Modeling of concrete cracking—A hybrid technique of using displacement discontinuity element method and direct boundary element method

This paper presents a new approach by making use of a hybrid method of using the displacement discontinuity element method and direct boundary element method to model concrete cracking by incorporating fictitious crack model. Fracture mechanics approach is followed using the Hillerborg's fictitious crack model. A boundary element based substructure method and a hybrid technique of using displacement discontinuity element method and direct boundary element method are compared in this paper. In order to represent the process zone ahead of the crack, closing forces are assumed to act in such a way that they obey a linear normal stress-crack opening displacement law. Plain concrete beams with and without initial crack under three-point loading were analyzed by both the methods. The numerical results obtained were shown to agree well with the results from existing finite element method. The model is capable of reproducing the whole range of load–deflection response including strain-softening and snap-back behavior as illustrated in the numerical examples.     

A numerical analysis of cracks emanating from a square hole in a rectangular plate under biaxial loads

This note concerns with stress intensity factors of cracks emanating from a square hole in rectangular plate under biaxial loads by means of the boundary element method which consists of the non-singular 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 corresponding left or right crack tip on top of the constant displacement discontinuity elements that cover the entire crack surface and the other boundary. The present numerical results illustrate that the present approach is very effective and accurate for calculating stress intensity factors of complicated cracks in a finite plate and can reveal the effect of the biaxial load and the cracked body geometry on stress intensity factors.     

AN IMPROVED NUMERICAL TECHNIQUE FOR SIMULATING THE GROWTH OF PLANAR FATIGUE CRACKS

Abstract— This paper describes a versatile technique for simulating the fatigue growth of a wide range of planar cracks of practical significance. Crack growth is predicted on a step-by-step basis from the Paris law using stress intensity factors calculated by the finite element method. The crack front is defined by a cubic spline curve from a set of nodes. Both the 1/4-node crack opening displacement and the three-dimensional J -integral (energy release rate) methods are used to calculate the stress intensity factors. Automatic remeshing of the finite element model to a new position which defines the new crack front enables the crack propagation to be followed. The accuracy and capability of this finite element simulation technique are demonstrated in this paper by the investigation of various problems of both theoretical and practical interest. These include the shape growth trend of an embedded initially penny-shaped defect and an embedded initially elliptical defect in an infinite body, the growth of a semi-elliptical surface crack in a finite thickness plate under tension and bending, the propagation of an internal crack in a round bar and the shape change of an external surface crack in a pressure vessel.     

Engineering critical assessment for offshore pipelines with 3-D elliptical embedded cracks

Offshore pipelines are usually constructed by girth welding method, while weldments may naturally contain surface or embedded defects. Most existing fracture assessment procedures based on the load-controlled method are not explicitly designed for situations with large plastic strain. In this paper, the nonlinear elastic plastic fracture response of pipeline girth weld with embedded cracks is investigated through 3-D finite element analysis combined with submodeling technique. Strain-based estimation formulas for crack tip opening displacement are proposed for the pure bending load as well as the bending combined with the internal pressure. Furthermore, comparisons of fracture assessment between BS7910 and finite element analysis in the current study are made for various crack sizes.     

General BE approach for three-dimensional dynamic fracture analysis

A general mixed boundary element approach for three-dimensional dynamic fracture mechanics problems is presented in this paper. A mixed traction-displacement integral equation formulation in the frequency domain is used. The hypersingular and strongly singular kernels are regularized by analytical transformations yielding an easy to implement BE approach. Nine-node quadrilateral and six-node triangular continuous quadratic elements are used for external boundaries and crack surfaces. The crack front elements have their mid node at one quarter of the element length allowing for a proper representation of the crack surface displacement. The present approach is intended for the frequency domain analysis of fracture mechanics problems of any general 3D geometry; i.e. boundless or bounded regions, single or multiple, surface or internal cracks. Transient dynamic problems are studied using the FFT algorithm. The numerical results presented show the robustness and accuracy of the approach which requires a reasonable number of elements and degrees of freedom.     

3D modelling of strong discontinuities in elastoplastic solids: fixed and rotating localization formulations

This paper is concerned with the incorporation of displacement discontinuities into a continuum theory of elastoplasticity for the modelling of localization processes such as cracking in brittle materials. Based on the strong discontinuity approach (SDA) (Computational Mechanics 1993; 12: 277–296) mesh objective 2D and 3D finite element formulations are developed using linear and quadratic 2D elements as well as 8‐noded 3D elements. In the formulation of the finite‐element model proposed in the paper, the analogy with standard formulations is emphasized. The parameter defining the amplitude of the displacement jump within the finite element is condensed out at the material level without employing the standard static condensation technique. This approach results in linearized constitutive equations formally identical to continuum models. Therefore, the standard return mapping algorithm is used to solve the non‐linear equations. In analogy to concepts used in continuum smeared crack models, a rotating formulation of the SDA is proposed in addition to the standard concept of fixed discontinuities. It is shown that the rotating localization approach reduces locking effects observed in analyses based on fixed localization directions. The applicability of the proposed SDA finite‐element model as well as its numerical performance is investigated by means of a three‐dimensional ultimate load analysis of a steel anchor embedded in a concrete block subjected to a shear force. Copyright © 2003 John Wiley & Sons, Ltd.     

