Impermeable Crack Model Research Articles

Theoretical Assessment of Cracking in Orthotropic Material under Symmetrical Heat Flow/Mechanical Loading

This paper studies a cracked orthotropic solid under symmetric heat flow, and symmetric mechanical loading is studied. A modified partially impermeable crack model is employed to simulate thermal load transfer. With the application of the Fourier transform technique and superposition theory, the related physical quantities and fracture parameters are obtained in explicit forms. The advantage of this paper is that the obtained solutions are explicitly closed. As a result, the calculation of the stress intensity factors of various cracks with different lengths becomes very convenient and fast. Some simple examples are used to demonstrate the method provided in this paper. The analysis results show the dimensionless thermal resistance (ϖc) between the upper and lower crack regions and the proposed coefficient (ε) greatly influence the related physical quantities and fracture parameters. In addition, the numerical analysis results also revealed that the calculated results of fracture parameters at the crack tip will not be physically meaningful unless certain conditions are met.

A thermal-medium crack model

In analyzing the fracture behavior of a cracked thermoelastic material, of much importance are the effects of thermal loadings on the crack growth. Under the consideration of a medium in an opening crack, a thermal-medium crack model is proposed in this paper. The heat flux at the crack surfaces is assumed to depend on the jumps of the temperature and the elastic displacement across the crack. The thermally permeable and impermeable crack models are the limiting cases of a thermal-medium one. The proposed crack model is applied to solve the problem of a Griffith crack in a transversely isotropic material under thermal and mechanical loadings. Using two introduced displacement functions and the Fourier transform technique, the thermoelastic field and the elastic T-stress are determined in explicit forms by using elementary functions. Numerical results are presented to show the effects of the thermal conductivity inside a crack and applied mechanical loadings on the heat flux at the crack faces, the jumps of temperature across the crack and mode-II stress intensity factor in graphics respectively. The obtained results reveal that the mode-II stress intensity factor for a thermal-medium crack in a thermoelastic material depends not only on applied thermal loadings but also on applied mechanical ones.

Transient response of two collinear dielectric cracks in a piezoelectric solid under inplane impacts

The paper is focused on the dynamic analysis of two collinear dielectric cracks in a piezoelectric material under the action of in-plane electromechanical impacts. Considering the dielectric permeability of crack interior, the electric displacements at the crack surfaces are governed by the jumps of electric potential and crack opening displacement across the cracks. The permeable and impermeable crack models are the limiting cases of the general one. The Laplace and Fourier transform techniques are further utilized to solve the mixed initial-boundary-value problem, and then to obtain the singular integral equations with Cauchy kernel, which are solved numerically. Dynamic intensity factors of stress, electric displacement and crack opening displacement are determined in time domain by means of a numerical inversion of the Laplace transform. Numerical results for PZT-5H are calculated to show the effects of the dielectric permeability inside the cracks, applied electric loadings and the geometry of the cracks on the fracture parameters in graphics. The observations reveal that based on the COD intensity factor, a positive electric field enhances the dynamic dielectric crack growth and a negative one impedes the dynamic dielectric crack growth in a piezoelectric solid.

Investigation of the critical states of dielectric cracks in functionally graded piezoelectric materials

A critical state for electromechanical loads that determines when the traditional impermeable (or permeable) crack model serves as the upper or the lower bound of the dielectric crack model, first proposed for homogeneous piezoelectric materials, is studied further for functionally graded piezoelectric materials (FGPMs) in the current work. The analytical formulations of a single crack and two interacting cracks in the FGPMs are derived by using Fourier transforms, and the resulting integral equations are solved with Chebyshev polynomials. Numerical simulations are conducted to show the effect of crack length, positions of two interacting cracks and material gradient of FGPMs at this critical state. Interesting results show that the combination of the material gradient and the crack length α a plays an important role in determining this critical state. Our solutions also reveal there may exist several critical states for two interacting cracks in the FGPMs.

Dynamic fatigue of cracked piezoelectric ceramics under electromechanical loading: three-point bending test and finite element analysis

