An exact and explicit treatment of an elliptic hole problem in thermopiezoelectric media

This paper presents an exact solution for the problem of an elliptic hole or a crack in a thermopiezoelectric solid. First, based on the extended version of Eshelby–Stroh's formulation, the generalized 2D problems of an elliptical hole in a thermopiezoelectric medium subject to uniform heat flow and mechanical–electrical loads at infinity are studied according to exact boundary conditions at the rim of the hole. The complex potentials in the medium and the electric field inside the hole are obtained in closed form, respectively. Then, when the hole degenerates into a crack, the explicit solutions for the field intensity factors near the crack tip and the electric field inside the crack are presented. It is shown that the singularities of all the field are dependent on the material constants, the applied heat load and mechanical loads at infinity, but not on the applied electric loads. It is also found that the electric field inside the crack is linearly variable, which is different from the result based on the impermeable crack model.     

Solution of a flat elliptical crack in an electrostrictive solid

Based on the assumption that the elastic strain of electrostrictive materials is a higher-order small quantity, this paper studies the 3D problem of an infinite electrostrictive solid with a flat elliptical crack which is electrically permeable. According to existing solutions of similar problems in pure elastic materials, with the displacement function method, we first derived explicit expression for displacement potential function and obtained stress field near the crack and open displacement of crack surface. Then, the general solution for the stress intensity factor was derived, and the corresponding solutions were also presented for a penny-shaped crack and a permeable line-crack as two special cases of the present problem. Finally, numerical results were given to discuss the effect of environment at infinity and electric field inside the crack on the stress-intensity factors.     

Interaction between bending and mobile charges in a piezoelectric semiconductor bimorph

We study the bending of a two-layer piezoelectric semiconductor plate (bimorph). The macroscopic theory of piezoelectric semiconductors is employed. A set of two-dimensional plate equations is derived from the three-dimensional equations. The plate equations exhibit direct couplings among bending, electric polarization along the plate thickness, and mobile charges. In the case of pure bending, a combination of physical and geometric parameters is identified which characterizes the strength of the interaction between the mechanical load and the distribution of mobile charges. In the bending of a rectangular plate under a distributed transverse mechanical load, it is shown that mobile charge distributions and potential barriers/wells develop in the plate. When the mechanical load is local and self-balanced, the induced carrier distributions and potential barriers/wells are also localized near the loading area. The results are fundamentally useful for mechanically manipulating mobile charges in piezoelectric semiconductor devices.     

An analytical model for electrode–ceramic interaction in multilayer piezoelectric actuators

The present paper develops an analytical model for multi-electrodes in multi-layered piezoelectric actuators, in which the electrodes are vertical to and terminated at the edges of the medium and electroelastic field concentrations ahead of the electrodes in the multilayer piezoelectric actuators are examined. By considering a representative unit in realistic multilayers, the problem is formulated in terms of electric potential between the electrode tips and results in a system of singular integral equations in which the electric potential is taken as unknown function. Effects are investigated of electrode spacing and piezoelectric coupling on the singular electroelastic fields at the electrode tips, and closed-form expressions are given for the electromechanical field near the electrode tips. Exact solution for un-coupled dielectrics is provided, where no piezoelectric coupling is present. The English text was polished by Yunming Chen.     

Electric field in the transverse cross section of the channel of an MHD generator

During the motion of a partially ionized gas in magnetohydrodynamic channels the distribution of the electrical conductivity is usually inhomogeneous due to the cooling of the plasma near the electrode walls. In Hall-type MHD generators with electrodes short-circuited in the transverse cross section of the channel the development of inhomogeneities results in a decrease of the efficiency of the MHD converter [1]. A two-dimensional electric field develops in the transverse section. Numerical computations of this effect for channels of rectangular cross section have been done in [2, 3], At the same time it is advisable to construct analytic solutions of model problems on the potential distribution in Hall channels, which would permit a qualitative analysis of the effect of the inhomogeneous conductivity on local and integral characteristics of the generators. In the present work an exact solution of the transverse two-dimensional problem is given for the case of a channel with elliptical cross section stretched along the magnetic field. The parametric model of the distribution of the electrical conductivity of boundary layer type has been used for obtaining the solution. The dependences of the electric field and the current and also of the integral electrical characteristics of the generator on the inhomogeneity parameters are analyzed.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 3–10, January–February, 1973.     

