Anti‐plane mechanical and in‐plane electrical fields for a finite electrode/punch on a finite piezoelectric layer

The problems of a surface electrode and a rigid punch on a finite piezoelectric layer are considered in this paper. The resultant force and the accumulated electric charge on the electrode/punch are prescribed. Closed‐form solutions for the electromechanical fields at the electrode/punch tip are obtained and are expressed in terms of the applied strain and electric field intensity factors. For infinite layer thickness, the strain and electric field intensity factors are obtained in closed‐form. For finite layer thickness, the strain and electric field intensity factors are obtained numerically by the singular integral equation technique. The effect of layer thickness on the electrode/punch tip fields is discussed. It is found that the field intensities at the electrode/punch tip can be reduced considerably by decreasing layer thickness. In addition to the single electrode/punch problem, this paper also provides a solution technique for two collinear surface electrodes/punches on a finite piezoelectric layer. The effect of the relative distance between the two electrodes/punches on the electromechanical fields in the piezoelectric layer is also discussed.     

The electrostatic problem for the SAW interdigital transducers inan external electric field. I. A general solution for a limited numberof electrodes

An exact solution for finding the surface charge and electric field distributions in interdigital transducers (IDTs) with a limited number N of electrodes is given. It is based on the Keldysh-Sedov solution to the mixed boundary problem of the analytic function theory. The IDT electrodes are placed on the plane boundary between two anisotropic dielectric media. The external electric field may arbitrarily vary along the structure. The solution contains N constants which may be found from the electrodes' connection conditions. For determining these constants a linear set of algebraic equations is obtained. The coefficients of this system are written in explicit form. The capacitance coefficients for a system of electrodes of different widths are obtained. For illustration purposes systems with one and two electrodes are considered in greater detail. In these cases the external electric field is assumed to vary harmonically along the structure with an arbitrary wavenumber. For one electrode the Fourier transform of the charge distribution is obtained in terms of the Bessel functions. For two electrodes of different widths a simple expression for the capacitance is found. The charge and electric field distributions are represented graphically for several wavenumbers and geometrical sizes of the electrode system. Section I contains a survey, including Russian literature, which is not well known in the west     

An intermediate crack model for flaws in piezoelectric solids

Summary The aim of this paper is to study a crack model for piezoelectric bodies. In recent years it became evident that the electrically impermeable or the perfect electric contact boundary conditions on the crack faces are inadequate for many physical situations, over- or underestimating the electric field influence on the propagation process. The crack model here investigated is intermediate between these two limit cases. The generalized plane problem for an infinite piezoelectric body with a central crack is converted into a system of integro-differential equations, then reduced to an integro-differential equation similar to Prandtl's equation of aerodynamics. For poled ceramics with transversely isotropic symmetry, the integral equation is numerically solved using quadrature formulae for both plane and antiplane states of deformation. The energy release rate calculated with the discrete solution is then compared with that given by the exact solution for an elliptic hole embedded in the infinite piezoelectric body. A range of values of the cavity thickness is found, for which the considered crack model is a good approximation of the exact two-body problem, while the impermeable and the perfect contact models are not appropriate.     

电磁功能梯度材料层合板中表面波的弥散特性

应用混合数值法研究电磁复合材料层合板中表面波的弥散特性。假设材料的材料常数和电、磁常数沿板厚方向呈线性变化,首先用线条元将电磁复合材料层合板划分为单元,建立单元的动力学微分方程,然后根据单元之间的连续条件将单元控制方程装配成系统的控制方程,将单元的位移向量表示成波动形式的解,得到波数域内的系统控制方程,求解系统的特征方程,得到波数与频率的关系即弥散关系,考察电、磁效应对波的弥散特性的影响。     

Electroelastic Analysis of a Piezoelectric Layer with Electrodes

The electroelastic analysis of a piezoelectric layer with two semi-infinite electrodes is made. The layer surfaces are clamped and a voltage is applied between the layer surfaces and the electrodes. By using the integral equation method, the electroelastic field is derived for the cases of crack-type electrodes and of rigid electrodes, respectively. It indicates that the concentration of electric field an electric displacement occurs near the electrode tips for both cases. However, the stresses are bounded for the former, and possess singularity for the latter. The field intensity factors and energy release rate are obtained in explicit form. The counterpart for a single semi-infinite electrode is given.     

