Abstract
Rate- and state-dependent friction law for velocity-step tests is analyzed from a thermodynamic point of view. A simple macroscopic non-equilibrium thermodynamic model with a single internal variable reproduces instantaneous jump and relaxation. Velocity weakening appears as a consequence of a plasticity related nonlinear coefficient. Permanent part of displacement corresponds to plastic strain, and relaxation effects are analogous to creep in thermodynamic rheology.
Similar content being viewed by others
References
Aki K, Richards PG (2002) Quantitative seismology, 2nd edn. University Science Books, Sausalito
Baumberger T, Caroli C (2006) Solid friction from stick-slip down to pinning and aging. Adv Phys 55:279–348. doi:10.1080/00018730600732186
Bowden FP, Tabor D (1950) The friction and lubrication of solids. Clarendon Press, Oxford
Brechet Y, Estrin Y (1994) The effect of strain rate sensitivity on dynamic friction of metals. Scr Metall et Mater 30:1449–1454. doi:10.1016/0956-716X(94)90244-5
Cimmelli VA, Jou D, Ruggeri T, Ván P (2014) Conceptual analysis of the entropy principle in continuum physics.
de Groot SR, Mazur P (1962) Non-equilibrium thermodynamics. North-Holland Publishing Company, Amsterdam
Dieterich JH (1972) Time-dependent friction in rocks. J Geophys Res 77:3690–3697. doi:10.1029/JB077i020p03690
Dieterich JH (1978) Time-dependent friction and the mechanics of stick-slip. Pure Appl Geophys 116:790–806. doi:10.1007/BF00876539
Dieterich JH (1979) Modeling of rock friction 1. experimental results and constitutive equations. J Geophys Res 84:2161–2168. doi:10.1029/JB084iB05p02161
Eyring H (1935) The activated complex in chemical reactions. J Chem Phys 3:107–115. doi:10.1063/1.1749604
Eyring H (1936) Viscosity, plasticity, and diffusion as examples of absolute reaction rates. J Chem Phys 4:283–291. doi:10.1063/1.1749836
Fülöp T (2008) Rheological circuit diagrams. In: Asszonyi Cs (ed) Material properties of isotropic continua, Pocket Library in Engineering Geology and Rock Mechanics, vol 6, chap 3, BME Publisher, Budapest, pp 93–120 (in Hungarian)
Fülöp T, Ván P (2012) Kinematic quantities of finite elastic and plastic deformations. Math Meth Appl Sci 35:1825–1841. doi:10.1002/mma.2558
Fülöp T, Asszonyi C, Ván P (2014) Distinguished rheological models in the framework of a thermodynamical internal variable theory
Heslot F, Baumberger T, Perrin B, Caroli B, Caroli C (1994) Creep, stick-slip, and dry-friction dynamics: experiments and a heuristic model. Phys Rev E 49:4973–4988. doi:10.1103/PhysRevE.49.4973
Houlsby GT, Puzrin AM (2006) Principles of hyperplasticity (An approach to plasticity theory based on thermodynamic principles). Springer, London
Kato N, Tullis TE (2001) A composite rate- and state-dependent law for rock friction. Geophys Res Lett 28:1103–1106. doi:10.1029/2000GL012060
Kawamura H, Hatano T, Kato N, Biswas S, Chakrabarti BK (2012) Statistical physics of fracture, friction, and earthquakes. Rev Mod Phys 84:839–884. doi:10.1103/RevModPhys.84.839
Lin W, Kuwahara Y, Satoh T, Shigematsu N, Kitagawa Y, Kiguchi T, Sato T, Tsukamoto H, Itaba S, Umeda Y, Seki Y, Takahashi M, Matsumoto N, Koizumi N (2010) A case study of 3d stress orientation determination in Shikoku Island and Kii Peninsula, Japan. In: Vrkljan I (ed) Rock engineering in difficult ground conditions (soft rock and karst). Balkema, London, pp 277–282
Linker MF, Dieterich JH (1992) Effects of variable normal stress on rock friction. J Geophys Res 97(B4):4923–4940
