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

Content

1. Introduction

2. A Heterogeneous Model of Stress Re-Distribution (SRD)

3. SRD Results of the Model

4. Summary

declustered earthquakes aftershocks complete catalog of earthquakes

Knopoff [PNAS, 2000, 97: 11880-11884]

the magnitude distribution of declustered earthquakesin Southern California

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)

G-R law : N (E1/ )-b b 1

characteristic b<=0.7

After Weatherley, Xia and Mora, (2000)

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

How stress re -distribution for heterogeneous media?

P = ? pr

Content

1. Introduction

2. A Heterogeneous Model of Stress Re-Distribution (SRD)

3. SRD Results of the Model

4. Summary

Heterogeneous Elastic -Brittle Model

• The same elastic modulus (E)

• Mesoscopically heterogeneous brittle fracture strength c follows a distribution

h(c)

*

*;1,)

*(

*

1;0

)(cc

ccq

c

c

c

c when

whenq

qh

i 0 I

0 5 100

0.5

1

meso-breaking strain

dist

ribu

tion

func

tion

h

q = 1.5

q = 2

q = 2.5

i 0 I

0 2 40

1

2

strain

stre

ssq = 1.5

q = 2

q = 2.5

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

02

rdr

d r

the elastic-brittle constitutive relation

*/)1)(21( cE

1*

- Model (3-D) --- D = 1 -

qr

1][

qrr

1]2)1[(

q1

][ rr

qr

1][ ,

Elastic-brittle constitutive relation ( 2 - D) when

Mixed-Model

qrr

1][ q

r 1][ ,

- Model

1

the balance equation leads to a non-linear ordinary differential equation of displacement u

0)1(2'

)31('

)1)(1('')1(2

2

r

uq

r

uvqq

u

uqu

dr

dur r

u

Governing Equation (Displacement u ) :

Content

1. Introduction

2. A Heterogeneous Model of Stress Re-Distribution (SRD)

3. SRD Results of the Model

4. Summary

pB2i2 2 qi 2

22 qi

1 1.5 20

1

2

3

q

p

qp

1

1122 qp )1(3

pr 3 - D 2 - D

=1/4 q p p 1 3 2 1.2 2.66 1.751.5 2.16 1.371.75 1.75 1.06

q 1.2

0 5 100.4

0.6

0.8

distance

circ

um

fere

nti

al s

tres

s

-modelmixed modelelastic

0 5 100

1

2

distance

radi

al s

tres

s

elastic-modelmixed model

q=1.2

q 1.5

0 5 100.4

0.6

0.8

distance

circ

um

fere

nti

al s

tres

s

-model

mixed model

elastic

0 5 100.4

0.6

0.8

distance

circ

umfe

rent

ial s

tres

s

-modelmixed model

elastic

q=1.75

FE Simulation, vs r =0.25, q=1.5

0 1 2 3 4 5 6 7 8

1.0

1.2

1.4

1.6

1.8

2.0

/P

r/a

r/a

step 2 steo 6 step 10 analytic response to elastic

and damaged elastic,respectively

loading steps 2

loading steps 6

loading steps 10

0 1 2 3 4 5 6 7 80.0

0.1

0.2

0.3

0.4

0.5

0.6

dam

age

frac

tion

r/a

damage field at different timedown->up:step 5,6,7,8,9,10

FE Simulation, D vs r =0.25, q=1.5

Damage field at successive loading steps 5-10

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.

More works are needed to justify the long-range SRD in heterogeneous media

1. Direct Simulations

2. Experimental Observations