岩石领域的纳米压痕技术应用

Application of nanoindentation technology in the field of rock materials

摘要

准确表征岩石和岩体的力学性质对抽水蓄能电站地下厂房、地下储气库及核废料处置库等工程的稳定性分析与评价具有重要意义。岩石作为矿物集合体，其强度和变形特性受矿物组成、胶结作用、界面形态等因素的共同影响。相较于宏观力学试验，纳米压痕技术所需试样尺寸小、流变试验时间短、易测得微观力学指标、可分离各矿物和胶结作用对岩石强度与变形的贡献，利于揭示岩石宏观力学行为的微观机理等优势，为岩石材料宏观-微观力学特性研究提供了新途径。首先介绍纳米压痕技术的基本原理及数据统计分析方法；其次分析矿物组成、结构、环境条件等影响因素下页岩、泥岩、花岗岩的纳米压痕力学特性，揭示了岩石的非均质性、各向异性、蠕变特性及尺寸效应；对比分析了5种均匀化理论分析方法、3种数值模拟方法对纳米力学指标跨尺度升级的适用性；最后，对纳米压痕技术在岩石领域的应用前景进行展望。

Abstract

Accurately characterizing the mechanical properties of rocks and rock masses is of great significance for the stability analysis and evaluation of projects such as underground powerhouses of pumped storage power stations, underground gas storage facilities, and nuclear waste repositories. As a mineral aggregate, the strength and deformation characteristics of rocks are jointly influenced by mineral composition, cementation, interface morphology, and other factors. Compared with macroscopic mechanical tests, nanoindentation technology offers advantages including a small sample size required, short rheological test duration, enabling easy measurement of microscopic mechanical indices, facilitating the separation of the contributions of individual minerals and cementation to rock strength and deformation, and supporting the elucidation of the microscopic mechanism underlying the macroscopic mechanical behaviors of rocks. It thus provides a novel approach for the study of macro-micro mechanical properties of rock materials. Firstly, the basic principles of nanoindentation technology and statistical data analysis methods are introduced. Secondly, the nanoindentation mechanical properties of shale, mudstone, and granite under the influence of factors such as mineral composition, structure, and environmental conditions are analyzed, thereby revealing the heterogeneity, anisotropy, creep characteristics, and size effect of rocks. The applicability of five homogenization theoretical methods and three numerical simulation methods for the cross-scale upscaling of nanomechanical indices is compared and analyzed. Finally, the application prospects of nanoindentation technology in the field of rock mechanics are outlined.

关键词

岩石 / 纳米压痕 / 弹性模量 / 断裂韧性 / 蠕变 / 层理 / 环境因素 / 尺度升级

Key words

rocks / nanoindentation / elastic modulus / fracture toughness / creep / bedding / environment factor / upscale

参考文献

[1] 兰恒星, 包含, 孙巍锋, 等. 岩体多尺度异质性及其力学行为[J]. 工程地质学报, 2022， 30（1）: 37-52.

[2] LAN Hengxing, BAO Han, SUN Weifeng, et al. Multi-scale heterogeneity of rock mass and its mechanical behavior[J]. Journal of engineering geology, 2022, 30(1): 37-52(in Chinese).

[3] WANG J F, DZIADKOWIEC J, LIU Y K, et al. Combining atomic force microscopy and nanoindentation helps characterizing in-situ mechanical properties of organic matter in shale[J]. International journal of coal geology, 2024, 281: 104406.

[4] CHAI Z Y, KANG T H, FENG G R. Effect of aqueous solution chemistry on the swelling of clayey rock[J]. Applied clay science, 2014, 93-94: 12-16.

[5] 迟广成, 伍月, 王海娇, 等. X射线荧光光谱分析技术在大理岩鉴定与分类中的应用[J]. 岩矿测试, 2018， 37（1）: 43-49.

[6] CHI Guangcheng, WU Yue, WANG Haijiao, et al. Application of X-ray fluorescence spectroscopy in identification and classification of marble[J]. Rock and mineral analysis, 2018, 37(1): 43-49(in Chinese).

