DYNAMIC BEHAVIOR OF TWO COLLINEAR PERMEABLE CRACKS IN A PIEZOELECTRIC LAYER BONDED TO TWO HALF SPACES

IntroductionIn the fracture mechanics studies for piezoelectric materials,differently electricboundary conditions at the crack surfaces have been proposed by many researchers.Forexample,for the sake of analytical simplification,the assumption that the cra…     

Two parallel limited-permeable mode-I cracks or four parallel limited-permeable mode-I cracks in the piezoelectric materials

In this paper, the basic solutions of two parallel mode-I cracks or four parallel mode-I cracks in the piezoelectric materials were investigated by means of the Schmidt method for the limited-permeable electric boundary conditions. The electric permittivity of air in the crack was considered. Through the Fourier transform, the problems can be solved with the help of two pairs of dual integral equations, in which the unknown variables were the jumps of the displacements across the crack surfaces, not the dislocation density functions. To solve the dual integral equations, the jumps of the displacements across the crack surfaces were directly expanded in a series of Jacobi polynomials. Finally, the effects of the distance between two parallel cracks, the distance between two collinear cracks and the electric boundary conditions on the stress and the electric intensity factors in the piezoelectric materials are analyzed. These results can be used for the strength evaluation of the piezoelectric materials with multi-cracks. The crack shielding effect is also present in the piezoelectric materials.     

Transient response of a magnetoelectroelastic solid with two collinear dielectric cracks under impacts

Dynamic analysis of two collinear electro-magnetically dielectric cracks in a piezoelectromagnetic material is made under in-plane magneto-electro-mechanical impacts. Generalized semi-permeable crack-face boundary conditions are proposed to simulate realistic opening cracks with dielectric. Ideal boundary conditions of a combination of electrically permeable or impermeable and magnetically permeable or impermeable assumptions are several limiting cases of the semi-permeable dielectric crack. Utilizing the Laplace and Fourier transforms, the mixed initial-boundary-value problem is reduced to solving singular integral equations with Cauchy kernel. Dynamic intensity factors of stress, electric displacement, magnetic induction and crack opening displacement (COD) near the inner and outer crack tips are determined in the Laplace transform domain. Numerical results for a special magnetoelectroelastic solid are calculated to show the influences of the dielectric permittivity and magnetic permeability inside the cracks on the crack-face electric displacement and magnetic induction. By means of a numerical inversion of the Laplace transform, the variations of the normalized intensity factors of stress and COD are discussed against applied magnetoelectric impact loadings and the geometry of the cracks for fully impermeable, vacuum, fully permeable cracks and shown in graphics.     

Interaction of elastic waves with a periodic array of collinear inplane cracks

A boundary integral equation method is applied to the study of the interaction of plane elastic waves with a periodic array of collinear inplane cracks. Numerical results are presented for the dynamic stress intensity factors. The effects of the wave type, wave frequency, wave incidence angle, and crack spacing on the dynamic stress intensity factors are analyzed in detail. The project supported by the Committee of Science and Technology of Shanghai and Tongji University     

Closed-form solutions for two anti-plane collinear cracks in a magnetoelectroelastic layer

In this paper we develop closed form solutions for anti-plane mechanical and in-plane electric and magnetic fields for two collinear cracks in magneto-electro-elastic layer of finite thickness under the conditions of permeable crack faces using integral transform method. The anti-plane mechanical shear or displacement and in-plane electrical and magnetic loading are applied to the top and bottom surfaces of the layer for the two cases considered. Expressions for shear stresses, electric displacements and magnetic inductions in the vicinity of the cracks are derived as well as intensity factors for two cracks in magneto-electro-elastic layer. Numerical results for stress intensity factors and energy release rate are shown graphically.     

Closed-form solution for two collinear cracks in a piezoelectric strip

Under the permeable electric boundary condition, the problem of two collinear anti-plane shear cracks lying at the mid-plane of a piezoelectric strip is investigated. By using the Fourier transform, the associated problem is reduced to a singular integral equation. Solving the resulting equation analytically, the electro-elastic field intensity factors and energy release rates at the inner and outer crack tips can be determined in explicit form. Numerical results for PZT-5H piezoelectric ceramic are also presented graphically. The results reveal that the effect of electric field on crack growth in piezoelectric materials is dependent on applied elastic displacement.     

