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A new way is proposed to simulate the crack propagation paths of the multi-cracked structure. Firstly, the complete displacement and stress fields of the multi-cracked structure are calculated by the extended boundary element method (XBEM) coupled with characteristic analysis of the crack tip. Secondly, based on the maximum circumferential stress criterion by considering the contribution of the non-singular stress terms, the crack initiation angles are obtained. Thirdly, the cracks propagate forward along the crack initiation angles to form a new multi-cracked structure. During this process, boundary element adaptive mesh technique is established and the XBEM is used to analyze the newly formed multi-cracked structure repeatedly. Finally, the crack propagation paths of the multi-cracked structure are obtained. Numerical examples show that the results obtained by the XBEM are in good agreement with the experimental results.

A new fast multipole boundary element method (FM-BEM) is proposed to analyze 2-D elastostatic problems by using linear and three-node quadratic elements. The use of higher-order elements in BEM analysis results in more complex forms of the integrands, in which the direct Gaussian quadrature is difficult to calculate the singular and nearly singular integrals. Herein, the complex notation is first introduced to simplify all integral formulations (including the near-field integrals) in FM-BEM for 2-D elasticity. In direct evaluation of the near-field integrals, the nearly singular integrals on linear elements are calculated by the analytic scheme, and those on quadratic elements are evaluated by a robust semi-analytical algorithm. Numerical examples show that the present method possesses higher accuracy than the FM-BEM with constant elements. The computed efficiency of FM-BEM with higher order elements for analyzing large scale problems is still O(N), where N is the number of linear system of equations. In particular, the proposed FM-BEM is available for solving thin structures.

For boundary element analysis of the orthotropic potential problems in thin structures, the higher order elements are expected to discretize the boundary. However, the use of the higher order elements leads to more complex forms of the integrands in boundary integral equations. The resulting nearly singular integrals on the higher order elements are difficult to be evaluated when the source point is very close to the integral element. In this paper, a semi-analytic algorithm is presented to evaluate the nearly singular integrals on the quadratic elements in two dimensional (2-D) orthotropic potential problems. By constructing the approximate singular integral kernels, the nearly singular integrals through subtraction technique are transformed into the sum of regular parts and singular parts. Then, the former are calculated by the conventional Gaussian quadrature and the latter are calculated by the analytical integral formulas. Numerical examples demonstrate that the present semi-analytic algorithm is efficient and accurate to calculate the nearly singular integrals on the quadratic elements. Especially, the BEM with the present semi-analytic algorithm is successfully applied to analyzing 2-D orthotropic potential problems in very thin structures.

A higher-order analysis of the stress singularity of the plane V-notches and cracks in power law hardening materials is studied. First the asymptotic displacement field in terms of radial coordinates at the notch tip is adopted. When the material near the notch tips arises in plastic deformation, the Von Mises yield criterion and the plastic‘total strain’theory are used. By introducing the displacement expressions into the governing differential equations of the plastic theory, it results in a set of the eigenvalue problem of nonlinear ordinary differential equations with the stress singularity orders and the associated eigenfunctions. Then the interpolating matrix method established by the first author is used to solve the eigenvalue problem by an iteration process. The several leading plastic stress singularity orders of plane V-notches and cracks have been obtained. Simultaneously the associated eigenvectors of the displacement and stress fields in the notch tip region have been determined with the same degree of accuracy. Finally two numerical examples have been given to illustrate the accuracy and the effectiveness of the present method to determine the singularity orders of the plastic V-notch and crack. Based on the computed results, some characteristics related to the stress singularity of the plastic notches and cracks are discussed.

The stress singularity degree associated to a V-notch has a great influence on the fatigue life of V-notched structure. The growth rate of the crack initiated at the tip of a V-notch depends on the stress singularity of the V-notch. The fatigue life accompanying with this small crack will represent a large amount of the total fatigue life. In this work, boundary element method (BEM) is used to study the propagation of the crack emanating from a V-notch tip under fatigue loading. A comparison of the fatigue life between the crack initiated from V-notch tip and a lateral crack is done by a crack propagation law until these two cracks have the same stress intensity factors (SIFs). The effect of initial crack length, notch opening angle and notch depth on the crack extension and propagation is analyzed. As an example of engineering application, the fatigue life of a welded joint is investigated by the present method. The influence of weld toe angle and initial crack length on the fatigue life of the welded structure is studied. Some suggestions are given as an attempt to improve the fatigue life of welded structures at the end.

A general algorithm is applied to the regularization of nearly singular integrals in the boundary element method of planar potential problems. For linear elements, the strongly singular and hypersingular integrals of the interior points very close to boundary were categorized into two forms. The factor leading to the singularity was transformed out of the integral representations with integration by parts, so non-singular regularized formulas were presented for the two forms of integrals. Furthermore, quadratic elements are used in addition to linear ones. The quadratic element very close to the internal point can be divided into two linear ones, so that the algorithm is still valid. Numerical examples demonstrate the effectiveness and accuracy of this algorithm. Especially for problems with curved boundaries, the combination of quadratic elements and linear elements can give more accurate results.

For the nearly singular integral of three-dimensional acoustic boundary element method (BEM), based on the 6-noded triangular isoparametric element, a new semi-analytical algorithm of three-dimensional high order element is proposed in this paper. Using Taylor expansion of trigonometric functions in the three dimensional acoustic fundamental solutions, the singular part of the fundamental solutions is separated. Based on the geometric characteristics of the 6-noded triangular element, an approximate singular kernel function is constructed which has the same singularity as singular integral kernel function. Subtracting the approximate kernel function from the kernel function of the singular integral, the latter is decomposed into a regular kernel function and an approximate singular kernel function. The integral of the regular kernel function can be calculated accurately by using the conventional Gauss numerical quadrature. The integral of the new singular part is calculated by the semi-analytic formula derived in this paper. In the surface of the integral element, the local coordinate system ρθ is established and the approximate singular integral is transformed into the integrals of variables ρ and θ which are already separated in ρθ system. The integral with respect to polar variable ρ is expressed by the analytic formulations first. Then the new singular integral which is a surface integral is transformed into the line integral with respect to variable θ , which can be evaluated by the Gaussian quadrature. Consequently, the new semi-analytic algorithm is established to calculate the nearly singular surface integrals in 3D acoustic BEM. Some examples are given in the last part of this paper to show the accuracy and the effectiveness of the present algorithm. The computed results demonstrate that the semi-analytic algorithm with high order element presented in this paper is more effective than linear regularization BEM to solve nearly singular integrals for 3D acoustic BEM.