Influence of specimen geometry and size-effect on the <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">R</Emphasis></Subscript>-curve based on the cohesive stress in concrete

Comparative study for determining the K _R -curves associated with the cohesive stress distribution for complete fracture process for two standard specimen geometries i.e., three-point bending test and compact tension test specimen geometries of concrete using analytical method and weight function approach is reported in the paper. The laboratory size specimen (100 ≤  D  ≤  400 mm) with initial-notch length/depth ratios 0.3 and 0.5 are considered in the investigation. The load-crack opening displacement curves for these specimens are obtained using well known version of Fictitious Crack Model (FCM). It is found from the numerical results that the weight function method improves computational efficiency without any appreciable error. The stability analysis on the K _R -curves and the influence of specimen geometry and the size-effect on the K _R -curves, the CTOD-curves and the process zone length during crack propagation of complete fracture process are also described.     

The Effect of Size on the Splitting Strength of Cubic Concrete Members

In the study of concrete fractures, split‐tension specimens, such as cylinders, cubes and diagonal cubes, are frequently preferred to beams. However, experimental investigations on concrete reveal that for the same specimen geometry, the nominal strength of specimen decreases with increasing specimen size. This phenomenon is named as the size effect in the fracture mechanics of concrete. Although nominal strength is also highly affected by the width of the distributed load in the split‐tension cylinder and cube specimens, this effect can be negligible within the practical range of the load‐distributed width in the diagonal cubes. However, the number of theoretical and experimental studies with diagonal split‐tension specimens is limited. Besides, a size effect formula for estimating the split‐tensile strength of the diagonal cube specimens has not been proposed. In this study, nine series of cube and diagonal cube specimens, with three different sizes but similar geometries, were tested under different load‐distributed widths. The ultimate loads obtained from the test results are analysed by the modified size effect law. Subsequently, prediction formulas are proposed, and they are compared with historical test data from the split‐cylinder specimens.     

Scaling of the fracture process zone in rock

The zone of microcracks surrounding a notch tip—the process zone—is a phenomenon observed in fracture of quasi-brittle materials, and the characterization of the process zone is the topic of the paper. Specimens of different sizes with a center notch fabricated from a granite of large grain (Rockville granite, average grain size of 10 mm), were tested in three-point bending. Acoustic emissions were recorded and locations of microcracks were determined up to peak load. The results show that both the length and width of the process zone increase with the increase of the specimen size. Furthermore, the suitability of a proposed theoretical relationship between the length and width of the process zone and specimen size was studied experimentally and numerically. The discrete element method with a tension softening contact bond model was used to investigate the development of the process zone with the specimen size. A synthetic rock composed of rigid circular particles that interact through normal and shear springs was tested in the numerical simulations. It was shown that the limiting specimen size, beyond which no further noticeable increase in the length of the process zone is observed, is significantly larger than the limiting specimen size beyond which the width of the process zone shows no size effect.     

Fracture toughness of 2-D carbon fibre reinforced carbon composites

The fracture toughness of 2-D woven carbon fibre reinforced carbon laminate has been evaluated by linear elastic fracture mechanics (LEFM),R-curve andJ-integral analysis using the single edge-notched bending (SENB) specimen of edge and flatwise geometries. The edgewise specimens failed by a small extension of the self similar crack whereas the flatwise specimens failed by delamination. The surface damage developing from the tip of the initial crack was revealed by the brittle lacquer coating technique and the zone shape varied with the specimen geometry, i.e. the loading axis relative to the woven layers. Acoustic emission (AE) was also used to monitor crack growth, and the total ring down count of AE was observed to increase as the initial crack length was decreased. Both the damage zone size and total AE counts were found to increase in two linear stages as a function of the square of the stress intensity factor,K.     

The effect of specimen size on the fracture energy and softening function of concrete

This paper examines the effect of specimen size on the fracture energy of concrete, G_F, as measured using the three-point bending test on pre-notched beams prescribed by RILEM TC-50 [1]. The concept of partial fracture energy is introduced and used to explain the observed size effect. The opening displacement at the top of the notch in the test specimen at the end of the test, ω, is affected by the size of the specimen, which in turn affects the measured value of the concrete fracture energy. In theory, when the specimen is large enough to allow the fracture process zone to develop fully,w _f will reach its critical value,w _c , and the effect of specimen size onG _F will be eliminated. The experimental results included here show that in reality the size of the specimen does affect the measurement ofG _F , even when the size is such that the fracture process zone develops fully. This may be due to local plastic deformation in the area around the loading point, which is particularly significant in larger specimens. It may also be due to differences in the influence of the boundary conditions of the test for different specimen sizes. In addition, a procedure is outlined for the determination of the softening function for concrete based on the fracture energy measured in RILEM tests, in which the specimens are small enough to ensure that the energy measured is actually due to fracture and not plastic deformation.     

