Piezoelectric Plane Research Articles

Irregular Inhomogeneities in an Anisotropic Piezoelectric Plane

This paper presents an analytical solution for the Eshelby problem of polygonal inhomogeneity in an anisotropic piezoelectric plane. By virtue of the equivalent body-force concept of eigenstrain, the induced elastic and piezoelectric fields in the corresponding inclusion are first expressed in terms of the line integral along its boundary with the integrand being the Green’s functions, which is carried out analytically. The Eshelby inhomogeneity relation for the elliptical shape is then extended to the polygonal inhomogeneity, with the final induced field involving only elementary functions with small steps of iteration. Numerical solutions are compared to the results obtained from other methods, which verified the accuracy of the proposed method. Finally, the solution is applied to a triangular and a rectangular quantum wire made of InAs within the semiconductor GaAs full-plane substrate.

Analytical solution of a semi-permeable crack in a 2D piezoelectric medium based on the PS model

This paper considers a straight but nonlinear crack in a two-dimensional piezoelectric plane. Different to the existing theoretical solution of the well-known polarization saturation (PS) model, we assume the crack to be semi-permeable. By introducing the dislocation density along the crack line, we derive the analytical solution for the field quantities. Numerical results show that the effect of different boundary conditions on the electric yielding zone and the stress intensity factor is significant and should not be ignored. The influence of the saturated electric displacement on the stress intensity factor and the electric displacement in the crack cavity is also demonstrated.

The unified equations to obtain the exact solutions of piezoelectric plane beam subjected to arbitrary loads

The unified equations to obtain the exact solutions for piezoelectric plane beam subjected to arbitrary mechanical and electrical loads with various ends supported conditions is founded by solving functional equations. Comparing this general method with traditional trial-and-error method, the most advantage is it can obtain the exact solutions directly and does not need to guess and modify the form of stress function or electric displacement function repeatedly. Firstly, the governing equation for piezoelectric plane beam is derived. The general solution for the governing equation is expressed by six unknown functions. Secondly, in terms of boundary conditions of the two longitudinal sides of the beam, six functional equations are yielded. These equations are simplified to derive the unified equations to solve the boundary value problems of piezoelectric plane beam. Finally, several examples show the correctness and generalization of this method.

Anti-plane dynamic hole–crack interaction in a functionally graded piezoelectric media

The anti-plane dynamic problem of a functionally graded piezoelectric plane containing a hole–crack system is treated by a non-hypersingular traction-based boundary integral equation method. The material parameters vary exponentially in the same manner in an arbitrary direction. The system is loaded by an incident SH-type wave, and impermeable boundary conditions are assumed. Using a frequency-dependent fundamental solution of the wave equation, the boundary value problem is transformed into a system of integro-differential equations along the boundary of the hole and on the crack line. Its numerical solution yields the dynamic stress intensity factors and stress concentration factors. A parametric study reveals their dependence on the hole–crack scenario and its geometry, characteristics of the dynamic load and magnitude and direction of material inhomogeneity.

Dynamic stress and electric field concentration in a functionally graded piezoelectric solid with a circular hole

AbstractThis work addresses the evaluation of the stress and electric field concentrations around a circular hole in a functionally graded piezoelectric plane subjected to anti‐plane elastic SH‐wave and in‐plane time‐harmonic electric load. All material parameters vary exponentially along a line of arbitrary orientation in the plane of the piezoelectric material under consideration. The computational tool is a non‐hypersingular traction based boundary integral equation method (BIEM). The kernel functions used in the BIEM are exact fundamental solutions that have been derived in previous work by the authors. Numerical solutions for the stress and electric field concentration factors (SCF and EFCF, respectively) around the perimeter of the hole are obtained. The simulation demonstrates the efficiency of the computational approach and its potential to reveal in an adequate way the dynamic stress and electric field distribution around the hole. Presented are results showing their dependence on various system parameters as e.g. the electro‐mechanical coupling, the type of the dynamic load and its characteristics, the wave‐hole and wave‐material interaction and the magnitude and direction of the material inhomogeneity.

