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.     

Green's functions for transversely isotropic piezoelectric multilayered half-spaces

This paper presents Green's functions for transversely isotropic piezoelectric and layered half-spaces. The surface of the half-space can be under general boundary conditions and a point source (point-force/point-charge) can be applied to the layered structure at any location. The Green's functions are obtained in terms of two systems of vector functions, combined with the propagator-matrix method. The most noticeable feature is that the homogeneous solution and propagator matrix are independent of the choice of the system of vector functions, and can therefore be treated in a unified manner. Since the physical-domain Green's functions involve improper integrals of Bessel functions, an adaptive Gauss-quadrature approach is applied to accelerate the convergence of the numerical integral. Typical numerical examples are presented for four different half-space models, and for both the spring-like and general traction-free boundary conditions. While the four half-space models are used to illustrate the effect of material stacking sequence and anisotropy, the spring-like boundary condition is chosen to show the effect of the spring constant on the Green's function solutions. In particular, it is observed that, when the spring constant is relatively large, the response curve can be completely different to that when it is small or when it is equal to zero, with the latter corresponding to the traction-free boundary condition.     

Green's functions for the isotropic or transversely isotropic space containing a circular crack

Summary Green's functions for an infinite three-dimensional elastic solid containing a circular crack are derived in terms of integrals of elementary functions. The solid is assumed to be either isotropic or transversely isotropic (with the crack being parallel to the plane isotropy).     

Green’s functions for transversely isotropic magnetoelectroelastic media

Based on the governing equations of transversely isotropic magnetoelectroelastic media, four general solutions on the cases of distinct eigenvalues and multiple eigenvalues are given and expressed in five mono-harmonic displacement functions. Then, based on these general solutions, employing the trial-and-error method, the three-dimensional Green’s functions of infinite, two-phase and semi-infinite magnetoelectroelastic media under point forces, point charge and magnetic monopole are all presented in terms of elementary functions for all cases of distinct eigenvalues and multiple eigenvalues. Numerical results are also presented.     

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.     

A fundamental solution for transversely isotropic and nonhomogeneous media

The purpose of this paper is to consider the concept of a ring of sources or forces using the integral transform techniques to derive the axisymmetric fundamental solution for nonhomogeneous transversely isotropic elastic media. Firstly, the formulation of the problem in homogeneous media to derive the fundamental solutions is shown. In the case of a nonhomogeneous medium, the shear modulus of the material varies with the z-coordinate exponentially.     

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.     

Dugdale model solutions for a penny-shaped crack in three-dimensional transversely isotropic piezoelectric media by boundary-integral equation method

Making use of the Displacement Discontinuity Boundary Integral Equation Method (DDBIEM), the dimension of the plastic zone at the tip of a penny-shaped crack in a three-dimensional elastic medium is determined by the application of the Dugdale model; Furthermore, the solutions for a penny-shaped crack in three-dimensional piezoelectric media are obtained by the use of the Dugdale-like model proposed by Gao et al.[Gao H, Zhang T, Tong P. Local and global energy release rates for an electrically yielded crack in a piezoelectric ceramic. J. Mech. Phys. Solids 1997;45:491–510], in which the electrical polarization is assumed to reach a saturation limit in a thin annular region in front of a crack while the mechanical stresses have the ordinary singularity.     

Deformational theory of plasticity of transversely isotropic media

We propose a version of the theory of plasticity of transversely isotropic media for the case of simple loading. Our version is based on the concept of yield surface. We use a quadratic condition of yield that takes into account the partial effects of equivalent stresses computed according to von Mises and Hill and according to Tresca on the plastic deformation of the material. In the general case, this condition can be interpreted as a singular surface in the space of stresses. On the basis of the assumptions concerning the linearity of trajectories of plastic deformation and their normality to the initial yield surface under simple loading as well as concerning the existence of a relationship between the introduced equivalent stresses and equivalent plastic strains independent of the type of the stressed state, we deduce reversible master equations of plasticity. The adequacy of the proposed model is confirmed by the good agreement between the results of numerical analysis and experimental data. Translated from Problemy Prochnosti, No. 1, pp. 5 – 14, January – February, 1998.     

Moving antiplane shear crack in transversely isotropic magnetoelectroelastic media

An antiplane shear strip crack moving uniformly in transversely isotopic magnetoelectroelastic media when subjected to representative non-constant crack-face loading conditions is studied. Readily calculable explicit closed-form representations are determined and discussed for the components of the stress, electric and magnetic fields created throughout the material. Representative numerical data are presented. Alternative boundary conditions for which corresponding analyses can be derived analogously are listed.     

Influence functions of the displacement discontinuity method for anisotropic bodies

In this study, the boundary element equations are obtained from the influence functions of a displacement discontinuity in an anisotropic elastic medium. For this purpose, Kelvin fundamental solutions for anisotropic media on infinite and semi-infinite planes are used to form dipoles from singular loads. Various combinations of these dipoles are used to obtain the influence functions of the displacement discontinuity. Boundary element equations are then derived analytically by the integration of these influence functions on a constant element which results in a linear system for unknown displacement discontinuities. The boundary integrals are calculated in closed form over constant elements. The obtained formulation is applied to a number of classical engineering problems.Tel.: +90-212-285-65-85, 90-212-285-37-07     

Asymmetric wave propagation in a transversely isotropic half-space in displacement potentials

With the aid of a complete representation using two displacement potentials, an efficient and accurate analytical derivation of the fundamental Green’s functions for a transversely isotropic elastic half-space subjected to an arbitrary, time-harmonic, finite, buried source is presented. The formulation includes a complete set of transformed stress-potential and displacement-potential relations that can be useful in a variety of elastodynamic as well as elastostatic problems. The present solutions are analytically in exact agreement with the existing solutions for a half-space with isotropic material properties. For the numerical evaluation of the inversion integrals, a quadrature scheme which gives the necessary account of the presence of singularities including branch points and pole on the path of integration is implemented. The reliability of the proposed numerical scheme is confirmed by comparisons with existing solutions.     

