Electric Displacement Intensity Factors Research Articles

Basic solution to four three-dimensional rectangular limited-permeable cracks in transversely isotropic magneto-electro-elastic material

The solution to four three-dimensional rectangular cracks in magneto-electro-elastic material is proposed by using the generalized Almansi’s theorem and the Schmidt method under limited-permeable boundary conditions. The problem is formulated through Fourier transform as three pairs of dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces. For solving the dual integral equations, the displacement jumps across the crack surfaces are directly expanded as a series of Jacobi polynomials. Finally, the effects of the electric permittivity, the magnetic permeability of the air inside the crack, the geometric shape of the rectangular crack and the distance among four rectangular cracks on the stress intensity factors, the electric displacement intensity factors and the magnetic flux intensity factors in magneto-electro-elastic material are presented and the magneto-electro-elastic coupling effects are also discussed.

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.

Analysis of a piezoelectric finite wedge under antiplane concentrated loads

This paper analyzes the antiplane problem of a finite piezoelectric wedge subjected to concentrated loads. The piezoelectric wedge is assumed to be transversely isotropic with the poling direction along the x3 direction. The concentrated loads considered here involve screw dislocations with the Burgers vectors parallel to the poling direction. In addition, a line force and a line charge are applied at the core of the dislocation. Four different boundary conditions on the radial edge and the circular edge are investigated and the concentrated loads can be located in the full domain of the finite wedge. The analytical derivation is based on the complex variable, analytical continuation and the conformal mapping methods. The derived complex potentials show that the stress and electric displacement fields display r1−λ type of singularity near the wedge crack-tip when the wedge angle is larger than π. The obtained solutions then are used to calculate the electric-elastic fields and the crack-tip stress and electric displacement intensity factors. The results are further degenerated to several specific cases and are agreed well with existing ones.

An axisymmetric problem of an embedded mixed-mode crack in a functionally graded magnetoelectroelastic infinite medium

This paper investigates the problem of an axisymmetric penny shaped crack embedded in an infinite functionally graded magneto electro elastic medium. The loading consists of magnetoelectromechanical loads applied on the crack surfaces assumed to be magneto electrically impermeable. The material’s gradient is parallel to the axisymmetric direction and is perpendicular to the crack plane. An anisotropic constitutive law is adopted to model the material behavior. The governing equations are converted analytically using Hankel transform into coupled singular integral equations, which are solved numerically to yield the crack tip stress, electric displacement and magnetic induction intensity factors. A similar problem but with a different crack morphology, that is a plane crack embedded in an infinite functionally graded magneto electro elastic medium, was considered by the authors in a previous work (Rekik et al., 2012) [25]. While the overall solution schemes look similar, the axisymmetric problem resulted in more mathematical complexities and let to different conclusions with respect to the influence of coupling between elastic, electric and magnetic effects. The main focus of this paper is to study the effect of material non-homogeneity on the fields’ intensity factors to understand further the behavior of graded magnetoelectroelastic materials containing penny shaped cracks and to inspect the effect of varying the crack geometry.

Effect of crack face residual surface stress on nanoscale fracture of piezoelectric materials

This paper investigates the problem of a nanoscale crack in piezoelectric nanomaterials by considering the effect of the residual surface stress on the crack surface. The solution of problem is obtained by the singular integral equation technique. The study of the fracture mechanics parameters confirms the fact that nanomaterials are crack-insensitive. It is also found that the effect of surface on the electric displacement intensity factor depends on the crack face electric boundary condition. Moreover, the electromechanical coupling fracture behavior of the piezoelectric materials is influenced by the residual surface stress on the entire crack surface.

The influence of Maxwell stresses on the fracture mechanics of piezoelectric materials

The influence of Maxwell stresses on the generalized 2D fracture mechanics problem of piezoelectric materials under combined mechanical and electric loads at infinity is studied. The electrically semi-permeable crack boundary condition is adopted in this paper. Based on the Stroh’s formalism, explicit and closed-form solutions of electric displacement inside the crack, stress and electric intensity factors are obtained. Numerical results are also given to discuss the effects of Maxwell stresses on the stress and electric displacement intensity factors when the interior of the crack and the surrounding space at infinity are filled with different dielectric medium. It is found that the stress intensity factor increases rapidly with increasing value of the applied electric displacement load for the case of the dielectric constant of the surrounding at infinity is smaller than that inside the crack. The electric displacement intensity factor always increases as the applied electric loads or the applied mechanical loads increase.

