Three-dimensional contact mechanics analysis of crack problems using the boundary element method

In this paper, a formulation based on the iterative-load incremental approach for the three-dimensional frictional contact mechanics analysis of fracture problems using the boundary element method (BEM), is presented. Special crack front elements are employed to provide a quick and direct means of obtaining the stress intensity factor. The veracity of the formulation is demonstrated with four crack problems. Three of these problems involve crack closure under bending loads, and the fourth is that of a pin-loaded rectangular plate with corner cracks at the pin-hole. The computed BEM solutions are compared, where possible, with those available in the literature, and there is generally good agreement between them. The numerical examples serve also to illustrate the need for a proper contact mechanics treatment to obtain accurate stress intensity factors for such problems.     

Probabilistic fracture mechanics by the boundary element method

In this paper, a new boundary element formulation is presented for probabilistic analysis of crack problems in context of linear elastic fracture mechanics. The method involves an implicit differentiation method for calculating fracture response derivatives with respect to random parameters and applies first-order reliability method to predicting the reliability of cracked structures. Direct differentiation is used to obtain the derivatives of fracture parameters with respect to crack size which is required for probabilistic analysis. Numerical examples are presented to illustrate the proposed method. The accuracy of the proposed method is demonstrated by comparing with either analytical solution or other established method.     

Flaw identification using the boundary element method

In this paper a new boundary element formulation is presented for the identification of the location and size of internal flaws in two-dimensional structures. An introduction to inverse analysis is given, with special reference to methods of flaw identification, along with a brief review of the optimization methods employed. Both the standard boundary element and the dual boundary element method are presented, with the dual boundary element method proposed as the basis for the new formulation. The flaw identification method is presented, along with the computation of the boundary displacement and traction derivatives and the specialized analytical integration used for cracked boundaries. Examples are given to demonstrate the accuracy of the sensitivity values and the performance of flaw location.     

An efficient method for the two-dimensional elastostatic boundary element method

In order to obtain accurate results and to reduce computation time, we have proposed in this paper a new strategic method, where quadratic elements are used at the corner points and linear elements at the points off the corner points. A computer program using this method has been developed and applied to several problems of various shapes. The usefulness of this method was illustrated by the application results.     

An incremental procedure for friction contact problems with the boundary element method

The contact problem between deformable bodies with the presence of friction is not a problem that can be solved linearly. The uncertainty of the final contact zone and/or the irreversible character of the friction itself are the causes of the nonlinearity. An incremental procedure is proposed to deal with this kind of problem, without restriction in the application of the load. The algorithm has been implemented using as solution tool the Boundary Element Method. The use of discontinuous elements makes the applications of boundary and contact conditions easier.     

Dual boundary element method for anisotropic dynamic fracture mechanics

In this work, the dual boundary element method formulation is developed for effective modelling of dynamic crack problems. The static fundamental solutions are used and the domain integral, which comes from the inertial term, is transformed into boundary integrals using the dual reciprocity technique. Dynamic stress intensity factors are computed from crack opening displacements. Comparisons are made with quasi‐isotropic as well as anisotropic results, using the sub‐region technique. Several examples are presented to assess the accuracy and efficiency of the proposed method. Copyright © 2004 John Wiley & Sons, Ltd.     

Exact integration in the boundary element method for two-dimensional elastostatic problems

This paper derives the exact integrations for the integrals in the boundary element analysis of two-dimensional elastostatics. For facilitation, the derivation is based on the simple forms of the fundamental functions by taking constant, discontinuous linear and discontinuous quadratic elements as examples. The efficiency and accuracy of the derived exact integrations are verified against five benchmark problems; the results indicate that the derived exact integrations significantly reduces the CPU time for forming the matrices of the boundary element analysis and solving the internal displacements.     

Automated simulation of fatigue crack propagation for two-dimensional linear elastic fracture mechanics problems by boundary element method

In this paper, automated simulation of multiple crack fatigue propagation for two-dimensional (2D) linear elastic fracture mechanics (LEFM) problems is developed by using boundary element method (BEM). The boundary element method is the displacement discontinuity method with crack-tip elements proposed by the author. Because of an intrinsic feature of the boundary element method, a general growth problem of multiple cracks can be solved in a single-region formulation. In the numerical simulation, for each increment of crack extension, remeshing of existing boundaries is not necessary. Local discretization on the incremental crack extension is performed easily. Further the new adding elements and the existing elements on the existing boundaries are employed to construct easily the total structural mesh representation. Here, the mixed-mode stress intensity factors are calculated by using the formulas based on the displacement fields around crack tip. The maximum circumferential stress theory is used to predict crack stability and direction of propagation at each step. The well-known Paris’ equation is extended to multiple crack case under mixed-mode loadings. Also, the user does not need to provide a desired crack length increment at the beginning of each simulation. The numerical examples are included to illustrate the validation of the numerical approach for fatigue growth simulation of multiple cracks for 2D LEFM problems.     

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.     

Using the boundary element method to solve rolling contact problems

A new approach to steady-state rolling, with and without force transmission, based on the boundary element method is presented. The proposed formulation solves the problem in a more general way than semi-analytical methods, with which it shares some approximations. The robustness and accuracy of the proposed method is reflected in the comparative analysis of the results obtained for three different types of rolling problems involving identical, dissimilar and tyred cylinders, respectively.     

