The evaluation of stress intensity factors in plate bending problems using the dual boundary element method

This paper discusses the application of the boundary element method for the determination of stress intensity factors in plate bending problems. A number of case studies having a range of plan forms, with different combinations of boundary conditions, crack configurations and loading conditions are presented to illustrate the effectiveness of the boundary element method for the fracture analysis of plates. Results of K_I, K_II and K_III stress intensity factors for linear elastic fracture mechanics are presented for the case studies considered. The J-integral method, the displacement extrapolation method, the quarter point approach and the stress extrapolation method have been used to determine the stress intensity factors. The boundary element results for the case studies considered in the paper have been compared with either analytical or finite element results and in all cases good agreement has been achieved.     

Stress intensity factors for cracks in thin plates

This paper presents stress intensity factor solutions for several crack configurations in plates. The loadings considered include internal pressure, and also combined bending and tension. The dual boundary element method is used to model the plate and mixed mode stress intensity factors are evaluated by a crack surface displacement extrapolation technique and the J-integral technique. Several cases including centre crack, edge crack and cracks emanating from a hole in finite width plates are presented.     

Crack Growth analysis of plates Loaded by bending and tension using dual boundary element method

This paper presents an extension of the dual boundary element method to analysis of crack growth in plates loaded in combine bending and tension. Five stress intensity factors, two for membrane behaviour and three for shear deformable plate bending are computed using the J-Integral technique. Crack growth processes are simulated with an incremental crack extension analysis based on the maximum principal stress criterion. The method is considered effective since no remeshing is required and the crack extension is modelled by adding new boundary elements to the previous crack boundaries. Several incremental crack growth analysis for different configurations and loadings are presented.     

Line-spring boundary element method for a surface cracked plate

In the published line-spring boundary element method, the effect of bending is not considered. Therefore it cannot be used to deal with the problem of a surface cracked plate. Taking the advantage of the line-spring model and the boundary element method the authors present a new line-spring boundary element method in which the effect of membrane force and bending moment in a Reissner plate is taken into account. The implementation of the method is discussed in detail. The stress intensity factors for several example problems concerning surface cracked plates are calculated. Comparisons are made with the Newman-Raju solutions. The results show that the proposed method is efficient.     

A variational technique for boundary element analysis of 3D fracture mechanics weight functions: static

The dual boundary element method coupled with the weight function technique is developed for the analysis of three-dimensional elastostatic fracture mechanics mixed-mode problems. The weight functions used to calculate the stress intensity factors are defined by the derivatives of traction and displacement for a reference problem. A knowledge of the weight functions allows the stress intensity factors for any loading on the boundary to be calculated by means of a simple boundary integration without singularities. Values of mixed-mode stress intensity factors are presented for an edge crack in a rectangular bar and a slant circular crack embedded in a cylindrical bar, for both uniform tensile and pure bending loads applied to the ends of the bars. © 1998 John Wiley & Sons, Ltd.     

Bending modified J–Q theory and crack-tip constraint quantification

It is well known that the J–Q theory can characterize the crack-tip fields and quantify constraint levels for various geometry and loading configurations in elastic–plastic materials, but it fails at bending-dominant large deformation. This drawback seriously restricts its applications to fracture constraint analysis. A modification of J–Q theory is developed as a three-term solution with an additional term to address the global bending stress to offset this restriction. The nonlinear bending stress is approximately linearized in the region of interest under large-scale yielding (LSY), with the linearization factor determined using a two-point matching method at each loading for a specific cracked geometry in bending. To validate the proposed solution, detailed elastic–plastic finite element analysis (FEA) is conducted under plane strain conditions for three conventional bending specimens with different crack lengths for X80 pipeline steel. These include single edge notched bend (SENB), single edge notched tension (SENT) and compact tension (CT) specimens from small-scale yielding (SSY) to LSY. Results show that the bending modified J–Q solution can well match FEA results of crack-tip stress fields for all bending specimens at all deformation levels from SSY to LSY, with the modified Q being a load- and distance-independent constraint parameter under LSY. Therefore, the modified parameter Q can be effectively used to quantify crack-tip constraint for bending geometries. Its application to fracture constraint analysis is demonstrated by determining constraint corrected J–R curves.     