Size effect on fracture resistance and fracture energy of concrete

A recent asymptotic approach dealing with the size effect on the fracture properties of a large plate is further developed to consider the influence of both the front and back free surfaces of small sized specimens. The new extension is applied to experimental results found in the literature, and good agreements have been found between the predictions and the fracture resistance and energy measured using geometrically similar specimens and specimens with identical size but different initial crack or notch lengths. The physics behind the size effect are discussed based on the modified asymptotic approach. It is found that both the specimen geometry and crack length contribute to the size effect on fracture properties besides its physical size. In particular, the ratio of a fracture process zone, size over its distance to a free surface plays a very important role.     

A proposed mixed-mode fracture specimen for wood under creep loadings

A mixed-mode fracture specimen was designed in this paper. This geometry is a judicious compromise between a modified Double Cantilever Beam specimen and Compact Tension Shear specimens. The main objective is to propose a specimen which traduces a stable crack growth during creep loading taking into account viscoelastic behaviour under mixed-mode loadings. The numerical design is based on the instantaneous response traduced by a crack growth stability zone. This zone is characterized by a decrease of the instantaneous energy release rate versus the crack length. In order to obtain a mixed-mode separation, the paper deals with the use of the M-integral approach implemented in finite element software, according to energetic fracture criterions. In these considerations, a numerical geometric optimization is operated for different mixed-mode ratios. Finally, a common specimen which provides to obtain fracture parameters, viscoelastic properties and creep crack growth process for different mixed-mode configurations is proposed.     

Size and geometry dependent tensile behavior of ultra-high-performance fiber-reinforced concrete

This research investigated direct tensile stress versus strain response of ultra-high-performance fiber-reinforced concrete (UHPFRC) with various sizes and geometries. The UHPFRC in this research contained 1% macro twisted and 1% micro smooth steel fibers by volume. The effects of gauge length, section area, volume and thickness of the specimens on the measured tensile response of the UHPFRC were experimentally discovered. The different sizes and geometries of specimens did not generate significant influence on the post cracking strength of UHPFRC whereas they produced clear effects on the strain capacity, energy absorption capacity and multiple cracking behavior of UHPFRC. The strain capacity, energy absorption capacity and the number of multiple micro cracks within unit length obviously decreased as the gauge length, section area and volume of UHPFRC specimens increased. In contrast, as the thickness of the specimen increased, different tendency was observed.     

Intralaminar toughness characterisation of unbalanced hybrid plain weave laminates

A numerical and experimental investigation on the mode-I intralaminar toughness of a hybrid plain weave composite laminate manufactured using resin infusion under flexible tooling (RIFT) process is presented in this paper. The pre-cracked geometries consisted of overheight compact tension (OCT), double edge notch (DEN) and centrally cracked four-point-bending (4PBT) test specimens. The position as well as the strain field ahead of the crack tip during the loading stage was determined using a digital speckle photogrammetry system. The limitation on the applicability of the standard data reduction schemes for the determination of intralaminar toughness of composite materials is presented and discussed. A methodology based on the numerical evaluation of the strain energy release rate using the J-integral method is proposed to derive new geometric correction functions for the determination of the stress intensity factor for composites. The method accounts for material anisotropy and finite specimen dimension effects regardless of the geometry. The approach has been validated for alternative non-standard specimen geometries. A comparison between different methods currently available for computing the intralaminar fracture toughness in composite laminates is presented and a good agreement between numerical and experimental results using the proposed methodology was obtained.     

Einfluß der Probengröße auf die bruchmechanischen Eigenschaften von Sinterstahl

Influence of Specimen Size on the Fracture Mechanical Behaviour of Sintered Steel Fracture mechanics testing was carried out with small and big specimens using high-temperature sintered Fe-2%Cu-2.5%Ni-alloys in the densities of ρ = 7.1 and 7.4 g/cm³. These steels are often used in the manufacturing of PM-parts. Due to the different dismensions the crack propagation is for the bigger sizes faster than for the smaller sizes. Also the conditional fracture toughness of the big specimens is superiour to the toughness of the small specimens. But under consideration of a plain strain state for the big specimens and of a plain stress state for the small specimens valid fracture toughness values being independent from the specimen size can be calculated applying linear-elastic fracture mechanics. These results were obtained for both densities investigated. The increase of the density delivers principally better fracture mechanical data. Hereby the relation of strength data with the microstructure is also discussed.     

