Nonlinear dynamic response and fatigue damage evolution for piezoelectric laminated plates with matrix cracks

Based on the Talreja’s damage model with tensor valued internal state variables, considering the effects of piezoelectricity, geometrical nonlinearity and the constitutive relations of the composite uni-ply plate with matrix cracks, the nonlinear dynamic equations of piezoelectric laminated plates with damage are established and the problems are solved by using the finite difference method. In numerical calculating examples, the nonlinear dynamics responses and the fatigue damage evolution curves of the plates are obtained, and the effects of piezoelectric materials, damage and external load on the nonlinear dynamic response and fatigue damage evolution of the plates are discussed.     

压电介质表面圆形电极边缘的电-弹场分析

基于线性压电理论，研究了表面附有圆形电极板的半无限压电介质电极边缘的电弹场奇异分布特性。利用Hankel变换技术，将空间轴对称混合边值问题简化为一组对偶积分方程的求解，并得到了应力分量和电位移分量的解析表达式。算例表明在电极边缘的邻域内存在剧烈的应力集中和电位移增强，足以造成介质的脆性断裂或介质击穿。并对这种类裂纹奇异性质进行了讨论。     

压电薄板板边导电裂纹尖端的力电场分析

采用有限元方法，分析了压电薄板板边不同长度导电裂纹尖端的力电场分布规律，发现导电裂纹尖端的应力场和电场强度存在明显的集中和奇异现象，集中和奇异的程度与裂纹长度有关。而且，在裂纹延长线上分别存在两点，这里的应力和电场对裂纹长度不太敏感，总等于无裂纹时薄板的均匀应力和均匀电场强度；同时，还研究了导电裂纹尖端的应力强度因子和电场强度因子对裂纹长度的依赖关系，发现在线性本构的前提下，导电裂纹尖端的应力强度因子与电场强度因子之间具有近似的线性关系。     

压电涂层-功能梯度压电界面层-基底结构的二维接触问题研究

在多层压电元件中,由于界面处材料成分和性质的突变,常常导致界面处应力集中,使得界面处出现开裂或蠕变现象,从而大大缩短了压电元件的使用寿命。功能梯度压电材料作为界面层,可有效的缓解界面材料不匹配导致的破坏。本文主要研究利用功能梯度压电材料界面层连接压电涂层和基底,分析三层结构在圆柱型压头作用下的力电响应。利用傅里叶积分变换技术,本文将压电涂层-功能梯度压电层-基底结构在刚性圆柱压头作用下的二维平面应变接触问题转化为带有柯西核的奇异积分方程。运用高斯-切比雪夫积分公式,将奇异积分方程转化为线性方程组并对其进行数值求解,得到压电涂层-功能梯度压电层-基底结构在圆柱形压头作用下的应力分布和电位移分布。数值结果表明,梯度压电材料参数的变化对结构中的力电响应具有重要的影响。本文研究结果对于利用功能梯度压电界面层消除界面处的应力不连续导致的界面破坏具有重要的理论指导意义,研究结果可为功能梯度压电材料界面层的设计提供帮助。     

Study on electroelastic field concentration around the electrode tip in multilayer ferroelectric actuators of two designs and their optimizations

Distribution of electromechanical field near electrode tips is closely related to the reliability of ferroelectric multilayer actuators. In this paper, the deformation and stress concentrations around the electrode tip in two multilayer actuator designs, partially and fully cofired, are investigated by means of experimental measurement and numerical simulations. The digital speckle correlation method (DSCM) is used to measure the full displacement field near the electrode tip with the high spatial resolution. The paths of electric breakdown and cracks initiated from the edge of electrodes were observed. With the proposed Double Gibbs free energy criterion, a fully coupled nonlinear electromechanical finite element method based on domain-switching mechanisms is developed and the simulation results agree well with the experiments. It is found that the crack-like “defects” in the partially cofired layered actuators, i.e. the interlayer gaps filled with soft insulating wax, can significantly reduce the maximum tensile stress level compared with that in “perfect” fully cofired actuators, which implies that the partially cofired design is more reliable than the fully cofired one. Further optimization on geometrical dimension of actuators is also carried out.     

COMBINED DAMAGE FRACTURE CRITERIA FOR PIEZOELECTRIC CERAMICS

Mechanical and electrical damages are introduced to study the fracture mechanics of piezoelectric ceramics in this paper. Two kinds of piezoelectric fracture criteria are established using the method of least squares combined with a damage analysis of the well-known piezoelectric fracture experiments of Park and Sun‘s. One is based on a linear combination of the mechanical and electrical damages and the other on their nonlinear combination. When the combined damage D is up to its critical value De, piezoelectric fracture occurs. It is found from the qualitative comparison of the numerical results with the experimental data that the nonlinearly combined damage fracture criterion can give a better prediction of piezoelectric fracture. And it is concluded from the nonlinearly combined damage fracture criterion that a negative electric field impedes fracture whereas the effect of a positive electric field on fracture depends on its magnitude.     

