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Coarse aggregates are often eliminated in ultra-high performance fibre reinforced concrete (UHPFRC) for the sake of homogeneity, however, this causes an impairment on impact resistance. The flexural performance of UHPFRC with coarse aggregates under different loading rates (0.2, 20 and 200 mm/min) is investigated here to clarify the flexure and energy absorption mechanism. The flexural behavior and crack propagation are measured, meanwhile, the fracture of coarse aggregates and the surface morphology of steel fiber are analysed. The results show the energy absorption tends to be more rate dependent than the first crack stress and flexural strength. An increase of crack propagation speed and multiple cracks are observed at higher loading rates. The percentage of fracture across coarse aggregate is 23%, 32% and 58% at loading rates of 0.2, 20 and 200 mm/min, respectively. Further, a rate-dependent model for predicting the fracture of coarse aggregates is proposed. The present results contribute to designing UHPFRC with enhanced flexural performance under different loading rates.

Fracture of normal strength concrete cylinder under static and dynamic loading is studied numerically. 3D finite element simulations are carried out at macro- and meso-scale. At meso-scale the analysis is performed with and without accounting for the interface zone (IZ) between aggregate and mortar. Aggregate is assumed to be linear elastic, and mortar is modeled using rate-dependent microplane model. To better understand behavior of concrete under dynamic fracture in compression, a parametric study is carried out to investigate the influence of the volume fraction of the aggregate, the role of IZ, the influence of confinement at the loading surface, the role of concrete quality and the influence of the size of the test specimen. The comparison between meso-scale and macro-scale analysis shows that the macroscopic analysis is principally able to account for the major effects related to dynamic fracture of concrete. Dynamic resistance of concrete in compression (apparent strength) depends on a number of parameters, and it is mainly influenced by the inertia effects that are closely related to the load-induced damage. Finally, it is pointed out that dynamic increase factor for compressive strength (CDIF), such as currently defined in design codes, for relatively high loading rates does not represent the true material strength.

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.

An analytical approach is proposed to predict the fracture parameters of coral aggregate concrete (CAC). Both the size-independent tensile strength and fracture toughness are related to the maximum fracture load linearly based on the boundary effect model by incorporating the average aggregate size. Moreover, an explicit expression is derived to correlate the maximum fracture load with the local fracture energy at the crack-tip region. The local fracture energy distribution and size-independent fracture energy are then determined by virtue of the maximum fracture load. Four groups of three-point-bending notched CAC beams are tested by considering two ages and two environmental conditions (immersion in seawater or not) and the initial crack length-to-beam depth ratios are set from 0.1 to 0.7 in each group. Results show that the failure modes of all the specimens are coral coarse aggregate fracture without interfacial debonding between the aggregate and surrounding mortar. The average values of tensile strength, fracture toughness and fracture energy can be obtained in each group by using the experimentally measured maximum fracture loads and the experimental scatters were analyzed based on normal distribution analysis. All the fracture parameters increase with the age and become larger if the specimens were immersed in seawater.

This study evaluates the relationship between fracture toughness and compressive strength of self-compacting lightweight concrete (SCLC). For this purpose, three mix designs with different w/c ratios were used, and three-point bending tests were done on 36 notched beams. The maximum loads of the beams were achieved, and size effect method was applied for analysing the results. The results of analyses displayed that the fracture toughness and mechanical properties were significantly increased as w/c ratio decreased. In fact, as the w/c ratio decreased from 0.47 to 0.37, the fracture toughness and compressive strength increased by 56% and 52%, respectively. Fracture toughness of mode I was compared with the compressive strength of SCLC. According to the experimental results, an acceptable relation is presented between the fracture toughness of mode I and compressive strength of SCLC in this paper.

The size-effect method for determining material fracture characteristics, as previously proposed by Bazant and extensively verified for normal strength concrete, is applied to typical high-strength concrete. Geometrically similar three-point bending specimens are tested and the measured peak load values are used to obtain the fracture energy, the fracture toughness, the effective length of the fracture process zone, and the effective critical crack-tip opening displacement. The brittleness of the material is shown to be objectively quantified through the size-effect method. Comparing the material fracture properties obtained with those of normal strength concrete shows that an increase of 160 percent in compressive strength causes: (1) an increase of fracture toughness by only about 25 percent, (2) a decrease of effective fracture process zone length by about 60 percent, and (3) more than doubling of the brittleness number, which may be an adverse feature that will need to be dealt with in design. The brittleness number, however, is still not high enough to permit the use of linear elastic fracture mechanics. The R-curves are demonstrated to derive according to the size-effect law exclusively from the maximum loads of specimens of various sizes and yield remarkably good predictions of the load-deflection curves.

Effects of graphite nanoplatelets (GNPs) and carbon nanofibers (CNFs) on mechanical properties of ultra-high-performance concrete (UHPC) are investigated. A non-proprietary UHPC mixture composed of 0.5% steel micro fibers, 5% silica fume, and 40% fly ash was used. The content of the nanomaterials ranged from 0 to 0.3% by weight of cementitious materials. The nanomaterials were dispersed using optimized surfactant content and ultra-sonification to ensure uniform dispersion in the UHPC mixture. As the content of nanomaterials is increased from 0 to 0.3%, the tensile strength and energy absorption capacity can be increased by 56% and 187%, respectively; the flexural strength and toughness can be increased by 59% and 276%, respectively. At 0.2% of GNPs, the UHPCs exhibited " strain-hardening " in tension and in flexure.

