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Modelling of earthquakes by using Rate-and-State friction laws on the Burridge-Knopoff spring-block system

Year 2019, Volume: 6 Issue: 2, 115 - 127, 01.11.2019
https://doi.org/10.9733/JGG.2019R0010.T

Abstract

Although physical
mechanism of earthquakes has not been clearly answered yet, it can be explained
substantially with friction laws when the slip event subjects to the brittle
crust. In such cases earthquakes are a result of frictional instability
accompanied by stick-and-slip motion. Earthquakes, foreshocks, aftershocks,
slow slip events have been modelled by using Rate-And-State Friction (RSF)
laws. In this study Dieterich, Ruina and Perrin type RSF laws have been studied
on the Burridge-Knopoff (BK) spring-block system, which was originally proposed
with a velocity dependent friction law. In order to comply with the reality,
fault geometry and its physical structure are chosen appropriate to the San
Andreas/Parkfield fault. Since the proposed systems are stiff nonlinear
dynamics, they are offered with numeric procedure adapted to solve stiff
differential equations. By applying stability analysis, the critical boundaries
between stable and unstable sliding (seismic cycle) are determined. The model
is simulated by tuning the RSF law parameters for unstable sliding regime. As a
result of the studies it has been found that, the magnitude of the slip event
is proportional to the distance from the curve which separates stable and
unstable sliding regimes. Besides, when system parameters deviates with a fixed
amount from the stability curve, the system shows the same dynamics. To the
best of my knowledge, the defined criterion is being published for the first
time within the scope of this work. This study will pave the way for further
researches of earthquake and weak triggering effects.

References

  • Barbot, S., Fialko, Y., & Bock, Y. (2009). Postseismic deformation due to the Mw 6.0 2004 Parkfield earthquake: Stress‐driven creep on a fault with spatially variable rate‐and‐state friction parameters. Journal of Geophysical Research: Solid Earth, 114(B7).
  • Belardinelli, M. E., Bizzarri, A., & Cocco, M. (2003). Earthquake triggering by static and dynamic stress changes. Journal of Geophysical Research: Solid Earth, 108(B3).
  • Burridge, R., & Knopoff, L. (1967). Model and theoretical seismicity. Bulletin of the seismological society of america, 57(3), 341-371.
  • Chang, S. H., Avouac, J. P., Barbot, S., & Lee, J. C. (2013). Spatially variable fault friction derived from dynamic modeling of aseismic afterslip due to the 2004 Parkfield earthquake. Journal of Geophysical Research: Solid Earth, 118(7), 3431-3447.
  • Dieterich, J. H. (1979). Modeling of rock friction: 1. Experimental results and constitutive equations. Journal of Geophysical Research: Solid Earth, 84(B5), 2161-2168.
  • Erickson, B., Birnir, B., & Lavallée, D. (2008). A model for aperiodicity in earthquakes. Nonlinear Processes in Geophysics, 15(1), 1-12.
  • Erickson, B., Birnir, B., & Lavallée, D. (2011). Periodicity, chaos and localization in a Burridge–Knopoff model of an earthquake with rate-and-state friction. Geophysical Journal International, 187(1), 178-198.
  • Gomberg, J., Blanpied, M. L., & Beeler, N. M. (1997). Transient triggering of near and distant earthquakes. Bulletin of the Seismological Society of America, 87(2), 294-309.
  • Gu, J. C., Rice, J. R., Ruina, A. L., & Simon, T. T. (1984). Slip motion and stability of a single degree of freedom elastic system with rate and state dependent friction. Journal of the Mechanics and Physics of Solids, 32(3), 167-196.
  • Helmstetter, A., & Shaw, B. E. (2009). Afterslip and aftershocks in the rate‐and‐state friction law. Journal of Geophysical Research: Solid Earth, 114(B1).
  • Johnson, K. M., Burgmann, R., & Larson, K. (2006). Frictional properties on the San Andreas fault near Parkfield, California, inferred from models of afterslip following the 2004 earthquake. Bulletin of the Seismological Society of America, 96(4B), S321-S338.
  • Kawamura, H., Ueda, Y., Kakui, S., Morimoto, S., & Yamamoto, T. (2017). Statistical properties of the one-dimensional Burridge-Knopoff model of earthquakes obeying the rate-and state-dependent friction law. Physical Review E, 95(4), 042122.
  • Marone, C. (1998). Laboratory-derived friction laws and their application to seismic faulting. Annual Review of Earth and Planetary Sciences, 26(1), 643-696.
  • 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. Journal of Geophysical Research: Solid Earth, 117(B2).
  • Nakatani, M. (2001). Conceptual and physical clarification of rate and state friction: Frictional sliding as a thermally activated rheology. Journal of Geophysical Research: Solid Earth, 106(B7), 13347-13380.
  • Perrin, G., Rice, J. R., & Zheng, G. (1995). Self-healing slip pulse on a frictional surface. Journal of the Mechanics and Physics of Solids, 43(9), 1461-1495.
  • Petzold, L. (1983). Automatic selection of methods for solving stiff and nonstiff systems of ordinary differential equations. SIAM journal on scientific and statistical computing, 4(1), 136-148.
  • Roy, M., & Marone, C. (1996). Earthquake nucleation on model faults with rate‐and state‐dependent friction: Effects of inertia. Journal of Geophysical Research: Solid Earth, 101(B6), 13919-13932.
  • Ruina, A. (1983). Slip instability and state variable friction laws. Journal of Geophysical Research: Solid Earth, 88(B12), 10359-10370.
  • Savage, J. C., & Langbein, J. (2008). Postearthquake relaxation after the 2004 M6 Parkfield, California, earthquake and rate‐and‐state friction. Journal of Geophysical Research: Solid Earth, 113(B10).
  • Scholz, C. H. (1998). Earthquakes and friction laws. Nature, 391(6662), 37.
  • Scholz, C. H. (2002). The mechanics of earthquakes and faulting. Cambridge university press.
  • Strogatz, S. H. (2018). Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. CRC Press.
  • Tullis, T. E. (1996). Rock friction and its implications for earthquake prediction examined via models of Parkfield earthquakes. Proceedings of the National Academy of Sciences, 93(9), 3803-3810.
  • Turcotte, D., & Schubert, G. (2014). Geodynamics. Cambridge university press.

