A novel fractal-order elasto-visco-plastic creep damage model and its engineering-scale creep parameter determination method

Abstract

The effective prediction and evaluation of the long-term stability of deep-buried tunnels are crucial for tunnel design, construction, and operation. The creep model is key to predicting time-dependent behavior, and the accuracy of time-dependent deformation predictions is determined by the creep parameters. This paper introduces a novel fractal-order elasto-visco-plastic creep damage (FEVPD) model that incorporates long-term strength into the damage evolution equation within the framework of continuum damage mechanics. The model effectively captures the three-stage creep behavior of various rock types and predicts their creep lifespans under different stress levels. The FEVPD model was implemented in FLAC3D using C++. Additionally, in determining the creep parameters of rock at the engineering scale to address the high computational cost of parameter inversion, an improved genetic algorithm was developed with adaptive perturbation, elitism, and dynamic mutation mechanisms. Application to field monitoring data from the Jinping II hydropower station tunnel demonstrated that the FEVPD model improved the prediction accuracy of time-dependent deformation by 32.68% compared to the classical Burgers–Mohr (CVISC) model. The enhanced inversion method also reduced the final error by 26.0% and 22.7% for the FEVPD and CVISC models, respectively, compared with the standard algorithm. Finally, this model was used to predict the long-term stability of the tunnel. The results provide a reliable and efficient framework for modeling and predicting creep behavior in deep rock engineering.

Keywords

Fractal-order derivatives / Creep model / Numerical simulation / Parameter inversion / Improved genetic algorithm

References

[1] Alessandro, C., & Eurípedes, V. (2023). A thermodynamic modeling of creep in rock salt. International Journal of Rock Mechanics and Mining Sciences, 162, 105298.

[2] An, J. S., Kang, K. N., Choi, J. Y., Sung, W. S., Suy, V., & Song, K. I. (2020). Tunnel back analysis based on differential evolution using stress and displacement. Advances in Civil Engineering, 2020(1), 8156573.

[3] Barla, G., Debernardi, D., & Sterpi, D. (2012). Time-dependent modeling of tunnels in squeezing conditions. International Journal of Geomechanics, 12(6), 697-710.

[4] Bonini, M., Debernardi, D., Barla, M., & Barla, G . (2007). The mechanical behaviour of clay shales and implications on the design of tunnels. Rock Mechanics and Rock Engineering, 42(2), 361-388.

[5] Cai, W., Chen, W., & Xu, W. X. (2016). Characterizing the creep of viscoelastic materials by fractal derivative models. International Journal of Non-Linear Mechanics, 87, 58-63.

[6] Chang, X. Y., Wang, H., & Zhang, Y. M. (2023). Back analysis of rock mass parameters in tunnel engineering using machine learning techniques. Computers and Geotechnics, 163, 105738.

[7] Chen, B. R., & Feng, X. T. (2008). Universal viscoelastoplastic combination model and its engineering applications. Chinese Journal of Rock Mechanics and Engineering, 27, 1028-1035 (in Chinese).

[8] Chen, B. R., Zhao, X. J., Feng, X. T., Zhao, H. B., & Wang, S. Y. (2013). Time-dependent damage constitutive model for the marble in the Jinping II hydropower station in China. Bulletin of Engineering Geology and the Environment, 73(2), 499-515.

[9] Cui, Q. Z., Hou, R. B., Li, Z. H., Du, F., Chen, X., Zhang, B. Y., & Li, L. L. (2025a). Time-dependent behavior of deep roadway surrounding rock considering damage induced by excavation and mining disturbances: Experiments, modeling, and simulation. International Journal of Mining Science and Technology (in press).

[10] Cui, Q. Z., Wu, F., Liu, J. F., Chen, J., Li, C. B., Gao, R. B., Gao, S., Liu, H. Q., & Ren, T. (2025b). A variable fractal-order rock salt creep model with full-time domain memory effect and its application in salt cavern hydrogen storage. Energy, 338, 138870.

