Evaluation of fracture response of Silesian granite specimens via Effective Crack Model Approach and finite element analysis

ScienceDirect Available online at www.sciencedirect.com Procedia Structural Integrity 23 (2019) 487–492 2452-3216 © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the scientific committee of the ICMSMF organizers 10.1016/j.prostr.2020.01.133 10.1016/j.prostr.2020.01.133 2452-3216 © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ) Peer-review under responsibility of the scientific committee of the ICMSMF organizers Available online at www.sciencedirect.com ScienceDirect Structural Integrity Procedia 00 (2019) 000–000 www.elsevier.com/locate/procedia 2452-3216 © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the scientific committee of the IC MSMF organizers. 9th International Conference on Materials Structure and Micromechanics of Fracture Evaluation of fracture response of Silesian granite specimens via Effective Crack Model approach and finite element analysis Lucie Malíkováa*, Leona Vavrob, Martin Vavrob, Zbyněk Keršnera aBrno University of Technology, Faculty of Civil Engineering, Veveří 331/95, 602 00 Brno, Czech Republic bThe Czech Academy of Sciences, Institute of Geonics, Studentská 1768, 708 00 Ostrava-Poruba, Czech Republic Abstract The paper is focused on the advanced evaluation of fracture tests of selected rock specimens. Assessment of fracture response of chevron notch cylindrical specimens of Silesian granite subjected to three-point bending test was performed. The effective crack model is used and academic software chevroncylinder (author Petr Frantík) based on the finite element method is applied for the analyses. Based on the described procedures, estimation of maximum values of the effective fracture toughness are established and discussed. © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the scientific committee of the IC MSMF organizers. Keywords: Granite; Fracture test; Chevron notch; Load–displacement diagram; Effective crack model 1. Introduction It is more and more evident that the procedures suggested in the well-known and commonly used linear elastic fracture mechanics (LEFM) cannot be used when the typical large zone exists ahead of the crack tip (where more complex non-linear fracture processes occurs). Such a phenomenon is typical for elastic-plastic or quasi-brittle materials. Rocks generally exhibit quasi-brittle behavior, see for instance Bažant and Planas (1998) or Shah et al. (1995). Thus, non-linear fracture models involving the cohesive nature of the crack propagation are necessary when * Corresponding author. Tel.: +420-541-147-381. E-mail address: [email protected] Available online at www.sciencedirect.com ScienceDirect Structural Integrity Procedia 00 (2019) 000–000 www.elsevier.com/locate/procedia 2452-3216 © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the scientific committee of the IC MSMF organizers. 9th International Conference on Materials Structure and Micromechanics of Fracture Evaluation of fracture response of Silesian granite specimens via Effective Crack Model approach and finite element analysis Lucie Malíkováa*, Leona Vavrob, Martin Vavrob, Zbyněk Keršnera aBrno University of Technology, Faculty of Civil Engineering, Veveří 331/95, 602 00 Brno, Czech Republic bThe Czech Academy of Sciences, Institute of Geonics, Studentská 1768, 708 00 Ostrava-Poruba, Czech Republic Abstract The paper is focused on the advanced evaluation of fracture tests of selected rock specimens. Assessment of fracture response of chevron notch cylindrical specimens of Silesian granite subjected to three-point bending test was performed. The effective crack model is used and academic software chevroncylinder (author Petr Frantík) based on the finite element method is applied for the analyses. Based on the described procedures, estimation of maximum values of the effective fracture toughness are established and discussed. © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the scientific committee of the IC MSMF organizers. Keywords: Granite; Fracture test; Chevron notch; Load–displacement diagram; Effective crack model 1. Introduction It is more and more evident that the procedures suggested in the well-known and commonly used linear elastic fracture mechanics (LEFM) cannot be used when the typical large zone exists ahead of the crack tip (where more complex non-linear fracture processes occurs). Such a phenomenon is typical for elastic-plastic or quasi-brittle materials. Rocks generally exhibit quasi-brittle behavior, see for instance Bažant and Planas (1998) or Shah et al. (1995). Thus, non-linear fracture models involving the cohesive nature of the crack propagation are necessary when * Corresponding author. Tel.: +420-541-147-381. E-mail