CONCEPTS: Fracture Stress Relationships for Coordinate Rotation Mohr Circle

1. Workshop on Megathrust Earthquakes and Tsunami Prof. Thorne Lay CONCEPTS: Fracture Stress Relationships for Coordinate Rotation Mohr Circle Coulomb Stress Changes – Earthquake Triggering Supplementary Reading (Optional, for more details/rigor) Lay and Wallace, Modern Global Seismology, Ch. 2 Stein and Wysession, An Introduction to Seismology, Earthquakes and Earth Structure.

2. After Ammo et al., Nature (2008) So, how to earthquakes ‘interact’ to produce aftershocks, doublets, migrations, etc.? Lay et al., JGR (2009)

3. (a) Rock usually is elastic until it undergoes brittle fracture when applied stress reaches f, if it is low temperature (b) Rock at high temperature usually undergoes plastic deformation when stress exceeds a yield stress 0. Some rock are plastic at low temperatures or for very long-term low level stress (folding). (c) Permanent strain results from plastic deformation when stress is raised to 0‘and released.

4. EXPERIMENT: STRESS APPLIED UNTIL ROCK BREAKS: STICK SLIP OCCURS When the fault forms, some of the stress is released and motion stops. If stress is reapplied, another stress drop and motion occur once stress reaches a certain level. As stress is reapplied, jerky sliding and stress release continues This pattern, called stick-slip, looks like a laboratory version of a sequence of earthquakes on a fault. By this analogy, the stress drop in an earthquake relieves only part of the total tectonic stress, and as the fault continues to be loaded by tectonic stress, occasional earthquakes occur. The analogy is strengthened by the fact that at higher temperatures (about 300° for granite), stick-slip does not occur Instead, stable sliding occurs on the fault, much as earthquakes do not occur at depths where the temperature exceeds a certain value. STICK SLIP STABLE SLIDING Brace and Byerlee, 1970

5. Earthquakes involve stress drop on the ruptured fault that affects stress in the surrounding medium Recall our discussion of the stress tensor

7. Trigonometry - identities: cos(2x) = cos2(x)-sin2(x) = cos2(x)-[1-cos2(x)] gives: cos2(x)=0.5[1+cos(2x)] cos(2x) = cos2(x)-sin2(x) = [1-sin2(x)]-sin2(x) gives: sin2(x)=0.5[1-cos(2x)] sin(2x)=2sin(x)cos(x)

8. SHEAR STRESS ON PLANE WITH NORMAL AT ANGLE q TO THE PRINCIPAL STRESS DIRECTION Note: compressive stress convention Positive in rock mechanics Negative in seismology (outward normal vector) NORMAL STRESS ON PLANE WITH NORMAL AT ANGLE q TO THE PRINCIPAL STRESS DIRECTION Arbitrary plane s1 q s2

10. With no internal friction, fracture occurs at an angle of 45º. With internal friction, fracture angle is 67.5º, and fault plane is closer (22.5 º) to the maximum compression (1 ) direction.

11. What do we know about friction? • Coulomb friction of solid rocks • rocks on rocks • No Eff Coefficient of friction Shear Stress = 0.6EffectiveNormal Stress + 0.5 kBar τ = μ σn + C Frictional shear stress Cohesion Effective normal stress = normal stress – pore pressure Shear Stress Byerlee, Pure and Applied Geoph, 1978 Normal Stress

12. Laboratory results for sliding on existing faults of various rock types find relation between normal stress on fault and shear stress required for sliding Byerlee's Law: Byerlee, 1978

13. Byerlee’s law, relating normal and shear stresses on a fault, can be used to infer principal stress as a function of depth. Assume the crust contains faults of all orientations, and the stresses cannot exceed the point where Mohr's circle is tangent to the frictional sliding line, or else sliding will occur

14. Laboratory experiments on rocks under compression show that fracture occurs when a critical combination of the absolute value of shear stress and the normal stress is exceeded. Higher normal stress requires higher shear stress for fracture This relation, the Coulomb-Mohr failure criterion, is |  | = o -n  where oand n are material properties known as the cohesive strength and coefficient of internal friction. INCREASED STRESS BREAKS ROCK WHEN MOHR’S CIRCLE REACHES COULOMB-MOHR FAILURE CRITERION Often: for given normal stress, rock breaks when shear stress high enough

15. MOHR’S CIRCLE AND COULOMB-MOHR FAILURE CRITERION DETERMINE Failure plane given by angle Failure stress given by point F

16. MOHR'S CIRCLE FOR SLIDING ON ROCK'S PREEXISTING FAULTS Preexisting faults have no cohesive strength Lower shear stress required for slip New fracture would form at an angle f given by fracture line. However, slip will occur on any preexisting fault with angle between S1 and S2 given by intersection of the circle with frictional sliding line. If so, as stress increases, sliding on existing fault is favored over new fault formation

17. PORE PRESSURE EFFECTS _ _ _ Water and other fluids are often present in rocks, especially in the upper crust.The fluid pressure, known as pore pressure, reduces the effect of the normal stress and allows sliding to take place at lower shear stresses. This is modeled by replacing normal stress  with the effective normal stress  =  - P f , where Pf is the pore fluid pressure. Similarly, taking into account pore pressure, effective principal stresses 1 = 1 - Pf2 = 2 - Pf are used in the fracture theory. The role of pore pressure in making sliding easier can be seen by trying to slide an object across a dry table and then wetting the table

21. Stress triggering may explain why successive earthquakes on a fault sometimes seem to have a coherent pattern. The 1999 Ms 7.4 Izmit earthquake on the North Anatolian fault appears to be part of a sequence of major (Ms 7) earthquakes over the past 60 years, which occurred successively further to the west and closer to the city of Istanbul The Izmit earthquake and the November 1999 Ms 7.1 Duzce earthquakes increased failure stress near Istanbul Parsons et al., 2000

23. Reasons to think that Byerlee’s Law is Problematic for an Earthquake Laboratory Experiments: High-Speed Friction is Low Gas gun experiments indicate μ= 0.2-0.4 for coseismic friction Yuan and Prakash, J. Mech. Phys. Solids, 2008

24. STRESS CONVENTIONS DIFFER FOR DIFFERENT COMMUNITIES. ENGINEERING CONVENTION HAS POSITIVE VALUE FOR COMPRESSIONAL STRESS: HENCE THE MOHR CIRCLES ARE FOR POSITIVE STRESS VALUES, BUT THEY MEAN THE SAME THING: