Anti-plane Mechanical Loading Research Articles

Variational principles and size-dependent bounds for piezoelectric inhomogeneous materials with piezoelectric coherent imperfect interfaces

In most practical situations, the interfaces between the constituent phases of composites or inhomogeneous materials are imperfect. The present work aims to establish the bounds for the effective properties of an inhomogeneous material with coherent imperfect interfaces in the setting of piezoelectricity. According to this imperfect interface model, the displacement and the electrical potential are continuous while the traction and the normal electrical displacement are discontinuous across an interface and proportional to the surface gradient of the displacement and the electrical potential, respectively, by the so-called Laplace–Young equation. To achieve this objective, firstly, the classical minimum potential principles of linear piezoelectricity are extended to such inhomogeneous material and to formally setting bounds for their effective piezoelectric properties. Secondly, we consider a transversely isotropic piezoelectric composite consisting of a matrix reinforced by cylindrical inhomogeneities via linearly piezoelectric coherent imperfect interface which is subjected to the uniform anti-plane mechanical load and in-plane electrical load boundary conditions. By taking simple trial strain and electrical displacement or simple trial traction and electrical potential couple fields, the first-order upper and lower bounds are explicitly derived for the corresponding elastic, piezoelectric and dielectric moduli of such composite by using the established variational principles. Finally, numerical results of the obtained bounds are provided to illustrate their size-dependence.

Analysis of three collinear antiplane interfacial cracks in dissimilar piezoelectric materials under non-self equilibrated electromechanical loadings on a center crack

The problem of three collinear interfacial cracks between two dissimilar transversely isotropic piezoelectric materials is considered under electromechanical loadings. The crack surfaces are assumed to be impermeable to the electric field. A single antiplane mechanical and inplane electrical loads are applied at a point on centred crack surface. The problem is formulated by the complex function method, and reduced to the vector Hilbert problem. By solving the problem, a closed form solution for the stress intensity and electric displacement intensity factor is obtained. This solution can be used as a Green’s function for different loading conditions.

Dynamic analysis of an interface crack between magnetoelectroelastic and functionally graded elastic layers under anti-plane mechanical and in-plane electro-magnetic loadings

The dynamic response of an interface crack between magnetoelectroelastic and functionally graded elastic layers under anti-plane shear and in-plane electric and magnetic impacts is investigated by the integral transform method. Fourier and Laplace transforms are applied to reduce the mixed boundary value problem of the interface crack to dual integral equations, which are expressed in terms of a Fredholm integral equation of the second kind. The singular stress, electric displacement and magnetic induction fields near the crack tip are obtained asymptotically, and the stress intensity factors (SIFs), electric displacement intensity factors (EDIFs) and magnetic induction intensity factors (MIIFs) are determined accordingly. Laplace inverse transform is applied to get the field intensities in time domain. Numerical results show how the dynamic SIF is influenced by the magnetoelectric interactions, the material properties, the electric and magnetic loadings, and the geometric size ratios and the functionally graded parameter.

Analysis of cracked functionally graded piezoelectric strip

The fracture behavior of a cracked strip under antiplane mechanical and inplane electrical loading is studied. A functionally graded piezoelectric strip with exponential material gradation is under consideration. The mechanical and electrical loading is combined via loading coupling factor. The problem of a graded piezoelectric strip containing a screw dislocation is solved. This solution results in stress and electric displacement components with Cauchy singularity. Based on the solution achieved for the dislocation, the distributed dislocation technique (DDT) is utilized to form any geometry of multiple cracks and analyze the behavior of a cracked strip under antiplane mechanical and inplane electrical loading. This technique is capable of the analysis of a strip with a system of interacting cracks. Several examples including strips with single crack, two straight cracks and two curved cracks are presented.

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.

Interaction between periodic cracks and periodic rigid-line inclusions in piezoelectric materials

Piezoelectric materials with a doubly periodic array of cracks and rigid-line inclusions under far-field antiplane mechanical load and inplane electric load are investigated. By employing the conformal mapping technique and the elliptical function theory, an exact solution of the whole-field stress and electrical displacement is obtained. The closed-form formulae for the stress and electrical displacement intensity factors at the tips of cracks and rigid-line inclusions and the effective electroelastic moduli of the composites are presented. Some existing solutions can be regarded as degenerated cases of the present results. Numerical examples are provided to show the interesting electroelastic interaction between multiple cracks and multiple rigid-line inclusions.

Three Collinear Antiplane Interfacial Cracks in Dissimilar Piezoelectric Materials

Three collinear impermeable interfacial cracks in bonded dissimilar transversely isotropic piezoelectric materials under electromechanical loadings is analyzed. A single antiplane mechanical and inplane electrical loads are applied at a point on crack surface. The problem is formulated by the complex function method, and reduced to the vector Hilbert problem. Solving this problem, a closed form solution for the stress intensity and electric displacement intensity factor is obtained. This solution can be used as a Green’s function for different loading conditions.

