Electroelastic Fields Research Articles

Fracture mechanics analysis of a composite piezoelectric strip with an internal semi-infinite electrode

IA piezoelectric strip with semi-infinite electrode is investigated. Two combinations of mechanical–electrical loadings are considered. They consist of the anti-plane deformation with in-plane electrical field and the in-plane electroelastic field. Based on the Fourier transform and the Wiener–Hopf technique, the electroelastic local stress fields are found to exhibit the square root singularity near the electrode tip. The energy density factor criterion is applied to examine the fracture behavior near the electrode tip.

Griffith crack moving in a piezoelectric strip

The dynamic problem of an impermeable crack of constant length 2a propagating along a piezoelectric ceramic strip is considered under the action of uniform anti-plane shear stress and uniform electric field. The integral transform technique is employed to reduce the mixed-boundary-value problem to a singular integral equation. For the case of a crack moving in the mid-plane, explicit analytic expressions for the electroelastic field and the field intensity factors are obtained, while for an eccentric crack moving along a piezoelectric strip, numerical results are determined via the Lobatto–Chebyshev collocation method for solving a resulting singular integral equation. The results reveal that the electric-displacement intensity factor is independent of the crack velocity, while other field intensity factors depend on the crack velocity when referred to the moving coordinate system. If the crack velocity vanishes, the present results reduce to those for a stationary crack in a piezoelectric strip. In contrast to the results for a stationary crack, applied stress gives rise to a singular electric field and applied electric field results in a singular stress for a moving crack in a piezoelectric strip.

A piezoelectric screw dislocation near a wedge-shaped bi-material interface

The electro-elastic stress field due to a piezoelectric screw dislocation near the tip of a wedge-shaped bi-material interface is derived. The screw dislocation is subjected to a line charge and a line force at the core. The explicit closed-form analytical solutions for the stress field are derived by means of the complex variable and conformal mapping methods. The stress and electric intensity factors of the wedge tip induced by the dislocation and the image force acting on the dislocation are also formulated and calculated. The influence of the wedge angle and the different bi-material constant combinations on the image force is discussed. Numerical results for three particular wedge angles are calculated and compared.

Electroelastic Analysis of an Internal Interface Crack in a Half-Plane Consisting of Two Bonded Dissimilar Piezoelectric Quarter-Planes

The problem of an interface crack in a half-plane consisting of two bonded dissimilar piezoelectric quarters is considered under antiplane shear and inplane electric loading. The problem is solved under the electrically permeable assumption for a crack. The integral transform technique is employed to reduce the problem to triple integral equations, which is further converted to a hypersingular integral equation for the crack sliding displacement. By solving the resulting equation analytically, the electroelastic field along the interface and the energy release rate are obtained in explicit form, respectively. Several examples are given to illustrate the influence of the material properties and the crack position on the energy release rate.

Homogeneous Solutions of a Skew-Symmetric Problem with Mixed Boundary Conditions for a Piezoceramic Layer

On the basis of a modified scheme of homogeneous solutions significantly simplifying the Lur'e scheme, we construct homogeneous solutions for a piezoceramic layer with mixed boundary conditions imposed on its faces. We consider an electroelastic state skew-symmetric about the median plane of the layer. It is shown that the corresponding homogeneous solutions do not contain biharmonic terms and the components of the conjugate electroelastic field in the layer have the form of Fourier series with respect to the coordinate x3.

846 電極を有する積層圧電アクチュエータの電気弾性場集中と分域回転

846 電極を有する積層圧電アクチュエータの電気弾性場集中と分域回転

Two-Dimensional Electroelastic Problem for a Multiply Connected Piezoelectric Body

In this paper, we present the basic relationships for the complex potentials of a two-dimensional electroelastic problem, their general representations for a multiply connected domain, expressions for stress, displacement, electrostatic field intensity and induction, and potential. A closed solution is found for a body with one elliptic cavity or one elliptic crack under the action at infinity of a constant electroelastic field or concentrated forces and charges

The principle of correspondence between elastic and piezoelectric problems

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).

A conducting arc crack between a circular piezoelectric inclusion and an unbounded matrix

The present paper investigates the problem of a conducting arc crack between a circular piezoelectric inclusion and an unbounded piezoelectric matrix. The original boundary value problem is reduced to a standard Riemann–Hilbert problem of vector form by means of analytical continuation. Explicit solutions for the stress singularities δ=−(1/2)±i ε are obtained, closed form solutions for the field potentials are then derived through adopting a decoupling procedure. In addition, explicit expressions for the field component distributions in the whole field and along the circular interface are also obtained. Different from the interface insulating crack, stresses, strains, electric displacements and electric fields at the crack tips all exhibit oscillatory singularities. We also define a complex electro-elastic field concentration vector to characterize the singular fields near the crack tips and derive a simple expression for the energy release rate, which is always positive, in terms of the field concentration vector. The condition for the disappearance of the index ε is also discussed. When the index ε is zero, we obtain conventionally defined electro-elastic intensity factors. The examples demonstrate the physical behavior and the correctness of the obtained solution.

