On the electrical boundary conditions on the crack surfaces in piezoelectric ceramics

This paper investigates a cracked piezoelectric ceramic under remote electro-mechanical loads. The ideal crack boundary conditions for electrically impermeable and permeable crack assumptions, and the deformed crack with a yet-to-be-determined crack shape are considered. The last is referred to as the “natural boundary condition (NBC)”. Closed-form solutions to the crack-tip field intensity factors are obtained. The analysis shows that traditional approaches to the electric boundary conditions on the crack faces, that is, either the impermeable crack assumption or the permeable crack assumption, produce significantly different results for the crack-tip quantities such as electric displacement intensity factor, energy release rate and crack opening displacement. There are also considerable differences between the results obtained from traditional impermeable and permeable crack analyses and those obtained from the proposed NBC. The difference increases with applied electric loads.     

Analysis of interlaminar stresses in general cross-ply laminates with distributed piezoelectric actuators

This paper represents an analytical solution to determine the interlaminar stresses of general cross-ply laminates with piezoelectric layers as actuators under transverse mechanical loads. The three-dimensional constitutive equations of piezoelectricity are considered and the governing equations are derived within the framework of second-order shear-deformation plate theory as a set of partial differential equations. These equations are solved for two kinds of boundary conditions and the unknown assumed functions of displacement field are found. Numerical results show that this approach can generally predict the behavior of interlaminar stresses. Also, they clearly indicate the singular behavior of interlaminar normal and shear stresses in the boundary region near the edges of the laminate.     

Linear electro-elastic fracture mechanics of piezoelectric materials

The concepts of linear elastic fracture mechanics, generalized to treat piezoelectric effects, are employed to study the influence of the electrical fields on the fracture behavior of piezoelectric materials. The method of distributed dislocations and electric dipoles, already existing in the literature, is used to calculate the electro-elastic fields and the energy-release rate for a finite crack embedded in an infinite piezoelectric medium which is subjected to both mechanical and electric loads. The energy-release rate expressions show that the electric fields generally tend to slow the crack growth. It is shown that the stress intensity factor criterion and the energy-release rate criterion differ when the energetics of the electric field is taken into account. The study of crack tip singular stress field yields a possible explanation for experimentally observed crack skewing in the presence of a strong electric field.     

The fracture behavior of functionally graded piezoelectric materials with dielectric cracks

This article provides a comprehensive investigation on the fracture behavior of cracked functionally graded piezoelectric materials (FGPMs). To account for the effect of dielectric medium inside the crack upon the fracture behavior of FGPMs, a dielectric crack model is used in this work, in which the electric boundary condition along crack surfaces is deformation-dependent and is nonlinear. The analytical formulations are developed using Fourier transform technique and solving the nonlinear singular equations using Chebyshev polynomials. A solution technique is developed to determine the desired deformation mode of the crack. Numerical simulations are given to show the effects of material gradient and the dielectric medium filling the crack upon the fracture behavior of FGPMs. The results obtained from this dielectric crack model clearly demonstrate how the transition between electrically impermeable and permeable crack models occurs with the change of crack opening displacement in response to the applied electromechanical loads. It is also observed that a critical state for the applied electromechanical loading exists for FGPMs that determines whether the impermeable (or permeable) crack model serves as the upper or lower bound for the dielectric crack model considering the effect of dielectric medium filling the crack.     

On the self-consistent, energetically consistent, and electrostatic traction approaches in piezoelectric fracture mechanics

In terms of crack opening, the present work studies the self-consistent, energetically consistent, and electrostatic traction approaches. In the self-consistent approach, crack will not open if no mechanical load is applied. The energetically consistent approach under a given electric field yields a threshold stress for crack opening and a bifurcation stress, which is higher than the threshold stress. Between the bifurcation and threshold points there are two solutions for crack opening. The electrostatic traction approach considers electrostatic tractions along crack faces and infinite boundaries. The electrostatic traction along infinite boundaries is equivalent to an additional mechanical load for a given electric field, which is tensile and promotes crack opening, if the dielectric constant of the infinite medium is smaller than that of the material. Energy release rate is also comprehensively analyzed in each of the three approaches.     

Crack repair performance of piezoelectric actuator estimated by slope continuity and fracture mechanics

The active repair performance of the piezoelectric actuator for the cracked structure is studied in this paper. As an extension of the author’s previous study, this paper discusses a special repair case and new results by using two repair criteria: the slope continuity criterion and fracture mechanics criterion. Crack contact analyses and fracture mechanics in the crack tip field are considered in the plane strain finite element analyses. From numerical results, it is concluded that the fracture mechanics criterion is better for defining the repair voltage. The crack tip field and crack contact condition must be considered to obtain correct results. However, it is not a good idea for the active repair to use higher applied voltage on the piezoelectric actuator.     

