A new constitutive model for the time-dependent behavior of rocks with consideration of damage parameter
Abstract
Deformation and time-dependent behavior of rocks are closely related to the stability and safety of underground structures and mines. In this paper, a numerical-analytical model is presented to investigate time-dependent damage and deformation of rocks under creep. The proposed model is obtained by combining the elastic-visco-plastic model based on the theory of over-stress and stress hardening law with the sub-critical crack growth model. The advantage of this model is that it is in incremental form and therefore can be implemented numerically. First, the governing equations of the model and its numerical computational algorithm are described. The proposed constitutive model is then implemented in the FLAC code using the FISH function. Determination of model parameters and calibration is done by various laboratory tests performed on a type of gypsum. The creep test was performed on gypsum under a stress of 13 MPa, which is equal to 70% of its compressive strength. After determining the parameters, by fitting the creep curve of the presented analyticalnumerical model, a good agreement is observed with the creep curve obtained from the laboratory data. It is also observed that during creep, the damage parameter and wing crack length increase.
Keyword : elastic-visco-plastic model, over-stress theory, sub-critical crack growth, creep test, stress hardening, wing crack
How to Cite
Ebrahimi Zohravi, S., Lakirouhani, A., Molladavoodi, H., Medzvieckas, J., & Kliukas, R. (2022). A new constitutive model for the time-dependent behavior of rocks with consideration of damage parameter. Journal of Civil Engineering and Management, 28(3), 223–231. https://doi.org/10.3846/jcem.2022.16609
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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Griggs, D. T., & Coles, N. E. (1954). Creep of single crystals of ice (SIPRE Report, 11). U.S. Army Snow, Ice, and Permafrost Research Establishment.
Gross, D., & Seelig, T. (2018). Fracture mechanics: With an introduction to micromechanics (3rd ed.). Springer. https://doi.org/10.1007/978-3-319-71090-7
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Kranz, R. L. (1979). Crack growth and development during creep of Barre granite. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 16(1), 23–35. https://doi.org/10.1016/0148-9062(79)90772-1
Kranz, R. L. (1980). The effects of confining pressure and stress difference on static fatigue of granite. Journal of Geophysical Research: Solid Earth, 85(B4), 1854–1866. https://doi.org/10.1029/JB085iB04p01854
Lakirouhani, A., & Hasanzadehshooiili, H. (2011). Review of rock strength criteria. In Proceedings of the 22nd World Mining Congress & Expo (pp. 473–482), Istanbul, Turkey.
Lemaitre, J., & Chaboche, J. (1990). Mechanics of solid materials. Cambridge University Press. https://doi.org/10.1017/CBO9781139167970
Lockner, D. (1993). Room temperature creep in saturated granite. Journal of Geophysical Research: Solid Earth, 98(B1), 475–487. https://doi.org/10.1029/92JB01828
Lockner, D. A., & Madden, T. R. (1991). A multiple-crack model of brittle fracture: 2. Time-dependent simulations. Journal of Geophysical Research: Solid Earth, 96(B12), 19643–19654. https://doi.org/10.1029/91JB01641
Lomnitz, C. (1956). Creep measurements in igneous rocks. The Journal of Geology, 64(5), 473–479. https://doi.org/10.1086/626379
Ma, L., Wang, M., Zhang, N., Fan, P., & Li, J. (2017). A variable-parameter creep damage model incorporating the effects of loading frequency for rock salt and its application in a bedded storage cavern. Rock Mechanics and Rock Engineering, 50, 2495. https://doi.org/10.1007/s00603-017-1236-9
Paliwal, B., & Ramesh, K. T. (2008). An interacting micro-crack damage model for failure of brittle materials under compression. Journal of the Mechanics and Physics of Solids, 56(3), 896–923. https://doi.org/10.1016/j.jmps.2007.06.012
