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.     

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.     

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.     

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.     

Dynamic steady-state crack propagation in a transversely isotropic viscoelastic body

The problem considered herein is the dynamic, subsonic, steady-state propagation of a semi-infinite, generalized plane strain crack in an infinite, transversely isotropic, linear viscoelastic body. The corresponding boundary value problem is considered initially for a general anisotropic, linear viscoelastic body and reduced via transform methods to a matrix Riemann–Hilbert problem. The general problem does not readily yield explicit closed form solutions, so attention is addressed to the special case of a transversely isotropic viscoelastic body whose principal axis of material symmetry is parallel to the crack edge. For this special case, the out-of-plane shear (Mode III), in-plane shear (Mode II) and in-plane opening (Mode I) modes uncouple. Explicit expressions are then constructed for all three Stress Intensity Factors (SIF). The analysis is valid for quite general forms for the relevant viscoelastic relaxation functions subject only to the thermodynamic restriction that work done in closed cycles be non-negative. As a special case, an analytical solution of the Mode I problem for a general isotropic linear viscoelastic material is obtained without the usual assumption of a constant Poissons ratio or exponential decay of the bulk and shear relaxation functions. The Mode I SIF is then calculated for a generalized standard linear solid with unequal mean relaxation times in bulk and shear leading to a non-constant Poissons ratio. Numerical simulations are performed for both point loading on the crack faces and for a uniform traction applied to a compact portion of the crack faces. In both cases, it is observed that the SIF can vanish for crack speeds well below the glassy Rayleigh wave speed. This phenomenon is not seen for Mode I cracks in elastic material or for Mode III cracks in viscoelastic material.     

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.     

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.     

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.     

Three-dimensional piezoelectric boundary element analysis of transversely isotropic half-space

In this paper, we present a boundary element method (BEM) solution technique for studying the three-dimensional transversely-isotropic piezoelectric half-space problems. The use of mixed alternative point force solutions for half and full-space problems presented are necessary to overcome the computation difficulties especially in the calculation of the derivatives with respect to z. Infinite boundary elements are introduced to model the surface of the half-space only when stresses at the internal points are required to be evaluated. The integration over the infinite boundary elements is bounded and some limitations of the infinite element construction are relaxed. Closed-form solutions for uniformly distributed mechanical and electrical loads acting on a circular area on the surface of half-space are derived. This theoretical work serves as a good verification tool for numerical computation. In this paper, the numerical and theoretical results show good agreement. Numerical analysis via the finite element method (FEM) is also carried out using the commercial solver ANSYS. These FEM results are used to verify against the accuracy of the BEM solution. Finally, numerical results for the case of Hertzian pressure applied to an imperfect half-space are presented. The effects of the coupled mechanical–electrical influences as well as the presence of voids are examined. This work was supported by NTU Academic Research Funds. The finite element simulation using the ANSYS code was conducted by Mr. Ji Ren. Also, the authors wish to acknowledge the journal editor and anonymous reviewers for their helpful suggestions and comments leading to improvement of the paper.     

Three-dimensional vibration analysis of a transversely isotropic piezoelectric cylindrical panel

Summary. In this work, based on three-dimensional piezoelectric elasticity, an exact analysis of the free vibrations of a simply supported, homogeneous, transversely isotropic cylindrical panel is presented. Three displacement potential functions are introduced so that the equations of motion and Gauss equation are uncoupled and simplified. It is noticed that a purely transverse (SH) mode is independent of piezoelectric effects and the rest of the motion. The equations for free vibration problems are further reduced to four second-order ordinary differential equations, after expanding the displacement and electric potential functions with an orthogonal series. The dispersion relations for an electrically shorted and charge free simply supported cylindrical panel with stress free edges have been obtained and discussed. A modified Bessel function solution with complex arguments is directly used for complex eigenvalues. In order to clarify the developed method and to compare the results to the existing areas, numerical examples are presented and the computed functions are illustrated graphically.     

