Some crack problems in transversely isotropic solids

Summary. Crack problems in transversely isotropic solids are reexamined from a new point of view. It is shown that, when the crack is on the isotropic plane, the asymptotic forms of the elastic crack-tip fields are identical with those in orthotropic media. The equivalent inclusion method in conjunction with Eshelbys S tensor of a strongly oblate spheroid in transversely isotropic materials is used to solve penny-shaped crack problems. The stress intensity factors corresponding to uniform tension and shear are determined, respectively. Griffiths energy criterion for brittle cracking and Irwins energy release rate are discussed in the present context. Finally, the weight function for an axisymmetrically loaded penny-shaped crack is derived. It is found that the axisymmetric weight function is independent of the material constants and is identical with the isotropic case.AcknowledgementThis work was supported in part by the National Science Council of Taiwan.     

Boundary element analysis of three-dimensional crack problems in two joined transversely isotropic solids

The present paper presents a boundary element analysis of penny-shaped crack problems in two joined transversely isotropic solids. The boundary element analysis is carried out by incorporating the fundamental singular solution for a concentrated point load in a transversely isotropic bi-material solid of infinite space into the conventional displacement boundary integral equations. The conventional multi-region method is used to analyze the crack problems. The traction-singular elements are employed to capture the singularity around the crack front. The values of the stress intensity factors are obtained by using crack opening displacements. The numerical scheme results are verified with the closed-form solutions available in the literature for a penny-shaped crack parallel to the plane of the isotropy of a homogeneous and transversely isotropic solid of infinite extent. The new problem of a penny-shaped crack perpendicular to the interface of a transversely isotropic bi-material solid is then examined in detail. The crack surfaces are subject to the three normal tractions and the uniform shear traction. The associated stress intensity factor values are obtained and analyzed. The present results can be used for the prediction of the stability of composite structures and the hydraulic fracturing in deep rock strata and reservoir engineering.     

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.     

Punch and crack problems in transversely isotropic bodies

An exact solution is proposed for the mixed boundary-value problem in a transversely isotropic half-space. Here, certain arbitrary shear tractions are prescribed inside a circular region, outside of which certain arbitrary tangential displacements are given. The normal stresses are supposed to be known all over the boundary. A particular case is considered, in detail, where normal stresses vanish all over the boundary with the shear tractions vanishing inside the circular region. A closed form expression is obtained for the tangential displacements inside the circular region directly through the displacements outside. As an example, a penny-shaped crack in an infinite transversely isotropic body is considered with arbitrary shear tractions acting on both sides of the crack. The formulae for the tangential displacements inside the circle and the shear stresses outside are obtained. Special cases where uniform shear and a concentrated tangential force arise are also discussed.     

Dynamic penny-shaped crack in a transversely isotropic material

The scattering of a harmonic longitudinal wave by a penny-shaped crack in a transversely isotropic material is investigated using the techniques of Hankel transform. The wave impinges normally on the crack surfaces. A complete contour integration is employed to simplify the expressions of the results. An exact expression of the dynamic stress-intensity factor is obtained as a function of the frequency factor and the anisotropic material constants. The normalized dynamic stress-intensity factor is shown to have different maximum values at different wave frequencies for the sample composite and metallic materials. The distortion of the dynamic crack shape and the displacement at the crack center are also shown to be dependent of the wave frequency and the anisotropy of the material.     

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.     

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.     

Penny-shaped crack in a transversely isotropic solid

An analytical solution is given for the displacement and stress distribution produced in the interior of a transversely isotropie solid containing a penny-shaped crack situated in an elastic symmetry plane and axially-loaded. Curves of numerical results are presented for the stress intensity factor and the normal displacement. They show the influence of this type of anisotropy.     

Diffusion of fluids through transversely isotropic solids

Summary In this paper, we investigate the problem of radial diffusion of fluids through a transversely isotropic hollow non-linearly elastic cylinder. The transversely isotropic cylinder is both sheared and stretched. We study in detail how shearing and stretching the cylinder affects the diffusion process. The influence of the anisotropy of the solid on diffusion is also determined. A comparison is made with previous work (cf. Gandhi, Rajagopal and Wineman [11]) on the radial diffusion of fluids through isotropic non-linearly elastic solids.     

Ductile penny-shaped crack in a transversely isotropic cylinder

The problem of a penny-shaped crack contained in a transversely isotropic cylinder of elastic perfectly-plastic material is considered for the case when the crack is extended by an axial load. The problem is reduced to solving numerically a Fredholm integral equation of the second kind for the width of the plastic zone. Graphical results are presented showing the effect of transverse isotropy upon the width of the plastic zone and these are compared with the results for isotropic materials.     

