Infinite Piezoelectric Medium Research Articles

Simplified numerical algorithms for generalized strip saturated two equal collinear cracks in piezoelectric media using distributed dislocation technique

AbstractIn this paper, we present the distributed dislocation technique (DDT) based numerical algorithms to study the generalized strip saturated (GSS) models for two equal collinear cracks in 2‐D finite and infinite piezoelectric media. Numerical studies for particular cases such as linear, quadratic and cubic strip saturated models are simulated by considering their equivalent forms based on the principle of superposition. Two equal collinear cracks problem and its particular case of coalesced zones are considered in 2‐D semipermeable piezoelectric media under arbitrary poling direction and in‐plane electromechanical loadings. The results of saturated zone lengths (inner and outer) and local stress intensity factors (at inner and outer tips) are evaluated numerically for infinite and finite domain problems. The results of infinite domain obtained using DDT are compared with the reference analytical solutions. A good agreement of the results shows the efficacy of the proposed algorithms based on DDT for solving GSS two equal collinear cracks problems in 2‐D finite/infinite piezoelectric media.

Complex variable solution for mode-III quadratically varying PS model in piezoelectric media

Complex variable technique and extended Stroh formalism approach are applied to develop the closed-form solutions for mode-III quadratically varying polarization saturation (PS) model in an infinite piezoelectric media. A semipermeable centre crack problem is considered for study under the influence of out of plane far-field mechanical loading and in-plane electrical displacement loading. Similar to PS model, the electrical yielding zones or saturated zones are considered just infront of the crack-tips in the form of a strip. But in place of constant PS condition imposed on yielding zones, the quadratically varying PS condition is considered for study. Mathematically, this problem reduces into simultaneous non-homogeneous Riemann-Hilbert problems which are solved by finding the solutions of involved singular integrals. Hence, some standard fracture parameters are derived in explicit forms such as saturated zone lengths, crack sliding displacement, crack potential drop and local stress intensity factor. Also, the comparative studies for quadratically varying PS model and PS model are presented graphically.

Wave Propagation in Piezoelectric Medium with the Flexoelectric Effect Considered

AbstractWith the consideration of the flexoelectric effect, the dispersion feature of plane waves propagating in infinite piezoelectric medium and the reflection amplitude ratio at a mechanically traction-free and dielectrically charge-free surface of semi-infinite piezoelectric half-space are studied in this paper. The flexoelectric effect is essentially the microstructure effect of heterogeneous piezoelectric medium. Besides, the strain gradient effect and the micro-inertial effect are also introduced to reflect the influences of the microstructure. Different from the classical piezoelectric medium, there are five kinds of coupled elastic waves in the microstructured piezoelectric medium. The influences of the flexoelectric effect, the strain gradient effect and the micro-inertial effect upon the dispersion feature of the coupled elastic waves and the reflection amplitude ratio are studied numerically. The numerical results are shown graphically and are validated by the energy conservation law. It is found that these microstructure effects have obvious influences upon the dispersion feature of the coupled elastic waves and therefore also affect the reflection energy partition at the mechanically free surface. Especially, the micro-inertial effect has fundamental influences on the dispersion properties of the coupled elastic waves.

Stress State of an Orthotropic Piezoelectric Material with an Elliptic Crack

The electric and stress state of an orthotropic electroelastic medium with an elliptic crack acted upon by mechanical and electric loads is analyzed. To solve the problem, the triple Fourier transform and Fourier-image of the Green function for an infinite orthotropic piezoelectric medium are used. The approach is tested by solving the problem for a crack located in the isotropy plane of a transversely isotropic piezoelectric material, for which the exact solution exists. The comparison of the results of calculations confirms the high efficiency of the approach. The distribution of stress intensity factors along the front of the crack in an orthotropic electroelastic material under uniform loading is studied numerically

