The stress intensity factors for pressed cracks in arbitrary shapes

A boundary element method (BEM) was specially developed for a crack under crack face pressure in arbitrary two-dimensional problems. It is based on the basic stress solutions for an infinite plane with a crack loaded by body forces and moment at arbitrary point, which were derived by Erdogan from the Kolosov-Muskhelishvili fundamental functions, and the basic solution for a crack in an infinite plate under crack surface pressure, so that the crack surface need not be modelled. Therefore, minimal modelling efforts are needed to obtain stress intensity factors with the method and its accuracy was established by comparing the obtained results with the exact SIF results and acceptable results for various problems of arbitrary shapes and loadings.     

Critical study of existing solutions for a penny-shaped interface crack, comparing with a new boundary element solution allowing for frictionless contact

A numerical study of the fundamental problem of a pressurized penny-shaped crack at the interface of two dissimilar half spaces is carried out allowing for the possibility of frictionless contact between crack faces. A new, highly accurate axi-symmetric formulation of the boundary element method (BEM) for the solution of elastic contact problems is employed. The correctness and accuracy of available predictions of different kinds for several key characteristics of the solution of this problem are checked. First, comparison of the BEM results for the near-tip contact length shows a very good agreement with some existing predictions. Second, the global solution obtained by BEM is compared with existing asymptotic solutions, obtained with both the open and the frictionless contact models. BEM results show that at the closest neighborhood to the crack tip the global solution of the problem is governed by the first term of the asymptotic solution of the frictionless contact model (up to a distance of the order of a fraction of the near-tip contact length). After a small transition region, in an adjacent surrounding zone whose extent is almost independent of the near-tip contact length, the global solution of the problem is governed by the first term of the asymptotic solution of the open model. As a result of the comparison presented, the regions in which the classical fracture parameters, stress intensity factor (SIF) and energy release rate, can be accurately obtained from the global numerical solution of a crack of this kind have been determined. Third, BEM results and previous estimations show certain discrepancies with a recently published closed form solution of the near-tip contact length and the mode II SIF of the frictionless contact model. A new closed form expression of this mode II SIF, derived from the asymptotic solution of the open model, is proposed in this paper.     

On the validity of conforming BEM algorithms for hypersingular boundary integral equations

The widely held notion that the use of standard conforming isoparametric boundary elements may not be used in the solution of hypersingular integral equations is investigated. It is demonstrated that for points on the boundary where the underlying field is C ^1,α continuous, a class of rigorous nonsingular conforming BEM algorithms may be applied. The justification for this class of algorithms is interpreted in terms of some recent criticism. It is shown that the numerical integration in these conforming BEM algorithms using relaxed regularization represents a finite approximation to the standard two-sided Hadamard finite part interpretation of hypersingular integrals. It is also shown that the integration schemes in this class of algorithms are not based upon one-sided finite part interpretations. As a result, the attendant ambiguities associated with the incorrect use of the one-sided interpretations in boundary integral equations pose no problem for this class of algorithms. Additionally, the distinction is made between the analytic discontinuities in the field which place limitations on the applicability of the conforming BEM and the discontinuities resulting from the use of piece-wise C ^1,α interpolations.     

Investigation in weighted function and optimization of isoparametric singular boundary elements' size in 3-d crack problem

In this paper, there are several measures being used to simplify the analytic method and improve the computational accuracy. We have emphatically studied: (1) An explicit weighted function expression in isoparametric singular BEM. (2) A reasonable collocation of 8-nodes isoparametric boundary elements and application of transition elements. (3) Optimization of the sizes of isoparametric singular elements. The study shows: (i) the deduced explicit weighted function expression of 8-nodes isoparametric BEM is good at setting up a general method of computing the SIF of 3-D crack problem under complex load, (ii) The reasonable collocation of isoparametric boundary elements and the optimization of the size of the singular boundary elements obviously improved the accuracy of numerical computation.     

A new boundary element method for mixed boundary value problems involving cracks and holes: Interactions between rigid inclusions and cracks

This paper derives a new boundary integral equation (BIE) formulation for plane elastic bodies containing cracks and holes and subjected to mixed displacement/ traction boundary conditions, and proposes a new boundary element method (BEM) based upon this formulation. The basic unknown in the formulation is a complex boundary function H(t), which is a linear combination of the boundary traction and boundary displacement density. The present BIE formulation can be related directly to Muskhelishvili's formalism. Singular interpolation functions of order r ^–1/2 (where r is the distance measured from the crack tip) are introduced such that singular integrand involved at the element level can be integrated analytically. By applying the BEM, the interaction between a rigid circular inclusion and a crack is investigated in details. Our results for the stress intensity factor are comparable with those given by Erdogan and Gupta (1975) and Gharpuray et al. (1990) for a crack emanating from a stiff inclusion, and with those by Erdogan et al. (1974) for a crack in the neighborhood of a stiff inclusion.     