address: [email protected] 488 Lucie Malíková et al. / Procedia Structural Integrity 23 (2019) 487–492 2 Lucie Malíková et al. / Structural Integrity Procedia 00 (2019) 000–000 the fracture response of the material is assessed, often the fracture energy and/or other softening parameters are utilized (Vavro et al. 2019). Fracture tests performed on rock material specimens exhibit several inconveniences that have to be dealt with. In this paper, the chevron bend (CB) specimen test was performed in order to estimate the range of the mode I fracture toughness. The CB test represents one of the methods used for testing the fracture properties of rocks (Ouchterlony 1988). Nevertheless, because a fracture process zone arises in rocks ahead of the crack tip, the initial notch length a0 seems not to be the enough representative parameter for evaluation of the fracture toughness. Therefore, a non-linear Effective Crack Model (ECM) is applied in this paper to estimate the maximum value of the fracture toughness involving a strongly non-linear fracture response of the rock. Calculations are carried out by means of the finite element method in the chevroncylinder academic software programmed in Java at the Brno University of Technology by Petr Frantík. 2. Material under the study – Silesian granite from the Černá Voda-Nový lom quarry The petrographic description and basic physical and mechanical properties of so-called Silesian granite from the Černá Voda-Nový lom quarry (ca 10 km N from the city of Jeseník) is introduced in this chapter. This rock was represented by light grey to grey, medium-grained biotite granite, typical in holocrystalline, equigranular, hypautomorphic to panxenomorphic granitic texture and massive structure (Fig. 1). Fig. 1. Macroscopic and microscopic view of so-called Silesian granite from the Černá Voda-Nový lom quarry with well visible equigranular texture and massive structure (left – polished surface of tested rock specimen, right – polished thin section, transmitted light, crossed polarizers). Mineral composition of the granite is relatively simple, felsic rock components are formed by quartz, K-feldspars and plagioclase, biotite is the basic mafic mineral. Quartz grains (ca 30%) are xenomorphic in crystal shape and ocassionally show undulose extinction. The average size (Md) of quarz is 0.68 mm, maximal grain size (Mmax) achieves 0.75 mm. Alkali feldspars (ca 40%, Md = 0.69 mm, Mmax to 1.0 mm) are represented by xenomorphic, sometimes perthitic ortoclase and xenomorphic to hypautomorphic microcline. Plagioclase grains (ca 25%, Md = 0.52 mm) are usually hypautomorphic and most frequently correspond to oligoclase. Intergrowths of plagioclase and quartz (myrmekite) are relatively frequent. Biotite comprise about 5% of the rock volume and occurs in the form of brown to red-brown hypidiomorphic flakes (Md = 0.38 mm, Mmax = 0.9 mm) with typical strong pleochroism and very rare chloritization. Accessory minerals include zircon, titanite, apatite, magnetite and rare allanite. Basic physical and mechanical properties of studied granite is presented in Table 1. Lucie Malíková et al. / Procedia Structural Integrity 23 (2019) 487–492 489 Lucie Malíková et al. / Structural Integrity Procedia 00 (2019) 000–000 3 Table 1. Basic physical and mechanical properties of so-called Silesian granite from the Černá Voda-Nový lom quarry according to various authors (n.d. not determined). Property Rybařík (1994) Koutník et al. (2015) Vavro (2015) Specific (mineralogical) density [kg/m3] 2680 2650 2680 Bulk density [kg/m3] 2610 2610 2620 Total porosity [%] 1.0 1.5 2.5 Water absorption capacity by weight [%] 0.54 0.41 0.31 Uniaxial compressive strength (dry sample) [MPa] 170 240 185 Flexural strength (dry sample) [MPa] 14.5 19.9 n.d. 3. Methodology In this paper, the chevron bend (CB) specimen tests were performed in order to estimate the maximum value of the mode I fracture toughness of the Silesian granite by means of the Effective Crack Model (ECM). The CB test represents one of the methods (for short overview see e.g. Wei et al. 2017) used in rock mechanics for testing the fracture properties of rocks. Together with the short rod (SR) test, cracked chevron notched Brazilian disc (CCNBD) test and notched semi-circular bending (SCB) test, the CB test has been suggested by the International Society for Rock Mechanics (Ouchterlony 1988, Fowell 1995, Kuruppu et al. 2014). Following equation for calculation of the mode I fracture toughness (LEFM) in the CB test is applied (Ouchterlony 1988): 𝐾𝐾𝐼𝐼𝐼𝐼 =𝐴𝐴𝑚𝑚𝑚𝑚𝑚𝑚𝐹𝐹𝑚𝑚𝑚𝑚𝑚𝑚/𝐷𝐷1.5 where 𝐴𝐴𝑚𝑚𝑚𝑚𝑚𝑚 = [1.835+7.15𝑚𝑚0 𝐷𝐷+9.85(𝑚𝑚0 𝐷𝐷)2]𝑆𝑆 𝐷𝐷 (1) In Eq. 1, Fmax is the maximum load, D is the diameter of the specimen, S is the span between the supports and a0 represents the length of the initial notch, see the schema in Fig. 2. Fig. 2. Schema of the chevron notch test and its cross-section with the typical dimensions. Nevertheless, when this relation is utilized, the values of the fracture toughness obtained are underestimated in comparison to the real values. Therefore, a non-linear ECM is introduced in this paper to estimate the maximum value of fracture toughness. 