HOLE PROBLEMS IN A CIRCULAR PIEZOELECTRIC PLATE

Due to the unique electric-mechanical coupling effects, piezoelectric materials are widely used in sensors and actuators. Not only in smart structures but also in structural health monitoring, the developments of piezoelectric materials combined with the micro electro mechanical system (MEMS) techniques make our lives totally different. This paper provides a series of numerical simulations about the circular piezoelectric plate with a hole in the plate by using the special boundary element method (BEM). The Green’s functions of BEM are obtained from the derivation of extended Stroh formalism. By this formalism, the analytical closedform solutions of hole in an infinite piezoelectric medium under various loading conditions are obtained. The results demonstrate the coupling effects with different boundary conditions in the circular plate, which will satisfy the requirements of the designers who need to use the products with the hole in the piezoelectric circular plate.

Microscale size effects on the electromechanical coupling in piezoelectric material for anti-plane problem

A continuum model of piezoelectric material with strain and electric gradient effects is proposed in this work. An energy variational principle with strain, strain gradient, electric field and electric field gradient as independent variables is presented to develop the equilibrium equations and all boundary conditions. Moreover, two internal length characteristics of the underlying microstructure are introduced in the constitutive equations. This model is employed to an anti-plane problem of the piezoelectric material, and a general solution is obtained in polar coordinates. Special attention is paid to an infinite piezoelectric medium with a microvoid. It is found that electromechanical fields predicted by the gradient model not only depend on the material parameters, but also on the relative size of the microvoid with respect to the scale of microstructure. More importantly, some new interesting phenomena are observed due to the gradient effects of strain and electric fields.

Anti-plane problems of piezoelectric material with a micro-void or micro-inclusion based on micromorphic electroelastic theory

The linear theory of micromorphic electroelasticity, which incorporate the coupled electromechanical behavior into the framework of micromorphic continuum theory, is used to solve the anti-plane problems of piezoelectric media with a micro-void or micro-inclusion in this paper. The electromechanical field solutions for a transversely isotropic piezoelectric medium are derived in the context of micromorphic electroelasticity and a generalized characteristic length is introduced to describe the size effect. Anti-plane problems of an infinite piezoelectric medium containing a micro-void or micro-inclusion are analyzed. Numerical results reveal that the mechanical and electric fields predicted by the present model highly depend on the relative size of the micro-void or micro-inclusion with respect to the generalized characteristic length, which is obviously different from the classical prediction.

Hybrid fundamental-solution-based FEM for piezoelectric materials

In this paper, a new type of hybrid finite element method (FEM), hybrid fundamental-solution-based FEM (HFS-FEM), is developed for analyzing plane piezoelectric problems by employing fundamental solutions (Green's functions) as internal interpolation functions. A modified variational functional used in the proposed model is first constructed, and then the assumed intra-element displacement fields satisfying a priori the governing equations of the problem are constructed by using a linear combination of fundamental solutions at a number of source points located outside the element domain. To ensure continuity of fields over inter-element boundaries, conventional shape functions are employed to construct the independent element frame displacement fields defined over the element boundary. The proposed methodology is assessed by several examples with different boundary conditions and is also used to investigate the phenomenon of stress concentration in infinite piezoelectric medium containing a hole under remote loading. The numerical results show that the proposed algorithm has good performance in numerical accuracy and mesh distortion insensitivity compared with analytical solutions and those from ABAQUS. In addition, some new insights on the stress concentration have been clarified and presented in the paper.

Dynamic Antiplane Behavior for a Quarter Infinite Piezoelectric Medium with a Subsurface Circular Inclusion

Dynamic antiplane behaviors are investigated theoretically in this paper for a quarter infinite piezoelectric medium with a subsurface circular inclusion. Based on complex variable and mirror image method, the expressions are obtained on dynamic stress concentration factor (DSCF) and electric field intensity concentration factor (EFICF) at the inclusion’s edge caused by the interaction between the inclusion and the right angle edge under time-harmonic anti-plane shearing. While some calculating cases are plotted, so as to show how the frequencies of incident wave, the piezoelectric material’s parameters and the structure’s geometry influence on DSCF and EFICF. The calculating results indicate that dynamic analyses are important to a quarter-infinite piezoelectric medium with defects at the surface vicinity.

