Lattice modelling of size effect in concrete strength

This paper uses a recently improved lattice network model to study the size effect in the strength of plain concrete structures. The several improvements made to the lattice network model are: (i) tension softening of the matrix phase is included in the material modelling; (ii) the structural response is modelled by incrementing the deformation rather than the load. This eliminates the need for introducing arbitrary scaling parameters in the beam element failure criteria and; (iii) a square rather than a triangular lattice beam network is found to be adequate for modelling concrete, thus greatly reducing the computational time.The improved square lattice network has been used to simulate the complete load-deformation response of notched three-point bend beams of different sizes with a view to checking the validity of several size effect models available in the literature. Lattice simulation was found to identify microcracking, crack branching, crack tortuosity and bridging, thus allowing the fracture process to be followed until complete failure. The improved lattice model predicted smooth structural response curves in excellent agreement with test results.The simulated nominal strengths also correlated very well with the test results, apart from that for the smallest beams (depth 38.1 mm). However, even in the relatively broad range of sizes (1:8) of the test beams, there was no clear evidence that one size effect model is superior to the other. In fact, rather surprisingly the test data would appear to be equally well described by all the available size effect models. The lattice simulations however indicated a trend which is better predicted by the multifractal scaling model.     

Boundary effect on concrete fracture and non-constant fracture energy distribution

This paper extends the local fracture energy concept of Hu and Wittmann [29] and [30], and proposes a bilinear model for boundary or size effect on the fracture properties of cementitious materials. The bilinear function used to approximate the non-constant local fracture energy distribution along a ligament is based on the assumption of the proportionality of the local fracture energy to the fracture process zone (FPZ) height and characterises the FPZ height reduction when approaching a specimen back boundary. The bilinear function consists of a horizontal straight line of the intrinsic fracture energy G_F and a declining straight line that reduces to zero at the back boundary. It is demonstrated that using the bilinear model, the size-independent fracture energy G_F can be estimated from the fracture energy data measured on laboratory-size specimens, and the intersection of these two linear functions, defined as the transition ligament, represents the influence of the back boundary on the fracture properties. It is also demonstrated that the specimen size alone is not sufficient to characterise the size effect in the fracture properties observed on laboratory-size specimens.     

Modelling of fracture process in concrete using a novel lattice model

Papers deals with simulations of fracture process in quasi-brittle materials like concrete with a novel lattice model. Concrete was described mainly as a three-phase material composed of aggregate, cement matrix and interfacial transition zones. The calculations were carried out for concrete specimens subject to uniaxial extension, shear and extension and three-point bending. Two-dimensional and three-dimensional simulations were performed. The advantages and disadvantages of the proposed model were outlined.     

Filler from crushed aggregate for concrete: Pore structure,specific surface,particle shape and size distribution

Characteristics of fine particles (0–125 μm diameter) from seven different crushed and natural sands from five different Norwegian rock types were determined. The results suggest that the same water absorption values, as determined by EN 1097-6 on coarser sand fractions, can be applied to the fines. The values of specific surface area measurements vary widely between different materials and between different measurement methods. BET measurements seem to be strongly affected by the mineralogical composition (presence of mica) and surface morphology (weathering) of the particles. Specific surface area calculated from the particle size distributions (PSD) is mainly dependent on the precision of the test methods in the size range below about 3–5 μm, because these small particles contain most of the surface area. Shape measurements by both Dynamic Image Analysis (DIA), which is a 2-D method, and X-ray microcomputed tomography (μCT), which is a 3-D method, have yielded similar relative length-to-thickness aspect ratios of the particles between different mineralogies, though with lower absolute values for DIA due to 2-D projection of 3-D quantities.     

Influence of microstructure of concrete on size/scale effects in tensile fracture

Size/scale effects on the fracture of concrete subjected to uniaxial tension are studied by means of analyses with the Delft lattice beam model and compared to recent experimental results. Using a simple local elastic-purely brittle material global softening behaviour is calculated. The effect of deterministic and statistical contributions to size effect is studied by implementing different degrees of heterogeneity to the lattices. They vary from ‘homogeneous’ regular triangular lattices to lattices with randomly varying beam length. Computer generated particle overlays are used to improve resemblance to real concretes. Trends in size effect on nominal strength and fracture energy are in close agreement with the experiments. The type of approach can be used for tuning macroscopic size/scale laws for concrete and related materials.     

On the significance and description of the size effect in multimodal fracture behavior. Experimental assessment on E-glass fibers

The existence of a size effect is an integral part of the physical phenomenon of brittle fracture. The literature dealing with the influence of fiber size on its ultimate strength has mainly focused on specimens whose failure is governed by a single category of flaw. In this work, we are discussing the size effect from a statistical point of view for fibers exhibiting multiple failure modes.     

