Analysis of cracked piezoelectric solids by a mixed three-dimensional BE approach

This paper has two main objectives in relation to the analysis of three-dimensional crack problems in piezoelectric solids. The first one is to present the formulation, effective implementation and numerical treatment of a mixed boundary element technique for the study of this type of problems. The numerical procedure is based on the use of extended displacement and extended traction integral equations for external and crack boundaries, respectively. The boundary element formulation is presented with particular emphasis on numerical aspects related to singular kernels regularization and evaluation of boundary integrals. Quadratic boundary elements and quarter-point boundary elements are implemented in a computer code. By using these elements, electric and stress intensity factors are directly computed from nodal values at quarter-point elements. The second purpose is to study several realistic piezoelectric crack problems for the first time. Unbounded and bounded cracked piezoelectric three-dimensional (3D) solids with different geometries are studied. Results presented in this paper can be used as a reference for future research. Prior to the analysis of problems whose solution was previously unknown, the technique is validated by solving some simple problems with known analytical or numerical solution. Then, more realistic crack problems of engineering interest have been analysed for the first time. In all cases, results for the solid deformed shape, the crack opening displacements and the extended stress intensity factor components, are shown.     

Boundary integral equation method for conductive cracks in two and three-dimensional transversely isotropic piezoelectric media

Using the fundamental solutions and the Somigliana identity of piezoelectric medium, the boundary integral equations are obtained for a conductive planar crack of arbitrary shape in three-dimensional transversely isotropic piezoelectric medium. The singular behaviors near the crack edge are studied by boundary integral equation approach, and the intensity factors are derived in terms of the displacement discontinuity and the electric displacement boundary value sum near the crack edge on crack faces. The boundary integral equations for two dimensional crack problems are deduced as a special case of infinite strip planar crack. Based on the analogy of the obtained boundary integral equations and those for cracks in conventional isotropic elastic material and for contact problem of half-space under the action of a rigid punch, an analysis method is proposed. As an example, the solution to conductive Griffith crack is derived.     

Three-dimensional fracture analysis in transversely isotropic solids

In this paper a boundary element formulation for three-dimensional crack problems in transversely isotropic bodies is presented. Quarter-point and singular quarter-point elements are implemented in a quadratic isoparametric element context. The point load fundamental solution for transversely isotropic media is implemented. Numerical solutions to several three-dimensional crack problems are obtained. The accuracy and robustness of the present approach for the analysis of fracture mechanics problems in transversely isotropic bodies are shown by comparison of some of the results obtained with existing analytical solutions. The approach is shown to be a simple and useful tool for the evaluation of stress intensity factors in transversely isotropic media.     

Hybrid extended displacement discontinuity-charge simulation method for analysis of cracks in 2D piezoelectric media

The extended displacement discontinuity method (EDDM) and the charge simulation method (CSM) are combined to develop an efficient approach for analysis of cracks in two-dimensional piezoelectric media. In the proposed hybrid EDD–CSM, the solution for an electrically impermeable crack is approximately expressed by a linear combination of fundamental solutions of the governing equations, which includes the extended point force fundamental solutions with the sources placed at chosen points outside the domain of the problem under consideration and the extended Crouch fundamental solutions with the extended displacement discontinuities placed on the crack. The coefficients of the fundamental solutions are determined by letting the approximated solution satisfy the conditions on the boundary of the domain and on the crack face. Furthermore, the hybrid EDD–CSM is applied to solve the problems of cracks under electrically permeable condition, as well as under semi-permeable conditions by using an iterative approach. Two important crack problems in fracture mechanics, the center cracks and the edge cracks in piezoelectric strips, are analyzed by the proposed method. The stress intensity factor and the electric displacement intensity factor are calculated. Meanwhile the effects of strip size and the electric boundary conditions on these intensity factors are studied.     

含界面边裂纹压电材料反平面问题的应力强度因子

研究了含界面边裂纹的不同压电介质组成的复合材料在反平面荷载和平面内电场作用下的电弹场,得到了级数形式的基本解和应力强度因子,最后用边界配置法求解了应力强度因子。结果表明,在外加剪切荷载的作用下,应力强度因子与外加电场无关。     

Fracture mechanics weight functions in three dimensions: Subtraction of fundamental fields

