The Interaction between a Piezoelectric Screw Dislocation Dipole and an Elliptic Blunt Crack in Elliptical Inhomogeneity

The interaction problem of a piezoelectric screw dislocation dipole with a confocal elliptic blunt crack in elliptical inhomogeneity subjected to remote anti-plane stress field and in-plane electric field is investigated by using the complex method of elasticity. The exact closed-form solutions of a series of quantities, such as singularity stress field, image force and image torque acting on the center of screw dislocation dipole, stress intensity factor and electric displacement intensity factor of crack tip, energy release rate, and generalized strain energy density are obtained. Then the influence laws of remote load, the dip angle of dislocation dipole, the size of blunt crack, and the material constants on the quantities are analyzed. The numerical results show that the image force, image torque, stress intensity factor, and electric displacement intensity factor make periodic variation as the dip angle of dislocation dipole; the energy release rate of crack tip is negative when subjected to pure electric field, however, it can be positive or negative when subjected to the combined action of mechanical field and electric field; the sharp crack is not easy to expand in some combined action of mechanical field and electric field.     

Stress intensity factors for cracks emanating from a triangular or square hole in an infinite plate by boundary elements

This paper concerns stress intensity factors of cracks emanating from a triangular or square hole in an infinite plate subjected to internal pressure calculated by means of a boundary element method, which consists of constant displacement discontinuity element presented by Crouch and Starfied and crack tip displacement discontinuity elements proposed by the author. In the boundary element implementation the left or the right crack tip displacement discontinuity element is placed locally at corresponding left or right crack tip on top of the constant displacement discontinuity elements that cover the entire crack surface and the other boundaries. Numerical examples are included to show that the method is very efficient and accurate for calculating stress intensity factors of plane elasticity crack problems. Specifically, the numerical results of stress intensity factors of cracks emanating from a triangular or square hole in an infinite plate subjected to internal pressure are given.     

The vibrational and buckling behaviors of piezoelectric nanobeams with surface effects

In this work, the influence of surface effects, including residual surface stress, surface elasticity and surface piezoelectricity, on the vibrational and buckling behaviors of piezoelectric nanobeams is investigated by using the Euler-Bernoulli beam theory. The surface effects are incorporated by applying the surface piezoelectricity model and the generalized Young-Laplace equations. The results demonstrate that surface effects play a significant role in predicting these behaviors. It is found that the influence of the residual surface stress and the surface piezoelectricity on the resonant frequencies and the critical electric potential for buckling is more prominent than the surface elasticity. The nanobeam boundary conditions are also found to influence the surface effects on these parameters. This study also shows that the resonant frequencies can be tuned by adjusting the applied electrical load. The present study is envisaged to provide useful insights for the design and applications of piezoelectric-beam-based nanodevices.     

Numerical analysis for piezoelectric crack under varied boundary conditions by optimized hybrid element method

In this article, a piezoelectric hybrid element is presented and optimized by penalty equilibrium approach, and special crack surface element is suggested for exactly implementing the boundary conditions on crack surface. An iteration technique is used to treat one of the electric boundary conditions. Then, a piezoelectric material with crack is numerically studied by the optimized hybrid element method, and the results are compared with the analytical solutions. The stress and the electrical displacement fields with different crack surface conditions are studied, and the influence to those fields arisen by the far field mechanical and electric loading is also studied.     

Numerical method of crack analysis in 2D finite magnetoelectroelastic media

The present paper extends the hybrid extended displacement discontinuity fundamental solution method (HEDD-FSM) (Eng Anal Bound Elem 33:592–600, 2009) to analysis of cracks in 2D finite magnetoelectroelastic media. The solution of the crack is expressed approximately by a linear combination of fundamental solutions of the governing equations, which includes the extended point force fundamental solutions with sources placed at chosen points outside the domain of the problem under consideration, and the extended Crouch fundamental solutions with extended displacement discontinuities placed on the crack. The coefficients of the fundamental solutions are determined by letting the approximated solution satisfy the prescribed boundary conditions on the boundary of the domain and on the crack face. The Crouch fundamental solution for a parabolic element at the crack tip is derived to model the square root variations of near tip fields. The extended stress intensity factors are calculated under different electric and magnetic boundary conditions.     

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.     

An effective method of stress intensity factor calculation for cracks emanating from a triangular or square hole under biaxial loads

This paper is concerned with stress intensity factors for cracks emanating from a triangular or square hole under biaxial loads by means of a new boundary element method. The boundary element method consists of the constant displacement discontinuity element presented by Crouch and Starfied and the crack‐tip displacement discontinuity elements proposed by the author. In the boundary element implementation, the left or the right crack‐tip displacement discontinuity element is placed locally at the corresponding left or right crack tip on top of the constant displacement discontinuity elements that cover the entire crack surface and the other boundaries. The method is called a Hybrid Displacement Discontinuity Method (HDDM). Numerical examples are included to show that the method is very efficient and accurate for calculating stress intensity factors for plane elastic crack problems. In addition, the present numerical results can reveal the effect of the biaxial loads on stress intensity factors.     