横观各向同性黏弹性沥青路面动力响应解析解

提出了移动荷载作用下各向异性层状地基上柔性路面结构动力响应分析的半解析解法。基于土-结构相互作用理论,建立了柔性路面-层状地基的动力耦合模型。对近场道路结构建立波数域有限元模型,基于精细积分及谱元法原理建立远场地基的谱元模型,利用子结构法来确定近-远场耦合界面处的边界条件,进而实现近场和远场结构的耦合,利用快速Fourier变换获得柔性路面在时空域的动力响应解答。该文方法既可以考虑道路结构的几何特性,也可以计入地基层状分布及各向异性特性的影响。近场有限域模型尺寸不会受到地基层数及层厚的影响,数值求解稳定。通过与解析解的对比,验证了提出方法的有效性与合理性。数值算例对比分析了道路-地基耦合模型与以往整体层状道路模型所得动力响应的差异,并讨论了道路结构层弹性模量的影响规律。

     

An elasticity solution for transversely isotropic,functionally graded circular plates

This article presents a new elasticity solution for transversely isotropic, functionally graded circular plates subject to axisymmetric loads. It is assumed that the material properties vary along the thickness of a circular plate according to an exponential form. By extending the displacement function presented by Plevako to the case of transversely isotropic material, we derived the governing equation of the problem studied. The displacement function was assumed as the sum of the Bessel function and polynomial function to obtain the analytical solution of a transversely isotropic, functionally graded circular plate under different boundary conditions. As a numerical example, the influence of the graded variations of the material properties on the displacements and stresses was studied. The results demonstrate that the graded variations have a significant effect on the mechanical behavior of a circular plate.     

Exact solution for axisymmetric deformation of laminated transversely isotropic annular plates

Summary Based on the three-dimensional theory of elasticity, this paper presents the state space equation for axisymmetric deformation of a laminated transversely isotropic annular plate. The finite Hankel transform is then introduced and applied to the state space equation. Four exact solutions corresponding to four specified boundary conditions are obtained and expressions for displacements and stresses are presented. Numerical results are finally compared with those obtained by the classical plate theory, the Reissner plate theory and the finite element method.     

Three‐dimensional fundamental solution for transversely isotropic piezothermoelastic material

Fundamental solutions play an important role in electroelastic analyses and numerical methods of piezoelectric material. However, most works available on this topic are on the case of identical temperature. We use the compact mono‐harmonic general solutions of transversely isotropic piezothermoelastic material to construct the three‐dimensional fundamental solution of a steady point heat source in an infinite piezothermoelastic material by four newly introduced mono‐harmonic functions. All components of coupled field are expressed in terms of elementary functions and are convenient to use. Numerical results for cadmium selenide are given graphically by contours. Copyright © 2008 John Wiley & Sons, Ltd.     

Three-dimensional linearized steady-state problems for micropolar viscous liquid media

Linearized steady-state boundary-value problems are posed within the micropolar (asymmetric) theory of liquid media, and analysis of their uniqueness is carried out.Belarusian State University, Minsk, Belarus. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 66, No. 5, pp. 617–621, May, 1994.     

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.     

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.     

General solution technique for transient thermoelasticity of transversely isotropic solids in cylindrical coordinates

Summary Goodier has proposed the thermoelastic potential function in order to analyze thermoelastic problems for isotropic solids. The thermoelastic problem can be reduced to the elastic problem by his technique. Elastic problems are in general analyzed by the generalized Boussinesq solutions and the Michell function. This paper discusses a new solution technique for thermoelastic problems of transversely isotropic solids in cylindrical coordinates. The present solution technique consists of five fundamental solutions which are developed from the Goodier's thermoelastic potential function, the generalized Boussinesq solutions and the Michell function. Considering the relations among the material constants of transverse isotropy, the present solution technique can be classified into two cases. One of them can be reduced to the three solution techniques above which are specifically for isotropic solids only. As an application of the present solution technique, a transient thermoelastic problem in a transversely isotropic cylinder with an external crack is analyzed.     

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.     

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.