Morphological aspects of fatigue crack propagation Part I—Computational procedure

In the present paper a simulation method is proposed for the evaluation of paths and lives of fatigue cracks. The simulation is based on an incremental crack extension procedure. At each increment the stress analysis ahead of a crack tip is carried out by the finite element method, and the next incremental crack-growth path is predicted by the first order perturbation method with the use of the local symmetry criterion. From the computational viewpoint, the step-by-step rezoning of finite element mesh subdivision is one of the most difficult processes of the simulation procedure. In order to overcome this difficulty, we shall use the modified quadtree method as an automatic mesh generation technique. Considerations are made for the proper mesh arrangement in the vicinity of a crack tip, where a special fine mesh pattern is embedded so that mixed mode stress intensity factors and the higher order coefficients of the near tip stress field parameters can accurately be obtained. Using the proposed method, we simulate the branched and curved fatigue crack growth in three-point-bending specimens. They show fairly good agreement with the experimental results. The simulation procedure is also applied to biaxially loaded cruciform joints.     

A quasi‐static crack growth simulation based on the singular ES‐FEM

In the edge‐based smoothed finite element method (ES‐FEM), one needs only the assumed displacement values (not the derivatives) on the boundary of the edge‐based smoothing domains to compute the stiffness matrix of the system. Adopting this important feature, a five‐node crack‐tip element is employed in this paper to produce a proper stress singularity near the crack tip based on a basic mesh of linear triangular elements that can be generated automatically for problems with complicated geometries. The singular ES‐FEM is then formulated and used to simulate the crack propagation in various settings, using a largely coarse mesh with a few layers of fine mesh near the crack tip. The results demonstrate that the singular ES‐FEM is much more accurate than X‐FEM and the existing FEM. Moreover, the excellent agreement between numerical results and the reference observations shows that the singular ES‐FEM offers an efficient and high‐quality solution for crack propagation problems. Copyright © 2011 John Wiley & Sons, Ltd.     

Mesh scalability in direct finite element simulation of brittle fracture

A new approach of dealing with mesh dependence in finite element modelling of fracture processes is introduced. In particular, in brittle fracture modelling, the stress concentration is mesh dependent as the results do not stabilise when refining the mesh. This paper presents an approach based on the explicit incorporation of mesh dependence into the computations. The dependence of the relevant stress is quantified on the finite elements at the crack tip upon the element size; when the dependence approaches a power law with the required accuracy, the mesh is called scalable. If the mesh is scalable and the exponent and pre-factor are known, then the results of the computations can be scaled to the size relevant to the scale of the physical microstructure of the material; the latter while not being modelled directly ultimately controls the fracture propagation. To illustrate this new approach, four 2D examples of a single straight crack loaded under tensile and shear tractions applied either to the external boundary or to the crack faces are considered. It is shown that combining the stresses at the crack tip computed using a set of similar meshes of different densities with the crack tip asymptotic allows accurate recovery of the stress intensity factors.     

承受双向等拉应力的平面应变I型裂纹线弹性断裂的I1准则与K准则一致性分析

采用线弹性有限元方法计算了承受双向等拉应力的平面应变I型裂纹的应力场,分析了裂纹尖端各应力分量间的关系,拟合了各非零应力分量关于裂纹半长度a和裂纹尖端最小网格尺寸l₁的函数,分析了应力第一不变量I₁与应力场强度因子K_I的相关性。结果表明,裂纹尖端各非零应力分量间存在稳定的比例关系;各非零应力分量值和加载应力的比值与裂纹半长度a的1/2次幂呈正比例关系、与裂纹尖端最小网格尺寸l₁的1/2次幂呈反比例关系;相同最小网格尺寸条件下,裂纹尖端的应力第一不变量与应力场强度因子的比值l₁/K_I为与加载应力和裂纹长度无关的常数,证明了承受双向等拉应力的平面应变I型裂纹线弹性断裂的I₁准则与K准则具有一致性。     

二维压电材料动强度因子的扩展有限元计算

采用扩展有限元求解二维弹性压电材料动断裂问题。扩展有限元的网格独立于裂纹,因此网格生成可大大地简化,且裂纹扩展时不需重构网格。采用相互作用积分技术计算动强度因子。比较了标准的力裂尖加强函数和力-电裂尖加强函数对动强度因子的影响,结果表明标准的力裂尖加强函数能有效地分析压电材料动断裂问题。分析了极化方向对动强度因子的影响。数值分析表明采用扩展有限元获得的动强度因子与其他数值方法解吻合得很好。     

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.     