3.1. Effective Crack Model ECM is a non-linear model following the basic idea that the fracture toughness estimation relates to the length of the notch a0, see for instance Karihaloo (1995). Nevertheless, the fracture process zone changes the conditions ahead of the crack tip and it is spoken about the effective crack length that is responsible for the quasi-brittle fracture 490 Lucie Malíková et al. / Procedia Structural Integrity 23 (2019) 487–492 4 Lucie Malíková et al. / Structural Integrity Procedia 00 (2019) 000–000 behavior and can be used for calculation of the effective fracture toughness. The idea is based on the theory that the effective crack length represents such a crack length that would correspond to the same stiffness if the ideally linear elastic material was considered. 3.2. Searching for the effective crack length via finite element method In order to be able to find the effective crack length, experiments on real specimens had to be carried out. Three granite specimens have been chosen for the study presented in this paper. From CB tests, load–displacements and/or load–CMOD (Crack Mouth Opening Displacement) curves have been obtained and the important values of the maximum loading force and corresponding CMOD can be together with basic dimensions of the specimens found in Tab. 2. Table 2. Basic dimensions of the Silesian granite specimens considered in this study. Parameter 11964/17 11964/19 11964/20 Diameter of the cylinder, D [mm] 49.5 49.3 49.3 Depth of the initial notch, a 0 [mm] 7.12 6.95 7.30 Span between the supports, S [mm] 163 163 163 Maximum loading force, F max [N] 2531.3 2547.4 2565.7 CMOD when F max occurs, CMOD Fmax [mm] 0.032035 0.027891 0.025517 The data from the measurements have been used as inputs for finite element (FE) simulations when the bending of the chevron notch specimen has been modelled. From FE simulations, dependences between the loading force and the CMOD have been obtained for several crack lengths a, see Fig. 2. Then, the loading force obtained from FE simulations for a = 0 was replaced by the value Fmax obtained experimentally and similarly the other values of the loading forces were recalculated according to the same ratio for increasing crack. The value of the CMOD for the zero crack was calculated from the initial stiffness obtained from the load-displacement experimental curve. Additionally, the CMOD values for other crack lengths could be calculated from the same ratio between CMOD from FE simulations and CMOD from experiment. Thus, a new load–CMOD diagram was obtained taking into account the idea described at the beginning about the equality of the stiffness of the real quasi-brittle specimen with an initial notch a0 and a virtual ideally elastic specimen with an effective crack length aeff. To find the value of aeff, the intersection point between the recalculated load-CMOD diagram and the line representing the stiffness when the Fmax is reached in the experiment is found and the corresponding effective crack length is calculated from the CMODeff value, see Fig. 3. More details about the approach can be found for instance in Halfar (2018) or Halfar et al. (2018). Lucie Malíková et al. / Procedia Structural Integrity 23 (2019) 487–492 491 Lucie Malíková et al. / Structural Integrity Procedia 00 (2019) 000–000 5 Fig. 3. Parts of F–CMOD diagram obtained for the granite specimen 11964/20 via the effective crack model: the intersection point brings the value of the CMODeff that can be recalculated to the aeff value. When the effective crack length is known, Eq. 1 for calculation of the maximum value of the fracture toughness can be applied in its modified form considering the effective crack length aeff as aeff = a0 + a: 𝐾𝐾𝐼𝐼𝐼𝐼,𝑒𝑒𝑒𝑒𝑒𝑒 =𝐴𝐴𝑚𝑚𝑚𝑚𝑚𝑚,𝑒𝑒𝑒𝑒𝑒𝑒𝐹𝐹𝑚𝑚𝑚𝑚𝑚𝑚/𝐷𝐷1.5 where 𝐴𝐴𝑚𝑚𝑚𝑚𝑚𝑚,𝑒𝑒𝑒𝑒𝑒𝑒 =[1.835+7.15𝑚𝑚𝑒𝑒𝑒𝑒𝑒𝑒 𝐷𝐷+9.85(𝑚𝑚𝑒𝑒𝑒𝑒𝑒𝑒 𝐷𝐷)2]𝑆𝑆 𝐷𝐷 (2) 4. Results and discussion Based on the combination of the experimental results and numerical simulations, the effective crack model was utilized to calculate the value of the effective fracture toughness that corresponds better to the non-linear fracture response observed on the Silesian granite and rocks generally. The results obtained can be seen in Tab. 3 together with several statistical values of the parameters calculated and it is proved that the effective fracture toughness calculated via the ECM exhibit much more higher values than the linear elastic fracture