Electro-mechanical pre-fracture zones for an electrically permeable interface crack in a piezoelectric bimaterial

A plane strain problem for two piezoelectric half-spaces adhered by a very thin isotropic interlayer with a crack under the action of remote mixed mode mechanical loading and electrical flux is considered. The crack is situated either at an interface or in the interlayer. It is assumed that the substrates are much stiffer than the intermediate layer. Therefore, pre-fracture zones (plastic or damage) arise at the crack continuations. Normal and shear stresses are assumed to be constant in this zones and to satisfy some material equation, which can be taken from theory or derived experimentally. Modeling the pre-fracture zones by the crack continuations with unknown cohesive stresses on their faces reduces the problem to elastic interface crack analysis leading to a Hilbert problem. This problem is solved exactly. The pre-fracture zone lengths and stresses in these zones are found from algebraical and transcendental equations. The latter are derived from the conditions of stress finiteness at the ends of pre-fracture zones and the material equations. The electrical displacement at any point of the pre-fracture zones is found in closed form as well. Particular cases of symmetrical loading and of equivalent properties of the upper and lower bimaterial components are considered. Numerical results corresponding to certain material combinations and interlayer material equations are presented and analysed. In the suggested model, any singularities connected with the crack are eliminated, i.e., all mechanical and electrical characteristics are limited in the near-crack tip region.     

Electrically permeable crack with contact zones between two piezoelectric materials

The paper addresses a plane problem for an infinite plane consisting of two different piezoceramic half-planes with an interfacial crack with smooth contact zones and subjected to the uniformly distributed electromechanical loading applied at infinity. Methods of complex-variable theory are used to reduce the problem to a Dirichlet-Riemann mixed homogeneous boundary-value problem. Its solution is found in closed form. A system with one crack that has one or two contact zones is calculated. Expressions for stresses, electric-flux density, and displacement discontinuities at the interface are written. Equations for the determination of the length of the contact zones and expressions for the stress intensity factors at the crack tips are derived __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 3, pp. 66–74, March 2008.     

Limited permeable crack in an interlayer between piezoelectric materials with different zones of electrical saturation and mechanical yielding

A plane problem for two identical piezoelectric semi-infinite spaces adhered by means of a thin isotropic interlayer is considered. It is assumed that a crack of a limited electric permeability occurs in the interlayer parallel to its faces. Combined electromechanical loading is prescribed at infinity. It is assumed that the interlayer is softer than the adherent materials. To avoid the singularities, which are typical for the Griffith crack model, two distinct zones – a zone of mechanical yielding and a zone of electrical saturation – of unknown lengths are introduced as crack continuations. These lengths can be essentially different, with the zone of mechanical yielding significantly longer or shorter than the zone of electrical saturation. Assuming that the interlayer thickness tends to zero, a constant normal stress is prescribed in the zone of mechanical yielding and a saturated electrical displacement is prescribed in the zone of electrical saturation. Outside of these zones, the semi-infinite spaces are assumed to be perfectly bonded. This formulation results in a linear fracture mechanics problem with unknown pre-fracture zone lengths. The problem, formulated mathematically by a system of two equations of linear relationship, is solved exactly. The unknown yield and saturated zones lengths are found from the conditions of finiteness of stress and electrical displacement at the ends of these zones for the both cases when the electrical saturated zone is longer and shorter than the zone of mechanical yielding. It is shown that the same equation as for the Griffith crack model can be used for the determination of the electrical displacement in the crack region. The main results of the paper are obtained in the form of simple analytical equations which are convenient for engineering applications. Some numerical illustrations in graphical and tabular form show dependencies of the pre-fracture zone lengths, the energy release rate, the mechanical displacement and electrical potential jumps on the electromechanical loading and the electrical permeability of the crack medium.     

功能梯度压电压磁材料中断裂问题分析

分析了功能梯度压电/压磁材料中裂纹在反平面剪切载荷下的断裂问题. 为了便于分析，假设材料性质沿着裂纹的法线方向呈指数变化. 利用Fourier变换，问题可以转化为对未知数是裂纹表面张开位移的一对对偶积分方程的求解，此对偶积分方程采用Schmidt方法求解. 最后分析了裂纹长度及表征功能梯度材料的参数βl对应力，电位移和磁通量强度因子的影响.     

