Transient piezothemoelasticity of a hexagonal plate of class 6 mm

Summary Piezoelastic materials entail piezoelectric polarization that is directly proportional to the applied strain. Therefore, piezoelastic materials have recently attracted considerable attention because of their potential use in intelligent structural systems. In this paper, we treat a transient problem of piezothermoelasticity in a hexagonal plate of class 6 mm subjected to transient temperature change. We analyze the transient problem by use of the potential function method, which can separate the coupling between the thermoelastic and electric fields. Numerical calculations for the transient problem are carried out for a cadmium selenide solid, and the results are presented graphically in comparison with those for a steady problem.     

On the boundary conditions for transversely isotropic piezoelectric plates

By invoking the theorem of work reciprocity for piezoelectric media, necessary conditions, which the prescribed edge data of the plate must fulfill in order that it should generate a decaying state within the plate, are established through generalizing the method proposed by Gregory and Wan. These decaying state conditions for the case of axisymmetric deformation of a transversely isotropic piezoelectric circular plate when stress and electric displacement conditions are imposed on the plate edge are derived explicitly, which are then used for the formulation of boundary conditions for the plate theory solution (or the interior solution). Also an analytical solution of the axisymmetric decaying state of transversely isotropic piezoelectric circular plates is derived. Furthermore, the corresponding necessary conditions for the axisymmetric deformation of elastic circular plates are indeed reproduced directly.     

Modelling of finite piezoelectric patches: Comparing an approximate power series expansion theory with exact theory

Plate equations for a plate consisting of one elastic layer and one piezoelectric layer with an applied electric voltage have previously been derived by use of power series expansions of the field variables in the thickness coordinate. These plate equations are here evaluated by the consideration of a time harmonic 2D vibration problem with finite layers. The boundary conditions at the sides of the layers then have to be considered. Numerical comparisons of the displacement field are made with solutions from two other theories; a solution with equivalent boundary conditions for a thin piezoelectric layer applied on an elastic plate and an exact solution based on Fourier series expansions. The two approximate theories are shown to be equally good and they both yield accurate results for low frequencies and thin plates.     

Green’s functions for multi-phase isotropic laminated plates

This work develops a series of Green’s functions for multi-phase Kirchhoff isotropic laminated plates. First, we derive the Green’s functions for a composite laminated plate composed of two bonded dissimilar isotropic laminated semi-infinite plates. Second, the obtained results for bimaterials are judiciously applied to obtain the Green’s function solution for a circular elastic inclusion embedded in an infinite isotropic laminated plate. Third, Green’s functions for a composite space composed of an arbitrary number of wedges of different isotropic laminated plates are derived. Finally, we derive Green’s functions for a laminated plate with an elliptical and a parabolic boundary, respectively.     

Plate-Like Smart Structures: Reduced Models and Numerical Simulations

In this paper asymptotic models describing the mechanical and electric equilibrium state of two types of smart structures are presented and justified. The first structure consists of an anisotropic elastic thin plate with two surface bonded anisotropic piezoelectric patches and the second one is an anisotropic elastic sandwich thin plate with an inserted anisotropic piezoelectric patch. The two unknowns of the corresponding asymptotic models, the mechanical displacements of the structures and the electric potentials of the patches, are partially decoupled. The former are the solution of modified Kirchhoff-Love plate models, while the latter can be derived as explicit functions of the mechanical displacements. Moreover, different formulas for the electric potential arise as a consequence of diverse electric boundary conditions. We report numerical simulations with these asymptotic models.     

Eccentric crack in a piezoelectric strip bonded to half planes

We consider the problem of determining the singular stresses and electric fields in a piezoelectric ceramic strip containing an eccentric Griffith crack off the centre line bonded to two elastic half planes under anti-plane shear loading using the continuous crack-face condition. Fourier transforms are used to reduce the problem to the solution of two pairs of dual integral equations, which are then expressed to a Fredholm integral equation of the second kind. Numerical values on the stress intensity factor and energy release rate are obtained.     

