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 efects 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.     

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.     

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.     

Coupled field state around three parallel non-symmetric cracks in a piezoelectric/piezomagnetic material plane

In this paper, the behavior of three parallel non-symmetric permeable cracks in a piezoelectric/piezomagnetic material plane subjected to anti-plane shear stress loading was studied by the Schmidt method. The problem was formulated through Fourier transform into three pairs of dual integral equations, in which unknown variables are jumps of displacements across the crack surfaces. To solve the dual integral equations, the jumps of displacements across the crack surfaces were directly expanded as a series of Jacobi polynomials. Finally, the relations among the electric displacement, the magnetic flux and the stress fields near the crack tips can be obtained. The results show that the stress, the electric displacement and the magnetic flux intensity factors at the crack tips depend on the lengths and spacing of cracks. It was also revealed that the crack shielding effect is present in piezoelectric/piezomagnetic materials.     

层状磁电复合材料界面共线裂纹平面问题分析

本文研究了面内电磁势载荷作用下双层压电压磁复合材料中共线界面裂纹问题．考虑了压电材料的导磁性质和压磁材料的介电性质,引入了界面电位移和磁感强度的连续性条件．利用Fourier 变换得到一组第二类Cauchy 型奇异积分方程．进一步导出了相应问题的应力强度因子、电位移强度因子和磁感强度强度因子的表达式,给出了应力强度因子的数值结果．结果表明电磁载荷会导致界面裂纹尖端I、II 混合型应力奇异性,同时还伴随着电位移和磁感强度的奇异性．比较了双裂纹左右端的应力强度因子,发现在面内极化方向上施加面内磁势载荷时共线裂纹内侧尖端区域的两个法向应力场发生互相干涉增强．     

Anti-Plane Shear Waves Scattering from a Partially Debonded Magneto-Electro-Elastic Circular Cylindrical Inhomogeneity

This article presents an analysis of the scattering of anti-plane shear waves from a single cylindrical inhomogeneity partially bonded to an unbounded magneto-electro-elastic matrix. The magneto-electric permeable boundary conditions are adopted. The crack opening displacement is represented by Chebyshev polynomials and a system of equations is derived and solved for the unknown coefficients. Some examples are calculated and the results are illustrated. The results show that the COD increases when the piezomagnetic coefficient of the inhomogeneity bonded to the piezoelectric matrix becomes larger, and that the COD decreases when the piezomagnetic coefficient of the matrix with the piezoelectric inhomogeneity increases.     

Exact solutions for magneto-electro-elastic laminates in cylindrical bending

Analytical solutions are derived for the cylindrical bending of multilayered, linear, and anisotropic magneto-electro-elastic plates under simple-supported edge conditions. We construct the general solution in terms of a simple formalism for any homogeneous layer, from which any physical quantities can be solved for the given boundary conditions. For multilayered plates, we derive the solution in terms of the propagator matrices. A special feature of cylindrical bending, which distinguishes itself from the three-dimensional plate problem, is that the associated eigenvalues for any homogeneous layer are independent of the sinusoidal mode, and thus need to be solved only once. Typical numerical examples are also presented for a piezomagnetic plate, a two-layered piezoelectric/piezomagnetic plate, and a four layered piezoelectric/piezomagnetic plate, with different span-to-thickness ratios. In particular, the piezoelectric and piezomagnetic fields show certain interesting features, which give guidance on the development of piezoelectric/piezomagnetic thin-plate theories. Furthermore, it is shown that the variations of the elastic, electric, and magnetic quantities with thickness depend strongly upon the material property and layering, which could be useful in the analysis and design of smart composite structures with sensors/actuators.     

An anti-plane propagating crack in a functionally graded piezoelectric strip bonded to a homogeneous piezoelectric strip

Summary A finite crack propagating at constant speed in a functionally graded piezoelectric strip (FGPS) bonded to a homogeneous piezoelectric strip is considered. It is assumed that the electroelastic material properties of the FGPS vary exponentially across the thickness of the strip, and that the bimaterial strip is under combined anti-plane mechanical shear and in-plane electrical loads. The analysis is conducted for the electrically unified crack boundary condition, which includes both the traditional permeable and the impermeable ones. By using the Fourier transform, the problem is reduced to the solution of Fredholm integral equations of the second kind. Numerical results for the stress intensity factor and the crack sliding displacement are presented to show the influences of the crack propagation speed, electric loads, FGPS gradation, crack length, electromechanical coupling coefficient, properties of the bonded homogeneous piezoelectric strip and crack location.     

