A new damage constitutive model for rock strain softening based on an improved Logistic function

Abstract

This study proposed a new and more flexible S-shaped rock damage evolution model from a phenomenological perspective based on an improved Logistic function to describe the characteristics of the rock strain softening and damage process. Simultaneously, it established a constitutive model capable of describing the entire process of rock pre-peak compaction and post-peak strain softening deformation, considering the nonlinear effects of the initial compaction stage of rocks, combined with damage mechanics theory and effective medium theory. In addition, this research verified the rationality of the constructed damage constitutive model using results from uniaxial and conventional triaxial compression tests on Miluo granite, yellow sandstone, mudstone, and glutenite. The results indicate that based on the improved Logistic function, the theoretical damage model accurately describes the entire evolution of damage characteristics during rock compression deformation, from maintenance through gradual onset, accelerated development to deceleration and termination, in a simple and unified expression. At the same time, the constructed constitutive model can accurately simulate the stress–strain process of different rock types under uniaxial and conventional triaxial compression, and the theoretical model curve closely aligns with experimental data. Compared to existing constitutive models, the proposed model has significant advantages. The damage model parameters a, r and β have clear physical meanings and interact competitively, where the three parameters collectively determine the shape of the theoretical stress-strain curve.

Keywords

rock mechanics / strain softening / improved Logistic function / S-shaped model / damage evolution / constitutive model

References

[1] HeM-c, XieH-p, PengS-p, et al.. Study on rock mechanics in deep mining engineering [J]. Chinese Journal of Rock Mechanics and Engineering, 2005, 24(16): 2803-2813(in Chinese)

[2] HeF-z, LiG-c, KanJ-g, et al.. Research progress on multi-scale damage of rock [J]. Coal Science and Technology, 2024, 52(10): 33-53(in Chinese)

[3] LiuD-q, WangZ, ZhangX-y. Characteristics of strain softening of rocks and its damage constitutive model [J]. Rock and Soil Mechanics, 2017, 38(10): 2901-2908(in Chinese)

[4] LiuD-q, HeM-c, CaiM. A damage model for modeling the complete stress–strain relations of brittle rocks under uniaxial compression [J]. International Journal of Damage Mechanics, 2018, 27(7): 1000-1019.

[5] EberhardtE, SteadD, StimpsonB. Quantifying progressive pre-peak brittle fracture damage in rock during uniaxial compression [J]. International Journal of Rock Mechanics and Mining Sciences, 1999, 36(3): 361-380.

[6] KimJ S, LeeK S, ChoW J, et al.. A comparative evaluation of stress–strain and acoustic emission methods for quantitative damage assessments of brittle rock [J]. Rock Mechanics and Rock Engineering, 2015, 48(2): 495-508.

[7] BruningT, KarakusM, NguyenG D, et al.. Experimental study on the damage evolution of brittle rock under triaxial confinement with full circumferential strain control [J]. Rock Mechanics and Rock Engineering, 2018, 51(11): 3321-3341.

[8] ZongY-j, HanL-j, WeiJ-j, et al.. Mechanical and damage evolution properties of sandstone under triaxial compression [J]. International Journal of Mining Science and Technology, 2016, 26(4): 601-607.

[9] ChenZ-q, HeC, MaG-y, et al.. Energy damage evolution mechanism of rock and its application to brittleness evaluation [J]. Rock Mechanics and Rock Engineering, 2019, 52(4): 1265-1274.

[10] GongF-q, ZhangP-l, LuoS, et al.. Theoretical damage characterisation and damage evolution process of intact rocks based on linear energy dissipation law under uniaxial compression [J]. International Journal of Rock Mechanics and Mining Sciences, 2021, 146104858.

[11] LiX, CaoW-g, SuY-h. A statistical damage constitutive model for softening behavior of rocks [J]. Engineering Geology, 2012, 143: 1-17.

