In-plane Electric Loading Research Articles

Exact solution for an elliptical cavity or crack in a reinforced magnetoelectroelastic material under anti-plane shear

An exact solution for an elliptical cavity embedded in a reinforced, magnetoelectroelastic material under anti-plane shear and in-plane electric and magnetic loading is provided in this paper. Explicit forms of the mechanical, electric and magnetic fields in the matrix, the reinforcement layer and the elliptical cavity are obtained by means of the method of complex potentials. The electromagnetic fields inside the elliptical cavity have been taken into consideration, and explicit solutions for a mode III crack, either impermeable or permeable, are obtained when the elliptical cavity is reduced to a crack. The change of energy due to the presence of the cavity and reinforcement layer has been derived. Some special cases, such as the elliptical cavity problem and crack problem are discussed in detail to illustrate the validity and versatility of the current formulation.

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.

Stress analysis of a functionally graded magneto-electro-elastic strip with multiple moving cracks

This paper investigates the linear steady state problem of several moving cracks in a functionally graded magneto-electro-elastic strip subjected to anti-plane mechanical and in-plane electric and magnetic loading. For simplicity, it is assumed that the properties of the strip vary continuously according to exponential functions along the thickness of the functionally graded piezoelectric piezomagnetic (FGPP) layer. By combining the dislocation method and integral transform technique, an exact solution in closed form to this problem is obtained. Electro-magneto-mechanical loads are applied on the crack surfaces, which are assumed to be magneto-electrically impermeable. Numerical examples are presented to show the interesting mechanical and electromagnetic coupling phenomena induced by multi-crack interactions. Finally, the effects of crack velocity, material constants, and geometric parameters upon the field intensity factors are studied. The results are useful for the design of the magneto-electro-elastic structures.

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.

Multiple moving interfacial cracks between two dissimilar piezoelectric layers under electromechanical loading

The dynamic problem of several moving cracks at the interface between two dissimilar piezoelectric materials is analyzed. The combined out-of-plane mechanical and in-plane electrical loads are applied to the layers. Fourier transforms are used to reduce the problem to a system of singular integral equations with simple Cauchy kernel. The integral equations are solved numerically by converting to a system of linear algebraic equations and by using a collocation technique. The results presented consist of the stress intensity factors and the electric displacement intensity factors. It is found that generally the field intensity factors increase with increasing crack propagation speed.

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.

Fracture analysis of a cracked orthotropic strip bonded to a magneto-electro-elastic layer

The problem of a magneto-electro-elastic coating bonded to an orthotropic strip with multiple cracks is analyzed. The problem is solved under anti-plane shear and in-plane electric and magnetic loading. First, an analytical solution is developed considering a single dislocation problem. Fourier transforms are applied to reduce problem to system of singular integral equations, in which the unknown variables are dislocation densities. The integral equations are of the Cauchy singular kinds which are solved to determine stress intensity factors at the crack tips. The results show that the stress intensity factors at the crack tips as influenced by the imperfect bonding coefficient, the crack geometry and material properties of the orthotropic substrate

Multilayered piezomagnetic/piezoelectric composites with periodic interfacial cracks subject to in-plane loading

A two-dimensional fracture problem of periodically distributed interfacial cracks in multilayered piezomagnetic/piezoelectric composites is studied under in-plane magnetic or electric loading. The magnetic permittivity of the piezoelectric material and the dielectric constant of the piezomagnetic material are considered. A system of singular integral equations of the second kind with a Cauchy kernel is obtained by means of Fourier transform and further solved by using Jacobi polynomials. The problem is solved in the real domain by constructing real fundamental solutions. The primary interfacial fracture mechanic parameters, such as the stress intensity factors (SIFs), the electric displacement intensity factors (EDIFs), the magnetic induction intensity factors (MIIFs) and the energy release rates (ERRs) are then obtained. It is found that a magnetic or electric loading normal to the crack surfaces can lead to a mixture of mode I and mode II type stress singularities at the crack tips. Numerical results show that increasing the thickness of the active layer will favor the crack initiation. Inversely, increasing the thickness of the passive layer will retard the crack initiation. Furthermore, the results indicate that the crack initiation can be inhibited by adjusting the direction of the applied magnetic or electric loading.

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.

The anti-plane solution for the edge cracks originating from an arbitrary hole in a piezoelectric material

In this paper, a general and simple way was found to solve the problem of an arbitrary hole with edge cracks in transversely isotropic piezoelectric materials based on the complex variable method and the method of numerical conformal mapping. Firstly, the approximate mapping function which maps the outside of the arbitrary hole and the cracks into the outside of a circular hole is derived after a series of conformal mapping process. Secondly, based on the assumption that the surface of the cracks and hole is electrically impermeable and traction-free, the approximate expressions for the complex potential, fields intensity factors and energy release rates are presented, respectively. Thirdly, under the in-plane electric loading together with the out-plane mechanical loading, the influences of the hole size, crack length and mechanical/electric loading on the fields intensity factors and energy release rates are analyzed. Finally, some particular holes with edge cracks are studied in numerical analysis. The result shows that, the mechanical loading always promotes crack growth, while the electric loading may retard crack growth.

Moving Dugdale crack along the interface of two dissimilar magnetoelectroelastic materials

In this paper, the concept of Dugdale crack model and Yoffe model is extended to propose a moving Dugdale interfacial crack model, and the interfacial crack between dissimilar magnetoelectroelastic materials under anti-plane shear and in-plane electric and magnetic loadings is investigated considering the magneto-electro-mechanical nonlinearity. It is assumed that the constant moving crack is magneto-electrically permeable and the length of the crack keeps constant. Fourier transform is applied to reduce the mixed boundary value problem of the crack to dual integral equations, which are solved exactly. The explicit expression of the size of the yield zone is derived, and the crack sliding displacement (CSD) is explicitly expressed. The result shows that the stress, electric and magnetic fields in the cracked magnetoelectroelastic material are no longer singular and the CSD is dependent on the loading, material properties and crack moving velocity. The current model can be reduced to the static interfacial crack case when the crack moving velocity is zero.

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.

Mode-I crack in a magnetoelectroelastic layer sandwiched by two elastic half-planes

In this paper we study the impermeable opening crack problem in a magnetoelectroelastic layer sandwiched between dissimilar elastic half-planes under in-plane mechanical, electric and magnetic loadings. Fourier transforms are applied to reduce the crack problem to dual integral equations, which can be further reduced to solving a system of simultaneous Fredholm integral equations. Numerical solutions of the integral equations can be obtained by using the collocation method, and the explicit form solution of the near crack-tip fields is obtained asymptotically. The field intensity factors are obtained and the effects of material properties, geometric size, and loadings are analyzed.