2D finite elements

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1- Time-indep model of rock fracture, damage, AE

• 2D finite elements
• constant strain rate
• (compression)
• confining pressure

• Behavior of each element
• E ? when ? reaches ?damage
• d models fractures at the micro scale
• Coulomb damage criterion
• C and E0 heterogeneous

Amitrano et al., GRL 1999 Amitrano, JGR 2003
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1- Time-indep model of rock fracture, damage, AE

• Macroscopic behavior
• damage localization along shear bands
• transition brittle/ductile as a function of ?
• power-law distribution of avalanche size

shear stress (strain)
damage events (time) for ?45
final damage (?)
Amitrano et al., GRL 1999 Amitrano, JGR 2003
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2- Time-dependent deformation creep
stress
?
strain
inelastic elastic, reversible
time
failure (?)
primary creep d?/dt ? Andrade law
tertiary creep d?/dt ?
d?/dt
secondary creep d?/dtconstant
tf(?)
time
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2-Time-to-failure laws review of creep
experiments on rocks

• Failure time
• tf ? with stress as a
• exp tfexp(-b?????
• or power law tf (???????with b20
• tf ? with T and CH20
• Mechanisms
• Subcritical crack growth
• due to stress corrosion at the crack tips
• Scholz, 72 Das Scholz, 81

stress ? / instantaneous strength ? ?
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2- Time-dependent numerical model

• Creep constant applied stress ?lt?i
• Each elements fails
• when stress ? ?i (Coulomb criterion)
• because of interactions between elements
• or when time failure time tf (?,?i) exp(-b
  ?/?i)
• After each damage event
• broken elements
• - E ? EE(1-d)E0(1-d)n
• - change C unif pdf
• - lifetime ? with damage tf tf(?,?i) E/E0

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• 2- Temporal evolution of strain, damage and
  energy rate

damage rate strain rate energy
rate
tertiary creep primary creep
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• 2- Distribution of damage events size

large??????? ???i0.95
??all???????????i0.25
t/tc
t/tc
-- P(W)W-1.5
-- P(W)W-1.5
Small stress narrow pdf P(W), typical energy ?
with time Large stress power-law pdf P(W)W-1.5
at all times
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• 2- Distribution of damage events size

Average energy as a function of time before
failure for ? stress Power law acceleration of
ltWgt before rupture ltWgt (t-tc)-0.3
ltWgt
time before rupture
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• 2- Damage localization

• Damage (t)
• localization during tertiary creep (tgttc/2)
• ????????????????????
• Final damage (?)
• Shear bands width
• ? with ?

strain (t)
damage 1-E/E0
energy (t)
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Rockfalls tertiary creep failure?
Power-law acceleration of earthquakes and energy
rates before collapse
Chalk cliff, Mesnival, France June 2002
time before collapse
Amitrano et al., GRL 2005
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• Conclusion

• Typical creep behavior
• - power-law relaxation during primary creep
• - acceleration during tertiary creep
• Power-law pdf of event sizes
• - only if integrated over all times for small ?
• - indep of times for large ?
• Damage localization during tertiary creep
• System always unstable (if tf E)
• - time of global failure tc local law tf(?,?i)