Document Type : Review Paper
Authors
Department of Mining Engineering, Amirkabir University of Technology, Tehran, Iran.
Abstract
The structure's response to the region's prevailing loading conditions guides the engineers in estimating the resilience of the structural materials and their reinforcement. One of the main concerns in designing rock structures is paying attention to the size effect phenomenon. The size effect influences the nominal strength, brittleness, load capacity, stress intensity factor, the characteristics of the fracture process zone at the crack tip, and the way and path of crack propagation. Therefore, studying the size effect law will make a guideline for correct decision-making, design, and implementation of efficient support systems. As a comprehensive review, this work investigates specimen size effect on the rock's mechanical and fracture properties. With a comprehensive look at this issue, it explains the essential points that help the engineers design rock structures. During the investigations carried out in this work, it is shown that the specimen size affects the fracture and mechanical properties of the rock. The severity of this phenomenon depends on various factors such as the brittleness index, the shape of the notch or crack length, and the size of the particles that create the rock. In concrete, it depends on the additive boosting materials in the concrete.
Keywords
[1]. Saxena, A. (1998). Nonlinear fracture mechanics for engineers. CRC press. pp. 23-49.
[2]. Goss, G., Alessi, D., Allen, D., Gehman, J., Brisbois, J., Kletke, S and Prosser, C. (2015). Unconventional wastewater management: a comparative review and analysis of hydraulic fracturing wastewater management practices across four North American basins. Canadian Water Network.
[3]. Li, X., Zhang, P., He, Z., Huang, Z., Cheng, M., and Guo, L. (2017). Identification of geological structure which induced heavy water and mud inrush in tunnel excavation: A case study on Lingjiao tunnel. Tunnelling and Underground Space Technology, 69, 203-208.
[4]. Lang, E.M., and Kovacs, Z. (2001). Size effect on shear strength measured by the ASTM method. Forest products journal, 51 (3): 49.
[5]. Al-Shayea, N.A. (2005). Crack propagation trajectories for rocks under mixed mode I–II fracture. Engineering Geology, 81 (1): 84-97.
[6]. Whittaker, B.N., Singh, R.N., and Sun, G. (1992). Rock fracture mechanics. Principles, design and applications. Pp. 40-86.
[7]. Bažant, Z.P. (2000). Size effect. International Journal of Solids and Structures, 37 (1-2): 69-80.
[8]. Bažant, Z.P., and Planas, J. (2019). Fracture and size effect in concrete and other quasibrittle materials. Routledge.
[9]. Ghamgosar, M., and Erarslan, N. (2016). Experimental and numerical studies on development of fracture process zone (FPZ) in rocks under cyclic and static loadings. Rock Mechanics and Rock Engineering, 49 (3): 893-908.
[10]. Anderson, T.L. (2017). Fracture mechanics: fundamentals and applications. CRC press.
[11]. Dehestani, A., Hosseini, M., and Beydokhti, A.T. (2020). Effect of wetting–drying cycles on mode I and mode II fracture toughness of cement mortar and concrete. Theoretical and Applied Fracture Mechanics, 106, 102448.
[12]. Harvey, C.M., Wood, J.D., Wang, S., and Watson, A. (2014). A novel method for the partition of mixed-mode fractures in 2D elastic laminated unidirectional composite beams. Composite Structures, 116, 589-594.
[13]. Alkılıçgil, Ç. (2010). Development of specimen geometries for mode I fracture toughness testing with disc type rock specimens.
[14]. Franklin, J. A., Zongqi, S., Atkinson, B.K., Meredith, P.C., Rummel, F., Mueller, W., and Bobrov, G. F. (1988). Suggested methods for determining the fracture toughness of rock. International Journal of Rock Mechanics and Mining & Geomechanics Abstracts, 25 (2).
[15]. Zhou, Y.X., Xia, K.W., Li, X.B., Li, H.B., Ma, G. W., Zhao, J. and Dai, F. (2011). Suggested methods for determining the dynamic strength parameters and mode-I fracture toughness of rock materials. In The ISRM Suggested Methods for Rock Characterization, Testing and Monitoring: 2007-2014 (pp. 35-44). Springer, Cham.
