On the energy balance behind frictional ruptures...Palmer and Rice, Proc. Roy. Soc., 1973 Barras et...
Transcript of On the energy balance behind frictional ruptures...Palmer and Rice, Proc. Roy. Soc., 1973 Barras et...
On the energy balance behind frictional ruptures
Fabian Barras1
1The Njord Centre, Department of Geosciences, University of Oslo, Norway
Motivation
Fabian Barras Slide 1
โข Earthquake ruptures are driven by the dynamic weakening of frictional strength along crustal faults
โข โLab-earthquakeโ experiments revealed the crack-like properties of frictional ruptures.
(e.g Xia et al., Science, 2004Passelรจgue et al., Science, 2013Svetlizky et al., Nature, 2014Rubino et al., Nature Comm., 2017)
This work investigates the rupture energy controlling the propagation of frictional ruptures by analogy with fracture mechanics
Context
Fabian Barras Slide 2
โข Shear stress does not drop to zero in the wake of frictional ruptures (unlike shear cracks)
โข The energy controlling the rupture dynamics (โก the rupture energy ๐บ๐บ๐๐) is only a subpart of the frictional dissipation
โข Slip-weakening friction model predicts that ๐บ๐บ๐๐ corresponds to the breakdown energy ๐ธ๐ธ๐ต๐ต๐ต๐ต (โก excess of frictional dissipation on top of the minimum stress)
What is the rupture energy for more sophisticated friction models?
Andrews, JGR, 1976Ida, JGR, 1972
Approach: dynamic simulation of frictional ruptures
Fabian Barras Slide 3
โข Elastodynamics boundary integral method
โข Friction law at the interface: two examples (model variables are written in red)
Rate-and-state friction
๐๐(๐ฅ๐ฅ) = ๐๐0 + ๐ด๐ด ln(1 + ๐ฃ๐ฃ(๐ฅ๐ฅ)๐ฃ๐ฃโ
) + ๐ต๐ต ln(1 + ๐๐(๐ฅ๐ฅ)๐๐โ
) ๐๐๐๐ ๐๐[s] can be related to the microcontacts area
(Baumberger and Caroli, Adv. In Phys., 2006) Empirical law representative of rock friction
experiments
Thermal pressurization across a layer of gouge at the interface
๐๐(๐ฅ๐ฅ) = ๐๐0 + ๐ด๐ด ln ๏ฟฝฬ๏ฟฝ๐พ(๐ฅ๐ฅ,๐ฆ๐ฆ)๏ฟฝฬ๏ฟฝ๐พโ
๐๐๐๐ โ ๐๐(๐ฅ๐ฅ,๐ฆ๐ฆ) ๏ฟฝฬ๏ฟฝ๐พ strain rate profile through the layer
depending on fluid press. ๐๐ and temp. ๐๐ Mature fault zone with a core filled with
fluid-saturated gouge
Geubelle and Rice, JMPS, 1995Breitenfeld and Geubelle, IJF, 1998
Dieterich, JGR, 1979Ruina, JGR, 1983
Bar Sinai et al., JGR, 2012 Rice et al., JGR, 2014Platt et al., JGR, 2014
Approach: dynamic simulation of frictional rupture
Fabian Barras Slide 3
โข Study the near-tip elastic fields to compute the energy released by the rupture
โข Compare it to the energy dissipation at the interface during the rupture
โข Elastodynamics boundary integral method
โข Friction law at the interface: two examples
Geubelle and Rice, JMPS, 1995Breitenfeld and Geubelle, IJF, 1998
Shear-fracture: a well-defined benchmark
Fabian Barras Slide 4
โข Demonstrate the approach for a shear crack simulated by slip-weakening cohesive law:
โข The near-tip singularity can be studied using fracture mechanics (LEFM) predictions for stress ๐๐and slip velocity ๐ฃ๐ฃ
โข ๐พ๐พ can be directly related to the energy release by unit crack advance ๐บ๐บ
๐พ๐พ2๐๐(๐ฅ๐ฅ โ ๐ฅ๐ฅ๐๐)
โ ๐๐ โ ๐ฃ๐ฃ๐๐๐ผ๐ผ๐ ๐ 2๐ฃ๐ฃ๐๐
Stress intensity factor (SIF)
Tip location Shear modulus/wave speedRupture speed
๐ผ๐ผ๐ ๐ 2 = 1 โ ๐ฃ๐ฃ๐๐2/๐๐๐ ๐ 2
๐บ๐บ =๐พ๐พ2
2๐๐๐ผ๐ผ๐ ๐
Freund, Dynamic Fracture, 1990
Shear-fracture: a well-defined benchmark
Fabian Barras Slide 4
โข Demonstrate the approach for a shear crack simulated by slip-weakening cohesive law:
โข The near-tip singularity can be studied using fracture mechanics (LEFM) predictions for stress ๐๐and slip velocity ๐ฃ๐ฃ
โข ๐พ๐พ can be directly related to the energy release by unit crack advance ๐บ๐บ
โข The fit of ๐๐ and ๐ฃ๐ฃ allows for estimating ๐บ๐บ
โข The rupture energy balance is verified ๐บ๐บ = ๐บ๐บ๐๐
๐พ๐พ2๐๐(๐ฅ๐ฅ โ ๐ฅ๐ฅ๐๐)
= ๐๐ = ๐ฃ๐ฃ๐๐๐ผ๐ผ๐ ๐ 2๐ฃ๐ฃ๐๐
๐บ๐บ =๐พ๐พ2
2๐๐๐ผ๐ผ๐ ๐ ๐ฃ๐ฃ๐๐
Letโs now study frictional ruptures
Fabian Barras Slide 5
โข As ๐๐ does not drop to zero, the frictional stress drop ฮ๐๐ = ๐๐ โ ๐๐๐๐shall be used in the analogy with LEFM (invoking the linearity of the bulk constitutive law)
โข Friction reaches a steady value of ๐๐๐๐ which can be predicted theoretically
โข The breakdown energy overestimates ๐บ๐บ !