This paper studies the dynamic fatigue or slow crack growth in piezoelectric ceramics under electromechanical loading by a combined numerical-experimental approach. Constant load-rate testing was conducted in three-point flexure using the single-edge precracked-beam specimens under zero and positive electric fields, and the effects of electric field and loading-rate on the fracture load and crack propagation were examined. A finite element analysis was also employed to calculate the energy release rate for the permeable, impermeable and open crack models, and the effect of electric field on the energy release rate was discussed. Crack propagation velocity versus energy release rate curves at various loading-rate were then estimated based on the finite element analysis using measured data. Piezoelectric ceramics of the lead zirconate titanate (PZT) class have been used for a number of years as sensors and actuators. The high mechanical stresses and intense electric fields in PZT ceramics can induce cracking that can lead to premature failure of the piezoelectric devices. The properties of PZT ceramics are also susceptible to degradation under electromechanical loading. Therefore, an understanding of piezoelectric fracture [Shindo et al. 2003; 2005] and fatigue [Cao and Evans 1994; Lynch et al. 1995] is a key issue for the efficient and reliable design of the piezoelectric devices. Shindo et al. [2007] conducted an experimental and analytical study of the static fatigue behavior of PZT ceramics under electromechanical loading. Narita et al. [2007] also reported experimental and numerical examination of the fatigue crack growth in PZT ceramics under a cyclic mechanical load and a constant electric field. Certain environments may affect formation and extension of cracks over time and at stress levels well below that which causes immediate failure to occur. This process is called dynamic fatigue or slow crack growth. Slow crack growth parameters have been estimated for only a few commercial ceramics and glass, and no one has investigated the resistance of piezoelectric ceramics to slow crack growth and the influence of electric field on the dynamic fatigue behavior. In this study, we report numerical and experimental examination of the dynamic fatigue or slow crack growth in piezoelectric ceramics under electromechanical loading. A crack was created normal to the poling direction. Constant load-rate testing was conducted in three-point flexure under zero and positive electric fields using single-edge precracked-beam method. A finite element analysis was also used to evaluate the energy release rate for the permeable, impermeable and open crack models, and the effect

The fracture behaviour of functionally graded piezoelectric materials containing collinear dielectric cracks

In this paper, the problem of two collinear cracks in functionally graded piezoelectric materials (FGPMs) under in-plane electromechanical loads is examined. The elastic, piezoelectric and dielectric constants of the FGPMs are assumed to vary continuously in space. The theoretical formulations are derived by using Fourier transforms and the resulting singular integral equations are solved with Chebyshev polynomials. A dielectric crack model with deformation-dependent electric boundary condition is adopted in the fracture analysis of FGPMs. Numerical simulations are made to show the effect of the dielectric medium, the material gradient and the geometry of interacting cracks upon the fracture parameters at crack tips. A critical state for applied electromechanical loading is identified, which determines whether the traditionally impermeable (or permeable) crack model serves as the upper or lower bound of the current dielectric crack model.

Effect of the electrical boundary condition at the crack face on the mode I energy release rate in piezoelectric ceramics

This paper discusses the crack behavior of rectangular piezoelectric ceramics under electromechanical loading. A finite element analysis was employed to calculate the fracture mechanics parameters such as mode I energy release rate for the permeable, impermeable, and open crack models. Recently proposed nonlinear electrically discharging crack model was also considered. The effect of applied electric fields on the fracture mechanics parameters was then examined under various electrical boundary conditions at the crack face.

Investigation of the crack problem in non-local piezoelectric materials under combined electromechanical loadings

The present investigation of the crack problem in piezoelectric materials is performed based on the non-local theory. After some manipulations, the impermeable crack, the permeable crack (the crack gap is full of NaCI solution), and the semi-permeable crack (the crack gap is full of air or silicon oil) are reduced to a uniform formulation by assuming the normal electric displacement on the crack surfaces to be an unknown variable. Thus, a triple integral equation with the unknown normal electric displacement is established. By using the Newton iterative method and solving the triple integral equation, it is found that the normal electric displacement on the crack surfaces is no longer a constant as determined by previous studies, rather, it depends upon the remote combined electromechanical loadings. Numerical results of the stresses and electric displacement fields show that there are no singularities at the crack tips so that the stresses remain finite. It is of great significance that the concrete electric boundary condition on the crack surfaces exerts significant influence on the near-tip fields and in this way plays an important role in evaluating the crack stability in the non-local piezoelectric materials. More specifically, the impermeable crack model always overestimates the finite stresses at the crack tips, whereas the permeable crack model always underestimates them.

On a moving dielectric crack in a piezoelectric interface with spatially varying properties

This paper provides a comprehensive theoretical analysis of a finite crack propagating with constant speed along an interface between two dissimilar piezoelectric media under inplane electromechanical loading. The interface is modeled as a graded piezoelectric layer with spatially varying properties (functionally graded piezoelectric materials, i.e., FGPMs). The analytical formulations are developed using Fourier transforms and the resulting singular integral equations are solved with Chebyshev polynomials. Using a dielectric crack model with deformation-dependent electric boundary condition, the dynamic stress intensity factors, electric displacement intensity factor, crack opening displacement ( COD) intensity factor, and energy release rate are derived to fully understand this inherent mixed mode dynamic fracture problem. Numerical simulations are made to show the effects of the material mismatch, the thickness of the interfacial layer, the crack position, and the crack speed upon the dynamic fracture behavior. A critical state for the electromechanical loading applied to the medium is identified, which determines whether the traditional impermeable (or permeable) crack model serves as the upper or lower bound for the dielectric model considering the effect of dielectric medium crack filling.