Green's functions in orthotropic thermoelastic diffusion media

The aim of the present paper is to study the Green's function in orthotropic thermoelastic diffusion media. With this objective, firstly the two-dimensional general solution in orthotropic thermoelastic diffusion media is derived. On the basis of general solution, the Green's function for a steady point heat source in the interior of semi-infinite orthotropic thermoelastic diffusion material is constructed by four newly introduced harmonic functions. The components of displacement, stress, temperature distribution and mass concentration are expressed in terms of elementary functions. From the present investigation, a special case of interest is also deduced, to depict the effect of diffusion on components of stress and temperature distribution.     

Some dynamic properties of incompressible,transversely isotropic elastic media

Summary This paper investigates various dynamic properties of incompressible, transversely isotropic elastic media. The propagation condition for such materials allows the wave speeds to be obtained in explicit form. An examination of the slowness surface and direction of energy flux as the extensional modulus along the fibre tends to infinity is then easily carried out. The paper also includes an investigation of the dynamic response of such materials to a particular line impulsive force. This is done using integral transforms. These transforms are invertible in closed form.     

Transducer-modeling in general transversely isotropic media via point-source-synthesis: Theory

Based on a theory of elastic wave propagation in arbitrarily oriented transversely isotropic media, which has been presented recently, the radiation characteristics of ultrasonic transducers in these media are determined. Using the directivity patterns for normal and transverse point sources on the free surface of such (semi-infinite) materials—the derivation is based on the reciprocity theorem—the radiated wave fields are obtained by the method of point-source-synthesis, i.e., by superposing the wave fields of numerous point sources located within the transducer aperture. Since ultrasonic inspection of anisotropic materials, especially weld material in nuclear power plants, suffers from the well-known effects of beam splitting, beam distortion, and beam skewing, valuable information in view of an optimized inspection is provided. Focusing on transversely isotropic weld material specimens, numerical evaluation is performed for several grain orientations with respect to the transducer-normal. The approach presented is particularly useful in view of an appropriate extension to inhomogeneous welds and the consideration of time-dependent RF-impulse functions.     

The method of fundamental solutions for three-dimensional elastostatic problems of transversely isotropic solids

This paper describes the method of fundamental solutions (MFS) to solve three-dimensional elastostatic problems of transversely isotropic solids. The desired solution is represented by a series of closed-form fundamental solutions, which are the displacement fields due to concentrated point forces acting on the transversely isotropic material. To obtain the unknown intensities of the fundamental solutions, the source points are properly located outside the computational domain and the boundary conditions are then collocated. Furthermore, the closed-form traction fields corresponding to the previously published point force solutions are reviewed and addressed explicitly in suitable forms for numerical implementations. Three numerical experiments including Dirichlet, Robin, and peanut-shaped-domain problems are carried out to validate the proposed method. It is found that the method performs well for all the three cases. Furthermore, a rescaling method is introduced to improve the accuracy of Robin problem with noisy boundary data. In the spirits of MFS, the present meshless method is free from numerical integrations as well as singularities.     

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.     

Axisymmetric crack terminating at the interface of transversely isotropic dissimilar media

In this paper, the axisymmetric elasticity problem of an infinitely long transversely isotropic solid cylinder imbedded in a transversely isotropic medium is considered. The cylinder contains an annular or a penny shaped crack subjected to uniform pressure on its surfaces. It is assumed that the cylinder is perfectly bonded to the medium. A singular integral equation of the first kind (whose unknown is the derivative of crack surface displacement) is derived by using Fourier and Hankel transforms. By performing an asymptotic analysis of the Fredholm kernel, the generalized Cauchy kernel associated with the case of `crack terminating at the interface' is derived. The stress singularity associated with this case is obtained. The singular integral equation is solved numerically for sample cases. Stress intensity factors are given for various crack geometries (internal annular and penny-shaped cracks, annular cracks and penny-shaped cracks terminating at the interface) for sample material pairs.     

Wave field simulation for heterogeneous transversely isotropic porous media with the JKD dynamic permeability

Poroelastic wave field in a 2D heterogeneous transversely isotropic porous medium is calculated. The Johnson-Koplik-Dashen (JKD) dynamic permeability is assumed with two scalar JKD permeability operators for vertical and horizontal direction, respectively. The time domain expression of drag force in the JKD model is expressed in terms of the shifted fractional derivative of the relative fluid velocity. A method for calculating the shifted fractional derivative without storing and integrating of the entire velocity histories is developed. By using the new method for calculating the shifted fractional derivative, the governing equations for the 2D transversely isotropic porous medium are reduced to a system of first-order differential equations for velocities, stresses, pore pressure and the quadrature variables associated with the drag forces. The spatial derivatives involved in the first-order differential equation system are calculated by the Fourier pseudospectral method, while the time derivatives of the system are discretized by a predictor-corrector method. For the demonstration of our method, some numerical results are given in the paper.