A mixed electric boundary value problem for a two-dimensional piezoelectric crack

In this paper, a mixed electric boundary value problem for a two-dimensional piezoelectric crack problem is presented, in the sense that the crack face is partly conducting and partly impermeable. By the analytical continuation method, the unknown electric charge distributions on the upper and lower conducting crack faces are reduced to two decoupled singular integral equations and then these two equations are converted into algebraic equations to find the full field solution. Though the results suggest that the stress intensity factors at the crack tip are identical to those of conventional piezoelectric materials, but the electric field and electric displacement are related to the electric boundary conditions on the crack faces. The electric field and electric displacement are singular not only at crack tips but also at the junctures between the impermeable part and conducting parts. Numerical results for the variations of the electric field, electric displacement field and J-integral with respect to the normalized impermeable crack length are shown. Some discussions for the energy release rate and the J-integral are made.     

Dynamic fracture behaviors of cracks in a functionally graded magneto-electro-elastic plate

In this paper the dynamic anti-plane problem for a functionally graded magneto-electro-elastic plate containing an internal or an edge crack parallel to the graded direction is investigated. The crack is assumed to be magneto-electrically impermeable. Integral transforms and dislocation density functions are employed to reduce the problem to Cauchy singular integral equations. Field intensity factors and energy release rate are derived, analyzed and partially calculated numerically. The effects of material graded index, loading combination parameter (including size and direction) and geometry criterion of the plate on the dynamic energy release rate are shown graphically. Numerical results indicate that increasing the graded index can all retard the crack extension, and that both the applied magnetic field loadings and electric field loadings play a dominant role in the dynamic fracture behaviors of crack tips.     

Arbitrarily oriented crack near interface in piezoelectric bimaterials

Arbitrarily oriented crack near interface in piezoelectric bimaterials is considered. After deriving the fundamental solution for an edge dislocation near the interface, the present problem can be expressed as a system of singular integral equations by modeling the crack as continuously distributed edge dislocations. In the paper, the dislocations are described by a density function defined on the crack line. By solving the singular integral equations numerically, the dislocation density function is determined. Then, the stress intensity factors (SIFs) and the electric displacement intensity factor (EDIF) at the crack tips are evaluated. Subsequently, the influences of the interface on crack tip SIFs, EDIF, and the mechanical strain energy release rate (MSERR) are investigated. The J-integral analysis in piezoelectric bimaterals is also performed. It is found that the path-independent of J₁-integral and the path-dependent of J₂-integral found in no-piezoelectric bimaterials are still valid in piezoelectric bimaterials.     

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.     

Antiplane internal crack normal to the edge of a functionally graded piezoelectric/piezomagnetic half plane

This paper deals with the antiplane magnetoelectroelastic problem of an internal crack normal to the edge of a functionally graded piezoelectric/piezomagnetic half plane. The properties of the material such as elastic modulus, piezoelectric constant, dielectric constant, piezomagnetic coefficient, magnetoelectric coefficient and magnetic permeability are assumed in exponential forms and vary along the crack direction. Fourier transforms are used to reduce the impermeable and permeable crack problems to a system of singular integral equations, which is solved numerically by using the Gauss-Chebyshev integration technique. The stress, electric displacement and magnetic induction intensity factors at the crack tips are determined numerically. The energy density theory is applied to study the effects of nonhomogeneous material parameter β, edge conditions, location of the crack and load ratios on the fracture behavior of the internal crack.     

The weight function for cracks in piezoelectrics

The weight function in fracture mechanics is the stress intensity factor at the tip of a crack in an elastic material due to a point load at an arbitrary location in the body containing the crack. For a piezoelectric material, this definition is extended to include the effect of point charges and the presence of an electric displacement intensity factor at the tip of the crack. Thus, the weight function permits the calculation of the crack tip intensity factors for an arbitrary distribution of applied loads and imposed electric charges. In this paper, the weight function for calculating the stress and electric displacement intensity factors for cracks in piezoelectric materials is formulated from Maxwell relationships among the energy release rate, the physical displacements and the electric potential as dependent variables and the applied loads and electric charges as independent variables. These Maxwell relationships arise as a result of an electric enthalpy for the body that can be formulated in terms of the applied loads and imposed electric charges. An electric enthalpy for a body containing an electrically impermeable crack can then be stated that accounts for the presence of loads and charges for a problem that has been solved previously plus the loads and charges associated with an unsolved problem for which the stress and electric displacement intensity factors are to be found. Differentiation of the electric enthalpy twice with respect to the applied loads (or imposed charges) and with respect to the crack length gives rise to Maxwell relationships for the derivative of the crack tip energy release rate with respect to the applied loads (or imposed charges) of the unsolved problem equal to the derivative of the physical displacements (or the electric potential) of the solved problem with respect to the crack length. The Irwin relationship for the crack tip energy release rate in terms of the crack tip intensity factors then allows the intensity factors for the unsolved problem to be formulated, thereby giving the desired weight function. The results are used to derive the weight function for an electrically impermeable Griffith crack in an infinite piezoelectric body, thereby giving the stress intensity factors and the electric displacement intensity factor due to a point load and a point charge anywhere in an infinite piezoelectric body. The use of the weight function to compute the electric displacement factor for an electrically permeable crack is then presented. Explicit results based on a previous analysis are given for a Griffith crack in an infinite body of PZT-5H poled orthogonally to the crack surfaces.     