[12] WenT, TangH-m, LiuY-r, et al.. Newly modified damage statistical constitutive model of rock based on impact factor [J]. Journal of China University of Mining & Technology, 2016, 45(1): 141-149(in Chinese)

[13] CaoW-g, MoR, LiX. Study on statistical constitutive model and determination of parameters of rock based on normal distribution [J]. Chinese Journal of Geotechnical Engineering, 2007, 29(5): 671-675(in Chinese)

[14] LiY-w, JiaD, RuiZ-h, et al.. Evaluation method of rock brittleness based on statistical constitutive relations for rock damage [J]. Journal of Petroleum Science and Engineering, 2017, 153: 123-132.

[15] WangJ-b, SongZ-p, ZhaoB-y, et al.. A study on the mechanical behavior and statistical damage constitutive model of sandstone [J]. Arabian Journal for Science and Engineering, 2018, 43(10): 5179-5192.

[16] XuX L, KarakusM. A coupled thermo-mechanical damage model for granite [J]. International Journal of Rock Mechanics and Mining Sciences, 2018, 103: 195-204.

[17] Vidana PathiranageiS, GratchevI. Coupled thermo-mechanical constitutive damage model for sandstone [J]. Journal of Rock Mechanics and Geotechnical Engineering, 2022, 14(6): 1710-1721.

[18] BianK, LiuJ, ZhangW, et al.. Mechanical behavior and damage constitutive model of rock subjected to water-weakening effect and uniaxial loading [J]. Rock Mechanics and Rock Engineering, 2019, 52(1): 97-106.

[19] BaiY, ShanR-l, JuY, et al.. Study on the mechanical properties and damage constitutive model of frozen weakly cemented red sandstone [J]. Cold Regions Science and Technology, 2020, 171102980.

[20] GaoF, XiongX, XuC-s, et al.. Mechanical property deterioration characteristics and a new constitutive model for rocks subjected to freeze-thaw weathering process [J]. International Journal of Rock Mechanics and Mining Sciences, 2021, 140104642.

[21] JiangW-t, LaiY-m, MaQ-g, et al.. Mechanical damage model and brittleness index of frozen rocks based on statistical damage theory [J]. Acta Geotechnica, 2023, 18(9): 4687-4713.

[22] JiangW-t, LaiY-m, YuF, et al.. Mechanical properties investigation and damage constitutive models of red sandstone subjected to freeze-thaw cycles [J]. Cold Regions Science and Technology, 2023, 207103776.

[23] WangJ, SongW-d, FuJ-x. A damage constitutive model and strength criterion of rock mass considering the dip angle of joints [J]. Chinese Journal of Rock Mechanics and Engineering, 2018, 37(10): 2253-2263(in Chinese)

[24] YanJ-b, ZouZ-x, GuoS-w, et al.. Mechanical behavior and damage constitutive model of granodiorite in a deep buried tunnel [J]. Bulletin of Engineering Geology and the Environment, 2022, 813118.

[25] LiS-n, XiaoJ, LiY, et al.. A new damage constitutive model of rock considering microscopic crack growth [J]. Chinese Journal of Rock Mechanics and Engineering, 2023, 42(3): 640-648(in Chinese)

[26] LiuD-q, GuoY-p, LiJ-y, et al.. Damage evolution and constitutive model of brittle rock under uniaxial compression based on acoustic emission [J]. Journal of China University of Mining & Technology, 2023, 52(4): 687-700(in Chinese)

[27] LiuD-q, GuoY-p, LiJ-y, et al.. Damage constitutive model for layered yellow sandstone based on dissipative energy evolution and its verification [J]. Chinese Journal of Engineering, 2024, 46(5): 784-799(in Chinese)

[28] ChenK, CudmaniR, OlarteA A P. Validation of a damage constitutive model based on Logistic model dedicated to the mechanical behavior of the rock [J]. Geomechanics and Geophysics for Geo-Energy and Geo-Resources, 2024, 10115.

