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.     

Mixed mode crack initiation in piezoelectric ceramic strip

When piezoelectric ceramics are subjected to mechanical and electrical load, they can fracture prematurely due to their brittle behavior. Hence, it is important to know the electro–elastic interaction and fracture behavior of piezoelectric materials. The problem of a through crack in a piezoelectric strip of finite thickness is studied in this paper. Fourier transforms are used to reduce the problem to the solution of singular integral equations. The model technique can solve for polarization in an arbitrary direction and material anisotropy. Numerical values of the crack-tip field amplification for a piezoelectric strip under in-plane electromechanical loading are obtained. Energy density factor criterion is applied to obtain the maximum of the minimum energy density and direction of crack initiation. The influence of crack length and crack position on stress intensity and energy density factors is discussed.     

Time-harmonic P-waves engulfing a rectangular limited-permeable crack in piezoelectric medium: Energy density and energy release

The dynamic behavior of a limited-permeable rectangular crack in a transversely isotropic piezoelectric material is impinged by to a P-wave. The generalized Almansi theorem and the Schmidt method are used to determine the stress intensity factor and energy density factor as the primary fracture criterion of failure. The mixed boundary value problem entails the evaluation of the appropriate crack edge stress singularities that are characteristics of the fundamental functions. The stress and electric displacement intensity factors are also used to find the energy release rate that can be computed numerically and compared with the results corresponding to those of the stress intensity factor, and energy density factor. Graphical presentation shows that the energy release rate is always negative for the boundary conditions considered while the energy density factors always remain positive. Under certain conditions, the stress and electric displacement intensity factors can be negative and subject to physical limitations. Piezoelectric material boundary value problem solutions should therefore be qualified by the application of failure criteria by fracture of otherwise, particularly when the mechanical and electrical energy can release by creating free surface at the macroscopic and microscopic scales. Negative energy release rate found for the piezoelectric medium in this work can be a case in point.Positive definiteness of the energy density factor can be applied to mutliscale fracture. This is not true for the stress intensity factor nor the energy release rate. Hence, crack initiation behavior for the permittivity of a rectangular crack due to the wave propagation effects may be studied. In particular, the initiation of micro-cracks may be identified with certain critical stress wave frequency band. Negative stress intensity factor may not enhance macrocracking but it does not exclude microcrack initiation.     

Crack initiation behavior of functionally graded piezoelectric material: prediction by the strain energy density criterion

Transient response of a functionally graded piezoelectric medium is considered for a through crack under the mixed-mode in-plane mechanical and electric load. Integral transforms and dislocation density functions are employed to reduce the problem to singular integral equations. The energy density factor criterion is applied to obtain the maximum of the minimum energy density factor. This determines the direction of crack initiation. Numerical results display the effects of material constants, loading combination parameter, mechanical loading angle and material gradient parameter on the possible fracture behavior.     

Limited permeable crack moving along the interface of a piezoelectric bi-material

A plane problem for a crack moving with a subsonic speed along the interface of two piezoelectric semi-infinite spaces is considered. The crack is assumed to be free from mechanical loading. The limited permeable electric condition with an account of electric traction is adopted at its faces. A uniformly distributed mixed mode mechanical loading and an electric flux are prescribed at infinity. The problem is reduced to the Riemann–Hilbert problem by means of introducing a moving coordinate system and assuming that the electric flux is uniformly distributed along the crack region. An exact solution of this problem is proposed. It permits to find in closed form all necessary electromechanical characteristics at the interface and to formulate the equation for the determination of the electric flux. Analysis of this equation confirms the correctness of the assumption concerning the uniform distribution of the electric flux in the crack region. The values of the electric flux are determined by solving the obtained equation. Thereafter, the stress and electric intensity factors as well as their asymptotic fields at the crack tip are also found. The particular case of a crack moving in a homogeneous piezoelectric material is considered. The values of the electric flux and the fracture parameters are found exactly in a simple form for this case. Also, a numerical analysis is performed for a crack propagating with a subsonic speed between PZT4 and PZT5 materials and for a crack moving in PZT4 material. The electric flux in the crack region, stress and electric intensity factors, crack opening and the energy release rate (ERR) are found as functions of the crack speed, loading and electric permeability of the crack medium. The influence of the electric traction on the crack faces upon the mentioned parameters is demonstrated.     