Magneto-electro-elastic antiplane analysis of a bimaterial BaTiO3–CoFe2O4 composite wedge with an interface crack

The antiplane analysis is made for a bimaterial BaTiO₃–CoFe₂O₄ composite wedge containing an interface crack. The coupled magneto-electro-elastic field is induced by the piezoelectric/piezomagnetic BaTiO₃–CoFe₂O₄ composite materials. For the crack problems, the intensity factors of stress, strain, electric displacement, electric field, magnetic induction and magnetic field at crack tips are derived analytically. Also, the energy density criterion is applied to predict the fracture behavior of the interface crack. The numerical results also show that the energy release rate for a crack in a single wedge is negative.     

Magnetic and electric poling effects associated with crack growth in BaTiO3–CoFe2O4 composite

Magnetoelectroelastic composite possesses the dual feature that the application of magnetic field induces electric polarization and electric field induces magnetization. The poling directions introduced magnetically and electrically can be different in addition to those for the applied magnetic and electric field. Their choices can influence the character of crack growth which could be enhanced or retarded. The details of how the directions of poling and applied field would affect crack initiation and growth are discussed in relation to the volume fraction of inclusions for a BaTiO₃–CoFe₂O₄ two phase composite. The multi-functional aspects of magnetoelectroelastic materials are involved since they entail multi-scaling features. Failure criteria that applies to isotropic elastic materials may not hold for composites exhibiting piezomagnetic and piezoelectric properties. For instance, a negative energy release rate has been obtained for cracks in piezoelectric materials.In view of what has been said with reference to the energy release rate approach, it is desirable to use the strain energy density function as a failure criterion, even if it is only for its positive definiteness character. Physically speaking, it is attractive to have a function that could rank the proportion of energy related to volume and shape change. They determine the proportion of the hard and soft phase of the composite and hence the volume fraction of the constituent. Strength and toughness parameters used for ranking isotropic and homogeneous materials will not apply for anisotropic and/or nonhomogeneous materials if these microstructure effects could not be suppressed to a lower scale and represented as an average at the macroscopic scale. Too much emphases cannot be placed on the need to clarify the multi-scaling aspects of piezoelectric and piezomagnetic materials. Their behavior as affected by the presence of crack-like defects should be understood prior to deciding whether the material characterization approach would be suitable. That is whether simplicity could justify at the expense of conceptual rigor. Much of this would depend on scaling the time and size related to loading and material structure interaction. The magnetoelectroelastic crack model selected in the work to follow perhaps will provide an insight into the complexicity of the state of affairs for treating the finer details of material behavior with rigor.The proposed test model shows that crack growth in the magnetoelectroelastic materials can be suppressed by increasing the magnitude of the piezomagnetic constants in relation to those for piezoelectricity. A more rational means of evaluating the resistance of materials against fracture is thus proposed, particularly when anisotropy and inhomogeneity might be present.     

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.     

Mode III crack problems for two bonded functionally graded piezoelectric materials

This paper investigates the singular electromechanical field near the crack tips of an internal crack. The crack is perpendicular to the interface formed by bonding two half planes of different functionally graded piezoelectric material. The properties of two materials, such as elastic modulus, piezoelectric constant and dielectric constant, are assumed in exponential forms and vary along the crack direction. The singular integral equations for impermeable and permeable cracks are derived and solved by using the Gauss–Chebyshev integration technique. It shows that the stresses and electrical displacements around the crack tips have the conventional square root singularity. The stress intensity and electric displacement intensity factors are highly affected by the material nonhomogeneity parameters β and γ. The solutions for some degenerated problems can also be obtained.     

Magnetostrictive/electrostrictive fracture of the piezomagnetic and piezoelectric layers in a multiferroic composite: Anti-plane case

The main purpose of the present work is to study the influences of magnetostriction, electrostriction and piezomagnetic/piezoelectric stiffening on the fracture behavior of a layered multiferroic composite. For comparison, it is assumed that there is a crack, parallel to the interface, in each layer. Methods of cosine transform and Cauchy singular integral equations are used to solve the crack problem. Numerical results of the stress intensity factor (SIF) are provided and the computational accuracy is demonstrated. Discussion on the numerical results indicates that the multiferroic composite consisting of cobalt ferrite and barium titanate layers are more prone to fracture under electric loading than under magnetic loading. In the case of magnetostriction, to increase the shear modulus of the piezomagnetic layer would raise the SIF; but to increase that of the piezoelectric layer would reduce the SIF; in the case of electrostriction, inverse results are obtained. Piezomagnetic stiffening can affect the SIF when the composite is under electrostriction; piezoelectric stiffening can influence the SIF if the composite is under magnetostriction. In addition, it is also revealed that two parallel equal cracks may shield each other even if an interface exists between them.     

