The Discontinuity‐Enriched Finite Element Method

We introduce a new methodology for modeling problems with both weak and strong discontinuities independently of the finite element discretization. At variance with the eXtended/Generalized Finite Element Method (X/GFEM), the new method, named the Discontinuity‐Enriched Finite Element Method (DE‐FEM), adds enriched degrees of freedom only to nodes created at the intersection between a discontinuity and edges of elements in the mesh. Although general, the method is demonstrated in the context of fracture mechanics, and its versatility is illustrated with a set of traction‐free and cohesive crack examples. We show that DE‐FEM recovers the same rate of convergence as the standard FEM with matching meshes, and we also compare the new approach to X/GFEM.     

Crack face contact for a hexahedral-based XFEM formulation

Taking into account arbitrary crack geometries, crack closure generally occurs independently of the load case. As the standard eXtended Finite Element Method (XFEM) does not prevent unphysical crack face penetration in this case, a formulation allowing for crack face contact is proposed in terms of a penalty formulation for normal contact. The discretization is developed for non-planar cracks intersecting hexahedral elements in an arbitrary manner. Typical problems of many crack face contact implementations within the XFEM, like locking or the introduction of additional degrees of freedom, are avoided by projecting the contact contribution onto the hexahedral element nodes. The method is tested by means of suitable numerical examples, finally presenting an application in form of a multiscale setup with arbitrarily arranged micro cracks in the vicinity of a macro crack front.     

Complete Tangent Stiffness for eXtended Finite Element Method by including crack growth parameters

The eXtended Finite Element Method (XFEM) is a useful tool for modeling the growth of discrete cracks in structures made of concrete and other quasi‐brittle and brittle materials. However, in a standard application of XFEM, the tangent stiffness is not complete. This is a result of not including the crack geometry parameters, such as the crack length and the crack direction directly in the virtual work formulation. For efficiency, it is essential to obtain a complete tangent stiffness. A new method in this work is presented to include an incremental form the crack growth parameters on equal terms with the degrees of freedom in the FEM‐equations. The complete tangential stiffness matrix is based on the virtual work together with the constitutive conditions at the crack tip. Introducing the crack growth parameters as direct unknowns, both equilibrium equations and the crack tip criterion can be handled within the same standard nonlinear iterations. This new solution strategy is believed to provide the modeling capabilities to deal with simultaneous growth of several cracks. A cohesive crack modeling is used. The method is applied to a partly cracked XFEM element of linear strain triangle type with the crack length as the unknown crack growth parameter. In this paper, two examples are given. The first example verifies the theory and the implementation. The second example is the benchmark test three point bending test, where the efficiency of the complete tangential behavior is shown. Copyright © 2013 John Wiley & Sons, Ltd.     

An embedded cohesive crack model for finite element analysis of mixed mode fracture of concrete*

An embedded cohesive crack model is proposed for the analysis of the mixed mode fracture of concrete in the framework of the Finite Element Method. Different models, based on the strong discontinuity approach, have been proposed in the last decade to simulate the fracture of concrete and other quasi‐brittle materials. This paper presents a simple embedded crack model based on the cohesive crack approach. The predominant local mode I crack growth of the cohesive materials is utilized and the cohesive softening curve (stress vs. crack opening) is implemented by means of a central force traction vector. The model only requires the elastic constants and the mode I softening curve. The need for a tracking algorithm is avoided using a consistent procedure for the selection of the separated nodes. Numerical simulations of well‐known experiments are presented to show the ability of the proposed model to simulate the mixed mode fracture of concrete.     

Performance of variable order singular elements in analyzing interfacial fractures

This work concerns the development of singular boundary elements and the investigation of their numerical performance in analyzing interfacial cracks. In the vicinity of such cracks arise singular stress fields with variable order of singularity depending on the material characterizing parameters. The development of these elements which approximate displacement and traction functions is accomplished through controlled relocation of the mid-side node determined by compatibility and continuity requirements which must obey shape functions. These elements were applied to simulate the elastic behavior of cracks which are perpendicular and terminate on the interface of a bimaterial structure. Their efficiency in conjunction to the boundary only element method, are demonstrated in crack opening displacement diagrams and crack tip stress tabulated results.     