[7] VAZQUEZ P, BENAVENTE D, MONTIEL D, et al. Mineralogical transformations in granitoids during heating at fire-related temperatures[J]. Applied sciences, 2022, 12(1): 188.

[8] BIAN K, PENG X C, ZHANG W, et al. 3D reconstruction of granite mesostructure based on electron backscatter diffraction tests[J]. Measurement, 2025, 251: 117282.

[9] 邬忠虎, 孟祥瑞, 蓝宝锋, 等. 基于CT试验的黔北凤冈地区牛蹄塘组含方解石脉页岩的力学行为研究[J]. 地学前缘, 2024， 31（5）: 117-129.

[10] WU Zhonghu, MENG Xiangrui, LAN Baofeng, et al. Mechanical behavior of calcite vein-bearing shale of the Niutitang formation in Fenggang area, northern Guizhou based on CT tests[J]. Earth science frontiers, 2024, 31(5): 117-129(in Chinese).

[11] 牛然, 蒋小伟, 闫永帅, 等. 基于核磁共振与波速测试的风化砂岩吸水性特征研究[J]. 水文地质工程地质, 2025， 52（2）: 63-71.

[12] NIU Ran, JIANG Xiaowei, YAN Yongshuai, et al. Characteristics of water absorption capacity of weathered sandstone based on nuclear magnetic resonance and wave velocity testing[J]. Hydrogeology & engineering geology, 2025, 52(2): 63-71(in Chinese).

[13] HU L B, WANG F Z, MENG Y X, et al. Inverse analysis of micro-mechanical parameters of rock-forming minerals based on nano-indentation technology[J]. Physica scripta, 2023, 98(5): 055003.

[14] LI C X, OSTADHASSAN M, ABARGHANI A, et al. Multi-scale evaluation of mechanical properties of the Bakken shale[J]. Journal of materials science, 2019, 54(3): 2133-2151.

[15] MIGHANI S, BERNABÉ Y, BOULENOUAR A, et al. Creep deformation in vaca muerta shale from nanoindentation to triaxial experiments[J]. Journal of geophysical research(solid earth), 2019, 124(8): 7842-7868.

[16] YU H Y, ZHANG Y H, LEBEDEV M, et al. Nanoscale geomechanical properties of Western Australian coal[J]. Journal of petroleum science and engineering, 2018, 162: 736-746.

[17] DOU Z H, GAO T Y, ZHAO Z H, et al. Effect of immersion duration on shear behavior of granite fractures[J]. Rock mechanics and rock engineering, 2021, 54(9): 4809-4823.

[18] STILWELL N A, TABOR D. Elastic recovery of conical indentations[J]. Proceedings of the Physical Society, 1961, 78(2): 169-179.

[19] LOUBET J L, GEORGES J M, MARCHESINI O, et al. Vickers indentation curves of magnesium oxide(MgO)[J]. Journal of tribology, 1984, 106(1): 43-48.

[20] DOERNER M F, NIX W D. A method for interpreting the data from depth-sensing indentation instruments[J]. Journal of materials research, 1986, 1(4): 601-609.

[21] OLIVER W C, PHARR G M. An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments[J]. Journal of materials research, 1992, 7(6): 1564-1583.

[22] CONSTANTINIDES G, RAVI CHANDRAN K S, ULM F J, et al. Grid indentation analysis of composite microstructure and mechanics: principles and validation[J]. Materials science and engineering: a, 2006, 430(1/2): 189-202.

[23] BOBKO C, ULM F J. The nano-mechanical morphology of shale[J]. Mechanics of materials, 2008, 40(4/5): 318-337.

[24] GATHIER B. Multiscale strength homogenization: application to shale nanoindentation[D]. Cambridge: Massachusetts Institute of Technology, 2008.