A methodology for crack tip mesh design

A methodology for crack tip mesh design is developed which consists of comparing the mesh geometric parameters against the accuracy of the finite element solution. By successive changes in the mesh parameters a near optimal mesh can be obtained. This was done here two-dimensional linear elastic single mode problems. The direct displacement extrapolation method for stress intensity factor estimation is used.     

XFEM fracture analysis of shells: The effect of crack tip enrichments

In this study the effect of crack tip enrichment functions in the extended finite element analysis of shells is investigated. Utilization of crack tip enrichments leads to reduction of the required number of elements, mesh independency and increased accuracy in computation of fracture mechanics parameters such as the stress intensity factor, the crack tip opening displacement and the crack tip opening angle. The procedure is verified by modeling various shell and plate problems and available benchmark tests. Also, effects of enrichments of in-plane, out-of-plane and rotational degrees of freedom and high order out-of-plane enrichments on different fracture modes are studied. Moreover, reduction of the dependency of crack tip opening angle on the element size in crack propagation problems is discussed.     

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.     

Application of conformal mapping for mesh generation in two-dimensional elasto-plastic fracture mechanics problems

Conformal mapping techniques may be used for the automatic generation of finite element meshes in two-dimensional fracture mechanics calculations. Using a Schwartz-Christonel conformal transformation, the conformal mapping leads to an optimum mesh in the crack region. This resultant mesh is refined around the crack tip, and becomes progressively more coarse at areas where regular stress fields are expected. The advantageous characteristics of the mapped mesh are verified by means of elasto-plastic finite-element analysis of a compact tension specimen and a center cracked plate.     

An extended finite element method with analytical enrichment for cohesive crack modeling

A recent approach to fracture modeling has combined the extended finite element method (XFEM) with cohesive zone models. Most studies have used simplified enrichment functions to represent the strong discontinuity but have lacked an analytical basis to represent the displacement gradients in the vicinity of the cohesive crack. In this study enrichment functions based upon an existing analytical investigation of the cohesive crack problem are proposed. These functions have the potential of representing displacement gradients in the vicinity of the cohesive crack and allow the crack to incrementally advance across each element. Key aspects of the corresponding numerical formulation and enrichment functions are discussed. A parameter study for a simple mode I model problem is presented to evaluate if quasi‐static crack propagation can be accurately followed with the proposed formulation. The effects of mesh refinement and mesh orientation are considered. Propagation of the cohesive zone tip and crack tip, time variation of the cohesive zone length, and crack profiles are examined. The analysis results indicate that the analytically based enrichment functions can accurately track the cohesive crack propagation of a mode I crack independent of mesh orientation. A mixed mode example further demonstrates the potential of the formulation. Copyright © 2008 John Wiley & Sons, Ltd.     

XFEM for direct evaluation of mixed mode SIFs in homogeneous and bi‐materials

The extended finite element method (XFEM) is improved to directly evaluate mixed mode stress intensity factors (SIFs) without extra post‐processing, for homogeneous materials as well as for bimaterials. This is achieved by enriching the finite element (FE) approximation of the nodes surrounding the crack tip with not only the first term but also the higher order terms of the crack tip asymptotic field using a partition of unity method (PUM). The crack faces behind the tip(s) are modelled independently of the mesh by displacement jump functions. The additional coefficients corresponding to the enrichments at the nodes of the elements surrounding the crack tip are forced to be equal by a penalty function method, thus ensuring that the displacement approximations reduce to the actual asymptotic fields adjacent to the crack tip. The numerical results so obtained are in excellent agreement with analytical and numerical results available in the literature. Copyright © 2004 John Wiley & Sons, Ltd.     

