Fracture of a rectangular piezoelectromagnetic body

The singular stress, electric fields and magnetic fields in a rectangular piezoelectromagnetic body containing a center Griffith crack under longitudinal shear are obtained. Fourier transforms and Fourier sine series are used to reduce the mixed boundary value problems of the crack, which is assumed to be impermeable, to dual integral equations. The solution of the dual integral equations is then expressed in terms of Fredholm integral equations of the second kind. Expressions for stresses, electric displacements and magnetic inductions in the vicinity of the crack tip are derived. Also obtained are the field intensity factors and the energy release rates. Numerical results obtained show that the geometry of the rectangular body have significant influence on the field intensity factors and the energy release rates.     

矩形压电体中反平面裂纹的电弹性场

基于线性压电理论,本文获得了含有中心反平面裂纹的矩形压电体中的奇异应力和电场。利用Fourier积分变换和Fourier正弦级数将电绝缘型裂纹问题化为对偶积分方程,并进一步归结为易于求解的第二类Fred-holm积分方程。获得了裂纹尖端应力、应变、电位移和电场的解析解,求得了裂纹尖端场的强度因子及能量释放率。分析了压电矩形体的几何尺寸对它们的影响。结果表明,对于电绝缘型裂纹,裂纹尖端附近的各个场变量都具有-1/2阶的奇异性,能量释放率与电荷载的方向及大小有关,并且有可能为负值。     

Impact response of an interface crack in a hybrid piezoelectric laminate

Summary  In a hybrid laminate containing an interfacial crack between piezoelectric and orthotropic layers, the dynamic field intensity factors and energy release rates are obtained for electro-mechanical impact loading. The analysis is performed within the framework of linear piezoelectricity. By using integral transform techniques, the problem is reduced to the solution of a Fredholm integral equation of the second kind, which is obtained from one pair of dual integral equations. Numerical results for the dynamic stress intensity factor show the influence of the geometry and electric field. Received 29 June 2001; accepted for publication 3 December 2001     

Transient response of a piezoelectric ceramic strip with an eccentric crack under electromechanical impacts

The transient response of a piezoelectric strip with an eccentric crack normal to the strip boundaries under applied electromechanical impacts is considered. By using the Laplace transform, the mixed initial-boundary-value problem is reduced to triple series equations, then to a singular integral equation of the first kind by introducing an auxiliary function. The Lobatto–Chebyshev collocation technique is adopted to solve numerically the resulting singular integral equation. Dynamic field intensity factors and energy release rate are obtained for both a permeable crack and an impermeable crack. The effects of the crack position and the material properties on the dynamic stress intensity factor are examined and numerical results are presented graphically.     

Dynamic fracture behaviors of cracks in a functionally graded magneto-electro-elastic plate

In this paper the dynamic anti-plane problem for a functionally graded magneto-electro-elastic plate containing an internal or an edge crack parallel to the graded direction is investigated. The crack is assumed to be magneto-electrically impermeable. Integral transforms and dislocation density functions are employed to reduce the problem to Cauchy singular integral equations. Field intensity factors and energy release rate are derived, analyzed and partially calculated numerically. The effects of material graded index, loading combination parameter (including size and direction) and geometry criterion of the plate on the dynamic energy release rate are shown graphically. Numerical results indicate that increasing the graded index can all retard the crack extension, and that both the applied magnetic field loadings and electric field loadings play a dominant role in the dynamic fracture behaviors of crack tips.     

Electroelastic Analysis of an Internal Interface Crack in a Half-Plane Consisting of Two Bonded Dissimilar Piezoelectric Quarter-Planes

The problem of an interface crack in a half-plane consisting of two bonded dissimilar piezoelectric quarters is considered under antiplane shear and inplane electric loading. The problem is solved under the electrically permeable assumption for a crack. The integral transform technique is employed to reduce the problem to triple integral equations, which is further converted to a hypersingular integral equation for the crack sliding displacement. By solving the resulting equation analytically, the electroelastic field along the interface and the energy release rate are obtained in explicit form, respectively. Several examples are given to illustrate the influence of the material properties and the crack position on the energy release rate.     

Anti-plane analysis on a finite crack in a one-dimensional hexagonal quasicrystal strip

The anti-plane fracture problem for a finite crack in a one-dimensional hexagonal quasicrystal strip is analyzed. By using Fourier transforms, the mixed boundary value problems are reduced to the dual integral equations. The solution of the dual integral equations is then expressed by the complete elliptic integrals of the first and the third kinds. The expressions for stress, strains, displacements and field intensity factors of the phonon and phason fields near the crack tip are obtained exactly. The path-independent integral derived by a conservation law equals the energy release rate, which can be used as a fracture criterion for a mode III fracture problem.     