Fracture mechanics analysis of anisotropic plates by the boundary element method

This paper presents an analysis of mixed mode fracture mechanics problems arising in anisotropic composite laminates. The boundary element method (BEM) and the J _k integral are presented as accurate techniques to compute the stress intensity factors K _I and K _II of two dimensional anisotropic bodies. Using function of a complex variable a decoupling procedure is derived to obtain the stress intensity factors. The procedure is based on the computation of the J ₁-integral and of the ratio of relative displacements at the crack faces, near the crack tip. Applications are presented for unidirectional and symmetric laminates of glass, boron and graphite-epoxy materials. Numerical examples of problems of pure mode I and mixed mode deformations are given, in order to demonstrate the accuracy of the method.     

An indirect boundary element method for three-dimensional dynamic fracture mechanics

Indirect boundary element methods (fictitious load and displacement discontinuity) have been developed for the analysis of three-dimensional elastostatic and elastodynamic fracture mechanics problems. A set of boundary integral equations for fictitious loads and displacement discontinuities have been derived. The stress intensity factors were obtained by the stress equivalent method for static loading. For dynamic loading the problem was studied in Laplace transform space where the numerical calculation procedure, for the stress intensity factor K_I(p), is the same: as that for the static problem. The Durbin inversion method for Laplace transforms was used to obtain the stress intensity factors in the time domain K_I(t). Results of this analysis are presented for a square bar, with either a rectangular or a circular crack, under static and dynamic loads.     

Shape sensitivity analysis of composites in contact using the boundary element method

This paper presents the application of the boundary element method to the shape sensitivity analysis of two-dimensional composite structures in contact. A directly differentiated form of boundary integral equation with respect to geometric design variables is used to calculate shape design sensitivities for anisotropic materials with frictionless contact. The selected design variables are the coordinates of the boundary points either in the contact or non-contact area. Three example problems with anisotropic material properties are presented to validate the applications of this formulation.     

An algorithm based on the boundary element method for problems in engineering mechanics

The solution to a two-dimensional problem using the boundary element method requires the evaluation of a line integral over the boundary. The integrand ot this line integral is a product of a known Green's function and an unknown function. A large number of Green's functions for two-dimensional problems can be represented by a linear combination of four singular functions. By approximating the unknown function by a linear combination of known polynomials, integrals are generated whose integrand is a product of the polynomiais and one of the four singular functions. To evaluate these integrals analytically, the boundary is approximated by a sum of straight-line segments. Recursive formulae are established which reduce the generality and the complexity of the integrands to simple expressions. Analytical forms for these simple expressions are found and are used for initiating the algorithm.     

A boundary element alternating method for two-dimensional mixed-mode fracture problems

A boundary element alternating method (BEAM) is presented for two dimensional fracture problems. An analytical solution for arbitrary polynomial normal and tangential pressure distributions applied to the crack faces of an embedded crack in an infinite plate is used as the fundamental solution in the alternating method. For the numerical part of the method the boundary element method is used. For problems of edge cracks a technique of utilizing finite elements with BEAM is presented to overcome the inherent singularity in boundary element stress calculation near the boundaries. Several computational aspects that make the algorithm efficient are presented. Finally the BEAM is applied to a variety of two-dimensional crack problems with different configurations and loadings to assess the validity of the method. The method gave accurate stress-intensity factors with minimal computing effort.     

The boundary element method for the solution of Stokes equations in two-dimensional domains

A boundary element method for the solution of Stokes equations governing creeping flow or Stokes flow in the interior of an arbitrary two-dimensional domain is presented. A procedure for introducing pressure data on the boundary of the domain is also included and the integral coefficients of the resulting linear algebraic equations are evaluated analytically. Calculations are performed in a circular domain using a variety of different boundary conditions, including a combination of the fluid velocity and the pressure. Results are presented both on the boundary and inside the solution domain in order to illustrate that the boundary element method developed here provides an efficient technique, in terms of accuracy and convergence, to investigate Stokes flow numerically.     

A finite element method for contact problems related to fracture mechanics

A finite element method for contact problems in crack mechanics is developed on the basis of the penalty function method. The method is successfully applied to three important problems in fracture mechanics: a crack propagated from a pin hole, a two-point supported specimen with an edge crack loaded by a stamp, and a thick plate with a through-wall crack under bending force.     

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.     

Crack growth analysis for multi-layered airframe structures by boundary element method

This paper describes an investigation into the application of the boundary element method (BEM) for analyses of aircraft structures. A new BEM using a multi-region formulation of plate bending and plane stress integral equations is developed. The method is used to study the behaviour of X-core and other similar sections subjected to out-of-plane loadings. Also investigated are behaviour of crack growth, which may be present on the skin. Comparison was made with finite element solutions for uncracked structures to show the accuracy and efficiency of proposed method.     

Crack propagation analysis of welded thin-walled joints using boundary element method

 Tube-to-plate nodal joints under cyclic bending are widely used in the road transport and agricultural industry. The square hollow sections (SHS) used in these constructions are thin-walled and cold formed, and they have thicknesses of less than 4 mm. Some fatigue failures have been observed. The weld undercut may affect the fatigue life of welded tubular joints especially for thin-walled sections. The undercut dimensions were measured using the silicon imprint technique. Modelling of thin-walled cruciform joints, as a simplification of welded tubular joints, is described in this paper to determine the effect of weld undercut on fatigue propagation life. The Boundary Element Analysis System Software (BEASY) is used. The results of the effect of weld toe undercut from this analysis are compared with results from previous research to determine the comparative reduction in fatigue life between thin-walled joints (T=3 mm) and those made of thicker sections (T=20 mm). The loss in fatigue strength of the thin-walled joints is found to be relatively more than that for thicker walled joints. A 3D model of a tube to plate T-joint is also modelled using the boundary element software, BEASY. The nodal joint consists of a square hollow section, 50×50×3 SHS, fillet welded to a 10-mm thick plate, and subjected to cyclic bending stress. Fatigue analyses are carried out and the results are compared with the only available S–N design curve.