Electro-elastic fields of piezoelectric materials with an elliptic hole under uniform internal shearing forces

The existing investigations on piezoelectric materials containing an elliptic hole mainly focus on remote uniform tensile loads. In order to have a better understanding of the fracture behavior of piezoelectric materials under different loading conditions, theoretical and numerical solutions are presented for an elliptic hole in transversely isotropic piezoelectric materials subjected to uniform internal shearing forces based on the complex potential approach. By solving ten variable linear equations, the analytical solutions inside and outside the hole satisfying the permeable electric boundary conditions are obtained. Taking PZT-4 ceramic into consideration, numerical results of electro-elastic fields along the edge of the hole and axes, and the electric displacements in the hole are presented. Comparison with stresses in transverse isotropic elastic materials shows that the hoop stress at the ends of major axis in two kinds of material equals zero for the various ratios of major to minor axis lengths; If the ratio is greater than 1, the hoop stress in piezoelectric materials is smaller than that in elastic materials, and if the ratio is smaller than 1, the hoop stress in piezoelectric materials is greater than that in elastic materials; When it is a circle hole, the shearing stress in two materials along axes is the same. The distribution of electric displacement components shows that the vertical electric displacement in the hole and along axes in the material is always zero though under the permeable electric boundary condition; The horizontal and vertical electric displacement components along the edge of the hole are symmetrical and antisymmetrical about horizontal axis, respectively. The stress and electric displacement distribution tends to zero at distances far from the elliptical hole, which conforms to the conclusion usually made on the basis of Saint-Venant’s principle. Unlike the existing work, uniform shearing forces acting on the edge of the hole, and the distribution of electro-elastic fields     

Fracture behavior of two non-symmetrical collinear cracks emanating from an elliptical hole in a piezoelectric material

Based on the Stroh-type formalism and the technique of conformal mapping, the problem of two non-symmetrical collinear cracks emanating from an elliptical hole in a piezoelectric solid is investigated under remotely uniform in-plane electric loading and anti-plane mechanical loading, which allowed us to take the electric field inside the hole and cracks into account. The analytical solutions of the field intensity factors and the energy release rate are presented in closed-form, which includes the extreme cases for an impermeable crack and a permeable crack. Numerical results are then presented to reveal the effects of geometrical parameters, crack permeability and combined mechanical and electric loadings on the energy release rate.     

压电材料平面渗透裂纹的机电耦合场

为提高压电传感器和换能器的品质,针对材料缺陷导致的压电元器件失效或不稳定,用复变函数的方法,结合椭圆形夹杂内的电场强度和电位移为常量这一早期研究结果,研究了压电材料平面电渗透裂纹的机电耦合场及其奇异性。解答表明,切向电场强度和法向电位移在裂纹尖端有由机械载荷引起的奇异,而与电载荷无关;应力强度因子与纯弹性材料结果一致。     

An electrically saturated anti-plane shear crack in a piezoelectric layer constrained between two elastic layers

Within the framework of linear elasticity and nonlinear electroelasticity, the electrical nonlinear behavior of an anti-plane shear crack in a piezoelectric ceramic layer constrained between two orthotropic layers is examined by using an electrical strip saturation model. The analysis is performed based on the electrical unified crack surface condition, which can describe all of the permeable, the impermeable, and the limited permeable ones. Fourier transforms are used to reduce the problem to the solutions of two pairs of dual integral equations, which are then expressed to two Fredholm integral equations of the second kind. Some numerical results for the energy release rates are presented to show the effects of the electrical crack condition parameter relating with elliptic flaw shape parameters and permittivity inside the crack, the electric field, the crack length, the electromechanical coupling coefficient, the crack position, and the bonded elastic materials.     

压电材料中螺型位错和偶极子与半椭圆槽表面裂纹的干涉效应

运用弹性复势方法,研究纵向剪切和面内电场共同作用下无限半平面压电材料中螺型位错和偶极子与半椭圆槽表面裂纹的电弹干涉效应,得到该问题复势函数的封闭形式解答,并由此导出广义应力场、裂纹尖端的广义应力强度因子以及作用在螺型位错上的位错力.算例结果表明:增大材料压电常数会相应增大位错力、位错对裂纹尖端的屏蔽和反屏蔽效应;增大位错离裂纹尖端的距离,位错力会相应减小;变化φ(偶极子臂与x轴正半轴夹角)值会出现一个改变位错偶极子对应力强度因子作用方向的临界值.     

Enriched finite element method for three-dimensional viscoelastic interface crack problems

The enriched finite element method is developed for three-dimensional problems of an interface crack between elastic and viscoelastic (including dissimilar viscoelastic) materials. According to the displacement fields of elastic interface crack, the displacement fields of viscoelastic interface crack are derived through the correspondence principle. By incorporating the displacement expressions into the displacement model of regular element, the incremental formulations of enriched element are derived. The stress intensity factors and strain energy release rates can be solved based on the enriched degree of freedoms. A 3-D through interface crack at the center of jointed dissimilar viscoelastic plate subjected to remote tension and a quarter-circular viscoelastic interface corner crack subjected to uniform thermal loading was investigated using the enriched finite element method. It is shown that the present solutions are consistent with the analytical solutions, which indicates the present method is correct and efficient.

