Beyond Elasticity stress, strain, time Don Weidner Stony Brook
Feb 24, 2016
Beyond Elasticitystress, strain, time
Don Weidner Stony Brook
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.
Deep Earthquake
Q, Vp,Vs
Rheology
Tomography
Phase Transitions
Thermoelastic
Convection
Seismic
Anisotropy
Earth’s mantle and
stress
Anelasticity
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
Rheology
• Elasticity: stress proportional to strain• Anelasticy: stress, strain relation depends on
time• Plasticity: strain not recoverable when stress is
removed
Example of non-elastic process
• Phase transformations can cause non-elastic volume change
From elasticity
• K=-V(dP/dV)• Vp = sqrt((K+4/3G)/rho)• Vs=sqrt(G/rho)• K/rho=Vp2-4/3Vs2
Adams-Williamson equation
∂ρ/∂z=ρg(ρ/K)
3
3.5
4
4.5
5
300 400 500 600 700 800 900Depth
Den
sity
, gm
cm
-3
• 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?
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
To model Velocity
• Phase diagram and Elasticity are not enough• Time scales of the phase transitions are also
important
Is the low velocity zone due to
OrMelting?
Melts?
From Hirschmann, 2000
10 20 30 40 500
50
100
150
200
1400 C
Pressure, kbars
10 20 30 40 500
50
100
150
200
3.00
3.10
3.20
3.30
1400 C
Pressure, kbars
10 20 30 40 500
50
100
150
200
3.00
3.10
3.20
3.30
1400 C
Pressure, kbars
spol
cpxopx
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
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
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
Measure Stress
Measure Deformation in situ
Deform at a constant slow rate
Challenges for Experiments
at deep Earth conditions of P and T
Measurement of Stress
= F/A
Measurement of Stress
= M*
X-rays define d, lattice spacings, and can be used to define elastic strain.
Ideal CircleLattice spacings for stressed sample
Stressed sample0 1000 2000
channel
coun
ts
det1det2det3det4
Conical Slits
Measure Stress
Measure Deformation in situ
Deform at a constant slow rate
Challenges for Experiments
at deep Earth conditions of P and T
Multi SSD
Press
Sample
Sample
gold foil
gold foil
Measure Stress
Measure Deformation in situ
Deform at a constant slow rate
Challenges for Experiments
at deep Earth conditions of P and T
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
-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
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
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
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
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
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