mechanics approach predicts. Table 3. Values of the effective crack length and effective fracture toughness obtained on the three specimens from Silesian granite. Parameter 11964/17 11964/19 11964/20 Arithmetic mean Standard deviation Depth of the initial notch, a 0 [mm] 7.12 6.95 7.30 7.12 0.14 Effective crack length, a eff [mm] 24.84 24.15 22.47 23.82 1.00 Ratio a eff /a 0 [%] 348.88 347.48 307.78 334.71 19.05 Fracture toughness, K IC [MPa.m1/2] 2.32 2.34 2.41 2.36 0.04 Effective fracture toughness, K Ic,eff [MPa.m1/2] 5.98 5.93 5.53 5.81 0.20 Ratio K Ic,eff /K Ic [%] 257.68 253.43 229.58 246.89 12.36 492 Lucie Malíková et al. / Procedia Structural Integrity 23 (2019) 487–492 6 Lucie Malíková et al. / Structural Integrity Procedia 00 (2019) 000–000 5. Conclusions The effective crack model (ECM) has been applied on three cylindrical specimens from Silesian granite that had been supplied by a chevron-type notch and loaded in three-point bending. The dimension of the real specimens as well as the load–CMOD curves measured during experiment have been used as inputs for finite element simulations. The non-linear ECM model enabled calculating of the maximum value of the effective fracture toughness and their comparison to the values of the classical fracture toughness obtained from purely elastic solution. It has been proved that when the non-linear ECM was applied on quasi-brittle materials (selected rock in the case of this paper), the more precise values of the fracture properties can be obtained contrary to the underestimated values of the fracture toughness calculated based on the classical linear elastic fracture mechanics. The aim of future works is to derive the appropriate function of geometry, which would allow determination of effective fracture toughness KIc,eff. Acknowledgements The financial support provided by the Czech Science Foundation (project No. 18-12289Y) is gratefully acknowledged. Presented work was also supported by the Ministry of Education, Youth and Sports of the Czech Republic (project No. NPU I – LO1406) and by the project for the long-term conceptual development of research organisations (RVO: 68145535). References Bažant, Z.P., Planas, J. 1998. Fracture and size effect in concrete and other quasibrittle materials. CRC Press, Boca Raton and London, 640 p. Fowel, RJ. 1995. Suggested method for determining mode I fracture toughness using cracked chevron notched Brazilian disc (CCNBD) specimens. International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts 32, 1, 57–64. Halfar, P. 2018. Evaluation of fracture test of selected core-drilled cylindrical specimen. Diploma Thesis. Brno University of Technology, 65 p. Halfar, P., Frantík, P., Šimonová, H., Daněk, P., Keršner, Z., Vavro, L., Vavro, M. 2018. Effective crack model application for analysis of chevronnotched cylindrical specimens fracture response. Structural Reliability and Modelling in Mechanics, Ostrava (in Czech). Karihaloo, B.L. 1995. Fracture mechanics and structural concrete. Longman Scientific an Technical, Essex. Koutník, P., Antoš, P., Hájková P., Martinec, P., Antošová, B., Ryšánek, P., Pacina, J., Šancer, J., Ščučka, J., Brůna, V., 2015. Decorative stones of Bohemia, Moravia and Czech Silesia. Jan Evangelista Purkyně University in Ústí na Labem, 495 p. (in Czech). Kuruppu, MD., Obara, Y., Ayatollahi, MR., Chong, KP., Funatsu, T. 2014. ISRM-Suggested method for determining the mode I static fracture toughness using semi-circular bend specimen. Rock Mechanics and Rock Engineering 47, 1, 267–274. Ouchterlony, F., 1988. Suggested methods for determining the fracture toughness of rock. International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts 25, 2, 71–96. Rybařík, V., 1994. Noble building and sculptural stone of the Czech Republic. Industrial Secondary School of Stonework and Sculpture in Hořice, 218 p. (in Czech). Shah, S.P., Swartz, S.E. and Ouyang, C., 1995. Fracture mechanics of structural concrete: applications of fracture mechanics to concrete, rock, and other quasi-brittle materials. John Wiley & Sons, Inc., New York, 588 p. Vavro, L. 2015. Fracture toughness of rock and its determination for geomechanical evaluation of rock and rock mass. Ph.D. Thesis. VSBTechnical University of Ostrava, 93 p. Vavro, L., Malíková, L., Frantík, P., Kubeš, P., Keršner, Z., Vavro, M. 2019. An advanced assessment of mechanical fracture parameters of sandstones depending on the internal rock texture features. Acta Geodynamica et Geomaterialia 16, 2, 157–168. Wei, MD., Dai, F., Xu, NW Zhao, T., Liu, Y. 2017. An experimental and theoretical assessment of semi-circular bend specimens with chevron and straight-through notches for mode I fracture toughness of rocks. International Journal of Rock Mechanics and Mining Sciences 99, 28–38.