Dynamic Stress Intensity Factor for a Radial Crack on a Circular Cavity in a Piezoelectric Medium

The problem of dynamic stress intensity factor is investigated theoretically in present paper for a radial crack on a circular cavity in an infinite piezoelectric medium, which is subjected to time-harmonic incident anti-plane shearing. First, a pair of electromechanically coupled Green’s functions are constructed which indicate the basic solutions for a semi-infinite piezoelectric medium with a semi-circular cavity. Second, based on the crack-division technique and conjunction technique, integral equations for the unknown stresses’ solution on the conjunction surface is established, which are related to the dynamic stress intensity factor at the crack tip. Third, the analytical expression on dynamic stress intensity factor at the crack tip is obtained. At last, some calculating cases are plotted to show how the frequencies of incident wave, the piezoelectric characteristic parameters of the material and the geometry of the crack and the circular cavity influence upon the dynamic stress intensity factors. While some of the calculating results are compared with the same situation about a straight crack and with static solutions.

Dynamic anti-plane interaction between an impermeable crack and a circular cavity in an infinite piezoelectric medium

Wave propagation in an infinite elastic piezoelectric medium with a circular cavity and an impermeable crack subjected to steady-state anti-plane shearing was studied based on Green’s function and the crack-division technique. Theoretical solutions were derived for the whole elastic displacement and electric potential field in the interaction between the circular cavity and the impermeable crack. Expressions were obtained on the dynamic stress concentration factor (DSCF) at the cavity’s edge, the dynamic stress intensity factor (DSIF) and the dynamic electric displacement intensity factor (DEDIF) at the crack tip. Numerical solutions were performed and plotted with different incident wave numbers, parameters of piezoelectric materials and geometries of the structure. Finally, some of the calculation results were compared with the case of dynamic anti-plane interaction of a permeable crack and a circular cavity in an infinite piezoelectric medium. This paper can provide a valuable reference for the design of piezoelectric actuators and sensors widely used in marine structures.

Dynamic Anti-Plane Behavior of the Interaction Between a Crack and a Circular Cavity in a Piezoelectric Medium

In this paper, the dynamic interaction is investigated theoretically between a crack and a circular cavity in an infinite piezoelectric medium under time-harmonic incident anti-plane shearing. The formulations are based on the method of complex variable and Green’s function. The resulting dynamic stress intensity factors at the crack’s tip and dynamic stress concentration factors at the cavity’s edge are obtained with crack-division technique. Numerical results are plotted to show how the frequencies of incident wave, the piezoelectric characteristic parameters of the material and the geometry of the crack and the circular cavity influence upon the dynamic stress intensity factors and dynamic stress concentration factors.

Steady-state vibrations of an unbounded linear piezoelectric medium

We consider fundamental (Dirichlet and Neumann-type) boundary value problems in a theory of generalized plane strain for the steady-state vibrations of an infinite piezoelectric medium with transversely isotropic symmetry (6 mm). Using integral equation methods with the appropriate Sommerfeld-type radiation conditions, we prove existence and uniqueness results for the corresponding exterior boundary value problems. Exact solutions are obtained in the form of integral potentials.

Dynamic anti-plane characteristic on an infinite piezoelectric medium with a movable rigid cylindrical inclusion

Scattering and dynamic stress concentrations of time harmonic SH-wave in an infinite elastic piezoelectric medium with a movable rigid cylindrical inclusion are studied in this paper with the help of complex variable and wave function expansion method. The relations that a movable rigid cylindrical inclusion depends on intensity of incident wave and electric field are revealed. The expressions of dynamic stress at the edge of the inclusion are obtained. Numerical calculations are made with different wave numbers and different piezoelectric characteristic parameters. The calculating results show that dynamic stress concentrations at the edge of the inclusion have linear dependence on the incident electric field. And dynamic analyses are very important for an infinite piezoelectric medium with a movable rigid cylindrical inclusion at larger piezoelectric characteristic parameters.

