Statistical prediction of fracture parameters of concrete and implications for choice of testing standard

This article shows how the fracture energy of concrete, as well as other fracture parameters such as the effective length of the fracture process zone, critical crack-tip opening displacement and the fracture toughness, can be approximately predicted from the standard compression strength, maximum aggregate size, water-cement ratio, and aggregate type (river or crushed). A database, consisting of 238 test data, is extracted from the literature and tabulated, and approximate mean prediction formulae calibrated by this very large data set are developed. A distinction is made between (a) the fracture energy, G_f, corresponding to the area under the initial tangent of the softening stress-separation curve of cohesive crack model, which governs the maximum loads of structures and is obtained by the size effect method (SEM) or related methods (Jenq-Shah two-parameter method and Karihaloo's effective crack model, ECM) and (b) the fracture energy, G_F, corresponding to the area under the complete stress-separation curve, which governs large postpeak deflections of structures and is obtained by the work-of-fracture method (WFM) proposed for concrete by Hillerborg. The coefficients of variation of the errors in the prediction formulae compared to the test data are calculated; they are 17.8% for G_f and 29.9% for G_F, the latter being 1.67 times higher than the former. Although the errors of the prediction formulae taking into account the differences among different concretes doubtless contribute significantly to the high values of these coefficients of variation, there is no reason for a bias of the statistics in favor of G_f or G_F. Thus, the statistics indicate that the fracture energy based on the measurements in the maximum load region is much less uncertain than that based on the measurement of the tail of the postpeak load-deflection curve. While both G_f and G_F are needed for accurate structural analysis, it follows that if the testing standard should measure, for the sake of simplicity, only one of these two fracture energies, then G_f is preferable.     

Determination of size-independent specific fracture energy of concrete mixes by the tri-linear model

The methods proposed by Elices and co-workers [1], [2], [3] and by Hu and Wittmann [4] are commonly used to determine the size-independent specific fracture energy (G_F) of concrete by correcting the size-dependent specific fracture energy (G_f) measured by the RILEM work-of-fracture method. In the boundary effect model of Hu and Wittmann [4], the change in the local fracture energy (g_f) is approximated by a bilinear function, whereas the method of Elices et al. [1] consists in determining the non-measured work-of-fracture by adjusting the tail of the P-δ curve that corresponds to the final part of the test. Acoustic emission (AE) experiments on notched specimens (Muralidhara et al. [5], [6]) have revealed that under loading the AE events follow approximately a tri-linear distribution; initially the number of events increases almost linearly reaching an extended plateau when the number of events remains nearly constant and eventually the number reduces as the crack approaches the back stress free boundary of the specimen, reminiscent of the development of R-curve in a finite size specimen. This paper exploits this observation and proposes a tri-linear model for the determination of the size-independent specific fracture energy for three different concrete mixes ranging in compressive strength from 57 to 122 MPa. Remarkably, it is found that the resulting size-independent specific fracture energy G_F determined by this tri-linear model and by the bi-linear model of Hu and Wittmann [4] is very nearly the same and independent of the size of the specimen.     

Substrate constraint and adhesive thickness effects on fracture toughness of adhesive joints

The adhesive thickness effect on fracture behaviour of adhesive joints has been studied using the boundary effect model recently developed for specimen size effect on fracture properties of concrete, and the essential work of fracture model for ligament (uncracked region) effect on largescale yield of bulk metals and polymers. The leading common mechanism responsible for the nonlinear elastic fracture mechanics behaviours, such as adhesive thickness effect of adhesive joints, specimen size effect of brittle heterogeneous materials and notch dependence of deeply notched metal and polymer specimens, is discussed. These two fracture mechanics models show that the height variation of a fracture process zone (FPZ) or a plastic zone is directly responsible for any change in fracture energy measurements such as the specific fracture energy G _f and the critical strain energy release rate G _c. Both models show that G _f is rapidly reduced when the crack-tip approaches the back-face boundary of a specimen because only a limited FPZ or plastic zone height h _FPZ can be developed in the boundary region. In the case of a thin adhesive joint, the development of a plastic zone height is limited by the thickness of the adhesive sandwiched between the upper and lower adherends or substrates. Consequently, a linear relationship between the adhesive joint toughness and adhesive thickness is established. Test results on adhesive joints from the literature are analysed and compared with the new adhesive joint failure model based on the two well-established fracture mechanics models developed for other material systems.     

