3D Frequency domain BEM for solving dipolar gradient elastic problems

A three dimensional (3D) boundary element method (BEM) for treating time harmonic problems in linear elastic structures exhibiting microstructure effects is presented. These microstructural effects are taken into account with the aid of the dipolar gradient elastic theory, which is the simplest dynamic version of Mindlins generalized elastic theory. A variational statement is established to determine all possible classical and non-classical (due to gradient terms) boundary conditions of the general boundary value problem. The dipolar gradient frequency domain elastodynamic fundamental solution is explicitly derived and used to construct the boundary integral representation of the solution with the aid of a reciprocal integral identity. In addition to a boundary integral representation for the displacement, a boundary integral representation for its normal derivative is also necessary for the complete formulation of a well posed problem. Surface quadratic quadrilateral boundary elements are employed and the discretization is restricted only to the boundary. The solution procedure is described in detail. A numerical example serves to illustrate the method and demonstrate its accuracy     

An advanced boundary element method for solving 2D and 3D static problems in Mindlin's strain‐gradient theory of elasticity

An advanced boundary element method (BEM) for solving two‐ (2D) and three‐dimensional (3D) problems in materials with microstructural effects is presented. The analysis is performed in the context of Mindlin's Form‐II gradient elastic theory. The fundamental solution of the equilibrium partial differential equation is explicitly derived. The integral representation of the problem, consisting of two boundary integral equations, one for displacements and the other for its normal derivative, is developed. The global boundary of the analyzed domain is discretized into quadratic line and quadrilateral elements for 2D and 3D problems, respectively. Representative 2D and 3D numerical examples are presented to illustrate the method, demonstrate its accuracy and efficiency and assess the gradient effect on the response. The importance of satisfying the correct boundary conditions in gradient elastic problems is illustrated with the solution of simple 2D problems. Copyright © 2010 John Wiley & Sons, Ltd.     

On the numerical solution of the Dirichlet initial boundary-value problem for the heat equation in the case of a torus

The initial-boundary-value problem for the heat equation in the case of a toroidal surface with Dirichlet boundary conditions is considered. This problem is reduced to a sequence of elleptic boundary-value problems by a Laguerre transformation. The special integral representation leads to boundary-integral equations of the first kind and the toroidal surface gives one-dimensional integral equations with a logarithmic singularity. The numerical solution is realized by a trigonometric quadrature method in cases of open or closed smooth boundaries. The results of some numerical experiments are presented.     

Static and harmonic BEM solutions of gradient elasticity problems with axisymmetry

An advanced boundary element method/fast Fourier transform (BEM/FFT) methodology for treating static and time harmonic axisymmetric problems in linear elastic structures exhibiting microstructure effects, is presented. These microstructure effects are taken into account with the aid of a simple strain gradient elastic theory proposed by Aifantis and co-workers [Aifantis (1992), Altan and Aifantis (1992), Ru and Aifantis (1993)]. Boundary integral representations of both static and dynamic gradient elastic problems are employed. Boundary quantities, classical and non-classical (due to gradient terms) boundary conditions are expanded in complex Fourier series in the circumferential direction and the problem is decomposed into a series of problems, which are solved by the BEM by discretizing only the surface generator of the axisymmetric body. The BEM integrations are performed by FFT in the circumferential directions simultaneously for all Fourier coefficients and by Gauss quadrature in the generator direction. All the strongly singular integrals are computed directly by employing highly accurate three-dimensional integration techniques. The Fourier transform solution is numerically inverted by the FFT to provide the final solution. The accuracy of the proposed boundary element methodology is demonstrated by means of representative numerical examples.The authors acknowledge with thanks for the support provided by I.K.Y. through the program IKYDA 2002 (scientific cooperation between the University of Patras, Greece and the Ruhr-University Bochum, Germany).     

