Infinite Piezoelectric Matrix Research Articles

Antiplane analysis for an elliptical inclusion in 1D hexagonal piezoelectric quasicrystal composites

An elliptical inclusion embedded in an infinite 1D hexagonal piezoelectric quasicrystal matrix is analysed in the framework of linear piezoelasticity of quasicrystals. Using the technique of conformal mapping, the closed-form solutions of the complex potentials, all the field quantities in the matrix and the inclusion are obtained under far-field antiplane mechanical loads of the phonon and phason fields and an inplane electrical load. Several special cases, such as a homogeneous material, a soft and permeable inclusion, an impermeable inclusion, a line inclusion, a rigid inclusion and a crack are reduced by the present results. Some numerical examples are provided to shows the variations of the stresses of the phonon and phason fields and the electric field with the shape of hole/inclusion, the dielectric permittivity and the distance from the hole/inclusion.

Closed-form solution for elliptical inclusion problem in antiplane piezoelectricity with far-field loading at an arbitrary angle

Abstract An elliptical piezoelectric inclusion embedded in an infinite piezoelectric matrix is analyzed in the framework of linear piezoelectricity. Using the conformal mapping technique, a closed-form solution is obtained for the case of a far-field antiplane mechanical load, τ 0 , and an inplane electrical load, E 0 , at an arbitrary angle β . The stress and electric field distribution patterns for different defect shapes, loading angles, and material constants are studied. The energy release rates of self-similarly expanding and rotating defects in the presence of an electric field are obtained using the generalized M - and L -integrals as a function of the loading angle. The physical significance of these results is discussed in terms of the stress and electric field distributions as well as the energy release rates.

Elliptical Inclusion Problem in Antiplane Piezoelectricity: Stress Concentrations and Energy Release Rates

An elliptical piezoelectric inclusion embedded in an infinite piezoelectric matrix is analyzed in the framework of antiplane piezoelectricity. The stress distribution patterns for different elliptical inclusion and cavity shapes with different material constants are discussed based on the closed-form solution obtained by the authors. It is found that the stress concentration at the interface of the inclusion and the matrix becomes significantly high when piezoelectric constants of the inclusion and matrix have opposite sign. The variation in the energy release rates of selfsimilarly expanding and rotating defects, expressed by the M-and L-integrals, respectively, are discussed as a function of the applied electric field. Information on the stress concentration and the energetics of such a system can be quite useful in predicting failures, hence, aid in properly designing piezoelectric electromechanical systems.

Piezoelectric screw dislocation in an arbitrarily shaped three-phase composite

Abstract Based on the methods of complex variable, conformal mapping, Faber series and Laurent series, the Green’s function for a coated inclusion of arbitrary shape embedded in an infinite piezoelectric matrix is obtained in this paper. The analytical complex potentials for all three regions can be expressed in series form with unknown coefficients. The continuity conditions of the interfaces are used to build up a set of linear equations to determine the unknown coefficients. After the unknown coefficients are solved, the stress, electric field and image force can be expressed explicitly. Numerical results are provided to show the effect of the inclusion shape, the material combinations on the electroelastic fields and image force calculated through the generalized Peach–Koehler formula. The solutions proposed here can be served as kernel functions to analyze the corresponding piezoelectric cracking problems.

Elliptical inclusion problem in antiplane piezoelectricity: Implications for fracture mechanics

An elliptical piezoelectric inclusion embedded in an infinite piezoelectric matrix is analyzed in the framework of linear piezoelectricity. Using the conformal mapping technique, a closed-form solution is obtained for the case of a far-field antiplane mechanical load and an inplane electrical load. The solution to a permeable elliptical hole problem is obtained as a limiting case of vanishing elastic modulus of the inclusion. This enables the study of the nature of crack tip electric field singularity which is shown to depend on the electrical boundary condition imposed on the crack faces. The energy release rate of a self-similarly expanding slender crack in the presence of electric fields is obtained by using the generalized M-integral. The energy release rate expression indicates that the electric field has a crack-arresting influence. This effect is inferred to have a more fundamental physical origin in the interaction between the applied electric field and the induced surface charges on the crack faces. An experimental result contradicting the theoretical prediction on the crack-arresting effect is also discussed.

