Fracture Behavior Of Piezoelectric Materials Research Articles

Phase field fracture model of transversely isotropic piezoelectric materials with thermal effect

Predicting the failure of piezoelectric actuators and sensors is of engineering and scientific significance. In this work, a new phase field computational framework with thermal effect is developed to characterize the fracture behavior of piezoelectric materials. The present model involves the thermo-electro-elastic coupling effect to quantify the variations of the electrical field, temperature evolution and mechanical deformation during the fracture process. The positive part of mechanical energy is solely employed as the crack driving force in order to describe the unsymmetric electrical and mechanical fracture behaviors. A robust and stable staggered algorithm, which is adopted to address multi-physics problems, is developed. Numerical calculations are carried out to show the thermal effect and the influence of the external electrical field on the fracture behaviors of piezoelectric materials. The thermal loading may promote or delay the crack propagation, while its effect on the fracture load is weak. The direction and magnitude of the external electrical fields may significantly affect the fracture loads. The present study may benefit future design of piezoelectric devices in practical engineering.

A Review On Fracture Mechanics In Piezoelectric Structures

This review paper summarizes an overview of the theoretical, experimental and numerical state of the art on the fracture behavior of piezoelectric materials. It assesses the macro-mechanics fracture behavior of piezoelectric materials under combined mechanical and electrical loading environment. It also encompasses the current stage of computational development which has been achieved for piezoelectric modeling. An attempt is made to summarize fracture behavior under electrical, mechanical or combined electro-mechanical loading and various methods of solution for different crack problems. It also discusses various fracture controlling parameters like intensity factors KI–KIV from near tip fields, electro-mechanical J-integral and Energy Release Rate without including in-depth physics. The presented state of the art will give the current status of the structural based research in piezo-electric materials; this will encourage young researchers to understand the concepts and contribute their innovative ideas in this interesting field.

Nanoscale anti-plane cracking of materials with consideration of bulk and surface piezoelectricity effects

The influence of surface effect, including surface elasticity and surface piezoelectricity, on the fracture behavior of piezoelectricmaterials with an anti-plane crack is studied.Based on the coupled surface and interface elasticity model, the solutions to the problem are obtained by applying the singular integral method. By comparing the solutions influenced by the surface piezoelectricity with those affected by the surface elasticity, it is found that the influence of the surface piezoelectricity on the crack opening displacement, the crack electric potential jump across the crack center, the crack tip stress and electric displacement intensity factors cannot be ignored. Under various electrical boundary conditions, the influence of surface piezoelectricity on the sliding displacement, crack tip stress and electric displacement intensity factors exhibits the same tendency. Besides, the influence of surface piezoelectricity on the electric displacement intensity factor is independent of the electrical boundary conditions, which is different from the results where only the surface elasticity is considered.

Electro-elastic fields of piezoelectric materials with an elliptic hole under uniform internal shearing forces

The existing investigations on piezoelectric materials containing an elliptic hole mainly focus on remote uniform tensile loads. In order to have a better understanding of the fracture behavior of piezoelectric materials under different loading conditions, theoretical and numerical solutions are presented for an elliptic hole in transversely isotropic piezoelectric materials subjected to uniform internal shearing forces based on the complex potential approach. By solving ten variable linear equations, the analytical solutions inside and outside the hole satisfying the permeable electric boundary conditions are obtained. Taking PZT-4 ceramic into consideration, numerical results of electro-elastic fields along the edge of the hole and axes, and the electric displacements in the hole are presented. Comparison with stresses in transverse isotropic elastic materials shows that the hoop stress at the ends of major axis in two kinds of material equals zero for the various ratios of major to minor axis lengths; If the ratio is greater than 1, the hoop stress in piezoelectric materials is smaller than that in elastic materials, and if the ratio is smaller than 1, the hoop stress in piezoelectric materials is greater than that in elastic materials; When it is a circle hole, the shearing stress in two materials along axes is the same. The distribution of electric displacement components shows that the vertical electric displacement in the hole and along axes in the material is always zero though under the permeable electric boundary condition; The horizontal and vertical electric displacement components along the edge of the hole are symmetrical and antisymmetrical about horizontal axis, respectively. The stress and electric displacement distribution tends to zero at distances far from the elliptical hole, which conforms to the conclusion usually made on the basis of Saint-Venant’s principle. Unlike the existing work, uniform shearing forces acting on the edge of the hole, and the distribution of electro-elastic fields inside and outside the elliptic hole are considered.

Transient electromechanical cracking of finite piezoelectric bodies

This article studies the problem of a finite crack in a finite piezoelectric medium subjected to transient electric and mechanical loads. Unlike past studies that mainly focused on the infinitesimal cracks, the current article considers the effect of the medium boundaries on the transient fracture behavior of piezoelectric materials. A system of singular integral equations is formulated in terms of the crack surface displacement and electric potential. Effects of crack length and medium size on the crack-tip field intensity factors are investigated in detail. Numerical results for the time-dependent stress and electric displacement intensity factors are obtained. Electrically impermeable crack assumption, electrically permeable crack assumption, and notch of finite height are investigated. Some new conclusions are drawn.