In this study, the interaction between cylindrical specimen made of homogeneous, isotropic, and linearly elastic material and loading jaws of any curvature is considered in the Brazilian test. It is assumed that the specimen is diametrically compressed by elliptic normal contact stresses. The frictional contact stresses between the specimen and platens are neglected. The analytical solution starts from the contact problem of the loading jaws of any curvature and cylindrical specimen. The contact width, corresponding loading angle (2θ0), and elliptical stresses obtained through solution of the contact problems are used as boundary conditions for a cylindrical specimen. The problem of the theory of elasticity for a cylinder is solved using Muskhelishvili’s method. In this method, the displacements and stresses are represented in terms of two analytical functions of a complex variable. In the main approaches, the nonlinear interaction between the loading bearing blocks and the specimen as well as the curvature of their surfaces and the elastic parameters of their materials are taken into account. Numerical examples are solved using MATLAB to demonstrate the influence of deformability, curvature of the specimen and platens on the distribution of the normal contact stresses as well as on the tensile and compressive stresses acting across the loaded diameter. Derived equations also allow calculating the modulus of elasticity, total deformation modulus and creep parameters of the specimen material based on the experimental data of radial contraction of the specimen.

This study explains from the fracture mechanics principles that the common size effect observed in fracture toughness and energy measurements of many engineering materials including concrete, fiber reinforced composites and even coarse-grained ceramics are in fact similar to what has been observed in the well-studied elastic–plastic fracture of metals. The conditions required for measurements of the material constant, the plane strain fracture toughness KIC, of metals are akin to those required to avoid such a size effect on the fracture toughness and energy of concrete and other composites. Using the common yield strength and plain strain fracture toughness criteria a reference crack length a∗ is defined in this study, which is then used to introduce a simple asymptotic function. The asymptotic analysis shows that the size effect on fracture toughness and energy of a heterogeneous material such as concrete will be inevitable if the relative crack measured by the crack ratio a/a∗ or the remaining ligament ratio is too close to one. This relative crack needs to be around 10 or even higher to avoid the size effect influence. Experimental results and previous models are compared with the current asymptotic analysis.

The fracture properties of concrete are highly influenced by the size effect and strain rate effect. To explore the size effect on the dynamic fracture properties of concrete, a fracture experiment on the three-point bending beam was carried out in this paper under four strain rates (10⁻⁵/s, 10⁻⁴/s, 10⁻³/s and 10⁻²/s) and three span to height ratios (2, 2.5 and 3). Then, the fracture characteristic parameters and mechanism were analyzed based on the experimental data combined with the digital image correlation (DIC) technology. The results indicate that the fracture properties of concrete have an obvious strain rate effect and size effect. The fracture load, fracture energy and crack initiation toughness all increase with an increasing strain rate, but decrease with an increasing span to height ratio. On the contrary, the unstable fracture toughness shows no obvious size effect or strain rate effect. Based on the DIC technology, the evolution of crack propagation length in different loading stages was calculated. It is shown that, under the strain rate of 10⁻⁵/s, 10⁻⁴/s, 10⁻³/s and 10⁻²/s, the rapid propagation of crack occurs in the period from Pre-70% to Pre-90%, from Peak load to Post-90%, from Post-90% to Post-80%, and from Post-90% to Post-80%, respectively. More specifically, the crack propagates most rapidly under the strain rate of 10⁻²/s. The rapid propagation period tends to be delayed with the increase of strain rate, and the strain rate effect is gradually weakened with the increase of span to height ratio. By carrying out in-depth research on the dynamic fracture failure mechanism of concrete, this paper provides reference for the future studies on the dynamic fracture properties of concrete.

The dynamic mechanical behavior of high-strength concrete is sensitive to aggregate properties, thus an accurate description of the effect of aggregate strength on the dynamic tensile mechanical properties of high-strength concrete is essential to evaluate the structural stability of concrete. In this paper, a concrete meso-scale modeling framework was proposed based on discrete element method (DEM). The meso-scale model considered the three-dimensional meso-structure of concrete (coarse aggregate, mortar and interfacial transition zone), and the crushable aggregate model with realistic morphological characteristics was generated by the “clump-cluster” method. Based on this, the effects of aggregate strength on dynamic tensile properties and damage behavior of high strength concrete were quantitatively studied by splitting tensile test. The simulation results reveal the influence of aggregate strength and strain rate on the tensile properties of concrete at both macroscopic and mesoscopic levels, including dynamic deformation behavior, microcracks propagation mode and the proportion of different kinds of microcracks at different loading stages. The results show that the aggregate strength plays an important role in the dynamic tensile properties of concrete. The failure behavior of concrete and the damage degree of different components under dynamic tensile loading are influenced by the ratio of aggregate strength to mortar strength.

Because of the heterogeneous and discontinuous properties in concrete, realistic fracture parameters were hardly obtained rationally based on traditional methods. The material heterogeneity and discontinuity may be more significant under the attack of high temperature. Therefore, the intention of this paper is to propose a predictive solution for fracture modeling of AASC (alkali-activated ground granulated blast furnace slag (GGBFS) and fly ash (FA) blended sea sand concrete) after exposure to elevated temperature. First, fracture test was performed on AASC after exposure to four high temperatures plus one room temperature. The fracture process and failure mechanism were analyzed and clarified at both macro- and micro-scales. Subsequently, an analytical model was presented to determine the fracture parameters of AASC by incorporating the material heterogeneity and discontinuity. The realistic tensile strength ft, fracture toughness KIC and fracture energy GF were then explicitly linked to the maximum fracture load Fmax by virtue of boundary effect model. Results show that the load-displacement curve becomes gentler and the failure mode is changed from trans-granular fracture to inter-granular fracture as the temperature increases. Once the Fmax is obtained from the test, the realistic ft, KIC and GF from each specimen can be predicted conveniently. The scatters in the predicted parameters would be clarified assisted by statistical analysis. As the temperature increases, the ft is reduced and the reduction becomes larger. But the GF shows insignificant variation until the temperature attains 400 °C, and apparently decreases as the temperature is further enhanced. Moreover, the predicted fracture parameters of AASC with higher GGBFS/FA mass ratio are larger than those of the other AASC below 400 °C. However, the former has higher strength loss than the latter. The crack resistance of AASC after exposure to high temperature can be well clarified based on the proposed predictive model.