Boundary element solution of 3-D wave scatter problems in a poroelastic medium

This study details the development of boundary integral equations suitable for treating problems involving the scatter of a plane harmonic wave by an inclusion embedded in an infinite poroelastic medium. The pore pressure-solid displacement form of the harmonic equations of motion are developed from the form of the equations originally presented by Biot. Fundamental solutions and a generalized reciprocal work relation are developed, and these are used to formulate a solution representation in terms of an integral over the inclusion surface. The corresponding boundary integral equations are developed in a form that is integrable in the usual sense, eliminating the need to evaluate Cauchy principal value integrals. These boundary integral equations are discretized and implemented into a boundary element computer program. The so-called forbidden frequency problem which causes computational difficulties in boundary integral treatments of wave scatter in elastic and acoustic media is shown to be absent in the poroelastic case. Numerical results obtained from the boundary element program are compared with analytical results for some test problems, and the program appears to produce accurate results at moderate frequencies.     

A meshless local boundary integral equation method for solving transient elastodynamic problems

A new local boundary integral equation (LBIE) method for solving two dimensional transient elastodynamic problems is proposed. The method utilizes, for its meshless implementation, nodal points spread over the analyzed domain and employs the moving least squares (MLS) approximation for the interpolation of the interior and boundary variables. On the global boundary, displacements and tractions are treated as independent variables. The local integral representation of displacements at each nodal point contains both surface and volume integrals, since it employs the simple elastostatic fundamental solution and considers the acceleration term as a body force. On the local boundaries, tractions are avoided with the aid of the elastostatic companion solution. The collocation of the local boundary/volume integral equations at all the interior and boundary nodes leads to a final system of ordinary differential equations, which is solved stepwise by the -Wilson finite difference scheme. Direct numerical techniques for the accurate evaluation of both surface and volume integrals are employed and presented in detail. All the strongly singular integrals are computed directly through highly accurate integration techniques. Three representative numerical examples that demonstrate the accuracy of the proposed methodology are provided.     

Numerical solution of elastic inclusion problem using complex variable boundary integral equation

Based on a complex variable boundary integral equation (CVBIE) suggested previously, this paper provides a numerical solution for the elastic inclusion problem using CVBIE. A dissimilar elastic inclusion is embedded in the infinite matrix. The original problem is decomposed into two problems. One is an interior boundary value problem (BVP) for the elastic inclusion, while the other is an exterior BVP for the matrix with notch. Both problems are connected by conventional boundary integral equations (BIEs) in complex variables. After performing discretization for the coupled BIEs, the inverse matrix technique is suggested to solve the relevant algebraic equations. Based on the properties of some integral operators, three ways for the inverse matrix technique are suggested. Several numerical examples are carried out to prove the efficiency of the suggested method.     

Boundary integral equation method for two dissimilar elastic inclusions in an infinite plate

This paper provides a numerical solution for an infinite plate containing two dissimilar elastic inclusions, which is based on complex variable boundary integral equation (CVBIE). The original problem is decomposed into two problems. One is an interior boundary value problem (BVP) for two elastic inclusions, while other is an exterior BVP for the matrix with notches. After performing discretization for the coupled boundary integral equations (BIEs), a system of algebraic equations is formulated. The inverse matrix technique is suggested to solve the relevant algebraic equations, which can avoid using the assembling of some matrices. Several numerical examples are carried out to prove the efficiency of suggested method and the hoop stress along the interface boundary is evaluated.     

Traction BIE solutions for flat cracks

The paper deals with the numerical solution techniques for the traction boundary integral equation (BIE), which describes the opening (and sliding) displacements of the surface of the traction loaded crack or arbitrary planform embedded in an elastic infinite body (buried crack problem). The traction BIE is a singular integral equation of the first kind for the displacement gradients. Its solution poses a number of numerical problems, such as the presence of derivatives of the unknown function in the integral equation, the modeling of the crack front displacement gradient singularity, and the regularization of the equation's singular kernels. All of the above problems have been addressed and solved. Details of the algorithm are provided. Numerical results of a number of crack configurations are presented, demonstrating high accuracy of the method.     