[25] 中华人民共和国国家质量监督检验检疫总局, 中国国家标准化管理委员会. 仪器化纳米压入试验方法通则: GB/T 22458-2008[S]. 北京: 中国标准出版社， 2008.

[26] OLIVER W C, PHARR G M. Measurement of hardness and elastic modulus by instrumented indentation: advances in understanding and refinements to methodology[J]. Journal of materials research, 2004, 19(1): 3-20.

[27] 张帆, 郭翰群, 赵建建, 等. 花岗岩微观力学性质试验研究[J]. 岩石力学与工程学报, 2017， 36（增刊2）: 3864-3872.

[28] ZHANG Fan, GUO Hanqun, ZHAO Jianjian, et al. Experimental study of micro-mechanical properties of granite[J]. Chinese journal of rock mechanics and engineering, 2017, 36(S2): 3864-3872(in Chinese).

[29] LAWN B R, EVANS A G, MARSHALL D B. Elastic/plastic indentation damage in ceramics: the median/radial crack system[J]. Journal of the American ceramic society, 1980, 63(9/10): 574-581.

[30] PALMQVIST S. A method to determine the toughness of brittle materials, especially hard metals[J]. Jernkontorets annaler, 1957, 141: 303-307.

[31] PHARR G M, HARDING D S, OLIVER W C. Measurement of fracture toughness in thin films and small volumes using nanoindentation methods[M]//NASTASI M, PARKIN D M, GLEITER H. Dordrecht: Springer Netherlands, 1993: 449-461.

[32] 张泰华, 杨业敏. 纳米硬度技术的发展和应用[J]. 力学进展, 2002， 32（3）: 349-364.

[33] ZHANG Taihua, YANG Yemin. Developments and applications of nano-hardness techniques[J]. Advances in mechanics, 2002, 32(3): 349-364(in Chinese).

[34] CHENG Y T, LI Z Y, CHENG C M. Scaling relationships for indentation measurements[J]. Philosophical magazine a, 2002, 82(10): 1821-1829.

[35] 何智海, 倪雅倩, 杜时贵, 等. 纳米压痕技术在岩石材料中的应用与研究进展[J]. 岩石力学与工程学报, 2022， 41（10）: 2045-2066.

[36] HE Zhihai, NI Yaqian, DU Shigui, et al. Application and research progress of nanoindentation technology in rock materials[J]. Chinese journal of rock mechanics and engineering, 2022, 41(10): 2045-2066(in Chinese).

[37] GANNEAU F P, CONSTANTINIDES G, ULM F J. Dual-indentation technique for the assessment of strength properties of cohesive-frictional materials[J]. International journal of solids and structures, 2006, 43(6): 1727-1745.

[38] LI M, CHEN W M, LIANG N G, et al. A numerical study of indentation using indenters of different geometry[J]. Journal of materials research, 2004, 19(1): 73-78.

[39] VANDAMME M, ULM F J. Nanogranular origin of concrete creep[J]. Proceedings of the National Academy of Sciences of the United States of America, 2009, 106(26): 10552-10557.

[40] VANDAMME M, ULM F J. Nanoindentation investigation of creep properties of calcium silicate hydrates[J]. Cement and concrete research, 2013, 52: 38-52.

[41] HACKNEY S A, AIFANTIS K E, TANGTRAKARN A, et al. Using the Kelvin-Voigt model for nanoindentation creep in Sn-C/PVDF nanocomposites[J]. Materials science and technology, 2012, 28(9/10): 1161-1166.

[42] SUDHARSHAN PHANI P, OLIVER W C, PHARR G M. On the measurement of power law creep parameters from instrumented indentation[J]. JOM, 2017, 69(11): 2229-2236.

[43] LUCAS B N, OLIVER W C. Indentation power-law creep of high-purity Indium[J]. Metallurgical and materials transactions a, 1999, 30(3): 601-610.

[44] MAIER V, DURST K, MUELLER J, et al. Nanoindentation strain-rate jump tests for determining the local strain-rate sensitivity in nanocrystalline Ni and ultrafine-grained Al[J]. Journal of materials research, 2011, 26(11): 1421-1430.