压电材料中反平面渗透裂纹的解

用复变函数的保角映射法,采用可渗透边界条件,研究了含裂纹的无限大压电材料在平面内电场和反平面荷载作用下的耦合场,得到了精确的解和场强度因子以及能量释放率。结果表明,电场强度在裂尖没有奇异性,应变、应力、电位移具有1/2阶的奇异性,能量释放率总是正的。     

Investigation of the interaction of two collinear cracks in anisotropic elasticity materials by means of the nonlocal theory

In this paper, the interaction of two collinear cracks in anisotropic elasticity materials subjected to an anti-plane shear loading is investigated by means of the nonlocal theory. By use of the Fourier transform, the problem can be solved with the help of a pair of triple integral equations, in which the unknown variable is the displacement on the crack surface. To solve the triple integral equations, the displacement on the crack surface is expanded in a series of Jacobi polynomials. Unlike the classical elasticity solutions, it is found that no stress singularity is present at the crack tip. The nonlocal elastic solutions yield a finite hoop stress at the crack tip, thus allowing us to use the maximum stress hypothesis as a fracture criterion. The magnitude of the finite stress field depends on the crack length, the distance between two cracks and the lattice parameter of materials.     

A numerical analysis of cracks emanating from a square hole in a rectangular plate under biaxial loads

This note concerns with stress intensity factors of cracks emanating from a square hole in rectangular plate under biaxial loads by means of the boundary element method which consists of the non-singular displacement discontinuity element presented by Crouch and Starfied and the crack tip displacement discontinuity elements proposed by the author. In the boundary element implementation the left or the right crack tip displacement discontinuity element is placed locally at corresponding left or right crack tip on top of the constant displacement discontinuity elements that cover the entire crack surface and the other boundary. The present numerical results illustrate that the present approach is very effective and accurate for calculating stress intensity factors of complicated cracks in a finite plate and can reveal the effect of the biaxial load and the cracked body geometry on stress intensity factors.     

Three-dimensional BEM for piezoelectric fracture analysis

A boundary element approach with quadratic isoparametric elements, quarter-point elements and singular quarter-point elements for three-dimensional crack problems in piezoelectric solids under mechanical and electrical loading conditions, is presented in this paper for the first time. The procedure is based on Deeg's fundamental solution for anisotropic piezoelectric materials, and the classical extended displacement boundary integral equation. Stress and electric displacement intensity factors are directly evaluated as system unknowns, and also as functions of the computed nodal displacements and electric potentials at crack faces. Special attention is paid to the fundamental solution evaluation. Several three-dimensional crack problems in transversely isotropic bodies under mechanical and electrical loading conditions are analysed. Numerical solutions computed for prismatic cracked 3D plate problems with a plane strain behaviour are in very good agreement with their corresponding 2D BE solutions. Results for a penny shape crack in a piezoelectric cylinder are presented for the first time. The proposed approach is shown to be a simple, robust and useful tool for stress and electric displacement intensity factors evaluation in piezoelectric media.     

Effect of electric boundary conditions on crack energy density and its derivatives for piezoelectric material

This paper deals with crack energy density (hereafter CED) as a possible fracture parameter in piezoelectricity under arbitrary electric boundary conditions on a notch-like crack surface. The definitions of CED and its derivatives are given first under exact boundary condition. Next, their path independent integrals are also derived and their approximate expressions are discussed under some restrictions on the crack surfaces. It is found that electrical terms along the notch-like crack surface do not vanish unlike in the case of impermeable crack. Then, we introduce evaluation methods of CED, and, with the help of the results of finite element analyses (FEA), we closely examine how electric boundary conditions along the notch surface and initial notch width influence CED and its derivatives. It is shown from the FEA results that because of the difficulties of computing path integral terms along the notch-like crack tip in the path independent expressions, the evaluation by the definitions of CED and its derivatives is preferable and more convenient than the evaluation of their path independent expressions. It is also found that all the parameters are significantly affected by both permittivity inside the electric inclusion and root radius of the notch. Finally, the possibility of mechanical CED as a governing fracture parameter is discussed.     

Electroelastic analysis of an electrically dielectric Griffith crack in a piezoelectric layer

The fracture analysis of an electrically dielectric Griffith crack embedded in a piezoelectric layer is made under in-plane electro-mechanical loadings. To simulate an opening crack full of a dielectric interior, the energetically consistent crack-face boundary conditions are utilized. Applying the Fourier transform technique, the boundary-value problem is reduced to solving two coupling singular integral equations. The intensity factors of stress, electric displacement, crack opening displacement (COD) and electric potential are further determined by the Lobatto-Chebyshev collocation method. The variations of the electric displacement at the crack surfaces are investigated by using the energetically consistent and semi-permeable boundary conditions respectively. The observations show that the electric displacement inside the crack is decreasing with an increase of the ratio between the crack length and piezoelectric layer width. Numerical computations are further carried out to compare the intensity factors of stress and electric potential, and the energy release rate using the energetically consistent boundary conditions with those using the semi-permeable boundary conditions. The obtained results reveal that the stress induced by a dielectric inside a crack has great effects on the stress intensity factor and energy release rate, but little influence on the electric potential difference across the crack.     