Effect of aggregate size on the fracture and mechanical properties of a simple concrete

The influence of the aggregate size on the fracture energy, tensile strength and elasticity modulus in different types of concrete are analyzed. For this purpose, nine simple cement-based composites have been designed, manufactured and tested, with one objective to provide experimental results that can be used as a benchmark for checking numerical models of concrete fracture, as this simple composite (a matrix, spherical aggregates of the same radius, and two types of matrix-aggregate interface) is amenable to modelling. All in all, 44 specimens were tested. From notched beam tests, values of the fracture energy and modulus of elasticity were obtained. The tensile stress was deduced from indirect standard tensile test. Data for bilinear softening functions extracted from the experimental measurements are also provided. Comparison with available experimental data is also included and discussed.     

Application of size effect to compressive strength of concrete members

It is important to consider the effect of size when estimating the ultimate strength of a concrete member under various loading conditions. Well known as the size effect, the strength of a member tends to decrease when its size increases. Therefore, in view of recent increased interest in the size effect of concrete this research focuses on the size effect of two main classes of compressive strength of concrete: pure axial compressive strength and flexural compressive strength. First, fracture mechanics type size effect on the compressive strength of cylindrical concrete specimens was studied, with the diameter, and the height/diameter ratio considered as the main parameters. Theoretical and statistical analyses were conducted, and a size effect equation was proposed to predict the compressive strength specimens. The proposed equation showed good agreement with the existing test results for concrete cylinders. Second, the size, length, and depth variations of a flexural compressive member have been studied experimentally. A series of C-shaped specimens subjected to axial compressive load and bending moment were tested. The shape of specimens and the test procedures used were similar to those by Hognestad and others. The test results are curve-fitted using Levenberg-Marquardt’s least squares method (LSM) to obtain parameters for the modified size effect law (MSEL) by Kim and co workers. The results of the analysis show that the effect of specimen size, length, and depth on ultimate strength is significant. Finally, more general parameters for MSEL are suggested.     

Determination of concrete fracture parameters based on two-parameter and size effect models using split-tension cubes

The notched beam specimens have been commonly used in concrete fracture. In this study, the splitting-cube specimens, which have some advantages - compactness and lightness - compared to the beams, were analyzed for the effective crack models: two-parameter model and size effect model. The linear elastic fracture mechanics formulas of the cube specimens namely the stress intensity factor, the crack mouth opening displacement, and the crack opening displacement profile were first determined for different load-distributed widths using the finite element method. Subsequently, four series of experimental studies on cubic, cylindrical, and beam specimens were performed. The statistical investigations indicated that the results of the split-cube tests look viable and very promising.     

Effects of grain size and thermodynamic energy on the lattice parameters of metallic nanomaterials

A thermodynamic method based on surface thermodynamics and atomic bond energy was developed to accurately investigate the lattice distortion rates of metallic nanomaterials. The results indicated that the lattice distortion rates of nanomaterials follow an inverse proportional relationship with the size, in good agreement with the experimental results. In this method, the anisotropy of the lattice distortion was a considerable issue. We found that the surface tension and Young’s modulus of the nanocrystals, compared with those of the bulk materials, change because of the lattice distortion and exhibit a linear relationship at the nanoscale. By defining a shape factor (ξ), the lattice distortion rates of nanoparticles, nanowires, and nanofilms were calculated. This method provides a new approach for the evaluation of the lattice distortion rates in nanomaterials.     

Probability distribution of energetic-statistical size effect in quasibrittle fracture

The physical sources of randomness in quasibrittle fracture described by the cohesive crack model are discussed and theoretical arguments for the basic form of the probability distribution are presented. The probability distribution of the size effect on the nominal strength of structures made of heterogeneous quasibrittle materials is derived, under certain simplifying assumptions, from the nonlocal generalization of Weibull theory. Attention is limited to structures of positive geometry failing at the initiation of macroscopic crack growth from a zone of distributed cracking. It is shown that, for small structures, which do not dwarf the fracture process zone (FPZ), the mean size effect is deterministic, agreeing with the energetic size effect theory, which describes the size effect due to stress redistribution and the associated energy release caused by finite size of the FPZ formed before failure. Material randomness governs the statistical distribution of the nominal strength of structure and, for very large structure sizes, also the mean. The large-size and small-size asymptotic properties of size effect are determined, and the reasons for the existence of intermediate asymptotics are pointed out. Asymptotic matching is then used to obtain an approximate closed-form analytical expression for the probability distribution of failure load for any structure size. For large sizes, the probability distribution converges to the Weibull distribution for the weakest link model, and for small sizes, it converges to the Gaussian distribution justified by Daniels' fiber bundle model. Comparisons with experimental data on the size-dependence of the modulus of rupture of concrete and laminates are shown. Monte Carlo simulations with finite elements are the subject of ongoing studies by Pang at Northwestern University to be reported later.     