[45] MAYO M J, SIEGEL R W, NARAYANASAMY A, et al. Mechanical properties of nanophase TiO₂ as determined by nanoindentation [J]. Journal of materials research, 1990, 5(5): 1073-1082.

[46] SCHOLZ C H. Mechanism of creep in brittle rock[J]. Journal of geophysical research, 1968, 73(10): 3295-3302.

[47] PETHICA J B, OLIVER W C. Mechanical properties of nanometre volumes of material: use of the elastic response of small area indentations[J]. MRS online proceedings library, 1988, 130: 13-23.

[48] LI X D, BHUSHAN B. A review of nanoindentation continuous stiffness measurement technique and its applications[J]. Materials characterization, 2002, 48(1): 11-36.

[49] PEARSON K. Contributions to the mathematical theory of evolution[J]. Philosophical transactions of the Royal Society of London a, 1894, 185: 71-110.

[50] SCHWARZ G. Estimating the dimension of a model[J]. The annals of statistics, 1978, 6(2): 461-464.

[51] HRBEK V, KOUDELKOVÁ V. Grid indentation and statistic deconvolution: limitations and accuracy[J]. Key engineering materials, 2017, 731: 15-22.

[52] ORTIZ-MEMBRADO L, BENÍTEZ R, LLANES L, et al. High-speed nanoindentation data analysis of WC-based cemented carbides using gaussian mixture model clustering and skew-normal mixture: beyond gaussian deconvolution[J]. International journal of refractory metals and hard materials, 2025, 126: 106917.

[53] LIU Y W, LIU A, LIU S M, et al. Nano-scale mechanical properties of constituent minerals in shales investigated by combined nanoindentation statistical analyses and SEM-EDS-XRD techniques[J]. International journal of rock mechanics and mining sciences, 2022, 159: 105187.

[54] VEYTSKIN Y B, TAMMINA V K, BOBKO C P, et al. Micromechanical characterization of shales through nanoindentation and energy dispersive x-ray spectrometry[J]. Geomechanics for energy and the environment, 2017, 9: 21-35.

[55] XIE X H, DENG H C, HU L X, et al. Assessing the effect of oriented structure characteristics of laminated shale on its mechanical behaviour with the aid of nano-indentation and FE-SEM techniques[J]. International journal of rock mechanics and mining sciences, 2024, 173: 105625.

[56] MA T S, LIU K Y, SU X, et al. Investigation on the anisotropy of meso-mechanical properties of shale rock using micro-indentation[J]. Bulletin of engineering geology and the environment, 2024, 83(1): 29.

[57] ZHANG M, ZHANG G Q, CHEN L, et al. The discrete fracture process zone of shale on micrometer scale[J]. International journal of rock mechanics and mining sciences, 2024, 175: 105679.

[58] SHI X, JIANG S, WANG Z X, et al. Application of nanoindentation technology for characterizing the mechanical properties of shale before and after supercritical CO₂ fluid treatment [J]. Journal of CO₂ utilization , 2020, 37: 158-172.

[59] ZENG L P, AKHONDZADEH H, IQBAL M A, et al. Effect of fluid-shale interactions on shales micromechanics: nanoindentation experiments and interpretation from geochemical perspective[J]. Journal of natural gas science and engineering, 2022, 101: 104545.

[60] 罗攀登, 赵兵, 冯禹菲, 等. 水软化作用下页岩微观结构和力学性质演化特性[J]. 金属矿山, 2024（8）: 29-34.

[61] LUO Pandeng, ZHAO Bing, FENG Yufei, et al. Evolution characteristics of shale rock microstructure and mechanical properties under the action of water softening[J]. Metal mine, 2024(8): 29-34(in Chinese).

[62] LU Y H, LI Y C, WU Y K, et al. Characterization of shale softening by large volume-based nanoindentation[J]. Rock mechanics and rock engineering, 2020, 53(3): 1393-1409.