A BEM analysis of a penny-shaped interface crack using the open and the frictionless contact models: Range of validity of various asymptotic solutions

The global elastic solution for the problem of a pressurized penny-shaped crack at the interface of two dissimilar half spaces has been numerically obtained employing the boundary element method (BEM). Using the Williams’ open model (for the whole range of feasible bi-material combinations), the comparison of the global BEM solution with an existing analytical asymptotic solution has shown: (i) that the extent of the zone in which the first term is dominant is always larger than the extent of the zone in which the interpenetrations take place and (ii) that, in the former zone, a recently proposed relation between the components of the complex stress intensity factor (SIF) and the components of the energy release rate (ERR) always yields accurate results. Consequently, the appearance of negative values of the normal contribution to the ERR in certain cases has been confirmed by the BEM solution, thus questioning the significance of the asymptotic results obtained from the open model in those cases. If the Comninou's frictionless contact model is employed, the ability of the BEM formulation employed to obtain accurate elastic solutions is shown by comparisons with an existing semi-analytical solution (for a particular bi-material combination).     

Analysis of cracked plates using an iterative hybrid technique of boundary element method and distributed dislocation method

An iterative hybrid technique of boundary element method (BEM) and distributed dislocation method (DDM) is introduced for solving two dimensional crack problems. The technique decomposes the problem into (n + 1) subsidiary problems where n is the number of crack branches. The required solution will be the sum of these (n + 1) solutions. The first subsidiary problem is to find the stress distribution induced in the plate in the absence of the crack using BEM. All of the remaining subsidiary problems, are stress disturbance ones that will be solved using DDM. The results will be added and compared with the boundary conditions of the original problem. Iteration will be performed between the plate boundaries and crack faces until all of the boundary conditions are satisfied.     

The BEM for plates of variable thickness on nonlinear biparametric elastic foundation. An analog equation solution

The BEM is developed for the analysis of plates with variable thickness resting on a nonlinear biparametric elastic foundation. The presented solution is achieved using the Analog Equation Method (AEM). According to the AEM the fourth-order partial differential equation with variable coefficients describing the response of the plate is converted to an equivalent linear problem for a plate with constant stiffness not resting on foundation and subjected only to an `appropriate' fictitious load under the same boundary conditions. The fictitious load is established using a technique based on the BEM and the solution of the actual problem is obtained from the known integral representation of the solution of the substitute problem, which is derived using the static fundamental solution of the biharmonic equation. The method is boundary-only in the sense that the discretization and the integration are performed only on the boundary. To illustrate the method and its efficiency, plates of various shapes are analyzed with linear and quadratic plate thickness variation laws resting on a nonlinear biparametric elastic foundation.     

Generalized Kelvin solution based boundary element method for crack problems in multilayered solids

This paper presents a new boundary element method (BEM) for linear elastic fracture mechanics in three-dimensional multilayered solids. The BEM is based on a generalized Kelvin solution. The generalized Kelvin solution is the fundamental singular solution for a multilayered elastic solid subject to point concentrated body-forces. For solving three-dimensional elastic crack problems in a finite region, a multi-region method is also employed in the present BEM. For crack problems in an infinite space, a large finite body is used to approximate the infinite body. In addition, eight-node traction-singular boundary elements are used in representing the displacements and tractions in the vicinity of a crack front. The incorporation of the generalized Kelvin solution into the boundary integral formulation has the advantages in elimination of the element discretization at the interfaces of different elastic layers. Three numerical examples are presented to illustrate the proposed method for the calculation of stress intensity factors for cracks in layered solids. The results obtained using the proposed method are well compared with the existing results available in the relevant literature.     

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.     

