含椭圆形夹杂的压电体平面应变问题

利用复变函数理论,对在无限远处均匀应力和电位移载荷作用下的含有椭圆形弹性夹杂的横观各向同性压电材料,作了力电分析．在该文有限元结果和前人相关理论解的基础上,提了出一个可接受的认为弹性夹杂体内的应力场为常应力场的假设．在采用了不导通电边界条件之后,获得了以复势形式表示的压电基体的和弹性夹杂体内部的应力场解．     

压电介质内裂纹问题的精确解

在压电介质断裂力学分析中,人们常假定裂纹面上的电位移法向分量为零,可是实验表明,这一假设将导致错误的结果。本文基于精确的电边界条件,并应用Stroh公式的方法,导出了含裂纹压电介质在无限远处均匀外载作用下二维问题的精确解。结果表明:(ⅰ)应力强度因子与各向同性材料相同,而电位移强度因子取决于材料常数和机械载荷,但与电载荷无关;(ⅱ)能量释放率大于纯弹性各向异性材料内的值,即总是正的,且与电载荷无关;(ⅲ)裂纹内所含空气的介电常数对介质内的场强无影响。     

Analytical solution for electromagnetothermoelastic behaviors of a functionally graded piezoelectric hollow cylinder

Analytical study for electromagnetothermoelastic behaviors of a hollow cylinder composed of functionally graded piezoelectric material (FGPM), placed in a uniform magnetic field, subjected to electric, thermal and mechanical loads are presented. For the case that the electric, magnetic, thermal and mechanical properties of the material obey an identical power law in the radial direction, exact solutions for electric displacement, stresses, electric potential and perturbation of magnetic field vector in the FGPM hollow cylinder are determined by using the infinitesimal theory of electromagnetothermoelasticity. Some useful discussions and numerical examples are presented to show the significant influence of material inhomogeneity, and adopting a certain value of the inhomogeneity parameter β and applying suitable electric, thermal and mechanical loads can optimize the FGPM hollow cylindrical structures. This will be of particular importance in modern engineering design.     

Effects of functionally graded interlayers on dispersion relations of shear horizontal waves in layered piezoelectric/piezomagnetic cylinders

The effects of functionally graded interlayers on dispersion relations of shear horizontal waves in layered piezoelectric/piezomagnetic cylinders are studied. First, the basic physical quantities of elastic waves in piezoelectric cylinder are derived by assuming that the SH waves propagate along the circumferential direction steadily. Then the transfer matrices of the functional graded interlayer and outer piezomagnetic cylinder are obtained by solving the state transfer equations with spatial-varying coefficients. Furthermore, making use of the electro-magnetic surface conditions of the outer cylinder, the dispersion relations for the shear horizontal waves in layered piezoelectric/piezomagnetic cylinders are obtained and the numerical results are shown graphically. Seven kinds of functionally graded interlayers and four kinds of electro-magnetic surface conditions are considered. It is found that the functionally graded interlayers have evident influences on the dispersion relations of shear horizontal waves in layered piezoelectric/piezomagnetic cylinders. The high order modes are more sensitive to the gradient interlayers while the low order modes are more sensitive to the electro-magnetic surface conditions.     

Torsional wave propagation in a finite inhomogeneous cylindrical shell under time-dependent shearing stress

Torsional wave motion in a finite hollow cylinder of piezoelectric material of (622) crystal class, under a time-dependent mechanical boundary condition is investigated. The inhomogeneity is restricted to the variations of density and other physical constants of the medium as a certain power of the radial distance. The expressions for the displacement and the electric potential of the present solution are compared with those under time-dependent electric boundary condition. Numerical values of the roots of the frequency equation for β-quartz are presented.     

