ENERGY BALANCE SIZE EFFECT MODEL OF COMPRESSIVE STRENGTH FOR QUASI-BRITTLE MATERIALS

Liu Xiaoyu; Yang Zheng; Zhang Huimei

doi:10.6052/0459-1879-21-460

Volume 54 Issue 6

May 2022

Turn off MathJax

Article Contents

Article Navigation > Chinese Journal of Theoretical and Applied Mechanics > 2022 > 54(6): 1613-1629

Liu Xiaoyu, Yang Zheng, Zhang Huimei. Energy balance size effect model of compressive strength for quasi-brittle materials. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(6): 1613-1629 doi: 10.6052/0459-1879-21-460

Citation:

Liu Xiaoyu, Yang Zheng, Zhang Huimei. Energy balance size effect model of compressive strength for quasi-brittle materials. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(6): 1613-1629 doi: 10.6052/0459-1879-21-460

Citation:

PDF( 4008 KB)

ENERGY BALANCE SIZE EFFECT MODEL OF COMPRESSIVE STRENGTH FOR QUASI-BRITTLE MATERIALS

doi: 10.6052/0459-1879-21-460

*.
College of Sciences, Xi'an University of Science and Technology, Xi’an 710600, China
†.
School of Human Settlement and Civil Engineering, Xi'an Jiaotong University, Xi’an 710049, China

Received Date: 2021-09-09
Accepted Date: 2022-04-02

Available Online: 2022-04-02

Publish Date: 2022-06-18

Abstract

Abstract

To address the problem of existing size effect models cannot reflect complete size effect of compressive strength and internal mechanism of quasi-brittle materials. In this paper, by analyzing energy input, storage, global-local energy dissipation during the failure process of quasi-brittle materials under uniaxial compression, mechanical model and bilinear nominal and true stress-strain curves are established to reflect global and local damage and describe the above energy evolution process respectively. On this basis, the expressions of input energy, stored elastic energy and global-local energy dissipation are determined when the nominal stress is the maximum. Finally, size effect model of compressive strength is established with energy balance principle. The energy balance size effect model of compressive strength can completely reflect the size effect of nominal compressive strength, namely with the increase of sample size, nominal compressive strength is the real strength when sample size is less than or equal to the size of local damage zone, and then gradually decreases, eventually tends to the elastic ultimate strength when the sample size approaches infinity. High to diameter ratio together with sample diameter can be taken into account in the energy balance size effect model of compressive strength. Its parameters, which can reflect the effect of real strength, elastic ultimate strength, nonlinear of nominal damage modulus, size and direction of local damage zone on nominal compressive strength size effect of quasi-brittle materials, have clear physical meaning. Experiment and numerical simulation data of various materials are utilized to validate and evaluate the energy balance size effect model of compressive strength and existing size effect models. The results indicate that the energy balance size effect model of compressive strength can well describe the nonlinear variation and internal mechanism of size effect of experiment and numerical simulation, and compared with the existing size effect models, its total average error is the smallest and less than 5%.
- quasi-brittle material,
- compressive strength,
- energy balance,
- size effect

FullText(HTML)

References(67)

