FEM analysis of concrete structures subjected to mode-I and mixed-mode loading conditions

In this study, the finite element method (FEM) for a body containing displacement discontinuity is used for the investigation of tensile fracture behavior under mode-I and mixed-mode loading conditions in concrete structures. A mechanical model for the tensile fracture behavior is reduced to a mathematical problem, and the analysis method is proposed. With the aid of this method, several factors which govern tensile fracture are examined, such as the unloading path in the tension-softening behavior and the transmission of shear stresses across crack surfaces. A plain concrete beam without a notch is analyzed by first neglecting and then taking into account the unloading path in the tension-softening behavior to demonstrate the phenomenon of cracking localization in mode-I crack growth. Pullout test specimens of practical significance are analyzed in order to study the crack growth phenomenon under mixed-mode loading conditions. Cases with and without lateral confinement are considered and the results obtained from the present analysis are compared with those obtained from available experimental data. A simple model for shear transfer across crack surfaces is established. By incorporating this model in the program, a pullout test specimen with lateral confinement is analyzed to examine the influence of shear transfer across crack surfaces on cracking localization.     

Numerical simulation of rapid crack propagation in viscoplastic materials

A finite element model for the simulation of fast crack propagation under dynamic Mode I load conditions is developed. The principal tools are an enhanced strain formulation and a quasi-contact algorithm. Crack propagation is controlled by the implemented fracture criterion. The model accounts for viscoplastic material behavior, dynamic effects and contact in the crack zone. The numerical results demonstrate that these physical effects are of major importance when load speed is increased to a high level and therefore they cannot be neglected.     

Homogenization-based multiscale crack modelling: From micro-diffusive damage to macro-cracks

The existence of a representative volume element (RVE) for a class of quasi-brittle materials having a random heterogeneous microstructure in tensile, shear and mixed mode loading is demonstrated by deriving traction–separation relations, which are objective with respect to RVE size. A computational homogenization based multiscale crack modelling framework, implemented in an FE² setting, for quasi-brittle solids with complex random microstructure is presented. The objectivity of the macroscopic response to the micro-sample size is shown by numerical simulations. Therefore, a homogenization scheme, which is objective with respect to macroscopic discretization and microscopic sample size, is devised. Numerical examples including a comparison with direct numerical simulation are given to demonstrate the performance of the proposed method.     

A hybrid finite element-scaled boundary finite element method for crack propagation modelling

This study develops a novel hybrid method that combines the finite element method (FEM) and the scaled boundary finite element method (SBFEM) for crack propagation modelling in brittle and quasi-brittle materials. A very simple yet flexible local remeshing procedure, solely based on the FE mesh, is used to accommodate crack propagation. The crack-tip FE mesh is then replaced by a SBFEM rosette. This enables direct extraction of accurate stress intensity factors (SIFs) from the semi-analytical displacement or stress solutions of the SBFEM, which are then used to evaluate the crack propagation criterion. The fracture process zones are modelled using nonlinear cohesive interface elements that are automatically inserted into the FE mesh as the cracks propagate. Both the FEM’s flexibility in remeshing multiple cracks and the SBFEM’s high accuracy in calculating SIFs are exploited. The efficiency of the hybrid method in calculating SIFs is first demonstrated in two problems with stationary cracks. Nonlinear cohesive crack propagation in three notched concrete beams is then modelled. The results compare well with experimental and numerical results available in the literature.     

二维圆环结构中弹性波传播的谱单元分析

为研究弹性波在结构中的波场特征,推导建立一种任意四边形二维谱单元,将其应用于二维圆环结构中的波传播行为模拟.以矩形平板结构中弹性导波传播分析为例,通过与传统有限元分析结果对比,验证所建立谱单元的有效性.有裂纹和无裂纹二维圆环结构谱单元模拟结果表明:弹性波在圆环结构中传播会发生频散;圆环结构表面的裂纹对瑞利波的传播影响最大;根据其反射波包位置可以确定损伤位置.     

