Electromechanical deformation and fracture of piezoelectric/ferroelectric materials

This review presents the progress and current status of the investigation on electromechamical deformation and fracture of piezoelectric/ferroelectric materials. An attempt is made to summarize a few fundamental aspects, which include electromechanical constitutive relations, piezoelectric micromechanics and electric fracture and fatigue, instead of describing all technological backgrounds, basic physics, experimental findings, and theoretical developments. A number of open questions and future prospective are presented. It is hoped that this review will encourage people to joint the exploration of this important and interesting field. The project supported by the National Natural Science Foundation of China (100025209)     

Fracture analysis of circular-arc interface cracks in piezoelectric materials

The anti-plane problem of N arc-shaped interfacial cracks between a circular piezoelectric inhomogeneity and an infinite piezoelectric matrix is investigated by means of the complex variable method. Cracks are assumed to be permeable and then explicit expressions are presented, respectively, for the electric field on the crack faces, the complex potentials in media and the intensity factors near the crack-tips. As examples, the corresponding solutions are obtained for a piezoelectric bimaterial system with one or two permeable arc-shaped interfacial cracks, respectively. Additionally, the solutions for the cases of impermeable cracks also are given by treating an impermeable crack as a particular case of a permeable crack. It is shown that for the case of permeable interfacial cracks, the electric field is jumpy ahead of the crack tips, and its intensity factor is always dependent on that of stress. Moreover all the field singularities are dependent not only on the applied mechanical load, but also on the applied electric load. However, for the case of a homogeneous material with permeable cracks, all the singular factors are related only to the applied stresses and material constants.     

Interacting multiple cracks in piezoelectric materials

In this paper, a method is presented to calculate the plane electro–elastic fields in piezoelectric materials with multiple cracks. The cracks may be distributed randomly in locations, orientations and sizes. In the method, each crack is treated as a continuous distributed dislocations with the density function to be determined according to the conditions of external loads and crack surfaces. Some numerical examples are given to show the interacting effect among multiple cracks.     

Electrically driven cracks in piezoelectric ceramics: experiments and fracture mechanics analysis

Piezoelectric systems like multilayer actuators are susceptible to damage by crack propagation induced by strain incompatibilities. These can arise under electric fields for example between the electroded and external regions. Such incompatibilities have been realised in thin rectangular model specimens from PZT-piezoelectric ceramics with top and bottom electrodes only close to one edge. Under an electric field, controlled crack propagation has been observed in situ in an optical microscope. The crack paths are reproducible with very high accuracy. Small electrode widths lead to straight cracks with two transitions between stable and unstable crack growth regions, while large electrode widths result in curved cracks with four transitions. Fracture mechanics analysis is able to explain the different crack paths. An iteration method is developed to simulate the curved crack propagation also for strong curvature of the crack paths using the finite element method. The computed crack contours exhibit excellent quantitative agreement with the experiment with respect to their shape, the stages of stable and unstable crack propagation and the transitions between them. Finally, also the crack length as a function of the electric field can be predicted.     

Coupled solution for anisotropic piezoelectric materials with cracks

The coupled elastic and electric fields for anisotropic piezoelectric materials with electrically permeable cracks are analyzed by using Stroh formula in anisotropic elasticity. It is shown from the solution that the tangent component of the electric field strength and the normal component of the electric displacement along the faces of cracks are all constants, and the electric field intensity and electric displacement have the singularity of type (1/2) at the crack tip. The energy release rate for crack propagation depends on both the stress intensity factor and material constants. The electric field intensity and electric displacement inside electrically permeable cracks are all constants.     

Interaction integrals for fracture analysis 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. Conservation integrals of J-type are derived based on the governing equations for piezoelectric media and the crack tip asymptotic fields of homogeneous piezoelectric medium as auxiliary fields. 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 appears in their derivation to compensate for the difference in the chosen crack tip asymptotic fields of homogeneous and functionally graded piezoelectric medium. The additional terms play an important role of ensuring domain independence of the presented interaction integrals. Comparison of the numerically evaluated intensity factors through the three consistent formulations with those obtained using displacement extrapolation method is presented by means of two examples.     

Out-of-plane interface cracks in dissimilar piezoelectric materials

Summary This paper investigates the problem of an anti-plane interfacial crack between two dissimilar piezoelectric material layers. A single crack is first considered. The effect of interaction of two collinear cracks in the medium on the field intensity factors is investigated. The solutions of several particular cases, including an infinite piezoelectric bi-material and a piezoelectric material bonded to an elastic medium, are given. The bi-material constants governing the behavior of the crack tip fields are identified. By considering the crack as a notch of finite thickness, it is shown that the thickness of the notch has a pronounced influence on the crack tip field. The results for the assumption of a permeable crack represent the limit case where the notch thickness is reduced to zero.BLW would like to thank the National Science Foundation of China (#10102004) and the City University of Hong Kong (DAG #7100219) for the support of this work. YGS also thanks the Multidiscipline Scientific Research Foundation Project (HIT. MD 2001. 39) of the Harbin Institute of Technology and the SRF for ROCS, SEM.accepted for publication 3 April 2003     

A domain-independent interaction integral for fracture analysis of nonhomogeneous piezoelectric materials

Piezoelectric materials and structures contain more or less electromechanical interfaces in engineering applications. It is difficult to obtain the fracture parameters efficiently of the piezoelectric materials with complex interfaces. This paper presents a domain-independent interaction integral for material nonhomogeneity and discontinuity which can be used for solving the stress intensity factors (SIFs) and the electric displacement intensity factor (EDIF) of piezoelectric materials with complex interfaces efficiently. The interaction integral is based on the J-integral by superimposition of two admissible states and the present formulation does not involve any derivatives of mechanical and electric properties. Moreover, it is proved that the interface in the integral domain does not affect the value of the interaction integral and thus, the present method does not require electromechanical parameters of piezoelectric materials to be continuous. The interaction integral method combined with the extended finite element method (XFEM) is used to investigate the influences of material continuity on the SIF and the EDIF and the results show that the material parameters and their first-order derivatives affect both the SIF and the EDIF greatly, while the higher-order derivatives affect both of them slightly.     

