Abstract
The recoverable strain of rock is completely classified as elastic strain in the conventional elastic-plastic theory, which often results in poor agreement between theoretical and experimental curves. This work proposes an improved elastoplastic model of rock materials considering the evolutions of crack deformation and elastic modulus to better characterize the nonlinear mechanical behavior of rock in the post-peak stage. In this model, the recoverable strain is assumed to be a combination of elastic and crack strain, and the constitutive relationship between crack strain and rock stress is deduced. Based on the proposed assumption, the evolutions of the mechanical parameters including strength parameters, elastic, plastic, and crack deformation parameters versus the plastic strain and confining stress were investigated. The developed elastoplastic model was validated by comparing the theoretical values with the results of the triaxial cyclic loading and unloading test. The theoretical calculation results show a good agreement with the laboratory test, which indicates that the improved elastoplastic model can effectively reflect the nonlinear mechanical behavior of the rock materials. The research results are expected to provide a valuable reference for further understanding the evolution of rock crack deformation.
摘要
在传统的弹塑性理论中, 岩石的可恢复应变被完全归纳为弹性应变, 这导致理论预测和实验曲线之间存在一定的偏差. 本文提 出了一种改进的岩石材料的弹塑性模型, 考虑了裂纹变形参数和弹性模量随塑性变形的演变, 以更好地表征岩石在峰后阶段的非线性 力学行为. 在该模型中, 假设可恢复应变是由弹性应变和裂纹应变组成, 并推导了岩石应力与裂纹应变之间的本构关系. 基于所提出的 假设, 研究了包括强度、弹性、塑性和裂纹变形参数在内的一系列力学参数与塑性剪应变和围压之间的演变关系. 通过将理论计算值 与三轴循环加卸载试验结果进行比较, 验证了该模型的准确性. 理论计算结果与室内试验吻合较好, 表明所提出的力学模型能够有效 地反映岩石材料的非线性力学行为. 研究结果有望为进一步了解岩石裂纹变形的演化提供有价值的参考.
Article PDF
Avoid common mistakes on your manuscript.
Data availability statement
All data, models, and codes that support the findings of this study are available from the corresponding author upon reasonable request.
References
X. G. Zhao, J. Wang, M. Cai, C. Cheng, L. K. Ma, R. Su, F. Zhao, and D. J. Li, Influence of unloading rate on the strainburst characteristics of Beishan granite under true-triaxial unloading conditions, Rock Mech. Rock Eng. 47, 467 (2014).
H. Xie, J. Lu, C. Li, M. Li, and M. Gao, Experimental study on the mechanical and failure behaviors of deep rock subjected to true triaxial stress: A review, Int. J. Min. Sci. Tech. 32, 915 (2022).
X. S. Liu, J. G. Ning, Y. L. Tan, and Q. H. Gu, Damage constitutive model based on energy dissipation for intact rock subjected to cyclic loading, Int. J. Rock Mech. Min. Sci. 85, 27 (2016).
X. Xu, M. Karakus, F. Gao, and Z. Zhang, Thermal damage constitutive model for rock considering damage threshold and residual strength, J. Cent. South Univ. 25, 2523 (2018).
H. Xie, M. Gao, R. Zhang, G. Peng, W. Wang, and A. Li, Study on the mechanical properties and mechanical response of coal mining at 1000 m or deeper, Rock Mech. Rock Eng. 52, 1475 (2019).
X. R. Liu, J. Liu, H. Feng, L. B. Du, and Z. L. Zhong, Experimental research on unloading mechanical properties of sandstone under different initial unloading levels and pore pressures (in Chinese), Chin. J. Geotechl. Eng. 40, 1143 (2018).
Q. Zhang, C. Li, X. Quan, Y. Wang, L. Yu, and B. Jiang, New true-triaxial rock strength criteria considering intrinsic material characteristics, Acta Mech. Sin. 34, 130 (2018).
Z. Han, J. Li, H. Wang, and J. Zhao, Initiation and propagation of a single internal 3D crack in brittle material under dynamic loads, Eng. Fract. Mech. 285, 109299 (2023).
Z. Han, D. Li, and X. Li, Effects of axial pre-force and loading rate on Mode I fracture behavior of granite, Int. J. Rock Mech. Min. Sci. 157, 105172 (2022).