Effect of loading condition,specimen geometry,size-effect and softening function on double-<Emphasis Type="BoldItalic">K</Emphasis> fracture parameters of concrete

This paper presents numerical investigation of the influence of the specimen geometry, loading condition, size-effect and softening function of concrete on double-K fracture parameters. The input data needed for computation of the double-K fracture parameters are obtained from the well-known version of Fictitious Crack Model (FCM). FCM is developed for three standard specimens: three-point bend test, compact tension specimen and four-point bend test of size range 100–600 mm at relative size of initial crack length 0.3. The analysis of numerical results shows some interesting behaviour of double-K fracture parameters.     

Numerical study of compact shear (Mode II) type test specimen geometry

The Iosipescu shear test specimen geometry has been investigated by a number of research workers in recent years with conflicting results. The paper describes a numerical study of a compact shear test specimen, based on the Iosipescu geometry, which is proposed to investigate size effects in shear failure. A range of geometries has been studied and the extreme cases are reported. Results are presented for the largest absolute principal stresses together with a detailed study of the stresses between and around the roots of the two notches introduced in the test specimens. The results for the largest absolute principal stresses show that tensile stresses are created at the roots of the two notches. These tensile stresses may result in Mode I failure and probably account for the Mode I or mixed mode fracture observed in tests using the Iosipescu geometry. The results for the distribution of stresses between the roots of the two notches show that deep notches increase the likelihood of shear fracture prior to tensile failure. Shallow notches give a stress distribution similar to that developed in the indirect tensile test and hence tensile failure is likely to precede shear failure in such cases. Further numerical and experimental work is proposed.      

Size dependence of concrete fracture energy determined by RILEM work-of-fracture method

The paper analyzes the size dependence of the fracture energy of concrete obtained according to the existing RILEM recommendation proposed by Hillerborg and based on the work-of-fracture method of Nakayama, Tattersal and Tappin, in which the energy dissipated at the fracture front is evaluated from the measured load-displacement curve. The analysis is based on the size effect law proposed by Baant, which has been shown to be applicable to the size ranges up to about 1:20 and apply in the same form for all specimen geometries. The analysis utilizes the previously developed method for calculating the R-curve from the size effect, and the load-deflection curve from the R-curve. The R-curve is dependent on the geometry of the specimen. The results show that the fracture energy according to the existing RILEM recommendation is not size-independent, as desired, but depends strongly on the specimen size. This dependence is even stronger than that of the R-curve. When the specimen size is extrapolated to infinity, the fracture energy according to the RILEM recommendation coincides with the fracture energy obtained by the size effect method. It is also found that, in fracture specimens of usual sizes, the pre-peak contribution of the work of the load to the fracture energy is relatively small. Finally, as a by-product, the analysis also verifies the fact that, in three-point bend fracture specimens, the fracture energy according to the RILEM definition is dependent on the notch depth.     

Numerical analysis of the width of fracture process zone in concrete beams

There are two ways of modelling cracking in concrete structures using a finite element method (FEM). In the first concept crack is considered as densely distributed throughout the finite area of element. The alternative approach assumes an isolated sharp interelement crack. The first concept of microcracked band is often used during numerical computations of concrete structures and a crack band model can be applied in design practice where relatively simple methods of calculations are preferred. In this model the width of microcracked zone, which is called the width of fracture process zone, is the additional parameter of tensile concrete.In this paper, the influence of the fracture process zone width on numerical calculations in case of bent concrete beams is analyzed. The results of numerical simulations are presented and compared with those obtained during the experiment. The crack band model proposed by Hillerborg [Cement and Concrete Research, 6 (1976) 773–782] has been used in numerical calculations and six different widths of fracture process zone have been modelled in the analyzed specimen. When comparing the obtained FEM-results their significant differences according to the width of fracture process zone have been observed. On the basis of the performed analysis it may be concluded that the width of fracture process zone taken in numerical simulations does have an influence on FEM-results.     

Effects of grain size and specimen geometry on the transformation and R-curve behaviour of 9Ce-TZP ceramics

Transformation and R-curve behaviour have been investigated in 9 mol% Ce-TZP ceramics with different grain sizes. Both single-edge notched beam and short double-cantilever beam specimens were tested to measure the crack-resistance curves. The size and shape of the transformation zone not only depend on grain size, but are also strongly influenced by the specimen geometry. This different transformation behaviour has led to different crack-resistance curves. These experimental results are discussed in terms of the thermodynamics of transformation, the effect of autocatalytic transformation, and fracture mechanics.     