A magneto-electro-elastic material with a penny-shaped crack subjected to temperature loading

Summary The analysis of intensity factors for a penny-shaped crack under thermal, mechanical, electrical and magnetic boundary conditions becomes a very important topic in fracture mechanics. An exact solution is derived for the problem of a penny-shaped crack in a magneto-electro-thermo-elastic material in a temperature field. The problem is analyzed within the framework of the theory of linear magneto-electro-thermo-elasticity. The coupling features of transversely isotropic magneto-electro-thermo-elastic solids are governed by a system of partial differential equations with respect to the elastic displacements, the electric potential, the magnetic potential and the temperature field. The heat conduction equation and equilibrium equations for an infinite magneto-electro-thermo-elastic media are solved by means of the Hankel integral transform. The mathematical formulations for the crack conditions are derived as a set of dual integral equations, which, in turn, are reduced to Abel's integral equation. Solution of Abel's integral equation is applied to derive the elastic, electric and magnetic fields as well as field intensity factors. The intensity factors of thermal stress, electric displacement and magnetic induction are derived explicitly for approximate (impermeable or permeable) and exact (a notch of finite thickness crack) conditions. Due to its explicitness, the solution is remarkable and should be of great interest in the magneto-electro-thermo-elastic material analysis and design.     

Modeling the response of polymer–ionic liquid electromechanical actuators

A model is derived for the electromechanical response of a porous membrane swollen with an ionic liquid and sandwiched between two nanoscale-thin electrodes under DC current. Bending of the membrane is induced by pressure in pores arising due to diffusion of ions through a network of nanochannels. Transport of ions is governed by the applied electric field and redox reactions at the surfaces of electrodes. Constitutive equations for the mechanical response of a porous medium and diffusion of ions are derived by means of the free energy imbalance inequality under an arbitrary deformation with finite strains. Under the assumption regarding small strains, but finite changes in concentrations of ions and the electrostatic potential, an explicit expression is developed for the curvature of the membrane. A steady-state solution to the Poisson–Nernst–Planck equations is obtained by means of the method of matched asymptotic expansions. Results of numerical analysis demonstrate the ability of the constitutive equations to describe observations. In particular, the model provides an explanation for bending to the anode and to the cathode and predicts qualitatively the effects of applied voltage, concentration of ionic liquid, and thickness of a membrane on its curvature.     

A two-dimensional model of glow discharge in view of vibrational excitation of molecular nitrogen

A two-dimensional model is suggested of dc glow discharge with parallel electrodes. The model includes equations for electron and ion concentrations, which are related to the Poisson equation for electric potential. The collision ionization and electron-ion recombination are described using empirical relations. Processes of vibrational excitation of N₂ molecules are considered. The vibrational kinetics of nitrogen are described in view of excitation by electron impact, vibrational exchange, and vibrational-translational relaxation. The finite-difference model used for solving kinetic equations is described. Results are obtained for a two-dimensional model of glow discharge in nitrogen at a pressure of 5 torr and an emf of 2000 V. The obtained fields of distribution of electron temperature and populations of vibrational levels of nitrogen are analyzed. The results enable one to estimate the fraction of electric field energy utilized for the excitation of vibrational states.     

Three-dimensional analysis of an antiparallel piezoelectric bimorph

Summary Three-dimensional electromechanical responses of a piezoelectric bimorph are studied. The bimorph is antiparallel in the sense that it consists of two identical, plate-like piezoelectric elements with opposite poling directions. Both the top and bottom surfaces of the bimorph are fully covered with negligibly thin conductive electrodes. By introducing a small parameter and using the transfer matrix method it is shown that a three-dimensional solution of the problem can be readily constructed, provided the solution to a set of two-dimensional equations very similar to those in the classic plate theory is obtainable. The three-dimensional solution satisfies all the field equations as well as the boundary conditions on the major surfaces and at the interface between the two piezoelectric plates. In many special cases, the electric edge condition can be fulfilled point by point, and thus the solution is exact in Saint-Venant's sense. The formulation and new analytical results for a strip-shaped cantilever bimorph under the action of applied voltage and end moment are presented.     

