Determination of the kink point in the bilinear softening model for concrete

The characterization of the softening curve from experimental results is essential for predicting the fracture behavior of quasi-brittle materials like concrete. Among various shapes (e.g. linear, exponential) to describe the softening behavior of concrete, the bilinear softening relationship has been extensively used and is the model of choice in this work. Currently, there is no consensus about the location of the kink point in the bilinear softening curve. In this study, the location of the kink point is proposed to be the stress at the critical crack tip opening displacement. Experimentally, the fracture parameters required to describe the bilinear softening curve can be determined with the “two-parameter fracture model” and the total work of fracture method based on a single concrete fracture test. The proposed location of the kink point compares well with the range of kink point locations reported in the literature, and is verified by plotting stress profiles along the expected fracture line obtained from numerical simulations with the cohesive zone model. Finally, prediction of experimental load versus crack mouth opening displacement curves validate the proposed location of the kink point for different concrete mixtures and also for geometrically similar specimens with the same concrete mixture. The experiments were performed on three-point bending specimens with concrete mixtures containing virgin coarse aggregate, recycled concrete coarse aggregate (RCA), and a 50-50 blend of RCA and virgin coarse aggregate. The verification and validation studies support the hypothesis of the kink point occurring at the critical crack tip opening displacement.     

Loading‐rate dependence of mode I crack growth in concrete

Mode I crack propagation process of concrete under relatively low loading rates which cover four orders of magnitude (0.2 μm/s to 2.0 mm/s) is investigated with three‐point bending (TPB) beams. All measured material properties exhibit rate sensitivity and follow a log‐linear relationship with the loading rate. A rate‐sensitive softening curve is established. The complete load‐crack mouth opening displacement (P‐CMOD) curve, crack propagation length, and fracture process zone (FPZ) length are simulated based on crack growth criterion with the fitted material parameters under those loading rates. Results show that the simulated P‐CMOD curves agree well with those of experimental measurements. It is clear that the peak load increases with the loading rate and so is the critical crack mouth opening displacement. Moreover, under the same load level, the length of the FPZ and the cohesive stress at the initial crack tip also increase with the increasing loading rate.     

混凝土I型裂缝扩展准则比较研究

针对混凝土I型裂缝扩展问题,分别采用以起裂韧度为参数的裂缝扩展准则、最大拉应力准则以及裂尖处应力强度因子为零的裂缝扩展准则,数值模拟了强度等级C20、C40、C60、C80和C100的混凝土三点弯曲梁裂缝扩展全过程,获取了试件的荷载-裂缝口张开位移(P-CMOD)曲线并与试验结果进行了比较。结果表明,三种准则中以起裂韧度为参数的裂缝扩展准则计算得到的峰值荷载及P-CMOD全曲线与试验结果差别最小。随着混凝土强度等级的提高,最大拉应力准则以及裂尖处应力强度因子为零的裂缝扩展准则计算出的P-CMOD曲线与试验结果相比均有较为明显的偏离,但以起裂韧度为参数的裂缝扩展准则计算结果与试验曲线更为吻合。试验与计算结果表明,以起裂韧度为参数的裂缝扩展准则更适用于不同强度混凝土材料的断裂分析。     

Experimental and numerical investigations on fracture process zone of rock–concrete interface

A crack propagation criterion for a rock–concrete interface is employed to investigate the evolution of the fracture process zone (FPZ) in rock–concrete composite beams under three‐point bending (TPB). According to the criterion, cracking initiates along the interface when the difference between the mode I stress intensity factor at the crack tip caused by external loading and the one caused by the cohesive stress acting on the fictitious crack surfaces reaches the initial fracture toughness of a rock–concrete interface. From the experimental results of the composite beams with various initial crack lengths but equal depths under TPB, the interface fracture parameters are determined. In addition, the FPZ evolution in a TPB specimen is investigated by using a digital image correlation technique. Thus, the fracture processes of the rock–concrete composite beams can be simulated by introducing the initial fracture criterion to determine the crack propagation. By comparing the load versus crack mouth opening displacement curves and FPZ evolution, the numerical and experimental results show a reasonable agreement, which verifies the numerical method developed in this study for analysing the crack propagation along the rock–concrete interface. Finally, based on the numerical results, the effect of ligament length on the FPZ evolution and the variations of the fracture model during crack propagation are discussed for the rock–concrete interface fracture under TPB. The results indicate that ligament length significantly affects the FPZ evolution at the rock–concrete interface under TPB and the stress intensity factor ratio of modes II to I is influenced by the specimen size during the propagation of the interfacial crack.     