Numerical simulation of domain switching in multilayer ferroelectric actuators

Micromechanical finite element methods are developed based on a nonlinear constitutive model of ferroelectric polycrystals. Electromechanical behaviors ahead of an internal electrode tip are numerically simulated in multilayer ferroelectric actuators. Around the electrode edge, the nonuniform electric field generates a concentration of stress due to the incompatible strain as well as spontaneous strain. The preferred domain switching enhances the concentration of residual stress and may cause the actuators to crack. An electrically permeable crack emanating from an internal electrode is analyzed. A large scale domain switching zone is found in the vicinity of crack tips. The larger the actuating strain and electric field are, the larger the switching zone will be. The size of switching zone even reaches the scale of crack length with increasing electromechanical loading.     

Analysis of piezoelectric ceramic multilayer actuators based on an electro-mechanical coupled meshless method

This paper presents an efficient meshless method for analyzing cracked piezoelectric structures subjected to mechanical and electrical loading. In this method, an element free Galerkin (EFG) formulation, an enriched basic function and some special shape functions that contain discontinuous derivatives are employed. Based on the moving least squares (MLS) interpolation approach, the EFG method is one of the promising methods for dealing with problems involving progressive crack growth. Since the method is meshless and no element connectivity data are needed, the burdensome remeshing procedure required in the conventional finite element method (FEM) is avoided. The numerical results show that the proposed method can yield an accurate near-tip stress field in an infinite piezoelectric plate containing an interior hole. In another example studying a ceramic multilayer actuator, the proposed model was found to be accurate in the simulation of stress and electric field concentrations arround the abrupt end of an internal electrode. The project supported by the National Natural Science Foundation of China (10025209, 10132010, and 90208002), and the Research Grants Council of the Hong Kong Special Administrative Region, China (HKU 7203/03E). The English text was polished by Yunming Chen.     

The two-dimensional electroelasticity problems for multiconnected bodies situated under electric potential difference action

Solutions of the two-dimensional electroelasticity problems for infinitely long piezoelectric cylinders with cavities and cracks under electrical potential difference action are obtained using the Lekhnitskii’s generalized complex potentials. Herewith, piezoelectric bodies under consideration are free from mechanical loads, some (or all) openings and maybe the exterior boundary of the cylinders are fully covered by thin electrodes. Voltage differences are supplied on the electroded surfaces, the rest boundary surfaces are charge-free. In this case, the complex potentials are investigated. Boundary conditions are satisfied by the least-squares method. There are carried out numerical investigations of behaviour of some basic electroelastic characteristics in a circular hollow cylinder and in an infinite body with two longitudinal cavities and a charge-free plane crack. New electromechanical regularities of influence of material properties, geometrical characteristics of considered regions on values of electroelastic state characteristics and the stress, electric displacements and tensions intensity factor $k_1$ are determined.     

Elastoplastic state of flexible cylindrical shells with two circular holes

The elastoplastic state of thin cylindrical shells with two equal circular holes is analyzed with allowance made for finite deflections. The shells are made of an isotropic homogeneous material. The load is internal pressure of given intensity. The distribution of stresses along the hole boundary and in the stress concentration zone (when holes are closely spaced) is analyzed by solving doubly nonlinear boundary-value problems. The results obtained are compared with the solutions that allow either for physical nonlinearity (plastic strains) or geometrical nonlinearity (finite deflections) and with the numerical solution of the linearly elastic problem. The stresses near the holes are analyzed for different distances between the holes and nonlinear factors.Translated from Prikladnaya Mekhanika, Vol. 40, No. 10, pp. 107–112, October 2004.     

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.     

THE PERFORATION PARAMETER OPTIMIZATION FOR MULTI FRACTURE INITIATION IN HYDRAULIC FRACTURING

针对复杂地质近井筒多裂缝起裂的难题, 结合岩体力学与损伤力学相关理论, 建立了近井筒地层流固——损伤力学模型, 采用数值求解获得了射孔壁面的应力及损伤状态, 计算表明:初始地应力最小值越小, 地层的起裂压力越低;射孔方位角越小, 起裂压力越低. 随着射孔方位角增大, 近井筒微裂缝为45° 夹角的拐折裂缝, 起裂压力增大, 基于此, 对于正断层及顺滑断层地应力类型, 取射孔方位角小于30°, 螺旋射孔夹角30°或45°, 对于逆断层地应力类型, 螺旋射孔夹角取45°, 可有效地实现多裂缝起裂.     