An analytical method for magnetothermoelastic analysis of functionally graded hollow cylinders

This paper considers magnetothermoelastic behavior of a functionally graded material (FGM) hollow cylinder, placed in a uniform magnetic field, subjected to thermal and mechanical loads. Exact solutions for stresses and perturbations of the magnetic field vector in FGM hollow cylinders is determined by using the infinitesimal theory of magnetothermoelasticity. Numerical results indicate that the inhomogeneous constants presented in the present study are useful parameters from a design point of view in that it can be tailored for specific applications to control the stress and perturbation of magnetic field vector distributions. This research is helpful for the optimum design annular cylindrical FGM sensors/actuators.     

A variational formulation for the torsional problem of piezoelastic beams

In this paper a variational formulation is presented for the torsional deformation of homogeneous, linear piezoelectric monoclinic beams. All results of the paper are based on a generalization of the Saint-Venant’s theory of uniform torsion of elastic beams to piezoelastic beams. Variational formulation uses the torsional and electric potential functions as the independent quantities of the considered variational functional. The mechanical meaning of the variational functional defined is also given. Examples illustrate the application of the presented variational formulation. Considered examples are the torsional problem of thin-walled piezoelastic beams with closed cross-section, and the torsion of hollow circular cylinders made of orthotropic piezoelectric material.     

Analysis of a dynamic contact problem for electro-viscoelastic cylinders

We consider a mathematical model which describes the antiplane shear deformations of a piezoelectric cylinder in frictional contact with a foundation. The process is mechanically dynamic and electrically static, the material behavior is described with a linearly electro-viscoelastic constitutive law, the contact is frictional and the foundation is assumed to be electrically conductive. Both the friction and the electrical conductivity condition on the contact surface are described with subdifferential boundary conditions. We derive a variational formulation of the problem which is of the form of a system coupling a second order hemivariational inequality for the displacement field with a time-dependent hemivariational inequality for the electric potential field. Then we prove the existence of a unique weak solution to the model. The proof is based on abstract results for second order evolutionary inclusions in Banach spaces. Finally, we present concrete examples of friction laws and electrical conductivity conditions for which our result is valid.     

Advanced Shell Element Formulations for Coupled Electromechanical Systems

With the significantly increasing applications of smart structures, piezoelectric material is widely used in branches of engineering sciences. Normally, the Finite Element Method is employed in the numerical analysis of these structures [2]. In this contribution, in order to avoid the locking effects and zero energy modes, the Assumed Natural Strain (ANS) Method [4] is implemented into four‐node piezoelectric shallow shell elements, by using the two‐field variational formulation in which displacements and electric potentials serve as independent variables and the three‐field variational formulation in which the dielectric displacement is taken as an independent variable additionally [3]. Moreover, a quadratic variation of the electric potential through the thickness direction is applied in the two‐field formulation. Numerical examples of piezoelectric sensors and actuators are presented, showing the behaviour of the shell elements by using different hybrid finite element formulations. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

含椭圆形夹杂的压电材料平面问题

应用复变函数的Ｆａｂｅｒ级数展开方法，本文研究了含椭圆形夹杂的压电材料平面问题，给出了问题的封闭解·解答表明，椭圆夹杂内的应力、应变、电场强度和电位移均为常量·通过算例，还讨论了正、逆压电效应在基体孔周处的机电行为·     

Non-local theory solution of two collinear mode-I cracks in piezoelectric materials

In this paper, the basic solution of two collinear cracks in a piezoelectric material plane subjected to a uniform tension loading is investigated by means of the non-local theory. Through the Fourier transform, the problem is solved with the help of two pairs of integral equations, in which the unknown variables are the jumps of displacements across the crack surfaces. To solve the integral equations, the jumps of displacements across the crack surfaces are directly expanded in a series of Jacobi polynomials. Numerical examples are provided to show the effects of the interaction of two cracks, the materials constants and the lattice parameter on the stress field and the electric displacement field near crack tips. Unlike the classical elasticity solution, it is found that no stress and electric displacement singularities are present at crack tips. The non-local elastic solutions yield a finite hoop stress at the crack tip, thus allowing us to using the maximum stress as a fracture criterion in piezoelectric materials.     