References

[1]	Hu X, Guan J, Wang Y, et al. Comparison of boundary and size effect models based on new developments. Engineering Fracture Mechanics, 2017, 175: 146-167 doi: 10.1016/j.engfracmech.2017.02.005
[2]	Vu CC, Weiss J, Ple O, et al. Revisiting statistical size effects on compressive failure of heterogeneous materials, with a special focus on concrete. Journal of the Mechanics and Physics of Solids, 2018, 121: 47-70 doi: 10.1016/j.jmps.2018.07.022
[3]	朱其志, 闵中泽, 王岩岩等. 粉砂岩三轴压缩试验中的试样尺寸效应研究. 岩石力学与工程学报, 2019, 38(S2): 3296-3303 (Zhu Qizhi, Min Zhongze, Wang Yanyan, et a. Study on the size effect of silty sandstone samples under conventional triaxial compression. Chinese Journal of Rock Mechanics and Engineering, 2019, 38(S2): 3296-3303 (in Chinese) Zhu Qizhi, Min Zhongze, Wang Yanyan, et a. Study on the size effect of silty sandstone samples under conventional triaxial compression. Chinese Journal of Rock Mechanics and Engineering, 2019, 38(S2): 3296-3303 (in Chinese))
[4]	Huang H, Shen J, Chen Q, et al. Estimation of REV for fractured rock masses based on Geological Strength Index. International Journal of Rock Mechanics and Mining Sciences, 2020, 126: 104179 doi: 10.1016/j.ijrmms.2019.104179
[5]	白以龙. 非均匀介质的通常和反常尺寸效应. 北京: 国家自然科学基金委结题报告, 2009 Bai Yilong. Normal and anomalous size effects of heterogeneous media. Beijing: Final Report to NNSFC, 2009 (in Chinese)
[6]	张后全, 徐建峰, 贺永年等. 灰岩单轴压缩实验室尺度效应研究. 岩石力学与工程学报, 2012, 31(S2): 3491-3496 (Zhang Houquan, Xu Jianfeng, He Yongnian, et a. Study of laboratory scale effect of limestone under uniaxial compression. Chinese Journal of Rock Mechanics and Engineering, 2012, 31(S2): 3491-3496 (in Chinese) Zhang Houquan, Xu Jianfeng, He Yongnian, et a. Study of laboratory scale effect of limestone under uniaxial compression. Chinese Journal of Rock Mechanics and Engineering, 2012, 31(S2): 3491-3496 (in Chinese))
[7]	程爱平, 戴顺意, 张玉山等. 胶结充填体损伤演化尺寸效应研究. 岩石力学与工程学报, 2019, 38(S1): 3054-3059 (Cheng Aiping, Dai Shunyi, Zhang Yushan, et a. Study on size effect of damage evolution of cemented backfill. Chinese Journal of Rock Mechanics and Engineering, 2019, 38(S1): 3054-3059 (in Chinese) Cheng Aiping, Dai Shunyi, Zhang Yushan, et a. Study on size effect of damage evolution of cemented backfill. Chinese Journal of Rock Mechanics and Engineering, 2019, 38(S1): 3054-3059 (in Chinese))
[8]	Zhai H, Masoumi H, Zoorabadi M, et al. Size-dependent behaviour of weak intact rocks. Rock Mechanics and Rock Engineering, 2020, 53(8): 3563-3587 doi: 10.1007/s00603-020-02117-z
[9]	伍法权, 乔磊, 管圣功等. 小尺寸岩样单轴压缩试验尺寸效应研究. 岩石力学与工程学报, 2021, 40(5): 856-873 (Wu Faquan, Qiao lei, Guan Shenggong, et al. Study on size effect of uniaxial compression tests of small size rock samples. Chinese Journal of Rock Mechanics and Engineering, 2021, 40(5): 856-873 (in Chinese) Wu Faquan, Qiao lei, Guan shenggong, et a. Study on size effect of uniaxial compression tests of small size rock samples. Chinese Journal of Rock Mechanics and Engineering, 2021, 40(5): 856-873 (in Chinese))
[10]	Bazant ZP. Size effect in blunt fracture: concrete, rock, metal. Journal of Engineering Mechanics, 1984, 110(4): 518-535 doi: 10.1061/(ASCE)0733-9399(1984)110:4(518)
[11]	Bazant ZP, Le JL. Probabilistic Mechanics of Quasibrittle Structures: Strength, Lifetime, and Size Effect. New York: Cambridge University Press, 2017
[12]	Carpinteri A. Fractal nature of material microstructure and size effects on apparent mechanical properties. Mechanics of Materials, 1994, 18(2): 89-101 doi: 10.1016/0167-6636(94)00008-5
[13]	Hu X, Wittmann F. Size effect on toughness induced by crack close to free surface. Engineering Fracture Mechanics, 2000, 65(2): 209-221
