beyond elasticity stress, strain, time
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Beyond Elasticity stress, strain, time. Don Weidner Stony Brook. From Don Anderson’s book ch. 14. - PowerPoint PPT PresentationTRANSCRIPT
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Beyond Elasticitystress, strain, time
Don Weidner Stony Brook
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From Don Anderson’s book ch. 14
• Real materials are not perfectly elastic. Stress and strain are not in phase, and strain is not a single- valued function of stress. Solids creep when a sufficiently high stress is applied, and the strain is a function of time.
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Deep Earthquake
Q, Vp,Vs
Rheology
Tomography
Phase Transitions
Thermoelastic
Convection
Seismic
Anisotropy
Earth’s mantle and
stress
Anelasticity
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Time scales
IN EARTH• Seismic waves1 sec – 1000 sec.• Earthquakes10 sec – 1000 sec• Plate
tectonics107 sec – 1016 sec
IN LAB• Acoustic
velocity10-9 sec – 10-6 sec• Rock mechanics1 msec – 1 msec• Ductile flow103 sec – 106 sec
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Rheology
• Elasticity: stress proportional to strain• Anelasticy: stress, strain relation depends on
time• Plasticity: strain not recoverable when stress is
removed
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Example of non-elastic process
• Phase transformations can cause non-elastic volume change
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From elasticity
• K=-V(dP/dV)• Vp = sqrt((K+4/3G)/rho)• Vs=sqrt(G/rho)• K/rho=Vp2-4/3Vs2
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Adams-Williamson equation
∂ρ/∂z=ρg(ρ/K)
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3
3.5
4
4.5
5
300 400 500 600 700 800 900Depth
Den
sity
, gm
cm
-3
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• Based on material properties:
0
2
4
6
8
10
12
14
250 750 1250 1750 2250 2750Depth, km
Vp
Vs
r
0
2
4
6
8
10
12
14
250 750 1250 1750 2250 2750Depth, km
0
2
4
6
8
10
12
14
TREAMPREM
Vp
Vs
r
• Disappearance of P660P reflection• Velocity jump (410, 660 Km) is smaller than mineral model • Gradient of the transition zone velocities are higher than mineral model• Is there a 520 discontinuity?
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Different time scale results in different velocity
Unrelaxed
High Vp, high Q
Relaxed
Low Vp, high Q
intermediate Vp, low Q
(Anderson, 1989) ω is seismic frequency; is time scale; Q is attenuation factor, c is velocity
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To model Velocity
• Phase diagram and Elasticity are not enough• Time scales of the phase transitions are also
important
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Is the low velocity zone due to
OrMelting?
Melts?
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From Hirschmann, 2000
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10 20 30 40 500
50
100
150
200
1400 C
Pressure, kbars
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10 20 30 40 500
50
100
150
200
3.00
3.10
3.20
3.30
1400 C
Pressure, kbars
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10 20 30 40 500
50
100
150
200
3.00
3.10
3.20
3.30
1400 C
Pressure, kbars
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spol
cpxopx
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Viscosity Profile of the Earth
1E+18
1E+23
1E+28
1E+33
1E+38
0 50 100 150 200 250 300 350 400depth, Km
visc
ocity
, Pa
s80myr
stress = 0.05 MPaV* = 5 cc/molpower-law creep (Li & Weidner, 2003)
0
500
1000
1500
2000
2500
0 100 200 300 400 500Depth (Km)
Tem
pera
ture
, K
80 Myr oceanic
(Master & Weidner, 2002)
(L. Li, thesis, 2003)
)/exp()()( * RTEA mdbn m
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Viscosity Profile of the Earth
1E+17
1E+19
1E+21
1E+23
0 50 100 150 200 250 300 350 400depth, Km
visc
ocity
, Pa
s
v=20v=15v=10v=5v=0
20Myr
80Myr
stress = 0.05 MPa
0
500
1000
1500
2000
2500
0 100 200 300 400 500Depth (Km)
Tem
pera
ture
, K
20 Myr oceanic
80 Myr oceanic
(Master & Weidner, 2002)
(L. Li, thesis, 2003)
)/)(exp()()( ** RTPVEA mdbn m
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Viscosity Profile of the Earth
1E+17
1E+19
1E+21
1E+23
1E+25
1E+27
1E+29
0 50 100 150 200 250 300 350 400depth, Km
visc
ocity
, Pa
s
v=20v=15v=10v=5v=0
20Myr
80Myr
Canadian Shield
stress = 0.05 MPa
0
500
1000
1500
2000
2500
0 100 200 300 400 500Depth (Km)
Tem
pera
ture
, K
Canadia shield T
20 Myr oceanic crust T
80 Myr oceanic crust T
(Master & Weidner, 2002)
(L. Li, thesis, 2003)
)/)(exp()()( ** RTPVEA mdbn m
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Measure Stress
Measure Deformation in situ
Deform at a constant slow rate
Challenges for Experiments
at deep Earth conditions of P and T
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Measurement of Stress
= F/A
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Measurement of Stress
= M*
X-rays define d, lattice spacings, and can be used to define elastic strain.