Crack kinking in a piezoelectric solid

The intrinsic coupling between the mechanical and the electric fields assigns a uniquefeature for the fracture in a piezoelectric solid. We model the kink of a crack by continuousdistribution of edge dislocations and electric dipoles. The problem admits an approach based onthe Stroh formalism. A set of coupled singular integral equations are derived for the dislocationand electric dipole density functions associated with a kinked crack. Numerical results indicatethat the crack tends to propagate in a straight line under a tensile stress and a positive electricfield. For a crack subjected to the mixed mode mechanical loading, a superimposed positiveelectric field tends to reduce the kink angle. The influence of the non-singular T-stress-chargeparallel to a crack is also investigated. It is shown that a transverse tensile stress or a positivetransverse electric field will lead to further deviation of the kinked crack from the crackextension line.     

Crack energy density in arbitrary direction for piezoelectrics and its characteristic features in electromechanical mixed mode crack

The concepts of crack energy density (CED) and its derivatives in arbitrary direction were established for piezoelectric material and, keeping their application to mixed mode fracture in mind, the characteristic features of them as fracture parameters were investigated based on the approximate equations for CED and its derivatives. That is, CED and its derivatives in arbitrary direction are defined first and separation into their each mode contribution is made. Subsequently, path independent integral expressions of them are derived, and then using them, approximate equations of each mode contribution of CED are obtained concretely for the case where linear singular solution is known. The resulting equations are then used to investigate the effects of electric field and electrical boundary condition on CED and its derivatives. An infinite piezoelectric plane with a crack inclined with respect to the poling direction is considered as a numerical example. Mode I contribution of mechanical CED is mainly employed as a possible fracture parameter for the study and it was shown that applied electric field significantly influences on fracture parameters especially for the impermeable crack perpendicular to the poling direction. The effect of electric field has the tendency to decrease as crack inclination angle increases. It was also found that, even for the impermeable crack perpendicular to the poling direction, crack propagation could be deviated from self-similar direction under a strong negative electric field, and this fact is qualitatively consistent with an existing experimental observation. For the ideally sharp crack with no width, impermeable and Hao and Shen type boundary conditions are admissible showing qualitative agreement with experimental results, but exact boundary condition is not suitable and finally consistent with permeable boundary condition.     

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.     

PZT-4紧凑拉伸试样的断裂分析

基于线性压电材料的复势理论,通过解析分析,导出了一种分析有限压电板裂纹问题的解析数值方法. 首先,计算了含中心裂纹有限板的断裂参数,与Woo和Wang的解析数值法(Int J Fract, 1993, 62: 203$\sim$218)相比较,表明该方法具有很高的精度和很好的计算效率. 随后,采用该方法和有限元法计算了PZT-4紧凑拉伸试样在绝缘裂纹面边界条件下断裂时的断裂参数,发现各断裂参数的临界值分散性很大,不能作为压电材料的单参数断裂准则. 进而,针对试样真实的裂隙形状,采用有限元法计算了裂隙尖端的应力、电位移场,比较了裂隙内介质的介电性能对裂隙尖端场的影响,计算了带微裂纹的真实裂隙模型的断裂参数并进行了理论分析.      

Permeable cracks between two dissimilar piezoelectric materials

An interfacial crack with electrically permeable surfaces between two dissimilar piezoelectric ceramics under electromechanical loading is investigated. An exact expression for singular stress and electric fields near the tip of a permeable crack between two dissimilar anisotropic piezoelectric media are obtained. The interfacial crack-tip fields are shown to consist of both an inverse square root singularity and a pair of oscillatory singularities. It is found that the singular fields near the permeable interfacial crack tip are uniquely characterized by the real valued stress intensity factors proposed in this paper. The energy release rate is obtained in terms of the stress intensity factors. The exact solution of stress and electric fields for a finite interfacial crack problem is also derived.     