Fast solution of the elasticity problem for a planar crack of arbitrary shape in 3D-anisotropic medium

A planar crack of arbitrary shape in a 3D-anisotropic elastic medium subjected to an arbitrary external stress field is considered. An efficient numerical method of the solution of the problem is proposed. The problem is reduced to an integral equation for the crack opening vector on the crack surface. For discretization of this equation, Gaussian (radial) approximation functions centered at a system of nodes that covers the crack surface are used. For such functions, the elements of the matrix of the discretized problem are calculated in a quasi analytical form that involves standard non-singular integrals. If the node grid is regular, the matrix of the discretized system has Teoplitz’s structure, and the Fast Fourier Transform algorithm may be used for the calculation of matrix-vector products with such a matrix. It accelerate substantially the process of the iterative solution of the discretized system. Examples of the solutions for a circular crack in a transversally isotropic elastic medium are presented.     

On Using Collocation in Three Dimensions and Solving a Model Semiconductor Problem

A research code has been written to solve an elliptic system of coupled nonlinear partial differential equations of conservation form on a rectangularly shaped three-dimensional domain. The code uses the method of collocation of Gauss points with tricubic Hermite piecewise continuous polynomial basis functions. The system of equations is solved by iteration. The system of nonlinear equations is linearized, and the system of linear equations is solved by iterative methods. When the matrix of the collocation equations is duly modified by using a scaled block-limited partial pivoting procedure of Gauss elimination, it is found that the rate of convergence of the iterative method is significantly improved and that a solution becomes possible. The code is used to solve Poisson’s equation for a model semiconductor problem. The electric potential distribution is calculated in a metal-oxide-semiconductor structure that is important to the fabrication of electron devices.     

APPLICATION OF THE COLLOCATION METHOD IN THREE DIMENSIONS TO A MODEL SEMICONDUCTOR PROBLEM

A research code has been written to solve an elliptic system of coupled non-linear partial differential equations of conservation form on a rectangularly shaped three-dimensional domain. The code uses the method of collocation of Gauss points with tricubic Hermite piecewise continuous polynomial basis functions. The system of equations is solved by iteration. The system of non-linear equations is linearized, and the system of linear equations is solved by iterative methods. When the matrix of the collocation equations is duly modified by using a scaled block-limited partial pivoting procedure of Gauss elimination, it is found that the rate of convergence of the iterative method is significantly improved and that a solution becomes possible. The code is used to solve Poisson's equation for a model semiconductor problem. The electric potential distribution is calculated in a metal-oxide-semiconductor structure that is important to the fabrication of electron devices.     

Transient analysis of a piezoelectric strip with a permeable crack under anti-plane impact loads

The problem of a through permeable crack situated in the mid-plane of a piezoelectric strip is considered under anti-plane impact loads for two cases. The first is that the strip boundaries are free of stresses and of electric displacements, and the second is that the strip boundaries are clamped rigid electrodes. The method adopted is to reduce the mixed initial-boundary value problem, by using integral transform techniques, to dual integral equations, which are further transformed into a Fredholm integral equation of the second kind by introducing an auxiliary function. The dynamic stress intensity factor and energy release rate in the Laplace transform domain are obtained in explicit form in terms of the auxiliary function. Some numerical results for the dynamic stress intensity factor are presented graphically in the physical space by using numerical techniques for solving the resulting Fredholm integral equation and inverting Laplace transform.     

Two collinear unequal cracks in a poled piezoelectric plane: Mode I case solved by a new approach of real fundamental solutions

The purpose of the present work is to study the problem of two collinear unequal cracks in a piezoelectric plane under mode I electromechanical loadings via a new approach. For the first time, real fundamental solutions are derived for in-plane piezoelectric governing equations. The cracks are simulated by continuously distributed generalized dislocations and Cauchy singular integral equations are established from the solution of a generalized point dislocation. Both the theorectical derivation and numerical computations are validated by the exact solution in a special case. Parametric studies are conducted to reveal the effects of crack space, crack length, electric loading and remanent electric displacement on energy release rate. It is found that negative electric displacement loading can decrease both the total energy release rate (TERR) and the mechanical strain energy release rate (MSERR), implying that it has a shielding effect on cracks definitely. Positive electric displacement loading can enhance MSERR, but meanwhile it can enhance or reduce TERR depending on the magnitude of the electric loading factor. The effect of a remanent electric displacement along the poling direction is equivalent to that of a positive electric field loading and should be considered in engineering design.     

