Boundary element analysis of curved cracked panels with adhesively bonded patches

A new boundary element formulation for analysis of curved cracked panels with adhesively bonded patches is presented in this paper. The effect of the adhesive layer is modelled by distributed body forces (i.e. two in‐plane forces, two moments and one out‐of‐plane force). A coupled boundary integral formulation of a shear deformable plate and two‐dimensional plane stress elasticity is used to determine bending and membrane forces along the adhesive layer taking into consideration the compatibility conditions in the patch area. Two numerical examples are presented to demonstrate the efficiency of the proposed method. It is shown that the out‐of‐plane bending behaviour and panel curvature have significant influence on the magnitude of the stress intensity factors. Copyright © 2003 John Wiley & Sons, Ltd.     

Boundary element analysis of dissimilar materials and interface crack

The highly-accurate BEM elastostatic program, which is especially useful for the analysis of dissimilar materials and interface cracks, is introduced in brief. By using this program, we can deal with the elastostatic poblems of isotropic or orthotropic dissimilar materials and also the bonded residual stress due to the mismatch of material constants. This paper shows some applications of the BEM program to the analysis of dissimilar materials and interface cracks considering the residual stress quantitatively, and also shows the method to evaluate the strength of dissimilar materials based on the interfacial fracture mechanics. Some experimental results and the evaluation on the strength of dissimilar materials are also presented.     

Boundary element based formulations for crack shape sensitivity analysis

The present paper addresses several BIE-based or BIE-oriented formulations for sensitivity analysis of integral functionals with respect to the geometrical shape of a crack. Functionals defined in terms of integrals over the external boundary of a cracked body and involving the solution of a frequency-domain boundary-value elastodynamic problem are considered, but the ideas presented in this paper are applicable, with the appropriate modifications, to other kinds of linear field equations as well. Both direct differentiation and adjoint problem techniques are addressed, with recourse to either collocation or symmetric Galerkin BIE formulations. After a review of some basic concepts about shape sensitivity and material differentiation, the derivative integral equations for the elastodynamic crack problem are discussed in connection with both collocation and symmetric Galerkin BIE formulations. Building upon these results, the direct differentiation and the adjoint solution approaches are then developed. In particular, the adjoint solution approach is presented in three different forms compatible with boundary element method (BEM) analysis of crack problems, based on the discretized collocation BEM equations, the symmetric Galerkin BEM equations and the direct and adjoint stress intensity factors, respectively. The paper closes with a few comments.     

Boundary element analysis of an integral equation for three-dmensional crack problems

In another paper, the authors proposed an integral equation for arbitrary shaped three-dimensional cracks. In the present paper, a discretization of this equation using a tensor formalism is formulated. This approach has the advantage of providing the displacement discontinuity vector in the local basis which varies as a function of the point of the crack surface. This also facilitates the computation of the stress intensity factors along the crack edge. Numerical examples reported for a circular crack and a semi-elliptical surface crack in a cylindrical bar show that one can obtain good results, using few Gaussian points and no singular elements.     

Boundary element analysis for an interface crack between dissimilar elastoplastic materials

The boundary element method (BEM) is presented for elastoplastic analysis of cracks between two dissimilar materials. The boundary integral equations and integral representation of stress rates are written in such a form that all integrals can be evaluated by the regular Gaussian quadrature rule. An advanced multidomain BEM formulation is suggested for the solution of analysed problems where the substantial reduction of stiffness matrix is observed. The elastoplastic behaviour is modelled through the use of an approximation for the plastic component of the stresses. The boundary and the yielding zone are discretized by elements with quadratic approximations. In numerical examples the path independence of the J- and L-integrals for a straight interface crack and a circular arc-shaped interface crack are investigated, respectively. The influence of the different values of Young's modulus on the J-integral, shape and size of plastic zones is treated too.     