Theoretical model of crack branching in magnetoelectric thermoelastic materials

Thermomagnetoelectroelastic crack branching of magnetoelectro thermoelastic materials is theoretically investigated based on Stroh formalism and continuous distribution of dislocation approach. The crack face boundary condition is assumed to be fully thermally, electrically and magnetically impermeable. Explicit Green’s functions for the interaction of a crack and a thermomagnetoelectroelastic dislocation (i.e., a thermal dislocation, a mechanical dislocation, an electric dipole and a magnetic dipole located at a same point) are presented. The problem is reduced to two sets of coupled singular integral equations with the thermal dislocation and magnetoelectroelastic dislocation densities along the branched crack line as the unknown variables. As a result, the formulations for the stress, electric displacement and magnetic induction intensity factors and energy release rate at the branched crack tip are expressed in terms of the dislocation density functions and the branch angle. Numerical results are presented to study the effect of applied thermal flux, electric field and magnetic field on the crack propagation path by using the maximum energy release rate criterion.     

Periodic cracks in a functionally graded piezoelectric layer bonded to a piezoelectric half-plane

Studied is the problem of a periodic array of cracks in a functionally graded piezoelectric strip bonded to a homogeneous piezoelectric material. The properties of the functionally graded piezoelectric strip, such as elastic modulus, piezoelectric constant and dielectric constant, are assumed in exponential forms and vary along the crack direction. The crack surface condition is assumed to be electrically impermeable or permeable. Integral transform and dislocation density functions are employed to reduce the problem to the solution of a system of singular integral equations. The effects of the periodic crack spacing, material constants and the geometry parameters on the stress intensity factor, the energy release ratio and the energy density factor are studied.     

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.     

A magnetoelectroelastic medium with an elliptical cavity under combined mechanical–electric–magnetic loading

The solution for an elliptical cavity in an infinite two-dimensional magnetoelectroelastic medium subject to remotely uniformly applied combined mechanical–electric–magnetic loadings is obtained by using the Stroh formalism and the exact boundary conditions along the surface of the cavity. By letting the minor-axis of the cavity to zero the solution for a crack is deduced. A self-consistent method is proposed to calculate the real crack opening under the combined mechanical–electric–magnetic loadings. The method requires that the crack opening is the minor-axis of the elliptical opening profile. Beside the real crack solution, four different extreme models, i.e., the impermeable crack, permeable crack, electrically impermeable and magnetically permeable crack and electrically permeable and magnetically impermeable crack, are discussed. An expression of the strain energy density factor is derived. Numerical results of the strain energy density at the crack tip are given for a BaTiO₃–CoFe₂O₄ composite with the piezoelectric BaTiO₃ material being the inclusion and the magnetostrictive CoFe₂O₄ material being the matrix. The effects of the proportion of the two phases, permeability of the crack to electric and magnetic fields, the electric and magnetic loadings on the strain energy density factor are discussed.     

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.     

Moving crack for functionally grated material in an infinite length strip under antiplane shear

Considered is a Yoffe crack in an infinite strip of functionally grated material (FGM) subjected to antiplane shear. The shear moduli in two directions of FGM are assumed to be of exponential form. The dynamic stress intensity factor and strain energy density factor at the crack tip are obtained by using integral transforms and dual-integral equations. The numerical results show that the decrease of the strain energy density factor varies with the shear moduli gradient, and the increase of the strain energy density factor varies with the increase of the moving crack speed. The ratio of shear moduli in material vertical orientation has a great influence on the strain energy density factor.     

The sliding interface crack with friction between elastic and rigid bodies

The paper analyzes the frictional sliding crack at the interface between a semi-infinite elastic body and a rigid one. It gives solutions in complex form for non-homogeneous loading at infinity and explicit solutions for polynomial loading at the interface. It is found that the singularities at the crack tips are different and that they are related to distinct kinematics at the crack tips. Firstly, we postulate that the geometry of the equilibrium crack with crack-tip positions b and a is determined by the conditions of square integrable stresses and continuous displacement at both crack tips. The crack geometry solution is not unique and is defined by any compatible pair (b,a) belonging to a quasi-elliptical curve. Then we prove that, for an equilibrium crack under given applied load, the “energy release rate” G_tip, defined at each crack tip by the J_ε-integral along a semi-circular path, centered at the crack tip, with vanishing radius ε, vanishes. For arbitrarily shaped paths embracing the whole crack, with end points on the unbroken zone, the J-integral is path-independent and has the significance of the rate, with respect to the crack length, of energy dissipated by friction on the crack.     