Crack Extension Resistance and Fracture Properties of Quasi-Brittle Softening Materials like Concrete Based on the Complete Process of Fracture

The crack extension resistance and fracture properties are studied in detail for quasi-brittle materials like concrete with a softening traction-separation law by investigating the complete fracture process. The computed samples are the three-point bending notched beams of concrete with different sizes tested by other researchers. The softening traction-separation law which was proposed by Reinhardt et al. based on direct tension tests for normal concrete materials was chosen in the computations. Different distribution shapes of the cohesive force on the fictitious crack zone were considered for the corresponding loading states. The computations were mainly based on the analytic solutions for this problem using Gauss–Chebyshev quadrature to achieve the integration which is singular at the integral boundary. The crack extension resistance curves in terms of stress intensity (K_R-curves) were determined by combining the crack initiation toughness that is the inherent toughness of the material needed to resist the crack initiation in the case that is in the lack of an extension of the main crack with the contribution due to the cohesive force along the fictitious crack zone during the complete processes of fracture. The situation of crack propagation can be judged by comparing K_R-curves of crack extension resistance with the stress intensity factor curves which were calculated using the lengths of the extending crack and the corresponding loads at each loading states, e.g., when the crack extension resistance curve(K_R-curve) is lower than the stress intensity factor curve, the crack propagation is stable; otherwise, it is unstable. In the computation, the obtained relationship between the crack tip opening displacement CTOD and the amount of crack extension for the complete fracture process is in agreement with the testing results of other researchers. This revised version was published online in August 2006 with corrections to the Cover Date.     

A comparative study on five approaches to evaluate double-K fracture toughness parameters of concrete and size effect analysis

The current paper presents a comprehensive comparison of double-K fracture toughness parameters of concrete evaluated using experimental method and four existing analytical methods. Fracture tests were carried out on compact tension wedge splitting specimens with various depths varying from 200 mm up to 1000 mm. In the analytical calculation, depending on the relationship between critical crack tip opening displacement and the abscissa value of turning point on bilinear softening curve, two different distributions of cohesive stress are considered along crack extension. Results show that four available analytical calculations yield almost the same values of double-K fracture toughness parameters and agree well with those obtained from the experiment, which confirms the consistency of five approaches. Size effect was discussed, including unstable fracture toughness, initiation fracture toughness, critical effective crack length, the length of critical fracture process zone and critical crack tip opening displacement.     

基于声发射技术的非标准自密实混凝土三点弯曲梁动态断裂特性

对不同初始缝高比的自密实混凝土(Self-compacting concrete,SCC)非标准三点弯曲梁开展不同加载速率下的断裂试验,获得其断裂的荷载-裂缝嘴张开口位移曲线及峰值荷载、断裂韧度、临界缝高比增量、弹性模量和柔度系数等断裂参数,结合Pearson相关性检验公式及加载速率效应模型,定量分析初始缝高比、加载速率与断裂参数间的相关性强弱及SCC断裂参数的加载速率效应。结果表明峰值荷载、断裂韧度及弹性模量均存在一定的加载速率效应,柔度系数仅与初始缝高比强相关,弹性模量和断裂韧度是材料的固有属性,不受初始缝高比影响。同时,基于声发射(Acoustic emission,AE)技术对SCC的损伤断裂过程、断裂边界效应及裂缝扩展模式进行分析,结果表明,AE参量能较好地反映混凝土断裂的三阶段特性及边界效应。裂缝的扩展首先以拉伸裂缝为主,剪切裂缝占比随着裂缝扩展过程逐渐增大。      

Process zone analysis for the single-edge notched specimen: Part II. Process zone growth and crack propagation

Process zone growth and crack propagation in the single-edge notched (SEN) specimen are studied using the relations among applied load, notional crack and process zone lengths, and crack opening displacement derived in the first part of this work [1]. Process zone growth is simulated by increasing the notional crack length while keeping the traction-free crack length constant. A model for crack propagation based on either critical crack tip opening displacement (CTOD) or critical process zone length, as criteria for traction-free crack extension is proposed. The influence of closing pressure distribution, initial traction-free crack length, and crack extension criterion on the behavior of load vs. CMOD curves is discussed. The present model can be used to model load-deformation behavior from initial loading through softening to failure of nonlinear materials, as is verified by comparing the theoretical and experimentally determined load vs. crack mouth opening (CMOD) curves for concrete beams.     