Stress intensity factor for a crack emanating from the root of a semi-circular edge notch in a tension plate

The stress intensity factor (SIF) for a crack radially emanating from the root of a semi-circular edge cut-out is determined theoretically. Firstly, the advantage of using finite Mellin type transform is demonstrated mainly to resolve the boundary conditions for a plate with an edge notch. Secondly, this method has been used to calculate the SIF of a crack in a semi-infinite tension plate, which involves the solution of a Weiner-Hopf integral equation. Finally these analytical results are verified through transmission photoelasticity experiments on model specimens. It is concluded that the SIF increases less rapidly compared to Bowie's problem [1] and there is no appreciable effect of the cut-out on the stress field close to the tip region when the crack size is about seven times the edge-notch radius (Fig. 1).     

BEM for crack‐inclusion problems of plane thermopiezoelectric solids

The problem of interactions between an inclusion and multiple cracks in a thermopiezoelectric solid is considered by boundary element method (BEM) in this paper. First of all, a BEM for the crack–inclusion problem is developed by way of potential variational principle, the concept of dislocation, and Green's function. In the BE model, the continuity condition of the interface between inclusion and matrix is satisfied, a priori, by the Green's function, and not involved in the boundary element equations. This is then followed by expressing the stress and electric displacement (SED) and elastic displacements and electric potential (EDEP) in terms of polynomials of complex variables ξ_t and ξ_k in the transformed ξ‐plane in order to simulate SED intensity factors by the BEM. The least‐squares method incorporating the BE formulation can, then, be used to calculate SED intensity factors directly. Numerical results for a piezoelectric plate with one inclusion and a crack are presented to illustrate the application of the proposed formulation. Copyright © 2000 John Wiley & Sons, Ltd.     

Periodic crack system in a layered elastic wedge

ABSTRACT

Three-dimensional periodic mode I crack problems for an elastic layered wedge are reduced to an integro-differential equation principal part of which is the same as in the classical Griffith problem for one crack in an elastic full space. The infinite system of elliptic cracks, with equal spaces between neighbouring cracks, is situated in the middle of the wedge parallel to its edge. The faces of the wedge with cracks are layered, with sliding support, by identical elastic incompressible wedge-shaped layers so that the system of three wedges have the same edge (for example, rubber–metal–rubber system). Three types of boundary conditions on the outer layered wedge faces are considered (sliding or rigid support, stress-free state). The regular asymptotic method is used to solve the problems effective for fairly big spaces between the cracks as well as for big distances between the cracks and the edge of the wedge. The stress intensity factor (SIF) is calculated for different geometric parameters. It is shown that the solution, including the SIF, for close cracks does not practically depend on the type of the outer boundary conditions.     

On the through-thickness crack with a curve front in center-cracked tension specimens

Three-dimensional finite element analyses are performed on through-thickness cracks with slightly wavy front in center-cracked plates. Considering there is an inherent relationship between the crack shape and the corresponding stress intensity factor (SIF) distribution of a crack, the curved configuration of the crack is determined using a heuristically derived iterative procedure if the SIF distribution function is known. Several simple SIF distribution functions, for instance the constant SIF distribution along the crack front, are assumed to determine the crack shape. Under the assumption that the rate of fatigue crack growth depends on the SIF range or the effective SIF range, possible effects of plate thickness, crack length and crack closure level gradient on the behaviour of crack tunneling are investigated. The stability of the curved shape of a through-thickness crack in fatigue is also discussed, i.e. whether a crack can maintain its shape satisfying the conditions of constant SIF distribution or other distribution along the crack front during fatigue growth. This study will be useful for a better understanding of the behaviour of crack tunneling and help to evaluate the validity of the two-dimensional linear elastic fracture mechanics in cracked plates.     

Modified crack closure integral using six-noded isoparametric quadrilateral singular elements

Six-noded, isoparametric serendipity type quadrilateral regular/singular elements are used for the estimation of stress intensity factors (SIF) in linear elastic fracture mechanics (LEFM) problems involving cracks in two-dimensional structural components. The square root singularity is achieved in the six-noded elements by moving the in-side nodes to the quarter point position. The modified crack closure integral (MCCI) method is adopted which could generate accurate estimates of SIF for a relatively coarse mesh. The equations for strain energy release rate and SIF are derived for mixed mode situations using six-noded quadrilateral elements at the crack tip. The model is validated by numerical studies for a centre crack in a finite plate under uniaxial tension, a single edge notched specimen under uniaxial tension, an inclined crack in a finite rectangular plate and cracks emanating from a pin-loaded lug (or lug attachment). The results compare very well with reference solutions available in the literature.     