Exact solutions for piezoelectric materials with an elliptic hole or a crack under uniform internal pressure

The existing investigations on piezoelectric materials containing an elliptic hole or a crack mainly focus on remote uniform tensile loads. In order to have a better understanding for the fracture behavior of piezoelectric materials under different loading conditions, theoretical and numerical solutions are presented for an elliptic hole or a crack in transversely isotropic piezoelectric materials subjected to uniform internal pressure and remote electro-mechanical loads. On the basis of the complex variable approach, analytical solutions of the elastic and electric fields inside and outside the defect are derived by satisfying permeable electric boundary condition at the surface of the elliptical hole. As an example of PZT-4 ceramics, numerical results of electro-elastic fields inside and outside the crack under various electric boundary conditions and electro-mechanical loads are given, and graphs of the electro-elastic fields in the vicinity of the crack tip are presented. The non-singular term is compared to the asymptotic one in the figures. It is shown that the dielectric constant of the air in the crack has no effect on the electric displacement component perpendicular to the crack, and the stresses in the piezoelectric material depend on the material properties and the mechanical loads on the crack surface and at infinity, but not on the electric loads at infinity. The figures obtained are strikingly similar to the available results. Unlike the existing work, the existence of electric fields inside an elliptic hole or a crack is considered, and the piezoelectric solid is subjected to complicated electro-mechanical loads.

A study of the effect of the electro-mechanical loading history on the fracture strength of piezoelectric material

Many studies to grasp and describe the fracture behavior of piezoelectric materials under electro-mechanical loading have been done. Although the crack energy density (CED) theory predicts that the mechanical and electrical CEDs can depend on the loading history, the effect of electro-mechanical loading history on the fracture strength of piezoelectric materials has not been studied. Therefore, in this paper, a fracture criterion based on the mechanical contribution of CED (CEDM) is introduced. Its applicability is studied by analyzing the results of three-point bending test regarding the loading path dependence of the fracture strength of piezoelectric ceramic. From the results of ( E → M) and ( M → E) tests for a C-21 piezoelectric ceramic specimen, it was found that the fracture behavior of piezoelectric ceramics depend on the loading history. The results further showed that the effect of the electric field on the fracture strength of piezoelectric ceramic in the ( M → E) test was larger than that in the ( E → M) test. Results from linear FE analyses, which assumed that the fracture-initiating load was the same, indicated that the CEDM values increased linearly from negative to positive, and the slopes of CEDM in descending order were: linear ( M → E) analysis > linear ( M, E) analysis > linear ( E → M) analysis.

Modeling the fracture behavior of piezoelectric materials using a gradual polarization switching model

Polarization switching experiments conducted on polycrystal piezoelectric materials indicate that the polarization switching is gradual (occurs over a range of applied electrical and mechanical loads). An empirical model that considers the gradual change in the polarization direction is proposed to capture the polarization switching phenomena and apply it to modeling the fracture behavior of piezoelectric materials. Changes in the dielectric constants (measured from a 90° polarization switching experiment) and the polarization direction are used to estimate the amount of domain switching for an applied electric field. To model polarization switching in a generalized state of stresses and electric fields, the applied electric field (during the experiment) is converted to the corresponding value of internal energy density causing polarization switching. Internal energy density is used as the parameter to estimate the amount of domain switching and the resulting gradual change in the polarization direction. The gradual polarization switching model computes changes in the material properties and spontaneous strain contributions (arising from crystal shape change in the switching domains) based on the current polarization direction. A finite element solution procedure that incorporates the gradual polarization switching model is developed and interfaced with ABAQUS. Stress–strain variations obtained from the compression experiments and FEA are compared for PZT-4 (Zr/Ti = 52/48) and PZT-5H (Zr/Ti = 53/47). The C(T) specimen that was used in Mode-I fracture experiments is analyzed with a slit crack assumption. Microscopic observations of the region near the crack tip of a C(T) fracture specimen is used to capture the actual crack geometry generated by the machining process. A finite element model with the actual crack geometry is used to investigate the influence of finite dimensions of the crack cavity on the size of the polarization switching zones in the vicinity of the crack tip and its effects on the opening stress and the stress intensity ahead of the crack tip.

