Exact solution of an external circular crack in a piezoelectric solid subjected to shear loading

A three-dimensional, exact analysis is presented in this paper for the problem of an external circular crack in a transversely isotropic piezoelectric medium subjected to arbitrary antisymmetric shear loading. A recently proposed general solution of three-dimensional piezoelectricity is employed. It is shown that four quasi harmonic functions involved in the general solution can be respresented by just one complex potential. Previous results in potential theory are then used to obtain the exact solution of the problem. For point shear loading, Green’s functions for the elastoelectric field are derived in terms of elementary functions. Project supported by Scientific Research Foundation for Returned Overseas Chinese Scholars, State Education Ministry and partly by National Natural Science Foundation of China (NSFC) (10002016).     

A PENNY-SHAPED CRACK IN AN INFINITE PIEZOELECTRIC BODY UNDER ANTISYMMETRIC POINT LOADS

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A penny-shaped crack in an infinite piezoelectric body under antisymmetric point loads

In this study, Fabrikant (1989, 1991)'s new results in potential theory were used to obtain the exact and complete solution for the problem of a penny-shaped crack in an infinite transversely isotropic piezoelectric body subjected to antisymmetric point loads (point charges and normal point forces); then the complete solution for the problem of one-sided loading of a penny-shaped crack was obtained by the superposition of the symmetric loading solution in Chen and Shioya (1999) and the antisymmetric one presented here; and then the reciprocity theorem of piezoelectric media was used to deal with the problem of interaction between arbitrarily located point forces and a point charge with a penny-shaped crack and obtained the exact expressions of the crack faces' normal displacement in terms of elementary functions and some non-singular integrals; and finally obtained the normal displacement of the positive and negative faces of the crack under many loading cases as shown in figures for an infinite PZT-4 piezoelectric ceramic body weakened by a penny-shaped crack. Project (19872060) supported by the Natural Science Foundation of China.     

The Reissner-Sagoci problem for transversely isotropic piezoelectric half-space

INTRODUCTION In isotropic elasticity a class of mixed boundary value problems for elastic half-space can be classified as axisymmetric torsion. In this case the elastic field in cylinder components takes a particularly simple form consisting of only one nonzero displacement and two nonzero stress components. For this loading case, Reissner and Sagoci (1944) were possibly the first to examine a mixed boundary problem between the tangential displacement and stress on the half-space surface.…     

Analytical modelling and free vibration analysis of piezoelectric bimorphs

INTRODUCTION Piezoelectric bimorphs (or benders), a special type of piezoelectric device, which can produce flexural deformation significantly larger than the length or thickness deformation of the individual piezoelectric layers, have been widely used as elec- troacoustic transducers, medical devices and micro- robot due to their characteristics of easy miniaturiza- tion, high positioning accuracy, sensitive response, and large displacement (e.g., Shirley and Hampton, 1978; Ha and Kim, 2…     

A Nonlinear Dynamical Theory of Non-Classical Plates

In this paper, a new nonlinear formulation of plates, including shear and rotatory inertia and transverse normal stress effects, is developed by means of general assumptions, of which the von Karman-type formulation and some thick plate theories are special cases. To keep the formulation fairly general, the problem addressed in this paper simultaneously includes: the effects of shear deformation according to the geometric deformation similarity of the cross-section, the rotatory inertia, and the transverse normal stress. The three-dimensional compatible equations are applied to derive the basic equations. Numerical results are given for linear and non-linear analysis of plates. This Work was supported by the Natural Science Fundation of Shanghai for Returning Scholars     

受拉平板裂纹尖端三维应力状态有限元分析

裂纹的应力应变状态对结构的断裂影响很大,是钢结构断裂分析和设计中研究的重点。由于一般构件的几何形状和荷载状态复杂,难于对裂纹尖端的应力应变状态得到理论解。本文利用ANSYS软件对受拉平板裂纹尖端三维应力状态进行了有限元分析,获得了钢结构裂纹尖端的应力分布规律。     

Hierarchical models for failure analysis of plates bent by distributed and localized transverse loadings

The failure analysis of simply supported, isotropic, square plates is addressed. Attention focuses on minimum failure load amplitudes and failure locations, von Mises＇ equivalent stress along the plate thickness is also addressed. Several distributed and localized loading conditions are considered. Loads act on the top of the plate. Bi-sinusoidal and uniform loads are taken into account for distributed loadings, while stepwise constant centric and off-centric loadings are addressed in the case of localized loadings. Analysis is performed considering plates whose length-to-thickness ratio a/h can be as high as 100 （thin plates） and as low as 2 （very thick plates）. Results are obtained via several 2D plate models. Classical theories （CTs） and higher order models are applied. Those theories are based on polynomial approximation of the displacement field. Among the higher order theories （HOTs） HOTsa models account for the transverse shear deformations, while HOTs models account for both transverse shear and transverse normal deformations. LHOTs represent a local application of the higher order theories. A layerwise approach is thus assumed： by means of mathematical interfaces, the plate is considered to be made of several fictitious layers. The exact 3D solution is presented in order to determine the accuracy of the results obtained via the 2D models. In this way a hierarchy among the 2D theories is established. CTs provide highly accurate results for a/h greater than 10 in the case of distributed loadings and greater than 20 for localized loadings. Results obtained via HOTs are highly accurate in the case of very thick plates for bi-sinusoidal and centric loadings. In the case of uniform and off-centric loadings a high gradient is present in the neighborhood of the plate top. In those cases, LHOTs yield results that match the exact solution.     

