Advertisement

Friction Versus Damage: Dynamic Self-similar Crack Growth Revisited

Conference paper
Part of the Mathematics for Industry book series (MFI, volume 34)

Abstract

Seismological observational studies have revealed that earthquakes exhibit dynamic self-similar crack growth constituting 50–90% of the shear wave velocity. Remarkably, the peak slip velocity defined on the crack surface is scale-invariant, even from M1 to M9 earthquakes. However, a classical self-similar crack model with a singularity does not satisfy all the observed properties above. In this chapter, we review these discrepancies and introduce friction and damage models to solve them, which have been proposed in several numerical studies. We show that velocity-dependent friction can fulfill some requirements of the observations, while slip- or time-dependent friction cannot. We finally discuss the theoretical equivalence of friction and damage model for a self-similar crack in terms of energetics, which has previously only been implied by numerical studies.

Notes

Acknowledgments

This work was supported by JSPS KAKENHI Grant No. 17H02857.

References

  1. 1.
    Aki, K.: 4. Generation and propagation of G waves from the Niigata earthquake of June 16, 1964. Part 2. Estimation of earthquake moment, released energy, and stress-strain drop from the G wave spectrum. Bulletin of Earthquake Research Institute. University of Tokyo (1966)Google Scholar
  2. 2.
    Andrews, D.J.: Rupture velocity of plane strain shear cracks. J. Geophys. Res. 81(32), 5679–5687 (1976)CrossRefGoogle Scholar
  3. 3.
    Andrews, D.J.: Rupture models with dynamically determined breakdown displacement. Bull. Seismol. Soc. Am. 94(3), 769–775 (2004)CrossRefGoogle Scholar
  4. 4.
    Andrews, D.J.: Rupture dynamics with energy loss outside the slip zone. J. Geophys. Res. 110(B1), B01307 (2005)Google Scholar
  5. 5.
    Barenblatt, G.I.: Scaling. Cambridge University Press, Cambridge (2003)CrossRefGoogle Scholar
  6. 6.
    Broberg, K.B.: Cracks and Fracture. Academic Press (1999)Google Scholar
  7. 7.
    Chester, F.M., Chester, J.S.: Ultracataclasite structure and friction processes of the Punchbowl fault, San Andreas system California. Tectonophysics 295(1–2), 199–221 (1998)CrossRefGoogle Scholar
  8. 8.
    Cochard, A., Madariaga, R.: Dynamic faulting under rate-dependent friction. Pure Appl. Geophys. PAGEOPH 142(3–4), 419–445 (1994)CrossRefGoogle Scholar
  9. 9.
    Dunham, E.M., Belanger, D., Cong, L., Kozdon, J.E.: Earthquake ruptures with strongly rate-weakening friction and off-fault plasticity, Part 1: planar faults. Bull. Seismol. Soc. Am. 101(5), 2296–2307 (2011)CrossRefGoogle Scholar
  10. 10.
    Faulkner, D.R., Mitchell, T.M., Healy, D., Heap, M.J.: Slip on “weak” faults by the rotation of regional stress in the fracture damage zone. Nature 444(7121), 922–5 (2006)CrossRefGoogle Scholar
  11. 11.
    Geller, R.J.: Scaling relations for earthquake source parameters and magnitudes. Bull. Seismol. Soc. Am. 66(5), 1501–1523 (1976). Retrieved fromGoogle Scholar
  12. 12.
    Hirano, S., Yagi, Y.: Dependence of seismic and radiated energy on shorter wavelength components. Geophys. J. Int. 1585–1592 (2017)Google Scholar
  13. 13.
    Hirano, S.: Integral representation and its applications in earthquake mechanics: a review, In: van Meurs, P. (eds.) Mathematical Analysis of Continuum Mechanics and Industrial Applications II. CoMFoS et al.: Mathematics for Industry, vol. 30. Springer, Singapore (2016)Google Scholar
  14. 14.
    Hirano, S. (accepted article).: Résumé: Energy estimation of earthquake faulting processes. In: Singularity and Asymptotic Behavior of Solutions For Partial Differential Equations with Conservation Law: RIMS Kôkyûroku BessatsuGoogle Scholar
  15. 15.
    Houston, H.: Influence of depth, focal mechanism, and tectonic setting on the shape and duration of earthquake source time functions. J. Geophys. Res.: Solid Earth 106(B6), 11137–11150 (2001)CrossRefGoogle Scholar
  16. 16.
    Ida, Y.: Cohesive force across the tip of a longitudinal-shear crack and Griffith’s specific surface energy. J. Geophys. Res. 77(20), 3796–3805 (1972)Google Scholar
  17. 17.
    Ide, S., Baltay, A., Beroza, G.C.: Shallow dynamic overshoot and energetic deep rupture in the 2011 Mw9.0 Tohoku-Oki earthquake. Science 332(6036), 1426–1429 (2011)Google Scholar
  18. 18.
    Kanamaori, H.: The radiated energy of the 2004 sumatra-andaman earthquake. Geophys. Monograph Ser. 170, 59–68 (2006)Google Scholar
  19. 19.
    Kawakata, H., Cho, A., Kiyama, T., Yanagidani, T., Kusunose, K., Shimada, M.: Three-dimensional observations of faulting process in westerly granite under uniaxial and triaxial conditions by X-ray CT Scan. Tectonophysics 313, 293–305 (1999)CrossRefGoogle Scholar
  20. 20.
    Meier, M.A., Heaton, T., Clinton, J.: Evidence for universal earthquake rupture initiation behavior. Geophys. Res. Lett. 43(15), 7991–7996 (2016)CrossRefGoogle Scholar
  21. 21.
    Ohnaka, M., Yamashita, T.: A cohesive zone model for dynamic shear faulting based on experimentally inferred constitutive relation and strong motion source parameters. J. Geophys. Res. 94(B4), 4089–4104 (1989)CrossRefGoogle Scholar
  22. 22.
    Palmer, A.C., Rice, J.R.: The growth of slip surfaces in the progressive failure of over-consolidated clay. Proc. R. Soc. A: Math., Phys. Eng. Sci. 332(1591), 527–548 (1973)zbMATHGoogle Scholar
  23. 23.
    Uchide, T., Ide, S.: Scaling of earthquake rupture growth in the Parkfield area: Self-similar growth and suppression by the finite seismogenic layer. J. Geophys. Res.: Solid Earth 115(11), 1–15 (2010)Google Scholar
  24. 24.
    Udías, A., Madariaga, R., Buforn, E.: Source Mechanisms of Earthquakes: Theory and Practice. Cambridge University Press, Cambridge (2014)CrossRefGoogle Scholar
  25. 25.
    Yamada, T., Mori, J.J., Ide, S., Abercrombie, R.E., Kawakata, H., Nakatani, M., Iio, Y., Ogasawara, H.: Stress drops and radiated seismic energies of microearthquakes in a South African gold mine. J. Geophys. Res.: Solid Earth 112(3), 1–12 (2007)Google Scholar
  26. 26.
    Yamashita, T.: Generation of microcracks by dynamic shear rupture and its effects on rupture growth and elastic wave radiation. Geophys. J. Int. 143(2), 395–406 (2000)Google Scholar
  27. 27.
    Yoshimitsu, N., Kawakata, H., Takahashi, N.: Magnitude \(-7\) level earthquakes: a new lower limit of self-similarity in seismic scaling relationships. Geophys. Res. Lett. 41, 4495–4502 (2014)Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  1. 1.College of Science and EngineeringRitsumeikan UniversityKusatsu, ShigaJapan

Personalised recommendations