Stress in piezoelectric hollow sphere with thermal gradient

The piezoelectric phenomenon has been exploited in science and engineering for decades. Recent advances in smart structures technology have led to a resurgence of interest in piezoelectricity, and in particular, in the solution of fundamental boundary value problems. In this paper, we develop an analytic solution to the axisymmetric problem of a radially polarized, spherically isotropic piezoelectric hollow sphere. The sphere is subjected to uniform internal pressure, or uniform external pressure, or both and thermal gradient. There is a constant thermal difference between its inner and outer surfaces. An analytic solution to the governing equilibrium equations (a coupled system of second-order ordinary differential equations) is obtained. On application of the boundary conditions, the problem is reduced to solving a system of linear algebraic equations. Finally, the stress distributions in the sphere are obtained numerically for two piezoceramics.     

Piezoelectric hollow cylinder with thermal gradient

The coupling nature of piezoelectric materials has acquired wide applications in electric-mechanical and electric devices. Recent advances in smart structures technology have led to a resurgence of interest in piezoelectricity, and in particular, in the solution of fundamental boundary value problems. In this paper, an analytic solution to the axisymmetric problem of a radially polarized, radially orthotropic piezoelectric hollow cylinder with thermal gradient is developed. An analytic solution to the governing equilibrium equations (a coupled system of second-order ordinary differential equations) is obtained. On application of the boundary conditions, the problem is reduced to solving a system of linear algebraic equations. The stress and potential field distributions in the cylinder are obtained numerically for two piezoceramics. It is shown that the hoop stresses in a cylinder composed of these materials can be decreased throughout the cross-section by applying an appropriate set of boundary conditions. This paper was recommended for publication in revised form by Associate Editor Jeong Sam Han Mahdi Saadatfar received a B. S. degree in Mechanical Engineering from University of Kashan 2006. He is currently a M.S student at the School of Mechanical Engineering at University of Tehran, Iran. He is currently researching about modeling of nanoindentation process in nanocomposites. Mr. Saadatfar’s research interests are in the area of piezoelectric Materials, Polymer/Clay nanocomposites and Finite element modeling. He has several published paper about piezoelectric materials and Finite element modeling of nanocomposites. Amin Shariat Razavi received a B.S degree in Mechanical Engineering from Kashan University in 2006. He is currently testing and examining an specific type of intelligent plasma cutting machine for process equipment that is designed by himself. Mr. Razavi’s research interests are smart materials and design of mechanical system.     

Free vibration of multi-layered spherically isotropic hollow spheres

A three-dimensional analysis is carried out for the free vibrations of a multi-layered spherically isotropic hollow sphere using a state-space method. By the introduction of three displacement functions and two stress functions, two independent state equations with varying coefficients are derived. Taylor's expansion theorem is then employed to obtain solutions to the two state equations and relationships between the state variables at the upper and lower surfaces of each lamina are established. A variable substitution technique is particularly used to make the derivation more natural and simpler. By virtue of the continuity conditions between two adjacent layers, two sets of linear algebraic equations about the boundary variables at the inner and outer surfaces of a multi-layered hollow sphere are obtained. Frequency equations for the free vibrations are then presented.     

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     

Three-dimensional solutions of multilayered piezoelectric hollow cylinders by an asymptotic approach

Three-dimensional (3D) solutions for the static analysis of multilayered piezoelectric hollow cylinders are presented by means of an asymptotic approach. Either applied load or applied electric potential on the lateral surfaces is considered. The edge conditions of the cylinders are considered as simple supports. In the present formulation, the 22 basic equations in a cylindrical coordinates system are firstly reduced to eight differential equations in terms of eight primary variables of elastic and electric fields. After the mathematical derivation of nondimensionalization, asymptotic expansion and successive integration, we obtained recurrent sets of governing equations for various order problems. In view of the recurrent property, the present asymptotic solutions can be obtained in a hierarchic manner and asymptotically approach the 3D piezoelectricity solutions. Several benchmark problems of single-layer and multilayered piezoelectric hollow cylinders are studied using the present asymptotic formulation.     

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。     

Closed-form vibration analysis of thick annular functionally graded plates with integrated piezoelectric layers

This paper employs an analytical method to analyze vibration of piezoelectric coupled thick annular functionally graded plates (FGPs) subjected to different combinations of soft simply supported, hard simply supported and clamped boundary conditions at the inner and outer edges of the annular plate on the basis of the Reddy's third-order shear deformation theory (TSDT). The properties of host plate are graded in the thickness direction according to a volume fraction power-law distribution. The distribution of electric potential along the thickness direction in the piezoelectric layer is assumed as a sinusoidal function so that the Maxwell static electricity equation is approximately satisfied. The differential equations of motion are solved analytically for various boundary conditions of the plate. In this study closed-form expressions for characteristic equations, displacement components of the plate and electric potential are derived for the first time in the literature. The present analysis is validated by comparing results with those in the literature and then natural frequencies of the piezoelectric coupled annular FG plate are presented in tabular and graphical forms for different thickness-radius ratios, inner-outer radius ratios, thickness of piezoelectric, material of piezoelectric, power index and boundary conditions.     

