Application of Hypersingular Integral Equation Method to a Three-Dimensional Crack in Piezoelectric Materials (original) (raw)
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Analysis of cracks in 3D piezoelectric media with various electrical boundary conditions
International Journal of Fracture, 2015
A weakly singular, symmetric Galerkin boundary element method capable of solving problems of isolated cracks in three-dimensional, linear anisotropic piezoelectric, infinite media with various types of crack-face boundary conditions including impermeable, permeable, semi-permeable, and the energetically consistent boundary condition introduced by Landis (Int J Solids Struct 41:6291-6315, 2004) is established. The key governing boundary integral equation used in the formulation possesses several crucial features including its desirable symmetric weak-form, weakly singular nature, and ability to treat general material anisotropy, arbitrary crack configurations and any type of boundary condition on the crack surface. The positive consequence of utilizing the singularityreduced integral equations in the modeling, is that all involved singular integrals can be interpreted in the sense of Riemann and their validity requires only continuous crack-face data allowing C 0-interpolation functions to be employed everywhere in the numerical dis
European Journal of Mechanics A-solids, 2010
The problem of an arbitrary number of arbitrarily oriented straight cracks in an infinitely long piezoelectric strip is considered here. The cracks are acted by suitably prescribed internal tractions and are assumed to be either electrically impermeable or permeable. A Green's function which satisfies the conditions on the parallel edges of the strip is derived using a Fourier transform technique and applied to formulate the electroelastic crack problem in terms of a system of hypersingular integral equations. Once the hypersingular integral equations are solved, quantities of practical interest, such as the crack tip stress and electric displacement intensity factors, can be easily computed. Some specific cases of the problem are examined.
A mixed electric boundary value problem for a two-dimensional piezoelectric crack
International Journal of Solids and Structures, 2003
In this paper, a mixed electric boundary value problem for a two-dimensional piezoelectric crack problem is presented, in the sense that the crack face is partly conducting and partly impermeable. By the analytical continuation method, the unknown electric charge distributions on the upper and lower conducting crack faces are reduced to two decoupled singular integral equations and then these two equations are converted into algebraic equations to find the full field solution. Though the results suggest that the stress intensity factors at the crack tip are identical to those of conventional piezoelectric materials, but the electric field and electric displacement are related to the electric boundary conditions on the crack faces. The electric field and electric displacement are singular not only at crack tips but also at the junctures between the impermeable part and conducting parts. Numerical results for the variations of the electric field, electric displacement field and J-integral with respect to the normalized impermeable crack length are shown. Some discussions for the energy release rate and the J-integral are made.
Nonsingular term effect on the fracture quantities of a crack in a piezoelectric medium
Engineering Fracture Mechanics, 2008
The static problem of a crack in a piezoelectric plate subjected to biaxial loading at infinity is analyzed. The aim of this paper is to estimate the influence of non-singular terms originated by the load biaxiality on the stress fields and on the elastic and electric displacements in the vicinity of the crack tip. An analytical method for seeking the electroelastic solution is proposed. The novel procedure involves a transformation of similarity induced by the fundamental matrix that enables to express the equations governing the problem in terms of complex potentials. The application of the boundary conditions leads then to the formulation of Hilbert problems whose solutions allow to obtain the generalized stress and displacement components. Numerical results and graphs are presented and discussed for various loading conditions. The non-singular solution is compared to the asymptotic one, generally considered in the literature when analyzing fracture problems. In particular, it is shown that the direction of incipient crack extension, sought through the maximum circumferential stress criterion, can be seen to deviate from the crack axis as the collinear load increases, although geometry and applied load are symmetric.
Journal of Applied Mechanics, 2009
In this paper, the M -integral is extended for calculating intensity factors for cracked piezoelectric ceramics using the exact boundary conditions on the crack faces. The poling direction is taken at an angle to the crack faces within the plane. Since an analytical solution exists, the problem of a finite length crack in an infinite body subjected to crack face traction and electric flux density is examined. In this case, poling is taken parallel to the crack faces. Numerical difficulties resulting from multiplication of large and small numbers were treated by normalizing the variables. This problem was solved with the M -integral and displacement-potential extrapolation methods. With this example, the superiority of the conservative integral is observed. The values for the intensity factor obtained with the M -integral are found to be more accurate than those found by means of the extrapolation method. In addition, a crack parallel to the poling direction in a four-point bend specimen subjected to both an applied load and an electric field was analyzed and different electric permittivity values in the crack gap were assumed. It is seen that the electric permittivity greatly influences the stress intensity factor K II and the electric flux density intensity factor K IV . The absolute value of these intensity factors increases with an increase in the value of the electric permittivity in the crack. The influence of the permittivity on K I is rather small.