A new procedure is presented for the determination of the fracture mechanics weight functions that are required for the evaluation of stress intensity factors in cracked solids. The procedure can be used with a standard three-dimensional boundary element code. The weight functions are proportional to the displacements on the boundary of the solid when the only loading is a pair of self-equilibrated point forces at the crack front. In previous work, the highly singular crack-tip fields that this loading produces have been modelled by replacing the crack front by a cylindrical cavity with appropriate displacement boundary conditions on the cavity walls. It is shown here that results are dependent on the cavity radius and that convergence of the results cannot be guaranteed. An alternative procedure, based on the substraction of fundamental fields (SFF), is demonstrated herein. The high-order singularities are removed from the field before the reduced problem is solved numerically using a standard boundary element method. Since the reduced problem is equivalent to an unloaded crack in a solìd subjected to boundary tractions, the usual quarter-point displacement elements and quarter-point traction singular elements can be used to improve the accuracy. Weight functions, so obtained, are used to evaluate stress intensity factors as a function of position on the crack front for a straight-fronted crack in a rectangular bar subjected to various loadings. Both edge and central cracks are considered and the validity of the technique is demonstrated by comparing the results with previously published values.     

Screw Dislocation Interacting with an Interfacial Edge Crack between Two Bonded Dissimilar Piezoelectric Wedges

This letter is concerned with an interfacial edge crack in a piezoelectric bimaterial wedge interacting with a screw dislocation under antiplane mechanical and in-plane electric loading. In addition to a discontinuous electric potential across the slip plane, the dislocation is subjected to a line-force and a line-charge at the tip. The out-of-plane displacement and electric potentials are obtained in closed-form based on conformal mapping technique and the solution for the screw dislocation in an infinite piezoelectric bimaterial with a semi-infinite interfacial crack. The intensity factors (IFs) and energy release rate (EER) are obtained explicitly. These solutions can be used as a base for constructing solutions for arbitrary coupled antiplane mechanical and in-plane electrical loadings.     

Two-dimensional time-harmonic BEM for cracked anisotropic solids

A mixed time-harmonic boundary element procedure for the analysis of two-dimensional dynamic problems in cracked solids of general anisotropy is presented. To the author's knowledge, no previous BE approach for time-harmonic two-dimensional crack problems in anisotropic solids exists. In the present work, the fundamental solution is split into the static singular part plus dynamic regular terms. Hypersingular integrals associated to the singular part in the traction boundary integral equation are transformed, by means of a simple change of variable, into regular ones plus very simple singular integrals with known analytical solution. Subsequently, only regular (frequency dependent) terms have to be added to the regularized static fundamental solution in order to solve the dynamic problem. The generality of this procedure permits the use of general straight or curved quadratic boundary elements. In particular, discontinuous quarter-point elements are used to represent the crack-tip behavior. Stress intensity factors are accurately computed from the nodal crack opening displacements at discontinuous quarter-point elements. The efficiency and robustness of the present time-harmonic BEM are verified numerically by several test examples. Results are also obtained for more complex configurations, not previously studied in the literature. They include curved crack geometry.     

Dynamic crack problems in three-dimensional transversely isotropic solids

In this paper, a general boundary element approach for three-dimensional dynamic crack problems in transversely isotropic bodies is presented for the first time. Quarter-point and singular quarter-point elements are implemented in a quadratic isoparametric element context. The procedure is based on the subdomain technique, the displacement integral representation for elastodynamic problems and the expressions of the time-harmonic point load fundamental solution for transversely isotropic media. Numerical results corresponding to cracks under the effects of impinging waves are presented. The accuracy of the present approach for the analysis of dynamic fracture mechanics problems in transversely isotropic solids is shown by comparison of the obtained results with existing solutions.     

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.     

Two collinear unequal cracks in a poled piezoelectric plane: Mode I case solved by a new approach of real fundamental solutions

The purpose of the present work is to study the problem of two collinear unequal cracks in a piezoelectric plane under mode I electromechanical loadings via a new approach. For the first time, real fundamental solutions are derived for in-plane piezoelectric governing equations. The cracks are simulated by continuously distributed generalized dislocations and Cauchy singular integral equations are established from the solution of a generalized point dislocation. Both the theorectical derivation and numerical computations are validated by the exact solution in a special case. Parametric studies are conducted to reveal the effects of crack space, crack length, electric loading and remanent electric displacement on energy release rate. It is found that negative electric displacement loading can decrease both the total energy release rate (TERR) and the mechanical strain energy release rate (MSERR), implying that it has a shielding effect on cracks definitely. Positive electric displacement loading can enhance MSERR, but meanwhile it can enhance or reduce TERR depending on the magnitude of the electric loading factor. The effect of a remanent electric displacement along the poling direction is equivalent to that of a positive electric field loading and should be considered in engineering design.     

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.     