Strain energy release rate determination of stress intensity factors by finite element methods

The determination of the Mode I stress intensity factors for selected crack configurations, using finite element methods and energy release rate principles, is the subject of this study. The crack configurations which were investigated are the double edge crack, the single edge crack and the center crack. The method of analysis utilized was the “Stiffness Derivative Method.” This approach relates the change in strain energy resulting from crack advancement, to the change in the stiffness matrix of the structure containing the crack. The results indicated that through mesh optimization and proper control of certain parameters including the crack advance increment, the crack tip element contour size and mesh refinement, an accurate solution can be calculated with a relatively coarse finite element mesh consisting entirely of contemporary elements. The numerically generated solutions are compared with analytical solutions with the results within 0.001% of each other for the double edge crack, 0.858% for the single edge crack and 2.021% for the center crack.     

A mesh objective method for modeling crack propagation using the smeared crack approach

This paper presents a mesh objective method for modeling crack propagation in brittle materials using a conventional finite element formulation. The primary shortcoming of the smeared crack approach is its pathological sensitivity to the mesh orientation, which is manifested by shear locking and stress field misalignment around the crack tip. Such undesirable characteristics preclude the ability to model arbitrary crack propagation at an angle through the mesh. Several techniques are developed to address these shortcomings. First, to preclude shear locking, a modified failure constitutive model is developed, which projects out the spurious stress increments as the crack opens. If a crack exists in an element, a crack tracking algorithm is used to identify the neighboring elements most likely to show crack continuation. This algorithm also identifies a crossover element when a crack passes through adjacent sides of an element. Then, the characteristic element length used in the constitutive equation is changed with the objective of providing the correct failure energy per unit crack length, a procedure called crossover scaling. The examples provided demonstrate that the developed methods work collectively to provide a simple and efficient method for modeling failure in brittle materials without mesh bias.     

Modelling multiple cohesive crack propagation using a finite element–scaled boundary finite element coupled method

This paper presents an extension of the recently-developed finite element–scaled boundary finite element (FEM–SBFEM) coupled method to model multiple crack propagation in concrete. The concrete bulk and fracture process zones are modelled using SBFEM and nonlinear cohesive interface finite elements (CIEs), respectively. The CIEs are automatically inserted into the SBFEM mesh as the cracks propagate. The algorithm previously devised for single crack propagation is augmented to model problems with multiple cracks and to allow cracks to initiate in an un-cracked SBFEM mesh. It also addresses crack propagation from one subdomain into another, as a result of partitioning a coarse SBFEM mesh, required for some mixed–mode problems. Each crack in the SBFEM mesh propagates when the sign of the Mode-I stress intensity factor at the crack tip turns positive from negative. Its propagation angle is determined using linear elastic fracture mechanics criteria. Three concrete beams involving multiple crack propagation are modelled. The predicted crack propagation patterns and load–displacement curves are in good agreement with data reported in literature.     

A moving cohesive interface model for fracture in creeping materials

We present a new cohesive interface model for quasi-static creep crack growth that is implemented within a moving-grid finite element model. A pseudo crack tip separates the cohesive process zone from the free surfaces of the crack. The moving-grid formulation models continuous crack advance by describing relative motion between the pseudo crack tip and the material. This eliminates the need for extensive mesh refinement away from the current crack-tip location and supports both transient and direct steady-state solutions. A traction-separation law determines the energetic properties of the decohesion process and generates a simple criterion for crack advance. The new formulation remedies a problem in earlier models which permit a crack to heal on unloading. Numerical examples demonstrate the moving cohesive interface model in studies of steady-state crack growth. Adaptive grid refinement is used to control the accuracy of the solution.     

Comments on Simulations of Fatigue Crack Propagation by Blunting and Re-sharpening: The Mesh Sensitivity

A selection of results of extensive analysis of mesh sensitivity of largedeformation elastoplastic finite element (FE) simulations of a crack under cyclic loading is presented. Notorious mesh sensitivity, which commences at spontaneous shear localization, is evidenced. This is argued to be not a mere numerical artefact, but a consequence of the inherent bifurcating behaviour of the boundary value problem solutions, where different mesh layouts and element technologies could trigger a variety of deformation patterns near the crack tip.