In-plane Electric Loading Research Articles

The interaction between a piezoelectric screw dislocation and collinear rigid lines in one-dimensional hexagonal quasicrystals

The interaction between a piezoelectric screw dislocation and collinear rigid lines in one-dimensional (1D) hexagonal quasicrystals (QCs) under anti-plane mechanical and in-plane electrical loading is analysed. Two idealized electrical assumptions, namely electrically conductive and dielectric, are considered. By using Muskhelishvilid’s complex transformation method, the explicit expressions for the field variables, the singularity of the field variables at the line tip and the force on the dislocation are obtained for a single rigid line. Based on the generalized Peach–Koehler formula, the generalized stress intensity factors and image force on dislocations in piezoelectric quasicrystal (PQC) are obtained. It is evident that the existence of phason field has an effect on image force.

Anti-plane fracture behavior of n nano-cracks emanating from a magnetoelectrically semi-permeable regular n-polygon nano-hole in magnetoelectroelastic materials

Based on Gurtin–Murdoch surface/interface model and complex potential theory, by constructing a new conformal mapping function, the fracture behavior on n nano-cracks emanating from a n-polygon nano-hole under far-field anti-plane mechanical load, inplane electrical load and magnetic load is studied. The analytical solutions of stress intensity factor, electric displacement intensity factor and magnetic induction intensity factor at the crack tip are given under the boundary conditions of magnetoelectrically semi-permeable. In addition, numerical examples show that when considering the surface effect, the stress field intensity factor, electric displacement intensity factor and magnetic induction intensity factor have obvious size-dependence. The results of this paper show that when the defect size increases to a certain extent, the influence of the surface effect begins to decrease, and finally tends to the results of classical elastic theory.

Fracture analysis of circular hole edge arbitrary position crack with surface effects in magnetoelectroelastic materials

The mechanical properties of magnetoelectroelastic (MEE) materials are brittle, which is easy to produce nano-defects (hole or crack). The micro-fields of nanoscale cracks and holes are size dependent significantly. Nanoscale cracks and holes interact with each other, and the interference effect is very obvious. In this paper, the Model III fracture behavior of nanoscale circular hole edge arbitrary position radial crack is studied under anti-plane mechanical load, in-plane electric displacement load and in-plane magnetic induction load. Based on the surface elasticity theory, the conformal mapping technique and the MEE elasticity theory, the MEE fields expressions of the nano-hole and nano-crack are obtained, and the analytical solution of generalized intensity factors (stress intensity factor, electric displacement intensity factor and magnetic induction intensity factor) at crack tip is given. The comparison between the present results and the existing research shows the correctness of the present method in this paper. The influences of the microstructure and coupling loads on the generalized intensity factors are discussed.

Fracture analysis on periodic radial cracks emanating from a nano-hole with surface effects in magnetoelectroelastic materials

In this paper, the fracture behavior of multiple radial cracks with periodic and uniform distribution around a nanoscale circular hole in a magnetoelectroelastic (MEE) solid is studied under anti-plane mechanical load, in-plane electric displacement load and magnetic induction load. Based on the surface elasticity theory, the analytical solutions of the MEE fields distribution and the MEE fields intensity factors at the crack tip are obtained by using the conformal mapping method, the boundary value theory of analytic function and the basic theory of MEE elastic solid. Considering that the defects are piezomagnetic phase or piezoelectric phase, size effects of the defects, cracks number, geometric parameters and applied MEE loads on the MEE fields intensity factors at crack tip are discussed. The size effect, interference effect and multi-fields load coupling effect of the MEE tip fields are studied in detail.

In-plane stress analysis in a cracked functionally graded piezoelectric plane

The fracture behavior of a cracked functionally graded piezoelectric plane was investigated under in-plane mechanical and electrical loading. The electro-elastic properties of the plane exponentially varied along the y-axis. Employing the Fourier transform, stress analysis containing a Volterra edge dislocation in a functionally graded piezoelectric plane was carried out. These solutions were used to derive a system of Cauchy singular integral equations for analyzing the straight and curved cracks. Linear, arc, parabolic and sine wave-shaped cracks were evaluated. The integral equations were numerically solved using the Lobatto–Chebyshev integration formula for dislocation density functions on the crack face which were employed to calculate field intensity factors. The effects of the shaped crack, loading parameters, and non-homogeneity parameter on the stress and electric intensity factors were considered.

