Anti-plane Mechanical Loading Research Articles

Mixed Electrical Conditions on Interface Inclusion in a Piezoelectric Bimaterial under Antiplane Mechanical and Inplane Electric Loadings

An absolutely rigid interface inclusion in a bimaterial piezoelectric space under the action of antiplane mechanical and in-plane electric loadings is analyzed. One zone of the inclusion is electrically insulated while the other part is electrically permeable. This problem is important for engineering application, but it has not been solved earlier in an analytical way. Presenting all electromechanical quantities via sectionally analytic vector functions, the combined Dirichlet-Riemann boundary value problem is formulated. An exact analytical solution of this problem is obtained. Closed form analytical expressions for electromechanical quantities at the interface are derived. Some of these values are also presented graphically along the corresponding parts of the material interface. Singular points of the shear strain and the electric displacement are found and the corresponding intensity factors are determined as well.

Interaction of a conductive crack and of an electrode at a piezoelectric bimaterial interface

The interaction of a conductive crack and an electrode at a piezoelectric bi-material interface is studied. The bimaterial is subjected to an in-plane electrical field parallel to the interface and an anti-plane mechanical loading. The problem is formulated and reduced, via the application of sectionally analytic vector functions, to a combined Dirichlet–Riemann boundary value problem. Simple analytical expressions for the stress, the electric field, and their intensity factors as well as for the crack faces' displacement jump are derived. Our numerical results illustrate the proposed approach and permit to draw some conclusions on the crack–electrode interaction.

Exact solution of two collinear cracks normal to the boundaries of a 1D layered hexagonal piezoelectric quasicrystal

The electroelastic problem related to two collinear cracks of equal length and normal to the boundaries of a one-dimensional hexagonal piezoelectric quasicrystal layer is analysed. By using the finite Fourier transform, a mixed boundary value problem is solved when antiplane mechanical loading and inplane electric loading are applied. The problem is reduce to triple series equations, which are then transformed to a singular integral equation. For uniform remote loading, an exact solution is obtained in closed form, and explicit expressions for the electroelastic field are determined. The intensity factors of the electroelastic field and the energy release rate at the inner and outer crack tips are given and presented graphically.

Effective electroelastic constants for three-phase confocal elliptical cylinder model in piezoelectric quasicrystal composites

A three-phase confocal elliptical cylinder model is proposed to analyze micromechanics of one-dimensional hexagonal piezoelectric quasicrystal (PQC) composites. Exact solutions of the phonon, phason, and electric fields are obtained by using the conformal mapping combined with the Laurent expansion technique when the model is subject to far-field anti-plane mechanical and in-plane electric loadings. The effective electroelastic constants of several different composites made up of PQC, quasicrystal (QC), and piezoelectric (PE) materials are predicted by the generalized self-consistent method. Numerical examples are conducted to show the effects of the volume fraction and the cross-sectional shape of inclusion (or fiber) on the effective electroelastic constants of these composites. Compared with other micromechanical methods, the generalized selfconsistent and Mori-Tanaka methods can predict the effective electroelastic constants of the composites consistently.

Analysis of multiple edge cracks in a non-homogeneous piezoelectric layer

This study is concerned with the treatment of the several edge cracks in a functionally graded piezoelectric (FGP) layer under anti-plane mechanical and in-plane electrical loading. The edge crack is assumed to be either electrically impermeable or permeable. The problem is formulated by using distributed dislocation technique. The integral equations are constructed for the analysis of a FGP layer, in which the unknown variables are dislocation densities. By use of the dislocation densities, the field intensity factors are calculated. Numerical examples are provided to show the effect of the location and orientations of edge crack upon the stress intensity factors.

An interface crack with partially electrically conductive crack faces under antiplane mechanical and in-plane electric loadings

An interface crack in a bimaterial piezoelectric space under the action of antiplane mechanical and in-plane electric loadings is analyzed. One zone of the crack faces is electrically conductive while the other part is electrically permeable. All electro-mechanical values are presented using sectionally-analytic vector-functions and a combined Dirichlet-Riemann boundary value problem is formulated. An exact analytical solution of this problem is obtained. Simple analytical expressions for the shear stress, electric field and also for mechanical displacement jump of the crack faces are derived. These values are also presented graphically along the corresponding parts of the material interface. Singular points of the shear stress, electric field and electric displacement jump are found. Their intensity factors are determined as well. Intensity factors variations with respect to the external electric field and different ratios between the electrically conductive and electrically permeable crack face zones are also demonstrated.

Effective properties of coated fibrous piezoelectric composites with spring-type interfaces under anti-plane mechanical and in-plane electrical loads

In this paper, we investigate the effective properties of three-phase (matrix/coating/fiber) cylindrical piezoelectric composites with imperfect interfaces under anti-plane mechanical and in-plane electrical loads. By using the electromechanically coupling spring-type interface model and the generalized self-consistent method (GSM), we analytically derived the effective electroelastic moduli. The present solutions include as special cases the three-phase cylindrical piezoelectric composites with perfect interfaces as well as the two-phase (matrix/fiber) case with imperfect or perfect interfaces. Selected calculations are graphically shown to demonstrate dependence of the effective moduli on the interfacial properties. The particular size-dependent characteristic due to the interfacial imperfection is also discussed.

