A stereological analysis of aggregate grading and size effect on concrete tensile strength

One of the most important topics in solid mechanics is the study of the so-called size effects, whose importance has been widely recognised during the last decades. Size effects are particularly strong in quasi-brittle (i.e., concrete-like) materials. In this paper we focus our attention on the tensile strength decrease associated with the size of concrete structures. An original explanation of this well-known size effect was proposed by the first Author based on the assumption of a fractal-like damage localisation at the mesostructural level. This hypothesis leads to a multifractal scaling law (MFSL) for concrete tensile strength. The present contribution provides a scaling law for concrete tensile strength based on its aggregate size distribution. Since the weakest link in normal strength concrete is represented by the interface between the cementitious matrix and the aggregates, it seems reasonable to look for a relationship between the aggregate grading and the material strength. Based on the hypothesis that the strength depends on the largest flaw, we compute the strength of a concrete specimen as a function of its size. Differently from other statistical approaches, we use a truncated distribution (namely the Füller distribution) in order to describe realistically the flaw population inside the specimen. Calculating the distribution of the largest flaw size by means of statistics of extremes, and relating it to the specimen size, we obtain a scaling law for concrete tensile strength whose trend strictly agrees with the MFSL. Finally, we pay particular attention to the computation of the power law exponent characterising the strength scaling at the smallest sizes and present a comparison with available experimental data.     

骨料粒径对混凝土动态拉伸强度及尺寸效应影响分析

骨料粒径是影响混凝土力学性能及破坏机理的重要因素。从细观角度出发,将混凝土看作由骨料颗粒、砂浆基质及界面过渡区组成的三相复合材料,考虑细观组分的应变率效应,建立了混凝土动态拉伸破坏行为研究的细观力学分析模型,模拟研究了不同骨料粒径下混凝土动态拉伸破坏行为,并揭示了动态拉伸强度的尺寸效应规律。研究表明:低应变率下骨料不发生破坏,骨料粒径对混凝土动态拉伸破坏模式及拉伸强度影响显著,且拉伸强度的尺寸效应随骨料粒径的减小而削弱;高应变率下裂缝将贯穿骨料,骨料粒径的大小对混凝土动态拉伸强度及尺寸效应影响可忽略。最后,结合应变率效应的影响机制,建立了混凝土拉伸强度的"静动态统一"尺寸效应理论公式,该公式可以较好描述各骨料粒径下混凝土动态拉伸强度与试件尺寸的定量关系。     

Probabilistic strength of {1 1 1} n-type silicon

The two-parameter Weibull strength distribution of {1 1 1} n-type silicon prismatic bars was determined in four-point bending and analyzed as a function of specimen size (width), loading rate, two different crystallographic orientations, and specimen orientation (polished or etched surface in tension). 100% fractography was performed to classify strength-limiting flaw types and to censor the strength data. All flaw types were extrinsic. Machining or cutting damage in the form of chipped edges (an edge-type flaw) was the dominant strength-limiting flaw when the polished surface was subjected to tensile stress, while a flat-bottomed etch pit (a surface-type flaw) was the dominant strength-limiting flaw when the etched surface was subjected to tensile stress. The censored Weibull strength distribution was independent of specimen width, loading rate (indicative of slow crack growth insusceptibility), and the two crystallographic orientations; however, it was dependent on specimen orientation. Pooling of the strength data was employed to tighten the confidence intervals about the censored parameters. The results from this study indicate that different extrinsic strength-limiting flaws and strength distributions will be operative depending on the manner in which a silicon component is stressed.     

骨料粒径对混凝土劈拉性能及尺寸效应影响的细观数值研究

混凝土材料宏观力学行为的非线性及尺寸效应根源于其内部组成的非均质性。考虑材料细观结构非均质性的影响,建立由骨料颗粒、砂浆基质和界面过渡区组成的混凝土细观尺度力学模型。对尺寸为150 mm、250 mm、350 mm和450 mm的混凝土立方体模型劈裂抗拉破坏行为进行细观数值模拟,探讨骨料粒径（最大粒径分别为:10 mm、20 mm、30 mm和40 mm）的影响机制,并与试验结果进行对比分析。结果表明:1） 混凝土材料的劈裂抗拉强度随着骨料粒径增大而略微降低,最大骨料粒径达到30 mm左右时,强度降低趋势变缓;2） 四种骨料粒径下混凝土立方体劈裂抗拉强度均存在尺寸效应现象,相比于大骨料试件,小骨料试件的破坏更具脆性,因而其尺寸效应更显著;3） 混凝土劈裂抗拉强度尺寸效应行为与 Ba?ant和Weibull提出的尺寸效应理论相吻合。     