Closed-form solution for a piezoelectric strip with two collinear cracks normal to the strip boundaries

The problem of two collinear cracks of equal length and normal to the strip boundaries in an infinitely long piezoelectric strip of finite width is analyzed. By using the Fourier series method, the mixed boundary value problem is reduced to triple series equations, which are then transformed to a singular integral equation. For four combined cases of uniform antiplane shear and uniform inplane electric loading at infinity, the solution is obtained in closed-form, and explicit expressions for the electroelastic field are determined. The formulae for calculating the intensity factors of the electroelastic field and the energy release rate at the inner and outer crack tips are given, respectively. Some special cases for the electroelastic field intensity factors and the energy release rate of the present results are discussed.

A micromechanics method to study the effect of domain switching on fracture behavior of polycrystalline ferroelectric ceramics

The effect of domain switching on anisotropic fracture behavior of polycrystalline ferroelectric ceramics was revealed on the basis of the micromechanics method. Firstly, the electroelastic field inside and outside an inclusion in an infinite ferroelectric ceramics is carried out by the way of Eshelby-Mori-Tanaka's theory and a statistical model, which accounts for the influence of domain switching. Further, the crack extension force (energyrelease rate) Gext for a penny-shape crack inside an effective polycrystalline ferroelectric ceramics is derived to estimate the averaged effect of domain switching on the fracture behavior of polycrystalline ferroelectric ceramics. The simulations of the crack extension force for a crack in a BaTiO3 ceramics are shown that the effect of domain switching must be taken into consideration while analyzing the fracture behavior of polycrystalline ferroelectric ceramics. These results also demonstrate that the influence of the applied electric field on the crack propagation is more profound at smaller mechanical loading and the applied electric field may enhance the crack extension in a sense, which are consistent with the experimental results.

Dynamic potentials and Green's functions of a quasi–plane piezoelectric medium with inclusion

The dynamic potentials of a two–dimensional (2D) quasi–plane piezoelectric infinite medium of transversely isotropic symmetry containing an inclusion of arbitrary shape is derived in terms of scalar solutions of the 2D Laplace and Helmholtz equations. Closed–form expressions for the space–frequency representation of this dynamic potential are obtained for the case when the spatial source distribution is characterized by a region occupied by a circular inclusion embedded in a quasi–plane transversely isotropic matrix. The results are used to solve the dynamic Eshelby problem of a circular inclusion (plane region with the same material characteristics as the matrix) undergoing uniform eigenstrain and eigenelectric field. In contrast to the static case, the dynamic electroelastic fields inside the circular inclusion are non–uniform in the space–frequency representation. The derived dynamic piezoelectric potentials are basic quantities for the description of the dynamic properties of micro–inhomogeneous quasi–plane piezoelectric material systems (e.g. fibre–reinforced piezocomposites).

On a moving Griffith crack in anisotropic piezoelectric solids

The generalized plane problem of a finite Griffith crack moving with constant velocity in an anisotropic piezoelectric material is investigated. The combined mechanical and electrical loads are applied at infinity. Based on the extended Stroh formalism, the closed-form expressions for the electroelastic fields are obtained in a concise way. Numerical results for PZT-4 piezoelectric ceramic are given graphically. The effects on the hoop stress of the velocity of the crack and the electrical to mechanical load ratios are analyzed. The propagation orientation of a moving crack is also predicted in terms of the criterion of the maximum tensile stress. When the crack speed vanishes, the results of the present paper are in good agreement with those given previously in the literature.

Propagation of electroacoustic waves in the transversely isotropic piezoelectric medium reinforced by randomly distributed cylindrical inhomogeneities

The propagation of electroacoustic waves in a piezoelectric medium containing a statistical ensemble of cylindrical fibers is considered. Both the matrix and the fibers consist of piezoelectric transversely isotropic material with symmetry axis parallel to the fiber axes. Special emphasis is given on the propagation of an electroacoustic axial shear wave polarized parallel to the axis of symmetry propagating in the direction normal to the fiber axis.The scattering problem of one isolated continuous fiber (“one-particle scattering problem”) is considered. By means of a Green’s function approach a system of coupled integral equations for the electroelastic field in the medium containing a single inhomogeneity (fiber) is solved in closed form in the long-wave approximation. The total scattering cross-section of this problem is obtained in closed form and is in accordance with the electroacoustic analogue of the optical theorem.The solution of the one-particle scattering problem is used to solve the homogenization problem for a random set of fibers by means of the self-consistent scheme of effective field method. Closed form expressions for the dynamic characteristics such as total cross-section, effective wave velocity and attenuation factor are obtained in the long-wave approximation.