Penny-shaped crack in transversely isotropic piezoelectric materials

Using a method of potential functions introduced successively to integrate the field equations of three-dimensional problems for transversely isotropic piezoelectric materials, we obtain the so-called general solution in which the displacement components and electric potential functions are represented by a singular function satisfying some special partial differential equations of 6th order. In order to analyse the mechanical-electric coupling behaviour of penny-shaped crack for above materials, another form of the general solution is obtained under cylindrical coordinate system by introducing three quasi-harmonic functions into the general equations obtained above. It is shown that both the two forms of the general solutions are complete. Furthermore, the mechanical-electric coupling behaviour of penny-shaped crack in transversely isotropic piezoelectric media is analysed under axisymmetric tensile loading case, and the crack-tip stress field and electric displacement field are obtained. The results show that the stress and the electric displacement components near the crack tip have (r ^−1/2) singularity. The project supported by the Natural Science Foundation of Shaanxi Province, China     

On anti-plane shear behavior of a Griffith permeable crack in piezoelectric materials by use of the non-local theory

In this paper, the non-local theory of elasticity is applied to obtain the behavior of a Griffith crack in the piezoelectric materials under anti-plane shear loading for permeable crack surface conditions. By means of the Fourier transform the problem can be solved with the help of a pair of dual integral equations with the unknown variable being the jump of the displacement across the crack surfaces. These equations are solved by the Schmidt method. Numerical examples are provided. Unlike the classical elasticity solutions, it is found that no stress and electric displacement singularity is present at the crack tip. The non-local elastic solutions yield a finite hoop stress at the crack tip, thus allowing for a fracture criterion based on the maximum stress hypothesis. The finite hoop stress at the crack tip depends on the crack length and the lattice parameter of the materials, respectively. The project supported by the National Natural Science Foundation of China (50232030 and 10172030)     

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.     

Analysis of a permeable interface crack in elastic dielectric/piezoelectric bimaterials

A permeable interface crack between elastic dielectric material and piezoelectric material is studied based on the extended Stroh’s formalism. Motivated by strong engineering demands to design new composite materials, the authors perform numerical analysis of interface crack tip singularities and the crack tip energy release rates for 35 types of dissimilar bimaterials, respectively, which are constructed by five kinds of elastic dielectric materials: Epoxy, Polymer, Al₂O₃, SiC, and Si₃N₄ and seven kinds of practical piezoelectric ceramics: PZT-4, BaTiO₃, PZT-5H, PZT-6B, PZT-7A, P-7, and PZT-PIC 151, respectively. The elastic dielectric material with much smaller permittivity than commercial piezoelectric ceramics is treated as a special transversely isotropic piezoelectric material with extremely small piezoelectricity. The present investigation shows that the structure of the singular field near the permeable interface crack tip consists of three singularities: and , which is quite different from that in the impermeable interface crack. It can be concluded that different far field loading cases have significant influence on the near-tip fracture behaviors of the permeable interface crack. Based on the present theoretical treatment and numerical analysis, the electric field induced crack growth is well explained, which provides a better understanding of the failure mechanism induced from interface crack growth in elastic dielectric/piezoelectric bimaterials. The project supported by the National Natural Science Foundation of China (10572110), Doctor Foundation of the Chinese Education Ministry and Doctorate Foundation of Xi’an Jiaotong University. The English text was polished by Yunming Chen.     

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.     

A moving mode-III crack in functionally graded piezoelectric material: permeable problem

In this paper a moving mode-III crack in functionally graded piezoelectric materials (FGPM) is studied. The crack surfaces are assumed to be permeable. The governing equations for FGPM are solved by means of Fourier cosine transform. The mathematical formulation for the permeable crack condition is derived as a set of dual integral equations, which, in turn, are reduced to a Fredholm integral equation of the second kind. The results obtained indicate that the stress intensity factor of moving crack in FGPM depends only on the mechanical loading. The gradient parameter of the FGPM and the moving velocity of the crack do have significant influence on the dynamic stress intensity factor.     

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.     

THE BEHAVIOR OF TWO PARALLEL SYMMETRIC PERMEABLE CRACKS IN PIEZOELECTRIC MATERIALS

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A 3-D rectangular permeable crack or two 3-D rectangular permeable cracks in a piezoelectric material

The solutions of a 3-D rectangular permeable crack and two 3-D rectangular permeable cracks in a piezoelectric material were investigated by using the generalized Almansi’s theorem and the Schmidt method. The problem was formulated through Fourier transform into three pairs of dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces. To solve the dual integral equations, the displacement jumps across the crack surfaces were directly expanded as a series of Jacobi polynomials. Finally, the effects of the shape of the rectangular crack and the distance between two rectangular cracks on the stress and electric displacement intensity factors in a piezoelectric material were analyzed. These results can be used for the strength and the coupling effect evaluation of cracked piezoelectric materials.     