Surface effects on the near-tip stress fields of a mode-II crack

On the physical nature, most crack tips are not ideally sharp but have a small curvature radius. Both surface energy and crack-root curvature affect the stress and displacement fields in the vicinity of the crack tip. In the present paper, a numerical method, which incorporates the effect of surface elasticity into the finite element method, is employed to study the surface effects on the mode-II crack tip fields. It is found that when the curvature radius of the crack root decreases to micro-/nanometers, surface elasticity has a significant influence on the stresses near the crack tip. For a mode-II crack, surface effects alter both the magnitude and position of the maximum stresses, as is different from a mode-I crack, in which case only the stress magnitude is influence by surface stresses.     

Near-tip fields and intensity factors for interfacial cracks in dissimilar anisotropic piezoelectric media

A complete form of stress and electric displacement fields in the vicinity of the tip of an interfacial crack, between two dissimilar anisotropic piezoelectric media, is derived by using the complex function theory. New definitions of real-valued stress and electric displacement intensity factors for the interfacial crack are proposed. These definitions are extensions of those for cracks in homogeneous piezoelectric media. Closed form solutions of the stress and electric displacement intensity factors for a semi-infinite crack as well as for a finite crack at the interface between two dissimilar piezoelectric media are also obtained by using the mutual integral.     

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.     

The Coulombic traction on the surfaces of an interface crack in dielectric/piezoelectric or metal/piezoelectric bimaterials

This paper deals with the Coulombic traction usually neglected, but inherently acting, on the surfaces of an interface crack in dielectric/piezoelectric or metal/piezoelectric bimaterials. The dielectric material phase is treated as a special kind of piezoelectric material with a little piezoelectricity, whereas the metal phase is treated as another special kind of piezoelectric material with an extremely large permittivity and an extremely small piezoelectricity. The permittivity of the medium inside the crack gap is accounted for either. The normal electric displacement component and the Coulombic traction on the crack surfaces are unknown, and are determined from a cubic equation deduced from the extended Stroh formula. Numerical results for the Coulombic traction in both kinds of bimaterials reveal that in most cases its magnitude is remarkable and cannot be entirely neglected when the applied electric field is higher. It is concluded that in most cases the Coulombic traction yields significant influence on the effective stress intensity factor at the crack tip and may influence the fracture behavior in such kinds of bimaterials. As compared to homogenous piezoelectric materials, the metal phase always decreases the Coulombic traction, whereas the dielectric phase decreases it under the negative electric field and increases it under the positive electric field. In all cases, BaTiO₃ always yields a much larger Coulombic traction than PZT-4.     

压电-压磁板条中反平面裂纹的电-磁-弹性分析

基于线性电磁弹性理论,获得了压电-压磁板条中反平面裂纹尖端附近的奇异应力、电场和磁场。假设裂纹位于和板条边界平行的中心位置,并且裂纹是电磁渗透型的。利用Fourier变换,将裂纹面的混合边值问题化为对偶积分方程,即而归结为第二类Fredholm积分方程。通过渐近分析,得到了裂纹尖端附近应力、应变、电位移、电场、磁场和磁感的封闭表达式。结果表明,对于电磁渗透裂纹,电场强度因子和磁场强度因子总为0;板条的宽度对应力强度因子有显著的影响;能量释放率总为正值。     

The scattering of harmonic elastic anti-plane shear waves by a Griffith crack in a piezoelectric material plane by using the non-local theory

In this paper, the dynamic behavior of a Griffith crack in a piezoelectric material plane under anti-plane shear waves is investigated by using the non-local theory for impermeable crack face conditions. For overcoming the mathematical difficulties, a one-dimensional non-local kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress and the electric displacement near the crack tips. By using the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations. These equations are solved using the Schmidt method. Contrary to the classical elasticity solution, it is found that no stress and electric displacement singularity is present near the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress near the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The finite hoop stress at the crack tip depends on the crack length, the circular frequency of incident wave and the lattice parameter. For comparison results between the non-local theory and the local theory for this problem, the same problem in the piezoelectric materials is also solved by using local theory.     

Investigation of anti-plane shear behavior of a Griffith permeable crack in functionally graded piezoelectric materials by use of the non-local theory

In this paper, the non-local theory of elasticity is applied to obtain the behavior of a Griffith crack in functionally graded piezoelectric materials under the anti-plane shear loading for the permeable electric boundary conditions. To make the analysis tractable, it is assumed that the material properties vary exponentially with coordinate vertical to the crack. By means of the Fourier transform, the problem can be solved with the help of a pair of dual-integral equations that the unknown variable is the jump of the displacement across the crack surfaces. These equations are solved by use of the Schmidt method. Numerical examples are provided. Unlike the classical elasticity solutions, it is found that no stress and electric displacement singularities are present near the crack tips. The non-local elastic solutions yield a finite hoop stress at the crack tips, thus allows us to using the maximum stress as a fracture criterion. The finite hoop stresses at the crack tips depend on the crack length, the functionally graded parameter and the lattice parameter of the materials, respectively.