[63] XU J J, TANG X H, WANG Z Z, et al. Investigating the softening of weak interlayers during landslides using nanoindentation experiments and simulations[J]. Engineering geology, 2020, 277: 105801.

[64] WANG Y B, PORTER D L, NALEWAY S E, et al. Thermo-mechanical characterization of shale using nanoindentation[J]. Scientific reports, 2021, 11(1): 18864.

[65] MEMON K R, ALI M, AWAN F U R, et al. Influence of cryogenic liquid nitrogen cooling and thermal shocks on petro-physical and morphological characteristics of Eagle Ford shale[J]. Journal of natural gas science and engineering, 2021, 96: 104313.

[66] XU J J, ZHANG Y H, RUTQVIST J, et al. Thermally induced microcracks in granite and their effect on the macroscale mechanical behavior[J]. Journal of geophysical research(solid earth), 2023, 128(1): e2022JB024920.

[67] LUO S M, LU Y H, WU Y K, et al. Cross-scale characterization of the elasticity of shales: statistical nanoindentation and data analytics[J]. Journal of the mechanics and physics of solids, 2020, 140: 103945.

[68] SUN C L, LI G C, ELGHARIB GOMAH M, et al. Meso-scale mechanical properties of mudstone investigated by nanoindentation[J]. Engineering fracture mechanics, 2020, 238: 107245.

[69] AUVRAY C, ARNOLD G, ARMAND G. Experimental study of elastic properties of different constituents of partially saturated argillite using nano-indentation tests[J]. Engineering geology, 2015, 191: 61-70.

[70] VALDES C C, HEIDARI Z. Application of nanoindentation for uncertainty assessment of elastic properties in mudrocks from micro-to well-log scales[J]. Geophysics, 2017, 82(6): D327-D339.

[71] GRAHAM S P, APLIN A C, ROUAINIA M, et al. A micromechanical investigation of diagenetically-induced changes to the anisotropic elastic properties of calcareous mudstones[J]. Geomechanics and geophysics for geo-energy and geo-resources, 2023, 9(1): 107.

[72] ZHANG F, XIE S Y, HU D W, et al. Effect of water content and structural anisotropy on mechanical property of claystone[J]. Applied clay science, 2012, 69: 79-86.

[73] SUN C L, LI G C, MA Z Q, et al. Polymineralic mudstone exhibits anisotropy due to the preferential orientation of kaolinite particles[J]. Rock mechanics and rock engineering, 2025, 58(8): 9023-9045.

[74] 李桂臣, 李菁华, 孙元田, 等. 泥岩多尺度模型与水岩作用特性研究进展[J]. 煤炭学报, 2022， 47（3）: 1138-1154.

[75] LI Guichen, LI Jinghua, SUN Yuantian, et al. Advance of multi-scale study on both analytic models and water-rock interaction characteristics of mudstone[J]. Journal of China Coal Society, 2022, 47(3): 1138-1154(in Chinese).

[76] 栗涵洁, 刘其明, 智慧文, 等. 不同类型钻井液浸泡对泥岩微观结构及纳米压痕硬度的影响差异性[J]. 中外能源, 2024， 29（9）: 48-55.

[77] LI Hanjie, LIU Qiming, ZHI Huiwen, et al. Differences in effects of drilling fluid soaking on microstructure and nanoindentation hardness of mudstones[J]. Sino-global energy, 2024, 29(9): 48-55(in Chinese).

[78] AUVRAY C, LAFRANCE N, BARTIER D. Elastic modulus of claystone evaluated by nano-/micro-indentation tests and meso-compression tests[J]. Journal of rock mechanics and geotechnical engineering, 2017, 9(1): 84-91.

[79] 徐鼎平, 柳秀洋, 徐怀胜, 等. 深埋花岗岩细观力学特性纳米压痕试验及参数均质化研究[J]. 中南大学学报（自然科学版）, 2021， 52（8）: 2761-2771.