弯扭组合载荷下圆管半椭圆表面裂纹应力强度因子的有限元分析

对表面裂纹复合型应力强度因子的研究一直是线弹性断裂力学中的重要课题,例如弯扭组合载荷下圆管半椭圆表面裂纹应力强度因子的计算,到现在也没有一个正确的分析解。考虑到裂尖的应力奇异性,在裂纹前沿手动设置三维奇异单元,用三维有限元法中的1/4点位移法计算弯扭组合载荷下圆管表面椭圆裂纹前沿的Ⅰ型、Ⅱ型和Ⅲ型应力强度因子,并分析其随裂纹深度增加时的变化规律。运用该方法计算了有关模型的应力强度因子,并与该模型的实验值进行了比较,计算结果和实验结果吻合良好。     

3-D dynamic interaction between a penny-shaped crack and a thin interlayer joining two elastic half-spaces

Three-dimensional (3-D) elastodynamic interaction between a penny-shaped crack and a thin elastic interlayer joining two elastic half-spaces is investigated by an improved boundary integral equation method or boundary element method. The penny-shaped crack is embedded in one of the half-spaces, perpendicular to the interlayer and subjected to a time-harmonic tensile loading on its surfaces. Effective “spring-like” boundary conditions are applied to approximate the effects of the thin layer in the mathematical model. Integral representations for the displacement and the stress components are derived by using modified Green’s functions, which satisfy the “spring-like” boundary conditions identically. Then, application of the dynamic loading condition on the crack-surfaces results in a boundary integral equation (BIE) for the crack-opening-displacement over the crack-surfaces only. A solution procedure is developed for solving the BIE numerically. Numerical results for the mode-I dynamic stress intensity factor (SIF) are presented and discussed to show the variations of the mode-I dynamic SIF with the angular coordinate of the crack-front points, the dimensionless wave number, the material mismatch and the crack-layer distance.     

An elastostatic FEM/BEM analysis of vertically loaded raft and piled raft foundation

In this paper, a static analysis of vertically loaded raft and piled raft foundations in smooth and continuous contact with the supporting soil is presented. In this approach the finite element method (FEM) and the boundary element method (BEM) are coupled: the bending plate is assumed to have linear elastic properties and is modelled by FEM while the soil is considered as an elastic half-space in the BEM. The pile is represented by a single element and the shear force along the shaft is interpolated by a quadratic function. The plate–soil interface is divided into triangular boundary elements (soil) also called cells and finite elements (plate) and the subgrade reaction is linearly interpolated across each cell. The subgrade tractions are eliminated from the FEM and BEM algebraic systems of equations, resulting in the governing system of equations for plate–pile–soil interaction problems. Numerical results are presented and they are close to those resulting from much more elaborate analyses.     

Analysis of dynamic interaction between an inclusion and a nearby moving crack by BEM

In this paper, the dynamic interaction between an inclusion and a nearby moving crack embedded in an elastic medium is studied by the boundary element method (BEM). To deal with this problem, the multi-region technique and two kinds of time-domain boundary integral equations (BIEs) are introduced. The system is divided into two parts along the interface between the inclusion and the matrix medium. Each part is linear, elastic, homogeneous and isotropic. The non-hypersingular traction boundary integral equation is applied on the crack surfaces; while the traditional displacement boundary integral equation is used on the interface and external boundaries. In the numerical solution procedure, square root shape functions are adopted as to describe the proper asymptotic behavior in the vicinity of the crack-tips. The crack growth is modeled by adding new elements of constant length to the moving crack tip, which is controlled by the fracture criterion based on the maximum circumferential stress. In each time step, the direction and the speed of the crack advance are evaluated. The numerical results of the crack growth path, speed, dynamic stress intensity factors (DSIFs) and dynamic interface tractions for various material combinations and geometries are presented. The effect of the inclusion on the moving crack is discussed.     

A general method for multiple crack problems in a finite plate

A novel method for the multiple crack problems in a finite plate is proposed in this paper. The basic stress functions of the solution consist of two parts. One is the Fredholm integral equation solution for the crack problem in an infinite plate, and the other is that of the weighted residual method for general plane problems. The combined stress functions are used in the analysis and the boundary conditions on the crack surfaces and the boundary are considered. After the coefficients of the functions have been determined, the stress intensity factors (SIF) at the crack tips can be calculated. Some numerical examples are given and it was observed that when the cracks are very short, the results compare very favorably with the existing results for an infinite plate. Furthermore, the influence of the boundary can be considered. This method can be used for arbitrary multiple crack problems in a finite plate.