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.     

Dielectric Breakdown Model for an Electrically Semi-Permeable Penny-Shaped Crack in Three-Dimensional Piezoelectric Media

The dielectric breakdown(DB) model for a penny-shaped crack under a semipermeable boundary condition in a three-dimensional piezoelectric medium is studied.An approximate analytical solution is derived by using the boundary integral equation with extended displacement discontinuity,and the corresponding boundary element method with double iterative approaches is developed to analyze the semi-permeable crack.The effect of electric boundary conditions on crack faces is discussed on the basis of DB model.By comparing the DB model with the polarization saturation(PS) model for different piezoelectric materials,some interesting phenomena related to the electric yielding zone and local J-integral are observed.     

A MIXED ELECTRIC BOUNDARY VALUE PROBLEM FOR AN ANTI－PLANE PIEZOELECTRIC CRACK

The analytical continuation method is adopted to solve a mixed electric boundary value problem for a piezoelectric medium under anti-plane deformation. The crack face is partly conductive and partly impermeable. The results show that the stress intensity factor is identical with the mode III stress intensity factor independent of the conducting length. But the electric field and the electric displacement are dependent on the electric boundary conditions on the crack faces and are singular not only at the crack tips but also at the junctures between the impermeable part and conducting portions.     

Effects of electric field on crack growth for a penny-shaped dielectric crack in a piezoelectric layer

The assumptions of impermeable and permeable cracks give rise to significant errors in determining electro-elastic behavior of a cracked piezoelectric material. The former simply imposes that the permittivity or electric displacement of the crack interior vanishes, and the latter neglects also the effects of the dielectric of an opening crack interior. Considering the presence of the dielectric of an opening crack interior and the permeability of the crack surfaces for electric field, this paper analyzes electro-elastic behavior induced by a penny-shaped dielectric crack in a piezoelectric ceramic layer. In the cases of prescribed displacement or prescribed stress at the layer surfaces, the Hankel transform technique is employed to reduce the problem to Fredholm integral equations with a parameter dependent nonlinearly on the unknown functions. For an infinite piezoelectric space, a closed-form solution can be derived explicitly, while for a piezoelectric layer, an iterative technique is suggested to solve the resulting nonlinear equations. Field intensity factors are obtained in terms of the solution of the equations. Numerical results of the crack opening displacement intensity factors are presented for a cracked PZT-5H layer and the effect of applied electric field on crack growth are examined for both cases. The results indicate that the fracture toughness of a piezoelectric ceramic is affected by the direction of applied electric fields, dependent on the elastic boundary conditions.     

A mixed electric boundary value problem for a two-dimensional piezoelectric crack

In this paper, a mixed electric boundary value problem for a two-dimensional piezoelectric crack problem is presented, in the sense that the crack face is partly conducting and partly impermeable. By the analytical continuation method, the unknown electric charge distributions on the upper and lower conducting crack faces are reduced to two decoupled singular integral equations and then these two equations are converted into algebraic equations to find the full field solution. Though the results suggest that the stress intensity factors at the crack tip are identical to those of conventional piezoelectric materials, but the electric field and electric displacement are related to the electric boundary conditions on the crack faces. The electric field and electric displacement are singular not only at crack tips but also at the junctures between the impermeable part and conducting parts. Numerical results for the variations of the electric field, electric displacement field and J-integral with respect to the normalized impermeable crack length are shown. Some discussions for the energy release rate and the J-integral are made.     

On the influence of electric boundary conditions on dynamic SIFs in piezoelectric materials

Dynamic stress intensity factors (SIFs) for a straight crack in a piezoelectric material under time-harmonic L- and SH-wave loading are determined for different electric boundary conditions. Impermeable, permeable and limited permeable cracks are compared. The problem is formulated and numerically solved using a nonhypersingular traction-based boundary integral equation method where the fundamental solution is obtained by Radon transform. A parametric study in the frequency domain shows the dependence of the SIFs on the choice of the electrical boundary conditions at the crack faces.     

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 effective electroelastic property of piezoelectric media with parallel dielectric cracks

Existing studies on the coupled electroelastic behaviour of cracked piezoelectric media have been based mostly on the electrically impermeable and permeable crack models. The current paper presents a study of the effective electroelastic property of piezoelectric media weakened by parallel cracks using a dielectric crack model with the electric boundary condition along the crack surfaces being governed by the opening displacement. The theoretical formulation is obtained using the dilute model of distributed cracks and the solution of a single dielectric crack problem. It is observed that the effective electroelastic property of cracked piezoelectric media is nonlinear and sensitive to loading conditions. Different modes of crack deformation are predicted and discussed. Attention is paid to the transition between electrically permeable and impermeable crack models.     