[11] Cui, Q. Z., Wu, F., Wang, L. T., Liu, J. F., Chen, J., Li, C. B., Liu, H. Q., & Gao, R. B. (2025c). A rock elasto-visco-plastic creep model fully considering the effect of time-dependent damage. Computers and Geotechnics, 187, 107490.

[12] Debernardi, D., & Barla, G. (2009). New viscoplastic model for design analysis of tunnels in squeezing conditions. Rock Mechanics and Rock Engineering, 42(2), 259-288.

[13] Feng, X. T., Chen, B. R., Yang, C. X., Zhou, H., & Ding, X. L. (2006). Identification of visco-elastic models for rocks using genetic programming coupled with the modified particle swarm optimization algorithm. International Journal of Rock Mechanics and Mining Sciences, 43(5), 789-801.

[14] Gao, R. B., Wu, F., Chen, J., Zhu, C., & Ji, C. X. (2022). Study on creep characteristics and constitutive model of typical argillaceous salt rock in energy storage caverns in China. Journal of Energy Storage, 50, 104248.

[15] Ghorbani, E., Moosavi, M., Hossaini, M. F., Assary, M., & Golabchi, Y. (2021). Determination of initial stress state and rock mass deformation modulus at Lavarak HEPP by back analysis using ant colony optimization and multivariable regression analysis. Bulletin of Engineering Geology and the Environment, 80(1), 429-442.

[16] Gong, Y. P., Kou, Y. G., Yue, Q., Zhuang, X. Y., Valizadeh, N., Qin, F., Wang, Q., & Rabczuk, T. (2025). A phase-field study on thermo-mechanical coupled damage evolution and failure mechanisms of sintered silver interconnections. Engineering Fracture Mechanics, 320, 111039.

[17] He, Q. C., Wu, F., & Gao, R. B. (2022). Nonlinear creep-damage constitutive model of surrounding rock in salt cavern reservoir. Journal of Energy Storage, 55, 105520.

[18] Herrmann, R. (2014). Fractional calculus: An introduction for physicists. Singapore: World Scientific Publishing.

[19] Holland, J. H. (1975). Adaptation in natural and artificial systems. Ann Arbor: University of Michigan Press.

[20] Hou, R. B., Cui, Q. Z., Guo, Y. Y., Shi, Y. K., & Fu, J. W. (2024). A new elasto-visco-plastic damage model and numerical simulation method used for time-dependent behavior prediction of deep tunnel. Computers and Geotechnics, 168, 106129.

[21] ITASCA (2017). FLAC3D (Version 6.0) users manual, Itasca Consulting. Minneapolis: Group Inc..

[22] Kabwe, E., Karakus, M., & Chanda, E. K. (2020a). Creep constitutive model considering the overstress theory with an associative viscoplastic flow rule. Computers and Geotechnics, 124, 103629.

[23] Kabwe, E., Karakus, M., & Chanda, E. K. (2020b). Isotropic damage constitutive model for time-dependent behaviour of tunnels in squeezing ground. Computers and Geotechnics, 127, 103738.

[24] Kachanov, L. M. (1958). Rupture time under creep conditions. International Journal of Fracture, 97(1), 11-18.

[25] Kilbas, A. A., Srivastava, H. M., & Trujillo, J. J. (2006). Theory and applications of fractional differential equations. Amsterdam: Elsevier Science B.V.

[26] Kong, F. C., Lu, D. C., Ma, C., Shen, C. P., Yang, X. D., & Du, X. L. (2023). Fractional viscoelastic solution of stratum displacement of a shallow tunnel under the surface slope condition. Underground Space, 10, 233-247.

[27] Lemaitre, J . (1996). A course on damage mechanics. Berlin: Springer.

[28] Li, H., Ma, H. L., Yang, C. H., Wang, X., & Daemen, J. J. K. (2024a). Creep-fatigue model of rock salt based on a fractal-order derivative considering thermo-mechanical damage. Computers and Geotechnics, 174, 106605.