Damage analysis of tetragonal perovskite structure ceramics implicated by asymptotic field solutions and boundary conditions

Cracking of ceramics with tetragonal perovskite grain structure is known to appear at different sites and scale level. The multiscale character of damage depends on the combined effects of electromechanical coupling, prevailing physical parameters and boundary conditions. These detail features are exhibited by application of the energy density criterion with judicious use of the mode I asymptotic and full field solution in the range of r/a=10⁻⁴ to 10⁻² where r and a are, respectively, the distance to the crack tip and half crack length. Very close to the stationary crack tip, bifurcation is predicted resembling the dislocation emission behavior invoked in the molecular dynamics model. At the macroscopic scale, crack growth is predicted to occur straight ahead with two yield zones to the sides. A multiscale feature of crack tip damage is provided for the first time. Numerical values of the relative distances and bifurcation angles are reported for the PZT-4 ceramic subjected to different electric field to applied stress ratio and boundary conditions that consist of the specification of electric field/mechanical stress, electric displacement/mechanical strain, and mixed conditions. To be emphasized is that the multiscale character of damage in piezoceramics does not appear in general. It occurs only for specific combinations of the external and internal field parameters, elastic/piezoelectric/dielectric constants and specified boundary conditions.     

Electromechanical impact of a crack in a functionally graded piezoelectric medium

The Mode-I transient response of a functionally graded piezoelectric medium is solved for a through crack under the in-plane mechanical and electric impact. Integral transforms and dislocation density functions are employed to reduce the problem to singular integral equations. Numerical results display the effects of the loading combination parameter λ and the material parameter βa on the dynamic stress intensity factor and electric displacement intensity factor. The energy density factor criterion is applied to obtain the maximum of the minimum energy density factor and the direction of crack initiation.     

Study of a propagating finite crack in functionally graded piezoelectric materials considering dielectric medium effect

This paper provides a study of the problem of a propagating finite crack under in-plane loading in functionally graded piezoelectric materials (FGPMs). The analytical formulations are developed by Fourier transforms and the resulting singular integral equations are solved by using Chebyshev polynomials. By using a dielectric crack model with deformation-dependent electric boundary condition, numerical simulations are made to show the effects of the dielectric medium, the gradient of material properties and the speed of crack propagation on the fracture parameters, such as the stress, electric displacement and crack opening displacement intensity factors. A critical state for the electromechanical loading applied to the FGPMs is observed, which determines whether the traditionally impermeable (or permeable) crack model serves as the upper or lower bound for the dielectric model. The validity of this dielectric crack model is also examined by comparing the results of different existing crack models.     

Periodic set of limited electrically permeable interface cracks with contact zones

Plane problem for an infinite space composed of two different piezoelectric or piezoelectric/dielectric semi-infinite spaces with a periodic set of limited electrically permeable interface cracks is considered. Uniformly distributed electromechanical loading is applied at infinity. The frictionless contact zones at the crack tips are taken into account. The problem is reduced to the combined Dirichlet–Riemann boundary value problem by means of the electromechanical factors presentation via sectionally analytic functions, assuming that the electric flux is uniformly distributed inside the cracks. An exact solution of the problem is proposed. It permits to find in a closed form all necessary electromechanical characteristics at the interface and to formulate the equation for the determination of the electric flux value. Analysis of this equation confirms the correctness of the assumption concerning the uniform distribution of the electric flux in the crack region.Formulae for stresses, electric displacement vector, elastic displacements and electric potential jump at the interface as well as the intensity factors at the crack tips are given. Equation for the contact zone length determination is presented. Calculations for certain material combinations are carried out. The influence of electric permeability of cracks on electromechanical fields and the fracture mechanical parameters is analyzed.     

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.     