Closed form solution for two unequal collinear semi-permeable straight cracks in a piezoelectric media

The problem of two unequal collinear straight cracks weakening a poled transversely isotropic piezoelectric ceramic is addressed under semi-permeable electric boundary conditions on the crack faces. The plate has been subjected to combined in-plane normal(to the faces of the cracks) mechanical and electric loads. Problem is formulated employing Stroh formalism and solved using complex variable technique. The elastic field, electric field and energy release rate are obtained in closed analytic form. A case study is presented for poled PZT-5H cracked plate to study the effect of prescribed mechanical load, electric load, inter-crack distance and crack lengths on crack arrest parameters stress intensity factor (SIF), electric displacement intensity factor (EDIF) and mechanical and total energy release rates (ERR). Moreover a comparative study is done of impermeable and semi-permeable crack face boundary conditions on SIF, EDIF and ERR, and results obtained is presented graphically. It is observed that the effect of dielectric medium in the crack gap cannot be ignored.     

Generalized 2D problem of piezoelectric media containing collinear cracks

The generalized 2D problem in piezoelectric media with collinear cracks is addressed based on Stroh's formulation and the exact electric boundary conditions on the crack faces. Exact solutions are obtained, respectively, for two special cases: one is that a piezoelectric solid withN collinear cracks is subjected to uniform loads at infinity, and the other is that a piezoelectric solid containing a single crack is subjected to a line load at an arbitrary point. It is shown when uniform loads are applied at infinity or on the crack faces that, the stress intensity factors are the same as those of isotropic materials, while the intensity factor of electric displacement is dependent on the material constants and the applied mechanical loads, but not on the applied electric loads. Moreover, it is found that the electric field inside any crack is not equal to zero, which is related to the material properties and applied mechanical-electric loads. The project supported by the National Natural Science Foundation of China (19772004)     

The dynamic behavior of two collinear interface cracks in magneto-electro-elastic materials

In this paper, the dynamic behavior of two collinear symmetric interface cracks between two dissimilar magneto-electro-elastic material half planes under the harmonic anti-plane shear waves loading is investigated by Schmidt method. By using the Fourier transform, the problem can be solved with a set of triple integral equations in which the unknown variable is the jump of the displacements across the crack surfaces. To solve the triple integral equations, the jump of the displacements across the crack surface is expanded in a series of Jacobi polynomials. Numerical solutions of the stress intensity factor, the electric displacement intensity factor and the magnetic flux intensity factor are given. The relations among the electric filed, the magnetic flux field and the stress field are obtained.     

SCATTERING OF THE HARMONIC STRESS WAVE BY CRACKS IN FUNCTIONALLY GRADED PIEZOELECTRIC MATERIALS

The present paper considers the scattering of the time harmonic stress wave by a single crack and two collinear cracks in functionally graded piezoelectric material （FGPM）. It is assumed that the properties of the FGPM vary continuously as an exponential function. By using the Fourier transform and defining the jumps of displacements and electric potential components across the crack surface as the unknown functions, two pairs of dual integral equations are derived. To solve the dual integral equations, the jumps of the displacement and electric potential components across the crack surface are expanded in a series of Jacobi polynomials. Numerical examples are provided to show the influences of material properties on the dynamic stress and the electric displacement intensity factors.     

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．     

THE BEHAVIOR OF TWO COLLINEAR CRACKS IN MAGNETO-ELECTRO-ELASTIC COMPOSITES UNDER ANTI-PLANE SHEAR STRESS LOADING

In this paper, the behavior of two collinear cracks in magneto-electro-elastic composite material under anti-plane shear stress loading is studied by the Schmidt method for permeable electric boundary conditions. By using the Fourier transform, the problem can be solved with a set of triple integral equations in which the unknown variable is the jump of displacements across the crack surfaces. In solving the triple integral equations, the unknown variable is expanded in a series of Jacobi polynomials. Numerical solutions are obtained. It is shown that the stress field is independent of the electric field and the magnetic flux.     

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.     

TRANSIENT RESPONSE OF COPLANAR INTERFACIAL CRACKS BETWEEN TWO DISSIMILAR PIEZOELECTRIC STRIPS UNDER ANTI-PLANE MECHANICAL AND IN-PLANE ELECTRICAL IMPACTS

The dynamic response of multiple coplanar interface cracks between two dissimilar piezoelectric strips subjected to mechanical and electrical impacts is investigated. Solutions to two kinds of electric boundary conditions on crack surfaces, i.e. electric impermeable and electric permeable, are obtained. Laplace and Fourier transforms and dislocation density functions are employed to reduce the mixed boundary value problem to Cauchy singular integral equations,which can be solved numerically. The effects of electrical load, geometry criterion of piezoelectric strips, relative location of cracks and material properties on the dynamic energy release rate are examined.     

