Long-range Stress Re-distribution Resulting from Damage in Heterogeneous Media
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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
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Content
1. Introduction
2. A Heterogeneous Model of Stress Re-Distribution (SRD)
3. SRD Results of the Model
4. Summary
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declustered earthquakes aftershocks complete catalog of earthquakes
Knopoff [PNAS, 2000, 97: 11880-11884]
the magnitude distribution of declustered earthquakesin Southern California
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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)
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G-R law : N (E1/ )-b b 1
characteristic b<=0.7
After Weatherley, Xia and Mora, (2000)
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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
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How stress re -distribution for heterogeneous media?
P = ? pr
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Content
1. Introduction
2. A Heterogeneous Model of Stress Re-Distribution (SRD)
3. SRD Results of the Model
4. Summary
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Heterogeneous Elastic -Brittle Model
• The same elastic modulus (E)
• Mesoscopically heterogeneous brittle fracture strength c follows a distribution
h(c)
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*
*;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
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i 0 I
0 2 40
1
2
strain
stre
ssq = 1.5
q = 2
q = 2.5
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Spherical(3-D) and cylindrical(2-D) configurations
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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
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][ rr
qr
1][ ,
Elastic-brittle constitutive relation ( 2 - D) when
Mixed-Model
qrr
1][ q
r 1][ ,
- Model
1
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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 ) :
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Content
1. Introduction
2. A Heterogeneous Model of Stress Re-Distribution (SRD)
3. SRD Results of the Model
4. Summary
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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
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q 1.2
0 5 100.4
0.6
0.8
distance
circ
um
fere
nti
al s
tres
s
-modelmixed modelelastic
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0 5 100
1
2
distance
radi
al s
tres
s
elastic-modelmixed model
q=1.2
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q 1.5
0 5 100.4
0.6
0.8
distance
circ
um
fere
nti
al s
tres
s
-model
mixed model
elastic
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0 5 100.4
0.6
0.8
distance
circ
umfe
rent
ial s
tres
s
-modelmixed model
elastic
q=1.75
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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
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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
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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.
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More works are needed to justify the long-range SRD in heterogeneous media
1. Direct Simulations
2. Experimental Observations
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