Magnetoelectroelastic analysis of an arbitrary shape inclusion undergoing eigenfields and remote loadings

Within the framework of the linear theory of magnetoelectroelasticity, the problem of an arbitrary shape inclusion in an entire plane under remotely uniform in-plane electromagnetic and anti-plane mechanical loadings is investigated in this paper. The inclusion is undergoing eigenfields and has different magnetoelectroelastic moduli as that of the matrix. Using complex variable and Faber series method, the complex potential for each component 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. Through solving these linear equations, one can obtain the complex potentials both inside the inclusion and in the matrix. Numerical results are provided and shown that different combinations of piezoelectric/piezomagnetic exhibit significant magnetoelectromechanical coupling effect that is not present in single-phase piezoelectric or piezomagnetic material. Applying appropriate remote loadings and eigenfields can effectively reduce stress level and increase magnetoelectric response around the inclusion.

횡등방 압전재료의 면외 계면균열문제

면외 기계적 하중 및 면내 전기적 하중하의 횡등방성 이종 압전재료에 대한 계면균열문제를 해석하였다. 복소함수를 도입하여 문제를 수식화 하고, 이로부터 Hilbert 문제를 구성하였다. Hilbert 문제를 풀므로써, 일반해를 얻었다. 일반해를 사용하여, 반무한 균열 혹은 한 개 유한균열 및 두 개 유한균열에 대한 폐형 해를 각각 구하였다. 이때 하중은 한 개의 집중 기계하중 및 전기적하중이 균열면에 작용한다. 이 문제는 기하조건만 동일하면, 임의의 하중에 대해서도 해를 얻을 수 있는 Green 함수로서 사용될 수 있다.

Magnetoelectric effect in coated fibrous magnetic–piezoelectric composites

In this study, the magnetoelectroelastic behavior is investigated for magnetic–piezoelectric composites with coated reinforcements under antiplane mechanical and in-plane electromagnetic loads. The coating is considered as a layer with properties different from those of the inclusion and the matrix. The developed model is based on the solution of the problem of the magnetoelectroelastic coated inclusion embedded in an infinite homogeneous medium obtained from two different approaches. The first is the complex variable method integrating the Laurent expansion method, and the second is a micromechanical approach based simultaneously on Green’s function techniques and the interfacial operators. The exact solution is obtained and used as background to determine the effective magnetoelectroelastic properties of the composite containing thinly coated inclusions embedded in a matrix through Mori–Tanaka’s model. Numerical investigations into magnetoelectroelastic moduli responsible for the magnetoelectric effect are presented as functions of the volume fraction and characteristics of the coated inclusions. Comparisons with existing models are presented for various fractions of the coated inclusions.

Fracture behavior of two non-symmetrical collinear cracks emanating from an elliptical hole in a piezoelectric material

Based on the Stroh-type formalism and the technique of conformal mapping, the problem of two nonsymmetrical collinear cracks emanating from an elliptical hole in a piezoelectric solid is investigated under remotely uniform in-plane electric loading and anti-plane mechanical loading, which allowed us to take the electric field inside the hole and cracks into account. The analytical solutions of the field intensity factors and the energy release rate are presented in closed-form, which includes the extreme cases for an impermeable crack and a permeable crack. Numerical results are then presented to reveal the effects of geometrical parameters, crack permeability and combined mechanical and electric loadings on the energy release rate.

Size-dependent effective electroelastic moduli of piezoelectric nanocomposites with interface effect

The problem of nanocomposite materials under far-field antiplane mechanical load and inplane electric load is investigated. Based on the theory of Gurtin–Murdoch surface/interface model, an exact solution is obtained for the inhomogeneity/matrix/equivalent medium model, in terms of which a generalized self-consistent approach is proposed for predicting the effective electroelastic moduli of nanocomposites. A closed-form solution of the effective electroelastic moduli is presented. The numerical results reveal that the effective electroelastic moduli are size dependent when the size of the inhomogeneity is on the order of nanometer. With the increase in the size of the inhomogeneity, the present solution approaches to the classical results obtained in the classical theory.

Energy density and release rate study of an elliptic-arc interface crack in piezoelectric solid under anti-plane shear

The anti-plane problem of an elliptical inhomogeneity with an interfacial crack in piezoelectric materials is investigated. The system is subjected to arbitrary singularity loads (point charge and anti-plane concentrated force) and remote anti-plane mechanical and in-plane electrical loads. Using the complex variable method, the explicit series form solutions for the complex potentials in the matrix and the inclusion regions are derived. The electroelastic field intensity factors, the corresponding energy release rates and the generalized strain energy density at the cracks tips are then provided. The influence of the aspect ratio of the ellipse, the crack geometry and the electromechanical coupling coefficient on the energy release rate and the strain energy density is discussed and shown in graphs. The results indicate that the energy release rate increases with increment of the aspect ratio of the ellipse and the influence of electromechanical coupling coefficient on the energy release rate is significant. The strain energy density decreases with increment of the aspect radio of the ellipse and it is always positive for the cases discussed. The energy release rate, however, can be negative when both mechanical and fields are applied.

