Multivariate Adaptive Regression Splines Model to Predict Fracture Characteristics of High Strength and Ultra High Strength Concrete Beams

This paper presents Multivariate Adaptive Regression Splines (MARS) model to predict the fracture characteristics of high strength and ultra high strength concrete beams. Fracture characteristics include fracture energy (GF), critical stress intensity factor (KIC) and critical crack tip opening displacement (CTODc). This paper also presents the details of development of MARS model to predict failure load (Pmax) of high strength concrete (HSC) and ultra high strength concrete (UHSC) beam specimens. Characterization of mix and testing of beams of high strength and ultra strength concrete have been described. Methodologies for evaluation of fracture energy, critical stress intensity factor and critical crack tip opening displacement have been outlined. MARS model has been developed by establishing a relationship between a set of predicators and dependent variables. MARS is based on a divide and conquers strategy partitioning the training data sets into separate regions; each gets its own regression line. Four MARS models have been developed by using MATLAB software for training and prediction of fracture parameters and failure load.MARS has been trained with about 70% of the total 87 data sets and tested with about 30% of the total data sets. It is observed from the studies that the predicted values of Pmax, GF, KIC and CTODC are in good agreement with those of the experimental values.     

基于虚拟裂缝模型的混凝土等效断裂韧度

本文采用虚拟裂缝模型,将临界裂缝尖端张开位移ＣＴＯＤｃ作为控制参数,利用三点弯曲梁试件通过迭代求得了混凝土裂缝亚临界扩展量的临界值△ａｃ,据此求得了混凝土起裂断裂韧度ＫｉｎｉＩｃ、等效断裂韧度ＫｕｎＩｃ值。计算结果表明,随着试件尺寸的增大,△ａｃ增大,但ＫｉｎｉＩｃ、ＫｕｎＩｃ值却是与试件尺寸无关的断裂参数。这表明线弹性断裂韧度准则可应用于混凝土结构的裂缝评定。     

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。     

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。     

Crack Growth Across a Strength Mismatched Bimaterial Interface

Crack growth across an interface between materials with different strength is examined by a cohesive zone model. The two materials have identical elastic properties but different fracture process properties, or different yield stresses, which is modeled by different cohesive stresses. The fracture criteria is a critical crack opening displacement. Load is represented by a stress intensity factor defining a remote square root singular stress field. The results show that the ratio between the cohesive stresses of the two materials primarily determines the behavior of the critical stress intensity factor. When the crack approaches a material with a higher cohesive stress the crack tip is shielded, but if the crack approaches a material with smaller critical crack opening displacement the maximum level of shielding is determined by the ratio between the critical crack opening displacements. When a crack approaches a material with a lower cohesive stress it is exposed to an amplified load. This revised version was published online in August 2006 with corrections to the Cover Date.     

泵头体高强钢材料的力学性能

为了进一步研究泵头体高强钢25Cr2Ni4MoV和30CrNi2MoV的力学性能,采用拉伸、冲击和裂纹尖端张开位移(CTOD)试验对其进行测试。结果表明:材料在常温下具有较好的综合力学性能;在静载荷条件下,具有优良的断裂韧度。     

Simplified equations for determining double‐K fracture parameters of concrete for 3‐point bending test

The paper presents simplified polynomial equations for determining the double‐K fracture parameters of concrete for 3‐point bending beams with variable strengths and material properties of concrete. The derived equations avoid complexities involved in computations of fracture parameters using existing analytical methods. The input data required for systematic computation in the study for deriving the nondimensional fracture parameters are obtained using a fictitious crack model. It is inferred that for a relative size of initial crack length, critical load and corresponding crack opening displacement maintain a linear relationship in their nondimensional forms. The value of critical mouth opening displacement can also be determined for known value of peak load using the derived nondimensional equation, thus avoiding the measurement of the crack mouth opening displacement in the experiment. Further, the derived polynomial equations predict the double‐K fracture parameters of concrete with negligible error as compared to those obtained based on experimental results.     

