Three-dimensional extended displacement discontinuity method for vertical cracks in transversely isotropic piezoelectric media

The conventional displacement discontinuity method is extended to study a vertical crack under electrically impermeable condition, running parallel to the poling direction and normal to the plane of isotropy in three-dimensional transversely isotropic piezoelectric media. The extended Green's functions specifically for extended point displacement discontinuities are derived based on the Green's functions of extended point forces and the Somigliana identity. The hyper-singular displacement discontinuity boundary integral equations are also derived. The asymptotical behavior near the crack tips along the crack front is studied and the ordinary 1/2 singularity is obtained at the tips. The extended field intensity factors are expressed in terms of the extended displacement discontinuity on crack faces. Numerical results on the extended field intensity factors for a vertical square crack are presented using the proposed extended displacement discontinuity method.     

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.     

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.     

Numerical method for nonlinear models of penny-shaped cracks in three-dimensional magnetoelectroelastic media

Green functions corresponding to uniformly distributed extended displacement discontinuities on an annular crack element in the isotropic plane of a three-dimensional transversely isotropic magnetoelectroelastic medium are derived. Using the obtained Green functions, an extended displacement discontinuity method is presented to analyze a penny-shaped crack under axisymmetric loadings. Using the electric and magnetic polarization saturation model and the electric and magnetic breakdown model, the electric and magnetic yielding zones, the extended displacement discontinuities, the extended stress intensity factors and the J-integral are numerically calculated. The accuracy and efficiency of the proposed method are demonstrated by comparing the numerical results with those obtained from analytical solutions.     

Three-dimensional analysis of piezoelectric/piezomagnetic elastic media

This paper presents an exact three-dimensional method of solution for a transversely isotropic piezoelectric/piezomagnetic elastic media. The control partial differential equation set that is expressed by the elastic displacement, electric potential and magnetism potential function are established. By means of drawing two-displacement potential functions the general solution for three-dimensional transversely isotropic magneto-electro-elastic is derived. As an illustrative example, the analysis solutions of a half space body of magneto-electro-elastic acted by a point force at the origin directed along the z-axis and a point charge and a point electric current at the origin are obtained.     

Analysis of an arbitrarily oriented crack in a finite piezoelectric plane via the hybrid extended displacement discontinuity-fundamental solution method

In this paper, we analyze an arbitrarily oriented crack in a finite two-dimensional piezoelectric medium with the polarization saturation model near the crack tip. We first derive the extended Green’s functions corresponding to the extended point-displacement discontinuities of an arbitrarily oriented crack based on the Green’s functions of the extended point forces and the Somigliana identity. Then, the extended field intensity factors and the local J-integral near the crack tip are expressed in terms of the extended displacement discontinuity on crack faces. Finally, the nonlinear hybrid extended displacement discontinuity-fundamental solution method is proposed to analyze an electrically nonlinear crack in a finite piezoelectric medium. Numerical examples are carried out for both linear and nonlinear fracture models of the crack under electrically impermeable boundary conditions. The influence of the crack orientation and geometric size on the fracture behaviors of the crack is investigated.     

Boundary element modeling of cracks in piezoelectric solids

This study focuses on the application of boundary element methods for linear fracture mechanics of two-dimensional piezoelectric solids. A complete set of piezoelectric Green's functions, based on the extended Lekhnitskii's formalism and distributed dislocation modeling, are presented. Special Green's functions are obtained for an infinite medium containing a conducting crack or an impermeable crack. The numerical solution of the boundary integral equation and the computation of fracture parameters are discussed. The concept of crack closure integral is utilized to calculate energy release rates. Accuracy of the boundary element solutions is confirmed by comparing with analytical solutions reported in the literature. The present scheme can be applied to study complex cracks such as branched cracks, forked cracks and microcrack clusters.     

Three-dimensional exact solutions for free vibrations of simply supported magneto-electro-elastic cylindrical panels

This work investigates the free vibrations of magneto-electro-elastic cylindrical panels based on three-dimensional theory. Firstly, the general solutions for transversely isotropic magneto-electro-elastic materials are introduced and the displacement functions in the general solutions are expanded in trigonometric functions along the circumferential and axial directions. Then an ordinary differential equation of the displacement functions in radial direction is derived and solved. As a result, the frequency equations are obtained through the traction-free conditions on the cylindrical surfaces of the panel as well as the electric and magnetic conditions. For the torsion and thickness-shear modes, the frequency equations in simpler forms are presented. It is found that the magneto-electro-elastic coupling effects disappeared in torsion vibration. Meanwhile, the frequencies of pure elastic materials and magneto-electro-elastic materials have an explicit relation for the thickness-shear modes. The aforementioned solutions satisfy all the governing equations and boundary conditions point by point and they are three-dimensionally exact. Finally the numerical example demonstrates the present method and is compared with those from finite element method. Parametric investigation is also conducted to show the behavior of free vibrations of cylindrical panels.     

