Coupled meshfree and fractal finite element method for mixed mode two‐dimensional crack problems

This paper presents a coupling technique for integrating the element‐free Galerkin method (EFGM) with the fractal finite element method (FFEM) for analyzing homogeneous, isotropic, and two‐dimensional linear‐elastic cracked structures subjected to mixed‐mode (modes I and II) loading conditions. FFEM is adopted for discretization of the domain close to the crack tip and EFGM is adopted in the rest of the domain. In the transition region interface elements are employed. The shape functions within interface elements which comprise both the EFG and the finite element (FE) shape functions, satisfies the consistency condition thus ensuring convergence of the proposed coupled EFGM–FFEM. The proposed method combines the best features of EFGM and FFEM, in the sense that no special enriched basis functions or no structured mesh with special FEs are necessary and no post‐processing (employing any path independent integrals) is needed to determine fracture parameters, such as stress‐intensity factors (SIFs) and T‐stress. The numerical results show that SIFs and T‐stress obtained using the proposed method are in excellent agreement with the reference solutions for the structural and crack geometries considered in the present study. Also, a parametric study is carried out to examine the effects of the integration order, the similarity ratio, the number of transformation terms, and the crack length to width ratio on the quality of the numerical solutions. A numerical example on mixed‐mode condition is presented to simulate crack propagation. As in the proposed coupled EFGM–FFEM at each increment during the crack propagation, the FFEM mesh (around the crack tip) is shifted as it is to the new updated position of the crack tip (such that FFEM mesh center coincides with the crack tip) and few meshless nodes are sprinkled in the location where the FFEM mesh was lying previously, crack‐propagation analysis can be dramatically simplified. Copyright © 2010 John Wiley & Sons, Ltd.     

The extended finite element method in thermoelastic fracture mechanics

The extended finite element method (XFEM) is applied to the simulation of thermally stressed, cracked solids. Both thermal and mechanical fields are enriched in the XFEM way in order to represent discontinuous temperature, heat flux, displacement, and traction across the crack surface, as well as singular heat flux and stress at the crack front. Consequently, the cracked thermomechanical problem may be solved on a mesh that is independent of the crack. Either adiabatic or isothermal condition is considered on the crack surface. In the second case, the temperature field is enriched such that it is continuous across the crack but with a discontinuous derivative and the temperature is enforced to the prescribed value by a penalty method. The stress intensity factors are extracted from the XFEM solution by an interaction integral in domain form with no crack face integration. The method is illustrated on several numerical examples (including a curvilinear crack, a propagating crack, and a three‐dimensional crack) and is compared with existing solutions. Copyright © 2007 John Wiley & Sons, Ltd.     

Modeling crack discontinuities without element‐partitioning in the extended finite element method

In this paper, we model crack discontinuities in two‐dimensional linear elastic continua using the extended finite element method without the need to partition an enriched element into a collection of triangles or quadrilaterals. For crack modeling in the extended finite element, the standard finite element approximation is enriched with a discontinuous function and the near‐tip crack functions. Each element that is fully cut by the crack is decomposed into two simple (convex or nonconvex) polygons, whereas the element that contains the crack tip is treated as a nonconvex polygon. On using Euler's homogeneous function theorem and Stokes's theorem to numerically integrate homogeneous functions on convex and nonconvex polygons, the exact contributions to the stiffness matrix from discontinuous enriched basis functions are computed. For contributions to the stiffness matrix from weakly singular integrals (because of enrichment with asymptotic crack‐tip functions), we only require a one‐dimensional quadrature rule along the edges of a polygon. Hence, neither element‐partitioning on either side of the crack discontinuity nor use of any cubature rule within an enriched element are needed. Structured finite element meshes consisting of rectangular elements, as well as unstructured triangular meshes, are used. We demonstrate the flexibility of the approach and its excellent accuracy in stress intensity factor computations for two‐dimensional crack problems. Copyright © 2016 John Wiley & Sons, Ltd.     