Hız-ve-durum sürtünme yasaları ve Burridge-Knopoff yay blok sistemi kullanılarak depremlerin dinamik modellenmesi

Year 2019, Volume: 6 Issue: 2, 115 - 127, 01.11.2019
https://doi.org/10.9733/JGG.2019R0010.T

Abstract

Depremlerin fiziksel oluşum
mekanizmaları henüz tam anlamıyla bilinememekle birlikte, kırılgan kabukta
gerçekleştiği durumda büyük ölçüde sürtünme yasaları ile açıklanabilmektedir.
Bu durumda depremler, tutma-bırakma hareketi sonucu oluşan sürtünme
kararsızlığının (frictional instability) bir sonucudur. Hız-ve-Durum yasaları (Rate-and-State
Friction law, RSF) ile doğadaki deprem olaylarına benzer artçı depremler, yavaş
depremler, sismik ve sismik olmayan hareketler modellenebilmektedir. Bu
çalışmada Dieterich, Ruina ve Perrin tipi RSF yasaları tek serbestlik dereceli
Burridge-Knopoff (BK) yay-blok sistemine entegre edilerek irdelenmiştir.
Modellemenin gerçekçi olması bakımından fay geometrisi ve fiziksel yapısı San Andreas/Parkfield
fayına uygun olarak belirlenmiştir. Çalışmada kullanılan dinamik sistemler
doğrusal olmayan sert (stiff) diferansiyel denklemlerden oluşmaktadır. Bu
nedenle önerilen modellerin doğrusal olmayan karakteri ile çözümü için nümerik
öneriler sunulmuştur. Modellere kararlılık analizi uygulanmış ve sistemin
sürtünme kararlılığı (sismik olmayan hareket) ve kararsızlığı (sismik döngü)
sergilediği kritik bölgeler belirlenmiştir. RSF parametre uzayı değiştirilerek
sadece sürtünme kararsızlığı sergilediği durumlar için sistem simüle
edilmiştir. Yapılan çalışmalar sonucu RSF yasalarından kaynaklı sistemin
oluşturacağı dinamiklerin büyüklüğünün kararlılık eğrisinden sapma ile orantılı
olduğu bulunmuştur. Bu eğriden sabit oranda sapma olduğunda ise sistemin aynı
dinamikleri sergilediği görülmüştür. Yapılan literatür taramasında, bulunan
ölçütün ilk kez bu çalışma kapsamında elde edildiği belirlenmiştir. Bu çalışma,
ileride depremleri tetikleyen güçsüz sinyallerin araştırılmasına fayda
sağlayacaktır.


