A new variable‐order singular boundary element for two‐dimensional stress analysis

A new variable‐order singular boundary element for two‐dimensional stress analysis is developed. This element is an extension of the basic three‐node quadratic boundary element with the shape functions enriched with variable‐order singular displacement and traction fields which are obtained from an asymptotic singularity analysis. Both the variable order of the singularity and the polar profile of the singular fields are incorporated into the singular element to enhance its accuracy. The enriched shape functions are also formulated such that the stress intensity factors appear as nodal unknowns at the singular node thereby enabling direct calculation instead of through indirect extrapolation or contour‐integral methods. Numerical examples involving crack, notch and corner problems in homogeneous materials and bimaterial systems show the singular element's great versatility and accuracy in solving a wide range of problems with various orders of singularities. The stress intensity factors which are obtained agree very well with those reported in the literature. Copyright © 2002 John Wiley & Sons, Ltd.     

Hyper-singular boundary element method for elastoplastic fracture mechanics analysis with large deformation

In this paper a two-dimensional hyper-singular boundary element method for elastoplastic fracture mechanics analysis with large deformation is presented. The proposed approach incorporates displacement and the traction boundary integral equations as well as finite deformation stress measures, and general crack problems can be solved with single-region formulations. Efficient regularization techniques are applied to the corresponding singular terms in displacement, displacement derivatives and traction boundary integral equations, according to the degree of singularity of the kernel functions. Within the numerical implementation of the hyper-singular boundary element formulation, crack tip and corners are modelled with discontinuous elements. Fracture measures are evaluated at each load increment, using the J-integral. Several cases studies with different boundary and loading conditions have been analysed. It has been shown that the new singularity removal technique and the non-linear elastoplastic formulation lead to accurate solutions.     

Taylor expansions for singular kernels in the boundary element method

The problem treated is the integration of singular functions which arise in three-dimensional isoparametric formulations of boundary integral equations. A Taylor expansion in the local parametric co-ordinates is developed for the singular integrand, so allowing singular terms to be integrated in closed form, even for curved surface elements. The remainder integral obtained by subtracting out the worst singularities is integrated by repeated Gaussian quadrature. Two groups of tests are presented. First, the accuracy of the integrations has been checked for plane parallelograms (for which exact solutions have been developed) and for curved elements on a sphere. Secondly, results from complete boundary element calculations based on point collocation have been compared with known analytical solutions to two problems; zonal surface harmonics on a sphere and the capacitance of an ellipsoid. The agreement obtained with few degrees-of-freedom suggests that errors which have previously been attributed to point collocation might have arisen in the numerical integration.     

Boundary element formulation for 3D transversely isotropic cracked bodies

The boundary traction integral representation is obtained in elasticity when the classical displacement representation is differentiated and combined according to Hooke's law. The use of both traction and displacement integral representations leads to a mixed (or dual) formulation of the BEM where the discretization effort for crack problems is much smaller than in the classical formulation. A boundary element analysis of three‐dimensional fracture mechanics problems of transversely isotropic solids based on the mixed formulation is presented in this paper. The hypersingular and strongly singular kernels appearing in the formulation are regularized by using two terms of the displacement series expansion and one term of the traction expansion, at the collocation point. All the remaining integrals are analytically evaluated or transformed by means of Stokes' theorem into regular or weakly singular integrals, which are numerically computed. The method is general and can be used for elements of any shape including quarter‐point crack front elements. No change of co‐ordinates is required for the integration. The formulation as presented in this paper is something as clear, general and easy to handle as the classical BE formulation. It is used in combination with three‐dimensional quadratic and quarter‐point elements to obtain accurate results for several different crack problems. Cracks in boundless and finite transversely isotropic domains are studied. The meshes are simple and include only discretization of the crack and the external boundary. The obtained results are in good agreement with those existing in the literature. Copyright © 2004 John Wiley & Sons, Ltd.     

On the calculation of boundary stresses with the Somigliana stress identity

The computation of boundary stresses by Boundary Element Method (BEM) is usually performed either by expressing the boundary tractions in a local co-ordinate system, calculating the remaining stresses by shape function differentiation and inserting into Hooke's law or recently also by solving the hypersingular integral equation for the stresses. While direct solution of the hypersingular integral equation, the so-called Somigliana stress identity, has been shown to be more reliable, the interpretation and numerical treatment of the hypersingularity causes a number of problems. In this paper, the limiting procedure in taking the load point to the boundary is carried out by leaving the boundary smooth and the contributions of all different types of singularities to the boundary integral equation are studied in detail. The hypersingular integral in the arising boundary integral equation is then reduced to a strongly singular one by considering a traction free rigid body motion. For the numerical treatment, an algorithm for multidimensional Cauchy Principal Value (CPV) integrals is extended that is applicable for the calculation of boundary stresses. Moreover, the shape of the surrounding of the singular point is studied in detail. A numerical example of elastostatics confirms the validity of the proposed method.     