Determination of stress intensity factors of 2D fracture mechanics problems through a new semi‐analytical method

This study presents a novel development of a new semi‐analytical method with diagonal coefficient matrices to model crack issues. Accurate stress intensity factors based on linear elastic fracture mechanics are extracted directly from the semi‐analytical method. In this method, only the boundaries of problems are discretized using specific subparametric elements and higher‐order Chebyshev mapping functions. Implementing the weighted residual method and using Clenshaw–Curtis numerical integration result in diagonal Euler's differential equations. Consequently, when the local coordinates origin is located at the crack tip, the stress intensity factors can be determined directly without further processing. In order to present infinite stress at the crack tip, a new form of nodal force function is proposed. Validity and accuracy of the proposed method is fully demonstrated through four benchmark problems, which are successfully modeled using a few numbers of degrees of freedom. The numerical results agree very well with the analytical solution, experimental outcomes and the results from existing numerical methods available in the literature.     

An effective method of stress intensity factor calculation for cracks emanating from a triangular or square hole under biaxial loads

This paper is concerned with stress intensity factors for cracks emanating from a triangular or square hole under biaxial loads by means of a new boundary element method. The boundary element method consists of the constant displacement discontinuity element presented by Crouch and Starfied and the crack‐tip displacement discontinuity elements proposed by the author. In the boundary element implementation, the left or the right crack‐tip displacement discontinuity element is placed locally at the corresponding left or right crack tip on top of the constant displacement discontinuity elements that cover the entire crack surface and the other boundaries. The method is called a Hybrid Displacement Discontinuity Method (HDDM). Numerical examples are included to show that the method is very efficient and accurate for calculating stress intensity factors for plane elastic crack problems. In addition, the present numerical results can reveal the effect of the biaxial loads on stress intensity factors.     

MLS‐based variable‐node elements compatible with quadratic interpolation. Part II: application for finite crack element

Two‐dimensional finite ‘crack’ elements for simulation of propagating cracks are developed using the moving least‐square (MLS) approximation. The mapping from the parental domain to the physical element domain is implicitly obtained from MLS approximation, with the shape functions and their derivatives calculated and saved only at the numerical integration points. The MLS‐based variable‐node elements are extended to construct the crack elements, which allow the discontinuity of crack faces and the crack‐tip singularity. The accuracy of the crack elements is checked by calculating the stress intensity factor under mode I loading. The crack elements turn out to be very efficient and accurate for simulating crack propagations, only with the minimal amount of element adjustment and node addition as the crack tip moves. Numerical results and comparison to the results from other works demonstrate the effectiveness and accuracy of the present scheme for the crack elements. Copyright © 2005 John Wiley & Sons, Ltd.     

Partition of unity enrichment for bimaterial interface cracks

Partition of unity enrichment techniques are developed for bimaterial interface cracks. A discontinuous function and the two‐dimensional near‐tip asymptotic displacement functions are added to the finite element approximation using the framework of partition of unity. This enables the domain to be modelled by finite elements without explicitly meshing the crack surfaces. The crack‐tip enrichment functions are chosen as those that span the asymptotic displacement fields for an interfacial crack. The concept of partition of unity facilitates the incorporation of the oscillatory nature of the singularity within a conforming finite element approximation. The mixed‐mode (complex) stress intensity factors for bimaterial interfacial cracks are numerically evaluated using the domain form of the interaction integral. Good agreement between the numerical results and the reference solutions for benchmark interfacial crack problems is realized. Copyright © 2004 John Wiley & Sons, Ltd.     

Boundary element‐free method for fracture analysis of 2‐D anisotropic piezoelectric solids

This paper considers a 2‐D fracture analysis of anisotropic piezoelectric solids by a boundary element‐free method. A traction boundary integral equation (BIE) that only involves the singular terms of order 1/r is first derived using integration by parts. New variables, namely, the tangential derivative of the extended displacement (the extended displacement density) for the general boundary and the tangential derivative of the extended crack opening displacement (the extended displacement dislocation density), are introduced to the equation so that solution to curved crack problems is possible. This resulted equation can be directly applied to general boundary and crack surface, and no separate treatments are necessary for the upper and lower surfaces of the crack. The extended displacement dislocation densities on the crack surface are expressed as the product of the characteristic terms and unknown weight functions, and the unknown weight functions are modelled using the moving least‐squares (MLS) approximation. The numerical scheme of the boundary element‐free method is established, and an effective numerical procedure is adopted to evaluate the singular integrals. The extended ‘stress intensity factors’ (SIFs) are computed for some selected example problems that contain straight or curved cracks, and good numerical results are obtained. Copyright © 2006 John Wiley & Sons, Ltd.     