Combined effect of shear and large deformation on stress intensity factors

An attempt has been made to study the influence of large deformation on the stress intensity factor in a cracked plate subjected to bending including shear deformation. It is assumed that singular terms for stress resultants and strains in the case of large deformation have the same angular distribution and order of singularity as in the case of a linear problem. With this in view the small deformation singular element has been used at the crack tip region surrounded by large deformation plate bending elements. The finite element analysis, based on total Lagrangian formulation combined with the modified Newton–Raphson technique, has been used to get numerical results. Several examples connected with large deformation of cracked plates subjected to bending are studied. Using the above technique stress intensity factors for linear and non-linear cases have been compared.     

Hybrid stress finite element analysis of bending of a plate with a through flaw

A hybrid stress finite element procedure for the solution of bending stress intensity factors of a plate with a through-the-thickness crack is presented. Reissner's sixth-order plate theory including the effects of transverse shear deformation is used. The dominant singular crack tip stress field is embedded in the crack tip singular elements and only regular polynomial functions are assumed in the far field elements. The stress intensity factors can be calculated directly from the crack tip singular stress solution functions. The effects of the plate thickness, the ratio between the crack size and the inplane dimension of the plate, and the singular element size on the stress intensity factor solution are investigated. The effects of the explicit enforcement of traction-free conditions along crack surfaces, which are the natural boundary conditions in the present hybrid stress finite element model, are also investigated. The numerical results of bending of a plate with a straight central crack compare favourably with analytical solutions. It is also found that the explicit enforcement of traction-free conditions along crack surfaces is mandatory to obtain meaningful results for the Mode I type of bending stress intensity factor.     

The finite element method by employing the singular element with concordant displacement at the crack tip

A new regular polygon singular element is presented here. Its displacement is divided into two parts—linear and singular, the latter including the stress intensity factors vanishes on the boundary of the singular element, so the continuity of the displacement is satisfied. Furthermore, each element in the singular stiffness matrix can be expressed in some more simple analytic formulas. The procedure of analysing and reasoning of this technique is given in this paper, with which the intensity factors of the three kinds of loading system—uniform tension, three-point bending and shearing—are calculated. The results of the calculation show that this method is successful.     

A numerical method for a void–crack interaction under cyclic loads

This paper presents a numerical method to model a general system containing cracks and voids in an infinite elastic plate under remote cyclic loads. By extending Bueckner’s principle suited for a crack to a general system containing cracks and voids, the original problem is divided into a homogeneous problem (the one without cracks and voids) subjected to remote loads and a void-crack problem in an unloaded body with applied tractions on the surfaces of cracks and voids. Thus, the results in terms of the stress intensity factors can be calculated by considering the latter problem, which is analyzed easily by using the hybrid displacement discontinuity method (a boundary element method). Further, a fatigue growth technique of a mixed-mode crack is combined with the numerical approach to simulating a void–crack interaction problem under cyclic loads. Test examples are included to illustrate that the numerical method is very simple and effective for analyzing a void–crack interaction problem.     

Determination of stress intensity factors for semielliptical surface cracks in the WWER-1000 reactor pressure vessel from the results of solving thermoelasticity boundary value problems based on the mixed mesh-projection scheme of the finite element method

Results of numerical analysis of stress intensity factors K_I for semielliptical surface cracks in the WWER-1000 reactor pressure vessel by emergency cooling simulation with known engineering procedures, the equivalent spatial integration and direct methods are presented. Engineering procedures employ the results of numerical solution of axially symmetric boundary value problems of thermoelasticity based on the mixed mesh-projection scheme of the finite element method implemented in the RELAX software. The three-dimensional K_I computations were performed with the SPACE software. __________ Translated from Problemy Prochnosti, No. 2, pp. 45–51, March–April, 2007.     

Multiple random crack propagation using a boundary element formulation

This paper proposes a boundary element method (BEM) model that is used for the analysis of multiple random crack growth by considering linear elastic fracture mechanics problems and structures subjected to fatigue. The formulation presented in this paper is based on the dual boundary element method, in which singular and hyper-singular integral equations are used. This technique avoids singularities of the resulting algebraic system of equations, despite the fact that the collocation points coincide for the two opposite crack faces. In fracture mechanics analyses, the displacement correlation technique is applied to evaluate stress intensity factors. The maximum circumferential stress theory is used to evaluate the propagation angle and the effective stress intensity factor. The fatigue model uses Paris’ law to predict structural life. Examples of simple and multi-fractured structures loaded until rupture are considered. These analyses demonstrate the robustness of the proposed model. In addition, the results indicate that this formulation is accurate and can model localisation and coalescence phenomena.     