Finite-part integral and boundary element method to solve three-dimensional crack problems in piezoelectric materials

Using the hypersingular integral equation method based on body force method, a planar crack in a three-dimensional transversely isotropic piezoelectric solid under mechanical and electrical loads is analyzed. This crack problem is reduced to solve a set of hypersingular integral equations. Compare with the crack problems in elastic isotropic materials, it is shown that for the impermeable crack, the intensity factors for piezoelectric materials can be obtained from those for elastic isotropic materials. Based on the exact analytical solution of the singular stresses and electrical displacements near the crack front, the numerical method of the hypersingular integral equation is proposed by the finite-part integral method and boundary element method, which the square root models of the displacement and electric potential discontinuities in elements near the crack front are applied. Finally, the numerical solutions of the stress and electric field intensity factors of some examples are given.     

Spherical-symmetric steady-state response of piezoelectric spherical shell under external excitation

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Analytical solution for the cylindrical bending vibration of piezoelectric composite plates

An analytical solution for the cylindrical bending vibrations of linear piezoelectric laminated plates is obtained by extending the Stroh formalism to the generalized plane strain vibrations of piezoelectric materials. The laminated plate consists of homogeneous elastic or piezoelectric laminae of arbitrary thickness and width. Fourier basis functions for the mechanical displacements and electric potential that identically satisfy the equations of motion and the charge equation of electrostatics are used to solve boundary value problems via the superposition principle. The coefficients in the infinite series solution are determined from the boundary conditions at the edges and continuity conditions at the interfaces between laminae, which are satisfied in the sense of Fourier series. The formulation admits different boundary conditions at the edges of the laminated piezoelectric composite plate. Results for laminated elastic plates with either distributed or segmented piezoelectric actuators are presented for different sets of boundary conditions at the edges.     

Accurate modelling of piezoelectric plates: single-layered plate

Summary  The paper presents an efficient two-dimensional approach to piezoelectric plates in the framework of linear theory of piezoelectricity. The approximation of the through-the-thickness variations accounts for the shear effects and a refinement of the electric potential. Using a variational formalism, electromechanically coupled plate equations are obtained for the generalized stress resultants as well as for the generalized electric inductions. The latter are deduced from the conservative electric charge equation, which plays a crucial role in the present model. Emphasis is placed on the boundary conditions at the plate faces. The model is used to examine some problems for cylindrical bending of a single simply supported plate. Number of situations are examined for a piezoelectric plate subject to (i) an applied electric potential, (ii) a surface density of force, and (iii) a surface density of electric charge. The through-thickness distributions of electromechanical quantities (displacements, stresses, electric potential and displacement) are obtained, and compared with results provided by finite element simulations and by a simplified plate model without shear effects. A good agreement is observed between the results coming from the present plate model and finite element computations, which ascertains the effectiveness of the proposed approach to piezoelectric plates. Received 17 July 2000; accepted for publication 26 September 2000     

Green’s functions of a surface-stiffened transversely isotropic half-space

Green’s functions of a transversely isotropic half-space overlaid by a thin coating layer are analytically obtained. The surface coating is modeled by a Kirchhoff thin plate perfectly bonded to the half-space. With the aid of superposition technique and employing appropriate displacement potential functions, the Green’s functions are expressed in two parts; (i) a closed-form part corresponding to the transversely isotropic half-space with surface kinematic constraints, and (ii) a numerically evaluated part reflecting the interaction between the half-space and the plate in the form of semi-infinite integrals. Some limiting cases of the problem such as surface-stiffened isotropic half-space, Boussinesq and Cerruti loadings, and extremely flexible and rigid plates are also studied. For the classical Cerruti problem in transversely isotropic materials, the effects of incompressibility are highlighted. Numerical results are provided to show the effects of material anisotropy, relative stiffness factor, and load buried depth. The obtained Green’s functions play a key role in treating further mixed-boundary-value problems in surface stiffened transversely isotropic half-spaces.     