Evaluation of fracture parameters of concrete from bending test using inverse analysis approach

In this study, an inverse analysis approach is developed to obtain the fracture parameters of concrete, including stress–crack opening relationship, cracking and tensile strength as well as fracture energy, from the results of a three-point bending test. Using this approach, the effects of coarse aggregate size (5–10, 10–16, 16–20 and 20–25 mm) and matrix strength (compressive strength of 40 and 80 MPa, respectively) on the fracture parameters are evaluated. For normal strength concrete, coarse aggregate size and cement matrix strength significantly influence the shape of σ–w curve. For a given total aggregate content, small aggregate size leads to a high tensile strength and a sharp post-peak stress drop. The smaller the coarse aggregate, the steeper is the post-peak σ–w curve. By contrast, in high strength concrete, a similar σ–w relationship is obtained for various aggregate sizes. The post-peak stress drop for high strength concrete is more abrupt than that for normal strength concrete. Also, the smaller the coarse aggregate size, the higher is the flexural strength. For both normal and high strength concrete, fracture energy and characteristic length are found to increase with increase of coarse aggregate size.     

Micromechanical modeling of crack-aggregate interaction in concrete materials

A new condition for crack penetration into the aggregate phase in concrete materials is developed based on numerical simulations. The numerical simulation utilizes a newly developed micromechanical model which considers the concrete internal structure as a three-phase material, viz. matrix, aggregate and interfaces between them. The micromechanical model is capable of capturing the entire load-deformation response of a concrete specimen under monotonic loading including softening. The new condition for crack penetration is developed based on a simple specimen configuration where a crack is driven towards an aggregate particle. Results from numerical simulations are implemented to relate the relative properties of both aggregate and matrix phases (represented by the characteristic length ratio) to their tensile strength ratio. It is shown that the tensile strength ratio between the aggregate and the matrix plays the dominant role in determining the penetration condition. Predictions based on this condition agree with direct tensile simulations using different specimen configuration.     

Non‐proportional size scaling of strength of concrete in uniaxial and biaxial loading conditions

A procedure for non‐proportional size scaling of the strength of concrete based on the weakest‐link statistics is proposed to synchronize strength data from specimens of different geometries and different loading modes. The procedure relies on proportional size scaling of strength to determine the parameters of the statistical model and often on finite element analysis to calculate the coefficient of the equivalent strength. The approach for non‐proportional size scaling is capable to synchronize the uniaxial strength data of concrete from uniaxial tensile specimens and 3‐point bending specimens, or the biaxial tensile strength data of circular plates in different loading mode. The non‐transference of the uniaxial strength data to the biaxial strength data is unclear in its mechanism but possibly due to the variation of statistical distribution of microcracks with stress states in different specimens.     

Experiments on concrete under uniaxial impact tensile loading

A problem of practical importance for designing of structural elements is discussed in this paper—the behaviour of concrete subjected to uniaxial impact tensile loading. The “Split Hopkinson Bar” technique was adopted for testing concrete in uniaxial tension at stress rates between 2 and 60 N/mm²/ms. A remarkable increase in tensile strength was observed due to high stress rate. The ratio of impact and static tensile strength varied between 1.33 and 2.34 for various concrete mixes. The influence of maximum aggregate size, water-cement ratio, cement content, cement type and quality, specimen humidity, static compressive strength and loading/casting direction upon the uniaxial impact tensile strength was studied. The high stress rate resulted in an increase of the modulus of elasticity of concrete in uniaxial tension. An explanation for the observed phenomena is suggested.     