[80] XU Dingping, LIU Xiuyang, XU Huaisheng, et al. Meso-mechanical properties of deep granite using nanoindentation test and homogenization approach[J]. Journal of Central South University(science and technology), 2021, 52(8): 2761-2771(in Chinese).

[81] XIE Z X, HUANG Z W, LI G S, et al. The effect of natural fractures on the failure mechanism of granite: implication for enhanced geothermal systems[J]. Engineering fracture mechanics, 2024, 299: 109938.

[82] WAN W K, LI C C, LIU S Q, et al. Changes in the nanomechanical properties of the constituent minerals in the ductile Fauske marble and the brittle kuru granite during progressive failure[J]. International journal of rock mechanics and mining sciences, 2024, 183: 105853.

[83] WHEELER J M, ARMSTRONG D E J, HEINZ W, et al. High temperature nanoindentation: the state of the art and future challenges[J]. Current opinion in solid state and materials science, 2015, 19(6): 354-366.

[84] LI N, ZHOU T, ZHAO M Y, et al. Thermal-shock-induced surface softening and conductivity evolution of hydraulic fracture in enhanced geothermal system[J]. Geothermics, 2022, 102: 102405.

[85] MO C K, ZHAO J L, ZHANG D X. Real-time measurement of mechanical behavior of granite during heating-cooling cycle: a mineralogical perspective[J]. Rock mechanics and rock engineering, 2022, 55(7): 4403-4422.

[86] LIU K Q, JIN Z J, ZAKHAROVA N, et al. Comparison of shale fracture toughness obtained from scratch test and nanoindentation test[J]. International journal of rock mechanics and mining sciences, 2023, 162: 105282.

[87] GUPTA I, SONDERGELD C, RAI C. Fracture toughness in shales using nano-indentation[J]. Journal of petroleum science and engineering, 2020, 191: 107222.

[88] 贾锁刚, 万有余, 王倩, 等. 页岩各向异性力学特性微观测试方法研究[J]. 地质力学学报, 2021， 27（1）: 10-18.

[89] JIA Suogang, WAN Youyu, WANG Qian, et al. Research on the micro-scale method for testing the mechanical anisotropy of shale[J]. Journal of geomechanics, 2021, 27(1): 10-18(in Chinese).

[90] 顾士坦, 逯英棋, 李文帅, 等. 基于纳米压痕的煤系泥岩细观力学及断裂性能试验研究[J]. 煤炭工程, 2023， 55（2）: 128-133.

[91] GU Shitan, LU Yingqi, LI Wenshuai, et al. Experiment on meso-mechanical and fracture characteristics of coal measures mudstone using nanoindentation[J]. Coal engineering, 2023, 55(2): 128-133(in Chinese).

[92] ZHENG J J, SONG Y Q, SHEN F X, et al. Study on mechanical properties of water-immersed mudstone based on nanoindentation tests[J]. Mining, metallurgy & exploration, 2024, 41(4): 2031-2046.

[93] MA Z Y, ZHANG C P, PATHEGAMA GAMAGE R, et al. Uncovering the creep deformation mechanism of rock-forming minerals using nanoindentation[J]. International journal of mining science and technology, 2022, 32(2): 283-294.

[94] THOM C A, GOLDSBY D L. Nanoindentation studies of plasticity and dislocation creep in halite[J]. Geosciences, 2019, 9(2): 79.

[95] LIU Y X, BURCH A C, BENNETT K C, et al. Bridging nanoindentation and triaxial creep tests on a shale[J]. Acta geotechnica, 2023, 18(12): 6475-6487.

[96] LIU K Q, OSTADHASSAN M, BUBACH B. Application of nanoindentation to characterize creep behavior of oil shales[J]. Journal of petroleum science and engineering, 2018, 167: 729-736.

