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