Entire Life Theoretical Model of Limestone under Unequal Cyclic Loading Based on the Expanding Theory of Thermodynamic System Analysis
Publication: International Journal of Geomechanics
Volume 23, Issue 8
Abstract
Breakage, Helmholtz free energy, and nonlinearity are involved in many fundamental phenomena of complex systems across natural sciences. However, a mathematical equation that can express the entire life cycle of complex natural objects (such as rock-like quasi-brittle materials) is lacking. We expand the material body system to an isolated compound thermodynamic system to establish an innovative theoretical model induced by coupling nonlinear separation of Helmholtz free energy and breakage evolution that can be used to express the extreme entire life model of rock under unequal amplitude loading and unloading cycle. We gain crucial insights into the life essence of limestone, which is termed “Negative Dissipation.” This study first shows that the change in the mechanical properties of quasi-brittle materials caused by the timely evolution of breakage can be represented by the nonlinear separation of Helmholtz free energy and negative dissipation. An analytical solution to the nonlinear separation variables of Helmholtz free energy is provided by combining the method of solving nonlinear partial differential equations in mathematics and thermodynamic law. An analytical solution of Helmholtz free energy considering nonlinearity and breakage is proposed, and an equation that can reflect the constitutive mechanics law of the entire life cycle of rock in the theoretical model is presented. Theoretical results are consistent with the experimental data obtained from limestone samples with different prefabricated cracks. This original study provides a theoretical foundation for the life model of complex natural objects for nonlinear breakage and an early warning investigation of rocks under various unprecedented conditions.
Practical Applications
This study first found that the change in the mechanical properties of complex natural objects caused by the timely evolution of breakage can be represented by the nonlinear separation of Helmholtz free energy. We establish an innovative theoretical model induced by coupling nonlinear separation of Helmholtz free energy and breakage evolution. The innovative theory includes three parts: (1) an analytical solution to the nonlinear separation variables of Helmholtz free energy, (2) an analytical solution of Helmholtz free energy considering nonlinearity and breakage, and (3) an equation of the theoretical model that can reflect the constitutive mechanics’ law of the entire life of quasi-brittle rocks is presented for the first time. This original research result provides the foundation for a more in-depth life cycle of quasi-brittle rocks to develop the theoretical basis for the nonlinear breakage and early warning research of materials under various unprecedented conditions. We report a route to material microstructure composition, which may open an alternative pathway to quasi-brittle materials, which may, in turn, open a door into the mysterious world of science.
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Acknowledgments
This work was supported by the Fundamental Research Funds for the Universities of Henan Province (No. NSFRF220443) and the Natural Science Foundation of China (No. 51404095).
Notation
The following symbols are used in this paper:
- B′, B
- breakage;
- E
- total internal energy;
- Er
- breakage energy;
- K
- weighting coefficient;
- Qk
- heat dissipated;
- S, Sh
- entropy;
- T
- temperature;
- U, Eh
- internal energy;
- displacement;
- Ω
- potential energy;
- Ψ
- Helmholtz free energy;
- Ψr
- elastic energy;
- Ψc
- energy consumed during the fracture process;
- Φ
- dissipated energy;
- increment dissipated energy;
- strain; and
- θr
- parameter of the Helmholtz free energy separation.
References
Baroni, S., S. D. Gironcoli, A. D. Corso, and P. Giannozzi. 2001. “Phonons and related crystal properties from density-functional perturbation theory.” Rev. Mod. Phys. 73: 515–562. https://doi.org/10.1103/RevModPhys.73.515.
Borja, R. I., K. M. Sama, and P. F. Sanz. 2003. “On the numerical integration of three-invariant elastoplastic constitutive models.” Comput. Methods Appl. Mech. Eng. 192 (9): 1227–1258. https://doi.org/10.1016/S0045-7825(02)00620-5.
Chang, C. S., and Y. B. Deng. 2022. “Energy equation and stress–dilatancy relationship for sand.” Acta Geotech. 17: 2675–2696. https://doi.org/10.1007/s11440-021-01389-1.
Chen, L., C. P. Wang, J. F. Liu, J. Liu, J. Wang, Y. Jia, and J. F. Shao. 2015. “Damage and plastic deformation modeling of Beishan granite under compressive stress conditions.” Rock Mech. Rock Eng. 48 (4): 1623–1633. https://doi.org/10.1007/s00603-014-0650-5.
Cui, B. Y., and E. Terentjev. 2020. “Comparison of the Helmholtz, Gibbs, and collective-modes methods to obtain nonaffine elastic constants.” J. Mech. Phys. Solids 140: 103954. https://doi.org/10.1016/j.jmps.2020.103954.