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.     

Formulation of a triangular finite element with an embedded interface via isoparametric mapping

 This paper presents the formulation of a triangular finite element with an embedded interface, designed for the simulation of discrete crack propagation processes. The element is developed within a displacement-based framework. Linear interpolation of the displacement discontinuities along the internal interface is assumed in order to ensure compatibility across inter-element boundaries. The proper representation of the rigid body motions and the solvability of the discretised version of the mechanical problem in point are specifically addressed. Finally, the element performance is illustrated through comparison with some alternative proposals. Received 24 January 2001     

A discrete strong discontinuity approach

In this paper, strong discontinuities are embedded in finite elements to describe fracture in quasi-brittle materials. A new formulation is presented in which global nodes are introduced along the crack path. The displacement jumps are transferred to the element nodes as a rigid body motion. This approach is compared to the discrete approach, in which interface elements are inserted to model discontinuities. The adopted embedded discontinuities and the interface elements share similar kinematics as well as the same numerical integration schemes. Thus, the present formulation is obtained within the framework of a discrete approach and this is why it is called the discrete strong discontinuity approach (DSDA). Numerical tests are considered, namely a shear band, a mode-I and a mixed-mode fracture examples and a failure test of a RC beam externally reinforced with a steel sheet. Results are compared with those obtained from analyses using interface elements and with experimental results. Finally, conclusions are drawn with respect to mesh independence and robustness of the method.     

Fracture mechanics analysis of cracked 2-D anisotropic media with a new formulation of the boundary element method

A new formulation of the boundary element method (BEM) is proposed in this paper to calculate stress intensity factors for cracked 2-D anisotropic materials. The most outstanding feature of this new approach is that the displacement and traction integral equations are collocated on the outside boundary of the problem (no-crack boundary) only and on one side of the crack surfaces only, respectively. Since the new BEM formulation uses displacements or tractions as unknowns on the outside boundary and displacement differences as unknowns on the crack surfaces, the formulation combines the best attributes of the traditional displacement BEM as well as the displacement discontinuity method (DDM). Compared with the recently proposed dual BEM, the present approach doesn't require dua elements and nodes on the crack surfaces, and further, it can be used for anisotropic media with cracks of any geometric shapes. Numerical examples of calculation of stress intensity factors were conducted, and excellent agreement with previously published results was obtained. The authors believe that the new BEM formulation presented in this paper will provide an alternative and yet efficient numerical technique for the study of cracked 2-D anisotropic media, and for the simulation of quasi-static crack propagation.     

Embedded representation of fracture in concrete with mixed finite elements

As an alternative to the smeared and discrete crack representations, an embedded representation of fracture for finite element analysis of concrete structures is presented. The three-field Hu–Washizu variational statement is extended to bodies with internal discontinuities. The extended variational statement is then utilized for formulating elements with a discontinuous displacement field. The new elements are capable of modelling different deformation modes of an internal discontinuity at the element level. The satisfactory performance of the embedded crack representation is verified by several case studies on concrete fracture.     

The dual boundary element method: Effective implementation for crack problems

The present paper is concerned with the effective numerical implementation of the two-dimensional dual boundary element method, for linear elastic crack problems. The dual equations of the method are the displacement and the traction boundary integral equations. When the displacement equation is applied on one of the crack surfaces and the traction equation on the other, general mixed-mode crack problems can be solved with a single-region formulation. Both crack surfaces are discretized with discontinuous quadratic boundary elements; this strategy not only automatically satisfies the necessary conditions for the existence of the finite-part integrals, which occur naturally, but also circumvents the problem of collocation at crack tips, crack kinks and crack-edge corners. Examples of geometries with edge, and embedded crack are analysed with the present method. Highly accurate results are obtained, when the stress intensity factor is evaluated with the J-integral technique. The accuracy and efficiency of the implementation described herein make this formulation ideal for the study of crack growth problems under mixed-mode conditions.     