Electro-elastic field of a piezoelectric quasicrystal medium containing two cylindrical inclusions

Considering the piezoelectric effect, the electro-elastic field of an infinite one-dimensional quasicrystal medium with two circular cylindrical inclusions is derived under antiplane shear and inplane electric loading. The boundary value problem of the composite material with circular cylindrical inclusions is analytically solved by the use of the conformal mapping technique and analytical continuation theory. The stresses in the phonon and phason fields and the electric displacements are obtained explicitly in the form of a power series both for the matrix and the inclusions. Some typical examples are analyzed to show the effect of the geometric parameters, material properties, and electro-mechanical loading on the electro-elastic fields in the matrix, inclusions, and interfaces. The limiting cases of circular cavities and rigid circular inclusions have also been investigated and discussed.

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.

Modelling and analysis of piezoelectric actuators with partially debonded adhesive layers

In the modelling of thin-sheet piezoelectric actuators, the bonding condition between the actuator and the host structure can place a significant influence on the behaviour of the actuator. This paper provides a comprehensive theoretical study of the electromechanical behaviour of a thin-sheet piezoelectric actuator bonded to a host structure through a partially debonded adhesive layer under in-plane electric loading. The focus is on the coupled effect of the debonding and adhesive layer on the interfacial stress. The piezoelectric actuator is characterized by a general electro-elastic Euler–Bernoulli beam model featuring both axial and bending deformation. The theoretical solution of the problem is formulated by using singular integral equations, in terms of the interfacial shear and normal stresses, which are solved by using Chebyshev polynomials. The introduction of the adhesive layer in the model changes the singular behaviour of the problem. The current singular integral equations are shown to be effective in solving for both singular and non-singular stresses. Typical examples have been provided to show the effect of the geometry and the material property on the local stress field along the interface.

A comprehensive electro-magneto-elastic buckling and bending analyses of three-layered doubly curved nanoshell, based on nonlocal three-dimensional theory

The present paper studies the three-dimensional magneto-electro-elastic bending and buckling analyses of three-layered doubly curved nanoshells based on nonlocal elasticity theory. The kinematic relations are developed based on two-variable sinusoidal transverse shear and thickness deformation theory. The transverse deflection is decomposed into bending, shear and thickness deformation components by using a two-variable sinusoidal shear deformation theory. The nanoshell is subjected to in-plane mechanical, electrical and magnetic loads in which these forces are accounted as external works in constitutive relations for buckling analysis. This introduces a variable transverse displacement along the thickness direction. The principle of virtual work is used to derive the governing equations of bending and buckling. Numerical results are obtained based on Navier’s technique for simply-supported boundary conditions. A parametric analysis is performed to investigate the influence of the nonlocal parameter, the applied electric and magnetic potentials on the magneto-electro-elastic bending and buckling responses of doubly curved nanoshells.

A novel method for measuring AFO stiffness using motion capture

A mathematical strip-saturation model is proposed to analyze the influence of changing poling direction on two hairline electrically semi-permeable collinear cracks embedded in a piezoelectric strip. The remote boundaries of the piezoelectric strip are subjected to anti-plane mechanical and in-plane electrical loads. Fourier series method is applied to transform the model into triple series equations which are then reduced to singular integral equations using the integral equation technique. The closed-form analytical expressions are derived for saturation zones length, crack-sliding displacement (CSD), crack opening potential drop (COPD), intensity factors (IFs), and energy release rate (ERR). Influence of the poling angle on CSD, COPD, and ERR are shown graphically for various cracked piezoelectric ceramics. In addition, the numerical study is also presented for analyzing the effect of inter-crack distance and prescribed electrical and mechanical loadings on the fracture parameters. All graphical results are analyzed and compared.

