Mechanics of Earthquakes and Faulting

Transcription

1 Mechanics of Earthquakes and Faulting Lecture 16, 9 Nov Energy Balance of dynamic rupture Crack tip stress field Frictional Rupture Fronts Meet in the lab (522 Deike) on Tuesday Nov. 14; See Problem Set #6 Exam 1 due tomorrow (10 Nov.)

2 Mode II crack propagation at speed Vr cohesive zone/slip weakening crack model for friction Shear Stress σ y σ o σ f f max scales as: f max = w/vr w = 7 24 Gu max r = 7 24 GD c w D c = 24 7 G w d c Cracked/Slipping zone Breakdown (cohesive) zone Intact, locked zone Slip

3 G is the fracture energy, Gc is the critical energy release rate for background see: L c is the crack length for unstable growth (and failure); it can be related to the critical patch size for frictional failure R c G is Shear modulus where C is a geometric constant and ν is poisson s ratio. For rupture propagation, Strength Excess S is an important parameter S = y 1 1 f N Propagating Rupture

4 Energy Balance: Specific fracture energy G c For a mode III crack, we can calculate a critical half length for crack extension note that G is shear modulus, sometimes written µ L c is of order 1-2 m for G c of 10 2 J/m 2. What about the stress concentration implied by such a model? How is the singularity resolved? (process zone? Irwin? ) θ See: Brittle Fracture Theory Predicts the Equation of Motion of Frictional Rupture Fronts, Ilya Svetlizky,David S. Kammer, Elsa Bayart, Gil Cohen, and Jay Fineberg, PRL 118, (2017) Classical shear cracks drive the onset of dry frictional motion, Svetlizky & Fineberg, Nature, See also: Johnson and Scholz, JGR, 1976.

5 Adhesive and Abrasive Wear: Fault gouge is wear material Chester et al., 2005 where T is gouge zone thickness, κ is a wear coefficient, D is slip, and h is material hardness This describes steady-state wear. But wear rate is generally higher during a run-in period. And what happens when the gouge zone thickness exceeds the surface roughness? We ll come back to this when we talk about fault growth and evolution.

6 Fault Growth and Development Fault gouge is wear material run in and steady-state wear rate Gouge Zone Thickness, T Fault offset, D This describes steady-state wear. But wear rate is generally higher during a run-in period.

7 Fault Growth and Development Fault gouge is wear material run in and steady-state wear rate Gouge Zone Thickness, T σ 2 > σ 1 σ 1 Fault offset, D This describes steady-state wear. But wear rate is generally higher during a run-in period.

8 Fault Growth and Development Fault gouge is wear material Gouge Zone Thickness, T run in and steady-state wear rate? Fault offset, D This describes steady-state wear. But wear rate is generally higher during a run-in period. And what happens when the gouge zone thickness exceeds the surface roughness?

9 Fault Growth and Development Fault gouge is wear material run in and steady-state wear rate Gouge Zone Thickness, T? Fault offset, D Scholz, 1987

10 Fault Growth and Development Fault Roughness Scholz, 1990

11 Scholz, 1990 Fault Growth and Development

12 Fault Growth and Development Cox and Scholz, JSG, 1988

13 Fault Growth and Development Fault zone width Tchalenko, GSA Bull., 1970

14 Fault zone roughness Ground (lab) surface Scholz, 1990

15 The critical slip distance for shear zones of finite width Fault zone of finite width; multiple, subparallel slip surfaces Surfaces obey rate and state friction Marone, C., Cocco, M., Richardson, E., and E. Tinti, 2009

16 Thermo-mechanics of faulting II San Andreas fault strength, heat flow. Consider: W f = τ v q If τ ~ 100 MPa and v is ~ 30 mm/year, then q is: 1e8 (N /m 2 ) 3e-2 (m/3e7s) = 1e-2 (J/s m 2 ) ~ 100 mw/m2. Problem of finding very low strength materials. Relates to the very broad question of the state of stress in the lithosphere? Byerlee s Law, Rangley experiments, Bore hole stress measurements, bore hole breakouts, earthquake focal mechanisms. Seismic stress drop vs. fault strength.

17 Fault Strength, State of Stress in the Lithosphere, and Earthquake Physics

18 Thermo-mechanics of faulting Fault strength, heat flow. Average Shear Stress Average slip velocity Consider shear heating: W f = τ v q If τ ~ 100 MPa and v is ~ 30 mm/year, then q is: 1e8 (N /m 2 ) 3e-2 (m/3e7s) = 1e-1 (J/s m 2 ) 100 mw/m 2

19 Fault Strength and State of Stress Data from Lachenbruch and Sass, 1980 Heat flow Stress orientations S A F e.g. Townend & Zoback, 2004;; Hickman & Zoback, 2004

20 Data from Lachenbruch and Sass, 1980 Fault Strength and State of Stress Heat flow Stress orientations Have been used to imply that the SAF is weak, µ 0.1. Inferred stress directions S A F σ 1 σ 1 S A F Predicted Observed e.g. Townend & Zoback, 2004;; Hickman & Zoback, 2004