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.

Solution for a Semi-Permeable Interface Crack in Elastic Dielectric/Piezoelectric Bimaterials

A semi-permeable interface crack in infinite elastic dielectric/piezoelectric bimaterials under combined electric and mechanical loading is studied by using the Stroh complex variable theory. Attention is focused on the influence induced from the permittivity of the medium inside the crack gap on the near-tip singularity and on the energy release rate (ERR). Thirty five kinds of such bimaterials are considered, which are constructed by five kinds of elastic dielectrics and seven kinds of piezoelectrics, respectively. Numerical results for the interface crack tip singularities are calculated. We demonstrate that, whatever the dielectric phase is much softer or much harder than the piezoelectric phase, the structure of the singular field near the semi-permeable interface crack tip in such bimaterials always consists of the singularity r−1∕2 and a pair of oscillatory singularities r−1∕2±iε. Calculated values of the oscillatory index ε for the 35 kinds of bimaterials are presented in tables, which are always within the range between 0.046 and 0.088. Energy analyses for five kinds of such bimaterials constructed by PZT-4 and the five kinds of elastic dielectrics are studied in more detail under four different cases: (i) the crack is electrically conducting, (ii) the crack gap is filled with air/vacuum, (iii) the crack gap is filled with silicon oil, and (iv) the crack is electrically impermeable. Detailed comparisons on the variable tendencies of the crack tip ERR against the applied electric field are given under some practical electromechanical loading levels. We conclude that the different values of the permittivity have no influence on the crack tip singularity but have significant influences on the crack tip ERR. We also conclude that the previous investigations under the impermeable crack model are incorrect since the results of the ERR for the impermeable crack show significant discrepancies from those for the semi-permeable crack, whereas the previous investigations under the conducting crack model may be accepted in a tolerant way since the results of the ERR show very small discrepancies from those for the semi-permeable crack, especially when the crack gap is filled with silicon oil. In all cases under consideration the curves of the ERR for silicon oil are more likely tending to those for the conducting crack rather than to those for air or vacuum. Finally, we conclude that the variable tendencies of the ERR against the applied electric field have an interesting load-dependent feature when the applied mechanical loading increases. This feature is due to the nonlinear relation between the normal electric displacement component and the applied electromechanical loadings from a quadratic equation.

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.

Transient Thermal Fracture Analysis of Transversely Isotropic Magneto-Electro-Elastic Materials

Modern materials such as magneto-electro-elastic materials are used in the development of smart structures. The magneto-electro-elastic materials possess the dual features that the application of electric field induces magnetization and magnetic field induces electric polarization. The theory of linear magneto-electro-elasticity is applied to solve transient thermal fracture in magneto-electro-elastic cylinder under sudden heating on its outer surface. The equilibrium equations are obtained from the constitutive equations. The governing partial differential equations are deduced by using equilibrium equations of elastic, electric and magnetic fields. The heat conduction equation is solved by separation of variable technique. Hankel transform is applied to solve elastic displacements, electric potential and magnetic potential. The problem is reduced into integral equation involving Bessel functions which is treated exactly using Abel's integral equation. Transient distributions of temperature, stress, displacement and magnetic inductions are derived for magneto-electro-elastic cylinder. Thermal stress, electric displacement and magnetic induction-intensity factors are obtained. The solutions are valid for both impermeable and permeable crack models. The studies are valuable for such material analysis and design.

2134 電場下における圧電セラミックスの三点曲げによる遅れ破壊挙動(S12-2 圧電材料,S12 先端材料システムの強度・機能評価とメゾメカニックス)

This paper investigates experimentally and analytically the delayed fracture in lead zirconate titanate (PZT) ceramics under electromechanical loading. Delayed fracture tests were conducted on single-edge precracked-beam specimens, and time-to-failure and fracture load under different DC electric fields were obtained. Possible mechanisms for delayed fracture were also discussed by scanning electron microscope (SEM) examination of the fracture surface of the PZT ceramics. Further, a nonlinear finite element analysis was employed to calculate the energy release rate for the permeable, impermeable and open crack models, and the effects of applied DC electric fields and localized polarization switching on the energy release rate are examined.