Delamination analysis of composites by new orthotropic bimaterial extended finite element method

The extended finite element method (XFEM) is further improved for fracture analysis of composite laminates containing interlaminar delaminations. New set of bimaterial orthotropic enrichment functions are developed and utilized in XFEM analysis of linear‐elastic fracture mechanics of layered composites. Interlaminar crack‐tip enrichment functions are derived from analytical asymptotic displacement fields around a traction‐free interfacial crack. Also, heaviside and weak discontinuity enrichment functions are utilized in modeling discontinuous fields across interface cracks and bimaterial weak discontinuities, respectively. In this procedure, elements containing a crack‐tip or strong/weak discontinuities are not required to conform to those geometries. In addition, the same mesh can be used to analyze different interlaminar cracks or delamination propagation. The domain interaction integral approach is also adopted in order to numerically evaluate the mixed‐mode stress intensity factors. A number of benchmark tests are simulated to assess the performance of the proposed approach and the results are compared with available reference results. Copyright © 2011 John Wiley & Sons, Ltd.     

Stress analysis around crack tips in finite strain problems using the eXtended finite element method

Fracture of rubber‐like materials is still an open problem. Indeed, it deals with modelling issues (crack growth law, bulk behaviour) and computational issues (robust crack growth in 2D and 3D, incompressibility). The present study focuses on the application of the eXtended Finite Element Method (X‐FEM) to large strain fracture mechanics for plane stress problems. Two important issues are investigated: the choice of the formulation used to solve the problem and the determination of suitable enrichment functions. It is demonstrated that the results obtained with the method are in good agreement with previously published works. Copyright © 2005 John Wiley & Sons, Ltd.     

Analysis of micro fracture in human Haversian cortical bone under transverse tension using extended physical imaging

We propose a procedure to investigate local stress intensity factors at the scale of the osteons in human Haversian cortical bone. The method combines a specific experimental setting for a three‐point bending millimetric specimen and a numerical method using the eXtended Finite Element Method (X‐FEM). The interface between the experimental setting and the numerical method is ensured through an imaging technique that analyses the light microscopy observations to import the geometrical heterogeneity of the Haversian microstructures, the boundary conditions and appearing crack discontinuities into the numerical model. The local mechanical elastic Young's moduli are measured by nano‐indentation, and the Poisson ratios are determined by an imaging technique of the stress–strain fields. The model is able to access three scales of measurement: the macro scale of the material level (mm), the micro scale inside the Haversian material for stress–strain fields (10–100µm), and the sub‐micro scale for the crack opening profiles (1–10µm ) and fracture parameters (stress intensity factors). The model is applied to several patients at different aging stages. Copyright © 2009 John Wiley & Sons, Ltd.     

Stress intensity factors computation for bending plates with extended finite element method

The modelization of bending plates with through‐the‐thickness cracks is investigated. We consider the Kirchhoff–Love plate model, which is valid for very thin plates. Reduced Hsieh–Clough–Tocher triangles and reduced Fraejis de Veubeke–Sanders quadrilaterals are used for the numerical discretization. We apply the eXtended Finite Element Method strategy: enrichment of the finite element space with the asymptotic bending singularities and with the discontinuity across the crack. The main point, addressed in this paper, is the numerical computation of stress intensity factors. For this, two strategies, direct estimate and J‐integral, are described and tested. Some practical rules, dealing with the choice of some numerical parameters, are underlined. Copyright © 2012 John Wiley & Sons, Ltd.     

线粘弹性断裂分析的增量加料有限元法

提出了一种用于解决线粘弹性断裂问题的增量加料有限元法。为了反映裂纹尖端的应力奇异性,在裂尖附近的应力奇异区采用若干四边形加料单元和过渡单元,非奇异区采用常规四边形单元,三种单元分区混合使用形成求解域网格划分。加料单元通过引入裂尖渐近位移场,构造出可以较好反映裂尖奇异性的单元位移模式,过渡单元在加料单元基础上引入调整函数构造单元位移模式,用于连接加料单元和常规单元,以消除加料单元和常规单元间位移不协调。基于Boltzmann叠加原理,推导了粘弹性材料的增量型本构关系,进而获得了增量加料有限元列式,并基于节点位移外推法计算粘弹性介质中裂纹应变能释放率。数值算例验证了该文方法的正确性和有效性。     

Nodal sensitivities as error estimates in computational mechanics

Summary This paper proposes the use of special sensitivities, called nodal sensitivities, as error indicators and estimators for numerical analysis in mechanics. Nodal sensitivities are defined as rates of change of response quantities with respect to nodal positions. Direct analytical differentiation is used to obtain the sensitivities, and the infinitesimal perturbations of the nodes are forced to lie along the elements. The idea proposed here can be used in conjunction with general purpose computational methods such as the Finite Element Method (FEM), the Boundary Element Method (BEM) or the Finite Difference Method (FDM); however, the BEM is the method of choice in this paper. The performance of the error indicators is evaluated through two numerical examples in linear elasticity.     