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.     

Anomalies associated with energy release parameters for cracks in piezoelectric materials

For a central crack in a piezoelectric plate, the mode-I stress intensity factor (K_I), electric displacement intensity factor (K_D), energy release rates (G, G_M) and energy density factor (S) are obtained from the finite element results. For the impermeable crack, the numerical results of K_I and K_D are coupled; this error is contrary to the uncoupled analytical solutions. The error has little effect on the total energy release rate G and energy density factor S, but in some cases, large errors in the mechanical energy release rate G_M are observed. G is global while SED is local. Also G is negative which defies physics where energy cannot be created while crack attempts to extend as implied by G. Computations should be made for the J-integral and also show that J becomes negative. What this shows is that the global fracture energy criterion is not suitable to address the local release of energy because it includes the overall energy which are irrelevant to fracture initiation being a local behavior. In addition, the case study shows that the energy density theory is the better fracture criterion for the piezoelectric material. According to the results of S, it retards the crack growth when the external electric field and piezoelectric poling are on opposite directions. This conclusion agrees with analytical and experimental evidence in the past references.     

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.     

Damage analysis and fracture criteria for piezoelectric ceramics

Both the mechanical and the electrical damages are introduced to study fracture mechanics of piezoelectric ceramics in this paper. Two kinds of piezoelectric fracture criteria are proposed by using the damage theory combined with the well-known piezoelectric fracture experiments of Park and Sun [Fracture criteria of piezoelectric ceramics, J. Am. Ceram. Soc. 78 (1995) 1475-1480]. One is based on a critical state of the mechanical damage and the other on a critical value of a proper linear combination of both the mechanical and the electrical damage variables. It is found that the fracture load predicted, which takes the mechanical damage into account only (mode 1), has greater deviation than predicted result by considering a proper linear combination of the mechanical and the electrical damages (mode 2). And the fracture criterion corresponding to mode 2 presented is shown to be superior to mode 1. It is also demonstrated that the mechanical damage has greater effect on fracture than the electrical damage.     

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.     

THE BEHAVIOR OF TWO PARALLEL SYMMETRIC PERMEABLE CRACKS IN PIEZOELECTRIC MATERIALS

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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.     

Interfacial cracks between piezoelectric and elastic materials under in-plane electric loading

A new experimental technique for accelerated fatigue crack growth tests was recently developed (Du et al., 2001). The technique, which uses piezoelectric actuators, enables application of cyclic loading at frequencies several orders higher than that by mechanical loading. However, the validity of this technique relies on the equivalence between piezoelectric and mechanical loading. In this paper, the behavior of an interfacial crack between a piezoelectric material and an elastic material under in-plane electric loading is studied. The displacement mismatch along a bonded interface due to electric potential loading on the piezoelectric material is modeled by inserting an array of uniformly distributed dislocations along the interface. By means of Fourier transformation methods, the governing equations are converted to an integral equation, which is then converted to a standard Hilbert problem. A closed form solution for stresses, electric field, and electric displacements along the bonded interface is obtained. The results agree very well with those obtained from numerical simulations. The results show that the closed form solution is accurate not only for far field distributions of stresses and electric variables, but also for the asymptotic distributions near the crack tip. The solution also suggests the likelihood of domain switching in the piezoelectric material near the crack tip, a process that may influence the interfacial fracture resistance.     

Thermal fracture analysis of nonhomogeneous piezoelectric materials using an interaction energy integral method

This paper presents a modified interaction energy integral method to analyze the thermal stress intensity factors (TSIFs) and electric displacement intensity factor (EDIF) in nonhomogeneous piezoelectric materials under thermal loading. This modified method is demonstrated to be domain-independent, even when the nonhomogeneous piezoelectric materials contain interfaces with thermo-electro-mechanical properties. As a result, the method is shown to be convenient for determining the TSIFs and EDIF in nonhomogeneous piezoelectric materials with interfaces. Several examples are shown, and they successfully verify the domain-independence of the present interaction energy integral. The study results also show that the mismatch of material properties can significantly influence the TSIFs and EDIF, particularly when the crack tip is close to the interface. Crack angles and temperature boundary conditions are also shown to significantly influence the TSIFs and EDIF.     

2D dynamic analysis of cracks and interface cracks in piezoelectric composites using the SBFEM

The dynamic stress and electric displacement intensity factors of impermeable cracks in homogeneous piezoelectric materials and interface cracks in piezoelectric bimaterials are evaluated by extending the scaled boundary finite element method (SBFEM). In this method, a piezoelectric plate is divided into polygons. Each polygon is treated as a scaled boundary finite element subdomain. Only the boundaries of the subdomains need to be discretized with line elements. The dynamic properties of a subdomain are represented by the high order stiffness and mass matrices obtained from a continued fraction solution, which is able to represent the high frequency response with only 3–4 terms per wavelength. The semi-analytical solutions model singular stress and electric displacement fields in the vicinity of crack tips accurately and efficiently. The dynamic stress and electric displacement intensity factors are evaluated directly from the scaled boundary finite element solutions. No asymptotic solution, local mesh refinement or other special treatments around a crack tip are required. Numerical examples are presented to verify the proposed technique with the analytical solutions and the results from the literature. The present results highlight the accuracy, simplicity and efficiency of the proposed technique.