混凝土软化本构关系对双K断裂参数的影响

混凝土裂缝扩展的双K 断裂准则，用于描述混凝土结构裂缝的起裂、稳定扩展和失稳断裂。其相 应的双K 断裂参数ini c K I 、un c K I 可通过简便的实验和基于虚拟裂缝扩展粘聚力的解析方法确定。已取得的 研究结果表明，起裂韧度ini c K I 的解析解与所确定的混凝土软化本构关系有关。对于简单实用的双线性软化 本构曲线而言，参数s s 、s w 、0 w 的取值直接影响软化本构关系的确定。根据本课题组新近所做楔入劈 拉实验，经分析对比得到了与实测结果相符合、为确定双K 断裂参数所使用的双线性软化本构曲线的相应 参数。     

Deterioration of fracture toughness of concrete under acid rain environment

Deterioration of fracture toughness of concrete caused by acid rain attack was investigated through laboratory-accelerated corrosion tests. Fracture toughness of the corroded concrete with different corrosion period was obtained by three point bending tests, and meanwhile the P-COMD curves and elastic modulus E of the corroded specimens with different corrosion degrees were also obtained. The experimental results showed that the deterioration of concrete surface is large and the corrosion depth increases with corrosion duration. The observation by scanning electron microscopy showed that the microstructures of crack tip are porous and loose due to acid corrosion, and many crisscross cracks appear at corrosion layer. Based on the experimental results, equivalent crack length in the fracture process zone and the effective fracture toughness K_IC were also obtained by theoretical analysis. Under the impact of acid corrosion, the concrete material physical and chemical properties changed significantly, which made the corresponding resistance stresses to crack propagation decrease. The curves of fracture toughness versus corrosion depth were obtained in this study.     

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.     

Effect of softening function on the cohesive crack fracture parameters of concrete CT specimen

The paper presents numerical study on the fracture parameters of concrete compact tension test specimens of different sizes using cohesive crack model. As softening function is the main ingredient of the cohesive crack model, seven numbers of widely used softening functions are incorporated in the model based on enhanced algorithm. It is found that the difference between the highest and the lowest peak loads obtained using various softening functions (except linear) is less than about 9%. The peak load predicted by the linear softening curve is about 16% larger than that of mean peak load predicted by other softening functions. The cohesive crack model with linear softening yields the fracture process zones lower by approximately 30–50% than those obtained by using the other softening relations for specimen size range 200–600 mm. The numerical results are further compared with a reference test result available in the literature. It is observed that some of softening relations (except linear) predict the experimentally obtained peak load up to 6% of accuracy whereas the linear softening curve overestimates it by about 30%. The numerically gained softening branch of load-displacement curves compare well with the experimental observation.     

不同软化曲线形状对裂缝扩展阻力G_R曲线的影响

最近基于裂缝粘聚力提出了描述裂缝扩展全过程阻力变化的GR曲线,所发展的解析解相关于骨料桥联咬合作用造成的非线性断裂过程区的能量耗散,故与该区域材料使用的拉伸软化本构关系即应力-裂缝张开口位移软化曲线有密切联系。鉴于此,该文采用4种不同的软化曲线研究了软化曲线形状对裂缝扩展阻力GR的影响。结果发现,GR曲线对软化曲线敏感,GR曲线的合理性依赖于软化曲线的准确性。在使用准确软化曲线的前提下,GR曲线的特征点与软化曲线的特征点存在相对应关系。     