Fracture characteristics of concrete at early ages

The purpose of this study is to experimentally investigate, at early ages, the fracture characteristics of concrete such as critical crack tip opening displacement, critical stress intensity factor, fracture energy, and bilinear softening curve based on the concepts of the effective-elastic crack model and the cohesive crack model.A wedge-splitting test for Mode I was performed on cubical specimens with an initial notch at the edge. By taking various strengths and ages, load-crack mouth opening displacement (CMOD) curves were obtained and these curves were evaluated by linear elastic fracture mechanics and finite element analysis.The results from the test and analysis indicate that critical crack tip opening displacement decreases and critical stress intensity factor and fracture energy increase with concrete ages from Day 1 to Day 28. By numerical analysis, four parameters of bilinear softening curves from Day 1 to Day 28 were obtained. In addition, it was observed that the parameters f_t and f₁ increase and the parameters w₁ and w_c decrease with increasing age. The obtained fracture parameters and bilinear softening curves at early ages may be used as a fracture criterion and an input data for finite element analysis of concrete at early ages.     

Fracture parameters for high-performance concrete

This paper reports and discusses the results of an experimental investigation on fracture properties of high-performance concrete, involving the tests of 115 three-point bend specimens with different compression strengths. The parameters were obtained by the work-of-fracture method and by the size effect method. For most series of tests the fracture energy was measured by the two methods, allowing a correlation between the values obtained by the two processes. A comparison with some results found in literature was performed. It was found that: (a) the ratio between the fracture energy measured by the work-of-fracture method (G_F) and by the size effect method (G_f) matched the value estimated by other researchers (G_F/G_f ≈ 2.5); (b) generally, G_F increases as the concrete compressive strength increases; and (c) the values obtained for G_f showed a slight trend to decrease with increasing compressive strength. The results obtained from the standard compressive strength tests indicate that the concrete strength was limited by the type and size of the coarse aggregate. The objective of this study is to provide some experimental data that can be useful in engineering practice for calibrating numerical constitutive models.     

Numerical simulation of the ENF test for the mode-II fracture characterization of bonded joints

In this work, the End Notched Flexure (ENF) test is analyzed in order to obtain the critical strain energy release rate in mode-II fracture of bonded joints. A cohesive model based on specially developed interface elements, including a linear softening damage process, is employed. The adequacy of the experimental ENF test is evaluated by numerical simulation. The objective is to compare the critical strain energy release rate in mode-II (G _{II c} ) obtained by different data reduction schemes with the real value which is an inputted parameter in the cohesive model. The effect of the Fracture Process Zone (FPZ) ahead of the crack tip is evaluated. A crack equivalent concept is proposed in order to account for the energy dissipated in the FPZ. A data reduction scheme avoiding the need to measure crack length is proposed. A good agreement with the inputted value of G _{II c} was obtained.     

Deterministic size effect in the strength of cracked concrete structures

This paper is concerned with identifying and quantifying the deterministic (as opposed to statistical) size effect in the strength of cracked concrete structures that is believed to be a result of stress discontinuities introduced by the cracks. For this, the strength of geometrically similar pre-cracked specimens of varying sizes made from three concrete mixes is measured in three-point bend and wedge splitting geometries. The true, size-independent specific fracture energy and the corresponding tension softening diagram of each of the three mixes are independently established in order to exclude their influence on the strength size effect. The test results show that the deterministic strength size effect weakens as the size of the crack reduces. This is confirmed by theoretical/computational studies based on the fictitious crack model in the range of sizes tested in the laboratory. The theoretical/computational model has been extended beyond this limited range to include cracked concrete structures in the size range 1 : 80. The computational results have been fitted by a simple strength size effect formula with appropriate asymptotic behaviour at both size extremes. The three unknown coefficients in this formula depend only on the size of the crack and they can be obtained by conducting tests on geometrically similar specimens of any shape but of varying sizes that can be conveniently handled in a laboratory. The three material properties of the concrete mix appearing in this formula, namely the Young modulus E, the direct tensile strength f_t and the size-independent specific fracture energy G_F must be independently measured.     

