Long-range Stress Re-distribution Resulting from Damage in Heterogeneous Media

1. ACES Meeting, May 5-10, 2002, Maui,Hawaii Long-range Stress Re-distribution Resulting from Damage in Heterogeneous Media Y.L.Bai a, Z.K. Jia a, F.J.Ke a, b and M.F.Xia a, c a State Key Laboratory for Non-linear Mechanics (LNM), Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, China b Department of Applied Physics, Beijing University of Aeronautics and Astronautics, Beijing 100083, China c Department of Physics, Peking University, Beijing 100871, China

2. Content 1. Introduction 2. A Heterogeneous Model of Stress Re-Distribution (SRD) 3. SRD Results of the Model 4. Summary

3. Knopoff [PNAS, 2000, 97: 11880-11884] the magnitude distribution of declustered earthquakes in Southern California declustered earthquakesaftershockscomplete catalog of earthquakes

4. Weatherley, Xia and Mora[AGU 2000] : • The interaction exponent (p in  1/rp ) determines the effective range • for strain re-distribution in the model. • The effective range decreases rapidly as the exponent (p) increases. • The event size-distribution illustrates • three different populations of events • (two-dimensional models): • Characteristic large events ( p < 1.5) • Power-law scaling events (1.5 < p  2.0 ) • Overdamped, no large events ( p > 2.0)

5. characteristic b<=0.7 G-R law : N (E1/ )-b b  1 After Weatherley, Xia and Mora, (2000)

6. Klein et al [AGU 2000] : • Linear elasticity yields long-range stress tensors for a variety of • geological applications • For a two-dimensional dislocation in a three-dimensional homogeneous • elastic medium, the magnitude of the stress tensor goes as  1/r3 • While geophysicists do not know the actual stress tensors for real faults, • they expect that long-range stress tensors, which are similar to the  • 1/r3 interaction, apply to faults • It is suspected that microcracks in a fault, as well as other “defects” • such as water, screen the  1/r3 interaction, leading to a proposed •  e-r/r3 interaction,where 1, implying a slow decay to the • long-range interaction over the fault’s extent

7. How stress re -distribution for heterogeneous media? P = ?

8. Content 1. Introduction 2. A Heterogeneous Model of Stress Re-Distribution (SRD) 3. SRD Results of the Model 4. Summary

9. Heterogeneous Elastic -Brittle Model • The same elastic modulus (E) • Mesoscopically heterogeneous brittle fracture strength c follows a distribution h(c)

10. q = 1.5 q = 2 q = 2.5

11. q = 1.5 q = 2 q = 2.5

12. Spherical(3-D) and cylindrical(2-D) configurations

13.  - Model (3-D) --- D = 1 - the elastic-brittle constitutive relation

14. Mixed-Model  - Model Elastic-brittle constitutive relation ( 2 - D) when

15. Governing Equation (Displacement u ) : the balance equation leads to a non-linear ordinary differential equation of displacement u

16. Content 1. Introduction 2. A Heterogeneous Model of Stress Re-Distribution (SRD) 3. SRD Results of the Model 4. Summary

17. 3 - D 2 - D =1/4 q p p 1 3 2 1.2 2.66 1.75 1.5 2.16 1.37 1.75 1.75 1.06

18. -model mixed model elastic

19. q=1.2 elastic -model mixed model

20. -model mixed model elastic

21. q=1.75 -model mixed model elastic

22. loading steps 2 loading steps 6 loading steps 10 FE Simulation,  vs r  =0.25, q=1.5

23. FE Simulation, D vs r  =0.25, q=1.5 Damage field at successive loading steps 5-10

24. 4. Summary In order to understand why a declustered or characteristic large earthquake may occur with a longer range stress re-distribution observed in CA simulations but aftershocks do not, we proposed a linear-elastic but heterogeneous-brittle model. The stress re-distribution in the heterogeneous-brittle medium implies a longer-range interaction. Therefore, it is supposed that the long-range stress re-distribution resulting from damage in heterogeneous media be a quite possible mechanism governing mainshocks.

25. More works are needed to justify the long-range SRD in heterogeneous media1. Direct Simulations2. Experimental Observations

26. END