Surface effects of electroelastic tip fields of multiple cracks emanating from a circular hole

The problem of piezoelectric solids containing a nanoscale multi-cracks cracked hole under antiplane mechanical load and inplane electric load is investigated theoretically. The surface electroelastic constants of the defects (crack and hole) are introduced based on the Gurtin-Murdoch surface model. An exact solution of the stress fields and electric displacement fields around the defects are obtained by constructing a conformal mapping function. Closed-form solution of the stress intensity factor and the electric displacement intensity factor at crack tips are presented. Numerical examples are provided to reveal the relationship between the electroelastic fields intensity factors and the defect size, the crack-hole interaction, and the applied mechanical-electrical loads.

Poling angle effect on two mode-III semi-permeable collinear cracks in a piezoelectric strip: Strip-saturation model

A mathematical strip-saturation model is proposed to analyze the influence of changing poling direction on two hairline electrically semi-permeable collinear cracks embedded in a piezoelectric strip. The remote boundaries of the piezoelectric strip are subjected to anti-plane mechanical and in-plane electrical loads. Fourier series method is applied to transform the model into triple series equations which are then reduced to singular integral equations using the integral equation technique. The closed-form analytical expressions are derived for saturation zones length, crack-sliding displacement (CSD), crack opening potential drop (COPD), intensity factors (IFs), and energy release rate (ERR). Influence of the poling angle on CSD, COPD, and ERR are shown graphically for various cracked piezoelectric ceramics. In addition, the numerical study is also presented for analyzing the effect of inter-crack distance and prescribed electrical and mechanical loadings on the fracture parameters. All graphical results are analyzed and compared.

In-Plane Analysis of an FGP Plane Weakened by Multiple Moving Cracks

In this paper, the analytical solution of an electric and Volterra edge dislocation in a functionally graded piezoelectric (FGP) medium is obtained by means of complex Fourier transform. The system is subjected to in-plane mechanical and electrical loading. The material properties of the medium vary exponentially with coordinating parallel to the crack. In this study, the rate of the gradual change of the shear moduli and mass density is assumed to be same. At first, the Volterra edge dislocation solutions are employed to derive singular integral equations in the form of Cauchy singularity for an FGP plane containing multiple horizontal moving cracks. Then, these equations are solved numerically to obtain dislocation density functions on moving crack surfaces. Finally, the effects of the crack moving velocity, material properties, electromechanical coupling factor and cracks arrangement on the normalized mode I and mode II stress intensity factors and electric displacement intensity factor are studied.

Fracture analysis on a cracked elliptical hole with surface effect in magnetoelectroelastic solid

We consider a micromechanical model of a cracked elliptical hole with surface effect in magnetoelectroelastic (MEE) solid subjected to anti-plane mechanical and in-plane electric and magnetic loads. By using the surface elasticity theory and conformal mapping technique of analytic functions, we present closed from solutions of the generalized MEE stresses fields around the defects (crack and elliptical hole) and the MEE fields intensity factors at the crack tip. Based on the rigorous solution, the fracture characteristics of the MEE solid weakened by nanoscale defects were studied. Numerical examples reveal the relationships of the MEE fields intensity factors with the size of the cracked elliptical hole, the crack-hole interaction, the shape of the elliptical hole and the applied mechanical-electrical-magnetical loads in this work.

Analysis of a mode III interface crack in a piezoelectric bimaterial based on the dielectric breakdown model

A mode III electrically conductive crack between two different piezoelectric materials under the action of anti-plane mechanical and in-plane electric loadings is analyzed. The strip dielectric breakdown (DB) model, which is free from the electric field singularity, is developed for this crack. According to this model, the electric field along a DB-zone situated in continuation of a crack is assumed to be equal to the electric breakdown strength. The DB-zone lengths are found from the condition of a finite electric field at the end point of such a zone. Using special representations of field variables via sectionally analytic functions, an inhomogeneous combined Dirichlet–Riemann boundary value problem is formulated and solved analytically. Explicit expressions for the shear stress, the electric field and the crack faces’ sliding displacement jump are derived. The stress intensity factor is determined as well. The dependencies of the mentioned values on the magnitude of the external electromechanical loading are presented.