Theoretical investigation of an elliptical crack in thermopiezoelectric material. Part I: Analytical development

An elliptical crack subjected to uniform temperature in a thermopiezoelectric solid is investigated in this two-part paper. Part I of the paper is devoted to analytical solutions of stresses and electric displacements near the crack front. The explicit expression of the field intensity factor is given out. The asymptotic fields near the crack front are presented as well.     

Nonlinear magnetoelastic coupling effects in a soft ferromagnetic material with a crack

In this paper, the magnetoelastic coupling effect in an infinite soft ferromagnetic material with a crack is analyzed. The nonlinear effect of magnetic field upon stress and the effect of the deformed crack configuration are taken into consideration. The coupling field is determined in the deformed configuration by regarding the deformed crack as an elliptical cylinder with its geometric coefficients, which are determined from a set of algebraic equations deduced from the displacements. The magnetic and stress fields near the crack tip are discussed for the case where both of the magnetic loading and the mechanical tension are present.     

双压电材料内含椭圆孔孔边界面裂纹的反平面问题

利用复变函数方法,研究了横观各向同性压电双材料中椭圆孔孔边界面裂纹的反平面问题．首先,利用保角变换函数将椭圆孔保角变换到一直线裂纹;其次,基于孔边及裂纹表面均电不可穿透并且自由的假设,利用Stroh公式分别得到了本问题的复势函数、裂尖场集中系数的解析表达式;最后,在面内电载荷及面外机械载荷的作用下,分析了椭圆孔尺寸、裂纹长度和外载对裂尖场集中系数的影响,并得到了一个有意义的结论：椭圆孔一边裂纹长度的改变对另一边裂纹裂尖场的影响有限,然而一旦椭圆孔退化为竖直裂纹,该影响将变得非常显著．     

压电材料中椭圆片状裂纹问题精确解的一种方法

在线性压电陶瓷本构关系和裂纹边界绝缘的框架下,用超奇异积分方程的方法对椭圆类片状裂纹问题进行了重新研究．超奇异积分方程中的未知位移间断和电势间断近似地表示为基本密度函数与多项式之积,其中基本密度函数反映了椭圆片状裂纹前沿电弹性场的奇异性,而多项式在均布载荷作用下可用一个常数来表达．引入椭球坐标系后,得到了均布载荷作用下未知位移间断和电势间断的解析解．使用这些解析解和电弹性场强度的定义,得到了裂纹前沿Ⅰ型、Ⅱ型和Ⅲ型应力强度因子以及电位移强度因子的精确表达式．法向均布载荷作用下的结果与现有精确解完全一致,切向均布载荷作用下的结果则尚未见有其它报道．     

弯曲波在含非圆孔洞无限压电薄板中的散射

基于线性压电动力学理论,采用波函数展开法、保角映射以及复变函数,对含非圆孔洞无限大压电薄板弹性波的散射及动应力集中问题进行了分析,给出了其动弯矩集中系数（DMCF）的解析表达式。为说明问题,以PZT-4为例,讨论了外加电场、椭圆孔长短半轴比、椭圆孔倾角以及入射波频率对含圆孔和椭圆孔无限大压电薄板弹性波散射的影响,并分别给出了无限压电薄板开圆孔和椭圆孔动弯矩集中系数的数值结果。     

The behavior of two non-symmetrical permeable cracks emanating from an elliptical hole in a piezoelectric solid

