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.     

Exact solution of a flat smooth punch on a piezoelectric half plane

This paper derives an exact solution for a flat smooth punch applied on a piezoelectric half plane. The piezoelectric solid occupies the lower half plane, and a flat rigid punch is applied on it. As the permittivity of the air (environment) is far less than that of the piezoelectric material, the electric induction of air may be neglected. The permittivity of the punch is also far less than that of the piezoelectric material and consequently the normal component of the electric displacement vanishes at the contact boundary. The exact solution is obtained by eigen-function representation and analytic continuation. The distribution of pressure under the punch has been found. The electric field along the surface of the lower half plane is extracted in a closed form.     

Penny-shaped crack in transversely isotropic piezoelectric materials

Using a method of potential functions introduced successively to integrate the field equations of three-dimensional problems for transversely isotropic piezoelectric materials, we obtain the so-called general solution in which the displacement components and electric potential functions are represented by a singular function satisfying some special partial differential equations of 6th order. In order to analyse the mechanical-electric coupling behaviour of penny-shaped crack for above materials, another form of the general solution is obtained under cylindrical coordinate system by introducing three quasi-harmonic functions into the general equations obtained above. It is shown that both the two forms of the general solutions are complete. Furthermore, the mechanical-electric coupling behaviour of penny-shaped crack in transversely isotropic piezoelectric media is analysed under axisymmetric tensile loading case, and the crack-tip stress field and electric displacement field are obtained. The results show that the stress and the electric displacement components near the crack tip have (r ^−1/2) singularity. The project supported by the Natural Science Foundation of Shaanxi Province, China     

A Second-Order Solution of Saint-Venant's Problem for a Piezoelectric Circular Bar Using Signorini's Perturbation Method

We study electromechanical deformations of a homogeneous transversely isotropic piezoelectric prismatic circular bar loaded only at the end faces. The constitutive relations for the material of the bar are taken to be quadratic in the displacement gradients and the electric field. It is found that the two end faces of the bar when twisted with no electric charge applied to them will exhibit a difference in the electric potential. Thus the piezoelectric cylinder could be used to measure the torque or the angular twist. This revised version was published online in August 2006 with corrections to the Cover Date.     

Elliptical crack and punch problems solved by integral equation method for piezoelectric media

The problem of an elliptical crack embedded in an unbounded transversely isotropic piezoelectric media with the crack-plane parallel to the plane of isotropy of the media and subjected to remote normal mechanical as well as electric loading is considered first. The problem has been successfully reduced to a pair of coupled integral equations that are suitable for the application of an integral equation method developed earlier for three-dimensional problems of LEFM. Solution to the mechanical displacement and electric potentials are obtained for prescribed uniform loadings and expressions for corresponding intensity factors and crack opening displacement are deduced. The above method has further been applied to solve the problem of a rigid flat-ended elliptical punch indenting a transversely isotropic piezoelectric half-space surface with the plane of isotropy parallel to the surface. Solutions to mechanical stress and electric displacement are obtained for prescribed constant normal displacement and constant electric potential interior to the elliptical region and expression for the total force required to maintain a prescribed indentation is deduced.     

Crack analysis in semi-infinite transversely isotropic piezoelectric solid. I. Green’s functions

Three-dimensional fundamental solutions corresponding to a unit point force and point electric charge are obtained for a semi-infinite transversely isotropic piezoelectric solid. The free boundary is parallel to the plane of isotropy. They can be used as the Green’s function for solving the problem of a flat circular crack near the free surface which will be dealt with in Part II of this work.     

A generalized JKR-model for two-dimensional adhesive contact of transversely isotropic piezoelectric half-space

The aim of the present paper is to investigate the adhesive behavior between a transversely isotropic piezoelectric half-space and a cylinder punch subjected to combined mechanical and electric loads under plane-strain condition. The effect of adhesion is described by using a generalized JKR-model which can account for the non-slip condition in the contact regions. Analytical function theory is employed to find the solution of the resulting singular integral equations. Our analysis shows that the adhesive contact behavior for different types of piezoelectric materials may be quite different. The results obtained in this paper may be helpful to understand the contact mechanics of piezoelectric materials at micro-scale.     

