而增大,电位移强度因子几乎不变;不同载荷作用下,裂纹尖端场强度因子随着裂纹长度改变时的变化趋势也不尽相同.研究结果可为压电准晶双材料的设计和制备提供一定的理论参考.

功能梯度压电压磁圆柱壳中纵向波的截止频率

采用解析方法研究材料参数沿径向变化的功能梯度压电压磁圆柱壳体中纵向导波的截止频率.建立了利用位移函数、电势函数和磁势函数表示的柱坐标系下功能梯度压电压磁圆柱壳中纵向波的力、电和磁耦合波动方程并表征齐次边界条件.在波数趋于0的条件下,采用Wentzel-Kramers-Brillouin(WKB)方法求解方程并推导得到梯度壳体中纵向导波截止频率的显式近似解析解.分别对比均质压电压磁圆柱壳体中纵向导波的截止频率的贝塞尔函数解和由功能梯度压电压磁圆柱壳体中纵向导波频散曲线所得的截止频率,验证了WKB解的高精确性.通过进一步解析推导简化可得:纵向波的截止频率是两个近似等差数列的集合;其公差分别与材料两种等效波速的倒数沿径向的积分成反比.数值算例表明,直接利用两个数列的公差表示截止频率具有足够的精度.研究对比分析了弹性参数、密度、电学参数、磁学参数和壳体厚度单独变化对截止频率的影响规律.其中弹性参数和密度梯度变化对截止频率影响较为明显;电学参数与磁学参数梯度变化对其截止频率影响较小;壳体厚度增加会导致截止频率降低.文中给出的分析方法与结论可适用于功能梯度弹性材料与功能梯度压电材料圆柱壳中纵向导波的截止频率,这些结论可为基于截止频率的非均质材料圆柱壳的超声无损检测提供理论依据.