     

Dynamic response of an anti-plane shear crack in a functionally graded piezoelectric strip

The dynamic response of a cracked functionally graded piezoelectric material (FGPM) under transient anti-plane shear mechanical and in-plane electrical loads is investigated in the present paper. It is assumed that the electroelastic material properties of the FGPM vary smoothly in the form of an exponential function along the thickness of the strip. The analysis is conducted on the basis of the unified (or natural) crack boundary condition which is related to the ellipsoidal crack parameters. By using the Laplace and Fourier transforms, the problem is reduced to the solutions of Fredholm integral equations of the second kind. Numerical results for the stress intensity factor and crack sliding displacement are presented to show the influences of the elliptic crack parameters, the electric field, FGPM gradation, crack length, and electromechanical coupling coefficient.     

Analysis of three collinear antiplane interfacial cracks in dissimilar piezoelectric materials under non-self equilibrated electromechanical loadings on a center crack

The problem of three collinear interfacial cracks between two dissimilar transversely isotropic piezoelectric materials is considered under electromechanical loadings. The crack surfaces are assumed to be impermeable to the electric field. A single antiplane mechanical and inplane electrical loads are applied at a point on centred crack surface. The problem is formulated by the complex function method, and reduced to the vector Hilbert problem. By solving the problem, a closed form solution for the stress intensity and electric displacement intensity factor is obtained. This solution can be used as a Green’s function for different loading conditions.     

Postbuckling of cross-ply laminated cylindrical shells with piezoelectric actuators under complex loading conditions

A postbuckling analysis is presented for a cross-ply laminated cylindrical shell with piezoelectric actuators subjected to the combined action of mechanical, electric and thermal loads. The temperature field considered is assumed to be a uniform distribution over the shell surface and through the shell thickness and the electric field is assumed to be the transverse component E_z only. The material properties are assumed to be independent of the temperature and the electric field. The governing equations are based on the classical shell theory with a von Kármán–Donnell-type of kinematic nonlinearity. The nonlinear prebuckling deformations and initial geometric imperfections of the shell are both taken into account. A boundary layer theory of shell buckling, which includes the effects of nonlinear prebuckling deformations, large deflections in the postbuckling range, and initial geometric imperfections of the shell, is extended to the case of hybrid laminated cylindrical shells. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of perfect and imperfect, cross-ply laminated cylindrical thin shells with fully covered or embedded piezoelectric actuators subjected to combined mechanical loading of external pressure and axial compression, and under different sets of thermal and electric loading conditions. The effects played by temperature rise, applied voltage, shell geometric parameter, stacking sequence, as well as initial geometric imperfections are studied.     

Mode I field intensity factors of infinitely long strip in piezoelectric media

We consider the problem of determining the singular stresses and electric fields in a piezoelectric ceramic strip containing a Griffith crack under in-plane normal loading within the framework of linear piezoelectricity. The potential theory method and Fourier transforms are used to reduce the problem to the solution of dual integral equations, which are then expressed to a Fredholm integral equation of the second kind. Numerical values on the field intensity factors are obtained, and the influences of the electric fields for PZT-6B piezoelectric ceramic are discussed.     

Eccentric crack in a piezoelectric strip under electro-mechanical loading

We consider the problem of determining the singular stresses and electric fields in a piezoelectric ceramic strip containing a Griffith eccentric crack off the center line under anti-plane shear loading with the theory of linear piezoelectricity. Fourier transforms are used to reduce the problem to the solution of two pairs of dual integral equations, which are then expressed to a Fredholm integral equation of the second kind. Numerical values on the stress intensity factor and the energy release rate are obtained, and the influences of the electric fields for piezoelectric ceramics are discussed.     

Stress and electric potential fields in piezoelectric smart spheres

Piezoelectric materials produce an electric field by deformation, and deform when subjected to an electric field. The coupling nature of piezoelectric materials has acquired wide applications in electric-mechanical and electric devices, including electric-mechanical actuators, sensors and structures. In this paper, a hollow sphere composed of a radially polarized spherically anisotropic piezoelectric material, e.g., PZT_5 or (Pb) (CoW) TiO₃ under internal or external uniform pressure and a constant potential difference between its inner and outer surfaces or combination of these loadings has been studied. Electrodes attached to the inner and outer surfaces of the sphere induce the potential difference. The governing equilibrium equations in radially polarized form are shown to reduce to a coupled system of second-order ordinary differential equations for the radial displacement and electric potential field. These differential equations are solved analytically for seven different sets of boundary conditions. The stress and the electric potential distributions in the sphere are discussed in detail for two piezoceramics, namely PZT_5 and (Pb) (CoW)TiO₃. It is shown that the hoop stresses in hollow sphere composed of these materials can be made virtually uniform across the thickness of the sphere by applying an appropriate set of boundary conditions.     