[29] MaK, RenF-q, WangH, et al.. Dynamic mechanical responses and freezing strengthening mechanism of frozen sandstone with single flaw: Insights from drop weight tests and numerical simulation [J]. Rock Mechanics and Rock Engineering, 2024, 57(2): 1263-1285.

[30] LiuZ-y, LiuX-l. Dissipative structure theory analysis of rock deformation and failure process and damage constitutive model [J]. Chinese Journal of Underground Space and Engineering, 2023, 19(1): 51-6078. (in Chinese)

[31] ZhengM-h, LiangY-p, StaatM, et al.. Discontinuous fracture behaviors and constitutive model of sandstone specimens containing non-parallel prefabricated fissures under uniaxial compression [J]. Theoretical and Applied Fracture Mechanics, 2024, 131104373.

[32] ZhanJ-w, ZhouY-l, WangY, et al.. Experimental study on physical damage and mechanical degradation of granite subjected to high-temperature cooling impact cycling [J]. Rock and Soil Mechanics, 2024, 45(8): 2362-2372. 2386

[33] HuL-h, LiY-c, LiangX, et al.. Rock damage and energy balance of strainbursts induced by low frequency seismic disturbance at high static stress [J]. Rock Mechanics and Rock Engineering, 2020, 53(11): 4857-4872.

[34] DuH-b, WangX-t, LiW-j, et al.. Dynamic responses and failure behaviors of submerged reefs containing pre-drilled hole subjected to repetitive impacting [J]. Ocean Engineering, 2024, 302117759.

[35] GuoY-p, LiuD-q, YangS-k, et al.. A new rock damage constitutive model based on usher function and its application to brittleness evaluation [J]. Rock Mechanics and Rock Engineering, 2025, 58(2): 1451-1472.

[36] LiuW, YinS-x, ThanhH V, et al.. Advancements and development trend in statistical damage constitutive models for rock: A comprehensive review [J]. Natural Hazards, 2025, 121(4): 3703-3744.

[37] CaoW-g, ZhangS, ZhaoM-h. Study on statistical damage constitutive model of rock based on new definition of damage [J]. Rock and Soil Mechanics, 2006, 27(1): 41-46(in Chinese)

[38] XieH-p, JuY, DongY-l. Discussion on the method of elastic module in the classic damage definition [J]. Mechanics and Practice, 1997, 19(2): 2-6(in Chinese)

[39] LiuD-q, GuoY-p, LingK, et al.. A whole process damage constitutive model for layered sandstone under uniaxial compression based on Logistic function [J]. Journal of Central South University, 2024, 31(7): 2411-2430.

[40] WangJ-b, LiuX-r, SongZ-p, et al.. A whole process creeping model of salt rock under uniaxial compression based on inverse S function [J]. Chinese Journal of Rock Mechanics and Engineering, 2018, 37(11): 2446-2459(in Chinese)

[41] WangJ-b, ZhangQ, SongZ-p, et al.. Creep properties and damage constitutive model of salt rock under uniaxial compression [J]. International Journal of Damage Mechanics, 2020, 29(6): 902-922.

[42] WangY-t, LiuX-p, MaoX-g, et al.. Research on prediction model of surface dynamic subsidence of mined-out region based on Usher time function [J]. Coal Science and Technology, 2021, 49(9): 145-151(in Chinese)

[43] TjørveK M C, TjørveE. The use of Gompertz models in growth analyses, and new Gompertz-model approach: An addition to the Unified-Richards family [J]. PLoS One, 2017, 126e0178691.

[44] HuangC-f, LiQ, WuS-c, et al.. Application of the richards model for settlement prediction based on a bidirectional difference-weighted least-squares method [J]. Arabian Journal for Science and Engineering, 2018, 43(10): 5057-5065.

