1. 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 ACES Meeting, May 5-10, 2002, Maui,Hawaii

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

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

4. Weatherley, Xia and Mora[AGU 2000] : The interaction exponent (p in  1/r p ) 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. G-R law : N  (E 1/  ) -b b  1 characteristic b<=0.7 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/r 3 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/r 3 interaction, apply to faults It is suspected that microcracks in a fault, as well as other “defects” such as water, screen the  1/r 3 interaction, leading to a proposed  e -  r /r 3 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. the elastic-brittle constitutive relation  - Model (3-D) --- D = 1 -

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

15. the balance equation leads to a non-linear ordinary differential equation of displacement u Governing Equation (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.  -model mixed model q=1.2

20.  -model mixed model elastic

21.  -model mixed model elastic q=1.75

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

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 media 1. Direct Simulations 2. Experimental Observations