Stress field near circular-arc interface crack tip based on electric saturation concept

Within the framework of nonlinear electroelasticity, the anti-plane problem of a circular-arc interfacial crack between a circular piezoelectric inhomogeneity and an infinite piezoelectric matrix subjected to a far-field uniform loading is investigated by an electrical strip saturation model, the complex variable method, and the method of analytical continuation. Explicit closed form expressions for the complex potentials in both the matrix and the inclusion, and the stress intensity factor at the crack tip are presented. Comparison with some related solutions based on the linear electroelastic theory shows the validity of the present solutions

On a semi-infinite crack penetrating a piezoelectric circular inhomogeneity with a viscous interface

We investigate a semi-infinite crack penetrating a piezoelectric circular inhomogeneity bonded to an infinite piezoelectric matrix through a linear viscous interface. The tip of the crack is at the center of the circular inhomogeneity. By means of the complex variable and conformal mapping methods, exact closed-form solutions in terms of elementary functions are derived for the following three loading cases: (i) nominal Mode-III stress and electric displacement intensity factors at infinity; (ii) a piezoelectric screw dislocation located in the unbounded matrix; and (iii) a piezoelectric screw dislocation located in the inhomogeneity. The time-dependent electroelastic field in the cracked composite system is obtained. Particularly the time-dependent stress and electric displacement intensity factors at the crack tip, jumps in the displacement and electric potential across the crack surfaces, displacement jump across the viscous interface, and image force acting on the piezoelectric screw dislocation are all derived. It is found that the value of the relaxation (or characteristic) time for this cracked composite system is just twice as that for the same fibrous composite system without crack. Finally, we extend the methods to the more general scenario where a semi-infinite wedge crack is within the inhomogeneity/matrix composite system with a viscous interface.

Interactions between N circular cylindrical inclusions in a piezoelectric matrix

This paper studies the interactions between N randomly-distributed cylindrical inclusions in a piezoelectric matrix. The inclusions are assumed to be perfectly bounded to the matrix, which is subjected to an anti-plane shear stress and an in-plane electric field at infinity. Based on the complex variable method, the complex potentials in the matrix and inside the inclusions are first obtained in form of power series, and then approximate solutions for electroelastic fields are derived. Numerical examples are presented to discuss the influences of the inclusion array, inclusion size and inclusion properties on couple fields in the matrix and inclusions. Solutions for the case of an infinite piezoelectric matrix with N circular holes or an infinite elastic matrix containing N circular piezoelectric fibers can also be obtained as special cases of the present work. It is shown that the electroelastic field distribution in a piezoelectric material with multiple inclusions is significantly different from that in the case of a single inclusion.

An Elliptic Thermopiezoelectric Inclusion Problem in An Anisotropic Thermopiezoelectric Medium

This study presents an analytical solution for an elliptical anisotropic piezoelectric inclusion embedded in an infinite anisotropic piezoelectric matrix subjected to arbitrary far-field uniform loadings by employing the Stroh formalism, the method of analytical continuation, the technique of conformal mapping, and the concept of superposition. Solutions of the temperature and stress functions either in the matrix or in the inclusion are expressed in complex matrix notation. It shows that the mechanic and electric loading leads to the appearance of the constant stress fields in the inclusion and the heat flux only leads to that of the linear stress fields. Comparison with some related works shows that the present solutions are valid and general.

Interfacial debonding of a circular inhomogeneity in piezoelectric–piezomagnetic composites under anti-plane mechanical and in-plane electromagnetic loading

This paper studies the problem of a partially debonded piezomagnetic circular inhomogeneity in an infinite piezoelectric matrix, or vice versa. The debonded parts are modeled by a set of circular-arc interfacial cracks. Crack surfaces are assumed to be electrically open and magnetically closed. The matrix is subjected to remote anti-plane mechanical and in-plane electromagnetic loadings. A general solution to the problem studied is obtained by employing the analytical continuation theory. Detailed analysis is made for a single circular-arc interfacial crack. The closed-form expressions of the complex potentials are obtained in both the inhomogeneity and the matrix. The intensity factors of stress, electric displacement and magnetic induction are also provided explicitly. For the case of piezoelectric matrix and piezomagnetic inclusion, the influence of crack geometry and the magnitudes of mechanical and electromagnetic loadings on the intensity factors are discussed in detail. In the case where anti-plane strain, in-plane electric and magnetic fields are applied simultaneously, the intensity factors only depend on the effective magneto-electro-elastic material constants, and whether the matrix is piezoelectric or piezomagnetic does not affect these intensity factors.