B. Sun, Z. Zhu, C. Shi, and Z. Luo, Dynamic mechanical behavior and fatigue damage evolution of sandstone under cyclic loading, Int. J. Rock Mech. Min. Sci. 94, 82 (2017).
X. Li, G. Si, J. Oh, I. Canbulat, R. Kong, and J. Zhang, A novel elastoplastic model for Yunnan sandstone under poly-axial loading, Int. J. Min. Sci. Tech. 33, 801 (2023).
K. Li, S. Yang, C. Liu, Y. Chen, G. Zhang, and Q. Ma, Mechanical response and microscopic damage mechanism of pre-flawed sandstone subjected to monotonic and multilevel cyclic loading: A laboratory-scale investigation, Int. J. Min. Sci. Tech. 33, 1487 (2023).
S. Q. Yang, B. Hu, and P. Xu, Study on the damage-softening constitutive model of rock and experimental verification, Acta Mech. Sin. 35, 786 (2019).
D. Li, H. Qi, C. Li, and C. Li, Investigating the effect of initial cracks on the mudstone mechanical behavior under uniaxial compression using FDEM, Acta Mech. Sin. 39, 422421 (2023).
D. Chang, J. Liu, and A. Xu, Constitutive modelling of a coarse sand under unfrozen and frozen states, Acta Mech. Sin. 39, 423108 (2023).
K. Fuenkajorn, and D. Phueakphum, Effects of cyclic loading on mechanical properties of Maha Sarakham salt, Eng. Geol. 112, 43 (2010).
H. Hu, B. Xia, Y. Luo, and J. Peng, Energy characteristics of sandstones with different crack angles under true triaxial cyclic loading and unloading, Energy Sci. Eng. 10, 1418 (2022).
X. Huang, C. Shi, H. Ruan, Y. Zhang, and W. Zhao, Stable crack propagation model of rock based on crack strain, Energies 15, 1885 (2022).
J. Zuo, Y. Chen, and X. Liu, Crack evolution behavior of rocks under confining pressures and its propagation model before peak stress, J. Cent. South Univ. 26, 3045 (2019).
Y. Zhao, L. Zhang, W. Wang, C. Pu, W. Wan, and J. Tang, Cracking and stress-strain behavior of rock-like material containing two flaws under uniaxial compression, Rock Mech. Rock Eng. 49, 2665 (2016).
H. Zhou, K. Zhang, X. T. Feng, J. F. Shao, and S. L. Qiu, Elastoplastic coupling mechanical model for brittle marble (in Chinese), Chin. J. Rock Mech. Eng. 29, 2398 (2010).
H. Zhou, F. J. Yang, C. Q. Zhang, R. C. Xu, and K. Zhang, An elastoplastic coupling mechanical model for marble considering confining pressure effect (in Chinese), Chin. J. Rock Mech. Eng. 31, 2389 (2012).
S. Chen, Z. Zhao, F. Feng, and M. Zhang, Stress evolution of deep surrounding rock under characteristics of bi-modulus and strength drop, J. Cent. South Univ. 29, 680 (2022).
J. Han, S. Li, S. Li, and W. Yang, A procedure of strain-softening model for elasto-plastic analysis of a circular opening considering elasto-plastic coupling, Tunn. Under. Sp. Tech. 37, 128 (2013).
Q. Zhang, C. Zhang, B. Jiang, N. Li, and Y. Wang, Elastoplastic coupling solution of circular openings in strain-softening rock mass considering pressure-dependent effect, Int. J. Geomech. 18, 04017132 (2018).
L. Cui, Q. Sheng, J. Zheng, M. Xie, and Y. Liu, A unified deterioration model for elastic modulus of rocks with coupling influence of plastic shear strain and confining stress, Rock Mech. Rock Eng. 55, 7409 (2022).
J. Ning, Q. Gu, J. Wang, S. Hu, Q. Ma, and P. Qiu, Strain-softening model for marble considering the degradation of elastic modulus, Arab. J. Sci. Eng. 43, 5635 (2018).
Y. N. Wang, L. C. Wang, and H. Z. Zhou, An experimental investigation and mechanical modeling of the combined action of confining stress and plastic strain in a rock mass, B. Eng. Geol. Environ. 81, 204 (2022).
M. Min, B. S. Jiang, M. M. Lu, J. K. Long, and Q. Zhang, An improved strain-softening model for Beishan granite considering the degradation of elastic modulus, Arab. J. Geosci. 13, 244 (2020).