Concrete 3D printing is getting increased attention in civil engineering. Fibers are generally added to reinforce printed concrete. Crack resistance of 3D printed fiber reinforced concrete in different directions need to be evaluated in a rational manner due to its anisotropic behavior. Therefore, fracture test was performed on printed and mold-cast concrete first in this paper, and failure mechanism were then examined. An analytical model was subsequently proposed based on the analysis of test results to predict the realistic tensile strength ft, fracture toughness KIC and fracture energy GF in different loading directions by incorporating material heterogeneity and discontinuity. The physical meaning of microstructure characteristic parameter reflecting the material heterogeneity was clarified according to the degree of fiber toughness. Closed-form solutions of ft, KIC and GF were obtained related to the maximum fracture load Fmax. The influence of loading direction on the predicted fracture parameters was analyzed and discussed. Results show that the ft, KIC and GF from specimens loaded perpendicularly to the printing direction were significantly larger than those parallel to the printing direction. Besides, the formers were also higher than the parameters from mold-cast specimens due to the fiber preferential alignment along the printing direction in printed concrete.

The size effect on the dynamic strength of concrete has been recognized, but the main causes of this phenomenon are not well understood. In this study, a mesoscale particle model was established to simulate the dynamic splitting tensile behavior of concrete and calibrated with experimental data. An equivalent momentum scheme based on mesoscale modeling was proposed for the quantitative division of the dynamic increase factors. The results showed that the inertia effect and multiple cracks improved the tensile strength, owing to the rate effects. However, the inertia factor played a dominant role, whereas the multiple crack factor minimally influenced the dynamic size effect of concrete.

This paper presents a comprehensive experimental investigation of the dynamic size effect and fracture characteristics of concrete. The study involves about two hundred concrete specimens with various maximum aggregate sizes subjected to tensile loading ranging over five orders of magnitude. The effect of structural size and aggregate size on the dynamic strength of concrete is explored using the Work of Fracture Method (WFM) and Size Effect Method (SEM). The experimental results indicate a considerable size effect in concrete subjected to dynamic loading, although the size effect on concrete nominal strength is weakened by increasing the loading rate and aggregate size.

Timely determination of the spatial distribution of post-earthquake relief forces and appropriate levels of response is a daunting task. This paper proposes a method based on building damage state and spatial characteristics analysis to assess the hotspot distribution of earthquake casualties and determine the dangerous gathering area. The method comprises three steps: (1) use the basic information of the buildings in the study area to calculate the building damage state and build the relationship with the casualties; (2) mesh the research area and establish the mapping relationship with the casualties caused by the building damage; (3) use grid data for spatial autocorrelation analysis (Moran's I) and hotspot analysis (Getis-Ord Gi*) to comprehensively assess the casualty cluster area. The feasibility of the proposed method was validated using the seismic damages of Longtoushan Town in the 2014 Ludian earthquake of China. The damage state of buildings and the number of casualties evaluated by this method coincide quite well with the actual earthquake results, and the casualty concentration area is determined. This method quantifies the degree of spatial risk aggregation of casualties and provides scientific guidance for disaster prevention planning and rescue.

To discuss the best performance price ratio of fiber-reinforced polymer (FRP) reinforced concrete with cracks under the dynamic load, the effect of FRP bonding layers on the fracture characteristics of concrete was studied by three-point bending tests. Additionally, the acoustic emission (AE) nondestructive testing technology was used to analyze its effect on the AE signal. The test results show that, the peak load of FRP reinforced concrete increases with the increase of FRP bonding layer. It is easy to form “super reinforcement failure” mode for FRP reinforced concrete beams bonded three-layer and four-layer. When the number of bonding layers is two, the double-K fracture toughness of FRP reinforced concrete reach the maximum value. The damage evolution curves show an inverted S-shape with loading time. With the increase of FRP bonding layers, the intensity of AE signal increases and the ring down times of high amplitude signal increase obviously. The AE cumulative energy and fracture energy increase with the increase of FRP bonding layers. The comprehensive analysis of RA–AF and b-value can be used to evaluate the damage state of FRP reinforced concrete.

The PeriDynamics (PD) model of material fracture can simulate the nucleation and propagation of cracks naturally with a simple bond breakage criterion. Combined with its advantages in multi-scale, the crack characteristics of concrete can be simulated from micro-scale. Therefore, the micro-calculation model of concrete was established by the spherical growth model based on the MATLAB-ABAQUS co-simulation method. Meanwhile, combined with the plastic softening characteristics of concrete, a quasi-plastic damage fracture constitutive model for concrete was established. Finally, the mode-I fracture test and two typical mixed-mode fracture tests were simulated by this model, and the PD-FEM (finite element method) method was used to reduce the calculation cost, in which the cracking regions were set as the PD model and other regions were set as the FEM model. The results show that the crack initiation and propagation of different calculation samples can be well described by the calculation model. Moreover, the proposed material model can better reflect the comprehensive mechanical behavior of concrete, and the crack path of the specimens is consistent with the test results. For the simulation of the I-II (tension shear) mixed fracture test, the crack paths of different calculation samples with small shear load are obviously different, and the range of crack paths can be predicted by the simulation of calculation samples. Furthermore, for the calculation samples with large shear load, the crack path and macro-behavior of different calculation samples are close to each other.

