Effect of Heterogeneity on Catastrophic Rupture

F.J.Ke a, b , H.L. Lia, M.F.Xia a, c and Y.L.Bai a

a State Key Laboratory for Non-linear Mechanics (LNM), Institute of Mechanics, Chinese Academy of Sciences, Beijing

100080, Chinab 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

Successful prediction (of earthquake)depends greatly on the heterogeneity of the area’s structure

------ Mogi

Damage Localization RuptureDamage accumulation catastrophic rupture

Content

1. Heterogeneous Elastic -Brittle Model2. Event Series prior to Rupture in Heterogeneous

Media in Mean Field Approximation3. Effect of Surrounding, Size Effect and Stress

Re-Distribution4. Network Simulations5. Concluding Remarks

1. Heterogeneous Elastic -Brittle Model

• Unique elastic behaviour (E)• Mesoscopically heterogeneous Brittle

Fracture Strength c

c follows Weibull distribution

])(exp[)(1

mcm

mc

c mw

m: Weibull Modulus

Weibull Distribution of Mesoscopic Strength c

0 0.5 1 1.5 20

1

2

3

4

w1n

w2n

w3n

cn

m=10m=5m=2

Brittle Fiber Ductile metalm 2 - 4 20

Heterogeneous Elastic -Brittle Model

Relation between Load(N) and Displacement(mm) of Sandstone, from Chinese Encyclopedia, Mechanics, p.529

0 1 20

0.2

0.4

0.6

n

En

Elastic - brittle modelm = 3

2. Event Series prior to Rupture in Heterogeneous Media

• Damage Localization (DL)• Maximum Stress (m, i.e. d/d=0) • Energy Release (ER and ERmax)• Surrounding and Size-effect • Stress Re-Distribution(SRD)• Catastrophic Rupture (d/d = -Ks)• Critical Sensitivity (S)

m=5

0 0.5 1 1.50

0.5

1

1.5

2

strain

stre

ss, d

amag

e an

d en

erge

rele

ase Rupture dER/d

when k=1

Damage D()localization

Strain0.000 .002 .004 .006 .008 .010

Stress

0.0

.1

.2

.3

.4

.5

m=5

3. Size-effect and Stress Re-Distribution

4. Network Simulations

Energy Release and Catastrophic Rupture (CR)

Load F

-Ks K

A Displacement

Displacement of surronding Displacement of body

Energy release within the body at CR ( )

Energy release from surrounding at CR ( )

Energy release due to damage D at a point A ( )

Surroundings

Km

SampleKs

-Km

m 3.591 implies catastrophic rupture (CR) for k=Ks/Km=1

• Catastrophic Rupture has a lower bound of Weibull modulus

mc = k * exp[( mc+1)/mc]

k = 1 mc = 3.59

Network Model vs. Mean Field ModelWeibull modulus: 2With elastic surroundings

Network Model vs. Mean Field ModelWeibull modulus: 5With elastic surroundingsk=1

0 1 2 3

0.0

.1

.2

.3

.4

Shear

• Size of Elastic Surrounding

Stiffness: Ks Km

Es EmLs Lm

Lm Surrounding

Ls Sample

Different kWeibull modulus: 5Black dash: rigid, Blue dash dot: k=1, Red solid:k=2

STRESS

STRAIN

2-D simulation, white: failed red : high stress

(Courtesy of YU Huaizhong)

•Stress Re-distribution due to heterogeneity and damage

STRESS

STRAIN

STRESS

STRAIN

STRESS

STRAIN

5. Concluding Remarks

Effects on Catastrophic Rupture owing to

• Surroundings• Size Effect• Stress Re-Distribution (SRD)

For accurate prediction of catastrophic rupture, there is a need of close look of the relationship between various effects and rupture.