Size effect of concrete members applied with flexural compressive stresses

In this study, two types of special experiments are carried out to understand flexural compressive strength size effect of concrete members. The first type is an ordinary cylindrical specimen (CS) with a fully penetrated and vertically standing plate type notch at the mid-height of the specimen, which is loaded in compression at the top surface (e.g., in the parallel direction to the notch length). The second type is a general double cantilever beam (DCB), which is compression loaded in axial direction (e.g., in the parallel direction of the notch). For CS, an adequate notch length is taken from the experimental results obtained from the compressive strength experiment of various initial notch lengths. The trial tests to select the effective initial notch length show that CS with an initial notch length approximately greater than four times the maximum aggregate size fails without an additional increased load and in stable manner under Mode I failure mechanism. Therefore, the initial notch length to the maximum aggregate size ratio of 4.0 is used for all size specimens. For DCB, the eccentricity of loading points with respect to the axial axis of each cantilever and the initial notch length are varied. In both specimens, the compressive loads apply flexural compressive stresses on the crack tip region of the specimens. These two types of specimens fail by Mode I crack opening mechanism. By testing 3 geometrically proportional size specimens for CS and DCB, the experimental datum for flexural compression size effect of concrete are obtained. Using the obtained flexural compressive strength size effect datum, regression analyses are performed using Levenberg-Marquardt's least square method (LSM) to suggest new parameters for the modified size effect law (MSEL). The analysis results show that size effect is apparent for flexural compressive strength of specimens with an initial notch. For CS, the effect of initial notch length on flexural compressive strength size effect is apparent. For DCB, flexural compressive size effect is dependent on the eccentricity of loading points with respect to the axial axis of the cantilever beam. In other words, if DCB specimen is applied with greater tensile stress at the crack tip, the size effect of concrete becomes more distinct. The results show that the flexural compressive strength size effect of initial notch length variation of DCB exists but directly dependent on the loading location. This is due to the fact that the sizes of fracture process zone (FPZ) of all DCB specimens are similar regardless of the differences in the specimen slenderness ratio, but the flexural compressive and tensile stress combinations resulting in stress concentration at the crack tip region has direct effect on size effect of concrete members.     

Determination of r‐curves of steel fiber reinforced concrete from size effect tests

Abstract

A developed size effect law for blunt fracture is used to determine the R‐Curves and related parameters of steel fiber reinforced concrete. Geometrically similar single‐edge notched beams of different sizes made of cementitious mixes at various fiber volume fractions and different maximum aggregate sizes were used for the tests. Fracture energy of concrete is identified by linear regression using the size effect law. From the experimentally calibrated size effect law, the R‐curve is obtained as the envelope of the family of fracture equilibrium curves for different specimen sizes.     

Effect of polystyrene aggregate size on strength and moisture migration characteristics of lightweight concrete

The aim of this study is to investigate the effect of polystyrene aggregate size on strength and moisture migration characteristics of lightweight concrete. The present study covers the use of expanded polystyrene (EPS) and un-expanded polystyrene (UEPS) beads as lightweight aggregate in concretes that contain fly ash as a supplementary cementitious material. Lightweight concrete with wide range of concrete densities (1000–1900 kg/m³) were studied mainly for compressive strength, split tensile strength, moisture migration and absorption. The results indicate that for comparable aggregate size and concrete density, concrete with UEPS aggregate exhibited 70% higher compressive strength than EPS aggregate. EPS aggregate concrete with small EPS aggregates showed higher compressive strength and the increase in compressive strength was more pronounced in low density concrete when compared with high density concrete. The UEPS aggregate concrete exhibited brittle failure similar to normal weight concrete (NWC), whereas, gradual failure was observed in EPS concrete. Moreover, the moisture migration and absorption results indicate that the EPS concrete containing bigger size and higher volumes of EPS aggregate show higher moisture migration and absorption.     

考虑细观组分影响的混凝土宏观力学性能理论预测模型

混凝土宏观力学性能与其内部细观结构构造密切相关。该文建立了一类能够考虑细观组分影响的混凝土宏观力学性能理论预测模型。首先,采用细观力学数值试验法对理论模型中的参数进行了标定;进而,基于该模型对混凝土断裂能和单轴抗拉强度在材料层次的尺寸效应行为进行了分析。结果表明:混凝土断裂能和单轴抗拉强度均随骨料级配（即最大骨料粒径）发生变化,且受到界面特性的影响。当界面过渡区力学性能相对薄弱时,混凝土强度较低,断裂能和单轴抗拉强度随骨料级配增大而呈现减小的趋势;当界面过渡区力学性能较强时,混凝土强度较高,断裂能和单轴抗拉强度随骨料级配增大亦呈现增大的趋势。计算结果与试验结果吻合良好,验证了该文建立的理论预测模型的准确性和合理性。     