Determination of size-independent specific fracture energy of concrete mixes by two methods

The RILEM work-of-fracture method for measuring the specific fracture energy of concrete from notched three-point bend specimens is still the most common method used throughout the world, despite the fact that the specific fracture energy so measured is known to vary with the size and shape of the test specimen. The reasons for this variation have also been known for nearly two decades, and two methods have been proposed in the literature to correct the measured size-dependent specific fracture energy (G_f) in order to obtain a size-independent value (G_F). It has also been proved recently, on the basis of a limited set of results on a single concrete mix with a compressive strength of 37 MPa, that when the size-dependent G_f measured by the RILEM method is corrected following either of these two methods, the resulting specific fracture energy G_F is very nearly the same and independent of the size of the specimen. In this paper, we will provide further evidence in support of this important conclusion using extensive independent test results of three different concrete mixes ranging in compressive strength from 57 to 122 MPa.     

Additive effect of micro-damage zones from nano-scale crack tip and nano-grains for fracture toughness measurements of 3Y-TZP zirconia ceramics

A nano-scale crack tip around 500 nm wide introduced by femtosecond laser still affects the accuracy of fracture toughness K_IC measurements of 3Y-TZP zirconia ceramics with average grain size G from 200 to 500 nm. A simple formula was proposed to estimate the additive effect of crack-tip damage zones from an infinitely sharp crack to a nano-scale blunt notch. The error in fracture toughness measurements is less than 8 % if the nano-scale crack-tip width < 0.5·G. The intrinsic K_IC can be deduced from the simple formula if the nano-scale crack tip > 0.5·G. This study shows the same K_IC was deduced from two different sets of 3Y-TZP measurements with nano- and micro-scale notches of 500 nm and 18 µm wide. Furthermore, the simple formula specifies the relation between the fracture toughness K_IC and intrinsic strength f_t via grain size G, which means K_IC can also be estimated from f_t and G without testing pre-cracked specimens. K_IC values of 3Y-TZP from specimens with and without pre-cracks were compared.     

Evolution of crack path and fracture surface with degradation in rubber-toughened epoxy adhesive joints: Application to open-faced specimens

The crack path and fracture surface in the mixed-mode fracture of two different rubber-toughened epoxy adhesives were evaluated using double-layered open-faced double cantilever beam (ODCB) specimens in which the primary adhesive layer had been environmentally aged. The crack path in the mixed-mode fracture of unaged ODCB specimens was unexpectedly in the secondary adhesive layer, and several hypotheses were examined to explain this. It was concluded that a reduced residual stress in the secondary adhesive layer produced stable crack growth in the secondary layer instead of the expected path in the primary layer. The average crack path depth, fracture surface roughness and maximum elevation in the fracture surface profiles were then measured using optical profilometry as a function of the degree of aging. The results showed a strong relationship between all these parameters and the critical strain energy release rate, G_cs, irrespective of the type of adhesive. In the case of adhesive A where significant irreversible degradation was observed, all these parameters varied approximately linearly with G_cs. In the case of adhesive B, aging did not result in permanent degradation (G_cs was unchanged) and so all these fracture surface parameters also remained unchanged after aging. The results indicate that quantifying fracture surface parameters as a post-failure analysis can be of use in the estimation of the fracture toughness at which a practical joint fails.     

Fracture of model concrete: 2. Fracture energy and characteristic length

The specific fracture energy G_F was measured in six types of simple concrete: all from the same matrix. The aggregates were spheres of the same diameter (strong aggregates, that debonded during concrete fracture, and weak aggregates, able to break); three kinds of matrix-aggregate interface (weak, intermediate and strong) were used. All in all, 55 test results are reported. These results are intended to be used as an experimental benchmark for checking numerical models of concrete fracture.A meso-level analysis of these results showed a correlation between the measured G_F values and the properties of the matrix, aggregates and interfaces, particularly with the actual area of the fracture surface. The strength of the matrix-aggregate interface correlates quite well with G_F, and concrete ductility, measured by means of the characteristic length, correlates also with the strength of the matrix-aggregate interface.     

