Transient response of a piezoelectric material with a semi-infinite mode-III crack under impact loads

The problem of a semi-infinite impermeable mode-III crack in a piezoelectric material is considered under the action of impact loads. For the case when a pair of concentrated anti-plane impact loads and electric displacements are exerted symmetrically on the upper and lower surfaces of the crack, the asymptotic electroelastic field ahead of the crack tip is determined in explicit form. The dynamic intensity factors of electroelastic field and dynamic mechanical strain energy release rate are obtained. The obtained results can be taken as fundamental solutions, from which general results may directly be evaluated by integration. The method adopted is to reduce the mixed initial-boundary value problem, by using the Laplace and Fourier transforms, into two simultaneous dual integral equations. One may be converted into an Abel's integral equation and the other into a singular integral equation with Cauchy kernel, and the solutions of both equations can be determined in closed-form, respectively. For some particular cases, the present results reduce to the previous results.     

Transient response of a cracked magnetoelectric material under the action of in-plane sudden impacts

Dynamic analysis of a crack embedded in a magnetoelectric material is made when subjected to in-plane mechanical, electric and magnetic impacts. The Laplace and Fourier transforms are applied to reduce the associated initial- and mixed-boundary value problem to dual integral equations, and then to singular integral equations with Cauchy kernel. By numerically solving the resulting equation, the dynamic field intensity factors as well as CODs, and energy release rates near the crack tip are evaluated and presented graphically. The effects of applied magnetic and electric impacts on crack growth are discussed. Obtained results show that, different from the static results, applied magnetic and electric impacts can strongly affect dynamic stress intensity factors.     

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.     

Transient response of a center crack in a functionally graded piezoelectric strip under electromechanical impact

The dynamic fracture problem for a functionally graded piezoelectric strip containing a center crack parallel to the free boundaries is considered in this study. It is assumed that the electroelastic properties of the medium vary continuously in the thickness direction, and that the strip is under in-plane mechanical and electric impact loadings. Integral transform techniques and dislocation density functions are employed to reduce the problem to the solutions of a system of singular integral equations. The dynamic stress and electric displacement intensity factors versus time are presented for various values of dimensionless parameters representing the crack size, the material nonhomogeneity and the loading combination.     

Effect of electromechanical coupling on the dynamic interaction of cracks in piezoelectric materials

Summary This article provides a comprehensive treatment of the dynamic interaction between two arbitrarily located and oriented cracks in a piezoelectric medium under steady-state inplane electrical and antiplane mechanical loads. Using an impermeable condition along the crack surfaces, a fundamental dynamic solution was developed for the single crack problem. In this fundamental solution, the single crack problem was treated using Fourier transform and the appropriate singular integral equations. The fundamental solution was then implemented into a pseudo-incident wave method to account for the interaction between the cracks. Numerical examples are provided to show the effect of the geometry of the cracks, the material constants, the frequency of the incident wave and the applied electrical field upon the dynamic stress intensity factors. The results show the significant effect of electromechanical coupling upon the stress intensity factor at the crack tip.     

Yoffe-type moving edge crack in a piezoelectric block

Summary We consider an anti-plane edge moving crack problem with the constant velocity in a piezoelectric ceramic block. The far-field anti-plane shear mechanical and in-plane electrical loads are applied to the piezoelectric block. It is expressed to a Fredholm integral equation of the second kind. Expressions for the dynamic field intensity factors and the dynamic energy release rate are obtained. The dynamic stress intensity factor and the dynamic energy release rate depend on the crack propagation speed. Numerical results for several piezoelectric materials are also presented.     

A magneto-electro-elastic material with a penny-shaped crack subjected to temperature loading

Summary The analysis of intensity factors for a penny-shaped crack under thermal, mechanical, electrical and magnetic boundary conditions becomes a very important topic in fracture mechanics. An exact solution is derived for the problem of a penny-shaped crack in a magneto-electro-thermo-elastic material in a temperature field. The problem is analyzed within the framework of the theory of linear magneto-electro-thermo-elasticity. The coupling features of transversely isotropic magneto-electro-thermo-elastic solids are governed by a system of partial differential equations with respect to the elastic displacements, the electric potential, the magnetic potential and the temperature field. The heat conduction equation and equilibrium equations for an infinite magneto-electro-thermo-elastic media are solved by means of the Hankel integral transform. The mathematical formulations for the crack conditions are derived as a set of dual integral equations, which, in turn, are reduced to Abel's integral equation. Solution of Abel's integral equation is applied to derive the elastic, electric and magnetic fields as well as field intensity factors. The intensity factors of thermal stress, electric displacement and magnetic induction are derived explicitly for approximate (impermeable or permeable) and exact (a notch of finite thickness crack) conditions. Due to its explicitness, the solution is remarkable and should be of great interest in the magneto-electro-thermo-elastic material analysis and design.     

