Extended displacement discontinuity Green's functions for three-dimensional transversely isotropic magneto-electro-elastic media and applications

A versatile method is presented to derive the extended displacement discontinuity Green's functions or fundamental solutions by using the integral equation method and the Green's functions of the extended point forces. In particular, the three-dimensional (3D) transversely isotropic magneto-electro-elastic problem is used to demonstrate the method. On this condition, the extended displacement discontinuities include the elastic displacement discontinuities, the electric potential discontinuity and the magnetic potential discontinuity, while the extended forces include the point forces, the point electric charge and the point electric current. Based on the obtained Green's functions, the extended Crouch fundamental solutions are derived and an extended displacement discontinuity method is developed for analysis of cracks in 3D magneto-electro-elastic media. The extended intensity factors of two coplanar and parallel rectangular cracks are calculated under impermeable boundary condition to illustrate the application, accuracy and efficiency of the proposed method.     

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.     

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.     

Two-dimensional elastodynamic displacement discontinuity method

A time-domain elastodynamic displacement discontinuity method is presented for analysing two-dimensional problems in solid mechanics. Linear, continuous in time and piecewise linear in space interpolation functions are assumed for the displacement discontinuities. Causal and analytical integrations are used throughout. The resulting numerical algorithms are time marching and implicit. The displacement discontinuity method can be used to model a class of traction boundary value problems, where the unknowns are the transient shear and normal discontinuities in displacement across the surface of a mathematical crack in an infinite, isotropic and homogeneous linear elastic medium.     

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.     

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.     

Computational modeling of strong and weak discontinuities using the extended finite element method

In this article, the extended finite element method is employed to solve problems, including weak and strong discontinuities. To this end, a level set framework is used to represent the discontinuities location, and the Heaviside and Branch function are included in the standard finite element method. The case of two arbitrary curved cracks is solved numerically and stress intensity factor (SIF) values at the crack tips are calculated based on the evaluation of the crack tip opening displacement. Afterwards, J-integral methodology is adopted to evaluate the SIFs for isotropic and anisotropic bi-material interface crack problems. Numerical results are verified with those presented in the literature.     

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.     

Ultrasonic 2D SH Crack Detection in a Layered Anisotropic Plate

The 2-D scattering problem of an internal crack in a layered anisotropic plate is considered in this paper. In the model, two ultrasonic SH probes are attached on the upper surface of the plate and the incoming displacement field is generated by one of the probes and the other probe is acting as a receiver. The transmitting and the receiving probe may be the same. The problem is solved by deriving the Green function for the layered plate and then using the integral representation for the total field to obtain an integral equation for the crack opening displacement. The integral equation is solved by expanding the crack opening displacement (COD) in Chebyshev functions. A crucial part of the method is the expansion of the Green function in a free space part, expressed in the crack coordinate system, and a reflection part, expressed in the plate coordinate system. The electrical signal response is calculated by an electromechanical reciprocity relation. Numerical examples are given for a transversally isotropic graphite-epoxy plate, where the symmetry axes are mutually perpendicular in the layers. The results are presented as A-scans, i.e. the electrical response as a function of time.     

Three Collinear Antiplane Interfacial Cracks in Dissimilar Piezoelectric Materials

Three collinear impermeable interfacial cracks in bonded dissimilar transversely isotropic piezoelectric materials under electromechanical loadings is analyzed. A single antiplane mechanical and inplane electrical loads are applied at a point on crack surface. The problem is formulated by the complex function method, and reduced to the vector Hilbert problem. Solving this 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.     

几何非线性扩展有限元法及其断裂力学应用

该文将扩展有限元方法应用到几何非线性及断裂力学问题中,并研制开发了扩展有限元Fortran程序。扩展有限元法其计算网格与不连续面相互独立,因此模拟移动的不连续面时无需对网格进行重新剖分。该文推导了几何非线性扩展有限元法的公式,在常规有限元位移模式中,基于单位分解的思想加进一个阶跃函数和二维渐近裂尖位移场,反映裂纹处位移的不连续性,并用2个水平集函数表示裂纹;采用拉格朗日描述方程建立了有限变形几何非线性扩展有限元方程;采用多点位移外推法计算裂纹应力强度因子并通过最小二乘法拟合得到更精确的结果。最后给出的大变形算例表明该文提出的几何非线性的断裂力学扩展有限元方法和相应的计算机程序是合理可行的,而且对于含裂纹及裂纹扩展的问题,扩展有限元法优于传统的有限元法。     

Solution of plane elasticity problems for orthotropic bodies with cracks by the displacement discontinuity method

 The force problem of fracture mechanics for orthotropic body is solved by boundary element method. The equations of plane strain state are used. The key point of this paper is obtaining the influence functions for finite segment with constant displacement discontinuities in the main axes of orthotropy. They have been deduced by means of two-dimensional Fourier transform and theory of generalized functions. The straight crack under the action of uniform tension in the infinity is considered by using obtained influence functions. The calculations were carried out for isotropic and two orthotropic materials. Received 6 November 2000     

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.     