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。     

引气混凝土气泡尺寸分布的三维重构

基于体视学和几何概率理论给出了引气混凝土三维气泡尺寸重构方法,由二维平面上气泡截面圆的直径分布计算气孔的实际尺寸分布,并生成了一个多尺度分布的立方体模型结构验证了该三维重构方法的合理性.然后,使用邻近粒子表面最近间距的解析解研究了气泡细度和混凝土含气量对邻近气泡表面最近间距平均值的影响,并与用传统方法得到的气泡间距因子进行了比较.结果表明,在含气量相同的条件下,用传统方法得到的气泡间距因子是邻近气泡表面最近间距平均值的3-4倍.该方法的给出,为从二维截面上获得的引气混凝土中的气泡截面圆信息获取实际气泡在三维空间中的气泡间距信息提供了依据.     

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.     

Reliability and scale effect in toughness of concrete structures

In the present work, the distribution of the random toughness characteristics (i.e. critical energy release rate, G_1c) has been evaluated on the basis of experimental observations. Fracture test results from three groups of geometrically similar concrete specimens of size (width×total depth×thickness), 420×420×50–1680×1680×200 mm³, made with different maximum aggregate size of 9.5, 19, 38, and 76 mm were analyzed using a recently proposed distribution of extremes. In applications of probability, it is important to use an appropriate distribution type and adequate techniques for estimating the parameters of distribution. In this study, a new type distribution of minima is employed for probability computations. It was noticed that the entropy of distribution increases with the crack length, i.e. the uncertainty of toughness, G_1c, value increases with crack length. A non-linear reduction of the maximum allowable splitting force with the defect size, a, was noticed. For large specimens, the maximum allowable splitting load is more sensitive to the required reliability level than that for small specimens. Reliability increases with aggregate size when all other conditions were constant.     

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.     

Size effect on the concrete cone pull-out load

In the present paper the failure mechanism and size effect of the concrete cone resistance is reviewed and studied. The influence of material and geometrical parameters on the failure mode and size effect is investigated. In the numerical studies the smeared crack finite element analysis, based on the microplane material model for concrete, was used. Both, experimental and numerical results show that there is a strong size effect on the nominal concrete cone pull-out strength. It is demonstrated that besides the embedment depth the scaling of the head of the stud as well as the scaling of the concrete member influence the nominal strength. This revised version was published online in July 2006 with corrections to the Cover Date.     

Age-dependent size effect and fracture characteristics of ultra-high performance concrete

This paper presents an investigation of the age-dependent size effect and fracture characteristics of ultra-high performance concrete (UHPC). The study is based on a unique set of experimental data connecting aging tests for two curing protocols of one size and size effect tests of one age. Both aging and size effect studies are performed on notched three-point bending tests. Experimental data are augmented by state-of-the-art simulations employing a recently developed discrete early-age computational framework. The framework is constructed by coupling a hygro-thermo-chemical (HTC) model and the Lattice Discrete Particle Model (LDPM) through a set of aging functions. The HTC component allows taking into account variable curing conditions and predicts the maturity of concrete. The mechanical component, LDPM, simulates the failure behavior of concrete at the length scale of major heterogeneities. After careful calibration and validation, the mesoscale HTC-LDPM model is uniquely posed to perform predictive simulations. The ultimate flexural strengths from experiments and simulations are analyzed by the cohesive size effect curves (CSEC) method, and the classical size effect law (SEL). The fracture energies obtained by LDPM, CSEC, SEL, and cohesive crack analyses are compared, and an aging formulation for fracture properties is proposed. Based on experiments, simulations, and size-effect analyses, the age-dependence of size effect and the robustness of analytical-size effect methods are evaluated.     

钢纤维增强超高性能混凝土抗压性能的细观数值模拟

利用LS-DYNA软件在细观层次上建立了三维钢纤维增强超高性能混凝土（Steel fiber reinforced ultra-high performance concrete,SF/UHPC）圆柱体试件有限元模型,对其轴心受压下的力学性能和裂缝发展进行了数值模拟。在验证细观数值模型的有效性和合理性的基础上进行参数分析,着重研究了钢纤维体积率、钢纤维长径比、形状效应和尺寸效应对超高性能钢纤维混凝土抗压强度、韧性和破坏形态的影响。最终,根据模拟结果拟合了超高性能钢纤维混凝土抗压强度计算公式。结果表明：三维超高性能钢纤维混凝土细观模型可以较好地模拟单轴受压应力条件下混凝土的静力性能和损伤破坏机制,所拟合的公式也能较好地预测超高性能钢纤维混凝土的抗压强度。