The dual boundary contour method for two-dimensional crack problems

This paper concerns the dual boundary contour method for solving two-dimensional crack problems. The formulation of the dual boundary contour method is presented. The crack surface is modeled by using continuous quadratic boundary elements. The traction boundary contour equation is applied for traction nodes on one of the crack surfaces and the displacement boundary contour equation is applied for displacement nodes on the opposite crack surface and noncrack boundaries. The direct calculation of the singular integrals arising in displacement BIEs is addressed. These singular integrals are accurately evaluated with potential functions. The singularity subtraction technique for determining the stress intensity factor K_I, K_II and the T-term are developed for mixed mode conditions. Some two-dimensional examples are presented and numerical results obtained by this approach are in very good agreement with the results of the previous papers. This revised version was published online in August 2006 with corrections to the Cover Date.     

General BE approach for three-dimensional dynamic fracture analysis

A general mixed boundary element approach for three-dimensional dynamic fracture mechanics problems is presented in this paper. A mixed traction-displacement integral equation formulation in the frequency domain is used. The hypersingular and strongly singular kernels are regularized by analytical transformations yielding an easy to implement BE approach. Nine-node quadrilateral and six-node triangular continuous quadratic elements are used for external boundaries and crack surfaces. The crack front elements have their mid node at one quarter of the element length allowing for a proper representation of the crack surface displacement. The present approach is intended for the frequency domain analysis of fracture mechanics problems of any general 3D geometry; i.e. boundless or bounded regions, single or multiple, surface or internal cracks. Transient dynamic problems are studied using the FFT algorithm. The numerical results presented show the robustness and accuracy of the approach which requires a reasonable number of elements and degrees of freedom.     

An investigation of dynamic interaction between multiple cracks and inclusions by TDBEM

In this paper, the dynamic interaction between multiple inclusions and cracks is studied by the time-domain boundary element method (TDBEM). To deal with this problem, two kinds of time-domain boundary integral equations together with the sub-region technique are applied. The cracked solid is divided into homogeneous and isotropic sub-regions bounded by the interfaces between the inclusions and the matrix. The non-hypersingular traction boundary integral equations are applied on the crack-surfaces; while the traditional displacement boundary integral equations are used on the interfaces and the exterior boundaries. In the numerical solution procedure, square-root shape functions are adopted for the crack-opening-displacements to describe the proper asymptotic behavior in the vicinity of the crack-tips. Numerical results for dynamic stress intensity factors are presented for various cases. The effects of the inclusion position, material combinations and multiple micro-cracks on the dynamic stress intensity factors are discussed.     

Singular boundary elements for the analysis of cracks in plane strain

Straight and curved cracks are modelled by direct formulation boundary elements, of geometry defined by Hermitian cubic shape functions. Displacement and traction are interpolated by the Hermitian functions, supplemented by singular functions which multiply stress intensity factors corresponding to the dominant modes of crack opening in which displacement is proportional to the square root of distance r from the crack tip, and subdominant modes in which it is proportional to r^1·5. The singular functions extend over many boundary elements on each crack face. A nodal collocation scheme is used, in which additional boundary integral equations are obtained by differentiation of the equation obtained from Betti's theorem. The hypersingular kernels of the equations so derived are integrated by consideration of trial displacement fields of subdomains lying to either side of the crack. Examples are shown of the analysis of buried and edge cracks, to demonstrate the effects of modelling subdominant modes and extending singular shape functions over many elements.     

Boundary-only element analysis of crack contact under thermal shock

This paper presents a multi-domain boundary integral equation approach of thermally excited crack surface interference observed under thermal transient conditions. According to this model, crack surface displacements and tractions are not free but subject to constraints simulating contact and prevailing overlapping of crack surfaces. The multi-domain approach allows the two faces of a crack to be modeled in independent sub-regions of the body, avoiding singularity difficulties and making it possible to analyze crack closure problems with contact stresses over part of the cracked faces. Crack-tip singularities are modeled through quarter-point elements. In order to approach the interference configuration, the interfacial traction distribution and solve the resulting equations, an iterative numerical procedure is applied. The numerical solution of this non-linear problem yields crack surface displacements and consequently the crack surface interference. Fracture parameters are evaluated from nodal displacements of singular elements utilizing proper formulas. Various results are illustrated and discussed for edge-cracks subjected to steady-state or thermal shock conditions.     

