Regularization of hypersingular integrals in 3-D fracture mechanics: Triangular BE,and piecewise-constant and piecewise-linear approximations

In this article the hypersingular integrals that arise when boundary integral equation (BIE) methods are used to solve fracture mechanics problems are considered. An approach for hypersingular integral regularization is based on the theory of distribution and Green's theorems. This approach is applied for regularization of the hypersingular integrals over triangular boundary elements (BEs) for the case of piecewise-constant and piecewise-linear approximations. The hypersingular integrals are transformed into regular contour integrals that can be easily calculated analytically.     

A direct traction BIE approach for three-dimensional crack problems

A boundary element (BE) approach based on the traction boundary integral equation for the general solution of three-dimensional (3D) crack problems is presented. The hypersingular and strongly singular integrals appearing in the formulation are analytically transformed to yield line and surface integrals which are at most weakly singular. Regularization and analytical transformation of the boundary integrals is done prior to any boundary discretization. The integration process does not require of any change of coordinates and the resulting integrals can be numerically evaluated in a simple and efficient way. In order to show the generality, simplicity and robustness of the proposed approach, different flat and curved crack problems in infinite and finite domains are analyzed. A simple BE discretization strategy is adopted. The results obtained using rather course meshes are very accurate. The emphasis of this paper is on the effective application of the proposed BE approach and it is pretended to contribute to the transformation of hypersingular boundary element formulation in something as clear, general and easy to handle as the classical formulation but much better suited for fracture mechanics problems.     

A general Boundary Element Analysis of 2-D Linear Elastic Fracture Mechanics

This paper presents a boundary element method (BEM) analysis of linear elastic fracture mechanics in two-dimensional solids. The most outstanding feature of this new analysis is that it is a single-domain method, and yet it is very accurate, efficient and versatile: Material properties in the medium can be anisotropic as well as isotropic. Problem domain can be finite, infinite or semi-infinite. Cracks can be of multiple, branched, internal or edged type with a straight or curved shape. Loading can be of in-plane or anti-plane, and can be applied along the no-crack boundary or crack surface. Furthermore, the body-force case can also be analyzed. The present BEM analysis is an extension of the work by Pan and Amadei (1996a) and is such that the displacement and traction integral equations are collocated, respectively, on the no-crack boundary and on one side of the crack surface. Since in this formulation the displacement and/or traction are used as unknowns on the no-crack boundary and the relative crack displacement (i.e. displacement discontinuity) as unknown on the crack surface, it possesses the advantages of both the traditional displacement BEM and the displacement discontinuity method (DDM) and yet gets rid of the disadvantages associated with these methods when modeling fracture mechanics problems. Numerical examples of calculation of stress intensity factors (SIFs) for various benchmark problems were conducted and excellent agreement with previously published results was obtained. This revised version was published online in August 2006 with corrections to the Cover Date.     

Exact boundary integral transformation of the thermoelastic domain integral in BEM for general 2D anisotropic elasticity

In the direct formulation of the boundary element method, body-force and thermal loads manifest themselves as additional volume integral terms in the boundary integral equation. The exact transformation of the volume integral associated with body-force loading into surface ones for two-dimensional elastostatics in general anisotropy, has only very recently been achieved. This paper extends the work to treat two-dimensional thermoelastic problems which, unlike in isotropic elasticity, pose additional complications in the formulation. The success of the exact volume-to-surface integral transformation and its implementation is illustrated with three examples. The present study restores the application of BEM to two-dimensional anisotropic elastostatics as a truly boundary solution technique even when thermal effects are involved.     

Boundary integral equations and the boundary element method for fracture mechanics analysis in 2D anisotropic thermoelasticity

This paper develops the Somigliana type boundary integral equations for fracture of anisotropic thermoelastic solids using the Stroh formalism and the theory of analytic functions. In the absence of body forces and internal heat sources, obtained integral equations contain only curvilinear integrals over the solid’s boundary and crack faces. Thus, the volume integration is eliminated and also there is no need to evaluate integrals over the contours in the mapped temperature domain as it was done before. In addition to finite solids, the case of an infinite anisotropic medium with a remote thermal load is also studied. The dual boundary element method for fracture of anisotropic thermoelastic solids is developed based on the obtained boundary integral equations. Presented numerical examples show the validity and efficiency of the obtained equations in the analysis of both finite and infinite solids with cracks.     

Boundary element analysis of moiré fields in anisotropic materials

A procedure for analyzing moiré fringe patterns using boundary elements is presented. The kernels of the boundary integrals are based on anisotropic elastic Green's functions developed for bimaterial problems. The interfacial boundary conditions are incorporated in the Green's functions so the interface does not require discretization. The bimaterial kernels are also appropriate for homogeneous problems as well as degenerate isotropic problems. The moiré fringe data provide full-field displacement information and are analyzed in a least-squares sense. The numerical procedure is shown to be a logical extension of the local collocation method developed for linear elastic fracture mechanics. An example is given to investigate convergence of the method, predictions of stress, and to investigate factors influencing the analysis. It is shown that moiré fields associated with both displacement components are needed for an accurate analysis.     