Finite size effects on a cracked magnetoelectroelastic layer sandwiched between two elastic layers

The crack problem of a magnetoelectroelastic layer of finite size sandwiched between two elastic layers under anti-plane shear and in-plane electric and magnetic loads is considered for impermeable and permeable crack surface conditions, respectively. Fourier transforms and Fourier sine series are used to reduce the mixed boundary value problem of the crack to dual integral equations, which are expressed in terms of Fredholm integral equations of the second kind. The stress field, electric field and magnetic field near the crack tip are obtained asymptotically, and the corresponding field intensity factors for impermeable and permeable cracks are determined. Numerical results show that the stress intensity factors, the electric displacement intensity factors and the magnetic induction intensity factors are influenced by the electric and magnetic loadings and the material properties. The finite size effects on the fracture behavior of the composite structure are investigated.

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.

The basic solution of a 3-d rectangular permeable crack in a piezoelectric/piezomagnetic composite material

The solution of a 3-D rectangular permeable crack in a piezoelectric/piezomagnetic composite material was investigated by using the generalized Almansi's theorem and the Schmidt method. The problem was formulated through Fourier transform into three pairs of dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces. To solve the dual integral equations, the displacement jumps across the crack surfaces were directly expanded as a series of Jacobi polynomials. Finally, the relations between the electric filed, the magnetic flux field and the stress field near the crack edges were obtained and the effects of the shape of the rectangular crack on the stress, the electric displacement and magnetic flux intensity factors in a piezoelectric/piezomagnetic composite material were analyzed.

T-Shaped Crack in a Piezoelectric Strip Under Thermo-Electro-Mechanical Loadings

The thermo-electro-mechanical fracture problem of a piezoelectric strip containing a T-shaped crack is considered in this study. The problem is solved for a piezoelectric strip that is under thermo-electro-mechanical loadings. The crack faces are supposed to be completely insulated thermally and electrically. By using both the principle of superposition and the Fourier transform techniques, the isothermal electromechanical problem is reduced to a system of singular integral equations, respectively. The outline of the procedure for obtaining the crack face traction is also presented. The singular integral equations are solved numerically. Numerical calculations are carried out, and detailed results are presented to illustrate the influence of the geometric parameters on the stress and electric displacement intensity factors.

Applicability of the crack faces thermoelectric boundary conditions for thermopiezoelectric materials

The applicability and effect of the crack surfaces thermoelectric boundary conditions in thermopiezoelectric fracture mechanics problem are discussed by using the finite thickness notch approach. The stress and electric displacement intensity factors at the notch tips, and thermal flux and electric displacement inside the notch are derived in closed-form. The numerical results are compared with the ideal crack solutions. It is found that the electrically impermeable crack boundary condition assumption is reasonable if the flaw in the material is a notch with finite width, and the thermal conductivity of air or vacuum inside the crack must be considered.

Toughening effect of ferroelectric ceramics induced by domain switching and dislocations

The multi-scale analysis of fracture toughness of ferroelectric ceramics under complicate mechanical–electrical coupling effect is carried out in this paper. The generalized stress intensity factor (SIF) arising from spontaneous strains and polarization transformation in switching domain zones is accurately obtained by using an extended Eshelby theory. Taking BaTiO3 ferroelectric ceramic for example, it is discovered that the crack propagation can be induced by domain switching arising from negative electrical field when the crack surface is parallel to the isotropic plane, and the obtained critical electric displacement intensity factor (EDIF) approximates closely to that obtained by the Green’s function method. Additionally, as pinning dislocations and slip dislocations can strongly influence properties of ferroelectric devices and induce the property degradation, it is necessary to investigate the dislocation toughening effects on fatigue and fracture mechanisms. The results show that the dislocation shielding and anti-shielding effects on mode II SIF, mode I SIF and EDIF are obviously different when a dislocation locates at a position near the crack tip. Through the calculation of the critical applied EDIF for crack propagation by using mechanical energy release rate (MERR) theory, it is discovered that the slip angles obviously influence fracture toughness, and the mode II SIF arising from dislocation has little influence on fracture toughness, however, the mode I SIF and EDIF arising from dislocation have great influences on fracture toughness.

Crack branching in thermopiezoelectric materials

Solutions are presented for an electrically impermeable crack branching out of the crack plane in a thermopiezoelectric medium under thermo-electro-mechanical loads based on Stroh formalism. Explicit Green’s functions for the interaction of a crack and a thermopiezoelectric dislocation (i.e., a thermal dislocation, a mechanical dislocation and an electric dipole located at the same point) are developed. The problem then can be expressed in terms of coupled singular integral equations for the thermopiezoelectric dislocation density functions associated with a branched crack. Some essential fracture mechanics parameters, such as stress and electric displacement intensity factors, and energy release rate at the branched crack tip are obtained. Numerical results are presented for the effect of applied thermal flux loads and electric field on the crack propagation path.