Fracture analysis of circular-arc interface cracks in piezoelectric materials

The anti-plane problem of N arc-shaped interfacial cracks between a circular piezoelectric inhomogeneity and an infinite piezoelectric matrix is investigated by means of the complex variable method. Cracks are assumed to be permeable and then explicit expressions are presented, respectively, for the electric field on the crack faces, the complex potentials in media and the intensity factors near the crack-tips. As examples, the corresponding solutions are obtained for a piezoelectric bimaterial system with one or two permeable arc-shaped interfacial cracks, respectively. Additionally, the solutions for the cases of impermeable cracks also are given by treating an impermeable crack as a particular case of a permeable crack. It is shown that for the case of permeable interfacial cracks, the electric field is jumpy ahead of the crack tips, and its intensity factor is always dependent on that of stress. Moreover all the field singularities are dependent not only on the applied mechanical load, but also on the applied electric load. However, for the case of a homogeneous material with permeable cracks, all the singular factors are related only to the applied stresses and material constants.

Scattering of Antiplane Shear Waves by a Circular Piezoelectric Inclusion Embedded in a Piezoelectric Medium Subjected to a Steady-State Electrical Load

The dynamic theory of antiplane piezoelectricity is applied to solve the problem of a circular piezoelectric inclusion embedded in an infinite piezoelectric matrix subjected to horizontally polarized shear waves and a steady-state inplane electrical load. The problem is formulated by means of the wave function expansion method. Numerical calculations are carried out for the dynamic stress and electric field concentrations, and the results are presented in graphical form.

On double circular inclusion problem in antiplane piezoelectricity

In this paper, an analytical solution in series form for the problem of double circular piezoelectric inclusions embedded in an infinite piezoelectric matrix is presented within the framework of linear theory of piezoelectricity. The matrix is subjected to remote electro-mechanical loading, and the three phase system is also subjected to the action of arbitrary singularities. The solution is obtained by applying complex potential approach in conjunction with the techniques of conformal mapping, analytical continuation, singularity analysis, Laurent's series expansion in an annular ring region and Cauchy integral formulae, etc. Based on the obtained complex potentials, explicit expressions for the stress and electric displacement in the matrix and the two circular inclusions are also derived. A numerical investigation for the case of remote loading is performed to illustrate the influence of a third phase on the system's electroelastic coupling behavior and also to verify the correctness and usefulness of the solution.

Thermoelectroelastic solution for elliptic inclusions and application to crack–inclusion problems

The thermoelectroelastic solution for an elliptic piezoelectric inclusion embedded in an infinite matrix and its application to crack–inclusion problems are investigated in this paper. By combining the method of Stroh's formalism, the technique of conformal mapping, the concept of perturbation and the method of analytical continuation, a general analytical thermoelectroelastic solution for an elliptic piezoelectric inclusion embedded in an infinite piezoelectric matrix subjected to thermal loading is obtained. The loading may be point heat source, temperature discontinuity, or uniform remote heat flow. Special cases when the inclusion becomes rigid or a hole are also investigated. As an application of the proposed solution, a system of singular integral equations is derived for analyzing crack–inclusion interactions. Numerical results for a piezoelectric plate with an elliptic inclusion and an inclined crack are given to illustrate the application of the proposed formulation.

Dynamic Response of a Piezoelectric Material with a Conducting Rigid Inclusion

The problem of a conducting rigid inclusion embedded in an infinite piezoelectric matrix is considered under the action of combined electromechanical impact loads. By using integral transform techniques, the mixed initial-boundary value problem for the case of anti-plane shear load and in-plane electric field is transformed into two systems of dual integral equations, the solutions of which give the singularity coefficients of electroelastic field near the inclusion tips in closed-form in the Laplace transform domain. Numerical results for the stress singularity coefficient in the physical space are presented graphically by numerically solving the resulting Fredholm integral equation and carrying out the numerical inversion of Laplace transform for a PZT-5H material with a conducting rigid line inclusion.