Impact response of a cracked soft ferromagnetic medium

A solution is given for the problem of an infinite soft ferromagnetic solid containing a central crack subjected to normal impact load. The solid is permeated by a uniform magnetostatic field normal to the crack surface. Laplace and Fourier transforms are employed to reduce the transient problem to the solution of integral equations in the Laplace transformed plane. A numerical Laplace inversion technique is used to compute the values of the dynamic stress-intensity factor, and the results are compared with the corresponding elastodynamic values to reveal the influence of magnetic field on the dynamic stress-intensity factor. The dynamic stress intensity factor is found to increase with increasing values of the magnetic field.With 4 Figures     

An Annular Crack in a Magnetoelectroelastic Layer

A flat annular crack in a magnetoelectroelastic layer subjected to mechanical, electric and magnetic loadings is investigated under magnetoelectrically impermeable boundary condition on the crack surface. Using Hankel transform technique, the mixed boundary value problem is reduced to a system of singular integral equations. With the aid of Gauss-Chebyshev integration technique, the integral equations are further reduced to a system of algebraic equations. The field intensity factor and energy release rate are determined. Numerical results reveal the effects of electric and magnetic loadings and crack configuration on crack propagation and growth.     

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.     

Dynamic fracture behavior of an internal interfacial crack between two dissimilar magneto-electro-elastic plates

In this paper, the anti-plane problem for an interfacial crack between two dissimilar magneto-electro-elastic plates subjected to anti-plane mechanical and in-plane magneto-electrical impact loadings is investigated. Four kinds of crack surface conditions are adopted: magneto-electrically impermeable (Case 1), magnetically impermeable and electrically permeable (Case 2), magnetically permeable and electrically impermeable (Case 3), and magneto-electrically permeable (Case 4). The position of the interfacial crack is arbitrary. The Laplace transform and finite Fourier transform techniques are employed to reduce the mixed boundary-value problem to triple trigonometric series equations in the Laplace transform domain. Then the dislocation density functions and proper replacements of the variables are introduced to reduce the 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 easily be solved. Field intensity factors and energy release rates are determined and discussed. The effects of loading combination parameters on dynamic energy release rate are plotted for Cases 1-3. On the other hand, since the magneto-electrically permeable condition is perhaps more physically reasonable for type III crack, the effect of the crack configuration on the dynamic fracture behavior of the crack tips is studied in detail for Case 4. The results could be useful for the design of multilayered magneto-electro-elastic structures and devices.     

The mixed-type integral equations for transient elastodynamic plane crack problems

In the present paper, by use of the boundary integral equation method and the techniques of Green fundamental solution and singularity analysis, the dynamic infinite plane crack problem is investigated. For the first time, the problem is reduced to solving a system of mixed-typed integral equations in Laplace transform domain. The equations consist of ordinary boundary integral equations along the outer boundary and Cauchy singular integral equations along the crack line. The equations obtained are strictly proved to be equivalent with the dual integral equations obtained by Sih in the special case of dynamic Griffith crack problem. The mixed-type integral equations can be solved by combining the numerical method of singular integral equation with the ordinary boundary element method. Further use the numerical method for Laplace transform, several typical examples are calculated and their dynamic stress intensity factors are obtained. The results show that the method proposed is successful and can be used to solve more complicated problems.     

Transient response of a crack in an anisotropic strip

Summary The problem of an anisotropic elastic strip containing a crack which is opened by stresses suddenly applied on the crack faces is considered here. The problem is reduced to a set of simultaneous Fredholm integral equations of the second kind which may be solved iteratively. Once the solutions of these integral equations are obtained, the dynamic stress intensity factors may be evaluated numerically. Numerical results are obtained for a particular transversely isotropic strip.With 1 Figure     

Dynamic SIF of interface crack between two dissimilar viscoelastic bodies under impact loading*

In this paper, the transient dynamic stress intensity factor (SIF) is determined for an interface crack between two dissimilar half-infinite isotropic viscoelastic bodies under impact loading. An anti-plane step loading is assumed to act suddenly on the surface of interface crack of finite length. The stress field incurred near the crack tip is analyzed. The integral transformation method and singular integral equation approach are used to get the solution. By virtue of the integral transformation method, the viscoelastic mixed boundary problem is reduced to a set of dual integral equations of crack open displacement function in the transformation domain. The dual integral equations can be further transformed into the first kind of Cauchy-type singular integral equation (SIE) by introduction of crack dislocation density function. A piecewise continuous function approach is adopted to get the numerical solution of SIE. Finally, numerical inverse integral transformation is performed and the dynamic SIF in transformation domain is recovered to that in time domain. The dynamic SIF during a small time-interval is evaluated, and the effects of the viscoelastic material parameters on dynamic SIF are analyzed.     