Interaction between a screw dislocation and an oblique edge crack in a half-infinite MEE solid

By using the complex variable function method, the interaction between a screw dislocation and an oblique edge crack in a half-infinite magnetoelectroelastic solid is investigated under remotely anti-plane mechanical and in-plane magnetoelectric loadings. The exact solutions in closed-form of the stress, electric displacement and magnetic induction are obtained by Cauchy integral formula and the technique of conformal mapping. The image force of the screw dislocation is then calculated by the generalized Peach–Koehler formula. The field intensity factors and energy release rate for the dislocation and crack are also derived explicitly. Numerical examples are provided to show the influences of the geometrical parameters of the crack and location of the dislocation on the field intensity factors, image force and energy release rate.

Three-Phase Cylinder Model of One-Dimensional Hexagonal Piezoelectric Quasi-Crystal Composites

A three-phase cylinder model (inclusion/matrix/composite) is proposed and analyzed for one-dimensional (1D) piezoelectric quasi-crystal composites. The exact closed-form solutions of the stresses of the phonon and phason fields and the electric field are derived under far-field antiplane mechanical and in-plane electric loadings via the Laurent expansion technique. Numerical results show that the thickness and material properties of the interphase layer can significantly affect the induced fields in the inclusion and interphase layer. Furthermore, the generalized self-consistent method is applied to predict analytically the effective moduli of the piezoelectric quasi-crystal composites. It is observed from the numerical examples that the effective moduli of piezoelectric quasi-crystal composites are very sensitive to the fiber volume fraction as well as to the individual material properties of the fiber and matrix. By comparing QC/PE with QC1/QC2, PE/QC, and PZT-7/epoxy, we found that using QC as fiber could, in general, enhance the effective properties, a conclusion which is in agreement with the recent experimental results.

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.

Screw dislocation near a piezoelectric oblique edge crack

The interaction between a screw dislocation and an oblique edge crack in a semi-infinite piezoelectric medium under remote anti-plane mechanical and in-plane electrical loadings is studied in this article. The screw dislocation suffers a finite discontinuity in the displacement and electric potential across the slip plane. The dislocation also has a line force and a line charge along its core. In the framework of linear piezoelectricity theory, the analytical solutions are derived from the complex variable and Schwarz–Christoffel transformation. Furthermore, the electric field, stress field, generalized stress intensity factor, image force and crack extension force are formulated exactly. Numerical calculations are then given graphically for studying the effects of crack length, crack angle and load position on these electric mechanical parameters. Besides, the dislocation solutions proposed here can be served as Green’s functions for the analyses of corresponding piezoelectric cracking or distributed loading problems.

A rigorous solution for the piezoelectric materials containing elliptic cavity or crack with surface effect

A theoretical study on the problem of piezoelectric materials with a nano elliptic cavity considering surface effect subjected to far‐field antiplane mechanical load and inplane electric load is reported. Surface effect is introduced based on the theory of Gurtin‐Murdoch surface/interface model. A rigorous whole‐field solution is presented by using the complex variable elasticity theory and the conformal mapping method, in terms of which closed‐form solution of the stress/electric displacement fields and the stress/electric displacement intensity factors is obtained. The results reveal that the stress/electric displacement fields and the stress/electric displacement intensity factors are size dependent when the size of the cavity/crack is on the order of nanometer. The present solution approaches to the classical electroelastic results with the increase of the size of the nano elliptic cavity and crack. The influences of the elliptic cavity shape ratio on the stress and electric displacement concentration factors are discussed.

면외변형하의 압전재료에 대한 침투 쐐기균열

In this study, we analyze the problem of wedge cracks, which are geometrically unsymmetrical in transversely piezoelectric materials. A single concentrated antiplane mechanical load and inplane electrical load are applied at the point of the wedge surface, while one concentrated antiplane load is applied at the crack surface. The crack surfaces are considered as permeable thin slits, where both the normal component of electric displacement and the electric potential are assumed to be continuous across these slits. Using Mellin transform method, the problem is formulated and the Wiener-Hopf equation is derived. By solving the equation, the solution is obtained in a closed form. The intensity factors of the stress and the electric displacement are obtained for any crack length as well as inclined and wedge angles. Based on the results, the intensity factors are independent of the applied electric loads. The electric displacement intensity factor is always dependent on that of stress intensity factor, while the electric field intensity factor is zero. In addition, the energy release rate is computed. These solutions can be used as fundamental solutions which can be superposed to arbitrary electromechanical loadings.

압전재료에 대한 면외하중하의 모서리 경사 균열

The occurrence of an inclined edge crack in transversely piezoelectric material is analyzed. Concentrated antiplane mechanical and inplane electrical loads are applied at the boundary and crack surface, respectively. The crack surfaces are assumed to be impermeable to the electric field. Using the Mellin transform with the introduction of a generalized displacement vector, the problem is formulated, and the Wiener-Hopf equation is derived. By solving the equation, the solution is obtained in a closed form. The intensity factors of the stress, the electric displacement, and the energy release rate are obtained for any crack length and inclination angle. These solutions can be used as fundamental solutions and can be superimposed to represent any arbitrary electromechanical loading.