Domínguez, C., et al. 2020. “Length scales of interfacial coupling between metal and insulator phases in oxides.” Nat. Mater. 19 (11): 1182–1188. https://doi.org/10.1038/s41563-020-0757-x.
Einav, I. 2007a. “Breakage mechanics— Part I: Theory.” J. Mech. Phys. Solids 55: 1274–1297. https://doi.org/10.1016/j.jmps.2006.11.003.
Einav, I. 2007b. “Breakage mechanics—Part II: Modelling granular materials.” J. Mech. Phys. Solids 55: 1298–1320. https://doi.org/10.1016/j.jmps.2006.11.004.
Freitas, R., M. Asta, and M. D. Koning. 2016. “Nonequilibrium free-energy calculation of solids using LAMMPS.” Comput. Mater. Sci. 112: 333–341. https://doi.org/10.1016/j.commatsci.2015.10.050.
Ghasemi, A., and W. Gao. 2020. “A method to predict energy barriers in stress modulated solid–solid phase transitions.” J. Mech. Phys. Solids 137: 103857. https://doi.org/10.1016/j.jmps.2019.103857.
Gong, F. Q., S. Luo, Q. Jiang, and L. Xu. 2022. “Theoretical verification of the rationality of strain energy storage index as rockburst criterion based on linear energy storage law.” J. Rock Mech. Geotech. Eng. 14 (6): 1737–1746. https://doi.org/10.1016/j.jrmge.2021.12.015.
Griffith, A. A. 1924. “The theory of rupture.” In Proc., 1st Int. Congress for Applied Mechanics, edited by C. B. Biezeno and J. M. Burgers, 55–63. Delft, Netherlands: International Union of Theoretical and Applied Mechanics.
Herbold, E. B., M. A. Homel, and M. B. Rubin. 2020. “A thermomechanical breakage model for shock-loaded granular media.” J. Mech. Phys. Solids 137: 103813. https://doi.org/10.1016/j.jmps.2019.103813.
Jiang, T., and J. F. Shao. 2012. “Micromechanical analysis of the nonlinear behavior of porous geomaterials based on the fast Fourier transform.” Comput Geotech. 46: 69–74. https://doi.org/10.1016/j.compgeo.2012.05.018.
Kittel, C. 2004. Introduction to solid state physics. Berkeley, CA: Univ. of California.
Kyrlidis, A., and R. A. Brown. 1993. “Model for density-functional thermodynamic perturbation analysis of Lennard–Jones solids.” Phys. Rev. E 47 (1): 427–438. https://doi.org/10.1103/PhysRevE.47.427.
Leblond, J. B., A. Karma, L. Ponson, and A. Vasudevan. 2019. “Configurational stability of a crack propagating in a material with mode-dependent fracture energy—Part I: Mixed-mode I + III.” J. Mech. Phys. Solids 126: 187–203. https://doi.org/10.1016/j.jmps.2019.02.007.
Li, J. L., et al. 2020. “Grain-orientation-engineered multilayer ceramic capacitors for energy storage applications.” Nat. Mater. 19 (9): 999–1005. https://doi.org/10.1038/s41563-020-0704-x.
Li, W. T., S. C. Li, X. D. Feng, S. C. Li, and C. Yuan. 2011. “Study of post-peak strain softening mechanical properties of rock based on Mohr–Coulomb criterion.” [In Chinese.] Chin. J. Rock Mech. Eng. 30 (7): 1460–1466.
Li, X. L., H. K. Chen, and J. H. Zhang. 2022. “Statistical damage model for whole deformation and failure process of rock considering initial void closure.” [In Chinese.] J. Southwest Jiaotong Univ. 57 (2): 314–321. https://doi.org/10.3969/j.issn.0258-2724.20200220.
Lieou, C., and C. Bronkhorst. 2020. “Thermodynamic theory of crystal plasticity: Formulation and application to polycrystal FCC copper.” J. Mech. Phys. Solids 138: 103905. https://doi.org/10.1016/j.jmps.2020.103905.
Liu, Y., and F. Dai. 2021. “A review of experimental and theoretical research on the deformation and failure behavior of rocks subjected to cyclic loading.” J. Rock Mech. Geotech. Eng. 13: 1203–1230. https://doi.org/10.1016/j.jrmge.2021.03.012.
Loghin, A., and P. F. Joseph. 2020. “Mixed mode fracture in power law hardening materials for plane stress.” J. Mech. Phys. Solids 139: 103890. https://doi.org/10.1016/j.jmps.2020.103890.