加层压电条界面裂纹的稳态扩展

研究当压电条同时与两个不同材料的弹性条粘接在一起,在反平面机械载荷及面内电载荷联合作用下,长度不变的有限Griffith 界面裂纹沿加层压电条界面以常速稳态扩展时裂尖的动态断裂问题。应用Fourier积分变换将问题化为以第二类Fredholm积分方程表示的对偶积分方程,导出了相应的动应力强度因子表达式。给出了动应力强度因子与裂纹传播速度、裂纹长度、压电条及弹性条厚度、电荷载大小及方向的关系曲线。研究结果对结构设计及结构失效的预防具有理论和应用价值。     

Surface crack problem for functionally graded magnetoelectroelastic coating-homogeneous elastic substrate system under anti-plane mechanical and in-plane electric and magnetic loading

A functionally graded magnetoelectroelastic material layer bonded to a homogeneous elastic substrate is investigated. The functionally graded magnetoelectroelastic layer contains a surface crack that is perpendicular to the surface of the medium. The structure is subjected to anti-plane mechanical and in-plane electric and magnetic loads, the crack problem involves the anti-plane elastic field coupled with the in-plane electric and magnetic field. The elastic layer can be an ideal insulator or an ideal conductor. 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 influences of the material gradient parameter and crack configuration on field intensity factors and energy release rates of the functionally graded magnetoelectroelastic coating-homogeneous elastic substrate structure.     

Axisymmetric elastodynamic response of a flat annular crack to normal impact waves

The axisymmetric response of a flat annular crack in an infinite medium subjected to normal impact load is investigated in this study. A step stress is applied to the crack surface. The singular solution is equivalent to solutions of the problem of diffraction of normally incident tension wave by a flat annular crack, and the problem of the sudden appearance of a flat annular crack in a uniform tensile stress field. Laplace and Hankel transforms are used to reduce the problem to the solution of a set of triple integral equations in the Laplace transform domain. These equations are solved by using a integral transform technique and the result is expressed in terms of a singular integral equation of the first kind with the kernel which is improved by means of a contour integration on the Riemann surface. A numerical Laplace inversion routine is used to recover the time dependence of the solution. Numerical results of the dynamic stress intensity factor are obtained to show the influence of inertia, the ratio of the inner radius to the outer one and Poisson's ratio on the load transmission to the crack tip.     

Closed-form solution for an eccentric anti-plane shear crack normal to the edges of a magnetoelectroelastic strip

Summary The problem of an anti-plane shear crack embedded in a magnetoelectroelastic strip is investigated. The crack is assumed to be normal to the strip edges. By using the finite Fourier transform, the associated mixed boundary-value problem is reduced to triple series equations, then to singular integral equations. Solving the resulting equations analytically, the field intensity factors and energy release rates at the crack tips can be determined in explicit form. The influences of applied electric and magnetic loadings on the normalized energy release rate and mechanical strain energy release rate are presented graphically. Obtained results reveal that applied electric and magnetic loadings affect crack growth, depending on their directions and adopted fracture criteria. The derived solution is applicable to other cases including two collinear cracks distributed symmetrically in a magnetoelectroelastic strip, and a periodic array of collinear cracks in a magnetoelectroelastic plane.     

压电复合材料中圆孔边Ⅲ型非对称裂纹的场强度因子

研究了压电复合材料中圆孔边4个非对称裂纹在远处受面内电载荷和面外力载荷共同作用下的断裂行为。利用复变函数方法和新映射函数将问题转化为Cauchy积分方程组。通过求解Cauchy积分方程组,得到了电非渗透型和电渗透型两种边界条件下裂纹尖端电弹性场和场强度因子的解析解。所得结果不仅可退化为已有解,而且可模拟出若干新的缺陷构型,如压电复合材料中圆孔边三裂纹、半无限压电复合材料中半圆孔边单裂纹及半无限压电体中边界裂纹。将所得结果与有限元结果进行比较,吻合很好,证实了文中方法的正确性和有效性。数值算例分析了缺陷的几何参数对场强度因子的影响规律。     

The thermal effect of piezoelectric medium for anti-plane problem under high electrical impact loading

In present paper, the anti-plane problem of thermal effect near crack tip region of piezoelectric material subjected to electrical impact loading is investigated by means of the integral transforms and the singular integral equations. By introducing the thermal power, the temperature field near crack tip is finally obtained on the basis of the hypotheses of the uncoupling between the thermal field and the electro-mechanical fields and the adiabatic approximation. The numerical results indicate that a high temperature field of small region near crack tip is deduced when high electrical impact load is applied. Moreover, the results show that the temperature field strongly depends on crack size. However, the thermal effect of mechanical impact comparing with electrical impact may almost be neglected.