Modelling fatigue crack propagation in CT specimens

Although there are a great number of numerical studies focused on the numerical simulation of crack shape evolution, a deeper understanding is required concerning the numerical parameters and the mathematical modelling. Therefore, the objectives of the paper are the study of the influence of numerical parameters, particularly the radial size of crack front elements and the magnitude of individual crack extensions, the mathematical modelling of crack propagation regimes, and the linking of crack shape changes with K distribution. A relatively simple through-crack geometry, the CT specimen, was studied and the numerical model was validated with experimental results with a good agreement. The K distribution along crack front was found to be the driving force for shape variations. Shape variations were found to be one order of magnitude lower than K variations.     

Boundary element analysis of fatigue crack propagation micromechanisms in austempered ductile iron

The Dual Boundary Element Method (DBEM) is used in this work to model the micro mechanics of fatigue crack propagation in austempered ductile iron (ADI). Emphasis is put in devising accurate procedures for the evaluation of the interaction effects between very close crack–microcrack arrays. Fracture parameters are computed via the so-called one-point displacement formula using special crack-tip elements. Crack propagation is modelled using an incremental crack extension analysis; with crack extensions calculated using a propagation law that accounts for the near-threshold regime. Obtained results are in agreement with experimental observations, providing evidence to fracture mechanics models proposed in the literature.     

Boundary element analysis of ductile crack growth using the traction-free fundamental solution

A new boundary element formulation in two-dimensional rate-independent plasticity is given. This new formulation uses a so-called traction-free fundamental solution so that the resulting boundary integral equation converges in the normal sense, and more important, a formal differentiation of the boundary integral equation leads to a valid integral representation for the in-plane stress component on the boundary. No finite difference approximation is needed to construct the stress recovery routine. The new boundary element method is then used to solve the problem of quasi-static ductile crack growth. Numerical simulations based on a set of experimental data have been carried out to evaluate a new path-independent integral,T* _M . TheT* _M ,-integral is a modified version of Atluri'sT*-integral. This modified version has an advantage of having a less singular domain integral near the crack flank so that it is numericaly preferable toT*.     

Finite element analysis of plates with curved edges

The application of the high precision triangular plate bending element to problems with curved boundaries is considered. Appropriate edge conditions for nodal points on these boundaries are derived. The error inherent in representing the shape of a curved boundary by a series of straight segments is found to be the limiting factor on accuracy, while the effect of approximations in the actual boundary conditions is minor. To overcome the first type of error, the high precision element is modified to include one curved edge. Substantial improvements in accuracy are obtained, as demonstrated in example calculations for circular and elliptical plates.     

Boundary element analysis of stress distribution around a crack in plane micropolar elasticity

In this paper, we use the boundary element method to find a semi-analytical solution to the problem of stress concentration around a crack in plane micropolar elasticity. We provide an example demonstrating the effect of material microstructure.     

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.     

Boundary element analysis of inclusions with corners

Boundary element method (BEM) has proven to have very good resolution of large stress gradients such as those that may arise at material interface and reentrant corners. There is, however, a paucity of literature in usage of BEM when the inclusion has a corner. The stress singularity at the corner creates numerical difficulties that need to be addressed. This paper describes: application of BEM to inclusion with and without corners; the numerical modeling difficulties; a methodology for calculation of eigenvalues and stress intensity factors without elaborate analytical expressions; and the future research that is needed for the growth of the boundary element methodology for application to inclusion problems. Numerical results for a rectangular inclusion with sharp and fillet corners that in the limit becomes a circular inclusion demonstrate the potential of the proposed methodology in the analysis of inclusion problems.     