Influence of temperature on plane stress ductile fracture of poly(ethylene terephthalate) film

Abstract

Double edge notched poly(ethylene terephthalate) (PET) specimens of varying ligament lengths and 0·125 mm thickness have been pulled to complete fracture between 23 and 160°C. Within this temperature range, propagation of the crack was always stable, producing load–displacement curves at various ligament lengths that were geometrically similar to one another. Essential work of fracture (EWF) analysis was used to study the effect of temperature on fracture toughness. A linear relationship was obtained between specific total work of fracture W _f and ligament length over the entire temperature range under consideration. The slope of the line, which is termed specific non-essential work of fracture βw _p , showed a maximum near the glass transition temperature of the material (T _g ≈ 93°C). Beyond this point, βw _p decreased sharply with increasing temperature. The intercept of the line at zero ligament length, which is referred to as specific essential work of fracture w _e , showed three types of variation with respect to temperature. Below T _g , w _e was found to be more or less independent of temperature; above T _g it increased with temperature and reached a maximum value at the end of the leathery region (～120°C); beyond which it decreased steadily.     

A laser irradiation disc test for fracture testing of refractory fine ceramics

The critical stress intensity factor, tensile strength and crack stability were analysed for a zirconia refractory by parameter identification based on a laser irradiation test and a finite element simulation. Furthermore, the results for the specific fracture energy were determined for different assumed cohesive behaviours. The tests were carried out on notched discs that were irradiated at their centres. During the tests, temperatures are recorded by a thermo vision camera and the crack propagation by an acoustic emission recorder. Acoustic emission allowed for the determination of the onset of crack initiation (time t₁) and unstable crack propagation (time t₂). A finite element model representing the geometry of the sample was built to determine the fracture mechanical parameters from t₁ and t₂. This method is believed to be favourable for rather brittle refractory ceramics, whereas for less brittle materials, a wedge splitting procedure according to Tschegg is considered more favourable.     

Elucidating the Crack Resistance of Alkali‐Activated Slag Mortars Using Coupled Fracture Tests and Image Correlation

The resistance of alkali silicate‐activated slag mortars to crack propagation is explored. With increasing SiO₂‐to‐alkali oxide ratio (M_s) of the activating solution (between 1.0 and 2.0), the flexural strengths, fracture energies, and the strain energy release rates (crack resistance, G_R) are noted to increase. The G_R values, especially of the systems with M_s of 1.5 and 2.0, are higher than that of ordinary portland cement (OPC) mortar. In contrast, the fracture process zone (FPZ) was observed to be smaller for the alkali‐activated slag mortars, with higher localized strains. Similarly, the FPZs also shrink with increasing M_s. These responses are related to the differences in the reaction products in these systems. The fundamental differences in the fracture response of these binder systems are elucidated through tracking the FPZ development. The crack extension‐crack tip opening displacement relations and its relationship with the inelastic strain energy release rates are also used to bring out the differences between the binder systems.     

On the measurement of fracture toughness of soft biogel

In this study, the rate dependent energy dissipation process and the fracture toughness of physical gels were investigated using agarose as a sample material. Both the J‐integral and Essential work of Fracture (EWF) methods were examined. To assess the quasi‐static fracture toughness of gels, linear regression was performed on critical J (J_c) values at different loading rates resulting in a quasi‐static J_c value of 6.5 J/m². This is close to the quasi‐static EWF value of 5.3 J/m² obtained by performing EWF tests at a quasi‐static loading rate (crosshead speed of less than 2 mm/min). Nearly constant crack propagation rates at low loading rates, regardless of crack length, suggest viscoplastic chain pull‐out is the fracture mechanism. At high loading rates failure was highly brittle, which is attributed to sufficient elastic energy accumulation to precipitate failure by chain scission. We conclude that in physical gels quasi‐static fracture toughness can be evaluated by both the J‐integral and EWF methods provided the effects of loading rate are investigated and accounted for. POLYM. ENG. SCI., 2011. © 2011 Society of Plastics Engineers     

Probabilistic prediction of mode Ⅰ fracture related to notch geometry and microstructure

The fracture stress (σ_N) of a notched specimen from polycrystalline material is influenced by notch geometry and material microstructure. In this paper, by analyzing measurements on notched specimens from materials with different average grain sizes (G), a unified model is proposed to describe notch geometry effect on σ_N and its relation to G. An analytical relation between stress concentration factor and notch geometry parameters is determined, and normal distribution theory is incorporated into the fracture model to describe the inevitable fracture scatter due to material heterogeneity and machining & testing errors. The results show that the model can well predict σ_N of notched specimens made from various polycrystalline materials.     