In order to study the effect of loading rate on fracture behavior of dam concrete, wedge splitting tests of various loading rates (0.1, 0.01, and 0.001 mm/s) are carried out on two kinds of full-graded dam concrete notched cubes with side lengths of 300 and 450 mm, respectively. Digital image correlation and acoustic emission technique are used to measure the deformation and acoustic emission parameters of the dam concrete. Test results show that: the peak load and fracture energy of dam concrete specimens increases with the increase of loading rate. And the higher the loading rate is, the fracture of concrete shows more obvious brittleness. Influenced by the boundary effect, the CTOD increases with the increasing of loading rate, however, the length of crack decreases as loading rate increases. With the loading rate increases, the energy mutation area is more obvious, while the accumulated acoustic emission energy is affected by both the loading rate and the maximum aggregate size. The number of acoustic emission three-dimensional locating points and the shear signal decrease with the increase of loading rate, which is attributed to that the faster the loading rate is, the less sufficient the development of micro cracks in concrete is. The test results can supply experimental data to the fracture mechanics of dam concrete.

A new type of concrete, namely alkali-activated slag and fly ash blended sea sand recycled aggregate concrete (AASRAC), is developed in this paper. By considering its future service under ocean environment, crack resistance of the new concrete should be emphasized and evaluated rationally. Similar to ordinary concrete, however, size effect would be inevitable in the determined tensile strength and fracture toughness based on traditional continuum mechanics. To address this issue, the intention of this paper is to propose an analytical fracture model to predict the tensile strength ft, fracture toughness KIC and energy GF by using tests on three-point-bending notched beams of AASRAC. The average aggregate size davg and two discrete numbers β and βw are introduced to represent the material heterogeneity and discontinuity, respectively. The critical effective crack propagation length and critical crack-tip opening displacement are quantified as the davg multiplied by β and βw, respectively, when the maximum applied load Fmax is reached. Closed-form solutions of the size-independent ft, KIC and GF are then obtained by using the experimentally measured Fmax based on boundary effect model. The means, and upper and lower bounds of the fracture parameters with 95% reliability are determined from the normal distribution analysis. Results show that the fracture parameters can be yielded with reasonable accuracy if β = 1.0 and βw = 1.0. The predicted ft, KIC and GF are higher as the slag-to-fly ash mass ratio increases in AASRAC. The predicted ft based on the proposed model is significantly larger than the traditional splitting tensile strength. The main failure mode of the tested AASRAC beam is that about 80% of the recycled coarse aggregates are fractured due to the improved interfacial micro-structures between the mortar and aggregates. The proposed model can give a rational design method for the new concrete.

This study presents a design methodology for concrete fracture properties, linking the average aggregate size dav to the tensile strength ft and fracture toughness KIC for a given cement grade and water/cement (W/C) ratio. Two different concrete mixes with dav = 3.5 and 7 mm were tested and analyzed. Similar peak load Pmax values were obtained under three-point-bending conditions, but with different “ductility”. Variations of ft & KIC with dav were established by a simple design formula, which can be used to tailor bulk concrete fracture properties for specific purposes. A statistical method was introduced to estimate the effective dav if only the maximum aggregate size dmax was given. Comprehensive fracture data in literature were analyzed to demonstrate the estimation of dav from dmax. The purposely designed concrete mixes and data in literature show the significance of dav design even after the cement grade and W/C ratio are fixed.

A closed-form fracture model with the consideration of the average aggregate size and crack-tip damage zone is used to analyse flexural strength of asphalt concrete using notched beams. Quasi-stable micro-crack formation and quasi-brittle fracture of these small samples with highly heterogeneous aggregate structures are formulated. While the apparent strength and fracture toughness of asphalt concrete based on Continuum Mechanics are size dependent, constant asymptotic “structural” tensile strength ft and “structural” fracture toughness KIC have been determined by this composite model from experimental results of notched beam specimens of any size with any notch length. Temperature (T) dependent ft and KIC measurements of asphalt concrete from −10 to + 23 °C, relevant to the real temperature environments, are then determined. The characteristic microstructure Cch in the model is the average aggregate size. The maximum failure load Pmax of asphalt concrete under three-point-bending (3-p-b) is linked directly to its constant “structural” tensile strength ft for a given temperature environment, independent of sample size and notch/crack length. In addition, although KIC is not valid for asphalt pavement with limited thickness, its influence is reflected through the “structural strength” criterion based on ft and Cch. This simple model can also be used as a design tool for asphalt concrete property optimization.

Combining the advantages of the size and the boundary effect models, an improved fracture model for determining the material parameters of concrete is proposed in this paper, considering concrete aggregate grading. The simple method for determining the fictitious crack growth length at peak loads of concrete specimens is identified through quantitative research. The independent fracture toughness and tensile strength of concrete can be determined with the maximum value of correlation coefficients in curve fitting, and the effect of maximum aggregate size, water to cement ratio, and coarse aggregate volume on fracture can be fully considered. A total of 300 self-compacting concrete specimens are used for analysis to verify the applicability and feasibility of the proposed model. The full structural fracture failure curves of concrete with upper and lower limits covering the experimental peak loads are established on the basis of the normal distribution methodology.