Dynamics response of complex defects near bimaterials interface by incident out-plane waves

Dynamics response of an elliptical cavity and a crack (on different sides) near bimaterials interface under incident out-plane waves is studied by applying the methods of complex variables and Green’s function. Firstly, based on “conjunction,” the analytical model is divided along the horizontal interface into an elastic half-plane possessing an elliptical cavity and a full elastic half-plane containing a crack. Using complex variables, the scattering displacement field of the half-plane containing an elliptical cavity under incident out-plane waves is then derived. According to the method of Green’s function, the corresponding Green’s functions of two half-planes impacted by an out-plane source load are further deduced. Combined with “crack division,” a crack at the full elastic the half-plane is created, and thus, expressions of displacement and stress are derived while the cavity coexists with the crack. Undetermined antiplane forces are loaded on the horizontal surfaces for conjunction of two sections and then solved by a series of Fredholm integral equations on account of continuity conditions of the interface. Finally, this paper focuses on the discussion of the influence law of different parameters on the dynamics response of complex defects near bimaterials interface by comprehensive numerical results.     

Electroelastic analysis for a Griffith crack interacting with a coated inclusion in piezoelectric solid

A Mode III Griffith crack interacting with a coated inclusion in piezoelectric media is investigated. The crack, the coated inclusion are embedded in an infinitely extended piezoelectric matrix media, with the crack being along the radial direction of the inclusion. In the study, three different piezoelectric material phases are involved: the inclusion, the coating layer, and the matrix. A far-field loading condition is considered. During the solution procedure, the crack is simulated as a continuous distribution of screw dislocations. By using the solution of a screw dislocation near a coated inclusion in piezoelectric media as the Green function, the problem is formulated into a set of singular integral equations, which are solved by numerical method. The stress and electric displacement intensity factors are derived in terms of the asymptotic values of the dislocation density functions evaluated from the integral equations. Numerical examples are given for various material constants combinations and geometric parameters.     

Three-dimensional steady-state general solution for transversely isotropic hygrothermopiezoelectric media and its application in fracture

The potential theory method is utilized to derive the steady-state, general solution for three-dimensional (3D) transversely isotropic, hygrothermopiezoelectric media in the present paper. Two displacement functions are introduced to simplify the governing equations. Employing the differential operator theory and superposition principle, all physical quantities can be expressed in terms of two functions, one satisfies a quasi-harmonic equation and the other satisfies a tenth-order partial differential equation. The obtained general solutions are in a very simple form and convenient to use in boundary value problems. As one example, the 3D fundamental solutions are presented for a steady point moisture source combined with a steady point heat source in the interior of an infinite, transversely isotropic, hygrothermopiezoelectric body. As another example, a flat crack embedded in an infinite, hygrothermopiezoelectric medium is investigated subjected to symmetric mechanical, electric, moisture and temperature loads on the crack faces. Specifically, for a penny-shaped crack under uniform combined loads, complete and exact solutions are given in terms of elementary functions, which serve as a benchmark for different kinds of numerical codes and approximate solutions.     

On the anisotropic piezoelastic contact problem for an elliptical punch

Summary This paper firstly conducts a systematic investigation of the problem of a rigid punch indenting an anisotropic piezoelectric half-space. The Fourier transform method is employed to the mixed boundary value problem. Using the principle of linear superposition, the resulting transformed (algebraic) equations, whose right-hand sides contain both pressure and electric displacement terms, can be solved by superposing the solutions of two sets of algebraic equations, one containing pressure and another containing electric displacement. For an arbitrarily shaped punch, two governing equations are derived, which can be solved numerically. In the case of transversely isotropic piezoelectric media, the two governing equations are corresponding with that given by others using potential theory. Particularly, when the punch has elliptic cross-section, and the pressure and electric displacement are given by some certain forms of polynomial functions, then the displacement and electric potential are prescribed by polynomial functions in the contact area. The parameters contained in it satisfy a set of linear algebraic equations, whose coefficients involve contour integrals. The problem of indentation by a smooth flat punch is examined for special orthotropic piezoelectric media, and some results obtained can be degenerated to the case of transversely isotropic piezoelectric media.     

A thermal-medium crack model

In analyzing the fracture behavior of a cracked thermoelastic material, of much importance are the effects of thermal loadings on the crack growth. Under the consideration of a medium in an opening crack, a thermal-medium crack model is proposed in this paper. The heat flux at the crack surfaces is assumed to depend on the jumps of the temperature and the elastic displacement across the crack. The thermally permeable and impermeable crack models are the limiting cases of a thermal-medium one. The proposed crack model is applied to solve the problem of a Griffith crack in a transversely isotropic material under thermal and mechanical loadings. Using two introduced displacement functions and the Fourier transform technique, the thermoelastic field and the elastic T-stress are determined in explicit forms by using elementary functions. Numerical results are presented to show the effects of the thermal conductivity inside a crack and applied mechanical loadings on the heat flux at the crack faces, the jumps of temperature across the crack and mode-II stress intensity factor in graphics respectively. The obtained results reveal that the mode-II stress intensity factor for a thermal-medium crack in a thermoelastic material depends not only on applied thermal loadings but also on applied mechanical ones.     

Numerical analysis of growing crack problems using particle discretization scheme

This paper presents the particle discretization scheme (PDS) to analyze brittle failure of solids. The scheme uses characteristic functions of Voronoi and Delaunay tessellations to discretize a function and its derivatives, respectively. A discretized function has numerous discontinuities so that these discontinuities are utilized as a candidate of crack path segment in modeling propagating cracks, without making any extra computation to accommodate new displacement discontinuities. When the scheme is implemented to a finite element method (FEM), the resulting stiffness matrix coincides with the one that is obtained by using linear elements. The accuracy of computing a stress intensity factor at a crack tip is examined. It is shown that the accuracy is better than that of a FEM with linear elements when the rotational degree of freedom is included in discretizing displacement functions. Three three‐dimensional growing crack problems are solved by means of the PDS and the results are presented. Copyright © 2009 John Wiley & Sons, Ltd.