Two collinear unequal cracks in a poled piezoelectric plane: Mode I case solved by a new approach of real fundamental solutions

The purpose of the present work is to study the problem of two collinear unequal cracks in a piezoelectric plane under mode I electromechanical loadings via a new approach. For the first time, real fundamental solutions are derived for in-plane piezoelectric governing equations. The cracks are simulated by continuously distributed generalized dislocations and Cauchy singular integral equations are established from the solution of a generalized point dislocation. Both the theorectical derivation and numerical computations are validated by the exact solution in a special case. Parametric studies are conducted to reveal the effects of crack space, crack length, electric loading and remanent electric displacement on energy release rate. It is found that negative electric displacement loading can decrease both the total energy release rate (TERR) and the mechanical strain energy release rate (MSERR), implying that it has a shielding effect on cracks definitely. Positive electric displacement loading can enhance MSERR, but meanwhile it can enhance or reduce TERR depending on the magnitude of the electric loading factor. The effect of a remanent electric displacement along the poling direction is equivalent to that of a positive electric field loading and should be considered in engineering design.

On contact zone models for an electrically limited permeable interface crack in a piezoelectric bimaterial

An electrically impermeable interface crack between two semi-infinite piezoelectric planes under remote mechanical tension-shear and electrical loading is studied. Assuming the stresses, strains and displacements are independent on the coordinate x2 the expressions for the elastic displacement and potential jumps as well as for the stresses and electrical displacement along the interface via a sectionally holomorphic vector function are found. Introducing an artificial contact zone at the right crack tip and assuming the materials possess the symmetry class 6 mm the problem is reduced for a wide range of bimaterial compounds to a combination of combined Dirichlet–Riemann and Hilbert boundary value problems which are solved analytically. From these solutions clear analytical expressions for characteristic mechanical and electrical parameters are derived. As particular cases of the above mentioned solution the classical (oscillating) and contact zone solutions are obtained. Further, a comparison with an associated solution for an electrically permeable crack has been performed. The fracture mechanical parameters for all models via the remote loads are found analytically and important relationships between these parameters are obtained. Due to these relationships an important algorithm of a numerical method applicable for the investigation of an interface crack in a finite sized piezoelectric bimaterial is suggested.

Interactions of multiple parallel symmetric permeable mode-III cracks in a piezoelectric material plane

In this paper, the interactions of multiple parallel symmetric and permeable finite length cracks in a piezoelectric material plane subjected to anti-plane shear stress loading were studied by the Schmidt method. The problem was formulated through Fourier transform into dual integral equations, in which the unknown variables are the jumps of displacements across the crack surfaces. To solve the dual integral equations, the jumps of displacements across the crack surfaces were directly expanded as a series of Jacobi polynomials. Finally, the relation between the electric field and the stress field near the crack tips was obtained. The results show that the stress and the electric displacement intensity factors at the crack tips depend on the lengths and spacing of the cracks. It is also revealed that the crack shielding effect presents in piezoelectric materials.

Mode III fracture problem of an arbitrarily oriented crack in an FGPM strip bonded to a homogeneous piezoelectric half plane

This paper studies the Mode III electric-elastic field of a cracked functionally graded piezoelectric strip bonded to a homogeneous piezoelectric half plane. The crack is oriented in arbitrary direction. The material properties of the strip vary along the strip thickness in exponential forms. By using the Fourier transform, the problem can be formulated to a system of singular integral equations and solved by applying the Gauss-Chebyshev integration formula. The effects come from the edge, crack orientations and the nonhomogeneous material parameter on intensity factors are discussed graphically.

Basic solution of a mode-I permeable crack in functionally graded piezoelectric materials

t. The basic solution of a mode-I finite length crack in an infinite functionally graded piezoelectric material plane was investigated by using the generalized Almansi's theorem and the Schmidt method. The problem was formulated through Fourier transform into two pairs of dual integral equations, in which unknown variables are jumps of displacements across the crack surfaces. To solve the dual integral equations, the jumps of displacements across the crack surfaces were directly expanded as a series of Jacobi polynomials. The solution of the present paper shows that the singular stresses and the singular electric displacements at the crack tips in functionally graded piezoelectric materials carry the same forms as those in homogeneous piezoelectric materials; however, the magnitudes of intensity factors depend on the gradient of functionally graded piezoelectric material properties.