Expanded Clay Foam Concrete (eC-FC) is a light-weight foam concrete with expanded clay aggregates, which is an important class of building materials, yet its tensile strength ft and fracture toughness KIC cannot be easily measured using small test samples due to its highly heterogeneous microstructures. This study presents a simple closed-form model for quasi-brittle fracture of heterogeneous solids confirmed by comprehensive three-point-bend (3-p-b) test results from 168 notched eC-FC specimens with eight group sample designs. One tensile strength ft was determined from individual group or all from the model initially developed for normal concrete, proving both the trans-granular fracture in eC-FC and inter-granular fracture in normal concrete can be modeled by this fracture model. Unidirectional compressive tests of the eC-FC were also performed, and tensile strength ft estimated from the compressive strength fc is consistent with ft from 3-p-b tests. Fracture toughness KIC was determined from the model using tensile strength ft and the average aggregate size G.

Concrete generally deforms and cracks in a non-uniform manner under drying-induced stress. This study used the lattice fracture model to simulate the drying-induced non-uniform deformations, stresses, and micro-crack propagation in concrete. Experiments were designed to validate the lattice fracture model, wherein the drying-induced non-uniform deformations and micro-crack patterns in concrete were measured using a digital image correlation technique and a fluorescent epoxy impregnation method, respectively. It was found that the simulated non-uniform deformations and micro-crack patterns were close to the experimental observations. The interaction mechanism between drying-induced non-uniform stresses and micro-cracks was analysed based on the validated lattice fracture model. The micro-cracks were found to cause stress concentration both in coarse aggregate and the mortar that covered coarse aggregate, which could lead to high micro-cracking risk as drying continues.

Quasi-brittle fracture properties of a medium-grain sandstone with an average grain size G around 0.3 ∼ 0.4 mm were investigated under three-point-bending (3-p-b) conditions. In total, 95 specimens were tested with the beam width W varying from 10 to 300 mm, or the specimen-size/grain-size ratio from 30 to 900. 45 medium-sized specimens (W = 30, 60, 100 mm) were tested first to determine the tensile strength ft, which was then used as a reference for tests of 8 large notched specimens (W = 300 mm) and 42 small un-notched specimens (W = 10 mm). Statistical fracture modelling, based on normal distributions and the characteristic microstructure measurement (the average grain size G in this study), was used to quantify the quasi-stable fictitious crack growth Δafic at the peak load Pmax and the characteristic crack length ach* defined by the bulk toughness and strength properties. The statistics-assisted modelling has changed the previous curve-fitting boundary effect model (BEM) to a predictive closed-form solution, providing a useful option when large scatters in experimental data and reliability in design need to be focused. The well-known size effect law (SEL) proposed for geometrically similar specimens was also used to fit the sandstone results, and compared with the closed-form BEM with built-in statistical functions. Applications of SEL and BEM and their key differences were explained.

To study the effect of loading rate on the fracture behavior, three-point bending fracture tests of concrete with loading rates of 0.0001 mm/s, 0.001 mm/s, 0.01 mm/s and 0.1 mm/s were carried out, respectively. And acoustic emission (AE) technology was adopted for real-time monitoring. The results show that the unable toughness of concrete has obvious rate effect. Meanwhile, there are two obvious inflection points in the curve of cumulative AE hits and the cumulative ringing count with the change of time, one of which may represent the starting point of concrete boundary effect. The number of AE events can represent the crack width of concrete fracture. It is found that with the increase of loading rate, the crack width and ductility of concrete decrease, while the quantity of shear crack of concrete increases. By comparing the fracture energy of concrete with the AE cumulative energy, both of them increase with the increase of loading rate, indicating that the AE cumulative energy can represent the change of fracture energy under different loading rates. Finally, the failure forms under different loading rates are analyzed by the change of b value. Based on the above research, AE technology can be used to study the fracture failure under low loading rate.

Ultra-High Performance Concrete (UHPC) is an innovative cement based material with superior strength for engineering structures. The mechanical properties of the cementitious composite depend on its exact composition and need to be determined experimentally. Whereas several standard tests for the static loading regime exist, dynamic data such as tensile strength and failure load are difficult to measure.
This contribution presents an experimental method to identify the material properties under impact loading. By means of a modified Hopkinson pressure bar, cylindrical UHPC specimens of different compositions without fiber are examined in strain rates of approximately 30s-1. Assuming uniaxial wave propagation, the resulting stress states are analyzed and the specimens’ dynamic elastic modulus, tensile strength and specific fracture energy are determined. Numerical fracture simulations in the sense of an inverse analysis prove the experimental approach valid. The agreement between both, experimental observations and numerical results, show that the obtained parameter are reliable and suitable for predictive simulations.

The high-rise buildings designed with a long lifetime may be exposed to one or more extreme hazards. Traditionally, specifications separately treated the multiple extreme hazards according to the controlling load case. Thus, the ability of high-rise buildings designed by the current codes to face the combined threats of earthquake and wind is rather vague. This paper presents a multihazard-based framework to assess the damage risk of a high-rise building subjected to earthquake and wind hazards separately and concurrently, which can be broken into three parts: the modeling of hazards, the structural fragility analysis and the damage probability computation. Firstly, based on the earthquake and wind data from 1971 to 2017 recorded in the Dali region of China, the hazard curves of single earthquake and wind, and the copula-based surface of bi-hazards are well established. Secondly, the multihazard-based fragility analysis of a high-rise building in Dali Prefecture is performed with the consideration of various load conditions. Lastly, upon completing the hazard models and fragility analyses, quantifications of the damage probabilities for the separate and concurrent hazards are determined directly. Numerical results indicate that the damage probability and contributions of each hazard circumstance are sensitive to damage severity. Furthermore, the damage probability induced by the bi-hazards dominates the total probability under most damage states conflicted with the common assumptions presented in the available researches. The comprehensive application highlights the necessity of examining the responses of high-rise buildings subjected to multihazard. The potential of the presented framework is of great help for decision-making.