Automatic LEFM crack propagation method based on local Lepp–Delaunay mesh refinement

A numerical method for 2D LEFM crack propagation simulation is presented. This uses a Lepp–Delaunay based mesh refinement algorithm for triangular meshes which allows both the generation of the initial mesh and the local modification of the current mesh as the crack propagates. For any triangle t, Lepp(t) (Longest Edge Propagation Path of t) is a finite, ordered list of increasing longest edge neighbor triangles, that allows to find a pair of triangles over which mesh refinement operations are easily and locally performed. This is particularly useful for fracture mechanics analysis, where high gradients of element size are needed. The crack propagation is simulated by using a finite element model for each crack propagation step, then the mesh near the crack tip is modified to take into account the crack advance. Stress intensify factors are calculated using the displacement extrapolation technique while the crack propagation angle is calculated using the maximum circumferential stress method. Empirical testing shows that the behavior of the method is in complete agreement with experimental results reported in the literature. Good results are obtained in terms of accuracy and mesh element size across the geometry during the process.     

Efficient numerical integration of an elastic–plastic damage law within a mixed velocity–pressure formulation

This study focuses on numerical integration of constitutive laws in numerical modeling of cold materials processing that involves large plastic strain together with ductile damage. A mixed velocity–pressure formulation is used to handle the incompressibility of plastic deformation. A Lemaitre damage model where dissipative phenomena are coupled is considered. Numerical aspects of the constitutive equations are addressed in detail. Three integration algorithms with different levels of coupling of damage with elastic–plastic behavior are presented and discussed in terms of accuracy and computational cost. The implicit gradient formulation with a non-local damage variable is used to regularize the localization phenomenon and thus to ensure the objectivity of numerical results for damage prediction problems. A tensile test on a plane plate specimen, where damage and plastic strain tend to localize in well-known shear bands, successfully shows both the objectivity and effectiveness of the developed approach.     

Nonlinear analysis of reinforced concrete hyperboloid cooling towers—I. Material model, finite element model and validation

This is the first part of a two-part series of papers in which the constitutive material modelling of reinforced concrete, in shell structures, which resist applied loads predominantly through membrane action, is presented. The material model includes the effects of tensile cracking, tension stiffening, compression softening, interface shear transfer, and change in material stiffness due to crack rotation. A four-noded isoparametric curved shell element has been used in the nonlinear finite element analysis. The results obtained by using the model for analysis of a shear wall panel subjected to in-plane loading have been compared with those from experimental investigation.     

巴西圆盘劈裂破坏的近场动力学建模与分析

基于非局部近场动力学（Peridynamics,PD）理论,对含预置裂纹的混凝土巴西圆盘劈裂破坏问题进行建模分析.将结构离散为包含混凝土材料信息的粒子,引入动态松弛、分级加载和失衡力守恒等粒子系统数值算法,构建可以自然模拟脆性裂纹扩展的PD算法体系.对含不同角度单预置中心裂纹巴西圆盘的裂纹扩展过程进行数值模拟,所得结果与试验结果吻合较好,验证所提出的模型和算法正确.进一步采用该方法模拟双预置裂纹巴西圆盘劈裂破坏过程中的裂纹扩展、交汇、贯通过程,通过将所得模拟结果与试验结果进行比较分析,探究该方法处理多裂纹扩展问题的可行性.     

Reliability analysis of brittle structures under random loadings

A method for the reliability analysis of brittle structures subjected to random loads is proposed. The method is based on the weakest-link hypothesis and Weibull statistics for brittle materials. Initial flaws with a given expected size are assumed to be distributed at random with a certain density per unit volume. Basic concepts in random vibration theory and fracture mechanics are utilized in evaluating stress statistics, crack propagation and strength degradation. A structure fails when the stress intensity at any flaw reaches a critical value for rapid crack propagation. The failure of the structure is modeled as the first exceedance in random vibration theory. The effects of multi-vibration modes on the failure probability of the structure are included in the formulation. The evaluation of stress distribution and the computation of failure probability can be accomplished in a finite element analysis. Numerical examples on the evaluation of lifetime reliabilities of structures are given to demonstrate the feasibility of the proposed method.     

