Propagation of an anti-plane moving crack in a functionally graded piezoelectric strip

Summary The propagation of an anti-plane moving crack in a functionally graded piezoelectric strip (FGPS) is studied in this paper. The governing equations for the proposed analysis are solved using Fourier cosine transform. The mixed boundary value problems of the anti-plane moving crack, which is assumed to be either impermeable or permeable, are formulated as dual integral equations. By appropriate transformations, the dual integral equations are reduced to Fredholm integral equations of the second kind. For the impermeable crack, the stress intensity factor (SIF) of the crack in the FGPS depends on both the mechanical and electric loading, whereas, the SIF for the permeable crack depends only on the mechanical loading. The results obtained show that the gradient parameter of the FGPS and the velocity of the crack have significant influence on the dynamic SIF.Support from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKU 7081/00E) is acknowledged. Support from the National Natural Science Foundation of China (Project No. 10072041) is also acknowledged.     

SCATTERING OF THE HARMONIC STRESS WAVE BY CRACKS IN FUNCTIONALLY GRADED PIEZOELECTRIC MATERIALS

The present paper considers the scattering of the time harmonic stress wave by a single crack and two collinear cracks in functionally graded piezoelectric material （FGPM）. It is assumed that the properties of the FGPM vary continuously as an exponential function. By using the Fourier transform and defining the jumps of displacements 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 polynomials. Numerical examples are provided to show the influences of material properties on the dynamic stress and the electric displacement intensity factors.     

Normal compression wave scattering by a permeable crack in a fluid-saturated poroelastic solid

A mathematical formulation is presented for the dynamic stress intensity factor (mode I) of a finite permeable crack subjected to a time-harmonic propagating longitudi-nal wave in an infinite poroelastic solid. In particular, the effect of the wave-induced fluid flow due to the presence of a liquid-saturated crack on the dynamic stress intensity fac-tor is analyzed. Fourier sine and cosine integral transforms in conjunction with Helmholtz potential theory are used to formulate the mixed boundary-value problem as dual inte-gral equations in the frequency domain. The dual integral equations are reduced to a Fredholm integral equation of the second kind. It is found that the stress intensity factor mono-tonically decreases with increasing frequency, decreasing the fastest when the crack width and the slow wave wavelength are of the same order. The characteristic frequency at which the stress intensity factor decays the fastest shifts to higher frequency values when the crack width decreases.     

Angle cracks in two bonded functionally graded piezoelectric material under anti-plane shear

Consider two bonded functionally graded piezoelectric material (FGPM) with finite height. Each material contains an arbitrary oriented crack. The material properties are assumed in exponential forms in the direction normal to the interface. The crack surface condition is assumed to be electrically impermeable or permeable. Using the Fourier transform technique, the problem can be reduced to a system of singular integral equations, which are then solved numerically by applying the Gauss-Chebyshev integration formula to obtain the stress intensity factors at the crack tips. Numerical calculations are carried out to obtain the energy density factor S and the energy release rate G. In impermeable case, the energy release rate has been shown to be negative as the electric loads are applied. The positive definite characteristic of the energy density factor makes it possible for predicting the fracture behavior of the cracked structure. The influences of the non-homogeneous parameters and crack orientation on the energy density factors at the crack tips are discussed in detail. The results show that the energy density factor at the crack tip will be increased when the crack tip is located within the softer material.     

A moving interface crack between two dissimilar functionally graded piezoelectric layers under electromechanical loading

The dynamic propagation of an interface crack between two dissimilar functionally graded piezoelectric material (FGPM) layers under anti-plane shear is analyzed using the integral transform method. The properties of the FGPM layers vary continuously along the thickness. The properties of the FGPM layers vary differently and the two layers are connected weak-discontinuously. A constant velocity Yoffe-type moving crack is considered. Fourier transform is used to reduce the problem to two sets of dual integral equations, which are then expressed to the Fredholm integral equations of the second kind. Numerical values on the dynamic energy release rate (DERR) are presented for the FGPM to show the effects on the electric loading, gradient of material properties, crack moving velocity, and thickness of layers. Followings are helpful to increase of the resistance of the interface crack propagation of FGPM: (a) certain direction and magnitude of the electric loading; (b) increase of the gradient of material properties; (c) increase of the material properties from the interface to the upper and lower free surface; (d) increase of the thickness of FGPM layer. The DERR increases or decreases with increase of the crack moving velocity.     

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.     

非均匀弹性材料中反平面的运动裂纹

用解析方法研究了非均匀弹性材料中反平面运动裂纹问题。首先采用余弦变换求解非均匀材料的基本方程，然后根据混合边值条件建立裂纹运动的对偶积分方程，再把对偶积分方程化为第二类Fredholm积分方程。给出了数值算例，计算结果表明材料的非均匀性对动应力强度因子有较大的影响。     

Pre-kinking of a moving crack in a magnetoelectroelastic material under in-plane loading

A constant moving crack in a magnetoelectroelastic material under in-plane mechanical, electric and magnetic loading is studied for impermeable crack surface boundary conditions. Fourier transform is employed to reduce the mixed boundary value problem of the crack to dual integral equations, which are solved exactly. Steady-state asymptotic fields near the crack tip are obtained in closed form and the corresponding field intensity factors are expressed explicitly. The crack speed influences the singular field distribution around the crack tip and the effects of electric and magnetic loading on the crack tip fields are discussed. The crack kinking phenomena is investigated using the maximum hoop stress intensity factor criterion. The magnitude of the maximum hoop stress intensity factor tends to increase as the crack speed increases.     

