P. Miarka et al.: MIXED-MODE FRACTURE ANALYSIS IN HIGH-PERFORMANCE CONCRETE USING A BRAZILIAN ... 233–238 MIXED-MODE FRACTURE ANALYSIS IN HIGH-PERFORMANCE CONCRETE USING A BRAZILIAN DISC TEST ANALIZA ME[ANEGA PRELOMA VISOKOKAKOVOSTNIH BETONOV Z BRAZILSKIM PREIZKUSOM Petr Miarka 1,2* , Stanislav Seitl 1,2 , Vlastimil Bílek 3 1 Brno University of Technology, Faculty of Civil Engineering, Institute of Structural Mechanics, Veveøí 331/95, 602 00 Brno, Czech Republic 2 Institute of Physics of Materials, Academy of Sciences of the Czech Republic, @i`kova 22, 616 62 Brno, Czech Republic 3 VSB-Technical University of Ostrava, Faculty of Civil Engineering, Department of Building Materials and Diagnostics of Structures, L. Podé{tì 1875/17, 708 33 Ostrava, Czech Republic Prejem rokopisa – received: 2018-07-24; sprejem za objavo – accepted for publication: 2018-11-06 doi:10.17222/mit.2018.161 This paper reports and discusses the results of an experimental investigation into the fracture mechanical properties of high-performance concrete (HPC) with a compressive strength higher than 100 MPa. In total, 6 specimens for a three-point bending test and 12 specimens of a Brazilian disc under a mixed-mode load were included in the experimental measurements. The fracture toughness was evaluated for mode I in the case of a three-point bend test and a Brazilian disc test, allowing a correlation between two geometrically different specimens. The fracture resistance under mixed mode I/II was analysed using the Brazilian disc test specimen with a central notch. To evaluate the fracture resistance under the mixed mode a generalised maximum tangential stress criterion was employed and the value of the critical distance rC is discussed as an appropriate HPC material characteristic. The last objective of this study was to provide some experimental data that can be useful in engineering practice for calibrating numerical constitutive models. Keywords BDCN, 3PBT, fracture mechanics, GMTS, HPC, mixed mode, critical distance Avtorji poro~ajo in razpravljajo o eksperimentalnih raziskavah lomno-mehanskih lastnosti visokokakovostnega betona (HPC; angl.: High Performance Concrete) s tla~no trdnostjo nad 100 MPa. V kompletu so izvedli trito~kovne upogibne preizkuse na {estih (6) vzorcih in na dvanajstih (12) vzorcih v obliki diskov so izvedli brazilski preizkus, ki predstavlja me{ano (natezno/tla~no) obremenitev. Dolo~ili so lomno `ilavost za na~in I v obeh primerih; to je trito~kovnega upogibnega preizkusa in brazilskega testa, upo{tevajo~ geometrijske lastnosti obeh vrst vzorcev. Odpornost proti lomu v me{anem na~inu I/II so analizirali s pomo~jo brazilskega preizkusa na vzorcih (diskih) s centralno zarezo. Da bi ovrednotili odpornost proti lomu v me{anem na~inu, so uporabili posplo{eni kriterij maksimalne tangencialne napetosti in ocenili, da je vrednost kriti~ne razdalje primeren kriterij za oceno lastnosti HPC. V zadnjem delu te {tudije so podali nekaj eksperimentalnih podatkov, ki bi lahko bili uporabni v in`enirski praksi za kalibracijo numeri~nih konstitutivnih modelov. Klju~ne besede: BDCN, 3PBT, lomna mehanika, GMTS, HPC, me{ani na~in, kriti~na razdalja 1 INTRODUCTION The sustainability of concrete structures assumes the optimised use of concrete. Instead of using high volumes of lower-strength concrete, a high-performance concrete (HPC) is considered as a sustainable solution to design subtle structures (thin elements) and therefore reduce the total consumption of material. The production of HPC consumes much less natural resources, i.e., raw materials for cement, aggregates, water, etc. To reduce the con- sumption of natural resources and to improve its mecha- nical characteristics the secondary pozzolans like silica fume, slag, fly ash, etc, are commonly used as a mineral admixtures into the HPC mixture. HPC is used where a weight reduction of structure is important or where architectural design demands smaller