Crack driving force and energy-momentum tensor in electroelastodynamic fracture

The energy flux integral and the energy-momentum tensor for studying the crack driving force in electroelastodynamic fracture are formulated within the framework of the nonlinear theory of coupled electric, thermal and mechanical fields based on fundamental principles of thermodynamics. This formulation lays a foundation for in-depth understanding of the fracture behavior of piezoelectric materials. Remarkably, the dynamic energy release rate thus obtained has an odd dependence on the electric displacement intensity factor for steady-state propagation of a conventional (unelectroded) crack with exact, electrically permeable, semi-permeable, or impermeable crack surface condition, which is in agreement with experimental evidence.

Thermal Effects of Cracked Piezoelectric Materials under Cycling Electric-Loading

It is still an open problem how the thermal effect influences the fracture behavior of piezoelectric materials especially under cycling electrical loading. Experimental observations have found that the fracture toughness of piezoelectric solids under electric loading may be greatly different from that under mechanical loading. A pronounced rise of temperature may be caused either by mechanical or by electric loading. In this paper, the thermal effects and energy dissipation mechanism in cracked piezoelectric materials under cyclic-electric-loading have been studied. The temperature rise is derived under the assumption of decoupling between thermal and electromechanical fields and the influences of frequency and the shape of electric wave on the temperature rise are quantitatively analyzed.

Dielectric breakdown model for a conductive crack and electrode in piezoelectric materials

The strip dielectric breakdown (DB) model introduced by Zhang and Gao [T.Y. Zhang, C.F. Gao, Fracture behavior of piezoelectric materials, Thero. Appl. Fract. Mech. 41 (2004) 339–379] is used to study the generalized 2D problem of a conductive crack and an electrode in an infinite piezoelectric material. The energy release rate and stress intensity factors are derived based on the Stroh formalism, and then they are applied as failure criteria to predict the critical fracture loads. It is found that the DB strip may take the shielding effect on a conductive crack or electrode. For the case of an electrode, the local energy release rate and stress intensity factor become zero when DB happens ahead of the electrode tip. For the case of a mode-I conductive crack in a transversely isotropic piezoelectric solid, the results based on the DB model show that the critical stress intensity factor linearly increases as the applied electric field parallel to the poling direction increases, while it linearly decreases as the applied electric field anti-parallel to the poling direction increases. Finally, the upper and lower bounds of the actual critical fracture loads are proposed for a conductive crack in a piezoelectric material under combined mechanical–electrical loads.

138 圧電材料の力学的CEDに及ぼす荷重履歴の影響(OS1-8 先端・機能材料の破壊現象(1),OS1 先端・機能材料のマルチスケール解析3)

Mechanical CED (Crack Energy Density) has been proposed as a parameter to describe fracture behavior of piezoelectric materials. In this paper, we show that it depends on the loading procedure, while the whole CED does not. Then, we evaluate fracture loads corresponding to experiments for PZT-4 by employing the fracture criterion based on mechanical CED and fundamentally study the effects of loading procedure on fracture load comparing the results with the experiment.

An analysis of the effect of domain switching on fracture behavior of piezoelectricsolids

Understanding the effect of an electric field on fracture behavior of piezoelectricmaterials is an important step towards a rational design of smart sensors andactuators. Domain switching has been observed to be the cause of significantnonlinearity in the response of piezoelectric materials subjected to mechanical andelectrical loads. While linear piezoelectricity is incapable of explaining the resultsof the experiments conducted on cracked piezoelectric specimens, a nonlinearanalysis considering domain switching is found to be necessary to describe some ofthe key observations of these experiments. In this paper, the response ofpiezoelectric solid is formulated by coupling thermal, electrical, and mechanicaleffects. The corresponding finite-element equations are derived and applied in thesolution of the piezoelectric centre crack problem, with different polingdirections, subjected to various mechanical and electrical loads. The effects ofdomain switching are evaluated on the near-tip stresses and the stressintensity factors. It is observed that a negative electric field increasesthe peak stress intensity factor while a positive electric field reduces it.

Fracture Mechanics of Piezoelectric Materials

Fracture Mechanics of Piezoelectric Materials

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.

The transient response of a piezoelectric strip with a vertical crack under electromechanical impact load

In this paper, the dynamic electromechanical response of a piezoelectric strip with a central crack vertical to the boundary was investigated. Based on the superposition principle and integral transform techniques, the present problem was reduced to the solution of two pairs of dual integral equations. To accommodate the finite size of the strip, two different Fourier transforms, about the x and y coordinates, were assumed. The solution was obtained in the Laplace transformed domain in terms of Fredholm integral equations of the second kind by a modified Copson’s method. The Laplace inversion was then carried out to obtain the resulting dynamic stress and electric displacement intensities. Unlike the findings observed in the static fracture behavior of piezoelectric materials, the present study indicates that the dynamic electric field will retard or promote the crack propagation at the different loading stages. Furthermore, the dynamic electric response is proportional to the electric impact and is independent of the applied mechanical impact. It is also shown that both the mechanical and electric response around the crack tip are greatly influenced by the free boundaries of the piezoelectric strip.

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.