Evaluation of Fracture Parameters by Double-G,Double-K Models and Crack Extension Resistance for High Strength and Ultra High Strength Concrete Beams

This paper presents the advanced analytical methodologies such as Double- G and Double - K models for fracture analysis of concrete specimens made up of high strength concrete (HSC, HSC1) and ultra high strength concrete. Brief details about characterization and experimentation of HSC, HSC1 and UHSC have been provided. Double-G model is based on energy concept and couples the Griffith's brittle fracture theory with the bridging softening property of concrete. The double-K fracture model is based on stress intensity factor approach. Various fracture parameters such as cohesive fracture toughness (K_Ic^c), unstable fracture toughness (K_Ic^un) and initiation fracture toughness (K_Icⁱⁿⁱ) have been evaluated based on linear elastic fracture mechanics and nonlinear fracture mechanics principles. Double-G and double-K method uses the secant compliance at the peak point of measured P-CMOD curves for determining the effective crack length. Bi-linear tension softening model has been employed to account for cohesive stresses ahead of the crack tip. From the studies, it is observed that the fracture parameters obtained by using double - G and double - K models are in good agreement with each other. Crack extension resistance has been estimated by using the fracture parameters obtained through double - K model. It is observed that the values of the crack extension resistance at the critical unstable point are almost equal to the values of the unstable fracture toughness K_Ic^un of the materials. The computed fracture parameters will be useful for crack growth study, remaining life and residual strength evaluation of concrete structural components.     

Crack growth in cement-based composites

A model that can be used to predict Mode I crack growth in cement-based composites is presented. The region ahead of a crack tip, where nonlinear deformations and aggregate interlock occur, is modeled as an extension of the actual stress-free crack subjected to a closing pressure that depends on the crack face displacements. In the case of concrete, crack propagation is assumed to occur when the crack opening displacement at the tip of the actual crack reaches a critical value. To predict results, the elastostatics problem of a layer containing a vertical edge crack was solved using a Green's function approach together with integral transform techniques. Stress intensity factors and crack opening displacements were obtained by numerically solving a singular integral equation. The closing pressure function and critical crack tip opening displacement were taken from experimental data for various materials, and the model was applied to the analysis of experiments performed on initially notched concrete and fiber-reinforced mortar beams.     

聚丙烯纤维水泥稳定碎石断裂性能试验研究

通过30个尺寸为100mm×100mm×515mm的聚丙烯纤维水泥稳定碎石和普通水泥稳定碎石三点弯曲试件断裂试验,探讨了聚丙烯纤维对水泥稳定碎石断裂韧度(KIC)、断裂能(GF)、临界裂缝嘴张开位移(CMODC)、临界裂缝尖端张开位移(CTODC)、极限裂缝嘴张开位移(CMODmax)和极限裂缝尖端张开位移(CTODmax)的影响。试验结果表明:聚丙烯纤维的掺入可增大水泥稳定碎石的断裂韧度、断裂能、临界裂缝嘴张开位移、极限裂缝嘴张开位移、临界裂缝尖端张开位移和极限裂缝尖端张开位移;随着聚丙烯纤维体积掺量的增加,断裂韧度、临界裂缝嘴张开位移和临界裂缝尖端位移的变化无明显规律,但断裂能、极限裂缝嘴张开位移和极限裂缝尖端位移基本上呈线性增加的。     

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.     

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.     

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.     

大初始缝高比混凝土试件双K断裂参数的试验研究

双K 断裂准则能够定量描述混凝土裂缝的起裂、稳定扩展和失稳断裂。该文采用混凝土三点弯曲梁试件,通过在试验中测得的起裂荷载P_ini、最大荷载P_max及临界裂缝口张开位移CMOD_C计算了初始缝高比为0.3~0.9共7组试件的起裂断裂韧度K_ICini 和失稳断裂韧度K_ICun 。结果表明,当初始缝高比为0.3~0.7时,混凝土裂缝扩展经历起裂、稳定扩展和失稳破坏3 个阶段,双K 参数均是与初始缝高比无关的材料参数;当初始缝高比大于或等于0.8 时,混凝土裂缝起裂后便进入失稳扩展阶段,起裂荷载即为最大荷载,且计算得到的K_ICini 仍与初始缝高比无关。因此,在确定K_ICini 时,仅需测得初始缝高比大于或等于0.8试件的P_max,将P_max作为P_ini直接计算得到K_ICini。同以往的试验方法相比,其结果更为准确且试验方法简单。     