On the screw dislocation in a functionally graded piezoelectric plane and half-plane

In this paper closed-form expressions of the electroelastic field induced by a piezoelectric screw dislocation in a functionally graded piezoelectric plane and half-plane are derived. The material properties are assumed to vary exponentially along the x and y-directions. The solution for a screw dislocation in a functionally graded piezoelectric plane is obtained through introduction of two generalized stress functions. The solution for a screw dislocation in a functionally graded piezoelectric half-plane is derived by using the method of image. It is also found that the interaction between a piezoelectric screw dislocation and a circular insulating hole in the functionally graded piezoelectric material can be solved by using series expansion method.

Crack energy density in arbitrary direction for piezoelectrics and its characteristic features in electromechanical mixed mode crack

The concepts of crack energy density (CED) and its derivatives in arbitrary direction were established for piezoelectric material and, keeping their application to mixed mode fracture in mind, the characteristic features of them as fracture parameters were investigated based on the approximate equations for CED and its derivatives. That is, CED and its derivatives in arbitrary direction are defined first and separation into their each mode contribution is made. Subsequently, path independent integral expressions of them are derived, and then using them, approximate equations of each mode contribution of CED are obtained concretely for the case where linear singular solution is known. The resulting equations are then used to investigate the effects of electric field and electrical boundary condition on CED and its derivatives. An infinite piezoelectric plane with a crack inclined with respect to the poling direction is considered as a numerical example. Mode I contribution of mechanical CED is mainly employed as a possible fracture parameter for the study and it was shown that applied electric field significantly influences on fracture parameters especially for the impermeable crack perpendicular to the poling direction. The effect of electric field has the tendency to decrease as crack inclination angle increases. It was also found that, even for the impermeable crack perpendicular to the poling direction, crack propagation could be deviated from self-similar direction under a strong negative electric field, and this fact is qualitatively consistent with an existing experimental observation. For the ideally sharp crack with no width, impermeable and Hao and Shen type boundary conditions are admissible showing qualitative agreement with experimental results, but exact boundary condition is not suitable and finally consistent with permeable boundary condition.

An analytically-numerical approach for the analysis of an interface crack with a contact zone in a piezoelectric bimaterial compound

An interface crack with an artificial contact zone at the right-hand side crack tip between two dissimilar finite-sized piezoelectric materials is considered under remote mixed-mode loading. To find the singular electromechanical field at the crack tip, an asymptotic solution is derived in connection with the conventional finite element method. For mechanical loads, the stress intensity factors at the singular points are obtained. As a particular case of this solution, the contact zone model (in Comninou’s sense) is derived. A simple transcendental equation and an asymptotic formula for the determination of the real contact zone length are derived. The dependencies of the contact zone lengths on external load coefficients are illustrated in graphical form. For a particular case of a short crack with respect to the dimensions of the bimaterial compound, the numerical results are compared to the exact analytical solutions, obtained for a piezoelectric bimaterial plane with an interface crack.

Piezoelasticity solutions for functionally graded piezoelectric beams

This paper considers the plane stress problem of generally anisotropic piezoelectric beamswith the coefficients of elastic compliance, piezoelectric and dielectric impermeability beingarbitrary functions of the thickness coordinate. Firstly, the partial differential equations forthe plane problem of anisotropic functionally graded piezoelectric materials are derived,which the stress function and electric displacement function satisfy. Secondly, the stressand electric displacement functions are assumed in forms of polynomials of thelongitudinal coordinate, so that the stress and electric displacement functions can beacquired through successive integrations. The analytical expressions of axial force,bending moment, shear force, displacements, electric displacements and electricpotential are then deduced. Thirdly, the stress and electric displacement functions areemployed to solve problems of functionally graded piezoelectric plane beams, withthe integral constants completely determined from boundary conditions. Twopiezoelasticity solutions are thus obtained, for cantilever beams subjected to shearforce and point charge applied at the free end, for cantilever beams subjected touniform load. These solutions can be easily degenerated into the piezoelasticitysolutions for homogeneous anisotropic piezoelectric beams. Finally, a numericalexample is presented to show the application of the proposed method to a specificcase.

