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.     

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.     

The method of fundamental solutions for nonlinear elliptic problems

A meshless method was presented, which couples the method of fundamental solutions (MFS) with radial basis functions (RBFs) and the analog equation method (AEM), to solve nonlinear problems. In this method, the AEM is used to convert the nonlinear governing equation into a corresponding linear inhomogeneous equation, so that a simpler fundamental solution can be employed. Then, the RBFs and the MFS are, respectively, used to construct the expressions of particular and homogeneous solution parts of the substitute equation, from which the approximate solution of the original problem and its derivatives involved in the governing equation are represented via the unknown coefficients. After satisfying all equations of the original problem at collocation points, a nonlinear system of equations can be obtained to determine all unknowns. Some numerical tests illustrate the efficiency of the method proposed.     

The method of fundamental solutions for inhomogeneous elliptic problems

We investigate the use of the Method of Fundamental Solutions (MFS) for solving inhomogeneous harmonic and biharmonic problems. These are transformed to homogeneous problems by subtracting a particular solution of the governing equation. This particular solution is taken to be a Newton potential and the resulting homogeneous problem is solved using the MFS. The numerical calculations indicate that accurate results can be obtained with relatively few degrees of freedom. Two methods for the special case where the inhomogeneous term is harmonic are also examined.     

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.     

The Almansi method of fundamental solutions for solving biharmonic problems

A new fundamental solutions method for the numerical solution of two-dimensional biharmonic problems is described. In this method, which is based on the Almansi representation of a biharmonic function in the plane, the approximate solution is expressed in terms of fundamental solutions of Laplace's equation, and is determined by a least squares fit of the boundary conditions. The results of numerical experiments which demonstrate the efficacy of the method are presented.     

The method of fundamental solutions for free surface Stefan problems

In this paper, free surface problems of Stefan-type for the parabolic heat equation are investigated using the method of fundamental solutions. The additional measurement necessary to determine the free surface could be a boundary temperature, a heat flux or an energy measurement. Both one- and two-phase flows are investigated. Numerical results are presented and discussed.     

The method of fundamental solutions for inverse 2D Stokes problems

A numerical scheme based on the method of fundamental solutions is proposed for the solution of two-dimensional boundary inverse Stokes problems, which involve over-specified or under-specified boundary conditions. The coefficients of the fundamental solutions for the inverse problems are determined by properly selecting the number of collocation points using all the known boundary values of the field variables. The boundary points of the inverse problems are collocated using the Stokeslet as the source points. Validation results obtained for two test cases of inverse Stokes flow in a circular cavity, without involving any iterative procedure, indicate the proposed method is able to predict results close to the analytical solutions. The effects of the number and the radius of the source points on the accuracy of numerical predictions have also been investigated. The capability of the method is demonstrated by solving different types of inverse problems obtained by assuming mixed combinations of field variables on varying number of under- and over-specified boundary segments.     

The method of fundamental solutions for problems of free vibrations of plates

In this paper a new boundary method for problems of free vibrations of plates is presented. The method is based on mathematically modelling of the physical response of a system to external excitation over a range of frequencies. The response amplitudes are then used to determine the resonant frequencies. So, contrary to the traditional scheme, the method described does not involve evaluation of determinants of linear systems. The method shows a high precision in simply and doubly connected domains. The results of the numerical experiments justifying the method are presented.     

The method of approximate fundamental solutions (MAFS) for Stefan problems

The paper presents a new meshless numerical technique for solving one and two-dimensional Stefan problems. The technique presented is based on the use of the delta-shaped functions and the method of approximate fundamental solutions (MAFS) first suggested for solving elliptic problems and heat equations in domains with fixed boundaries. The one-dimensional problems in the plane and cylindrical geometries are considered. The numerical examples are presented and the results are compared with the analytical solutions. The comparison shows that the method presented provides a very high precision in determining the position of the moving boundary even for degenerate and singular problems when a region initially has zero thickness. The same technique was developed for 2D Stefan problems with completely or partially unknown boundary.     

Bifurcation of cavitation solutions for incompressible transversely isotropic hyper-elastic materials

In this paper, the bifurcation problem of void formation and growth in a solid circular cylinder, composed of an incompressible, transversely isotropic hyper-elastic material, under a uniform radial tensile boundary dead load and an axial stretch is examined. At first, the deformation of the cylinder, containing an undetermined parameter-the void radius, is given by using the condition of incompressibility of the material. Then the exact analytic formulas to determine the critical load and the bifurcation values for the parameter are obtained by solving the differential equation for the deformation function. Thus, an analytic solution for bifurcation problems in incompressible anisotropic hyper-elastic materials is obtained. The solution depends on the degree of anisotropy of the material. It shows that the bifurcation may occur locally to the right or to the left, depending on the degree of anisotropy, and the condition for the bifurcation to the right or to the left is discussed. The stress distributions subsequent to the cavitation are given and the jumping and concentration of stresses are discussed. The stability of solutions is discussed through comparison of the associated potential energies. The bifurcation to the left is a `snap cavitation'. The growth of a pre-existing void in the cylinder is also observed. The results for a similar problem in three dimensions were obtained by Polignone and Horgan.     

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.     

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.     

The boundary element method based on the three-dimensional elastostatic fundamental solution for the orthotropic multilayered space: Application to composite materials

In this work a specialization of the boundary element technique, which makes use of the three-dimensional fundamental solution for a point load acting in the interior of an infinite orthotropic multilayered space is presented. This new formulation is specially suited for the treatment of three-dimensional composite laminated material problems whose geometries contain many parallel planar surfaces-interfaces-.Two, three-dimensional, problems are numerically analyzed by using the above procedure. These are the problems of pressurized circular and elliptical holes in infinite laminated plates. The accuracy of our scheme is established by direct comparison of our results with available published data.     

Green's function solutions for semi-infinite transversely isotropic materials

The Green's function problem of a semi-infinite transversely isotropic medium with the plane boundary parallel to the plane of isotropy is solved by using the potential function method. The Green's function solutions are expressed in terms of harmonic and bi-harmonic functions which are obtained by the separation of variables method. Closed form solutions for point forces applied in the interior of the medium are obtained. The present solution reduces to Sveklo's results when the point force is normal to the plane of isotropy and $\text{√(C}{}_{11}\text{C}{}_{33}\text{)-C}{}_{13}\text{-2C}{}_{44}\text{≠ 0}$. The Green's function solutions of Michell, Lekhnitzki and Hu, which deal with point forces applied at the free surface of a half-plane and $\text{√(C}{}_{11}\text{C}{}_{33}\text{)-C}{}_{13}\text{-2C}{}_{44}\text{≠ 0}$, can also be reproduced from the present approach. Furthermore, the present solution can be reduced to the results of Mindlin for semi-infinite isotropic materials by suitable substitutions of elastic constants.