Modelling dynamic crack propagation using the scaled boundary finite element method

This study presents a novel application of the scaled boundary finite element method (SBFEM) to model dynamic crack propagation problems. Accurate dynamic stress intensity factors are extracted directly from the semi‐analytical solutions of SBFEM. They are then used in the dynamic fracture criteria to determine the crack‐tip position, velocity and propagation direction. A simple, yet flexible remeshing algorithm is used to accommodate crack propagation. Three dynamic crack propagation problems that include mode‐I and mix‐mode fracture are modelled. The results show good agreement with experimental and numerical results available in the literature. It is found that the developed method offers some advantages over conventional FEM in terms of accuracy, efficiency and ease of implementation. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical evaluation of the quarter-point crack tip element

This paper attempts to answer two commonly raised questions during the preparation of a finite element mesh, for the linear elastic fracture analysis of cracked structure: how to set up the finite element mesh around the crack tip, and what level of accuracy is to be expected from such a modelling. Two test problems, with known analytical expressions for their stress intensity factors, are analysed by the finite element method using the isoparametric quadratic singular element. The modified parameters were the order of integration, aspect ratio, number of elements surrounding the crack tip, use of transition elements, the singular element length over the total crack length, the symmetry of the mesh around the crack tip. Based on these analyses, a data base is created and various plots produced. The results are interpreted, the accuracy evaluated and recommendations drawn. Contrary to previous reports, it is found that the computed stress intensity factor (SIF) remains within engineering accuracy (10 per cent) throughout a large range of l/a (singular element length over crack length) for problems with a uniform non-singular stress distribution ahead of the crack tip (i.e. double edge notch), and l/a should be less than 0·1 for problems with a non-singular stress gradient (i.e three-point bend). Also, it is found that the best results are achieved by using at least four singular elements around the crack tip, with their internal angles around 45 degrees, and a reduced (2 × 2) numerical integration.     

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.     

A super‐element for crack analysis in the time domain

A super‐element for the dynamic analysis of two‐dimensional crack problems is developed based on the scaled boundary finite‐element method. The boundary of the super‐element containing a crack tip is discretized with line elements. The governing partial differential equations formulated in the scaled boundary co‐ordinates are transformed to ordinary differential equations in the frequency domain by applying the Galerkin's weighted residual technique. The displacements in the radial direction from the crack tip to a point on the boundary are solved analytically without any a priori assumption. The scaled boundary finite‐element formulation leads to symmetric static stiffness and mass matrices. The super‐element can be coupled seamlessly with standard finite elements. The transient response is evaluated directly in the time domain using a standard time‐integration scheme. The stress field, including the singularity around the crack tip, is expressed semi‐analytically. The stress intensity factors are evaluated without directly addressing singular functions, as the limit in their definitions is performed analytically. Copyright © 2004 John Wiley & Sons, Ltd.     

Stress intensity factors for cracks emanating from a triangular or square hole in an infinite plate by boundary elements

This paper concerns stress intensity factors of cracks emanating from a triangular or square hole in an infinite plate subjected to internal pressure calculated by means of a boundary element method, which consists of constant displacement discontinuity element presented by Crouch and Starfied and crack tip displacement discontinuity elements proposed by the author. In the boundary element implementation the left or the right crack tip displacement discontinuity element is placed locally at corresponding left or right crack tip on top of the constant displacement discontinuity elements that cover the entire crack surface and the other boundaries. Numerical examples are included to show that the method is very efficient and accurate for calculating stress intensity factors of plane elasticity crack problems. Specifically, the numerical results of stress intensity factors of cracks emanating from a triangular or square hole in an infinite plate subjected to internal pressure are given.     

A new variable‐order singular boundary element for two‐dimensional stress analysis

A new variable‐order singular boundary element for two‐dimensional stress analysis is developed. This element is an extension of the basic three‐node quadratic boundary element with the shape functions enriched with variable‐order singular displacement and traction fields which are obtained from an asymptotic singularity analysis. Both the variable order of the singularity and the polar profile of the singular fields are incorporated into the singular element to enhance its accuracy. The enriched shape functions are also formulated such that the stress intensity factors appear as nodal unknowns at the singular node thereby enabling direct calculation instead of through indirect extrapolation or contour‐integral methods. Numerical examples involving crack, notch and corner problems in homogeneous materials and bimaterial systems show the singular element's great versatility and accuracy in solving a wide range of problems with various orders of singularities. The stress intensity factors which are obtained agree very well with those reported in the literature. Copyright © 2002 John Wiley & Sons, Ltd.     