Coupled behaviour of interacting dielectric cracks in piezoelectric materials

Existing studies indicate that the commonly used electrically impermeable and permeable crack models may be inadequate in evaluating the fracture behaviour of piezoelectric materials in some cases. In this paper, a dielectric crack model based on the ‘real’ electric boundary condition is used to study the electromechanical behaviour of interacting cracks arbitrarily oriented in an infinite piezoelectric medium. The electric boundary condition along the crack surfaces is governed by the opening displacement of the cracks. The formulation of this nonlinear problem is based on modelling the cracks using distributed dislocations and solving the resulting nonlinear singular integral equations using Chebyshev polynomials. Numerical simulation is conducted to show the effect of crack orientation, crack interaction and electric boundary condition upon the fracture behaviour of cracked piezoelectric media.

Green's functions for anti-plane problems in piezoelectric media with a finite crack

Based on the complex variable method and the perturbation technique, Green's functions for the anti-plane problems in an infinite piezoelectric medium with a crack under remote uniform loadings are derived. The analysis is conducted on the electrically unified crack boundary condition with the introduction of the electric crack condition parameter that can describe all the electric crack boundary conditions. The two ideal crack boundary conditions, namely, the electrically impermeable and permeable crack assumptions are obtained as two special cases for the current solution. The explicit expressions of the mechanical and electrical fields produced by a line-force, a line-charge and a screw dislocation are derived and the field intensity factors are calculated. The image forces on the dislocation due to the crack and the remote uniform loadings are computed as functions of dislocation position and material constant combinations. Numerical examples are performed to show how the electric crack condition parameter affects the field intensity factors and the force on the dislocation.

A line dislocation interacting with a semi-infinite crack in piezoelectric solid

Closed-form analytical solutions are presented for the physical problem of a semi-infinite crack interacting with a line dislocation under the loading of a line force and a line charge in two-dimensional infinite anisotropic piezoelectric medium. The crack can be a conventional Griffith crack or an anti-crack (a rigid line inhomogeneity). Using the extended Stroh formalism and perturbation technique, the explicit expressions of the field intensity factors and the image force on the dislocation are computed as functions of dislocation location and material constants. The results are discussed and compared with those from special cases existed in the literature. The analytical solutions obtained can be applied to studying interacting cracks and crack branching problems in piezoelectric solids.

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.

Measuring fracture toughness of piezoceramics by vickers indentation under the influence of electric fields

The Vickers indentation technique was used to determine the apparent fracture toughness of PZT-4 piezoceramic under the influence of electric fields. The toughness was measured in terms of the mechanical strain energy release rate. Using the analytical mechanical strain energy release rate for an infinite piezoelectric medium with a center crack and far field in-plane mechanical and electrical loadings, the mechanical strain energy release rate was obtained for Vickers indentation. The effect of the electric field was included. The proposed formula was evaluated with the experimental results. It was found that the fracture toughness depended on the indentation force. Vickers indentation tests were also performed on depoled PZT-4 specimens; these test results indicated that the fracture toughness of depoled PZT-4 was also dependent on the indentation force. To eliminate the indentation force dependence, a modified stress intensity formula based on Vickers indentation was proposed.

Two piezoelectric circular cylindrical inclusions in the infinite piezoelectric medium

The electro–elastic field of the infinite piezoelectric medium with two piezoelectric circular cylindrical inclusions is derived under the antiplane line force and inplane line charge. 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 antiplane line force and inplane line charge. 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 considered in this paper.