压电陶瓷中圆币形裂纹在横向剪力下的机—电耦合行为

以弹性位移分量和电热函数基本未知量时，横观各向同性压电介质三维问题的场方程可化为四个联立的二阶线性偏微分方程组，本文导出了用四个调和函数表示位移分量及电势函数的表达式，即得到了该场方程的势函数能通解，作为通解的应用举例，文中求解了圆币形裂纹受横向剪切载荷下圆币形裂纹的尖端场及应力、电位移强度因子均具有明显的机－电耦合性质，而应力和电位移分量在裂尖仍具有－１／２的奇异性。     

压电介质中受拉伸与弯曲联合作用的圆币形裂纹问题

以弹性位移分量和电势函数为基本未知量时，横观各向同性压电介质非轴对称三维问题的控制微分方程是四个二阶线性偏微分方程相联立的方程组。本文导出了用四个调和函数表示位移及电势的该方程组的势函数通解。作为通解的应用举例，文中求解了压电陶瓷材料中受拉伸与弯曲联合作用的圆币形裂纹问题，得到了裂纹尖端附近应力场及电位移场的解析表达式。结果表明裂尖场以及应力强度因子和电位移强度因子均表现出复杂的机－电耦合行为。     

热释电材料问题的通解与界面裂纹

该文讨论了热释电材料中的热弹性问题的一般解，进而求解了共线界面裂纹问题．利用Ｓｔｒｏｈ方法，把热释电材料的热弹性界面裂纹问题化为一向量形式的Ｈｉｌｂｅｒｔ问题，求出这一Ｈｉｌｂｅｒｔ问题的通解，进而求得了热释电材料热弹性界面裂纹的闭合解，得到了温度、热流、位移、电势、应力和电位移的全场解，得到了裂纹张开位移及电势差的精确表达式．在此基础上，还求得了均匀热释电体中单个热弹性裂纹裂尖场，单个界面裂纹裂尖场以及点热源与界面裂纹的作用．此外，该文还对界面裂纹顶点附近的端部场作了渐近分析．     

十次对称二维准晶中的热应力分析

推导了当考虑热效应时十次对称二维准晶体平面应变问题的通解表示．作为应用，采用所获得的通解首先得到了十次对称二维准晶体中的一个点热源所引起的声子场和相位子场，给出了点热源所引起的声子场和相位子场应力分量的解析表达式；接着获得了在均匀热流作用下十次对称二维准晶体中-绝缘椭圆孔洞所引起的热应力问题的弹性解答，给出了沿椭圆边界环向应力分布的解析表达式；当椭圆的短轴趋于零时，则获得了裂纹问题的解答，给出了应力强度因子、裂纹表面张开位移及能量释放率的解析表达式；推导了在任意热载荷作用下裂尖附近的渐近场．     

压电体中裂纹与孤立电偶极子的相互作用

研究压电体裂纹与电偶极子的相互作用,得到问题的闭合解,包括应力-电位移场,裂纹张开位移和电势差,以及裂尖应力强度因子,结果表明,电偶极子的方向对裂纹场的影响可由压电体各向异性方向函数表示;当电偶极子位于裂尖附近时,在原点取在裂尖的局部极坐标系中电偶极子位置的极角对裂尖场的影响可由各向异性方向函数表示,电偶极子引起的裂尖应力强度因子与其距裂尖的距离的-3/2次幂成正比。     

应用半权函数法计算界面裂纹的应力强度因子

应用半权函数法求解双材料界面裂纹的应力强度因子，得到以半权函数对参考位移与应力加权积分的形式表示的应力强度因子。针对特征值为复数λ的双材料界面裂纹裂尖应力和位移场，设置与之对应特征值为-λ的位移函数，即半权函数。半权函数的应力函数满足平衡方程，应力应变关系，界面的连续条件以及在裂纹面上面力为0；半权函数与裂纹体的几何尺寸无关，对边界条件没有要求。由功的互等定理得到应力强度因子KⅠ和KⅡ的积分形式表达式。本文计算了多种情况下界面裂纹应力强度因子的算例，与文献结果符合得很好。由于裂尖应力的振荡奇异性已经在积分中避免，只需考虑绕裂尖远场的任意路径上位移和应力，即使采用该路径上较粗糙的参考解也可以得到较精确的结果。     