A new generation of boundary element methods in fracture mechanics

In this paper the dual boundary element methods for the analysis of crack problems in fracture mechanics is presented. The formulations described include: elastostatic, thermoelastic, elastoplastic and elastodynamic. Also presented are formulations relating to anisotropic and concrete materials. Particular attention is given to crack growth modelling. Examples are presented to demonstrate the capability and robustness of this new generation of boundary element methods. This revised version was published online in July 2006 with corrections to the Cover Date.     

Indirect finite element evaluation of two-dimensional finite part integral using Fourier transformation

A broad class of engineering problems in fracture mechanics, thermal/fluid transport and electromagnetic theory involve the evaluation of two-dimensional finite part integrals of the form A method for evaluation of such integrals is developed by deriving an equivalent integral using Fourier transformation. This equivalent integral does not involve a kernel with singular behaviour. Consequently, standard numerical integration methodologies with conventional analytical evaluation techniques can be used in the finite element computations. The accuracy and convergence of the developed numerical procedure are successfully demonstrated by numerical examples for planar fracture geometries.     

A dual-reciprocity boundary element method for axisymmetric thermoelastostatic analysis of nonhomogeneous materials

The problem of determining the axisymmetric time-independent temperature and thermoelastic displacement and stress fields in a nonhomogeneous material is solved numerically by using a dual-reciprocity boundary element technique. Interpolating functions that are bounded in the solution domain but that are in relatively simple elementary forms for easy computation are constructed for treating the domain integrals in the dual-reciprocity boundary element formulation. The proposed numerical approach is successfully applied to solve several specific problems.     

Some identities for fundamental solutions and their applications to weakly-singular boundary element formulations

Some integral identities for the fundamental solutions of potential and elastostatic problems are established in this paper. With these identities it is shown that the conventional boundary integral equation (BIE), which is generally expressed in terms of singular integrals in the sense of the Cauchy principal value (CPV), and the derivative BIE, which is similarly expressed in terms of hypersingular integrals in the sense of the Hadamard finite-part (HFP), can both be written as weakly-singular integral equations in a systematic approach. Discretization of the weakly-singular BIE leads to the weakly-singular boundary element formulation equivalent to the method of using the rigid body displacement to determine the diagonal submatrices, which involve the CPV terms and the geometric matrix C, in the conventional BEM. The discretization of the weakly-singular derivative BIE possesses a similar feature, i.e. no CPV and HFP are involved. All these suggest that the practice of calculating CPV or HFP (for boundary integrals) and the geometric matrix C, either analytically or numerically, is unnecessary in the BEM. The approach developed in this paper is applicable to other problems such as plate bending, acoustics and elastodynamics.     

Elastodynamic problems by meshless local integral method: Analytical formulation

In this paper, analytical forms of integrals in the meshless local integral equation method in the Laplace space are derived and implemented for elastodynamic problems. The meshless approximation based on the radial basis function (RBF) is employed for implementation of displacements. A weak form of governing equations with a unit test function is transformed into local integral equations. A completed set of the local boundary integrals are obtained in closed form. As the closed form of the local boundary integrals are obtained, there are no domain or boundary integrals to be calculated numerically. Several examples including dynamic fracture mechanics problems are presented to demonstrate the accuracy of the proposed method in comparison with analytical solutions and the boundary element method.     

Thermoelastic fracture mechanics with regularized hypersingular boundary integral equations

This paper is concerned with thermoelastic fracture mechanics, in three-dimensional linear elasticity, using hypersingular boundary integral equations (HBIEs). The HBIEs are regularized by employing modes of deformation (or “simple solutions”). In addition to rigid-body and linear displacement modes, which have been used before for isothermal linear elastic fracture mechanics (LEFM), a new mode is employed here. This new mode is a thermal one in which the body is elevated to a uniform temperature but fully constrained (i.e., zero prescribed displacements) on its bounding surface. An existing isothermal LEFM computer code called BES is extended in this work to include thermoelastic terms. Some numerical results are presented from this new code.     

Boundary-only element solutions of 2D and 3D nonlinear and nonhomogeneous elastic problems

This paper presents a robust boundary element method (BEM) that can be used to solve elastic problems with nonlinearly varying material parameters, such as the functionally graded material (FGM) and damage mechanics problems. The main feature of this method is that no internal cells are required to evaluate domain integrals appearing in the conventional integral equations derived for these problems, and very few internal points are needed to improve the computational accuracy. In addition, one of the basic field quantities used in the boundary integral equations is normalized by the material parameter. As a result, no gradients of the field quantities are involved in the integral equations. Another advantage of using the normalized quantities is that no material parameters are included in the boundary integrals, so that a unified equation form can be established for multi-region problems which have different material parameters. This is very efficient for solving composite structural problems.     