Numerical solution of polarization saturation/dielectric breakdown model in 2D finite piezoelectric media

The polarization saturation (PS) model [Gao, H., Barnett, D.M., 1996. An invariance property of local energy release rates in a strip saturation model of piezoelectric fracture. Int. J. Fract. 79, R25–R29; Gao, H., Zhang, T.Y., Tong, P., 1997. Local and global energy release rates for an electrically yielded crack in a piezoelectric ceramic. J. Mech. Phys. Solids 45, 491–510], and the dielectric breakdown (DB) model [Zhang, T.Y., Zhao, M.H., Cao, C.F., 2005. The strip dielectric breakdown model. Int. J. Fract. 132, 311–327] explain very well some experimental observations of fracture of piezoelectric ceramics. In this paper, the nonlinear hybrid extended displacement discontinuity-fundamental solution method (NLHEDD-FSM) is presented for numerical analysis of both the PS and DB models of two-dimensional (2D) finite piezoelectric media under impermeable and semi-permeable electric boundary conditions. In this NLHEDD-FSM, the solution is expressed approximately by a linear combination of fundamental solutions of the governing equations, which includes the extended point force fundamental solutions with sources placed at chosen points outside the domain of the problem under consideration, and the extended Crouch fundamental solutions with extended displacement discontinuities placed on the crack and the electric yielding zone. The coefficients of the fundamental solutions are determined by letting the approximated solution satisfy certain conditions on the boundary of the domain, on the crack face and the electric yielding zone. The zero electric displacement intensity factor in the PS model or the zero electric field strength intensity factor in the DB model at the outer tips of the electric yielding zone is used as a supplementary condition to determine the size of the electric yielding zone. Iteration approaches are adopted in the NLHEDD-FSM. The electric yielding zone is determined, and the extended intensity factors and the local J-integral are calculated for center cracks in piezoelectric strips. The effects of finite domain size, saturation property and different electric boundary conditions, as well as different models on the electric yielding zone and the local J-integral, are studied.     

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.     

热压电介质中的渗透型周期裂纹问题

利用Stroh公式,研究了含共线周期裂纹热的压电介质的广义二维问题。该工作有两个特征：一是裂纹被建模为具有渗透表面的缝隙,并假设为跨越上下表面时,电场的切向分量和电位移的法向分量是连续的;另一个特征是,机－电载荷和热载荷被假设作用在无限远处,而不是在裂纹表面。基于这两个假设,我们获得了有关场强因子,以及裂纹内电场的相当简洁的表达式。结果表明：①在裂纹内电场是线性变化的,②电位移的奇异性总是取决于应力的奇异性.③所有场的奇异性与所加的电载荷无关。     

On the dynamic behaviour of a piezoelectric laminate with multiple interfacial collinear cracks

This paper investigates the dynamic behaviour of a piezoelectric laminate containing multiple interfacial collinear cracks subjected to steady-state electro-mechanical loads. Both the permeable and impermeable boundary conditions are examined and discussed. Based on the use of integral transform techniques, the problem is reduced to a set of singular integral equations, which can be solved using Chebyshev polynomial expansions. Numerical results are provided to show the effect of the geometry of interacting collinear cracks, the applied electric fields, the electric boundary conditions along the crack faces and the loading frequency on the resulting dynamic stress intensity and electric displacement intensity factors.     

Nonlinear behaviour of interacting dielectric cracks in piezoelectric materials

Existing studies on the fracture of cracked piezoelectric materials have been limited mostly to the electrically impermeable and permeable crack models, which represent the limiting cases of the physical boundary condition along the crack surfaces. This paper presents a study on the electromechanical behaviour of interacting dielectric cracks in piezoelectric materials. The cracks are filled with dielectric media and, as the result, the electric boundary condition along the crack surfaces is governed by the opening displacement of the cracks. The formulation of this nonlinear problem is based on simulating the cracks using distributed dislocations and solving the resulting nonlinear singular integral equations. Multiple deformation modes are observed. A solution technique is developed to determine the desired deformation mode of the interacting cracks. Numerical results are given to show the effect of the interaction between parallel cracks. Attention is paid to the transition between permeable and impermeable models with increasing crack opening.     