Arbitrarily oriented crack near interface in piezoelectric bimaterials

Arbitrarily oriented crack near interface in piezoelectric bimaterials is considered. After deriving the fundamental solution for an edge dislocation near the interface, the present problem can be expressed as a system of singular integral equations by modeling the crack as continuously distributed edge dislocations. In the paper, the dislocations are described by a density function defined on the crack line. By solving the singular integral equations numerically, the dislocation density function is determined. Then, the stress intensity factors (SIFs) and the electric displacement intensity factor (EDIF) at the crack tips are evaluated. Subsequently, the influences of the interface on crack tip SIFs, EDIF, and the mechanical strain energy release rate (MSERR) are investigated. The J-integral analysis in piezoelectric bimaterals is also performed. It is found that the path-independent of J₁-integral and the path-dependent of J₂-integral found in no-piezoelectric bimaterials are still valid in piezoelectric bimaterials.     

Fracture of piezoelectromagnetic materials

The crack problem in a medium possessing coupled piezoelectric, piezomagnetic and magnetoelectric effects is considered. A conservative integral is derived based on the governing equations for magnetoelectroelastic media. Closed-form solution is obtained for an anti-plane crack in an infinite medium. The conservative integral is used to obtain the path-independent integral near the crack tip. Expressions for stresses, electric displacements and magnetic inductions in the vicinity of a crack tip are derived. It is found that the path-independent integral around the crack tip equals the energy release rate. In the absence of applied mechanical loads, the energy release rate is always negative.     

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.     

压电压磁复合材料中裂纹对反平面简谐弹性波的散射问题

分析了压电压磁复合材料中裂纹对反平面简谐弹性波的散射问题。利用傅立叶变换,使问题的求解转换为对一对以裂纹表面上的位移差为未知变量的对偶积分方程的求解。为了求解对偶积分方程,把裂纹面上的位移差展开为雅可比多项式形式,进而得到了裂纹长度、入射波波速及入射波频率对裂纹应力强度因子的影响。从数值结果可以看出,压电压磁复合材料中可导通裂纹的反平面问题的动应力奇异性与一般弹性材料中的反平面断裂问题动应力奇异性相同。     

Effective multi-field properties of electro-magneto-thermoelastic composites estimated by finite element method approach

A finite element approach based on the micromechanics was performed to estimate the multi-field properties of electro-magneto-thermoelastic composites. The thermal field and the involved pyroelectric and pyromagnetic effect of the multi-phase composite materials were taken into account in the investigation and implemented in the finite element modeling. The multifields related to the electric field, magnetic field, deformation and temperature field, as well as their coupling effects of the smart composites under periodic boundary conditions were obtained numerically. Especially, by means of the homogenization approximation, the effective thermal expansion coefficients, pyroelectric coefficients, pyromagnetic coefficients and other elastic, electric,and magnetic properties for the piezoelectric material, piezomagnetic material and magnetoelectric material were calculated, respectively. Some results are compared to the theoretical predictions by the well-known Mori-Tanaka method to show good agreements.     

Solutions to a limited-permeable crack or two limited-permeable collinear cracks in piezoelectric/piezomagnetic materials

The solutions of a limited-permeable crack (case I) or two collinear limited-permeable cracks (case II) in piezoelectric/piezomagnetic materials subjected to a uniform tension loading were investigated in this paper using the generalized Almansi’s theorem. At the same time, the electric permittivity and the magnetic permeability of air in crack were firstly considered. Through the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations, in which the unknown variables were jumps of displacements across crack surfaces, not the dislocation density functions or the complex variable functions. To solve the dual integral equations, the jumps of displacements across crack surfaces were directly expanded in a series of Jacobi polynomials to obtain the relations among electric displacement intensity factors, magnetic flux intensity factors and stress intensity factors at crack tips.     

Scattering of harmonic anti-plane shear waves by an interface crack in magneto-electro-elastic composites

IntroductionCompositematerialconsistingofapiezoelectricphaseandapiezomagneticphasehasdrawnsignificantinterestinrecentyears,duetotherapiddevelopmentinadaptivematerialsystems .Itshowsaremarkablylargemagnetoelectriccoefficient,thecouplingcoefficientbetweenst…