A general exact transformation of body-force volume integral in BEM for 2D anisotropic elasticity

In a previous study (Zhang, Tan and Afagh, 1995), the present authors successfully transformed the body-force volume integrals in BEM for 2D anisotropic elasticity, to boundary ones. This restores the BEM as a truly boundary solution process for treating anisotropic bodies involving body forces. However, the formulation is valid only for problem domains which are geometrically convex and simply connected. This paper presents a general and exact transformation of the bodyforce volume integrals in BEM to line integrals for 2D anisotropic elasticity, in which the above-mentioned restriction on the geometry of the domain is eliminated. The successful implementation of the formulation is demonstrated by three practical examples.     

A direct approach for boundary integral equations with high‐order singularities

Boundary integral equations with extremely singular (i.e., more than hypersingular) kernels would be useful in several fields of applied mechanics, particularly when second‐ and third‐order derivatives of the primary variable are required. However, their definition and numerical treatment pose several problems. In this paper, it is shown how to obtain these boundary integral equations with still unnamed singularities and, moreover, how to efficiently and reliably compute all the singular integrals. This is done by extending in full generality the so‐called direct approach. Only for definiteness, the method is presented for the analysis of the deflection of thin elastic plates. Numerical results concerning integrals with singularities up to order r⁻⁴ are presented to validate the proposed algorithm. Copyright © 2000 John Wiley & Sons, Ltd.     

Numerical treatment of finite part integrals in 2-D boundary element analysis with application infracture mechanics

For the solution of problems in fracture mechanics by the boundary element method usually the subregion technique is employed to decouple the crack surfaces. In this paper a different procedure is presented. By using the displacement boundary integral equation on one side of the crack surface and the hypersingular traction boundary integral equation on the opposite side, one can renounce the subregion technique.An essential point when applying the traction boundary integral equation is the treatment of the thus arising hypersingular integrals. Two methods for their numerical computation are presented, both based on the finite part concept. One may either scale the integrals properly and use a specific quadrature rule, or one may apply the definition formula for finite part integrals and transform the resulting regular integrals into the usual element coordinate system afterwards. While the former method is restricted to linear or circular approximations of the boundary geometry, the latter one allows for arbitrary curved (e.g. isoparametric) elements. Two numerical examples are enclosed to demonstrate the accuracy of the two boundary integral equations technique compared with the subregion technique.     

The multiple Reciprocity boundary element method in elasticity: A new approach for transforming domain integrals to the boundary

This paper presents a new boundary element approach to transform domain integrals into equivalent boundary integrals. The technique, called the Multiple Reciprocity Method, is applied to 2-D elasticity problems and operates on domain integrals resulting from different types of body forces such as gravitational and centrifugal forces, as well as loadings due to linear and quadratic temperature distributions. Numerical examples are presented to demonstrate the accuracy and efficiency of the method.     

Three-dimensional dynamic fracture analysis with the dual reciprocity method in Laplace domain

In this paper, the dual reciprocity boundary element method in the Laplace domain has been developed for the analysis of three-dimensional elastodynamic fracture mechanics mixed-mode problems. The boundary element method is used to calculate the unknowns of transformed boundary displacement and traction and the domain integrals in the elastodynamic equation are transformed into boundary integrals by the use of the dual reciprocity method. The transformed dynamic stress intensity factors are determined by the crack opening displacement (COD) directly in the Laplace domain. By using Durbin's inversion technique, the dynamic stress intensity factors in the time domain are obtained. Several numerical examples are presented to demonstrate the good agreement with existing solutions.     

Dual boundary element method for three-dimensional thermoelastic crack problems

This paper describes the formulation and numerical implementation of the three-dimensional dual boundary element method (DBEM) for the thermoelastic analysis of mixed-mode crack problems in linear elastic fracture mechanics. The DBEM incorporates two pairs of independent boundary integral equations; namely the temperature and displacement, and the flux and traction equations. In this technique, one pair is applied on one of the crack faces and the other pair on the opposite one. On non-crack boundaries, the temperature and displacement equations are applied. This revised version was published online in July 2006 with corrections to the Cover Date.     

Static and dynamic contact/impact problems using fictitious forces

A new finite element model for the contact/impact problem is presented, and both its static and dynamic implementations are described. In this geometrical non-linear formulation, contact (from node to surface) is simulated through fictitious equivalent pressure along the boundary. Contrary to most existing models, this formulation entails relatively few matrix decompositions and thus is computationally inexpensive. The model is first assessed through some classical contact problems, and is subsequently applied to the fracture mechanics based analysis of a cracked dam under seismic excitation.