A unified generalized thermoelasticity; solution for cylinders and spheres

A new unified formulation for the generalized theories of the coupled thermoelasticity based on the Lord–Shulman, Green–Lindsay, and Green–Naghdi models is proposed in this paper. The unified form of the governing equations is presented by introducing the unifier parameters. The formulations are derived and given for the anisotropic heterogeneous materials. The unified equations are reduced for the isotropic and homogeneous materials. Transforming the governing equations into the Laplace domain, they are analytically solved in the space domain for a hollow sphere and cylinder, where a parameter is introduced to consolidate the solution for the sphere and cylinder in a unified form. A thermal shock load is applied to the inner surface of the sphere and cylinder and the results are presented using a numerical inversion technique of the Laplace transform. The results are validated with the known data in the literature.     

Exact solutions for piezoelectric materials with an elliptic hole or a crack under uniform internal pressure

The existing investigations on piezoelectric materials containing an elliptic hole or a crack mainly focus on remote uniform tensile loads.In order to have a better understanding for the fracture behavior of piezoelectric materials under different loading conditions,theoretical and numerical solutions are presented for an elliptic hole or a crack in transversely isotropic piezoelectric materials subjected to uniform internal pressure and remote electro-mechanical loads.On the basis of the complex variable approach,analytical solutions of the elastic and electric fields inside and outside the defect are derived by satisfying permeable electric boundary condition at the surface of the elliptical hole.As an example of PZT-4 ceramics,numerical results of electro-elastic fields inside and outside the crack under various electric boundary conditions and electro-mechanical loads are given,and graphs of the electro-elastic fields in the vicinity of the crack tip are presented.The non-singular term is compared to the asymptotic one in the figures.It is shown that the dielectric constant of the air in the crack has no effect on the electric displacement component perpendicular to the crack,and the stresses in the piezoelectric material depend on the material properties and the mechanical loads on the crack surface and at infinity,but not on the electric loads at infinity.The figures obtained are strikingly similar to the available results.Unlike the existing work,the existence of electric fields inside an elliptic hole or a crack is considered,and the piezoelectric solid is subjected to complicated electro-mechanical loads.     

MATHEMATIC MODEL AND ANALYTIC SOLUTION FOR CYLINDER SUBJECT TO UNEVEN PRESSURES

According to the inverse solution of elasticity mechanics, a stress function is constructed which meets the space Inharmonic equation, this stress functions is about cubic function pressure on the inner and outer surfaces of cylinder. When borderline condition that is predigested according to the Saint-Venant's theory is joined, an equation suit is constructed which meets both the biharmonic equations and the boundary conditions. Furthermore, its analytic solution is deduced with Matlab. When this theory is applied to hydraulic bulging rollers, the experimental results inosculate with the theoretic calculation. Simultaneously, the limit along the axis invariable direction is given and the famous Lame solution can be induced from this limit. The above work paves the way for mathematic model building of hollow cylinder and for the analytic solution of hollow cylinder with randomly uneven pressure.     

介电泳效应磨粒流抛光中的电极形态

针对薄壁陶瓷工件内表面抛光,提出一种基于介电泳效应的磨粒流抛光方法。将非均匀电场布置于陶瓷工件外表面,极化磨粒,实现陶瓷工件内表面高效抛光。仿真分析发现:电极间隙比为2时,SiC磨粒具有最好的介电泳效应,参与抛光的磨粒最多。陶瓷工件初始内表面粗糙度值Ra为(208±5)nm时,抛光10 h后,无介电泳效应的磨粒流抛光工件内表面粗糙度值Ra为51 nm,有介电泳效应的磨粒流抛光工件内表面粗糙度值Ra为23 nm。     

Static electromechanical response of smart composite/sandwich plates using an efficient finite element with physical and electric nodes

An efficient quadrilateral element, recently developed by the authors, based on an improved zigzag theory is assessed for the static electromechanical response of hybrid plates with electroded piezoelectric sensors and actuators. The theory accounts for the transverse normal strain due to the electric field in the approximation of deflection. The electric potential is approximated as layerwise quadratic across the thickness. By introducing an electric node in the element for the electric potentials of the electroded piezoelectric surfaces, the equipotential condition of such surfaces is modelled very efficiently. The electric potential degrees of freedom (DOF) corresponding to the quadratic component of the electric potential distribution are associated with the four physical nodes to allow for the inplane electric field induced due to direct piezoelectric effect. The requirement of C¹ continuity of interpolation functions of the deflection is circumvented by employing the improved discrete Kirchhoff constraint technique. The finite element (FE) formulation is validated by comparing the results with other available results in the literature. Comparison of the present results for static response of a variety of piezoelectric bimorph, hybrid composite and sandwich plates, with 3D analytical and FE solutions and those of other available elements establishes that the present element is accurate, robust and computationally efficient.     