采用峰值荷载法确定全级配水工混凝土断裂参数

该文采用峰值荷载法研究了全级配水工混凝土试件的断裂参数。通过有限元方法,得到了适用于全级配水工混凝土的非标准楔入劈拉试件的应力强度因子K、裂缝嘴张开口位移CMOD计算公式以及裂缝张开位移COD与CMOD比值COD/CMOD的相关曲线。基于相关断裂试验,通过该文所提公式得到了全级配水工混凝土的等效断裂韧度与临界裂缝尖端张开口位移CTOD_c等断裂参数。研究结果表明:峰值荷载法可应用于截面尺寸不同而缝高比相同的全级配水工混凝土试件,峰值荷载法只需实测各试件的峰值荷载,就可方便确定全级配水工混凝土的断裂参数。     

Crack tip opening displacement in micro-cracked concrete by an embedded optical fiber sensor

Crack tip opening displacement (CTOD) is an important parameter that is often employed in characterization of fracture in engineering materials. Due to inherent difficulties in direct determination of CTOD, the crack mouth opening displacement (CMOD) has been measured during fracture tests of concrete beams and then related to CTOD. Analytical schemes are used for the determination of CTOD from the measured crack mouth opening displacements, which leads to the determination of other fracture parameters of importance. The research presented here describes development of an experimental technique for direct determination of CTOD based on embedded fiber optic sensors. The transduction mechanism is based on the correlation between the applied strain and the light intensity variation of speckle patterns generated at the output end of the multimode optical fiber due to mode redistribution.The measured displacements, CTOD, and CMOD, are compared and the validity of analytical schemes in estimation of CTOD is evaluated.     

砼双K断裂参数的实用解析方法

采用标准三点弯曲梁砼试件,利用试验中测得的最大荷载maxP及对应的裂缝口张开位移CMODc计算了砼裂缝亚临界扩展量caD。提出了由虚拟裂缝闭合力)(xs产生的应力强度因子cKI的实用解析方法,进而计算了砼双K断裂参数iniKIC及unKIC。采用这一方法,只需在试验中测得最大荷载maxP及对应的裂缝口张开位移CMODc、砼弹性模量及抗拉强度即可确定双K断裂参数并且无需对cKI进行数值积分。大量的计算分析表明,这一实用解析方法不但可使计算过程进一步简化,而且具有很高的精度。     

Effect of acid corrosion on crack propagation of concrete beams

The effect of acid corrosion on crack propagation of concrete beams was theoretically studied by the method of crack extension resistance curve. Based on this method, a calculation approach was proposed to determine fracture stress intensity factors in crack propagation of concrete beams. Loop iteration analysis was carried out to calculate maximum bearing capacity load, unstable crack toughness, resistance toughness curve, cohesive toughness curve and load–crack mouth opening displacement. Both bilinear and nonlinear softening traction–separation curves were adopted for each of these calculation parameters. The analysis results of each showed the effect of acid corrosion degrees. The influence of acid corrosion on fracture properties was discussed through the calculated results of cohesive toughness curves. These five kinds of simulated results were basically consistent, before the load attained the maximum value. However, with further crack propagation, cohesive toughness of nonlinear softening model was significantly larger than that of bilinear softening model, and the descending branch of P–CMOD curve by nonlinear law is higher than that by bilinear law. To validate the approach, tests of specimens under six different corrosion periods were experimentally studied, using three-point bending notched concrete beams soaked in sulphuric acid solution. The Double-K fracture parameters were investigated based on the test results, and load–crack mouth opening displacement curves for different acid conditions were obtained using synchronous sampling of a load sensor and clip-gauge. Numerical results by bilinear softening model showed a good correlation with the experimental ones.     

Prediction of Fracture Parameters of High Strength and Ultra-High Strength Concrete Beams using Minimax Probability Machine Regression and Extreme Learning Machine

This paper deals with the development of models for prediction of facture parameters, namely, fracture energy and ultimate load of high strength and ultra high strength concrete based on Minimax Probability Machine Regression (MPMR) and Extreme Learning Machine (ELM). MPMR is developed based on Minimax Probability Machine Classification (MPMC). ELM is the modified version of Single Hidden Layer Feed Foreword Network (SLFN). MPMR and ELM has been used as regression techniques. Mathematical models have been developed in the form of relation between several input variables such as beam dimensions, water cement ratio, compressive strength, split tensile strength, notch depth, and modulus of elasticity and output is fracture energy and ultimate load A total of 87 data sets (input-output pairs) are used, 61 of which are used to train the model and 26 are used to test the models. The data-sets used in this study are derived from experimental results. A comparative study has been presented between the developed MPMR and ELM models. The results showed that the developed models give reasonable performance for prediction of fracture energy and ultimate load.