Punch problem for piezoelectric ceramic half-plane

Based on the theory of linear piezoelectricity, this study presents an exact solution for a two dimensional indentation on a piezoelectric ceramic half-plane with different contact conditions. The flat-ended indenters are assumed to be rigid. Besides, they can either be insulating or conducting. In addition, different contact conditions, including frictionless, frictional, and adhesive punches are investigated. Lekhnitskii's formulism and Fourier transforms are used to obtain the Green's function of a piezoelectric ceramic half-plane subjected to a point loading. Utilizing Green's half-plane function, we obtain the three integral equations by connecting generalized displacement gradients at the surface and surface loading. Both uncoupled and coupled integral equations can be transformed into a Fredholm integral equation. The analytical closed form solutions of the contact forces and the electric charges under the indenter can be derived by solving the Fredholm integral equations. Once the distributions of the contact forces and the electric charges on the surface are known, the electroelastic response in the half-plane can also be obtained.     

Numerical evaluation of harmonic Green's functions for triclinic half‐space with embedded sources—Part II: a 3D model

The displacement and stress Green's functions for a 3D triclinic half‐space with embedded harmonic point load is considered. The resulting displacement and stress fields are expressed in terms of triple Fourier integrals. The first integral was evaluated using contour integration and the 3D Green's functions were obtained as a superposition of 2D results over the azimuthal angle. The resulting algorithm developed for evaluation of the Green's functions avoids repeated calculations of the same quantities and it utilizes the vectorized manipulation within MATLAB environment. The algorithm places no restriction on material properties, frequency and location of source and observation points. Extensive testing of the numerical results was performed for both displacement and stress. The tests confirm the accuracy of the numerical results. Copyright © 2006 John Wiley & Sons, Ltd.     

Analytical solution for functionally graded magneto-electro-elastic plane beams

This paper investigates the plane stress problem of generally anisotropic magneto-electro-elastic beams with the coefficients of elastic compliance, piezoelectricity, dielectric impermeability, piezomagnetism, magnetoelectricity, and magnetic permeability being arbitrary functions of the thickness coordinate. Firstly, partial differential equations governing stress function, electric displacement function and magnetic induction function are derived for plane problems of anisotropic functionally graded magneto-electro-elastic materials. Secondly, these functions are assumed in forms of polynomials in the longitudinal coordinate and can be acquired through a successive integral approach. The analytical expressions of axial force, bending moment, shear force, average electric displacement, average magnetic induction, displacements, electric potential and magnetic potential are then deduced. Thirdly, problems of functionally graded magneto-electro-elastic plane beams are considered, with integral constants being completely determinable from boundary conditions. A series of analytical solutions are thus obtained, including the solutions for beams under tension and pure bending, for cantilever beams subjected to shear force applied at the free end, and for cantilever beams subjected to uniform load. These solutions can be easily degenerated into the solutions for homogenous anisotropic magneto-electro-elastic beams. Finally, a numerical example is presented to show the application of the proposed method.     

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.     

The method of fundamental solutions for fracture mechanics—Reissner's plate application

The application of the method of fundamental solutions (MFS), a mesh-free technique, to solve cracked Reissner's plates is discussed in this work. Here, the numerical Green's function (NGF) previously developed by the authors is used as the fundamental solution required by the method. Stress intensity factors or the related force intensity factors are obtained using the generalized crack openings at a single point near the tip, computed through a summation of the fundamental generalized openings at that point weighted by their influence factors. Despite the ill-conditioning of the equations system, which may require appropriate handling to solve (such as the singular value decomposition method), examples show good results for problems with embedded cracks. The method can be a good option to evaluate stress intensity factors of given problems due to its simple and intuitive implementation.     

STRESS INTENSITY FACTORS FOR CRACKS IN NOTCHED FINITE PLATES SUBJECTED TO BIAXIAL LOADING

Stress intensity factors were calculated, based on Bueckner's principle for cracks in both infinite and finite plates with notches subjected to biaxial loading. Approximate Green's functions have been obtained by modifying two existing Green's functions, originally for unnotched plates. Values of stress intensity factors calculated using Bueckner's principle with the approximate Green's functions are in good agreement with published stress intensity factors for cracks in both infinite and finite plates containing a circular notch or an elliptical notch, previously found by the method of boundary collocation.     