Fracture behaviour of single crystal silicon microstructures

The fracture behaviour of single crystal silicon (SCSi) microstructures is analysed based on microme-chanical torsional and tensile experiments. The uniaxial testpieces are characterised by the presence of sharp not-ches at the gauge length extremities. The critical loading conditions are reproduced in a finite element model in order to identify the analogies of the failure conditions in tension and torsion. The stress field in the vicinity of the notch tip (were cracks originate) is analyzed, and fracture mechanics parameters are determined. In the finite element model a crack, reproducing the failure process observed in the experiments, is included. The crack area is incrementally increased and the energy release rate for the critical loading conditions in tension and torsion is calculated. Based on these results a failure criterion is formulated along with a procedure for the mechanical integrity analysis of SCSi microstructures of arbitrary shape and loading conditions.     

Design of maintenance schedules for fatigue-prone metallic components using reliability-based optimization

This contribution focuses on the design of optimal maintenance schedules for metallic structures prone to develop fatigue cracks. The crack propagation phenomenon is addressed using a fracture mechanics approach. The problem of maintenance scheduling is addressed within the framework of reliability-based optimization (RBO). Thus, it is possible to minimize the costs associated with maintenance and eventual failure while explicitly considering uncertainties in the crack propagation phenomenon and inspection activities. The underlying RBO problem is solved using an efficient method recently developed by the authors. A numerical example demonstrating the application of the proposed approach is presented.     

Limit load analysis of thick-walled concrete structures - a finite element approach to fracture

The paper illustrates the interaction of constitutive modelling and finite element solution techniques for limit load prediction of concrete structures.On the constitutive side, an engineering model of concrete fracture is developed in which the Mohr-Coulomb criterion is augmented by tension cut-off to describe incipient failure. Upon intersection with the stress path the failure surface collapses for brittle behaviour according to one of three softening rules — no-tension, no-cohesion, and no-friction. The stress transfer accompanying the energy dissipation during local failure is modeled by several fracture rules which are examined with regard to ultimate load prediction.On the numerical side the effect of finite element idealization is studied first as far as ultimate load convergence is concerned. Subsequently, incremental tangential and initial load techniques are compared together with the effect of step size.Limit load analyses of a thick-walled concrete ring and a lined concrete reactor closure conclude the paper along with engineering examples.     

Numerical simulation of multiple 3D fracture propagation using arbitrary meshes

This paper describes a mesh-independent finite element based method for propagating fractures in three dimensions. The iterative algorithm automatically grows fractures in a 3D brittle medium represented by an isotropic linear elastic matrix. Growth is controlled by an input failure and propagation criterion. The geometry and mesh are stored separately, and mesh refinement is topologically guided. Propagation results in the modification of crack geometry, as opposed to changes in the mesh, as the arbitrary tetrahedral mesh adapts to the evolving geometry. Stress intensity factors are computed using the volumetric J Integral on a virtual piecewise cylinder. Modal stress intensity factors are computed using the decomposition method. Mesh and cylinder size effects are studied, as is computational efficiency. A through-going crack embedded in a thick slab, and a horizontal and inclined penny-shape crack, are used to validate the accuracy of the method. The predicted stress intensity factors are in good agreement with analytical solutions. For six integration points per tip segment, integration local to single tips, and a cylinder radius that adapts to the local geometric conditions, results agree with analytical solutions with less than 5% deviation from experimental results.     

Underground rockfall stability analysis using the numerical manifold method

The present study applies the numerical manifold method (NMM) as a tool to investigate the rockfall hazard in underground engineering. The crack evolution technique with crack initiation and propagation criterion, which has been successfully applied to handle cracking problems in rocks, is used in this study. A rockbolt element is introduced, which is first validated by a simple case. The mechanism of the rockbolt in reinforcing a layered rock mass is then investigated through a four-layered rock beam example. The developed NMM is then used to investigate the rockfall instability caused by either natural joints or mining induced fractures in an underground power station house or a tunnel. The results illustrate that the developed NMM can not only capture the entire dynamic process of the rockfall but also locate the keyblock successfully. As such, corresponding reinforcement methods can be chosen reasonably.     

Simulation of delamination in stringer stiffened fiber-reinforced composite shells

Fiber-reinforced composites are often used for high performance lightweight structures. For an enhanced exploitation of material reserves, fracture mechanisms should be taken into consideration. In this work, delamination and skin-stringer separation are examined in the framework of the finite element method. A cohesive interface element is used which is written in stress-strain relationships. The cohesive law rests upon a Smith-Ferrante type free energy function. It is edited so that only tensile normal or shear stresses provoke damage and contact is accounted for by an additional penalty term. Some numerical examples show the applicability of the proposed model.     