Indentation of punches into a piezoceramic body: two-dimensional contact problem of electroelasticity

The paper establishes the relationship between the solutions of the static contact problems of elasticity (no friction) for an isotropic half-plane and problems of electroelasticity for a transversely isotropic piezoelectric half-plane with the boundary perpendicular to the polarization axis. This allows finding the contact characteristics in the electroelastic case from the known elastic solution, without the need to solve the electroelastic problem. The contact problems of electroelasticity for different types of wedge-shaped punches (flat punch with rounded one or two edges, half-parabolic punch, and a periodic system of punches) are solved as examples Translated from Prikladnaya Mekhanika, Vol. 44, No. 11, pp. 55–70, November 2008.     

Exact Electromagnetothermoelastic Solution for a Transversely Isotropic Piezoelectric Hollow Sphere Subjected to Arbitrary Thermal Shock

This paper presents analytical study for electromagnetothermoelastic transient behavior of a transversely isotropic hollow sphere, placed in a uniform magnetic field, subjected to arbitrary thermal shock. Exact solutions for the transient responses of stresses, perturbation of magnetic field vector, electric displacement and electric potential in the transversely isotropic piezoelectric hollow sphere are obtained by means of the Hankel transform, the Laplace transform and their inverse transforms. An interpolation method is used to solve the Volterra integral equation of the second kind caused by interactions among electric, magnetic, thermal and elastic fields. From the sample numerical calculations, it is seen that the present method is suitable for the transversely isotropic hollow sphere, placed in a uniform magnetic field, subjected to arbitrary thermal shock. Finally, the result can be used as a reference to solve other transient coupling problems of electromagnetothermoelasticity.     

功能梯度压电材料圆环在位移和反对称电势边值条件下的解析解

采用了在径向极化情况下横观各向同性的线性本构关系,考虑了材料性质沿径向的梯度分布,对功能梯度压电材料圆环在给定的位移和电势边界条件下,导出了问题的一般解．推导了外壁固定、接地,内壁沿垂向有一微小位移、电势为反对称分布问题的解析解,并计算了该问题在位移和电势作用情况下的位移、电势在不同梯度分布时的数值结果．     

Indentation responses of piezoelectric films

Frictionless indentation responses of transversely isotropic piezoelectric film/rigid substrate systems under circular cylindrical indenter (i.e., punch), conical indenter (i.e., cone), and spherical indenter (i.e., sphere) are investigated. Both insulating and conducting indenters are considered. The technique of Hankel transformation is employed to derive the corresponding dual integral equations for the mixed boundary value indentation problems. For the two limiting cases of infinitely thick and infinitely thin piezoelectric films, closed-form solutions are obtained. For piezoelectric films of finite thickness, a numerical method is constructed to solve the dual integral equations and semi-empirical models having only two unknown parameters are proposed for the responses of indentation force, electric charge and electric potential, and contact radius. With the two parameters inferred from the numerical results, the semi-empirical formulae are found to provide good estimates of the indentation responses for the two limiting cases of infinitely thick and thin piezoelectric films, as well as those in between. The inferred parameters in the proposed semi-empirical formulae for normalized indentation force and electric charge are checked against four different piezoelectric materials and are found to be insensitive to the selection of piezoelectric materials. It is believed that the proposed semi-empirical indentation formulae are useful in developing experimental indentation techniques to extract the material properties of piezoelectric films.     

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.     

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.     

Axisymmetric bonded punch problem: A complete solution

Summary Explicit expressions are derived for the field of stresses and displacements around an axisymmetric punch bonded to a transversely isotropic elastic half-space. The method is based on the new results in potential theory obtained by the author earlier. A flat centrally loaded circular punch is considered as an example. Specific computations were performed in order to compare the elastic field in the vicinity of a bonded punch with similar parameters for a smooth punch.

---

Eine vollständige Lösung des Stempelproblems bei Axialsymmetrie und Haften

Übersieht Für die Spannungs- und Verschiebungsfelder eines transversal-isotropn elastischen Halbraums, der unter Haftbedingung von einem axialsymmetrischen Stempel belastet ist, werden explizite Formeln hergeleitet. Die dazu benutzte Methode beruht auf neueren Ergebnissen des Autors in der Potentialtheorie. Als Beispiel wird ein zentrisch belasteter, flacher Kreisstempel behandelt. Als numerische Ergebnisse werden die Verschiebungsfelder in der Umgebung eines haftenden und glatten Stempels bei gleichen Parameterbedingungen zum Vergleich vorgestellt.