and thinner bearing structural elements for a required esthetical value. The use of optimized structural ele- ments made from HPC leads to a reduction in the total amount of transported material, which results in an overall cost reduction for the structure. An experimental study of the structural behaviour of HPC and very-high-performance concrete is presented in 1 ; however, the HPC has a compressive strength limited to 84 MPa. A pilot study with a focus on the fracture mechanical parameters (specific fracture energy) of five different kinds of HPC is mentioned in 2 ,b u tt h ec o m - pressive strength is only 88 MPa of all the tested HPCs. The influence of various coarse aggregates for high- strength concrete on the fracture energy and strength properties is shown in 3 , but the compressive strength of the studied HPC was only 90 MPa. The aim of long-term research is to design and prepare concrete mixtures suitable for precast elements, which result to total weight reduction. Typically pro- duced precast elements are light frames and beams of small-span bridges. These subtle structural elements have to be designed with a greater attention to durability, fatigue behaviour and fracture behaviour. This is one of the reasons why fracture mechanics is employed in this Materiali in tehnologije / Materials and technology 53 (2019) 2, 233–238 233 UDK 620.1:666.97 ISSN 1580-2949 Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 53(2)233(2019) *Corresponding author e-mail: Petr.Miarka@vut.cz contribution, especially for the fracture behaviour of HPC under the mixed-mode I/II loading conditions. All recent studies are focused on HPC with a compressive strength of less than 100 MPa, thus it is useful to see how HPC behaves when the compressive strength is higher than 100 MPa. This contribution aims to introduce the mechanical characteristics and the fracture resistance of HPC under the mixed mode I/II using a Brazilian disc test with a central notch (BDCN) 4 . To evaluate the fracture resist- ance under the mixed mode a generalised maximum tangential stress (GMTS) 5 criterion was employed. The discussion of an appropriate value for the critical dist- ance r C with use as a material characteristic is presented. 2 THEORETICAL BACKGROUND This contribution is based on linear elastic fracture mechanics. The linear elastic fracture mechanics concept uses the stress field in the close vicinity of the crack tip described by the Williams’ expansion. 6 This expansion is an infinite power series, originally derived for a homo- genous elastic isotropic cracked body. The stress field for mode I and mode II can be described with the following equation: i j ij ij ij K r f K r fT Or =++ + I I II I 22 ππ ,,, () () (,) (1) where ij represents the stress tensor components, K I , K II are the stress-intensity factors (SIFs) for mode I and mode II, respectively, f ij , () I , f ij , () II are known shape functions for mode I and mode II, usually written for BDCN 7 as Y I and Y II , T (or T-stress) represents the second term independent of r, O ij represents the higher-order terms, and r, are the polar coordinates (with the origin at the crack tip; the crack faces lie along the x-axis). 2.1 Brazilian disc test with a central notch (BDCN) The BDCN is a specimen with a circular cross-sec- tion made from a cylinder with a notch in the middle of the specimen (Figure 1a). The test performed on the BDCN specimen is carried out under relatively simple experimental conditions (Figure 1b), using only the testing press with a sufficient load capacity. The evaluation of the fracture parameters for modes I, II and mixed mode I/II is made by inclining the notch at an angle against the load position (Figure 1a). The SIFs for a finite specimen in shape of a BDCN and the polar angle = 0° can be calculated by following Equations (2–3): 8 () K Pa RB a R YaR II = − π 1 1 /, (2) () K Pa RB a R YaR II = − π 1 1 /, (3) where P is the compressive load, a is the initial notch length, R is the radius of the disc (D/2), B is the disc’s thickness, is the inclination angle and Y I (a/R, ), Y II (a/R, ) are the dimensionless shape functions for mode I and mode II, respectively. The geometry func- tions Y I and Y II used in Equation (2) and (3) can be found in 7,9 . To calculate the T-stress, a direct extra- polation method 10 was used, for the polar angle =0° the following Equation (4) was used: T r xx yy =− → lim ( ) 0 (4) where xx and yy are the stress components in front of the crack tip in the direction for = 0°. 