An exact solution of crack problems in piezoelectric materials

IntroductionInrecentyearscrackproblemsinpiezoelectricmaterialhavereceivedmuchattention.Manytheoreticalanalyseshavebeengivenby[1～16].Itshouldbe,however,notedthatalltheaboveanalysesarebasedonaso-calledimpermeablecrackassumphon,i.e.thecrackfacesareassumedtobeimpermeabletoelectricfield,sotheelectricdisplacementvanishesinsidethecrack.Usingthisassumption,onewillobtainthefollowingresultS[2'3'5,6'9'16]=whentheelectricloadsaresolelyaPPliedatinLfinity,theelectricdisplacementissquare-rootsingularatthe…     

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.     

Non-local theory solution for a Mode I crack in piezoelectric materials

In this paper, the non-local theory of elasticity is applied to obtain the behavior of a Griffith crack in the piezoelectric materials subjected to a uniform tension loading. The permittivity of the air in the crack is considered. By means of the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations, in which the unknown variables are the jumps of the displacements across the crack surfaces. To solve the dual integral equations, the jumps of the displacements across the crack surfaces are expanded in a series of Jacobi polynomials. Numerical examples are provided to show the effects of the crack length, the materials constants, the electric boundary conditions and the lattice parameter on the stress and the electric displacement fields near the crack tips. It can be obtained that the effects of the electric boundary conditions on the electric displacement fields are large. Unlike the classical elasticity solutions, it is found that no stress and electric displacement singularities are present at the crack tips. The non-local elastic solutions yield a finite hoop stress at the crack tips, thus allowing us to use the maximum stress as a fracture criterion.     

A mode III crack in functionally graded piezoelectric materials

This paper considers the mode III crack problem in functionally graded piezoelectric materials. The mechanical and the electrical properties of the medium are considered for a class of functional forms for which the equilibrium equations have an analytical solution. The problem is solved by means of singular integral equation technique. Both a single crack and a series of collinear cracks are investigated. The results are plotted to show the effect of the material inhomogeneity on the stress and the electric displacement intensity factors.     

An interface crack with contact zones in a piezoelectric/piezomagnetic bimaterial

An interface crack with a frictionless contact zone at the right crack tip between two semi-infinite piezoelectric/piezomagnetic spaces under the action of a remote mechanical loading, magnetic and electric fluxes as well as concentrated forces at the crack faces is considered. Assuming that all fields are independent on the coordinate x ₂ co-directed with the crack front, the stresses, the electrical and the magnetic fluxes as well as the derivatives of the jumps of the displacements, the electrical and magnetic potentials are presented via a set of analytic functions in the (x ₁, x ₃)-plane with a cut along the crack region. Two cases of magneto-electric conditions at the crack faces are considered. The first case assumes that the crack is electrically and magnetically permeable, and in the second case the crack is assumed electrically permeable while the open part of the crack is magnetically impermeable. For both these cases due to the above-mentioned representation the combined Dirichlet–Riemann boundary value problems have been formulated and solved exactly. Stress, electric and magnetic induction intensity factors are found in a simple analytical form. Transcendental equations and a closed form analytical formula for the determination of the real contact zone length have been derived for both cases of magnetic conditions in the crack region. For a numerical illustration of the obtained results a bimaterial BaTiO₃–CoFe₂O₄ with different volume fractions of BaTiO₃ has been used, and the influence of the mechanical loading and the intensity of the magnetic flux upon the contact zone length and the associated intensity factors as well as the energy release rate has been shown.     

Mode III crack problems for two bonded functionally graded piezoelectric materials

This paper investigates the singular electromechanical field near the crack tips of an internal crack. The crack is perpendicular to the interface formed by bonding two half planes of different functionally graded piezoelectric material. The properties of two materials, such as elastic modulus, piezoelectric constant and dielectric constant, are assumed in exponential forms and vary along the crack direction. The singular integral equations for impermeable and permeable cracks are derived and solved by using the Gauss–Chebyshev integration technique. It shows that the stresses and electrical displacements around the crack tips have the conventional square root singularity. The stress intensity and electric displacement intensity factors are highly affected by the material nonhomogeneity parameters β and γ. The solutions for some degenerated problems can also be obtained.     

Transient anti-plane crack problem for two bonded functionally graded piezoelectric materials

In this paper, the dynamic anti-plane crack problem for two bonded functionally graded piezoelectric materials is considered. The crack is perpendicular to the interface and assumed to be electrically impermeable or permeable. Integral transforms are employed to reduce the problem to Cauchy singular equations that can be solved numerically. The effects of the loading parameter λ, material constants and the geometry parameters on the stress intensity factor and the energy density factor are studied. It is found that for the impermeable case, the normalized dynamic stress intensity factor may increase or decrease in different time domains determined by the sign and magnitude of λ.