The electro-elastic field of the infinite piezoelectric medium with two piezoelectric circular cylindrical inclusions

The electro-elastic field of the infinite piezoelectric medium with two piezoelectric circular cylindrical inclusions is derived under the antiplane shear stresses and inplane electric fields. The analytical solution is obtained. The proposed method is based upon the use of conformal mapping and the theorem of analytic continuation. From the results obtained, it can be found that the electro-elastic field depends on the material constants of individual phases, the geometric parameters of the system and the applied antiplane shear stresses and electric fields at infinity. In addition, the specific cases when two circular cylindrical inclusions are tangent to each other and they are holes and/or rigid ones, are also studied in this paper.

A Screw Dislocation Interacting with an Interfacial Crack in Two Dissimilar Piezoelectric Media

In this paper the interaction of a screw dislocation with a semi-infinite interfacial crack of two dissimilar piezoelectric materials is considered. The expressions for the electroelastic fields, intensity factors, and image force acting on a screw dislocation are given in explicit form in terms of complex functions. When one of two materials is purely elastic, the influence of electromechanical coupling on the stress intensity factor and the image force is analyzed in detail. In addition, the results obtained in the present paper can be reduced to several special cases that have been reported previously in the literature.

Fracture analysis of piezoelectric materials with flat ellipsoidal cracks

The objective of this study is to investigate a three-dimensional, infinitely extended, anisotropic piezoelectric solid containing a flat ellipsoidal crack with emphasis laced on when one of the principal axes of the crack becomes zero. Utilizing the equivalent inclusion method, the electroelastic fields around the crack are obtained explicitly by treating the elastic moduli and piezoelectric coefficients of the flat ellipsoidal inclusion as zero, and neglecting the permittivity. With resulting strain, stress, electric field, and electric displacement, the interaction energy between the crack and electromechanical loads is calculated. Based on the Griffith fracture criterion, the fracture stresses and critical electric displacement of the crack respectively subjected to a simple tension, an in-plane shear, an out-of-plane shear, and an electric displacement are obtained in closed forms by utilizing the obtained interaction energy. It is shown that the resulting fracture stresses can be reduced to those for uncoupled linear elastic fracture mechanics when piezoelectric coupling is absent and the material is isotropic. Numerical results for a three-dimensional PIC-151 piezoelectric solid with a flat ellipsoidal crack are given to demonstrate the application of the proposed formulation. Analysis results also indicate that the electroelastic coupling nature reduces the fracture loads and consequently impedes crack growth.

2D electro-elastic fields in a piezoelectric layer–substrate structure

The coupled electro-elastic fields are found in the piezoelectric layer–substrate structure of unrestricted anisotropy containing straight-line sources in the interior or at the surface of the structure. The line sources in the interior are parallel to the surface, which is supposed to be free of traction or clamped and, electrically, metallized, free of charge or adjoined to an isotropic dielectric medium. The boundary problems of prescribed distributions of loads, displacements, electric potential or charge density at the surface are also considered. All obtained solutions are presented in a form of convergent Fourier integrals. The integrands are implicitly expressed in terms of the eigenvalues and the eigenvectors of the generalized Stroh matrix.

Decoupled formulation of piezoelectric elasticity under generalized plane deformation and its application to wedge problems

Abstract This paper presents the formulation of piezoelectric elasticity under generalized plane deformation derived from the three-dimensional theory. There are four decoupled classes in the generalized plane deformation formulation, i.e. when l 3 ( μ )= l 2 * ( μ )=0, l 3 ( μ )= l 3 * ( μ )=0, l 3 * ( μ )= l 2 * ( μ )=0 or l 3 ( μ )= l 3 * ( μ )= l 2 * ( μ )=0. Only the inplane fields of the first class and the antiplane field of the second class include the piezoelectric effect. Several examples of wedge problem often encountered in smart structures, such as sensors or actuators are studied to examine the stress singularity near the apex of the structure. The bonded materials to the PZT-4 wedge are PZT-5, graphite/epoxy or aluminum (conductor). The influencing factors on the singular behavior of the electro-elastic fields include the wedge angle, material type, poling direction, and the boundary and interface conditions. The numerical results of the first case are compared with Xu's graphs and some comments are made in detail. In addition, some new results regarding the antiplane stress singularity of the second class are obtained via the case study. The coupled singularity solutions under generalized plane deformation are also investigated to seek the conditions of the weakest or vanishing singular stress fields.

Electroelastic field for an impermeable anti-plane shear crack in a piezoelectric ceramics plate

Electroelastic behavior of a cracked piezoelectric ceramics plate subjected to four cases of combined mechanical-electrical loads is analyzed. The integral transform method is applied to convert the problem involving an impermeable anti-plane crack to dual integral equations. Solving the resulting equations, the explicit analytic expressions for electroelastic field along the crack line and the intensity factors of relevant quantities near the crack tip and the mechanical strain energy release rate are obtained. The known results for an infinite piezoelectric ceramics plane containing an impermeable anti-plane crack are recovered from the present results only if the thickness of the plate h → ∞.