A new method for modelling cohesive cracks using finite elements

A model which allows the introduction of displacements jumps to conventional finite elements is developed. The path of the discontinuity is completely independent of the mesh structure. Unlike so‐called ‘embedded discontinuity’ models, which are based on incompatible strain modes, there is no restriction on the type of underlying solid finite element that can be used and displacement jumps are continuous across element boundaries. Using finite element shape functions as partitions of unity, the displacement jump across a crack is represented by extra degrees of freedom at existing nodes. To model fracture in quasi‐brittle heterogeneous materials, a cohesive crack model is used. Numerical simulations illustrate the ability of the method to objectively simulate fracture with unstructured meshes. Copyright © 2001 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.     

Strong displacement discontinuities and Lagrange multipliers in the analysis of finite displacement fracture problems

The finite displacement formulation of a quadrilateral element containing an embedded displacement discontinuity is presented. The formulation is based on the kinematically optimal technique, and some known defects of this technique are addressed. Lagrange multipliers are adopted to ensure the correct crack closure prior to initiation. Six additional degrees of freedom in each element allow the representation of the two states of the crack. A classical enhanced strain technique is employed to improve the bending performance of the element. Numerical examples illustrate both the robustness and the accuracy of the proposed solutions.     

基于单元嵌入技术和弹性比拟的含裂纹混凝土三维渗流模拟方法

混凝土结构在服役过程中受到力学荷载和环境因素的影响会产生裂纹,裂纹的产生、扩展和连通会显著影响混凝土的渗透性。该文将混凝土看作由基体和裂纹夹杂二相组成的复合材料,采用单元嵌入技术,有效地解决了基于连续网格的实体裂纹模型存在的网格协调问题。考虑到弹性问题与渗流问题在数学上的相似性,通过对含裂纹混凝土位移场和应力场的弹性比拟,得到相应的渗流场。在计算过程中,分别研究了渗透系数、网格尺寸及裂纹张开度对渗流场的影响,以及该文方法对多裂纹体的适用性。计算结果表明:该文的方法与实体裂纹模型的渗流场计算结果吻合,保证了结果的正确性。同时也降低了三维裂纹建模的难度,提高了计算的效率。     

垂直界面裂纹应力强度因子的加料有限元分析

为求解裂尖位于界面上的垂直双材料界面裂纹应力强度因子,发展了一种加料有限元方法。该方法应用Williams本征函数展开和线性变换方法求解裂尖渐进位移场,将该位移场加入常规单元位移模式中,得到加料垂直界面裂纹单元和过渡单元的位移模式,给出加料有限元方程。建立了典型垂直界面裂纹平面问题的加料有限元模型,求解加料有限元方程直接得到应力强度因子,与文献结果对比表明该方法具有较高的精度,可方便地推广应用于垂直界面裂纹的计算分析。     

On elastic–plastic fracture behavior of a bi-layered composite plate with a sub-interface crack under mixed mode loading

A generalized Irwin model is proposed to investigate elastic–plastic fracture behavior of a bi-layered composite plate with a sub-interface crack under combined tension and shear loading. The dependence of the stress intensity factors, the plastic zone size, the effective stress intensity factor and the crack tip opening displacement on the crack depth h, the Dundurs’ parameters and the phase angle θ is discussed in detail. Numerical results show that in most cases, if the crack is embedded in a stiffer material, when the crack is close to the interface, the plastic zone size and the crack tip opening displacement will increase. On the contrary, if the crack is embedded in a softer material, when the crack is close to the interface, the plastic zone size and the crack tip opening displacement will decrease.     

Numerical simulation of dynamic fracture using finite elements with embedded discontinuities

This paper presents the extension of some finite elements with embedded strong discontinuities to the fully transient range with the focus on dynamic fracture. Cracks and shear bands are modeled in this setting as discontinuities of the displacement field, the so-called strong discontinuities, propagating through the continuum. These discontinuities are embedded into the finite elements through the proper enhancement of the discrete strain field of the element. General elements, like displacement or assumed strain based elements, can be considered in this framework, capturing sharply the kinematics of the discontinuity for all these cases. The local character of the enhancement (local in the sense of defined at the element level, independently for each element) allows the static condensation of the different local parameters considered in the definition of the displacement jumps. All these features lead to an efficient formulation for the modeling of fracture in solids, very easily incorporated in an existing general finite element code due to its modularity. We investigate in this paper the use of this finite element formulation for the special challenges that the dynamic range leads to. Specifically, we consider the modeling of failure mode transitions in ductile materials and crack branching in brittle solids. To illustrate the performance of the proposed formulation, we present a series of numerical simulations of these cases with detailed comparisons with experimental and other numerical results reported in the literature. We conclude that these finite element methods handle well these dynamic problems, still maintaining the aforementioned features of computational efficiency and modularity.