[16]. Kuruppu, M.D., Obara, Y., Ayatollahi, M.R., Chong, K.P., and 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.
[17]. Bažant, Z.P. (1999). Size effect on structural strength: a review. Archive of applied Mechanics, 69, 703-725.
[18]. Carpinteri, A., Chiaia, B., and Ferro, G. (1995). Size effects on nominal tensile strength of concrete structures: multifractality of material ligaments and dimensional transition from order to disorder. Materials and Structures, 28, 311-317.
[19]. Bažant, Z.P., and Yu, Q. (2009). Universal size effect law and effect of crack depth on quasi-brittle structure strength. Journal of engineering mechanics, 135 (2): 78-84.
[20]. Kumar, S., and Barai, S.V. (2010). Size-effect prediction from the double-K fracture model for notched concrete beam. International Journal of Damage Mechanics, 19 (4): 473-497.
[21]. Carpinteri, A. (2021). Size-scale transition from ductile to brittle failure. In Fracture and Complexity: One Century since Griffith’s Milestone (pp. 343-405). Dordrecht: Springer Netherlands.
[22]. Xie, Q., Liu, X., Li, S., Du, k., Gong, F., and Li, X. (2022). Prediction of Mode I Fracture Toughness of Shale Specimens by Different Fracture Theories Considering Size Effect. Rock Mechanics and Rock Engineering, 1-18.
[23]. Nguyen, D.L., Kim, D.J., Ryu, G.S., and Koh, K.T. (2013). Size effect on flexural behavior of ultra-high-performance hybrid fiber-reinforced concrete. Composites Part B: Engineering, 45 (1): 1104-1116.
[24]. Bahaldini, M., (2016). Investigating the effect of scale on tensile strength in the Brazilian test, Journal of Engineering geology, 319-342.
[25]. Kong, X., Liu, Q., and Lu, H. (2021). Effects of rock specimen size on mechanical properties in laboratory testing. Journal of Geotechnical and Geoenvironmental Engineering, 147 (5): 04021013.
[26]. Usol’Tseva, O.M., and Tsoi, P.A. (2021, April). The influence of size effect on strength and deformation characteristics of different types of rock samples. In IOP Conference Series: Earth and Environmental Science (Vol. 720, No. 1, p. 012089). IOP Publishing.
[27]. Zi, G., Kim, J., and Bazant, Z.P. (2014). Size effect on biaxial flexural strength of concrete. ACI Materials Journal, 111 (3): 319-326.
[28] Lin, H., Xiong, W., and Yan, Q. (2015). Three-dimensional effect of tensile strength in the standard Brazilian test considering contact length. ASTM International.
[29]. Kaklis, K.N., Maurigiannakis, S.P., Agioutantis, Z.G., Stathogianni, F.K., and Steiakakis, E.K. (2015, July). Experimental investigation of the size effect on the mechanical properties on two natural building stones. In Proceedings of 8th GRACM international congress on computational mechanics, Volos, Greece.
[30]. Al-Rkaby, A.H., and Alafandi, Z.M. (2015). Size effect on the unconfined compressive strength and Modulus of elasticity of limestone rock. Electronic Journal of Geotechnical Engineering, 20 (12): 1393-1401.
[31]. Khosravi, A., Simon, R., and Rivard, P. (2017). The shape effect on the morphology of the fracture surface induced by the Brazilian test. International Journal of Rock Mechanics and Mining Sciences, 93, 201-209.
[32]. Li, Y. (2018). A review of shear and tensile strengths of the Malan Loess in China. Engineering Geology, 236, 4-10.
[33]. Fládr, J., and Bílý, P. (2018). Specimen size effect on compressive and flexural strength of high-strength fibre-reinforced concrete containing coarse aggregate. Composites Part B: Engineering, 138, 77-86.
[34]. Yu, Q., Zhu, W., Ranjith, P.G., and Shao, S. (2018). Numerical simulation and interpretation of the grain size effect on rock strength. Geomechanics and Geophysics for Geo-energy and Geo-resources, 4, 157-173.
[35]. Zhai, H., Masoumi, H., Zoorabadi, M., and Canbulat, I. (2020). Size-dependent behaviour of weak intact rocks. Rock Mechanics and Rock Engineering, 53(8): 3563-3587.