A general solution procedure for fracture mechanics weight function evalution based on the boundary element method

This paper reports on the development of an efficient and accurate means for the direct computation of crack surface weight functions for two dimensional fracture mechanics analysis. Weight functions are mathematical representations which can be used to efficiently calculate stress intensity factors for a variety of crack loading and boundary conditions. The method is inherently capable of handling mixed-mode problems. The weight function capability is especially important for problems of fatigue crack growth modeling where the efficient calculation of stress intensity factors is crucial.The basis of the new formulation and numerical solution method is the boundary element method (BEM), as implemented for two dimensional fracture mechanics analysis. The paper will review the analytical formulation of the new BEM, the numerical solution algorithm, and a limited number of validation examples.     

基于X-SBFEM的裂纹体非网格重剖分耦合模型研究

扩展有限元法利用了非网格重剖分技术,但需要基于裂尖解析解构造复杂的插值基函数,计算精度受网格疏密和插值基函数等因素影响。比例边界有限元法则在求解无限域和裂尖奇异性问题优势明显,两者衔接于有限元法理论内,可建立一种结合二者优势的断裂耦合数值模型。该文从虚功原理出发,利用位移协调与力平衡机制,提出了一种断裂计算的新方法X-SBFEM,达到了扩展有限元模拟裂纹主体、比例边界有限元模拟裂尖的目的。在数值算例中,通过边裂纹和混合型裂纹的应力强度因子计算,并与理论解对比,验证了该方法的准确性和有效性。     

Analysis of three-dimensional interface cracks using enriched finite elements

Many important interface crack problems are inherently three-dimensional in nature, e.g., debonding of laminated structures at corners and holes. In an effort to accurately analyze three-dimensional interface fracture problems, an efficient computational technique was developed that utilizes enriched crack tip elements containing the correct interface crack tip asymptotic behavior. In the enriched element formulation, the stress intensity factors K _I, K _II, and K _III are treated as additional degrees of freedom and are obtained directly during the finite element solution phase. In this study, the results that should be of greatest interest are obtained for semi-circular surface and quarter-circular corner cracks. Solutions are generated for uniform remote tension and uniform thermal loading, over a wide range of bimaterial combinations. Of particular interest are the free surface effects, and the influence of Dundurs’ material parameters on the strain energy release rate magnitudes and corresponding phase angles.     

Intensity of stress singularity for the circumferential V‐shape corner front of a three‐dimensional diamond‐like defect

Three‐dimensional diamond‐like defects with circumferential V‐shape corner fronts are often contained in engineering materials. In this paper, generalized stress intensity factors are calculated for this type of defect using a modified advanced finite element method. A super corner front element model in the global coordinates is established to capture the stress singularities along the circumferential corner front. Three‐dimensional numerical series eigen‐solutions in the element have been transformed from asymptotic expressions in the local curvilinear coordinates. The element is suitable for a sharp V‐shape corner with arbitrary opening and inclination angle. Singular stress fields near various shapes of diamond‐like defects are systematically investigated. The interaction of an embedded defect with free surface or another identical defect is also investigated. The numerical results can be used as stress intensity parameters to predict fatigue strength at circumferential corner front of a diamond‐like defect.     

线弹性断裂力学问题的扩展自然单元法

为了更加有效地求解线弹性断裂问题,提出了扩展自然单元法。该方法基于单位分解的思想,在自然单元法的位移模式中加入扩展项表征不连续位移场和裂纹尖端奇异场。通过水平集方法确定裂纹面和裂纹尖端区域,并基于虚位移原理推导了平衡方程的离散线性方程。由于自然单元法的形函数满足Kronecker delta函数性质,本质边界条件易于施加。混合模式裂纹的应力强度因子由相互作用能量积分方法计算。数值算例结果表明扩展自然单元法可以方便地求解线弹性断裂力学问题。