压电材料中三维I型断裂力学分析

讨论了不可导通情况下三维横观各向刚性压电材料中受拉伸和电载荷作用的平片裂纹Ⅰ型断裂力学问题．使用自限部分概念，从二维线性压电理论出发，严格得到了一组以裂纹面位移间断和电势间断为未知变量的超奇异积分方程组；应用二维超奇异积分的主部分析法，从理论上分析得到了裂纹前沿应力和电势奇性指数以及应力和电位移奇性场，从而找到了以裂纹面位移间断和电势间断表示的应力和电位移强度因子、能量释放率表达式；为所得到的超奇异积分方程组建立了数值法，并用此计算了若干典型的平片裂纹问题，数值结果令人满意．     

压电材料中三维I型断裂力学分析

讨论了不可导通情况下三维横观各向同性压电材料中受拉伸和电载荷作用的平片裂纹I型断裂力学问题. 使用有限部积分概念，从三维线性压电理论出发，严格得到了一组以裂纹面位移间断和电势间断为未知变量的超奇异积分方程组；应用二维超奇异积分的主部分析法，从理论上分析得到了裂纹前沿应力和电势奇性指数以及应力和电位移奇性场，从而找到了以裂纹面位移间断和电势间断表示的应力和电位移强度因子、能量释放率表达式；为所得到的超奇异积分方程组建立了数值法，并用此计算了若干典型的平片裂纹问题，数值结果令人满意.     

双压电体界面上的电偶极子和裂纹5

求得了压电体双材料界面上的孤立二维电偶极子的解析解，结果表明某点电偶极子激发的应力－电位移场与该点到电偶极子的距离的平方成反比。研究了压电体双材料界面上的电偶极子对裂纹的作用，得到了问题的闭合解。在电偶极子的作用下，界面裂纹裂尖近区应力－电位移仍具有r^-1/2 iεα的振荡奇异性，文中求得裂尖应力强度因子，当电偶极子距裂尖距离ρ很近时，裂尖应力强度因子与ρ^-3/2-iεα成比例。     

压电材料平面裂纹尖端场的杂交应力有限元分析

基于复势理论和杂交变分原理建立了一种适用于力电耦合分析的杂交应力有限元模 型. 给出了建立刚度矩阵的主要公式和推导过程，单元内的位移场和应力场采用满足平 衡方程的复变函数级数解，假设的复变函数级数解事先精确满足裂纹的无应力和电位移法向 分量为零的条件，单元外边界的位移场假设按抛物线变化, 单元的刚度矩阵采用Gauss积分的方法得出. 通过对力电耦合裂尖场的数值计算验证了程序 的正确性和单元的有效性，同时也用所得结果校验了理论解.     

Penny-shaped crack in transversely isotropic piezoelectric materials

Using a method of potential functions introduced successively to integrate the field equations of three-dimensional problems for transversely isotropic piezoelectric materials, we obtain the so-called general solution in which the displacement components and electric potential functions are represented by a singular function satisfying some special partial differential equations of 6th order. In order to analyse the mechanical-electric coupling behaviour of penny-shaped crack for above materials, another form of the general solution is obtained under cylindrical coordinate system by introducing three quasi-harmonic functions into the general equations obtained above. It is shown that both the two forms of the general solutions are complete. Furthermore, the mechanical-electric coupling behaviour of penny-shaped crack in transversely isotropic piezoelectric media is analysed under axisymmetric tensile loading case, and the crack-tip stress field and electric displacement field are obtained. The results show that the stress and the electric displacement components near the crack tip have (r ^−1/2) singularity. The project supported by the Natural Science Foundation of Shaanxi Province, China     

功能梯度压电材料反平面裂纹问题

基于三维弹性理论和压电理论，导出了材料系数在横观各向同性平面内梯度分布的压电体的状态方程，进而对材料系数指数函数规律分布的半无限大压电体中的反平面裂纹问题进行了求解，利用Fourier变换给出了半无限大压电体中位移，应力，电势及电位移的解析表达式，并求得了裂纹尖端的应力强度因子和电位移强度因子，分析了不同的非均匀材料系数及几何尺寸对它们的影响。     

Three-dimensional analytical solution for a rotating disc of functionally graded materials with transverse isotropy

Based on the basic equations for axisymmetric problems of transversely isotropic elastic materials, the displacement components are expressed in terms of polynomials of the radial coordinate with the five involved coefficients, named as displacement functions in this paper, being undetermined functions of the axial (thickness) coordinate. Five equations governing the displacement functions are then derived. It is shown that the displacement functions can be found through progressive integration by incorporating the boundary conditions. Thus a three-dimensional analytical solution is obtained for a transversely isotropic functionally graded disc rotating at a constant angular velocity.The solution can be degenerated into that for an isotropic functionally graded rotating disc. A prominent feature of this solution is that the material properties can be arbitrary functions of the axial coordinate. Thus, the solution for a homogeneous transversely isotropic rotating disc is just a special case that can be easily derived. An example is finally considered for a special functionally graded material, and numerical results shows that the material inhomogeneity has a remarkable effect on the elastic field.     