A direct traction boundary integral equation method for three-dimension crack problems in infinite and finite domains

This paper presents a direct traction boundary integral equation method (TBIEM) for three-dimensional crack problems. The TBIEM is based on the traction boundary integral equation (TBIE). The TBIE is collocated on both the external boundary and one of the crack surfaces. The displacements and tractions are used as unknowns on the external boundary and the relative crack opening displacements (CODs) are introduced as unknowns on the crack surface. In our implementation, all the surfaces of the considered structure are discretized into discontinuous elements to satisfy the continuity requirement for the existence of finite-part integrals, and special crack-front elements are constructed to capture the crack-tip behavior. To calculate the finite-part integrals, an adaptive singular integral technique is proposed. The stress intensity factors (SIFs) are computed through a modified COD extrapolation method. Numerical examples of SIFs computation are presented to demonstrate the accuracy and efficiency of our method.     

Analysis of cracked piezoelectric solids by a mixed three-dimensional BE approach

This paper has two main objectives in relation to the analysis of three-dimensional crack problems in piezoelectric solids. The first one is to present the formulation, effective implementation and numerical treatment of a mixed boundary element technique for the study of this type of problems. The numerical procedure is based on the use of extended displacement and extended traction integral equations for external and crack boundaries, respectively. The boundary element formulation is presented with particular emphasis on numerical aspects related to singular kernels regularization and evaluation of boundary integrals. Quadratic boundary elements and quarter-point boundary elements are implemented in a computer code. By using these elements, electric and stress intensity factors are directly computed from nodal values at quarter-point elements. The second purpose is to study several realistic piezoelectric crack problems for the first time. Unbounded and bounded cracked piezoelectric three-dimensional (3D) solids with different geometries are studied. Results presented in this paper can be used as a reference for future research. Prior to the analysis of problems whose solution was previously unknown, the technique is validated by solving some simple problems with known analytical or numerical solution. Then, more realistic crack problems of engineering interest have been analysed for the first time. In all cases, results for the solid deformed shape, the crack opening displacements and the extended stress intensity factor components, are shown.     

Three-dimensional singular boundary elements for corner and edge singularities in potential problems

It is well known that the spatial derivative of the potential field governed by the Laplace and Poisson equations can become infinite at corners (in two and three dimensions) and edges (in three dimensions). Conventional elements in the finite element and boundary element methods do not give accurate results at these singular locations. This paper describes the formulation and implementation of new singular elements for three-dimensional boundary element analysis of corner and edge singularities in potential problems. Unlike the standard element, the singular element shape functions incorporate the correct singular behavior at the edges and corners, specifically the eigenvalues, in the formulation. The singular elements are used to solve some numerical examples in electrostatics, and it is shown that they can improve the accuracy of the results for capacitance and electrostatic forces quite significantly. The effects of the size of the singular elements are also investigated.     

Implementation of thermoelastic forces in boundary element analysis of elastic contact and fracture mechanics problems

This paper presents a pseudo-body-force approach multi-domain boundary integral equation method for the analysis of thermoelastic and body-force type elastic contact and fracture mechanics problems. Using this approach only the boundaries of the bodies involved have to be discretized. The transformation of the domain integrals due to body-force and pseudo-force to their equivalent boundary integrals are shown. Also, it is shown that by employing the initial strain approach the same set of equivalent boundary integrals would be obtained. Isoparametric quadratic elements are employed to represent the geometries and the functions. This two-dimensional BEM thermoelastic implementation can be found very simple and can be applied to both harmonic and nonharmonic temperature distributions. The accuracy is asserted by applying it to several thermoelastic fracture mechanics and contact problems.     

SOLVING TRANSIENT DYNAMIC CRACK PROBLEMS BY THE HYPERSINGULAR BOUNDARY ELEMENT METHOD

Abstract— The subject of hypersingular boundary integral equations is a rapidly developing topic due to the advantages which this kind of formulation offers compared to the standard boundary integral method. The hypersingular formulation is particularly well suited for fracture mechanics problems, where there are important gradients of the stress field and singularities. This formulation for time domain antiplane problems has been recently addressed by the authors and in the present paper, the formulation for time domain plane problems is presented and applied for the first time. A mixed Boundary Element approach based on the standard integral equation and the hypersingular integral equation is developed. The mixed formulation allows for a very simple discretization of the problem, where no subregion is needed. Conforming quadratic elements are used for the crack and the external boundaries. The hypersingular integral equation is used for collocation points within the crack elements, while the standard integral representation is used for the external boundaries. Several examples with different crack geometries are studied to illustrate the possibilities of the method. The Stress Intensity Factor (S.I.F.) is very accurately computed from the crack tip opening displacements along the crack tip element. The results show that the proposed approach for S.I.F. evaluation is simple and produces accurate solutions.     