Time-domain BEM for transient interfacial crack problems in anisotropic piezoelectric bi-materials

A time-domain boundary element method (BEM) together with the sub-domain technique is applied to study transient response of interfacial cracks in piecewise homogeneous, anisotropic and linear piezoelectric bi-materials under electrical and mechanical impacts. The present time-domain BEM uses a quadrature formula for the temporal discretization to approximate the convolution integrals and a collocation method for the spatial discretization. Quadratic quarter-point elements are implemented at the tips of the interface cracks. To determine the real or complex dynamic stress intensity factors and the dynamic electrical displacement intensity factor of the interfacial cracks, an explicit extrapolating formula in a typical state of the crack plane perpendicular to the poling direction is presented in this paper. Numerical examples are presented; and the effects of the load combination and material combination on dynamic intensity factors and dynamic energy release rate are discussed.     

Time-domain BEM for three-dimensional fracture mechanics

The present paper deals with time-domain analysis of three-dimensional transient dynamic crack problems. The time-domain formulation of the boundary element method for 3-D elastodynamic problems is used. Quarter-point and singular quarter-point elements represent displacements and tractions, respectively, near the crack front. Special attention is paid to integration and algorithms to preserve stability. Cracks in finite and unbounded regions under single and mixed mode dynamic loading conditions are studied. To the authors’ knowledge, no previous BE approach for 3-D elastodynamic crack problems based on the time-domain displacement representation exists.     

压电材料中渗透性周期共线反平面裂纹问题

本文利用复变函数方法,借助于Riemann-Schwarz延拓技术和保形映照方法,研究了渗透性边界条件下周期共线反平面裂纹问题,获得了解的表达式,得到了力学和电学强度因子。结果表明在裂纹尖端应力和电位移的奇异性都与远场作用的应力载荷和裂纹长度有关,其中应力的奇异性与材料无关,电位移的奇异性则与材料有关,电载荷对裂尖的奇异性没有影响。最后,运用数值算例,给出周期裂纹间的干涉效应和裂纹的尺度效应。     

Transient response of an insulating crack near to the interface between two piezoelectric half-planes under electromechanical impacts by BEM

A time-domain boundary element method (BEM) together with the sub-domain technique is applied to study transient response of an insulating crack near to the interface between two anisotropic piezoelectric half-planes under electromechanical impacts. The present time-domain BEM uses a quadrature formula for the temporal discretization to approximate the convolution integrals and a collocation method for the spatial discretization. Quadratic quarter-point elements are implemented at the crack tip. A displacement extrapolation technique is used to determine the dynamic stress intensity factors (DSIFs) and the dynamic electrical displacement intensity factor. Numerical examples are presented to show the effects of load combination, geometric configuration and material combination on dynamic intensity factors and dynamic energy release rate.     

On contact zone models for an electrically impermeable interface crack in a piezoelectric bimaterial

An electrically impermeable interface crack between two semi-infinite piezoelectric planes under remote mechanical tension-shear and electrical loading is studied. Assuming the stresses, strains and displacements are independent on the coordinate x ₂ the expressions for the elastic displacement and potential jumps as well as for the stresses and electrical displacement along the interface via a sectionally holomorphic vector function are found. Introducing an artificial contact zone at the right crack tip and assuming the materials possess the symmetry class 6 mm the problem is reduced for a wide range of bimaterial compounds to a combination of combined Dirichlet–Riemann and Hilbert boundary value problems which are solved analytically. From these solutions clear analytical expressions for characteristic mechanical and electrical parameters are derived. As particular cases of the above mentioned solution the classical (oscillating) and contact zone solutions are obtained. Further, a comparison with an associated solution for an electrically permeable crack has been performed. The fracture mechanical parameters for all models via the remote loads are found analytically and important relationships between these parameters are obtained. Due to these relationships an important algorithm of a numerical method applicable for the investigation of an interface crack in a finite sized piezoelectric bimaterial is suggested.     

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。     

Stress-intensity factor computation in three dimensions with quarter-point elements

A consistent method for computing stress-intensity factors from three-dimensional quarter-point element nodal displacements is presented. The method is generalized to permit functional evaluation of stress-intensity factors along the crack front. Embedded, surface, and corner crack problems are solved using the proposed technique. Results are compared to previous finite element and boundary element solutions. The comparison shows that use of the functional evaluation technique allows a dramatic decrease in problem size while still maintaining engineering accuracy. Next, a three-dimensional stress-intensity factor calibration of an unusual specimen configuration is presented. By taking advantage of the proposed technique, the calibration was performed with little difference in cost over the more usual two-dimensional approach. Moreover, the three-dimensional solution revealed intersting behaviour that would have been undetected by a two-dimensional solution. Finally, the results of a study on optimum size of the quarter-point element are presented. Surprisingly, Poisson ratio is shown to have marked effect on optimum element size.     

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.