Finite element analysis of crack mouth opening displacement compliance in crack length evaluation for clamped single edge tension specimens

In the unloading compliance method developed for clamped single edge tension (SE(T)) specimens, six crack mouth opening displacement (CMOD)‐based compliance equations (i.e. a/W = f(BCE′)) were proposed for the crack length evaluation without clearly clarifying the corresponding predictive accuracies. In addition, the effective elastic modulus (E_e) that reflects the actual state of stress should also be introduced in the crack length evaluation for SE(T) specimens, because the actual state of stress in the remaining ligament of the test specimen is neither plane stress (E) nor plane strain (E′). In this study, two‐dimensional (2D) plane strain and three‐dimensional (3D) finite element analyses (FEAs) are carried out to investigate predictive accuracies of the six compliance equations. In both 2D and 3D FEA, specimens with a wide range of crack lengths and geometric configurations are included. For a given specimen, the value of E_e that presents the equivalent stress state in the remaining ligament is calculated on the basis of 3D FEA data. A set of formulae for the clamped SE(T) specimen is proposed that allows to evaluate E_e from the corresponding CMOD compliance. This approach is verified using numerical data. The observations of the numerical verification suggest that the use of E_e instead of E or E′ in CMOD‐based compliance equations markedly improves the accuracy of the predicted crack length for clamped SE(T) specimens.     

Finite element modelling and analysis of crack shape evolution in mode-I fatigue Middle Cracked Tension specimens

A numerical procedure was employed to study the shape evolution of fatigue cracks in Middle Cracked Tension specimens. This iterative procedure consists of a 3D finite element analysis to obtain the displacement field in the cracked body, calculation of stress intensity factors along crack front and definition of local crack advances considering the Paris law. Numerical predictions were compared with experimental crack shapes with a good agreement. The evolution of crack shape was analysed for different propagation conditions considering robust dependent parameters. Two main propagation stages were identified: an initial transient stage highly dependent on initial crack shape and a stable stage where the crack follows preferred paths. Mathematical models were proposed for transient and stable stages consisting of exponential and polynomial functions, respectively. The transition between both stages was defined considering two criteria: the rate of shape variation and the distance to stable shape. Finally, the crack shape change was linked with the distribution of stress intensity factor along crack front.     

Routine FE determination of stress intensity factors at curved crack fronts using an influence function technique

A remarkably simple and accurate one-step application of the finite element (FE) method is suggested as a means for the engineer's routine determination of stress intensity factors in linear fracture mechanics for complicated non-symmetric geometries with three-dimensional states of stress and curved crack fronts. The vector-valued influence functions (Green functions) used here are a special kind of weight functions. Mode separation is inherent to the present procedure. Numerical examples demonstrate the versatility of the method. Accuracies within 1% are easily achieved. Detailed guidance to the design of the FE mesh at the crack front is given. Any standard FE code can be used, without requirements for special finite or boundary elements. In retrospect, the present method can be seen as a rather trivial calculation technique which has been made feasible and attractive by the capabilities of today's computers and softwares.     

Boundary element crack closure calculation of three-dimensional stress intensity factors

A general method for boundary element-crack closure integral calculation of three-dimensional stress intensity factors is presented. An equation for the strain energy release rate in terms of products of nodal values of tractions and displacements is obtained. Embedded and surface cracks of modes I, II, and III are analyzed using the proposed method. The multidomain boundary element technique is introduced so that the crack surface geometry is correctly modeled and the unsymmetrical boundary conditions for mode's II and III crack analysis are handled conveniently. Conventional quadrilateral elements are sufficient for this method and the selection of the size of the crack front elements is independent of the crack mode and geometry. For all of the examples demonstrated in this paper, 54 boundary elements are used, and the most suitable ratio of the width of the crack front elements to the crack depth is 1/10 and the calculation error is kept within ±1.5 percent. Compared to existing analytical and finite element solutions the boundary element-crack closure integral method is very efficient and accurate and it can be easily applied to general three-dimensional crack problems.     

Boundary element analysis of dielectric waveguides

We apply the boundary element method to the analysis of optical waveguides. After summarizing constant and linear element algorithms for both two- and three-dimensional simulations, we introduce a new recursive series procedure for constructing the diagonal matrix elements. We then demonstrate that our method can be employed to minimize the reflectivity of optical waveguide antireflection coatings with both straight and angled facets.