三维压电介质界面裂纹的边界积分-微分方程

基于三维两相横观各向同性压电介质的基本解和压电介质的Somigliana恒等式，利用发散积分的有限部理论，建立以裂纹面上的不连续位移和不连续电势为基本未知量的三维压电介质界面裂纹问题的超奇异积分－微分方程组，其中的积分核具有O（1/r^2）阶的奇异性。当两相材料退化为均质材料或单相材料时，方程组中的微分项的系数为零，从而积分－微分方程组退化为已有的均质压电材料的超奇异边界积分方程。     

Anti-plane shear crack in a functionally gradient piezoelectric layer bonded to dissimilar half spaces

In this paper, the problem of a crack located in a functionally gradient piezoelectric interlayer between two dissimilar homogeneous piezoelectric half-planes being subjected to an anti-plane mechanical loading and an in-plane electric loading is considered. The material properties of the interlayer, such as the elastic stiffness, piezoelectric constant and dielectric constant, are assumed to vary continuously along the thickness of the interlayer, and the crack surface condition is assumed to be impermeable or permeable. By using the Fourier transform, the problem is first reduced to two pairs of dual integral equations and then into a Fredholm integral equation of the second kind. Numerical calculations are carried out, and the effects of crack geometric parameters on the stress intensity factor and the energy release rate are shown graphically.     

Stress field near circular-arc interface crack tip based on electric saturation concept

Within the framework of nonlinear electroelasticity, the anti-plane problem of a circular-arc interfacial crack between a circular piezoelectric inhomogeneity and an infinite piezoelectric matrix subjected to a far-field uniform loading is investigated by an electrical strip saturation model, the complex variable method, and the method of analytical continuation. Explicit closed form expressions for the complex potentials in both the matrix and the inclusion, and the stress intensity factor at the crack tip are presented. Comparison with some related solutions based on the linear electroelastic theory shows the validity of the present solutions     

弹性半空间中平片裂纹问题的超奇异积分方程方法

使用边界积分方程方法,在有限部积分的意义下,将弹性半空间中垂于自边界面的平片裂纹归结为一组以裂纹面位移间数为示知函数的超奇异积仞氖限部积分蜞 建立了数值人出了用裂纹面位移间尖力强度因子的公式。通过对圆形、菜和矩形等贡型的平片裂纹问题的计算,分析了自由边界面对裂纹前沿应力强度因子的影响。     

Exact solution of a compositionally graded piezoelectric layer under uniform stretch, bending and twisting

Electromechanical responses of compositionally graded piezoelectric layers are analysed. The layers consist of polycrystalline piezoelectric ceramics poled along the thickness direction, and thus exhibit material symmetry of a hexagonal crystal in class 6mm. Two cases for layers (i) covered by surface electrodes and (ii) without surface electrodes are considered. Analytical solutions are exact in Saint Venant's sense for both cases. However, solutions are obtained for layers subjected to uniform mechanical loads, including stretch, bending and twisting. Numerical results to show the effects of different compositional gradients are presented.     

稳态温度场中螺旋位错与圆弧裂纹的交互作用

研究压电材料在稳态温度场下螺旋位错与圆弧裂纹的交互作用.综合运用复变函数分区全纯理论、解析函数奇性主部分析方法、解析延拓原理、Cauchy型积分以及Riemann边值问题求解方法,导出各复势函数之间的解析关系,进一步得到特殊情况下所讨论问题的封闭解,并解出像力随温度梯度和位错位置变化的表达式.数值分析结果表明,温度梯度越大,像力变化越明显.     

压电介质中裂纹与邻近微孔洞的交互作用

基于线性压电理论，文中对裂纹和微孔洞的交互作用进行有限元计算分析。结果表明，(1)电场对裂纹启裂起促进还是抑止作用，与外加电场方向和裂纹面电学边界条件密切相关。(2)电-力载荷比、电学边界条件和微孔洞裂纹之间的韧带长度对裂纹前缘应力场均有一定影响，但无论对于何种裂纹电学边界条件(可导或绝缘)，裂纹前端的微孔洞只有在韧带长度小于2倍微孔洞直径时，才对裂纹前缘应力场有明显的影响。(3)在一定的电-力加载比和一定的韧带长度范围内，微孔洞对裂纹的启裂有屏蔽作用，且微孔洞对导通裂纹的屏蔽作用比绝缘裂纹要明显得多。此外，文中还给出韧带上张开应力的分布。     

On the stress concentration around a hole in an infinite plate subject to a uniform load at infinity

In this paper stress concentration around a hole in an infinite plate that is subjected to a uniform load at infinity is considered. The stress is calculated by using a modified Muskhelishvili complex variable method. The method is illustrated by several examples of stress distribution around polygonal holes of a complex geometry utilizing the Schwartz-Chistoffel mapping function.