Anti-plane crack problem of a functionally graded piezoelectric materials strip with arbitrarily distributed properties

This paper treats the anti-plane crack problem of a functionally graded piezoelectric material (FGPM) strip with arbitrarily distributed properties. It is assumed that the elastic stiffness, piezoelectric constant, and dielectric constant of the FGPM vary continuously along the thickness of the medium, and the strip is under anti-plane mechanical and in-plane electric impact loadings. By using the Fourier transform and defining unknown discontinuous functions across the crack surfaces, the anti-plane crack problem of FGPM is reduced to a group of singular integral equations, which are solved numerically. The stress and electric displacement intensity factors are presented, and the influences of the nonhomogeneous and geometric parameters on stress and electric displacement intensity factors are also included.

New analytical solutions for modified polarization saturation models in piezoelectric materials

In this paper, we present new analytical solutions for modified polarization saturation (PS) models for arbitrary polarized and semipermeable 2-D piezoelectric media. The PS model is modified to various other non-linear fracture models by varying the normal electric displacement saturated conditions in place of a constant saturated value. These variations are hereby defined as interpolating linear, quadratic and cubic type’s polynomials into saturated value interpolated on the basis of possible electric displacement values at the centre of the crack, actual crack tip and saturated zone tip. To present the analytical study, a centre crack problem in 2-D semipermeable piezoelectric media under arbitrary poling direction and combined in-plane mechanical and electric displacement loadings is considered. Using extended Stroh formalism and complex variable approach, these models are mathematically reduced into generalized non-homogeneous Riemann–Hilbert problems. Applying Riemann–Hilbert technique, the solution of these problems is obtained in terms of generalized complex potential functions representing traction forces and electric displacement components. These lead to obtain the explicit expressions for saturated zone length, crack opening displacement, crack opening potential and local intensity factor in each case. Also, the results of saturation zones and local intensity factor are obtained in each case and compared with the established numerical results. The studies show that saturated zone length increases with increasing electric loading for each case and also depends on the polynomial varying saturation condition. For a particular electric loading applied within the range of critical value, it increases with increasing degree of polynomial type saturated condition i.e. from constant to cubic type’s normal electric displacement saturated condition.

Interaction of Screw Dislocation with Edge Interfacial Crack on Fine-Grained Piezoelectric Coating/Substrate

The interaction between micro- and macrocracks in a fine-grained piezoelectric coating/substrate under remote antiplane mechanical and in-plane electrical loadings was studied. The principle of superposition and a mapping function method was used to transform the fine-grained coating/substrate structure containing the screw dislocation and the edge interfacial crack into the right semi-infinite plane piezoelectric bimaterial with screw dislocation to simplify the problem. Furthermore, the electric field, displacement field, intensity factors, and image force of these two problems were established. In addition, numerical calculations were then given graphically to study the effects of the elastic modulus of the material, the size of the crack, the thickness of the coating, and the screw dislocation angle on the edge interface crack and dislocation.

A screw dislocation near a non-elliptical piezoelectric inhomogeneity with internal uniform electroelastic field

Conformal mapping and analytic continuation are employed to prove the existence of an internal uniform electroelastic field inside a non-elliptical piezoelectric inhomogeneity interacting with a screw dislocation. We focus specifically on the case when the piezoelectric matrix surrounding the inhomogeneity is subjected to uniform remote anti-plane mechanical and in-plane electrical loading and a constraint is imposed between the remote loading and the screw dislocation. The constraint can be expressed in a relatively simple decoupled form by utilizing orthogonality relationships between two corresponding eigenvectors. The internal uniform electroelastic field is found to be independent of the presence of the screw dislocation; moreover, it can be expressed in decoupled form.

Interaction between a completely coated semi-infinite insulating crack and a piezoelectric screw dislocation

We derive analytic solutions to the problem of a completely coated semi-infinite insulating crack interacting with a screw dislocation under the assumption of anti-plane mechanical and in-plane electrical loading in a piezoelectric composite. The screw dislocation can be located either in the piezoelectric matrix or in the piezoelectric coating itself. We obtain explicit expressions for the resultant stress and electric displacement intensity factors, the resultant crack extension force and the image force acting on the screw dislocation. By employing orthogonality relations for two corresponding eigenvectors, we avoid tedious matrix operations in the development of the final expressions for the resultant field intensity factors, resultant crack extension force and image force.