Analysis of moiré data for near-interface cracks

The analysis of moiré data obtained in bimaterials with near-interface cracks is examined. To extract stress intensity factors, a collocation-type method is developed where Westergaard crack-tip expansions are used for displacements in the cracked portion of the bimaterial, expansions from the method of fundamental solutions are used for displacements in the uncracked portion of the bimaterial, and continuity conditions at the interface are used to couple the two expansions. Proof-of-principle numerical experiments performed on synthetic data from a boundary element analysis of a cracked bimaterial successfully demonstrated the analysis method. Mixed-mode stress intensity factors were then determined from actual moiré data obtained in a copper–tungsten specimen.     

Dynamic interaction of parallel cracks in elastic bodies

We study time dependences of the stress intensity factors in an infinite isotropic elastic body with parallel cracks whose faces are subjected to impact loads. The original problem is reduced to the solution of a system of boundary integral equations with a parameter corresponding to the wave number in the decomposition of the dynamic process into the set of monochromatic components. The numerical reconstruction of the dependences of unknown quantities on time is carried out for two circular cracks loaded by tensile forces in the form of the Heaviside function. The presence of stress intensity factors of all three types is a distinctive feature of the considered problem. Ya. S. Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lvov, Ukraine. Translated from Problemy Prochnosti, No. 3, pp. 94–102, May–June, 1998.     

Calculation of stress intensity factors based on mode I crack opening integral parameters

We present stress intensity factor assessment using nodal displacements of the crack surfaces determined by the finite element method for cracked bodies. The equation is solved by expanding the crack opening displacement in the Chebyshev function, where crack front asymptotic behavior corresponds to the regulations of the linear elastic fracture mechanics. Results of the stress intensity factor calculations are obtained for test problems with analytical solution. Crack opening displacements are defined with the help of the 3D SPACE software package designed to model mixed variational formulation of the finite element method for displacements and strains of the thermoelastic boundary value problems. Translated from Problemy Prochnosti, No. 6, pp. 122–127, November–December, 2008.     

Use of discontinuous boundary elements for fracture mechanics analysis

The boundary element method (BEM) has long been considered a suitable technique for the analysis of fracture mechanics problems. While research is being published showing how advanced BEM formulations can reduce the data preparation time and produce accurate fracture results, it is still very simple to use more conventional direct boundary element methods to find accurate stress intensity factors very quickly and easily. This is made possible with discontinuous elements. This paper describes modeling and post-processing techniques which can assist the designer in this process. The result are compared against the charts published by Rooke & Cartwright, and show an excellent agreement. The method offers a very quick and accurate method of predicting fracture properties of real-life, non-standard geometries.     

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.     

Time-domain boundary element analysis of dynamic near-tip fields for impact-loaded collinear cracks

A time-domain boundary integral equation method has been developed to calculate elastodynamic fields generated by the incidence of stress (or displacement) pulses on single cracks and systems of two collinear cracks. The system of boundary integral equations has been cast in a form which is amenable to solution by the boundary element method in conjunction with a time-stepping technique. Particular attention has been devoted to dynamic overshoots of the stress intensity factors. Elastodynamic stress intensity factors for pulse incidence on a single crack have been computed as function of time, and they have been compared with results of other authors. For collinear macrocrack-microcrack configurations the stress intensity factors at both tips of the macrocrack have been computed as functions of time for various values of the crack spacing and the relative size of the microcrack.     

Measurement of Specific Fracture Energy and Surface Tension of Brittle Materials in Powder Form

Studies of the influence of specimen geometry and size–effect on the K _R –curves and the related fracture parameters were carried out by the authors (Kumar and Barai 2008b). The present paper is a supplementary contribution and reports interesting results related to the effect of the loading condition and size–effect studies on the K _R –curves associated with the cohesive stress distribution for complete fracture process, the double–K fracture parameters, the CTOD–curves and the process zone length using two different loading conditions (i.e., three–point bending test and four–point bending test). The laboratory size specimen with initial–notch length/depth ratios 0.3 and 0.5 are considered in the work. The load–crack opening displacement curves for these loading conditions are obtained using well known version of fictitious crack model.     

Line-spring model for surface cracks in a reissner plate

In this paper the line-spring model developed by Rice and Levy for a surface crack in elastic plates is reconsidered. The problem is formulated by using Reissner's plate bending theory. For the plane strain problem of a strip containing an edge crack and subjected to tension and bending new expressions for stress intensity factors are used which are valid up to a depth-to-thickness ratio of 0.8. The stress intensity factors for a semi-elliptic and a rectangular crack are calculated. Considering the simplicity of the technique and the severity of the underlying assumptions, the results compare rather well with the existing finite element solutions.