During the fracture of concrete the formation of microcracks is a continuous process where new microcracks are formed in the vicinity of the macrocrack. As the fracture grows, these microcracks interact with the macrocrack and disrupt its smooth opening. In fracture mechanics based approaches; a smooth stress-crack opening softening behaviour is generally used to describe the local fracture process, where the effect of micro-macro crack interactions is lacking. This paper presents firstly an experimental study on the interaction between microcracking and the macrocrack opening (COD) in concrete using digital image correlation and acoustic emission techniques. The study reveals a transient behaviour of macrocrack opening as microcracks are formed in the surrounding. Based on the experimental results, the paper presents a new approach based on fracture mechanics to consider the effects of micro-macro crack interaction on the stress-crack opening softening behaviour. Local fracture energy is determined using the new softening model with micro-macro crack interaction for different geometrically similar sizes of the beams and at different locations on the crack profile.

Maximum fracture loads (Pmax) of small notched granite specimens under three-point-bending (3-p-b) conditions can be easily measured with any notch/size ratio. In this study, we report a simple closed-form solution of a non-Linear Elastic Fracture Mechanics (non-LEFM) model emphasizing the influence of average grain size G on quasi-brittle fracture of granite. This simple analytical solution containing the grain size G can be conveniently used to determine granite tensile strength ft and fracture toughness KIC from Pmax measurements of small notched 3-p-b specimens of geometry dissimilarity. The span/width (S/W) ratios of small 3-p-b specimens can vary, e.g. 2.5 or 4. The notch/width (a0/W) can also vary, e.g. the notch a0 can be as short as the average grain size (G), or close to width W. However, specimens with α-ratio (=a0/W) around 0.2 are recommended to minimize the boundary influence/effect from both the front and back specimen boundaries as proven by the Boundary Effect Model (BEM). Blue granite with the average grain size around 2 (mm) was selected to test the new method. Total 64 granite samples from four different groups (different 3-p-b sample designs) were tested, with W = 27, 40 and 70 (mm), S/W = 2.5, 4.0 and a0 = 4, 6 and 8 (mm). The tensile strength ft and fracture toughness KIC estimated from every group is fairly close to the values determined from the entire population of 64 tests. Therefore, tests from any specimen group of given geometry and size are sufficient. Estimations for G = 1.5 and 2.5 (mm) were also provided and compared with those for G = 2 (mm) to show the grain size influence. Advantages and disadvantages of BEM and well-known SEL (size effect law) are also discussed using the granite results.

The deformation measurement system was built with a CCD camera and a high-speed camera. Experimental tests were conducted on rectangular granite specimens with pre-cracks of mode-I under different loading rates. The speckle images captured in the process of experiments were analysed by the digital speckle correlation method (DSCM). We studied the speed of crack growth, crack-tip opening displacement, crack tip opening angle and crack stress intensity factor (SIF) of granite specimens with pre-cracks of mode-I under different loading rates. The initial speed of the crack growth of granite specimens increases linearly with the loading rate. With the increase of loading rate, the maximum speed crack increased rapidly at first and then slowly. At different loading rates, the crack-tip opening displacement of granite specimens showed three stages of evolution: nonlinear slow growth,rapid growth, and linear growth. In the linear growth stage, the slope of the curve increased with the increase of loading rate. With the increase of loading rate, the initial crack opening angle of granite specimen decreases gradually and then stabilises to 0.1°~0.13°. Before the initial fracture of granite specimens, the SIF value increased exponentially with the increase of loading rate.

This study investigates the effects of colloidal nano-silica content on the fracture parameters and brittleness of self-compacting semi-lightweight concrete. For this purpose, eight mix compositions were prepared with various contents of colloidal nano-silica (0%, 1%, 3%, and 5%) and two water to binder ratios of 0.35 and 0.45. A total of 96 notched beams of different sizes were made and three-point bending tests were conducted on them. The fracture parameters of all semi-lightweight concrete mixes were specified and analyzed by size effect method. The results of this research indicated that using colloidal nano-silica in semi-lightweight concrete and decreasing water to binder ratio from 0.45 to 0.35, (1) fracture energy (Gf) and fracture toughness (KIC) of semi-lightweight concrete increased, (2) the brittleness number of semi-lightweight concrete increased, (3) the best effects of colloidal nano-silica on the mechanical properties and the fracture parameters of semi-lightweight concrete were obtained when cement was replaced by 3% of colloidal nano-silica, and (4) failure behavior of the semi-lightweight concrete moved to the strength criterion.

Loading rate effect on crack propagation in ultra-high-performance fiber-reinforced concrete (UHPFRCs) was investigated using a pre-notched three-point bending specimen in an improved-strain energy frame impact machine (I-SEFIM) and image processing techniques. The crack velocity of up to 984 m/s and the crack initiation strain rate of up to 271 s⁻¹ were observed. Crack velocity in UHPFRCs increased as the applied strain rate increased. Fiber reinforcements significantly affected on the crack velocity in the UHPFRC at static rates, but slightly did at high strain rates. There is a strong correlation between the strain-rate sensitivity and the dynamic crack growth characteristics of UHPFRCs.