基于扩展有限元法的钢筋混凝土梁复合断裂过程模拟研究

针对钢筋混凝土梁裂纹扩展问题,基于扩展有限元法,建立了预置裂纹的简支混凝土梁三维模型,用粘聚裂纹模型描述裂纹面间的力学行为,采用线性的软化曲线表示裂纹尖端断裂过程区的应变软化行为,分别对素混凝土梁和钢筋混凝土梁的复合断裂过程进行模拟,分析了纵向钢筋对裂纹扩展路径、荷载-挠度和荷载 -CMOD (裂缝开口处张开位移)曲线的影响,并与文献中的试验结果进行对比,计算结果与试验结果吻合良好,展示了扩展有限元法在结构断裂破坏分析方面的独特优势。     

小尺寸混凝土试件双K断裂参数试验研究

采用最大尺寸为680mm×160mm×40mm的标准三点弯曲梁试件,利用在初始裂缝两侧粘贴电阻应变片并利用混凝土裂缝扩展到此处时其应变回缩的方法测得了起裂荷载Pini,在此基础上根据Pini及初始缝长a0得到了起裂断裂韧度KIiCni;根据在试验中测得的最大荷载Pmax及对应的裂缝口张开位移CMODC计算了混凝土等效裂缝长度aC,据此计算了失稳断裂韧度KIuCn。结果表明:采用电阻应变片法可准确测定混凝土的起裂荷载Pini,且方法简单。试验结果还表明:在本试验范围内,三点弯曲梁法测得的混凝土双K断裂参数KIiCni、KIuCn与试件高度无关,进一步说明了混凝土双K断裂参数可以作为描述混凝土裂缝扩展的断裂参数。     

Modelling crack propagation in reinforced concrete using a hybrid finite element–scaled boundary finite element method

A previously developed hybrid finite element–scaled boundary finite element method (FEM–SBFEM) is extended to model multiple cohesive crack propagation in reinforced concrete. This hybrid method can efficiently extract accurate stress intensity factors from the semi-analytical solutions of SBFEM and is also flexible in remeshing multiple cracks. Crack propagation in the concrete bulk is modelled by automatically inserted cohesive interface elements with nonlinear softening laws. The concrete–reinforcement interaction is also modelled by cohesive interface elements. The bond shear stress–slip relation of CEB-FIP Model Code 90 and an empirical confining stress–crack opening relation are used to characterise slip and split failure at the concrete–reinforcement interface, respectively. Three RC beams were simulated. The numerical results agreed well with both experimental and numerical results available in the literature. Parametric studies demonstrated the importance of modelling both slip and split failure mechanisms at the concrete–reinforcement interface.     

Size effect and inverse analysis in concrete fracture

This paper summarizes the basic experimental and numerical results supporting an easy procedure to determine up to two fracture parameters based on numerically computed size effect curves. Furthermore, it supplies closed-form expressions to determine the initial linear segment approximation of the (stress vs. crack opening) softening curve of cohesive crack models for concrete, based only on the peak loads determined in splitting-tension (Brazilian) tests and in three-point-bending test on notched specimens. Knowledge of the initial segment, although not enough to describe all the fracture process of concrete structures, is enough to predict the fracture behavior of unnotched concrete structures prior and around the peak load. The same is true for notched structures provided their size is less than a limiting size, approximately defined in the paper. This revised version was published online in July 2006 with corrections to the Cover Date.     

Analytical and experimental investigations on the fracture behavior of hybrid fiber reinforced concrete

In the present study, Mode-I fracture tests of hybrid fiber reinforced concrete (HFRC) composite beams were conducted and the fracture properties and other post peak strength characteristics of the HFRC composites were evaluated and analyzed. The HFRC composite was produced using three types of fibers namely steel, Kevlar and polypropylene. A total of 27 HFRC composite beam specimens were cast and tested using the RILEM recommended three point bending test. The main variables were the fiber volume content and combinations of different fibers. The load versus crack mouth opening displacement (CMOD) curves of HFRC composite beams were obtained. Inverse analysis was carried out to determine the tensile strength and crack opening relationship. Analytical models based on comprehensive reinforcing index were developed for determining the influence of the fibers on fracture energy, flexural tensile strength, equivalent tensile strengths and residual tensile strengths of HFRC composites. Based on the experimental results and inverse analysis, a model for predicting the tensile softening diagram of HFRC composite mixes was also developed. The analytical models show conformity with the experimental results.