Stress intensity factors for cracks of arbitrary shape near an interfacial boundary

Modes I, II and III stress intensity factors for a crack of arbitrary planar shape near a bimaterial interface are calculated. The solution utilizes the body-force method and requires Green's functions for perfectly bonded elastic half-spaces. The formulation leads to a system of two-dimensional singular integral equations whose solutions represent the three modes of crack opening displacement. Numerical examples of a semicircular or semielliptical crack terminating at the interface and circular or elliptical cracks contained in one material are given for both internal pressure and farfield tension.     

Numerical Green's functions for some electroelastic crack problems

A plane electroelastic problem involving planar cracks in a piezoelectric body is considered. The deformation of the body is assumed to be independent of time and one of the Cartesian coordinates. The cracks are traction free and are electrically either permeable or impermeable. Numerical Green's functions which satisfy the boundary conditions on the cracks are derived using the hypersingular integral approach and applied to obtain a boundary integral solution for the electroelastic crack problem considered here. As the conditions on the cracks are built into the Green's functions, the boundary integral solution does not contain integrals over the cracks. It is used to derive a boundary element procedure for computing the crack tip stress and electrical displacement intensity factors.     

Dislocation and point-force-based approach to the special Green's Function BEM for elliptic hole and crack problems in two dimensions

In this paper we give the theoretical foundation for a dislocation and point-force-based approach to the special Green's function boundary element method and formulate, as an example, the special Green's function boundary element method for elliptic hole and crack problems. The crack is treated as a particular case of the elliptic hole. We adopt a physical interpretation of Somigliana's identity and formulate the boundary element method in terms of distributions of point forces and dislocation dipoles in the infinite domain with an elliptic hole. There is no need to model the hole by the boundary elements since the traction free boundary condition there for the point force and the dislocation dipole is automatically satisfied. The Green's functions are derived following the Muskhelishvili complex variable formalism and the boundary element method is formulated using complex variables. All the boundary integrals, including the formula for the stress intensity factor for the crack, are evaluated analytically to give a simple yet accurate special Green's function boundary element method. The numerical results obtained for the stress concentration and intensity factors are extremely accurate. © 1997 John Wiley & Sons, Ltd.     

A boundary integral approach for plane analysis of electrically semi-permeable planar cracks in a piezoelectric solid

A plane electro-elastostatic problem involving arbitrarily located planar stress free cracks which are electrically semi-permeable is considered. Through the use of the numerical Green's function for impermeable cracks, the problem is formulated in terms of boundary integral equations which are solved numerically by a boundary element procedure together with a predictor–corrector method. The crack tip stress and electric displacement intensity factors can be easily computed once the boundary integral equations are properly solved.     

An indirect elastodynamic boundary element method with analytic bases

An indirect time‐domain boundary element method (BEM) is presented here for the treatment of 2D elastodynamic problems. The approximated solution in this method is formulated as a linear combination of a set of particular solutions, which are called bases. The displacement and stress fields of a basis are analytically derived by means of solving Lame's displacement potentials. A semi‐collocation method is proposed to be the time‐stepping algorithm. This method is equivalent to a displacement discontinuity method with piecewise linear discontinuities in both space and time. The resulting time‐stepping scheme is explicit. The BEM is implemented to solve three numerical examples, Lamb's problem, half‐plane with a buried crack and Selberg's problem. Though Lamb's problem is considered a difficult problem for numerical methods, the current numerical results for the surface displacements show accurately the characteristics of the Rayleigh wave. This method is efficient and accurate. Copyright © 2003 John Wiley & Sons, Ltd.     

BEM for crack‐inclusion problems of plane thermopiezoelectric solids

The problem of interactions between an inclusion and multiple cracks in a thermopiezoelectric solid is considered by boundary element method (BEM) in this paper. First of all, a BEM for the crack–inclusion problem is developed by way of potential variational principle, the concept of dislocation, and Green's function. In the BE model, the continuity condition of the interface between inclusion and matrix is satisfied, a priori, by the Green's function, and not involved in the boundary element equations. This is then followed by expressing the stress and electric displacement (SED) and elastic displacements and electric potential (EDEP) in terms of polynomials of complex variables ξ_t and ξ_k in the transformed ξ‐plane in order to simulate SED intensity factors by the BEM. The least‐squares method incorporating the BE formulation can, then, be used to calculate SED intensity factors directly. Numerical results for a piezoelectric plate with one inclusion and a crack are presented to illustrate the application of the proposed formulation. Copyright © 2000 John Wiley & Sons, Ltd.