O. Pourhosseini, and M. Shabanimashcool, Development of an elastoplastic constitutive model for intact rocks, Int. J. Rock Mech. Min. Sci. 66, 1 (2014).
X. Wu, Y. Jiang, and Z. Guan, A modified strain-softening model with multi-post-peak behaviours and its application in circular tunnel, Eng. Geol. 240, 21 (2018).
X. G. Zhao, P. F. Li, L. K. Ma, R. Su, and J. Wang, Damage and dilation characteristics of deep granite at Beishan under cyclic loading-unloading conditions (in Chinese), Chin. J. Rock Mech. Eng. 33, 1740 (2014).
P. F. Li, X. G. Zhao, Z. Guo, L. K. Ma, L. Chen, and J. Wang, Variation of strength parameters of Beishan granite under triaxial compression (in Chinese), Chin. J. Rock Mech. Eng. 36, 1599 (2017).
X. G. Zhao, and M. Cai, A mobilized dilation angle model for rocks, Int. J. Rock Mech. Min. Sci. 47, 368 (2010).
J. C. Jin, C. X. She, and P. Y. Shang, Study on strength parameters and dilation angle evolution models in hard rock elasto-plastic deformation and failure process (in Chinese), Rock Soil Mech. 40, 4401 (2019).
S. Miao, Z. Liu, X. Zhao, L. Ma, Y. Zheng, and D. Xia, Plastic and damage energy dissipation characteristics and damage evolution of Beishan granite under triaxial cyclic loading, Int. J. Rock Mech. Min. Sci. 174, 105644 (2024).
Y. Song, S. Q. Yang, K. S. Li, P. F. Yin, and P. Z. Pan, Mechanical behavior and fracture evolution mechanism of composite rock under triaxial compression: Insights from three-dimensional DEM modeling, Rock Mech. Rock Eng. 56, 7673 (2023).
Y. Yang, W. Li, Q. Wang, W. Chen, and K. Zhou, Experimental study on mechanical behavior and permeability evolution of weakly cemented sandstone under unloading conditions, B. Eng. Geol. Environ. 83, 115 (2024).
J. Q. Guo, W. F. Huang, X. R. Liu, J. H. Chen, and J. G. Jiang, Rock dilation criteria development based on releasable strain energy (in Chinese), J. China Coal. Soc. 44, 2094 (2019).
G. Walton, J. Arzúa, L. R. Alejano, and M. S. Diederichs, A laboratory-testing-based study on the strength, deformability, and dilatancy of carbonate rocks at low confinement, Rock Mech. Rock Eng. 48, 941 (2015).
P. A. Vermeer, Non-associated plasticity for soils, concrete and rock, in: Physics of Dry Granular Media (Springer, Dordrecht, 1998), pp. 163–196.
Q. Meng, J. F. Liu, L. Ren, H. Pu, and Y. Chen, Experimental study on rock strength and deformation characteristics under triaxial cyclic loading and unloading conditions, Rock Mech. Rock Eng. 54, 777 (2021).
B. Fu, L. Hu, and C. Tang, Experimental and numerical investigations on crack development and mechanical behavior of marble under uniaxial cyclic loading compression, Int. J. Rock Mech. Min. Sci. 130, 104289 (2020).
X. P. Zhou, H. Cheng, and Y. F. Feng, An experimental study of crack coalescence behaviour in rock-like materials containing multiple flaws under uniaxial compression, Rock Mech. Rock Eng. 47, 1961 (2014).
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant No. 52074269), the Postgraduate Research & Practice Innovation Program of Jiangsu Province, and the Graduate Innovation Program of China University of Mining and Technology (Grant No. 2024WLKXJ202).
Author information
Authors and Affiliations
Contributions
Author contributions Qiuxin Gu designed the research and wrote the first draft of the manuscript. Qiang Zhang helped organize the manuscript. Yapeng Li, Peinan Wu, and Guilei Han revised and edited the final version.
Corresponding author
Ethics declarations
Conflict of interest On behalf of all authors, the corresponding author states that there is no conflict of interest.
Rights and permissions
About this article
Cite this article
Gu, Q., Zhang, Q., Li, Y. et al. An improved elastoplastic model for rocks and application to cyclic loading and unloading triaxial compression tests. Acta Mech. Sin. 40, 424053 (2024). https://doi.org/10.1007/s10409-024-24053-x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10409-024-24053-x