     

The principle of correspondence between elastic and piezoelectric problems

Summary  A correspondence principle is established between elastic and piezoelectric problems for transversely isotropic materials, in such a way that the knowledge of an elastic solution yields fully coupled electro–elastic fields for the corresponding piezoelectric problem, provided the elastic solution is written in a certain form. The implementation of this principle is illustrated by constructing, in a routine way, several piezoelectric solutions involving crack and punch problems (one of them has not been solved previously). Received 12 Feburary 2002; accepted for publication 29 April 2002     

Circular crack in a transversely isotropic piezoelectric space under point forces and point charges

In this paper, two kinds of circular crack including external circular crack and penny-shaped crack in a transversely isotropic piezoelectric space are considered. Firstly, we obtain the solution to the problem of an external circular crack in a transversely isotropic piezoelectric space subjected to antisymmetric normal point forces and point charges. Based on this, the solution of one-sided loading of an external circular crack is constructed. Secondly, the real shape of an external circular crack and the opening displacement of a penny-shaped crack under an arbitrary point force and point charge are further obtained. At last, the results are presented in a graphical form. The project supported by the National Natural Science Foundation of China (19872060 and 69982009) and the Postdoctoral Foundation of China     

Derivation of Plate and Rod Equations for a Piezoelectric Body from a Mixed Three-Dimensional Variational Principle

We use a mixed 3-dimensional variational principle to derive 2-dimensional equations for an anisotropic plate-like piezoelectric body and one-dimensional equations for an anisotropic beam-like piezoelectric body. The formulation accounts for double forces without moments which may change the thickness of the plate and deform the cross-section of the rod. The dependence of the bending rigidities of a transversely isotropic plate upon the angle between the normal to the midsurface and the direction of transverse isotropy is exhibited. The plate equations are used to study the cylindrical deformations of a transversely isotropic plate due to equal and opposite charges applied to its top and bottom surfaces. It is also found that a piezoelectric circular rod with axis of transverse isotropy not coincident with its centroidal axis and subjected to electric charges at the end faces is deformed into a non-prismatic body.     

A moving piezoelectric screw dislocation

A moving screw dislocation in a transversely isotropic piezoelectric material is investigated in detail. The analytical expressions for displacement, electric potential, stresses, electric displacements, strains, and electric fields are obtained. When the velocity of the dislocation is zero, or when the piezoelectric constant is zero, the results derived in this paper can reduce to known results.     

Electro-mechanical frictionless contact behavior of a functionally graded piezoelectric layered half-plane under a rigid punch

The frictionless contact problem of a functionally graded piezoelectric layered half-plane in-plane strain state under the action of a rigid flat or cylindrical punch is investigated in this paper. It is assumed that the punch is a perfect electrical conductor with a constant potential. The electro-elastic properties of the functionally graded piezoelectric materials (FGPMs) vary exponentially along the thickness direction. The problem is reduced to a pair of coupled Cauchy singular integral equations by using the Fourier integral transform technique and then is numerically solved to determine the contact pressure, surface electric charge distribution, normal stress and electric displacement fields. For a flat punch, the normal stress intensity factor and electric displacement intensity factor are also given to quantitatively characterize the singularity behavior at the punch ends. Numerical results show that both material property gradient of the FGPM layer and punch geometry have a significant influence on the contact performance of the FGPM layered half-plane.     

压电介质中受拉伸与弯曲联合作用的圆币形裂纹问题

以弹性位移分量和电势函数为基本未知量时，横观各向同性压电介质非轴对称三维问题的控制微分方程是四个二阶线性偏微分方程相联立的方程组。本文导出了用四个调和函数表示位移及电势的该方程组的势函数通解。作为通解的应用举例，文中求解了压电陶瓷材料中受拉伸与弯曲联合作用的圆币形裂纹问题，得到了裂纹尖端附近应力场及电位移场的解析表达式。结果表明裂尖场以及应力强度因子和电位移强度因子均表现出复杂的机－电耦合行为。