Olson, J. (1993). Joint pattern development: Effects of subcritical crack growth and mechanical crack interaction. Journal of Geophysical Research, 98, 12251–12265. https://doi.org/10.1029/93JB00779
Pellet, F., Hajdu, A., Deleruyelle, F., & Besnus, F. (2005). A viscoplastic model including anisotropic damage for the time dependent behaviour of rock. International Journal for Numerical and Analytical Methods in Geomechanics, 29(9), 941–970. https://doi.org/10.1002/nag.450
Perzyna, P. (1966). Fundamental problems in viscoplasticity. Advances in Applied Mechanics, 9, 246–377. https://doi.org/10.1016/S0065-2156(08)70009-7
Robertson, E. C. (1955). Experimental study of the strength of rocks. GSA Bulletin, 66(10), 1275–1314. https://doi.org/10.1130/0016-7606(1955)66[1275:ESOTSO]2.0.CO;2
Shao, J.-F., Chau, K. T., & Feng, X. T. (2006). Modeling of anisotropic damage and creep deformation in brittle rocks. International Journal of Rock Mechanics and Mining Sciences, 43, 82–592. https://doi.org/10.1016/j.ijrmms.2005.10.004
Sterpi, D., & Gioda, G. (2009). Visco-plastic behaviour around advancing tunnels in squeezing rock. Rock Mechanics and Rock Engineering, 42(2), 319–339. https://doi.org/10.1007/s00603-007-0137-8
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. https://doi.org/10.1016/j.tust.2018.12.019
Ashby, M. F., & Hallam, S. D. (1986). The failure of brittle solids containing small cracks under compressive stress states. Acta Metallurgica, 34(3), 497–510. https://doi.org/10.1016/0001-6160(86)90086-6
Aydan, Ö. (2016). Time-dependency in rock mechanics and rock engineering (1st ed.). CRC Press. https://doi.org/10.1201/9781315375151
Aydan, Ö., Ito, T., Özbay, U., Kwasniewski, M., Shariar, K., Okuno, T., Özgenoğlu, A., Malan, D. F., & Okada, T. (2014). ISRM suggested methods for determining the creep characteristics of rock. Rock Mechanics and Rock Engineering, 47(1), 275–290. https://doi.org/10.1007/s00603-013-0520-6
Debernardi, D., & Barla, G. (2009). New viscoplastic model for design analysis of tunnels in squeezing conditions. Rock Mechanics and Rock Engineering, 42(2), 259. https://doi.org/10.1007/s00603-009-0174-6
Feng, W.-l., Qiao, C., Niu, S., Yang, Z., & Wang, T. (2020). An improved nonlinear damage model of rocks considering initial damage and damage evolution. International Journal of Damage Mechanics, 29, 1117–1137. https://doi.org/10.1177%2F1056789520909531
Gioda, G., & Cividini, A. (1996). Numerical methods for the analysis of tunnel performance in squeezing rocks. Rock Mechanics and Rock Engineering, 29(4), 171–193. https://doi.org/10.1007/BF01042531
Goodman, R. E. (1989). Introduction to rock mechanics (2nd ed.). John Wiley & Sons Ltd.
Griggs, D. T., & Coles, N. E. (1954). Creep of single crystals of ice (SIPRE Report, 11). U.S. Army Snow, Ice, and Permafrost Research Establishment.
Gross, D., & Seelig, T. (2018). Fracture mechanics: With an introduction to micromechanics (3rd ed.). Springer. https://doi.org/10.1007/978-3-319-71090-7
Hasanzadehshooiili, H., Lakirouhani, A., & Medzvieckas, J. (2012). Evaluating elastic-plastic behaviour of rock materials using hoek–brown failure criterion. Journal of Civil Engineering and Management, 18(3), 402–407. https://doi.org/10.3846/13923730.2012.693535
Horii, H., & Nemat-Nasser, S. (1986). Brittle failure in compression: splitting faulting and brittle-ductile transition. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 319, 337–374. https://doi.org/10.1098/rsta.1986.0101
Hou, R., Zhang, K., Tao, J., Xue, X., & Chen, Y. (2019). A nonlinear creep damage coupled model for rock considering the effect of initial damage. Rock Mechanics and Rock Engineering, 52(5), 1275–1285. https://doi.org/10.1007/s00603-018-1626-7
Huang, M., Zhan, J. W., Xu, C. S., & Jiang, S. (2020). New creep constitutive model for soft rocks and its application in the prediction of time-dependent deformation in tunnels. International Journal of Geomechanics, 20(7), 04020096. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001663
Itasca Consulting Group, Inc. (2019). FLAC — Fast Lagrangian analysis of continua (Ver. 8.1). Itasca.