21 SAFOD The San Andreas Fault Observatory at Depth Is the San Andreas anomalously weak?

22 SAF - Geology Based on Zoback et al., EOS, 2010

23 Frictional Strength, SAFOD Phase III Core Carpenter, Marone, and Saffer, Nature Geoscience, 2011 Carpenter, Saffer and Marone, Geology, 2012

24 Weak Fault in a Strong Crust Carpenter, Saffer, and Marone. Geology, 2012

25 Seismic Spectra & Earthquake Scaling laws. Aki, Scaling law of seismic spectrum, JGR, 72, , Hanks, b Values and ω -γ seismic source models: implications for tectonic stress variations along active crustal fault zones and the estimation of high-frequency strong ground motion, JGR, 84, , Scaling and Self-Similarity of Earthquake Rupture: Implications for Rupture Dynamics and the Mode of Rupture Propagation 0 Self-similar: Are small earthquakes the same as large ones? Do small ones become large ones or are large eq s different from the start? 1 Geometric self-similarity: aspect ratio of rupture area 2 Physical self-similarity: stress drop, seismic strain, scaling of slip with rupture dimension 3 Observation of constant b-value over a wide range of inferred source dimension. 4 Same physical processes operate during shear rupture of very small (lab scale, mining induced seismicity) and very large earthquakes? 5 Expectation of scaling break if rupture physics/dynamics change in at a critical size (or slip velocity, etc.). Shimazaki result. (Fig. 4.12). Length-Moment scaling and transition at L 60km (Romanowicz, 1992; Scholz, 1994). 6 Gutenberg-Richter frequency-magnitude scaling, b-values. 7 G-R scaling, b-value data. Single-fault versus fault population. G-R versus characteristic earthquake model. 8 Crack vs. slip-pulse models

26 Earthquake Source Properties, Spectra, Scaling, Self-similarity Displacement and acceleration source spectra. Spectra: zero-frequency intercept (M o ), corner frequency (ω o or f c ), high frequency decay (ω γ ), maximum (observed, emitted) frequency f max ω -n ω-square model, ω -2 ω-cube model, ω -3 log u at R ω ο Far-field body-wave spectra and relation to source slip function log freq. (ω) Displacement waveform for P & S waves: In general, very complex. Ω(x, t) and Ω(ω) depend on slip function, azimuth to observer and relative importance of nucleation and stopping phases

27 Scaling and Self-Similarity Are small earthquakes the same as large ones? 1 Geometric self-similarity: rupture aspect ratio 2 Physical self-similarity: stress drop, seismic strain, scaling of slip with rupture dimension Circular ruptures (small) Hanks, 1977 Abercrombie & Leary, 1993

28 Earthquake Source Properties, Spectra, Scaling, Self-similarity ω -3 log u at R ω ο log freq. (ω) ω-cube model, ω -3 Similarity condition M o α L 3 ω o α L -1 Ω(0) α ω o -3 This defines a scaling law. Spectral curves differ by a constant factor at a given period (e.g., 20 s), but they have the same high-freq. asymptote This behavior is expected when the nucleation phase is responsible for the high-freq. asymptote --but consider problem of time domain implication for amplitude (M b decreases with M o )

29 Seismic Source Spectra. Saturation occurs for large events, particularly saturation of M s (T=20 s) Corner frequency, Brune Stress drop. Aki, 1967

30 Earthquake Source Properties, Spectra, Scaling, Self-similarity ω -2 log u at R ω ο ω-square model, ω -2 Two possible explanations log freq. (ω) 1)!Similarity condition (not-similarity) M o α L 2 ω o α L -1 Ω(0) α ω o -2 2) Have similarity condition in terms of nucleation, but high-freq. asymptote is produced by stopping phase if rupture stops very abruptly

31 Earthquake Source Properties, Spectra, Scaling, Self-similarity Displacement and acceleration source spectra. Spectra: zero-frequency intercept (M o ), corner frequency (ω o or f c ), high frequency decay (ω γ ), maximum (observed, emitted) frequency f max f max log u at R ω ο ω -n log a at R ω ο log freq. (ω) log freq. (ω) Aki, Scaling law of seismic spectrum, JGR, 72, , Hanks, b Values and ω -γ seismic source models: implications for tectonic stress variations along active crustal fault zones and the estimation of high-frequency strong ground motion, JGR, 84, , 1979.

32 Earthquake Source Properties, Spectra, Scaling, Self-similarity Source spectra for two events of equal stress drop: omega cube model Large and Small Eq L log u at R ω ο L ω -3 S ω ο S f max log freq. (ω) High-freq. spectral properties: produced by rupture growth, represent nucleation and enlargement log a at R ω ο L ω ο S log freq. (ω)

33 Source spectra for two events of equal stress drop: omega square model Large and Small Eq L log u at R ω ο L ω -2 S ω ο S L f max log freq. (ω) High-freq. spectral properties: produced by rupture growth, represent nucleation and enlargement log a at R ω ο L ω ο S S f max log freq. (ω)

34 Earthquake Source Properties, Spectra, Scaling, Self-similarity Relation between source (a) displacement (b) velocity (c) acceleration history and asymptotic behavior of spectrum