Based on the Stroh-type formalism, we present a concise analytic method to solve the problem of complicated defects in piezoelectric materials. Using this method 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. The exact solutions of the field intensity factors and the energy release rate are presented in closed-form under the permeable electric boundary condition. With the variation of the geometrical parameters, the present results can be reduced to the well-known results of a mode-III crack in piezoelectric materials. Moreover, new special models used for simulating more practical defects in a piezoelectric solid are obtained, such as two symmetrical edge cracks and single edge crack emanating from an elliptical hole or circular hole, T-shaped crack, cross-shaped crack, and semi-infinite plane with an edge crack. Numerical results are then presented to reveal the effects of geometrical parameters and the applied mechanical loading on the field intensity factors and the energy release rate.     

陶瓷基复合材料基体唇形裂纹失效规律研究

建立横向拉伸载荷下的唇形裂纹数学模型,采用复变函数的方法,通过保角映射,推导了唇形裂纹尖端应力场和位移场的解析解,建立了唇形裂纹的应力强度因子准则和最大能量释放率准则,结合算例分析陶瓷基复合材料基体唇形裂纹的几何参数、外载荷和纤维分布对失效准则的影响规律.结果 表明,(1)裂纹尖端应力场和位移场的解析解与有限元计算结果进行对比,验证了方法的有效性;(2)相较于Griffith裂纹和椭圆裂纹,基于唇形裂纹的失效准则对裂纹尖端的敏感性更高,适用于预测脆性陶瓷基体裂纹的扩展;(3)对于不同几何参数的唇形裂纹,采用最大能量释放率准则的基体裂纹的扩展速率要大于应力强度因子准则.     

含任意椭圆核各向异性板杂交应力有限元

基于含椭圆核有限大各向异性板弹性问题的复变函数级数解,应用杂交变分原理建立了一种与常规有限元相协调的含任意椭圆核各向异性板杂交应力有限元.单元内的应力场和位移场采用满足平衡方程、几何方程与物理方程的复变函数级数解,假设的复变函数级数解精确满足椭圆核边界处的位移协调条件和应力连续条件,单元外边界上的位移场按常规有限元位移场假设,单元内椭圆核的长轴可以与材料主轴不重合.单元刚度矩阵采用Gauss积分求得,并给出了建立刚度矩阵的主要公式和推倒过程.数值计算结果表明该单元具有计算精度高、计算工作量小等优点.     

A small-scale magnetic-yielding model for an infinite magnetostrictive plane with a crack-like flaw

The effect of an external magnetic field on the fracture toughness of magnetostrictive materials has been investigated by determining the local stress fields around the tip of a very slender elliptical flaw embedded in an infinite magnetostrictive plane subjected to magnetic loading, based on the assumption of linear magnetization. In this paper, the above-mentioned analytical approach is extended to develop a small-scale magnetic-yielding model. The magnetic saturation zone is constructed and the distributions of magnetic field and magnetization are obtained around the tip of a slender elliptical crack. Based on the complex potential theory, the stress field is obtained in the vicinity of the tip of the slender elliptical crack by implementing the continuity conditions of displacement and resultant force at the interface between the magnetic saturation and magnetoelastic zones. The stress fields near the tip of the slender elliptical crack are obtained for two kinds of soft ferromagnetic materials each with a small induction magnetostrictive modulus. The theoretical results obtained show that the stresses in the neighborhood of a crack-tip are finite even when the elliptical crack reduces to a sharp crack, and are much smaller than the yield stress or the nominal fracture stress of the material. This suggests that, generally, the magnetic field has no obvious effects on the apparent fracture toughness of soft ferromagnetic materials, which is in agreement with the existing experimental results published in the existing literature. In addition, the theoretical analysis illustrates that no crack is magnetically impermeable, and the corresponding boundary conditions are inappropriate for fracture analysis of soft ferromagnetic materials.     