[97] WANG J F, LIU Y K, YANG C, et al. Upscaling the creep behavior of clay-rich and quartz-rich shales from nanoindentation measurements: application to the Wufeng-Longmaxi shale, China[J]. International journal of rock mechanics and mining sciences, 2023, 171: 105580.

[98] SHARMA P, PRAKASH R, ABEDI S. Effect of temperature on nano-and microscale creep properties of organic-rich shales[J]. Journal of petroleum science and engineering, 2019, 175: 375-388.

[99] 陈玉丽, 马勇, 潘飞, 等. 多尺度复合材料力学研究进展[J]. 固体力学学报, 2018， 39（ 1）: 1-68.

[100] CHEN Yuli, MA Yong, PAN Fei, et al. Research progress in multi-scale mechanics of composite materials[J]. Chinese journal of solid mechanics, 2018, 39(1): 1-68(in Chinese).

[101] KUSTER G T, TOKSOZ M N. Velocity and attenuation of seismic waves in two-phase media[J]. Geophysics, 1974, 39(5): 587-606.

[102] BERRYMAN J G, BERGE P A. Critique of two explicit schemes for estimating elastic properties of multiphase composites[J]. Mechanics of materials, 1996, 22(2): 149-164.

[103] ALIGHOLI S, KHANDELWAL M. Computing elastic moduli of igneous rocks using modal composition and effective medium theory[J]. Geosciences, 2022, 12(11): 413.

[104] HERSHEY A V. The elasticity of an isotropic aggregate of anisotropic cubic crystals[J]. Journal of applied mechanics, 1954, 21(3): 236-240.

[105] HILL R. Continuum micro-mechanics of elastoplastic polycrystals[J]. Journal of the mechanics and physics of solids, 1965, 13(2): 89-101.

[106] 沈观林, 胡更开, 刘彬. 复合材料力学[M]. 2版. 北京: 清华大学出版社， 2013.

[107] ABOU-CHAKRA GUÉRY A, CORMERY F, SHAO J F, et al. A comparative micromechanical analysis of the effective properties of a geomaterial: effect of mineralogical compositions[J]. Computers and geotechnics, 2010, 37(5): 585-593.

[108] MORI T, TANAKA K. Average stress in matrix and average elastic energy of materials with misfitting inclusions[J]. Acta metallurgica, 1973, 21(5): 571-574.

[109] BENVENISTE Y. On the Mori-Tanaka's method in cracked bodies[J]. Mechanics research communications, 1986, 13(4): 193-201.

[110] HAN T C, YAN H, FU L Y, et al. Applicability of cross-property differential effective medium model to the joint elastic-electrical properties of reservoir sandstones[J]. Geophysical prospecting, 2022, 70(7): 1243-1251.

[111] ZHAO J L, ZHANG W, ZHANG D X, et al. Influence of geochemical features on the mechanical properties of organic matter in shale[J]. Journal of geophysical research(solid earth), 2020, 125(9): e2020JB019809.

[112] FJÆR E, HOLT R M, HORSRUD P, et al. Chapter 1 elasticity[M]// FJÆR E, HOLT R M, HORSRUD P, et al. Petroleum Related Rock Mechanics. 2nd ed. Amsterdam: Elsevier, 2008: 1-54.

[113] LEI M, DANG F N, XUE H B, et al. Evaluation method of granite multiscale mechanical properties based on nanoindentation technology[J]. Geofluids, 2021, 2021: 6745900.

[114] WANG J F, YANG C, GUO Y, et al. Examining the reliability of current micro-and nano-indentation-based rock mechanical upscaling schemes: a comprehensive comparison with uniaxial/triaxial macroscopic mechanical testing[J]. Geomechanics and geophysics for geo-energy and geo-resources, 2025, 11(1): 28.

[115] SADEGHPOUR F, DARKHAL A, GAO Y F, et al. Comparison of geomechanical upscaling methods for prediction of elastic modulus of heterogeneous media[J]. Geoenergy science and engineering, 2024, 239: 212915.