A semi-analytical solution method for problems of cohesive fracture and some of its applications

Cohesive zone models are extensively used for the failure load estimates for structure elements with cracks. This paper focuses on some features of the models associated with the failure load and size of the cohesive zone predictions. For simplicity, considered is a mode I crack in an infinite plane under symmetrical tensile stresses. A traction–separation law is prescribed in the crack process zone. It is assumed by the problem statement that the crack faces close smoothly. This requirement is satisfied numerically by a formulation of the modified boundary conditions. The critical state of a plate with a cohesive crack is analyzed using singular integral equations. A numerical procedure is proposed to solve the obtained systems of integral equations and inequalities. The presented solution is in agreement with other published results for some limiting cases. Thus, an effective methodology is devised to solve crack mechanics problems within the framework of a cohesive zone model. Using this methodology, some problems are solved to illustrate the (i) influence of shape parameters of traction–separation law on the failure load, (ii) ability to account for contact stress for contacting crack faces, (iii) influence of getting rid of stress finiteness condition in the problem statement.     

HYPERSINGULAR BEM FOR TRANSIENT ELASTODYNAMICS

The topic of hypersingular boundary integral equations is a rapidly developing one due to the advantages which this kind of formulation offers compared to the standard boundary integral one. In this paper the hypersingular formulation is developed for time-domain antiplane elastodynamic problems. Firstly, the gradient representation is found from the displacement one, removing the strong singularities (Dirac's delta functions) which arise due to the differentiation process. The gradient representation is carried to the boundary through a limiting process and the resulting equation is shown to be consistent with the static formulation. Next, the numerical treatment of the traction boundary integral equation and its application to crack problems are presented. For the boundary discretization, conforming quadratic elements are tested, which are introduced in this paper for the first time, and it is shown that the results are very good in spite of the lesser number of unknowns of this approach in comparison to the non-conforming element alternative. A procedure is devised to numerically perform the hypersingular integrals that is both accurate and versatile. Several crack problems are solved to show the possibilities of the method. To this end both straight and curved elements are employed as well as regular and distorted quarter point elements.     

Three-dimensional boundary element analysis of electromagnetic wave scattering by penetrable bodies

A frequency domain boundary element methodology of solving three dimensional electromagnetic wave scattering problems by dielectric particles is reported. The method utilizes a computationally attractive surface integral equation containing only weakly and strongly singular integrals in the contrast to most formulations involving not only strongly singular but hypersingular integrals as well. The main advantage of this integral equation is the fact that its strongly singular part is similar to the one appearing in the corresponding integral equation of dynamic elasticity. Thus, well known advanced integration techniques used successfully in elastic scattering problems can be directly applied to the present analysis. Both continuous and discontinuous quadratic elements are employed in order to accurately treat dielectric scatterers with smooth and piecewise smooth boundaries. Numerical examples dealing with three dimensional electromagnetic wave scattering problems demonstrate the accuracy and efficiency of the proposed boundary element formulation.     

Limiting forms for surface singularity distributions when the field point is on the surface

Scalar and vector mathematical identities involving an integral of singularities distributed over a surface and sometimes over a field can be employed to define field values of a quantity of interest. As the volume excluding the singular point from the field tends to zero, the field value is derived. The expressions that result become singular as the point of interest in the field approaches the boundary. Derivation of limiting integral expressions as the field point tends to the surface having a distribution of first and second degree singularities is the main task reported. The limiting expressions for vector values require evaluation as generalized Cauchy Principal-Value Integrals for which some aspect of symmetry in a local region excluding the singularity is required. A contribution from the integral over the local region doubles the value of the identities at a point on the boundary. For a doublet distribution, a singular term arises from the local-region integration that cancels a similar singularity in the integral over the remaining surface. This local contribution for doublets depends explicitly upon the shape of the local region as well as non-orthogonality of the surface coordinate axes. The resulting expressions for surface integrals reproduce known relations for line integrals in two-dimensional fields.     

Dynamic element discretization method for solving 2D traction boundary integral equations

A sufficient condition for the existence of element singular integral of the traction boundary integral equation for elastic problems requires that the tangential derivatives of the boundary displacements are Hölder continuous at collocation points. This condition is violated if a collocation point is at the junction between two standard conforming boundary elements even if the field variables themselves are Hölder continuous there. Various methods are proposed to overcome this difficulty. Some of them are rather complicated and others are too different from the conventional boundary element method. A dynamic element discretization method to overcome this difficulty is proposed in this work. This method is novel and very simple: the form of the standard traction boundary integral equation remains the same; the standard conforming isoparametric elements are still used and all collocation points are located in the interior of elements where the continuity requirements are satisfied. For boundary elements with boundary points where the field variables themselves are singular, such as crack tips, corners and other boundary points where the stress tensors are not unique, a general procedure to construct special elements has been developed in this paper. Highly accurate numerical results for various typical examples have been obtained.