[14]	Wang Y, Hu X. Determination of tensile strength and fracture toughness of granite using notched three-point-bend samples. Rock Mechanics and Rock Engineering, 2017, 50(1): 17-28 doi: 10.1007/s00603-016-1098-6
[15]	Lockner DA, Byerlee JD, Kuksenko V, et al. Quasi-static fault growth and shear fracture energy in granite. Nature, 1991, 350(6313): 39-42 doi: 10.1038/350039a0
[16]	Renard F, McBeck JA, Kandula N, et al. Volumetric and shear processes in crystalline rock approaching faulting. Proceedings of the National Academy of Sciences of the United States of America, 2019, 116(33): 16234-16239 doi: 10.1073/pnas.1902994116
[17]	Kim JK. Size effect in concrete specimens with dissimilar initial cracks. Magazine of Concrete Research, 1990, 42(153): 233-238 doi: 10.1680/macr.1990.42.153.233
[18]	Chen P, Liu C, Wang Y. Size effect on peak axial strain and stress-strain behavior of concrete subjected to axial compression. Construction and Building Materials, 2018, 188: 645-655 doi: 10.1016/j.conbuildmat.2018.08.072
[19]	Jin L, Yu W, Du X, et al. Meso-scale modelling of the size effect on dynamic compressive failure of concrete under different strain rates. International Journal of Impact Engineering, 2019, 125: 1-12 doi: 10.1016/j.ijimpeng.2018.10.011
[20]	Bazant ZP, Xiang Y. Size effect in compression fracture: splitting crack band propagation. Journal of Engineering Mechanics ASCE, 1997, 123(2): 162-172 doi: 10.1061/(ASCE)0733-9399(1997)123:2(162)
[21]	张颖. 混凝土细观数值模拟及尺寸效应研究. [硕士论文]. 沈阳: 哈尔滨工业大学, 2017 Zhang Ying. Research on meso-scale modeling and size effect of concrete. [Master Thesis]. Shenyang: Harbin Institute of Technology, 2017 (in Chinese))
[22]	Horii H, Nemat-Nasser S. Compression-induced microcrack growth in brittle solids: axial splitting and shear failure. Journal of Geophysical Research Solid Earth, 1985, 90(B4): 3105-3125 doi: 10.1029/JB090iB04p03105
[23]	Bobet A, Einstein HH. Fracture coalescence in rock-type materials under uniaxial and biaxial compression. International Journal of Rock Mechanics and Mining Sciences, 1998, 35(7): 863-888 doi: 10.1016/S0148-9062(98)00005-9
[24]	Yang SQ, Jiang YZ, Xu WY, et al. Experimental investigation on strength and failure behavior of pre-cracked marble under conventional triaxial compression. International Journal of Solids and Structures, 2008, 45(17): 4796-4819 doi: 10.1016/j.ijsolstr.2008.04.023
[25]	Cheng Y, Wong LNY. Microscopic characterization of tensile and shear fracturing in progressive failure in marble. Journal of Geophysical Research, 2017, 123(1): 204-225
[26]	Bazant ZP, Lin FBB, Lippmann H. Fracture energy release and size effect in borehole breakout. International Journal for Numerical and Analytical Methods iIn Geomechanics, 1993, 17(1): 1-14 doi: 10.1002/nag.1610170102
[27]	Muciaccia G, Rosati G, Luzio GD. Compressive failure and size effect in plain concrete cylindrical specimens. Construction and Building Materials, 2017, 137: 185-194 doi: 10.1016/j.conbuildmat.2017.01.057
[28]	苏捷, 刘伟, 史才军等. 超高性能混凝土立方体抗压强度尺寸效应. 硅酸盐学报, 2021, 49(2): 305-311 (Su Jie, Liu Wei, Shi Caijun, et al. Scale effect of cubic compressive strength of ultra-high performance concrete. Journal of the Chinese Ceramic Society, 2021, 49(2): 305-311 (in Chinese) Su Jie, Liu Wei, Shi Caijun, et al. Scale effect of cubic compressive strength of ultra-high performance concrete. Journal of the Chinese Ceramic Society, 2021, 49(02): 305-311 (in Chinese))
[29]	杨高升. 岩石材料尺寸效应的研究. [硕士论文]. 兰州: 兰州大学, 2018 Yang Gaosheng. Study on size effect of rock material. [Master Thesis]. Lanzhou: Lanzhou University, 2018 (in Chinese))