Exact solutions of two semi-infinite collinear cracks in piezoelectric strip

Using the complex variable function method and the conformal mapping technique,the fracture problem of two semi-infinite collinear cracks in a piezoelectric strip is studied under the anti-plane shear stress and the in-plane electric load on the partial crack surface.Analytic solutions of the field intensity factors and the mechanical strain energy release rate are derived under the assumption that the surfaces of the crack are electrically impermeable.The results can be reduced to the well-known solutio...     

Dynamic internal crack problem of a functionally graded magneto-electro-elastic strip

In this paper the dynamic anti-plane problem for a functionally graded magneto-electro-elastic strip containing an internal crack perpendicular to the boundary is investigated. The crack is assumed to be either magneto-electrically impermeable or permeable. Integral transforms and dislocation density functions are employed to reduce the problem to Cauchy singular integral equations. Numerical results show the effects of loading combination parameter, material gradient parameter and crack configuration on the dynamic response. With the magneto-electrically permeable assumption, both the magnetical and electrical impacts have no contribution to the crack tip field singularity. However, with the impermeable assumption, both the applied magnetical loads and electrical loads play a dominant role in the dynamic fracture behavior of crack tips. And for the two kinds of crack surface conditions, increasing the graded index can all retard the crack extension.     

Fracture behavior of a bonded magneto-electro-elastic rectangular plate with an interface crack

In this paper the anti-plane problem for an interface crack between two dissimilar magneto-electro-elastic plates subjected to anti-plane mechanical and in-plane magneto-electrical loads is investigated. The interface crack is assumed to be either magneto-electrically impermeable or permeable, and the position of the interface crack is arbitrary. The finite Fourier transform method is employed to reduce the mixed boundary-value problem to triple trigonometric series equations. The dislocation density functions and proper replacement of the variables are introduced to reduce these series equations to a standard Cauchy singular integral equation of the first kind. The resulting integral equation together with the corresponding single-valued condition is approximated as a system of linear algebra equations which can be easily solved. Field intensity factors and energy release rates are determined numerically and discussed in detail. Numerical results show the effects of crack configuration and loading combination parameters on the fracture behaviors of crack tips according to energy release rate criterion. The study of this problem is expected to have applications to the investigation of dynamic fracture properties of magneto-electro-elastic materials with cracks.     

Dynamic anti-plane shear of a cracked piezoelectric ceramic

The dynamic theory of antiplane piezoelectricity is applied to solve the problem of a line crack subjected to horizontally polarized shear waves in an arbitrary direction. The problem is formulated by means of integral transforms and reduced to the solution of a Fredholm integral equation of the second kind. The path-independent integral G is extended here to include piezoelectric effects, and is evaluated at the crack tip to obtain the dynamic energy release rate. Numerical calculations are carried out for the dynamic stress intensity factor and energy release rate. The material is piezoelectric ceramic.     

Mode III crack problem in a functionally graded magneto-electro-elastic strip

Considering the material properties to be one-dimensionally dependent, this paper studied an anti-plane problem for an embedded crack and edge crack perpendicular to the boundary of a functionally graded magneto-electro-elastic strip. The crack is assumed to be either magneto-electrically impermeable or permeable. Integral transform and dislocation density functions are employed to reduce the problem to the solution of a system of singular integral equations. Numerical results show the effects of the loading combination parameter, material gradient parameter and crack configuration on the field intensity factors and the energy release rates of the functionally graded magneto-electro-elastic strip.     

矩形压电体中反平面裂纹的电弹性场

基于线性压电理论,本文获得了含有中心反平面裂纹的矩形压电体中的奇异应力和电场。利用Fourier积分变换和Fourier正弦级数将电绝缘型裂纹问题化为对偶积分方程,并进一步归结为易于求解的第二类Fred-holm积分方程。获得了裂纹尖端应力、应变、电位移和电场的解析解,求得了裂纹尖端场的强度因子及能量释放率。分析了压电矩形体的几何尺寸对它们的影响。结果表明,对于电绝缘型裂纹,裂纹尖端附近的各个场变量都具有-1/2阶的奇异性,能量释放率与电荷载的方向及大小有关,并且有可能为负值。     

Dynamic crack tip fields and dynamic crack propagation characteristics of anisotropic material

Based on mechanics of anisotropic material, the dynamic crack propagation problem of I/II mixed mode crack in an infinite anisotropic body is investigated. Expressions of dynamic stress intensity factors for modes I and II crack are obtained. Components of dynamic stress and dynamic displacements around the crack tip are derived. The strain energy density theory is used to predict the dynamic crack extension angle. The critical strain energy density is determined by the strength parameters of anisotropic materials. The obtained dynamic crack tip fields are unified and applicable to the analysis of the crack tip fields of anisotropic material, orthotropic material and isotropic material under dynamic or static load. The obtained results show Crack propagation characteristics are represented by the mechanical properties of anisotropic material, i.e., crack propagation velocity M and fiber direction α. In particular, the fiber direction α and the crack propagation velocity M give greater influence on the variations of the stress fields and displacement fields. Fracture angle is found to depend not only on the crack propagation but also on the anisotropic character of the material.