Engineering Analysis with Boundary Elements, 2014
The polarization saturation (PS) model and the dielectric breakdown (DB) model are both used, under the electrically impermeable crack assumption, to analyze penny-shaped cracks in the isotropic plane of three-dimensional (3D) infinite piezoelectric solids. Using the extended displacement discontinuity integral equation method, we obtained analytical solutions for the size of the electric yielding zone, the extended displacement discontinuities, the extended field intensity factor and the J-integral. Integrating the Green function for the point extended displacement discontinuity provided constant element fundamental solutions. These solutions correspond to an annular crack element applied with uniformly distributed extended displacement discontinuities in the transversely isotropic plane of a 3D piezoelectric medium. Using the obtained Green functions, the extended displacement discontinuity boundary element method was developed to analyze the PS model and DB model for penny-shaped cracks. The numerical method was validated by the analytical solution. Both the analytical results and numerical results show that the PS and the DB models give equivalent solutions for nonlinear fracture analysis of 3D piezoelectric materials, even though they are based on two physically different grounds.
Time–Harmonic Behaviour of Cracked Piezoelectric Solid by Boundary Integral Equation Method
Journal of Theoretical and Applied Mechanics, 2014
Anti-plane cracked functionally graded finite piezoelectric solid under time-harmonic elecro-mechanical load is studied by a nonhypersingular traction boundary integral equation method (BIEM). Exponentially varying material properties are considered. Numerical solutions are obtained by using Mathematica. The dependance of the intensity factors (IF) -mechanical stress intensity factor (SIF) and electrical field intensity factor (FIF) on the inhomogeneous material parameters, on the type and frequency of the dynamic load and on the crack position are analyzed by numerical illustrative examples.
On a plane crack in piezoelectric solids
International Journal of Solids and Structures, 2001
A new analytical solution for a piezoelectric plane with an elliptical void is derived by removing the commonly held assumptions that the void boundary is impermeable and a void axis is perpendicular to the poling direction. The approach of Lekhnitskii's complex potential functions is used in the derivation. Applicability of the common practice of reducing a void solution to a crack solution is examined. It is shown that a recently reported solution for exact electric boundary conditions is actually the well known solution for a permeable crack. A uni®ed formulation for plane cracks containing air or vacuum is then developed to account for dierent electric boundary conditions. Crack closure is taken into consideration in the analysis. The in¯uence of electric boundary conditions and crack orientation on fracture parameters is discussed.
Analysis of interface crack in piezoelectric materials using extended finite element method
Mechanics of Advanced Materials and Structures, 2018
In the present work, static impermeable crack at interface of piezoelectric materials is analyzed using an extended finite element method. An edge crack and center crack at the interface of PZT-5H and PZT-4 for k-class singularity of piezoelectric materials have been analyzed under mechanical and electromechanical loadings. Enrichment functions for k-class are used for determining the asymptotic fields near the crack tip. K I and K IV are determined using the interaction integral. The effect on fracture parameters of major crack due to minor cracks, holes, and combination of minor cracks and holes has been investigated. The variation in intensity factors with elastic field and electric displacement has been analyzed.
Theoretical and Applied Fracture Mechanics, 2007
Using Green's functions, the extended general displacement solutions of a three-dimensional crack problem in anisotropic electro-magneto-elastic (EME) bimaterials under extended loads are analyzed by the boundary element method. Then, the crack problem is reduced to solving a set of hypersingular integral equations (HIE) coupled with boundary integral equations. The singularity of the extended displacement discontinuities around the crack front terminating at the interface is analyzed by the main-part analysis method of HIE, and the exact analytical solutions of the extended singular stresses and extended stress intensity factors (SIFs) near the crack front in anisotropic EME bimaterials are given. Also, the numerical method of the HIE for a rectangular crack subjected to extended loads is put forward with the extended crack opening dislocation approximated by the product of basic density functions and polynomials. At last, numerical solutions of the extended SIFs of some examples are obtained.