Shape optimization for 2-D mixed-mode fracture using Extended FEM (XFEM) and Level Set Method (LSM)

Weight and service life are often the two most important considerations in design of structural components. This research incorporates a novel crack propagation analysis technique into shape optimization framework to support design of 2-D structural components under mixed-mode fracture for: (1) maximum service life, subject to an upper limit on volume, and (2) minimum weight subject to specified minimum service life. In both cases, structural performance measures are selected as constraints and CAD dimensions are employed as shape design variables. Fracture parameters, such as crack growth rate and crack growth direction are computed using extended finite element method (XFEM) and level set method (LSM). XFEM employs special enrichment functions to incorporate the discontinuity of structural responses caused by the crack surfaces and crack tip fields into finite element approximation. The LSM utilizes level set functions to track the crack during the crack propagation analysis. As a result, this method does not require highly refined mesh around the crack tip nor re-mesh to conform to the geometric shape of the crack when it propagates, which makes the method extremely attractive for crack propagation analysis. An accurate and efficient semi-analytical design sensitivity analysis (DSA) method is developed for calculating gradients of fracture parameters. Two different approaches—a batch-mode, gradient-based, nonlinear algorithm and an interactive what-if analysis—are used for optimization. An engine connecting rod example is used to demonstrate the feasibility of the proposed method.     

Computational service life estimation of contacting mechanical elements in regard to pitting

A computational model for determining the service life of contacting surfaces in regard to surface pitting is presented. The model considers the material fatigue process leading to pitting, i.e. the conditions required for the short fatigue crack propagation originating from the initial crack in a single material grain. In view of small crack lengths observed in surface pitting, the simulation takes into account the short crack growth theory. The stress field in the contact area and the required functional relationship between the stress intensity factor and the crack length are determined by the finite element method. An equivalent model of two contacting cylinders is used for numerical simulations of crack propagation in the contact area. On the basis of numerical results, and with consideration of some particular material parameters, the probable service life period of contacting surfaces is estimated for surface curvatures and loadings that are most commonly encountered in engineering practice.     

An efficient model of mixed-mode delamination in laminated composites including bridging mechanisms

A refined model is developed to analyse delamination in composite laminates accounting for bridging stresses at crack faces. The analysis adopts a first-order shear deformable layer-wise kinematics for the laminate and an interface model simulating mixed-mode fracture in the presence of a bridged delamination. A penalised interface simulates adhesion between layers and provides energy release rates through its strain energy density while a two-parameter softening interface with a limit displacement models bridging stresses. Delamination evolution analysis is performed by solving the non-linear boundary value problem resulting from a stress analysis coupled with opportune propagation conditions. Numerical examples are presented for composite laminates subjected to both pure mode and mixed-mode loading conditions and the results are compared with those obtained adopting classic delamination models. Analytical formulae for energy release rate evaluation are also proposed to carry out an investigation of the main factors governing accuracy in predicting delamination growth. The proposed approach captures important effects which are not included in classic delamination models. The accuracy of the model is assessed by comparisons with 2D finite element results obtained by using delamination interface elements. The finite element model agrees well with results obtained by using the proposed approach.     

X-FEM a good candidate for energy conservation in simulation of brittle dynamic crack propagation

This paper is devoted to the simulation of dynamic brittle crack propagation in an isotropic medium. It focuses on cases where the crack deviates from a straight-line trajectory and goes through stop-and-restart stages. Our argument is that standard methods such as element deletion or remeshing, although easy to use and implement, are not robust tools for this type of simulation essentially because they do not enable one to assess local energy conservation. Standard cohesive zone models behave much better when the crack’s path is known in advance, but are difficult to use when the crack’s path is unknown. The simplest method which consists in placing the cohesive segments along the sides of the finite elements leads to crack trajectories which are mesh-sensitive. The adaptive cohesive element formulation, which adds new cohesive elements when the crack propagates, is shown to have the proper energy conservation properties during remeshing. We show that the X-FEM is a good candidate for the simulation of complex dynamic crack propagation. A two-dimensional version of the proposed X-FEM approach is validated against dynamic experiments on a brittle isotropic plate.