A new two-dimensional model for electro-mechanical response of thick laminated piezoelectric actuator

This paper investigates the electro-mechanical behaviour of a thick, laminated actuator with piezoelectric and isotropic lamina under externally applied electric loading using a new two-dimensional computational model. The elastic core is relatively thick and thus it is modelled by Timoshenko thick-beam theory. Although the piezoelectric lamina is a beam-like layer, it is formulated via a two-dimensional model because of not only the strong electro-mechanical coupling, but also of the presence of a two-dimensional electric field. It is shown in this paper that a one-dimensional model for the piezoelectric beam-like layer is inadequate. The piezoelectric model is constructed within the scope of linear piezoelectricity. The actuation response is induced through the application of external electric voltage. Under the strong coupling of elasticity and electricity, the strain energy and work of electric potential are presented. The electro-mechanical response of the laminated Timoshenko beam is formulated and determined via a variational energy principle. Numerical examples presented illustrate convincing comparison with finite element solutions and existing published data. New numerical solutions are also presented to investigate the geometric effect on the electro-mechanical bending behaviour.     

Green''s functions for transversely isotropic piezoelectric solids

Closed-form expressions are obtained for the infinite-body Green's functions for a transversely isotropic piezoelectric medium. The four Green's functions represent the coupled elastic and electric response to an applied point force or point charge. The Green's functions are obtained using a formulation where the three displacements and the electric potential are derivable from two potential functions. When piezoelectric coupling is absent, the results reduce to those for uncoupled elasticity and electrostatics.     

On the micro-mechanical study of 1–3 type piezoelectric composites with semi-coupled thermo-electro-elastic effects

This paper deals with a two-dimensional generalized plane strain micro-mechanical model to simulate semi-coupled thermo-electro-elastic behavior of transversely polarized piezoelectric fibrous composites. The solution domain includes a representative volume element (RVE) consists of a long piezoelectric fiber surrounded by corresponding matrix in a square array arrangement. Fibers have orthotropic and/or transversely isotropic properties while are perfectly bonded to the isotropic matrix. In addition, the constituents are assumed to have linear thermo-electro-elastic behavior. The virtual form of equilibrium equations has been extended to cover the semi-coupled thermo-electro-elastic loading by using appropriate constitutive relations. The element-free Galerkin method is employed to discretize the governing equations in terms of three main primary variables including, displacements, electric potential and temperature. The performance of the present micro-mechanical study reveals close agreement compared with other techniques available in the literature. Based on the present study, ample results are addressed to provide an insight into the effects of the local fields, i.e. displacement, electric potential, electric field, and stress distributions within the RVE for the specific fiber volume fraction.     

自由端受集中力作用下压电悬臂梁弯曲问题解析解

本文对由横观各向同性压电介质构成的悬臂梁，在自由端受集中力作用下的弯曲问题进行了研究。首先根据问题的特点，得到简化的线弹性压电悬臂梁的基本方程。然后根据正交各向异性材料悬臂梁应力分布特点，采用逆解法，建立了该问题的应力函数与电势分布函数，进而得到精确多项式解析解。该解析解形式简单，便于应用。文中对自由端受集中力的常规材料和压电材料悬臂梁的挠度也进行了比较。     

Closed form solution for two unequal collinear semi-permeable straight cracks in a piezoelectric media

The problem of two unequal collinear straight cracks weakening a poled transversely isotropic piezoelectric ceramic is addressed under semi-permeable electric boundary conditions on the crack faces. The plate has been subjected to combined in-plane normal(to the faces of the cracks) mechanical and electric loads. Problem is formulated employing Stroh formalism and solved using complex variable technique. The elastic field, electric field and energy release rate are obtained in closed analytic form. A case study is presented for poled PZT-5H cracked plate to study the effect of prescribed mechanical load, electric load, inter-crack distance and crack lengths on crack arrest parameters stress intensity factor (SIF), electric displacement intensity factor (EDIF) and mechanical and total energy release rates (ERR). Moreover a comparative study is done of impermeable and semi-permeable crack face boundary conditions on SIF, EDIF and ERR, and results obtained is presented graphically. It is observed that the effect of dielectric medium in the crack gap cannot be ignored.     