[116] BHATTACHARYA A K, NIX W D. Finite element simulation of indentation experiments[J]. International journal of solids and structures, 1988, 24(9): 881-891.

[117] SWADENER J G, GEORGE E P, PHARR G M. The correlation of the indentation size effect measured with indenters of various shapes[J]. Journal of the mechanics and physics of solids, 2002, 50(4): 681-694.

[118] STELMASHENKO N A, WALLS M G, BROWN L M, et al. Micro-indentations on W and Mo oriented single crystals: an STM study[J]. Acta metallurgica et materialia, 1993, 41(10): 2855-2865.

[119] 李敏, 梁乃刚, 张泰华, 等. 纳米压痕过程的三维有限元数值试验研究[J]. 力学学报, 2003， 35（3）: 257-264.

[120] LI Min, LIANG Naigang, ZHANG Taihua, et al. 3D finite element simulation of the nanoindentation process[J]. Acta mechanica Sinica, 2003, 35(3): 257-264(in Chinese).

[121] 李明耀, 彭磊, 左建平, 等. 基于DIP-FFT数值方法的花岗岩多尺度力学特性研究[J]. 岩石力学与工程学报, 2022， 41（11）: 2254-2267.

[122] LI Mingyao, PENG Lei, ZUO Jianping, et al. Study on multi-scale mechanical properties of granite based on DIP-FFT numerical method[J]. Chinese journal of rock mechanics and engineering, 2022, 41(11): 2254-2267(in Chinese).

[123] 张子星, 杨海霞, 邵婷悦. 无定形水化硅酸钙纳米压痕模拟及卸载段处理[J]. 力学季刊, 2021， 42（4）: 731-742.

[124] ZHANG Zixing, YANG Haixia, SHAO Tingyue. Simulation of nanoindentation and analysis of unloading section of amorphous calcium silicate hydrate(C-S-H)[J]. Chinese quarterly of mechanics, 2021, 42(4): 731-742(in Chinese).

[125] 孟筠青, 牛家兴, 夏捃凯, 等. 纳米尺度下煤的力学性质及破坏机制研究[J]. 岩石力学与工程学报, 2020， 39（1）: 84-92.

[126] MENG Yunqing, NIU Jiaxing, XIA Junkai, et al. Study on mechanical properties and failure mechanisms of coal at the nanometer scale[J]. Chinese journal of rock mechanics and engineering, 2020, 39(1): 84-92(in Chinese).

[127] ZHANG Y H, WONG L N Y. A review of numerical techniques approaching microstructures of crystalline rocks[J]. Computers & geosciences, 2018, 115: 167-187.

[128] YANG X, ZHANG Y, LI G, et al. Mesoscopic modeling approach and application based on rock thin slices and nanoindentation[J]. Computers and geotechnics, 2024, 165: 105875.

[129] TANG X H, ZHANG Y H, XU J J, et al. Determining young's modulus of granite using accurate grain-based modeling with microscale rock mechanical experiments[J]. International journal of rock mechanics and mining sciences, 2022, 157: 105167.

[130] LIU Y W, LIU Q S, FENG G, et al. Upscaling mechanical properties of shale obtained by nanoindentation to macroscale using accurate grain-based modeling(AGBM)[J]. Energy, 2025, 314: 134126.

[131] 齐姝霞, 刘一苇, 唐旭海, 等. 基于纳米压痕的玄武岩微观弹性模量测定及参数尺度升级研究[J]. 岩土工程学报, 2025， 47（9）: 1956-1966.

[132] QI Shuxia, LIU Yiwei, TANG Xuhai, et al. Determination of microscopic elastic modulus of basalt based on nanoindentation experiment and study on parameter upscaling methods[J]. Chinese journal of geotechnical engineering, 2025, 47(9): 1956-1966(in Chinese).

基金资助

国家自然科学基金资助项目(52009107)

国家重点研发计划课题资助项目(2022YFC3003404-02)

陕西农林职业技术大学国基金培育项目(GZ2023-007)