Two Parallel Cracks in Arbitrary Positions of a Piezoelectric Material Strip Under Thermo-Electric Loadings

In this article, thermo-electro-elastic fracture behavior of two parallel cracks in arbitrary positions of a piezoelectric material strip under thermo-electric loadings is considered. The crack faces are assumed to be insulated thermally and electrically. Fourier transform techniques are used to reduce the mixed boundary value problems to two systems of singular integral equations. Numerical calculations are carried out, and detailed results are presented to illustrate the influence of the geometric parameters on the stress and electric displacement intensity factors. The results for the temperature and electro-elastic fields are also included.

Analysis of a subinterface crack in piezoelectric bimaterials with the extended finite element method

In this work, a subinterface crack problem in piezoelectric bimaterials is analyzed by the extended finite element method (XFEM). Associated with the level set method, the XFEM enables us to accurately capture the singularities at the crack-tips. The fracture parameters consisting of the mechanical stress intensity factors and the electrical displacement intensity factor are evaluated by using the asymptotic crack-tip fields derived from the generalized Stroh’s formalism and the interaction integral. Numerical examples for an electrically impermeable subinterface crack are presented and discussed to reveal various aspects including the effects of the crack distance to the interface, the crack inclination, the poling direction, the loading conditions, the basis enrichment functions, the enrichment strategies, the domain of the J-integral computation, etc. on the field intensity factors. Convergence study in the energy norm and in the intensity factors is additionally presented. To assess the accuracy of the proposed approach, the results obtained by the XFEM are compared with the analytical reference solutions available in the literature and excellent agreements are found.

Dynamic response of planar cracks in an infinitely long piezoelectric strip

The problem of determining the electro-elastic fields around an array of parallel planar cracks in an infinitely long piezoelectric strip is considered. The cracks, acted upon by dynamic loads, are either electrically impermeable or permeable. A semi-analytic method based on the theory of exponential Fourier transformation is presented for solving the problem in the Laplace transform domain. The Laplace transforms of the jumps in the displacements and electric potential across opposite crack faces are determined by solving a system of hypersingular integral equations. Once these displacement and electric potential jumps are obtained, the displacements and electric potential and other physical quantities of interest, such as the crack tip stress and electric displacement intensity factors, can be recovered with the help of a suitable algorithm for inverting Laplace transforms. The stress and electric displacement intensity factors are computed for some specific cases of the problem.

Analysis of generalized dynamic intensity factors of cracked magnetoelectroelastic solids by X-FEM

An investigation of the generalized dynamic intensity factors (GDIFs) of cracked homogeneous and linear magnetoelectroelastic (MEE) solids using the extended finite element method (X-FEM) is presented. Stationary straight and curved cracks in two-dimensional (2D) MEE solids with impermeable electromagnetic crack-face boundary conditions under coupled electro-magneto-mechanical impact loads are investigated. The effects of various aspects including mesh sensitivity; combined dynamic impact loads; time-step size; material polarization directions and interaction cracks on the GDIFs are numerically studied. A dynamic X-FEM computer code, integrated with Newmark time integration scheme and the level set method to accurately capture the crack geometry, is developed. The eight-fold enrichment functions particularly suitable for cracks in MEE materials are adopted to appropriately describe the singular fields at the crack-tips. To assess the dynamic stress, electric displacement and magnetic induction intensity factors accurately and efficiently, domain-form of the integration integral taking the inertial effect into account in conjunction with the asymptotic near crack-tip fields in MEE materials is presented. Several numerical examples are shown to confirm the accuracy of the proposed approach, and the numerical results are thus investigated, compared and discussed in detail.

Analysis of cracks and notches in piezoelectric composites using scaled boundary finite element method

This paper presents a semi-analytical technique to analyze two-dimensional cracks and notches inside piezoelectric composites. The proposed technique is based on the so-called scaled boundary finite element method. The use of high-order spectral elements leads to high accuracy and fast convergence. By using the proposed technique, the stress and electric displacement intensity factors KI, KII and KIV are evaluated directly from the solutions of the singular stress and electric displacement fields. Meanwhile, the orders of singularity are also investigated. The numerical studies are conducted to demonstrate the simplicity of the proposed technique in handling fracture problems of piezoelectric composites. The present results also show the high accuracy of the proposed technique can achieve.

Transient electromechanical cracking of finite piezoelectric bodies

This article studies the problem of a finite crack in a finite piezoelectric medium subjected to transient electric and mechanical loads. Unlike past studies that mainly focused on the infinitesimal cracks, the current article considers the effect of the medium boundaries on the transient fracture behavior of piezoelectric materials. A system of singular integral equations is formulated in terms of the crack surface displacement and electric potential. Effects of crack length and medium size on the crack-tip field intensity factors are investigated in detail. Numerical results for the time-dependent stress and electric displacement intensity factors are obtained. Electrically impermeable crack assumption, electrically permeable crack assumption, and notch of finite height are investigated. Some new conclusions are drawn.

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.