[29] Li, J., Hou, S. B., Jiang, Y. L., Fu, H. L., & Hu, K. X. (2025). A constitutive model of rock based on strain cone surface and the parameter calibration method with high efficiency. Computers and Geotechnics, 177, 106876.

[30] Li, T. B., Chen, C., Peng, F., Ma, C. C., Li, M., & Wang, Y. X. (2024b). Creep damage constitutive model of rock based on the mechanisms of crack-initiated damage and extended damage. Underground Space, 18, 295-313.

[31] Liang, C., Liu, J. F., Yang, J. X., Xu, H. N., Chen, Z. W., & Ran, L. N. (2024). A creep model for ultra-deep salt rock considering thermal-mechanical damage under triaxial stress conditions. Journal of Rock Mechanics and Geotechnical Engineering, 16(2), 588-596.

[32] Liu, C., Zhang, D. L., Zhang, S. L., Fang, Q., & Sun, Z. Y. (2023). Long-term mechanical analysis of tunnel structures in rheological rock considering the degradation of primary lining. Underground Space, 10, 217-232.

[33] Liu, J., Zeng, X. H., Zheng, W. Q., Lai, H. P., Wang, Y. L., Wang, F., & Qiu, H. (2025). Dynamic behavior of a running crack crossing mortar-granite interface with different interface inclinations. Engineering Fracture Mechanics, 314, 110705.

[34] Luo, W., Le, J. L., Rasoolinejad, M., & Bažant, Z. P. (2021). Coefficient of variation of shear strength of RC beams and size effect. Journal of Engineering Mechanics, 147(2), 04020144.

[35] Lyu, C., Liu, J. F., Ren, Y., Liang, C., & Liao, Y. L. (2021). Study on very long-term creep tests and nonlinear creep-damage constitutive model of salt rock. International Journal of Rock Mechanics and Mining Sciences, 146, 104873.

[36] Mahmoudi, M., & Rajabi, A. M. (2020). Application of numerical back analysis for determination of soil mass specifications during tunnel construction. Arabian Journal of Geosciences, 13(19), 990.

[37] Majdi, A., & Beiki, M. (2010). Evolving neural network using a genetic algorithm for predicting the deformation modulus of rock masses. International Journal of Rock Mechanics and Mining Sciences, 47(2), 246-253.

[38] Marquardt, D. W. (1963). An algorithm for least-squares estimation of nonlinear parameters. Journal of the Society for Industrial and Applied Mathematics, 11(2), 431-441.

[39] Perzyna, P. (1966). Fundamental problems in viscoplasticity. Advances in Applied Mechanics, 9, 243-377.

[40] Samaniego, E., Anitescu, C., Goswami, S., Nguyen-Thanh, V. M., Guo, H., Hamdia, K., Zhuang, X., & Rabczuk, T. (2020). An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications. Computer Methods in Applied Mechanics and Engineering, 362, 112790.

[41] Sun, J. L., Wang, F., Wang, X. L., & Wu, X. (2021). A quantitative evaluation method based on back analysis and the double-strength reduction optimization method for tunnel stability. Advances in Civil Engineering, 2021, 1-14.

[42] Wang, C., Ma, G. W., Zhao, J., & Soh, C. K. (2004). Identification of dynamic rock properties using a genetic algorithm. International Journal of Rock Mechanics and Mining Sciences, 41, 490-495.

[43] Wang, H. B., Wang, F. M., Guo, C. C., Qin, L., Liu, J., & Qu, T. M. (2025). Experimental investigation on the failure characteristic and synergistic load-bearing mechanism of multi-layer linings for deep soft rock tunnels. Underground Space, 20, 259-276.

[44] Wang, J. B., Zhang, Q., Song, Z. P., Feng, S. J., & Zhang, Y. W. (2022). Nonlinear creep model of salt rock used for displacement prediction of salt cavern gas storage. Journal of Energy Storage, 48, 103951.