[36]. Han, Q., Gao, Y., and Zhang, Y. (2021). Experimental Study of Size Effects on the Deformation Strength and Failure Characteristics of Hard Rocks under True Triaxial Compression. Advances in Civil Engineering, 2021.
[37]. Zhao, Y., Mishra, B., Shi, Q., and Zhao, G. (2022). A Size-Dependent Bonded-Particle Model for Transversely Isotropic Rock and its Application in Studying the Size Effect of Shale. Geotechnical and Geological Engineering, 1-17.
[38]. Cao, X., Ren, Y.C., Qian, K., Fu, F., Deng, X.F., and Zhang, W.J. (2022). Size effect on flexural behavior of ultra-high-performance concrete beams with different reinforcement. In Structures Vol. 41, 969-981.
[39]. Asadi, P., and Fakhimi, A. (2023). Bonded particle modeling of grain size effect on tensile and compressive strengths of rock under static and dynamic loading. Advanced Powder Technology, 34 (5): 104013.
[40]. Aliha, M.R.M., Bahmani, A., and Akhondi, S. (2016). Mixed mode fracture toughness testing of PMMA with different three-point bend type specimens. European Journal of Mechanics-A/Solids, 58, 148-162.
[41]. Hu, X., and Duan, K. (2008). Size effect and quasi-brittle fracture: the role of FPZ. International Journal of Fracture, 154, 3-14.
[42]. Tarokh, A., Makhnenko, R.Y., Fakhimi, A., and Labuz, J.F. (2017). Scaling of the fracture process zone in rock. International Journal of Fracture, 204 (2): 191-204.
[43]. Fakhimi, A., and Tarokh, A. (2013). Process zone and size effect in fracture testing of rock. International Journal of Rock Mechanics and Mining Sciences, 60, 95-102.
[44]. Zhang, S., Wang, L., and Gao, M. (2019). Experimental investigation of the size effect of the mode I static fracture toughness of limestone. Advances in Civil Engineering, 2019.
[45]. Ayatollahi, M.R., and Akbardoost, J. (2013). Size effects in mode II brittle fracture of rocks. Engineering Fracture Mechanics, 112, 165-180.
[46]. Bidadi, J., Akbardoost, J., and Aliha, M.R.M. (2020). Thickness effect on the mode III fracture resistance and fracture path of rock using ENDB specimens. Fatigue & Fracture of Engineering Materials & Structures, 43 (2): 277-291.
[47]. Aliha, M.R.M., Ayatollahi, M.R., Smith, D.J., and Pavier, M.J. (2010). Geometry and size effects on fracture trajectory in a limestone rock under mixed mode loading. Engineering Fracture Mechanics, 77 (11): 2200-2212.
[48]. Khoramishad, H., Akbardoost, J., and Ayatollahi, M. R. (2014). Size effects on parameters of cohesive zone model in mode I fracture of limestone. International journal of damage mechanics, 23 (4): 588-605.
[49]. Ayatollahi, M.R., and Akbardoost, J. (2014). Size and geometry effects on rock fracture toughness: mode I fracture. Rock mechanics and rock engineering, 47 (2): 677-687.
[50]. Aliha, M.R.M., Bahmani, A., and Akhondi, S. (2015). Determination of mode III fracture toughness for different materials using a new designed test configuration. Materials & Design, 86, 863-871.
[51]. Akbardoost, J. (2016). Investigating the effect of specimen size on the dynamic fracture of cracked rocks. Iranian Journal of Mining Engineering, 11 (31): 91-101.
[52]. Ayatollahi, M.R., Akbardoost, J., and Berto, F. (2016). Size effects on mixed-mode fracture behavior of polygranular graphite. Carbon, 103, 394-403.
[53]. Akbardoost, J., and Rastin, A. (2016). Scaling effect on the mixed-mode fracture path of rock materials. Физическая мезомеханика, 19 (4): 82-91.
[54]. Jeong, S.S., Nakamura, K., Yoshioka, S., Obara, Y., and Kataoka, M. (2017). Fracture toughness of granite measured using micro to macro scale specimens. Procedia engineering, 191, 761-767.