三维分数阶对流色散方程的数值解法

对一个在有限范围内含有变系数的三维分数阶对流色散方程进行对流项与扩散项施行不同的近似处理,从而构建一种交替方向的隐式Euler方法求解该方程,并探讨该方法的稳定性、相容性和收敛性,最后给出一个含有准确解的数值例子以验证该方法的有效性.     

理想流体与三维复函数

通过引进三维复函数的概念和其性质,推广了二维解析函数,将空间理想流体问题求解归结为该函数的边值问题．从而为解决该类问题提供了一种新的方法．     

Burgers-mKdV方程的一类精确解

直接假设Burgers-mKdV方程ut+6u2ux+μuxx+δuxxx=0的精确解的一种形式,将求解Burgers-mKdV方程的问题转化为一个代数方程组的求解,获得了Burgers-mKdV方程的一类精确解.     

Time-dependent response of laminated isotropic strips with viscoelastic interfaces

INTRODUCTION A traditional premise in the theory of com-posites is the continuity of tractions and displace-ments at constituent interfaces. However, the in-terfaces are generally weaker than the plies inlaminates, which then frequently suffer from failure(such as delamination, interlaminar slip, etc.) due tohigh stress concentration at the interfaces. Beforethe final failure, these interfaces are usuallyweakened and become imperfect due to the emer-gence of microcracks. There are nu…     

Time-dependent response of laminated isotropic strips with viscoelastic interfaces

The two dimensional problem of simply supported laminated isotropic strips with viscoelastic interfaces and under static loading was studied. Exact solution was derived based on the exact elasticity equation and the Kelvin-Voigt viscoelastic interfacial model. Numerical computations were performed for a strip consisting of three layers of equal thickness. Results indicated that the response of the laminate was very sensitive to the presence of viscoelastic interfaces.     

剪跨比对BFRP筋混凝土梁抗剪性能影响的研究

玄武岩纤维具有优异的耐腐蚀性能,采用玄武岩纤维材料（BFRP）筋代替钢筋是一种解决钢筋锈蚀问题的潜在方法。通过6根试验梁,研究不同剪跨比下BFRP筋混凝土粱的抗剪性能。试验表明,BFRP筋混凝土梁呈脆性破坏,破坏位置集中在分配梁加载点附近;随着剪跨比的减小,斜截面初裂荷载逐渐提高,且逐渐接近于极限荷载;混凝土梁开裂前,挠度增长缓慢,开裂后增长较快,表现为荷载-挠度曲线呈折现变化。     

Guest Review:Potential theory method for 3D crack and contact problems of multi-field coupled media:A survey

This paper presents an overview of the recent progress of potential theory method in the analysis of mixed boundary value problems mainly stemming from three-dimensional crack or contact problems of multi-field coupled media.This method was used to derive a series of exact three dimensional solutions which should be of great theoretical significance because most of them usually cannot be derived by other methods such as the transform method and the trial-and-error method. Further, many solutions are obtained in terms of elementary functions that enable us to treat more complicated problems easily. It is pointed out here that the method is usually only applicable to media characterizing transverse isotropy,from which, however, the results for the isotropic case can be readily obtained.     

Modeling of sensor function for piezoelectric bender elements

An analytical sandwich beam model for piezoelectric bender elements is derived based on the first-order shear deformation theory （FSDT）, which assumes a single rotation angle for the whole cross-section and a quadratic distribution for coupled electric potential in piezoelectric layers. Shear coefficient is introduced to correct the effect of transverse shear strain on shear force and the electric displacement integration. Static and free vibration analyses of simply-supported bender elements are carried out for the sensor function. The results illustrate the high accuracy of the present model compared with the exact 2D solutions.     

Guest Review:Potential theory method for 3D crack and contact problems of multi-field coupled media: A survey

This paper presents an overview of the recent progress of potential theory method in the analysis of mixed boundary value problems mainly stemming from three-dimensional crack or contact problems of multi-field coupled media. This method was used to derive a series of exact three dimensional solutions which should be of great theoretical significance because most of them usually cannot be derived by other methods such as the transform method and the trial-and-error method. Further, many solutions are obtained in terms of elementary functions that enable us to treat more complicated problems easily. It is pointed out here that the method is usually only applicable to media characterizing transverse isotropy, from which, however, the results for the isotropic case can be readily obtained.     

具有无反射型势的Dirac方程和Klein-Gordon方程的散射态

本文在三维无反射型标量势和矢量势相等的条件下，给出了Dirac方程和KleinGordon方程的S波散射态精确解．     

一种推广的宝石分埋问题

提出一个新的宝石分埋问题,对《宝石分埋问题的一种解法》一文中的宝石分埋问题由二维空间推广到三维空间,得到了一种新的推广的宝石分埋问题.利用直积循环群作为解决推广的宝石分埋问题的一种数学模型,并给出了一个具体的推广的宝石分埋问题的一种解法.     

Pure bending of simply supported circular plate of transversely isotropic functionally graded material

INTRODUCTION In recent years, functionally graded materials (FGMs) have attracted more and more attention. Due to their continuously varying material properties in space on the macroscopic scale, FGMs, therefore, are usually superior to conventional fiber-matrix materi-als in mechanical behavior, especially for perform-ance under thermal loading. Now FGMs have been widely used in various fields including electronics, chemistry, optics, biomedicine, etc. Heretofore, volumes of literatu…