Towards a fracture mechanics for brittle piezoelectric and dielectric materials

A theoretical fracture mechanics for brittle piezoelectric and dielectric materials is developed consistent with standard features of elasticity and dielectricity. The influence of electric field and mechanical loading is considered in this approach and a Griffith style energy balance is used to establish the relevant energy release rates. Results are given for a finite crack in an infinite isotropic dielectric and for steady state cracking in a piezoelectric strip. In the latter problem, the effect of charge separation in the material and discharge in the crack are considered. Observations of crack behavior in piezoelectrics under combined mechanical and electrical load are discussed to assess which features of the theory are useful.     

Evaluation of fracture parameters in continuously nonhomogeneous piezoelectric solids

A contour integral method is developed for computation of stress intensity and electric intensity factors for cracks in continuously nonhomogeneous piezoelectric body under a transient dynamic load. It is shown that the asymptotic fields in the crack-tip vicinity in a continuously nonhomogeneos medium is the same as in a homogeneous one. A meshless method based on the local Petrov-Galerkin approach is applied for computation of physical fields occurring in the contour integral expressions of intensity factors. A unit step function is used as the test functions in the local weak-form. This leads to local integral equations (LBIEs) involving only contour-integrals on the surfaces of subdomains. The moving least-squares (MLS) method is adopted for approximating the physical quantities in the LBIEs. The accuracy of the present method for computing the stress intensity factors (SIF) and electrical displacement intensity factors (EDIF) are discussed by comparison with available analytical or numerical solutions.     

Discussion on ‘influence of applied electric field and mechanical boundary condition on the stress distribution at the crack tip in piezoelectric materials’

Using the finite element results, Kumar and Singh [3] have amazingly pointed out that the applied electric field can strongly influence the mechanical boundary condition. Due to the importance of the applied electric field, the applications of various kinds of electric boundary conditions have been discussed in order that one can correctly make use of both electric and mechanical boundary conditions. The influence of the applied electric field has also been proved on the basis of analytic method.     

Fracture mechanics of piezoelectric materials

This paper presents an analysis of crack problems in homogeneous piezoelectrics or on the interfaces between two dissimilar piezoelectric materials based on the continuity of normal electric displacement and electric potential across the crack faces. The explicit analytic solutions are obtained for a single crack in an infinite piezoelectric or on the interface of piezoelectric bimaterials. For homogeneous materials it is found that the normal electric displacement D₂, induced by the crack, is constant along the crack faces which depends only on the remote applied stress fields. Within the crack slit, the perturbed electric fields induced by the crack are also constant and not affected by the applied electric displacement fields. For bimaterials, generally speaking, an interface crack exhibits oscillatory behavior and the normal electric displacement D₂ is a complex function along the crack faces. However, for bimaterials, having certain symmetry, in which an interface crack displays no oscillatory behavior, it is observed that the normal electric displacement D₂ is also constant along the crack faces and the electric field E₂ has the singularity ahead of the crack tip and has a jump across the interface. Energy release rates are established for homogeneous materials and bimaterials having certain symmetry. Both the crack front parallel to the poling axis and perpendicular to the poling axis are discussed. It is revealed that the energy release rates are always positive for stable materials and the applied electric displacements have no contribution to the energy release rates. This revised version was published online in July 2006 with corrections to the Cover Date.     

Bound theorem and implementation of dual finite elements for fracture assessment of piezoelectric materials

Owing to the complexity of piezoelectric crack problems, derivation of closed-form solutions is virtually impossible and numerical solutions are largely resorted to. Hence, the upper and lower bound estimation for piezoelectric fracture parameters are of theoretical and practical values. An alternative assessment approach for electromechanical coupling cracked system is suggested in the paper. The dual path-independent integrals and the related bound theorems are presented. Consequently, the dual piezoelectric finite elements are formulated and implemented in bound analysis.     

Dynamic fracture mechanics study of an electrically impermeable mode III crack in a transversely isotropic piezoelectric material under pure electric load

The dynamic response of an electrically impermeable Mode III crack in a transversely isotropic piezoelectric material under pure electric load is investigated by treating the electric loading process as a transient impact load, which may be more appropriate to mimic the real service environment of piezoelectric materials. The stress intensity factor, the mechanical energy release rate, and the total energy release rate are derived and expressed as a function of time for a given applied electric load. The theoretical results indicate that a purely electric load can fracture the piezoelectric material if the stress intensity factor or the mechanical energy release rate is used as a failure criterion.     

Time-domain BEM for three-dimensional fracture mechanics

The present paper deals with time-domain analysis of three-dimensional transient dynamic crack problems. The time-domain formulation of the boundary element method for 3-D elastodynamic problems is used. Quarter-point and singular quarter-point elements represent displacements and tractions, respectively, near the crack front. Special attention is paid to integration and algorithms to preserve stability. Cracks in finite and unbounded regions under single and mixed mode dynamic loading conditions are studied. To the authors’ knowledge, no previous BE approach for 3-D elastodynamic crack problems based on the time-domain displacement representation exists.     

