The central crack problem for a functionally graded piezoelectric strip

In this paper the dynamic anti-plane problem for a functionally graded piezoelectric strip containing a central crack vertical to the boundary is considered. The crack is assumed to be 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 permeable assumption, the electric impact has no contribution to the crack tip field singularity. With the impermeable assumption, the direction of applied electric impact loading plays a great role in the behavior of dynamic stress intensity factor, and the existence of electric load always enhances the crack propagation. However, the crack is easier to propagate under the negative electric load than that under the positive electric load.     

Transient analysis of a piezoelectric strip with a permeable crack under anti-plane impact loads

The problem of a through permeable crack situated in the mid-plane of a piezoelectric strip is considered under anti-plane impact loads for two cases. The first is that the strip boundaries are free of stresses and of electric displacements, and the second is that the strip boundaries are clamped rigid electrodes. The method adopted is to reduce the mixed initial-boundary value problem, by using integral transform techniques, to dual integral equations, which are further transformed into a Fredholm integral equation of the second kind by introducing an auxiliary function. The dynamic stress intensity factor and energy release rate in the Laplace transform domain are obtained in explicit form in terms of the auxiliary function. Some numerical results for the dynamic stress intensity factor are presented graphically in the physical space by using numerical techniques for solving the resulting Fredholm integral equation and inverting Laplace transform.     

压电材料中反平面渗透裂纹的解

用复变函数的保角映射法,采用可渗透边界条件,研究了含裂纹的无限大压电材料在平面内电场和反平面荷载作用下的耦合场,得到了精确的解和场强度因子以及能量释放率。结果表明,电场强度在裂尖没有奇异性,应变、应力、电位移具有1/2阶的奇异性,能量释放率总是正的。     

Crack arrest model for a piezoelectric plate — A generalised Dugdale model

The constant search for new materials has provided impetus to research in piezoelectric materials. An anti-plane problem for a cracked unbounded two-dimensional poled piezoelectric plate has been investigated. The crack rims open on account of shear mechanical forces applied at the remote boundary and in-plane electric displacement field prescribed at the infinite boundary. Thus the crack yields both mechanically and electrically. Consequently, a plastic zone and a saturation zone protrude ahead of each tip of the crack. These developed zones are in turn closed by applying yield point shear stress at the rims of plastic zone and normal closing saturation limit displacement on the rims of saturation zones. Two cases are investigated when (i) the developed saturation zone length exceeds that of the developed plastic zone, and (ii) saturation zone length is smaller than that of the plastic zone. Fourier integral transform method is used in each case to obtain the length of plastic zone and saturation zone. Closed form analytic expressions are obtained in each case. Crack opening displacement and potential drop across the rims of the crack are also obtained. The effect of mechanical loads on crack closure in the presence of electric field is investigated and vice-versa. Also energy release rate expressions are obtained for both the cases.     

The scattering of harmonic elastic anti-plane shear waves by a Griffith crack in a piezoelectric material plane by using the non-local theory

In this paper, the dynamic behavior of a Griffith crack in a piezoelectric material plane under anti-plane shear waves is investigated by using the non-local theory for impermeable crack face conditions. For overcoming the mathematical difficulties, a one-dimensional non-local kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress and the electric displacement near the crack tips. By using the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations. These equations are solved using the Schmidt method. Contrary to the classical elasticity solution, it is found that no stress and electric displacement singularity is present near the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress near the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The finite hoop stress at the crack tip depends on the crack length, the circular frequency of incident wave and the lattice parameter. For comparison results between the non-local theory and the local theory for this problem, the same problem in the piezoelectric materials is also solved by using local theory.     

Dynamic fracture mechanics study of an electrically impermeable mode III crack in a transversely isotropic piezoelectric material under pure electric load

The dynamic response of an electrically impermeable Mode III crack in a transversely isotropic piezoelectric material under pure electric load is investigated by treating the electric loading process as a transient impact load, which may be more appropriate to mimic the real service environment of piezoelectric materials. The stress intensity factor, the mechanical energy release rate, and the total energy release rate are derived and expressed as a function of time for a given applied electric load. The theoretical results indicate that a purely electric load can fracture the piezoelectric material if the stress intensity factor or the mechanical energy release rate is used as a failure criterion.     

压电材料中渗透性周期共线反平面裂纹问题

本文利用复变函数方法,借助于Riemann-Schwarz延拓技术和保形映照方法,研究了渗透性边界条件下周期共线反平面裂纹问题,获得了解的表达式,得到了力学和电学强度因子。结果表明在裂纹尖端应力和电位移的奇异性都与远场作用的应力载荷和裂纹长度有关,其中应力的奇异性与材料无关,电位移的奇异性则与材料有关,电载荷对裂尖的奇异性没有影响。最后,运用数值算例,给出周期裂纹间的干涉效应和裂纹的尺度效应。     

The Interaction between a Piezoelectric Screw Dislocation Dipole and an Elliptic Blunt Crack in Elliptical Inhomogeneity

