Beyond Elasticity stress, strain, time

1. Beyond Elasticitystress, strain, time Don Weidner Stony Brook

2. 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.

3. Deep Earthquake Rheology Tomography Phase Transitions Thermoelastic Convection Seismic Anisotropy Anelasticity Earth’s mantle and stress Q, Vp,Vs

4. IN EARTH Seismic waves 1 sec – 1000 sec. Earthquakes 10 sec – 1000 sec Plate tectonics 107 sec – 1016 sec IN LAB Acoustic velocity 10-9 sec – 10-6 sec Rock mechanics 1 msec – 1 msec Ductile flow 103 sec – 106 sec Time scales

5. Rheology • Elasticity: stress proportional to strain • Anelasticy: stress, strain relation depends on time • Plasticity: strain not recoverable when stress is removed

6. Example of non-elastic process • Phase transformations can cause non-elastic volume change

7. From elasticity • K=-V(dP/dV) • Vp = sqrt((K+4/3G)/rho) • Vs=sqrt(G/rho) • K/rho=Vp2-4/3Vs2

8. Adams-Williamson equation ∂ρ/∂z=ρg(ρ/K)

10. Based on material properties: • 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?

11. Different time scale results in different velocity intermediate Vp, low Q Relaxed Low Vp, high Q Unrelaxed High Vp, high Q (Anderson, 1989) ω is seismic frequency;  is time scale; Q is attenuation factor, c is velocity

12. To model Velocity • Phase diagram and Elasticity are not enough • Time scales of the phase transitions are also important

13. Melts? Is the low velocity zone due to Or Melting?

14. From Hirschmann, 2000

18. sp ol opx cpx

19. Viscosity Profile of the Earth (L. Li, thesis, 2003)

20. Viscosity Profile of the Earth (L. Li, thesis, 2003)

21. Viscosity Profile of the Earth (L. Li, thesis, 2003)

22. Challenges for Experiments at deep Earth conditions of P and T Measure Deformation in situ Measure Stress Deform at a constant slow rate

23. Measurement of Stress s = F/A

24. Measurement of Stress s = M*e X-rays define d, lattice spacings, and can be used to define elastic strain.

25. Ideal Circle Stressed sample Lattice spacings for stressed sample

26. Conical Slits

28. Challenges for Experiments at deep Earth conditions of P and T Measure Deformation in situ Measure Stress Deform at a constant slow rate

29. Multi SSD Press Sample gold foil Sample gold foil

30. Challenges for Experiments at deep Earth conditions of P and T Measure Deformation in situ Measure Stress Deform at a constant slow rate

36. Measurement of Stress by Proxy

37. Forced oscillation on MgO and Al2O3 T= 800 oC P = 5GPa Frequency = 10-100mHz

38. Li Li et al 2009

39. MgO Li Li et al 2009

40. Measure Amplitude of Diffraction Peaks with Time and Temperature 700 C

41. Measure Amplitude of Diffraction Peaks with Time and Temperature

42. Measure Amplitude of Diffraction Peaks with Time and Temperature