A Note On A Loaded Piezoelectric Transformer

Due to the coupling of mechanical and electrical quantities in a piezoceramic material, it is possible to transform a harmonic input voltage to a harmonic output voltage by using a piezoceramic transformer. The advantage of such a transformer is the avoidance of magnetic fields and the very simple assembly. The piezoelectric transformer considered here consists of several parts. Three brass and two piezoceramic rods, that are glued together. The purpose of the system is to transform a determined input voltage to a determined ouput voltage with a high efficiency. Therefore, one of the piezoceramics is excited harmonically with a frequency close to a resonance frequency of the system. According to the electromechanical coupling in piezoceramics, the transformer oscillates with the excitation frequency. Due to the piezoelectric effect an electric displacement or an electric voltage is generated between the electrodes of the second piezoceramic. The amplitude of the output voltage depends on the geometry and the load connected between the electrodes of the second piezoceramic. In this paper a theoretical model to determine the gain and input impedance for such a transformer is derived and the results are compared with experiments.     

Bending of sinusoidal functionally graded piezoelectric plate under an in-plane magnetic field

This work investigates the bending of a simply supported functionally graded piezoelectric plate under an in-plane magnetic field. The extended sinusoidal plate theory for piezoelectric plate is adopted. The governing equations are derived by the principle of the virtual work considering the Lorentz magnetic force obtained from the Maxwell's relation. The effect of magnetic field, electric loading and gradient index on the displacement, electric potential, stress and electric displacement are numerically presented and discussed in detail. These conclusions will be of particular interest to the future analysis of piezoelectric plate in magnetic field.     

Transient response of two collinear dielectric cracks in a piezoelectric solid under inplane impacts

The paper is focused on the dynamic analysis of two collinear dielectric cracks in a piezoelectric material under the action of in-plane electromechanical impacts. Considering the dielectric permeability of crack interior, the electric displacements at the crack surfaces are governed by the jumps of electric potential and crack opening displacement across the cracks. The permeable and impermeable crack models are the limiting cases of the general one. The Laplace and Fourier transform techniques are further utilized to solve the mixed initial-boundary-value problem, and then to obtain the singular integral equations with Cauchy kernel, which are solved numerically. Dynamic intensity factors of stress, electric displacement and crack opening displacement are determined in time domain by means of a numerical inversion of the Laplace transform. Numerical results for PZT-5H are calculated to show the effects of the dielectric permeability inside the cracks, applied electric loadings and the geometry of the cracks on the fracture parameters in graphics. The observations reveal that based on the COD intensity factor, a positive electric field enhances the dynamic dielectric crack growth and a negative one impedes the dynamic dielectric crack growth in a piezoelectric solid.     

A mixed finite element formulation for piezoelectric shells

A finite element formulation to analyze piezoelectric shell problems is presented. A reference surface of the shell is modelled with a four node element. Each node possesses six mechanical degrees of freedom, three displacements and three rotations, and one electric degree of freedom, which is the difference of the electric potential in thickness direction. The formulation is based on the mixed field variational principle of Hu-Washizu. The independent fields are displacements u , electric potential φ, strains E , electric field E , stresses S and dielectric displacements D . The mixed formulation allows an interpolation of the strains and the electric field in thickness direction. Accordingly a three-dimensional material law is incorporated in the variational formulation. It is remarked that no simplification regarding the constitutive law is assumed. The formulation allows the consideration of arbitrary constitutive relations. The normal zero stress condition and the normal zero dielectric displacement condition are enforced by the independent stress and dielectric displacement fields. They are defined as zero in thickness direction. The present shell element fulfills the important patch tests: the in-plane, bending and shear test. Some numerical examples demonstrate the applicability of the present piezoelectric shell element. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Asymptotic analysis of interactions between highly conducting cylinders

We consider the conduction of heat or electricity between a pair of equal, touching cylinders embedded in a matrix. We solve exactly a problem in which each cylinder contains a heat source, and analyse the solution asymptotically for the case in which the conductivity of the fibres is much greater than that of the matrix. Using this analysis, we assess the accuracy of approximate schemes for calculating the interactions between the fibres of a composite material. We conclude that the approximate solutions are better at predicting global quantities, such as total flux, than they are at predicting pointwise quantities such as the temperature field.     