层合圆柱壳体表层局部分层屈曲及其扩展

应用能量法研究了对称铺层的闭合圆柱形层合壳体的表层局部分层屈曲问题；表层分层形式有三角形，椭圆和双纽线；分析了母层壳，子层壳的几何参数，物理参数同分层屈曲临界应变值之间的关系；给出了子层分层为椭圆的分层扩展的可能发生方向。     

三相压电复合本构模型中的弧形界面裂纹

深入研究了三相同心圆柱压电复合本构模型中的弧形绝缘界面裂纹问题。采用复势方法获得了该问题的级数形式的解答,并给出了应力、应变、电位移和电场强度等物理量在全场及界面上的分布,同时推导了裂尖处广义强度因子及裂面张开位移和裂面上电势差的表达式。具体计算表明该级数解答收敛迅速,同时显示出第三相混杂区的影响是不能忽略的。由于裂尖处应力奇异性为－1／2,则这种解答不会出现平面应变状态下界面裂纹裂尖处的振荡奇异性,从而不会产生违反物理实际的裂面相互嵌入现象,则该弹性解答也是建立了坚实的物理基础之上。     

导电圆柱薄壳的热磁弹性效应分析

研究了磁场环境中受机械载荷作用的导电圆柱薄壳的热磁弹性问题．首先,根据电动力学方程和广义Ohm定律,得到了导电薄壳电流密度的分布,考虑到Joule热效应及热平衡方程,得到了导电薄壳的温度分布．其次,通过几何方程、物理方程、运动方程和电动力学方程导出了导电薄壳在机械场、电磁场以及温度场作用下的基本方程．最后,采用差分法及准线性化方法,得到了可以应用离散正交法求解的准线性微分方程组．对于导电圆柱薄壳,得到了Lorentz力表达式,并且推导了温度场积分特征值．讨论了导电圆柱薄壳应力、温度及变形随外加电磁参量的变化规律,并通过实例证实了可以通过改变电、磁、力场的参数来实现对薄壳的应力、应变、温度的控制．     

Electroelastic field concentrations ahead of electrodes in multilayer piezoelectric actuators: experiment and finite element simulation

The effects of applied voltage on the electroelastic field concentrations ahead of electrodes in multilayer piezoelectric actuators were examined in a combined experimental and numerical investigation. Experiments were performed to measure the strain near internal and surface electrodes at various electrical loading conditions. The finite element method was also used to solve the coupled electro-elastic boundary value problem. The strain, stress and electric displacement concentrations were calculated and a non-linear behavior induced by localized polarization switching was discussed. A comparison of strain concentration was made between experiment and simulation.     

A unified treatment of the elastic elliptical inclusion under antiplane shear

Summary A generalized and unified treatment is presented for the antiplane problem of an elastic elliptical inclusion undergoing uniform eigenstrains and subjected to arbitrary loading in the surrounding matrix. The general solution to the problem is obtained through the use of conformal mapping technique and Laurent series expansion of the associated complex potentials. The resulting elastic fields are derived explicitly in both transformed and physical planes for the inclusion and the surrounding matrix. These relations are universal in the sense of being independent of any particular loading as well as the geometry of the matrix. The complete field solutions are provided for an elliptical inclusion under uniform loading at inifinity, and for a screw dislocation interacting with the elastic elliptical inclusion.     

Three-dimensional analysis of defects in a piezoelectric material

In this paper, the Green's function technique is used to develop a solution of an infinite, piezoelectric medium containing either an ellipsoidal cavity or a flat elliptical crack. The coupled elastic and electric fields both inside the cavity and on the boundary of the cavity are obtained, and the stress intensity factor and the electric field intensity factor are also obtained for an elliptical crack. It is found that; (1) the coupled elastic and electric fields inside the cavity keep uniform when the external elastic field and electric field are constant; (2) the behavior of the stress and electric field components in the neighborhood of the crack tip shows the classical type of singularity. The project supported by National Natural Science Foundation of China