Active macro‐zone approach for incremental elastoplastic‐contact analysis

The symmetric boundary element method, based on the Galerkin hypotheses, has found an application in the nonlinear analysis of plasticity and in contact‐detachment problems, but both dealt with separately. In this paper, we want to treat these complex phenomena together as a linear complementarity problem. A mixed variable multidomain approach is utilized in which the substructures are distinguished into macroelements, where elastic behavior is assumed, and bem‐elements, where it is possible that plastic strains may occur. Elasticity equations are written for all the substructures, and regularity conditions in weighted (weak) form on the boundary sides and in the nodes (strong) between contiguous substructures have to be introduced, in order to attain the solving equation system governing the elastoplastic‐contact/detachment problem. The elastoplasticity is solved by incremental analysis, called for active macro‐zones, and uses the well‐known concept of self‐equilibrium stress field here shown in a discrete form through the introduction of the influence matrix (self‐stress matrix). The solution of the frictionless contact/detachment problem was performed using a strategy based on the consistent formulation of the classical Signorini equations rewritten in discrete form by utilizing boundary nodal quantities as check elements in the zones of potential contact or detachment. Copyright © 2013 John Wiley & Sons, Ltd.     

Electroelastic analysis of an interface anti-plane shear crack in a layered piezoelectric plate

The problem of an anti-plane interface crack in a layered piezoelectric plate composed of two bonded dissimilar piezoelectric ceramic layers subjected to applied voltage is considered. It is assumed that the crack is either impermeable or permeable. An integral transform technique is employed to reduce the problem considered to dual integral equations, then to a Fredholm integral equation by introducing an auxiliary function. Field intensity factors and energy release rate are obtained in explicit form in terms of the auxiliary function. In particular, by solving analytically a resulting singular integral equation, they are determined explicitly in terms of given electromechanical loadings for the case of two bonded layers of equal thickness. Some numerical results are presented graphically to show the influence of the geometric parameters on the field intensity factors and the energy release rate.     

Solving the steady‐state groundwater flow equation for finite linear aquifers using a generalized Fourier series approach in two‐dimensional domains

A method to solve steady linear groundwater flow problems using generalized Fourier Series is developed and particularized for multiple Fourier series in two‐dimensional domains. It leads to a linear vector equation whose solution provides a finite number of generalized Fourier coefficients approximating the hydraulic head field. Its implementation is shown and two relevant properties are found for the system matrix. It is always symmetric and, once computed, if additional Fourier terms are needed for a better approximation of the hydraulic head field, previously computed matrix elements remain invariant, i.e. only new rows and columns are added to the system matrix. The method is demonstrated in three simple cases with different geometries and transmissivity fields, where solutions are compared with analytical and finite element method results. Thus, the method is verified as an alternative to other flow solvers. Additionally, it provides a direct way to obtain the spectral form of the flow equation solution, given a spectral representation of transmissivity, and can be easily extended to obtain continuous velocity fields and their approximated spectral expressions. Copyright © 2009 John Wiley & Sons, Ltd.     

A general constitutive equation of an ER suspension based on the internal variable theory

Summary.  A microstructural constitutive theory of ER suspensions was formulated in this investigation. The framework was based on the internal variable theory and the mechanism analysis. The ER suspension consists of fine particles with high dielectric constant and the supporting fluid. Under the action of the electric field, the polarized particles will aggregate together to form the chain-like structures along the direction of the electric field. As the size and orientation of the particle aggregates are volatile, and they adjust according to the applied electric field and strain rate, the energy conservation equation and the force equilibrium equation were thus established to determine the orientation and size of the aggregates. Following that, a three-dimensional, explicit form of the constitutive equation was derived based on the interaction energy and the dissipation function of the system. The response of the system under the action of a simple shearing load was considered and discussed in detail. It is found that the shear-thinning viscosity of an ER suspension is well approximated by the power-law ∝ (Mn)^−0.82. Received February 15, 2001; revised May 7, 2002 Published online: June 12, 2003     

Modeling charge transfer dynamics and electric field distribution in glasses during poling and electrostimulated diffusion

A generalized problem of positive ion transfer in a dielectric under the action of an electric field in the presence of diffusion and a flux of other positive charge carriers with different mobilities entering into the medium is considered based on the diffusion equation and the Poisson equation. A strict system of equations is written and its numerical solution is obtained in cases where the ions and other carriers possess (i) significantly and (ii) insignificantly different mobilities, which correspond to the poling of glasses and the electrostimulated ion exchange, respectively. The obtained temporal dependences of the depth of a modified region and the electric field (in normalized coordinates) well agree with data available for particular cases.