On the elimination of quadrature subcells for discontinuous functions in the eXtended Finite‐Element Method

The introduction of discontinuous/non‐differentiable functions in the eXtended Finite‐Element Method allows to model discontinuities independent of the mesh structure. However, to compute the stiffness matrix of the elements intersected by the discontinuity, a subdivision of the elements into quadrature subcells aligned with the discontinuity line is commonly adopted. In the paper, it is shown how standard Gauss quadrature can be used in the elements containing the discontinuity without splitting the elements into subcells or introducing any additional approximation. The technique is illustrated and developed in one, two and three dimensions for crack and material discontinuity problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Stress states and growth behavior of bridged cracks at creep regime

In this study growth behavior of bridged cracks, resulting from the growth of pre-nucleated creep cavities with diffusional and dislocation-assisted mechanisms, is investigated numerically. The elements bridging the crack are assumed to be elastic; the bridging behavior ranges from full development of the bridging zones to failure of the bridging elements during the course of crack growth. The results indicate that the bridging traction significantly relaxes even with the overall creep deformation alone. The rate of this relaxation is not influenced by the rate of crack growth. However, the rate of change in the bridging zone length or the density of the bridging elements in the bridging zone strongly affects both the maximum value and the distribution of the traction in the bridging zone. A much weaker stress singularity than the ones described by K or C^* was found ahead of the bridged cracks in the creep regime. In this weak singularity region the cavities, located at increasing distance from the crack tip, grow at similar high rates to each other.     

Combined extended and superimposed finite element method for cracks

A combination of the extended finite element method (XFEM) and the mesh superposition method (s‐version FEM) for modelling of stationary and growing cracks is presented. The near‐tip field is modelled by superimposed quarter point elements on an overlaid mesh and the rest of the discontinuity is implicitly described by a step function on partition of unity. The two displacement fields are matched through a transition region. The method can robustly deal with stationary crack and crack growth. It simplifies the numerical integration of the weak form in the Galerkin method as compared to the s‐version FEM. Numerical experiments are provided to demonstrate the effectiveness and robustness of the proposed method. Copyright © 2004 John Wiley & Sons, Ltd.     

Time domain boundary element method for dynamic stress intensity factor computations

This paper presents a procedure for transient dynamic stress intensity factor computations using traction singular quarter-point boundary elements in combination with the direct time domain formulation of the Boundary Element Method. The stress intensity factors are computed directly from the traction nodal values at the crack tip. Several examples of finite cracks in finite domains under mode-I and mixed mode dynamic loading conditions are presented. The computed stress intensity factors are represented versus time and compared with those obtained by other authors using different methods. The agreement is very good. The results are reliable and little mesh dependent. These facts allow for the analysis of dynamic crack problems with simple boundary discretizations. The versatile procedure presented can be easily applied to problems with complex geometry which include one or several cracks.     

用扩展有限元方法模拟混凝土的复合型开裂过程

用扩展有限元法对混凝土梁复合型开裂过程进行了数值模拟。裂纹面间的力学行为采用粘聚裂纹模型来描述,通过引入切向保留刚度考虑剪力分量的影响。开裂方向的计算采用了一种简化的最大切向应力准则。对Arrea和Ingraffea的混凝土梁复合开裂实验进行了数值模拟。计算给出了裂纹萌生、扩展的过程及破坏形态,并获得了与实验结果对比良好的荷载-裂纹开口滑移曲线。结果表明,扩展有限元法通过附加特定的位移模式,使裂纹两侧不连位移场的表达独立于网格划分,是一种能够模拟准脆性材料复合开裂问题的有效方法。     

基于水平集算法的扩展有限元方法研究

扩展有限元是一种以单位分解思想为基础,在常规有限元位移中加入跳跃函数和渐近位移场函数,以处理不连续问题的数值方法。将水平集算法应用到裂纹界面的描述及加强单元类型的判别,并与扩展有限元相结合,用于分析材料断裂问题。相比传统有限元,有限元网格与裂纹面位置相互独立,不需满足裂纹为单元边、裂尖为单元节点和在裂纹附近进行高密度的...     

Modeling bimaterial interface cracks using the numerical manifold method

The numerical manifold method (NMM) is explored for simulations of bimaterial interface cracks. Two special types of physical covers with customized cover functions are introduced to describe the weak discontinuity across the material interface, the partially cracked elements as well as the interface crack tip singularity. Three typical bimaterial crack problems are simulated. The mixed-mode stress intensity factors are evaluated by the virtue of the domain form of the interaction integral and compared with the available reference solutions. Good agreements have demonstrated the validity and accuracy of the developed program.