Non-local theory solution of two collinear mode-I cracks in piezoelectric materials

In this paper, the basic solution of two collinear cracks in a piezoelectric material plane subjected to a uniform tension loading is investigated by means of the non-local theory. Through the Fourier transform, the problem is solved with the help of two pairs of integral equations, in which the unknown variables are the jumps of displacements across the crack surfaces. To solve the integral equations, the jumps of displacements across the crack surfaces are directly expanded in a series of Jacobi polynomials. Numerical examples are provided to show the effects of the interaction of two cracks, the materials constants and the lattice parameter on the stress field and the electric displacement field near crack tips. Unlike the classical elasticity solution, it is found that no stress and electric displacement singularities are present at crack tips. The non-local elastic solutions yield a finite hoop stress at the crack tip, thus allowing us to using the maximum stress as a fracture criterion in piezoelectric materials.

Multi-region Trefftz boundary element method for fracture analysis in plane piezoelectricity

This paper presents a multi-region Trefftz boundary element method for fracture analysis in plane piezoelectricity. To model the sub-region that contains the crack, a special set of Trefftz functions that satisfy the traction-free and charge-free conditions along the crack faces are constructed. To model the remaining sub-regions, the basic set of Trefftz functions co-derived previously by the authors are employed. With the two sets of Trefftz functions, the multi-region Trefftz boundary element method is formulated by point collocation. The special set of Trefftz functions exempts all the boundary treatment of the crack faces and enables the direct determination of the electromechanical intensity factors. Numerical examples are presented to illustrate the efficacy of the formulation.

Trefftz solutions for piezoelectricity by Lekhnitskii's formalism and boundary‐collocation method

AbstractIn this paper, a solution procedure for plane piezoelectricity is developed by Trefftz boundary‐collocation method. Starting with the general plane piezoelectricity solution derived by Lekhnitskii's formalism, the basic sets of Trefftz functions which satisfy the homogeneous governing equations are derived. Moreover, special sets of Trefftz functions are derived for impermeable elliptical voids, impermeable sharp cracks and permeable sharp cracks with arbitrary orientations with respect to the material poling direction. The functions in the special sets satisfy not only the homogeneous governing equations but also the boundary conditions at the peripheries of the pertinent defects. By adopting Trefftz functions as the trial functions, multi‐domain Trefftz boundary‐collocation method is formulated. Numerical examples are presented to illustrate the efficacy of the formulation. Copyright © 2005 John Wiley & Sons, Ltd.

Polynomial solutions to piezoelectric beams (II)—Analytical solutions to typical problems

For the orthoropic piezoelectric plane problem, a series of piezoelectric beams is solved and the corresponding analytical solutions are obtained with the trial-and-error method on the basis of the general solution in the case of three distinct eigenvalues, in which all displacements, electrical potential, stresses and electrical displacements are expressed by three displacement functions in terms of harmonic polynomials. These problems are cantilever beam with cross force and point charge at free end, cantilever beam and simply-supported beam subjected to uniform loads on the upper and lower surfaces, and cantilever beam subjected to linear electrical potential.

Polynomial solutions to piezoelectric beams (I)—Several exact solutions

For the orthotropic piezoelectric plane problem, a series of piezoelectric beams is solved and the corresponding exact solutions are obtained with the trial-and-error method on the basis of the general solution in the case of three distinct eigenvalues, in which all displacements, electrical potential, stresses and electrical displacements are expressed by three displacement functions in terms of harmonic polynomials. These problems are rectangular beams having rigid body displacements and identical electrical potential, rectangular beams under uniform tension and electric displacement as well as pure shearing and pure bending, beams of two free ends subjected to uniform electrical potential on the upper and lower surfaces.

Singularity analysis at the vertex of polygonal quantum wire inclusions

Singular behavior near the vertex/corner of polygonal inclusions in anisotropic piezoelectric planes is identified and investigated. Using a recently derived exact closed-form solution, the elastic and electric fields associated with regular polygon-shaped quantum wires are analyzed in detail, revealing the effect of factors such as material orientation, type of eigenstrain, and number of sides of polygons on the singular behavior. Our study shows that by adjusting these factors properly, one could avoid the singular behavior or reduce the level of singularity, which could be of significance in the design and fabrication of the semiconductor devices.