The Bueckner work conjugate integral for a permeable crack in piezoelectric materials

The Bueckner work conjugate integral for cracked piezoelectric materials is studied. The analysis is based on the permeable crack model to avoid the unphysically impermeable crack assumption. It is proved that the values of the Bueckner integrals are dependent on the normal electric displacement component on the crack surfaces and the coordinates of the starting point and the end point of the integral contour on the lower and upper crack faces, respectively. This means that, strictly speaking, the Bueckner integral for the permeable crack model is path-dependent unless a special condition is satisfied, i.e., the horizontal coordinates of the starting point and the end points of the integral contour are the same. The present conclusions are quite different from those derived from the impermeable crack model, where the Bueckner integral is strictly path-independent. The universal relation between the Bueckner integral and the J-integral existing in the impermeable crack model is also proved not to be always valid for the present permeable model unless the special integral contour is taken. This implies that the crack electric boundary condition in piezoelectric materials yields some significant influence on the properties of the Bueckner work conjugate integral.

A new approach to the pseudo-orthogonal properties of eigenfunction expansion form of the crack-tip complex potential function in anisotropic and piezoelectric fracture mechanics

Abstract Over the past twenty years, the well-known weight function theory based on the Bueckner work conjugate integral has been widely used to calculate crack tip fracture dominant parameter such as the stress intensity factor, the energy release rate (or the J -integral) and the T -stress in various kinds of cracked materials (e.g. isotropic materials, anisotropic materials and piezoelectric materials). Meanwhile, the pseudo-orthogonal property of the eigenfunction expansion form of the crack tip stress complex potential function has been proved to play a very important role in the theory. In this paper, we provide a new approach to establish the pseudo-orthogonal properties for crack problems in anisotropic and/or piezoelectric materials. In the latter case associated with mechanical-electric coupling, the electrical boundary conditions under both impermeable and permeable crack models are considered. The approach developed is much simpler than the classical complex variable separation technique proposed by previous researchers and hence the cumbersome and lengthy manipulations are avoided. Moreover, it is shown that, unlike previous works, the orthogonal properties of the material characteristic matrices A and B induced by the Stroh theory are no longer necessary in establishing the pseudo-orthogonal properties of eigenfunction expansion form in cracked piezoelectric materials. The approach can be easily extended to treat many other different crack problems concerning the Bueckner integral involving several complex arguments.

Double torsion testing and finite element analysis for determining the electric fracture properties of piezoelectric ceramics

This paper presents the results of an experimental and numerical investigation in electric fracture behavior of composite [Pb(Zr,Ti)O3] double torsion (DT) specimens. DT tests were conducted on a commercial piezoelectric ceramic bonded between two metals. Fracture loads under different electric fields were obtained from the experiment. Nonlinear three-dimensional finite element analysis was also employed to calculate the energy release rate for DT specimens based on the exact (permeable) and approximate (impermeable) crack models. The effects of applied electric field and domain switching on the energy release rate are discussed, and the model predictions are compared with the results of the experiments.

The effective electroelastic property of cracked piezoelectric media

This paper presents a study on the effective electroelastic property of piezoelectric media with parallel or randomly distributed cracks. The theoretical formulation is derived using the dilute model of distributed cracks and the solution of a single dielectric crack problem, in which the electric boundary condition along the crack surfaces is governed by the crack opening displacement. It is observed that the effective electroelastic property of such cracked piezoelectric media is nonlinear and sensitive to loading conditions. Numerical simulations are conducted to show the effects of crack distribution and electric boundary condition upon the effective electroelastic property. The transition between the commonly used electrically permeable and impermeable crack models is studied.

Coupled behaviour of interacting dielectric cracks in piezoelectric materials

Existing studies indicate that the commonly used electrically impermeable and permeable crack models may be inadequate in evaluating the fracture behaviour of piezoelectric materials in some cases. In this paper, a dielectric crack model based on the ‘real’ electric boundary condition is used to study the electromechanical behaviour of interacting cracks arbitrarily oriented in an infinite piezoelectric medium. 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 modelling the cracks using distributed dislocations and solving the resulting nonlinear singular integral equations using Chebyshev polynomials. Numerical simulation is conducted to show the effect of crack orientation, crack interaction and electric boundary condition upon the fracture behaviour of cracked piezoelectric media.

Electroelastic fracture mechanics analysis of central active piezoelectric transformer

The theory of linear piezoelectricity is applied to solve the plane strain electroelastic problem of a central active piezoelectric transformer in presence of a crack located normal to the interfaces. The ends of the crack are situated at equal distances away from the interfaces. The problem of an infinite central active piezoelectric transformer is formulated by means of integral transforms and reduced to the solution of a system of Fredholm integral equations of the second kind. Numerical calculations are carried out, and the fracture mechanics parameters such as stress intensity factor, energy release rate and energy density factor based on both permeable and impermeable crack models are shown graphically and compared.