References

  • Barbot, S., Fialko, Y., & Bock, Y. (2009). Postseismic deformation due to the Mw 6.0 2004 Parkfield earthquake: Stress‐driven creep on a fault with spatially variable rate‐and‐state friction parameters. Journal of Geophysical Research: Solid Earth, 114(B7).
  • Belardinelli, M. E., Bizzarri, A., & Cocco, M. (2003). Earthquake triggering by static and dynamic stress changes. Journal of Geophysical Research: Solid Earth, 108(B3).
  • Burridge, R., & Knopoff, L. (1967). Model and theoretical seismicity. Bulletin of the seismological society of america, 57(3), 341-371.
  • Chang, S. H., Avouac, J. P., Barbot, S., & Lee, J. C. (2013). Spatially variable fault friction derived from dynamic modeling of aseismic afterslip due to the 2004 Parkfield earthquake. Journal of Geophysical Research: Solid Earth, 118(7), 3431-3447.
  • Dieterich, J. H. (1979). Modeling of rock friction: 1. Experimental results and constitutive equations. Journal of Geophysical Research: Solid Earth, 84(B5), 2161-2168.
  • Erickson, B., Birnir, B., & Lavallée, D. (2008). A model for aperiodicity in earthquakes. Nonlinear Processes in Geophysics, 15(1), 1-12.
  • Erickson, B., Birnir, B., & Lavallée, D. (2011). Periodicity, chaos and localization in a Burridge–Knopoff model of an earthquake with rate-and-state friction. Geophysical Journal International, 187(1), 178-198.
  • Gomberg, J., Blanpied, M. L., & Beeler, N. M. (1997). Transient triggering of near and distant earthquakes. Bulletin of the Seismological Society of America, 87(2), 294-309.
  • Gu, J. C., Rice, J. R., Ruina, A. L., & Simon, T. T. (1984). Slip motion and stability of a single degree of freedom elastic system with rate and state dependent friction. Journal of the Mechanics and Physics of Solids, 32(3), 167-196.
  • Helmstetter, A., & Shaw, B. E. (2009). Afterslip and aftershocks in the rate‐and‐state friction law. Journal of Geophysical Research: Solid Earth, 114(B1).
  • Johnson, K. M., Burgmann, R., & Larson, K. (2006). Frictional properties on the San Andreas fault near Parkfield, California, inferred from models of afterslip following the 2004 earthquake. Bulletin of the Seismological Society of America, 96(4B), S321-S338.
  • Kawamura, H., Ueda, Y., Kakui, S., Morimoto, S., & Yamamoto, T. (2017). Statistical properties of the one-dimensional Burridge-Knopoff model of earthquakes obeying the rate-and state-dependent friction law. Physical Review E, 95(4), 042122.
  • Marone, C. (1998). Laboratory-derived friction laws and their application to seismic faulting. Annual Review of Earth and Planetary Sciences, 26(1), 643-696.
  • 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. Journal of Geophysical Research: Solid Earth, 117(B2).
  • Nakatani, M. (2001). Conceptual and physical clarification of rate and state friction: Frictional sliding as a thermally activated rheology. Journal of Geophysical Research: Solid Earth, 106(B7), 13347-13380.
  • Perrin, G., Rice, J. R., & Zheng, G. (1995). Self-healing slip pulse on a frictional surface. Journal of the Mechanics and Physics of Solids, 43(9), 1461-1495.
  • Petzold, L. (1983). Automatic selection of methods for solving stiff and nonstiff systems of ordinary differential equations. SIAM journal on scientific and statistical computing, 4(1), 136-148.
  • Roy, M., & Marone, C. (1996). Earthquake nucleation on model faults with rate‐and state‐dependent friction: Effects of inertia. Journal of Geophysical Research: Solid Earth, 101(B6), 13919-13932.
  • Ruina, A. (1983). Slip instability and state variable friction laws. Journal of Geophysical Research: Solid Earth, 88(B12), 10359-10370.
  • Savage, J. C., & Langbein, J. (2008). Postearthquake relaxation after the 2004 M6 Parkfield, California, earthquake and rate‐and‐state friction. Journal of Geophysical Research: Solid Earth, 113(B10).
  • Scholz, C. H. (1998). Earthquakes and friction laws. Nature, 391(6662), 37.
  • Scholz, C. H. (2002). The mechanics of earthquakes and faulting. Cambridge university press.
  • Strogatz, S. H. (2018). Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. CRC Press.
  • Tullis, T. E. (1996). Rock friction and its implications for earthquake prediction examined via models of Parkfield earthquakes. Proceedings of the National Academy of Sciences, 93(9), 3803-3810.
  • Turcotte, D., & Schubert, G. (2014). Geodynamics. Cambridge university press.
There are 25 citations in total.