Mair K, Marone C (1999) Friction of simulated fault gouge for a wide range of velocities and normal stresses. J Geophys Res 104(B12):28899–28914
Marone C (1998) Laboratory-derived friction laws and their application to seismic faulting. Annu Rev Earth Planet Sci 26:643–696. doi:10.1146/annurev.earth.26.1.643
Marone C, Raleigh CB, Scholz CH (1990) Frictional behavior and constitutive modeling of simulated fault gouge. J Geophys Res 95:7007–7025
Matolcsi T (2005) Ordinary thermodynamics. Akadémiai Kiadó (Publishing House of the Hungarian Academy of Sciences), Budapest
Matsuki K (2008) Anelastic strain recovery compliance of rocks and its application to in situ stress measurement. Int J Rock Mech Min Sci 45:952–965. doi:10.1016/j.ijrmms.2007.10.005
Maugin G (1999) The thermomechanics of nonlinear irreversible behaviors: an introduction. World Scientific, Singapore-New Jersey-London-Hong Kong
Maugin GA, Muschik W (1994) Thermodynamics with internal variables. part i. Gen concepts. J Non-Equilib Thermodyn 19:217–249. doi:10.1515/jnet.1994.19.3.217
Maugin GA, Muschik W (1994) Thermodynamics with internal variables. Part II Applications. J Non-Equilib Thermodyn 19:250–289. doi:10.1515/jnet.1994.19.3.250
Nagata K, Nakatani M, Yoshida S (2012) A revised rate- and state-dependent friction law obtained by constraining constitutive and evolution laws separately with laboratory data. J Geophys Res 117:B02314. doi:10.1029/2011JB008818
Nakatani M (2001) Conceptual and physical clarification of rate and state friction: frictional sliding as a thermally activated rheology. J Geophys Res 106:13,347–13,380. doi:10.1029/2000JB900453
Perrin G, Rice JR, Zheng G (1995) Self-healing slip pulse on a frictional surface. J Mech Phys Solids 43:1461–1495. doi:10.1016/0022-5096(95)00036-I
Polrier J-P (1985) Creep of Crystals. Cambridge University Press, New York
Putelat T, Dawes JHP, Willis JR (2011) On the microphysical foundations of rate-and-state friction. J Mech Phys Solids 59:1062–1075. doi:10.1016/j.jmps.2011.02.002
Ruina A (1983) Slip instability and state variable friction laws. J Geophys Res 88:10359–10370. doi:10.1029/JB088iB12p10359
Rusinko A, Rusinko K (2011) Plasticity and Creep of Metals. Springer, Berlin
Ván P (2010) Thermodynamics of plasticity. In: Fülöp T (ed) Time and space derivatives in material laws, Pocket Library in Engineering Geology and Rock Mechanics, Vol. 10, BME Publisher, Budapest, pp 15–50 (in Hungarian)
Vásárhelyi B (1998) Influence of normal load on joint dilatation rate. Rock Mech Rock Eng 31(2):117–123
Vásárhelyi B, Ván P (2006) Shearing test with continuously increasing normal stress. Period Polytech Ser Civil Eng 50(2):171–180
Verhás J (1997) Thermodynamics and Rheology. Akadémiai Kiadó and Kluwer Academic Publisher, Budapest
Ziegler H (1983) An introduction to thermomechanics, 2nd edn. North-Holland Publishing Company, Amsterdam-New York-Oxford
Acknowledgments
The authors thank to Tamás Fülöp for valuable discussions. This manuscript was greatly improved by the comments of anonymous reviewers. The work was supported by the Grant Otka K81161 and K104260. N. Mitsui was supported by Canon Foundation in Europe.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Mitsui, N., Ván, P. Thermodynamic aspects of rock friction. Acta Geod Geophys 49, 135–146 (2014). https://doi.org/10.1007/s40328-014-0048-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40328-014-0048-6