Electroelastic field for an impermeable anti-plane shear crack in a piezoelectric ceramics plate

Electroelastic behavior of a cracked piezoelectric ceramics plate subjected to four cases of combined mechanical-electrical loads is analyzed. The integral transform method is applied to convert the problem involving an impermeable anti-plane crack to dual integral equations. Solving the resulting equations, the explicit analytic expressions for electroelastic field along the crack line and the intensity factors of relevant quantities near the crack tip and the mechanical strain energy release rate are obtained. The known results for an infinite piezoelectric ceramics plane containing an impermeable anti-plane crack are recovered from the present results only if the thickness of the plate h → ∞. Biography: LI Xian-fang (1964-)     

Dynamic crack propagation in a magneto-electro-elastic solid subjected to mixed loads: Transient Mode-III problem

The transient response of a Mode-III crack propagating in a magneto-electro-elastic solid subjected to mixed loads is investigated through solving the corresponding boundary-initial-value problem in both the cracked solid region and the interior fluid region with treatment of electro-magnetically permeable and impermeable crack face conditions in a unified way. The closed-form results for the dynamic field intensity factors are used to evaluate the dynamic energy release rate through the crack-tip dynamic contour integral. The permeability of the interior fluid region relative to the cracked solid region significantly affects the magneto-electro-mechanical coupling coefficient in the Bleustein–Gulyaev wave function and, consequently, the horizontal shear surface wave speed, the dynamic field intensity factors and the dynamic energy release rate. It is revealed from dynamic fracture mechanics analysis that the dynamic energy release rate thus obtained has an odd dependence on the dynamic electric displacement intensity factor and the dynamic magnetic induction intensity factor. It is also found that the horizontal shear surface wave speed provides the limiting velocity for the propagation of a Mode-III crack in a magneto-electro-elastic solid when there is only applied traction loading.     

Fracture behavior of a bonded magneto-electro-elastic rectangular plate with an interface crack

In this paper the anti-plane problem for an interface crack between two dissimilar magneto-electro-elastic plates subjected to anti-plane mechanical and in-plane magneto-electrical loads is investigated. The interface crack is assumed to be either magneto-electrically impermeable or permeable, and the position of the interface crack is arbitrary. The finite Fourier transform method is employed to reduce the mixed boundary-value problem to triple trigonometric series equations. The dislocation density functions and proper replacement of the variables are introduced to reduce these series equations to a standard Cauchy singular integral equation of the first kind. The resulting integral equation together with the corresponding single-valued condition is approximated as a system of linear algebra equations which can be easily solved. Field intensity factors and energy release rates are determined numerically and discussed in detail. Numerical results show the effects of crack configuration and loading combination parameters on the fracture behaviors of crack tips according to energy release rate criterion. The study of this problem is expected to have applications to the investigation of dynamic fracture properties of magneto-electro-elastic materials with cracks.     

折线型裂纹对SH波的动力响应

利用Fourier积分变换方法,得出了无限平面中用裂纹位错密度函数表示的单裂纹散射场．根据无穷积分的性质,把单裂纹的散射场分解为奇异部分和有界部分．利用单裂纹的散射场建立了折线裂纹在SH波作用下的Cauchy型奇异积分方程．根据折线裂纹散射场和所得的积分方程讨论了裂纹在折点处的奇性应力及折点处的奇性应力指数．利用所得的奇性应力定义了折点处的应力强度因子．对所得Cauchy型奇积分方程的数值求解,可得裂纹端点和折点处的动应力强度因子。     

Crack kinking in piezoelectric materials

A solution is presented for a class of two-dimensional electroelastic branched crack problems. Explicit Green's function for an interface crack subject to an edge dislocation is developed using the extended Stroh formulation allowing the branched crack problem to be expressed in terms of coupled singular integral equations. The integral equations are obtained by the method that models a kink as a continuous distribution of edge dislocations, and the dislocation density function is defined on the line of the branch crack only. Competition between crack extension along the interface and kinking into the substrate is investigated using the integral equations and the maximum energy release rate criterion. Numerical results are presented to show the effect of electric field on the path of crack extension. The work was supported by the Australian Research Council through a Queen Elizabeth II fellowship and by the Australian Academy of Science through the J.G. Russell Award.     