Dynamic behavior of unequal parallel permeable interface multi-cracks in a piezoelectric layer bonded to two piezoelectric materials half planes

This study is concerned with the treatment of the dynamic behavior of interacting cracks in a piezoelectric layer bonded to two dissimilar piezoelectric half planes subjected to harmonic anti-plane shear waves. The permeable electric boundary condition is considered. By use of the Fourier transform technique, the problem can be solved with the help of two pairs of dual integral equations in which the unknown variables are the jumps of the displacements across the crack surfaces. To solve the dual integral equations, the jumps of the displacements across the crack surfaces are expanded in two series of Jacobi polynomials. The electromechanical behavior of two pairs of unequal parallel cracks was determined. Numerical examples are provided to show the effects of the geometry of the cracks, the frequency of the incident waves and materials properties upon the dynamic stress intensity factors (DSIFs) and the electric displacement intensity factors.     

Interfacial coplanar cracks in piezoelectric bi-material systems under pure mechanical impact loading

In this paper, we examine the dynamic behaviour of different piezoelectric bi-material combinations containing two interfacial cracks subjected to mechanical impact loading. The problem is formulated in terms of integral transforms techniques and the collocation method to obtain the solution for the resulting singular integral equation in the transformed plane. Laplace inversion was then used to obtain the resulting dynamic stress intensity factors in the physical plane. Numerical examples are provided for five different types of piezoelectric bi-material systems to illustrate the effect of the presence of collinear interacting cracks and the different material combinations upon the resulting dynamic stress intensity factors.     

Closed-form solution for a piezoelectric strip with two collinear cracks normal to the strip boundaries

The problem of two collinear cracks of equal length and normal to the strip boundaries in an infinitely long piezoelectric strip of finite width is analyzed. By using the Fourier series method, the mixed boundary value problem is reduced to triple series equations, which are then transformed to a singular integral equation. For four combined cases of uniform antiplane shear and uniform inplane electric loading at infinity, the solution is obtained in closed-form, and explicit expressions for the electroelastic field are determined. The formulae for calculating the intensity factors of the electroelastic field and the energy release rate at the inner and outer crack tips are given, respectively. Some special cases for the electroelastic field intensity factors and the energy release rate of the present results are discussed.     

The stress intensity factors near the tips of two collinear cracks in composite under in-plane shear

In this paper, the stress-intensity factors for two collinear cracks in a composite bonded by an isotropic and an anisotropic half-plane were calculated. The cracks are paralell to the interface, and the crack surfaces are loaded by uniform shear stresses. By using Fourier transform, the mixed boundary value problem is reduced to a set of singular integral equations. For solving the integral equations, the crack surface displacements are expanded in triangular series and the unknown coefficients in the series are determined by the Schmidt method. The stress intensity factors for the cracks in the boron-fibre plastics and aluminium joined composite and in carbon-fibre reinforced plastics were calculated numerically.     

Basic solution of two parallel non-symmetric permeable cracks in piezoelectric materials

The behavior of two parallel non-symmetric cracks in piezoelectric materials subjected to the anti-plane shear loading was studied by the Schmidt method for the permeable crack electric boundary conditions. Through the Fourier transform, the present problem can be solved with two pairs of dual integral equations ip which the unknown variables are the jumps of displacements across crack surfaces. To solve the dual integral equations, the jumps of displacements across crack surfaces were directly expanded in a series of Jacobi polynomials. Finally, the relations between electric displacement intensity factors and stress intensity factors at crack tips can be obtained. Numerical examples are provided to show the effect of the distance between two cracks upon stress and electric displacement intensity factors at crack tips. Contrary to the impermeable crack surface condition solution, it is found that electric displacement intensity factors for the permeable crack surface conditions are much smaller than those for the impermeable crack surface conditions. At the same time, it can be found that the crack shielding effect is also present in the piezoelectric materials.     

Antiplane mechanical and inplane electric time-dependent load applied to two coplanar cracks in piezoelectric ceramic material

This work is concerned with the dynamic response of two coplanar cracks in a piezoelectric ceramic under antiplane mechanical and inplane electric time-dependent load. The cracks are assumed to act either as an insulator or as a conductor. Laplace and Fourier transforms are used to reduce the mixed boundary value problems to Cauchy-type singular integral equations in Laplace transform domain. A numerical Laplace inversion algorithm is used to determine the dynamic stress and electric displacement factors that depend on time and geometry. A normalized equivalent parameter describing the ratio of the equivalent magnitude of electric load to that of mechanical load is introduced in the numerical computation of the dynamic stress intensity factor (DSIF) which has a similar trend as that for the pure elastic material. The results show that the dynamic electric field will impede or enhance crack propagation in a piezoelectric ceramic material at different stages of the dynamic electromechanical load. Moreover, the electromechanical response is greatly affected by the ratio of the crack length to the ligament between the cracks. The stress and electric displacement intensity factor can be combined by the energy density factor or function to address the fracture of piezoelectric materials under the combined influence of electromechanical loading.