A Piezoelectric Finite Shell Element for the Geometrically Nonlinear Analysis of Sensors

A geometrically nonlinear finite element formulation to analyze piezoelectric shell structures is presented. The formulation is based on the mixed field variational functional of Hu–Washizu. Within this variational principle the independent fields are displacements, electric potential, strains, electric field, stresses and dielectric displacements. The mixed formulation allows an interpolation of the strains and the electric field through the shell thickness, which is an essential advantage when using a three dimensional material law. It is remarked that no simplification regarding the constitutive relation is assumed. The normal zero stress condition and the normal zero dielectric displacement condition are enforced by the independent resultant stress and resultant dielectric displacement fields. The shell structure is modeled by a reference surface with a four node element. Each node possesses six mechanical degrees of freedom, three displacements and three rotations, and one electrical degree of freedom, which is the difference of the electric potential through the shell thickness. The developed mixed hybrid shell element fulfills the in–plane, bending and shear patch tests, which have been adopted for coupled field problems. A numerical investigation of a smart antenna demonstrates the applicability of the piezoelectric shell element under the consideration of geometrical nonlinearity. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Four-node Finite Element for Piezoelectric Shell Structures in Convective Co-ordinates

In this contribution, an isoparametric piezoelectric shell element is presented which is based on convective coordinates and which allows for the analysis of arbitrary shell geometries. A two-field variation formulation [1, 2] is used in which the displacements and the electric potentials serve as independent variables. Especially, for thin-walled structures under certain boundary conditions and load cases, the displacement based element tend to shear and membrane locking. In order to avoid this poor behaviour, the Assumed Natural Strain (ANS) method [3] is introduced into the piezoelectric shell element. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

两压电介质之间的界面夹杂问题

应用Stroh理论,研究了两压电介质之间的刚性介电线夹杂问题。首先该问题被化为Hilbert问题,然后分别给出了压电介质内的复势函数解、夹杂内的电场解和夹杂尖端场的解析表达式。结果表明,在夹杂尖端附近,所有的场变量均呈现奇异性和振荡性,且其强度取决于介质的材料常数和无限场远处的应变。此外,结果还表明,当从夹杂内部趋近夹杂尖端时,夹杂内的电场也呈现奇异性和振荡性。     

A strip yield model solution for an internally cracked piezoelectric strip

An analysis of the crack closure and fatigue crack growth rate have been carried out for an infinitely long poled piezoelectric ceramic strip weakened by a straight hair line internal crack. The ceramic under consideration is assumed to be mechanically more brittle. The crack faces are perpendicular to the poled direction of the strip. The crack faces open in Mode-I deformation on account of in-plane tension applied to the edges of the strip together with either an in-plane electric displacement prescribed on edges of the strip or a uniform constant electric field prescribed on its edges. As a result, a yield zone is formed ahead of each tip of the crack. The yield zones developed are then arrested by applying a normal, cohesive, linearly varying yield point-stress to their rims. For each case, the Fourier transform method is used to find a solution. The resulting integral equations are solved numerically. Expressions are derived for the crack opening displacement and the crack growth rate. The variations in these quantities are plotted in relation to the affecting parameters, viz., the strip thickness, the yield zone length, the electric displacement, and material constants. A case study is presented graphically for PZT-4, PZT-5H, and BaTiO₃ ceramics. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 44, No. 5, pp. 647–664, September–October, 2008.