The interaction problem of a piezoelectric screw dislocation dipole with a confocal elliptic blunt crack in elliptical inhomogeneity subjected to remote anti-plane stress field and in-plane electric field is investigated by using the complex method of elasticity. The exact closed-form solutions of a series of quantities, such as singularity stress field, image force and image torque acting on the center of screw dislocation dipole, stress intensity factor and electric displacement intensity factor of crack tip, energy release rate, and generalized strain energy density are obtained. Then the influence laws of remote load, the dip angle of dislocation dipole, the size of blunt crack, and the material constants on the quantities are analyzed. The numerical results show that the image force, image torque, stress intensity factor, and electric displacement intensity factor make periodic variation as the dip angle of dislocation dipole; the energy release rate of crack tip is negative when subjected to pure electric field, however, it can be positive or negative when subjected to the combined action of mechanical field and electric field; the sharp crack is not easy to expand in some combined action of mechanical field and electric field.     

有限高磁电弹性体中双半动态与静态裂纹分析

狭长体中的裂纹是断裂力学中经常采用的研究模型。含有共线无限长裂纹的条形磁电弹性体,当面内的力电磁和反平面的剪应力作用在左边裂纹尖端附近的一段裂纹面上时,往往会产生动态断裂。利用复变函数法中的拱形变换公式,导出了磁电全非渗透型边界条件下左裂纹尖端动态的应力强度因子以及机械应变能释放率的解析解。当运动速度趋于零时退化为静止状态下的解。通过数值算例分析了断裂机理,讨论了静止状态下狭长体和裂纹的几何尺寸、外力、电场和磁场分别对能量释放率的影响,为相关器件的设计与制造提供了帮助。     

含界面边裂纹压电材料反平面问题的应力强度因子

研究了含界面边裂纹的不同压电介质组成的复合材料在反平面荷载和平面内电场作用下的电弹场,得到了级数形式的基本解和应力强度因子,最后用边界配置法求解了应力强度因子.结果表明,在外加剪切荷载的作用下,应力强度因子与外加电场无关.     

Edge cracked piezoelectric ceramic block under electromechanical impact loading

The dynamic field intensity factors and energy release rates in a piezoelectric ceramic block containing an edge crack with the condition of continuous electric crack faces under electromechanical impact loading are obtained. Integral transform method is used to reduce the problem to two pairs of dual integral equations, which are then expressed to an Fredholm integral equation of the second kind. Numerical values on the dynamic stress intensity factor and dynamic energy release rate are obtained to show the influence of the geometry and electric field.     

Impact response of a finite crack in a finite strip under anti-plane shear

The response of a through-the thickness crack with finite dimensions to impact in a finite elastic strip is investigated in this study. The elastic strip is assumed to be subjected to anti-plane shear deformation. Laplace and Fourier transform were used to formulate the mixed boundary value problem. The dynamic stress intensity factor and crack opening displacement are obtained as a function of time and the strip width to crack length ratio, h/a. The results indicate that the intensity of the crack-tip stress field reaches a peak very quickly and then decreases in magnitude oscillating about the static value. In general, the dynamic stress intensity factor is higher for small $\text{h/a}$. Similar behavior has also been found for the crack surface displacement.     

The M-integral for calculating intensity factors of an impermeable crack in a piezoelectric material

In this study, a conservative integral is derived for calculating the intensity factors associated with piezoelectric material for an impermeable crack. This is an extension of the M-integral or interaction energy integral for mode separation in mechanical problems. In addition, the method of displacement extrapolation is extended for this application as a check on results obtained with the conservative integral. Poling is assumed parallel, perpendicular and at an arbitrary angle with respect to the crack plane, as well as parallel to the crack front. In the latter case, a three-dimensional treatment is required for the conservative integral which is beyond the scope of this investigation. The asymptotic fields are obtained; these include stress, electric, displacement and electric flux density fields which are used as auxiliary solutions for the M-integral.Several benchmark problems are examined to demonstrate the accuracy of the methods. Numerical difficulties encountered resulting from multiplication of large and small numbers were solved by normalizing the variables. Since an analytical solution exists, a finite length crack in an infinite body is also considered. Finally, a four point bend specimen subjected to both an applied load and an electric field is presented for a crack parallel, perpendicular and at an angle to the poling direction. It is seen that neglecting the piezoelectric effect in calculating stress intensity factors may lead to errors.     

Antiplane Permeable Edge Cracks in a Piezoelectric Strip of Finite Width

A piezoelectric strip with permeable edge cracks normal to the strip boundaries is analyzed. Under uniform antiplane mechanical shear and inplane electric loading, the distribution of the entire electroelastic field in a cracked piezoelectric strip is determined in explicit analytic form via the conformal mapping technique. It is found that the strain and the electric displacement exhibit the same singularity as the stress near the crack tips, while the electric field is always uniform. The field intensity factors and the energy release rate are independent of the applied electric load for prescribed stress, and related to the applied electric load for prescribed strain.     

