A Self-Similar Crack Expansion method for three-dimensional cracks in an infinite or semi-infinite medium

The Self-Similar Crack Expansion (SSCE) method is used to calculate stress intensity factors for three-dimensional cracks in an infinite medium or semi-infinite medium by the boundary integral element technique, whereby, the stress intensity factors at crack tips are determined by calculating the crack-opening displacements over the crack surface. For elements on the crack surface, regular integrals and singular integrals are precisely evaluated based on closed form expressions, which improves the accuracy. Examples show that this method yields very accurate results for stress intensity factors of penny-shaped cracks and elliptical cracks in the full space, with errors of less than 1% as compared with analytical solutions. The stress intensity factors of subsurface cracks are in good agreement with other analytical solutions.     

三维间断位移法及强奇异和超奇异积分的处理方法

从积分方程Somigliana等式出发,导出三维状态下单位位错集度的基本解.在此基础上,建立了边界积分方程,并给出了其离散形式.对强奇异和超奇异积分,采用了Hadamard定义的有限部分积分来处理.最后,给出了计算裂纹应力强度因子的算例,并与解析解进行了比较,证实了该方法的有效性.     

Analysis of three-dimensional edge cracks under tensile loading

Three-dimensional edge cracks are analyzed using the Self-Similar Crack Expansion (SSCE) method with a boundary integral equation technique. The boundary integral equations for surface cracks in a half space are presented based on a half space Green's function (Mindlin, 1936). By using the SSCE method, the stress intensity factors are determined by the crack-opening displacement over the crack surface. In discrete boundary integral equations, the regular and singular integrals on the crack surface elements are evaluated by an analytical method, and the closed form expressions of the integrals are given for subsurface cracks and edge crakcs. This globally numerical and locally analytical method improves the solution accuracy and computational effort. Numerical results for edge cracks under tensile loading with various geometries, such as rectangular cracks, elliptical cracks, and semi-circular cracks, are presented using the SSCE method. Results for stress intensity factors of those surface breaking cracks are in good agreement with other numerical and analytical solutions.     

Fracture mechanics analysis of curved stiffened panels using BEM

A dual boundary element method is developed for a analysis of reinforced cracked shallow shells. Boundary integral equations are derived from the Betti’s reciprocal theorem for a cracked shallow shell with transverse frames and longitudinal stiffeners. The effect of frames and stiffeners are treated as a distribution of line body forces. The radial basis function is used to transform domain integrals to boundary integrals. Stress intensity factors are evaluated from crack opening displacements. The effect of curvature on the stress intensity factors is illustrated by numerical examples. Three examples are presented to demonstrate the accuracy of this method compared with solutions obtained using the finite element method.     

APPLICATION OF THE HYPERSINGULAR INTEGRAL EQUATIONS METHOD TO THE INTERACTION BETWEEN TWO PARALLEL PLANAR CRACKS IN 3-D ELASTICITY

By using the analytic theory of hypersingular integral equations in three-dimensional fracture mechanics, the interactions between two parallel planar cracksunder arbitrary loads are investigated. According to the concepts and method of finite-part integrals, a set of hypersingular integral equations is derived, in which theunknown functions are the displacement discontinuities of the crack surfaces. Then itsnumerical method is proposed by combining the finite-part integral method with theboundary element method. Based on the above results, the method for calculating thestress intensity factors with the displacement discontinuities of the crack surfaces ispresented. Finally, several typical examples are calculated and the numerical resultsare satisfactory.     

Interaction integrals for fracture analysis of functionally graded piezoelectric materials

This paper presents domain form of the interaction integrals based on three independent formulations for computation of the stress intensity factors and electric displacement intensity factor for cracks in functionally graded piezoelectric materials. Conservation integrals of J-type are derived based on the governing equations for piezoelectric media and the crack tip asymptotic fields of homogeneous piezoelectric medium as auxiliary fields. Each of the formulation differs in the way auxiliary fields are imposed in the evaluation of interaction integral and each of them results in a consistent form of the interaction integral in the sense that extra terms naturally appears in their derivation to compensate for the difference in the chosen crack tip asymptotic fields of homogeneous and functionally graded piezoelectric medium. The additional terms play an important role of ensuring domain independence of the presented interaction integrals. Comparison of the numerically evaluated intensity factors through the three consistent formulations with those obtained using displacement extrapolation method is presented by means of two examples.     