压电薄板切口尖端前沿力电损伤场分析

基于Yang等建立的三维各向异性压电损伤本构模型,给出了广义平面应力压电损伤问题的本构方程和有限元平衡方程,运用有限元方法分析了张开角和深度对切口前沿力电损伤分布的影响规律。结果表明,张开角对力损伤的影响较小,并且几乎不改变力电损伤的区域尺寸,但是深度对力电损伤的大小和区域尺寸都有很大的影响。     

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.     

Aerodynamic optimization of the flat-plate leading edge for experimental studies of laminar and transitional boundary layers

This work is concerned with the design of a leading edge for a flat-plate model used to study laminar and transitional boundary layers. For this study, the flow over the complete boundary-layer model, including leading edge, flat section, and trailing-edge flap, is modeled. The effect of important geometrical features of the leading edge on the resulting pressure distribution, starting from the well-known symmetric modified super ellipse, is investigated. A minimal pressure gradient on the measurement side of the plate is achieved using an asymmetrical configuration of modified super ellipses, with a thickness ratio of 7/24. An aerodynamic shape optimization is performed to obtain a novel leading edge shape that greatly reduces the length of the non-zero pressure gradient region and the adverse pressure gradient region compared to geometries defined by ellipses. Wind tunnel testing is used to validate the numerical solutions.     

含与不含晶界空穴的双晶体蠕变行为研究

基于晶体滑移理论,建立了各向异性镍基合金双晶体的蠕变本构模型和蠕变寿命预测模型,通过MARC用户子程序CRPLAW将上述本构模型进行了有限元实现,并对双晶体蠕变行为进行了计算分析,考虑了:(1)晶体取向的影响;(2)垂直、倾斜和平行于外载方向的三种位向晶界情况;(3)晶界处引进空间空穴的影响。结果表明,双晶体上特别是微空穴和晶界附近区域的蠕变应力应变呈现不同的变化规律,对此晶粒晶体取向和晶界位向有较大的影响;微空穴的存在削弱了双晶体的承载能力,显著地影响了双晶体蠕变持久寿命;相同条件下,垂直晶界对双晶体模型的蠕变损伤影响最为强烈,倾斜晶界次之,平行晶界最小;微空穴的生长与晶界位向和晶体取向有强烈的依赖关系,其中垂直晶界更有利于晶体滑移和微空穴生长。     

Frictional sliding modes along an interface between identical elastic plates subject to shear impact loading

Frictional sliding along an interface between two identical isotropic elastic plates under impact shear loading is investigated experimentally and numerically. The plates are held together by a compressive stress and one plate is subject to edge impact near the interface. The experiments exhibit both a crack-like and a pulse-like mode of sliding. Plane stress finite element calculations modeling the experimental configuration are carried out, with the interface characterized by a rate and state dependent frictional law. A variety of sliding modes are obtained in the calculations depending on the impact velocity, the initial compressive stress and the values of interface variables. For low values of the initial compressive stress and impact velocity, sliding occurs in a crack-like mode. For higher values of the initial compressive stress and/or impact velocity, sliding takes place in a pulse-like mode. One pulse-like mode involves well-separated pulses with the pulse amplitude increasing with propagation distance. Another pulse-like mode involves a pulse train of essentially constant amplitude. The propagation speed of the leading pulse (or of the tip of the crack-like sliding region) is near the longitudinal wave speed and never less than times the shear wave speed. Supersonic trailing pulses are seen both experimentally and computationally. The trends in the calculations are compared with those seen in the experiments.     

固定网格下基于光滑变形隐式曲线的孔形优化设计方法

孔形优化设计是减缓开孔结构孔边应力集中的有效手段，其对应力分析精度和孔边曲线的几何表达能力都有着较高的要求．论文针对现有固定网格孔形优化方法分析精度较低和/或孔边曲线变形能力有限等不足，采用有限胞元方法在固定网格下对开孔结构进行高精度的力学性能分析，采用能够自由光滑变形的隐式曲线描述待优化的孔洞边界，进而建立优化模型并推导出相应的解析灵敏度计算公式，形成了固定网格下开孔结构孔形优化设计新框架．通过对不同载荷边界条件下的开孔平板结构进行孔形优化设计，表明本方法无需网格更新、灵敏度推导简便、力学分析精度高且优化设计空间大，能够有效降低孔边应力集中程度．     

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。