๐บ๐บ < ๐ธ๐ธ๐ต๐ต๐ต๐ต(๐ฅ๐ฅ) = ๏ฟฝ ๐๐ ๐ฅ๐ฅ โ ๐๐๐๐๐๐๐๐ ๐๐๐๐(๐ฅ๐ฅ)
Palmer and Rice, Proc. Roy. Soc., 1973
Barras et al., PRX, 2019
๐ฃ๐ฃ๐๐Case 1: Rate-and-state friction
Letโs now study frictional ruptures
Fabian Barras Slide 5
๐บ๐บ๐๐ =1๐ฃ๐ฃ๐๐
๏ฟฝ๏ฟฝฬ๏ฟฝ๐<0
๐๐ ๐ฅ๐ฅ โ ๐๐๐๐ ๐ฃ๐ฃ ๐ฅ๐ฅ ๐๐๐ฅ๐ฅ
Case 1: Rate-and-state friction
โข As ๐๐ does not drop to zero, the frictional stress drop ฮ๐๐ = ๐๐ โ ๐๐๐๐shall be used in the analogy with LEFM (invoking the linearity of the bulk constitutive law)
โข Friction reaches a steady value of ๐๐๐๐ which can be predicted theoretically
โข The breakdown energy overestimates ๐บ๐บ !
โข A better estimate of ๐บ๐บ๐๐ can be obtained by integrating only the near-tip dissipation
Palmer and Rice, Proc. Roy. Soc., 1973
Barras et al., PRX, 2019
Estimated ๐บ๐บ๐๐ is in good agreement with theoretical predictions (e.g. Ampuero and Rubin, JGR, 2008)
๐ฃ๐ฃ๐๐
Letโs now study frictional ruptures
Fabian Barras Slide 6
โข Crack-like rupture fronts emerge from intense shear-localization across the gouge layer
โข The computed energy release rate ๐บ๐บ is also much smaller than ๐ธ๐ธ๐ต๐ต๐ต๐ต.
โข No obvious value for ๐๐๐๐ as friction keeps weakening after the rupture due to pressurization. Its value is also determined during the fit before computing ฮ๐๐ = ๐๐ โ ๐๐๐๐
Case 2: Thermal pressurization
๐๐๐๐ emerging from fit
๐ฃ๐ฃ๐๐
Different friction laws, but generic observations
Fabian Barras Slide 7
โข Rupture energy ๐บ๐บ๐๐ = ๐บ๐บ is only a fraction of ๐ธ๐ธ๐ต๐ต๐ต๐ต
โข Rupture energy is associated to the rapid weakening of friction immediately in the wake of the rupture
โข The subsequent long-term evolution of friction after the rupture does not enter the rupture energy budget and significantly differs along the interface
โข A critical displacement ๐๐๐๐ can be associated to this transition
Evolution of ๐ธ๐ธ๐ต๐ต๐ต๐ต during the rupture integrated at different locations along the
interface
Thermal pressurization
Rate-and-state
๐๐๐๐ can be rationalized from the physics behind the two friction laws
Fabian Barras Slide 8
๐๐๐๐ arises when the most intense shear-localization is observed within the gouge
๐๐๐๐ arises when ๐๐ reaches its minimum, i.e. when the area of underlying microcontact reaches a minimum value after the rupture
Rate-and-state Thermal pressurization
Strain rate evolution across the gouge during the rupture
Summary and outlook
Fabian Barras Slide 9
โข The energy balance driving the propagation of frictional rupture is studied for two different types of frictional interface: o Rate-and-state interface (rock friction experiments) o Fluid-saturated granular layer (representative of fault cores filled with gouge)
โข The rupture energy ๐บ๐บ๐๐ is a small fraction of the total breakdown energy observed during the rupture
โข ๐บ๐บ๐๐ corresponds to the near-tip dissipation associated with the rapid drop of frictional stress, whose extend is rationalized using the physics behind each friction law
โข The energy partition inherited from slip-weakening friction (๐บ๐บ๐๐ = ๐ธ๐ธ๐ต๐ต๐ต๐ต) appears to be the exception rather than the rule
โข Implications for the energy budget of earthquakesTinti et al., JGR, 2005
Abercombie and Rice, JGI, 2005Chester et al., Nature, 2005
Nielsen et al., GRL, 2016
Thank you!
Fabian Barras Slide 10
โข Jean-Franรงois Molinari, Thibault Roch (EPFL Lausanne)โข Eran Bouchbinder, Michael Aldam (Weizmann Institute of Science)โข Efim A. Brener (Forschungszentrum Jรผlich)
โข Nicolas Brantut (UCL, London)
Collaborators during this work
Referencesโข F. Barras, M. Aldam, T. Roch, E.A. Brener, E. Bouchbinder, J.-F. Molinari,
Emergence of Cracklike Behavior of Frictional Rupture: The Origin of Stress Drops, Physical Review X, 9, 041043, 2019
โข F. Barras, M. Aldam, T. Roch, E.A. Brener, E. Bouchbinder, J.-F. Molinari, The emergence of crack-like behavior of frictional rupture: Edge singularity and energy balance, Earth and Planetary Science Letters, Vol. 531, 115978, 2020
โข F. Barras, N. Brantut, Earthquake rupture driven by shear localization within the fault gouge, In preparation, 2020