Lu, D. C., X. L. Du, G. S. Wang, A. N. Zhou, and A. K. Li. 2016. “A three dimensional elastoplastic constitutive model for concrete.” Comput. Struct. 163: 41–55. https://doi.org/10.1016/j.compstruc.2015.10.003.
Ma, J. F., X. You, S. Sun, X. X. Wang, S. Qin, and S. F. Sui. 2020. “Structural basis of energy transfer in Porphyridium purpureum phycobilisome.” Nature 579: 146–185. https://doi.org/10.1038/s41586-020-2020-7.
Manjunath, G. L., and B. Jha. 2019. “Geomechanical characterization of Gondwana shale across nano–micro–meso scales.” Int. J. Rock Mech. Min. Sci. 119: 35–45. https://doi.org/10.1016/j.ijrmms.2019.04.003.
Mao, S. T., P. Z. Pan, P. Konicek, P. Y. Yu, and K. L. Liu. 2021. “Rock damage and fracturing induced by high static stress and slightly dynamic disturbance with acoustic emission and digital image correlation techniques.” J. Rock Mech. Geotech. 13: 1002–1019. https://doi.org/10.1016/j.jrmge.2021.05.001.
Marinelli, F., and G. Buscarnera. 2019. “Anisotropic breakage mechanics: From stored energy to yielding in transversely isotropic granular rocks.” J. Mech. Phys. Solids 129: 1–18. https://doi.org/10.1016/j.jmps.2019.04.013.
Nguyen, G. D., A. M. Korsunsky, and J. P. Belnoue. 2015. “A nonlocal coupled damage-plasticity model for the analysis of ductile failure.” Int. J. Plast. 64: 56–75. https://doi.org/10.1016/j.ijplas.2014.08.001.
Petryk, H. 2020. “A quasi-extremal energy principle for non-potential problems in rate-independent plasticity.” J. Mech. Phys. Solids 136: 103691. https://doi.org/10.1016/j.jmps.2019.103691.
Saez, P., S. Eppell, R. Ballarini, and J. R. Matas. 2020. “A complementary energy approach accommodates scale differences in soft tissues.” J. Mech. Phys. Solids 138: 103895. https://doi.org/10.1016/j.jmps.2020.103895.
Sallarès, V., and C. Ranero. 2019. “Upper-plate rigidity determines depth varying rupture behaviour of megathrust earthquakes.” Nature 576: 96–114. https://doi.org/10.1038/s41586-019-1784-0.
Schrödinger, E. 1940. What is life?: With mind and matter and autobiographical sketches. Cambridge, MA: Cambridge University Press.
Shen, H., Y. Q. Liu, B. Liu, H. B. Li, D. H. Wu, and B. Peng. 2021. “Nonlinear theoretical model for describing shear mechanical behaviors of rock joints.” [In Chinese.] Chin. J. Rock Mech. Eng. 40 (12): 2421–2433. https://doi.org/10.13722/j.cnki.jrme.2021.0661.
Szilard, L. 1929. “On the decrease of entropy in a thermodynamic system by the intervention of intelligent beings.” J. Soc. Gen. Syst. Res. 53: 840–856.
Tang, H. D. 2020. “Multi-scale crack propagation and damage acceleration during uniaxial compression of marble.” Int. J. Rock Mech. Min. Sci. 131: 104330. https://doi.org/10.1016/j.ijrmms.2020.104330.
Tang, H. D., and M. L. Zhu. 2020. “Timely spread characteristics of micro- and meso-scale fractures based on SEM experiment.” Environ. Earth Sci. 79: 484. https://doi.org/10.1007/s12665-020-09224-8.
Tang, H. D., and M. L. Zhu. 2021. “Multiple expansion and non-linear breakage mechanism of local micro-fractures in limestone with energy storage and separation under cyclic loading.” J. Constr. Build. Mater. 292: 123469. https://doi.org/10.1016/j.conbuildmat.2021.123469.
Vasoya, M., B. Kondori, A. A. Benzerga, and A. Needleman. 2020. “Energy dissipation rate and kinetic relations for Eshelby transformations.” J. Mech. Phys. Solids 136: 103699. https://doi.org/10.1016/j.jmps.2019.103699.
Vasudevan, A., L. Ponson, A. Karma, and J. B. Leblond. 2020. “Configurational stability of a crack propagating in a material with mode-dependent fracture energy—Part II: Drift of fracture facets in mixed-mode I + II + III.” J. Mech. Phys. Solids 137: 103894. https://doi.org/10.1016/j.jmps.2020.103894.
Wang, H. T., M. M. He, Z. Q. Zhang, and J. W. Zhu. 2022. “Determination of the constant mi in the Hoek–Brown criterion of rock based on drilling parameters.” Int. J. Min. Sci. Technol. 32 (4): 747–759. https://doi.org/10.1016/j.ijmst.2022.06.002.