Effect of initial stress on the propagation behavior of Love waves in a layered piezoelectric structure

The propagation behavior of Love waves in a layered piezoelectric structure with an initial stress is investigated in this article. It involves a thin piezoelectric layer bonded perfectly to an elastic substrate. Solutions of the mechanical displacement and electrical potential function are obtained for the piezoelectric layer and elastic substrate by solving the coupled electromechanical field equations. The phase velocity equations of the Love wave propagation and the stress fields in the layered piezoelectric structure are obtained for electrical open and short cases on the free surface, respectively. The effect of the initial stress on the phase velocity, the stress fields and the coupled electromechanical factor are discussed, respectively. Three sets of piezoelectric layer–elastic substrate systems are considered, i.e. BaTiO₃ ceramic layer–borosilicate glass substrate, PZT-5H ceramic layer–borosilicate glass substrate, and PZT-5H ceramic layer–SiO₂ glass substrate. It is seen that the phase velocity of the Love wave propagation decreases with the increase of the magnitude of the initial stress. The coupled electromechanical factor increases remarkably, as the magnitude of the initial the stress is greater than 100 MPa. This is useful for the design of acoustic surface wave devices.     

The scattering of P-waves by a piezoelectric particle with FGPM interfacial layers in a polymer matrix

Propagation of P-wave in an unbounded elastic polymer medium which contains a set of nested concentric spherical piezoelectric inhomogeneities is formulated. The polymer matrix is made of Epoxy and is isotropic; each phase of the inhomogeneity is made of a different piezoelectric material and is radially polarized and has spherical isotropy. Note that the individual phases are homogeneous, and all interfaces are perfectly bonded. The scattered displacement and electric potentials in the matrix are expressed in terms of spherical wave vector functions and Legendre functions, respectively. The transmitted displacement and electric potentials within each phase of the piezoelectric particle are expressed in terms of Legendre functions. The equations of motion and electrostatics in each phase of the piezoelectric inhomogeneity lead to a system of coupled second order differential equations, which is solved using the generalized Frobenius series. The present theory is extended to the case where the core of the inhomogeneity is made of PZT-4 and its coating is made of functionally graded piezoelectric material (FGPM) whose microstructural composition varies smoothly from PZT-4 at the core–coating interface to Epoxy at the coating–matrix interface. The effects of different types of variation in the electro-mechanical properties of FGPM on scattering cross-section and other electro-mechanical fields are addressed. The present theory is valid for arbitrary coating thickness, and arbitrary frequencies.     

Mechanics of indentation for piezoelectric thin films on elastic substrate

Frictionless normal indentation problem of rigid flat-ended cylindrical, conical and spherical indenters on piezoelectric film, which is either in frictionless contact with or perfectly bonded to an elastic half-space (substrate), is investigated. Both conducting and insulating indenters are considered. With Hankel transform, the general solutions of the homogeneous governing equations for the piezoelectric layer and the elastic half-space are presented. Using the boundary conditions for a vertical point force or a point electric charge, and the boundary conditions on the film/substrate interface, the Green’s functions can be obtained by solving sets of simultaneous linear algebraic equations. The solution of the indentation problem is obtained by integrating these Green’s functions over the contact area with unknown surface tractions or electric charge distribution, which will be determined from the boundary conditions on the contact surface between the indenter and the film. The solution is expressed in terms of dual integral equations that are converted to a Fredholm integral equation of the second kind and solved numerically. Numerical examples are also presented. The comparison between two film/substrate bonding conditions is made. It shows that the indentation rigidity of the film/substrate system is lower when the film is in frictionless contact with the substrate. The effects of the Young’s modulus and Poisson’s ratio of the elastic substrate, indenter electrical condition and indenter prescribed electric potential on the indentation responses are presented.     

Electroelastic analysis for a piezoelectric layer with surface electrodes

An electroelastic analysis of a transverse isotropic piezoelectric layer with surface electrodes is made. The piezoelectric layer is infinite long along the poling direction, and the top surface is perfectly bonded to a rigid electrode. The problem is solved via the conformal mapping technique for two cases of elastic boundary conditions on the bottom surface with two spaced electrodes, and the distribution of the electrostatic field in the entire piezoelectric layer is determined in an explicit analytic form, respectively. It is found that for the bottom surface electrodes with vanishing stiffness, the induced strain is singular, but no stress. Instead, for the bottom electrodes with stiffness as infinity, the induced stress is singular, but no strain.