[30]	Weiss J, Girard L, Gimbert F, et al. Finite statistical size effects on compressive strength. Proceedings of the National Academy of Sciences of the United States of America, 2014, 111(17): 6231-6236 doi: 10.1073/pnas.1403500111
[31]	Bieniawski ZT. The effect of specimen size on compressive strength of coal. International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts, 1968, 5(4): 325-335 doi: 10.1016/0148-9062(68)90004-1
[32]	Hoek E, Brown ET. Underground Excavations in Rock. London: Institution of Mining and Metallurgy, 1980
[33]	刘宝琛, 张家生, 杜奇中等. 岩石抗压强度的尺寸效应. 岩石力学与工程学报, 1998, 17(6): 611-614 (Liu Baochen, Zhang Jiasheng, Du Qizhong, et al. Size effect of rock compressive strength. Chinese Journal of Rock Mechanics and Engineering, 1998, 17(6): 611-614 (in Chinese) doi: 10.3321/j.issn:1000-6915.1998.06.001 Liu B, Zhang J, Du Q, et al. Size effect of rock compressive strength. Chinese Journal of Rock Mechanics and Engineering, 1998, 17(6): 611-614 (in Chinese)). doi: 10.3321/j.issn:1000-6915.1998.06.001
[34]	杨圣奇, 苏承东, 徐卫亚. 岩石材料尺寸效应的试验和理论研究. 工程力学, 2005, 22(4): 112-118 (Yang Shengqi, Su Chengdong, Xu Weiya, et al. Experimental and theoretical study of size effect of rock material. Engineering Mechanics, 2005, 22(4): 112-118 (in Chinese) doi: 10.3969/j.issn.1000-4750.2005.04.022 Yang Shengqi, Su Chengdong, Xu Weiya, et al. Experimental and theoretical study of size effect of rock material. Engineering Mechanics, 2005, 22(4): 112-118 (in Chinese)) doi: 10.3969/j.issn.1000-4750.2005.04.022
[35]	Darlington WJ, Ranjith PG, Choi SK. The effect of specimen size on strength and other properties in laboratory testing of rock and rock-like cementitious brittle materials. Rock Mechanics Rock Engineering, 2011, 44(5): 513 doi: 10.1007/s00603-011-0161-6
[36]	Song H, Jiang Y, Elsworth D, et al. Scale effects and strength anisotropy in coal. International Journal of Coal Geology, 2018, 195: 37-46 doi: 10.1016/j.coal.2018.05.006
[37]	Lei WS, Qian G, Yu Z, et al. Statistical size scaling of compressive strength of quasi-brittle materials incorporating specimen length-to-diameter ratio effect. Theoretical and Applied Fracture Mechanics, 2019, 104: 102345
[38]	Delonca A, Vallejos JA. Incorporating scale effect into a failure criterion for predicting stress-induced overbreak around excavations. International Journal of Rock Mechanics and Mining Sciences, 2020, 127: 104213
[39]	Kong X, Liu Q, Lu H. Effects of rock specimen size on mechanical properties in laboratory testing. Journal of Geotechnical and Geoenvironmental Engineering, 2021, 147(5): 04021013 doi: 10.1061/(ASCE)GT.1943-5606.0002478
[40]	谢和平, 鞠杨, 黎立云. 基于能量耗散与释放原理的岩石强度与整体破坏准则. 岩石力学与工程学报, 2005, 24(17): 3003-3010 (Xie Heping, Ju Yang, Li Liyun. Criteria for strength and structural failure of rocks based on energy dissipation and energy release principles. Chinese Journal of Rock Mechanics and Engineering, 2005, 24(17): 3003-3010 (in Chinese) doi: 10.3321/j.issn:1000-6915.2005.17.001 Xie Heping, JU Yang, Li Liyun. Criteria for strength and structural failure of rocks based on energy dissipation and energy release principles. Chinese Journal of Rock Mechanics and Engineering, 2005, 24(17): 3003-3010 (in Chinese)) doi: 10.3321/j.issn:1000-6915.2005.17.001
[41]	张志镇. 岩石变形破坏过程中的能量演化机制. [博士论文]. 北京: 中国矿业大学, 2013 Zhang Zhizhen. Energy evolution mechanism during rock deformation and failure. [PhD Thesis]. Beijing: China University of Mining and Technology, 2013 (in Chinese))
[42]	Huang D, Li Y. Conversion of strain energy in triaxial unloading tests on marble. International Journal of Rock Mechanics and Mining Sciences, 2014, 66: 160-168 doi: 10.1016/j.ijrmms.2013.12.001