SELF-SIMILAR CRACK EXPANSION METHOD FOR THE STRESS INTENSITY FACTOR ANALYSIS

The Self-Similar Crack Expansion (SSCE) method is proposed to evaluate stress intensi-ty factors at crack tips, whereby stress intensity factors of a crack can be determined by the crackopening displacement over the crack, not just by the local displacement around the crack tip. The crackexpansion rate is estimated by taking advantage of the crack self-similarity. Therefore, the accuracy ofthe calculation is improved. The singular integrals on crack tip elements are also analyzed and are pre-cisely evaluated in terms of a special integral analysis. Combination of these two techniques greatly in-creases the accuracy in estimating the stress distribution around the crack tip. A variety of two-dimen-sional cracks, such as subsurface cracks, edge cracks, and their interactions are calculated in terms ofthe self-similar expansion rate. Solutions are satisfied with errors less than 0.5% as compared with theanalytical solutions. Based on the calculations of the crack interactions, a theory for crack interactionsis proposed such that for a group of aligned cracks the summation of the square of SIFs at the right tipsof cracks is always equal to that at the left tips of cracks. This theory was proved by the mehtod ofSelf-Similar Crack Expansion in this paper.     

含界面裂纹的不可压缩双金属胶接件的解析变分－守恒积分解法

本文根据平面问题的复变函数理论推导了含界面裂纹双金属胶接件满足微分方程、开裂界面边界条件与未开裂界面连续条件的应力与位移本征函数展开式，并建立了不可压缩双金属界面裂纹的复合型守恒积分及其与应力强度因子之关系，进而利用分区广义变分原理满足其余边界条件确定包含应力强度因子在内的展开式系数，得到守恒积分并求出应力强度因子．数值计算表明，沿不同回路的在恒积分具有很好的守恒性而且由这两种方法所得应力强度因子具有很好的一致性．     

三维有限体平片裂纹的超奇异积分方程与边界元法

利用Ｓｏｍｉｇｌｉａｎａ公式及有限部积分的概念，导出了含任意平片裂纹三维有限体问题的超奇异积分方程组，并联合使用有限部积分与边界元法，建立了数值求解方法．在裂纹前沿附近单元，采用与理论分析一致的平方根位移模型，以提高数值结果的精度．最后计算了若干典型例子的应力强度因子．     

Finite-part integral and boundary element method for interaction between two parallel planar cracks in three-dimensional finite bodies

By using the Somigliana representation and the concepts of finite-part integrals, a set of hypersingular integral equations of the interaction between two parallel planar cracks in a three-dimensional finite body subjected to arbitrary loads is derived, and then its numerical method is proposed by the finite-part integral method combined with the boundary element method. According to the analytic theory of hypersingular integral equations, the square root models of displacement discontinuities in the elements near the crack front are applied, and thus the computational precision is raised. Based on this, the stress intensity factors can be directly calculated. Finally, the stress intensity factors of several typical interaction problems are calculated.     

三维断裂力学的超奇异积分方程方法

本文利用有限部积分的概念和方法,严格地证明了三维弹性体中受任意载荷作用的平片裂纹问题的超奇异积分方程组,并对未知解的性态作了理论分析,得到了性态指数,在此基础上通过主部分析,精确地求得了裂纹前沿光滑点附近的奇性应力场,从而找到了以裂纹面位移间断(位错)表示的应力强度因子表达式,最后对所得的超奇异积分方程组建立了数值法,并用此计算了若干典型的平片裂纹问题,数值结果令人满意。     

An effective boundary element method for analysis of crack problems in a plane elastic plate

A simple and effective boundary element method for stress intensity factor calculation for crack problems in a plane elastic plate is presented. The boundary element method consists of the constant displacement discontinuity element presented by Crouch and Starfield and the crack-tip displacement discontinuity elements proposed by YAN Xiangqiao. In the boundary element implementation the left or the right crack-tip displacement discontinuity element was placed locally at the corresponding left or right each crack tip on top of the constant displacement discontinuity elements that cover the entire crack surface and the other boundaries. Test examples (i. e. , a center crack in an infinite plate under tension, a circular hole and a crack in an infinite plate under tension) are included to illustrate that the numerical approach is very simple and accurate for stress intensity factor calculation of plane elasticity crack problems. In addition, specifically, the stress intensity factors of branching cracks emanating from a square hole in a rectangular plate under biaxial loads were analysed. These numerical results indicate the present numerical approach is very effective for calculating stress intensity factors of complex cracks in a 2-D finite body, and are used to reveal the effect of the biaxial loads and the cracked body geometry on stress intensity factors.     

Mixed-mode stress intensity factors of 3D interface crack in fully coupled electromagnetothermoelastic multiphase composites

This contribution presents an extended hypersingular intergro-differential equation (E-HIDE) method for modeling the 3D interface crack problem in fully coupled electromagnetothermoelastic anisotropic multiphase composites under extended electro-magneto-thermo-elastic coupled loads through theoretical analysis and numerical simulations. First, based on the extended boundary element method, the 3D interface crack problem is reduced to solving a set of E-HIDEs coupled with extended boundary integral equations, in which the unknown functions are the extended displacement discontinuities. Then, the behavior of the extended singular stress indices around the interface crack front terminating at the interface is analyzed by the extended main-part analysis. The extended stress intensity factors near the crack front are defined. In addition, a numerical method for a 3D interface crack problem subjected to extended loads is proposed, in which the extended displacement discontinuities are approximated by the product of basic density functions and polynomials. Finally, the radiation distribution of extended stress intensity factors at the interface crack surface are calculated, and the results are presented toward demonstrating the applicability of the proposed method.     