On the essential work of fracture method: Energy partitioning of the fracture process in iPP films

Summary The fracture properties of an iPP are investigated by the EWF method. A separation between crack initiation and propagation fracture parameters is done by splitting the total energy of the load-displacement curves in two. The influence of the DDENT specimen height and the test rate on these different parameters is studied, obtaining that varying the height has no influence in the range 40 to 80mm, but changing the crosshead speed (2 to 100mm/min) has an effect on the fracture parameters. It is interesting to note that the “Initiation Specific Essential Work” (w_e ^I) seems not to be sensible to the stress-state transition. Received: 20 October 1998/Revised version: 1 December 1998/Accepted: 7 December 1998     

Composite bonded joints under mode I fatigue loading

Experimental investigation on fatigue behavior of carbon-epoxy composite bonded joints under mode I loading was performed in this work. The objective is to evaluate the performance of different data reduction schemes to obtain the energy release rate (G_I) in the fatigue crack growth (FCG) rate using double cantilever beam (DCB) specimens. This law relates the evolution of the crack along time as a function of the energy release rate (G_I) and is generally composed of three different regions: damage nucleation, stable propagation and abrupt final failure. The second phase corresponding to stable propagation leads to a linear trend on the Paris law representation (log-log scale) and must be well characterized to define the fatigue behavior of the structure. During fatigue tests the classical methods require rigorous monitoring of the crack length during its propagation, which is cumbersome and not easy to perform in some materials. In this work, an alternative data reduction scheme based on specimen compliance and crack equivalent concept is proposed to overcome this difficulty. The results provided by the proposed method, namely Compliance Based Beam Method (CBBM), are compared to the ones obtained from the polynomial and Beam on Elastic Foundation Method (BEFM), both of which require crack monitoring. The first is a compliance calibration method that fits a third-order polynomial curve to the experimental results (compliance (C) versus crack length (a)). The second one uses the beam theory to establish the C=f(a) relationship taking into account the properties of the adhesive. One additional advantage can be pointed to the proposed CBBM relative to the other classical methods. In fact, the equivalent crack is related to the specimen compliance, thus taking into account the influence of fracture process zone on specimen behavior. This issue is particularly important when adhesives with some ductility are being characterized in fatigue tests.     

Temperature dependence of crack initiation fracture toughness of various nanoparticles filled polyamide 66

In the present study, the crack initiation fracture toughness of various nanoparticles filled polyamide 66 was investigated in a broad temperature range (23-120 °C) by using an essential work of fracture (EWF) approach. Four types of spherical nanoparticles, i.e. two types of TiO₂ (21 nm, with/without surface modification), SiO₂ (13 nm) and Al₂O₃ (13 nm), were selected with a constant volume content of 1% in nanocomposites, which were compounded using a twin-screw-extruder. The addition of nanoparticles led to an enhanced specific EWF item at most test temperatures at the cost of the reduction of the non-EWF item. The value of the specific EWF was also estimated by a crack opening displacement method. Associated with SEM fractograph analysis, it was clear that two basic factors, i.e. crack tip blunting and net section stress, finally determined the EWF value. With the addition of nanoparticles, the item of crack tip blunting was increased at most temperature range, which may be incidental with the formation of numerous dimples and sub-dimples induced by nanoparticles; while the item of net section stress was correlated with the particle distribution, especially at room temperature, which was notably decreased in case of poor nanoparticle distribution.     

Determination of the fracture energy in polymeric films by in situ photoelasticimetry on double edge notch specimen

The fracture toughness is key parameters to select polymeric films. The essential work of fracture (EWF) is a phenomenological but efficient way to characterize this resistance to fracture. One can gain valuable information on the resistance to perforation and propagation of flaws. A new technique was developed to better understanding the EWF experiments. A tensile test combined to photoelasticimetry allows following in situ the geometry and amount of plastic deformation on double edge notched specimen. The EWF parameters are determined when the plastic deformation appears constant, so when the fracture energy W_f only contributes to rupture filament. This new methodology requires just a single sample, whereas at least five specimens are required for general method. It will help characterize expensive polymeric films or reveal the heterogeneous behavior, for instance after polymer ageing. © 2015 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2016 , 132, 42854.