BEM analysis of wave scattering in transversely isotropic solids

A boundary element approach for wave propagation problems in transversely isotropic solids is developed in this paper. The procedure is based on the well‐known formulation for time‐harmonic elasticity and a new version of a recently obtained fundamental solution for transversely isotropic media. The fundamental solution is transformed to obtain new expressions which can be efficiently evaluated at any point. This fact allows for a drastic reduction of the computation time and makes possible the implementation of a general purpose three‐dimensional quadratic element code. To show the simplicity and accuracy of the approach, the diffraction of waves by a spherical cavity and the interaction between two cavities in a boundless domain are studied. The computed results show a very good agreement with the analytical solution in the simple case where such solution exists. Other geometries can be studied without difficulty. Copyright © 1999 John Wiley & Sons, Ltd.     

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.     

A transversely isotropic composite with a penny-shaped crack

Summary We consider the problem of determining the stress intensity factor and the crack energy in a transversely isotropic composite medium, containing a penny-shaped crack. We assume that the crack surface is perpendicular to the bond face and the crack is opened by constant internal pressure. By use of integral transform, we reduce the problem to solving a Fredholm integral equation of the second kind. Numerical results are given for the combination of some practical materials such as magnesium and cadmium. The effect of transverse isotropy upon the stress intensity factor, the crack energy and the deformation on the crack surface is discussed.

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Ein transversal, isotropes, komposites Medium mit einem münzenförmigen Riß

Zusammenfassung Das Problem der Bestimmung des Spannungsintensitätsfaktors und der Rißenergie, in einem transversalen, isotropen, kompositen Medium mit einem münzenförmigen Riß, wird betrachtet. Es wird vorausgesetzt, daß die Rißoberfläche normal zur Verbundfläche liegt, und der Riß sich durch konstanten inneren Druck öffnet. Durch Anwendung einer Integraltransformation, wird das Problem auf die Lösung einer Fredholmschen Integralgleichung zweiter Art reduziert. Numerische Ergebnisse werden für die Kombination einiger Materialien, wie Magnesium und Cadmium angegeben. Der Einfluß der transversalen Isotropie auf den Spannungskonzentrationsfaktor, die Rißenergie und die Deformation an der Rißoberfläche werden diskutiert.

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This work is supported by the Board of Scientific and Industrial Research, Orissa (India).     

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.     

Annular crack surrounding an elastic fibre in transversely isotropic elasticity

The axisymmetric problem of an infinitely long transversely isotropic elastic fibre perfectly bonded to a dissimilar transversely isotropic elastic matrix containing an annular crack is considered. The annular crack, surrounding the fibre, is subjected to prescribed longitudinal tension. A potential function approach is used to find the solution of the basic equations. The mixed boundary value problem is reduced to the solution of a singular integral equation, which is further reduced, by using Chebyshev polynomials, to a system of algebraic equations.     

Penny-shaped crack in a transversely isotropic plate of finite thickness

The effects of the material anisotropy on the stress intensity factor and on the crack shape are investigated for a penny-shaped crack in a transversely isotropic plate of finite thickness. The surfaces of the crack are subjected to uniform pressures. The plate surfaces are free from stresses for case I while smooth-clamp conditions are prescribed on the plate surfaces for case II. The techniques of Hankel transforms are used to obtain solutions for both cases. The solutions are largely written in terms of the sum and difference of the characteristic roots so that the results can easily be seen as real-value functions for both real and complex roots.Exact expressions for the stress intensity factor and the crack-shape function are obtained as products of dimensional quantities and nondimensional functions which are the stress intensity correction factor and the normalized crack shape function. The nondimensional functions were calculated numerically for three different typical materials which involved both real and complex characteristic roots. The numerical results clearly reveal the effects of the material anisotropy on the stress intensity factor and on the opening of the crack.

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Résumé On a étudié les effets de l'anisotropie du matériau sur le facteur d'intensité de contrainte et sur la forme d'une fissure dans le cas d'une fissuration en demi-lune située dans une plaque transversalement isotrope d'épaisseur finie. Les surfaces de la fissure ont été soumises à des pressions uniformes. Les surfaces de la plaque étaient libres de contrainte dans le cas I tandis que l'on prévoyait des conditions correspondant à un clamage léger sur les surfaces de la plaque dans un cas II. Les techniques de transformées de Hankel ont été utlisées pour obtenir les solutions dans les deux cas. Les solutions ont été exprimées en terme de somme et de différence de racines caractéristiques, de sorte que les résultats peuvent aisément être déduits comme des fonctions à valeur réelle de racine réelle et de racine complexe.Les expressions exactes pour le facteur d'intensité de contrainte et pour la fonction de forme de la fissure ont été obtenues comme les produits de fonctions à quantité dimensionnelle et non dimensionnelle qui sont le facteur de correction de l'intensité de contrainte et une fonction de forme de la fissure normalisée. Les fonctions sans dimension ont été calculées par voie numérique dans le cas de trois matériaux différents et typiques, mettant en oeuvre des racines caractéristiques réelles et des racines caractéristiques complexes. Les résultats numériques ont montré clairement les effets de l'anisotropie des matériaux sur le facteur d'intensité de contrainte et sur l'ouverture de la fissure.