Details

Primary Language Turkish
Subjects Geological Sciences and Engineering (Other)
Journal Section Articles
Authors

Eyüp Sopacı 0000-0001-7265-4511

Publication Date November 1, 2019
Submission Date May 24, 2019
Published in Issue Year 2019 Volume: 6 Issue: 2

Cite

APA Sopacı, E. (2019). Hız-ve-durum sürtünme yasaları ve Burridge-Knopoff yay blok sistemi kullanılarak depremlerin dinamik modellenmesi. Jeodezi Ve Jeoinformasyon Dergisi, 6(2), 115-127. https://doi.org/10.9733/JGG.2019R0010.T
AMA Sopacı E. Hız-ve-durum sürtünme yasaları ve Burridge-Knopoff yay blok sistemi kullanılarak depremlerin dinamik modellenmesi. hkmojjd. November 2019;6(2):115-127. doi:10.9733/JGG.2019R0010.T
Chicago Sopacı, Eyüp. “Hız-Ve-Durum sürtünme Yasaları Ve Burridge-Knopoff Yay Blok Sistemi kullanılarak Depremlerin Dinamik Modellenmesi”. Jeodezi Ve Jeoinformasyon Dergisi 6, no. 2 (November 2019): 115-27. https://doi.org/10.9733/JGG.2019R0010.T.
EndNote Sopacı E (November 1, 2019) Hız-ve-durum sürtünme yasaları ve Burridge-Knopoff yay blok sistemi kullanılarak depremlerin dinamik modellenmesi. Jeodezi ve Jeoinformasyon Dergisi 6 2 115–127.
IEEE E. Sopacı, “Hız-ve-durum sürtünme yasaları ve Burridge-Knopoff yay blok sistemi kullanılarak depremlerin dinamik modellenmesi”, hkmojjd, vol. 6, no. 2, pp. 115–127, 2019, doi: 10.9733/JGG.2019R0010.T.
ISNAD Sopacı, Eyüp. “Hız-Ve-Durum sürtünme Yasaları Ve Burridge-Knopoff Yay Blok Sistemi kullanılarak Depremlerin Dinamik Modellenmesi”. Jeodezi ve Jeoinformasyon Dergisi 6/2 (November 2019), 115-127. https://doi.org/10.9733/JGG.2019R0010.T.
JAMA Sopacı E. Hız-ve-durum sürtünme yasaları ve Burridge-Knopoff yay blok sistemi kullanılarak depremlerin dinamik modellenmesi. hkmojjd. 2019;6:115–127.
MLA Sopacı, Eyüp. “Hız-Ve-Durum sürtünme Yasaları Ve Burridge-Knopoff Yay Blok Sistemi kullanılarak Depremlerin Dinamik Modellenmesi”. Jeodezi Ve Jeoinformasyon Dergisi, vol. 6, no. 2, 2019, pp. 115-27, doi:10.9733/JGG.2019R0010.T.
Vancouver Sopacı E. Hız-ve-durum sürtünme yasaları ve Burridge-Knopoff yay blok sistemi kullanılarak depremlerin dinamik modellenmesi. hkmojjd. 2019;6(2):115-27.