2.2 GMTS Criterion There are several criteria for predicting the onset of the mixed mode I/II fracture of a brittle material, such as the strain energy density (SED) criterion, 11 the averaged strain energy (ASED) criterion 12 and the extended maxi- mum tangential strain (EMTSN) criterion. 13 A com- monly used criterion on the BDCN specimen is the maximum tangential stress (MTS) criterion. 14 However, this criterion conservatively predicts the onset of mixed mode I/II fracture. This disadvantage leads to the deve- lopment of the GMTS criterion. The GMTS criterion has been recently used with an accurate prediction of fracture resistance by Aliha et al. for PMMA, 15 Seitl et al. for concrete C 50/60 9,16 and Hou et al. 17 for mortar and concrete. =− ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ + ++ 1 2 3 2 2 21 πr KK TO r cos cos sin sin ( / II 2 ) (5) The GMTS criterion uses two terms of the Williams’ expansion (SIFs and T-stress) in the series for . The higher-order terms O(r 1/2 ) are often negligible near the crack tip and only three terms K I , K II and the T-stress are used in the further analysis. According to the GMTS P. Miarka et al.: MIXED-MODE FRACTURE ANALYSIS IN HIGH-PERFORMANCE CONCRETE USING A BRAZILIAN ... 234 Materiali in tehnologije / Materials and technology 53 (2019) 2, 233–238 Figure 1: Brazilian disc with central notch: a) principle of testing and b) actual test setup criterion, the onset of fracture is the angle of the maxi- mum tangential stress 0 and can be determined from: ∂ ∂ = 0 and ∂ ∂ 2 0 < (6) The assumption mentioned in Equation (6) leads into: [] KK T r II I c sin ( cos cos sin 3 16 3 20 − −= π (7) The crack-initiation angle 0 is then used for the evaluation of the beginning of the mixed mode I/II on the BDCN specimen. While in the MTS criterion the crack-initiation angle 0 is a function of only SIFs, in the case of GMTS the angle 0 depends on SIFs, T-stress and the critical distance r C , where the GMTS criterion is applied. 15 The critical distance r C and the angle 0 are then considered as a material constant. 2.3 Application of the GMTS on THE Brazilian Disc Specimen Pure mode I fracture initiation appears when K I = K IC (fracture toughness), K II = 0 and 0 = 0°. The fracture toughness K IC can be expressed as K IC = 2πr c and Equation (7) leads into Equation (8): KKK rT c IC I II c = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ + + cos cos sin sin 2 2 3 2 2π (8) where K IC is the materials’ fracture toughness, Equation 8 shows that the angle 0 for any combination of modes I and II depends on K I , K II , T, and r C . The critical distance r C can be evaluated from the recommended Equations (9) for plane stress and plane strain, respect- ively 8 . Equations (9) give the value of r C for predefined boundary conditions, hence the material behaviour could be different. r K r K tt c IC c IC , = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 1 2 1 6 22 ππ (9) where K IC is the fracture toughness and t is the tensile strength. 3 MATERIALS AND METHODS The studied material was kind of high-performance concrete (HPC) that was designed with the intent to produce subtle elements. The maximum size of the aggregate was chosen to be 8 mm. The aggregates were composed of natural sand 0/4 mm and crushed high- quality granite 4/8 mm. Portland cement CEM I 42.5 R was used with three mineral admixtures. The first, was metakaolin, with strong pozzolanic properties. 