Analysis of a screw dislocation inside an elliptical inhomogeneity in piezoelectric solids

This paper formulates and examines the electro-elastic coupling effects resulting from the presence of a screw dislocation inside an elliptical piezoelectric inhomogeneity embedded in an infinite piezoelectric matrix. The general solution to this problem is obtained by conformal mapping and Laurent series expansion of the corresponding complex potentials. The appropriate expressions of the field potentials and the field components are given explicitly in both the inhomogeneity and the surrounding matrix using a perturbation technique. The internal energy and the force on the dislocation are computed and several specific examples are provided to illustrate the validity and versatility of the developed formulations.

Electro-elastic interaction between a screw dislocation and an elliptical inhomogeneity in piezoelectric materials

This article is concerned with the electro-elastic interaction between a dislocation and an elliptical piezoelectric inhomogeneity in an infinite piezoelectric matrix. The matrix is subjected to remote antiplane shear and inplane electric fields. The explicit expressions of the complex potentials are derived in both the inhomogeneity and the surrounding matrix using conformal mapping and the perturbation techniques. The results reveal that when the inhomogeneity reduces to a cavity, the electric field strength in both the cavity and the matrix is not affected by the dislocation. In addition, the results also show that the electric field strength is uniform in the cavity. In the case of a slit crack, the electric field strength in the matrix becomes uniform along the slit and in the matrix, while the stress and the electric displacements show the traditional square root singularity at the crack tip.

Stress and induction field of a spheroidal inclusion or a penny-shaped crack in a transversely isotropic piezo-electric material

Exact closed-form solutions are obtained for the stress and induction field of a spheroidal piezo-electric inclusion in an infinite piezo-electric matrix subjected to spatially homogeneous mechanical and electrical loadings far away from the inclusion. Three types of loading are considered: axisymmetric, in-plane and out-of-plane shear. A limiting case of this solution allows us to determine the stress and induction field of a penny-shaped crack in a piezo-electric material. Closed-form expressions for the stress and induction intensity factors of a penny-shaped crack are obtained.

Electromechanical Properties of Porous Piezoelectric Ceramics

A theoretical approach is forwarded to predict the electromechanical properties of porous piezoelectric ceramics. The analysis is able to account for the effects of porosity shape and concentration and is applicable to piezoelectric ceramics of arbitrary material symmetry. By coupling the exact solution for a single ellipsoidal pore embedded in an infinite piezoelectric matrix with an effective medium approximation, the theory considers, in an approximate manner, interaction effects at finite porosity concentrations. The theoretical estimates are developed using a matrix formulation which enables all elastic, dielectric, and piezoelectric moduli of the porous solid to be readily computed. Numerical results are presented to illustrate the effects of the shape and concentration of the porosity on the effective electroelastic moduli and transducer parameters of practical importance. Particular attention is devoted to assessing the sensitivity of the effective electromechanical properties to the accuracy of the input data. Finally, theoretical estimates are shown to be in good agreement with existing experimental results for porous piezoelectric ceramics with various microstructural geometries.

Circular inclusion problem in antiplane piezoelectricity

A circular piezoelectric inclusion embedded in an infinite piezoelectric matrix is analysed in the framework of linear piezoelectricity. A closed-form solution is obtained for the case of a far-field antiplane mechanical load and a far-field inplane electrical load. It is shown that in modeling cavities, imposing an impermeable boundary condition is a good approximation provided that the piezoelectric material has high dielectric constant and strong electro-elastic coupling. Stress and electric field concentrations are also studied. It is shown that a high electric field can be induced in the inclusion under a mechanical load when the matrix and the inclusion are poled in the opposite directions. The path-independent M-integral of elastostatics, generalized to take piezoelectric effect into account, is used to study the energetics of a self-similarly expanding piezoelectric inclusion.