Specimen size effect during tensile testing of an unreinforced polymer

Various specimen sizes of an unreinforced polymer, Hercules 3501 -6 thermosetting epoxy, were subjected to a tensile test. The general specimen geometry was a rectangular dog-bone shape with constant gauge length, but with each specimen size having a different crosssectional area. These cross-sectional areas were obtained by varying the thickness of the epoxy during casting, and the gauge section width during grinding. The resulting failure surfaces of the specimens were observed and photographed using scanning electron microscopy. The results indicate that failure stress, dimensions of the critical flaw which caused failure, and a quantity which is proportional to the fracture toughness, are all correlated with specimen size.     

Notch sensitivity and brittleness in fracture testing of S2 columnar freshwater ice

Two loading configurations (four-point-bend, three-point-bend) were used in the laboratory at Clarkson to test for the fracture toughness of carefully grown S2 columnar freshwater ice. For one specific crack orientation and one grain size, the crack length was varied ranging from very short to very deep. The crack length effects were studied in this way for three specimen sizes (the in-plane dimensions of the specimen size were geometrically scaled; the specimen thickness was essentially constant). These crack length and specimen size tests are primarily directed towards designing fracture toughness tests for ice that both satisfy small scale yielding requirements and provide material properties (in the sense of (1)) — toughness values independent of the size and geometry of the specimen. Considerations of sufficient notch sensitivity in terms of brittleness numbers provide a means to determine the necessary specimen size. The results reported in this paper suggest that the specimen sizes used in testing S2 ice to date have largely been sub-size.     

Fracture toughness of plain concrete from three-point bend specimens

A simple method is proposed for determining the fracture toughness of plain concrete from three-point bend specimens, based on the concept of effective notch depth to account for the non-linear pre-peak load-deflection behaviour. The fracture toughness so determined is shown not to depend on the specimen size. The method improves an earlier version of the effective crack model in several ways. First, it is no longer necessary to linearize the pre-peak non-linearity, thereby eliminating the inaccurate task of establishing the limit of elastic response. Secondly, regression expressions for determining the effective notch depth should be far more accurate because they are based on an analysis of not only the authors’ test data but that of several researchers around the world. Thirdly, these expressions do not depend on the size of the test specimen. It is shown that the predictions of the effective crack model are in good agreement with two other non-linear process zone models, as far as three-point notched beam specimens are concerned.     

用能量方法研究混凝土断裂过程区的力学性能

准脆性混凝土自由裂缝前缘断裂过程区的发展与其非线性断裂特征及尺寸效应现象密切相关。它的物理力学行为的量化分析对理解混凝土断裂破坏机理和建立适用于混凝土结构裂缝稳定分析和安全评估断裂准则尤为重要,一直是混凝土断裂力学研究的核心问题。该文依据Hillerborg给出的断裂能定义,给出了计算单位长度断裂过程区发展能量耗散的通用表达式。以三点弯曲梁为例,采用非线性软化本构关系,进一步给出了计算此平均能量耗散的具体步骤及对应的公式。在根据实测的三点弯曲梁的断裂能回归拟合了特征裂缝张开位移w0后,计算了每个试件整个断裂全过程中不同荷载时刻断裂过程区耗能的平均值。结果表明:随着裂缝扩展,断裂过程区能量耗散的变化和试件尺寸无关,可描述断裂过程区混凝土材料的力学性能。     

Size and geometry effects on ductile rupture of notched bars in a C-Mn steel: experiments and modelling

The aim of this work was to investigate the effect of specimen size and geometry on ductile fracture of a C-Mn steel with high sulphur content. Uniaxial tensile tests were conducted at 300°C on axisymmetric notched specimens having different sizes and geometries. Geometry effects were studied using specimens with various notch radii, thus inducing different stress triaxiality levels. Size effects were evidenced using homothetic samples having the same geometry. Results show that ductility is reduced on specimens with sharp notches (which is a common observation). As specimen size increases, mean ductility as well as scatter are reduced (showing a clear size effect). In order to predict rupture, locally coupled (post-processing type) and fully coupled (continuum damage mechanics) Finite Element models were used. They are based on the plastic criteria introduced by Gurson and Rousselier. In order to model size effect (decrease of ductility and scatter), initial distribution of inclusion volume fractions, measured by quantitative metallography, was accounted for in the simulations. Comparison of experiments with simulations showed that both model types could predict mean values of ductility and scatter. This revised version was published online in August 2006 with corrections to the Cover Date.