The improved Brazilian test is a popular method for indirectly measuring rock mechanical parameters, in which the current assumption of a uniform contact pressure distribution on the disc-jaw interface would result in measurement deviations because the real distribution is non-uniform. This investigation examines the influence of pressure distribution and friction on the determination of mechanical properties such as tensile strength σt and Young's modulus E by both theoretical analyses and experiments. For a Brazilian disc under arbitrary distributions of normal and tangential loads, the power series expansion technique is used to find the analytical solutions of the full-field displacements and stresses. The related formulae for calculating the tensile strength σt and Young's modulus E are derived. Eight load distributions with identical resultant forces are considered. The results show that the series solution converges in the form of a negative power law. Based on the Griffith failure criterion, the minimum contact angles that make the crack initiate at the center of the disc are given for different load distributions. The analysis indicates that these distributions have a significant influence on Young's modulus E and, if the contact angle is smaller, on tensile strength σt. The deviation caused by ignoring friction is also discussed. The Brazilian test is implemented for shale disc samples. The tensile strength σt and Young's modulus E of shale are determined indirectly using the theoretical formulae discussed here and experimental data.

This study introduces a new algorithm to determine size independent values of fracture energy, fracture toughness, and fracture process zone length in three-point bending specimens with shallow to deep notches. By using the exact beam theory, a concept of equivalent notch length is introduced for specimens with no notches in order to predict the peak loads with acceptable precisions. Moreover, the method considers the variations of fracture process zone length and effects of higher order terms of stress field in each specimen size. In this paper, it was demonstrated that the use of some recently developed size effect laws raises some concerns due to the use of nonlinear regression analysis. By using a comprehensive fracture test data, provided by Hoover and Bazant, the algorithm has been assessed. It could be concluded that the proposed algorithm can facilitate a powerful tool for size effect study of three-point bending specimens with different notch lengths.

The effects of water subjected to an electromagnetic field on the fracture parameters and mechanical properties of self-compacting lightweight concrete (SCLC) were investigated. The test variables were the magnetic field intensity (MFI) used to treat the water and the water/cement (w/c) ratio. Eight mix compositions with various MFIs and two w/c ratios (0·37 and 0·42) were considered. For each w/c ratio, the nominal maximum aggregate size and all mix designs were constant, while four magnetic fields were considered. Three-point bending tests were conducted on 96 notched beams and the results were analysed by means of the size effect method. Satisfactory results were achieved in terms of the fracture parameters and the mechanical properties of the SCLC. By increasing the MFI, the results indicated that: (a) the mechanical properties, initial fracture energy and fracture toughness increased at different rates, which can be attributed to the positive effects of the magnetic field on water clusters; (b) the effective length of the fracture process zone increased, illustrating an improvement in ductility of the SCLC specimens; (c) the desired design criterion of the SCLC samples approached the strength criterion.

This paper investigates crack speed in ultra-high performance concrete (UHPC) using pre-notched three-point bending specimens. The experimental parameters are fiber volume fraction and rate of loading. A hydraulic servo-controlled testing machine is used to apply lower notch tip strain rates, in the range of 0.025–1.0 1/s, while a newly developed impact testing system is used to achieve higher notch tip strain rates, ranging from 6.8 to 41.1 1/s. A high-speed camera is used to record images of the UHPC specimens during testing. Notch tip strain and crack speed are computed from the images, which show that crack speed increases asymptotically as the crack initiation strain rate increases. Crack speeds of up to 514 m/s were achieved at the lower notch tip strain rates and up to 1454 m/s for the higher notch tip strain rates. The achieved relationships are incorporated into a recently proposed crack-velocity dependent dynamic fracture model. The model is validated using published experimental data and used to show that, like conventional concrete, the strain rate sensitivity of UHPC is strongly associated with the characteristics of dynamic crack growth.

The fracture front in concrete, as well as rock, is blunted by a zone of microcracking, and in ductile metals by a zone of yielding. This blunting causes deviations from the structural size effect known from linear elastic fracture mechanics (LEFM). The size effect is studied first for concrete or rock structures, using dimensional analysis and illustrative examples. Fracture is considered to be caused by propagation of a crack band that has a fixed width at its front relative to the aggregate size. The analysis rests on the hypothesis that the energy release caused by fracture depends on both the length and the area of the crack band. The size effect is shown to consist in a smooth transition from the strength criterion for small sizes to LEFM for large sizes, and the nominal stress σN at failure is found to decline as (1 + λ/λ0)−1/2 in which λ0 = constant and λ = relative structure size. This function is verified by Walsh's test data. If reinforcement is present at the fracture front and behaves elastically, the decline of σN is of the same type but is shifted to larger sizes; however, if the reinforcement yields, the decline of σN stops. It is also noted that some known size effects which have been attributed to random strength variations within the structure should be explained by fracture mechanics, which gives a very different extrapolation to large structures. Finally, exploiting the fact that in metals the size of the yielding zone at the fracture front is approximately constant, it is shown by dimensional analysis that elastic-plastic fracture causes a similar size effect.

Experiments have consistently shown that the tensile strength of concrete increases with increasing strain rate. The reasons for this phenomenon are not yet well understood and several hypotheses have been proposed in the past to explain it. This study offers additional insight through the application of dynamic fracture mechanics. The relationship between crack velocity and strain rate of concrete is first investigated using a cohesive zone model and fitted to available experimental data. The obtained relationship is then implemented into two different versions of crack-speed dependent dynamic fracture models. Both models show that computed strength versus strain rate responses compare favorably to well-established test data, suggesting that strain rate sensitivity is strongly associated with the characteristics of dynamic crack growth and inertial effects at the boundaries of the crack. A constitutive modeling scheme that incorporates the obtained dynamic fracture models into a meso-mechanical model is also proposed to predict stress–strain behavior of concrete under dynamic tensile loading. Comparisons between model predictions and published experimental data are provided to show the accuracy of the proposed framework.