[55]. Ghasemi-Ghalebahman, A., Akbardoost, J., and Ghaffari, Y. (2017). Evaluation of size effect on mixed-mode fracture behavior of epoxy/silica nanocomposites. The Journal of Strain Analysis for Engineering Design, 52(4): 239-248.
[56]. Moazzami, M., Ayatollahi, M.R., and Akhavan-Safar, A. (2020). Assessment of the fracture process zone in rocks using digital image correlation technique: The role of mode-mixity, size, geometry and material. International Journal of Damage Mechanics, 29 (4): 646-666.
[57]. Ayatollahi, M.R., Ajdani, A., Akhavan-Safar, A., and da Silva, L.F.M. (2019). Effect of notch length and pre-crack size on mode II fracture energy of brittle adhesives. Engineering Fracture Mechanics, 212, 123-135.
[58]. Zhang, S., An, D., Zhang, X., Yu, B., and Wang, H. (2021). Research on size effect of fracture toughness of sandstone using the center-cracked circular disc samples. Engineering Fracture Mechanics, 251, 107777.
[59]. Muñoz-Ibáñez, A., Delgado-Martín, J., and Juncosa-Rivera, R. (2021). Size effect and other effects on mode I fracture toughness using two testing methods. International Journal of Rock Mechanics and Mining Sciences, 143, 104785.
[60]. Nejati, M., Bahrami, B., Ayatollahi, M.R., and Driesner, T. (2021). On the anisotropy of shear fracture toughness in rocks. Theoretical and Applied Fracture Mechanics, 113, 102946.
[61]. Zhang, J., Liu, X., Wu, Z., Yu, H., and Fang, Q. (2022). Fracture properties of steel fiber reinforced concrete: Size effect study via mesoscale modelling approach. Engineering Fracture Mechanics, 260, 108193.
[62]. Wang, Y., Hu, S., and Sun, X. (2022). Experimental investigation on the elastic modulus and fracture properties of basalt fiber–reinforced fly ash geopolymer concrete. Construction and Building Materials, 338, 127570.
[63]. Xie, Q., Liu, X., Li, S., Du, K., Gong, F., and Li, X. (2022). Prediction of mode I fracture toughness of shale specimens by different fracture theories considering size effect. Rock Mechanics and Rock Engineering, 55 (11): 7289-7306.
[64]. Pijaudier-Cabot, G., Hajimohammadi, A., Nouailletas, O., La Borderie, C., Padin, A., and Mathieu, J.P. (2022). Determination of the fracture energy of rocks from size effect tests: Application to shales and carbonate rocks. Engineering Fracture Mechanics, 271, 108630.
[65]. Li, C., Yang, D., Xie, H., Ren, L., and Wang, J. (2023). Size effect of fracture characteristics for anisotropic quasi-brittle geomaterials. International Journal of Mining Science and Technology, 33 (2): 201-213.
[66]. Davis, P., Bryski, E., and Kirane, K. (2023, January). Size Effect Analysis and Characterization of Quasibrittle Fracture of Sandstone Rocks. In Fracture, Fatigue, Failure and Damage Evolution, Volume 3: Proceedings of the 2022 Annual Conference on Experimental and Applied Mechanics (pp. 1-11). Cham: Springer International Publishing.
[67]. Karimi, H.R., Aliha, M.R.M., Ebneabbasi, P., Salehi, S.M., Khedri, E., and Haghighatpour, P.J. (2023). Mode I and mode II fracture toughness and fracture energy of cement concrete containing different percentages of coarse and fine recycled tire rubber granules. Theoretical and Applied Fracture Mechanics, 123, 103722.
[68]. Daneshfar, M., Hassani, A., Aliha, M.R.M., and Sadowski, T. (2023). Assessment of the Specimen Size Effect on the Fracture Energy of Macro-Synthetic-Fiber-Reinforced Concrete. Materials, 16 (2): 673.
[69]. Ayatollahi, M.R., and Akbardoost, J. (2012). Size effects on fracture toughness of quasi-brittle materials–A new approach. Engineering Fracture Mechanics, 92, 89-100.
[70]. Hosseini, M., Dolatshahi, A.R., and Ramezani, E. (2022). Effect of Acid Rain on Physical and Mechanical Properties of Concrete Containing Micro-Silica and Limestone Powder. Journal of Mining and Environment, 13(1): 185-200.