Analytical solution of multiple moving cracks in functionally graded piezoelectric strip

The dynamic behaviors of several moving cracks in a functionally graded piezoelectric (FGP) strip subjected to anti-plane mechanical loading and in-plane electrical loading are investigated. For the first time, the distributed dislocation technique is used to construct the integral equations for FGP materials, in which the unknown variables are the dislocation densities. With the dislocation densities, the field intensity factors are determined. Moreover, the effects of the speed of the crack propagation on the field intensity factors are studied. Several examples are solved, and the numerical results for the stress intensity factor and the electric displacement intensity factor are presented graphically finally.

Homogenization of fibrous piezoelectric composites with general imperfect interfaces under anti-plane mechanical and in-plane electrical loadings

The present paper aims mainly to estimate the size-dependent effective properties of fibrous piezoelectric composites with general imperfect interfaces. The interface model used states that the displacement, traction, electric potential, and normal electric displacement all suffer jumps across an interface. In addition, it can degenerate into the well-known special ones by employing appropriate high-contrast interfacial parameters. To achieve our objective, an auxiliary inhomogeneity problem of a circular fiber embedded in an infinite cylindrical reference phase via general imperfect interface under anti-plane mechanical and in-plane electrical boundary conditions is analytically solved. This solution allows us to apply the well-known micromechanical schemes such as the dilute, Mori–Tanaka to obtain the closed-form expressions for the size-dependent overall properties of composites under consideration. Some numerical examples are provided for illustrating the features of the obtained general results.

Interface energy effect on electromechanical response of piezoelectric composites with an arbitrary nano-inclusion under anti-plane shear

For piezoelectric composites with nano-inclusions, the interface energy around the nano-inclusions will play important roles in predicting the electromechanical response under different loadings. In this paper, the interface energy effect on the electromechanical response of piezoelectric composites with a coated nano-inclusion of arbitrary shape under anti-plane mechanical and in-plane electric loadings is studied, and the analytical solutions of the stress and electric displacement are presented. Combining the Laurent series expansion technique and conformal mapping method, the general solutions of complex potentials are expressed. The unknown coefficients are determined by satisfying the boundary conditions with consideration of surface/interface effect. Some examples such as triangle and quasi-square shapes of nano-inclusions are given to show the distribution of internal stress and electric field. It is found that the interface energy effect shows significant variation with the shapes of nano-inclusions, and the effect in the case of circular shape is the smallest. The interface effect on the stress and electric field at the inner and outer boundaries is also examined.

Fracture analysis of an orthotropic strip with imperfect piezoelectric coating containing multiple defects

This paper presents an analytical model for the analysis of an orthotropic strip with piezoelectric coating weakened by multiple defects. The medium is subjected to anti-plane mechanical and in-plane electrical loading. Using the dislocation method, the problem can be reduced to a system of singular integral equations, in which the unknown variables are dislocation densities. The integral equations are of the Cauchy singular kind and are solved numerically to determine stress intensity factors at the crack tips and hoop stress on the cavity. The obtained results are analyzed to evaluate the dependence of the stress intensity factors and hoop stress on the imperfect bonding coefficient, the defect geometry and material properties of the orthotropic substrate.

Analysis of cracked piezoelectric layer with imperfect non-homogeneous orthotropic coating

The fracture problem for a medium composed of a cracked piezoelectric strip with functionally graded orthotropic coating is studied. The layer is subjected to anti-plane mechanical and in-plane electrical loading. In this paper, we first address, the problem of a screw dislocation located in a substrate which is imperfectly bonded to the coating. Then, in order to model the cracked piezoelectric layer, by means of the dislocation solution, we construct integral equations for the layer, in which the unknown variables are dislocation densities. These unknowns are determined through satisfaction of the boundary conditions on the crack faces. By use of the dislocation densities, the field intensity factors are determined. Several examples are presented to demonstrate the applicability of the proposed solution.

Variational principles and size-dependent bounds for piezoelectric inhomogeneous materials with piezoelectric spring–layer imperfect interfaces

The piezoelectric spring–layer interface model is widely used in describing some imperfect interfaces frequently involved in piezoelectric inhomogeneous materials. Typically, it is appropriate for modeling a thin, soft and low conducting interface between two bulk phases. This model stipulates that, across an interface, the displacement and electric potential are discontinuous while the traction and normal electric displacement are continuous and proportional to the displacement and electric potential jumps, respectively. In this work, the classical minimum potential principles of linear piezoelectricity are extended to piezoelectric inhomogeneous materials with piezoelectric spring–layer imperfect interfaces, and to investigating the interface effects on their effective properties. By choosing simple admissible displacement–electric displacement and stress–electric potential coupling fields, the extended Voigt and Reuss bounds can explicitly be derived for the corresponding effective properties of a transversely isotropic fiber reinforced composite which is subjected to remotely uniform in-plane electric loading and anti-plane mechanical loading. Numerical results are provided to illustrate the size-dependent features of the obtained Voigt and Reuss bounds.