Multi-interface cracks in a piezoelectric layer bonded to two half-piezoelectric spaces under anti-plane shear loading

The behavior of four parallel symmetry permeable interface cracks in a piezoelectric layer bonded to two half-piezoelectric spaces under anti-plane shear loading is investigated. By using the Fourier transform, the problem can be solved with the help of two pairs of triple integral equations. These equations are solved by the Schmidt method. This process is quite different from that papers adopted previously. The normalized stress and electrical displacement intensity factors are determined for different geometric and property parameters for permeable crack surface conditions. Numerical examples are provided to show the effect of the geometry of the interacting cracks, the thickness and the materials constants of the piezoelectric layer upon the stress and electric displacement intensity factors of the cracks. It is found that the electric displacement intensity factors for the permeable crack surface conditions are much smaller than the results for the impermeable crack surface conditions.     

THE BEHAVIOR OF TWO PARALLEL SYMMETRIC PERMEABLE CRACKS IN PIEZOELECTRIC MATERIALS

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Dynamic behavior of two unequal parallel permeable interface cracks in a piezoelectric layer bonded to two half piezoelectric materials planes

IntroductionDuetotheintrinsicelectro_mechanicalcouplingbehavior,piezoelectricmaterialsareveryusefulinelectronicdevices.However,mostpiezoelectricmaterialsarebrittlesuchasceramicsandcrystals.Therefore ,piezoelectricmaterialshaveatendencytodevelopcriticalcracksduringthemanufacturingandthepolingprocesses.So ,itisimportanttostudytheelectro_elasticinteractionandfracturebehaviorsofpiezoelectricmaterials.Theincreasingattentiontothestudyofcrackproblemsinpiezoelectricmaterialshasledtoalotofsignificantw…     

Basic solutions of multiple parallel symmetric mode-III cracks in functionally graded piezoelectric/piezomagnetic material plane

The Schmidt method is adopted to investigate the fracture problem of multiple parallel symmetric and permeable finite length mode-III cracks in a functionally graded piezoelectric/piezomagnetic material plane. This problem is formulated into dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces. In order to obtain the dual integral equations, the displacement jumps across the crack surfaces are directly expanded as a series of Jacobi polynomials. The results show that the stress, the electric displacement, and the magnetic flux intensity factors of cracks depend on the crack length, the functionally graded parameter, and the distance among the multiple parallel cracks. The crack shielding effect is also obviously presented in a functionally graded piezoelectric/piezomagnetic material plane with mul- tiple parallel symmetric mode-III cracks.     

折线型裂纹对SH波的动力响应

利用Fourier积分变换方法,得出了无限平面中用裂纹位错密度函数表示的单裂纹散射场．根据无穷积分的性质,把单裂纹的散射场分解为奇异部分和有界部分．利用单裂纹的散射场建立了折线裂纹在SH波作用下的Cauchy型奇异积分方程．根据折线裂纹散射场和所得的积分方程讨论了裂纹在折点处的奇性应力及折点处的奇性应力指数．利用所得的奇性应力定义了折点处的应力强度因子．对所得Cauchy型奇积分方程的数值求解,可得裂纹端点和折点处的动应力强度因子。     

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。     

Analytical solution of multiple moving cracks in functionally graded piezoelectric strip

The dynamic behaviors of several moving cracks in a functionally graded piezoelectric (FGP) strip subjected to anti-plane mechanical loading and in-plane electrical loading are investigated. For the first time, the distributed dislocation technique is used to construct the integral equations for FGP materials, in which the unknown variables are the dislocation densities. With the dislocation densities, the field intensity factors are determined. Moreover, the effects of the speed of the crack propagation on the field intensity factors are studied. Several examples are solved, and the numerical results for the stress intensity factor and the electric displacement intensity factor are presented graphically finally.     

THE BASIC SOLUTION OF A 3-D RECTANGULAR PERMEABLE CRACK IN A PIEZOELECTRIC/PIEZOMAGNETIC COMPOSITE MATERIAL

The solution of a 3-D rectangular permeable crack in a piezoelectric/piezomagnetic composite material was 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 relations between the electric filed,the magnetic flux field and the stress field near the crack edges were obtained and the efects of the shape of the rectangular crack on the stress,the electric displacement and magnetic flux intensity factors in a piezoelectric/piezomagnetic composite material were analyzed.     

MULTIPLE PARALLEL SYMMETRIC PERMEABLE MODEL-Ⅲ CRACKS IN A PIEZOELECTRIC/PIEZOMAGNETIC COMPOSITE MATERIAL PLANE

In this paper, the interactions of multiple parallel symmetric and permeable finite length cracks in a piezoelectric/piezomagnetic material plane subjected to anti-plane shear stress loading are studied by the Schmidt method.The problem is formulated through Fourier transform into 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 are directly expanded as a series of Jacobi polynomials.Finally, the relation between the electric field, the magnetic flux field and the stress field near the crack tips is obtained.The results show that the stress, the electric displacement and the magnetic flux intensity factors at the crack tips depend on the length and spacing of the cracks.It is also revealed that the crack shielding effect presents in piezoelectric/piezomagnetic materials.     

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.     

Moving interfacial crack between piezoelectric ceramic and elastic layers

An analysis is performed for the problem of a finite Griffith crack moving with constant velocity along the interface of a two-layered strip composed of a piezoelectric ceramic and an elastic layers. The combined out-of-plane mechanical and in-plane electrical loads are applied to the strip. Fourier transforms are used to reduce the problem to a pair of dual integral equations, which is then expressed in terms of a Fredholm integral equation of the second kind. The dynamic stress intensity factor(DSIF) is determined, and numerical results show that DSIF depends on the crack length, the ratio of stiffness and thickness, and the magnitude and direction of electrical loads as well as the crack speed. In case that the crack moves along the interface of piezoelectric and elastic half planes, DSIF is independent of the crack speed.     

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.