[45] Wu, F., Ji, C. X., Liu, J., Gao, R. B., Li, C. B., Zou, Q. L., & Chen, J. (2023a). Study on visco-elastoplastic fractional creep model of surrounding rock of salt cavern gas storage. Journal of Energy Storage, 67, 107606.

[46] Wu, F., Yu, Q. L., & Liu, C. W. (2022). Creep characteristics and constitutive model of bio-based concrete in aqueous environment. Construction and Building Materials, 320, 126213.

[47] Wu, J., Wang, W., Cao, Y. J., Liu, S. F., Zhang, Q., & Chu, W. J. (2023b). A novel nonlinear fractional viscoelastic-viscoplastic damage creep model for rock-like geomaterials. Computers and Geotechnics, 163, 105726.

[48] Yang, C., Zhu, Z. J., Wang, J., Deng, H. F., Huang, D., & Tang, J. T. (2024). Damaged creep constitutive models of rock-like materials containing multiple flaws and their relationships to those of the intact materials. Theoretical and Applied Fracture Mechanics, 130, 104306.

[49] Yang, W. D., Wang, X. P., Ivanović, A., & Zhang, X. (2023). Coupled analytical solutions for circular tunnels considering rock creep effects and time-dependent anchoring forces in prestressed bolts. Tunnelling and Underground Space Technology, 134, 104954.

[50] Yin, W., Wang, X. Y., Zheng, S., Zhang, K., & Zhang, F. R. (2024). Triaxial creep test and damage model study of layered red sandstone under freeze-thaw cycles. Case Studies in Construction Materials, 21, e03785.

[51] Zhang, J. Z., Zhou, X. P., & Yin, P. (2019). Visco-plastic deformation analysis of rock tunnels based on fractional derivatives. Tunnelling and Underground Space Technology, 85, 209-219.

[52] Zhao, C. W., Zhou, H., Zhang, C. Q., Liu, W. B., Li, L. Y., Liu, Z. B., & Huo, Z. Y. (2025). Creep-slip behaviors of surrounding rocks in a water diversion tunnel considering strike-slip and dip-slip scenarios. Tunnelling and Underground Space Technology, 158, 106405.

[53] Zhao, Y. L., Wang, Y. X., Wang, W. J., Wan, W., & Tang, J. Z. (2017a). Modeling of non-linear rheological behavior of hard rock using triaxial rheological experiment. International Journal of Rock Mechanics and Mining Sciences, 93, 66-75.

[54] Zhao, Y. L., Zhang, L. Y., Wang, W. J., Wan, W., Li, S. Q., Ma, W. H., & Wang, Y. X. (2017b). Creep behavior of intact and cracked limestone under multi-level loading and unloading cycles. Rock Mechanics and Rock Engineering, 50(6), 1409-1424.

[55] Zheng, C. T., Zhu, X. Y., & Zhang, Z. Q. (2024). Damage evolution of tunnel lining under creep action considering interlayer effect in gently inclined layered surrounding rock. Engineering Failure Analysis, 162, 108392.

[56] Zheng, Y., Chen, C. X., Meng, F., Liu, T. T., & Xia, K. Z. (2020). Assessing the stability of rock slopes with respect to flexural toppling failure using a limit equilibrium model and genetic algorithm. Computers and Geotechnics, 124, 103619.

[57] Zhou, J. X., Zhang, J. W., Wang, J. A., Li, F., & Zhou, Y. (2022). Research on nonlinear damage hardening creep model of soft surrounding rock under the stress of deep coal resources mining. Energy Reports, 8, 1493-1507.

[58] Zienkiewicz, O. C., & Cormeau, I. C. (1974). Visco-plasticity-plasticity and creep in elastic solids-a unified numerical solution approach. International Journal for Numerical Methods in Engineering, 8(4), 821-845.