Scattering of the harmonic anti-plane shear waves by a crack in functionally graded piezoelectric materials

In the present paper, the dynamic behavior of a Griffth crack in the functionally graded piezoelectric material (FGPM) is investigated. It is assumed that the elastic stiffness, piezoelectric constant, dielectric permittivity and mass density of the FGPM vary continuously as an exponential function, and that FGPM is under the anti-plane mechanical loading and in-plane electrical loading. By using the Fourier transform and defining the jumps of displacement and electric potential components across the crack surface as the unknown functions, two pairs of dual integral equations are derived. To solve the dual integral equations, the jumps of the displacement and electric potential components across the crack surface are expanded in a series of Jacobi polynomial. Numerical examples are provided to show the effects of material properties on the stress and the electric displacement intensity factors.     

Dynamic behavior of a crack in a functionally graded piezoelectric strip bonded to two dissimilar half piezoelectric material planes

Summary. The dynamic behavior of a crack in a functionally graded piezoelectric material (FGPM) strip bonded to two half dissimilar piezoelectric material planes subjected to combined harmonic anti-plane shear wave and in-plane electrical loading was studied under the limited permeable and permeable electric boundary conditions. It was assumed that the elastic stiffness, piezoelectric constant and dielectric permittivity of the functionally graded piezoelectric layer vary continuously along the thickness of the strip. By using the Fourier transform, the problem can be solved with a set of dual integral equations in which the unknown variables are the jumps of the displacements and the electric potentials across the crack surfaces. In solving the dual integral equations, the jumps of the displacements and the electric potentials across the crack surfaces were expanded in a series of Jacobi polynomials. Numerical results illustrate the effects of the gradient parameter of FGPM, electric loading, wave number, thickness of FGPM strip and electric boundary conditions on the dynamic stress intensity factors (SIFs).     

Antiplane permeable edge cracks in a piezoelectric strip of finite width

A priezoelectric strip with permeable edge cracks normal to the strip boundaries is analyzed. Under uniform antiplane mechanical shear and inplane electric loading, the distribution of the entire electroelastic field in a cracked piezoelectric strip is determined in explicit analytic form via the conformal mapping technique. It is found that the strain and the electric displacement exhibit the same singularity as the stress near the crack tips, while the electric field is always uniform. The field intensity factors and the energy release rate are independent of the applied electric load for prescribed stress, and related to the applied electric load for prescribed strain.     

Dynamic response for non-homogeneous piezoelectric medium with multiple cracks

A general procedure to analyze the dynamic response of non-homogeneous piezoelectric medium containing some non-collinear cracks is developed. It is assumed that all the material properties only depend on the coordinates y (along the thickness direction). The assumption is made that the non-homogeneous medium is composed of numerous laminae with their surfaces perpendicular to the thick direction. The solution method is based upon the Fourier and Laplace transforms to reduce the boundary value problem to a system of generalized singularity equations in the Laplace transform domain. The singular integral equations for the problem are derived and numerically solved by weight residual value method. The time-dependent full field solutions are obtained in the time domain. As numerical illustration, the stress and electric displacement intensity factors for a three-layer plate specimen with two cracks are presented. It is found that the stress and electric fields are coupled in the crack plane ahead of the crack tip for non-homogenous piezoelectric materials.     

Dynamic stress intensity factors of two 3D rectangular permeable cracks in a transversely isotropic piezoelectric material under a time‐harmonic elastic P‐wave

The dynamic behavior of two 3D rectangular permeable cracks in a transversely isotropic piezoelectric material is investigated under an incident harmonic stress wave by using the generalized Almansi's theorem and the Schmidt method. The problem is formulated through double Fourier transform into three pairs of dual integral equations with the displacement jumps across the crack surfaces as the unknown variables. To solve the dual integral equations, the displacement jumps across the crack surfaces are directly expanded as a series of Jacobi polynomials. Finally, the relations among the dynamic stress field and the dynamic electric displacement filed near the crack edges are obtained, and the effects of the shape of the rectangular crack, the characteristics of the harmonic wave, and the distance between two rectangular cracks on the stress and the electric intensity factors in a piezoelectric composite material are analyzed. Copyright © 2015 John Wiley & Sons, Ltd.     

压电复合材料中幂函数型曲线裂纹的反平面问题

通过构造新保角映射, 利用Stroh公式研究了远场受反平面剪应力和面内电载荷共同作用下无限大压电复合材料中幂函数型曲线裂纹的断裂行为。给出了电不可渗透边界条件下裂纹尖端场强度因子和机械应变能释放率的解析解。该解析解在幂函数的幂次为零时, 可退化为已有文献中无限大压电复合材料含直线裂纹的结果, 证明了其合理性。由解析解可知, 裂纹几何形状一定时, 电场分布将不受机械载荷的影响。最后, 通过数值算例讨论了幂函数的幂次、 系数及其在 x₁轴上的投影长度对机械应变能释放率的影响。结果表明, 当压电体仅受 x₂方向载荷作用时, 对于给定幂次与开口的曲线裂纹, 在 x₁轴上的投影长度存在一临界值使其最容易开裂; 而对于给定投影长度与幂次的曲线裂纹, 开口越大裂纹越容易扩展。