Analytic investigation of effect of electric field on elasto-plastic response of a functionally graded piezoelectric hollow sphere

In this work, the effect of electric potential on the mechanical (Stresses, strains, displacement) and electrical (electrical displacement and intensity) response of a Functionally graded piezoelectric (FGP) hollow sphere is analytically investigated. The sphere is under the action of internal/external pressure and temperature gradient as well. The inhomogeneity is based on power law in radial direction. The analysis is done in two parts: elastic response and plastic response, using Tresca yield criterion. It is shown by illustrative example that under internal pressure and assumed model parameters, the commencement of plastic region is from outside surface towards inside in the plastic zone is extended with the increase of electric potential. Interestingly, radial stress and displacement have an extreme not on the boundaries, but on the inside.     

Nonlocal electro-thermal transverse vibration of embedded fluid-conveying DWBNNTs

Electro-thermal transverse vibration of fluid-conveying double-walled boron nitride nanotubes (DWBNNTs) embedded in an elastic medium such as polyvinylidene fluoride (PVDF) which is a piezoelectric polymer is investigated. The elastic medium is simulated as a spring and van der Waals (vdW) forces between inner and outer nanotubes are also taken into account. Zigzag structure of boron nitride nanotubes (BNNTs) is described based on the nonlocal continuum piezoelasticity cylindrical shell theory, and Hamilton??s principle is employed to derive the corresponding higher-order equations of motion. In this model, DWBNNTs are placed in uniform temperature and electric field, the latter being applied through attached electrodes at both ends. Having considered the small scale effect, aspect ratio (L/R), densities of fluid and elastic medium, four different cases of loading are assumed in this study, including: a) direct voltage and heating (DVH), b) direct voltage and cooling (DVC), c) reverse voltage and heating (RVH), and d) reverse voltage and cooling (RVC). Numerical results indicate that increasing nonlocal parameter (e ₀ a), for the four above mentioned cases, decreases the critical flow velocity of fluid. The results could be used in design of nano-electro-mechanical devices for measuring density of a fluid such as blood flowing through such nanotubes with great applications in medical fields.     

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.     

压电悬臂梁采收低频振动能的理论分析与仿真

采收环境振动能量为无线网络传感器供电是近年来研究的热点,目前还没有一个完整的理论和解决方案.文中设计了一种压电悬臂梁结构的环境振动能量采收装置,研究了悬臂梁压电振子结构受激励后产生电荷量与频率的关系,并进行了ANSYS仿真,得出了最佳的机电耦合模型和压电悬臂梁几何尺寸对固有频率的影响的关系.为采收环境低频振动能量,实现网络传感器自供电装置提供了设计的理论依据.     

Vibration characteristics of a piezoelectric disk laminated with an elastic disk

This paper theoretically and experimentally deals with the vibration characteristics of a piezoelectric disk polarized in the thickness direction and laminated with an elastic disk. Axisymmetric vibration modes include radial and axial motions. Theoretically, in this study, the differential equations of piezoelectric motions were derived in terms of radial and axial displacements and electric potential. The differential equations of elastic motions were expressed in terms of radial and axial displacements. Solving the governing equations and boundary conditions for a coupled structure produced characteristic equations that provided natural frequencies and mode shapes. Experimentally, the natural frequencies were measured using an impedance analyzer and the radial in-plane motions of the fundamental mode were measured using an in-plane laser interferometer. The results of the theoretical analysis were compared with those of a finiteelement analysis and experiments; moreover, the theoretical analysis was verified on the basis of this comparison. It appeared that the natural frequencies of the radial modes of the piezoelectric disk were not affected by the elastic disk; however, those of the thickness modes were reduced by the elastic disk owing to the added mass effect.     

Axisymmetric response of circular plates with piezoelectric layers: an exact solution

Electromechanical responses of symmetric circular laminates consisting of piezoelectric layers are studied, and the influence of surface and interlayer electrodes are involved. The laminates are traction-free on the top and bottom surfaces, but may be subjected to external forces at the lateral edge and to voltages applied across certain layers. Under axisymmetric deformation conditions, an approximate model which employs Kirchhoff hypothesis and incorporates the charge equation of electrostatic is established. Then, a closed-form three-dimensional solution of the laminates is generated in a very straightforward manner by the solution of the approximate model. The three-dimensional solution fulfills all field equations and interface or surface conditions as well as the specified electric edge boundary conditions; the only restriction is that the mechanical edge boundary conditions are satisfied in an average manner, rather than point by point. Thus, according to Saint-Venant's principle the proposed solution is exact in the interior region of the laminates.     

三维压电介质中不同电边界条件的裂纹分析

在不可导通、可导通和半可导通等三种电边界条件下，系统研究并给出三维无限横观各向同性压电介质中平行于各向同性面的、任意形状的平片裂纹在任意载荷作用下的不连续位移和不连续电势边界积分方程方法。无论平片裂纹的形状如何，结果都表明：对不可导通裂纹，应力强度因子只与机械载荷有关，电位移强度因子只与电载荷有关；而可导通裂纹的应力强度因子和电位移强度因子只与机械载荷有关，电位移强度因子由机械载荷通过压电效应产生。半可导通裂纹所对应的边界积分方程组为非线性方程组，文中给出一种迭代解法。在均布的机械载荷和电载荷作用下。裂纹腔内的电位移为一常数，只与外加载荷有关，而与裂纹形状无关。