A finite crack in a semi-infinite strip of a graded piezoelectric material under electric loading

In this paper, the mixed-mode crack problem for a functionally graded piezoelectric material (FGPM) strip is considered. It is assumed that the electroelastic properties of the strip vary continuously along the thickness of the strip, and that the strip is under in-plane electric loading. The problem is formulated in terms of a system of singular integral equations. The stress and electric displacement intensity factors are presented for various values of dimensionless parameters representing the crack size, the crack location, and the material nonhomogeneity.     

Functionally graded piezoelectric strip with a penny-shaped crack under electromechanical loadings

In this paper, the mixed-mode penny-shaped crack problem for a functionally graded piezoelectric material (FGPM) strip is considered. It is assumed that the electroelastic properties of the strip vary continuously along the thickness of the strip, and that the strip is under in-plane electromechanical loadings. The problem is formulated in terms of a system of singular integral equations. The stress and electric displacement intensity factors are presented for various values of dimensionless parameters representing the crack size, the crack location, and the material nonhomogeneity.     

Investigation of the Dynamic Behavior of a Griffith Crack in a Piezoelectric Material Strip Subjected to the Harmonic Elastic Anti-plane Shear Waves by Use of the Non-local Theory

In this paper, the dynamic behavior of a Griffith crack in a piezoelectric material strip subjected to the harmonic anti-plane shear waves is investigated by use of the non-local theory for impermeable crack surface conditions. To overcome 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 at the crack tip. By means of 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 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 thickness of the strip, the circular frequency of incident wave and the lattice parameter.     

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...     

Dielectric crack problem for a magnetoelectroelastic strip with functionally graded properties

The paper presents a fracture analysis for an electromagnetically dielectric crack in a functionally graded magnetoelectroelastic strip. It is considered that the material properties are varying exponentially along the width direction. Under the assumption of the in-plane magneto-electro-mechanical loadings, the dielectric crack is simulated by using the semi-permeable crack-face boundary conditions. The Fourier transform technique is applied to solve the boundary-value problem and four coupling singular integral equations are determined. A nonlinear system of algebraic equations is further derived and solved numerically to determine the electromagnetic field inside the crack. Then the field intensity factors of stress, electric displacement, and magnetic induction are given. Through the numerical computations, the effects of the material non-homogeneity and the permeability of crack interior on the electric displacement and the magnetic induction at the crack faces are studied. The variations of the intensity factors of stress, electric displacement, and magnetic induction versus the geometry of the crack, the strip width, and the material non-homogeneity are presented in graphics respectively.     

Griffith crack moving in a piezoelectric strip

Summary  The dynamic problem of an impermeable crack of constant length 2a propagating along a piezoelectric ceramic strip is considered under the action of uniform anti-plane shear stress and uniform electric field. The integral transform technique is employed to reduce the mixed-boundary-value problem to a singular integral equation. For the case of a crack moving in the mid-plane, explicit analytic expressions for the electroelastic field and the field intensity factors are obtained, while for an eccentric crack moving along a piezoelectric strip, numerical results are determined via the Lobatto–Chebyshev collocation method for solving a resulting singular integral equation. The results reveal that the electric-displacement intensity factor is independent of the crack velocity, while other field intensity factors depend on the crack velocity when referred to the moving coordinate system. If the crack velocity vanishes, the present results reduce to those for a stationary crack in a piezoelectric strip. In contrast to the results for a stationary crack, applied stress gives rise to a singular electric field and applied electric field results in a singular stress for a moving crack in a piezoelectric strip. Received 14 August 2001; accepted for publication 24 September 2002 The author is indebted to the AAM Reviewers for their helpful suggestions for improving this paper. The work was supported by the National Natural Science Foundation of China under Grant 70272043.     

Anti-plane analysis of semi-infinite crack in piezoelectric strip

Using the complex variable function method and the technique of the conformal mapping, the fracture problem of a semi-infinite crack in a piezoelectric strip is studied under the anti-plane shear stress and the in-plane electric load. The analytic solutions of the field intensity factors and the mechanical strain energy release rate are presented under the assumption that the surface of the crack is electrically impermeable. When the height of the strip tends to infinity, the analytic solutions of an infinitely large piezoelectric solid with a semi-infinite crack are obtained. Moreover, the present results can be reduced to the well-known solutions for a purely elastic material in the absence of the electric loading. In addition, numerical examples are given to show the influences of the loaded crack length, the height of the strip, and the applied mechanical/electric loads on the mechanical strain energy release rate.     