粗骨料与钢纤维对超高性能混凝土单轴拉伸性能的影响

为了提高含粗骨料超高性能混凝土(Ultra-high performance concrete,UHPC)的单轴拉伸性能,采用单轴拉伸试验和图像分析技术分别研究了粗骨料掺量、颗粒粒径对含粗骨料UHPC单轴拉伸性能和钢纤维在UHPC体系中分散性能的影响规律。结果表明,随着粗骨料掺量及颗粒粒径的增大,钢纤维在UHPC体系中的分散系数和取向系数显著降低,含粗骨料UHPC的单轴拉伸初裂强度、裂后强度和耗能也随之减小。根据粗骨料颗粒最大粒径与钢纤维体积分数、直径间的匹配关系式(Dmax=3df/(Vf)0.5),采用纤维混杂可以充分发挥多尺度纤维与具有不同粒径分布的骨料间的分级匹配关系;粗骨料体积分数和颗粒最大粒径分别为10%和10mm时,采用平直钢纤维(直径0.12mm、长度10mm、体积掺量1.2%)和端钩钢纤维(直径0.35 mm、长度20mm、体积掺量1.8%)混杂实现了含粗骨料UHPC的单轴拉伸性能的提升,其裂后强度和耗能分别为8.69 MPa和11.10J。     

NUMERICAL SMEARED FRACTURE ANALYSIS: NONLOCAL MICROCRACK INTERACTION APPROACH

A recently proposed new nonlocal concept based on microcrack interactions is discussed, its implementation in a smeared cracking finite element code for concrete is presented, numerical studies are reported, and comparisons with experimental results are made. The nonlocality is not merely a mathematical device to prevent excessive spurious localization into a zone of zero volume but is a necessary physical consequence of microcrack interactions. Since the constitutive law itself is strictly local, the new nonlocal concept can be combined with any type of constitutive law for strain-softening nonlocal damage, which is here chosen to be the microplane model. A simple method is formulated to approximately identify the material parameters in the model from the basic characteristics of concrete such as the tensile strength, fracture energy and maximum aggregate size. The results of finite element analysis are shown to be mesh insensitive, and good convergence is obtained. Cracking damage is found to localize into a volume whose size and shape depend on the macroscopic concrete properties as well as the current stress–strain state. Although the damage is considered to be tensile on the microlevel, due solely to mode I microcracks, the new nonlocal model can describe well not only mode I fracture tests but also complex shear-dominated and mixed-mode types of failure such a diagonal shear, and can do so for the same values of material parameters (which was not the case for previous nonlocal models). Most importantly, the new nonlocal model can correctly capture the size effect of quasibrittle fracture, in approximate agreement with Bažant's size effect law.     

Scaling of quasibrittle fracture: hypotheses of invasive and lacunar fractality, their critique and Weibull connection

Considerable progress has been achieved in fractal characterization of the properties of crack surfaces in quasibrittle materials such as concrete, rock, ice, ceramics and composites. Recently, fractality of cracks or microcracks was proposed as the explanation of the observed size effect on the nominal strength of structures. This explanation, though, has rested merely on intuitive analogy and geometric reasoning, and did not take into account the mechanics of crack propagation. In this paper, the energy-based asymptotic analysis of scaling presented in the preceding companion paper in this issue [1] is extended to the effect of fractality on scaling. First, attention is focused on the propagation of fractal crack curves (invasive fractals). The modifications of the scaling law caused by crack fractality are derived, both for quasibrittle failures after large stable crack growth and for failures at the initiation of a fractal crack in the boundary layer near the surface. Second, attention is focused on discrete fractal distribution of microcracks (lacunar fractals), which is shown to lead to an analogy with Weibull's statistical theory of size effect due to material strength randomness. The predictions ensuing from the fractal hypothesis, either invasive or lacunar, disagree with the experimentally confirmed asymptotic characteristics of the size effect in quasibrittle structures. It is also pointed out that considering the crack curve as a self-similar fractal conflicts with kinematics. This can be remedied by considering the crack to be an affine fractal. It is concluded that the fractal characteristics of either the fracture surface or the microcracking at the fracture front cannot have a significant influence on the law of scaling of failure loads, although they can affect the fracture characteristics. Walter P. Murphy, Professor| of This revised version was published online in July 2006 with corrections to the Cover Date.     

Effect of Coarse Aggregate Size on Relationship between Stress and Crack Opening in Normal and High Strength Concretes

Fine and coarse aggregates play an important role in the fracture of concrete. However, quantitative information available on the effect of the coarse aggregate size on the fracture properties of concrete is still limited. In the present paper, the effect of coarse aggregate size (single grade of 5～10, 10～16, 16～20 and 20～25 mm) on stress-crack opening (σ-w) relation in normal and high strength concretes (compressive strength of 40 and 80 MPa, respectively)was studied. The investigation was based on three-point bending tests implemented by fictitious crack analysis. The result shows that coarse aggregate size and cement matrix strength significantly influence the shape of σ-w curve.For a given total aggregate content, in normal strength concrete, smaller size of aggregate leads to a high tensile strength and a sharp stress drop after the peak stress. The smaller the coarse aggregate, the steeper the σ-w curve.By contrast, in high strength concrete, the effect of aggregate size on σ-w relation almost vanishes. A similar σ-w relation is obtained for the concrete except for the case of 20～25 mm coarse aggregate size. The stress drop after the peak stress is more significant for high strength concrete than that for normal strength concrete. Meanwhile, the smaller the coarse aggregate size, the higher the flexural strength. Fracture energy and characteristic length increase with increasing coarse aggregate size in both normal and high strength concretes.     