[43]	宫凤强, 罗松, 李夕兵等. 红砂岩张拉破坏过程中的线性储能和耗能规律. 岩石力学与工程学报, 2018, 37(2): 352-363 (Gong Fengqiang, Luo Song, Li Xibing, et al. Linear energy storage and dissipation rule of red sandstone materials during the tensile failure process. Chinese Journal of Rock Mechanics and Engineering, 2018, 37(2): 352-363 (in Chinese) Gong Feng-qiang, Luo Song, Li Xi-bing, et al. Linear energy storage and dissipation rule of red sandstone materials during the tensile failure process. Chinese Journal of Rock Mechanics and Engineering, 2018, 37(2): 352-363 (in Chinese))
[44]	Schubnel A, Thompson BD, Fortin J, et al. Fluid induced rupture experiment on Fontainebleau sandstone: Premonitory activity, rupture propagation, and aftershocks. Geophysical Research Letters, 2007, 34(19): L19307
[45]	Zhao XG, Cai M, Wang J, et al. Damage stress and acoustic emission characteristics of the Beishan granite. International Journal of Rock Mechanics and Mining Sciences, 2013, 64: 258-269
[46]	Cartwright-Taylor A, Main IG, Butler IB, et al. Catastrophic failure: How and when? insights from 4-D in situ X-ray microtomography. Journal of Geophysical Research Solid Earth, 2020, 125(8): 1-30
[47]	Healy D, Jones RR, Holdsworth RE. New insights into the development of brittle shear fractures from a 3-D numerical model of microcrack interaction. Earth and Planetary Science Letters, 2006, 249(1): 14-28
[48]	Shang J. Rupture of veined granite in polyaxial compression: insights from three-dimensional discrete element method modeling. Journal of Geophysical Research Solid Earth, 2020, 125(2): 1-25
[49]	Xu T, Fu TF, Heap MJ, et al. Mesoscopic damage and fracturing of heterogeneous brittle rocks based on three-dimensional polycrystalline discrete element method. Rock Mechanics and Rock Engineering, 2020, 53: 5389-5409
[50]	尤明庆, 苏承东. 大理岩试样的长度对单轴压缩试验的影响. 岩石力学与工程学报, 2004, 23(22): 3754-3760 (You Mingqing, Su Chengdong. Effect of length of fine and coarse crystal marble specimens on uniaxial compression tests. Chinese Journal of Rock Mechanics and Engineering, 2004, 23(22): 3754-3760 (in Chinese) doi: 10.3321/j.issn:1000-6915.2004.22.003 You Mingqing, Su Chengdong. Effect of length of fine and coarse crystal marble specimens on uniaxial compression tests. Chinese Journal of Rock Mechanics and Engineering, 2004, 23(22): 3754-3760 (in Chinese)) doi: 10.3321/j.issn:1000-6915.2004.22.003
[51]	杨圣奇, 徐卫亚, 苏承东. 考虑尺寸效应的岩石损伤统计本构模型研究. 岩石力学与工程学报, 2005(24): 4484-4490 (Yang Shengqi, Xu Weiya, Su Chengdong, et al. Study on statistical damage constitutive model of rock considering scale effect. Chinese Journal of Rock Mechanics and Engineering, 2005(24): 4484-4490 (in Chinese) doi: 10.3321/j.issn:1000-6915.2005.24.014 Yang Shengqi, Xu Weiya, Su Chengdong, et al. Study on statistical damage constitutive model of rock considering scale effect. Chinese Journal of Rock Mechanics and Engineering, 2005, (24): 4484-4490 (in Chinese)) doi: 10.3321/j.issn:1000-6915.2005.24.014
[52]	闵明. 北山花岗岩高温力学特性试验研究. [博士论文]. 徐州: 中国矿业大学, 2019 Min Ming. Experimental study on high temperature mechanical properties of beishan granite. [PhD Thesis]. Xuzhou: China University of Mining and Technology, 2019 (in Chinese))
[53]	Wong LNY, Meng F, Guo T, et al. The role of load control modes in determination of mechanical properties of granite. Rock Mechanics and Rock Engineering, 2020, 53(2): 539-522 doi: 10.1007/s00603-019-01924-3
[54]	Quinones J, Arzúa J, Alejano L, et al. Analysis of size effects on the geomechanical parameters of intact granite samples under unconfined conditions. Acta Geotechnica, 2017, 12(6): 1229-1242 doi: 10.1007/s11440-017-0531-7