Kemeny, J. (1991). A model for non-linear rock deformation under compression due to sub-critical crack growth. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 28, 459–467. https://doi.org/10.1016/0148-9062%2891%2991121-7
Ko, T. Y., Einstein, H., & Kemeny, J. (2006). Crack coalescence in brittle material under cyclic loading. In Proceedings of the 41st U.S. Rock Mechanics Symposium (Paper ARMA 06-930). Golden, CO.
Ko, T. Y., & Kemeny, J. (2013). Determination of the subcritical crack growth parameters in rocks using the constant stress-rate test. International Journal of Rock Mechanics and Mining Sciences, 59, 166–178. https://doi.org/10.1016/j.ijrmms.2012.11.006
Ko, T. Y., & Lee, S. S. (2020). Characteristics of crack growth in rock-like materials under monotonic and cyclic loading conditions. Applied Sciences, 10(2), 719. https://doi.org/10.3390/app10020719
Kranz, R. L. (1979). Crack growth and development during creep of Barre granite. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 16(1), 23–35. https://doi.org/10.1016/0148-9062(79)90772-1
Kranz, R. L. (1980). The effects of confining pressure and stress difference on static fatigue of granite. Journal of Geophysical Research: Solid Earth, 85(B4), 1854–1866. https://doi.org/10.1029/JB085iB04p01854
Lakirouhani, A., & Hasanzadehshooiili, H. (2011). Review of rock strength criteria. In Proceedings of the 22nd World Mining Congress & Expo (pp. 473–482), Istanbul, Turkey.
Lemaitre, J., & Chaboche, J. (1990). Mechanics of solid materials. Cambridge University Press. https://doi.org/10.1017/CBO9781139167970
Lockner, D. (1993). Room temperature creep in saturated granite. Journal of Geophysical Research: Solid Earth, 98(B1), 475–487. https://doi.org/10.1029/92JB01828
Lockner, D. A., & Madden, T. R. (1991). A multiple-crack model of brittle fracture: 2. Time-dependent simulations. Journal of Geophysical Research: Solid Earth, 96(B12), 19643–19654. https://doi.org/10.1029/91JB01641
Lomnitz, C. (1956). Creep measurements in igneous rocks. The Journal of Geology, 64(5), 473–479. https://doi.org/10.1086/626379
Ma, L., Wang, M., Zhang, N., Fan, P., & Li, J. (2017). A variable-parameter creep damage model incorporating the effects of loading frequency for rock salt and its application in a bedded storage cavern. Rock Mechanics and Rock Engineering, 50, 2495. https://doi.org/10.1007/s00603-017-1236-9
Paliwal, B., & Ramesh, K. T. (2008). An interacting micro-crack damage model for failure of brittle materials under compression. Journal of the Mechanics and Physics of Solids, 56(3), 896–923. https://doi.org/10.1016/j.jmps.2007.06.012
Olson, J. (1993). Joint pattern development: Effects of subcritical crack growth and mechanical crack interaction. Journal of Geophysical Research, 98, 12251–12265. https://doi.org/10.1029/93JB00779
Pellet, F., Hajdu, A., Deleruyelle, F., & Besnus, F. (2005). A viscoplastic model including anisotropic damage for the time dependent behaviour of rock. International Journal for Numerical and Analytical Methods in Geomechanics, 29(9), 941–970. https://doi.org/10.1002/nag.450
Perzyna, P. (1966). Fundamental problems in viscoplasticity. Advances in Applied Mechanics, 9, 246–377. https://doi.org/10.1016/S0065-2156(08)70009-7
Robertson, E. C. (1955). Experimental study of the strength of rocks. GSA Bulletin, 66(10), 1275–1314. https://doi.org/10.1130/0016-7606(1955)66[1275:ESOTSO]2.0.CO;2
Shao, J.-F., Chau, K. T., & Feng, X. T. (2006). Modeling of anisotropic damage and creep deformation in brittle rocks. International Journal of Rock Mechanics and Mining Sciences, 43, 82–592. https://doi.org/10.1016/j.ijrmms.2005.10.004
Sterpi, D., & Gioda, G. (2009). Visco-plastic behaviour around advancing tunnels in squeezing rock. Rock Mechanics and Rock Engineering, 42(2), 319–339. https://doi.org/10.1007/s00603-007-0137-8
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. https://doi.org/10.1016/j.tust.2018.12.019