[71]. Hosseini, M., Dolatshahi, A., and Ramezani, E. (2022). Effect of sodium sulfate and chlorine ion on the properties of concrete containing micro-silica, concrete containing zeolite powder and its comparison with ordinary concrete. Journal of Mining Engineering, 17 (57): 55-67.
[72]. Sun, W., Wang, L., and Wang, Y. (2017). Mechanical properties of rock materials with related to mineralogical characteristics and grain size through experimental investigation: a comprehensive review. Frontiers of Structural and Civil Engineering, 11, 322-328.
[73]. Hemmati, A., Ghafoori, M., Moomivand, H., and Lashkaripour, G.R. (2020). The effect of mineralogy and textural characteristics on the strength of crystalline igneous rocks using image-based textural quantification. Engineering Geology, 266, 105467.
[74]. Alneasan, M., and Behnia, M. (2021). An experimental investigation on tensile fracturing of brittle rocks by considering the effect of grain size and mineralogical composition. International Journal of Rock Mechanics and Mining Sciences, 137, 104570.
[75]. Zhang, C., and Wang, M. (2022). A critical review of breakthrough pressure for tight rocks and relevant factors. Journal of Natural Gas Science and Engineering, 104456.
[76]. Torabi, A.R., Jabbari, M., and Akbardoost, J. (2020). Scaling effects on notch fracture toughness of graphite specimens under mode I loading. Engineering Fracture Mechanics, 235, 107153.
[77]. Aslani, M., Akbardoost, J., and Berto, F. (2022). A stress-based approach for considering the size effect on the mixed mode fracture behavior of rock. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 236(21): 10841-10851.
[78]. Shah, K.S., Mohd Hashim, M.H., Rehman, H., and Ariffin, K.S. (2021). Application of Stochastic Simulation in Assessing Effect of Particle Morphology on Fracture Characteristics of Sandstone. Journal of Mining and Environment, 12(4): 969-986.
[79]. Golewski, G.L., and Sadowski, T. (2016). Macroscopic evaluation of fracture processes in fly ash concrete. In Solid State Phenomena (Vol. 254, pp. 188-193). Trans Tech Publications Ltd.
[80]. Golewski, G.L., and Sadowski, T. (2016). A study of mode III fracture toughness in young and mature concrete with fly ash additive. In Solid State Phenomena (Vol. 254, pp. 120-125). Trans Tech Publications Ltd.
[81]. Golewski, G.L. (2023). Combined Effect of Coal Fly Ash (CFA) and Nanosilica (nS) on the Strength Parameters and Microstructural Properties of Eco-Friendly Concrete. Energies, 16 (1): 452.
[82]. Golewski, G.L. (2022). Comparative measurements of fracture toughgness combined with visual analysis of cracks propagation using the DIC technique of concretes based on cement matrix with a highly diversified composition. Theoretical and Applied Fracture Mechanics, 121, 103553.
[83]. Helwany, S. (2007). Applied soil mechanics with ABAQUS applications. John Wiley & Sons. pp 152-187.
[84]. Michał, S., and Andrzej, W. (2015). Calibration of the CDP model parameters in Abaqus. World Congr Adv Struct Eng Mech (ASEM 15): Incheon Korea.
[85]. Xia, Y., Zhou, H., Zhang, C., He, S., Gao, Y., and Wang, P. (2022). The evaluation of rock brittleness and its application: a review study. European Journal of Environmental and Civil Engineering, 26 (1): 239-279.
[86]. Meng, F., Wong, L.N.Y., and Zhou, H. (2021). Rock brittleness indices and their applications to different fields of rock engineering: A review. Journal of Rock Mechanics and Geotechnical Engineering, 13 (1): 221-247.
[87]. Kang, G., Ning, Y., Chen, P., Pang, S., and Shao, Y. (2022). Comprehensive simulations of rock fracturing with pre-existing cracks by the numerical manifold method. Acta Geotechnica, 17 (3): 857-876.
[88]. Ning, Y., Lu, Q., and Liu, X. (2023). Fracturing failure simulations of rock discs with pre-existing cracks by numerical manifold method. Engineering Analysis with Boundary Elements, 148, 389-400.
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