Polynomial approach for modeling a piezoelectric disc resonator partially covered with electrodes

The frequency spectrum of a partially metallized piezoelectric disc resonator was studied using Legendre polynomials. The formulation, based on three-dimensional equations of linear elasticity, takes into account the high contrast between the electroded and non-electroded regions. The mechanical displacement components and the electrical potential were expanded in a double series of orthonormal functions and were introduced into the equations governing wave propagation in piezoelectric media. The boundary and continuity conditions were automatically incorporated into the equations of motion by assuming position-dependent physical material constants or delta-functions. The incorporation of electrical sources is illustrated. Structure symmetry was used to reduce the number of unknowns. The vibration characteristics of the piezoelectric discs were analyzed using a three-dimensional modelling approach with modal and harmonic analyses. The numerical results are presented as resonance and anti-resonance frequencies, electric input admittance, electromechanical coupling coefficient and field profiles of fully and partially metallized PIC151 and PZT5A resonator discs. In order to validate our model, the results obtained were compared with those published previously and those obtained using an analytical approach.     

横观各向同性压电材料中裂纹问题的边界元法

采用Somigiliana公式给出了三维横观各向同性压电材料中的非渗漏裂纹问题的一般解和超奇异积分方程,其中未知函数为裂纹面上的位移间断和电势间断．在此基础上,使用有限部积分和边界元结合的方法,建立了超奇异积分方程的数值求解方法,并给出了一些典型数值算例的应力强度因子和电位移强度因子的数值结果,结果令人满意．     

A coupled electromechanical analysis of a piezoelectric layer bonded to an elastic substrate: Part I,development of governing equations

This two-part contribution presents a novel and efficient method to analyze the two-dimensional (2-D) electromechanical fields of a piezoelectric layer bonded to an elastic substrate, which takes into account the fully coupled electromechanical behavior. In Part I, Hellinger–Reissner variational principle for elasticity is extended to electromechanical problems of the bimaterial, and is utilized to obtain the governing equations for the problems concerned. The 2-D electromechanical field quantities in the piezoelectric layer are expanded in the thickness-coordinate with seven one-dimensional (1-D) unknown functions. Such an expansion satisfies exactly the mechanical equilibrium equations, Gauss law, the constitutive equations, two of the three displacement–strain relations as well as one of the two electric field-electric potential relations. For the substrate the fundamental solutions of a half-plane subjected to a vertical or horizontal concentrated force on the surface are used. Two differential equations and two singular integro-differential equations of four unknown functions, the axial force, N, the moment, M, the average and the first moment of electric displacement, D₀ and D₁, as well as the associated boundary conditions have been derived rigorously from the stationary conditions of Hellinger–Reissner variational functional. In contrast to the thin film/substrate theory that ignores the interfacial normal stress the present one can predict both the interfacial shear and normal stresses, the latter one is believed to control the delamination initiation.     

A coupled zigzag theory for the dynamics of piezoelectric hybrid cross-ply plates

A recently developed coupled third-order zigzag theory for the statics of piezoelectric hybrid cross-ply plates is extended to dynamics. The theory combines a third-order zigzag approximation for the in-plane displacements and a sub-layerwise linear approximation for the electric potential, considering all components of the electric field. The nonuniform variation of the transverse displacement due to the piezoelectric field is accounted for. The conditions for the absence of shear traction at the top and bottom surfaces and continuity of transverse shear stresses in the presence of electromechanical loading are satisfied exactly, thereby reducing the number of displacement variables to five, which is the same as in a first- or third-order equivalent single-layer theory. The governing equations of motion are derived from the extended Hamilton's principle. The theory is assessed by comparing the Navier solutions for the free and forced harmonic vibration response of simply supported plates with the exact three-dimensional piezoelasticity solutions. Comparisons for hybrid test, composite and sandwich plates establish that the present theory is quite accurate for the dynamic response of moderately thick plates.     

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.