ELECTROELASTIC FIELD FOR AN IMPERMEABLE ANTI-PLANE SHEAR CRACK IN A PIEZOELECTRIC CERAMICS PLATE

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Dynamic fracture analysis of an annular interfacial crack between dissimilar magnetoelectroelastic layers

Transient response of an annular interfacial crack between dissimilar magnetoelectroelastic layers under impacts is investigated. On the crack surface, magnetoelectrically impermeable boundary condition is adopted. Using Laplace and Hankel transform techniques, the mixed boundary value problem is reduced to a system of singular integral equations. The integral equations are further reduced to a system of algebraic equations with the aid of Jacobi polynomials. The dynamic field intensity factor and dynamic energy release rate are determined. Numerical results reveal the effects of electric or magnetic loadings and material parameters of composite on crack propagation and growth.     

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.     

Moving eccentric crack in a piezoelectric strip bonded to elastic materials

Summary  The steady-state of a propagation eccentric crack in a piezoelectric ceramic strip bonded between two elastic materials under combined anti-plane mechanical shear and in-plane electrical loadings is considered in this paper. The analysis based on the integral transform approach is conducted on the permeable crack condition. Field intensity factors and energy release rate are obtained in terms of a Fredholm integral equation of the second kind. It is shown for this geometry that the crack propagation speed has influence on the dynamic energy release rate. The initial crack branching angle for a PZT-5H piezoceramic structure is predicted by the maximum energy release rate criterion. Received 23 January 2001; accepted for publication 18 October 2001     

Dynamic analysis of a penny-shaped dielectric crack in a magnetoelectroelastic solid under impacts

The transient response of a magneto-electro-elastic material with a penny-shaped dielectric crack subjected to in-plane magneto-electro-mechanical impacts is made. To simulate an opening crack with a dielectric interior, the crack-face electromagnetic boundary conditions are supposed to depend on the crack opening displacement and the jumps of electric and magnetic potentials across the crack. Four ideal crack-face electromagnetic boundary conditions involving a combination of electrically permeable or impermeable and magnetically permeable or impermeable assumptions can be reduced. The Laplace and Hankel transform techniques are further utilized to solve the mixed initial-boundary-value problem. Three coupling Fredholm integral equations are obtained and solved by the composite Simpson's rule. Dynamic field intensity factors of stress, electric displacement, magnetic induction, crack opening displacement (COD), electric potential and magnetic potential are given in the Laplace transform domain. By means of a numerical inversion of the Laplace transform, numerical results are calculated to show the variations of the physical parameters of concern versus the normalized time in graphics. The effects of applied electric and magnetic loads on the dynamic intensity factors of stress and COD, and the dynamic energy release rate for a BaTiO₃-CoFe₂O₄ composite with a penny-shaped vacuum crack are discussed in detail.     

FRACTURE ANALYSIS OF AN ANNULAR CRACK IN A PIEZOELECTRIC LAYER

A flat annular crack in a piezoelectric layer subjected to electroelastic loadings is investigated under electrically impermeable boundary condition on the crack surface. Using Hankel transform technique, the mixed boundary value problem is reduced to a system of singular integral equations. With the aid of Gauss-Chebyshev integration technique, the integral equations are further reduced to a system of algebraic equations. The field intensity factor and energy release rate are determined. Numerical results reveal the effects of electric loadings and crack configuration on crack propagation and growth. The results seem useful for design of the piezoelectric structures and devices of high performance.     

A dielectric crack in a functionally graded piezoelectric layer

Considering the dielectric effects inside a crack, the problem of an electrically dielectric crack in a functionally graded piezoelectric layer is addressed in this paper. The energetically consistent crack-face boundary conditions are utilized to analyze the effects of a dielectric of crack interior. Applying the Fourier transform technique, the boundary-value problem is reduced to solving three coupling singular equations. Then a system of non-linear algebraic equations is obtained and the field intensity factors along with the energy release rate are given. Numerical results show the differences of the electric displacement inside a crack, the stress and electric displacement intensity factors and the energy release rate using the permeable, impermeable, semi-permeable and energetically consistent boundary conditions respectively. The effects of the material non-homogeneity, the applied electric field and the discharge field of crack interior on the electrostatic traction acting on the crack faces and the energy release rate are further studied through the energetically consistent boundary conditions.     

Electroelastic analysis of a piezoelectric cylindrical fiber with a penny-shaped crack embedded in a matrix

The electroelastic response of a penny-shaped crack in a piezoelectric cylindrical fiber embedded in an elastic matrix is investigated in this study. Fourier and Hankel transforms are used to reduce the problem to the solution of a pair of dual integral equations. They are then reduced to a Fredholm integral equation of the second kind. Numerical values on the stress intensity factor, energy release rate and energy density factor for piezoelectric composites are obtained to show the influence of applied electric fields.