On Cauchy singular integrals and stress intensity factors for 2D mode I cracks in an infinite solid

This paper comments on some of the different numerical techniques commonly employed in evaluating Cauchy singular integrals of the first kind; e.g. as pertaining to 2D through cracks in a brittle material undergoing Mode I loading. In addition, a different more direct method is proposed here. Also, two different ways to calculate the stress intensity factor (K_I) are contrasted. The accuracy attained by the different methods in calculating K_I, and the factors affecting the calculation, are compared. Finally, comments on calculating the stress field of a 2D crack and important considerations are presented.     

The hybrid crack-tip element approach to thermo-elastic cracks

The stress intensity factors for a thermal crack problem are computed directly from the crack tip singular field as it was embedded in a specialized crack-tip element. The method gives accurate results without modeling a very refined mesh near the crack tip and without evaluating additional line and/or area integrals for the “modified” path independent integrals as suggested by most current methods for thermoelastic cracks.     

同震断层三维裂纹扩展波动时域超奇异积分法

应用波动时域超奇异积分法将P波、S波和磁电热弹多场耦合作用下同震断层任意形状三维裂纹扩展问题转化为求解以广义位移间断率为未知函数的超奇异积分方程组问题;定义了广义应力强度因子,得到裂纹前沿广义奇异应力增量解析表达式;应用波动时域有限部积分概念及体积力法,为超奇异积分方程组建立了数值求解方法,编制了FORTRAN程序,以三维矩形裂纹扩展问题为例,通过典型算例,研究了广义应力强度因子随裂纹位置变化规律;分析了同震断层裂纹扩展中力、磁、电场辐射规律.      

Computing bounds to mixed-mode stress intensity factors in elasticity

A bounding procedure combined with an effective error bound method for linear functionals of the displacements and a simple two points displacement extrapolation method is presented to compute the lower and upper bounds to the stress intensity factors in elastic fracture problems. First, the displacements of two nodes (or node pairs) on the crack edges are used to construct the linear extrapolation to obtain the stress intensity factors at the crack tip, so that stress intensity factors are explicitly expressed as linear functionals of the displacements. Then, a posteriori bounding method is utilized to compute the bounds to the stress intensity factors. Finally, the bounding procedure is verified by a mixed-mode homogenous elastic fracture problem and a bimaterial interface crack problem.     

三维有限体中两平行裂纹干扰问题的有限部积分与边界元法

利用Ｓｏｍｉｇｌｉａｎａ公式及有限部积分的概念，导出含两平行平片裂纹三维有限体裂纹干扰问题的超奇异积分方程组，联合使用有限部积分与边界元法，建立了数值求解方法，为提高数值计算结果的精度，在裂纹前疝附近单元，采用平方根位移模型，并在此基础雌出直接计算应力强度因子的公式，最后计算若干典型例子裂纹前沿的应力强度因子。     

MESHLESS METHOD FOR 2D MIXED-MODE CRACK PROPAGATION BASED ON VORONOI CELL

A meshless method integrated with linear elastic fracture mechanics (LEFM) is presented for 2D mixed-mode crack propagation analysis. The domain is divided automatically into sub-domains based on Voronoi cells, which are used for quadrature for the potential energy. The continuous crack propagation is simulated with an incremental crack-extension method which assumes a piecewise linear discretization of the unknown crack path. For each increment of the crack extension, the meshless method is applied to carry out a stress analysis of the cracked structure. The J-integral, which can be decomposed into mode I and mode II for mixed-mode crack, is used for the evaluation of the stress intensity factors (SIFs). The crack-propagation direction, predicted on an incremental basis, is computed by a criterion defined in terms of the SIFs. The flowchart of the proposed procedure is presented and two numerical problems are analyzed with this method. The meshless results agree well with the experimental ones, which validates the accuracy and efficiency of the method.     

Weight functions for interface cracks in dissimilar anisotropic materials

Bueckner‘s work conjugate integral customarily adopted for linear elastic materials is established for an interface crack in dissimilar anisotropic materials. The difficulties in separating Stroh‘s six complex arguments involved in the integral for the dissimilar materials are overcome and then the explicit function representations of the integral are given and studied in detail. It is found that the pseudo-orthogonal properties of the eigenfunction expansion form (EEF) for a crack presented previously in isotropic elastic cases, in isotopic bimaterial cases, and in orthotropic cases are also valid in the present dissimilar arbitrary anisotropic cases. The relation between Bueckner‘s work conjugate integral and the J-integral in these cases is obtained by introducing a complementary stressdisplacement state. Finally, some useful path-independent integrals and weight functions are proposed for calculating the crack tip parameters such as the stress intensity factors.