功能梯度压电带中裂纹对SH波的散射

研究功能梯度压电带中裂纹对SH波的散射问题,为了便于分析,材料性质假定为指数模型,并假设裂纹面上的边界条件为电渗透型的.根据压电理论得到压电体的状态方程,利用Fourier积分变换,问题转化为对偶积分方程的求解.用Copson方法求解积分方程.求得了裂纹尖端动应力强度因子、电位移强度因子的解析表达式,最后数值结果显示了标准动应力强度因子与入射波数、材料参数、带宽、波数以及入射角之间的关系.     

Electrical nonlinear anti-plane shear crack in a functionally graded piezoelectric strip

The electrical nonlinear behavior of an anti-plane shear crack in a functionally graded piezoelectric strip is studied by using the strip saturation model within the framework of linear electroelasticity. The analysis is conducted on the electrically unified crack boundary condition with the introduction of the electric crack condition parameter that can describe all the electric crack boundary condition in accordance with the aspect ratio of an ellipsoidal crack and the permittivity inside the crack, in particular, including traditional permeable and impermeable crack boundary conditions. The resulting mixed boundary value problem is analysed and near tip field is obtained by using the integral transform techniques. Numerical results for the normalized five kinds of energy release rates under the small scale electric saturation condition are presented and compared to show the influences of the electric crack condition parameter with the variation of the ellipsoidal crack parameters, electric loads, functionally graded piezoelectric material gradation, crack length, electromechanical coupling coefficient, and crack location. It reveals that there are considerable differences between the results obtained from the traditional electric crack models and those obtained from the current unified crack model.     

Mode III eccentric crack in a functionally graded piezoelectric strip

This paper studies the internal crack problem located within one functionally graded piezoelectric strip. One crack is normal to the edge of the strip and the material properties vary along the direction of crack length. Three different boundary conditions and both impermeable and permeable cases are discussed. The problem can be reduced to a system of singular integral equations and solved by using the Gauss–Chebyshev formulas. The results show that the edge boundary conditions and the nonhomogeneous parameter significantly control the magnitudes of stress and electric displacement intensity factors.     

Transient response of a cracked piezoelectric strip under arbitrary electro-mechanical impact

By using the well-developed integral transform methodology, the dynamic response of stress and electric displacement around a finite crack in an infinite piezoelectric strip are investigated under arbitrary dynamic anti-plane loads. The dynamic stress intensity factors and electric displacement are obtained analytically. It is shown that the dynamic crack-tip stress and electric field still have a square-root singularity. Numerical computations for the dynamic stress intensity factor show that the electric load has a significant influence on the dynamic response of stress field. The higher the ratio of the crack length to the width of the strip, the higher the peak value of the dynamic stress intensity factor is. On the other hand, the dynamic response of the electric field is determined solely by the applied electric load. The electric field will promote or retard the propagation of the crack depending on the time elapse since the application of the external electro-mechanical loads. The project supported by the National Natural Science Foundation of China and the Post-Doctor Science Foundation of China     

A Crack with an Electric Displacement Saturation Zone in an Electrostrictive Material

A crack with an electric displacement saturation zone in an electrostrictive material under purely electric loading is analyzed. A strip saturation model is here employed to investigate the effect of the electrical polarization saturation on electric fields and elastic fields. A closed form solution of electric fields and elastic fields for the crack with the strip saturation zone is obtained by using the complex function theory. It is found that the K _I -dominant region is very small compared to the strip saturation zone. The generalized Dugdale zone model is also employed in order to investigate the effect of the saturation zone shape on the stress intensity factor. Using the body force analogy, the stress intensity factor for the asymptotic problem of a crack with an elliptical saturation zone is evaluated numerically.     

An anti-plane propagating crack in a functionally graded piezoelectric strip bonded to a homogeneous piezoelectric strip

Summary A finite crack propagating at constant speed in a functionally graded piezoelectric strip (FGPS) bonded to a homogeneous piezoelectric strip is considered. It is assumed that the electroelastic material properties of the FGPS vary exponentially across the thickness of the strip, and that the bimaterial strip is under combined anti-plane mechanical shear and in-plane electrical loads. The analysis is conducted for the electrically unified crack boundary condition, which includes both the traditional permeable and the impermeable ones. By using the Fourier transform, the problem is reduced to the solution of Fredholm integral equations of the second kind. Numerical results for the stress intensity factor and the crack sliding displacement are presented to show the influences of the crack propagation speed, electric loads, FGPS gradation, crack length, electromechanical coupling coefficient, properties of the bonded homogeneous piezoelectric strip and crack location.     