18 The second and third admixture were chosen to reach synergy in ternary binders based on experiments, see, for example. 19 Generally, the binder consist 81 % of CEM I 42.5 R, 9.5 % of metakaolin, 7.5 % of granulated blast- furnace slag (GBFS) and 2.5 % of limestone. The water-to-binder ratio was w/b = 0.22 with drinkable water. A polycarboxylate-based superplasticizer was selected based on its compatibility with cement. The concrete was mixed in a volume of 0.7 m 3 and poured into moulds. 3.1 Mechanical characteristics of the HPC The concrete was tested at ages of 1 d and 28 d, some other characteristics – especially non-destructive – were tested in a later age. Due to mixing the concrete in an industrial mixer, the mechanical properties, especially the compressive strengths, were lower compared to the laboratory tests. The measured mechanical properties from the tests were the 150-mm-cube compressive strength was 65.2±1.4 MPa at the age of 24 h and 106.2±2.5 MPa at the age of 28 d, the flexural strength at the age of 28 d was 9.3±0.9 MPa 20 and static modulus of elasticity measured on cylinders 150 mm × 300 mm was 41.0±0.6 GPa. 21 This is in accordance with Carras- quillo’s et al. formula: 21 E = 3320 f c 12 / + 6900 [MPa] (10) where E is the Young’s modulus and f c is the compres- sive strength. The Young’s modulus was obtained from Equation (10) by using f c = 106.2 MPa is E = 40.91 GPa. However, the measured/calculated values of E were lower than ex- pected with respect to compressive strength according to 3 – an absence of the coarse aggregates could be a reason. The indirect tensile strength on an unnotched Brazilian disc test 4 was measured using three samples. During the indirect tensile-strength test, the sample is attached bet- ween two load stripes and loaded with a speed of the upper support equal to 0.025 mm/s. The maximum load at the fracture was measured. The data were evaluated by following Equation (11): f P BD t c = 2 π [MPa] (11) where P C is the maximum compressive load, D is the diameter of the disc and B is the disc’s thickness. The introduced mechanical parameters (f c,cube , f ct , E, f t )ofthe HPC are listed with a standard deviation in Table 1. Table 1: Mechanical parameters with a standard deviation of the tested concrete at 28 days. Compressive strength f c,cube (MPa) 106.2 ± 2.5 Young’s modulus E (GPa) 41.0 ± 0.6 Flexural strength f ct (MPa) 9.3 ± 0.9 Indirect tensile strength f t (MPa) 6.43 ± 0.3 3.4 Fracture mechanical parameters for the normal mode Firstly, the fracture mechanical properties (FMPs) were tested on a notched beam with dimensions of P. Miarka et al.: MIXED-MODE FRACTURE ANALYSIS IN HIGH-PERFORMANCE CONCRETE USING A BRAZILIAN ... Materiali in tehnologije / Materials and technology 53 (2019) 2, 233–238 235 80 mm × 80 mm × 480 mm, in accordance with Kari- haloo’s and Nallathambi’s effective crack model 22 and afterwards on a BDCN specimen. The FMPs measured from the beams were: modulus of elasticity E = 46.5±1.2 GPa, which is similar to the values from the mechanical measurement, see Table 1, effective crack length a ef = 15.5±2.8 mm, effective fracture toughness K ICe = 1.74±0.18 MPam 1/2 and fracture energy G C = 65±11 Jm –2 . 3.4 Method of measurement of fracture mechanical parameters for mixed mode The machine for the BDCN tests has a maximum loading capacity of 200 kN, and the speed of the induced displacement of the upper support was equal to 0.025 mm/s. The BDCN specimens were prepared from standardized cylinders with diameter D = 150 mm, normally used for testing material for compressive strength. The initial notches were prepared with a water-jet cutter; this technique provides a straight through notch. The initial notch length was selected as 2a = 60 mm (to have a length of the ligament area five times the size of the maximum