自密实钢纤维轻骨料混凝土的早期性能与损伤分析

在自密实轻骨料混凝土基础之上掺入钢纤维配制出自密实钢纤维轻骨料混凝土,分析了自密实钢纤维轻骨料混凝土的抗压强度、抗拉强度等主要力学性能以及收缩、抗碳化等耐久性能,并与普通骨料自密实混凝土进行对比分析。探讨了钢纤维对于改善自密实轻骨料混凝土损伤所起的作用及其机理。结果表明:掺入钢纤维后自密实轻骨料混凝土的抗压强度增大,劈拉强度明显提高,收缩及抗碳化能力也有明显改善。与普通骨料混凝土相比,自密实钢纤维轻骨料混凝土初始裂缝的产生与发展得到有效抑制。     

Capillary suction model for characterizing salt scaling resistance of concrete containing GGBFS

This paper presents a rational method to characterize the freeze–thaw salt scaling performance of concrete based on capillary suction forces. The testing program included evaluating the concrete’s chemical chloride binding capacity, degree of hydration, total porosity, compressive strength, sorptivity and de-icer salt scaling resistance conducted at both, 28 days and 2 years. Mixtures consisted of paste and concrete containing 0–60% GGBFS as cement replacement, and a water-to-binder ratio of 0.31 or 0.38. The proposed capillary suction model was adapted to include the concrete’s chloride binding capacity, pore characteristics, and age. Results from the capillary suction model are found to correlate with experimentally measured ionic sorption coefficients. Results revealed that the capillary suction depth decreases with increasing percentages of GGBFS due to blocking of capillary pores as a result of the chemical interaction between the saline solution and the binder material. A freeze–thaw salt scaling resistance factor (F/T SSRF) is proposed which accounts for the concrete’s degree of hydration, sorptivity and tensile strength and is shown to correlate with the concrete’s cumulative mass loss tested in accordance with MTO LS-412. Accordingly, the salt scaling performance of concrete containing GGBFS is influenced by the combined effect of the concrete’s pore size distribution of the exposed surface, chloride binding capacity, degree of hydration, and tensile strength.     

低应变率下混凝土动态拉伸破坏尺寸效应细观模拟

在混凝土静态破坏尺寸效应方面已取得了较完善的成果,而在动态破坏尺寸效应方面,包括其产生机制及对应的尺寸效应律的研究则非常匮乏。为探讨动态荷载作用下混凝土尺寸效应行为,从细观角度出发,结合混凝土细观结构特征,考虑动态加载下细观组分应变率效应的影响,建立了混凝土破坏行为研究的细观力学分析模型与方法。以双边缺口混凝土试件为例,对其在低应变率（10-5 s-1~1 s-1）下混凝土动态拉伸破坏行为及尺寸效应进行细观数值模拟,并分析了应变率效应对动态破坏尺寸效应的影响。最后,结合应变率效应对强度及尺寸效应的影响规律—“强度增强效应”与“尺寸效应削弱效应”,在静态破坏尺寸效应律的基础上,建立了混凝土拉伸强度的“静动态统一”尺寸效应理论公式,并验证了理论公式的准确性和合理性。     

The probability of hertzian fracture

The indentation strength of brittle solids is traditionally characterized by Auerbach's law, which predicts a linear relationship between the load required to initiate a Hertzian cone crack and the radius of a spherical indentor. This paper reviews both the energy balance and flaw statistical explanations of Auerbach's law. It is shown that Auerbach's law in the strictest sense only applies to well-abraded specimens. A novel application of Weibull statistics is presented which allows the distribution of fracture loads to be predicted for any specimen surface condition for a given indentor size. The indentation strength of a brittle solid, for both spherical and cylindrical indentors, is shown to be influenced by both its surface flaw statistics and the degree of interfacial friction. It is observed that the indentation strength of soda-lime glass is increased by a factor of about three times that expected for frictionless contact, and that for a fully bonded indentor, conical fractures cannot occur.