On the influence of the electric permeability on an interface crack in a piezoelectric bimaterial compound

An interface crack between two semi-infinite piezoelectric spaces under the action of remote mixed mode loading and electric flux is considered. The properties of the materials, loading and crack geometry admit to consider a two-dimensional problem in the plane perpendicular to the crack front. The crack is assumed to be free from mechanical loading and the limited permeable electric condition holds true. Assuming the electric flux is constant along the crack area, using the known presentations of all electromechanical fields via a piecewise holomorphic vector function, the problem is reduced to a vector Hilbert problem and solved in an analytical way. Clear analytical expressions for stresses and electric displacement as well as for stress and electric intensity factors are derived. As a particular case, a crack in a homogeneous piezoelectric material is considered and exact analytical formulae are presented for this case. The numerical analysis of the obtained formulae showed that for small values of the electric flux the model of a completely permeable crack can be used for any real crack permeability’s. The validity of such an approximation decreases with increase in the mechanical loading and especially of the electric flux.     

Pre-kinking of a moving crack in a magnetoelectroelastic material under in-plane loading

A constant moving crack in a magnetoelectroelastic material under in-plane mechanical, electric and magnetic loading is studied for impermeable crack surface boundary conditions. Fourier transform is employed to reduce the mixed boundary value problem of the crack to dual integral equations, which are solved exactly. Steady-state asymptotic fields near the crack tip are obtained in closed form and the corresponding field intensity factors are expressed explicitly. The crack speed influences the singular field distribution around the crack tip and the effects of electric and magnetic loading on the crack tip fields are discussed. The crack kinking phenomena is investigated using the maximum hoop stress intensity factor criterion. The magnitude of the maximum hoop stress intensity factor tends to increase as the crack speed increases.     

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.     

广义平均形状改变能密度断裂准则及应用

基于形状改变能为材料起裂扩展控制参量的物理事实及裂纹尖端断裂控制区能量平均概念,提出了计及裂纹尖端应力场特征级数展开奇异项和常数项的广义平均形状改变能密度(GADSED)准则,建立了Ⅰ-Ⅱ-Ⅲ混合型裂纹断裂判据计算式,为工程结构疲劳断裂评估提供了新选择.基于提出的GADSED准则,系统分析了T应力对裂纹断裂判据的影响,结果表明:当参数|Bα|数值相同时:负值T应力的Ⅰ-Ⅱ混合型裂纹更易起裂扩展,正值T应力的K_Ⅰf值高于负值T应力的K_Ⅰf值,正负T应力的K_Ⅱf值相同;T应力在区间0Ⅰf值(3%以内),T应力降低了K_Ⅱf值.根据GADSED准则完成了双轴疲劳载荷平板表面Ⅰ型裂纹扩展寿命预报,结果表明:基于GADSED准则的裂纹扩展寿命预报值约为传统方法预报值的40.7%,可为工程实际评估提供借鉴.     

Re-examination of the PKHS crack model in piezoelectric materials

In this paper, the PKHS crack model is re-examined in detail. Some novel features and better understandings are presented. It is found that the normal crack opening displacement jump (NCODJ) across the crack is always extremely small, whereas the electric potential drop (EPD) across the crack is very large for PZT-4 or PZT-5H under 20 MPa and a varying electric field from −0.5 MV/m to 0.5 MV/m. It is also found that the permittivities of the medium inside the crack gap influence the NCODJ and the EPD significantly. Of great significance is that the crack tip energy release rate (ERR) does not always show an even functional dependence on the applied electric field as previous researchers reported. Its variable tendencies (even or odd) depend on the mechanical loading levels, which are called as the load dependence feature here. In PZT-4, this load-dependence feature induced from the increasing mechanical loading may lead to a transformation of the crack tip ERR from an even functional dependence to an odd one when the applied electric loading varies within the range mentioned above, whereas in PZT-5H no such feature exists. This implies that different mechanical loading levels may yield different fracture behaviors when using the crack tip ERR as a fracture criterion. Moreover, the permittivities of medium inside the PKHS crack gap influence the variable tendencies of the ERR significantly. Thus, a useful addition to previous works is provided and a reasonable explanation for the fundamental discrepancy between previous theoretical predictions and experimental observations in piezoelectric fracture can be given. That is, previous experiments were performed for quite different piezoelectric materials, under quite different mechanical loading levels, and with quite different media inside crack gaps.