功能梯度压电板条中电绝缘型运动裂纹的电弹性场

基于三维弹性理论和压电理论，对材料系数按指数函数规律分布的功能梯度压电板条中的反平面运动裂纹问题进行了求解。利用Fourier积分变换方法将电绝缘型运动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Fredholm积分方程。通过渐近分析，获得了裂纹尖端应力、应变、电位移和电场的解析解，给出了裂纹尖端场各个变量的角分布函数，并求得了裂纹尖端场的强度因子，分析了压电材料物性梯度参数、几何尺寸及裂纹运动速度对它们的影响。结果表明，对于电绝缘型裂纹，功能梯度压电板条中运动裂纹尖端附近的各个场变量都具有-1／2阶的奇异性；当裂纹运动速度增大时，裂纹扩展的方向会偏离裂纹面。     

Dynamic behavior of two parallel symmetric permeable cracks in a piezoelectric strip

The dynamic behavior of two parallel symmetric cracks in a piezoelectric strip under harmonic anti-plane shear waves is studied using the Schmidt method for permeable crack surface conditions. The cracks are parallel to the edge of the strip. By means of 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. The results show that the stress and the electric displacement intensity factors depend on the geometry of the cracks, the frequency of incident waves, the distance between cracks and the thickness of the strip. It is also found that the electric displacement intensity factors for the permeable crack surface conditions are much smaller than those for the impermeable crack surface conditions. Project supported by the Post Doctoral Science Foundation of Heilongjiang Province, the Natural Science Foundation of Heilongjiang Province, the National Science Foundation with the Excellent Young Investigator Award (No. 19725209) and the Scientific Research Foundation of Harbin Institute of Technology (HIT.2000.30).     

Transient response of a piezoelectric ceramic strip with an eccentric crack under electromechanical impacts

The transient response of a piezoelectric strip with an eccentric crack normal to the strip boundaries under applied electromechanical impacts is considered. By using the Laplace transform, the mixed initial-boundary-value problem is reduced to triple series equations, then to a singular integral equation of the first kind by introducing an auxiliary function. The Lobatto–Chebyshev collocation technique is adopted to solve numerically the resulting singular integral equation. Dynamic field intensity factors and energy release rate are obtained for both a permeable crack and an impermeable crack. The effects of the crack position and the material properties on the dynamic stress intensity factor are examined and numerical results are presented graphically.     

Torsional impact response of a penny-shaped crack in a functional graded strip

The torsional impact response of a penny-shaped crack in a nonhomogeneous strip is considered. The shear modulus is assumed to be functionally graded such that the mathematics is tractable. Laplace and Hankel transforms were used to reduce the problem to solving a Fredholm integral equation. The crack tip stress field is obtained by considering the asymptotic behavior of Bessel function. Explicit expressions of both the dynamic stress intensity factor and the energy density factor were derived. And it is shown that, as crack driving force, they are equivalent for the present crack problem. Investigated are the effects of material nonhomogeneity and strip‘s highness on the dynamic fracture behavior.Numerical results reveal that the peak of the dynamic stress intensity factor can be suppressed by increasing the nonhomogeneity parameter of the shear modulus, and that the dynamic behavior varies little with the adjusting of the strip‘ s highness.     

Transient response of a cracked magnetoelectroelastic strip under anti-plane impact

The transient anti-plane problem of a magnetoelectroelastic strip containing a crack vertical to the boundary is considered. Singular integral equations for the impermeable crack are obtained by using Fourier and Laplace transforms. Numerical results show the effects of the relative loading parameters κ_D and κ_B, and the crack configuration on the dynamic fracture behavior. The results obtained indicate that for the impermeable crack, the electric and magnetic impacts have significant influences on the dynamic stress intensity factor and the dynamic energy density factor.     

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.     

A piezoelectric material strip with a crack perpendicular to its boundary surfaces

A cracked piezoelectric material strip under combining mechanical and electrical loads is considered. The crack is vertical to the top and bottom edges of the strip. The edges of the strip are parallel to the x-axis and perpendicular to the z-axis. When a piezoelectric ceramic is poled, it exhibits transversely isotropic behavior. Among many possible poled axis orientations, a particular orientation when the poling direction lies parallel to x-axis is examined in this paper. Both impermeable crack and permeable crack assumptions are considered. Numerical results are included for three kinds of fracture mechanics specimens, namely an edge-cracked strip, a double edge-cracked strip, and a center-cracked strip, subjected to uniform tensions and uniform electric displacement loads simultaneously, at the far ends. In addition, an edge-cracked strip under pure bending and uniform electric displacement loads at the far ends is also investigated in this paper.