[55]	Liang Z, Wu N, Li Y, et al. Numerical study on anisotropy of the representative elementary volume of strength and deformability of jointed rock masses. Rock Mechanics and Rock Engineering, 2019, 52(11): 4387-4402 doi: 10.1007/s00603-019-01859-9
[56]	吕龙龙, 宋丽, 廖红建等. 红层软岩单轴抗压强度的尺寸效应. 长江科学院院报, 2016, 33(9): 78-82 (Lü Longlong, Song Li, Liao Hongjian, et al. Size effect on uniaxial compressive strength of red bed soft rock. Journal of Yangtze River Scientific Research Institute, 2016, 33(9): 78-82 (in Chinese) doi: 10.11988/ckyyb.20150567 Lu Longlong, Song Li, Liao Hongjian, et al. Size effect on uniaxial compressive strength of red bed soft rock. Journal of Yangtze River Scientific Research Institute, 2016, 33(9): 78-82 (in Chinese)) doi: 10.11988/ckyyb.20150567
[57]	Yu Q, Shan Z W, Li J, et al. Strong crystal size effect on deformation twinning. Nature, 2010, 463(7279): 335-338 doi: 10.1038/nature08692
[58]	Peng J, Wong LNY, Teh CI. A re-examination of slenderness ratio effect on rock strength: Insights from DEM grain-based modelling. Engineering Geology, 2018, 246: 245-254 doi: 10.1016/j.enggeo.2018.10.003
[59]	Scholtès L, Donzé FV, Khanal M. Scale effects on strength of geomaterials, case study: Coal. Journal of the Mechanics Physics of Solids, 2011, 59(5): 1131-1146 doi: 10.1016/j.jmps.2011.01.009
[60]	Zhang Q, Zhu H, Zhang L, et al. Study of scale effect on intact rock strength using particle flow modeling. International Journal of Rock Mechanics and Mining Sciences, 2011, 48(8): 1320-1328 doi: 10.1016/j.ijrmms.2011.09.016
[61]	刘丹, 黄曼, 洪陈杰等. 基于代表性取样的节理岩体抗压强度尺寸效应试验研究. 岩石力学与工程学报, 2021, 40(4): 766-776 (Liu Dan, Huang Man, Hong Chenjie, et al. Experimental study on size effect of compressive strength of jointed rock mass based on representative sampling. Chinese Journal of Rock Mechanics and Engineering, 2021, 40(4): 766-776 (in Chinese) Liu Dan, Huang Man, Hong Chen Jie, et al. Experimental study on size effect of compressive strength of jointed rock mass based on representative sampling. Chinese Journal of Rock Mechanics and Engineering, 2021, 40(04): 766-776 (in Chinese))
[62]	刘小宇, 杨政, 张慧梅. 不同高径比和试样尺寸准脆性材料抗压强度试验和数值模拟数据. https://www.researchgate.net/publication/354461441 Liu Xiaoyu, Yang Zheng, Zhang Huimei. Experiment and numerical simulation data of compressive strength for quasi-brittle materials with different height to diameter ratios and sample sizes. https://www.researchgate.net/publication/354461441 (in Chinese))
[63]	Hawkins A. Aspects of rock strength. Bulletin of Engineering Geology and the Environment, 1998, 57(1): 17-30 doi: 10.1007/s100640050017
[64]	Mogi K. Experimental Rock Mechanics. London: Taylor & Francis, 2006
[65]	梁正召, 张永彬, 唐世斌等. 岩体尺寸效应及其特征参数计算. 岩石力学与工程学报, 2013, 32(6): 1157-1166 (Liang Zhengzhao, Zhang Yongbin, Tang Shibin, et al. Size effect of rock messes and associated representative element properties. Chinese Journal of Rock Mechanics and Engineering, 2013, 32(6): 1157-1166 (in Chinese) doi: 10.3969/j.issn.1000-6915.2013.06.009 Liang Zhengzhao, Zhang Yongbin, Tang Shibin, et al. Size effect of rock messes and associated representative element properties. Chinese Journal of Rock Mechanics and Engineering, 2013, 32(6): 1157-1166 (in Chinese)) doi: 10.3969/j.issn.1000-6915.2013.06.009
[66]	刘洪雨. 基于应变梯度理论的混凝土塑性损伤模型及抗压强度尺寸效应研究. [硕士论文]. 北京: 北京交通大学, 2019 Liu Hongyu. Plastic damage model of concrete based on strain gradient theory and size effect of compressive strength. [Master Thesis]. Beijing: Beijing Jiaotong University, 2019 (in Chinese))
[67]	Tuncay E, Hasancebi N. The effect of length to diameter ratio of test specimens on the uniaxial compressive strength of rock. Bulletin of Engineering Geology and the Environment, 2009, 68(4): 491-497 doi: 10.1007/s10064-009-0227-9