Transient dynamic analysis of a fluid-saturated porous gradient elastic column

The dynamic response of a fluid-saturated porous gradient elastic column to a transient disturbance is determined analytically and numerically. The basic dynamic theory of a fluid-saturated poroelastic medium due to Biot is modified by replacing the classical linear elastic model of the solid skeleton by the simple gradient elastic model of Mindlin with just one elastic constant (internal length scale) in addition to the classical ones. Thus, the new theory, which is presently restricted to the one-dimensional case, can take into account the microstructural effects of the solid skeleton. After the establishment of appropriate boundary and initial conditions, the one-dimensional dynamic column problem is solved analytically with the aid of the Laplace transform with respect to time. The time domain response is finally obtained by a numerical inversion of the transformed solution. The effect of the solid microstructure on the response is assessed and discussed.     

Three-dimensional structural vibration analysis by the Dual Reciprocity BEM

The dual reciprocity boundary element method (DR/BEM) is employed for the analysis of free and forced vibrations of three-dimensional elastic solids. Use of the elastostatic fundamental solution in the integral formulation of elastodynamics creates an inertial volume integral in addition to the boundary ones. This volume integral is transformed into a surface integral by invoking the reciprocal theorem. A general analytical method is described for the closed form determination of the particular solutions of the displacement and traction tensors corresponding to any radial basis function employed in the transformation process. The simple but effective 1+r radial basis function is used in the applications of this paper. Quadratic continuous and discontinuous 9-noded boundary elements are used in the analysis. Free vibrations are studied by solving the corresponding eigenvalue problem iteratively. Harmonic forced vibration problems are solved directly in the frequency domain. Transient forced vibration problems are solved by integrating the equations of motion stepwise with the aid of various algorithms. Interior collection points are also used for assessing the accuracy of the method. Two numerical examples involving free and forced vibrations of a sphere and a cube are presented in detail.     

A direct boundary integral equation formulation for the Oseen flow past a two-dimensional cylinder of arbitrary cross-section

Summary A novel boundary integral formulation is presented for the direct solution of the classical problem of slow flow past a two-dimensional cylinder of arbitrary cross section in an unbounded viscous medium, the equations of motion having first been linearised by the Oseen approximation. It is shown how the governing partial differential equations of motion, together with the no-slip boundary conditions on the cylinder, may be reformulated as a pair of coupled integral equations of the second kind, which may be manipulated further to yield the lift and drag coefficients explicitly, as well as flow characteristics anywhere in the flowfield.The present formulation requires a non-iterative numerical solution procedure which is applicable to low Reynolds number flows. The method is not restricted in its ability to deal with complicated cylinder geometries, as the discretisation of only the cylinder surface is required.Results of the present method are shown to be in good agreement with those of previous analytical and numerical investigations.With 2 Figures     

On integral equation approaches to the mechanics of fibre-crack interaction

The paper examines the problem of a penny-shaped crack which is formed by the development of a crack in both the fibre and the matrix of a composite consisting of an isolated elastic fibre located in an elastic matrix of infinite extent. The composite region is subjected to a uniform strain field in the direction of the fibre. The paper presents two integral-equation based approaches for the analysis of the problem. The first approach considers the formulation of the complete integral equations governing the associated elasticity problem for a two material region. The second approach considers the boundary integral equation formulation of the problem. Both methods entail the numerical solution of the governing integral equations. The solutions to these integral equations are used to evaluate the stress intensity factor at the boundary of the penny-shaped crack.     

A 2D time-domain BEM for transient wave scattering analysis by a crack in anisotropic solids

A two-dimensional (2D) time-domain boundary element method (BEM) is presented in this paper for transient analysis of elastic wave scattering by a crack in homogeneous, anisotropic and linearly elastic solids. A traction boundary integral equation formulation is applied to solve the arising initial-boundary value problem. A numerical solution procedure is developed to solve the time-domain boundary integral equations. A collocation method is used for the temporal discretization, while a Galerkin-method is adopted for the spatial discretization of the boundary integral equations. Since the hypersingular boundary integral equations are first regularized to weakly singular ones, no special integration technique is needed in the present method. Special attention of the analysis is devoted to the computation of the scattered wave fields. Numerical examples are given to show the accuracy and the reliability of the present time-domain BEM. The effects of the material anisotropy on the transient wave scattering characteristics are investigated.     