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Ideal CircleLattice spacings for stressed sample
Stressed sample0 1000 2000
channel
coun
ts
det1det2det3det4
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Conical Slits
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Measure Stress
Measure Deformation in situ
Deform at a constant slow rate
Challenges for Experiments
at deep Earth conditions of P and T
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Multi SSD
Press
Sample
Sample
gold foil
gold foil
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Measure Stress
Measure Deformation in situ
Deform at a constant slow rate
Challenges for Experiments
at deep Earth conditions of P and T
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Measurement of Stress by Proxy
0
3000
6000
9000
12000
400 600 800 1000 1200 1400Channel
Intensity, count
det1det2det3det4det5det6det7det8det9det10
-1
-0.5
0
0.5
1
0 360 720 1080 1440
time, s
stress oscillation amplitude.
Sinusoidal Oscillation
XRD collection
X-ray radiograph collection
Figure 6. Synchronized X-ray diffraction and X-ray radiograph during sinusoidal stress oscillation. Shown is for stress oscillation with a period of 1440 second. A diffraction data (shown on the top left) was collected every 120 seconds and include the energy dispersive X-ray diffraction pattern for 10 detectors. The ten detectors are distributed around a circle at a fixed two theta (arranged as shown in upper right). The patterns collected by detector 1 and detector 9 are along the unique stress axis. An X-ray radiograph is also collected every 120 second.
Active detector elements, det1-10
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-2.E-03
-1.E-03
0.E+00
1.E-03
2.E-03
0 500 1000 1500
time, s
stra
in mgoal2o3
Forced oscillation on MgO and Al2O3
T= 800 oCP = 5GPaFrequency = 10-100mHz
-1.E-03
-5.E-04
0.E+00
5.E-04
1.E-03
0 50 100 150
time, s
stra
in mgoal2o3
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0
3000
6000
9000
12000
400 600 800 1000 1200 1400Channel
Inte
nsity
, cou
nt
det1det2det3det4det5det6det7det8det9det10
-1
-0.5
0
0.5
1
0 360 720 1080 1440
time, s
stre
ss o
scill
atio
n am
plitu
de.
Sinusoidal Oscillation
XRD collection
X-ray radiograph collection
Figure 6. Synchronized X-ray diffraction and X-ray radiograph during sinusoidal stress oscillation. Shown is for stress oscillation with a period of 1440 second. A diffraction data (shown on the top left) was collected every 120 seconds and include the energy dispersive X-ray diffraction pattern for 10 detectors. The ten detectors are distributed around a circle at a fixed two theta (arranged as shown in upper right). The patterns collected by detector 1 and detector 9 are along the unique stress axis. An X-ray radiograph is also collected every 120 second.
Active detector elements, det1-10
Li Li et al 2009
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500
600
700
800
900
1000
1100
1200
36000 41000 46000 51000 56000 61000 66000Time, seconds
Tem
pera
ture
, C
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Stre
ss, G
Pa
Temperaturestress-200stress-220stress-111
Figure 6. Stress and temperature as a function of time determined from each of three diffraction peaks. Temperature was ramped down as the DDIA rams were driven by a sinusoidal signal. Zero stress corresponds to hydrostatic pressure and positive stress is extensional.
Li Li et al 2009
MgO
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amplitude fractional change
-0.5
-0.3
-0.1
0.1
0.3
0.5
54000 56000 58000 60000 62000 64000 66000
D A/A
-1
-0.5
0
0.5
1
Stre
ss, G
Pa
positive stress is extension
Stress
3 point averages
Time, seconds
Measure Amplitude of Diffraction Peaks with Time and Temperature
700 C
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amplitude fractional change
-0.5
-0.3
-0.1
0.1
0.3
0.5
54000 56000 58000 60000 62000 64000 66000
D A/A
-1
-0.5
0
0.5
1
Stre
ss, G
Pa
positive stress is extension
[111] Stress
3 point averages
Time, seconds
Measure Amplitude of Diffraction Peaks with Time and Temperature
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amplitude fractional change
-0.5
-0.3
-0.1
0.1
0.3
0.5
54000 56000 58000 60000 62000 64000 66000
D A/A
-1
-0.5
0
0.5
1
Stre
ss, G
Pa
positive stress is extension
[111][200]
[220]
Stress
3 point averages
Time, seconds
Measure Amplitude of Diffraction Peaks with Time and Temperature