aggregate in front of each crack tip). The BDCN specimens with an average rela- tive notch length a/R = 0.4 were inclined against the loading positions under the selected angles = {0°; 5°; 10°; 15°; 20°; 25.25°}. Table 2 gives an overview of the mean values of the specimen dimensions. Table 2: Dimensions of the tested BDCN specimens Specimen no. Inclina- tion angle (°) Diameter D (mm) Thickness B (mm) Notch length 2a (mm) a/R (-) 6_2_02 0 149.09 29.43 59.70 0.400 6_2_01 0 149.15 29.99 59.44 0.399 6_2_05 5 149.23 28.35 59.91 0.401 6_2_10 10 149.32 28.48 59.27 0.396 6_2_11 10 149.01 27.57 60.13 0.403 6_2_08 15 149.18 28.09 60.06 0.403 6_2_09 15 149.28 28.70 59.96 0.402 6_2_06 20 149.21 28.33 60.01 0.402 6_2_07 20 149.12 28.45 60.03 0.403 6_2_03 25.2 149.18 28.45 59.81 0.400 6_2_04 25.2 149.23 28.96 59.93 0.402 4 MIXED-MODE RESISTANCE CURVES AND DISSCUSION The evaluated value of the fracture toughness from the BDCN specimen test is K IC,BDCN = 1.106±0.06 MPa m 1/2 . This result is lower by 45% than the K IC measured on notched beams geometry and the difference is cause by the geometry effect. This geometry effect was recently discussed for lime stone, 23 for C 50/60, 16 for normal strength concrete (NSC) and high-strength concrete (HSC). 24 To evaluate the fracture resistance curve of the HPC under the mixed I/II, BDCN specimens were tested with various inclination angles . The measured values of the fracture force Pc and the corresponding evaluated values of SIFs for mode I – K I and for mode II – K II are pre- sented in Figure 2. The fracture resistance for both modes is expressed with the ratio K I /K IC and K II /K IC . This ratio is obtained from Equation (8) by dividing the whole expression by K I and K II , respectively. Figure 3 gives an overview for the mixed mode I/II fracture resistance of the HPC for various critical distances r C . The critical distance used for the evaluation were: MTS – r C = 0 mm, GMTS plane strain – r C = 1.56 mm, GMTS plane stress – r C = 4.67 mm and maximum fine aggregate size – r C = 4 mm. From Figure 3 it is clear that the r C for the plane- strain boundary condition gives reliable results, where the K I is dominant in the fracture process, i.e., r C = 1.56 mm, while r C for a maximum aggregate size of 4 mm gives reliable results, when the K II is dominant. Recently, it was shown by researchers in 16 that the r C for C 50/60 could also be used for plane-strain boundary P. Miarka et al.: MIXED-MODE FRACTURE ANALYSIS IN HIGH-PERFORMANCE CONCRETE USING A BRAZILIAN ... 236 Materiali in tehnologije / Materials and technology 53 (2019) 2, 233–238 Figure 3: Mixed-mode I/II fracture-toughness diagram for the relative crack length a/R = 0.4 Figure 2: Fracture forces and values of SIFs for selected angles used for the BDCN measurement for a/R = 0.4 conditions in whole fracture process, i.e., r C = 1.559 mm. In the work 17 , the r C was 1.38 mm and 2.54 mm for the low-strength mortar and concrete, respectively. The cri- tical distance for two types of limestone was presented in 23 and r C varies from 2.6 mm to 5.2 mm. The experimental work performed on the Brazilian disc with a central notch is verified by a numerical simu- lation using a concrete damaged plasticity model. 25 5 CONCLUSIONS In this contribution a general overview of the mecha- nical and fracture properties of HPC are presented and discussed. The main focus was given to the fracture resistance under the mixed mode I/II by using the GMTS criterion. The influence of various critical distances r C was studied, and it can be concluded that the r C for the plane-strain boundary condition r C = 1.56 mm gives more reasonable results for K I dominance, i.e., for the ratio K I /K IC from 0.6 to 1.2. However, where the K II is dominant, i.e., for a K I /K IC ratio lower than 0.4, r C should be used with respect to the maximum aggregate size, i.e., 4 mm. 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