Edge crack in an elastic layer resting on Winkler foundation

The paper deals with the stress analysis near a crack tip in an elastic layer resting on Winkler foundation. The edge crack is assumed to be normal to the lower boundary plane. The upper surface of the layer is loaded by given forces normal to the boundary. The considered problem is solved by using the method of Fourier transforms and dual integral equations, which are reduced to a Fredholm integral equation of the second kind. The stress intensity factor is given in the term of solution of the Fredholm integral equation and some numerical results are presented.     

ELASTOHYDRODYNAMIC LUBRICATION ANALYSIS OF LAYERED LINE CONTACT BY THE BOUNDARY ELEMENT METHOD

This paper presents a numerical routine to compute the contact characteristics of elastomer layered cylinders lubricated by isoviscous liquids. The indentation of the elastic layer is calculated from boundary integral equations which are solved by linear and quadratic boundary element methods for a finite plane model and a circular representation of the junction. The hydrodynamic equation is also transformed into a boundary integral equation and solved by Simpson's rule. Some factors which possibly affect numerical accuracy are examined. Examples for finite plane and circular layer are analysed with reference to parameters for printing press roller contact, in which results are obtained for the indentation, film thickness and liquid pressure, as well as internal stresses through the simultaneous solution of the elasticity and hydrodynamic equations. The results show that high precision is easily achieved and the method is efficient for such layered problems.     

Micromechanical modelling of size effects in failure of porous elastic solids using first order plane strain gradient elasticity

In this paper a first order porous strain gradient elasticity model is presented. The constitutive equations have been obtained by higher order homogenization and the model is used with a failure criterion in order to discuss size effects in failure of porous elastic solids. The model contains two microstructural parameters namely the void volume fraction and the half void spacing. After an extended numerical validation of the porous strain gradient elasticity model, the boundary value problem of a plate with a hole under bi- and uniaxial remote tension is investigated. The numerical simulations have been performed varying both microstructural parameters in order to study the influence of different microstructural dimensions on the onset of macroscopic failure. The numerical results show that the presented model is able to predict size effects and that size effects in failure do not only depend on the microstructural properties but also on the macroscopic geometry, loading conditions, and the failure mechanism.     

Thermal stresses near a penny-shaped crack in an elastic sphere embedded in an infinite elastic space

This paper gives an analysis of the distribution of thermal stresses in a sphere which is bonded to an infinite elastic medium. The thermal and the elastic properties of the sphere and the elastic infinite medium are assumed to be different. The penny-shaped crack lies on the diametral plane of the sphere and the centre of the crack is the centre of the sphere. By making a suitable representation of the temperature function, the heat conduction problem is reduced to the solution of a Fredholm integral equation of the second kind. Using suitable solution of the thermoelastic displacement differential equation, the problem is then reduced to the solution of a Fredholm integral equation, in which the solution of the earlier integral equation arising from heat conduction problem occurs as a known function. Numerical solutions of these two Fredholm integral equations are obtained. These solutions are used to evaluate numerical values for the stress intensity factors. These values are displayed graphically.     

Surface crack problem for functionally graded magnetoelectroelastic coating-homogeneous elastic substrate system under anti-plane mechanical and in-plane electric and magnetic loading

A functionally graded magnetoelectroelastic material layer bonded to a homogeneous elastic substrate is investigated. The functionally graded magnetoelectroelastic layer contains a surface crack that is perpendicular to the surface of the medium. The structure is subjected to anti-plane mechanical and in-plane electric and magnetic loads, the crack problem involves the anti-plane elastic field coupled with the in-plane electric and magnetic field. The elastic layer can be an ideal insulator or an ideal conductor. Integral transform and dislocation density functions are employed to reduce the problem to the solution of a system of singular integral equations. Numerical results show the influences of the material gradient parameter and crack configuration on field intensity factors and energy release rates of the functionally graded magnetoelectroelastic coating-homogeneous elastic substrate structure.