Exact solution to the antiplane problem of doubly periodic conducting rigid line inclusions of unequal size in piezoelectric materials

AbstractThe problem of a doubly periodic array of conducting rigid line inclusions in piezoelectric materials under far‐field antiplane mechanical load and inplane electric load is investigated, where the fundamental cell contains four rigid line inclusions of unequal size. An exact solution to the problem is presented by employing the conformal mapping technique and the elliptical function theory. The closed form formulae for the stress and electrical displacement intensity factors at the rigid line inclusion tip and the effective electroelastic moduli of such composites are derived. Many new and existing solutions can be regarded as the special or degenerated cases. Numerical examples are provided to show the interesting electroelastic interaction phenomenon induced by multiple rigid line inclusions. The present work is helpful in understanding the optimization mechanism of some naturally occurring composites and in designing novel materials by direct engineering of their microstructure.

The fracture behavior of multiple cracks emanating from a circular hole in piezoelectric materials

Using the complex variable method and a new conformal mapping, the fracture behavior of multiple cracks emanating from a circular hole in piezoelectric materials is considered under remotely uniform in-plane electric and anti-plane mechanical loadings. The analytic solutions of the field intensity factors and the energy release rate are presented by taking the effect of dielectric permittivity into consideration. Known results can be derived as special cases from the general solutions. The results are illustrated with plots, showing that the piezoelectric material containing three radial cracks spaced equally at 120° apart originating from a circular hole is the easiest to fail for all cases of cracks originating from a circular hole under in-plane electric and anti-plane mechanical loadings. Moreover, if there exist multiple cracks (n ≥ 3) originating from a circular hole in piezoelectric materials, an increase in the number of cracks can enhance the reliability of these materials. The star-shaped cracks possess similar characteristics.

Anti-plane analysis of multiple cracks originating from a circular hole in a magnetoelectroelastic solid

The anti-plane problem of multiple cracks originating from a circular hole in a magnetoelectroelastic solid is investigated under remotely uniform anti-plane mechanical loading and in-plane electromagnetic loadings. The boundary value problem is reduced to a Cauchy integral equation by a new mapping function and the complex variable method, which is further solved exactly. The analytic expressions of the complex potentials, the field intensity factors and the energy release rate are derived in closed-form under the assumption that the surfaces of the cracks and hole are both electrically and magnetically impermeable. The effects of crack configurations and combined loadings on the energy release rate are shown graphically. Several useful results which may have potential applications to the design and fracture analysis of magnetoelectroelastic structures are given.

The behavior of two non-symmetrical permeable cracks emanating from an elliptical hole in a piezoelectric solid

Based on the Stroh-type formalism, we present a concise analytic method to solve the problem of complicated defects in piezoelectric materials. Using this method and the technique of conformal mapping, the problem of two non-symmetrical collinear cracks emanating from an elliptical hole in a piezoelectric solid is investigated under remotely uniform in-plane electric loading and anti-plane mechanical loading. The exact solutions of the field intensity factors and the energy release rate are presented in closed-form under the permeable electric boundary condition. With the variation of the geometrical parameters, the present results can be reduced to the well-known results of a mode-III crack in piezoelectric materials. Moreover, new special models used for simulating more practical defects in a piezoelectric solid are obtained, such as two symmetrical edge cracks and single edge crack emanating from an elliptical hole or circular hole, T-shaped crack, cross-shaped crack, and semi-infinite plane with an edge crack. Numerical results are then presented to reveal the effects of geometrical parameters and the applied mechanical loading on the field intensity factors and the energy release rate.

Piezoelectric study for a three-phase composite containing arbitrary inclusion

This paper analyzes a coated inclusion of arbitrary shape embedded in an infinite plate subjected to anti-plane mechanical and in-plane electrical loadings. Based on the methods of complex variable, conformal mapping, Faber series and Laurent series, the complex potential for each component 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. The exact solutions are obtained after the complex potentials are solved. By truncating the infinite system of linear equations at finite N terms, some numerical examples are provided to show the effect of the material mismatch and the shape of inclusion on the interfacial stress and elastic field. The accuracy of the linear systems is also discussed in the article.

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.

Mode III arbitrarily oriented crack in an FGPM medium

This paper investigates the fracture behavior of an arbitrarily oriented crack in a functionally graded piezoelectric material subjected to antiplane mechanical and inplane electric loads. The material properties are assumed to vary exponentially. After applying the Fourier transform to the displacement and electrical potential, a system of singular integral equations is formulated and solved by applying the Gauss‐Chebyshev integration formula. The results show the effects of crack orientations and the nonhomogeneous material parameter on the stress and electrical displacement intensity factors.