[45] ZhangJ, AiC, LiY-w, et al.. Brittleness evaluation index based on energy variation in the whole process of rock failure [J]. Chinese Journal of Rock Mechanics and Engineering, 2017, 36(6): 1326-1340(in Chinese)

[46] PengJ, RongG, CaiM, et al.. A model for characterizing crack closure effect of rocks [J]. Engineering Geology, 2015, 189: 48-57.

[47] ZhangC, CaoW-g, XuZ, et al.. Initial macrodeformation simulation and determination method of microcrack closure stress for rock [J]. Rock and Soil Mechanics, 2018, 39(4): 1281-12881301

[48] ZuoJ-p, ChenY, LiuX-l. Crack evolution behavior of rocks under confining pressures and its propagation model before peak stress [J]. Journal of Central South University, 2019, 26(11): 3045-3056.

[49] LiX-l, ChenH-k, ZhangJ-h. Statistical damage model for whole deformation and failure process of rock considering initial void closure [J]. Journal of Southwest Jiaotong University, 2022, 57(2): 314-321(in Chinese)

[50] WangT-n, ZhaiY, GaoH, et al.. A novel binary effective medium model to describe the prepeak stress-strain relationship of combined bodies of rock-like material and rock [J]. International Journal of Mining Science and Technology, 2023, 33(5): 601-616.

[51] ZhengY-n, JiaC-j, LeiM-f, et al.. Investigation of the constitutive damage model of rock under the coupled effect of freeze–thaw cycles and loading [J]. Rock Mechanics and Rock Engineering, 2024, 57(3): 1861-1879.

[52] LiuX-s, NingJ-g, TanY-l, et al.. Damage constitutive model based on energy dissipation for intact rock subjected to cyclic loading [J]. International Journal of Rock Mechanics and Mining Sciences, 2016, 85: 27-32.

[53] GongF-q, ZhangP-l, XuL. Damage constitutive model of brittle rock under uniaxial compression based on linear energy dissipation law [J]. International Journal of Rock Mechanics and Mining Sciences, 2022, 160105273.

[54] XiongX, ZhouK-p, GaoF, et al.. A new damage constitutive model for frozen-thawed sandstone under triaxial conditions: Considering the characteristics of pre-peak compaction and post-peak residual strength [J]. International Journal of Damage Mechanics, 2024, 33(3): 193-222.

[55] FengW-l, QiaoC-s, WangT, et al.. Strain-softening composite damage model of rock under thermal environment [J]. Bulletin of Engineering Geology and the Environment, 2020, 79(8): 4321-4333.

[56] ChenK, CudmaniR, OlarteA A P. Mechanical impairment characteristics and a novel constitutive model for rocks subjected to uniaxial loading process [J]. International Journal of Damage Mechanics, 2024, 33(7): 497-526.

[57] WangK, JiangY-f, XuC. Mechanical properties and statistical damage model of coal with different moisture contents under uniaxial compression [J]. Chinese Journal of Rock Mechanics and Engineering, 2018, 37(5): 1070-1079(in Chinese)

[58] RenC-h, YuJ, LiuX-y, et al.. Cyclic constitutive equations of rock with coupled damage induced by compaction and cracking [J]. International Journal of Mining Science and Technology, 2022, 32(5): 1153-1165.

[59] JiangH-p, JiangA-n, YangX-r, et al.. Experimental investigation and statistical damage constitutive model on layered slate under thermal-mechanical condition [J]. Natural Resources Research, 2022, 31(1): 443-461.

[60] XuX-l, GaoF, ZhangZ-z. Thermo-mechanical coupling damage constitutive model of rock based on the Hoek-Brown strength criterion [J]. International Journal of Damage Mechanics, 2018, 27(8): 1213-1230.

[61] ChenK. Constitutive model of rock triaxial damage based on the rock strength statistics [J]. International Journal of Damage Mechanics, 2020, 29(10): 1487-1511.

[62] LiL-c, ZhaiM-y, ZhangL-y, et al.. Brittleness evaluation of glutenite based on energy balance and damage evolution [J]. Energies, 2019, 12183421.

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