Wang, Y., W. H. Tan, D. Q. Liu, Z. Q. Hou, and C. H. Li. 2019. “On anisotropic fracture evolution and energy mechanism during marble failure under uniaxial deformation.” Rock Mech. Rock Eng. 52 (10): 3567–3583. https://doi.org/10.1007/s00603-019-01829-1.
Wang, Y., Q. H. Zhao, Y. G. Xiao, and Z. Q. Hou. 2020. “Influence of time-lagged unloading paths on fracture behaviors of marble using energy analysis and post-test CT visualization.” Environ. Earth Sci. 79: 217. https://doi.org/10.1007/s12665-020-08945-0.
Xia, K. Z., C. X. Chen, K. Yang, C. Sun, X. Liu, and Y. Zhou. 2022a. “Influence of relative humidity on the non-linear failure and stability of gypsum mines.” Eur J Environ Civ Eng. 26 (4): 1518–1543. https://doi.org/1-26.10.1080/19648189.2020.1715850.
Xia, K. Z., C. X. Chen, T. Wang, Y. Zheng, and Y. Wang. 2022b. “Estimating the geological strength index and disturbance factor in the Hoek–Brown criterion using the acoustic wave velocity in the rock mass.” Eng. Geol. 306: 106745. https://doi.org/10.1016/j.enggeo.2022.106745.
Yildiz, D., M. Kisiel, U. Gysin, O. Gürlü, and E. Meyer. 2019. “Mechanical dissipation via image potential states on a topological insulator surface.” Nat. Mater. 18 (11): 1201–1207. https://doi.org/10.1038/s41563-019-0492-3.
You, M. Q., and Z. A. Hua. 2002. “Energy analysis on failure process of rock specimens.” [In Chinese.] Chin. J. Rock Mech. Eng. 21 (6): 778–781.
Yuan, X. P., H. Y. Liu, and Z. Q. Wang. 2013. “An interacting crack mechanics based model for elastoplastic damage model of rock-like materials under compression.” Int. J. Rock Mech. Min. Sci. 58: 92–102. https://doi.org/10.1016/j.ijrmms.2012.09.007.
Zhang, J. Z., and X. P. Zhou. 2020. “AE event rate characteristics of flawed granite: From damage stress to ultimate failure.” Geophys. J. Int. 222 (2): 795–814. https://doi.org/10.1093/gji/ggaa207.
Zhang, Y. D., and G. Buscarnera. 2018. “Breakage mechanics for granular materials in surface-reactive environments.” J. Mech. Phys. Solids 112: 89–108. https://doi.org/10.1016/j.jmps.2017.11.008.
Zhang, Y. D., G. Buscarnera, and I. Einav. 2016. “Grain size dependence of yielding in granular soils interpreted using fracture mechanics, breakage mechanics and Weibull statistics.” Géotechnique 66: 149–160. https://doi.org/http://dx.doi.org/10.1680/jgeot.15.P.119.
Zhao, L. Y., W. L. Zhang, Y. M. Lai, F. J. Niu, Q. Z. Zhu, and J. F. Shao. 2021. “A heuristic elastoplastic damage constitutive modeling method for geomaterials: From strength criterion to analytical full-spectrum stress–strain curves.” Int. J. Geomech. 21 (2): 04020255. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001902.
Zhou, H. W., Z. H. Wang, C. S. Wang, and J. F. Liu. 2019. “On acoustic emission and post-peak energy evolution in Beishan granite under cyclic loading.” Rock Mech. Rock Eng. 52 (1): 283–288. https://doi.org/10.1007/s00603-018-1614-y.
Zong, M., Y. Zu, J. Guo, Z. Zhang, J. Liu, Y. Ge, J. Liu, and L. Su 2021. “Broadband nonlinear optical response of graphdiyne for mid-infrared solid-state lasers.” Sci. China Phys. Mech. Astron. 64: 294214. https://doi.org/10.1007/s11433-021-1720-3.
Zuo, G. Z., M. Linares, T. Upreti, and M. Kemerink. 2019. “General rule for the energy of water-induced traps in organic semiconductors.” Nat. Mater. 18 (6): 588–596. https://doi.org/10.1038/s41563-019-0347-y.
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Published In
International Journal of Geomechanics
Volume 23 • Issue 8 • August 2023
Copyright
© 2023 American Society of Civil Engineers.
History
Received: Jun 29, 2022
Accepted: Feb 20, 2023
Published online: Jun 13, 2023
Published in print: Aug 1, 2023
Discussion open until: Nov 13, 2023
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