Analysis of multiple axisymmetric annular cracks in a piezoelectric medium

An axisymmetric annular electric dislocation is defined. The solution of axisymmetric electric and Volterra climb and glide dislocations in an infinite transversely isotropic piezoelectric domain is obtained by means of Hankel transforms. The distributed dislocation technique is used to construct integral equations for a system of co-axial annular cracks with so-called permeable and impermeable electric boundary conditions on the crack faces where the domain is under axisymmetric electromechanical loading. These equations are solved numerically to obtain dislocation densities on the crack surfaces. The dislocation densities are employed to determine field intensity factors for a system of interacting annular and/or penny-shaped cracks.     

Transient response of a piezoelectric ceramic with two coplanar cracks under electromechanical impact

The transient response of two coplanar cracks in a piezoelectric ceramic under antiplane mechanical and inplane electric impacting loads is investigated in the present paper. Laplace and Fourier transforms are used to reduce the mixed boundary value problems to Cauchy-type singular integral equations in Laplace transform domain, which are solved numerically. The dynamic stress and electric displacement factors are obtained as the functions of time and geometry parameters. The present study shows that the presence of the dynamic electric field will impede or enhance the propagation of the crack in piezoelectric ceramics at different stages of the dynamic electromechanical load. Moreover, the electromechanical response is greatly affected by the ratio of the space of the cracks and the crack length.     

Finite element analyses of cracks in piezoelectric structures: a survey

Piezoelectric materials have widespread applications in modern technical areas such as mechatronics, smart structures or microsystem technology, where they serve as sensors or actuators. For the assessment of strength and reliability of piezoelectric structures under combined electrical and mechanical loading, the existence of cracklike defects plays an important role. Meanwhile, piezoelectric fracture mechanics has been established quite well, but its application to realistic crack configurations and loading situations in piezoelectric structures requires the use of numerical techniques as finite element methods (FEM) or boundary element methods (BEM). The aim of this paper is to review the state of the art of FEM to compute the coupled electromechanical boundary value problem of cracks in 2D and 3D piezoelectric structures under static and dynamic loading. In order to calculate the relevant fracture parameters very precisely and efficiently, the numerical treatment must account for the singularity of the mechanical and electrical fields at crack tips. The following specialized techniques are presented in detail (1) special singular crack tip elements, (2) determination of intensity factors K _I –K _IV from near tip fields, (3) modified crack closure integral, (4) computation of the electromechanical J-integral, and (5) exploitation of interaction integrals. Special emphasis is devoted to a realistic modeling of the dielectric medium inside the crack, leading to specific electric crack face boundary conditions. The accuracy, efficiency, and applicability of these techniques are examined by various example problems and discussed with respect to their advantages and drawbacks for practical applications.     

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.     

Strip-electromechanical model solution for piezoelectric plate cut along two semi-permeable collinear cracks

A strip electric saturation and mechanical yielding model solution is proposed for a piezoelectric plate cut along two equal collinear semi-permeable mode-I cracks with electrical polarization reaching a saturation limit and normal stress reaching a yield stress along a line segment in front of the cracks. By using Stroh formalism and complex variable technique, we derived the analytical solution for the field quantities. Three different situations are investigated when developed electrical saturation zone is bigger/smaller or equal to the developed mechanical yield zone. Numerical results show that the effect of different electric boundary conditions on the crack opening displacement and crack opening potential drop is significant and should not be ignored. The influence of electric load displacement on the energy release rate is also investigated for PZT-4, PZT-5H and BaTiO₃ ceramics, and it may assists for the correct choosing of ceramic for specific job.     

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.     

An exact solution of crack problems in piezoelectric materials

IntroductionInrecentyearscrackproblemsinpiezoelectricmaterialhavereceivedmuchattention.Manytheoreticalanalyseshavebeengivenby[1～16].Itshouldbe,however,notedthatalltheaboveanalysesarebasedonaso-calledimpermeablecrackassumphon,i.e.thecrackfacesareassumedtobeimpermeabletoelectricfield,sotheelectricdisplacementvanishesinsidethecrack.Usingthisassumption,onewillobtainthefollowingresultS[2'3'5,6'9'16]=whentheelectricloadsaresolelyaPPliedatinLfinity,theelectricdisplacementissquare-rootsingularatthe…