Notched plain concrete beams loaded by impact hammer are numerically studied. The numerical and experimental results are compared in terms of load-deflection response, rate dependent tensile strength and rate dependent fracture energy. Moreover, the effect of impact velocity on reaction, strain rate, crack opening rate and crack velocity is predicted numerically and compared with the experimental results. The numerical model realistically captures the experimentally observed behavior of the notched plain concrete beams under dynamic loads. It is pointed out that to evaluate the true rate dependent material properties, such as tensile strength and fracture energy, inertia have to be filtered out otherwise for higher strain rates the material properties are significantly overestimated.

The free water content of concrete underlies the various physical mechanisms that shape its mechanical behaviour. This Paper attempts to show that properties from creep to dynamic behaviour can be explained with reference to the process of cracking. Assumptions are made concerning the physical mechanisms involved, with a view to understanding what happens inside the material rather than to give rise to quantitative predictions.

This paper investigates the specimen size effect on the dynamic response of plain concrete. The report is based upon experimental data by the writers and others and considers results from creep tests on beams, beams under flexural impact, and cylinders under axial impact loading. Size effect is examined using Bazant's size effect law and the multifractal scaling law, and both scaling models are able to capture the size effect on strength. For fracture energy, on the other hand, the size effect manifests itself only at impact rates. Under quasi-static loading, plain concrete in compression is less sensitive to the specimen size. But under impact, the compressive response appears to be more size dependent than flexure. However, upon accounting for the stress rate effects, the flexural response depicts a more significant size effect, similar to that seen at quasi-static rates.

Attempts to apply linear elastic fracture mechanics (LEFM) to concrete have been made for several years. Several investigators have reported that when fracture toughness, Kk, is evaluated from notched specimens using conventional LEFM (measured peak load and initial notch length) a significant size effect is observed. This size effect has been attributed to nonlinear slow crack growth occurring prior to the peak load. A two parameter fracture model is proposed to include this nonlinear slow crack growth. Critical stress intensity factor, Kic, is calculated at the tip of the effective crack. The critical effective crack extension is dictated by the elastic critical crack tip opening displacement, CTODc. Tests on notched beam specimens showed that the proposed fracture criteria to be size independent. The proposed model can be used to calculate the maximum load (for Mode I failure) of a structure of an arbitrary geometry. The validity of the model is demonstrated by an accurate simulation of the experimentally observed results of tension and beam tests.

The behavior of concrete structures is strongly influenced by the loading rate. Compared to quasi-static loading concrete loaded by impact loading acts in a different way. First, there is a strain-rate influence on strength, stiffness, and ductility, and, second, there are inertia forces activated. Both influences are clearly demonstrated in experiments. Moreover, for concrete structures, which exhibit damage and fracture phenomena, the failure mode and cracking pattern depend on loading rate. In general, there is a tendency that with the increase of loading rate the failure mode changes from mode-I to mixed mode. Furthermore, theoretical and experimental investigations indicate that after the crack reaches critical speed of propagation there is crack branching. The present paper focuses on 3D finite-element study of the crack propagation of the concrete compact tension specimen. The rate sensitive microplane model is used as a constitutive law for concrete. The strain-rate influence is captured by the activation energy theory. Inertia forces are implicitly accounted for through dynamic finite element analysis. The results of the study show that the fracture of the specimen strongly depends on the loading rate. For relatively low loading rates there is a single crack due to the mode-I fracture. However, with the increase of loading rate crack branching is observed. Up to certain threshold (critical) loading rate the maximal crack velocity increases with increase of loading rate, however, for higher loading rates maximal velocity of the crack propagation becomes independent of the loading rate. The critical crack velocity at the onset of crack branching is found to be approximately 500m/s.

This paper presents improved expressions for the calculation of effective notch depth in three-point bend notched specimens used for the determination of the fracture toughness of plain concrete. The improvement is achieved in two ways. First, by using the exact elasticity solution for the midspan deflection of an unnotched beam. Second, by calculating the additional midspan deflection due to the presence of the notch from the corresponding expression for the stress intensity factor of a three-point bend specimen. The predictions of the improved effective crack model are shown to be in good agreement with those of the two-parameter model.

The practical performance of Gc-tests is analyzed. The conditions of stability for a three-point bend test on a notched beam are calculated by using the fictitious crack model. The results are presented in Fig. 1.Gc-values for different concrete qualities are determined from stable three-point bend tests on the notched beams. The results in this paper imply that Gc is strongly influenced by the quality of the aggregate, the water-cement-ratio and the age of the concrete (Fig. 4–8).ZusammenfassungNous traitons ici l'exécution pratique des essais pour la détermination du Gc. Les conditions de stabilité d'un essai de flexion à trois points effectué sur une poutre entaillée sont calculées en utilisant pour cet effet un modéle noveau de la mécanique de rupture. Les résultats obtenus sont représentés en Fig. 1.Les valeurs Gc se rapportant à diverses qualités de béton sont déterminées à partir des essais stables de flexion à trois points effectués sur des poutres entaillées. Dans cette étude, les résultats font suggèrent que le Gc est fortement influencé par la qualité de l'agrégat, le rapport eau-ciment et l'age du béton (Fig. 4–8).