Exact and variational theorems for fracture mechanics of composites with residual stresses, traction-loaded cracks, and imperfect interfaces

By partitioning the total stresses in a damaged composite into either mechanical and residual stresses or into initial and perturbation stresses, it was possible to derive several exact results for the energy release rate due to crack growth. These general results automatically include the effects of residual stresses, traction-loaded cracks, and imperfect interfaces. The exact energy release rate results were expressed in terms of exact solutions to reduced composite stress analysis problems. By considering the common situation where the initial stresses are known exactly, but the perturbation stresses are only known approximately, it was possible to derive rigorous upper and lower bounds to the energy release rate for crack growth. Some of the new fracture mechanics equations were applied to crack closure calculations, to fiber fracture and interfacial debonding in the fragmentation test, and to microcracking in composite laminates.     

基于Maxwell的磁流变阻尼器的磁路有限元分析

系统地分析了磁流变阻尼器磁路的材料选择，设计计算方法，以及在设计时应该注意的因素。应用Maxwell电磁场分析软件，对所建立模型进行了仿真分析。分析发现，对活塞进行倒角处理后，倒角部分漏磁进一步增加，降低了磁路的利用率，但是在间隙内部磁场分布相对均匀，所以合理地选择阻尼通道的长度和间隙的大小是至关重要的。     

Interaction integrals for thermal fracture of functionally graded piezoelectric materials

This paper presents domain form of the interaction integrals based on three independent formulations for computation of the stress intensity factors and electric displacement intensity factor for cracks in functionally graded piezoelectric materials subjected to steady-state thermal loading. Each of the formulation differs in the way auxiliary fields are imposed in the evaluation of interaction integral and each of them results in a consistent form of the interaction integral in the sense that extra terms naturally appear in their derivation to compensate for the difference in the chosen crack tip asymptotic fields of homogeneous and functionally graded piezoelectric medium.     

Influence of mechanical stresses on partial discharge in a piezoelectric solid containing cavities

For a piezoelectric solid with an air-filled elliptic cavity, the two-dimensional problems of partial discharge inside the cavity are studied when the solid is subjected to combined mechanical stress and electric field. Based on the law of Paschen, the influence of mechanical stress on partial discharge is discussed. It is shown from the obtained results that the applied mechanical stress can retard or enhance the occurrence of partial discharge, which is dependent on the signs of applied electric fields. It is also found that for a mathematical crack (with zero initial width), partial discharge may not happen even when the significantly high mechanical and electric loadings are applied within the range of practical interest.     

On the effects of an electric field on the fracture toughness of poled piezoelectric ceramics

Central crack specimens have been used to study the effects of an applied electric field on the fracture toughness of poled soft lead zirconate titanate ceramics (PZT-5). The ultrasonic lapping technique was used to machine the central crack of the specimens used. The present experimental study illustrated that changing the field from negative to positive reduced the fracture toughness of a specimen subjected to an applied electric field. These experimental results were in agreement with the observations made by Park and Sun (J. Am. Ceram. Soc. 78 (1995) 1475) and Heyer et al. (Acta Mater. 46 (1998) 6615). Finite element results were employed to compare with the experimental data. Four fracture criteria, i.e. total energy release rate, mechanical energy release rate, local energy release rate and strain energy release rate, were compared with the experimental results. It was found the local energy release rate, which was defined based on a strip electric saturation model, was in broad agreement with the experimental observations. Moreover, it was found that the strain energy density criterion was unable to describe the effect of electric field on fracture toughness when a large electric field was applied.     

Electromechanical model of periodic cracks in piezoelectric materials

The electro-elastic problem for a periodic array of cracks in a piezoelectric medium subjected to coupled electro-mechanical loads is investigated. The mixed boundary value problem, which is formulated directly in terms of the crack surface displacements and electrical potentials, results in a system of hyper-singular integral equations in which the unknown functions are the crack surface displacement and electric potential. Numerical results include the crack surface displacement and the stress and electric intensity factors for the entire range of possible periodic crack spacing and medium size. The central contribution of this paper is the development of an analytical model that predicts crack-spacing effect. The resulting model is validated by a 2D finite element analysis.     

A finite element modelling for directly determining intensity factors of piezoelectric materials with cracks

Recently, authors(Cao et al., Acta Aeronautica et Astronautica Sinica 25(5): 470–472, 2004) extended the singular crack element originally introduced by Wang et al. (Eng Fract Mech 37(6):1195–1201, 1990) for evaluating the stress intensity factors (SIFs). Extensive studies have proved the versatility and accuracy of the element. This study is to show the versatility of the element for piezoelectric materials. In this paper, electric potential and displacement fields near a crack tip of piezoelectric materials are first used to construct a finite element version for directly determining intensity factors of piezoelectric materials with cracks. A singular finite element is constituted and a new method to calculate intensity factors of piezoelectric materials with cracks is obtained without any post-processing procedures. Detailed derivations are given and the results obtained with present method are good agreement with those of theoretical results, the FEM data by ANSYS and singular electromechanical crack tip elements. The results to the different selections of the structural dimensions are carried out. Numerical examples demonstrate the accuracy and validity of the novel element of present method.